equation b/w muzzle energy and powder burn energy

[khuengo]

Gentlemen,
I am looking for an equation that shows the relation between muzzle energy of the bullet (kinetic energy) and powder burn energy of the powder. The equation should be like this:

Ep = Eb + Er + Ef ....

where
Ep= burn energy of powder, which is a function of burning rate and weight
Eb= kinetic energy of bullet (=mv^2/2) = mass of bullet x square of velocity divided by two
Er= recoil energy
Ef = fiction in the barrel

Thanks

[Unclenick]

The formula you are looking for is to calculate what is called ballistic efficiency. It is simply the percentage of the potential energy stored in the bullet that is translated to the bullet. The force resulting from propellant gas pressure pushes backward on the gun, gets a small amount of the energy, and some is expended straining the chamber walls and is ultimately lost as heat, and some becomes heat due to barrel friction. A big one that few people keep in mind is that the mass of the propellant gas and some of the powder that is still burning get accelerated down the tube as well, and that requires a chunk of the energy. This portion gets bigger as muzzle velocity gets higher.

QuickLOAD internal ballistics software and probably others will calculate ballistic efficiency up to the point the bullet leaves the muzzle for a given load. For your equation form to work, this efficiency would have to be constant, but it varies from the teens to the forty percent range depending on the component combination and gun. As you can imagine, this number winds up depending on whether all the powder burns in the barrel or not (often it doesn't all burn in the tube). It also depends on the shape of the pressure curve, which affects how much gas pressure keeps working on the bullet to accelerate it. Only in the case of the standard velocity .22 rimfire, that I am aware of, does this get used fairly completely. Beyond about 18 inches of barrel no further significant bullet acceleration occurs.

Ballistic efficiency at the muzzle isn't the end of the story, either. For most high power rifles, the muzzle gas jet continues to push on the bullet base a little for up to 6 feet. A bullet usually has to travel 10-15 feet before its velocity drops back to what it was right at the muzzle. I don't know of anything other than Doppler bullet radar that can tell you what the peak velocity produced by a particular shot actually is?

In the end, there is no really simple solution. If you have the stored energy of the powder (typically not published), it is usually stated as the powder's heat of explosion in kilojoules/kilogram. And if you know the weight of the powder charge and have muzzle energy of the bullet, then dividing the latter by the product of the former and the weight of the charge then the result multiplied by 100 will give you the % ballistic efficiency.

100 x Muzzle Energy/(E of explosion x Weight of charge) = ballistic efficiency

If you want to start with the powder's stored energy and arrive at all the rest, you have to know how to calculate the pressure curve over the time the bullet is in the barrel and integrate under that curve to get average accelerating force, then subtract friction and calculate the bullet's resulting speed and muzzle energy. This is a differential equations problem since the bullet weight affects the acceleration and the acceleration affects the time the bullet spends in the barrel. It is a constantly changing equilibrium until the bullet leaves. Even after you have the seven necessary characteristics of a powder, you still have the barrel length and bullet weight and case volume and frictional and start pressure influences to allow for. It isn't a trivial calculation and there is more than one model available. If you want to persue it you'll likely need to read a book on the subject. Hartmut Broemel, QuickLOAD's author, recommends Theory of the Interior Ballistics of Guns, by J. Corner, John Wiley & Sons, pub. The added bump from the gas jet after the bullet exits the muzzle would also be quite complex to calculate, and given that it is lost again 10-15 feet from the muzzle, it is of little practical interest.

Nick

[khuengo]

Thank you Nick for your explanation. Btw, I came across this one

A pound of single-base rifle powder has an energy content of about 1,246,000 ft-lbs of energy or about 178 ft-lbs per grain of powder. In actual practice only a fraction of this energy is available to accelerate the bullet. Julian Hatcher in HATCHER'S NOTEBOOK reported the energy distribution for the Browning Machine Rifle as follows:

Heat to cartridge case 4%
Kinetic energy to bullet 29%
Kinetic energy to gases 19%
Heat to barrel 22%
Heat to gases 19%
Heat to bullet friction 7%
===================
Total: 100%

You will note that the energy imparted to the bullet is only about 29% of the total powder energy available. While this is typical of many small arms cartridges, actual efficiencies may range from 17 to 37 percent or more. The actual efficiency is basically a function of expansion ratio and charge weight to bullet weight ratio.

http://www.loadammo.com/WhatIsBallistics.htm

[Unclenick]

Yes. A number of assumptions are built into that approximation. Not just 100% powder burn, but also the rate at which the powder burns affects ballistic efficiency. If I look at the 30-06 firing the 175 grain Sierra MatchKing (very close to the 172 grain military match FMJ bullet, which was actually specified to be 174.5 grains + 0.0/-3.0 grians) over 46.5 grains of IMR 4895 (the last Lake City M72 match loading used), the ballistic efficiency in a 24" barrel (M1 Garand length, about which Hatcher wrote much), I get a ballistic efficiency of 30.4%. IMR 4895 is a single-base powder with 1,302,000 ft-lbf/lb stored energy. This is all quite close to Hatcher's conditions. 99.87% of the powder burns in the tube, according to QuickLOAD.

If I increase the charge by 3.5 grains to 50 grains (a load up in the maximum range), the added pressure burns the powder 100% when the bullet has travelled 20.4 inches down the tube, and ballistic efficiency increases to 31.9%. If I drop the load down 3.5 grains to 43 grains, 98.73% of the powder burns while the bullet is still in the barrel, and ballistic efficiency drops to 28.7%. If I shorten the barrel to 20", only 97.46% of the powder has time to burn, and ballistic efficiency drops to 26.1%.

These are just to give you some idea: 17 grains of the very fast Vihtavuori N310 in the .308 under a 175 grain Sierra MK produces a ballistic efficiency of 44.0% in a 30 inch Palma Barrel. 40 grains of the very slow IMR 7828 SSC under the 175 grain bullet in the 30-06 with a 20" barrel produces a ballistic efficiency of 14.5%. There is quite a range out there.

Nick