Here's a straightforward consideration of the question of barrel length and percentage of powder burned from An Essay on Shooting by Wm. Cleator in 1789. He references the Royal Society study and conclusions.
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From the theory we have just given a detail of, it appears that the superior range of long barrels was founded intirely upon the opinion, that the powder fires gradually all the time it is passing through the barrel. The conclusion necessarily drawn from this, was, that the larger the charge, the longer time it was in kindling; and hence, that for any assigned charge, the barrel must be of a proportional length, so that the powder may be completely inflamed just as the ball or shot are about to quit the muzzle: The converse of this proposition was, that in every piece there is a certain charge which will be all fired just at the parting of the ball or shot from the mouth of the piece, and that any addition of powder will not take fire, but will only serve, by its weight, to diminish the action of the rest, and consequently, to ****** the velocity of the shot or ball.
If this matter be considered with mathematical rigour, it must certainly be allowed, that, as the fire is not at once applied to every particle of the powder, the whole of the charge cannot be inflamed at the same instant; but it is equally certain, that the progress of the inflammation is so rapid, that the powder is completely kindled before the ball or shot arrives at the mouth of the shortest barrel ever made use of.
To determine this, a number of experiments were made by a committee of the Royal Society, so long ago as the year 1743* and by these it was shewn, that when a barrel was shortened so much that the ball placed before the powder was upon a level with the muzzle, the unfired powder, collected from the discharge, by means of a cloth spread before the piece, weighed but one twelfth of the charge. This was analysed, and found to contain less saltpetre than an equal portion of the same powder did. Hence, and from the extreme minuteness of the particles collected, there was reason to believe, that the grains to which they belonged were less susceptible of inflammation than the rest, owing, perhaps, to some inequality in their mixture: what served in a great measure to prove this, was, that when the charge, and consequently the heat generated during the explosion, was increased, the quantity of unfired powder collected was proportionally smaller. And we may therefore safely conclude, that the powder is completely inflamed before the ball or shot arrives at the mouth of the shortest barrel ever employed. [Italics]
Having now removed the ground upon which the superior range of long barrels was formerly rested, it is necessary that we should endeavour to shew upon what circumstances it does really depend, and to what extent it is found to take place.
The elastic fluid produced by the firing of gunpowder is found, by experiment, to occupy, when cooled to the temperature of the atmosphere, a space 244 times greater than that taken up by the powder from which it was obtained. But from the heat generated during the explosion, this elastic fluid is rarefied to upwards of four times its former bulk. The expansive force of this fluid, therefore, is at the moment of inflammation, one thousand times greater than that of common air, or, which is the same, than the pressure of the atmosphere; or, supposing the powder to have occupied the space of one cubic inch, its expansive force, when fired, is equal to that which would be exerted by one thousand cubic inches of common air compressed into the space of one inch.
As the velocity with which the flame of gunpowder expands when uncompressed, is much greater than that with which the ball, or shot, moves forward, the flame must continue to press upon the ball, and add to its velocity, until it quits the mouth of the piece. This pressure, however, becomes less and less, as the ball proceeds, and ceases entirely when it leaves the muzzle, in consequence of the flame being then allowed to expand itself laterally. Thus, for example, if the charge of powder takes up one inch of the barrel, and the whole length of the barrel be thirty inches, then, when the ball arrives at the muzzle of the piece, the inflamed powder (whose expansive effort is in proportion to the smallness of the space it occupies) extends through thirty times the space it did when the ball began to move, and consequently presses forward with but one thirtieth part the force it possessed at first. Moreover, although the velocity of the bullet is continually increased by this pressure of the inflamed powder, its acceleration becomes less and less as it proceeds through the barrel; for besides that the quantity of the pressure diminishes as the flame expands, the bullet continuing to move faster and faster, must receive continually less and less addition of impulse from the flame pressing behind it. Hence, if two pieces of the fame bore, but of different lengths are charged with the fame quantity of powder, the longer piece will, strictly speaking, communicate the greater velocity and force to its ball, or shot. But as the inflammation of the powder has been shewn to be nearly instantaneous, and as the increase of acceleration which the ball or shot receives after the first impulse of the powder upon it, is not very considerable; it follows that the force with which two barrels of the fame bore, and with the same charge, throw their ball or shot will be nearly the same, unless their lengths be extremely disproportionate.
*Philosoph. Transact. No. 465 p. 172
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Spence
the author attempts to establish a given amount of powder will inpart same total energy to a lighter or heavier projectile in a very short barrel. If powder was burning partially a heavier projectile, spending more time in the barrel should have much more muzzle energy than 4% observed.
