Friction comes into play only for rifled barrels, where the projectile engages the rifling. The frictional force is presumably constant throughout the length of the barrel, whereas the propulsive force declines as the projectile moves down-barrel. It's possible to show that if you assume constant burn rate, isothermal expansion, constant frictional force along barrel, and optimal projectile length (powder completely burnt, and frictional force equal propulsive force, just as projectile reaches muzzle), the length at which the frictional force equals the propulsive force must be proportional to the mass of the projectile and the square of the muzzle speed, and inversely proportional to the frictional force in the barrel (this results from combining Denny equations N7.6, N7.9, and N8.2). Since the length of the region of contact is kept small, the frictional force at any given moment should be proportional to the circumference of the bore and thus to the diameter.
For smoothbores, interior collisions slow down the projectile, but as noted in the discussion of "windage," they should be less common as the diameter increases.
Like friction, outside atmospheric force is a constant resistive force, but it's proportional to the area and thus to the square of the diameter of the bore.
Powder charge. The expectation was that up to a point, increasing the powder charge (relative to the shot weight) would increase muzzle velocity. Obviously, once the projectile left the muzzle, any unconsumed powder would fail to provide any further boost to its speed.
There was great controversy, however, as to whether a point could be reached where any further increase in charge would actually reduce the muzzle velocity. Robins was insistent that this could not possibly be the case. However, Benton (130) reported a progressive decrease in muzzle velocity for a 36-pounder firing charges ranging from 36–77 pounds, and Farrow (289) suggests that the charge yielding the maximum velocity is half to two-thirds projectile weight. Still, it's not clear from the underlying physics why this diminution should occur.
Even if there weren't diminishing returns vis-a-vis muzzle velocity, the amount of powder used would be constrained by the size of the powder chamber, fear of bursting the gun, and the recoil.
It has also been reported that the maximum velocity charge increases with the length of the gun. This makes sense as, for the same rate of acceleration, it gives more time for useful consumption of powder.(Simpson 177).
Multi-Chamber Guns
High-Low Pressure Gun. These have a divided propellant chamber, with two compartments separated by a plate with holes. The powder is ignited in the first compartment, generating a high pressure. Because of the constricted communication with the second, the pressure there is lower, resulting in a lower muzzle velocity but also a lower recoil. If you are wondering why not just use a conventional gun with a low powder charge, it's because the high pressure results in a better "burn" curve, and only the first chamber needs a thick wall. The concept was first implemented in the Panzerabwehrwerfer 600 (1945) and copied in the British Limbo depth charge launcher (1955) and later the American M79 grenade launcher.
Lyman-Haskell Multicharge Gun. The American government tested a 6-inch multicharge gun in 1883. This had five powder chambers, one at the breech, and the remaining four distributed along the length of the bore. The charge at the breech was smaller than the others, and the nearer the powder chamber to the muzzle, the faster burning the powder used. The theory was that the pressure created by their deflagration would also be distributed, allowing one to achieve a much higher muzzle velocity without overstressing the barrel. The breech charge was ignited in the usual way and the other charges by the passage of the combustion gases propelling the projectile.
The multicharge gun, with 119 pounds of powder, accelerated a 111-pound projectile to a muzzle velocity of 2004 fps, but with a barrel pressure of only 31,550 psi. In contrast, the Krupp 5.9 inch gun used a single charge to propel a 112.2 pound shell, achieving a muzzle velocity of 1676 fps with a pressure of 40,320 psi. So what's the catch, other than the profligate use of powder?
Well, the multicharge gun was 25 feet long (50 calibers). It therefore was quite heavy (25 tons), whereas the Krupp gun weighed only 3.8. The Ordinance Board was of the opinion that the multicharge gun's performance should be compared, not to guns of equal caliber, but to those of equal weight. While the 10-inch gun, weighing 18 tons, had a lower muzzle velocity (1400 psi), its 400 pound projectile carried greater kinetic energy and, in the Board's opinion, was likely to have more penetrating power. (Walker; Haskell).
Propellants (Gunpowder, etc.)
Gunpowder is a mixture of saltpeter, charcoal and sulfur. The saltpeter (potassium nitrate) is an oxidant, and it burns the charcoal (carbon), forming carbon dioxide gas. The sulfur combines with the potassium ion of the saltpeter, forming potassium sulfide, and in the process generates a lot of heat. Since the heated gas is confined by the gun, that results in an increase in pressure. And that's what pushes the projectile out. It also stresses the barrel, so you can't use too much powder and how much can be used depends on its burn rate.
Among the down-timers, there's no consensus as to the proper formula for gunpowder (black powder). Just one master gunner, Peter Whitehorne (1560), presented 20 different recipes, with saltpeter content of 16–84 %, charcoal of 8-64 %, and sulfur of 8-28 %. (Walton 123). EB11/Gunpowder says that the following formula was used in Britain in 1647: 66.6 % saltpeter, 16.6 % charcoal, 16.6 % sulfur. By 1781, the proportions were 75-15-10, the ones given in H. Beam Piper's Lord Kalvan of Otherwhen. Other formulae of possible interest included 52.2-26.1-21.7 (Germany 1596), 68.3-23.2–8.5 (Denmark 1608). 75.6-13.6-10.8 (France 1650), and 73-17-10 (Sweden 1697). Even in the nineteenth century, different countries had different preferences, with saltpeter 70–80 %, charcoal 11–18.5 %, and sulfur 9.5-13 %. (Beauchant 149). The proportions given in the modern EB (2002CD) is 75-14-11. The Medieval Gunpowder Research Group, using a replica of the Loshult Gun, found that muzzle velocity peaked at a saltpeter content of about 72 %. (MGRG2).
The burn rate is proportional to the burning surface. It thus is dependent in part on initial particle ("grain") size; the smaller the particles, the greater the total surface area for a given weight of powder, and the faster the "deflagration" reaction. However, the reaction shouldn't be too fast; you want it to continue until the projectile reaches the muzzle. Thus, the grain size must be matched to the barrel length; muskets used finer powder than did "great guns." The term "powder" became a bit misleading; the "grains" can be several inches in diameter-please look at Fig. 1 in EB11/Gunpowder.
The quality of gunpowder has improved over time. In 1587, gunners used "serpentine," which was floury. Because of the small particle size, it was necessary to leave part of the powder chamber empty, to provide oxygen. The powder also absorbed moisture readily. The charge for a culverin was equal to the shot weight, and for a cannon, half that weight.
By 1625, "corned" powder, which was granulated, was common. (Lavery 135). The size of the grains could be controlled by sieving. In 1673, a culverin used a two-thirds shot weight charge, and a cannon, one-half.
Improvements were also made in the preparation of the components of gunpowder. By 1740, the charges ranged from 40 % for a 42-pounder to 66 % for a 9-pounder. (Id.) In 1783, "cylinder powder" was introduced, although it didn't come into common use until 1803 (Rodger 421). It incorporated a better grade of charcoal. The wood was placed in cast iron cylinders, and heated over a stove, rather than charred in a kiln. (Id.; Douglas 201). This permitted reducing the standard charge to one-third the weight of the ball for ordinary guns, and a mere 8 % for carronades (Lavery). The method is described by EB11/Gunpowder but without discussion of its advantages over former practice.
You could use less if you were trying to conserve powder, or were hoping to produce more splinters if the shot didn't hole the target. A one-sixth charge is sufficient to "drive a ball from any large gun through the side of a ship at 1100 yards" but for a 24-pounder would require twice the elevation as a one-
third charge, thus reducing accuracy. (Douglas 54).
In the mid-nineteenth century, the increase in gun size led to incompatibility with the ordinary black powder; it burned too quickly, creating conditions that strained the gun. A slower-burning "brown powder," described in EB11/Gunpowder, was introduced.
The shape of the grains is also significant. Normally, as deflagration continues, the particles are consumed inward, reducing the total burning surface and thus reducing the burn (regressive burn). This is experienced with all solid grains, whether they be spheres, cylinders or plates.
In 1860, what EB11 calls "shaped powders" were introduced. The grains had one or more perforations so they were consumed both inward and outward, resulting in a constant (neutral burn) or even increasing (progressive burn) burn rate. EB11 describes how they were made.
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In the late-nineteenth century, gunpowder was largely replaced by nitrocellulose-based propellants (the so-called "smokeless powders"). These produced less smoke and flash, burned progressively, and caused less erosion to the barrels. They are classified as being single-base (nitrocellulose) or double-base (nitrocellulose combined with nitroglycerin or some other liquid organic nitrate).
Ballistite (1887) was 40 % nitrocellulose and 60 % nitroglycerin (EB2002CD/explosive). Cordite was similar; 37 % nitrocellulose, 58 % nitroglycerin, 5 % Vaseline (Rinker 34) or later 65-30-5 (EB11/Cordite). EB11 doesn't say anything about stabilizers, but EB2002CD suggests diphenylamine.
A member of a Civil War reenactment group would probably be familiar with Pyrodex, which was developed in the 1970s. It's essentially black powder with various additives so it burns more cleanly-less fouling of the bore, less smoke. However, the formula of Pyrodex is proprietary, and the person who developed it (Powlak) lost his life in the process.
The decomposition products of black powder are 43 % gaseous and 57 % solid, the latter being responsible for the smoke of the proverbial "smoking gun." In contrast, modern smokeless powder is more than 99 % gaseous. Gases can be accelerated to higher velocities than solids, for a given internal pressure. Consequently, black powder has a low "specific impulse" (pounds thrust produced per pound propellant burned per second)-~50–70 seconds-whereas double base powders provide ~180–210 seconds. (Guilmartin 300).
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Average muzzle velocities increased over the nineteenth century, from 1575 fps for ordinary black powder, to 2133 fps for prismatic powder and 2225 fps for early (1885) smokeless powder. (Breyer 38).
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With black powder, the principal manufacturing considerations were "strength", freedom from fouling, and proneness to deterioration. These were affected by composition, density, moisture content, and grain size, shape, hardness and "glazing."
Until 1868, powder density was measured by "cubing"-weighing it in a box of standardized volume. This was improved upon by the mercury densimeter. (Farrow 313).
There was no quantitative test for hardness; the grain would be broken between finger and thumb. (Smith).
Powder strength will vary from manufacturer to manufacturer, from lot to lot, and even from barrel to barrel. (Dahlgren 180). Even at the end of the black powder era, powder manufacture was an art, not a science. In 1881, 150,000 pounds of Westphalian Company prismatic powder was rejected because it didn't meet the standards; the representative blamed it on manufacturing "during very cold weather." (Buchanan 325).
Powder strength was originally tested by setting a small amount afire in the open air, and observing the results. Eprouvettes ("provers"), which ignited the powder in a confined space simulating a cannon barrel, provided more useful data (von Malitz 163ff). An early eprouvette was described by William Bourne (1578). This was essentially a box with a hinged and ratcheted lid and a small fuse hole. A set quantity of powder was placed inside, and set off. The force of the combustion gases would drive the lid upward, and the lid would be kept from dropping back into place by the ratchet. The angle reached by the lid was a measure of the strength of the powder.
In Bourne's eprouvette, the propulsive force was resisted by the weight of the lid, but some later devices used a spring mechanism, and Du Me's eprouvette (1702) employed water resistance. Also, some were engineered so the propelled object moved linearly rather than angularly. And, instead of measuring that movement, one could measure the eprouvette's recoil.
Most of the eprouvettes just worked on an indicator object like Bourne's lid, but the mortar eprouvette actually fired a projectile at a fixed angle, usually 45°, so the power was inferred from the range achieved.
In storage, gunpowder could become damp, and once its moisture content rises above 1 %, it begins losing explosive power. (Kelly 59). Keeping it away from seawater isn't enough because it can actually absorb water vapor. (Douglas 199). It follows that what's needed is airtight storage or, if that isn't possible, storage together with some desiccant.
Powder was examined for dampness and if damp, it was dried. This was a ticklish operation as the drying could melt the sulfur or even explode the grains. (200). If the powder were past redemption, one could at least attempt to recover the saltpeter, which was a rare and valuable commodity in Europe (201).
Alternative Propellants
Steam. Jacob Perkins received a British patent in 1824 for a steam gun. This was no "paper patent"; in an 1825 demonstration an 800 (or 900) psi boiler projected one ounce musket balls out a barrel, achieving penetration of quarter-inch iron plate and eleven inches of pine at a range of 35 yards. Moreover, he developed a rapid-fire gravity feed enhancement. (Smith). The rapid fire version was later a major attraction at the National Gallery of Popular Science (1832). Some outrageous claims were made for how fast it was, but I am inclined to believe the ten balls/minute that Perkins' son asserted in 1861. (Bruce 138).
In 1828, Perkins designed for the French a 1500 psi steam gun, with a barrel six feet long and three inches caliber, firing four pound balls. It worked, but its range was only half that of a conventional cannon of the same caliber. Not only was the barrel pressure much lower than in a "powder" gun, it may have suffered much more acutely from bore-windage because of that difference. Another problem was weight; the 1825 model had a five-ton boiler. (BPHS).
EB11/Explosives alludes to the Winans (Dickinson) steam gun, built for the Confederacy. It was never put to use; Mythbusters Episode 93 suggests that it would have gotten off five rounds a second and had a maximum range of 700 yards, but expressed doubt that the impact velocity beyond point blank range was high enough to be lethal.
The concept of using steam to throw a projectile wasn't new; Leonardo da Vinci had speculated that Archimedes had used a steam cannon at the siege of Syracuse, and drew one. In 2006, an MIT team figured out how to implement Leonardo's concept. They were deliberately coy about the particulars of steam generation, but they built a steam cannon designed for 3,500-4,000 psi, and fired a one pound projectile (i.e., equivalent to that of a robinet) with a muzzle velocity over 300 m/s. The bore was 2 feet long and 1.5 inches diameter. They were able to fire one round every two minutes. (MIT).
While I am sure the projectiles made a satisfying whizz, the fact remains that "steamer" muzzle velocity is low compared to that achieved with powder guns. A bit of a back-of-the-envelope calculation puts this into perspective. Let's assume that we have a large enough reservoir of steam so that we can maintain constant pressure. Let's also assume that the barrel is horizontal (so we can ignore gravity), frictionless, and without windage. If so, the projectile accelerates at a constant rate and the muzzle velocity will therefore be
sqrt (2*L*P*A/m),
where L is bore length, P pressure, A cross-sectional area of the projectile and m mass. For a standard projectile (one pound, one inch diameter) this reduces to
24.56 sqrt(LP) (L inches, P psi).
Perkins' 1828 gun thus has a theoretical ideal muzzle velocity of 1345 fps, and the MIT gun, 951. But note that the actual muzzle velocity for the MIT gun was a bit less than a third of th
e theoretical value.
Compressed Air. The blowgun is the earliest compressed air weapon, limited in propulsive force by the ratio of the volume of air one can huff (about 60 cubic inches) to the bore volume of the blowgun (14 cubic inches for a six footer with a half inch caliber). (Gurstelle 142). A "pneumatic rifle was built at the beginning of the seventeenth century," and some Austrian jaegers carried the model 1780 rifle (300 m/s muzzle velocity), which was a great weapon for covert operations against French occupation forces. (Rossi 232).
The USS Vesuvius (1888) carried three 15-inch pneumatic "dynamite" guns. Compressed air from a 1000 psi reservoir was fed into the barrels, which were only 55 feet long (!), partially below deck, and mounted at a fixed elevation of 16 degrees. Range was changed by adjusting the pressure. The guns couldn't be traversed; you aimed the ship to aim the gun. The guns fired finned projectiles filled with up to 600 pounds of dynamite; this high explosive was too sensitive to be used in an ordinary gun and indeed even the muzzle velocity of the pneumatic gun had to be limited. The maximum range was 5,000 yards, with a subcaliber (6") shell. (NAVWEAPS; NAVSOURCE; Hamilton; Clark).
Because of the low pressure, a 20-inch gun could have a steel or aluminum bronze barrel that was one half an inch thick. In trials, the gun had good accuracy, and could fire about one round a minute. (Zalinski). The Zalinski gun is discussed approvingly in EB11/Pneumatic Gun.
The secret to understanding the dynamite gun is to think of it, not as a gun, but as a torpedo launch system. Ship armor had reached the point at which ordinary shells weren't reliably penetrating it. The projectiles fired by the dynamite gun were conceptualized as "aerial torpedoes," traveling faster and farther than any underwater torpedo and exploding underwater against the unarmored bilge of the enemy craft (Parkerson 83).
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