Pressure Measurement. In 1842, U.S. Army Chief of Ordnance George Bomford "had holes drilled at regular intervals along a cannon barrel. Pistol barrels were then fastened into the holes, each loaded with a bullet. Opposite each barrel he placed … a ballistic pendulum" (see below). This allowed him to generate a projectile displacement-pressure curve. That, in turn, permitted the design of guns to have metal exactly where it was needed. It's not clear to me how much attention the rather hidebound Navy paid to these newfangled Army notions, but by 1850 John Dahlgren had designed a 135-pounder shell gun with a soda bottle shape. (Park 113).
A somewhat less Rube Goldbergesque sensor was the Rodman indenting gauge (1858). Its tube, like Bomford's pistol barrel, fitted into a drilled hole in the barrel wall, and the expanding gases moved a piston with a gas check, which in turn moved a knife that indented a copper disc. The depth of the indentation was compared to that achieved with a matching disk (from same copper bar) and knife using a standard testing machine. (VNEM). In the Noble crusher gauge (1860), the Rodman disc was replaced with a cylinder of copper, resulting in the pressure being expressed as so many "copper units of pressure" (CUP). For guns developing lesser pressures, lead cylinders were used. (EB11/Ballistics; Barnett 195ff; Buchanan 306).(The Rodman or Noble gauge could be inserted behind the cartridge, but this had its own limitations.)
We've been focusing on pressure, but the deflagration also results in an increase in temperature. The temperature can reach 5,550°F, and barrel steel melts at 2,500°F (Rinker 62). Fortunately, the projectile is only in the barrel for something like ten thousandths of a second. Still, gun barrels can definitely overheat. It's therefore very important that barrels have a high thermal conductivity so heat is dissipated quickly.
Gustavus Adolphus experimented in the 1620s with "leather cannon" for field use. This was actually a thin copper barrel with leather wrapped around it and bound with wire, cord and canvas; we know this because one prototype (test-shot in 1628) survived. (Brzezinski 18). Leather-like ceramics, glass and plastic-is a poor conductor of heat, and the leather cannon had a tendency to overheat and burst; they were superseded by bronze pieces. A Mythbusters version fired a cannonball at 450 mph, but blew out its breech in the process (episode 141).
Guns often were designed with separate powder chambers; these were narrower than the bore (to reduce stress) but communicated with it. They could be cylindrical, spherical or conical in shape. Spherical chambers offered the greatest muzzle velocity, but were difficult to construct, load and clean, and strained the gun most. Conical offered the worst muzzle velocity, so cylindrical became the happy compromise. (Jeffers 98).
The maximum quantity of powder that could be used in the gun was limited by the gun's bursting strength and the size of its powder chamber). One pound of 1820s powder occupied 30 cubic inches. (Beauchant 104).
Barrels can suffer permanent bore expansion as a result of exceeding the "elastic limit," catastrophic rupture, gas leakage, fatigue (micro-cracking), and erosion/wear. Barrels can be inspected for deterioration in a number of ways, including measuring the bore diameter deep inside the barrel with a long-handled inside caliper, and visually inspecting it with a borescope. A rigid borescope would be something like a periscope with a magnifier and a light attachment. A flexible borescope uses optical fibers and thus requires a higher tech level.
Internal Ballistics and Windage
Bore-windage had several effects. First, gas could escape around the ball, reducing the effective pressure driving the projectile. This reduced muzzle velocity and wasted energy, but also eased the stresses on the gun barrel. Secondly, as the ball progressed down-bore, it would glance off the walls of the bore. Each bounce drains some of the kinetic energy of the ball, thus further reducing muzzle velocity. Also, the direction and spin the ball emerged from the muzzle would be dictated by its last bounce. Obviously, this affected range and accuracy. All the bouncing around was also bad for the gun barrels. (Douglas 81).
The direct energy loss from escaped gas is proportional to the ratio of the annular area to the bore's cross-sectional area. Since windage is small, this ratio is roughly inversely proportional to the bore diameter. By my analysis, the indirect loss, from inelastic collision with the bore wall, will be proportional to 1-r? where r is the "coefficient of restitution" (the kinetic energy after collision as fraction of that before collision, for each collision) and n is the average number of bounces, which is proportional to the barrel length divided by the windage (as diameter difference).
Well, that's all theoretical. In practice, Douglas (70) says that one-quarter to one-half of the force of the powder was lost in consequence of the early-nineteenth-century standard windage. Douglas urged that windage should just be a fixed allowance, rather than one proportional to the gun's caliber. Only the degree of expansion due to heat, he reasoned, would be dependent on caliber (amounting to 1/70th caliber at white heat); rusting of the shot and fouling of the bore wouldn't be. He suggested reducing windage to 0.1–0.15 inches. (74ff).
The maximum pressure usually obtained in the late-nineteenth century was 15 tsi in rifled guns and 3 in smoothbores-this shows how much difference windage makes! (Barnett 196).
Muzzle Velocity
Range is definitely a function of muzzle velocity. The following table shows expected ranges for an early-nineteenth-century 24-pounder fired at a 45° elevation:
(Douglas 43).
Suppose that the work done by the powder in moving the projectile down the bore is proportional to the powder charge. If so, then the kinetic energy obtained must be proportional to the charge, and the muzzle velocity is then proportional to the square root of the powder charge relative to the weight of the shot. (Sladen) and this was generally assumed by early-nineteenth-century writers on gunnery (Beauchant 45; Douglas 53, 57).
In 1828, Beauchant proposed the following formula:
MV = 1600 sqrt (2 powder weight / shot weight)
This leads to the following results:
(Beauchant 45, 133)
This rule is probably good enough for our purposes, although I suspect that "1600" is a bit high for 1630s guns. Assuming powder quality is 75 % of early-nineteenth-century levels, we could use "1200" instead.
However, the work done on the projectile per pound of powder is not really constant regardless of the charge. It's dependent on the expansion ratio of the full bore relative to the initial charge volume, and thus depends on the length of the bore and the size of the charge. (Sladen 32).
Grantville has the Encyclopedia Britannica 9th edition, and its "Gunmaking" article provides Noble's table of the theoretical maximum work done by gunpowder per pound of charge ("specific work"), as a function of the expansion ratio. One can therefore calculate the expansion ratio, interpolate the "work/pound" from the table, and plug it into this formula:
Vmuzz= sqrt (2 g k e wp/ ws)
where g is gravitational acceleration (322 fps), k is the specific work per pound, e is the efficiency of the powder relative to the theoretical maximum, wpis the weight of the powder, and wsthe weight of the shot.