An alternative to the percussion cap is the friction tube. This is described in detail in EB11/Ammunition; essentially this is a T-shaped device, the vertical branch communicates with the vent hole, and the horizontal branch contains a copper friction bar surrounded by a "friction composition." The lanyard causes the friction bar to be pulled out, igniting the composition which ignites the powder in the vertical tube.
The great advantage of electric ignition was that it reduced the firing interval. However, the reliability of the power source is the sticking point. While it would be possible to have the ship carry a generator and run lines from it to all the guns, that would mean that a shot that took out the generator rendered them useless. Hence, each gun must have its own battery. And developing a working battery itself took some time. An 1894 article (Morgan) noted that while electrical ignition had until then been in limited use, the Bureau of Ordnance had recently adopted a zinc-carbon dry battery as well as a new electric primer design. By WW I, electric ignition was the norm, with percussion systems as backup.
Internal Ballistics
The term "ballistics" was coined by Marin Mersenne in 1644. Ballistics may be divided into three broad categories: interior (internal) ballistics, explaining what happens inside the gun barrel; exterior ballistics, describing the flight of the projectile through the air; and terminal ballistics, dealing with its penetration of the target.
Gun designers manipulate the internal ballistics of a gun so that it projects the desired projectile at the desired muzzle speed without bursting the gun.
A good propellant is one whose ingredients react very quickly ("deflagrate") to form a gas. At normal temperature and pressure, this gas would occupy a much greater volume than the original ingredients; but initially the volume of the gas is limited by that of the propellant (powder) chamber, and so there is instead an increase in pressure. The deflagration reaction also generates heat, which further increases pressure.
As the reaction continues, the pressure reaches the point that it's sufficient to overcome the friction holding the projectile in place ("shot-start force"), and it starts traveling down-bore.
The propulsive force on the projectile is the pressure times the area of the projectile base; the acceleration it feels is the force divided by the mass of the projectile. For spherical shot, the base area is proportional to the square of the diameter, and the mass to the cube, so acceleration is inversely proportional to the diameter.
You can cheat to some degree by using a sabot as a gas check. The sabot ensures that the pressure is exerted on the largest possible area, but the projectile may be subcaliber (narrower than the bore) and therefore less massive.
For a given caliber and shape, stone projectiles have a lower sectional density (mass/frontal area) than lead or iron and that means that for a given barrel length, they require less force to accelerate them to a given muzzle velocity. Less force means less pressure which means a smaller charge. Stone projectiles can therefore be fired from lighter cannon than metal ones of the same weight. As of the early-seventeenth century, stone throwers (pedreros) were being phased out in Europe, but they remained popular in the Ottoman Empire.
As the projectile moves, the volume available to the gas increases, which tends to reduce the pressure. On the other hand, if the deflagration reaction is still going on, the newly-produced gas and heat will tend to increase the pressure. One can thus draw a pressure-time or pressure-travel curve, and the location of its peak will depend on the specific characteristics of the powder. Likewise one can draw velocity-time or velocity-travel curves for the projectile. Sample curves appear in EB11/Ballistics.
Once the powder is completely consumed, the propulsive force on the projectile can only decrease as it moves down-bore, and once that propulsive force is less than the resisting forces, any further travel will reduce the projectile's speed.
After the projectile exits the muzzle, the pressure on it drops precipitously, although it may experience a brief period of additional acceleration from the escaping gases.
Internal Ballistics: Barrel Stress
A gun barrel, in essence, is a pressure vessel, containing the gases generated by the rapid combustion of the powder. We want to design the barrel so the gun is safe to fire, without inordinately increasing its weight and cost.
One way to do this was to put the metal where it was needed, i.e., where the pressure was greatest. It was certainly known to the down-timers that the thickness had to be greater at the breech than at the muzzle-they figured twice as much (see Manucy 37 for the detailed thickness variation)-but they had no quantitative knowledge of the pressure variation. That was revealed by nineteenth-century experimentation, as discussed below.
It is also important not to impair the barrel's function. Until the mid-nineteenth century, it was customary to bedeck cast cannon with a variety of ornamentation. However, these protuberances acted as "stress raisers," weakening the gun. (Hazlett 147-8, 221).
For a thin-walled (thickness not more than one-tenth diameter) cylindrical pressure vessel, the hoop stress is pressure * radius / thickness, in which case thickness should be proportional to bore diameter if pressure held constant. However, a cannon can't be considered thin-walled; the cannon of the Santissimo Sacramento (sunk 1668) had a maximum barrel thickness about equal to the bore diameter. (Guilmartin).
The thick-walled tube hoop stress formula, if external pressure is ignored, is
(Ri2*p/(Ro2-Ri2)) * (1+ Ro2/r2)
with
Ri inner radius
Ro outer radius
p internal pressure
r radius at which stress is calculated. (Labossier; McEvily 53).
So, at r=Ri, stress is
p*(Ro2+Ri2)/(Ro2-Ri2)
and at r=Ro, it is
2*p*Ri2/(Ro2-Ri2)
It's immediately evident that the stress is greater at the inside radius (Ri) than at the outside radius (Ro); the gun will crack first on the inside and the crack will grow each time the gun is fired.
Wall thickness of course is Ro-Ri and bore diameter is 2*Ri. When increasing the barrel thickness, each additional layer decreases the stress inflicted by a given internal pressure, but with diminishing returns (Table 2–1):
If we express Ro as k*Ri, so thickness is (k-1)*Ri, then the inside stress is proportional to
(k2+ 1)
– -
(k2- 1).
The even more complex "Gunmaker's formula," for built-up guns (see part 1), appears in EB11/Ordinance.
I mentioned pre-stressing in connection with cannon manufacture; this is to "make the outer layers of metal in the barrel bear a greater proportion of the bursting load." (Payne 264).
****
With real guns, the pressure varies according to the position of the projectile. In 1861, the distance-pressure curve for a 42-pounder with the powder of that time might feature a maximum pressure of 45,000 psi, dropping to one-tenth that by the time the shot exited the muzzle. (Bruce 138). Hence thickness (and thus weight) can be reduced if you know how pressure varies.
It is possible to control the curve to some degree by powder design. A progressive powder (burn rate increases with time) reduces the peak pressure, and thus the required barrel strength. This also reduces barrel wear, which tends to be more dependent on peak pressure than average pressure (Rinker 43). And it's likely to provide the highest exit velocity. On the other hand, if you have a short barrel, then use a degressive propellant, so you develop high projectile velocity quickly. (ES310).
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 ne
wfangled 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.
The efficiency of a gun is less than 100 % because some energy will be lost by heating the barrel and, for rifled guns, in rotating the projectile. EB9 suggests an efficiency ("factor of effect) ranging from 0.6–0.65 for field guns up to 0.85-0.95 for heavy guns.
In the nineteenth century, the "gold standard" for predicting muzzle velocity was Sarrau's monomial (for quick powders) or binomial (for slow powders) approximation. This had a couple of adjustable parameters to account for the differences between powders and these were determined by measuring the muzzle velocity for the same powder fired in two dissimilar guns. You could then apply the same formula to any other gun using the same powder.
The Sarrau formula was available in a few texts for general readers, including Johnson's Universal Cyclopaedia (1895), and the Encyclopedia Britannica 10th edition (the 1902 supplement to the 9th edition), but these are not, as far as I know, among the books that traveled back with Grantville.
Optimal Bore Length. Bore length is probably 92–94 % of the length of the piece. (Douglas 293). In general, the longer the effective bore (from projectile starting position to muzzle), the greater the muzzle velocity for a given powder charge; the muzzle velocity in turn determines range and penetrating power.
But there are definitely diminishing returns. Experiments have been conducted in which a barrel is successively cut down and the new muzzle velocity
determined. In 1862, Benton (130) reported that for a small change in length of a 12-pounder, the velocity was in fact proportional to the fourth root of the length. Another writer says that it's proportional to something between the square and cube root of the length of the bore (Douglas 101). I have seen the opinion expressed that there was no advantage to making a sixteenth-century gun longer than ten feet (Rodger 215). A 24-pounder of Douglas' time would have a bore length of 5.5–8.92 feet (293).
Theory also predicts diminishing returns. If the powder burns at a constant rate, slowly enough so the last of the powder is consumed just as the projectile exits the barrel, and the gas expansion is isothermal, the muzzle velocity will be proportional to the cube root of the barrel length. If we instead assume that the gas expansion is adiabatic (no heat lost), then it will be proportional to the fifth root of the length. (Denny 183-5). Note that this requires that the powder charge be proportional to the length of the barrel, which was not usually the case!
The analysis above is for black powder propellant; with smokeless powder, the pressure-position curve is different, and the optimal barrel length depends on the shape of the curve. (Denny 67ff, 188ff).
The advantage of increasing length is not so much the increased muzzle velocity, but rather that one can then use a slower-burning powder and thus reduce the maximum pressure-permitting reduction of barrel thickness and increasing barrel life. (Sladen 34).
Is there a limit beyond which increasing length has no effect or even reduces muzzle velocity? If the force propelling the projectile merely diminished as it traveled down bore, then there would be no bore length at which muzzle velocity was maximized, merely diminishing returns from lengthening it. But there is such a length, because the projectile's movement faces opposition even as the propulsive forces decline.
Benton (128) suggested three opposing forces: (1) friction, (2) inelastic collision, and (3) the pressure of the air in front of the projectile, and urged that if the length is increased too much, keeping the charge constant, the muzzle velocity will decrease.
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