[A continuation of "Velocities and Pressures in Guns," Vol. XIV, No. 2.]
In a paper entitled "Velocities and Pressures in Guns," Vol. XIV, No. 2, the action of powder in guns was treated mathematically, with the tacit assumption of a certain premise which, though not new, being in fact the very foundation of Messrs. Noble and Abel's treatment of the work done by powder in guns, at first appears doubtful. The question is whether or not a progressive powder, if all burnt in a gun, and the gases produced have gained mechanical and thermal equilibrium by the time the projectile reaches the muzzle, will give as great a muzzle velocity, weight for weight, as a powder all of which is burnt before the projectile begins to move. This point can be elucidated in several ways, assuming, as in that paper, no radiation or conduction of heat to the gun or between the portions of the gas. In other words, the portions of gas are assumed to expand adiabatically.
One comment has been made and may be noticed here, namely: if, immediately around the surface of the burning powder grains, gas at high temperature and pressure exists, and the rest of the gas is, or may be considered, in thermal and mechanical equilibrium at a lower temperature and pressure, then the lower pressure, being that of the gas directly in rear of the projectile, would be the force accelerating it. Looking into the subject, however, we see that if only the low pressure exists at the base of the projectile, the high pressure must be somewhere else, as at the face of the breech-block; and if the low pressure may be considered by itself as accelerating the projectile in one direction, so may the high as accelerating the gun in the opposite direction. Newton's law on the equality of action and reaction would then not apply, remembering, of course, that the mean forward motion of the powder gas (a motion partly action and partly reaction) is limited by the relative motion of the projectile and gun.
To proceed, suppose that we have a quantity of gas occupying a non-conducting cylinder OMCD, Fig. 1, that the pressure of this gas is represented in Fig. 2 by LB and the volume by OD. Interpose a non-conducting piston AB between the two halves of the volume of gas.
Now suppose that we push the piston AB to the position A1B1 (Fig. 1). The pressure of the gas to the left of the piston will increase from LB (Fig. 2) to NB1, following the pressure-volume law pvn = povon, where n equals c’/c, the ratio of the specific heats of the gas, or approximately 1.4. The work done in compressing the left half will then be represented (in proper units) by an area NB1BL (supposing no friction). Part of this work, LBB1K, will be done by the other half of the gas. The external work necessary to push AB to the position A1B1 will therefore be represented by NKL. If now the piston at A1B1 is released it will start towards AB, and when it reaches that position it will have a vis viva represented by NKL, will pass the position AB, and will then encounter retarding pressures equal to those formerly accelerating it. It will be brought to rest at the position A2B2, determined by BB1 = BB2, will then return to A1B1, then back again, and so on, as long as there is no external work done.
Now connect the piston by means of a piston-rod, or otherwise, with the outside, so that it may do external work. It will come to rest when it has done the external work represented by NKL, and not until then. It will then be in its original position AB, and the pressure on each side of it will be the same as originally, so that if it were removed the gas would be in thermal and mechanical equilibrium.
Evidently, while the piston is in motion from B1 to B2, the cylinder head CD will be subject in succession to all the pressures represented by the ordinates of the curve KLN1, and not alone to the pressure B1K, with which the pressure to the right of the piston begins.
If now the mass of the piston is made smaller, its velocity at B will be increased, its mechanical energy or vis viva still remaining KLN. In the limiting case, where the mass of the piston is infinitesimal, its velocity will be infinite, the mean velocity for the travel of the piston will be infinite, and the distance B1B2 will be traversed in an infinitesimal time. Suppose that we take this limiting mass for the piston, and suppose, simultaneously with its release, that we release the cylinder head CD, allowing it free motion along OD, the mass of CD being supposed appreciable. Evidently CD will start under an accelerating force KB1, and in an infinitesimal time will be subject to all the pressures KB1 to N1B2. The piston AB will make an infinite number of vibrations in a short finite time, and will finally take up a mean position between O and D. The piston AB might then be removed, as, its velocity being finite, it will have no finite vis viva, the gas will be in equilibrium, and all the energy represented by NKL will have been transmitted to CD. The piston of infinitesimal mass of course does not exist; more properly a piston should be supposed which by its mass and motion would represent the mass and motion of the gas itself In this case equilibrium would be gained, though not so quickly as with the infinitesimal piston. A little explanation possibly will be necessary to connect the assumed case with the case of fired gunpowder.
A charge of gunpowder is not all ignited at once. The particles of gas have a certain temperature and definite tension at the instant of combustion. Suppose a small quantity of the powder ignited instantaneously, and suppose for the sake of simplicity, in lieu of other gas, that gas from just the same small weight of powder (expanded against pressure) already fills the powder chamber. The pressure of the last may be represented above by B1K, and the pressure of the newly burnt powder by B1N. The piston AB represents the outside surface of the last burnt powder gas, this gas being supposed to expand without loss of heat by radiation or conduction. CD represents the base of the projectile supposed free to move when the powder is burned. When the two parts of the powder gas are in equilibrium, the energy KLN will have been transferred from the last burned gas to the projectile. This will be in excess of the work that would be done by the same expansion in total volume, assuming both portions originally in equilibrium at the pressure LB. It will be the same with each successive portion, so that when all arrive at equilibrium the pressure will be the same as if the combustion of all had been simultaneous.
It is necessary to remember that the pressure of a gas is exerted equally in all directions, and that it does just the same quantity of work in expanding from one volume to another in all directions as in expanding in only one. The newly formed gas at the surface of the grain expands in all directions, and the vibrating piston AB ordinarily corresponds exactly with a vibrating spherical (oval or ring) surface around the grain, the gas inside of which finally gains equilibrium with the remainder of the gas in the gun.
In deducing the general equation ("Velocities and Pressures in Guns") of motion in the bore of a gun, a mean temperature T is assumed to exist in the gases, such that the pressure may be represented by P = Rx(T/V) Now, while powder is burning in a gun no such equilibrium does exist, though the pressure P is fixed and definite; given P, we can very readily find a mean temperature for the gas if we solve this equation.
Given the work already done, we can also find a value for T. The question is, when no equilibrium really exists, are both answers the same, or should they be? To answer this let us look at Fig. 2. The pressure KB1, would furnish the temperature of the expanded gas, and by subtracting this from the temperature of combustion of powder, the T work already done, KB1, would equal Rx(T/V), where x is the weight of the expanded gas (as in "Velocities and Pressures in Guns"). Is the pressure KB1 that this T determines the accelerating pressure? No, unless all the powder gas is in equilibrium; as before stated, the accelerating pressure is a compound of all the ordinates from B2N1 to B1K. If, however, no powder is burning, all the portions of gas being supposed at the same temperature and therefore the same pressure, it is. In other words, the general equation is true at any point w, the weight of powder burned and in equilibrium, becomes constant at that point.
We assume, in mathematically treating this subject, that each elementary portion of the powder gas gains equilibrium in the element of time after burning. This is simply one way of saying that the powder gas gains equilibrium just as fast as it is evolved.
Imagine, to look at the subject in another way, the bore of a gun subdivided into a number of smaller gun-bores of equal size, and suppose each similarly loaded with its proportion of powder, the same for each of the smaller calibers. Now, imagine that these bores are so small that we can consider in each the combustion of the small quantity of powder contained as instantaneous. Further, suppose the bullets are of such different sizes that, though the small charges are ignited at different times, the bullets will all reach the muzzle of their respective barrels at the same time. Ignite the charges in succession, as determined by the size of the bullets. The pressure in each barrel at the muzzle will be independent of the time of arriving there, following the law pvn = C, where C is some constant and will evidently be the same in all the barrels. The work done in each barrel will be precisely the same; and the total work for all the barrels will be precisely the same as if all the powder had burnt instantaneously in the original gun and then expanded to the muzzle. If now the size of the bullets be changed, the same equality will hold, provided that each bullet reaches the muzzle at the same time. There would then be an equilibrium of pressure throughout the gas if at the muzzle all the barrels were suddenly merged in one.
If, however, the bullets had not been properly proportioned there would be no such equilibrium, some of the projectiles would arrive late at the muzzle, and when merged the tensions throughout the gas would be variable. It is thus plain, if equilibrium is gained, that only one fixed quantity of work will be done for a fixed expansion of the same mass of powder gas (of fixed quality). Some light is thrown on the subject by Professor J. Clerk Maxwell in his "Theory of Heat" (edition of 1883), page 189, where he says: "If the system under consideration consists of a number of bodies at different pressures and temperatures, contained within a vessel from which neither matter nor heat can escape, then the amount of energy converted into work will be greatest when the system is reduced to thermal and mechanical equilibrium by the following process.
1st. Let each of the bodies be brought to the same temperature by expansion or compression, without communication of heat.
2d. The bodies being now at the same temperature, let those which exert the greatest pressure be allowed to expand and to compress those which exert less pressure, till the pressures of all the bodies in the vessel are equal, the process being conducted so slowly that the temperatures of all the bodies remain sensibly equal to each other throughout the process.
During the first part of this process, in which there is no communication of heat between the bodies, the entropy of each body remains constant. During the second part the bodies are all at the same temperature, and therefore the communication of heat from one body to another diminishes the entropy of the one body as much as it increases that of the other, so that the sum of the entropy remains constant. Hence the total entropy of the system remains the same from the beginning to the end of the process. The work done against mechanical resistance during the establishment of thermal and mechanical equilibrium is greater when the process is conducted in this way, than when conduction of heat is allowed to take place between bodies at sensibly different temperatures."
The case of powder in guns as we are treating it is precisely similar to the case cited, with the exception that the 2d expansion does not take place, as the 1st will bring the portions of gas (treated as the above-mentioned bodies) to the same temperature and same pressure at the same time. But the entropy of the powder gas will determine its temperature at the muzzle, and therefore the work done; in short, it would determine the isentropic curve, according to which the gases would finally expand in the case of either instantaneous or progressive combustion.
M. Sarrau is probably correct in his statement that for any particular gun, charge, projectile, shape of powder grain, and material of powder, there is a size of grain that gives a maximum velocity, or that does in the gun the maximum amount of useful work. It is impossible, however, to discuss this maximum if we regard only primary causes. According to the most favorable treatment, the progressive powder can at best only equal the instantaneous. The maximum powder then cannot be much better than any other powder completely burned and equalized in a gun. If the loss of velocity due to secondary causes is less with the progressive powder than with the instantaneous, we might expect the most progressive powder which can be completely burned in a gun to bring this loss to a minimum; in other words, the powder, all of which is completely burnt just as the projectile leaves the muzzle, cannot be far removed from the maximum powder. If then we make (l/lT) = KY1 = 1 (page 403, Proceedings U. S. Naval Institute, Vol. XIV, No. 2), and solve for r, we will have very nearly the time of burning of the grain of maximum powder in open air. All grains smaller than this will be of quick powders, all grains larger, slow powders, the other conditions of loading remaining the same.
M. Sarrau's method of finding the maximum powder by means of the first derivative of an empirical value of v (formula 15), which is true for slow powders only, is of course inadmissible. Generally now in guns maximum powders cannot be used, as the pressures would be too high.
Messrs. Noble and Abel really treat only of quick powders in guns. In finding the velocity due to powders of the same material, they use for any gun a certain factor of effect, which they multiply into the theoretical velocity (according to their method) due to the weight of the charge.
They tacitly assume as a fundamental principle, without question or attempt at proof, that the muzzle velocity of a certain projectile will be the same in the same gun if the weight of charge and the material are the same, no matter what the form or size of the grains may be. This of course can be true only when all of the charge is burnt in the gun.
M. Sarrau in his later works really treats, on the other hand, of slow powders only. From certain premises he obtains an equation of motion, equation 1 (p. 318, Proceedings Naval Institute, Vol. XIV, No. 2), and for w substitutes a function of the travel of the shot. When, however, w becomes equal to W, or when all the powder is burnt in the gun, it is plain that w ceases to be a function of the travel of the shot, it being then a constant. M. Sarrau's treatment then will not hold for quick powders or those which are entirely burned in the gun. In "Velocities and Pressures in Guns" it is readily seen that M. Sarrau's velocity formula for slow powders is deduced and certain numerical functions calculated, identical in value with corresponding ones of M. Sarrau, starting with Messrs. Noble and Abel's fundamental principle for velocities with quick powders. The point of difference of these authorities is that each practically omits treatment of the half of the subject treated by the other, and that the value of the exponent n is assumed by M. Sarrau as 1.4, and by Messrs. Noble and Abel as slightly greater than I.
Similar guns are those of which the bores are, or may be considered, similar right cylinders. They are similarly loaded when the volumes of powder chambers and the weights of the charges and projectiles vary as the cubes of the calibers, and the powder grains are similar in figure, their corresponding linear dimensions being proportional to the calibers. It is readily seen that "in similar guns similarly loaded, the velocities and pressures corresponding to distances passed over proportional to the caliber are equal."
Taking formula (4), (page 400, Vol. XIV, No. 2, Proceedings Naval Institute), we see that the quantity in parenthesis is abstract, and the same for the same travel in two similar guns similarly loaded; it is readily seen that the term outside the parenthesis reduces to the same quantity in the two cases, when w equals W the full weight of charge; that is, when all the powder is burnt, as with quick powders.
For slow powders w = WaKY1(1 – ?KY1 + µK2Y12) (page 403, "Velocities and Pressures in Guns"); w for the two guns is proportional (in place of equal) to W, and the principle is proved for v as before. The proof for pressures is similar.
In the formula pvn = a constant, which represents the instantaneous powder process, I will state that I do not consider n — 1.4 as final, or necessarily the best value; n is greater than 1 and probably not greater than 1.4. Tables could just as readily be calculated for 1.2 or any other value as for 1.4. This latter is simply true for the perfect gas. Nor is it certain absolutely that the velocity of combustion varies as the square root of the pressure (static). Any other power of the static pressure could be followed out just as readily, however, if further experiment should indicate another value. These values, however, lead directly to the best (probably) existing empirical formula for velocity, and therefore cannot be very far from the truth.
It then appears that the accelerating pressure when powder is burning in a gun is composed of two quantities, one of which we may call the natural pressure, a static pressure corresponding, to take a familiar case, somewhat with the barometric pressure of the atmosphere; the other quantity depends on the rate of evolution of the powder gas, and corresponds almost exactly with the pressure on a surface due to a wind. Suppose a very low barometric pressure forward of a shell, a high barometric pressure in rear with a strong wind blowing directly into the chamber of the gun, and the case is not dissimilar to what really must take place when a progressive powder is burnt in a gun. When the powder is all burnt, to carry on the similarity, the wind will almost instantly die out and the gas will exert a pressure due to its weight, volume and temperature alone, a purely static pressure.
The quantity depending on the rate of evolution of the powder gas plays a still more important part with explosive compounds than with powder. If a finite quantity of explosive could be burned and equalized (as regards temperature and pressure) in an infinitesimal time, the pressure would be infinite. This is not realized, but is approached, when explosives are detonated. It undoubtedly partly, if not entirely, accounts for the great differences observed between the results of primary and secondary explosions of the same explosives.
It must be remembered that there are a number of different ways in which the subject of powder in guns may be treated, and in choosing any particular way the reasons that lead to that choice should be given. No theory or practice can make a progressive powder do more work than an instantaneous powder, though experiment seems to show that the former may do more of useful work. The theory chosen makes them equal as to work, and the experimental method of determining coefficients (by means of velocities in two guns) assumes them equal as regards useful work. In this latter the method is slightly in error. A factor of effect may be involved in the above-mentioned coefficients (A and B of formula (15) or M and N of (20), "Velocities and Pressures in Guns"). If this factor is found to vary slightly with the size of guns, the coefficients for any powder for a certain gun should be determined from guns of approximately the same size. For fairly approximate values of the coefficients to cover a number of guns of different sizes, guns of nearly the extreme sizes should be chosen. Dissimilar guns or dissimilar loadings should always be chosen, in order to make the ratios of the two terms of the formula as different as possible for the different guns, thus permitting a more accurate solution where the given quantities are at best only approximate.
Many powders now-a-days are not uniformly dense, being most dense on the surface and correspondingly slow burning, and least dense in the interior. If the velocity of combustion of powder varied inversely as the density, the formulas for velocity and pressure as deduced would apply equally well to these powders, provided that layers of uniform density were regularly arranged in the powder grains. This law, however, does not hold, and this fact may be expected to cause some error in the use of the formulas of "Velocities and Pressures in Guns."
It may be remarked that with slow-burning powders, such as is German cocoa powder when used in the 6-inch B.L.R., there is an interior layer in the grain that might as well be non-explosive, so far as giving velocity to the projectile is concerned.
A part of the work done by powder is done on the gun itself. In fact, the gun is only a projectile of certain mass (or virtual mass), and a pressure and velocity curve could readily be constructed for the travel of the gun while the projectile is in the gun, just as for the travel of the projectile during the same time. The work done on the gun in this way is of course small relatively, and taking account of it would generally be a useless refinement.
It may be stated, in closing, that no method dealing with pressures in guns as purely static can account for the velocities obtained. The pressures are partly static. The resistance to the projectile in air is quite similar to the pressure in the gun, and, as well as the resistance to a ship moving through the water, is very largely dynamic. The greater the velocity with which the explosive gas is evolved, the greater the pressure; just as in the air, the greater the velocity of the projectile (or the more quickly the air ahead is compressed an-d that in rear expanded), the greater is the resistance.