In this paper are presented, (1) certain new ways of regarding the quantities (muzzle energy)/(pressure), (muzzle energy)/(weight of powder), (muzzle energy)/(weight of gun) together with a new measure of the progressiveness of powder ; and (2) the apparently best method of obtaining muzzle and chase pressures in guns.
I.
For all guns made and tested at the present time, the maximum pressure upon the breech plug and the muzzle velocity are ascertained by direct measurement with pressure gauges and chronographs. The first of these quantities, it is not infrequently asserted, is not correctly given by the methods generally followed ; and the second is universally admitted to be slightly greater than the velocity of the projectile at the instant that it cleared the muzzle ; for the reason that the shot is, for a time after clearing the muzzle, enveloped in powder gas whose velocity is greater than its own. The correctness of the maximum pressure given when powder is the explosive used and with ordinary methods, has, however, been experimentally shown by Sarrau and Vieille, in the Etude sur L'Emploi des Manom6tres a Ecrasement; the authors having ascertained by experiment that, when a copper cylinder is subjected to a given maximum pressure, whether rapidly applied by a Rodman or other similar machine, or reached with the quickness ordinary in guns, the shortening will be the same. We have then, always, the correct maximum pressure to which the breech of the gun has been subjected, and a velocity which is greater than the true muzzle velocity. From these data, and others which are available, the values of the three quantities already referred to are derived.
It will conduce to clearness to examine these three quantities, and ascertain their dimensional values if possible. In (muzzle energy)/(pressure) , the pressure taken is always that on unit area. If, however, we take the total pressure upon the projectile s base, (muzzle energy)/(pressure) is a length; that which the gun must have in order that, by acting continuously along it, the maximum pressure may produce the known muzzle energy. This length is always in practice less than the length of travel of the projectile in the gun ; and the ratio of this effective length, as it may be called, to the travel of the shot is the only available true measure of the progressiveness of the powder when only the data referred to are had ; perfect progressiveness being reached by the maintenance of a uniform pressure along the gun's length. In this case the ratio above suggested— (effective length)/(length of travel)—is evidently numerically equal to unity.
As long as the weights of gun, projectile, and powder do not change, the quantities muzzle-energy per pound of gun and per pound of powder will increase or decrease together ; when, by any alteration in other circumstances of loading, the form of the pressure curve is altered. The latter change requires, of course, that the external form of the gun should be changed. If we suppose that the form of the pressure curve may be altered as we please, and also permit of variation in the weights of gun, projectile, and charge, it becomes evident that increase in energy per pound of gun is directly opposed to increase in energy per pound of powder. For, if we wish to increase energy per pound of gun, we must dispose the available weight of gun in the best manner possible. To do this, since the hooptension which is transmitted to any point in the mass of a thick hollow cyhnder, when subjected to inside stress, diminishes as the square of the radius as we go outwards, the weight must be put at as short a radius as possible, or the gun must be a cylinder. If then the safe pressure acts uniformly throughout the gun's length, the greatest attainable value of energy per pound of gun will be reached. Also it is clear that, since in any actual case where the above conditions are realized, a part of the powder will be blown out unburned, and the final tension of the gas be high, energy per pound of powder must be relatively small. The greatest value which energy per pound of gun can theoretically reach will be attained when the gun's chase is a very long and very thin cylinder, and is numerically about 9000 foot-pounds per pound of gun, with a gun built of steel whose elastic strength is 45,000 pounds per square inch. One-eleventh of this limiting value would be a large figure to reach in practice. It is to be observed, however, that it is more precise to use the weight of the gun without the chamber in statements of energy per pound of gun, as this part of the gun alone directly produces muzzle-energy. The figures just given have been so arrived at.
To increase energy per pound of powder, on the other hand, we must use the powder gas expansively, so to speak. The pressure must be allowed to reach a high maximum, and the gun must be long enough to allow the gas to expand to a low pressure. In this case the gun must be very thick at the breech, and may thin away towards the muzzle ; the lengthwise section of its exterior being concave outwards. Evidently the conditions which the gun and charge must fulfill in this case are totally different from those of the former case.
It thus appears that, looking at the question broadly, the endeavor to attain high values of both these quantities leads to compromises and a mediocre result in each.
Unless the difficulty of stowing and carrying large charges becomes of commanding importance, it is sound practice to endeavor to increase energy per pound of gun by variation of the form of the pressure curve and outside form of the gun, with a secondary regard only to energy per pound of powder. As energy per pound of gun is increased by causing the pressure curve to approach a straight line parallel to the axis of the gun, the effective length will approach in value to the length of travel m the gun, or the ratio (mean pressure)/(maximum pressure) approach unity. The endeavor to make the mean pressure approach the maximum pressure has led to the use of enlarged chambers in guns, and it is interesting to note that the effect of this feature of practice upon the weight of the gun is small. For it is easy to show from the formulae ordinarily used for computing the strength of guns, that if we neglect the weight of the breech plug, or consider the chamber open at both ends, its weight for a constant internal volume (or constant weight of charge) and constant strength is independent of the internal radius. In other words, as between two chambers whose internal volume and safe maximum pressure are the same, the only difference in their weights will be that of their breech plugs. This is evidently the only fair way of making the comparison in question.
II.
There is at present no universally accepted method of determining muzzle and chase pressures in guns, and it is intended here to present shortly the methods of determining them by Sarrau's equations and to advocate the adoption of these. Mr. Anderson, C. E., of England, in his Howard Lectures on the Conversion of Heat into Useful Work, has recently presented a method in which he assumes the temperature of the powder gas at the instant the projectile clears the muzzle ; and from the further assumption that the temperature and pressure of the gas are connected by the well-known laws which apply to more simple gases, deduces the muzzle pressure. It is obvious that we might as well assume the pressure at once, and the method cannot be admitted to be of practical use. Mr. Anderson's advocacy of S6bert's methods is well judged; but the application of these to practice has hardly begun, Sebert himself giving very few; while his reasoning does not contain fewer unsatisfactory approximations than Sarrau's.
It may be said that little or nothing is known of the muzzle and chase pressures in most guns; and yet this knowledge is most essential, with regard to both the safety of the gun and the intelligent study of the behavior of the powder. It will be admitted that if we have the law connecting the velocity of the projectile with the distance passed over, we can find the pressure upon the shot's base throughout any distance where this law holds good.
It is to be noted that the velocity which will be used in (1), probably in the way of the calculation from experimental results of the constants which it contains, is, as has already been remarked,, slightly too great; and consequently the pressure which we shall derive from (2) will probably be too small. Also, the value of the pressure derived from (2) is, assuming (as we cannot do better) the velocity to be the true value of the velocity which the shot had at the instant its base cleared the muzzle, the pressure upon the shot's base at that point. And similarly, the value of P derived for any particular value of u is the pressure upon the shot's base. It remains to examine whether this is the greatest pressure which acts at the point considered during the phenomenon of explosion, for this greatest pressure is what the gun-maker wishes to ascertain.
Suppose then we have found a certain pressure P1 at a point u1 from (2). The pressure in the gas as we pass backwards to the breech from u1 must increase, because the velocity of the gas decreases. We have then some pressure P1 at the breech which is greater than P1. The difference between P1 and P1’ is not known, but the latter is generally estimated to be from 25 to 50 per cent, the greater. Nor is the form of the line of pressures connecting P1 and P1' known ; but it is extremely improbable that it is concave towards. The accuracy, though not as close as could be desired, is yet tolerable. The pressures given by (5) are smaller than those given by (3), by .18, .18, .20 of themselves. The equation (3) would of course be used whenever the two constants could be determined; and so closely does it agree with the results of practice when u is widely varied, that its credibility must be held to be of a high order.
In all cases, after a pressure curve is drawn by the method above, in the neighborhood of the muzzle and chase, and near the breech by methods given in Sarrau's work, and exemplified in Elastic Strength of Guns, already referred to, we have a valuable check on it, in that the area under it (BHKC in the first figure) must be equal to the muzzle energy of the shot. For it is evident that the curve of pressures with which we are dealing is the locus of those pressures which have given the shot motion. If we could ascertain the locus of the mean pressure in the gaseous mass behind the shot at each instant, the area under this second curve would be the whole work done in the gun on the charge and projectile. In two separate and distinct pressure curves which the writer of this article laid down conjointly with Lieutenant R. R. Ingersoll, U. S. N., this check was used by applying Simpson's Rules to the calculation of the area under the former curve, and was found to satisfy with remarkable accuracy.