[From Interior Ballistics]
Let MN (see Figure) represent the burning surface of a powder grain (smokeless or otherwise). From this, as shown in the figure, issues a stream of powder-gas normally. Denote the pressure of the powder-gas at the instant of formation by F, and the pressure of the gaseous surrounding medium, outside the issuing streams of gas by P. Also place p equal to the varying pressure in the stream, and p equal to the variable mass of unit volume, in it. Suppose the cross-section of the stream to be unity, and assume anywhere in it a lamina of thickness ds and at right angles to it. The measure of dp, the accelerating force of the lamina, will be (mass X acceleration)
The initial velocity of the issuing gas is o, and we will denote the final velocity by Vl.
Integrating between proper limits, combining constants in one when possible, and remembering that the velocity increases as the pressure decreases.
This is the velocity with which the gas passes a point in the stream where the pressure is P. The mass of unit volume of the gas at the same time will be determined by and the mass of gas that flows past the above point in unit of time will be.
But if the density of the powder grain is constant the mass of gas formed, which will be the same as that which passes any point, will be proportional to the velocity of combustion. We may, therefore, write for the velocity of combustion,
If F could be increased the velocity of combustion might be increased indefinitely. If P is constant, we know that is; therefore F is constant. The surface of the powder grain next the heated gas is evidently in a state of unstable equilibrium, held in check by the pressure, and "if the pressure on the surface is too great, it will remain in that state. As the pressure is lessened by expansion of the gas, the reaction already begun on the surface is completed and gas is again produced, raising the pressure till again too great, stopping the reaction as before, and so on.
In order to keep the mass of gas flowing past a fixed point in the stream constant, the density of the power being variable, the "velocity of combustion must vary inversely as the density." This law was first enunciated, as the result of experiment, by General Piobert.
This agrees with the experiments of M. de Saint Robert. He filled a lead tube with gunpowder, drawing the tube out and cutting it into equal lengths. These lengths were burned at different altitudes above the sea. From these experiments, M. de Saint Robert concluded that the velocity of combustion of gunpowder varied as the 73 power of the pressure of the surrounding medium. Captain Castan verified the increase of velocity of combustion with pressure by means of a tube containing powder and having a hole for escape of the gas. By changing the size of the hole he changed the pressure.
As P begins to be appreciable comparatively with F, the quantity in parentheses may be represented approximately by the product of a constant and a small negative power of P. This power increasing (negatively) is still quite small for the pressures employed in guns. As a sufficient approximation we may, therefore, employ a principle which is undoubtedly the result of the closest observation, that of M. Sarrau, namely: The velocity of combustion of gunpowder (in guns) varies as the square root of the pressure of the surrounding medium.
It may be noted in the deduction of (1) that no account is taken of the non-gaseous residue of gunpowder. The close agreement of the formula with M. de Saint Robert's experiments, however, seems to indicate that the consideration of the residue would effect a needless complication of the subject. Moreover, this consideration would be impossible unless the residue were supposed formed in its entirety at the instant of combustion, and there is no real reason to suppose that it is not a result of expansion. For smokeless powder, and for other explosives leaving no residue, in process of combustion as distinguished from detonation, the above equations should hold with exactness.
(Additional formulas are available in the PDF.)