The Le Duc method of internal ballistics originated from a course of lectures by Captain Le Duc published in 1905. The method was adopted as a standard for the U.S. Navy well into WW1. The system is almost entirely empirical and is based on a simple equation that approximates the velocity-space curve for a projectile in a gun.
V = a*x/(x+b)
x is the distance of bullet travel
v is the velocity of the bullet
a, and b are parameters of the model.
This model is about as far as the usual gun magazines get into internal ballistics. It is convenient to use the model to figure the effect of gun barrel length on the bullet velocity given 2 velocity - distance points that can be used to determine the parameters a and b. It is also possible to solve for the parameters given a peak pressure, velocity, and distance point, since the above equation can be differentiated to give the acceleration which is proportional to pressure. dv/dt = v*dv/dx = a^2*b*x/(x+b)^3
The peak acceleration/pressure point can be deduced by taking the derivative and setting it to 0.
V'' = (a^b^2 - 2*a^2*b*x)/(x+b)^4
so x = b/2 when v'' = 0. This means peak pressure occurs at x = b/2.
Substituting b/2 for x in the equation for v' gives v' = 4*a^2/(27*b) at peak pressure. From F=ma we write
Ps*A = m*v'
where Ps is the pressure on the base of the shot (bullet) and A is the area of the bore, and m is the bullet mass. The relationship between bullet pressure and gun breech pressure is Pb (breech) = (1+C/(2*m))*Ps as derived in the classical theory section, where C is the powder charge mass. The expression for breech pressure, adding in bore friction Fr would then be:
Pb = (1+C/(2m))*m/A*[a^2*b*x/(x+b)^3 + Fr] and the peak pressure at x=b/2 would then be:
Pb(max) = (1+C/(2m))*m/A*[4*a^2/(27*b) + Fr]
This equation coupled with the original velocity equation makes it possible to tabulate v vs. x given a peak pressure and velocity/distance point. Consult the program section of the home page to pick up a free LECUC program which does this. Once you know a and b, it is possible to graph the pressure-space curve for the system with the above equation for Pb as below for a typical .223 Rem load.
The Le Duc model outlined in "Internal Ballistics" by Hunt, goes further by giving expressions for determining "a" and "b" in terms of the loading conditions. "a" is related to the "potential" of the gun, since "a" is the maximum velocity as x becomes very large. The potential is defined in the classical theory section as the maximum velocity obtainable by a theoretical gun of infinite barrel length with no heat or frictional losses.
v = sqrt( 2*F*C/(w*(gamma-1)))
where gamma is the heat capacity ratio of the gases, and
F is the force constant of the powder,
C is the charge weight.
The old Le Duc formula for "a" included a loading density factor now considered somewhat fallacious and is:
a = 6823*(C/w)^1/2*(27.68*C/K0)^(1/2*(gamma-1))
where K0 is the chamber volume in cubic inches, and C and w are in lbs. gamma = 7/6 here.
It is hard to relate "b" to any physical calculation, but it was assumed that the distance to peak pressure was proportional to some function of powder quickness, initial air space, and some power of the chamber volume and projectile weight:
b = q*(1-C/(d*K0))*(K0/w)^2/3
where q is the powder quickness factor, and
d is the powder density
If we plug these expressions into the peak pressure equation, dropping the constants:
Ps = C^(7/6)*w^(2/3)/(A*q*(1-C/(d*K0))*K0^(5/6))
This suggests that peak pressure is proportional to charge to the power of a little more than 2, assuming the (1-C/(d*K0)) factor in the denominator gives the effect of increasing pressure by a factor of "C". More modern models give a charge dependence of C to the third or fourth power.