Homer Powley has worked in the Internal Ballistics field and has developed a lot of experience with small arms ballistics. He is famous for creating the "Powley Computer for Handloaders" which is a set of slide rule like devices for computing optimum loads for a generalized gun given the charge to bullet mass ratio, case capacity and expansion ratio. The computer calculates the IMR series powder type and charge to use, and gives muzzle velocity. The calculator assumes a fixed loading density of 86% of the case volume and a peak pressure of 45K psi. The second calculator relates chamber pressure to velocity, given the expansion ratio, case capacity and charge.
The Powley equations may be found in the NRA Handloading manual and in the BASIC program POWVER22:
Note: recently a reader has brought to my attention that the BASIC program above has mistakes. I have verified the mistakes and that it seems to differ from the slide rule version, but have not determined correct fixes yet. Frankly I would rather spend my time on more advanced models, so if someone wants to check this out I'll be grateful.
The first calculation is based on a pressure/ powder quickness formula that is reminiscent of the Le Duc formulation. Neglecting the constants the formula is:
P = w*(C/w)^0.6/(A*q*(1-C/(d*K0))
where q is the powder quickness factor, and d is the powder density of 1.62 gm/cc. C is set at 86% of the case volume, P is set to 45Kpsi, and the equation is solved for q, which is then related to the IMR powder type. The Le Duc formulation, using the velocity potential formula for the "a" parameter is:
P = w*(C/w)/(A*q*(1-C/(d*K0)*(K0/w)^(2/3))
The difference is Powley drops the K0/w term in the denominator, and uses a C/w ratio raised to the 0.6 power instead of 1.0. The charge dependence of this pressure formula is even lower than that of the Le Duc method, however wisely, no attempt is made to use this formula for determining charge to pressure dependence of the loading. The formula is used only to relate bullet weight and powder quickness at a given pressure and loading density. The second calculator/equation is used to relate pressure, charge, and velocity.
The second equation, from the NRA Handloading book is:
V^2 = K*C*(1-R^(-1/4))/(w+C/3)
where v is the velocity,
K is a constant,
R is the expansion ratio
w is the bullet weight,
C is the charge weight.
The classical theory section derives the formula for the energy of an adiabatic expansion of a gas. The equation is:
½*w1*v^2 = Pi*Vi/(1-gamma)*[ (V/Vi)^(1-gamma) -1 ]
where Pi and Vi are the initial Pressure and Volume respectively, and V is the final volume. Gamma is typically 1.25 for powder gases. V/Vi is the expansion ratio, Pi = F*C/Vi if all the charge is burned at the initial volume, so we have:
½*w1*v^2 = K*C*(1-R^(-1/4)) where K is a constant. w1 = w + C/3 to account for the (1+C/(3w)) average to bullet base pressure ratio.
In summary the Powley velocity equation is a simple instantaneous burn, frictionless, adiabatic expansion solution, where the constant K is adjusted to suit an average of experimental results.