Help With Understanding Thrust Calculations (rifle muzzle brake)

[Ozen]

TL;DR Summary

I have two equations for thrust on a muzzle brake, however they each yield different results, which is correct?

Let me start off by saying I decided to post this in the aerospace engineering thread because it directly deals with thrust, even though it is not for a plane or similar.

I have two equations that I can use to calculate the force on a baffle of a muzzle brake, Equation 1, from The Engineering Design Handbook Series, and Equation 2, from Armament Engineering, a computer aided approach. But both yield different results with the same inputs (even when converted to there correct counterpart. Equation 1 yields significantly higher results than equation 2. Below is a run-down on solving them both for the same given system, note that they don't use the same variables always.

Givens:

Ae = .000029 m^2 (bore area)
mc = .00169 kg (charge mass)
mp = .00402 kg (projectile mass)
Vt = 9.2E-06 m^3 (16in barrel volume); 3.5E-06 m^3 (7in barrel volume)
vo = 892 m/s (muzzle velocity); 650 m/s
pe = 5.52E+07 PA; 1.17E+08 PA (chamber pressure)
lambda = 1.64 (factor)
Ct = 2.21 (correction factor)
RT0 = 700119 m^2/s^2 (gas constant * average temperature of gas at shot ejection)
pmz = pe * (1 - (mc / (2 * mp +mc)) (pressure at muzzle exit)

Equation 1:

Fb = 0.26 * lambda * Ct * (mc / Vt) * Ae * RT0 * (1 + (mc / 6 * mp))
***Units must be in imperial, all were converted to their correct correspondent before computing

Solved:

16in barrel: Fb = 9385 N
7in barrel: Fb = 29389 N

Doesn't this seem rather odd? Almost triple the force for a system that has lower velocity, lower volume, and higher pressure.

Equation 2:

Fb = pmz * Ae * lambda * Ct
***Units must be metric, same as above

Solved:

16in barrel: Fb = 4794 N
7in barrel: Fb = 10162 N

My Question:
Which equation should I use? I lean towards Equation 2 since that seem more realistic, but if that is wrong, the material may yield and break when I would test a design. Or worse, the design could work for testing but have a bad fatigue life and end up breaking not long after.

***Supplemental question: Why do shorter barrel systems produce more thrust on the baffle than the longer barrel systems?

 

[jrmichler]

Why do shorter barrel systems produce more thrust on the baffle than the longer barrel systems?

Both short and long barrels will have the same peak combustion pressure, but the longer barrel puts more of that energy into bullet velocity. The pressure is lower at the point where the bullet leaves the muzzle with the longer barrel.