From: sfaber@ihlpb.att.com (Steven R Faber)
Newsgroups: rec.guns
Subject: Barrel Wear
Date: 11 Nov 92 20:53:00 GMT
Organization: AT&T


In a previous post Bart Bobbit proposed an empirical formula for
determining barrel wear.  I wanted to try and correlate it with
some theory or calculations, and this is the result:

First the premise is that wear that determines the life
of the barrel is due to gas leakage around the bullet causing
chamber throat erosion before it engraves into the bore and seals
the gases.  I don't know how this can be related to shorter barrel
life for rapid firing, unless it is due to hotter gases because of 
a hotter barrel.

The "Thermodynamics of Firearms" by C. S.Robinson reference provides
the formulas needed in the section that computes the back pressure 
due to pushing a column of air out of a gun bore.

The Fanning formula gives velocity^2 = dP/dL*g*d/(2*f*rho)
where dP/dL is the pressure gradient, g= acceleration of gravity,
d is the diameter of the pipe or in this case is 2* the width of
the annulus between bullet and case, f is a friction factor that
is a function of Reynold's number (an impirical function when
in the turbulent vs laminar flow region) and rho is the density
of the gas which can be determined from the ideal gas formula
using the average molecular weight of the combustion gases.
The ideal gas correction factors were not used in this computation.
Maybe later if I can find a reliable way to figure them out.

The Reynolds number depends on d,v,rho and the viscosity of the gases
which can be determined from the Sutherland formula averaging the 
viscosities of the constituent combustion product gases.
The velocity could then be determined by iterating several times
until the equation converges on a solution.

The friction factor could use more researching since it depends, I
think on the type of surface and heat conductivity etc, but I just
used the equation given in the reference.

Given the velocity, we need to compute the pressure vs time until
the bullet seals in the bore.  To do this I assumed a linear
rate of increase in pressure derived from some published pressure curves.
The time can then be computed by t^3=6*m*F/(K*A) where k is the pressure
rate if increase, and F is the freebore distance of the bullet, m is mass
and A the area of the bullet cross section.

30-06  150gr FMJBT  48gr 3031  
    k         t        P        d       L      F     v          G
200psi/usec  140usec  28000psi  .008in .35in .124in 5817ft/sec 13.3gr/msec
gas lost	erosion
 .92 grains	236 gr/in^2

The above is the result for a boattail bullet.  G is the mass flow rate
of gas at the final pressure P when the bullet is sealing. v is
the velocity of the gas leak.  The calculation was done first by
calculating the G for a number of pressures up the curve, and summing,but
it was found that G was amazingly linear with P, so the total gas loss
can be computed from the average using the final G above.
Erosion would then be the gas flux (rho *G ) integrated over time.

Next is the result for the same weight M2 square base bullet.
Everything is the same except L, the length of the annulus is greater
(.51in) causing more resistance to gas flow.
L    v        G           gas lost   erosion
.51  4755   10.9gr/msec    .76 gr    195gr/in^2

So if when shooting boattails the barrel lasts 3000 rounds,
the square base bullets should give about 20% longer life or 3600 rounds
as predicted above.

The following is the first case using a slower powder:

30-06  150gr FMJBT  53gr 4064
    k         t        P        d       L      F     v          G
174psi/usec  147usec  25580psi  .008in .35in .124in           12.05gr/msec
gas lost	erosion
 .89 grains	227 gr/in^2

The pressure slope is lower, time higher, but P is lower giving overall
lower gas loss and erosion, but only by about 4%.

Next is the calculation for .223 Rem 55gr FMJBT  27gr AA2230

    k         t        P        d       L      F     v          G
166psi/usec  123usec  20420psi  .012in .265in .1in 8444ft/sec 15.6gr/msec
gas lost	erosion
 .96 grains	221 gr/in^2

Note that the gap in the annulus is greater with this gun even though
the caliber is smaller.  The pressure did not get as high in this case,
but the leak is even greater than the 30-06.  The erosion is almost
comparable (the density of gas is lower in this case).

According to another source the mass flow rate for an eccentric
annulus is up to 2.5 times that of the concentric case, so the
total leak could be quite a bit more if some eccentricity occurs.

It is easy to see how the peak chamber pressure of the .223 Rem 
above could drastically increase  if the bullet is seated out
at the lands.

Well, I can't really correlate much with Bart's idea since there
seem to be a lot of variables that don't scale well with caliber,
but I hope it gives some insight into the problem.

Steve Faber