Newsgroups: sci.military.moderated
From: higgins@me.mecheng.mcgill.ca (Andrew Higgins)
Subject: Re: Maximum effective length of 120mm tank gun barrel
Date: Thu, 12 Mar 1998 14:05:16 GMT

In article <EosA73.13p@ranger.daytonoh.ncr.com>, roger@police.tas.gov.au
(Roger Fleming) wrote:

>careysub@earthling.net (Carey Sublette) wrote:
>[...]
>>u = 2c/(gamma - 1)
>>
>>where u is the velocity and gamma is the specific heat ratio of the gas.
>[...]
>>Thus u has a limiting value of eight times the speed of sound in the same
>>combustion gases. The speed of sound in hot dense combustion gases will be
>>much higher than the 330 m/sec of air at 0 C (273 K). Since the combusion
>>gases are something like a factor of ten hotter than air[...]
>
>Carey, I always appreciate your informed and reasoned posts, but there seems
>to be a problem in this analysis. By the time the expanding gas (expanding
>adiabatically along a frictionless tube into vacuum) has reached its limiting
>velocity (or at least, very closely approached it) it will have cooled and
>expanded considerably. In fact, under these idealised conditions, the
>limiting temperature and density are both zero. So what value of c should be
>used?

The *initial* value of c, before the projectile and gases have started to
move, is the correct value to use in this analysis.

This is a classic mistake people make in analysis of unsteady flows (such
as internal ballistics).  Since the maximum velocity, as Carey pointed
out, is

     u = 2c/(gamma - 1)

some people are tempted to say that the maximum projectile velocity is
Mach 5, since u/c = 2/(gamma -1) = 5 for gamma = 1.4.  However, c here
refers to the initial sound speed of the propellant gases, before any
expansion has occurred.  It is more properly labeled "c0."  Once the
projectile begins to accelerate, the gases expand and cool, and the
projectile velocity with respect to the gas immediately behind it (Mach
number) could be much greater than 5.  The projectile's Mach number with
respect to the air is another matter entirely.

In unsteady flows, you must be careful to reference everything to the same
initial state:  in guns, that is the initial conditions of the propellant
gas in the breech.  In real guns, this becomes complicated since the
projectile begins to move *before* all the propellant is consumed and the
maximum breech pressure can occur *after* projectile starts moving.
Carey's equation applies to a "preburned propellant" gun where all the
powder is converted to gas while the projectile is still locked in place.

If you have access to a good engineering library, see:

     A. Seigel, "The Theory of High Speed Guns," AGARDograph 91, 1965

for an excellent introduction to internal ballistics.
--
     Andrew J. Higgins            Department of Mechanical Eng.
     Shock Wave Physics Group     McGill University
     higgins@mecheng.mcgill.ca    Montreal, Quebec