MODIFIED FOR SMOKELESS POWDER.
The Sarrau binomial formula for velocity, appearing in the various publications on interior ballistics,is based on experimental data obtained, using black powder. It has not been found applicable for use with smokeless powder.
About two years ago sufficient data appeared to me to be available in the records of the Naval Proving Ground for a revision of constants of this formula. Using this data, I worked out a formula, which has been found reliable for all densities of loading from 0.3 to 0.7, and is in many respects more elastic and convenient than others now in use.
The method was as follows:
Record of a lot of powder fired in two guns with different chamber capacities, all other ballistic elements remaining constant—
Gun. | Charge. | Velocity. | Chamber Capacity. |
A | 300 lbs. | 2510 f. s. | 16974 cu. in. |
B | 340 “ | 2860 “ | 14970 “ |
Then
(340/300)x (14970/16974)y=(2860/2510), (1)
and
x log a + y log b =log c. (2)
Adopting the approximations of Sarrau in the general equation*
V=A(fa/r) ? ¼+r S1/2+2r u r /W¼C1-2r ?r-1/4 [1-B ?/r (Wu)½/c], (3)
*NAVAL INSTITUTE, Vol. X, No. 5, page 167.
in which
x=1/4 +r,
y=1/2-2r,
y=1-2x.
Then in (2)
x(log a-2log b)=log c – log b
x=log c – log b/log a -2log b.
Solving, (1) gives
x=.68
y=-.36
r=.43
Substituting the value of r, in equation (3), we have
V=A(fa/r) ½ ?.68 S.36 u.43 /W¼C.14 ?.18 [1-B ?/r (Wu)½/c].
?.18 is sufficiently near a constant.
Taking the nearest approximation of exponents gives
V=A ?.7 u.4 /W¼C.14 ?.18 [1-B ?/r (Wu)½/c],
in which
?=charge in pounds.
u=travel of shell in pounds.
Q=weight of shell in pounds.
S=chamber capacity in cubic inches.
C=caliber in feet.
Representing gun constants by
a=u.4/W¼ S3/8C1/8
?=(Wu) ½/C,
gives the working formula
V=A ?.7a[1-B?].
The following results are fair samples of what this form will do:
Powder. | Fired in. | Velocity Calculated. | Velocity Found. |
I. H. X. Lot I | 8”.35 | 2097 | 2100 |
I. H. X. Lot I | 13”.35 | 2005 | 2000 |
I. H. X. Lot I | 7”.45 | 2687 | 2700 |
I. H. X. Lot I | 8”.45 | 2586 | 2600 |
I. H. L. Lot I | 6”.40 | 2159 | 2150 |
I. H. L. Lot I | 4”.50 | 2622 | 2600 |
I. H. L. Lot I | 6”.50 | 2584 | 2600 |
I. H. L. Lot I | 5”.50 | 2683 | 2700 |
I. H. C. A. Lot II | 10”.30 | 2015 | 2000 |
I. H. C. A. Lot II | 6”.50 | 2615 | 2600 |
D. S. B. Lot 9 | 6”.50 | 2586 | 2600 |
D. S. B. Lot 9 | 5”.40 | 2318 | 2300 |
D. S. A. Lot 2 | 4”.40 | 2005 | 2000 |
L. R. B. Lot 1 | 6”.50 | 2180 | 2181 |
I. H. D. D. Lot 2 | 8”.45 VI | 2750 | 2740 |
Having found the velocity formula accurate, a graphic method was sought to show relations between constants A and B and the web thickness of standard powders.
Sarrau's formula for maximum pressure:*
P=Ka4 (?W)½/C2 (??/?-?)½
K=constant.
a=variable depending on quickness on powder.
?=weight of charge in pounds.
W=weight of shell in pounds.
?=density of powder.
?=density of loading.
C=caliber in feet.
was found to agree reasonably well with experimental results for maximum pressures by gauges. With medium densities of loading variations were not more than 0.3 tons between calculated and experimental results in a large number of cases.
Having found the value of Ka2 for a number of powders from the firing records, the value a=1 was assumed for a particular powder and the values a, A and B plotted. Faired curves for these points are shown on the appended plate. The web thickness curve is for standard navy powder of 1910, multi-perforated grain. In using the formula, the black ignition charge is neglected, w being the weight of smokeless powder only.
*NAVAL INSTITUTE, Vol. X, No. 1, page 142.
While the approximations assumed may be open to criticism, the velocity formula has been particularly successful in predicting charges for unusual velocities and for new types of guns. The pressure formula is not considered sufficiently reliable for general use.