by supposing the gun under consideration, together with its shell and charge, to be expanded or contracted symmetrically until its calibre is equal to I inch. Corresponding to G, W, S, M, and L, for the gun calibre d in. we shall have for the standard gun
V and P will remain unchanged.
The working formulae for muzzle velocity and maximum pressure will be based on the assumption that they can be put in the form
(1) V = K, Gi Si' Wi" Mi" Li'
(2) P = K p Gi Wi-Mi-Li'
Where g, s, w, m, and / are empirical indices, positive or negative integral or fractional, and K v , K p empirical constants. The values of the indices denoted by the same letter are different in (i) and (2).
These formulae are to be regarded as purely empirical, but with a due appreciation of their possibilities and limitations they will be found very useful working tools.
Assuming as the result of experience, suitable values for the indices, we can from known firing results (muzzle velocity and maxi- mum pressure) for a certain gun, charge, and shell evaluate K and K p of (i) and (2) by simple calculation with a table of logarithms. Then by a reverse process, using the values of K. and K p so obtained, we can calculate the muzzle velocity and maximum pressure to be expected with another gun, shell, and charge (of the same propellant made up of grains of the same form), inserting the appropriate values of d, Si, Wi. Mi and LI in (i) and (2).
The following tables have been drawn up as a guide to values of the indices which will be found suitable for guns, at any rate for trial purposes, with the following propellants :
Long cords of Cordite M.D. (M.D.).
Long tubes of Cordite M.D.T. (M.D.T.).
Short tubes or m.p. grains of nitrocellulose (N.C.T.).
TABLE I.
| A table should appear at this position in the text. See Help:Table for formatting instructions. |
Indices for Muzzle Velocity.
g-
i.
W.
m.
I.
-0-25
+0-2
-0-4
M.D.T. +07
N.C.T. \ , M.D. \ +' 6
M.D.T. \ N.C.T. / --3
M.D. -0-15
TABLE II.
Indices for Maximum Pressure.
g-
w.
m.
I.
M.D.T.-i-oo N.C.T. -i-io M.D. -1-15
+0-6
M.D.T. +1-8
N.C.T. \ , M.D. / + 1 ' 6
M.D.T. \ N.C.T. / - I- 4
M.D. -0-85
An example to illustrate the uses of these tables will now be given :
A 6-in. gun, chamber capacity 1600 in., shot travel 250 in., fires a loo-lb. shell with a charge of 25 Ib. of M.D., diam. of cord 0-2 in., gives a muzzle velocity of 2650 f/s with maximum pressure of 16 ton/in 2 . What muzzle velocity and maximum pressure may be expected from a 5-in. gun, chamber capacity 600 in., shot travel 140 in. with a 55-lb. proj. shell and a charge of 10 Ib. M.D., diam. of cord 0-12 inches?
For the 6-in. gun we have:
G = i6oo Gi=74
8=250 81-41-6
W = ioo Wi= 0-463
L=o-2 Li =0-033.
V = 26so P = i6
From (i) and Table I.
V = K d ' 25 Si ' 2 Wi ' 4 Mi ' 6 Li ' 15 . Taking logarithms and rearranging,
log. K,=log. V+o-25 log. Gi+o-6 log. i /Mi 0-2 log. Si 0-4 log. i/Wi -0-15 log. i/Li
whence
log. K, = 3-524.
Similarly from (2) and Table II.
P = K p Gi Wi Mi LI whence
Then for the 5-in. gun: G=6oo 8 = 140
and
= io = o-i2
G,=4-8
Si =28 Wi=o-44 Mi =0-08
Li =0-024.
log. V=log. K.+0-2 log. Si+o-4 log. 81+0-15 log. i/Li-o-25 log. Gi 0-6 log. i /Mi
which, using the value of log. K found for the 6-in. gun, gives
f/s.
Similarly using the value of log. K p found for the 6-in. gun we get for the 5-in. gun
P = l8-6 ton/in 2 .
It must not be inferred from this that for any propellant we can arrive at values of K v and K p and the indices g, s, w, m, I, which will reproduce the firing results in all circumstances. Investigations to determine such fixed values once for all, will soon lead to disappoint- ment. It must be remembered that we have only embodied in the formulae differences in weight, calibre, chamber capacity, shot travel, weight of shell, weight of charge, and dimensions of the propellant. We have not taken into account any of the other causes of variation touched on above.
When we analyze firing results by means of (l) and (2) all these neglected factors are as it were embodied in the values of K c and K p arrived at, and these values of K, and K,, and also the values of the indices are only suitable for application in other cases in which the effects of the neglected factors are proportionally similar.
The " density of loading," ' and the position of the point of com- plete combustion of the charge will also have an influence, and an adjustment of constants and indices may be necessary for widely different densities of loading, and according as to whether the charge is completely consumed well back in the gun, or whether there is a proportion of the charge still unburnt when the shell leaves the muzzle.
The values of the indices in Tables I. and II. are adjusted for the average conditions of modern practice, 2 and if the above warning is kept in mind and the formulae used in an intelligent manner they will, as already stated, be found extremely useful working tools.
If only a few of the data vary it is not necessary to work with the complete formulae (i) and (2). Thus if we are dealing with the same gun and shell and the same propellant of the same form and size, and only wish to investigate the effect on the muzzle velocity of differ- ences in weight of the charge, we need not introduce the standard gun and work out the constant K, but may write
XI
V"
where the muzzle velocity V is known for a charge of weight M', and we want to find the velocity V" for a charge M". Again if we are dealing with differences in weight of both charge and shell we may employ
Yl/ivr\m /w\
V" \M") \W"/ '
As an example a gun gives m.v. 2500 f/s with full charge 12 Ib. M.D.T., what will the m.v. be with a 3/4 charge of 9 Ib.?
Here ' = 2500 for M' = i2, and we have to find V" for M" = 9 from
0-7
we have
Therefore
Yl- /Ml\ V" \M"j
Yl = /! V
V" U /
V"
?52? =2050 f/s.
1-22
When the variations in the data are comparatively small the monomial formulae may be replaced by a simple percentage approxi- mation which will give sufficient accuracy while reducing the calculations to little more than easy mental arithmetic. The follow- ing tables derived from the indices already employed with the monomial formulae give the information necessary for such per- centage calculations.
1 The density of loading is defined as the " ratio of the weight of the charge to the weight of a volume of water just sufficient to fill the chamber." This is given by 27-7 M/G. The greater the density of loading, the less the " initial air space " (the volume of the cham- ber not actually occupied by the grains of the charge).
2 These indices are suitable for ordnance. For rifles they require considerable modification, see Hardcastle " Monomial Formulas for Pressure and Velocity for Ordnance and Small Arms," Royal Artillery Journal, vol. xlii.
