AR ffIECINICAL INTELLIGENCE0* TRANSLATION

(Title Unelaseti ed)INTz2I0I' BALLISTICS

by

E. . Serevryakov

state Printing House of .heDefense Industry

Moscow, 19)i9, 2nd Ea tion

672 Pages

(Part 2 of 10 Parts,

Pape 8"&172)

MATEAIR TCHNICL NELLIGNCE ETE

WRI~H*PATESU AIUOCEBSOCHI2Of

F~TS-.7OHIO B sAvailjable COPYDi~tributior'i of thi:-;~i i

u~nlimited. It ri be reloaoudto the "Le~a ir h1.ooj, Depart, n.I

Reproduced by th of ;o-nmrcol for galu to theCLEARiNGHOUSE

for Federal Scientific & Technical CenelU'I public.Informaiion Springfield Va. 22151

m il 1 411

I-~~~~V '-- --.- v -

MMOR ALLISICS

BYI

STATE FINTI OSRO BALISISEIXSM

.4 .3? ...

CIUAPTYR 711 - "CALCULAI ING "!M IFAT LIST TO Ti:EWALI.S DURING URNING 01' IO OMI, 1,N A CIa1,.1J CIAV,1iQ(

When powder iti btirned in a closed chanber (in a bomb), t portitn

of the heat energy is lost on heating the walls of the bomb. As a

-- ,result, the pressure p. developed by the gases is somewhat lower than

the theoretical pr-ssure; the latter would be obtained If all the

thermal energy emitted during the combustion of .the given powder were

utilized to increase the pressure of the gases.

This loss of heat depends on a number of loading factors.

Experiments conducted by professor S.P. Vukolov in the Naval

Technical Research Laboratory back in 1895 and 1896 had shown that

at La.- 0.20 the pressures p. developed in a standard bomb and in a

bomb whose interior surface was lined with a thin layer of nonconductive

mica were different. The resultant pressures were: 2033 kg/cm 2 iu

the bomb wtthout the mica layer- and 2202 kg/cm 2 in the bomb containir

tL~e mica, the difference - 169 kg/cm 2 - comprises about 8%.

It was stated prev'.ously(in the theory of powder combustion), in

the discussion of problems relating to powder ignition in a gun,'that

the contact of the powder grains with the cool surface of the chamber

slows down the ignition process.

The following simple experiment in the open air will show this to

be true. If a strip of powder is zlamped vertically in a locksmith's

metal vise and iguited from the top, the process of burning will be

:rrested-upou reaching the vise. The jt-ip will be extinguished :I

because a considerable portion of the heat is t1ken up by the cold

met 1.

F-TS-7327-RE" 84,I-

" .4

it i*-j.known thnc a rionsi,,ul taneous ignition distorts te initial

yhane of the grain and c'tuses d,:vla't-ions from tbe ideal law of

coubus tion. Cozsequently, the transfer qf heat to the walls should

have an effect not only on the mngnitu/e of pressure, but also on/

the character'of its development.

By disregarding loss-ei in a bomb due to heat transfer we commit

an error in the determination pm and, consequently, aL error in the

magnityde of energy f and covolume a determined on '41e basis of

experiments with a manometric bomb.

In view of this heat transfer, Nobel's formula will hold true

for pressures above 1000 kg/cm2 . At lower pressures, corresponding'.,:

PD'to. charging densities of4 <0.10, the points in the diagram

versus pm will fal .below the s'traight line relationship expressed

Pby the known equation f . f 4 ap3.

Systematic experiments conducted by assistant A.I. Kokhanov in

1933-,-at charging densities of from 0.015 to 0.20, have shown that a

hyperbolic curve abc, approaching asymptotically the stfaight line dePm

(fig. 16), is obtained in the system of coordinates E pm instead

of a straight line. The smaller the value of A , the greater is the

deviation from the theoretical relat'nship,

The results obtained! in determining the powder energy f and

covolume a may therefore differ, depending on the charging densities

at which the tests were conducted. The higher the value of 6 , the

greater will be the magnitude of f and the smaller the covolume a

(points b and c). And, conversely, small values of,- shold produce

a small energy and a large covolume (points a and b).

F-TS-7327-RE 85

_Q&, 40-.

Our experimeits have also disclosed the foliouing.

If a powder of the same composi'tion but of different thickness

is burned in a bomb at the same value of &, the nmaxriuw prescure p

will, be the lower tL-e thicker the powder. Therefore, the energy

produced by thick powders,.determined without taking the heat losses

into consideration, will also be the smaller, the .thicker the powder.

This can be explained by.the fact that a thick powder burns longer

and, consequently, a larger portion of the heat is transmitted to the

walls of the bomb.

If an identical powder is burned at the same A in bombs of

different'sizes, the value of p will be higher in the larger bomb,

because the surface per unit weight of powder charge is smaller in

the larger bomb, and hence the heat losses will be smaller in it.

All of these facts confirm the presence of cooling through the

walls of the bomb. The considerable number of tests conducted by

various investigators made it' possible to-determine quantitatively

the corrections to be Introduced in the charging of powders of various

thickness under different conditions of loading, in order to obtain

pressures corrected for heat transfer. Although these corrections

are not final, nevertheless their Introduction served to explain the

phenomena discussed aboVe-; Miuraur's correction may be considered

to be the best founded among corrections of this type.

F -TS-7327-RE 86

-

oiS

HIC REPRODUCIBLE

Pil

A 1 1'

Fig. 16 - - as a Function of P , According

to Kokhanov.

According to Miuraur the heat loss through transfer is pro-

portional to the number of collisions between the powder gas molecules

and the walls of the bombp and th4.s number is in-turn proportional

to the surface of the bomb S4 , pressure p and time t. For a

fluctuating pressure, the loss is proportional to $ and pdt. Also,

because )pdt does not depend on &, the loss of heat &Q through

* the surface of walls Si is constant for any powder charge Ci ort .

This was confirmed by Miuraur who conducted tests at different

values ofSA. Inasmuch as the total quantity of the heat evolved is

proportional to the weight of the charge'i, the relative loss Q

is inversely proportional totj or.6. Consequently, .2Qis pro-J' Qportional to pdt.

The magnitude of maximal pressure at a given value of Ciis

I- governed by the volume of the bomb(*). Actually, an n% pressure

(*) Consequently, magnitudesf and a will also depend on the volumeof the homO, in which the combustion takes place, while the magnitudeof 5 pdt should not change because of cooling through the walls ofthe bomb.

F-TS-7327-R 87

'i -.

drop entails a corre-sponding pro)otgatlot of the period of burnig;

tho curve "p,t becortes distorted, but the itrea I pdt remalls unchangegd.

All this was checked and verified by our tests with bomibs of

different sizes. The following important deduction can be reachedel

from the above: the rate of burning u1 - can be determined

pdt

with the same degree of accuracy in both large and small. bombs,

regardless of the heat lost through the walls.

Experiments for calculating heat losses. For quantitative

determination of heat losses Miuraur conducted experiments at A- 0.20

in a bomb measuring 150 cm3by volume. In one case the powder was

burned under normal conditions, and in another a steel, trough-shaped

insert with projections or ridges was inserted into the bomb ia such

amanner tht its exterior surface did not come in contact witb the

surface of the bomb (fig. 17).

In the first instance, when the cooled surface SI was equal to

the surface of the bomb Si. the obtained pressure was p,; in the

second case, when the cooled surface S2 was larger, i.e., when it

consisted of the surface of the bomb S and the surface of the insert

SBK (S2 - S + 5BKn), the resultant pressure p2 was lower. The pressure

difference & p - p1 - P2 resulted in consequence of the surface area

difference S2 - S1 -BKJ .

In-order to determine the pressure in the absence of cooling,

calculations were made of such an increment Lp'l, which corresponded

to a change in the surface areaLAS equal to the surface area of the

bomb Si:

. Se S6- - or Ap' AP

. SBK ) BKF-TS-7327-RE 88

i! ..... -

-

In this mnanner, pressure p1 + Ap' would corre.ipond to the

• urf;ict. S - 0, i.e., it would correspond to the cofldition in which

io-,,.; d , e to hea t t r~insfer were a1bsen t

Ki4owlng p, and ,Ap', Miuraur determin.'d the relative correction

for _ p' APm T i'acntnfor heat transfer - -- , which was equal to A- in a constant..... p P T1

volu e.

Such tests conducted wirh a large numbor of powders of varied

thicknesses and properties established the significant dependence of

pressure lo-. -=-=k on the time of powder burning at A - 0.20, thePm ~2/

magnitude S6 /w j in these tests being equal to 7.774 cm2/g- 77.74 dm 2/kg.

The experiments were conducted with cylindrical crushers.' Very

strong igniters of gunpowder were used to determine the actual powder• c2

burning time tk . The igniters developed a pressure of Pn ow 250 kg/cm

which provided the crusher with small residual compression.

A LTThe obtained data was plotted, and the curve -T Pm . or-%

Pm TIas a function of the time of burning. tk, was termed "curve C"

(fig. 18).

In order to determine the lossee due to heat transfer under

conditions other than those discussed, the powder must first be

tested at A - 0.20 and the time of burning tk found, and the magnitude

CM - then determined from.curve C. Losses under other

conditions (in a different bomb and 4t a different density of loading),

can be found by means of the following forrjula:

A LT CM% S6 C% 6 1 (

m - M (17)

P5 T1 7.774 CJ 7.774 W A

Cy, depends on the thickness -nd nintur of thoe powde r, while Sl/w

characterizes the relative-surface of the boiab (exposed surface of

F-TS-7327-RE 89

uonbh) and cain bc catlcul;ted 'Is the exposed surt;,ce of :a urj:i -ur ;;

cy itnde r ( 2 #i - d, .'n;,'h -I('); t. u'' Ict rIz 'ir -uad It , ,t iU.,,I:I,

SC _ 2 + .. .3.Wo 1 d :• d c

The relative surface of the bomb diu,inishes ab the disnicter d anI

length 2c are incraetd. When volume W0 is increased 8 tinses, SdW 0

diminishes approximately by one half.

Formiula (17) shois that An a given bo~ub increases as A is

reduced and hence the tosm-.; are partic,larly hig/ at low values of A.

Fig. 17 - Test Arrangement Fig. 13 -Cure .for Calculating Losses .due Characterizing the Losses Due

to Heat Transfer. to Heat Trarsfer (according"to Miuraur).

* Ordinate: pressure loss in 5; .abscissa: milliseconas; 1)curve "C.'

Large igniters are not necessary when working with conical

crushers, because almuost instantaneous ignition results at the low

value of p 100-120 kg/cm2 .

Therefore, Miuraur's data ciinnot be directlt applied to our test

results and to powder tests for determining heat transfer, without

conducting special tests. However, using the theoretical formula as

a baiis, it is possible to compute the burning time at d - 0.20 and

PB o 250 kg/cm 2 and to determine the order of the coefLtcient values

C4 for powders of various thicknesses, in order to introduce the

necessary correction for heat loss.

The table presented below contains the results of sucil calculations,

F-TS-7327-RE 90 "-

=i i9

ub 1nt i 'd t Iof CIIIV* C.

wat, de.- Le 11J (t fr ow formju I,:

t 2.303t logP13

whereeC

7. "I 1 - 1 1

u. f A Ul P

(for a powder with a cons~tant burning surface area) (Table 8).

Table 8 - Values 0of coefficient CM for pyroxylin powders

Thicknass of powder 0.30 0.30 0.40 1.00 2.00 4.002e1 in mm flegme.

Rate'of combustion u9 o 70 80 75 72 "62dm -1-9 107 !

isec dIm2

'Coefficient I 1220 950 813 305 146 69

,Burning time in milliseconds,

;At L - 0.20 and po - 250 kg/cm 1.76 2.26 2.64 7.00 14.6 31.0

jHeat transfer coefficient I1 11.5 2.0 2.6 4.0 .0 6.1

Corrections for heat loss during burning of powder in a bomb

can be introduced by the aid of this table.

Let us introduce the correction for heat loss when determining

f and a.

We shall assume that tests were conducted with a bomb having !

volume W0 - 78.5 cm3 at two loading densities A 1 " 0.15

and z- 2 0.25. The powder is 1 mm thick.

* F-TS-7327-RE 91

--1~ Y7.'-

Pa1 l 1435 kg/cm , P12 " 2760 kg/c"2 2 - Pml 1325. We

determine f and a without taking the heat-transfer losses into

consideration:

Pal Pm2 P,2 Pml 1409570 - 11,020; 2 - - 1450.

- - 1.096 dm 3/kg1325

f 951 000 - 1.096 143,300 -.800,000 kg-dm/kg,

We now introduce the correction for heat losses in pressures Pm

and P.20 and determine the'corrected values of energy fo and covolume

!10.

Apa S6 1 CM%

p To &, 7.774

According to the table, CM - 4%.

ror a bomb of volume W0 - 78.5 cm3

I 5 1

6 1 - 0.1673; 0.1673 ' 4 - 0.6692.

10 7.774 2

S4 CM% p 5 apm - 9.6692 * 9570 - 64 kg/cm2

~"3 \w0 ~ ~ m2 -0.6692 .11,040 -74g/2

P Pm1 +AP01 - 1435 + 64- 1499 kg/cm2 Pi- 10,000;

.\ I2 P2 11$I340PM Pm2 + &P2 " 27G0 + 74 2834 kg/cm -13

F-TS-1327- 92

F2

Po2 - "1335 kg/c2;

Pz,2 PnII1340;

a.1340 3/kg 1.01 - 1.004 din 1.00.

0 1335

f - 1,00tO00 - 1.004 1499 - 849,500 kg-dni/kg - 850,000 kg-dm/kg.0

As can be seen from the above results, the introduttion of a

correction for the heat' lost has increased the energy of the powder

from f - 800,000 to 850,000 kg-dm/kg, i.e., by 6.25.0 nnd reduced the

magnitude of covolumo from 1.096 to 1 dm3 /kg, i.e., by 9.6%.

The same powder burned in a bomb having a volume of 25.25 cm , and

at the same loading density, will give pml g 1405 kg/cm2 and p.2 2725

kgc2kg/cm On the b~sis of this data, the resultant magnitudes of energy

f and covolume ax will be:

f - 774,000 kg-dm/kg and a - 1.16 dm3/kg.

For this bomb

S6 /

- . 1.89 cm2/c30

Corrections in pressure for heat transfer will be as follows:

APmi - 95 kg/cm2 and LP 32 " 110 kg/cm ;

f - 850.000 and a -1.00.

In this way, magnitudes f and a for a bomb of small volume are

obtained with a.larger percentage of error: the energy will be

F-TS-7327-RE 93

'sri~ and the covolurne will~be higher than their true values.

- - As shown by Kokhanov 's experim~ents, these errors are extremely

=- l'arge for very low loading densities and thick powders.

The introduction of corrections for heat transfer losses by the

;-k ab-Ove method provides practically identical values of f and a for alfP

lo6ading densities, and transforms the hyperbola ..! pm obtained inAKokhanov's experiment's into a straight line, as; it should be &J

The iable presented below lists values of magnitudes -, andI wo

S1 ~ for manometric bombr of the most typical dimensions (Table 9).

Table 9

d 2.2 3.0 4.4 4.4 3.0 4.4 8.0

-m 217 1.89 18.30 1206 1.46 110 0r.ps53

0 _ __ __ _

St 0.279 0.243 0.1673 0.1495 0.184j0.135 0.0682'

Curve C~ is expressed as a function of the burning time tk of

powder at A 0.20.

* Inasmuch as the burning time of powder t kis diirectly proportional- I

to the total pressure impulse pdtv the dependence of C as a

0.

function Of Ik can be plotted independently of the loading density.

Work of this type van conducted by U.I. Samarina at the Naval Artillery

-~Research Institute in 1938, at which time experiments, were repeated

wvith-bombs with znd without steel inserts. The pressure was recorded

IP. FTS-7327PE 94

by njeans of oonica.1- crushers. The values of A--- - CA' were plotted

as a function of 1 k (fig. 19).. The form uf the curve differs some-

what from curve C~tk, . /

~, I

/ I'

Fig. 19 - Curve CA, Characterizing Losses due to Heat Transfer ON

(according to the Naval Artillery Research Institute data).

Bomb tests for determining f and a involve heat losses in the

form of heat transfer to walls of the bomb, and, therefore, pressures

p and 2 contain an error in their determined values, pand

being inversely proportional to magnitudes 1 - a&, and 1 - ct2, andt P! 6P2 S 1 CM%

6P2 Pj; but -- 2 because -P .0 - where C

is the heat transfer coefficient.

Correctioas for heat losses in the powder energy f and the

magnitude of covolume can be determined after introducing corrections

for beat losses in the values of pressures p1 and P2 , by adding the

equalities (7), (8),. (9) and (10), taking into consideration equality

(a) (see Chapter I):

A2 + 1 1 .2 2

6f 6fI + f2 A 2 A1 (P 2 - Pl2 6p 1 : 6P2 )2 6p - (2P Pl - p62

.2 " 61 P2 Pl P26 P -f. 1 2

P2 - Pl A2 &1 (P2 -P)P2P 1 P2 P

F-TS-7327-RE 95

A ram;

Fialy, A P2

P2p P1 p2-1 2_________ (18)

A1 A2Ja JaI + ~g2 l) (P 26p1 -p 1

6p2) -

P26Pf 1O2 r.f Pi P2 (19)

When the corrections for heat transfer are taken into ,accountj

p 7.774 W A0*

SeIFor a giyen-powder, C. conat, and 'for a given bomb, I corist,

only A changes. -W

Designating

* CM~ S6

7.774 V0

we get-

-- D JP2

P1 '61 P2 "2

jSubstituting t.hese expressions In formulas (18) and (19), we obtain:

1 (20)

F-TS-7327-RE 96

r2i1 1

- --- fD-" (21)

P2 Pl P " p2

Let us apply the example presented above.

Example. 1 mm thick nowder is burned in a bomb having a volume

of 78.5 cm3 ; the powder coefficient is C 471; 'and the value of

for this bomb - 1.30 cm2/cm3 .W0

S6 C %- 0.1673 • 4 . 0.6692% - 0.006692 . D.T 0 7.774

P

1 - 0.15; pl 1433;.-i = 9:570 Without correction for heattransfer lossesf - 800,000 kg-dm/kg;

P2 m3A 0.25; p 2 - 2756; 11,020 a 1.096 dm /kg.

We shall now calculate the corrections for f and a taking the heat-

transfer into account:

2756 1433

0.15 0.25- 0.6692 - 0.6692 18,350 - 5730 -6.4;

f 2756 - 1433 1323

fO 0 f (1 800,000 1.064 - 852,000 kg-dm/kg;

2 2dc -- f D 0-o,000 •0.006692 -- 0.1085;

P1P2 143,300 " 275j600

a cia O + ba - 1,096 0.1085 - 0.9875'2' 0.99.F-TS-7327-RE 97

T.o

CHAPTE.R IV - THEv LAW OF GAS FORMATION

I. DEFINITIONK

The study of the law of gas fcrmati. under the simplest conditions,

inFU a constant volume,. permits the application of the obtained relation-

-.9hps to the determination of pressure in the bore of a weapon when the

;-tgu-i's fired, iee., under conditions of variable volume.

The general pyrostatics formulaC

fw*4 fc4'

p 1 ,

shlows-that the magnitude of gaspressure at given conditions of

loading (V,, , f, a, 6) Is governed by the amount of the burned,OK

portion of the chargej' where wt jis the gravimetric inflow of

powder gases at a given instant,, and fw? jis the inflow of the energy

contained in this quantity of gas.

-keeping In mind that Wqj varies slightly under conditions prevailing

-In a manometric bomb, it may be said that the pressure is almost

proportional to the burned portion of the charge t at the given jpowder energy f andthe given loading density.*

Correspondingly, the nature of pressure increase in time .Runderdt

the same conditions of loading is also determined in 'the main by the

change of magnitude d+with time.dt

40 The law of gas formation is a term defining the change of the

magnitude of +I'with time and of its derivative L1known'as the "ratedc'

: of gas formation" or "'volumetric rate of burning."

F-TS-7327-RE -98

An analysis of thsis magnitude dmnikes it possible to determineI dt

the means by which the gas inflow during burning of powder can be

regulated,

2. ILATE OF GAS FO,ATION

We shallk4drive the formula for the rate of gas formation, based

on the burning of powder in parallel layers.

Let us assume that the burning of a powder grain of arbitra.ry

shape proceeds in concentric layers at a constant rate in all directions.

At some instant the grain, whose initial volume is Al and whose

initial surface is Si, has a volume A and a surface S (fig. .).

We shall also assume thu. a layer of thickness de is burned during the

time interval dt. Then, the volume burned during the time interval

dt will be expressed by formula:

d A Sde,cr

whencee

cr " Sde;

0

Acr Sde

AcA 1

Differentiating both sides of this equality vith respect tc t,

we gst:

F-TS-7327-RE 99

i ... . .... . . . .. ..

d4' Sde S

d ,dt dt t dt A

• . Mutiplying and dividng the right side by S1 , we get:

- /dt "l s

//

The first two mUltiples in the right side of the expressiodepend

on the geometry of the powder grain:

the initial exposed area of tho powder grain or the specific

surface per unit grain volume at the start of burning; it

will be shown later that this area depends *on the form and

the dimensions of the grain;

. - - the relative surface of a powder grain; it varies during

burning and depends only on the form of the grain and on

the relative thickness Of -thet-rnt powder layer, but not-on

its absolute dimensions.

As will be shown later' the third multiple, the linear rate of

deburningti - T dependson the type of powder' the pressure under

which the powder burns, and on its temperature.

In order to determine the rate of burning ir an actual tiest bomb

,under variable pressure, it is necessary to know the thickness

variation of the burning layer per unit of time, and to this end.it is

necessary "to establish the purely geometrical relation between the

.thickpess of the burned layer and the volume of the powder gases in

their various forms.

F-TS-1327-RE 100

MI I. .. . . - - . . ' . . ... . - t' -- a,'.-,- - , '~ , ,Fe. e ,

GRAPHIC NOT REPRDUCIBLE

Fig. 20 - Diagram of Powder Burning in Parallel Layers.

If the pressure curve plotted as i function of time-, or a table of

the values of p versus t is available from the bomb test, the values

of T,, t can be calculated by means of the general pyrostatics formula

(or from special tables), following which the geometrical law of

burning can be applied to establish the relationship between the

thickness of the powder, the value of * and S, and to derive theS1 1

dependence of the initial exposure --- on the form and dimensions ofAl

the grains. The numerical differentiation of t and e with respect

to time t will permit obtaining from the experiment the rate of gas

formation and the linear rate of burning, as well as their variation,

during the burning of the powder.

The establishment of all these relationships will facilitate the

analysis of those factors which can be used to control the quantity

and the intensity of gas formation during the burning of powder and

therefore, will permit the control of the phenomena of a gun discharge.

F-TS-7327-RE 101

- ... . .. . . . 4 p- ', r c, . .'t , , - - . ,

3. THE EFFECT PRODUCED BY THE GEOMETRY OF A POWDER GRAIN ON

GAS FORMATION

"" The inflow of gases per unit of time for a powder of a given type

a(f # s o, u1) can be regulated by the loading density or by the

* dimensions and the shape of the powder grains.

The effect of the geometry of the given grafns on the rate of

gas form'ation depending on the thickness of the powder burned at the

given instant, can be determined by means of the basic conditions of

the geometrical law of combustion.

The geometrical law of combustion permits determining the

.relationship between the relative thickness of the powder burned at

the given instant z - e the burnt portion of the grain A -

*~~ +A ben th voue, h

(A. being the volume of the burnt portion of the powder), and the

relative surface of the powder - S/S1 at the same instant. The

dependence of the roduct 1 on the shape and the diensionss S o

of the powder (gra n) can be established at the same time, which

product enters the formula for determining the rate of gas formation

and has a great effect on the law governing the gas pressure development

during a discharge,

A. Relation Between the Burnt Portion of Powder !' and the

Relative Thickness of Powder z, Consumed at the Same Instant

tw (Inflow of Gases)

Investigations ow that the dependence of on z for all forms of

powder is expressed b a formula of the same type(*)s

(*) In deriving this f rmula for one grain, as well as for an entirecharge composed of man 'grains, it is assumed that all grains of the

.chaxge have strictly Id ntical dimensions and are regular in form, thelatter being bounded by parallel planes, intersecting at right anglesor by surfaces of revol tion whose axes coincide or are parallel toeach other.

F-TS-7327-RE 102

(1 ( Xz Lz 2 ), (22)

where L, , tx are shape characteristics - constanf values depending

on the shape of the grain; they possess a particular numerical value*

for each grain shape inherent to the given grain form.

The thickness of the burnt layer e varies during burning frora

0 to e,; the relative thickness z varies from 0 to 1; and the relative4

volume fluCtuates between 0 and 1. / n

We shall now derive the dependence of y on z for strip powder (a Zc

parallelepiped with three different dimensions) (fig. 21). We shall

introduce the designations: 2e for the thickness of the strip, 2b

for the width of tie strip, and 2c fo. its length:

2e, 2e1-a; -s

2b 2c 0

I s - " ' - -- -

Fig. 21 - Burning Diagram of a Powder Strip.

Magnitudes a and characterize the span of the strip in thickness

and length.* Furthermiore, inasmuch as all the dimensions become reduced

in all directions by a magnitude 2e during the full burn'ing of the2e1 I

powder, a - 2 represents the relative reduction of the strip in

2elwidth and 5 2c represents the relative reduction of its length

during the full burning period of the powder.

F-TS-7327-RE 103

Inasmuch as

2e < 2b <2c,1

I> a> 1> 0.

Let us assume that at the given instant a layer of powder of

thickness e will be burned on all sidea. The volume burned can be

determined more easily as the differer.ce, between the initial volume

A1 and the remaining volume A .

We will have (see fig. 21):

Ac A,.' - ^ocr Aoc ":t -

A, Al

A -2e 1 2b 2c;.

(2e - 2e) (2b - 2e) (2c - 2e);

--~r e- 1 e be c - e0

AI e b1(

but

* • el e e e I

e b be c c e1

then

'ocr . 1-z) (1-%z) (1- Pz).!

F-TS-7327-RE 104

&

Upon removing the parentheses we obtain;

2 2

A 1

Substitu'ing this expression in the formula fort, we find:

(1 + a + 5)z - (a + s + aQ)z 2 + agz3 '

or, reducing it to the general form of equation (22), we get:

.... + + "B aB 2

+ a+)zI a+ +a+ l

Introducing the designations

+a - + + ~ -S=A; a: - it (23)

we obt in a general type formula:

= (1,.! + )Lz + z2).

At the end of burning att z '. 1 4- 1, and formula (23) assumes

an equality in the form

1 - x (1+ A + gL), (24)

which must be satisfied by the umerical characteristics 9 and IL.

F-TS-7327-RE 105

o -

This equality serves to verify the values of the characteristics

calculated by ateans of formula (23).

The characteristicsX, A and p. depend on the ratio of the

-dimensions, the-shorter and the narrower the strip, the greater is

a and f; the' longer the strip, the closer isie to unity and ) and ji

to zero, 4 -=

2. The Law Governing the Change of the Powder Surface When thePowder is Burned.

Formula (22) is a general formula for all powder shapes; the

difference will be only in the numerical values of the characteristics

','• IL and g. Using this formula as a basis, we shall derive a form-

ulafor depicting the relative surface - and the initiai exposure

, characterizing the effect of the shape and dimensions of the

powder on the rate of gas formation.

Differentiating t with respect to z, we find:

= ( 1 + 2Xz + 34z2 ). (25)

Inasmuch as

• dt de .

dz dt de dz'

and

1 u, dt I de de

dt A 1 " s de u 8z de el ,

e,

F-TS-7327-RE 106

-JA

then

4 S1 S el S1 Sdd - u ---e

dz A s u A S

Substituting this expression in the left side of equation (25),

we obtain /

Sl e S . (I + 2z + 3z) (26)

1)s

At the start of burning z - 0, S .- S , S - 1, and equation (26) at1 S1

the start of burning will be written as follows:

S1 eI (27)

A 1

Dividing each termof (26) by (27), we will find the desired

relationship:

S 2,6= = 1 + 2Az + 3z. (28)

S1

At the start of burning, at z - 0,

6- 1;

at the end of burning, at z 1,

SKG - - 1 + 2A+ 3g.K S1

F-TS-7327-RE 107

-- | ..... ... ... - =-; --- '" : - r . = - - ' -, .-

4' -A.4 ' ' - -

V

While the powder burnh, and zvarkt-s from 0 to 1, the change of e I

w1-11 mainly depend on the manpitude and sigu of the characteristic k,

'since ± is small compared to A.

If the surface of the powder diminishes while burning (strip, .d/be,

Sbar,<O0) the shape of such a powder is c'lled regressive; if the

-surface area becomes greater during burning (powder with multiple

-perforationsA> 0), the powder is called progressive.

The 9 function depends on X and g. i.e., on the shape of the

grain and Its dimensional ratio, rather than on the absolute dimensionsof the .powder. The greater the value cf IAJ, the greater will be

', the surface change of' the powder in burning.

From equality (27) we obtain an expression for the initial exposure

:which 'is of-great importance in controlling the rate of gas'formation:"

" --(29)

equality shows that the initial burning area of the powder depends

-on its shape (characteristic t), as well as on its dimensions (el). jThe smaller the value of el, the thinner is the powder, and the

•gre.ater will be the quan.tity of gases which it will evolve per unit

of time. _

Substituting (28) in (25), we obtain formula

.. dz

which will be used in plotting the graph for ' z.

F-TS-7327-RE . 108=29-

C. Deterini,1tion of P z for Other Grain Shapes.

a) Tubular grain %;ith a single perforation (fig. 22).

Des ignating:

2e - web thickne*ss1

D outside diameter

d - inside diameter

2c - length of tube;

2e, e- (which is the same as for strip powder);

2c c

2I 2(D2 - d2) c;

1 4

" 4T f(D - 2e) - (d + 2e) 2 . (2c - 2e);

2 2Ao. "(D - 2e) -(d + 2e)J-7 (c - e)

AD 2 d d2 ""C

GRAPHIC NOT REPRODUCIBLE

I: I

Fig. 22 - Burning of a Tubular Gr'ain.

.1. /

F-TS-7327-RE 109

t -7n s

. -. . .

Removing the parentheses and bearing in miind that

D -d -2 *2e~,

e el

- M- - a *-c - e c

we get:

EoZD2 d2 2(D ;.d2 7 . ( d . e(e-c e

IV ( (D d 2-2

Z) (I IS I+Oz+ Oz

1

Assuming I + 13 1K, - and g 0, we gtonceagainagerl

type formula

-where the term 2~ smsig eas i 0

dixnsfon n wdthdoes not change i.n burn zig. This is equivalent

In uc a as,,the-characteristics ofsrppowder will be

.7'-TS-7327-RE 110

X- 1+0+ + -1 f ;

- 0,

i.e., the same as those obtained for tubular powder.

It is of interest to note that characteristics 9 and A for a

tubular grain do not depend on the diameter, but, rather, on the web

thickness 2e1 and the length of the tube 2c.

b) Grain shapes which are derivatives of strip powder

It can be easily seen that the following grain shRpes can be

obtained from a strip as special cases of the latter:

1) square rod: 2b - 2c; a - 3;

2) square slab: 2e - 2b; a - 1; > 0;

3) cube: 2e 2b -.2c; a - 1 - 1.

The characteristics of these shapes, as well as of strips and

tubes, are given below in Table 10; alsv given in this table are the

values of 6 at the end of powder burning.

F -TS-7327-RE ill

- -Table 10

Powder Shape jC 1 + 2

.- Tube (infinitely' 1 + 0 0 1 -1-wide strip) I+ + _ _I_ +

+ c ++a G__ 1( a) 0I-)Strip. + + 1+ + a+ ,• l~a+ l+a+0Il~+

20 + 12 02 (12-)Square Rod 1+ 2 -

I + 25 1 + 25 1 + 2:3

I + 2D ,Square Zlab 2 + + 0

2+0 2+01AA

ube 3 -1 0._ _._ 3 1

The data presented in the table shows that all the grain charac-

_ teristics are increased ir. changing over from a tubular to a cube shape: j(o.' * increases from 1 to 3, I'I - from a small fraction -to I, L - from 0

to 1/3. r1 S,

-Since A characterizes the initial surface area for a given

pbde rt-hctkness 2el, the increase of X shows that in changing over

froma strip to a cabe;" with the web thickness 2e remaining the same,

-the initial surface area is increased almost three times whereas the

simultaneous increase of X indicates a more drastic reduction of the

. surface area.

The diagram in fig. 23 clarifies the above: the heavy broken

A - .-"llne divides the str.ip into square rods, the dotted line divides it

-into square slabs, while the dot-and-dash line divides it into cubes

*..; 7f the same thickness as that of the strip..

------ FTS-7327-RE. 112

. ..... .-... - - - -

~77T t -~-. " ---- '- - - -'2r

• ° )

Increasing the tube le,.%h 'k.11 ulti:,.ately result in a powder

grain with a constant burnling surface, 1or which/ • /

o s , - o, - ; /

consequently, the relations between. z and 6, z will be expressed

by the following formulas:

4'- z; - .

GRAPHIC NOT REPRODUCIBLE

Fig. 23 - A Strip Divided into Der, Vi Z"Iapps

The some law governing the burning of powder will apply to tubular

powder inhibited at its end,.

Examples of Calculating the Characteristics of Powder Shapes

1. Strip powder Cf (SP); dimensions in millimeters: 1 by 18

by 300.

2e, I"a -... 0.05555; X 1 1+ a + 1 - 1.0589;

2b 18

2e, -(_ +________90-1 0.00333; -( + + a3) 0.05907 - 0.05575;-

2c 300 .1.0589

aB 0.00019a5 - 0.00019; L _ - - - 0.00018;

1.0589

K +,2% + 3i - 1 - 0.11150 + 0.00054 0.889;

F-TS-7327-RE , 113

I

_ 1.0589 2/ 2-'-! . . .. .. 2 .1178 nan /$SIM .-,A 0.50

For the same powder reduced to 40 um: in lengh, the charac-

teristics will change as follows:

tr- 1.086; X- -0.0758; ± - 0.00129; LK 0.852; 7--2.161.

2, A tubular grain having the same wall thickness and length as

the strip powder:

2. -1; 2c -300;1

a=, 0.00333; -O.300

1 "0.00333- 1 + 5 - 1.00333; 1 ,, - - - 0.00332;

1 + 1 1.0033

Ad -'.. 1 + 2A. 1 - 0.09664 - 0.9934;S1

~1~ 1.0033_ 2.0066.A," 1

The same tube 40 ma in length:

X 6 1.025; A - 0.0244;

2 ; 0.9512.

F-TS-7327-RE 114

Coiapared to a strip of the sam~e leng~th, a tube has a sm~aller

effective surface area and a lowvr degree of regr.e-sion. The surface

area and regression in thc burning area increase with decrease in

length.

If we calculate the characteristics for shapes having the form

of solids of reVolution, we will find that their characteristics are

expressed by precisely the same ratios as given in the table of

characteristics for parallelepipeds; the only changes occur in theI

values of a and t, which are-expressed in terms of the dimensions of

the solids of revolution.

Table 11, below, contains shape characteristics for solids of

revolution in the form of a solid rod, sphere and tube.

Table 11 - Shape Characteristics of Solids of Revolution

Easic dimensions .e e a F5+la~a - - a x + a + --

Y1b l+ a + 1e 12b 2c _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

t I Sphere

2R 2R 2R 1 1 1 3 -

Solid cylinder (rod)

2R 2R 2c 10- R 5J 2 + 1U- J. I I 1,2+I32+ 5

Solid cylinder whose diameter equals its height

R I I I I I I i I

Cylindrical plate (pel et)

A 2R 2R 1 + Z3 1 +25" II- IIL -_I + '

Kr2 + 2 ubF-IS-7327-RE 115

L~~~7.M ME I'~~4 ~W~

Tnble 11 (Coiit'd.)

Ring-shaped pell1et

2c 00 R .r 0 2e 5 0 1l +~ 0 -

(2*c) -r+

A comparison of the data in Tables 10 and 11 will show that the

characteristics X, A. g of some of the grain shapes listed in these

tables are identical, and that therefore the law governing the change

.of volume Tand surface area C; as a function of the relative .burned

thickness z is the same. Such shapes are termed equivalent shapes.

They are exemplified, for example, by a cube, a sphere, a cylinder

whose height equals Its diameter, or by square and round slabs, and

square and round bars.

D. Graphical Illustration of the Relationships Between

'r' L Z..

Knowing the general expressions for the inflow of gases

jm tZ(l +X z + z 2

and for the law governing the change of relative surface

C; 1 + 2;L z + 3tiz 2,

we can by assigning apecific values to z, calculate the corresponding

values of e; and *and plot a graph for the investigate d regressive

* grain shapes.

These diagrams are termed "progressive data sheets."

F-TS-727-RE 116

0C

We shall consider the following shapes: " - tube; 2 - strip;

3 - square plate; 4 - a solid slab; 5 - cube or sphere.

a) N, z diagram (fig. 24) (progressive dat.i sheet).

Since A is negative and ji is small in all the shapes under

consideration, curves 4, z will alwayri lie..blow the horizontal dotted

line 1-1, which corresponds to a powder with a constant burning surface

area.

At the start of burning, when z - 0, 6- 1 for all powder shapes. fAt the end of burning, when z - 1, the values of z will vary forKf

different grain shapes. The expressions for these values are given in

Table 10. Inasmucn as for a tubular grain p - 0, the expression

6, 1 + 2Xz will depict a straight line with an angular coefficient

2 , whereL < 0; the straight line 1 is very close to the horizontal

1-1.

For strip powder, as well as all the other powder shapes,

I 1 + 2Az + 3Lz2 whereX? < 0.

2A + '6z; 6 >0.

dz dz2

Consequently, curve G , z is convex with respect to the z-axis.

At the start of burning

2X <0.

Since the value of 1 is small for a strip, the curvature of the

line is very slight; the values of )L and p increase as the shape

F-TS-7327-RE 117

r" " ..... -, a _ _ := . u. 7--. _ --= r~a ..-. -_ . _ ..... . . . . . . . . . . . -. r

becomes more regressive, the angle of inclination of the.tangent at

the origin of the coordinate axes 'icreases, as does the curvature of

the curve itself. In the case of a solid slab the curve has a9

, pronounce downward bulge, and passes through zero (Gg 0) when

zs 1 -(curve 4).

In the case of a cube the downward slope and the convexity are

maximuM, when K - Os z - 1 (curve 5). /

Thus the diagram shows that the cube is the most regressive of the

five powder shapes considered here; the surface of the cube diminishes

abruptly at the very start of burning and approaches zero at the end

of burning.* The least regressive powder shape is the tube, whose

burn ng (surface) area remains practically constant at all times

(reduction does not exceed 1%). Powder in the shape of a cube, at a

given thickness of elements, gives off a maximum Quantity of gases

per -unit of time at the start of burning; this quantity diminishes

rapidly with burning. On the other hand, the quantity of gases given

off by a tubular powder grain remains practically constant. B

b) 4, z. Diagram (fig. 25): !A

+mrz(l +AZ+ z2).

Previously, we had the exjres ion (9): -

X" - (1 + 2Az + 3z 2 ) 0;

dz

when z O - and (X -m .

_ -TS71327-RE 118 "

- , . •

!-

GRAPHIC NOT REPRODUCIBLE

//

F1g024 - Change inarea duig Fig. 25 -The effect of grainregressiyve burning of powders shape on gas inflow for re-of'fdif fe ren t shapes T f (z) . gressive powders *-F(z).

d_ is the tangent of the slope angle of curve 4, z with thedz.

abscissa, while the shape characteristic e represents the slope angle

of the curve at the origin of the coordinates.

When z =0 O, -I;when z i- .

All the curves are located within a square whose sides equal

unity.*

All the curves originate at the origin of the coordinates and pass

through point i- 1, z 1 1; the tangent of the angle of inclination

at the origin is F-' - . Further variation of the angle of

inclination is characterized by the value of G. Inasmuch as 0

diminishes in all the powder shapes considered here, the angle of

inclination of all the curves diminishes also, and hence all the -

curves are convex upwards. In the case of a tubular graiu' e approaches

unity and the change of 6 is small; curve 1 practically merges with

a diagonal drawn from the origin of the coordinate system. For a

cube,'Xe- 3 is maximum; curve 5 has the greatest angle of inclination

at the origin' and the greatest variation of this angle, corresponds

to the variation of G * In the case of a cube and a slab -O

F-TS-7327-RE 119

:......*.

472

," " :ind hence the curves are tangent to the horizontal line 1-1 when

- _ The arrangement of the remaining curves 2, 3, and 4 is obvioun

! a-n does not require any explanation.

-The diagram shows that for a given value of z the portion of the

*."Aburt grain will be-the larger, the more regressive is the powder

and the greater is a ..For example, in the case of tubular grPin,

at the -Instant the first half of the thickness (z - 0.5) is burned,

the burnt portion of volume t will be equal to 0.5005, and in the

case of a cube, when Z - 0.5, j- 8 0.875.8

Consequently, a more regressive powder gives off a larger quantity

* *tof gaies during the first half of the burning process, and a smaller

such quantity, during the second half.

. c) The_*, %' diagram has the greatest practical value, because

it can be more easily compared with experimental data when evaluating

the pressure curves obtained in the burning of powder in a manometric

bomb. j is determine.d from the value of p by the aid of\ the general

formula of pyrostatics, while 0 goes into Li obtained br the numericaldt

differentiation of the dependence of on t. When this ata is

available, a comparison can be made of the theoretical and the

experimental results. i

Inasmuch as the equation for 4, z is a third power e uation,

and s, z is a second power equation, z is usually not elim nated when

determining the dependence of 6 , t; rather, by assigning d finite

values to z, the corresponding values of t and 6 are comput d, and

* *the results are then plotted on the G , diagram.

FI-TS-7327-RE 120

MMIS , INRIMPRim,0 1

GRAPHIC NOT REPRODUCIBIZHow will the e , z diagram change after it is transformed into a

-" diagrama (fig. 26)?

Fig. 1 - . of the Shape of RegressivePowder Grains on the Change of the Powder

Area During Burning - - c? (t).

Similarly to the 5, z diagram, 9- 1 when 0 0. At the end

of burning when j 1, cK will have the same values as when z 1.

Consequently, the position of the initial and final points will not

change.

Since all curves of ', z are situated above the diagonal dividing

the square t , z in half, the magnitude * > z will correspond to some

value of z, and this magnitude will be the greater, the more re-

gressive is curve ',, z..

Therefore, when Y is replaced by z, all the points on curvesz (see fig. 24), while remaining at the same height, will shift

to the right and the amount of displacement.will be the greater, the

more regressive is curve T, z or G, z (see fig. 26).

d) A binomial formula for the relationship ', z.

The examples given here for calculating the characteristics Je.

X, I for strip type powder show that g is very small for strip and

plate type grains, and that the term gz 2 does not appreciably affect

the law governing the variation of and 6. Therefore, in order to

F-TS-7327-RE 121

'", ' ," 'tJ - - "- °

simplify the expression subsequently entering the ratther complex

formulas of pyrodynamics in the solution of the basic problem, a

binomial formula is used to express J, for strip type powders without

( I mpairing the accuracy, by neglecting the teri 4tz 2 in paren-thesis.

The influence of the neglected term pz2 is comapsated for by

changing the remaining characteristics K and A, based on the following

considerations.

Having a complete triuonlal formula

4, - atz(1 + Az + Vlz 2 )

with known characteristics, we shall replace it by a binomial formula

with new characteristics 9, and XI:

/= jIz(1 + Xz).

In both equations 4'- 0 when z - 0. We shall establish the following

conditions in order to determine coefficients )e and A 1) when

z ;- I (end of burning), the.blnomial-formula must give us - , and

2) when z - 0.5, the-value-of tdetermined by means of the binomial

formula must have the same vi.lue as T found by means of the tri-

'--nomi'al-formula--at the same value of z - 0.5.

We thus obtain.a system of two equations with two unknown- coef-

fticients -1 and A1 :

when z -1

j1 + + ) - 1 - Je1(1. + A

when z - 0.5

--- - - -

F-TS-7327-RI- . 122

.... . ~. *_i , .-,

or M - __ __

22 4/ 2 \ 2)

Solving this system, we get t-12J: ,

1 2 k2

We can obtain from the first equation of the system

1

1 2

3

2

When the values of the characteristics and A are determined__1 1

in this manner, the second degree curve , z and the third degree

* curve , z will coincide, at the starting point z - 0, then at z - 0.5,

and finally at the end point z - 1.

Thus, in the case of strip-type powder, the curves practically

coincide also at the intermediate points, when the values of 01 and

A1 are chosen as above.

F-TS-7327-RE 123

" I.

;- *; Thus, the binomial formula can be used for strip nnd plate type

powders also in the future

t~J '" 'z(1 + A Z).

Similarly, we shall have for the surface ratio

/ G.II+ 2A z./1

Eliinating z from this system of equations, we get the dependence of

on on in the following form:

,.l +4;-- 11

?or g given value of this relationship permits a direct calculation

of the corresponding value of G.

Hereafter, we shall drop the indexes of the characteristics 0

and Xf

4. PROGRESSIVE-BURNING POWDERS

A. General Data

In all the regressive types of powder considered here, excepting

tubular powders, the surface area always diminishes when the powder is

burned, because burning proceeds inside the grain in concentric

layers. A tubular grain is the exception in this respect: the

* surface of the perforation Is displaced in burning from the axis of the

tube outwards, thus increasing its area,, and hence partly compensates

F-TS-7327-RE 124

• 2t-

* * N ,

GRAPHIC NOT REPRODUCIBLEfor the reduction of the exterior surface area. The tubular type of

powder would represent n powder of constant burning area bordering

between progressive and regressive forms of powder, if the tube were

not to burn from its enrls and remain unchanged in length.

* - . ., .,-'.-*) +

Fig. 27 - Grain with Seven Perforations:

a - before burning;b - at the instant of decomposition.

At the start of burning the total surface of the tube of length

2c will be expressed by the following formula, if the end-areas and

reduction in length are disregarded:

Si - 2R2c + 2fr • 2c- 2(R + r) 2c.

When the thickness burnt on the inside and the outside of the tube is

e, the surface at that instant and at the same tube length will be

S - 2n(R - e)2c + 21r(r + e)2c - 2,r(R + r2c;

consequently, the surface krea S - S const, because the reduction

of the exterior surface is compensated for by the equivalent nlargement

of the interior surface.

F-TS-7327-RE 125

I

Under such conditions the constancy of the surface area does not

depend on the diameter of the perforation. This property-of the

perforation surface to increase in burning indicates a means for

obtaining a grain of the progressive type. This will require the use

of grains with several perforations, whereby the increased surface area

of the latter will compensate for the reduction of the outside area.

The grain with 7 perforations is based on this principle; the

centrally disposed perforation along the axis of the grain compensates

the decrease in the outside area, and the six radially disposedperforations in the vertices of a regular hexagon serve to increase

the area during burning. The outside surface is located at a distance

of 2e from the perforations. In such an arrangement (fig. 27a) the

web thickness, i.e., the distance between the centrally located

perforation and the outer perforations, as well as between the latter

and the outer surface, will be the same, so that all the webs will

burn simultaneously. *

Burning from the perforation centers proceeds along concentric

cylindrical surfaces, forming circles in section; when the latter

converge and the thickness eI is burned 'n all directions, the grain

disintegrates into 12 rods of irregular cross section (slivers): 6

inner, small rods and 6 outer larger rods (fig. 27b). These products

of decomposition burn with a sharp reduction of the burning surface,

similarly to a solid slab, and provide even greater regression because

of the presence of sharp, rapidly burning projecting angles. Thus

all progressive powder grains with several perforations, whose

initial burning is accompanied by increased surface area disintegrate

F-TS-7327-RE 126

aeGRAPHIC NOT REPRODUCIBLE.at some instant into regressively burning products of. decomposition.

This is the undesirable characteristic of powders of the progressive

type.

A grain with 7 oerforations usually has a standard dimension

ratio: the web thickness "e between the perforations themselves as/

well as between the latter and (he outer wall must be the same and

equal to twice the diameter of the perforation (or the perforation

diameter d must'be equal to half ok the web thickness e ):

2rI - 2d;

accordingly, the outside grain diameter is

D 4- 2e + 3d- lld-le.

The length of the grain 2c is not great, it is usually equal to (2-2.5)

D or (20-25)d.

Fig. 28 - Products of Decomposition of a GrainWith 7' Perforations.

At this dimension ratio, the surface increase at the time of

decomposition amounts to about 37% (Ss/S- - 1.37,,where Ss is the

surface being burned at the instant decompos.ition occurs). At the

same time, if the perforations are spaced correctly, about 85% of the

F-TS-7327-RE 127

? y -- t~

grain will be burned "('s 0.85). Consequently, about 15', o. the

grain is burned regressively with a sharp reduction of the surface

area. If the spacing of the perforations is irregular, decomposition

-does not take place at the same instant: webs of the least thickness

a' are burned first, followed by gradual burning of the thicker webs;

& portion of the grain undergoes progressive burning and the remaining

portion suffers a sharp reduction of its area. The maximum value of

S8 is smaller than in a normal grain, and this corresponds .also to a -

lower value of js.

If we inscribe a circle in the outer prism of decomposition (fig.

28), its radiusi/'- 0.1772(d + 2e) L/127; in the case of standard

dimension p - 0.1772 ".3e I - 0.5316e1 ' 0.532e1 .

The radius of the circle ir-icribed in the inner prisms o.

decompositionp'- 0.07*14(d + 2el) - 0.2322ei 0.232ei. Therefore,

at the end of burning of the grain as a whole, when the burning surfaces

merge at the center of tho outside prisms of decomposition, the

thickness burned will be e- e1 + ,'- e1 + 0.532eI - 1.532e

Thus the burning of progressi2e powders is subdivided into two

sharply differing phases: 1) prior to decomposition z varies from

0 to 1, tvaries from 0 tots <1; burning proceeds with a gradually

increasing area; 2) after decomposition z changes from 1 to

1+P

zK = 1 +fl/el, changes fromj to 1; burning is progressive.

Some authcrs (V.E. Slukhotsky) consider it more expedient to

consider z- 1 when t- 1; then, at the instant of decomposition,

Z< 1 (z - 0.653 for standnrd dimensions).

F-TS-7327-RE 128

-,

Disintegration and regressive burning of the powder in the

second,phase are the defects of powders of the progressive type. In

order to decrease the products of decoposition, a great many grain

shapes were suggested by various authorities, of which the most

inttresting ones are the shapes suggested by Walsh and Kisnemsky. 4.

Walsh's grain is an improved grain with 7 perforations, whose.

outer wall is not/a continuous (single) cylindric.l surface of

£H.ameter D - lld, but, rather, consists of six cylindrical surfaces

circumscribed about the axis of each perforation of radius

r d + 2e (in the case of standard dimensions r - 2.5d - 2.5e1 ) ,2

The cross section of such a grain is as shown in fig. 29a, and at the

instant of decomposition its section takes on the form shown in

fig. 29b.

GRAPHIC NOT REPRODUCIBLEI I

F.ig. 29 alsh's Grain with 7 Fig. 30 Burning of Kisnemsky'sPerfora tions. Grain.

I Before burning; b) at theinstant of decomposition.

A grain of this type is often called a "shaped grain."

The dimensions of the products of decomposition in such a grain

are considerably smaller than in an ordinary grain with 7 perforations.

F-TS-7327-RE 129

Calculations show that s 0.95, i.e., that 95% of the grain undergoes

progressive burning, and only 5% is regressive. Also, for the sapie

dimensions d, 2eI and 2c as in a standard grain, the surface area

.:increase Is the same, or 37%, so that 6 1 1.37.

* -The Kisnemzky grain was designed for the purpose of eliminating

--the products of. decomposition on the onehand, and for obtAining a

-grain of "greater progressivity" on thecother. It is in the iorm of a

square slab with square perforations djsposed in two mutually 'T

V perpendicular directions (fig. 30a). According to Kisnemsky, the

square perforation should burn in parallel layers and retain its

" square form,.so that burning would terminate without decomposition.Furthermore, a large-number of narrow perforations should provide a

highly progressive grain (according to calculations 6 -2). This

is the powder recommended by Kisnemsky for obtaining ext':a-long-range

firing; the KOCAPTOfl (Kosartop) (Special Artillery Research Committee),

headed by V.M. Trofimov, was the body engaged in the investigation of

this type of powder at the time. All the theoretical work on the

performance of this powder was carried out by V.A. Pashkevich.

It was found that Kisnemsky's estimates were not justified either

with regard to burning without decomposition, or with regard to high

progressivity. Tests had shown that the square-shaped perforation

does not retain its shape, because burning from the corners of the

square proceeds at the same rate as along a normal to the side of the

- square (there is no reason to believe that the powder would burn more

- " - -rapidly along the diagonal than in other directions); it proceeds from

each corner in concentric circles, whose radius equals the thickness

F-TS-7327-RE 130 -

Mal.

of the burnt layer in the direction of normals to the sides of the

square. As a result, the cross section is that of a square with

rounded corners, and ait the instint of decoirposition the shape of the

grain appears as in fig. 30b.

The products of decoinposition amounting to 10% remain between

the perforations. Hence Y. is not 1 but is about 0.90 (90% of the

grain burns progressively). v - (V2-)e 1 , because e I + p is theS -

diagonal of a square with side e•.11

For reasons which will be discussed later, the powder suggested

by Kisnemsky did not show the high progressive quality anticipated

by him.

GRAPHIC NOT REPRODUCIBLE

Fig. 31 - A Grain of Kisneirsky's PowderBefore and After Disintegration (Ob-

tained in Firing).

Figure 31, on the left, is a Kisnenisky grain with 35 perforations

before burning; on the right is the same grain incompletely burned and

ejected from the gun after firing. The photograph clearly shows the

products of the starting disintegration having the form of the "ace

of diamonds" in cross section; the perforations are almost round.

F-TS-7327-RE 131

--

GRAPHIC NOT REPRODUCIBLE

.41.4

'Fig. 32 - Transformation of a Siaare Perfo-

ration into a Round One.

Figure 32 is a large grain with a single perforation: a) before

burning; b and c) intermediate states. Inasmuch as the side of the

square perforation is small in comparison with the web thickness,

the perforation is transformed into a circle.

'.B Determination of the Characteristics of Progres'sive Powders

a) First phase, first method.

The law governing burning and the change of the surface ofj

progressive type powders is expressed by the samegeneral formulas:

j-tz(l +)z + tz 2 ),

~ ~GmI + 2X2z + 31iz2 ,

,as-for regressive types; the difference is only in the numerical

values of the characteristics .X, X and L aid in their signs

.(X.> ._O S0). The-derivation of the characteristics equations

Is obtained similarly to regressive powders, and only the obtained

expressions themselves Are somewhat more complex than for regressive

powders. ..

~T-.27-RE 132

_ _ _ __ ___ __ _ _ __7_ _ _ _

We shall introduce the following dvsigrinations: 2c - grain length,

2e - web thccn, s; d - diameter of perforation; D - grain dia~meter;

n - nunber of perforations.

The burned portion of the .charge is:

AOCT 2e,

2c Pc

D +nd D2 -nd 2" 'l; "QI"

2c (2c)2

It can be easily seen that the magnitude l1 - D + nd/2c is the

ratio of the perimeter of the grain's cross section to a circumference

whose diameter equals the length of the tube 2c. The magnitude

D2 -nd2

Q1 is the ratio between the base area and the area-of a(2C)2

circle whose diameter is 2c.

The initial grain volume is

" IYD 2 nd2)2c.

The unburned portion of the grain by volume at a given insta.t

is

AOCT Z-(D - 2e) 2 - n(d + 2e)2 7 (2c--2e)--,

0 4

AOCT -D2 nd2 - 2(D + nd) 2e - (n - 1) (2e) 2 J (2c 2e)22 2e

A I D - nd2 2c

F-TS-7327-RE 133

Bearing in mind that

2c 2c 2eI

-.- 2c 2c~ 2e' -M5

and.dividing each term in the parenthesis by the denominator, we get:

A2(D+nd) (n 1) 2

7 "22 i " + 2 - n d 2 nd 2 2 "

or - substituting I 2c for'V + nd and QI (2c) 2 for D - nd , we

AOCT 2 nI1 *2c -2e n- 1 2S"-'--- "-- - (2e) (1 - z) -

(l (2c)2 ql(2c) 2 I

- -TT i~r~ 222) (I_

j- )

( n--2TTI

O- - -Z.-

" n+ 1 13k-T) ( 1 n 33in + LA ' + - z

/ 2

QjQ 1+2%T Ql+2TT1

F-TS-7327-RE 134

.;-- 1 i1i:zZ~

I "

We thus get a general type formula:

z + Az + z2)

where

(,),Q12T1 (*

n - 1 2T1 2(3 - 11)R- ; when n -7 A-

Q1+ 2T 1 Ql+2Tgl

(n-.)S 62

Q I Qz+V Q+2T

These formulas show that for powders with many perforations the

characteristic ii <0 and hence the convexity of curve 6 , z is directed

upwards (when n - 1, ,t - 0). The sign of L depends on the difference

(n - 1) - 2T1l, and in ordinary powders with 7 perforations. of

standard dimensions X) > 0. When the grain is shortened X may become

equal to zero: this will occur when n - I - 2TT1 0 or n - 1- 2 D+nd2c

- 0, whence

(*) These formulas are suitable not only to progressive powders wheren > 1, but also for regressive shapes - solids of revolution, for example:,for n - 1 (tube) and n - 0 (round slab without perforations).

Hence, they are general formulas: thus, for example, for a tubewhen n - 1, X < 0, pt - 0, when n - 0 (solid slab)? < 0, Vt > 01 as wasestablished in the previous formulas.

F-TS-7327-RE 135

* i

N * *& '-* ... *' d . - - * , ... .* -- -- -... .. . -' -- .N' i

D+nd n-I

2c 2

Under such conditions, perforated powder is no longer progressive.

" . In particular, for a grain of standard cross section with 7

perforations, whose D - lid, the condi-tion A - 0 is satisfied when

2c - 6d - 3 2el,

and since*-< O the area of such a short grain will diminish when

burned.

A study of the A coefficient shows that progressivity increases

with the number of perforations, if the diameter of the perforations

at the same web thickness decreases and the length of the slab

increases.

The expression for the surface (area) change will have the

following general. form:

I'+ 2Xz +31gz *

In the case of rectangular shapes, such as the Kisnemsky grain,

the magnitude" T is the ratio between the perimeter of the cross !

section and thi perimeter of a square with side 2c equal to the

length of the slab, and Q, is the ratio between the cross-sectional

area of the grain and the area of the .same square with side 2c.

If we call the side of a square slab Al, the side of the square

perforation a, the length of the slab 2c and the number of perforations

n2 (n horizontal and vertical rows),

F-TS-7327-RE 136

- _ _ _ __i_ _ _ _ _-

i ~ ~

2 2 2(A1 + A1 - n a1 2e 1

2c (c) 2c

These formulas hold true for Walsh's grain as well, but inas-

much as the cross section of this grain is more complex, the formulas

for "I and Q, are likewise more complex.

The outside wall of Walsh's grain can b.e considered consisting

of six arcs whose lengths equal 1/3 of a circumference of diameter

JI - d + 2 * 2e and which are described from the centers of six1 1

perforations, and six arcs each measuring 1/6 of a circumference of

diameter dI described from the outside vertices of equilateral

triangles with side a1 - 2eI 4 dl, whose other two vertices lie in

the center of the perforations (fig. 33).

In such a case the cross-sectional area ST of the grain consists

of the following elements:

1) 12 triangles with side a1 less three sectors of a circle of

diameter dI at their apexes;

2) six sectors each measuring 1/3 of a circle of diameter 61

less six sectors each equivalent to 1/3 of a circle of diameter d.

S~l3( a2 3 2>+!~3s- 12 a- 3 1 4 d 1)+6 -32 - d )

Tr"'J

l48d1 + 281)

F-TS-7327-RE - 137

• I

The perimeter P of. the grain will consist of:

1) six arcs each nieasurirg 1/3 of a circumference of diatneter S1;

2) seven circles of diameter dl;

3) six arcs each measuring 1/6 of a circumference of diameter

dI (in the "ngles included in the outside web.

P m r(I + 7d + 6 i aw- Ir( + 8d) - 2r(c 1+ 4dl). IThen:

2e1

2c 1

p .2(S1.+ 4d1 )

3 7r aI+ 26 1 8d I

2 (2c,) 2

2c1

00

i - 4

J1

! Fig. 33- Diagram of Walsh's; Grain. "." :

-F-TS-7327-RE 138---:-

'K''

!,t~d ...... . ....____= '- "I~l

b) A bi~~. lfor;.' ~I for the s(-cond phase.

Sinice, for a st-andird grain i th 7 pf-rforations the charactf-rist ic

11i s s-l1,the law goveri1ng burning t - f(z) can also be expressed

with sufficient ;a&-cuiacy by mieans of a binomiial forc.ula:.

The characteristics Of and 17L will be found under the condition

that when z -1, 4' ', and w en z - 0.5, the value of -1i accordingI S/

to a trinomijal formula would be equal to the value~ of * according to

the binomial one. We thus obtain-a system of two equations as for

regressive powders:

when z -1

j(1 + X. + 10) - js i1 + A 1 );

when z -0.5

__ ~ 1+ ~4) 2 2..+ )Solving it we get

and then

(instead of a- -:1 fo r regressive powders).

F -TS-7327-RE 139

Example. Compute the shape characteristics of progressive powders.

Grain with 7 perforations.

2e1 -1; d - 0.5; D - 5.5; 2c - 12.5 mm

2e.1" 1 D + 7d 9- - - - -0..0 8 ;tI 1 "1 " d - - 0.720;

2c 12.5 2c 12.5

D D - 7d2 30.25 - 7 * 0.25 28.50Q " - =0.1824;

(2c)2 156.25 156.25

Q, + 2T - 0.1824 + 2 • 0.720 - 1.6224;

Q, + 2 1 1.6224S- - 0.08 - 0.712;

QI 0.1824

(n - 1) - 2TI 6 - 1.44X * 0.08 - 0.225;Q+ 2T,1.622

- - (n- )01 6 •0.0064

*(lhA"&- - - " ll0.0237;Q0 + 2 1 22 1.622

-e(1 + X + )-0.712(l + 0.225 - 0.0237) -0.712 •1.2013

-0.855;

4 ......... 6s 1 2 + 34t + 0.45 -0.0711 -1.379.

F-TS-7321-R E 140 "-

In the case of the bitioinial for,:ula d- xz(o + 7LlZ)

(. )e ( 0 .712 1 - 10071-0.712 *1.0118 -0.720;

2 [2

is 0.855''s -1 -- -1 - 0.1873;1 0.720

-6 + A ) 0.720(l + 0.1873) - 0.855;

s 1 + 2?L - 1.375.

It can be seen that the difference between the values of s

corresponding to the instant of disintegration obtained by trinomial

and binomial formulas would be very small (0.004), so that in practice

the binomial formula can be used for powders with 7 perforations in

the first phase of burning.

c) Second phase (after disintegration)

The products obtained after decomposition in the form of small

prisms of triangular cross section (with curved sides) burn regressively

with rapid surface reduction, similarly to k square or round slab.

A detailed investigation made by G.V. Oppokov Z-13-7, using

the assumption that burning proceeds in strictly parallel layers,

rovides general expressions: one - prior to burningof the thinner

,nside prisms, and the other - until the end of burning of the outside

risms. Using the formulas suggested by him as a basis, Oppokov

F TS-7327-RE 141

-" i

prepared a table for the values of as a function of z when the small

and large prisms are burned and then when only the, remainders of the flarge prisms are burned. An analysis of his data shows that the change

in the surface area of the products of decomposition almost corresponds

.to the change in the surface of a cube, i.e., its burning is more-

--regressive thau in the case of a prismatic slab.

-- The binomial formula can be used for determining the relationship

't,-'z in the second phase. Transferring the origin of the coordinates

to the point of decomposition (z8 I, T-rs) we will have

- s -3C 2 (z -Z ) C + X 2 )Z - l)J..., (30)

with z varying from 1 to zKj

We shall determine O2 and A2 by imposing the following requirements:

( 1) when- zK, z ymust equal unity according to formula (30);

2) at the end of burning when z - zK the surface s must become

equal to zero.

._ -Differentiating equation (30) with respect to z, we get:

d4 . 1 = 2 11 + 21 2 (z :--)J. . ..

A,

The second requirement-will be satisfied if:

S1+ 2 X2(ZK -1) -0.

The first requirement will be written thus;

4-TS-7327-R l2

- 2

I - s '(z - 1) 7T1 +A (z.- 1)J.

Is (ZK + 2JK

Solving this equation we find that

-1 2( - )'A 2 "' -22 (zK-l) z K

Calculations by means of these formulas show that for a standard

grain with 7 perforations when zK - 1.532 Ts 0.855:.

1.I 2 - 0.145A . --0.94; 2 - - 0.545;

2 2 • 0.532 2 0.532

for Walsh's grain when zK - 1.232 and .,s - 0.95

-1 2 * 0.05

A2 - -2.16; X2 M=2 0.05 - 0.432.

2 0.232 0.232

C. Graphic Representation of Relations G- z, f- z. G-P

for Progressive Shapes

In order to construct diagrams z; S -z; S -t for progressivL-S1 S1

shapes, we shall resort to the following general formulas:

X j-az(1 +';Lz + gz 2 ) and -S - 1 + 2Az +3gz2

Si

which are applicable also to these shapes, but now when z - 1

F-TS-7327-RE - 143

! .~- .

- s < 1; the end of burning occurs when z >1, the coefficient

-- 0 (not only for grains with 7-perforations, but also for the

•Kisnemsky grain.)_:- eodig ." _/W..i.-A-- 6iC~odingly, the diagra i1l appear as shown in fig. 34 and 35:

. "I -- grain with 7 perforations;

b11. - rain with shaped outer web;

/ III - Kisne.,3ky's grain. /

GRAPHIC NOT REPRODUCIBLE

" Fig. 34 - *- f(z) relationship Fig. 35 - 1(z) relationship.for progressive grains, for progressive grains.

Inasmuch as IL < 0, the convexity of the S/Sl, z curves is directed

upards. When constructing an S/s diagram as a function of.,, the

'-corresponng"points o te , diagram will. £e" displaced to the

'left (the revers-e of regressive grains), this displacement being the

smaller the greater is the value of z; hence the S/S1, curves willj .. -- '

-lkewise remain convex upwards as do the S/Sl, z curves.(fig. 36).

a . /

F-TS-7327-RE 144

_4I

-. 4 44-.-n.W-. ' 4* 3

R"APHIC NOT REPRODUCIBL

• -

Fig. 36 - 6 f2(j) at'.::'.ion for Progressive Powders.

a) Inhibited powders of high progressivity

Inhibited powders constitute ohe of the forms of highly progressive

powders.

These powders first appeared in Russia soon after the appearance

of Kisnemsky's powders and were developed jointly with the latter.

This problem was studied by O.G. Filippov, an instructor at the Artillery

Academy, in 1920-1928.

Inhibited powder is obtained fromi ordinary tubular powder, whose

outside surface is coated with a specia,. nonburning substance.

When such a powder is ignited, only the inside surface of the

grain burns, which surface increases in proportion to the diameters

ratio D0 : do0 When the diameter of the perforation equals the

thickness of the--tube

SK Do

SI d0

*The resulting progressiveness is greater than in a Kisnemsky

grain with 36 perforations.

F-TS-7327-RE 145

TM

i ,1

* GRAPHIC NOT REPRODUCIBLE I

SFig.37 Inhibited Tubular Powder.

-Notwithstanding the apparent simplicity of this idea and the

A° :th*4retical possibility of obtaining-high progressivity, the practical

• ': allatonof same .was found to be very difficult, because the

-, itbhibitor layer must not burn and must be capable of protecting the

"ter tube surface from burning. At the same time it must be

/ gutfficiebtly st-rong to withstand abrasion when shaken and be .strong

:;h! oijh,.not to be torn off the surface by the action of the gases.

/" :I ' The undesirable property of inhibited powders is greater smoking

':.,hen fired, due to the disintegration of the inhibitor at the instant

, Ihi-shill is ejected from the bore of-the gun.

S -b) Characteristics of inhibited powders.

The outside surface and ends of the powder are inhibited and

dttburn.•,-ealIa contradistinction to an ordinary tube, bhe web thickness of

..tn. nhibited powder burns in one direction only and should therefore

beou noted by e rather than by el which is the usual designation.

,hn fidetplying the general method, ote get:

F-T5. ?327-RE 146

h shellis ejectedfromthe o e. ,

A1 2 ! o -t 2ei) 2 d 2 2c-"2d 2e + 4e2 7-T2cd e +2e4 0 1 - 0 2' 1 124 4

A f7 = -(d 0 + 2e) 2e) 2_2c 7= 2cd 0 e 1 + eI -(doe + e

2 ee e I z +-

AOCT doe + e 2 '- d0 do

A del+ e 1 e e1I d- d

1++- d o o

Comparing it with the usual formu)a - z(1 +\z), we get.

el 1 1

0 +

do

0

2e, + 2eI DoS - I + 2Az-1+- z; " 1 + - zdo=S1 do0Kd0 d

When the tube thickness el equals Its diameter d

F-TS-7327-RE 147

S~-.....----. - .

- 1; ~~- -- - - - 0 . 5 ; G K - 1 + 2 - 3 .

'By reducing the diameter of the perforation with the web thickness

---remiining the same, the geometric progressiveness can be considerably

-increased:

d A -2;.Je ; -1 + 2A 5

3

- CHAPTER V - BURNING RATE

. T The burning rate of powder mainly depends on its properties and

temperature, and the pressure and temperature of the gases surroundingS _ .

-Inasmuch as the temperature of the gases formed during burning

of powder as yet does not lend itself to experimental determination,

* the burning rate is usually expressed as a function of its properties

Siand gs .pressure which is known from experiment at any given instant

-of time.

The functional dependence of the burning rate u on pressure of

the form u - f(p) is known as the "burning rat law," and this

law Is expressed by various empirical fonmulas as given by different

authors.

Experimental detervination of the burning rate of powder is

* possible on the basis of a test curve depicting pressure as a function

of time; this involves the use of the fundamental-relationship of

the geometrical la of burning giving the relation between and

Z e/e I .. -.

F-TS-7327-RE 148

! --~

For determiinltg th#- burnii,,: r.te, use is usually mad- ol strip,

plate or tubular powder, of uniform thickness:; its dimensions are

carefully measured, the iweal vjles of 2el, 2b, 2c or 2e,, DO, doIt 02c are determined, the charactG~istics ct R and X are calculated

(using" the binomial formula), and a graph is constructed for Y-f(z).The powder is then burned in a manometric bomb using a strong I£ i

igniter, to insure simultaneous ignition along tle entire powder

surface so as not to impair the initial dimesions used for calculati .g

the J and W characteristics.

The analysis of the bomb test data is conducted in the following 7manner.

Having obtained from the bomb test a curve of pressure p as a

function of time t, and upon determining for this powder on the basis

of the derived relationships the law governing the variation of

with z or e a constructing the corresponding , z orj, e didgram,

we.can determine the rate of burning u at the given pressure. Indeed,

knowing the values of p, we can determine by means of the general

pyrostatics formula or from tables the values of p, for. which we take

the values of e from the 4 , e diagram. The difference between the

neighboring values of e will give the increment A e for the tire

interval+&t. known from measurement of the p, t curve; the &6/At

ratio gives the burning rate u at p which, is an average value for

the given section. Thus, having at our disposal calculated data in

the form of a table, we can determine the variation in the burning

rate and of the given powder due to change of pressure p (Table 12).

F-TS-7327-RE 1 49 1- j'. . j.... . " .

Table 12 -Determining the B~urning Rate u f (p)

Auxiliary t&eble ordiagram4 compAled Table Derived frow Test Dataon the basis ofthe geometric lawofI burning_______ _____

1_ - 12 3 4 5__ 7

ze LePt p Pcp ccordingt Ae WU

Auxiliary ttDiag ram

0 0 0 Q BIP 0' 0 Ae i- 9

2 -e0

z t" e t p -(P, +p) e he" Ul2 mef -e'

We ji eq .l -9 ft

e ti PM el0

Plttn th obaie vaue of 0tpp(C ranTasa

onagahw a idtecageo( ihcag fpesr ,an thu *een the bunn raelw

F-T0-021* 0 100

1I

The for,.ula. ,ost often used for determining the burning rate

are the following.

a) Vieille's formula (exponential equation)

u - ApV,

where A and V depend on the nature of the powder, and, in particular,

v may be equal to unity.

The smaller the value ofy , the less sqnsitive is the powder to

pressure changes.

For smokeless powders Vieille used V - 2/3, and for ordinary

black powders , - 1/2. Our tests with slowly burning black powders

for a time fuze gave the value of "- 1/5.

G.A. Zabudsky used V- 0.93 for pyroxyline powders. Some authors

use ) - 1 for cordites and V - 1.07 for ballistite.

b) Binomial formula

u -a + bp.

it was'first used by Prof. S.P. Vukolov .(1891-1897) in the Naval

Technical Laboratory, and then by .olf (1903) and Prof. I.P. Grave

(1904). Muiraur employed this formula considerably later (1930-1935).

In his thesis (1904) I.P. Grav6 Z-14_J compared formula u - ApP

with formula u - ap + b by analyzing a large number of bomb tests

conducteI by himself and others and arrived at the following conclusion:

"Both fo ,uiulas can be considered equally valid for expressing the

law gove r ing the change of burning rate under varying pressure,

because t e mean errors obtained with the use of these formulas are

Y-TS-7327- IE 151

-.,. ~-

generally the same, and both formulas give practically identical

results."

This deduction whlch appears strange at first glance, namely,

that a parabola (semicubical) and a straight line not passing through

'the origin give identically accurate results, can be explained as

follows.

The first tests in bombs .since the year 1880 were conducted with

the use of cylindrical crushers which do not permit recording press'ures

below 300-400 kg/cm 2 (fig. 38). The test data (points) are usually

scattered to a certain extent and do not lie on a definite line. As

a result, some of the investigators drew a parabola u - Apo through

these points, and others drew a straight line u - ap + b which does

not pass through the origin of the coordinates (curve 2).

The conclusion arrived at by Prof. Grave' confirms the fact that

( both lines pass sufficiently close to the points plotted on the

2basis of tests at pressures exceeding 400 kg/cm

The relationship u - f(p) could not be obtained experimentally at

low pressures .( - 400 kg/cm2 ) at that time, and the question of the

true relationship continued to remain open.

c)Formula u - Ap...

Charbonier (1908) first accepted the burning rate law in its

general form u - Apos and then, on the basis of his own analysis of

test curves p, t obtained in bomb tests arrived at the conclusion

that for French strip-type B powders Ycan be taken equal to o - 1.

F-TS-7327-RE 152

47 j~..u---'--'---'I

j i= !:Ij:i _ -.. .. . . . . . . . . . .

Thii law was accepted also 15y our Prof. N.F. Drozdov in hi's

thesis /-15 "7 in the year 1910.

In 1913, Schmitz burned tubular po~ders in a large Krupp bomb,

using an elastic bar with an optical method of recording pressures

inste s of a crusher. He succeeded in obtaining a full curve of the

pressure increase in the bomb from the start to the end of powder

burning. In order to evaluate the accuracy of either burning rate

law, he introduced a new criterion, and proved by means of an attual

test the justness of the burning rate law in the form u - Ap.

Fig. 38 - Dependence of Burning Rate on Pressure.

The criterion determining the justness of this law confirmed

by experiment is presented below.

We shall assume that the following law holds true:

u- Ap; 0since the burning rate is a ratio between the increment of the

burned thickness de and the corresponding time element dt, i.e.,

de

t h e n - -

de/dt - Ap;

de - Ap-dt.

F-TS-7327-RE 153

I ntegrtng-, we__get:

" ':" e t t"

dem e. A pdt or pdt "-, - --

a 0

Burning. terminates when thickness e is burned. The full time14 of burning will be tK, and we will have:

ii A pdt

0

wbence /

pdt el cont-(for the given powder).• A0

Magnitudes e and A characterize the dimensions and nature of

the powder and do not depend on .the con~d~jt-ins of loading.

It follows: that if the burning rate law u - Ap is correct,

the pressure impulse of the powder gases depends only on the burned

thickness of the powder* on the burning rate coefficient, and on

the characteristic properties of the powder,, and does not depend on

the loading density

The full pressure imoulse during the full time of burbing equals

half of the thickness e1 of the burning layer divided by the burning

'rate coefficient A, and does not depend on the loading density.

F-TS-'7327-RE 154

GRAPHIC NOT REPRODUCIBLE

Fig. 39 - po t Curves for Different Loading Densities.

When conducting tests with tubular powder of one kind in a Krupp

bomb, with A varying from 0.12 to 0.26,, and upon measuring the area

under the pressure curves p - fsL), Schmitz had fouind that the areas

t K4pdt found by experiment are actually equal to one another, which

0finding confirms the validity of the u - Ap law.

Figure 39 shows the form and arrangem~ent of pA t curves.

The greater the loading density, the smaller is the burning time

and the higher is the gas pressure p, t curve.

Were the u -de/dt - ap + b law valid, then, after transformation,

we would bave:

do - apdt /r b-dt

and

5pdtm-e1 /a-0

Inasmuch as t K decreases when n.increases, then, according to

the u, - ap + b law, the full gas pressure ir.pulse should become

greater with Increase of loading density.

When applying the u ApI law, where 10 I an analogous

F-TS-7327-RE' 155

deduction Is obtained.

tKSchmitz's tests have shown that pdt does not dtpend on the

0A loading densityo which served to prove the validity of the u - Ap

"* '= law. ." -. .

'These tests had attracted the attention of many investigators

u- and provided data for the verification of theoretical deduction and

lnvestigations. The latter include the work of Muiraur on the study

of the burning rates of colloidal powders (1927-1928) and of

calculating the amount of heat transferred to. the walls during burning

of powder (1924-1925).

U.S. Serebriakov's tests with pyroxyline and nitroglycerine powders

confirmed the validity of the u - Ap law. These tests will be

discussed in greater detail in section III dealing with the physical

law of burning. •

Thus it may be assued that the value of u as the rate of

penetration of the burning reaction inside the grain is directly

-proportional to preskure, i.e., it is expressed by the formula:

u - Ap.

Here A can be express d as the ratio between the burning rate at

p - I (weshall desig ate .it by u1 ) and the magnitude of this pressure

• p-i: •

A- ul/1

(the subscript "1" iadi ates that this burning rate refers to pressure

p -i ).

P(

F-TS-7327-RE 156

This formula can be rewritten thus: L

u -u 1 * 2 up1 1

when p " 1 u - u1*

The dimensionality of the value of u can be seen from the

equality

u dm kg Ip sec dm 2

Ii.e., this represents the rate referred to unit pressure.

Similarly to powder energy f and covolume a, the magnitude uI

constitutes a fundamental ballistic characteristic of powder and,

similarly to I and a, depends on the physical and chemical properties

of the powder.

The value of u for pyroxyline powders varies from 0.0000060 to

0.0000090 dm/sec : kg/dm2 . The thicker the powder, the greater is

the content of volatiles and the slower is the burning of the powder.

The higher the nitrogen content in pyroxyline, the more rapid is

the burning. In nitroglycerine powders uI depends in the main on the

nitroglycerine content itself, and the greater its content the more

rapid is the burning. The admixture of dinitro-derivatives in

powders in a nonvolatile solvent usually reduces the burning rate.

Varying of the conteut of volatiles by +1% lowers the burning

rate of pyroxyline powders by 10-12%.

The following empirical formula for determining the burning

rate of pyroxyline powders, introduced for the first time by

F-TS-7327-RE 157

.~.*4 4 * *,. .,~ *

- - -.- ~- - --- .p

IN-.?~ Drozdov /.162 in our country, appears in American liter~iture:

-4 0o .0025E

* ( -48 (180 -,to) + 2561h + 909.5h'

where C - 9,4O0 (N -6.37) -powder energy in kg-rn/kg;

h.- content of volatile substtnces in %,removed, by six hours- *of drying (moisture);

h ik conltent of residual sovn n ntrevovedafesihours of drying; sleti o fe i

X --nitrogen content in*%

This formula clearly shows the effect of various individual

factors-on the burning rate, but does not give sufficient satisfactory

results as regards our own domestic powders.

-the following formula is better adopted to our pyroxyline powders

(1 - and is more convenient for performing the necessary calculations:

- 0.175(N -63)ma kg

0.42C - to) + 3h + h' sec m0.04(2

0.175 1 0-4(N -6.37) dm kg

0.04(220 - to) + 3h + hsec IM2

j.where 220 is4 the ignition temperature of 'the powder;

S 700,000(N- 6.37 kg-dm/kt;).

* F-TS-7327-RE 158

o.°

Letin assuied o.l the basis of the kiunetic theory of gases that

the bur:iing of powder is a process in which the powder v,oleculis

are split by the impact of gas molecules, aud offers the follouiiig

formula for determining u 1 : I2C2

/ C2

u - (i+ l)e P

pP

where g - acceleration of gravity;

6 mphysicl density of powder;

Cp - probable velocity of molecules of'gases formed in burningof powder;

C - velocity of active gas molecules, whose kinetic energy issufficient to split off at least one molecule when thesurface of the powder undergoes an impact.

c and .cI depend on the nature of the powder and of the gases

formed during its combustion.

Schmitz had conducted his tests at loading densities& of from

0.12 to 0.26. Later tests had shown that at very low loading densitiestK

(,0 .015) the integral pdt is a linearly decreasing function of0

time:

pdt S - Tt.

0

As was shown above, such a relationship is obtained under the

burning rate law u - ap + b.

F-TS -7327-RE 159

-/.-

E2

In the tests conducted by Y.E. Serebriakov /-5_ and A.I. Eokhaziov

it was shown that the integral pdt changes with increase of the0

time of burning only in the case of powders of considerable thickness

( ('2e l > 0.5); ia the.case of very thiL powders, the full impulse, even

-at low loading densities, does not depend on the loading density.

This shows •that the speed of the procesa governing the heating of the

entire powder mass is of importance, and that the increase of the

powder temperature increases the burning rate u and reduces the value

of the integral

tK

3 pt el/U1 9

0

In order to determine the effect of heating on thc burning rate

of powder under a given constant pressure, tests were conducted with

powder strips burned at different temperatures in open air. It was

found that the time of burning of a strip of a given length varies

from 14.1 seconds at t - 15 0 C to 9.4 seconds at t - 500 C, and hence

the rate of burning increases 1 1/2 times. The integral Ipdt - pa tK

was reduced in the same proportion, where pa is atmospheric pressure.

The effect of heating on the burning rate of powder can be

confirmed by the following tests.

If several charges of the same density are burned in succession

in a bomb without cooling, the latter becomes quite hot. The powder

inside the bomb becomes heated also, because the time between charging

.and the end of burning is considerable (several minutes). The value

of the integral becomes smaller with each successive test, which

F-TS-7327-RE 160

~2 _ __ ___ _

IT

condition points at an increasing rate of burning u1 for the same

powder thickness.

Thebe tests permit tne conclusion that the reduction. of the

integral IK with decrease of L for thick powders when A < 0.10 is

the result of heating of the powder mass under the condition of

relatively slow burning, whereby the degree of heating and hence the

increase in the value of u, is the greater, the smaller the value of

Z , i.e., the slower is the burning of the powder at low pressures.

Inasmuch as the integral of IK decreases with the decrease of tL,

the above will be theoretically valid if the burning rate laws u - Ap

and u - ap + b are adhered to. Tests conducted by M.E. Serebriakov

(1932) with powders with hard solvents showed that at pressures

Pm> 1000 kg/cm 2 the linear law u - Ap can be applied to determine

the burning rate, and that at pressures pm < 1000 kg/cm 2 the law0.82

expressed by the forruula u - Alp will apply.

In analyzing later tests conducted by Prof. Yu. A. Pobedonostsev

in bombs with nozzles at very low pressures (5 to 250 atm), Prof.

Ya.M. Shapiro arrived at the relationship u - 0.37 p0.7 which,

seemingly, is contradictory to the relationship u - Ap.

Actually, as was shown above, the decrease of the integral pdt

at small values of & can be explained also when applying the u -ulp

law by t.he increase of the burning rate uI due to heating of the

powder. This explanation s founded on the theory of Prof. Ya.B.

Zeldovich mentioned earlie'.

In any case a mcre accurate evaluation of either expression for

the burning rate law requir s further investigations (see Section I1).

F-TS-7327 -RE 161

41

Inasinuch as powders' i: gun barrels burn unde~r high presues

and udrhigh laigdniistefloigbrigr~eIwna

be considered valid for such pow,4ers:t

-'U umulpe

Going back to the formula for expressing the rate of gas form~ation, Iwe can now write it as follows:

d- S SuIp (31)

or

lS P-s(32)dt e1 Sl IK

2 K

whelre -and Sdepend on the geometry of the powder;1

'.u-As the burning rate of powder when p m1; it characterizesI~-the-nature of the powder and the degree to which it is heated;

p is the pressure at which the powder is burned; it characterizesthe influence of the surroiinding medium on the powder anddepends onA, f,, .,4.

CHAPTER VI - PIESSURE VARIATION AS A 'FUNCTION OF TIME

We havt derived above the following formulas: a) a formula for

' determining the rate ~of gas formation

I!Idt A1 5 el A

F,-TS-7327-RE 162

* Room

and b) the general pyrostatics'for-raila which lakes the igniter into

account.

4-, f, /1 P Pip + ., (34)

.where A 6. a is the relative free space in

-1 0.the bomb in which the powder is burned.

It is necessary to determine the dependence of the pressure change

on time when the powder is burned in a constant volume, i.e., to give

an analytical expression for the relationship between p -f(t)

dp/it and the full time of burning tK.

Differentiating equation (34) with respect to t, we get after

simple transformations:A + A) adp ( dy fA d

dt 2dt (1 at.) 2dt

1 - -is the value of A+ at the start of burning;

1 - aA the same'at the end of burning,

A > -A -1 n

It may be assumed with sufficient accuracy for practical purposes

that when 'cp 1/2

F-TS-7327-RE 163

FI -/ " , -.. c. -.-----

-,. K , - ___,___ 1.

- A2

Then,.making use of the relation (33), we get:

dp. fA d+. fA ul~L )E - :.. . .. sp. (35): ,dt' I -aA dt 1 a4. •l

In obrder to simplify further derivations, we shall consider a

*pwder whose, surface area changes little *hen burned, so that it may

'be* assumed that 6 " -Gc" const.cpA To surh powders belong the tube and the strip, for which the

binomial relationship j-bez(l +*Az) is valid, whereby for the end

Of burning (z = 1, I1)

IC 1 - )(l +X);

• " I+1+2

cp - 2

Therefore

xdc- it (1 + ) - i,

and for tubular or (to a lesser degree) strip powder

dp f ul 11 - PBd 6p - --lp. p (36)

4 dt/ I a eI I.

F-TS-1327-RE 164

We shall introduce the deb1g9:4L1on

el 1 -K

tm - " -secj; (37)u fI pm PB

dp _ P (38)

dt r

Upon separating the variables:

dp. dt

P 13

Integrating, we get

p t

dP I" dt o nP . tPdt or

PB 0

whence on the one hand

t 2.303rlog PB "(39)PB

and for the end of burning

t 2.303t log (40)

F-TS-7327-RE.

onl the other hand

t t

P e or p p~e ,(41)

and, or the end of burning

-pm pe .(42)

'Thus, all the required relationships are derived:

t 2.303:rl::og ;

dp _Pm1.- PB fA u1

dt IX m e, .

IK ellc1- - - secj7.

The magnitude 'r ______ constitutes the burning time-. ~p U( 3 PB) umn

of the powder if it burned at constant pressure pm Btrogot(.The full time of burning at a given loading density is proportional

to v and lgp./P8 , in other words, it is directly proportional to the

r-TS-7327-RE . 166

thickness of the poAder as;d irs.] proportional to the etwrg) of the

powder f and rate of burning ul, a.d .ecreases with the increase of

f&and P., (since the valuc of p1 in th#- deno,.iuator of r is more

effective than in the nu, erator tnder the logarithm); it also decreases

with the increase of the igniter pressure p.,

For tubular powder the relation p - f(t) serves as a characteristic

curve, whose slope .ngle (dp/dt - tanc) must continuously increase

in proportion to the gas pressure during the entire combustion process,

the rate of increase being the higher, the greater are the values of

f, &, u1 and the smaller the powder thickness ei. At the end of

burning the slope angle must be maximum (see figs. 39-42).

We shall derive the relation for the powder gas pressure impulse

on the basis of formula (41)

t Tt V

pdt PB e dt PpBt e d

0 0 0

B) el P -PB (3

- pBT(e - 1) -c(.p - PB) ( 43m - (4)

For the end of burning p Pm and

tK

pdt-- . (44)U-

0

The derived formulas confirm the general laws. of powder burning

F-TS-7327-U. 167

and show that the pressures the titae tor c onlplete co,.bustion and theS /

rate of pressure increase depend on the ballistic characteristics and- density of loading. Thus, it is precisely the ballistic characteristics

* L -,u, Ul the shape and dimensions of the powder (x , G, el) and the J

-P' losding density 4 that can be used to control the magnitude and rate- of zpressure increase of the gases evolved during the burning of- .

S, powder in a constant volume and to regulate the phenomenon of powder

buining and gas formation.j Example. Determine the maximum pressure and time of burning of

.tubular powder (2e- 1.00 am) when m 0.25;

- "~- -~- " 9 00,000. kg-dm/kg; a - 1.00 dm3/kg;

n1 - 0.0000075 dm/sec : kg/dm2, igniter pressure PB 50 kg/cm2 ;

" Pm PB + - .- 5000 + 900000 • 0.25 305,000 kg/d 2 - 3050 kg/cm 2 ;•1- a I-1.0 • 0.25

el 0.005I]m0.0 0- 667 kg/dm2 •sec,uI 0.0000075

IK 667- -66 - 0.002223 sec;

- P 300000

K 2.303 logP ,..2.303 0.002223 log 3050 2.303 .002223

Ps 50•

• 1.7853 - 0.00913 sec.

F-TS-7327-RE 168

01! FI

t

Curves p -pB e fo'r the given powder will have the following

form:

a) at different A and the same PB (fig. 40)

tK tg t K

pdt " pdt " pdt l

0 0 0

b) at the same A and different pB (fig. 41)

P m P B - PB Pm Pm PB PB

pdt- pdt - pdt.

0 0 0

Fig. 40 - ppt Curves at Different Fig. 41 - pt t Curves at DifferentValues of & and the Same Value of Values of PB a.,d the Same Value of

PBO

( "For powders of the same kind but varying thickness el < el < e ]' .

-- at the, same loading density and the same igniter pressure PB, curves

pp t will have the form shown in fig. 42, where

F-TS-7327-RE 169

1WM M12

N41.~

t e e

,K; K t

f~: r

,Vif-i*in odri salbmsoflmtdeonain(ai

-lnt o obadisdaee o xedn -) h cre

ressure~~ inras-r ralin hratr

eyer ~ t :h tode is :undi ea lon bob(mogb

thit poebrin powder In sfalleobs of li20ited5 elongtieo(aiog

* ofpres re increased, tre grdutl in ctharcessre eodrbh

cuhrin'roeae, l the s ta0.0 o powders is. mmni atohibcb(lklgb

th ~mpressure 'vae~i rcred byrame~e take 230shp0f ae.z

orde of and.k in theef

case. of thick powders it the same value o~ A the pressure was found

to be -4000 kg/cM2

GR1APHIC NO REPRODUCIBLEFig. 42 - pt t Curves at Vario s.Values of

It the crusher cones areplaced at the en s of a long bomb and

F-TS-7327-RE ~- 170

ZMM ME 1 I

* :/' ~ V'al-

the powder is ignited fron- the side at one of the ends, the curves

of the pressure growth at both ends will appear unve-like in form

if the charge is not dittributed uniioiwily., whereby the Ixiurd of

one curve will correspond to the minimum of the other at the same

instant of time. These tests show that a wave-like process of pressure

distribution occurs in a bomb, where the pressure waves are reflected

from one end of the closed pipe to the other.

All of the tests described above were conducted by Vieille in

a special long bomb, using cylindrical crushers for recording the

pressure growth at both ends. We had verified Vieille's deductions

on a similar test set-up using conical crushers.

When the charge concentrated at one end of the bomb is ignited,

a localized pressure increase occurs due to gases becoming separated

or detached from the burning surface. This pressure increase is

the greater, the larger the surface area of the charge, i.e., the

thinner the powder. Due to the action of the localized pressure, the

gases begin to move to the opposite end of the bonb in the form of

a stream whose 'rate of flow increases rapidly, thus tending to form

a vacuum at the point of the start of burning. Upon reaching the

opposite end, the gas stream will stop abruptly at its forward end

and its kinetic energy will be expended in compressing this part of

the bomb until the pressure developed in it exceeds the pressure at

the pl-ace occupied by the charge. The movement of the stream will

then be reversed and the burning of the charge will proceed more

intensely under the pressure of the gas stream, creating once again

a localized pressure increase. This phenomenon will then be repeated.

F-TS-7327-RE 171

..............................................,/

At the end of burning of the charge the Oheniomenon takes oti the

character of P' damped oscilIjiting gas motion inside the bomb*'

Siila~r conditions mlay occur in the 1,-re of a guni barrel. There-

'tore,. when the loading density is low. and the chamber is not completely

Ai1ed with powder, it is recommonidd-Io__uniformly distribute the

~-- chatrge along the entire length of the chamber.

VM72 7

11, j1 111 19 1 1 1 11 1 1