Optimization of Gas Guns pTlC of Gas Guns pTlC 'I. JUN 23, 1981 ... a TIE U N C L A S S I FI ED. ... These guns have inside barrel diameters of 2, ...

0 HDL-CR-81-723-1LEU•()

may 1981

kill Performance Calculations andOptimization of Gas Guns pTlC

'I. JUN 23, 1981

Arnold E. SeigelSE CT3302 Pauline UriveChevy Chase,. MD 20015

Under cq~trectOAAG33.7O4723

U S. At-my Electronics Re9searchand Development Command

Harry Diamond LaborstorietA-deiphi, MD 20783

00!

~ 6

The findings in this report are notto be construed as an official Departmentof the Army positicn unless so designatedby other authorized documents.

Citation of manufacturers' or tradenamoes does not constitute an official in-'dorsement or approval of the use thereof.

Destroy this report wheit it is nolonger needed. Do not return it to theoriginator.

UNCLASSIFIEDSKC3IJ•UY CLASSIFICATION OF THIS PAOC (Won Daoe &Mmt•f

SREPORT DOCUMENTATION PAGE READ ,..TRUCT.ONSBEFORE COMPLETING FORM

'HLCR-81-723-1NNE _A~SINO~ ~itNsCTLGku~t

, 5f.iAELrf 5%''* £ 'Vrm OOCOVIRW

/Performance Calculations and Optimization! Contractor's ýep.t.of Gas Guns, -_ t " .....7'"

Arnold Seige OAAG39-76-M-723 /

51

KF0 MING ORGANIZATION MAMIE ANO ArowIs- AO. PROGRAM lýEIEMT. PROJ6CT. TASKCARILA 4 WOR&NYRIT MUM0ftI3302 Pauline Drive Program ElementChejy Chase, MD 20015 6.21..20.A

1:1I. COI" jLNG OPPICZ MAMIE AND ADOS IL '- "F. . AI -.

Harry Diamond Laboratories May 6,98 J2800 Powder Hill Road 13I. Mu~aef OF *&GsMAdephi, Maryland 20783 45

14L MO4GiTOR1140 A4SNMCV MAMIE # AOO*USVfl 4OUGM 4*N Cmm9tiS4 aOlteejZ 16, SICUURT'( CLASS. (.1110.l .06w)

Unclassified

14. 30711014 VIYUCHN $AbNI~f* als X"04")

Approved for public release; distribution unlimited

HDL Project: A77886ORCMS Code: 612120, H25001_..DA Project. Hi26,. 2J

Compressed gas guns are used at the Harry Diamond Laboratories(HDL) to test fu:es and fuze components. Calculations were madeto describe the performance of both existing RDL gas guns andplanned future ROL gas guns. * In the latter case, chamner diameterand chamber length were varied to evaluate their effect oa pro-jectile velocity. These results permit selection of an optimumchamber geometry.

do " 0* 10011 is 0... a TIEO D ̀ J AW " U N C L A S S I FI ED

CONTENTS

NOMENCLATURE . ............................. 5

1. INTRODUCTION . ......................... . 7

1.1 Use of Gas Guns at Harry Diamond Laboratories ....... ...... .. 71.2 Preburned Propellant Gun ................ . .................. 7

2. CALCULATIONS ON H DL GAS GUNS ....... ..... .................... 8

2.1 Assumptions Made in Calculations ..... ....... ................ 92.2 Methods Used to Calculate Gas Gun Performance ...... .......... 9

3. RESULTS OF CALCULATIONS ................. ....................... 10

3.1 Existing HDL Guns ................ ........................ 103.2 Future HDL Guns ............. ........................ ... 103.3 Effects which Decrease Projectile Velocity ...... ........... 17

Appendix Description o the reurne Propellant Gun ........ 29

Appendix B. The Gas Dyuamics Equations for a Chambered PropellantGas Gun .............. ....................... ...... 31

Appendix C. Calculations by Electronic Computing Machines ......... ... 37

Appendix D. The Special Solution of Pidduck-Kent ....... ............ 39

DISTRIBUTION ................ ..... ............................ .. 45

ILLUSTRATIONS

Figure Page1 HDL Preburned Propellant Gas Gun. .... ................. 72 HDL Gas Gun................ ..................... 73 2-Inch Gun, Air Driven ......... ..................... .... 114 2-Inch Gun, Helium Driven ................ .................. II5 Comparison of Air and Helium Drivers for 2-Inch Gun ........ ... 12

6 Velocity as Function of Breech Position for Fixed Length(3-Inch) Air Driven Gun .................... 12

7 Velocity as Function of Breech Position for Fixed Length(3-Inch) Helium Driven Gun ................................... 13

8 Comparison of Air and Helium Drivers for Variable BreechPosition in Fixed Length 3-Inch Gun ........ .......... . . 13

9 Velocity as Function of Breech Position for Fixed Length(4-Inch) Nitrogen Driven Gun ..... .................. .. 14

10 Velocity as Function of Breech Position for Fixed Length(4-Inch) Helium Driven Gun ............. ................... 14

11 Effect of Chambrage on 4-Inch Nitrogen Driven Gun .... ....... 15

3

CONTENTS (cont.)

Figure Page12 Expected Characteristics of 7-Inch Infinite Reservoir Gun 1613 Predicted Performance of 7-Inch Gun with 100-Foot Driver

Section ........ ............. ............... ........... 1614 Velocity as Function of Breech Position for Fixed Length

3-Inch and 9-Inch Launchers .. .. ... ......................... 18

15 Velocity as Function of Breech Position for Fixed LengthAir Gun, D0 /D -- ......... ..... ........... .. . 19

16 Velocity as Function of Breech Position for Fixed LengthAir Gun, Do/D 1 - 2 ...... ...................... 19

17 Velocity as Function of Breech Position for Fixed LengthAir Gun, Max Acceleration - 320 g& .................. .... 20

18 Velocity as Function of Breech Position for Fixed LengthAir Gun, Max Acceleration - 1280 g ................. ..... 20

19 Velocity as Function of Breech Position for Fixed LengthAir Gun, Max Acceleration - 3200 g ................. .... 21

20 Velocity as Function of Breech Position for Fixed LengthHelium Gun, D /D - . ........ ... .... ... . .. ... . 22

21 Velocity as Funciion of Breech Position for Fixed LengthHelium Gun, Do/D1 - 2 .................... .............. 22

22 Velocity as Function of Breech Pesition for Fixed LengthHelium Gun, Max Acceleration - 320 g ... ........... .... 23

23 Velocity as Function of Breech Position for Fixed LengthHelium Gun, Max Acceleration - 1200 g. .......... 23

24 Velocity as Function of Breech Position for Fixed LengthHelium Gun, Max Acceleration - 3200 g ......... ... 24

25 Velocity as Function of Breech Position for Fixed Length GasGuns, Max Acceleration 3-0 g25............... 25

26 Velocity as Function of Breech Position for Fixed Length Gas

Guns, Max Acceleration - 1280 g& .... .............. 2527 Velocity as Function of Breech Position for Fixed Length Gas

Guns, Max Acceleration 3200g ......... 26

Table

I Existing HDL Gas Gunas .................... ,

11 Future HDL Gas Guns ........ ......... ..................... 9

4

NOMENCLATURE

a Sound speed of driver gas.A Local gun cross-sectional area.b Covolume.D Inside diameter.E Internal energy of driver gas in control volume.g Initial (maximum) projectile acceleration.G Mass of driver gas initially in reservoir.h Enthalpy of driver gas.L Gun barrel length.m Mass of driver gab in control volume.M Projectile mass.n Arbitrary integer.N Degrees of freedom of driver gas.P Gas pressure.R Gas constant.t Time.T Gas temperature.u Gas Velocity.u p Projectile Mach number based on initial gas sound speed (u p/a ).

v Specific volume.X Dimensional position coordinate along gun axis.i p Nondimensional gun barrel length (P A IL/Ma 2).

Z Fractional pin position in gun. (X /(X +L)]

Ca Parameter of Pidduck Kent solution.y Driver gas adiabatic constant (5/3 for helium; 7/5 for air and nitrogen).P Local instaneous driver gas density.

0 Initial gas density in reservoir.

a Raimann function of driver gas.

Subscripts

c Control voluoe.0, o Initial state in chamber at gun.i Position in barrel at exit of transittao. section.1 Gas barrel. !p Projectile.

RE: Contract DAAG-39-7644-723There should be four digits at end of con-tract but contract office at HDL wnable toverify. The records have been retired per"Ms. Dydak, contract office and Mr. Curshack,project officer

1. INTRODUCTION

*1.1 Use of Gas Guns at Harry Diamond Laboratories

Gas guns are used at the Harry Diamond Laboratories (HDL) toaccelerate ordnance items to velocities typical of artillery projectiles,i.e., 500 to 4000 ft/s.

For certain test considerations, it is necessary to achieve therequired velocity with a limited acceleration or g force exerted on theprojectile while in the gun. Therefore, long barreled guns operating atlow-pressures are used. These guns are located in fixed length rooms.The characteristics of the guns must be known to configure them properlyand to maximize the achievable velocity imparted to the fuze or fuzecomponent for a particular acceleration level.

1.2 Preburned Propellant Gun

The gas guns at HDL are classed as preburned propellant (PP) guns.PP guns are described in appendix A. Schematically, they appear assketched in figure 4. (See Nomenclature page for meanings of symbols.)

-RDUALQB ABRUPT.* '• :COMPRESSED .GAS TRANST1ONSECTION

Tm M BARREL

0,0o PO ao

EVACUATEDIFKF•RED

Figure 1. HDL Preburned Propellant Gas Gun,

In the HDL gas gun, the compressed gas (nitrogen, air, or helium) is atroom temperature (T - 20"C); one type of PP gun used at HDL is shown infigure 2.

EVACUATED BARRELP"

ATMOSPHEE DU

figure 2. HffL Gas Gun.

7

htCd=MQ E" MWMJIOT n "

2. CALCULATIONS ON HDL GAS GUNS

Specifically, two tasks were undertaken. One was the calculation ofthe performance of various configurations of four existing HDL gas guns.These guns have inside barrel diameters of 2, 3, 4, and 7 in. Table Ilists data pertaining to these existing HDL gas guns.

TABLE I. EXISTING HDL GAS GUNS

Barrel Chamber Projectile Propelling gas*

D1 L + x D x M Type Max P(in!) . (ft)° (in?) (fs ())

2 32 3.2 ft 3 300 to 1000 Helium or air 100

2 32 2 Infinite** 300 to 1000 Air 14.7

3 97 3 20 1000 to 2000 Helium or air 600

4 100 4 12, 24, or 400 to 3000 Helium or air 60036

7 100 or 7 Infinite** 400 to 10,000 Air 14.7314

7 314 7 100"** 400 to 10,000 Helium or air 100

*T 0 room temperature.

**Projectile initially positioned at beginning of barrel with back of

"projectile subjected to atmosphere.***Possible configuration.

l8

The second task was the calculation of the performance of andoptimization of two future planned HDL gas guns. One gun will havea 3-in. inside barrel diameter; the other will have a 9-in. insidebarrel diameter. Table II lists data pertaining to these future HDLgas guns.

I

TABLE II. FUTURE HDL GAS GUNS

Overall length Barrel Chamber Max Max P /M Propelling(L + xo) D D acceleration gas. 0 1 0 based on may a

(ft) -(in.) acceleration

200 3 To be 3200 g 1 psi/gm Helium oroptimized air

200 9 To be 3200 g 9 psi/gm Helium or-optimized air

2.1 Assumptions Made in Calculations

The following assumptions were made in the calculations.

(1) The compressed propellant gas behaves as an ideal gas.(2) The compressed propellant gas expands isentropicall'.

(Thus, gaseous frictional and heat-transfer effects areassumed negligible.)

(3) Projectile friction is negligible.(4) There is no gas leakage around the projectile.(5) The pressure in front of the projectile is negligible.

These assumptions are discussed further in section 3.3.

2.2 Methods Used to Calculate Gas Gun Performance

The following two mathods were used to calculate the performanceof the UDL gas guns:

(I) The plots that were used were dimensionles¶ projectilevelocity, u , versus noudimensional length, x , plots. They wereobtained b7Papplyilg the method of characterigtics for the constantdiameter chamber and the constant diameter barrel. Steady flow equationswere used in the transition section. The calculations were done byelectronic computing machines.

IA. E. Seigel, The Theory of High Speed Guns, Agardograph 91 (May 1965),obtainable from the National Technical Information Service, DefenseDocumentation Center, Springfield, VA, AD 475660.

9

The gas dynamics equations that were used are described in appendixB.

Additional information relative to the plots obtained from theelectronic computers is discussed in appendix C.

(2) The second method used was the Pidduck-Kent Special Solution.This closed-form solution is an approximation to the wave solution. Itis particularly accurate when applied to low G/M cases. (G is the gasmass, and H is the projectile mass.) This procedure was used to calculateperformance in cases where the dimensionless u versus x plots weredifficult to read and where G/M was less thanul/4. AppeRdix D describesthe Pidduck-Kent Special Solution.

3. RESULTS OF CALCULATIONS

3.1 Existing HDL Guns

The calculated results for the existing HDL guns are presentedas plots in figurej 3 to 13.

Figures 3 to 5 (2-in. gun) show projectile velocity (up) as afunction of initial pressure (P.), projsctile mass (M). and type of gas(helium or air).

Figures 6 to 8 (3-in, gun) show the effects on projectile velocityof varying chamber length (xo), M, and gas (helium or air).

Figures 9 to U1 (4-in. gun) show the offects on projectile velocityof varying %o, diameter ratio (Do/D1). and gas (helium or air).

Finally, figures 12 and 13 show the effects on projectile velocityof varying M, length of barrel, and gas (helium or air).

3.2 Future HDL Guns

The calculated results for the future 1DL guns are presented as"plots of up versus xo in figures 14 to 27.

It was established here that the guns with 3- and 9-in.-dimeterbarrels could both be represented by the same up versui to plots because

up- u%/a - (it, G/I, Do/Dl, given gas),

10

C4AMIEMI VOLUMEs

I IN.

PAI

U. fit$S

low

I ¶

60 '. mj-m o o

104 ms

Figure~~~0 4. 2-#%1u;Hei Die

1I*A'fCA 'VOLUO.E 3 OT

war

tft

300

3m OaV

II4Figure 5. Comparison of Air and hielium Drivers for 2-Inch G4n

"It

•tos

T & * Ti " \e f o r f i e

Ib --*vg

im-S

Figure 6. Velocity as Function of Breeth Postion for fed

ur LeAgth (3-Inch) Air Driven Gun

12

U0 F T11.

M ISOO GRAMS

1400

1 0 M o 2000 G RA M• S

S1300

1000

P6000 S

700 -

400

20- I I I JI . . J

0 1 t0 Is 20 21 30 33 40 46

X0, FT

Figure 7. Velocity as Function of Breech Position for Fixed

Length (3-Inch) Helium Driven Gun

F -S----- "ILIUM

.130/AIR

M 1000 13AAMS ...

1400- // o

k I /30, 100 OPAUS

IS.

1100 /

•!• ,~~~~~oo -///• . £>• /i7/~/ /--'"/

• 13,

to0 PSI

600 IO

10 to 200 so

Xe. FT-

Figure 8. Comparison of Air and Helium Drivers for Variable

Breech Position in Fixed Length 3-Inch Gun13

01 A IN.

•V1

00 S00 PSI'-

0U00.X0Q -- ---

100 F

1600

UFI& 100

S1.414

"000

10 le 30 *4

X*. PY

Figure 9. Velocity as Fur.tion of Breech Position for

Fixed Length (4-Inch) Nitrogen Driven Gun

Ut-0 .''r A- IN 7 L%0,-,0 0

2"a0

I-t

Itoo

Figure 10. Velocity as Function o. Breech Position for Fixed

Length (4-Inch) Helium Driven Gun

14

lowo

IZOO -

tw~o - G-

600 PS

400 0

~00

1002

Ot

Figureý 11. Effect of Chembrage (chamber) on 4-Inch NitrogenDriven Gkin

15

U,. FYIS

NAMBIENTAIR -"-U,

12001-P S

A 1100

LIwo1000

700 -L * 14 fT

600

f..1001 •I . t

too

XM AW w e lo 00 601 C" I."l 20

U~~~M GRI MAW.1

Figure 12. Expected characteristics of 7-Inch GnfiniteReservoir Gun

IN6

Ito* V

Figur 13 rdctdPromac f7InhGnwth10Fo

1016

where

Rp AyIp/Mao 2 ,

G/M- pOAoXo/M = (YPoAlxo/Mao 2 ) (Do,/D1 ).

Since the gun length, L, for both the 3- and 9-in. guns is 200 ft,

L - 200 - xo.

Also, the acceleration may be expressed as

From the above equations, one obtains

up- Up (g, X1, Do/D 1 , given gas).

Thus, both the 3- and 9-in. guns are represented by the same plots forthe case of equal & force.

Figure 14 shows a general arrangement of the 3- and 9-in. gas guns.This figure shows that three values for the maximum g force will befeatured (3200, 1280 and 320 g).

Figures 15 to 19 are plots of up versus xo for the three values ofg force, varying Do/D 1, and using compressed air as the propelling gas.Figures 20 to 24 are similar plots using compressed helium.

Figures 25 to 27 compare results obtained for compressed air andhelium by showing up versus xo for varying Do/D 1 ratios for the threeg values.

From the plots in these figures one may optimize the performanceof the 3- and 9-in. 8as guns by proper selection of the gun geometry.For example, from ligure 27, if maximum velocity were desired at a maxi-mum acceleration of 3200 g, one would design a gun with a Do/D 1 - 2 andx0 W 30 ft or so. One should check to make certain that the quantity ofgas emerging from the barrel behind the projectile would not interferewith the experimental setup and could be taken care of safely.

3.3 Effects which Decrease Projectile Velocity

In the calculations, these were assumed:

(1) The compressed propellant gas is an ideal gas. Actually,the gas behavior is nonideal; as a result, the pro ectile velocity willbe less than calculated, as has been demonstrated. Nevertheless, theexact equation of state could be used to obtain the correct projectilevelocity.

1A. E. Seigel, The Theory of High Speed Guns, Agardograph 91 (Mty 1965),obtainable from the National Technical Information Service, DefenseDocumentation Center, Springfield, VA, AD 475660.

17

E INI VU L

00 .",O* I.

Ph~ > VIPIA )~UO

L Ipj II~IUUI IIP

MAXIMUMeoiy sfntino rec oiio o ixdLntACCI ~ 3-Inch too 9-ICh4 L4aAM1 NUNcheS

P,-18

; 0

1 "

•41

0•

I ,

0 ...

!/''x,r• ---

Sk a

u . r2i1206

US.

01.4

II

2W

* 230 -

I toIs to2 311 A as *A 44 w i 1 1 is 11 i

Figure 17. Velocity as Function of Breech Position'for FixedLength A4ir Gun, Max AUcaleration 320 g

tooo

i-

O- *

Figure 18. Velocity as uvction of Breech Position for FixedAeWngth Air Gun, Max Acceleration - 1280 g

20

U,. 4A

12tot

0 o 320

V40,tjU lt tw

S #, ~o

olct Cceit as a

o

0

4-4

0 V

> ;7C*14

00

~4

al4A

00

0 •

Ax

to

-4

I I 1

*4

f-p2

22N

0

'a

(0

0

040

0y40

-1

v4

' 44

0

o Q

.-4• 0 1

C~4

I t I I23!

1I i

•41 *

.'

OGr

|WOO

low-

Figur 24. Velcit a$Fnto fBrehPsto

U my

"ais

'• F? ---

i'i Figure 24. Velocity a8 Functiou of Breech Position

for Fixed Length Helium Gun, HaxAcceleration 3200 g

24

U. IFT 1 0 -

W|LIUIAU//.P, /

HILIUM10

1000 v

j/to0 /'m J

3w

0, _________________

a I 3q I H 30 z It 60 1 i

Figure 25. Velocity as Function of Breech Positionfor Fixed Length Gas Guns. Max Acceleration 320 g

U0,

/ .I.4U'U-

44/ "/,_ / /

//

/4ý 01-a

/

Figure 26. Velocity as Function of Breech Position for Fixed

15o0

Slim - 1100

/ /;ic // ..

/ /

•Lu~ • . Ve'ocity &a Function of Breer-h Posit ion

for Fixed Length Gas GuIas,' ,H-.a

Acceleration - 200

26

Lm.

(2) The compressed propellar. gas expands isentropically.In fact, the gaseous frictio::,) effects and heat-transfer effects dodecrease the projectile velo.i.y. The quantitative decrease is afunction of up/ao, as has been shown. 1

(3) Projectile friction is negligible. Whether or not thisis true depends on the precision specified in fabrication of barrel andprojectile as well as on the type of projectile material used.

(4) There is no gas leakage around the projectile. Again,this depends on the projectile and barrel construction, materials, andseals.

(5) The presswre in f.-nt of the projectileý is negligible.This may be assured by evacuating the barrel.

tDepending on the above effects in the region of velocities of

the HDL gas guns, the experimenta.l velocities will range from 2 to 12 percent* of the calculated LX'eoretical velocities.

1A. E. Seiiel, The Theory of High L,)eed Guns, Agardograph 91 (May "965),obtainable from the Nationa. Technical information Service, DefenseDocumentation Center Springfield, VA, tD h75660.

27

APPENDIX A.--DESCRIPTION OF THE PREBURNED PROPELLANT GUN

Here is described the gun system in which the propellant hasbeen completely reacted before the projectile is allowed to move. Thisgun system is termed a preburned propellant (PP) gun. The gunconsists of a chamber of diameter Do Joined by means of a transition6ection to a barrel of diameter D1 . The projectile is positionedinitially so that its back end is at the beginning of the barrel section.Immediately before the projectile begins to move, the reacted propellantproduces a gas in the chamber at an initial and peak pressure, P.; soundspeed, ao; temperature, T0 , etc. (See fig. A-I.) (In the case of HarryDiamond Laboratories (DL) guns, the zero subscript values are the initialcund.itions of the compressed gas).

COMPRESSED GASOR GRADUAL OR ABRUPT

0 Po ao , _To

I ___L

CHAMBERFigure A-I. IQDL Preburned Propellant Gas Cun

When the chamber diameter is greater than the barrel diameter (Do/D 1 > 1).the gun is desc-ibad as a "chambered" gun, or a gun with "chambrage."When the chamber diameter is equal to that at the barrel, the gun isdescribed as "having no chambrage," or as a "constant diameter gun."

In practice, a PP gun may employ a diaphragm to separate thepropella.it in the chamber from the projectile; this diaphragm isruptured when the propellant has completed its reaction. Anotherpossibility is the use of a "shear disc" around the projectile itself: thedisc shears when the reaction has been completed. The HDL ppgun uses as a propellant a nonreacting gas (such as compressed helium orair). A retractable pin restrains the projectile.

29

J14M&EWIM Iva ELANS4OT 71M

In a PP gun, the projectile is restricted from movement until thepressure has reached a peak value; it can be shown that, after theprojectile is released, the pressure behind the projectile decreasesas the projectile increases in velocity and moves along the barrel.(See fig. A'2). .

Pp P

Figure A-2. Pressure versus Velocity in Gun Barrel.

30

APPENDIX B.-THE GAS DYNAMICS EQUATIONS FOR A CHAMBRED PROPELLANT GAS GUNU

To determine analytically the behavior of the expanding propellant

gas in a chambered gun, the assumption is made that the flow is isentropic.

The one-dimensional characteristic equations are applicable to theconstant diameter chamber and are applicable to the constant diameterbarrel:

S (u + a) + (u + a) j-(u + a) - ,3 t - (u--a -- 0

where the sound speec4 aand the Riemann function, a, are defined by

a2 -

do - (dp/aP)s

With the notation

SD : • + (u + a)

the characteristic equations become

D(u t a) (B-1)Dt

(See fig. B-I).

TRANSITON

St u±o)o)D( 0) 0

- -x, DtDt

Figure B-1. Gas Dynamics for Chambered Propellant Gas Gun

31

The gas flow in the transition section, which joins the chamber tothe bore, is actually a two-dimensional unsteady flow. However, it isnot feasible to solve the two-dimensional unsteady equations. Thereare two possible approximate methods of'treating the flow through thetransition section. The first method is to assume that the change inarea from the chamber to the barrel occurs gradually so that the flowmay be assumed tc be one-dimensional. Then, the one-dimensionalcharacteristic method can be applied to this change in area section.The characteristic equations become, for the change in area section,

D (u+ o c )audA (B-2)

D(u + 0) a (u + a) + (u + a) a (u+ ) - A dx'Dt at T-- -- d

where u is the gas velocity, a is the sound speed, a is the Riemannfunction, and A is the cross-sectional area of the gas layer at

position x in time t. These equations require a tedious numericalprocedure to solve and are generally not suitable for hand computation.However, the quantity uta, in contrast to the constant diameter case,does not remain constant for disturbances in the transition section.

The second approach, one chosen here as being more convenientand a good approximation to the actual situation, is to assume thefollowing: At any given time, the rate of change of mass and energywithin the transition section is negligible relative to the differencesbetween the exit and entrance fluxes of these quantities; thus, thechanges due to variations of time are assumed negligible relative tothose due to the variations in position within the control volume.This assumption is made clear by taking as a control volume the tran-sition section as shown in figure B-2.

o D1

. L• "•CONTROL VOLUME

Figure B-2. Variations of Time, Mass, and Energy.

32

Then the applicable equations of continuity and energy are, respec-tively,

Cno (PuA)c - (PuA)i (B-3)ConVol

and

ConVol c

where m and E are the mass and the internal energy in the transition

section. By our assumption above, the two unsteady terms on the leftside of equations (B-3) and (B-4) are negligible.

It is observed that, if the transition is rather sudden, thecontrol volume is small; hence, since the unsteady terms on the left side ofthese equations are proportional to the magnitude of the controlvolume, the unsteady terms are necessarily small. Thus, in the caseof a sudden transition, the assumption above is automatically valid.

With this assumption, the equations which are applicable to relatethe conditions at the entrance of a transition section to those at the

Sexit of the transition section are the quasi-steady equations ofcontinuity and energy. Thus, at each instant of time, the applicableequations are

+ " hi + u12 function of time (B-5)

JcucAc PiuiAi * function of time. 'B-6)

Since the flow has been assumed isentropic, the thermodynamicrelation between enthalpy and pressure is

dh- (dp/p)0 (B-7)

andhi- hca f dol/ (B-8)

Pc

33

iI

Equation (5) becomes

ui 2 _ Uc2 fPc dp/po .B- 9 )

2 Pi

It may be shown from equations (B-3) and (B-4) that the ust of thequasi-steady flow equations to describe the gas flow between thechamber and the barrel of the gun yields a larger projectile velocity thanwuld be yielded by the use of the actually applicable unsteadyequations. However, experimental results from a chambered PPgun by Seigel and Dawson have demonstrated that the differencewas small enough to be unmeasurable. These experiments were made witha gun using room temperature air at about 3000 lb/in.2 as a propellant.The gun had a 0.50-in.-diameter barre; which could be joined to variouschambers of 30-deg half-angle taper. The projectiles were 1-gmplastic projectiles and were sheared by the compressed air in thechamber. The measured projectile velocities were compared with thetheoretically predicted velocities based on the use of the quasi-steadyequations above.

The comparison showed that the quasi-steady flow approximation izthe transition section yields good agreement with experiment.

Figure B-t3.• gaas the. characterlstics of a chambered PP gun inaction.,

t --Figure B-3. Prebuzned Propellant Gun in Action.

54

Characteristics may be drawn in the transition section by fairingthem from the known conditions at the inlet to the known conditions atthe exit. The simple wave region in the chamber for which u + a - ois denoted by the letters A, B, and C.

With equations (B-1), (B-6), and (B-8) and the isentropic equationsof state of the gas, it is possible to calculate quantitatively thebehavior of the projectile in a PP chambered gun.

For the RDL gas guns, the ideal equation of state was used:

p a PRT$ (B-10)

p- a Y PO/oY" (B-I)

From these equations, the sound speed and the Riemann function become

a2 . yp/p - yRTI (B-12)

a - 2a/(y - 1)' (B-13)

where a is taken to be zero at a - 0. The enthalpy for an idealPgs is

Sh- a2 /(v - 1) - (y - 1) 02/4. (B-14)

35

APPENDIX C.--CALCULATIONS BY ELECTRONIC COMPUTING MACHINES

The method of characteristics as outlined in Appendix B may-benumerically applied to calculate the performance of a preburned propellant (PP)gun system. However, in the cases where the chamber is not effectivelyinfinite in length, hand calculation becomes extremely lengthy and tedious.Further, the accuracy of the calculated results depends on the spacing ofthe numerical points. The greater the spacing, the greater the error;'hand calculation, particularly, does not allow small spacing. Thus,calculating by electronic computing machines offers great advantagescompared with hand calculation. Not only is much time saved, butaccuracy may be substantially increased.

Calculations have been obtained by electronic computing machines.The results may be expressed in terms of dimensionless plots of up versusxp, or they may be in terms of other dimensionless variables. Thus, ap ot of dimensionless projectile velocity versus projectile travel for agiven geometry (i.e., for a given Do/D 1 and a given G/M) has been foundconvenient. The results of computations t.ade for the U.S. Naval OrdnanceLaboratory at the Naval Weapons Laboratory on electronic computingmachines have been published for a PP ideal gas gun. 1 The plots infigure C-I present curves of up/a0 versus Xp for varying values of G/Mand a given Do/D 1 and y (as adapted from Agardograph 91, fig. 201).

800U Y * 1 .4

Xpa p•AAI /MaX -

Figure C-i. Curves for Preburned Propellant Ideal Gas Gun Behavior.

1A. E. Seigel, The Theory of High Speed Guns, Agardograph 91 (May 1965),fig. 20 and 21, obtainable from the National Technical Information Service,Defense Documentation Center, Springfield, VA, AD 475660.

S~37 •

M= WPL LM-MFL

I

The curve marked G/M = = is the infinite chamber length case.

The above curves are replotted in figure C-2 as u•/ao versus PoAl/la.o2

for varying values of Do/D 1 and a given G/M and y (as adapted from Agar ograph91, fig. 211).

Up 1

a0

Xp= PoA 1 X/Mea

Figure C-2. Replotted Curves for Preburned Propellant Ideal GasGun Behavior.

Figures 20 and 21 of Agardograph 911 thus present the entire performanceof a projectile in a PP ideal gas gun with chambrage (chamber).

'A. E. Seigel, The Theory of High Speed Guns, Agardograph 91 (May 1965),fig. 20 and 21, obtainable from the National Technical Information Service.,Defense Documentation Center, Springfield, VA, AD 475660.

38

APPENDIX D.--THE SPECIAL SOLUTION OF PIDDUCK-K1NT

The classical LaGrange Problem of Internal Ballistics presentsthe situation where a projectile initially at rest in a constant cross-sectional area gun is propelled by a propellant which burns instantan-eously. This is the problem of the preburned propellant (PP) gun. Theprocess is considered to be one-dimensional, frictionless, and adiabatic.LaGrange initiated the study of this problem in 1793 when he presentedan approximate solution to the problem. After LaGrange had initiatedthe study of the LaGrange Problem, Hugoniot extended Riemann's theoryof waves of finite amplitude and applied it to the problem; he solvedit to the point where the first expansion disturbance shed by theprojectile reached te breech. Gossot and Louisville went stillfurther and followed the first expansion disturbance after it had beenreflected from the breech back to the projectile. The culmination ofthis method of attack (which did not use the method of characteristics)was the complete solution by Love in 1921 up to the first disturbancetraveling back and toward the breech for the third time.

Love replaced the system of hyperbolic quasi-linear partialdifferential equations which describe the problem by a single partialdifferential equation of second order for one single dependent variableand solved it separately for each wavelet. His solution containedlengthy and involved computations and was valid only for a Noble-Abelgas (with isentropic relation p(v - b)y constant), whose ratio of

suecific heats was of the form

y M.2n+li

2n - 1

where n is an integer.

Pidd'ick noted, from the results he had calculated with Love'sequations, that the ratio of the breech pressure to the pressure of thegas directly behind the projectile oscillated as shown in figure D-1.

PPBREECH

~PROJECTILE

PX /XPx .L UMITING VALUE

pTIME -4Figure D-1. Osclllation of Frictionless Gun.

39

:I

This oscillation results from the lowering of the pressure occurring asthe first disturbance reflects back and forth between breech andprojectile. Pidduck found that the oscillations damped out and thath the pressure ratio approached a certain limiting value. He thendeduced a "special solution" to the governing differential equationswhich indeed did yield the condition that the ratio p( p is aconstant, not only in a limit but at all'times. This solution, ananalytic one, did not satisfy the initial conditions of the LaGrangeProblem; the initial conditions for the special solution were anonuniform distribution of density and pressure. Pidduck and alllater investigators have suspected, but not proved, that the accuratesolution to the LaGrange Problem approached the special solution inthe limit of large travel.

The special solution has also been derived by Kent and by Vintiand Kravitz (see also Corner). It is often referred to as "ThePidduck-Kent Special Solution" or "Pidduck Special Solution." Theessential results are

U - X + XO

UP X (D-1~)

a0 )Px--xo -l• Y/(Y-l)"-"- ( - , ( -)Pp,

or, for a y I 1 gas,

P= eCLO (D-3)pp.I

where ao and ao depend on G/M and y, as shown below and plotted infigure D-2.

_(1 y ?u I -oY ll2 (yul)

Y0 0 ~ U

(D-4)

Or, for a Y - 1 gas,

G zoe~o f e" d. (D-5)0

40

i . i... ... ... ' .. ... .i: . ... .I .. . .. ....

YI

-~lt'A-': :: - '!- ij4 4.l-

__ F.

0.4 4

0.0

&sA

Figuare D-2. Pidduck-rkent Special Solution.

41

For small G/M, ao may be approximated as

G(y-i 3y 2,+ . D6ao0 M(2y) 6y M'

The projectile velocity is obtained as

S(D-T)

and, for Y - 1,

( o1/2up Va ge g P

0 /- 1 (D-8)

where

Thus, Oigure D-2 may be used with equations (D-7) to (D-9)to calculate the projectile velocity for any gun, even a chamberedgun, although the solution was derived for a Do/D 1 = 1 gkm. Then, forthe chambered -un, to should be replaced by xOAo/Al in equations

(D-7) and (04-).

The above results may be deduced for a DoiD! = I gun with acovolume propellant gas, and these results have been applied "s anapproximation even to the case of a chambered gun with a covoluepropellant gas. In the chambered covolumme caze, the sg velocity,&.0, in all the equations above should be replaced by 'yRTo C2 ypo v o"band xo should be replaced by (Aoxo - bG)/A 1.

12

it is found that, when G/M becomes infinite, ao approaches 1.In this case, for the Do/DI = I gun, the projectile velocity becomes,from equation (D-7),

2a% 1/2 (D-10)

U P- l C p X0 / Jy-Y I + o

which, for infinite travel distance, becomes equal to 2ao/(y - 1); thisis, as it should be, the escape velocity for a Do/D1 = i, x0 cc gun.

The special solution applies to the Do/DI - 1 gun in which,initially, there is a pressure gradient in the propellant gas; theLaGrange ballistics problem (the PP gun), however, assumes no gradientsinitially. Pidduck and later investigators suspected, but never proved,that the special solution approaches the accurate solution in the limitof large travel. The results of calculations made on the electroniccomputing machines seem to confirm their suspicion.

The computed results indicate that, indeed, the special solutionis an amazingly good approximation for the finite chamber length PPgun for any DO/D 1 ; this is particularly true for G/M values less than 1/4.

43

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