Performance analysis of UTIAS implosion-driven hypervelocity launcher

JANUARY 1968

PERFORMANCE ANALYSIS OF UTIAS

IMPLOSION-DRlVEN HYPERVELOCITY LAUNCHER

by

P.A.L. Sevray

PERFORMANCE ANALYSIS OF urIAS

IMPLOSION-DRIVEN HYPERVELOCITY LAUNCHER

by

P. A. L. Sevray

Manuscript received October 1967

,.

ACKNOWLEDGEMENTS

I wish to express my thanks to Dr. G. N. Patterson for the opportunity to conduct this work at the Institute for Aerospace Studies.

I wish also to thank Dr. I. I. Glass who conceived the Implosion Driven Launcher and suggested this problem. Ris continuous interest and help-ful advice are most appreciated.

Special thanks are due to Dr. R. F. Flagg and W. O. GraL for many useful discussions and assistance in completing this work.

I am very grateful to the "Conseil des Arts du Canada" for a scholarship which gave me this opportunity öf studying in Canada.

The computer time provided by the Institute of Computer Science is acknowledged with thanks .

•

The present research was financially supported by the Aerospace Research Laboratory of the United States Air Force under Contract No.

SUMMARY

A performance analysis of the UTIAS Implosion-Driven Hypervelocity Launcher is presented which is based on a numeri cal code using the artificial viscosity technique. The numerical calculations are inviscid as no losses are taken into account and the analytical results yield performances that are at least

50

percent greater than the available experimental data. From the re-sults obtained numerically, detailed wave-diagrams of the processes in the launcher aregiven. The launching conditions at the base of the projectile are emphasized.

An optimization study of the existing launcher (8 in. dia.) hemispherical implosion c~amber and several barrel sizes has also been made followed by an analysis .~ what should be the optimum parameters of a second

/

generation launcher that' is able to launchs::: one-inch diameter projectiles to hypervelocities. In order to make this calculation more realistic, a number of points should be studied and these are outlined in the conclusions.

1. 2.

3.

4.

5.

6.

7.

8.

9·

TABLE OF CONTENTS NOTATION INTRODUCTION

DESCRIPTION OF THE CODE

2.1 Mathematical Formulation of the Problem 2.2 Numerical Integration

2.3 Limits of the Code

CONTINUATION OF PREVIOUS CALCULATIONS 3.1 Initial Conditions of the Calculation 3.2 Res~lts on the Implosion Wave

3.3 Performance 3.4 Comparisons

PETN CALCULATIONS 4.1

4.2

Equations of State and Detonation Scheme Choice of Representation by Zones

4.2.1 4.2.2 4.2.3

EquaL_ >_ Distance Zones Equal Mass Zones

Scheme Adopted

Performance Calculations Corresponding with Actual Experiments Characteristics of Three Cases

Analysis of the Case 200 g PETN and 200 psi, 2H2 + 02 Comparison of Three Cases

CALCULATIONS WITHOUT EXPLOSIVE

OPl;IMIZATION FOR THE 8-INCH DIAMETER CHAMBER 6.1 Varying Parameters

6.2 Variation of the Weight of Explosive

6.3 Variation of the Initial Pressure of the Gas 6.4 Influence of the Projectile

6.5 Conclusions for the 8-Inch Diameter Chamber INFLUENCE OF THE CHAMBER SIZE

v 1 1 2

4

4

5 B 5

6

6

6

6

8

8

10 11 12~ 12 12 13 14 15 15 15 16 17 18 18 7.1 Effect of the Explosive Weight in the 30 in. dia. Chamber 18

7.1.1 Effect of the Initial Pressure 18

7.2 Similarity

7.3 Use of Similarity

7.4 Optimum Projectile for the 8 in. Chamber

OPTIMUM CONDITIpNS FOR A ONE-INCH DIAMETER ONE-CALIBER PROJECTILE IN THE 30-INCH bIAMETER CHAMBER

CONCLUSIONS REFERENCES TAB LES FIGURES

19

20 21 22 22 23

A C E P Q R T U d m Note on units: NOTATION area

artificial viscosi ty con~tant specific internal energy pressure

artificial viscosity pressure radius

time, temperature projectile velocity thickness of explosive mass

radius of the barrel volume

All the results in pressure are given in bars: . 1 bar

=

14.5 psi

All the results in velocity are given in km/sec: 1 km/s 3300 ft/sec (3280 ft/sec}

1. INTRODUCTION

The object of a hypervelocity launcher is to launch projectiles at high veloeities, typically in the order of

50,000

ft/sec and beydnd, 'such

that experiments on planetary entry and micrometeoroid impact can be made

possible. .

In order to achieve such veloeities, in

1959,

Prof. l.I. Glass,

Institute for Aerospace Studies, suggested the use of spherical, imploding

shock waves to create a region of very high pressure and temperature to drive the projectile (Ref. 1).

In Fig. l i s illustrated the operating principle of the launcher.

It consists of a hemispherical chamber and a barrel such that the entrance of the barrel coincides with the center of the hemisphere. The projectile is 10-cated at the entrance of the barrel. The interior of the chamber ~s lined with a thin layer of explosive and the remainder of the chamber is fill~d with a combustible gas mixture, typically stoichiometrie hydrogen and oxygen. The gas

is ignited at the origin in s~ch a way th at a detonation wave propagates into

it (Fig. la). The detonation wave ignites the explosive and astrong

imp10d-inging shock wave is generated (Fig. lb). This implosion wave creates a region of high pressure and temperature near the origin and the projectile is accelerat-ed into the barrel (Fig. lc) .

As a contin~ation of the work done on this problem in the past years (Refs. 2,

3,

and

4),

it was decided to obtain a numerical model of this

launcher, in order to be able to analyse and design a second generation launcher and to optimize its parameters. This is the object of the subsequent sections,

where such a numerical code ia described with its limitations and the problems arising during its use. In this process, a number of numerical results were obtained to show the influence of the weight of explosive, the initial pressure of the gas and the dimension of the chamber on the optimization program.

Finally, a review is presented of the points that should be examined in order to bring the mqdel closer to ·treality~L . .

2. DESCRIPTION OF THE CODEt

The numerical program used in this report is based on the"Q-method" as established by von Neumann and R~chtmyer to solve shock problems

(Ref.

5).

The equations are written in Lagrangian form.

The present work was initiated by the one-dimensional planar code for a light-gas gun, prepared at NOL (Ref.

6).

The extension was to

con-sider the coupling of a one-dimensional spherical geometry for the chamber with a planar geometry in the barrel. It was also intended to introduce a region of

explosive with an appropriate equation of state. In this study, the work by

Brode et al (Ref. 7) on the numerical treatment of the general type of hydro-dynamic motion was very helpful. The code was written in Fortran IV and two adaptations were made, respectively, for the IBM

7094,

Institute of Computer

In order to check the results, the same problem of preparà.ng " a numerical code for analysing the UTIAS Implosion-Driven Hypervelocity Launcher was studied in parallel by Piacesi (Ref.

8).

When a comparison of the two sets

of results was made, some difficulties appeared regarding the numerical treat-ment of the discontinuous geometry at the origin of the UTIAS Launcher. This will be explained later on.

Most of the calculations in this report were do ne with a stoi-chiometrie mixture of hydrogen and oxygen, as the gas used in the chamber, and with pentaerythrite tetranitrate (PETN) as the explosive. The two equa-tions of state for the gas and the explosive were both programmed by Piacesi, and were used without any modification in this report.

2.1 Mathematical Formulation of the Problem

To start the calculations, the configuration is represented as in Fig. 2. The chamber is divided into two regions, the explosive and the gas, each having its own equation of state. Each region is then divided into zones and mass-points containing one half the mass of two adjacent zones are assumed to be at the interface of these zones.

In Lagrangian motion, the mass-points are considered as constant along their paths and the calculations follow their motion with time. That is, we will determine at a certain time the position of the interfaces of the zones they represent.

The pro~lem is then to solve a set of nonlinear partial differ-ential equations expressing the conservation of mass, momentum and energy as follows: conservation of mass: 1

₌

V = 5v 5m p 5U - A 5 (p + = 5T 5m conservation of momentum: 5E _{_(p + Q) 5V} 5T = 5T conservation of energy: equation of state: where V v = U P Q A E m T = E

=

E (p, V) specific volume volume partiele velocity pressure

artificial viscosity pressure area

specific internal energy mass

time

( 1)

Q) ₍₂₎

The term Q, which is added to the pressure, is introduced to diffuse a shock front and thus eliminates discontinuities in the hydrodynamic parameters. The artificial viscosity term is defined as:

Q

₌

c

[~ ~

r

if 5U

<

0 (5)

V 5x

.. ~.:";

Q

=

0 if 5U» 0

5x - (6)

The value of Q is then calculated every time a zone is compressed. The constant C determines the spreading of shocks. In all the calculations a constant

value was used for C, that is, C

= 4

for the gas mixture, and C =

6

for the explosive.

In order to integrate this set of differential equations, they

must'~be transformed into a set of finite difference equations which approximate them. The solution is then obtained by repeating the calculation of these differences over the set of finite mass-points, for a given small time incre-ment. In the finite difference form, the two independent variables, time and distance are represented by N, which is the number of the cycle in the calcu-lation, and by J, which is the label carried by a mass-point. The value of J increases from the wallof the chamber to the base of the projectile.

The equations become:

conservation of mass: v(J, J+l) i m

-I-J+.!..

-2

v(J, J+l) volume between the interfaces J and J + 1)

conservation of momentum: DU =

r(

P + Q) 1 - (p + Q) l ' ]

DT

_{J- 2 J+ 2}

[u;+~

. N+.!.. 2 N + 1 _{C U} J

2J

2 with A N J if U J

<

UJ_l QJ 1 [ N+l _{] x ,} po/2 2

~.!..

V 1 + J-₂

_-2

(po is the initial density)

Q

o

(8)

conservation of e~ergy:

,

[~JN+

__

1~

+

~N

'

N+l.] y-.:

PJ-~

+

QJ-~

x (11) equation of state: (12) 2.2 Numerical Integration

Using the above equations, each variable of state V, P,

Q,

E is calculated for every zone at each cycle on the calculation. This enables us to know the velocity and' the new position of each mass point at a given time.

Figure

3

shows the way the differences are calculated in the integration, that is, the illustration of the labels J-~, J, J+~, N-~, N, N+~ in the equations in their finite difference form. The position of any of the variables on the graph represents when and where it is calculated.

The progress from one step to another is made by the choice of a time increment 6T. This time step is determined according to limitations due to stability criteria. The first stability requirement is the Courant condition: the time increment 6T must be small enough that sound signals from one mass point will not have had time to reach the next mass point in that length of time. The other stability requirement takes into account the fact that the artificial viscosity changes the nature of the differential equations from that of a wave equation to that of a diffusion type equation.

From Ref.

7,

it can be written:

1

(13)

2.3 Limits of the Code

Before we examine any illustration of the use of this code, it should be noted that this calculation represents an ideal hydrodynamic motion. That is, radiation, conduction, and viscous losses are not taken into account. It will be shown that the conditions reached in the Launcher are such that dissipative terms are very important. This is probably the main reason why the results we obtained in performance were always about

50%

higher than the experi-mental ones.

Other limitations due to the particular geometry of the Launcher and the equations of state available will be pointed out later on.

4

, ~ ... ~ '~f·

3. CONTINUATION OF PREVIOUS CALCULATIONS

In order to obtain some insight into the operation of the

pre-sent Launcher, Brode (Ref. 7) performed several numerical calculations. A de-tailed analysis of these results can be found in Ref. 4. The object of the present calculations was to take one of Brode's cases for implosions in a hemis-phere and extend them to include the barrel and the projectile as coupled to the chamber. This also provided a check on the present numerical treatment of the process in the chamber with the results obtained by Brode.

3.1 Initial Conditions of the Calculation a)

b)

c)

a sphere 20 cm radi~s is filled with a mixture of 2 H2 + 02 + 7He at 100 psi and the gas is assumed as perfect (7

=

1.67)

the barrel is 0.22 inches in diameter and the projectile (poly-ethylene) has a mass of 170 rog.

the explosive liner is 0.1 inch thick, consists of TNT, at a density of 1.5 g/cc.

These

state for TNT can be

conditions are summarized given by (Ref. 7),

in Table 1. The eqpation of

p where, 2.556T V E = al

=

10·9575 a₂

=

-288.237 a 3

=

2343.39 a4 -7799.83 a 5 9236.55

8

.5

x 10

-4

+ T

4

+ 1200 T5 7.73 x lO-3T

T in 104 oK, V in cm3/g,p in 1010 ergs/cm3 and E in 1010 ergs/go

3.2 Results on the Implosion Wave

(14)

(15)

The calculation was initiated at time 50 ~s in Dr. Brode's re-snlts for this case. That is, we start our calculation when the explosive has detonated and thus introduce the pressure profile (Fig. 4) in the chamber that Dr. Brode has obtained at that time.

Figure 5 compares the wave diagram that was obtained in the pre-sent calculations with that shown in Ref. 4. We can conclude that as far as the implosion wave is concerned the two numerical programs give the same result

although there is a slight difference in time (this is due to an improper choice of the time T = 0, when the detonation wave reflects on the wall.

3.3 Performance

The interest of this calculation lies in Fig. 6 where the tempera-ture and pressure at the base of the pro~ectile and its loc al velocity are shown. The velocity (km/sec), the temperature ( K) and the pressure (bars) appears as functions of time (microseconds). The final conditions at the end of the 5 ft barrel are listed in Table. 1. To each rise in pressure corresponds an in-crease in velocity. It should be noted that, af ter about two feet the

pro-jectile velocity is almost constant. The temperature profile has the same shape as the pressure profile but the amplitude of the variations is less and the temperature remains at a very high value. An examination of the pressure pro-file poses the question whether or not the oscillations are caused by numerical effects. The next step was to compare the results with those of Piacesi.

It will be seen that the performance results for the TNT case should be taken with considerable reserve.

3.4 Comparisons

As mentioned previously, Piacesi (Ref. 8) studied the same pro-blem. The two sets of results are compared on Fig. 7. There is a large ence in the final velocity (24 km/s instead of 16 km/s) and also a large differ-ence in.the pressure profile, although the number of pulses in both cases is the same. The two' codes used were practically the same and the calculations differed only in the number of zones used: 7-in the explosive and 20 in the gas were used by Piacesi, and 21 in the explosive and 19 in the gas in the pre-sent case. Using the same number of zones as Piacesi, the agreement was good. (Fig.

8).

Therefore this was an indication of the failure of the code to simu-late the whole process in the launcher. The result should not vary very much with different representations of the global mass of fluid. The reason for this variation, however, was found later on, while studying cases where the explosive was PETN and the gas a mixture of 2H2 + 02. An attempt to correct this problem was made, as will be shown subsequently.

4. ·~TN CALCULATIONS

The above calculation with TNT was only a test for the numerical code but in order to make comparisons with actual experiments, we had to use a different chamber size (8 in. dia.), a stoichiometrie mixture of hydrogen and oxygen for the gas, and PETN as the explosive. The two equations of state for ~~N and _2H2 + 02 were provided by' Piacesi (Ref.

8).

4.1 Equations of State and Detonation Scheme

The equation of staté for the gas is an extrapolation of the real gas calculations of Moffat (Ref. 9).

It is expressed as:

E

=

6.57 pV + 974.0 (pV)2 _

a

[0.101 x 10- 3

n40.0 + (pV)4 Ln _{1.013 x 103} p - 0.2325 x 10-3J (16)

where,

ex

=

0

for pV<:5:; 1.

0465

ex

8600

pV -

9000

for

1.0465

<

pV ~

3.488

ex

=

2.

x

103

for

3.488

<

pV

From Ref.

9,

this equation is representative of the

2H

₂

+ O

2

mixture in the

range:

1600

0K

<

T

<

6000

0K

0.01

<

p

<

1000

atm.

As we will see, the pressure and temperature reached in the launcher are currently far beyond these limits and consequently this equation of state is only an approximation of the behaviour of the gas. A systematic error will result. Moreover, the temperature could not be expressed directly in terms of internal energy and pressure and was deduced from the ideal relation: pV

=

RT. T~us the temperatures are only'indicative of the actual temperature variations. From Ref.

10,

we should write pV

=

Z RT to take in account

dissociation and ionization but

Z

is not known. In addition, at high den-sities,

Z

will also change as a result of van der WaäTh forces and the finite size of the molecules.

The equation of state for PETN is an exact fit to the experi-mental data of a

50%

mixture of PETN and TNT.

P (E,V}

=

A(p) + B(p) E for P in megabars and E in megabars - cc/g

where, A(p)

=

0.002164

p4 +

2.0755

e-

6 .

O/p

B(p)

.35P

p

=

po(density)/V. Detonation

In programming these equations of state, Piacesi introduced a detonation scheme, as it was necessary this time to start the calculation at the actual time zero. The detonation scheme is the one described by Wilkins in Ref. H.

An internal energy, equal to the chemical energy plus the energy remaining in the gas when the specific volume goes to infinity, is given

initially to each zone. The pressure is considered to be zero everywhere. The equation of state for the pressure becomes,

P

=

P(E,V) x f where,f is the burnt fraction, f

=

l=;V,

CJ

with V ' being the Chapman-Jouguet specific volume and V the specific volume:J

The values of Vcj that were used were:

7

v .

explosi ve

CJ 0.7872

V _{. cj} (2H ₂ +O~_c:. ) 0.54 (from Ref. 12)

The chemical energy of PETN was taken as 1.255 kcal/g and the initial density 0.58 gm/cc. For the gas, (Ref. 12) the chemical energy was taken as 0.157 kcal/ cc and the initial density was deduced from an ideal gas calculation.

To start the detonation, f is given the value 1 at the point of ignition. When the detonation is ended, f is equal to 1 everywhere. It should be noted that giving an initia1 pressure of zero before the calculation is only a computing device. In fact the actual initial pressure is taken into account as the initial density and initial specific energy for the gas are calculated according to this initial pressure.

4.2 Choice of Representation by Zones

The conclusion from the first numerical experiment, using TNT was that the code had to be modified in some way, since the results should not be dependent on the numerical methods. To find the reason why there were large differences in performance when various numbers of zones were used, various calculations were made for the same case: 200 g of explosive (PETN), an initial pressure of 200 psi for the gas mixture (2H2 + 02)' and using the dimensional characteristics of the existing chamber (Table 2).

4.2.1 Equa1.:_c." Distance Zones

It first appeared that increasing the number of zones would increase considerably the computing time. For example, on the IBM 1130, the computing time was almost doub1ed with 30 zones (10 in the explosive, 20 in the gas) as compared to 20 zones (5 in the explosive, 15 in the gas). In fact, the particular problem presented by the change in geometry of the UTIAS

Hyper-velocity Launcher became apparent while trying to reduce this computing time. It is worth noting the way a continuous mass of fluid is divided into a number of zones of finite mass. As seen in Fig. 2, the two regions, explosive and gas are both divided in zones at equal distance from one another. We then follow the motion of mass-points representing the mass contained in one half of two adjacent zones •. If we use zones at equal distance, it is clear that the mass-points become monotonically smaller from the wall .to the origin of the chamber. Af ter the exp10sive has detonated, a strong shock propagates towards the origin, so that the zones are highly compressed. From the num-erical point of view, in the region close to the origin, this means that the mass-points crowd close to one another.

Due to the discontinuity at the origin between the spherical and the cylindrical geometries, a number of numerical effects occur there,

a) The calculation slows down as two mass-points become nearly adjacent, this is due to the Courant condition (Sec. 2.2).

b) As the projectile starts moving down the barrel, it is possible to have two mass-points such that one is in the chamber and the other is in the barrel as shown in Fig. 9a. To keep the spherical symmetry,

c)

any interface in the chamber should be considered as a hemispherical shell, until we reach the origin of the chamber. The pressure is calculated at mid distance between the two mass-points and is assumed to be the same in that portion. Thus, the force exerted on mass-point

J is less than that exerted on mass-point J + 1, as the area in the barrel on which the pressure is exerted is constant while that in the

chamber tends to zero at the origine The gas would therefore artificially flow into the chamber, rather than flow out, as in the actual case.

The one-dimensional program is unable to represent the

two-dimensional flow pattern in the region of the origine In fact as long as one uses this one-dimensional code, no rigorous solution exists and we have to choose the best approximation. The following approxi-mation was made: whenever a mass-point is located in the chamber, in the region defined by the radius of the barrel and the origin, the area at this point was taken as

1.5

~ r~, as an average value as shown below. This affects the momentum equatlon

DU

DT

MASS

AREAf~f

J

but the error did not seem to affect the global process very much. If we consider a reasonably large number of zones, the pressure in two adjacent zones did not differ, too much. Moreover, the value of the projectile base rb is small compared to the·radius of the chamber and thus the overall the error in the calculation of the area, assumed as

1.5

~ r~,is smalle

In any event, the actual area of the interface lies between 2 ~ r~ for a hemisphere, and ~ r~, for a plate, but the pressure would not be the same everywhere.

When the initial zones are at equal distance fr om one another, we also observed oscillations in the pressure, either at the origin or at the base of the projectile (see Fig. 10). Let us consider Fig. 9b and

suppose that when the imploding shock reaches the origin, various mass-points are p!.lshed very close to the orlgln. When we calculate the volume occupied by these masses, we see tha~ it cah become very small

as we approach the origine With radial symmetry in the chamber, 2~ ( 3 3)

v

=

~ RJ+l - RJ . The specific volume is then calculated, knowing our constant mass:

Volume between.1J and J + 1

~ (mass

J + massJ+l )

If we have several mass points in the si tuation of Fig •.. 9b, the specific volume can tend to zero and then small errors in V. will affect very much the computation of the artificial viscosity

(Q

=

C($U)2 )

(20)

(21)

This will in turn affect the pressure and energy in the equations of state. As a result we observe oscillations in the value of the pressure and depending of the number of zones in the calculation, these errors will be more or less amplified.

It would be worthwhile to modify somehow the scheme for cal cu-lating the volume around the origin, as we did for the area.

Un-fortunately, any change in the calculation of the volume would violate the law of conservation of mass and we have just noted the importance of errors on the specific volume.

In order to avoid this problem, it is better to find another choice for the mass-points in the initial setup •. Choosing zones at equal distance, does not seem to be the best solution, since mass-points of very small mass exist in the region of the origin and there is a

tendency of these points to crowd at the origin when the gas is compressed.

Figure 10 shows the results in pressure at the back of the pro-jectile for the case 200 g of PETN, 200 psi when this scheme of initial zones at equal distance was used. Two calculations are represented, one with 5 zones in the explosive and 15 zones in the gas, the other with 10 zones in the explo-sive and 20 zones in the gas. First of all, the observation can be made that, as in the previous calculation with TNT (Sec.

3.4),

the results differ widely with the number of zones used and cannot be tolerated. When the number of zones is small, the oscillations in the pressure profile are very large. If the num-ber of zones is increased, the calculation is more accurate and there are less oscillations in pressure. However, in the latter case, because of the Courant condition on the very small last zone, the computing time is extremelylong. (This is the reason why we do not have the end of the calculation in that case;

4

hours were already expended on the IBM 1130~) 4.2.2 Equal Mass Zones

The first attempt of preventing the above oscillations from occurring was to use equal mass zones. The idea was to obtain zones large enough so that the mass-points would remain at a reasonable distance from one another in the region of the origin, when the implosion wave compresses the gas. It was noticed that this kind of solution resulted in a lack of de-tail as to what was happening in the barrel and consequently a dispersion of the final results for various calculations.

It was noted that the pressure is calculated at mid distance be-tween two mass-points and is assumed to be the same bebe-tween these points. Due to the very small diameter of the barrel compared with that of the chamber, the use of equal mass zones would cause a unit mass of gas to occupy a long distance in the barrel. If a shock would travel into the last zone for example, (the one next to the projectile), it would remain in this zone for a long period

of time and the pressure at the base of the projectile would be unaffected for that period, giving rise to large errors.

Consider Table 2, where the various calculations done on the same case of 200 g PETN and 200 psi of 2H2 + _{O2 are listed. Owing to a lack of}

detail in the barrel, there is a large dispersion in the final velocity when various numbers of zones are used. The pressure profiles corresponding to these calculations are shown in Fig. 11. The rise in pressure corresponds to the arrival of the implosion wave. With a greater number of zones, the drop in pressure af ter the reflection of the wave on the projectile is faster. A more accurate calculation results from having more zones to represent the gas out-flow into the barrel. The shorter ~ _ time taken for a shock wave to travel in one zone gives a larger drop in base pressure.

It is interesting to note that we have suppressed the

oscilla-tions in the pressure profile and if there is still a difference in the result

with the number of zones used, it is smaller than for zones at equal distanceo

However, it can be seen that apart from a lack of accuracy in the calculation

of the drop in pressure9 the profile obtained with 10 zones in the explosive and 20 zones in the gas, and the one with 15 zones in the explosive and 30 zones in

the gas, are quite closeo

It can be concluded that by using zones of equal mass we have improved the codeo However, it was still necessary to increase the accuracy

of the calculation in the barrel when the gas flowed outo

40203

Scheme Adopted

Usually, the total gas outflow in the barrel is of the order of

1/5 the initial mass of gaso In order to improve the scheme using equal mass

zones, we have to represent this outflow adequately in the calculationo For

this purpose the following scheme was usedo In the initial setup, the zones

are not chosen uniformly throughout the gaso As 1/5 of the gas is going to

expand in the barrel over a long distrance, more zones are needed in this portion

than for the 4/5 remaining in the chambero That is why the 4/5 of the mass of

the gas are represented with N zones and the remaining 1/5 by the same number

of zones No (Typically, N

=

20, gives sufficient accuracyo) The N zones in the

first portion (4/5) are all of equal mass. To obtain a better result, the

N zones of the second portion are redivided in two, so that N/2 zones of equal

mass present 3/20 of the total mass of gas and N/2 zones represent 1/200 In

the explosive, only zones of equal mass are chosen in Region N (Figo 12)0

This scheme used for representing the zones is empiricalo It has the advantage

of the zones of equal mass and avoids a discontinuity at the origine At the same

time there are enough zones in the barrel to provide sufficient detail on gas

outflow and conditions at the base of the projectileo

Another i mpr ove me nt was made in the code, in order to increase

the number of zones during the actual computationo It was noted that the

barrel is of small diameter so that a small volume of gas can still occupy a

long distance in the barrelo For this reason, whenever the last zone (next to

the projectile) is too long, it is divided in two, automatically. For practical

reasons, considering the computing time and the length of the barrel, this last

zone was split in two whenever its length was greater than

3

cmo

To illustrate the reason why this scheme was finally adopted,

we have represented in Figo 12, the pressure and velocity versus distance in the barrel obtai~ed in two calculations. The case treated is still the same,

200 g of FETN, initial pressure 200 psi, _2H2 + 02, with 45 zones in one calculat-ion and 70 in the other. Despite the increased number of zones, we have the same result for velocity and pressure and no oscillations in the pressure

pro-file was observed. This does not mean that the numerical code is an ideal representation of the launcher. It would be better still to consider a

two-dimensional flow in coupling the chamber and the barrel around the origine

However, whatever the error introduced (it is certainly small compared to the

systematic error introduced when we neglect radiative, viscous and other losses),

the result is no longer dependent on the numerical treatment itself and the

num-ber of zones usedo It is therefore now possible to compare the performance

4.3 Performance Calculations Corresponding with Actual Experiments

4.3.1 Characteristics of Three Cases

The char~cteristics of the three cases corresponding with experi-ments performed by Flagg4 in the existing chamber (8 in. dia.)are listed in Table 3. The barrel is 5 ft. long and 5/16 in. (0.793 cm). In all three cases, the initial pressure of the stoichiometrie mixturelwas 200 psi. In Case 1, the weight of the explosive was 200 g (PETN) and 100 g for Cases 2 and 3. The polyethylene projectile was the same in Cases 1 and 2 and had a weight of 0.356 g: Case 3 had a magnesium projectile (0. 712g) •

4.3.2 Analysis of the Case 200 g PETN and 200 psi, 2H2 + 02

A d~tailed wave-diagram for this case is shown in Fig. 13. The gas is ignited at the origin and a detonation wave propagates towards the peri-phery. It travels with a velocity of approximately 3 km/s, which is the result obtained in Ref. 3 for such conditions (stoichiometrie mixture of H2 and O2 at

200 psi).However, in the computations a jump in pressure across the detonation front, of 20 times the initial pressure as predicted. in Ref. 12, was not observ-ed. The reason for this is probably due to the large radial thickness occupied by one zone compared to the thickness of the detonation front. When more

zones are used in the calculation, then larger values for the pressure jump are observed just behind the detonation front. (In a more accurate calculation -see Sec. 7, for a 30-in. dia. chamber, 25 kg _{of PETN and 200 psi, 2H2} +

d

_{2 ,} a ratio of 20 was observed, whereas in the present case, the maximum ratio was 15.)

There is an increase in the velocity (slope) of the detonation wave when it enters the explosive. As soon as the explosive starts detonating, the gas is compressed and an imploding shock wave forms. The detonation wave reflects at the wallof the chamber and gives rise to a second imploding shock wave. This shock wave intersects the contact surface separating the explosive gases from the driver gas and later on overtakes the first shock wave. These

interactions give ri se to rarefaction waves and contact surfaces as explained in Ref. 10. 'They have been represented sChematically, since it is impossi~le

to detect them from the numerical results.

Consequently, there is then only one combined imploding shock hitting the projectile, which starts it moving impulsively into the barrel.

(Note the difference in scale for distances in the chamber .and in the barrel.) The shock wave reflects from the back of the projectile and encounters the con-tact surface. Part of it is transmitted through the contact surface and part of it is reflected. This reflected shock reaches the origin and is partially transmitted into the barrel to give a second pulse to the projectile.

We see on the diagram that the process goes on in the chamber but now there is no transmitted shock strong enough to catch up with the projec-tile. It should be noted that this wave diagram is strongly dependent on the initial conditions. Therefore, the present one is valua~le only for achamber of

8

in. dia. with a 200 g liner of PETN and a gas mixture of 2 H₂ + O_{2 at}

an initial pressure of 200 psi. However, the general pattern is going to be similar for all explosive calculations. The two important facts about the diagram

are the intersections of the two imploding shock waves before they reach

reflection of the shock wave at the contact surface. Whether the two imploding shock waves will intersect or not depends on the mass of explosive and on the radius of the chamber. The influence of this will be seen in the optimization study.

Figure 14 shows the velocity profile of the projectile and the pressure and temperature at the base of the projectile for the case 200 gPETN and 200 psi, 2H2 + 02. If we follow the velocity versus time, we note first a rise fr om 0 to 0.06 km/s at an almost constant acceleration . . This corresponds on the wave diagram to the period when the detonation wave travels from the origin to the wall and the imploding shock travels from the wall to the origin. The pressure behind the projectile is that of the gaseous mixture af ter detona-tion ( 65 bars in ihis case) and i t is almost ct>.nstant. The imploding sho§k hits the projectile and we observe a very large acceleration (up to 2.8 x 10 g's), a :ise in pressure of up to 2.xl05 bars, ~nd a jump in v~locity of up to 12 km/s,

wh~le the temperature reaches values as h~gh as 3.6 x 10 oK.

We see that these conditions are f~r beyond the applicability of the equation of state used for the 2H2 + 02, and certainly too high to deduce the temperature from the thermally perfect equation of state, pV

=

RT. The second pulse shown on the wave diagram can also be se en on the pressure pro-file, as af ter about 120 ~s the pressure stops decaying and rises again. It is less easy to detect the corresponding change on the velocity profile, as we use a logarithmic scale • . In fact, af ter 2 feet of travel in the barrel, the velocity of the projectile is almost constant.

It is of interest to note the mass outflow of gas into the barrel during the launch period. In this case of 200 g PETN and 200 psi, 2H2 + _O2, the mass outflow was 2~16 g for an initial mass of gas of 12 g, so that only 1/6 of the gas flows out into the barrel. In order to explain this small out-flow we should check the conditions at the origin during the launch. Among the parameters calculated -(llcbrJ.,stabili ty'~purPQses) <tHlc..this pbogtam-!l!S! the~speed of

sound, which has limitations similar to the temperature. At very high pressures and temperatures generated at the origin, the flow is nearly always subsonic, as shown in Fig. 14, where piots are given of the speed. of sound at the origin and the velocity of the mass point located in the chamber, next to the origin, versus time. It can be seen that the flow is supersonic only during short

periods corresponding to the arrival at the origin of the imploding shock waves. The speed of sound, except for the implosion phases, is almost constant and

re-mains at h~h values. .

Comparison of Three Cases

The performancesobtained in these three cases are listed in Table

3.

The experimental results (Cases 2 and

3),

available from Ref. 4, show large deviations. The numerical predictions are about 45% higher than those obtained experimentally. This indicates that large losses probably occur as a result of radiation, ablation and conduction at high temperatures and pressures.

Figure 15 shows the velocity profiles for the three cases: Velocity is plotted versus time, and the corresponding positions of the

pro-jectile are also given. We can note from the results of cases 2 and

3

that

,doubling the mass of the projectile reduces the velocity by a factor of .~, approximately.· This means that the kinetic energy transferred to the projectile

is the same for fixed launching conditions. A comparison of cases 1 and 2, showing the influence of the weight of explosive, will be made in the optimi-zation study.

5. CALCULATIONS WITHOUT EXPLOSlVE

Reference 3 contains an experimental study of the performance of the illIAS Hypervelocity Launcher, when no explosive is used, that is launchings using only detonation drivers. It was therefore worthwhile to make a calcu-lation of the corresponding cases where a stoichiometrie mixture of hydrogen and oxygen was used as a detonation driver. The geometry was as follows: a chamber 8 in. dia. with a barrel 6 ft. in length and 0.22 in. (0.56 cm) dia. The calculation was performed using 20 zones in the same code. As we will see, the gas outflow in this case is very small, and to be consistent with the idea explained above, we chose to represent 1/40 of the mass of gas by 10 zones and 39/40 by 10 zones. _{Figure 16 is a typical wave diagram (2H2} + _O2

at

200 psi) deduced from these numerical results.

The process in the chamber is now simpler as it only consists of a succession of reflections of the shock wave on the wallof the chamber and at the origine Every time the shock wave reaches the origin, there is a shock transmitted through the barrel which will eventually catch', up with the pro-jectile. However, on this diagram we have not represented the interaction of the wave with perturbations (see ~elow) created behind the detonation front, in order to simplify the drawing.

As time increases, the shock wave weakens and so does the trans-mitted shock in the barrel, in our case we observe that only two shocks re ach the projectile when it is travelling down the barrel. The corresponding pro-files of velocity , pressure, and temperature anè,shown in Fig. 17, where U, P, T are functions of t~me or distance. The maximum pressure on the base of the projectile is 4 x 10 bars and it is produced by the first imploding shock wave.

Initially there was a mass of gas of 14.3 g, and the total mass outflow when the projectile reaches the end of the barrel is 0.37 g, or

approximately 1/80 of the initial mass of gas. It is seen that the gas out-flow is an order of magnitude smaller than in the explosive cases. Again, the flow is subsonic at the origin except during the period of the arrival of the implosion shock. On Fig. 17, is also shown the pressure profiles at the origin of the chamber. This profile is an image of the wave diàgram as one follows the motion of the wave. From this curve we can deduce that the pseudo-period of the oseillations of the wave is about 100 ~s which is the result observed in Fig. 16. It should be noted that in Ref. 13, the measured cyele time for such conditions was 80 ~s. It ean be seen from the pressure profile at the origin that the amplitude of the wave is decaying with time. There are therefore two reasons for the flow to be mainly subsonic, the short duration of the high pressures at each impulse at the origin and their decay, while the relatively high speed of sound, like the temperature, remains almost constant. It should be noted that when the projectile is still at rest, there are oscillations in the pressure profile at the origin. These are due to the perturbations created behind the detonation wave.

Several caleulations were made using different initial pressures in the gas, all other condittons remaining the same, and a comparison is made

in Fig. 18 with the experimental results observed by Watson. We note that for the case 100 psi, the ideal final velocity is 55% higher than the experimental one, and the difference goes increasing with the initial pressure. An increase in the initial pressure means an increase in the gas energy and corresponding

increases in the peak presspre and temperature.

There are then two reasons why this ideal calculation is less accurate: the losses (radiation and viscosity) increase, and the peak conditions are increasingly less accurate as far as the application of the equation of

state used for 2H2 + _{O2 is concerned.}

6. OPI'IMIZATION FOR THE 8-INCH DIAMETER CHAMBER

6.1 Varying Parameters

The optimization study was done with 2H2 + _{O2 for the gas and} PETN as the explosive. The object of the optimization was to find the best launching conditions using the above numerical code in order to assist in assessing the design of the second generation hypervelocity launcher.

A detailed study was made of the existing launcher, which

con-sists of a hemisphere 8 inches in diameter and a barrel 0.312 inches in diameter and 5 feet long. The projectile was made of polyethylene and had a mass of 0.356 g.

Keeping the dàmensional parameters fixed, the effect of the variation of the weight of explosive and the initial pressure of the gas was investigated.

6.2 Variation of the Weight of Explosive

The effect of changing the weight of explosive was studied in the range 0-600 g of PETN, (the chamber can contain up to 1250 g), when the initial pressure of the gas is kept at the constant value of 200 psi. The per-formance curves obtained versus the weight of explosive are shown in Fig. 19. We can consider three different regions in the range 0-600 g.

Region 1-0-200 g

From 0 to 200 g, we observe a practically linear increase in the final velocity with the weight of explosive. The type of wave-diagram is shown in detail in Fig. 20 • . The two imploding shock waves, formed duringand af ter the complete burning of the explosive, intersect before they reach the projectile. Consequently, the pressure profile at the base of the projectile consists of a

single impulse and very high peak pressures are reached.

Region 11 - 200-500 g

This is the region of maximum performance. We note on the wave diagram of Fig. 21 that the two imploding shock waves no longer intersect, and this results in two distinct impulses, (1) and (2), in the pressure profile at the base of the projectile. We observe that from 200 g to 400 g, these two impulses become more and more separated in time. There is also a third impulse due to the reflection of the shock at the contact surf ace and depending on the particular conditions (position of the waves, contact surface, projectile in the barrel) its effect on the pressure profile. (It is particularly important for

first impulse decays in strength while the second one increases. This gives us a means of optimizing the weight of explosive such that the two pulses be of the same amplitude.

Region 111 - 500 g - 600 g

If we further increase the weight of explosive, the final

velocity decreases. The wave-diagram in Fig. 22 illustrates the case of 500 g of PETN. The intersection between the second imploding shock and the reflect-ed shock occurs at the contact surface. If we go beyond 500 g of EETN, the intersection occurs in the explosive gas. The effect of this is to reduce the strength of the second impulse in the pressure profile and thus reduce the final performance.

From this analysis, the importance of the position of the contact surface becomes evident. In fact, the whole process depends on the time taken by the detonation wave to propagate into the explosive. Although this has not been studied in this report, an optimization could be made on the initial den-sity of the explosive, such that we could.~tin~ explosive and remain in

region 11 of high performances, simply by increasing the density of the explosive. Another problem would arise, however, due to the high pressures that would be reached at the base of the projectile. If for ex~mple, we con-sider that 105 bars is a limit from the point of view of the projectile inte-grity, only three pressure profiles among those drawn on Figs. 20, 21 and 22 would meet this limit: 300 g, 500 g and 600 g of PETN. In order to reduce the charnber of high-explosive loads, 300 g of PETN would be the best explosive weight and it would lead to an ideal velocity of 15 km/s for the 0.356 g pro-jectile.

6.3 VaXiäti6n of the Initial Pressure of the Gas

Keeping the foregoing division into regions of the mass of explo-sive the effect of the initial pressure of the gas in each region was investi-gated.

Region I - 0-200 g PETN

The study of the· variation of explosive loading was done for an initial gas pressure of 200 psi in the 2H2 + O_{2 mixture (Fig. 19). Recall that} region ~I was characterized by a single pulse on the projectile. tomparing with Fig. 20, it is seen that the effects of lowering the pressure as shown in Figs. 23a and 23b, for an initial pressure of 100 psi, we observe that now the projectile has two distinct impulses and the peak pressure due to the first implosion is higher than for the 200 psi case. The effect of lowering the

gas pressure is to increase the strength of the implosion and also to accentuate very much the strength of the reflected shock at the contact surface. This reflected shock gives rise to a significant impulse at the back of the pro-jectile.

If we now increase the initial pressure (Cases C on Fig. 23a and 23b where 600 psi were used) we observe the reverse effect, the implosion decays in strength but the pressure profile tends to spread over time so that the performance is almost the same as for the 200 psi case.

For optimization purposes we can make the following statements~

an initial pressure of 100 psi increases considerably the performance but gen-erates higher peak pressure at the base of the projectile. The use of an init·ial pressure greater than 200 psi does not affect the performance too much and per-mits us to lower the peak pressure in the projectile.

Region II - 200 - 500 g PETN

This region (Figs. 19 and 21) was characterized by two impulses

in the pressure profile at the base of the projectile for an initial pressure of

200 psi, _2H2 + _{O2 . The effect of lowering the initial pressure to 100 psi can} be seen on curve A in Fig. 24a. If we compare it with the profile obtained at

200 psi, the two impulses are now clearly separated and we observe that the second implosion gives rise to a very high pressure (6 x 105 bars). If we now increase the initial pressure to 600 psi, the two implosion waves hit the pro-jectile at the same time and we obtain a profile typical of region I.

The same observations can be made fr om Fig. 24b for the case 400 g PETN, a decrease in the initial pressure accentuates the second peak in pressure while an increase in the initial pressure gives a lower value.i flor this peak'.l_ 'l!he~"first implosion in that case of 400 g PETN is not very much affected by changes in the initial pressure.

Region III - 500 - 600 g PETN

For 200 psi, from Figs. 19 and 22, the pressure profile consists of two distinct impulses. We find these two impulses on Fig. 25a, and while the

magnitude of the first implosion is not affected greatly by the pressure of the gas, the second peak can be changed very much. From an initial pressure of 100 psi to 600 psi, the magnitude of the second peak varies from

3

x 105 bars to

8 x 10

3

bars. The same applies to Fig. 25b for the case of 600 g PETN. The results are summarized in Fig. 26.

6.4 Influence of the Projectile

In the above analysis, the size and weight of the projectile

were fixed. The following problem arises: given a certain chamber size, which is the most suitable projectile~ Section 6.4 attempts to answer this question using three calculations with different types of projectiles. Fixing the initial conditions in the chaober at 400 g PETN, and 400 psi 2H2 + 02, the projectile was kept at one caliber in the three cases. lts diameter was varied from 0.31 to 0.52 and 0.62 inches polyethylene (0.92

glee).

The three different pressure profiles are plotted in Fig. 27. The first impulse, corresponding to the arrival of the first implosion wave when the projectile is still at the origin, has the same amplitude for all three projectiles. This is not the case for the sub~e­

quent impulses, for, depending on the mass launched, the second and third

im-p~lse at the base of the projectile vary considerably. If the projectile is heavy, the second shock wave catches up to it sooner and the pressure is higher. The result is that, although we launch projectiles of very different masses

(0.356 g to 2.85 g) the final velocity does not decay too much. However, if we consider that the pressure should be kept as low as possible, in that particular case a projectile of 1 g would be the best. In Sec. 7.4 the best projectile corresponding to the 8 inch chamber, from both points of view of final per-formance and the pressure generated at the base, will be reconsidered.

6.5 Conclusions for the 8-Inch Diameter Chamber

The above calculations of the effects of various initial gas pressures and various explosive loadings given an indication of the best

launching conditions. In order to launch a

5/16

in. dia. projectile using the 8 in. dia. chamber, we arrive at the following conclusions: The initial pressure for the gas should be kept at 200 psi. . Lower initial pressures give rise to very high final pressures and temperatures at the base of the projectile.

Higher initial pressures reduce the performance with no significant improvement

from the point of view of the integrity problem of the projectile. (We have

summarized all the results in Table 4.)

If a pressure of 105 bars is considered as a limit for

projec-tile integrity, an ideal velocity in the range 14-16 km/s can be obtained by using explosive loadings of either 300 g or 500 g of PETN. In fact, from the calculations in Sec. 6.4, it seems that a furthér improvement on the launching

conditions could be made with an appropriate projectile (see Sec. 7.4).

7. INFLUENCE OF THE CHAMBER SIZE

From the analysis in Ref. 14 a 30 in. dia. hemispherical chamber

was adopted for the second generation launcher, UTIAS Implosion-Driven Hyper-velocity Launcher, Mark Ir. It was of int.eres.t to investigate the performance

attainable using such achamber size. The type of projectile used was the same as in Ref. 14 that is, a one-caliber, 1 in. dia. polyethylene (density

1

g/cc)

projectile, weighing 12.65 g.

7.1 Effect of the Explosive Weight in the 30 in. dia. Chamber

The barrel length (6 feet) and the projectile mass (12.65 g)

were fixed and the initial gas pressure was kept at 200 _{psi 2H2} + 02 and the effect of increasing the mass of explosive was investigated. The performance curve

obtained is shown in Fig. 28 and again we can make a division into three regions as in subsection 4.2. In Region I, the wave diagram is characterized by the intersection of the two imploding waves before they reach the projectile and consequently the pressure profile on the projectile is a single impulse (Fig.

29a) .

In Region 11, the imploding waves arrive at the orlgln at different times. Depending on the strength of the reflected wave at the con-tact surface, there can be two or three impulses on the pressure profile at the base of the projectile (Fig. 29b).

In Region 111, the second implosion intersects with the reflect-ed wave (formreflect-ed by the reflection of the first implosion on the projectile), inside the explosive gas. There are always two p~lses and, at least in the range studied 25 Kg - 35 Kg, the performance decreases with increasing explo-sive weight (Fig. 29c).

7.1.1 Effect of the Initial Pressure

A few calculations were made in order to check the effect of changing the initial pressure in the region of interest: 10 Kg - 25 Kg of

PETN. Thus on Fig. 30 are represented three profiles obtained with an initial pressure of 400 psi.

Like in the

8

in. dia. chamber, an increase in the initial pressure lowers the peaks in pressure but also the performance. However, the result for 15 Kg of PETN might appear strange, as we have a much better per-formance for 400 psi than for 200 pSi,2H2 + _{O2 , It is worth recalling} ~hat the gas pressure effects the velocity of propagation of the shock waves. For the case 15 Kg, _{PETN, 200 psi, 2H2} + _{O2 , the wave diagram is such that two separate} shock waves hit the projectile; for the case 15 kg, 400 psi, the waves overtake before they reach the projectile and all the energy is transferred in a single pulse, the final performance is then higher. In fact, if we consider the two performance curves on Fig. 30, it is ~een that the curve for 400 psi is as if it were translated to the right af\the 2Çl0,j:Ei;cUli've. With a higher initial

press~e in the gas, one therefore needs more explosive to obtain the same

performance. 7.2 Similarity

The notion of similarity is important for launcher design and this idea appears from the above study. For the two chambers (8 in. dia and 30 in. dia.) we have obtained the same result of the existence of three performance regions when we vary the weight of explosive. In fact, if the thickness of explosive determine~the type of wave ·diagram for a given chaffiber size, it was to be expected that in two chambers of radius Rl and R2, explosive thicknesses such that dl/d2

=

R1/R2

should give the same type of wave diagram and the same type of pressure profile at the base of the projectile. Table 5 shows the weight of explosive, its corresponding thickness, and the ratio

diR,

for the various cases studied in the two chaffibers.

The limits of the various regions determined by the weight of explosive occur for some values of

diR.

Moreover, if we consider a given

ratio

diR,

say for example

diR

=

0.53 which corresponds to the cases 200 g ~TN, in the

8

in. dia. chamber and 10 Kg in the 30 in. dia. chamber, we observe that the pressure profile at the base of the projectile (subsection 4.2 and 7.1) has the same shape (one single pulse of practically the same amplitude, 2 x 105 bars) .

The same observation can be made for other cases where

diR

is the same.

If we now represent the final performance as a function of

diR

instead of the weight of explosive, we obtain similar curves for the two cham-bers (Fig. 31). We have also shown in Fig. 31 some of the results obtained in Ref. 14 for a 20 in. dia. chamber for the same 1 in. dia., single-caliber

pro-jectile. For these three chambers, the limits between the regions are common and the best performance is obtained in region 11, for a value of

diR

practically equal to 0.1. Thus, the similarity between chambers of different size enables us to conclude that, given a certain chamber, its optimum per-formance öccurs for a l~yer of explosive whose thickness is one-tenth of the chamber radius.

7.3

,

Use of Similarity

In our search for the optimum chamber size we can state the pro-blem as follows: . In order to launch a projectile of l.0 in. ,dia. one-caliber long, weighing 12.65 g, determine the optimum chamber size to yield a high

~inal velocity and a pressure profile at the base of the projectile that 'will

lower the maximum accéleration. The variable parameters are as follows: the

initial pressure of the gas, the weight of explosive, and the chamber size. From the above study of the

8

in. dia. chamber and the 30 in. dia. chamber, we can state that an initial pressure of 200 psi can be con-sidered as the optimum.

It was seen that the explosive weight and the charnber size can

be related in a single parameter

diR.

For the same values of

diR

we have

similar wave-diagrams and pressure profiles at the base of the projectile. As a result of the performance shown in Fig. 31, it seems reasonable to choose a

value of

d/R

,

=

0.1, since for the differ~nt cases studied this gives an optimum

value. Consequently, we end up with a single parameter to vary, namely, the

chamber radius R. With diR

=

0.1, we determine the corresponding weight of

explosive, and similar wave diagrams in all cases.

An analysis was done by varying the chamber radius from

7.5

in.

to 30 inches. ' Table

6

shows the corresponding weights of the explosive; Fig.

32 shows pressure profiles versus time (same scale for all cases); Fig. 33 shows

the final performance versus c,hamber radius.

Considering the pressure profiles, we note fr om curve (1) to (5)

that the first rise in pressure due to the arrival of the first implosion wave when the projectile is still at the origin, is practically the same in ampli-tude although it spreads over time. The second pulse however decays very much

from (1) to (5). , If we consider Fig. 34, the velocity profile versus distance

in the barrel for these five cases, the effects of the two impulses appears

clearly. From cruve (1) to curve

(5),

the second shock catches up with the

projectile at a greater distance in the barrel until, :' ': curve

(5),

where it does not reach the projectile in the barrel length considered.

The optimum chamber for launching the one in. dia., single-calibre projectile is either a chamber of 30 in. dia. or a chamber of 40 in. dia.

Considering that we obtain practically the same performance and the same peak pressure in the 30 in. dia. chamber as in the 40 in. dia. chamber, it is appar-ent that the smalle st chamber is desirable, since it needs only 18.5· kg of

PETN instead of 44 kg in the 40 in. dia. chamber. Furthermore i t would be

easier to design and manufacture and much less costly.

Therefore, using the similarity between various chambers when

diR

=

const., gives a means of comparing the performance and the pressure peaks

when launching a given projectile for different sizes of charnbers.

It is worth noting that the existence of an optimum in Fig.

33

may seem strange as it implies that, for a fixed projectile weight, using a larger chamber and consequently more explosive, we obtain a lower performance. This is due to the peculiar conditions in the launcher when 1/10 radius of

exp~osive is used. Two shock waves hit the projectile in that case, one when