The longshot free-piston cycle: Part 1 - theory

H

FOR FLUID DYNAMICS

Technica1 Note 51

THE LONGSHOT FREE-PISTON CYCLE Part I - Theory by K.R. ENKENHUS and C. PARAZZOLI RHODE-SAINT-GENESE, BELGIUM November 1968

Technica1 Note 51

THE LONGSHOT FREE-PISTON CYCLE Part I - Theory by K.R. ENKENHUS and C. PARAZZOLI November 1968

1. 2.

3.

SUMMARY _iii

NOTATION v

LIST OF FIGURES vii

INTRODUCTION 1

DISCUSSION OF THE FREE-PISTON CYCLE 6 2~1 Description of the Lopgshot Piston Cycle

and the Basis for the Analysis 6 202 The Influence of Real Gas Effects on the

Piston Cycle

THEORY 13

301 Assumptions 13

3.2 The Pressure Acting on the Rear of the

Piston 14

3.3 The Pressure Acting on the Front of the

Piston 18

3.4 Solution of the Equation of Motion of

the Piston 19

3.5 Correction of Piston Speed to Allow for

Chamberage Effects 27

30

6

Ca1culation of the Final Pressure and

Temperature in the Reservoir 31 307 Summary of the Calculation Procedure 34 4. COMPARISON OF THE ANALYTICAL SOLUTION WITH

EXACT NUMERICAL CALCULATIONS 37

4.1 Accurcay of the Analytical Solution for

a Perfect Gas 37

40 2 Accurcay of the Analytical Solution for

Dense Nitrogen 38

PREDICTED LONGSHOT PERFORMANCE 41

5.1 Attainable Reservoir Pressure and

Temperature 41

5.2 Volume of Gas Compressed by the Piston

- Decay Rates 41

6.

CONCLUSIONS REFERENCES

APPENDIX I - The Thermodynamic Behaviour

of a Van der Waals gas

APPENDIX 11- Approximate Expressions for the Thermodynamics Broperties of a

Moderately Dense Van der Waals Gas with Constant Specific Heats.

44

45

SUMMARY

An approximate analytical solution is presented for the free-piston cycle in the VKI Longshot Hypersonic Tunnel. The Longshot differs from a conventional gun tunnel in that a heavy piston is used to compress the nitrogen test gas, which is th en trapped at the end of the barrel at very high pressure by the closing of a system of check valves as the piston re-bounds. Real gas effects are accounted for by assuming that the nitrogen behaves as a Van der Waals gas with variabIe specific heats.

Acknowledgement

The authors wish to th ank Mme N. Roels for assist-ance in computer programming, arid Mlle Jo Stuyck for prepar-ing the manuscript.

a A D e g h M

=

M' Mil P R s So v x z a 8 y NOTATION sound speed cross-sectional area diameter

internal energy per unit mass acceleration due to gravity enthalpy per unit mass

piston Mach No _.

=

vp /a4 when accelerating into a _v vacuum

limiting piston Mach No

=

_{v p =/a4}

piston perturbation Mach No

=

v~/~ in phase I piston perturbation Mach No

=

v

p

/a4 in phase 11 primary shock Mach No

=

WS/al

pressure

gas constant per unit mass entropy per unit mass

entropy of a perfect gas at temperature Tand unit pressure

entropy of a perfect gas at standard state (po.To) time temperature flow velocity piston speed volume piston weight shock speed

distance along barrel from start of piston motion variabIe related to Mv defined by Eq.(30)

intermolecular attractive force constant in Van der Waals equation

intermolecular repulsive force constant in Van der Waals equation

ratio of specific heats in a thermally and calorically perfect gas

r(T) Tl P cr 1 2 3 4 5 B f i m JI ~ c

function defined by Eq.(5) factor defined by Eq.(18) density

Riemann function

characteristic vibrational temperature

W

piston mass per unit barrel cross-sectional area= ~

gAl

Subscripts

initial value in barrel

value behind primary shock or ahead of piston value immediately behind piston

initial value in driver gas

value behind first reflected shock value at entrence to barrel

value when piston comes to rest

value when piston begins to decelerate value at match point where Mv=M~

refers to piston sonic value

value for infinite piston travel value with chamberage

2 3

4

5 6

7

8

9

,LIST OF FIGURES

Longshot Compression Cycle

Comparison Between Measured and Predicted Pressure Decay in the Longshot Reservoir

Effect of Supply Conditions on Longshot ,Reservoir Pressure Decay

Wave Diagram of the Longshot Free-Piston Cycle The Longshot Free-Piston Hypersonic Tunnel Linearized Solution of Piston Motion

Comparison of Analytical and Numerical Predictions of Piston Speed for Dense Nitrogen

Predicted Longshot Performance

1. INTRODUCTION

The extreme environmental conditions encountered during atmospheric re-entry make flight simulation in a con-tinuous hypersonic tunnel very difficult. (See, for example, Refs. 1 and 2). Intermittent facilities, with running times of

the order of milliseconds, have therefore been widely used because they offer a number of important advantages. The first of these is that the gas supplying the nozzle can be raised to high pressures and temperatures by purely aerodynamic means, such as shock wave or piston compression, thereby eliminating the expensive gas pressurizing and heating equipment which, furthermore, requires very high power during continuous

opera-tiono Secondly, the problems associated with the selection of

materials and the engineering design of vessels for the contain-ment of high-pressure, high-temperature gases are minimized. A

cold driver gas at moderate pressures can provide the relatively

small volume of hot, high-pressure gas sufficient for intermit-tent operation. Finally, the nozzle throat heating problem, which places a limit on low-altitude, high-Mach number simula-tion in a continuous facility, is also essentially overcome in spite of the extremely high local heating rates, because the testing time is so short.

Although a wide variety of intermittent facilities have been developed, fRef. 3) all of them fail to provide com-plete simulation of the aerodynamic environment during the flight of a typical long-range ballistic missile. The reason is the well-known one that it is impossible at present to provide an air supply at a temperature of up to say, 70000_K which is simultaneously at a pressure of the order of 10,000 atmospheres or more necessary to reproduce low altitude, high Mach number flight conditions in the nozzle test section. Thus we find, on the one hand, a facility such as the shock tunnel,

~n which a very hot gas supply at moderate pressures ~s gener-ated by shock compression. This facility can produce a hyper-sonic flow with real gas effects, but only at moderate Reynolds numbers which generally result in a laminar boundary layer on the model. We find,on the other hand, the gun tunnel, in which a pistonis ·used ·to compress the gas to high pressures but only moderately ·hightemperatures. This facility produces a non-reacting 'hypersonic flow but at the high Reynolds numbers necessary for 'the 'generation of a turbulent boundary ,layer on the model.

The "VKI 'Longshot free-piston hypersonic tunnel is an

intermittent facility designed specifically for the attainment of very high Reynolds number hypersonic flows (Ref. 4). It differs from the conventional gun tunnel in that a heavy~ston

is used to compress the nitrogen test gas, which is then trapped in a reservoir at the end of the barrel at very high pressure by the closing 'of a system of check valves as the piston re-bounds. (See Fig~ '1). Typical nozzle supply conditions achieved

in current ·tests 'are '30;OOO psi and 2150 o K, resulting in a

Reynolds number 'of 3.4xl0 6 per foot at ·Mach '15. The feasibility of extending operation to a pressure of 200,000 psi has been established in tests of a 41 mm pilot facility by Republic Aviation "(Refö 5). The realization of such pressures in Long-shot would -be of the 'greatest significance, since with the gas only heated 'sufficiently to avoid condensation,a Reynolds number of 1.8xl08 _{per foot .would .be .obtained .at .MaC.h 15, and}

1.2xl0 7 .per 'foot .at .Mach .24. This would permit the generation of turbulent 'boundary layers on re-entry body models in the entire regime which is most critical for re-entryo

The 'primary dis~dvantage of the -Longshot type of facility is 'that the supply conditions decay during the run as the gas trapped in the relatively small reservoir volume

flows o~t ' thenozzle. Typical pressure decay rates with the

present facility 'configuration are of-the order of I to 5

percent per 'millisecond. depending on the throat size used.

This problem has -been'studied in detail in Ref.

6.

It was found

that the m!asured -pressure decay rate was in good agreement with

the theory for -the 'quasi-steady adiabatic flow of a dense

(Abel-Noble) 'gas. 'when the 'theory was 'correctedby a small

amount to allow "for 'the effects of heat transfer losses to

the reservoir walls (Fig. 2). The pressure decay rate under any operating conditions may be approximately predicted using

curves such 'as these shown in Fig. 3. In this figure, the

pres-sure ratio PO/POi (where Poi ,is the initial pressure in the

reservoir) 'is plotted versus a dimensionless time t i t . t is a

characteristic time equal to V/A /yRT o .• The theory for a _H ~

perfect gas collapses on a sngle curve (the uppermost curve

in the figure) when the results are plotted in this manner,

since the decay rate is directly proportional to the nozzle

throat, area AH and the initial sound speed aOi=/yRTO~, and

inversely proportional to the reservoir volume V. At suffic-iently high supply pressures, the decay rate increases marked-ly due to dense gas effects, as shown by the other curves on

the figurei which are for various values of POi up to 10~000

atmospheres. The decay rate th en also depends on the imtial temperature TOi. The curves shown in Fig. 3 are for T oi=2500oK.

The foregoing discussion indicates that, while it

app,ears to be mechanically feasible to extend Longshot

oper-ation to pressures of the order of 200,000 psi in order to

achie~e very high Reynolds numbers, the rate of decay of test

conditions may perhaps be excessive unless an effieient driver is used, which is capable of compressing a relatively large

mass of gas to high pressures and te~ratureso There is

there-fore a need for a relatively aCCurate analysis of the piston

This need was recognized by Humphrey et al at Republic Aviation, who_ developed an elaborate computer program for the free piston cycle which performed a numerical solution of the characteristics equations for a one-dimensional unsteady flow (Ref.

5).

Good agreement with experimental measurements was not obtained, however, because of the assumption of a perfect gas. As mentioned above, direct evidence of the importance of dense gas effects has been obtained in the study of the reservoir pressure decay problem. Previous investigations of light gas guns operating at high pressures have also amply-demonstrated the importance of taking into account dense real gas ef~ect~ in the calculation of the projectile speed (see,for example, the review by Siegel, Ref. 7).

The course of action which first suggested itself was to modify the available program utilizing the method of charac-t~ristics to include real gas effects. This possibility was rejected for three reasons. First, the program would, in fact, have to be almost entirely rewritten; secondly, the rewritten program would be much too large for the IBM 1130 computer avail-able at VKI; and finally, even with a larger machine, the

computing time vould probably be much too long to allov

extensive performance calculations to be economically carried out.

Faced vith these considerations, the-plan adopted was to attempt to derive an approximate solution for the piston cycle which avoided the use of the method of characteristics. Such a solution can be obtained by the judicious use of assump-tions which are valid for a heavy piston facility of the Long-shot type. The purpose of this report is to present this analysis. The validity of the theoretical model has been

experiments not only justify the present approach, but also indicate the value of an analytical solution in obtaining an understanding of the relative importance of the various

20 DISCUSSION OF THE FREE-PISTON CYCLE

2010 Description of the Longshot Piston Cycle and the Basis for the Analysis

The important features of the Longshot operation may be understood by studying the schematic drawing in Figo 1 and the x-t diagram in Figo 40 A photograph of the facility is shown in Fig. 5. The driver section is a5-inch bore steel tube 20 feet long; the barrel is 3 inches in internal diameter and 89.5

feet long. Before firing~ the 5 lb nylon piston is held by an

aluminium shear disc at the junction of the driver section and

the barrelo At the present time. nitrogen at ambient temperature

is used both as the driver and test gaso Typical initial pres-sures are 15,000 psi in the driver and 48 psi in the barrelo

When the shear disc ruptures j the piston is accelerated to a

speed of about 2000 feet/sec by the time i t approaches the end of the barrelo It is then rapidly brought to rest and beg ins to rebound as the pressure ahead of it rises to a high value o The test gas is forced through a system of 48 small check valves into the reservoir located at the downstream end of the barrelo

These valves close as the piston begins to reboud~ trapping

the gas in the reservoir at peak pressure and temperature

( ~30,000 psi and 2l500

K)0 In the present configurationi the

reservoir volume is 1903 cubic incheso A plastic plug is placed

in the nozzle throat to permit the test section to be evacuated

before a run. When the reservoir pressure suddenly rises j this

plug is blown out and the nitrogen begins to flow through the

conical hypersonic nozzleo Condensation-free flows at Mach

numbers ranging from 14 to 27 may be generated using different throat sizeso

The details of the unsteady wave system and the nature of the flow processes in the piston cycle will now be discussed

(see Fig.

4).

As the piston accelerates, the compression waves formed ahead of it coalesce to form a shock which is partially reflected fromfue valve faces and partially transmitted into the reservoiro The piston begins to decelerate when struck by the first reflected shock. A series of additional shock reflec-tions occur between the front of the piston and the valve faces, and also in the interior of the reservoir, before the check

valves finally close. These multiple shock reflections and the frictional losses in the flow through the check valves produce an entropy rise in the test gas. However, most of the pressure rise, which takes place near the end of the stroke, is the result of a nearly isentropic compression process.

As pointed out by Stalker, (Ref.

9),

the final conditions in the reservoir are primarily governed by the equation of conservation of energy. Stated in words, the in-crease in the internal energy of the test gas as the piston moves from an initial position xi (say, the position where it is struck by the first reflected~ock and begins to decelerate) to the final position xf where it comes to rest i must equal the kinetic energy of the piston at xi. plus the work done by the driver gas on the piston in the displacement from xi to xf~ Since the pressure on the rear of the piston is relatively low, the work done by the driver gas during the final phase of the motion is small, and the increase in inter~al energy of the test gas is approximately equal to the loss in kinetic energy of the piston.

This information is, however, insufficient for deter-mining the final oondition of the gas in the reservoir unless the associated entropy rise is known o However, previous cal-culations have shown (Ref. 5) th at most of the entropy increase

due to multiple shock reflections occurs across the pr~nary and first reflected waves. Because only about 20% of the gas com-pressed by the piston is passed into the reservoir, the flow in the check valves is sUbsonic, and the associated losses are

small~ The final conditions in the reservoir are therefore very

nearly the same as in the remaining gas ahead of the piston. This suggests that the final state of the gas in the reservoir may be determined with sufficient accuracy by only taking into account the entropy rise across the primary and first reflected shock, neglecting the presence of the check valves, and deter-mining the temperature and pressure resulting from an isentropic compression in which the increase in internal energy of the test gas is that demanded by the law of conservation of energy.

As discussed above, the kinetic energy of the piston near the end of the stroke plays a primary role in determining the final conditions in the reservoir. To determine the piston speed, it is necessary to integrate the equation of motion, taking into account the varying pressures on the front and rear faces, and, if necessary, allowing for bore friction.

As the piston accelerates, the pressure on its base falls and an unsteady expansion wave moves upstream into the driver gas as shown in Fig.

4.

Due to the presence of chamber-age, this expansion wave is partially refle~ted downstream again as a compression wave at the junction between the barrel and driver tube, and partially transmitted upstream as a weak-ened rarefaction. This weakweak-ened wave is reflected from the up-stream end of the driver as a rarefaction wave which again moves downstream, but in Longshot i t does not overtake the piston

during its stroke. The Longshot may therefore be analyz~d. in

the Same manner as a chambered gun with effectively infinite chamber length.

Losses in the valves will not be negligible if a larger reservoir than· the present one (of 19.4 in 3 volume) is used.

If the barrel and driver tube were of equal size, the pressure acting on the base of the piston would at all times be related to the initial driver pressure by the equation for a

simple wave regiono The presence of chamberage produces an

in-crease in the speed of the piston because the compression

reflected downstream fr om the barrel-driver junction makes the base pressure on the piston higher than would otherwise be the

caseg The effect of chamberage can only be properly evaluated.

howe'ver, by relatL vely complex analyses (see Ref 0 7, pp 28-51) 8

This suggests that, in a simplified treatment of the problem, the piston velocity should first be calculated neglecting chamberage, and then an empirical correction factor can be applied to allow for the increase in piston speed due to

chamberage which is based on the more exact solutions referred to above.

The pressure acting on the front of the piston is also not so easily determined, particularly during the early stages of the motion before trecompression waves which are generated coalesce into a shock. The co-ordinates at which the shock forms relative to the initial position of the piston are, for

a perfect gas (see Ref. 7, p 54)

tshoCk • 284/< Y4+1)

(;:Q)

o

x = a t

shock 4 shock

where the initial acceleration of the piston is approximately

(::p)o

=

g:;A

since the pressure ahead of it is negligible compared with the

These equations show that the shock forms quite quickly in Longshot. Using the typical conditions p~=15,OOOpsi (2.l6xl06

lb/ft 2 ), a4=1148 ft/sec (corresponding to a nitrogen driver tem-perature of 70°F), with a piston weight Wp =5 lbs, and a barrel cross sectional area A=.049l ft2 (corresponding to a3-inch bore) I it is found that t s hock=1.40xlO-3 seconds and Xshock=1.6l feet.

It is reasonable, therefore, to make the assumption th at the shock is present during the enti~e motion. The further simplification can then be made that the pressure acting on the front of the piston is the same as that immediately behind the shock, and that the strength of the shock is such that the piston speed is equal to t~particle velocity behind the shock front. Very l i t t l e error in determining the piston speed is incurred by these assumptions, even if they are not very ac-curate, because during most of the motion the pressure in front of the piston is quite small compared to the pressure behind it.

2.2. The Influence of Real Gas Effects on the Piston Cycle

The p~eceeding general discussion will now be extended to consider the influence of real (dense) gas effects on the piston cycle. Although~aborate equations might be used to describe the behaviour of a dense gas, the present analysis employs the well-known Van der Waals gas model which is tract-able for analysis. The required thermodynamic properties of this gas are derived in Appendix I.

Across the expansion wave formed in the driver gas, the pressure ralls from the order of 15,000 psi in the driver chamber to a relatively low value at the base of the piston during the later stage of the motion. With an unheated driver gas, the temperature correspond~ngly varies from ambient value

in the chamber to rather low values immediately behind the piston. Under these conditions, the driver gas will behave as a dense gas without vibrational excitation, and a model such as a Van der Waals gas with a constant value of cv, the specific heat at constant volume, is appropriate. (The value of cp varies, however, since cp-cv~R for a dense gas). 'For calculations with a preheated driver gas, i t is necessary to account for both a variable cp and cv'

Across the incident and reflected shocks, the assump-tion of a perfect gas is adequate because the temperatures and pressures are not higp.

In the final isentropic compress10n near the end of the piston stroke, both very high pressures and temperatures are obtained. It is therefore necessary to employ a gas model such as a Van der Waals gas with variable specific heats.

The expected change in piston speed produced by dense gas effects may be explainéd in a simple way, In ~n unchambered gun, the increase in gas velocity across a simple expansion wave is

dv = ~ _ap

Because the acoustic impedence ap of a dense gas is greater ttian that of a perfect gas, the velocity of the piston will be less 1n a dense gas.

In a chambered gas, an increase in piston speed is obtained because a part of the upstream-moving rarefaction is returned to the piston as a compression wave. The efect of chamberage-is discussed in detail in Ref. 7, p 125-149. The piston velocity increase obtained with a perfect gas driver has been determined in this reference. The correct ion is ex-pected to be larger in a -dense gas for the following reason.

The increase ~n gas velocity in the transition section between the driver and the barrel is

d(f.)

=

Because the density of a real gas at a given pressure and temperature is lower than th at of a perfect gas, there is a greater increase of velocity in the transition section in the real gas case. The correction is calculated by an approximate method in the present report.

Before turning to the detailed theoretical formulation of the Longshot piston cycle problem, a final word about the evaluation of theoreûcal models may be given. As has been seen, the calculation of the piston cycle naturally divides itself into two steps (1) The determination of the piston speed near the end of the stroke; and (2) The calculation of the final tem-perature and pressure in the reservoir for a given piston speed. The influence of chamberage and d.ense gas effects in the driver gas may therefore be evaluated by comparing with experiment the piston speed calculated for four cases, i.e., for a perfect and a real gas, and with and without chamberage. Then,real gas ef-fects during the final part of the compression strpke may be separately evaluated by comparing measured reservoir conditions with calculations for a given piston speed, made for a perfect gas and a real gas. In this way, a great deal of insight will be obtained into the influence of the various physical mech-anisms involved.

3. THEORY

3.1. Assumptions

From the considerations discussed in the Introduction, the fOllowing assumptions are adopted in the

analysis:-1. The piston speed is calculated first neglecting chamberage, then a correct ion factor is applied to the piston speed near the end of the stroke.

2. The pressure acting on the re ar of the piston (for the case of no chamberage) is obtained using the expression for sn expansion wave. In the real gas case, with an unheated

driver, the formula for a Van der Waals gas with a constant value of Cv is used.

3. The pressure in front of the piston is given by the perfect gas formula for the pressure across a normal shock, the strength of~ich is such that the particle speed behind the wave equals the speed of the piston.

4.

Bore friction is neglected.

5.

Only the entropy rise across the incident and first reflected shock in the test gas is taken into account, and the

cor-responding state of the gas is determined using perfect gas relations. Losses in the check valve system are neglected, and the state of the gas in the reservoir and in front of the piston is assumed to be the same.

6.

The final part of the piston stroke consists of an isentropic compression to the reservoir conditions Po and To. Real gas calculations are made assuming a Van der Waals gas with variable specific heats.

3.2. The Pressure Acting on the Rear of the Piston

If one neglects chamberage, the pressure P3 acting on

the rear of the piston is related to the driver pressure P4 by the formula which bolds for a simple wave, namely

=

JP4 v_p

=

a4-a3

P3

~ ap

where a is the Riemann function,

a

=

JP

~

_ap

o

For a Van der Waals, which obeys the equation of state

the isentrope is, in the general case with variable specific heats (see Appendix I)

sO

i

=

~

_{tn(T/T o )+r(T)-r(To)-tn(p+a}p2 _{)+tnpo+,r= constant (4)}

is the contribution to the entropy due to the vibrational

excitation of the gas. s~iR is the entropy at the standard

state (Po, To) andfue constant on the right of Eq.(4) is the value of the dimensionless entropy in the driver reservoir,

By eliminating T from Eq.(4) using the equation of state (Eq.(3)), a single equation in pand presults which can be used to calculate the density p=p{p) required for the numer-ical integration of the Riemann function.

The sound speed is given by the expression (see Appendix I) [ 2ap _

(.1.

+ dr)

(l!)

= p+ap2 2T ~ ap p (

_2'!

7 +

_dT

d

r) (a

_{\.ap p - p+ ap 2}

T)

1

t

(6)

The derivative dr/dT is gotten from Eq.(5), while (aT/ap)p and

(aT/ap) are obtained fr om the equation of state, Eq.(3). The

p

values of these derivatives are

ev/T dr (e /T) 2e v

dT

= _(eV/T

_~

₂ T e -1

C*)

1

[a{1-2BP)-~

J

=

-

_R P ( 8 ) and

C*)

1 1

=

_'R

(- - B) P P

T must,of course, be eliminated from Eqo(7) using the equation of state.

Eqs.(2)-{9) may be used in Eq.{l) for the numerical calculation of the pressure on the back of the piston as a function of piston speed for a Van der Waals gas with variable specific heats. Symbolically, there is obtained the relation

(10)

In the special case where the driver gas is cold enough so that the vibrational energy is negligible, r(T)=O, and the isentrppic (Eq.(4» immediately reduces, with the aid of Eq.(3) to

(11) where Y4=(cvd+R)/R=7/5. This Y is not to be confused with the true ratio of specific heats cp/cv' which is v.ariable.

iaentrope

a=

The sound speed is obtainable by differentiating the

1

=

C~p)'

_[

1+ ap2

p

-I-Sp

The value of the acoustic impedance, ap, required in the Riemann function cr is thus immediately expressible as a function of pand P, and Eq.(ll) allows p to be determined as a function of p. The numerical integration of Eq.(l) th en leads to the appropriate form of Eqo(lO), which relates the pressure on the rear of the piston to its speed.

For applications to a facility with moderate driver pressures. it is possible to obtain an approximate analytical expression for the Riemann function by making the assumption that the quantities ap2/p and

SP

appearing in Van der Waals equation are small compared with unity. (See Appendix 11). A binomial expansion of the terms involving these quantities

in the expression for t'he acoustic impedance is carried out, and only first order terms are retained. The quantities ap2/p

and

SP

are then approximated by their perfect gas values, i.e.,

-and

1

ap

=

;~:

(f;t

This leads to the following linearized equation for the acoustic impedance, in which the first factor is the value for a perfect gas: 1 Y,++l 2" 2-Y4 2 Y,+ _a(y,+-l)

P'[l+(?4)Y4)

y,+p,+

(f4J

ap =

RY4

1+ _{2Y4( RT 4)2} (14)

The insertion of this formula into Eq.(3) yields the following approximate expression for the Riemann function (see

Appendix 11): Y4-l

2Y'::-cr

=

Y~-l

_-JY4RT

(t)

1-a(y₄-l) 2Y4(RT 4 )2

This expression reduces to the perfect gas result when a=O. It will be noted that the correction for dense gas effects

is entirely due to the~termolecular attractive force coefficient

~ I

a and does not depend on the repulsive coefficient,

S.

One

addi-tional very useful approximation may now be made by noting that as P/P4 varies from unity to a low value, the coefficient in the curly brackets varies from 1.25 to 1.00 for Y,+=7/5. Calculations

have shown th at the most satisfactory agreement with an exact numerical solution is obtained when the coefficient is assumed to be unityo When this approximation is made, and Eqo(15) is inserted into Eqo(2)j there is obtained the following explicit form of Eqo(lO) : 2ya.

-P·G -

~J

Ya.- l Ya.- l P3

=

_2n

where aa.

=

IYa.RTa.

~s the perfect gas sound speed j and

n

=

1 -a(Ya.- l )(5-Ya.) 4Ya.(3-Ya.) Pa.

(16)

(18)

is a first order correction factor for dense gas effectso It will be seen that, since n < 1, dense gas effects reduce the

piston speed for a given pressure ratio P3/Pa..

3030 The Pressure Acting on the Front of the Piston

The pressure acting on the front of the piston is, from perfect gas normal shock relations (see Ref. 10, p

46,

Eq.(23»

where the shock speed is given by (Ref. 10, p

46,

Eqo(22»

For a strong shock (Ws~=)

A1though the strong shock assumption gives on1y an approximate1y correct result, it is sufficient1y accurate for the present purpose, since the pressure acting on the front of the piston is much lower than on the rear for most of the stroke. Eqs.(19) and (21) thus yie1d the desired re1ation

(22a)

The exact expression is obtainab1e from Eqs.(41) and (42) which are given in section 3.4.3. It is

P2

~

PI [1+

It is seen that Eq.{22b) reduces to Eq.{22a) when Yl+1 v ~ ~al 2 »1. (22b)

3.4. Solution of the Equation of Motion of the Piston

The equation of motion of the ~ston (neg1ecting friction) is W dv ~ -.la g dt W dv =~v

--...:2=

g P dx P

This equation may be integrated numerically af ter the insertion of the values of P3 and P2 as functions of v p from Eqs.(lO) or

(16) and (22), respectively.

However, an analytical solution can be obtained for a moderately dense driver gas by linearizing the equation of motion in a manner demonstrated for a perfect gas by Stalker

(Ref.

9).

The assumptions required for the linearization are

that:-(1) During the first phase of the piston motion, (O~xp~xm) the piston speed differs by only a small amount, v~ _{, from the speed v pv which it would attain if accelerating} into a vacuum.

(2) During the second phase of the motion,(xm~xp~xi) the piston speed differs by a small amount v; from the limiting velocity vp~ which it would attain in a barrel of infinite

length.

Here, x m is the match point at which vpv=vp~. A

continuous solution for the piston motion is obtained by matching the perturbation velocities v

p

and v

p

at x=xm. The above ideas are shown graphically in Fig.

6.

Xi is the point at which the piston face is struck by the first reflected shock and begins to decelerate.

3.4.2. ~2!~~!2~_2!_~~=_~~~~~!2~_2!_~2~!2~_~~!!~~

Phase I

During the first phase of the motion, the equation of motion of the piston is

(24)

v pv ' v

p'

based on the speed of sound in a perfect driver gas (Eq.(17».

The equation of motion of the piston if it were acce1erating into a vacuum is. using Eqs.(16). (23) and (24). with P2=M'=O. 2Y4 dv.pv dM 2 dM P4

[1-

Y4- 1

]Y4=l

v v

_

_dt

.

=a4

_{d t}

= a4Mv

-

_dx =

-

_w

-

_2n Mv p (25)

where w, = Wp/gAl is the piston mass per unit cross-sectiona1 area. The distance travelled b~ the piston when it has attained a Mach number Mv is

wa~

I

MV [ . Y4-1 xp =

P4

1 -

""'"'2rï

o

Integrating by parts. u

=

wa2 ₄

'P'4'"

Mv and There resu1ts· 2wa~n

=

_{(Y4+1 Jt)4} dv

=

(26)

The acce1eration of the piston when the pressure

• This equation app1ies to a perfect gas when n=l. It is in-correct1y written in Ref.

90

acting on the front face is included is, from Eqs.(16), (22), (23) and (24),

dv

P4

{[l-

Y4- l ~

"

-dt w. 2n 2Y4

<VM')

Y4-1_

~

1 + Yl

<:'

+l

t:)

\Mv+MryJ}

<

27) where, by Eq.(24), dv

r

(28)

The insertion of Eq.(28) into Eq.(27), and expression of dMv/dt by Eq.(25) leads to the following differential

equation: Y4+l [ Y4-l 1 -

"'""2rï

= -

Yn

4

[1-

Y~~l

MvJ Y4-1 M'

where ~ consists of terms in the products MvM', (M')2 and those of higher order resulting fr om the binomial expansion of the first term on the right-hand side of Eq.(27). These terms may

be expected to be small because the initial pressure ratio P4/Pl is large and (M')2«M'«1. These assumptions are indeed found to be valid in the Longshot piston eycle, as subsequent calculations have shown. By neglecting ~, and transforming to the variable

Eq.(29) becomes the first-order 1inear differentia1 equation -2ya. dM' 2ya. M'

d'Z - y;;:T

z

_[

2n2Yl(Yl+1)~aa.,2

] Ya.-

i

1 + - ) (1_z)2 z (Ya.-1 )2 al 2n l' 1 =

y;::ï

P4

The solution of this equation is

- fPdz [ J P d Z

1

M' = e IQe +constant where -2y

p(z)

= (Ya.- 1

1z

and -2ya.

-2n 2 Yl(Yl+1) 2

J

Y4-1

Q(z)

= - - 1 + 2" PI [ ( : : ) (l-z)2

Ya.-1 Pa. _{(Ya.- 1 )2}

Integration of Eq.(31), and app1ication of the boundary condition M'=O at z=l, i . e . , that the perturbation velocity is zero at the start of the motion (Mv=O) leads to the resu1tH

Y4+1 ₂

PI Y4-

1

3Ya.+1 - Y4-

1

3Y4+1

( 31-)

Y4- 3 Y4:.'1 t2"2Y! (YI +1)

(~Y~z-2n

M'=

-

+ z _Y4+3 z

3Ya.+1 1'4 (YIi-1)2 Y4+1

H The-perfect gas form of this equation, obtained with n=l, is

The piston speed vp=a~(Mv+M') in the ~irst phase o~

the motion is given as a ~unction o~ its position xp by Eqs.

(26), (30) and (32). In per~orming calculations, Mv is the

natural independent variable, and correspondin~ values o~ z,

M', Mv, xp and v p may be ~ound in a direct manner.

3.4.3. ~2~~~!2~_~!_~~=_~~~!~!~~_~!_~~~!2~_~~!!~§

Phase 11

During the second phase o~ the motion, the ~eed o~

the piston is

v p = vp +v~ = a~(Mm+M")

m

vPm is the asymptotic speed, approached aa Xp.m,

which is equal to the speed o~ the contact sur~ace in a shock

tube under similar initial conditions. v; is a small perturo·

ation ~rom this value. Mand M" are piston Mach numbers cor-_m

responding to the _{velocities v Pm and} v;~ respectively, based

on the initial per~ect gas value o~ the sound speed in the

dr'i ver gas.

The equation o~ motion o~ the piston during phase 11

is, ~ rom E q s • (1 6 ), (22), (2 3) a n d (3 3 ) , a2₄ (M +M") co d(M +M") co

=

~

ç

r

l _ Y4-l w

II

2n PI P4

This equation may be linearized by neglecting M" compared with

Mm in the coe~~icient o~ _{dM"/dx p ' and retaining only terms in}

on the right hand side. The following differential equation with constant coefficients in then obtained:

where

dM"

ëiX

p

+ CIM" = C2

and C2 is composed of the remaining constant terms in the linearized form of Eq.(34).

M"

=

where C' is .the constant of integration. The boundary condition is that M"+O when xp+~' i.e., the perturbation speed vanishes for infinite piston travel. Therefore C2/CIC'=O. The solution must also be matched to the piston speed resulting from the motion in Phase I. If the match point x m is chosen to be the piston position for which Mv=M~. then M~=M~. The Bolution for the piston perturbation speed in Phase 11 is therefore

(38)

X_{m is given by Eq.(26), in which Mv is set equal to} M~. (The determination of M~ is discussed below). zm may ~hen be gotten from Eq.(30), and M~ from Eq.(32).

The limiting piston Mach number, M , is obtainable

~

from the equation of motion (Eq.(34» by setting the accelera-tion

(i.e

~

.

the left hand side) and the perturbation speed M" equal to zero. It is then found that M is given as a function _~

of the initial pressure ratio across the piston by the imp1icit equation

(39a)

It maybe noted that the nu~erator of this expression is P2/Pl (Eq.(22) and the denominator is P2/P4=P3/P4 (Eq.(16». A some-what more accurate va1ue of M may therefore be obtained if an _co expression for P2/Pl is used which does not assume that the shock is strong. In terms of the shock Mach number Ms=1vs/al' this expression is (see Eq.(2.19) P 16 of Ref. 11)

P2

-

_PI

=

2YIM~-(Yl-1) _Yl+1 where Ms is re1ated to M by _co 1+ (40) (41)

Eq.(41) is readily derived from Eq.(2.29) P 25 of Ref. 11. If the va1ue of Ms from Eq.(41) is now inserted into Eq.(40), thus forming a revised va1ue for the numerator of Eq.(34), the resu1t is

P4 PI =

(::yM;

[1+

1

+(rYl~~ja4MJ]

2

shock is

3.5. Correction of Piston Speed to Allow for Chamberage Effects .

The piston speed when it is ~ruck by the reflected

(42)

where M~ is given by Eq.(39), and M~ by Eq.(38), with xi

conveniently taken equal to the total barrel length xl, since the piston speed varies but little near the end of the stroke, and xi~xl.

A correction must now be applied to this value of piston speed in order to allow for chamberage effects~

3.5.1. Effect of Chamberage with a Perfect Gas

---In the case of a perfect gas, the most straightfor-ward way of accounting for chamberage is to make use of the

so-called sonic approximation (see Refo 7, p 47). In this

approxi-mation, sonic flow is assumed from the driver reservoir into the barrel from the beginning of the motion, whereas in fact this condition is only asymptotically approached. It can then be shown that the effect of chamberage is equivalent to in-creases in the initial pressure and sound speed of the driver gas, given by 2Y4 p' 4

=

P4

(

(~

) ( t

1

Y)JY4-I

1 + ...;.- -1 1-

~

...

,

r

1~f5:!.

.1)

~-G~Y1

(43) and _{a 4}

₌

a4

Accurate numerical calculations of the effect of chamberage with a perfect gas have been made in Ref. 7, and

it is possible to approximate the results by the simple formula

vpwith chamberage=

(44)

Y4Vp

which is valid for ~ 3.

a4

We now proceed to estimate how this result will be influence by the high density of the driver gas. To do so. it may reasonably be postulated that the increase in piston speed in a chambered gun with a dense driver gas is proportional to the increase infue limiting piston speed with infinite

chamberage.

The equations which hold in the transition sect~on

with infinite chamberage and effectively infinite chamber length are (see Ref. 7. p

40.

and the sketch below)

(simple wave in barrel)

(energy conservation in the transition section) (sonic flow at barrel

entrance)

(45a)

(45b)

Since the pressure in the driver section is ~igh and the temperature is norma11y at àmbient va1ue, the assumptions th at ap2/p and Sp«l in Van der Waals equation are not generally valid. An ~ytica1 solution for rea1 gas effects on chamberage may neverthe1ess be obtained for the more simp1ified gas model represented by the Abe1-Nob1e equation of state

(46)

which represents the behaviour of a very dense gas with reason-ab1e accuracy. It can be shown that the Riemann function, en-tha1py and entropy of an Abel-Nob1e gas are given exactly by setting a=O in Eqs.(A2-8), (A2-10) and (A2-12), without any restriction on the magnitude. of Sp, but with the proviso that it may be approximated by the perfect ~as va1ue given in Eq. (13).

From Eqs.(45b), (45c), (A2-10), and (A2-12) there then resu1ts the fol1owing equation for PB/ P4

Y4

y;:-:r

-[ 1

+~(2)*J2

RT4 P4

(47)

The perfect gas sOlution, obtained by setting S=O, is Y4

-PB Y4-1 ( Y4 2 +1 ) (48)

-

= P4

This value may be inserted without significant error into those terms in Eq.(47} which have 8 as a coefficient. The solution which isthen gotten is

Y4-1

(~)

Y4 •

The limiting piston speed is given by Eq.(45}, with vB=aB expressed using Eq.(A2-l0}, and oB using Eq.(A2-8}. It is

PB where

P4 ~s obtained from Eq.(49}

The perfect gas solution, gotten by putting 8=0 in Eqs.(49} and (50) is the well-known result

(50)

A calculation made for Longshot for P4=1000 atm, T4=530oK show th at (vPm )

/(V

Pm ) = 1.07. This probably overestimates the

real perf

correction, because only repulsive intermolecular forces are accounted for in the Abel-Noble gas model.

Using Eqs.(44}, (49), (50) and (51), the velocity of the piston is córrected for chamberage by multiplying by fue factor

3.6. Calculation of the Final Pressure and Temperature in the Reservoir _:

The deceleration of the piston ~egins at the position x=xi when it is struck by the first reflected shock. Since this occurs very close to the end of the barrel, the piston speed at this point may be calculated with negligib1e error by setting xi=xl (the total barrellength) in the equations given in section 3.4.3.

The first step in the determination of the final conditions in the reservoir consists of finding the state of thegas behind the first ref1ected shock, using perfect gas relations. The second step is then to obtain the fina1 con-ditions resulting from the isentropic compression of a Van der Waals gas with variable specific heats, which is terminated when the piston comes to rest.

The pressure in the gas behind the first reflected shock is (see Eq.(2.37). p 27 of Ref. 11»

The corresponding temperature is (ibid, Eq.(2.38»

In these equations.

Ms-~

xp=Xi~Xl. It is given by Eq.(41), M +M"

co i·

( 53b)

is the shock Mach number at in which M is rep1aced by

The positions xi and xf. at which the pieton is struck by the first reflected shock. and finally comes to rest. respect.

i~~ly. are related to initial conditions in the barrel by the

continuity equation

The value of xi is immediately determinable as

where use has been made of the perfect gas law to expr-e1J'1! PI

lp

5

in terms of the correspondi~g pressures and temperatures.

The value of xf. and the final conditions po. To. PO. in the reservoir are found using the continuity equation (Eq~5~)

the equation of state (Eq.(31D. the equation for the isent~ope

(Eq.(4». and the law of conservation of energy. which. for unit barrel cross-sectional area. may be written

The first term in this equation is the work done by the pressure force acting on the rear of the piston; this pres-sure is as~umed to remain constant at the value P3i. The second term is the kinetic energy of the piston. The energy represented by these two terms is converted into the increase in internal energy AE of the confined gas. Ae is the increase in internal energy per unit mass, and PIXI is the mass of the gas. For a Van der Waals gas with variable specific heats, the internal energy per unit mass is (see Appendix I)

Under the assumption that the ~inal compression is isentropic, Eq.(4) 'yields

1.n

G(~

'\

512 !:.i(l-P 0

S) ]

+

~ TS~

Po

I-ps/'

r(TO) - r(TS) i

=

0

(58)

,

, The densities

Ps

and

Po

appearing in Eqs.(57) and (58) can be expressed in terms of PI using Eq. (54). The energy balancé equation, Eq.(56), then becomes

+

Ra

_v

and the isentrope (Eq.(58», is solved for xf to obtain

(60)

Substitution o~ Eq.(60) into Eq.(59) then results in an implicit equation in the single unknown Ta, which may be solved to obtain the reservoir temperature. Eq.(60) then gives xf, and Eq.(54) yields the reservoir density

Po.

The final reservoir pressure Po is then calculated ~rom the equation o~ state, Eq.(3).

The solution for a Van der Waals gas with constant specific heats is gotten py setting the vibrational terms in r(T) equal to z~ro. The per~ect gas result follows whenboth r(T), a and S are zero.

The perfect gas solution for the reservoir pressure and temperatureis" found to be

1

-Ta

T~

( P4

[(~) (~J

(l-Gt) Yl

)+

~

WVPi 2

~J]

Yi

=

"fi

1+(YI-1) PI P4 x I (61 ) and YI Po

(~,~) [(~ )(~)

J

Y

1-1

-

_P4

=

(62) l/Yl

In Eq.(61). the term (PsJpo) «1 and may usua11y be

l.

neg1ected. In ract. the work done by the pressure rorce P3i on the rear or the piston is sma11, and thus to a good approxi-mation, the reservoir temperature is

(YI-l)wVPi2

2PIxi (63)

3.7.

Summary of the Calcu1ation Procedure

The calculation procedure used for the analytic solution of the Longshot piston cyc1e is summarized be1ow.

Basic Step Quantity 1. Input Data _XI,D,Wp

<l1,81,Rl,Sv I <l4, 84 ,Rtt ,BV4 PI,Tl,P4,T4 2. pre1iminary al,a4 Ca1cu1ations n M 00 x m Equation 17 18 39b Eq.(26) with M =M _v 00 Remarks faci1ity geometry test gas properties driver gas properties initia1 conditions in test and driver gases

initia1 (perfect gas) sound speed in test and driver gases

dense gas parameter 1imiting piston Mach No. match point between

Basic Step 3. Phase I Ca1culations Quantity (v_p ) Cl) per:f F xp(Mv ) z (M_{v )} M' (M_{v )} v p =Fv _{c P}

4.

Phase 11 xm~xp~xl Ca1cu1ations M"(x ) _pi v =Fv Pc P Equation 36

49,50

51

52 26 30 32

24

16 22

38

33 16 22 Remarks a constant

1imiting piston speed -Abe1-Noble gas, ~n­

:finite chamberage limiting piston speed -per:fect gas, in:finite chamberage

chamberage correction :factor inc1uding dense gas e:f:fects Mv is the independent variabIe piston position interrnediate variabIe piston perturbation Mach Noo

piston velocity (no chamberage)

piston velocity correc~

ed :for charnberage

pressure behind piston pressure ahead o:f

piston

piston position is in-dependent variabIe piston perturbation Mach No.

piston velocity (no chamberage)

piston velocity correc~

ed :for charnberage) pressure behind piston pressure ahead o:f piston

Basic Step 5. Ca1cu1ation of eonditions when piston is struck by first reflectr. ed shock 6. Ca1cu1ation of conditions in reservoir Quantity x· _~ Mpi MSi P Si lp 1 T S

./T

1 ~ P3i x. ~ Xi' P3i'

Ps·,Ts·

_{~ ~} TO Po Po Equation =a4vp (x.) _{c ~} Eq.(41) with M =M Ol) P 53a 53b 16 55 Remarks

assume xi=xl to begin with

piston Mach No.

primary shock Mach N~

pressure ratio across ref1ected shock

temp. ratio across

re-f1ected shock

pressure behind piston piston position when struck by first re-f1ected shock

Repeat ca1cu1ations unti1 convergence Fina1 va1ues

simultaneous reservoir temperature solution of Eqs.(59) and

(60)

60

54 3 position at which piston comes to rest reservoir density reservoir pressure

4. COMPARISON OF THE ANALYTICAL SOLUTION VITH 'E~ACT NUMERI CAL CALCULATIONS

4.1. Accuracy of the Analytical Solution for a Perfect Gas

The analytical solution déscribed in this report and summarized in Section 3.7 employs a linearized equation of motion for the piston. The linearization is accomplished by assuming that, during the first phase of the motion, the speed of the piston is a small perturbation from the value it would have if accelerating into a vacuum. During the second phase of of themotion, it is assumed that the piston speed differs only slightly from the limiting speed achieved af ter travelling an infinite distance. The accuracy of the linearized theory was checked by comparing it with the results of a direct numerical solution obtained using a fourth-order Runga-Kutta integration technique.

The initial conditions chosen for the calculation correspond te Longshot operation with nitregen driver and test gases, with a 48 psi initial pressure in the barrel and a

12,000 psi driver pressure, and a 5lb pistonoChamberage was assumed equal to unity.

The input data was as follows :-Barrel length

Barrel (and driver) dia. Piston weight

Specific heat ratio Gas constant Initial conditions Xl = 89.5 ft Dl

=

D4

=

0.25 ft W_p = 5 lb YI

=

Y4

=

1.40 R

=

1776.0 ft-lbs/slugOR PI

=

6.92xl03 lb/ft2 P4

=

1.730xl06 lb/ft2 Tl

=

T4

=

528°R

The piston Mach number is plotted vs distance along the barrel in Fig. 7. It can be seen that the perturbation Mach numbers M', M" in phases I and II, respectively, are small, as required by the linearized theory. The piston Mach number given by an exact numerical integration agrees within 1% with the linearized solution.

A comparison o~ the results given by the analytical and numerical solutions is given below. The agreement is seen to be quite satis~actory. The analytical solution predicts values of reservoir pressure and temperature which are slight-ly low because the piston speed v

Pi is also low.

Quantity Analytical Numerical % Error in

x. J. v_Pi PSi Ts· _J. xf PO Ta

Solution S;olution Analytical Solution

(ft) 79.654 79.719 -0.08 (ft /sec) 2100.9 2122.2 -1. 01 (lb/ft 2 ) 2.349xlO S 2.383xlO

s

-1.45 (OR) _{1972.2 1987.5 -0.78} (ft) 88.289 88.313 -0.03 (1 b / ft 2 ) 4.419xl06 4.564xl06 -3.18 (OR) _{4560.9 4620.1 -1.28}

4.2. Accuracy of the Analytical Solution for Dense Nitrogen

In this report, the analytical solution for the piston motion, first obtained for a perfect gas by Stalker in Ref. 9, has been extended to a moderately dense Van der Waals driver gas. The assumptions involved are that

SP

and ap2/p are small compared to unity in the Van der Waals equation of~ate,

The valid~ty of the analytica~ solution was checked by a calculation for Longshot operation with nitrogen test and

driver gases for the same test conditions as listed in Section 4.1., except th at the value of driver pressure was taken to be

p~ = 15,000 psi (2.l6xl06 lb/ft 2 ). This is the maximum value obtainable in Longshot. The Van der Waals constants used were a = 9.87xlOS 1~Lslui2ft~,

a

= 0.716 ft3_{/slug, and}

_e

v = 6l02°R. A plot of the piston speed vs position along the barrel is shown in Fig. 8. At xi = 76.8 ft, (the position at which the piston is truck by the first reflected shock, ne-glecting chamberage) the predicted piston speeds compare as follows :

analytical solution vPi = 1616.3 ft/sec numerical solution vPi = 1660.5 ft/sec error in analytical solution -2.7

%

This is a remarkably small difference when it is considered that the assumption of moderate gas density is not weIl sa-tisfied at a driver pressure of 15,000 ps~, e.g., ap~

=

0069 and is not small compared with unity.

The piston speed must, in any case, be corrected by the chamberage factor F, given by Eq.(52), which is not known with certainty and must ultimately come fr om experiment. The analytical solutionfuerefore provides an adequate prediction of piston velocity for practical purposes. since any error in the piston speed given by the analytical solution could be in-corporated into the chamberage factor.

The final conditions in the reservoir were calculated using the theoretical value of chamberage factor for the given test conditions, namely F=1.23. This led to the following final results :

vPi (ft/sec) 1992.6 pSi (lb/ft2 ) 2f075xl0s Ts-_l. (OR) 1845.1 xf ( ft.) 87.959 Po (lb /ft2 ) 3.861xl06 To (OR) 3678.0

As a check on the analytical solution for the final compression process for a vibrationally-excited Van der Waals gas, a numerical solution was obtained by direct integration of the flow equations. For the same value of v pi ' identical results we re obtained.

5.PREDICTED LONGSHOT PERFORMANCE

5.1. Attainab1e Reservoir Pressure and Temperature

The reservoir supp1y pressures and temperatures ob-tainab1e in Longshot with nitrogen test and driver gases have been ca1cu1ated and are shown in Fig.8. The driver pressure is assumed to be 15,000 psi. It is seen that, for a given initia1 barrel pressure, both the supp1y pressure and temperature are

increased by using a heavier piston. At a given supp1y pressure, the temperature may be independent1y varied by adjusting the barrel pressure and piston weight. The theory prediets th at supp1y pressures over 200,000 psi, and temperatures from be10w 15000 _{K to over 3000}0 _{K can be achieved. To obtain the maximum}

va1ue of test section Reyno1ds number, operation with a supp1y temperature which is on1y sufficient1y high to avoid condens-ation is desirab1e. It is a1so seen that the typica1 current operating condition does not begin to exploit the fu11 poten-tialities of the faci1ity for high Reyno1ds number testing.

5.2. Volume of Gas Compressed by the Piston - Decay Rates

As discussed in the Introduction, the supp1y condi-tions in Longshot decay as the gas trapped in the reservoir exhausts through the nozz1e. The rate of decay is inverse1y

proportiona1 to the reservoir volume, so th at this volume shou1d be as large as possib1e.

The fina1 volume Vf of the gas compressed by the piston is shown in Fig.9. For typica1 current operating condi-tions (P4=15,000 psi, Pl=48 psi, Wp =5 1b) this volume is

130.5 in30 Since the volume of the reservoir is 19.4 in3 , on1y 15% of the gas is stored in the reservoir. Fig.9 shows

th at much 1arger va1ues o~ V~ can be obtained by using a heavier piston together with a higher initia1 barrel pressure. It is c1~ar that the reservoir volume shou1d be great1y increased to take advantage o~ thiso However, as the reservoir volume is in-creased beyond some ~raction o~ V~, va1ve ~low losses rnay be expected to become signi~icanto

In order to estimate the decay rate o~ supp1y pressure when operating at high supp1y pressures, consider as an examp1e M=19 condensation thresho1d operation at po=10,OOO atm. The condensation criterion o~ Re~.13 indicates that the supp1y temperature required is 18000_{K H}

0 Fig.8 shows that with a

15,000 psi driver pressure these conditions are approximate1y achieved using a 45 1b piston with an initia1 barrel pressure o~ 200 psi. Fig.9 indicates that the ~ina1 volume o~ the com-pressed gas is V~=224 in3 • As a conservative estimate, let us assume that' 1/4 o~ this gas, or 56 in3 , can be stored in the reservoir without signi~icant losses. Assume that the nozz1e exit diameter is 1 ~oot; then ~or the isentropic expansion o~ rea1 nitrogen ~rom the given supp1y conditions, A =2.2x10-4 ~t2

H

{dH=0.2 in}. The characteristic decay time is T=V/I~y~R~T~O-A =73.2 ms.

H

From Fig.3, the time required ~or the supp1y pressure to ~a11 to 1/2 the initia1 va1ue is t/T=0.3, i.e., t=22 ms. The Reyno1ds number under these conditions is initia11y 27 mi11ion per ~ooto This examp1e demonstrates the ~easibi1ity o~ operating Longshot at very high Mach and Reyno1ds numbers with reasonab1e decay rates.

H A Mollier chart ~or dense nitrogen was used to determine

5.3.

Experimental Measurements

The theory presented in this report is compared with experimental measurements in Ref.8, which represents a continu-ation of the Longshot piston cycle studies. The agreement be-tween theory and experiment has been found to be quite satis-factory over a fairly wide range of operating conditions (driver pressures from 5000 to 15,000 psi, barrel pressures from 50 to

200 psi, and piston weights fr om 5 to 15 lbs). The only dis-crepancy found was that the theoretical chamberage correction factor F is not in very good agreement with experimental meas-urements. This is attributed to (a) the complicated geometry at the barrel-driver junction in Longshot, and (b) the use of an Abel-Noble gas model to predict the dense gas effects on chamber-age, which ignores attractive intermolecular forces.

60

CONCLUSIONS

The approximate analytical solutidn for the Longshot free-piston cycle presented in this report has been shown to agree well with more exact numerical calculationso The theory is also in good accord with experimental measurements (see Refo

8), except that the theoretical predictiori of the increase in piston speed due to chamberage is not entirely satisfactoryo

Calculations of Longshot performance show that it should be possible to obtain supply pressures of over 200,000 psi and supply temperatures from below l5000

K to over 30000

K by using pistons weighing from 5 to 50 lbs o The supply temperature obtained at a given supply pressure can be independently varied by changing the initial barrel pressure as well as the piston weighto

Calculations made with the present theory show that only a small fraction of the gas compressed by the piston is stored in the reservoiro This is the primary reason why losses in check valves are negligible under current operating condi-tionso The rate-of-decay of supply conditions could be greatly decreased by using a larger reservoiri but then valve flow

cal-culations should be carried out to determine the losses which whill be incurred o

REFERENCES

1. ENKENHUS, K.R., "The Flight Environment of Long-Range Bal-listic Missiles and Glide Vehicles", NAVORD Report 6745,

u.s.

Naval Ordnance Laboratory, Silver Spring, Maryland~ Oet. 1959.

2. ENKENHUS, K.R., HARRIS, E.L., GLOWACKI, W.J., MAHER, E.F., CERRETA, P.A., "Aerodynamic Design of a Continuous Hyper-velocity Wind Tunnel", NOLTR 62-20, U.S. Naval Ordnanee Laboratory, Silver Spring, Maryland, March 1962.

3. ENKENHUS, K.R., "Intermittent Faeilities", VKI Short Course on Physics of High Temperature Gases, Rhode-Saint-Genèse, Belgium, April 24-28, 1967.

4. PERRY, R., "The Longshot Type of High-Reynolds-Number Tunnel", Third Hypervelocity Teehniques Symposium, Denver,

Colorado, March 1964.

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