Theoretical performance study of the free-piston shock tube

FOR FLUID DYNAMICS

TECHNICAL NOTE

42

THEORETICAL PERFORMANCE STUDY OF THE FREE-PISTON SHOCK TUBE

by

K.R. ENKENHUS

RHODE-SAINT-GENESE, BELGIUM

TECHNICAL NOTE 42

THEORETICAL PERFORMANCE STUDY OF THE FREE-PISTON SHOCK TUBE

by

K.R. ENKENHUS

SUMMARY

Formulas and charts are presented for predicting th. performance of a free-piston shock tube using helium or hydrogen as the driver gas and real air as the test gas. The use of a piston to adiabatically compress the test gas in what is other-wise a conventional shock tube affords the practical possibility of obtaining shock Mach numbers above 40 in air. with temperatures well above l2,000oK behind the incident shock.

ACKNOWLEDGEMENTS

The author wishes to thank Miss J. Stuyck for the

TABLE OF CONTENTS SUMMARY i TABLE OF CONTENTS i i LIST OF SYMBOLS iv LIST OF FIGURES vi 1. INTRODUCTION 1 2. THEORY 2 2.1. Basic Considerations 2 2.2. Assumptions

4

2.3. The Piston Compression

6

2.4. The Shock Tube Equations 10

3. RESULTS AND DISCUSSION 15

3.1. Basic Performance Charts 15

3.2. Piston Shock Tube Performance Characteristics 16

3.3. Performance Limitations 19

CONCLUSIONS 23

REFERENCES 25

a A C DB' g Ma m p Ro t.t t.t t.x tc T v x Xc Xm.in a y À p Ds LIST OF SYMBOLS sound speed

chamberage area ratio

=

_{(DB/D s} )2

chamberage factor defined by Eq.(12)

diameter of barrel and shock tube, respectively ac cel erat ion due to gravi ty

Mach number of primary shock

=

WS/a_l molecular weight

pressure

universal gas constant

testing time behind incident shock

testing time behind incident s~ock per unit driven length

time af ter rupture of ,shock tube diaphragm that re-flected rarefaction wave head' overtakes contact surfac e

temperature velocity volume

piston weight

speed of incident and reflected shock, respectively distanc e

distance downstream of shock tube diaphragm at which reflected rarefaction wave head overtakes contact surfac e

minimum length of shock tube, measured from diaphragm to end of tube

parameter defined in Eq.(20) ratio of specific heats

compression ratio

=

_{V4i /V 4} density

o

1 2 3 1+

I+i

perf max min Subscripts reservoir

initial value in downstream section of shock tube value behind incident shock

value behind contact surface

initial value upstream of shock tube diaphragm when diaphragm bursts

initial value in barrel perfect gas value

maximum value

la lb lc 2a, b 3a, b 4a, b 5a, b 6 7a, b 8a, b 9a, b 10a, b 11a, b 12a, b 13a, b 14 15 LIST OF FIGURES

The piston shock tube x-t diagram

Chamberage geometry

Ratio of driver pressure (po) to initial shock tube

pressure (PI) VB shock Mach No. (He/Real Air)

Ratio of initial barrel pressure (P4i) to initial

shock tube pressure (PI) vs shock Mach No. (He/Real Air) Diaphragm pressure ratio vs shock Mach No. (He/Real Air) Ratio of distance from diaphragm to test section (xc) to barrel length (Cx4i) for maximum running time behind

inc ident shock. (He /Real Air)

Ratio of minimum shock tube 1ength (xmon ) to distance

from diaphragm to observation station rxc) formaximum running time

Dimensionless maximum running time behind incident

shock vs shock Mach No. (He/Real Air)

Running time behind incident shock per unit driven 1ength

Ratio of driver pressure (po) to initia1 shock tube

pressure (PI) vs shock Mach No. (H.2/Real Air)

Ratio of initia1 barrel pressure (P4i) to initia1 shock

tube pressure (PI) vs shock Mach No. (H2/Real Air)

Diaphragm pressure ratio vs shock Mach No. _{(H 2 /Real Air})

Ratio of distance from diaphragm to test section (xc)

to barrel length (Cx4i) for maximum running time behind

incident shock.(H2/Real Air)

Dimension1ess maximum running time behind incident

shock vs shock Mach No. (H2/Real Air)

Pressure ratio across normal shock vs shock Mach No.

16

17

18

19 20 21 22

T~mperature ratio across normal shock vs shock Mach No. Performance of piston shock tube using He/air

Performance of piston shock tube using H2/Air

Effect of driver gas on piston shock tube performance Comparison of~ston shock tube performance wit h He and HZ driver gases

Temperature achieved behind incident shock with He in barrel

Temperature achieved behind incident shock with Hz in barrel

1. INTRODUCTION

The shock tube has been widely used in the study of the chemical kinetics and radiative properties of high temp-erature gases (~ee, for example, Refs. 1-4). It is well known that the ~equirements for producing very strong shocks, i.e., high temperatures, are a high initial diaphragm pressure ratio and a high sound speed in the driver gas. The latter requirement has been met by using a light driver gas, and by preheating i t , eithe'r electrical1y or by combustion. Another method of pre-heating the driver gas and simu1taneously raising it to high pressures is to subject it to an adiabatic compression by m~ans

of a heavy piston. This is the principle emp10yed in the piston shock tube (Refs. 5-17), the deve10pment of which was pioneered by R.J. Stalker.

This report presents idealized theoretical performance for a free-piston driven shock tube. Design charts are given for helium and hydrogen, treated as perfect gases, driving real air. The effects of the boundary layer on theshock tube walls, and mixing or chemica1 reactions at the interface,are not considered. Real gas effects in air are only approximate1y accounted for,

2. THEORY

2.1. Basic Considerations

A schematic diagram of a free-piston shock tube, and the wave system which is generated during its operation, are shown in Fig. 1.

The facility consists of three main parts : a reservoir, containing the gas which propels the piston; a barrel, in which the pistan travels and compresses the driver gas; and a con-ventional shock tube, which is attached to the downstream end

of the barrel. There are t~o diaphragms, the first separating

the reservoir from the barrel, and the second separating the high and low pressure sections of the shock tube.

The facility is prepared for operation by placing the piston at the upstream end of the barrel, and charging the

reservoir with, say, air or nitrogen tQ a pressure Po; rilling the barrel with a light driver gas to a pressure P4i; and

filling the downstream end of the shock tube with the test gas to a pressure Pl. The piston stroke is initiated by rupturing the diaphragm between the reservoir and the barrel. This is conveniently done at a desired value of PO by using a double-diaphragm arrangement. (In some facilities, the double-diaphragm is ómitted and the piston is held in place mechanically until the test begins).

The initial temperatures of the gases in the barrel

and shock tube, T4i and Tl' respectively, are normally ambient

temperature. The quasi-steady adiabatic compression of the driver gas by the piston results in a final pressure and

values P4i and T4i. The final pressure P4 in the barrel is controlled by the strength of the shock tube diaphragm. The initial pressures PO and P4i are selected so that the piston comes to rest just as the shock tube diaphragm ruptures. The basic parameter governing the tinal conditions in the driver gas is the volumetric compression ratio,

When the second diaphragm breaks, a shock wave propogates down the shock tube and eventually retlects trom the downstream end, while a rarefaction wave moves upstream and reflects from the piston face. The wave system generated, which is shown in Fig. lb, is exactly the same as in a

conventional shock tube. The strength of the shock wave depends primarily on the compression ratio À and the initial diaphragm

pressure ratio, P4/Pl.

If high compression ratios are used to obtain very elevated pressures and temperatures in the driver gas, then the distance X4 between the piston face and the shock tube diaphragm when the diaphragm ruptures will be smalle If X4 is less than a few shock tube diameters, there is a possibility that the flow generated behind the incident shock as the dia-phragm opens wili be non-uniform and disturbed. In addit10n,

if X4 is smal'l , the test ing time flt b ehind t he inc i dent shoc k may be un~uly limited by the overtaking of the cQntact surtace by the reflected rarefaction wave he ad at a relatively short distance Xc downstre~m of the diaphragm. (See' Fig. lb).: As is well known, the maximum testing time (flt)max is ·obtained by locating the observation section at xc. The minimum shock tube length, xmin' is that value for which the shock reflected from

the end of the tube arrives at the contact surface at the same time as the reflected rarefaction wave head.

In order to extend the shock tube testing time and

produce a better quality flow, it is desirable to increase X4

by employing chamberage, i.e., by making the diameter of the

shock tube less than that of the barrel, as sho~n in Fig. lc.

The maximum advantage is obtained in the case where the piston comes to rest at the very end of the barrel. If the piston comes to rest earlier than this, then x4 will not be as large

tor a given compression ratio, and, furthermore, a compression

wave will be reflected downstream again as soon as the upstream-moving rarefaction wave reaches the junction between the barrel and the shock tube.

It might at first be supposed that the tact that the

driver length X4 is relatively small in the piston shock tube

facility would lead to severe limitations in testing time behind the incident shock. We shall see that for high shock strengths this is not the case, because the gas in the region behind the contact surface is very cold; the reflected rarefaction wave head overtakes the contact surface with a relative speed equal to the local sound velocity in the gas, which is relatively low.

2.2. Assumptions

The following assumptions are made in the analysis 1. The reservoir is large enough, and the piston heavy enough,

so that the piston velocity is low Bnd the pressure on the back face of the piston remains essentially constant at the

2. The gas in the barrel undergoes a quasi-steady isentropic compression. (The low piston yelocity insures that no shocks are produced, but ther~ will a small loss of heat to the barrel walls).

3. The piston is frictionless.

4.

The driver gas behaves as a perfect gas with a constant ratio of specific heats during the compression in the barrel, and as well in the subsequent unsteady expansion

into the shock tube. This is an excellent assumption for an inert monatomic gas such as helium, but is not such a good assumption for hydrogen which is heated to high temp-eratures by the use of large compression ratios. Even in hydrogen, real gas effects will be minimized because the pressure is very high at the same time that the temperature is high. At very large values of P4, compressibility effects may be significant.

5. Ideal, inviscid ~ave systems are generated in the shock tube. Some departures from ideality may occur is X4 is too small, or if P4 is so large that very thick shock tube diaphragms must be employed. The most significant departures from

ideality will, in most cases, result from the viscous

boundary layer on the shock tube walls when operating at low values of th~ initial shock tube pressure, Pl ' or with a shock tube of high length/diameter ratio. When very strong shocks are produced, the temperature will also decay between the shock front and the contact surface due to radiation losses.

6.

Real gas effects in the test gas will be accounted for only in an approximate manner. The test gas is assumed to be air at an initial temperature of 300 o K.

7.

The analysis is valid for the case where the piston stops at the end of the barrel. In this case, the wave systems in the shock tube are the same as if no .chamberage were present •.

Theeffect of area chanse on performance is discussed in

Ref~

16.

2.3. The Piston Compression .

The notation used in the analysis of the. piston stroke ~s shown in Fig. lc. The equation of motion of the piston is

~ dvp

g dt ( 2 )

where according to assumption 1, Po remains constant during thEl stroke.

When the pis~on face is at a distance x from the shock tube diaphragm, the volume V of the driver gas in the barrel is related to the initial volume V~i by

For a quasi-steady isentropic compression (assumption~

the instantaneous pressure in the driver gas is

( 4 )

When Eqs. (3) and

(4)

are used in Eq. (2), the equation of motion of the piston becomes, in integral form,

J

v [

{va..)Yj

V Po-p,i

T

dV

Iti

This leads to the following expression for the piston velocity, v p , in terms of the instantaneous dxiver gas volume ,V:

v

=

"

The piston comes to rest again when V=Va.. Eq.

(5)

then indicates that the compression ratio ~=Va.i/Va. is related to the initial pressures in the reservoir and barrel by

( 6)

The max~mum piston speed is obtained by setting the

derivative dVp/dV equal to zero. It is found that the volume of the driver gas corresponding to maximum piston speed is

The corresponding position of the piston is given by Eq. (3). At high compression ratios, PO»Ya. i according to Eq.(6) and Eqs.

(7)

and (3) then indicate that the maximum piston speed occurs near the end of the stroke.

and (7) as

where

The maximum piston speed may be written~ from Eqs.(5)

v

Pvac 1 - _{'Va.- l Po} 'Va. (pl+i)ia. -

~

1 - -_'V 1 ₁₊

2gV . P 0 4l.

w

P

( 8)

is the velocity the piston would attain in accelerating to the end of an evacuated barrel of volume Va.i under a constant driving pressure PO. This value of velocity is obtained with an infinite compression ratio.

The final pres sur e in the barr el is ~ by Eqs. (1) and

( 4 ) ,

Since the driver gas l.S assumed to obey the perfect

gas law

pV =nROT (n

=

no. of mol~s) (10)

the final temperature is , from Eqs. (9) and (10) ,

(11)

Eqs. (9) and (11) show that the use of a driver gas

pressures and temperatures for a given compression ratios; i.e., a monatomic gas gives higher values than a diatomic gas.

The effect of chamberag~ in increasing the length x4

of the column of compressed driver gas may be deduced from the geometry of Fig. Ic. With chamberage, the compression ratio may be written À =

c

x4i X4 where C =

_{(1 -}

-

A-I _{: : i ) /}

_{(1 -}

A-I

Xs)

(12)

A A x4 and A = (DB/D_{s ) 2}

is the area ratio of the contraction.

The maximum value of C is produced when the piston stops at the very end of the barrel. It is

C = A

r

_{l - AA-}l Xs

1

L

X4i

It should be noted that C~A for large compression

ratios (more precisely, when the volume of the high pressure

section of the shock tube is small compared with the volum~

of the barrel). The value of x4 for a given compression rat io

is, according to Eq.

(12),

C times as great in a facility with chamberage as in one with a barrel and shock tube of equal

diameter: Or, putting this another way, a chambered facil ity

is equivalent to an unchambered one with C times as long a barrel; both compress the driver gas into the same length, x4.

2.4. The Shock Tube Eguations

Equations derived in this section approximately

re-present the behavior of real air in a s~ock tube. The equations

developed are modifications of the perfect gas relations which are given, for examp1e, in Ref. 2.

The pressure ratio across the primary shock in real

air at an initia1 temperature of 3000_{K is shown in Fig. 14,which}

is taken from Ref. 18. The curves are represented with sufficient accuracy by the formu1a

P2

=

1.1

(P2)

=

1.1

Pl Pl perf (14)

The pressure ratio across the rarefaction wave (for

a perfect driver gas, (assumption 4) is

The velocity V2 behind the i ncident shock in real .air

is, according to Fig. 2 of Ref. 18, approximate1y given by

(16)

From Eqs.(14)-(16), it is th~n found that the initial

diaphragm pres~ure ratio is related tothe shock Mach number, Ms

Pit

(2Y

1M_S

2_(y,-1) )

t

(4-1)

al

~s

;slY4~1

-

= 1.1 1-1.1 - (17)

Pl Yl+1 . Y·l +1 alt

The initial sound speed in the driver and test gases is given by the perfect' gas re1ation

(18 )

since the hot driver gas is, by hypothesis, perfect, whi1e the air test gas is initia11y at ambient temperat~re. The initial sound spëe~ ratio appearing in Eq.(17) is then, upon inserting Eq.(18) into Eq.(11),

(YIt- 1 )

al

C,m4T,) -

2

a 4 = _{Y4ml T4i} À (19 )

where in the usual case, T4i = Tl

The maximum va1ue of shock Mach number, _{(Ms )lim' is} obtained with an infinite diaphragm pressure ratio. Eq.(17)

shows that

Note that (Ms )l"

=

a in practice. l.m

As mentioned in section 2.1, the testing time behind

surface by the ref1ected rarefaction wave head, as shown in

Fig. lb. Under the assumption that perfect gas re1ations ~old

in the driver gas, the time af ter diaphragm burst th at this

,event occurs is (see p. 61-62 of Ref. 2) Y~+l 2x~

~

+ Y~-l

~(y.-1

)

tc = M3 a~ 2 (21 ) where Y₄-1

-

2y~

-1]

M3 = 2

[(P2 P, )

Y4-1 PI p~ (22)

is the Mach number in the region immediate1y behind the contact surface .

The time required for the rarefacti?n wave head to

overtake the contact surface is obtained by inserting Eqs.

(22),

(14), (17) and (19) into Eq. (21). It is, in dimension1ess form

=

2

x

(23 )

The distance downstream of ~he diaphram at

which the ref1ected rarefaction wave head overtakes the contact

surface is Xc

=

vztc' Eq. (16) yie1ds

The testing time

=

(6t) _max behind the incident shock a distance Xc downstream of the diaphragm is (see Fig. lb)

The shock wave continuity equation is

Eliminating the shock speed Ws from Eq. (25) using

Eq.

( 26) ,

we find that

X _{PI PI}

(6t

k

= c

-

'-

= _te

(27)

ax v2 P2 P2

The testing time in real air will be shorter than that in perfect air because Pl/P2 is smaller in the former. For the purpose of estimating testing time, the density ratio, which is shown in Fig. 15, may be approximated for Ms ~

5

by

= 6 + (0. 59 -

o.

09 log 10 PI ) (M s - 5 ),

mm (28)

where PI is the initial shock tube pr~ssure in millimeters of mm

mercury. For ~p~ii6ation in later calculations, the maximum testing time behind the incident shock is conveniently expressed in the dimens ionless form

al (6t) _max

(altCy~2

)

(29)

CX 4i

=

CX 4 i ~

As mentioned in section 2.1, to obtain the maximum testing time the observation section should be located a

length,xmin' is then the value for which the reflected shock meets the contact surface at the same instant as the reflected rarefaction wave head (see Fig. lb). It can be shown that, in general,

x .

ml.n

=

t _c + x /W_{c R}

('30 )

where Ws ' WR, are the speeds of the incident and reflected shocks, respectively. Eq. (30) can be written in the form

W

s 1 + W

R

For real air, Fig. 2•

4-8,

p. 271 of Ref. 1 shows that the ratio of reflected to incident shock speeds is approximated by W .

I

W R

₌

(0.6 + 0.2 loglO Pl ) Ms + 0.12 s mm

for Ms~5. Although this formula is not very accurate, it is adequate for the present application.

Finally, it is of interest to know the testing time behind the incident shock per unit driven length. It is, in

dimensionless form,

3. RESULTS AND DISCUSSION

3.1. Basic Performance Charts

An IBM 1130 Fortran IV computer program was written which calculates piston shock tube performance for any perfect

driver and test gases, or for any perfect gas driving real air as approximated by the formulas in the Theory section. Only the

results obtained for helium and hy~~ogen driving real air are

presented in this report. The calculations were made using the

shock Mac~ number Ms and compression ratio À as the independent

variables.

The basic performance charts for helium driving real

air are presented in Figs. 2 to

8.

The results with a hydrogen

driver are shown in Figs.

9

to 13. The dimensionless quantities

which are plotted as functions of Ms and À are :

Po _P4i P4 x _c x _min al(~t) _max al ~t and

CX4i

,

_{CX 4i}

,

~x

PI PI PI x_c '

When g~ven the initial shock tube pressure, _{PI ,} the

initial sound speed a l , and the geometrical parametern C and

x4i' (see Fig. lc and Eqs. (12) and (13)) the above ratios

yield,respectively, the reservoir pressure; the initial barrel pressure; the final barrel pressure; the distance downstream of

the diaphragm for observing the maximum testing t ime behind the

incident shock; the minimum shock tube length for obtaining the

max~mum testing time; the value of the max~mum testing time;

and the testing time behind the incident shock per unit driven

3.2. Piston Shock Tube Performance Characteristics

The performance characteristics of any piston shock tube facility employing the aforementioned gases may be developed using Figs. 2 to 16. Performance charts with helium and hydrogen drivers are presented in Figs. 17 and 18, respectively. In each figure, the ordinate is the shock Mach number, ,Ms, and the

abscissa is the compression ratio, À.

By far the highest pressure ~n the piston shock tube is the final driver pressure, P4' The maximum performance is

therefore limited by the maximum allowable value of this pressure. I In 'Figs. 17 and 18, P4 is assumed to be 4000 atm. However, a

very high order of performance is still possible with a pressure an order of magnitude smaller than this value.

Once P4 is specified, the reservo~r pressure, Po, and the initial barrel pressure, P4i' dep end only on the compression ratio À. Scales for these quantities are therefore shown along the bottom of the graphs. It should be noted that very modest

values of PO and P4i are sufficient. For example, with a helium driver (Fig. 17), compression ratios from 18 to 180 can be

achieved with 32 <Po <300 atm and 0.7 <P4i <30 atm. With a hydrogen driver, Po and P4iare only a li~tl~' larger.

The shock Mach number achieved with a given compression ratio and a fixed value of P4 depends on the initial shock tube pressure, Pl' In figs. 17 and 18, five curves are drawn for values of Pl ranging from 1 to 10- 4 atm. The maximum shock Mach

number achieved at a compression ratio of 100 and Pl=10-4 atm is 38.5 with a helium driver and 39.3 with hydrogen - there is notmuch difference in the performance of the two gases at this compression ratio.

The 1imiting Mach ·number at À=100 is, according to

Eq. (20), 44.8 with helium an~ 51.8 with hydrogene The fu11

increase in performance th~or~tical1y realizab1e with a

i

hydrogen driver is not obtainab1e at Pl=10-~ atm.(p~/Pl=4x107)

because the diaphragm pressure ratio is not high enough. For I

example, Eq. (17) shows that a compression ratio of 2.55x10 ll

wou1d :b'e required to obtain a shock Mach number of 50 us ing

hydrog~n.

A comparison of Figs. 17 and 18 shows that consider-ably higher shock Mach numbers can be obtained at low compression ratios with H2 than with He. For examp1e, at À=20, Pl=10- 4atm., Ms = 29.2 with a hydrogen driver, but on1y 23.2 with He. At high compression ratios the situation is actua11y reversed. The reason for this can be seen from Eqs. (19) and (20). The 1imiting Mach number is approximate1y

1

1.1 À

(Y ~.-l ) 2

When this equation is app1ied to helium and hydrogen driving air, with T4i = Tl' one obtains

1

(Ms)l"

_'"

9.57 À3 for He

l.m

1 (35)

(Ms)l" _lom

_'"

20.7 À5 for H2

At sufficiently high compression ratios, the 1imiting Mach number with a helium driver wi11 exceed that with hydrogen

since (M s )l" increases more rapid1y with the compression ratio 1

l.m

for helium. The underlying reason for this is that, as shown by

compression ratio in a monatomic gas (Y4=1.667) than in a di-atomie gas (Y4 ~1.40). Hydrogen has a larger multiplying

coefficient in Eqs. (35), i.e., it is superior at low compression ratios, because it has an inherently ~igher sound speed.

The exact values of (M s )1. for perfect and real air,

1m

calculated from Eq. (20), are shown for various driver gases in Fig. 19. At this point we mayalso note from Eq. (34) that real gas effects lower the limiting shock Mach number by a factor of approximately 1.1. It may be seen that nitrogen and argon are very poor choices for the driver gas. The superiority of hydrogen over helium at low compression ratios is evident. Helium gives a higher limiting shock Mach number at cPJ!lPression ratios .above about 300. In practice, with a finite rather than an infinite diaphragm pressure ratio, such as P4/Pl=4xl07 , (P4=4000 atm,., Pl=10-4 atm) the shock Mach number obtained with a helium driver exceeds that obtained with hydrogen at compression ratios above 115, as may be seen by comparing Figs. 17 and 18.

The testing time behind the incident shock, 6t/6x, in

microsecond per foot of driven length, is also plotted in Figs. 17 and 18. The values range from about 1 to 10 ~s/ft. The maxi-mum testing time (6t)max cannot be increased to any desired value by using a longer shock tube because the observation station must be at or upstream of the position where the

re-flected rarefaction vave head overtakes the contact surface. Curves are plotted for (6t)max=10 and 100 microseconds, assum-ing x4i =

6

ft. and C = 5.43. The value of C used is the limiting value = A at large compression ratios {see Eq. (13» for a 3.5" barrel driving a 1.5" shock tube. The variation of C with

compression ratio was not taken into account in plotting the curves of {6t)max.

The distance Xc downstream of the diaphragm at which the test section"would have to be located in order to achieve the maximum testing time behind the incident shock may be rather small at high compression ratios and large values of the initial shock tube pressure P l' A curve for Xc = 6 ft. is shown in Figs. 17 and 18, assuming x'd=6 ft., C=5.43. At low 'values of Pl, Xc

is much larger ,_{(i.e., the curve for Xc /CX 4i=1 corresponds to}

Xc = 32.6 ft.), but a long shock tube cannot be used unless it is of quite a large diameter, in order to avoid excessi've

deterioration of performance due to the effects of the boundary layer on the shock tube walls.

The minimum shock tube length for maximum testing time, xmin' is not shown in Figs. 17 and 18, since it only exceeds Xc by a few percent, as shown in Fig. 6. The real gas value of xmin/xc depends only on Ms and Pl; the dependenee on

Pl is very weak except at low shock Mach numbers. Only a single, means curve has been shown. xciXmin is smaller in real a~r than in perfect air because the speed of the reflected shock is much lower.

3.3. Performance Limitations

The operating reg ion of a small piston shock tube usi ng helium or hydrogen to drive real air is shown in Fig. 20. The

~ollowing parameters limit the performance which can be achi eved and thus fix the boundaries of operation

1. The maximum compression ratio, À _max • For the particular

facility being considered DB=3.5", Ds=1.5", x4i=6 ft., C=5.43 so that x4=4", or 2.67 shock tube diameters at À=lOO. Higher compression ratios would probably result in an unacceptably small value of x4' Higher compression ratios could easily be

a~hieved in a facility with a longer barrel or greater chamber-age.

2. Tbe maximum driver pressure, (P4)max' A value o~ 4000 atm. is assumed. However, the per~ormance is still excellent at much lower values o~ P4' For example, Figs. 3 and 10 show that at P4=500 atm., Pl=10-4 atm., and a compression ratio

o~ 100, Ms=35.6 with He .and 35.4 with H2' as compared with Ms=38.5 and 39.3, respectively, at P4=4ooo atm.

3. The minimum shock tube initial pressure, (Pl)min' A value of 10- 4 atm. is assumed. Tbe question is one o~ avoiding excessive viscous e~fects; the lowest values are permissible with a large diameter, relatively short shock tube.

4. Tbe minimum allowable value of xc' One of the operating boundaries in Fig. 21 is taken as Xc = x 4i = 6 ~t. for the small facility under consideration. The observation station must be located at a distance downstream of tbe shock tube d'Îaphragm ~ xc'

5. Tbe maximum available reservoir pressure, (PO)max' This value was taken to be 300 atm. since an air supply is readily avail-able which provides this pressure. Much lower values of Po

(and, correspondingly, of P4i and P4) would s t i l l lead to a high order o~ performance.

6. The minimum allowable running time per unit driver length, (öt/öx)min' The boundary shown was arbitrarily selected as 1 ~s/ft, About all that can be said here is that measurements become increasingly difficult at higher shock speeds because the time available is so short.

Some comments will now be made concerning the operating boundaries shown in Fig. 20. This figure shows that, for a given

value of (P4)max, (Pl)min determines the maximum shock Mach number attainable with any given compression ratio. The maximum

shock Mach number increases markedly as the compression ratio is raised, particularly with a helium driver. At high compression ratios (or with very low valuesof PI) the avai~able running time per foot of_ driven length may limit the maximum useful value of shock Mach number. Such a limitation can be overcome by using a sufficiently large shock tube. The overtaking of the contact surface by the reflected rarefaction wave head places a limit on the maximum initial pressure at which a given shock Mach number can be achieved. At low shock tube pressures, Xc is so large that this phenomenon is no problem.

Regarding the choice of driver gas, hydrogen provides considerably higher shock Mach numbers at low compression ratios but is actually somewhat poorer at high compression ratios. Since the amount of driver gas required per~ot is small, there is no reason why helium should not be preferred,especially when opera-ting safety is also taken. mto account.

The temperatures achievable behind the ,incident shock with helium and hydrogen drivers are shown in Figs. 21 and 22, respectively. The values, which are derived using Fig. 16,which is taken from Ref. 18, range from about 70000 _{K to over 12,OOOoK.}

Data was not readily available for 'p16ttingisotherms above 12,OOOoK, so theyhave beenomitted. It is clear that the

piston shock tube is readily capable of generating very high enthalpy flows, and opens up some exciting possibilities for new research in this area.

CONCLUSIONS

The performance analysis has indicate that the piston shock tube is a versatile tool for generating high enthalpy flows. The use of a piston to adiabatically compress the driver gas in what is otherwise a conventional shock tube affords the practical possibility of attaining shock Mach numbers above 40 in air, with temperature well above l2,000oK behind the incident shock.

Helium and hydrogen make the best driver gases.

Although hydrogen offers better performance at low compression ratios, helium may be somewhat better for generating the high-est shock Mach numbers using high compression ratios. The usa-ble upper limit of shock Mach number will be determined by the viscous effects, which become prominent at the low initial shock tube pressures which are required.

The performance of the piston shock tube is in most respects governed by the same considerations as a conventional shock tube. At very high shock Mach numbers, the testing time tends to become rat her short unless a very long tube is used. The fact that the driver gas is initially compressed into a short column does not in itself tend to limit the running time at high shock Mach numbers.

Very modest pressures are requ~ired to operate a piston shock tube, and very high performance can be obtained in a relatively small facility. Those facts make the piston shock tube economical to construct and operate.

REFERENCES

1. GLASS, l . I . , HALL, J.G. : "Shock Tubes".

Handbook of Supersonic Aerodynamics, Vol. 6, Section 18, NAVORD Report 1488, Dec. 1959.

2. GAYDON, A.G., HURLE, I.R. : "The Shock Tube in High-Temp-erature Chemical Physics".

Reinhold Pub.Co., N.Y., 1963.

3. GREENE, E.F., TOENNIES, J.P. "Chemical Reactions in Shock Wavesll

•

Edward Arnold (Pub.) Ltd."London, 1964.

4. ENKENHUS, K.R. : "Intermittent Facilities".

VKI Short Course on High Temperature Gas Physics, April 24-25, 1967.

5. STALKER, R.J. : "An !nvestigation of Free Piston Compression of Shock Tube Driver Gas".

Div. of Mechanical Engineering, National Research Council, Canada, Report MT-44, 1961.

6. STALKER, R.J. : IIThe Free Piston Shock Tube".

Aeronautical Quaterly, XVII, Part

4,

p 351-370, 1966.

7. GRIEF, R. : lI'l'he Free Piston Shock Tube".

Ph.D. Thesis, Harvard University, 1962.

8. GRIEF, R., BRYSON, A.E. "Measurments 1n a Free Piston Shock Tube".

9. WILLIARD, J.I>!. : "Design and Performance of the JPL Free-Piston Shock Tube".

Vol. I, Fifth Hypervelocity Techniques Symposium, U. of Denver, March 1967, p. 157-205.

10. ROFFE, G.A. : "The Free Piston "Shock Tube Driver A preliminary Theoretical Study".

Tech. Report NÓ.32-560, Jet Propulsion Laboratory, Pasadena, Cal., Dec. 15,1963.

11. BABINEAUX, T.L., RIALE, B.R. : "preliminary Performance of a Free Piston Shock Tube Driver".

Space Prog~ams Summary No.37-32, Vol. IV, p 101,

Jet Propulsion Laboratory, Pasadena, Cal. Feb-March 65.

12. BABINEAUX, T.L., RIALE, B.R. : "Preliminary Performance of a Free Piston Shock Tube".

Space Programs Summary No.37-34, Vol. IV, Jet

Propulsion Laboratory, Pasadena, Cal., June-July 1965.

13. WILLIARD, J.W. : "Performance Evaluation of a 12" Diameter Free-Piston Shock Tube".

Space Programs Summary NO.37-41, Vol. IV, Jet

Propulsion Laboratory, Pasadena, Cal., Aug-Sept. 1966.

14. WILSON, J. L., SCHOFIELD, D., REGAN, J. D. : "~lasma Flow in an Electromagnetic Shock Tube and in a Compress ion Shock Tube".

Ministry of Aviation (Gt. Br.) ARC Report C.P.No.866,1966.

15. STALKER, R.J. : "Isentropic Compression of Shock Tube Driver Gas".

16. STALKER, R.J. : "A~ea. Change with a Free Piston Shock Tube".

AIAA Jour., Vo1~ 2, p 396, 1964.

17. PRATT, M.J. : "The Deve10pment at Cranfie1d of a Free Piston Compression Shock Tube".

The .Co11ege of Aeronáutics, Cranfie1d, CoA Report

No .194, Feb 1967.

18. LEWIS, C.H., BURGESS, E.G. l I l : "Charts of Normal Shock Wave Properties in Imperfect Air".

PISTON

IST DIAPHRAGM

o

DIAPHRAGM

PISTON STOPS HERE

Fig. 1a THE PISTON SHOCK TUBE

REFlECTED ~ RAREFACT ION tc~

__

~~~

________

~

__

~~ ~ u ~

z

~ IJ')

~ ~--~~---+-~

_Xc . "min Fig, 1b x - t diagram

Fig,

1c

CHAMBERAGE GEOMETRV

He/REAL AIR

10-2 L _ _ .J.... _ _ ..l... _ _ ....L _ _ ~:-_--:~_--:

15 20 25 30

10

SHOCK MACH NO.

Fig. 20 RATIO OF DRIVER PRESSURE (po) TO INITIAL SHOCK TUBE PRES5URE (p, ) VS SHOCK MACH NO.

1~c---r----~----,----r~

COMPRESSION RATIO: 5

I

He/ REAL AIR

I

1Ö3L-____ -L ____ ~ ______ ~ ____ ~ ____ _:~--__:

o ro ~ ro ~ ~

SHOCK MACH NO. M.

Fig. 30 RATIO OF INITIAL BARREL PRESSURE (I\l' TO INITIAL SHOCK TUBE PRESSURE (p,' YS SHOCK MACH NO.

l1L _ _ -L _ _ -L_-~:----:~----:~--~

30 35 40 45 50 55 60

SHOCK MACH NO. M.

Flg.2b RATIO OF DRIVER PRES5URE (Po) TO INITlAL SHOCK TUBE PRESSURE (P, ' VS SHOCK MACH NO.

RATIO

I

H./REAL AIR

I

30 35 . 40 45 50 55 60

(

SHOCK MACH NO. M.

Fig. Jb RATIO OF INITIAL BARREL PRESSURE (P4l) To INITIAL SHOCI TUBE PRESSURE (p,) VS SHOCK MACH NO.

10 COMPRESSION RATIO. 5 10 1~ ____ ~ ____ ~ ____ ~ ____ ~ ____ ~ ____ ~ o 5 10 15 20 25 30 SHOCK MACH NO. M.

Fig. 40 OIAPHRAGM PRES5URE RATIO VS SHOCK MACH NO,

COMPRESSION RATIO = 5 10 20

I

HejREAL AIR

I

-3 10 L-____ ~ ______ L-____ ~ ______ ~ ____ ~~ ____ ~ 10 15 20 25 30 SHOCK MACH NO. M.

Fig. 50 RATIO OF DISTANCE FROM DIAPHRAGM TO TEST SECTION (re) T_{O BARREL LENGTH(cr4},) "OR MAXIMUM RUNNING TIME BEHIND INCIDENT SHOCK.

RATIO

10JL-____ ~ ______ L-____ ~ ______ ~ ____ ~ ______ ~

JO 35 40 45 50 55 60

SHOCK MACH NO.

Fig. 4b DIAPHRAGM PRESSURE RATIO VS SHOCK MACH NO.

I

H. jREAL AIR

I

-2

10 L-____ ~ ____ ~ ____ ~ ____ ~~ ____ ~ __ ~

30 35 40 45 50 55 60 SHOCK MACH NO. M.

Flg, Sb RATIO OF DI5TANCE FROM DIAPHRAGM TO TEST SECTION (re) TO BARREL LENGTH(Cr4i' FOR MAXIMUM RUNNING TIME BEHIND INCIDENT SHOCK.

Xmin -rc-1.08 \, - - -PERfECT AIR - - R E A l AIR " ...

_---

-',04 1.02 1.00 ... _ _ .1..-_ _ .1..-_ _ ... _ _ -'-_ _ - ' - _ - - ' 10 20 '30 40 50 60

SHOCK MACH NO. M.

f;g. 6 RATIO Of MINIMUM SHOCK TUBE LENGTH ('mln)

TO DI5TANCE fROM DIAPHRAGM TO OB5ERVATION 5TATION (xC> fOR MAXIMUM RUNNING

TIME-102.,....---;:====~_1_1_{~ I ,} _11 100~-..---.---.--I-II-TI-1 10 -I 10 a,(61)mox CX4i -2 10 lÖ3 10-4 150 175 lÖ5[ '200 I fig. 70 10 IS 20 25 5HOCK MACH NO. lol.

DIMENSIONlE55 MAXIMUM RUNNING TIME BEHIND INCIDENT 5HOCK V5 SHOCK MACH NO.

30 10 COMPRE5510N '40 10-1 a, (6 Il max ex, i lÖ5L' ______ ~~ __ _L _ _ _ _ ~~ _ _ _ _ ~ _ _ _ _ ~ _ _ _ _ __J 30 35 40 45 50 55

SHOCK MACH NO. M.

fig. 7b DIMENSIONlESS MAXIMUM RUNNING TIME BEHIND

INCIDENT 5HOCK VS SHOCK MACH NO.

10.2 At a1iï IÖ3 *T,8300·K 10.4 0 10 15 20 25 30 SHOCK MACH NUMSEI! M.

Fig, 8a RUNNING TIME BEHIND INCIDENT SHOCK PER UNIT oRIVEN LENGTH. 1o7r-____________________________ , -__ - . ____ - , Po Pl 2 10 10 ·1 10 ·2 COMPRESSION RATIO. 5 10 ~----~---~,~0----~1~5----~2~0~----~2~5----~30.

SHOCK MACH NO. M.

:; z ~ z Ol > ! :!; t: ';Ir $ ! 0-" z Z z :0 ca:

FIG. 9 •. RATIO OF DRIVER PRESSURE IPo I TO INIT lAL SHOCK TUBE PRESSURE IP, I VS SHOCK MACH NO.

20 10 I I 1 •• .1 .4 .2

•

z

i

20 ~ Z 10.2 Ol > iS a At l i l IItCmrn.HIöI _~ t: ~ $ 10.3 Ol J t-I!> ! z .4 ~ .2 * Tl' 300"K 10.4 0.1 30 35 40 45 50 55 60 SHOCK MACH NUMBER lol.

Fig. eb RUNNING TIME BEHIND INCIDENT SHOCK PER UNIT ORIVEN lENG TH.

102~----~~----~----~----~~----~----~ _{~ 35 40 45 50 55' 60}

SHOCK MACH NO. M.

FIG.9b RATIO OF DRIVER PRESSUREIPo'TO INITlAL SHOCK TUBE PRESSURE lP, I VS SHOCK MACH NO.

10

-1 10

-3

10 OL---~----~1~0---~15~----~2~0----~2~5----~30

SHOCK MACH NO. Ms

FIG.l0. RATIO OF INI TIA!. BARREL PRESSURE I P4i) TO INITIA!. SHOCK TUBE PRESSURE lP,) VS SHOCK MACH NO.

108r-______________________________ - , ____ ~--~ 4 10 3 10 10 COMPRESSION RATIO • S 10 15 20 25 30

SHOCK MACH NO. M5

FIG.!l. DIAPHRAGM PRESSURE RATIO VS. SHOCK MACH NO.

10

30 35 40 45 sa 55 60

SHOCK MACH NO, Ms

FIG. lOb RATIO OF INITlAL BARREL PRESSURE IP4i) TO INITIA!.

3 10

30

SHOCK TUBE PRESSURE lP, ) VS SHOCK MACH NO.

I

H2/REAL AIR

I

35 45 50 55 60

SHOCK. MACH NO.

10 -I 10 - ) COMPRESSION RATIO.

I

H2/REAl AIR

I

10 ~----~---~,0~----~1~5----~2"0~----~25~----~)0 2 SHOCK MA C H NO.

FIG 120 RATIO OF DISTANCE FROM DIAPHRAGM TO TEST SECTION (xc) TO BARREL LENG TH (Cl") FOR MAXIMUM

RUNNING TIME BEHINO INCIDENT SHOCK

10 ~---~~ 10 -3 10 -4 10 -5 10 ~----~~----~10~---,~5---~20~----~2-5---J30

S HOC K MACH NO. lol.

FIG.130 DIMENSIONLESS MAXIMUM RUNNING TIME BEHIND INCIDENT SHOCK VS SHOCK MACH NO.

-2 10 3~0~--~3*5~----~40~----~4~5----~5*0~--~5~5~--~60 10 -I 10 O,(dt)mol --c:i4i -4 10

SHOCK MACH NO. M.

FIG 12b RATIO OF DISTANCE FROM DIAPHRAGM TO TEST SECTION (Xc) TO BARREL LENGTH(CI4i) FOR MAXIMUM RUNNING TIME BEHIND INCIDENT SHOCK.

-5

10 ~.~----~35~---~40~---4~5---5~0---5~5---~60

SHOCK MAC H NO. lol.

FIG 13b DIMENSIONlESS MAXIMUM RUNNING TIME BEHI NO INCIDENT. SHOCK VS SHOCK MACH NO.

1000. 1 7 ] 100 100 400 P2 P;- 200 1001

I

J

o Fig. 14 5' 10 15 20 2S

SHOCK MACH NO. lol.

PRESSURE RATIO ACROSS NORMAL SHOCK VS

5HOCK MACH NO.

~ P, 20 15 10 P, MM. HG. .01 _---f~ .. m.!.-G~'i.::J~2.':J---il ,,,,,,- , 10 15 20 25 30 SI10CK MACH NO. M.

Fig. 15 DENSITV RATIO ACROSS NORMAL SHOCK

WAVE VS SHOCK MACH NUMBER.

40 35 30 T2 T, 25 20 15 10 5

,

I

,

,

I

,

,

§I

;:.;

_,

.", ~,

,

...

,

Ij

_,

,

,

,

,

I I I I I

:

I I, o~, ____ ~ __ ~ ____ ~ ____ ~ __ ~ __ ~ o ~ ~ ~ 25 » SHOCK MACH 110. Ms

Fig. 16 TEMPEIIATUIIE· ... TlO ACROSS _MAL SHOCK VS SHOCK MACH HUNSER •

40 H"/REAl AIR Plo = 4000 ATM C = 5.43 35L ",30 ,; ó z 25 :I: u ct ~ x u 20 0 :I: lil 15 4 3 2,5 1.8 1,6 1,4 1.2 0,9 0,8

5L BARREL PRESSURE _P4l (ATM)

1000 300 200 140 100 80 70 60 55 50 45 40 35 DRIVER PRES5URE Po (ATM)

0

0 20 40 60 80 100 120 140 160 COMPRESSION RATIO À

Fig. 17 PERFORMANCE OF PISTON SHOCK TUBE USING HELIUM I AI R

Of 180

'"

::r x U o :I: lil 40 35 30 10 H₂/REAl AIR Plo =4000 ATM C = 5.43 ",'" 'f..'\'1 ...

2feG

~ .. '-),.-_ _ _ _ _ _ _

,

,,'o~~

d":-~

~>'

",> / _{_ ; , t ' -}/ " " , ,t;l': / " / / c.~,;. ... ' / "

"

~~/.,

,

, / 5~00100 40 20 10 9 8 7 6 5

BARREl PRES5URE Plo; (ATM)

I I I I I I

ol

1000 500 300 200 140 100 90 80 70 DRIVER PRES5URE ~ (ATM) 0 20 40 60 80 100 120

COMPRESSION RATIO '\

4

60

140 160

Fig, 18 PERFORMANCE OF PISTON SHOCK TUBE USING H2/AIR

3

50

60 _{T4I •} TI

E ₅₀ -_{_ •• _. REAL GAS} PERFECT GAS

•

Z 40

,

111:

"

W

,"

CD Z

,"

i ₃₀

_,

,,' l: ~,. U

"

oe

,

z ... ' 20

..

'

~

..

' u " §!

....

11) N2 ~ 10 :::E A 0 5 10 20 40 60 80 100 200 COMPRESSION RATIO À

FIG. 19 EFFECT OF DRIVER GAS ON PISTON SHOCK TUBE PERFORMANCE

40 40 At:l~/FT7 At :IIJ.S/F7 35 35 Ol Ol 30 :E 30 :E 0 z 2~s/Fl 0 z :I: :I: 25 u 25 H2/ REAL AIR u

'"

'"

:::E :::E _i. OPERATING !IC .... 3 slFl !IC _u

'"

REGION u 0 0 0 :I: 20 :I: 20 0 111 i. _.... 111

...

M

'"

0 0 Q, 0

...

• 15 0 ₁₅ Q, 10~ __ ~ ____ ~ ____ ~ ____ ~ __ ~ o 20 40 60 80 100 lO~ __ ~~ __ ~ ____ ~ ____ ~ __ ~ o 20 1.0 60 80 100

COMPRESSION RATIO À COtoIPRESSION RA T 10 À

FIG. 20 COMPARISON OF PISTON SHOCK TUBE PERFORMANCE WtTH H. AND H2 DRIVER GASES

~

I

H 2 1 REAL AIR

I

I

I

He/REAL AIR

I

~

3sl 35 L _,tl .. \0 ~,\~. ~ ~,

/

111

i30~

_{~ ~OD~}

~ 30

n'

₀ 0

/ ,

~o.nro_ ~ z """"""= JO,OOQoK z ~ 25 IJ .l:»

.

u ct ~ c( ~ ~ :l: ct CID X Cl X 0 U Cl U Cl 0 C") 020 Cl 0 ~ 20 " 0 ~ .;,l _70: Cl) ~ Cl) ~ ~ ct Cl 15 L Cl C")~~ _{/ 15} 11 0 ~ 10 I I I I I I 10 0 20 40 60 80 100 0 20 40 60 80 100 COMPRESSION RATIO ). COMPRESSION RATIO ).

FIG.21 TEMPERATURE ACHIEVED BEHIND I~IDENT FIG.22 TEMPERATURE ACHIEVED BEHIND INCIDENT