On the pressure decay rate in the longshot reservoir

FOR FLUID DYNAMICS

TECHNICAL HOTE 40

ON THE PRESSURE DECAY RATE

IN THE LONGSHOT RESERVOIR

by

Kurt R. ENKENHUS

TECHNICAL NOTE 40

ON THE PRESSUR~ DECAY RATE

IN THE LONG SHOT RESERVOIR

by

TABLE OF CONTENTS SUMMARY 0 0 0 0 0 0 0 0 0 0 0

..

• 0 0 0 •

..

0 iii ACKNOWLEDGEMENTS 0

..

0 0

..

..

• 0 0 0 0 • 0 0 iv NOTATION

..

0 0 0 0

..

0' • 0 •

..

• 0 0 0 0

..

0 v 10 INTRODUCTION 0 0 0

..

0 0 0 0 •

.. ..

0 0

..

0 0 1 2 0 THEORY 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 2 .. 1 Assumptions _•

_..

0 • _~ 0 • 0

..

0 0 • 0 •

•

5 202 Ana1yais 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 30 EXPERIMENTAL RESU~TS 0 0 0 0

..

..

0 0 0 0 0 0 19 4 0 CONCLUSIONS 0 _{0 0 0} 0 0 0 0

..

0 0 0 0 0 0 0 2 5

5 0 RECOMMENDATIONS FOR FUTURE WORK _{0 0- 0 0}

0 0 0 2 7

REFERENCES 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 29

SUMMARY

In the Longshot free-piston shock tunnel, the gas compressed by the piston is trapped in a reservoir at peak

pressure conditions by the closing of a series of check valves o The adiabàtic pressure dec~y of the dense nitrogen test gas due to sonic flow through the nozzle throat is calculated for a perfect gas and a real gaso The real gas behaviour is approxi-mated by an Abel-Noble equation of state whose coefficients are determined from a Mollier chart for dense~ high tempera~ure _, nitrogeno Analytical expressions are obtained for the pressure decay rate of a perfect gas, and for an Abel-Noble gas under the assumption that the covolume is small compared with the total volumeo The analytical solution for the Abel-Noble gas is in remarkably good agreement with the exact numerical

,

solution even at pressures as high as 10,000 atmosphereso

Experimental measurements of the Longshot pressure decay rate were made with nitrogen at an initial pressure and temperature of 35,000 pai and 19000 _K

e respectivelyo One run was conducted with the nozzle throat open i the second with the throat blockedo From the second run it was possible to deter-mine the contribution of heat 10Bses and leaks to the total

decay rateo It is found that the measured pressure ~ecay rate with the throat open is in excellent agreement with the theo-retically predicted value for the adiabatic flow of a real gasi corrected by a small amQunt

_,

to allow for the effects ot heat

loss and leaks o

Charts are presented which permit the preS8ure decay rat, to be approxim~tely predicted over a wide range of operating conditions with nitrogeno These charts do not include a correction for the effect of heat losseB~

ACKNOWLEDGEMENTS

The experimental work in the Longshot free-piston

shock tunnel presented in this report was carried out by

Professor BoEo Richards, with the assistance of JoLo Royen

and F. Vandebroeck. We also wieh to thank Mme Roels for

carrying out the computer programmingo The interest shown in

thi. work by Mr RoO. Dietz, Director of VoKolo, is gratetully

a A b Cl,C2 Ca,Ca. C ,C P v e m M P R t T s v , V Y p t oi 0 H

•

B NOTATION sound speed area

covolume per unit mass of gas

constants defined in Eqso

(50),(53),(54)

and

(57),

respecti~ly

specific heats at constant pressure and volume, res-pectively

internal energy per unit mass

enthalpy per unit mass

functions of y defined in Eqso

(46),(47)

and

(50),

respectively

.I!!

• 2

Mach number preisure

gas constant per unit mass time

temperature

entropy per unit mass flow velocity

reser/Y"oir volume

ratio of specific heats density

charact eristic time

=

V/A*' {yRTOi

Subscripts

reservoir conditions at t

=

0 reservoir conditions at time t throat value (at M

=

1)

value at large molecular separation

10 INTRODUCTION

In the Long.hot free-piston shock tunnel (Ref o 1) the gas compressed by the piston is stored in a reservoir at the peak pressure achievedo A series of check valves first open to admit gas to the re ervoir during the compression stroke~ then close a. the piston reboundso The supply pre8sure and temperature drop during the run as the test gas flows through ~he nozzle

throato Although very high Reynolds numbers can be achieved by trapping the gas at peak pressure~which is in the neighbourhood of 35,000 psi~ it was observed that the r~servoir pressure decay rate was quite rapid with the

~

n diameter nozzle throat

em-ployed in the initial runso The pres8ure fell to one-half the initial value in leas than 10 millisecondso

The objectives ofthe work reported here were (a) to theoretically examine the rate of pressure decay in the reservoiri

taking into account the compreslibility effects in the dense nitrogen test gas which are present at the very high supply

pressurel at which Longshot opera~eSt (b) to compare theoretical I

predictions with experiment, (c) to aSBelS the importance of flow non-idealities, such as heat losses to the reservoir walls~

and leaks, and (d) to prepare a series of charts from which the reservoir pressure decay rate might be approximately predicted over a wide range of Longshot operating conditionso It is con-cluded that a moderate ~reBsure decay rate can be achieved by employing a smaller nozzle throat o

An idealized model of the decay proces. is obtained by assuming that the reservoir is adiabatici and that the dense

gas expands iBentropically through the nozzle throat in a quasi-steady fashiono By quasi-steady it is meant that the charac-teristic time required for the pressure to fall to

1

of its

initial value is very much longer than the time required for a sonic disturbance to traverse the reservoir. Under these as-sumptions, the flow variables at the nozzle throat are related to the corresponding values in the reservoir by the steady, isentropic flow equations for a dense gas.

In practice, the measured pressure decay rate will be somewhat greater than the value calculated using this idealized model due to heat losses to the reservoir walls, and due to a

small amount of leakage through the closed check valves~ However,

it will be seen that when an experimentally-determined correc-tion tor the latter effect. is applied to the idealized solucorrec-tion, the predicted pressure decay rate agrees well with experimental

measurementso

A

numeri cal solution of the pressure decay problem for

dense air has been obtained by Pinkus (R~f. 2) using an equation

of state due to Hirschfelder, et al. (RefQ

3).

This equation of

state includes not only compressibility effects per se, but also the effect of vibrational excitation on real air propertieso Pinkuso predictions cannot be applied with confidence to

Longshot measUrement~ because the test gas is nitrogen rather

than airo Furthermorei we have found that the equation of state

of Refo

3

is not accurate at pressures above a few thousand

atmospheres and temperatures of 20000K or abo.eo

In this report is presented an ana~ytical sol ut ion

for an Abel-Noble gas with constant specific heats, which has the equation of state

1

p(- -

_p b)

=

RT and the isentrope

1

P Y

(1 _

_p b)

=

constant (2)

This gas model is range of flow conditions if by fitting the equations to

surprisingly accurate over a limited _, the'value~ ,of band y are obtained the appropriate region of a Mollier charto The Abel-Noble gas model has been found to be

satis-factory for predicting the perfor~ance , of high speed guns (Ref o 4, po 138-149)0 In particu1ar, Seigèl «Refso

5, 6)

has shown that isentropic datafor dense gases may ~e fitted by the equation

8-2

"'T

p

(1 -

f)

=

K

p

where

B,

f and K are functions of the entropy, and hence are constants during an isentropic expansiono Eqo (3) is the same as Eqo (2)i where 8 • 2y/(y-l)i f

=

band the constant is Ko Although Eqo (3) was applied by Seigel only up to pressures of 6000 atm and tempera~ures below 423°K, he suggests that the

same equation should be satisfactory at much higher temperatureso

For examp1e. Bjork (Refo ~) was able to fit hydrogen data up

to pressures of 100iOOO psi and temperatures of l2,OOOoK by

the more restrictive equation 1

n

~ = Kg (4)

p

where n and KI are functions of entropy on1yo

In summarYi the use of a simple equation of state is particu1arly attractive for the reservoir pressure decay

prob1em because the pressure and temperature vary on1y by a

factor of two in going from stagnation to sonic conditions, so

that a 10ca1 fit to the more comp1ete representation given by

a M011ier chart may be expected to be rather accurate. The use

of a simp1e gas mode1 a1so provides the pOBsibi1ity of obtaining an ana1ytica1 so1ution to the prob1emo

20 THEORY

20 1 Assumptions

The following assumptions are made in the idealized apalysis~

(à) There is no heat loss from the gas to the reservoir walls, (b) The expansion to the nozzle throat is isentropic and

quasi-steady,

(c) The volume of the reservoir is large compared to that of the contract ion section leading ~o the nozzle throat,so that we may make the approximation that the entire mass of gas which is trapped in the reservoir volume V is at a uniform con~ition initially and at later times,

(d) The gas is an Abel-Noble gas with constant specific heats~ (Comparisons will also be made with perfect gas behaviour)~

202 Analysis

A perfect gas analysis will be carried out first~ since results are readily obtained in closed forme

The notation used is shown in Fig o 10 Let the sub-scripts

(0

)

,

(

K

)

denot e conditions in the reservoir and at the nozzle throat, respectively~ at time to The initial reser-voir conditions at t = 0 are denoted by the subscript (Oi)e Then, the continuity of mass flow equation is

where V is the reservoir volume, and

A.

is the nozzle throat area. The variation of reservoir gas density with time may be written where and t

-

T

.

-v

ao

_1.

.

A

_•

is a characteristic decay timeo

(6)

(T)

( 8 )

From the steady flow isentropic relations for a perfect gas, namely

t~ere is obtained. setting M

=

1 1

y:ï

2

=

(y;r)

a 11 - I : aO 1/ 2 2 (Y+ï)

(10)

The sound speed in the reservoir is re1ated to the

initia1 va1ue, according to Eqo

(9)0

by

- =

ao·

_l. (12)

wherei for a perfect gas0

that

a o. • {yR T 00

l. l.

SUbstituting Eqso (10) to (12) into Eqo

(6)0

we find

.·Po [ 1 2

2{~:lj

-

=

1 +

l.::.:.!:. ( - )

PO' _l. 2 y+1

~

-2 1'-1 t

-

T

(14)

Fr01l1 Eqo (9) il the corresponding pressure decay is

PO

--

Po 0

l.

where, 'by Eqs' (8") and (13)e

V

T = ~~;;=

A {yRTO 0

K l.

20202 Abe1-Nob1e gas model

---(16)

completely determined by the equation ot state and the assumed

properties at large molecular separation (Refo

8)0

The internal

energy per unit mass is

The enthalpy is

The entropy is

s

=

J

p (pR - (ll)

)~

-R1npRT + s

aT P p2 00

o

The specitic heat at constant volume is

o

The specitic heat at constant pressure is

(ap/3T)2

c

=

c

+

. . ! . e

p v p2 (ap/ap)T

(18)

(20) (21 )

In these equations. eoo~ SIlO and C _{- v} are the internal

00

energy. entropy at unit pressure. and specific heat at constant

volume. respectivelY0 per unit mass at large molecular

separa-tiono The value ot s ₀₀ is

fT (Cv +R) dT +R1n Po + SO _{(22 )} s

=

T

00 ₀ 00 To

where

So

is the entropy at some stand~~d conditions (po~To)~

. The integrals are to be evaluated at a constant temperature Tc The values o~ the derivatives which appear may be ~ound ~rom the equation o~ stateo

The internal energy

j

at large molecular separation T

e

_..,

=

_J

C dT + eO ₍₂₃₎

Vo

0

0

where eg is the zero point energy~ which may be neglected in the present analysiso FOT an Abel-Noble gas~ which has the equation o~ st ate (Eqo (l»

pc

l ..,

_p b) • RT

we thus obtain ~rom Eqs (17) to (2l)~

e

_•

_e. h

_•

e

_..,

+.E.

p 8 = -R tn p + s

..,

C _v

•

Cv> ( 00 C

•

C + R P v

_•

(24)

I~ we now aS8ume that the gas has a constant value

o~ the:Bpeci~ic heat Cv.., at large molecular separation(io~Oi

under per~ect gas conditions), th en by Eqs (22) and (23), 8 _.., •

Cc

_v,''' + R) tn T +Rtn Po + 88

e

_..,

_{• C T}

v •

(26)

and the isentrope is

(C +R) 1n T - R 1n p • constant

vOl>

With the aid of the equation of state, the isentrope can be written in the form

(28)

We can then obtain the sound speed from the

defini-tion a 2 • (ap/ap)s as

a 2 •

.ll?

(-:-~)

P 1-DP (29)

The entha1py may be expressed as

h •

~.Eo

(1 -

~)

y-l p y

(30)

In Eqs

(28-30)

y • Cv Cv +R GO C ~ C is the (constant) Ol> v

ratio of specific heats of the gaso

Note that a more accurate gas model would have been

obtained if we had assumed Cv depended on temperature due to

CD

vibrationa1 excitationo In a diatomic gas, for examp1e,

2 8 /T 5 R (8 v/T) e v

c

_v CD =

"2

R

+

--e-/'J":!T---2-(e Y - 1)

(31)

wpere

e

_y is the characteristic vibrational temperatureo In our

~(corres-ponding to zero vibrationa1 excitation). but a va1ue obtained

by titting Eqo (28) loca11y to a Mollier chart. correspq~ding

to some average. constant. va1ue ot _{vibrationa1 excitation o}

The va1ues ot band y were obtained trom the latest

AEDC Mollier chart tor dense nitrogen (Ret o

9)0

The cO~olume

was calculated direct1y trom the equation of state according to the relation

RT

b •

!

-P P

using values of P. pand T read trom the chart o It is plotted

as a tunction of pressure tor T

=

2500 0K in Fig o 20 The

varia-tion of b with tempèrature was found to be smal1 enough 80

that a sing~e curve is adequate over the range of temperatures

of interest. trom 10000K to 3000 oKo

The value of y was found fr om the thermodynamic

pro-perties at two neighbouring points along an isentrope on the

Mollier chartso From Eqs (1) and (28). it can be shown that

y •

1

-1

R.n{T2/T l)

R.n{P2/Pl)

The value ot y obtained from the Mollier chart for

pressur~s trom 1,000 to 15.000 atmospheres and temperatures

from 10000K to 3000 0K was 103 t 0020 Since it is ditticult t~

find the ratio of specific heats accurate1Yi the value ot 103

202g3 ~r~s~u~e_d~c~y_r~t~ !o~ ih~ !b~l=N~ble_g~s

Numerical solution

---During the expansion of the gas from the reservoir to the throat. the total enthalpy

hO = h +

~

v2

is conserved. and at t, he nozzle throat v2 u a2 g

Inser~ion

ot

H, ' H

Eqs (29) and (30) into Eqo (34) then yiel,_I ds

v , p. bp yp. 1 Po bp 0

~ _{y-l PH}

(1 -

--!)

_y + - - - ( _{2p l-bp)} •

-I-

_y-l

--(1- ---)

_{Po y} ~35) _"

• H

At the nozzle throat. Eqo (28) becomes

A simul taBeous sol ution 0""f Eqs (35) and (.36) then gives the throat density and pressure

CP

iP ) in terms of the

K K

supply v~lues (P00PO)O The pressure may be eliminated from

( 5) • • • y-l P H . • ,

Eqo 3 by mult1ply1ng 1t by - - andsubst1tut1ng for

y p.

PH/PO in the repulting expression from Eqo (36). When the equa~ion thus obtained is solved tor th& density ratio, we obtain

1

(Y~l)

y:r

1

Y=ï

This is still an implicit equation in p which, in

•

the case of the numerical solution, was ,solved by an iterative

procedure using Newton 9 s methodo

The sound speed at the throat may be obtained by

inserting Eqo

(29)

into Eqo

(35):

2 a

•

Y=ï

which in turn yields

a 11 - = ao 2 a +

-!. •

2 2bp 1-

---!

+ y

From Eqs

(29)

and (28) we find

2

aO bpo

--- (1-

_y-l ---)(l-bpo) _y

1/2

(38)

Eqo

(29),

together with the equation of state, Eqo

(1),

show that

{yRTO·

1

(40)

Eqo

(6)

may now be numerically integr~d to obt~in

the d~nsity in the reservoir as a function of timeo The va~ue

P a aO

of g(-po) • (-!)(...!!.)(-)po is derived using Eqs (~7). (38)

Po ao aOi

(Eq. (28»

Po

(41 )

- =

202.4 Pressure decay rate for the Abel-Noble Gas

---!n.!lz.ti·c.!l_$~l.!!tio!!.

Since the termsin bp appearing in the expressions

p a 11

fo~

-!o

and

-!

(Eqs

(37)

and

(38),

respectively) nepresent

cor-P

ao

.

'

rections to the perfect gas law, it is a reasonable approxi-mation to use Eq.

(16)

and set

1

Y-ï

2

bp. Cl (;t;+l) bpo (42)

Eqs

(37)

and

(38),

together with Eqo (39), then yield pand a as functions of Po alone~' At moderate densi tie,s, for

11 11

which the assumption bp « 1 is valid, an analytical sol ut ion is now obtainable by linearizmg the equationso By perform~ng binomial expansions of Eqs

(

37), (38)

and

(39).

and retaining only first order terms, we get

a 11 - = ao - = ao· ₁ 1 y-l 2

(Y+I)

(1 + KIbpo)

(44)

where and 1

y:T

Y

2_1_(~) (y2+y_2) y+1 ' Kl

=

---y (,\,-1) 1

2°

y+1-(-) K2 • _~_:!I:-+_l'--2y m

=

y+1 2 (46) (47) (48)

The variation of density with time is gotten by

substituting Eqs (43-48) into EqQ (6) and performing the

indi-cated iDtegrationo The resu1t is

Po

- = PO > 1 1+{m 2-1 )Ksbp -m 0' 1, +(m-1')CI

.1

T -1

m:r

where T is given by Eqo (16) (ioeo, the equation has been written

in the form which retains the perfect gas value of T) and

(5Q)

Since PO appears on the right hand side of Eqo (49)

by the perfect gas formula, Eqo

(14)0

The explicit dependence

o~ the reservoir density o~. time tor an Abel-Noble gas is t~en

P()

PO:" •

_1. [ x+l

~

1 2 2(X- l )

~

. 1+ X-l K3 b p 0 + .I.:.:.(_) (l-bp 00 )

ol

3-x

i 2 X+l 1. 't

The corresponding pressure is obtainable from Eqo

(41)0

2.20

5

- -

Presentation of theoretical results

-

-

-

-

--

-

",

-

-

--

-

-

- -

-(51)

A comparison ot the n~merical and analytical solutions

tor an Abel-Nob~e gas is shown in Figo

30

The test gas is

nit'rogen at an initial temperature TO

i • 2500 oKo y was assumed

to be

103,

and the value ot the covolUme b given by Fig~ 2

was usedo Tbe pressure ratio PO/POi is plótted as a tunction ot

the dimensionless time t/'t tor various initial reservoir

pres-sures up to 10,000 atm o The perte ct gas sOlution, given by Eqo

(15)

is the curve tor pOi

=

00 The perfect gas solution bolds

tor any initial reservoir temperature

toi,

since the int1uence

ot TO

i is incorporated in the characteristic time, 'to It is

seen that tbe analytical solution for an Abel-Noble gas, given

by Eqs

(51)

and

(41),

agrees rather well with the exact

numeri-cal solution even at POi

=

10,000 atmo This agreement was not

anticipated at such a high pressure, because the assumption that

bp « 1 is th en no longer valido It is the result of a series

ot~ compensating errors o The pressure decay becomes more rapid as the initial pressure is raised.

Figs 4a,' 4b and 4e present the reservoir pressure

ratio po/poi as a function of time , in milliseconds for

initial reservoir gas temperatures of 2000 o

K.

25000

_K

_{and 3000}0

_K

respectivelyo The calculations. made by the eKact numerical

method, are for a reservoir volume of 1904 in 3 and a throat

diameter of 7/32 ino - the values currently being used in

Longshoto To use these curves tor other values of reservoi~

volume an d throat size i the time scale ne,ed only be multiplied

by the "running time multiplying factor" shown in Figo

50

This

factor, which is derived from Eqo (16), is

where V is

F

=

in in3 and d is in incheso

•

It Vill be noted from Fig o

4

that the decay rate is

,

not very strongly influe~ced by th~ initia~ gas temperatureo

The pressure decays more rapidly for a perfect gas (po., + 0)

;l-as the temperature is raised; however, at 15,000 atm, the trend is actually reversed o

The dimerisio~less initial decay rate.

is plotted as a function of the initial pressure pOi in Fig o 60

It is seen that the value of this decay rate parameter is

inde-pendent of the in~tial ~emperature TO_i for a perfect gaso For a

rml gas,the dimensionless initial decay rate increases almost

linearly vith pressure, and decreases with increasing initial

The initial pressure decay rate, in percent per

millisecond, is plotted vs initial pressure pOi in Figo

7.

tor

Longshot cónditions - V • 1904 in3~ d

_•

• 7/32 ino Ag~in. curves •

are ,ehown tor initial reservoir tem~eratures ot 20~OoK, 25000_K and 3000oKo In order to obtain the corresponding initial decay rate tor any other values ot V and d., i t is only necessary to divide by the tactor' given in Fig.

5.

The curves given in Figs

4

and

7

require turther correction to allow tor the ettect ot heat losses and leaks in practical applications.

In'order to check the theory, two experimental tests vere made in the Longshot free-piston shock tunne~ - one, a normal run. the second. with the thTDat blockedo'

The oscilloscope trace tJf reservoir pressure vs time

m

the normal run is reproduced in Figo

80

A calorimeter-type heat

transfer gauge mounted in the test section indicàted that the

.

~otal temperature in the reservoir at the start óf the run was To· • 1900 o K .. The nominal test section Mach number v~s 200 In

1 ,

order to determine the value of pOi' the trace of,. Figo 8 vas

replotted on semi-log paper'as shovn in Figo

90

Fig o

9

shows

that pOi

=

35.000 psio

Thel pressure trace obtained with the thrQat blocked is

I

shown in Figo 100 Although i t was attempted to dupli~ate the

conditions of the previous run. the ini'tial pressne vas lover (PO i m 30,000 psi) because the diaphragm which releases the

piston broke at a slightly lower driver pressureo It is ditfi-cult to determine the pressure during the first few milliseconds because of ringing oftbegaugeo When the pe1rcent pressure decay is plotted 'vs time as shown in Fig o llt there is evidence that the decay rate is more rapid during the first ~ milliseconds because the check valves are still in the process

ot

closingo Af ter

4

milliseconds. the decay ratet which may be attributed to the

combined eftects ~t heat loss to the reservoir walls and leaks in the check valv~s. becomes constant at a value ot 00 42% per millisecond o

In order to obtain furthe~ information on the pheno-mena occuring in the reservoir, an ~dditional oscilioscope pre~­ sure trace was taken on a time sweep of 1 cm/seco THis trace is

reproduced in Fig o 120 For the first 150 milliseconds or so the decay is an order of magnitude more rapid than'for ~he rest of the 6 second periodo Af ter 5 seconds th~ pressure in the rese.r-voir equalizes with that of the large volume of gas trapped in the barrel o The rapid decay ceases when the reservoir pressure has fallen trom 30,000 to l8~000 psi, ioeo, to 00

6

of the

initial value.

If we accept for the moment the hypot~eBiB that the leak rate is small (which we shall prove presently) then the density in the reservoir varies slowly with 'time, and from the equation of state

(Eqo

(1», the temperature in the reservoir at the end of the 150 millisecond period would be (for p •

constant)

,

To • 9 ( _po )

6

0

4

0 To- • 00 x 1900 K • 11 0 K

Po.

1 3.

What kind of heat transfer mechanism could account for a small rate of heat loss once the temperature has fallen to this value ? The answer is radiationo If it is assumed that

. 4

the heat loss var1es as T o ' then at 150 me the heat transfer rate would have fallen to 0064 = 0 013 of the initial valueo

,The corresponding rate of reservoir pressure decay would be

(assuming the initial decay rate to be entirely due to heat

loss) 00 42 x 013

=

0005% per millisecondo As we shall see

presently, the pressure decay rate due to leaks and convection or conduc~ion is

0

0

04%

per millisecondo The natp~e of the curve

therefore changes as soon as the twodecay mechanisms have rates which are comparable in valueo

At some time af ter 150 milliseconds, the decay rate due to leaks and convection or conduction will become completely

dominant. For example, once the reservoir temperature has fal~en

to

~

the initial value, the pressure decay rate due to radiative

heat loss would be

~

x

(OQ42

~ 0

0

04)

=

0.0047%

per millisecond

3

- an order of magnitude less than the decay due to leaks~ etco

Now let us examine the pressure decay af ter

150

ms in

detailo Once the radiative heat loss rate ~ecomes small, it can

be assumed that the reservoir temperature continues to fall due to conduction or free convectiono

It is assumed that the governing, equation for heat

transfer is

where C2 is a constant and T _wa 11« _' Toe

For a viscous leak, the mass flux is dp 0

- V --- D C3(PO-P )

dt B

where C3 is another tonstant, and PB is the barrel pressureo

By logarithmic differentiation of Eqo

(1),

there

results

dPO Po dp 0 Po dTo

- =

dt po(l-bpo)

dt' +

To

d't"

(55

)

dPO

the differential equation

d-t

It will now be assumed that the variation of the _.

_.

quant1ty in the square brackets is s.mall, so that one may write

where C4 is a constant, equal to some ave rage value of the

bracketed terms on the right hand side of Eqo

(56)0

The solution of Eqo

(57)

is

-C4(t-t')

(58)

= e

w

where the reservoir pressure is Po at some time tWo

Eqo

(58)

indieates that a plot of (~O-PB) vs t on semi-log paper should yield a straight linea That this is so is

shown in Figo 130 From the figure it is determined that the,

rate of fall of reservoir pressure due to ûhe combined effects

of leaks and conductiQ~ or convection heat losses is 0004% per

millisecond, or about "one-tenth of the initial rate due to

radiation heat losses and leakso We note from Figo 13 that

the decay rate increases as PO-PB + 00 This phenomenon can be

attributed -to the fact that the term

C2/~- ~)in

Eqo

(56)

becomes lar.ge as Po + PB = 2600 psio We see that this term is

beginning to have an ef'f'ect w~en PO-PB • 4000 psi" i. e

o.

when

the va~ue of' the term is 1 065 C20 Ät P ,.PB710,OOO psî. th~ ~eat

lOBS tèrm is 1026 C20 We can speculáte that _{. .} C4 ~ie äpproximately

,

constant over a, considerable range

- C3 RT

o

,~

'

.

"

term 2 decreases v1th_t1me

V(l-bpo)

-time'o .

of preesure, because the leak

C2 '

-vhile increases with

PB

1

-Po

It should be mentioned that the "viggle" in the

pressure' trace in Figo 12 at a time of about 002 seconde ha8

not been explaine~, but i t may be due to gauge drif't o Obviously,

the pressure should have decreased monatonicallyo The f'igure

was plotted so that the final pressure for t > 5 seconds

agreed with the final barrel presBure measured in other runso

We are nov at last in a position to compare the

measured pres8ure decay 'rate during a normal run vith

theore-tical pred~ctionso This is shown in Figo 140 The circlee

re-present the measured decaJ ~ate ,as obtained from Figo

8

(or

Figo

9)0

The solid cufve gives the adiabatic theory (Eqs (51)

and 41»0 The dashed curve' gives the decay rate~ corrected for

the lo~~ of' 0042% per millisecond in reservoir pressure due to heat losses and leaks o It is seen that the corrected data are in excellent agreement with theoretical predictionso

Finally, i t may be mentioned that the initial rate of' temperature decay may be inferred from the measurements if the leak rate is neglectedo Eq; (55) then yields, f'or p • constant,

1 dTO

= - -

_{TO dt}

ioeo. that the percentage fall in reservoir temperature per

unit time is the same al that of the pressure. or 0042% per

milliseconde At TO- • 1900 o K. this means the initial temperature

1

decay rate is 80000

40

CONCLUSIONS

An approximat~ a~alytical solution for the adi abatic

preseure decay of a dense (Abel-Noble) gas in a reservoir with a sonie orifiee has been obtained whieh is in good agreement

\

wit~ the exact n~erical solutionQ

The measured pressure decay rate in the Longshot free-piston shock tunnel reservoir with nitrogen at 35000 psi

and 19000_{K is in excellent ag~eement with the theory, providing}

allowance ia made for the small additional pressure decay due to heat losses to the reservoir walls and leaks in the check valveso

The initial pressure decay rate due to radiation

heat losses under the above test "Conditio.ns was about 004%

per millisecondo The corresponding res.rvoir temperature decay

rate is of the order of 80000_{K per second o Af ter ~50}

milli-seconde, convection or conduction becomes the dominant heat los. mechanismo The pressure decay rate attribuable to these forms of heat transfer plus leaks in the check valves was only 0 004% per millisecondo It was not possible to establish the

leak rate itselfe but it is obviously small, say, of the order

of OoO~%· per mil lisecondo

The pressure decay rate becomes higher as the

,\

reservoir pres.ure i~ increased~ according to the theory, due

to compressibility effectso Curves ar~ presented which permit

the pressure decay rate to be approximately predicted over A

50

RECOMMENDATIONS FOR FUTURE WORK

The rapid reservoir pressure decay rate presently

"

.

obeerved in Longlhot will be eliminated by employing a nozzle with ·a throat diameter of ab.out half the present valueo The

decay rate will then be reduced by a factor of four, to about 2% per millisecond o This will also require a reduction in

noz~le exi~ aize from 2 ft to 1 foot o The latter size is still

quite adequa.:te for test purposes o

Further tests are required to determine the decay of temperature with time i~ the relervoiro Since radiation appearl to be initially the dominant heat l~ss mechanism_i the temperature decay rate may be expected to increase iapidly with operating temperature~ and~ indeed. may be anticipated to plaee a limit on th-_{e gas temperatures which can be achcieved o}

It is planned to use thermocoupleB~ and an ultrasonic device for measuring sound speed in the reservoi~ in order to obtain further information on the temp~rature~

REFERENCES

1. BOISON, JoC.: Longshot I HyperBonic Free-Piston Tunne10 Repub1ic Aviation Corpo, Farmingda1e, Lolo, Rept o NO RAC 1884A, Deco 1963.

2. PINKUS, 0.: Thermo~ynamic Properties ~nd Dynamics of Rea1

Air trom Subcritica1 Temperatures to 150qOKo

Repub1ic Aviation Corpo, Farmingda1e, ·L.I., Rept o

~o RAC 2716, 31 Dec. 19640

3. HIRSCHFELDER, JoOO j BUHLER, RoJo, McGEE, H.A. Jr, SUTTON,JoR e :

Genera1ized Equation of State. _{tor Gases and Liquids o}

Industria1

&

EngineeringChemistry, Vol. 50, pe 375

March 1958.

40 SEIGEL, AoE.: The Theory of High Speed Guns o AGARDograph 91, March 19650

50 SEIGEL, AoEe :. The Rapid Expansion ofCompressed Gases

behind a Pistono

Pho Do Thesisj Univo of Amsterdam, Jan o 19520

60 SEIGEL, AoE.: A Convenient and Accurate Semi-Empirica1

Entropiq Equation tor Use in Interna1 Ba111stics Ca1cu1ationso

NAVORD Rept 2695, US Nava1 Ordnance Laboratory, Si1ver Spring, Mary1and, Feb o 19530

7. BJORK, ReLo : The .Atomic Hydrogen Gun.

8.

ROSSINI, FoDo. Ed.: Thermodynamics and Physics of Mattero

High Speed Aerodynamics and Jet Propulsion. Voll.o I,

Princeton Univo Press,

i9550

9.

BRAHINSKY, HoSo, NORTHCUTT, 00 : AEDC Mollier Diagram tor

Equilibrium Nitrogeno

Fig, 1 SON IC FLOW PR08LEM w

1.0

~ 0 ~

_,

(!) 0.8 ::J ...J I./')

-M ~ lI... .0 0.6 0 41 E :J 0 0.4 > 0 u 0.2

o

2 4 6 8 104 2 pr4?Ssure p (AT M)

0

...

_c

'-GI L-:J 11\ 11\ GI L-~ \...

.-

₀ > \... GI 11\ GI a:: 0.9 numerical solution 0.8

---

_---

analytical solution _ 2Y 0.7

~

Y·l

J

y-1 1+ y-1 (_2_) 2CY-1) _t_ (perfect gas) 2 Y+1 1: 0.6 P.. (AlM) 0.5 OL

0.4

0.3 0,2 o~ ~

____

~

____

~

____

~

______

~

____

~

____

~

____

~

____

~

o

0.2 0.4 0.6 0,8 1,0 1.2 1.4 1.6 Dimensionless time

~

Fig. 3 COMPARISON OF NUMERICAL AND ANALYll CAL SOLUTIONS

a.

0/

rf

...

o 0.9

-

o

L. Cl L.. ~ lil lil CII L.. a. L. o > L.. QI lil CII 0:: 0.8

0.7

0.6

0.5 0,4

0.3

0.2

Fig. 40 RESERVOIR PRESSURE OECAY

To '

=

2000·K I d * = l I N 32 3 V

=

19.4 IN PO' (AlM I 0.1, ____________ ~ ____ ~ ____ ~ ____ ~ ____ ~ ____ ~ ____ ~~ __ ~ ____ ~

o

2 4 6 8 10 12 14 16 18 20 Time t (mi1\iseconds)

a.°la.°-0

-

a '-C1I '-::J VI 0.9 : 0.8

'-a.

~ o > '-C1I VI C1I a:

0.7

0.6 0.5 0.4 0,3 0.2

fig. 4b RESERVOIR PRESSURE DECAY (CONTINUEO)

T_o' :: 25000 K l 7 d

_*

=

- I N 32 • V:19,41N 3 DOi (ATM) 0.11

::J

o

2 4 6 8 10 12 14 16 18 20 Time t (milliseconds)

°10"

a. a. ·0

-

o L-CIJ L-':J \11 111 CIJ L-a. L-o > L-CII 111 CIJ

a:

0.9 Fig. 4c RESERVOrR PRESSURE DECAV (conctuded)

O.S

I

_{\ \\\ \.}

"

"

•

T_o-= 3000 K I 0.7L

\ \\\

'"

"

"

d*=3~

IN V= 19.4 IN 3

I

_\

_\\\.

"

'"

_~

"

0.6

I

_\

_\\"

" "'"

_~

P Oi (AlM)

-

-

-

-

-0.5 0.4 0.3 0.2 O~LI ____ ~ ____ ~ ____ ~ ______ L-____ L-____ ~ ____ ~ ____ -L~==~

____

-J

o

2 4 6 8 10 12 14 16 18 20 Time t (miltiseconds)

...

_u C

....

Cl c: ₂₀

.-

>--

a.

.-..

--

::J E

-CII

z

E 10

.-

-Ol ~ c: 41

.-

_c:

-

C1I c: _E ::J ₀ a:: '0

-

0 0 ~ .t:. ~ 1~~ __ ~ __ ~ ________ ~~ ______ ~ __ ~~ ______ ~ Conditions 1 n fig. 4 20 30 40 50 60 70 Reservoir volume V (I N3) Fig. 5 CHART FOR CALCULATING RUNNING TIME FOR

cfÎci

_

+

'tJ 1"0 IV Ö

_...

>. a u Cl> "'0 Cl> ... :J lil lil Cl>

...

Q. '6

..

'ë o 30 u IV lil ~ 25 ... Cl> Cl C 20 IV U Q; Q. Cl. ë _... 15 >0 0 U CII "0 10 ~ :J lil lil IV ... Cl Ö 5

..

c 0 5000 10000 15000 Pressure P Oi (AT M)

Fig, 6 OIMENSIONLESS INITIAL OECAV RATE IN THE RESERVOIR

2

Fig. 7

4

d. = 3~ IN,

V = 19.4 IN3

For decay rate with other values of d* and V, divide by factor

given Jn Fig.5

6 8 10 12 14

Pressure POL (ATMx10-3) INITIAL PRESSURE OECAV RATE

\i'i e:. ~o Ol ... -;, lil lil GI ... Q. ... '0 >

...

Ol lil Ol a:: \ii e:. Q.0 ~ lil lil ~ Q. ... 'g ... Ol lil Ol a:: 50 40 30 20 10 8 6 4 2 1 40 30 20 10 o Run Ne .3. 7 d ... : TI" IN V: 19.4 I N3 TOL: 1900' K -10 o 10 20 30 40 TIme t (mill1seconds)

Fig, 8 REPRODUCTION OF OSCILLOSCOPE TRACE OF LONGSHOT RESERVOIR

PRESSURE VS TIME.

-

POi = 35,000 pst

" ,

0 5 10 15 20 25

Tim9 t (m ill i seconds) Fig,9 SEMI-LOG PLOT OF PRESSURE FROM Fig. 8

>. 0 u Ol "0 Ol

...

:J lil lil Ol

...

0.

-

c CII u

...

CII n.

e

::J lil lil Ol

..

Q.

..

·ö >

..

CII lil CII Ir 15 10 5

o

40 30 20 10 o~~~~

____

~

____

~

______

~

____

~

____

~ -10

o

Fig. 10 2 10 20 30 40 50 Time t (milliseconds)

REPRODUCTION OF 05CILL05COPE TRACE OF LONG5HOT RE5ERVOIR PRE55URE V5 TIME WITH THROAT CL05ED

Oecay due to heat 1055.

tea ks = 0.42 "10 I M 5

4 6 8 10 12 14

Time t (milliseconds)

Fig. 11 OETERMINATION OF PRES5URE DECAY RATE WITH THROAT CL05ED (DATA FROM Fig.10)

0.0 40 Go

..

:l 11\ 11\ Go Co 30

..

'ö >

..

~ 20 Go 0:: 10 Radiation + leaksllg : 0,42 ·1./M5

~uction

or convection + leaksAE.: _{, p .} 004'1, IM" _., o 6 ~ 40 ;n a. 30 o.aJ I 0.° 20 11> L. :l 11\ 11\ 11> L. 0. ä: 10 L. L. ₈ 0 .0 I 6 L. 'ö > L. C) 11\ ₄ C) 0:: 2 Tl

m.

(s IJconds) Fig, 12 REPRODUCTION OF DSCILL05COPE TRACE

OF RE5ERVOIR PRE55URE OECAV WITH THROAT BLOCKED OVER EXTENDED

o TIME PERIOD, radiation • leaks conduction or convection • '"aks '= 0,04 '1.1 ms 2 3

,

ca.

,

\

_\ 4 ~

\

\ \ \ 5 6 T I me (seconds)

Fig 13 DETERMINATION OF PRES5URE DECAV RATE DUE TO LEAKS

°

..

0 '-CII ~ :J lil 11' CII

L-a.

0,9

,

0.8 0,7

0.6

POi =. 35JOOO psi

"

T

_{o; ::}

1900 K

0.5

V =. 19.1 in 3

d

_*=-/n

7.

16

o longshot experiment ~ (iaCt

"-..2os

1.,

',~~ ~Jo

"

_"

'

...

"

" ,

0.4~.

____

~

__

~~

__

- * ____ ~ ____ ~ __ ~ ____ ~ ____ ~ __ ~~ __ . .

o

2

4

6

8

10

Fig, 14

Time milliseconds

COMPARISON OF MEASURED PRESSURE DECAY IN