Mixing of thermal molecular jets produced from knudsen effusion

~

I

J. I

..

von

KARMAN INSTITUTE

FOR FLUID DYNAMICS

TECR !SC

IE

71-2

._<)

MIXING OF THERMAL MOLECULAR JETS PRODUCED FROM KNUDSEN EFFUSION

by T. YTREHUS. J • J. SMOLDEREN. J. F. WENDT RHODE-SAINT-GENESE, BELGIUM JUNE 1971

von KARMAN INSTITUTE FOR FLUID DYNAMICS

PREPRINT 71-2

MIXING OF THERMAL MOLECULAR JETS

PRODUCED FROM KNUDSEN EFFUSION

T. YTREHUS. J.J. SMOLDEREN.

J. F. WENDT

Paper presented at "3ème Symposium International sur les Jets Molêculaires" 29 June - 2 July 1971. Cannes

SUMMARY

The flow resulting from a mixture of a large number of thermal aolecular jets in a region of finite pressure is studied. The theoretical problem is treated by a moment method based on a particular form of the molecular velocity distribution function.

The anal~sis indicates that the resulting one-dimensional flow velocity can attain a slightly supersonic value. The velocity depends on only one parameter (the pressure-porosity parameter) which is naturally introduced into the calculation.

In the non-equilibrium region near the plane containing the orifices.the gas experiences a one-dimensional expansion. accompanied by a slight reduction in temperature.

The theoretical results concerning this region are con-firmed by measuremenuof the molecular flux near the orifice plane.

SOMMAIRE

L'écoulement résultant du mélange d'un grand nombre de jets moléculaires thermiques dansune enceinte à pression finie est étudié. Le probl~me théo~ique est traité par une méthode de moments. basée sur une forme particulière de la

fonction de distribution des vitesses moléculaires.

L'analyse indique que la vitesse de l'écoulement unidimensionnel résultant peut atteindre une valeur légère-ment supersonique. Cette vitesse dépend d'un seul paramètre

(param~tre de pression-porosité) qui s'introduit naturelle-ment dans les calculs.

Dans la région de non-équilibre. voisine du plan contenant les orifices. le gaz subit une détente unidimen-sionnelle accompagnée d'une lég~re diminution de température.

Les résultats theoriques concernant cette région Bont étayés par des mesures de flux moléculaires vers le plan des orifices.

TABLE OF CONTENTS

NOMENCLATURE

_•

_{• •}

_•

_•

_•

_{• • •}

_•

1

INTRODUCTION _{• •}

_•

_•

_{• • •}

_•

_•

_{• • •} 2

THEORY

_•

_•

_•

_•

_•

_•

_{• •}

_•

_•

_•

_•

_•

_{• • •} 3

THE FINAL FLOW FIELD _{• •}

_•

_•

_{• • •}

_•

_•

_•

7

THE KINETIC FLOW FIELD

_•

_{• • •}

_•

_•

_•

9

MEASUREHEN TS _{• •}

_{• •}

_{• • •}

_•

_{• • • •} 11

CONCLUSIONS _{• • •}

_•

_•

_•

_{• •}

_•

_•

_•

13

REFEREN C.ES _•

_{• •}

_•

_•

_{• •}

_{• • •}

14

APPENDIX

_•

_{• •}

_{• •}

_•

_{• • • •}

15

a f m n p q u x z L R S T I 3 - I -NOMENCLATURE

amplitude functions for different modes of the distribution function

molecular distribution function molècular mass

number density pressure

fractional porosity

-bulk ve loc i ty

distance in flow direction

pressure-porosity parameter (eq.

9)

relaxation length (eq. ~4)

gas constant per unit mass

speed ratio ( )

12RT₃ absolute temperature

absolute molecular velocity molecular mean free path

Subscripts

effusion mode downstream mode

- 2 -INTRODUCTION

The Knudsen effusion of gas from one single orifice into a vacuum. is a well understood. nearly trivial, example of molecular flow and is treated in most standard text books on kinetic theory of gases. It is well known that such a flow field will never approach equilibrium. but rather will remain a completely free molecular flow as there are practically no molecular collisions taking place in the gas.

When several molecular jets - all of Knudsen effusion type - are allbwed to interact with each other and with a

downstream atmosphere of finite pres8ure, the resulting flow field is very much different fr om the one above. Severa~ kinds of molecular collisions must then be considered. and the gas will tend to approach a state of local equilibrium. This is particularly true if a purely one dimensional flow process

is considered, in which case the final flow field a160 must be one dimensionale and the final state is discribed by an asymptotically local uniform Maxwellian distribution function. Such a flow field might be of some practical use as it exhibits a high degree of uniformity over considerable spatial exten-sions, at the same time as the density level is very low and velocities are moderate. The commonly encountered difficulty in producing uniform flow fields at extremely low densities using contoured nozzles. might therefore be circumvented by

adapting a flow capability based on the above principle.

In the present study more attention will be paid to the processes occuring in the zone where the molecular jets are mixing together and the interactions between the jets and the final flow are occuring. It will be assumed that the jets are emanating from aporous. thin wall. separating the test section from the stagnation chamber (fig. 1). and that this wall is the only ~olid boundary in the prohlem. By assuming a diffuse reflection for the molecules scattered by the wall. one can guess reasonable expressions for the distribution

- 3

-A moment method can then be used to obtain an approx-imate soln.tion of the Boltzmann equation, by selecting such a representation for the distribution function. In this sen se our theoretical approach is thus reminiscent of the Mott-Smith treatment tor the shock wave structure problem (refs. 1 and 2).

THEORY

The effusion from the orifices in the porous wall

~s assumed to be of free molecular type, which means that the eftusing molecules will have a half range Maxwellian distri-bution in velocity space. Furthermore, we consider a great number of orifices. uniformly distributed on the wall, and introduce therefore the porosity, q; of the wall instead of dealing with isolated jets.

We will assume that the distribution tunction ~s a linear combination of terms representing various molecular groups:

(1)

.

'*

t i ~s the half-range Maxwell~an mode , f₂ represents the molecules scattered back towards the wall, and f3 is the

Maxwellian distribution describing molecules in the downstream equilibrium state. The amplitude functions a.(x) will be

~

determined from suitable moment equations derived trom the Boltzmann equation.

'*

The molecules reflected trom the wall are also incorporated into th~ term since the wall temperature is taken to be equal to the stagnation temperature.

4

-AreasonabIe choice tor f2 consists in taking the

half range of the downstream equilibrium distribution corres-ponding to negative x-component of molecular velocity. For f3 one can then take the positive velocity branch of the same function (Note: In ref. 3 f2 was based upon the wall effusive mode. and f3 was the complete downstream distribution function. Th~ approach was considerably simpIer than the present one. but suffeIed from some serious shortcomings. and will not be dealt with in this paper although some of its results will be referred to for comparison).

The assumed distribution eq. ( I) thus has the following explicit expression

f(x) = adx) fl + &3(x)f3 • T + _{a3(x)f 3} :ot +

with

~

nj exp [-

,2.< 2.'j

x ;t z ( 21TRT

o )

3/2 _2RT

_o

f l

=

<

l

0

f;

n, exp

[-(, _u,)2.,2.<j

_{x ;t z} fi _'3

₌

(21TRT 3 )3/Z 2RT3

1

0

rc

n,

_{exp [-}

(,

x

_u,)2.'2.,~

;t z + ( 21T R T 3 ) 3/2 2RT 3 f3

=

l

₀

where the density nl is given by

t _x > 0

t _x < 0

~x > 0

ç; < 0 !IC

A distribution function of the above form will - when substituted into the Boltzmann equation - account for collisions

5

-between molecules in each of the three modes. and for colli-sions taking place between molecules from one mode and partners from any of the other two modes.

The boundary conditions. which the amplitude func-tions will have to satisfy. are:

x

=

0: x

₌

00: al

=

1 + roF' al

=

°

t t 1 ( 4 ) a3

=

ro _{a 3}

=

+ + a3

=

°

a3

=

1 ~

where ro lS an unknown parameter. and FO is a function of the

final speed ratio given in the Appendix. The simple relation t

which 18 seen to exist between al and a3 at x

=

0. results from

the fact that the flux of molecules impinging upon the wal 1 must be equal to the flux of molecules reemitted from the wall.

Simp Ie considerations indicate that tour moment equations derived from the Boltzmann equation must be used in order to define a solution to the rroblem. i.e •• to give the quantities al(x). aj(x). a;(x). rOt and the final flow condi-tions expressed by n3' T 3 and u3'

The three classical conservation equations for mass. momentum and energy willof course be used but an additional equation will be required. We will use the moment equation for

the quantity

,2

as the fourth equatio~ To evaluate this~ last

x

eqU&.tioD .• a specific molecular interaction potential must be

selected. We will consider the case of Maxwell molecules. i.e •• molecules which repel each other according to an inverse fifth

power force law. because this is the only interaction law ,:

which allows the moment collision terms to be expressed in simple form. In addition. one quantity at downstream infinity say the pressure level - must be known a priori in order tor the problem to be weil posed.

6

-The Boltzmann equation for a one dimensional steady state can be written as

~

II

=

J(ff)

x ax

where J(ff) is the non linear integral collision term (see for instance. refa.

4. 5

and

6).

The conservation equations are then formally written as

(6 )

The collision term does not contribute to these moment equations as is well known from ~inetic theory (refa.

4.5.6).

Using the results for Maxwell molecules given 1n ref.

6.

the moment equation for the non-conserved quantity may be written as

~2

X

where the bulk quantities - like the density n and the element

T of the viscous stress tensor - are subject to standard

xx

definitions (see for instance refs. 5 and 6). The above equa-tions are thus worked out by introducing the distribution function and its ranges in velocity space :from eqs. (2) and (3).

7

-THE FINAL FLOW FIELD

The final downstream state ,of the flow and the quan-tity rOt which characterizes 'the number of mo~ecules impinging upon the wall. are obtained from the conservation equations af ter one trivial integration in physieal spaee and three non-trivial integrations in the veloeity spaee have been performed. Using the downstream boundary conditions on the a's (eq.

4).

and introdueing the effusive number. momentum. and energy fluxes as sealing quantities. the resulting conservation equations

are written as (8a) + 1

Gt

7 ₁

Gi

1 _{(4S 2 +2)} al + a3 _z + a3

.

=

Z z (8b) + 1 _H+ T _{l H}T 2lT _S2(S2+

₁₎

al + a3 + a3

-

.

₌

---z2 z2 z2 2 (8e)

Here the quantities F' s • G , s • and H a r e eomplieated ' s funetions of the downstream

given ~n the Appendix. z is

speed ratio. the pressure

S

=

u3/12RT3~ and are -porosity parameter given by

Po

z

=

q

-

P

P3

and is the only adjustable parameter entering the analysis. When the eonditions on the a's at x

=

0 are introdueed into eqs.

(8).

the following relations are readily obtained

Z2(Ft(4S2+2) + Gt) _ Z(Ht+FT21TS 2 (S2+

~))

+ Ht (4s2 +2) _ Gt2nS 2 (S2+ ~)

=

0 4S2+2-z T t z _{F +G l}

(10 )

( 11)

A solution to eq.

(10)

will yield the final speed ratio as a

8

-and from the definitions of S -and z-and the equation of mass conservation, the remaining downstream quantities are easily obtained.

On fig. 3 are shown the final densities and

tempera-tures which will result under different flow conditions. We

note that the present theory prediets the correct limiting

T _{3 . •}

value

T;

= I, as the speed rat~o goes to zero, wh~le the

theory reported in ref. 3 gave 0.80 for this limit.

It will be noted that this analysis prediets a mono-tonie increase in the speed ratio as the pressure-porosity

parameter is increased. A close~ examination of the solution,

however, shows that the Boltzmann H-theorem is violated for

values of z above 5.15, corresponding to va~ues of S above

1.19, and the state of the final flow ~s therefore restricted

by

o

< S < 1.19

These considerations therefore seem to indicate that the as-sumptions expressed by eqs. (2) and (3) for f have only limited validity and certainly do not apply at high speed ratios.

The above results are true for any molecular inter-action law, since they are derived using only the conservation equations.

9

-THE KINETIC FLOW FIELD

To obtain information on the details in the flow close to the perforated wall, where kinetic effects are predo-min&nt, we must solve fcr the x-dependence of the amplitude

functions, a's, to obtain the resulting distribution function according to eq. (2). This requires the differential eq. (=l)

to be expressed in terms of the a's and solved together with the previous eqs.

(W.

Although eqs.(W are three equations in

+ t

the three unknown al, a3 and a3, that system alone does not define a solution because of the compatibility cnndition. eq. ~O~ which means that the system is indeterminate. This result is also obvious from simple physical reasoning.

It ~s thus convenient first to express eq.(7) in terms of the three amplitude functions. perform the indicated

integrations in velocity space. and then use either two of the three eqs. (8) to eliminate two of the a' s in favour of

. . +

the tÀ~rd, say a3. We thus get the following differential equation for a;:

- M

crx-

=

121.1

.!. ...

(a3-I ) (a3+ r 2) (12)

t

da3 1Twlw2

where the parameters M. WI. W2. and r2 all are rather compli-cated. positive functions of the speed ratio. S. (the details of which will be given in a VKI paper in progress). With the boundary conditions taken into account. the complete solution for the amplitude functions - and thereby_ the distribution function - is now easily obtained. and is written as follows

r2 + ra

-

r2 I-ra e

(

) -xl

L a3 =

(

) -xl

L r2 +ra + l-ro e (13a) a3

-

ra + a

₌

3 1

-

ra (13 b ) al = ( Ft

-

l+FT)(a_I-ra f ₃

-

1 ) (13c)

1

L

=

10

-The "relaxation length", L. is given by TlWlwz(l+rz)

12MÀl (14)

and will be of the same magnitUde as Àl - except at very low values of the speed ratio - where À3 is a more suitable sealing length.

Some typieal values of the parameters entering the above expressions are given in the table below

TABLE 1 S .2 .4 .6 .8 1.0 ro 1.059 1.271 1.928 3.862 9.440 rz 1.636 2.189 3.440 7.364 14.439 À 1

-

_L 8.531 3.290 1.873 1.365 1.195

The amplitude functions are plotted on tig.

4

for the case of S

=

0.5. and the spatial evolution is seen to agree well with what should be expected from the physics of the

problem.

Since the distribution function has been determined.

°any gas dynamic quantity of interest may be computed in the kinetic flow region. In particular. we find for the number density and the bulk temperature

and

(16)

- 11

-1

= -

,

.... ... :t

and the functions Tand T depend on x, through - - , and the u

u ₃

speed ratio as shown in the Appendix. A plot of the density and temperature variation through the kinetic region is given on fig, 5, again for a typical case of S

=

0.5.

HEASUREMENTS

A flow field was produced in the VKI Low Density

Wind Tunnel (ref,

7)

by injecting air from the settling chamber through a thin perforated foil into the test section at ambient tunnel pressure of lower limit 1 ~Hg, and at stagnation pres-sures from 10 ~Hg up to about 100 ~Hg. The porosity of the

wall was 12%,obtained by orifices of 1 mm diameter, which makes the Knudsen number, based upon stagnation quantities and orifice diameter, range from one to ten.

Speed ratio measurements were made in the final flow, using free molecule probes as described in ref. 8. The results from these tests are shown on fig. 2, but some reser-vat ion must be taken as to the accuracy of the measurements because of some small leaks which developed in the system. Nevertheless, there is a reasonably good agreement between the measured speed ratios and those predicted by the the~ry.

Measurements were also made of the particle flux scattered back towards the perforated wall. An orifice probe was instalIed in the wall itself without disturbing the flow, and the pressure in the probe was measured for different flow conditions. The pressure readings, p, are simply related to the number flux by

- 12 -~

ra

₌

I

~x f~ (18) mRTo 2n ~ <0 x X=O

af ter it has been assumed that the temperature inside the probe is the same as the stagnation - and wall - temperature.

~ qpa

According to the present theory we then have

On fig.

6

both theoretical and experiment al values for the back scattered flux are shown. and they are seen to be in good agreement with each other. For the highest measured values of z - or of S - some discrepancies are found to occur. an effect which might be due to deviation from free molecular effusion in this region. It might alBo - to some extent - be due to the breakdown of the theoretical analysis as S

13

-CONCLUSIONS

The mixing of thermal molecular jets produced by Knudsen effusion is seen to result in a final. local equi-librium flow with speed ratio in the subsonic and low super-sonic range. Experimentally. the upper limit in speed ratio has not as yet been determined. under purely one dimensional flow conditions.

For the kinetic flow region. a linea~ combination of half range Maxwellians was selected to represent the

distribution function. This leads to a solution which appears to be qualitatively reasonable. The measurements of the flux of molecules scattered back towands the wall. give some

14

-REFERENCES

1. MOTT-SMITH, H.M.: The solution of the Boltzmann equation

-for a shock wave.

Phys. Rev. 82, 1951, pp 885-.

2. MUCKENFUSS, C.: Some aspects of shock structure according to the bimodal model.

Phys. of Fluids, vol. 5, 1962, No 11. 3. YTREHUS, T.: Effusive flows at low densities.

VKI PR 70-279, 1970.

4. SMOLDEREN, J.J.: Evolution of equations of gas flows. Progress in Aeronautical Sciences, voli 6, 1965; Pergamon Press.

5.

KOGAN, M.N.: Rarefied gas dynamics. Plenum Press, 1969.

6.

VINCENTI, W.G. & KRUGER, Ch.H.: Introduction to physical gas dynamics.

Wiley, 1965.

7. WENDT, J.F.: Low density facilities.

VKI Short course on Low Density Gas Dynamics, Jan. 1967.

8. ENKENHUS, K.R.: Pressure probes at very low densities. UTIAS, Report No

43,

1957.

15

-APPENDIX

The fo11owing functions have been introduced in deve10ping the resu1ts of thispaper

...

₃ u _S2 nl _(~) 2 T

=

_2'

z

-

4 + S2 u3 n3 u3 Se _S2 u 2 ... 1 1 2(1- ~) T

=

I 1 + (1- --) s2(11 erf(S)) u3

_r;

u3

The error function and the i.ncomp1ete gamma function occuring above are defined in the usua1 way as

erf(S)

₌

2 JS e _t2 dt

;;

0

y(~.S2)

Z

(

_{-t t}

_1/

_{2 dt}

=

e

in

0

STAGNAT'ON

K

CD

EQUllIBR IUM MOLECULAR

no

I

*E:~0

"3

~G)

T_O

~

T₃ CHAMBER

I

FLOW U3

K

CD

FLOW

..

"""

•

{}n~

PUMPS

CD

EFFUSION FROM WALL ORIFICE

CD

INTERACTION BETWEEN GAS AND WALL

CD

GAS IN LOCAL EOU III BRIUM

Fig. 1

SCHEMATIC REPRESENTATION OF EFFUSIVE

1.5

r---,---.---~--___:::;~__:J ~~ ~ ~~~ ~"

""

lII'~m 1/')1,0 t---+---+---.."'-""'-=;=---+---I o

-

o

~ "U ()I ~ .5t---~~---+---~---I 11\ REF. 3 PRESENT THEORY a MEASURED VALUES

0---~2---~~---6---~8

pressure.porocity parameter z Fig 2

SPEED RATIO IN THE FINAL FLOW VS.

1,5 r - - - r - - - r - - - r - - - r - - - , , - - - , _ _ _ RH 3 --- PRESENT THEORY

-c

gl,O~-~-~---~---~--~r----~ CT E o ~

...

-

'" .5 J---.----+--~

-c

~ o '0

o

2 3 speed ratio S Fig 3

DENSITY AND TEMPERATURE IN THE FINAL FLOW VS SPEED RATIO

· l,S a---- - - . -- - - , - - - , - -- - , - - - -- ----, V' C

o

-Q. E·S~--~~4_---+~---4---~----~

o

.5

15

2D

25

downstrcam distancc xl À, Fig t.

AMPLITUDE FUNCTIONS FOR THE DIFFERENT MODES

--

c

o ::» 1,3 P'

1.0

~ ::» .0

0,7

1\

Is=0.51

~

~n)

~

r---~

\

I\!.!

T) 0,5 1,5 2,0 2,5 downstr<lom distonc<l xl)., Fig. 5

NUMBER DENSITY AND TEMPERATURE AS FUNCTIONS OF DOWNSTREAM DISTANCE

1.5

o

-

Q5

)( ~

-o

i'

,

,

\

~

~

_\

----

REF 3 PRESENT lHEORY

•

Kn >3 MEA 5 UREMENTS

•

Kn < 3

~

,,~

...

_~

--

,.,

..

~----

~---

t---

---2

3

,

5

6

₇

pressure porocity parameter z

Fig.

G

FLUX OF MOLECULES SCATTERED INTO THE WAlL FOR