Langmuir probe measurements in the stagnation point boundary layer of a blunt-nosed body in a supersonic plasma flow

LANGMUIR PROBE MEASUREMENTS IN THE ST AGNA TION POINT BOUNDARY LAYER OF A BLUNT-NOSED BODY IN A SUPERSONIC

PLASMA FLOW

JUNE, 1963

by

A. A. Sonin

T[CH~!iScm: IIOG~SCHOOl DELFT VL.IEGiUIGBOUW KUNDE

maUOrHEEK

1

NO\~

1963

,

LANGMUIR PROBE MEASUREMENTS IN THE STAGNATION POINT BOUNDARY LAYER OF A BLUNT-NOSED BODY IN A SUPERSONIC

PLASMA FLOW

JUNE. 1963

by A. A. Sonin

UTIA TECHNICAL NOTE NO. 58 AFOSR -2430

ACKNOWLEDGEMENTS

The author is indebted to Dr. G. N. Patterson for the oppor-tunity of carrying out this research at the Institute of Aerophysics.

Sincere thanks are due to Dr. J. H. deLeeuw, whose super-vision, suggestions and stimulating discussions have left their marks on this work. Many helpful discussions with J . B. French are also gratefully ack-nowledged.

The financial assistance of the Defence Research Board of Canada and the United States Air Force Office of Scientific Research made this study possible.

SUMMARY

Langmuir probes have been used to traverse the boundary layer at the stagnation point of a blunt-nosed body in a supersonic plasma stream) and the experimental profiles of electron number density compared to the predictions of theory. The agreement of experiment and theory has been shown to be fairly good) although the lack of an accurate method of temperature measurement prevented a conclusive comparison.

TABLE OF CONTENTS

Page

NOTATION v

1. INTRODUCTION 1

2. THE DISTRIBUTION OF ELECTRON AND ION NUMBER 1 DENSITY IN THE STAGNATION POINT BOUNDARY LAYER

2.1 Review of the Theory of the Larninar Boundary Layer at 1 the Stagnation Point of a Cool Blunt Body

2.2 Solution of the Sirnplified Equations 6

3. THE LANGMUIR PROBE

4. DESCRIPTION OF EXPERIMENTAL WORK 5. RESULTS AND DISCUSSION

5. 1 Experirnental Results 5.2 Cornparison with Theory 5.-3 Discussion

6. CONC L UDING REMARKS

APPENDIX A - A Note on Arnbipolar Diffusion REFERENCES TABLE I FIGURES 1 to 14 8b 10 11 11 12 16 19 20 22

A c D D e -" E f g h J k 1 m M n P Pr R NOTATION area (of a probe's collecting surface)

concentration by weight of electron-ion pairs most probable thermal speed

specific heat

frozen specific heat of mixture diameter of body

ambipolar diffusion coefficient diffusion coefficient of ions magnitude of electronic charge electric field vector

dimensionless stream function (Eq. 2. 15) dimensionless frozen stagnation enthalpy static enthalpy per unit mass

frozen stagnation enthalpy per unit mass total current (to probe)

Boltzmann' s constant =

efL

~r

particle mass Mach number number density pressure Prandtl nurnber specific gas constant

s Sc t T u v

v

x.y

z Greek Letters (Eq. 2. 33)

Schmidt number. based on the ambipolar diffusion coefficient Da

time

temperature

velocity component in the x direction (Fig. 1) velocity component in the y direction (Fig. 1) voltage (of probe relative to the plasma potential) free stream velocity

(with subscripts

+. -

or!) drift velocity vector relative to neutral gas

coordinates (Fig. 1)

dimensionless concentration of electron-ion pairs.

coefficient of recombination - dUe

- ëIX

(Eq. 2. 30) boundary layer thickness permittivity of free space defined by Eq. (2.1l) Debye length (Eq. 5.7)

'coefficient of therm al conductivity

(with subscripts

+

or - ) coefficient of mobility coefficient of viscosity

defined by Eq. (2. lO) mass density

Subscripts e w 1 2 +

+

potential field

stream function (Eq. 2. 14)

boundary layer edge wall of body

free stream

conditions behind normal shock wave ions electrons electron-ion pairs

T[CH.

~S(!.~ nOG~sC!Il)()f

DElFT

VLlëGTUIGBOU\V::U DE

B BlIOIHEEK

1. INTRODUCTION

The Langmuir probe has found widespread application in ex-perimental work with rarefied ionized gases. lts trustworthiness has

gene rally been accepted.in stationary gases, but in high speed plasma flows it is expected that aerodynamic aspects introduce an additional degree of complexity into the interpretation of its readings. Talbot (1) has proposed

a method of ion density measurement in hypersonic plasrnas which, in order to provide the static conditions for which probe theories have been developed, employs a collecting surface imbedded in the stagnation point of a

blunt-nosed body, while French (2) has investigated the behaviour of small Lang-muir probes in supersonic rarefied plasmas.

This paper describes some experiments in which small probes similar to the ones described by French were employed to determine the dis-tribution of electron density in the stagnation point boundary layer of a blunt body in a supersonic plasma flow. The results are compared to theory. 2. THE DISTRIBUTION OF ELECTRON AND ION NUMBER DENSITY IN

THE STAGNATION POINT BOUNDARY LAYER

Consider a body travelling at supersonic speed through an ionized gas. If the electrons and ions recombine upon striking the body, the distribution of charged particles in its immediate vicinity will be governed largely by the mechanism by which these particles move through the

boun-ary layer to the wall.

Talbot (Ref. 1) has considered this problem for the stagna-tion point region of a blunt body; this section is a review of his theory.

2. 1 Review of the Theory of the Laminar Boundary Layer at the Stagnation Point of a Cool Blunt Body

The general steady state equations for the laminar boundary layer at the stagnation point of a blunt-nosed body of revolution have been formulated by Lees (3), Fay and Riddell (4) and others. These allow for a multi-component gas in the boundary layer. Talbot (1) pointed out that the equations can be particularly simply applied to an ionized monatornic gas in which the diffusion of electrons and ions is ambipolar, for the charged particles then possess a common drift motion and number density and can consequently be treated together as one species, lIe l ec tron-ion pairs", which drift relative to the neutral parent gas with a drift velocity

(2. 1)

where Da is the arnbipolar diffusion coefficient and c is the concentration by weight of the electron-ion pairs,

c

=-

P+

+

p-

C::

P+

_{(2. 2)}

p

p

(the subscript

+

refers to ions, - to electrons, and "±" to electron-ion pairs).

p

being the density. In using Eq. (2.1), it has been implicitly assumed that all species in the mixture have a common temperature .(for a discussion of this assumption, see Appendix A). Note that ambipolar diffusion can exist only if there is no net flow of current, and this implies that the body in this case must be electrically floating.

If we specialize further to the case in which the rate of re-combination in the gas phase is sufficiently slow for the flow to be considered frozen, we may treat the gas in the boundary layer as a mixture of two non-reacting species, atoms and electron-ion pairs. The latter species moves toward the wall of the body by a process of diffusion through the neutral atoms and, although no recombination occurs in the boundary layer proper, the

electrons and ions are assumed to recombine upon striking the wall and are thereafter removed from the problem. (We restrict our consideration to cases where the degree of ionization is small and the body is cool and acts as the heat sink necessary to absorb the heat of recombination). Note that the two-component analogy breaks down within the ion sheath (in which ions and electrons are no longer paired) which forms on the surface of the

electrically floating body; however, the presence of this sheath may be neglected in the analysis if its thickness is very small compared to that of the boundary layer.

Following Refs. (1) and {3), we write the boundary layer equa-tions appropriate to our assurnpequa-tions in terrns of the intrinsic coordinates shown on Fig. 1:

Continuity of mass:

(2. 3)

Conservation of electron-ion pairs:

(2. 4)

Momentum:

(2. 5)

where h

=

static enthalpy per unit mass

hR = enthalpy of electron-ion recombination ÀT

=

gas thermal conductivity

and the subscript (e) refers to conditions at the outer edge of the boundary layer.

Variations in the static enthalpy can be separated into those due to changes in Tand those due to changes in c by writing

where

Cp =

L.

_{c.C p} = frozen specific heat

. 1 1 i

(2. 7)

For frozen flow, it is convenient to introduce a frozen stag-nation enthalpy, h_s' which excludes the energy which would be released by recombination of electron-ion pairs,

(2. 8)

If the momentum equation is multiplied by u and added to the energy equation, one obtains another form of the energy equation which can be expressed in terms of the frozen stagnation enthalpy as

f

r

\A.:~hs

+

\r

~l

==

.:L

[)-.T

I

~~~

\1

L

ex

0't:l

o~ ~~j

where Pr

=f-:

Cp

>-T

~

f

/._1

~~U,5]

-

ë>~

LiA\

Pr

-'I

~

27

is the Prandtl number.

(2. 9)

To solve these equations, we follow the approach used by Lees (3) to obtain similar solutions

*.

We introduce the dimensionless quantities

dimensionless frozen stagnation enthalpy

'Z..

-=

_{dimensionless conc. of electron-ion pairs}

where the subscript e refers to conditions at the outer edge of the boundary layer, and assume that u e> r, and all thermodynamic properties at the

bound-ary layer edge depend on the x coordinate only. New independent variables are introduc ed: ~

~(X)::

S

Ik

pa

lA.e

r'

Jx

'}

7

(""'0)

=

~~

f

f

cllt

o (2. 10) (2.11)

The equations are transformed to these variables by means of the relations

~'d

=

1~ ~e ~'l.

(2. 13)

In addition, we define the customary stream function

'f(~,\()

by

(2. 14)

and a dimensionless stream function f by

(2. 15)

which, it is clear, obeys the relation

~

_ p I

-=

___..\Á-0'1. - ..,...

\Ae. (2. 16) In terms of these variables, the equations become

Energy:

o

Pair conservation:

fz.'

4- (

te.

Zl)'

=

D

where Sc

=

Schmidt number. based on the ambipolar diffusion coefficient.

(2. 18)

(2. 19)

Further simplifications can be made in the stagnation region in order to throw these equations into particularly manageable forms.

Firstly. when p~

/fw«

1. a good first approximation is obtained by neglecting the last term in the momentum equation. This is equivalent to the assumption Tw/T e

<<.

1 (cool wall). Sefondly. the last term in the

energy equation can be neglected since ue /hse« 1(it is always zero if Pr

=

1). Lastly. the values of Q.. Sc. and Pr are assumed to be constant throughout the boundary layer. Following Lees. we set! equal to unity. The lirnitations of these approximations are discussed by Lees (3) and Goulard (5). for example.

In the first approximation, the stagnation point boundary layer equations thus reduce to

Momentum:

Energy:

Conservation:

~

"( vt)

-+

Pr

f

(1)

~'(

1)

=

0

z.

lI(

Yl) -\- Sc.

f(vt)

:z'(rt)

=

0

with the boundary conditions

t{c)

-=

-Ç'(O)

=

0

oa

(0')

«I

"%(0)

= 0

-f'(~)

=

I

41(00)::1

Z(OÓ)

=

I

(2. 20) (2.21) (2 .. 22) (2. 23)

The condition z(O)

=

0 implies that the rate of catalytic recombination of electrons and ions at the wal1 is infinitely fast. an assumption which is in keeping with a continuum boundary layer . .

2.2 Solution of the Simplified Equations

A solution for the electron-ion number density can be obtained as follows:

Since ~« 2 2

and hence

Cp T in the stagnation region.

~

=

Vls

~

1$

___

T

Y\~e \;;e..

Te..

The electron-ion number density is thus given by

'(\±.

:::::.-:ze

Yl)

ü1:i-)~

~

(VI.)

(2. 24)

where zand gare the solutions of Eqs. (2.21) and (2.22). Since the latter are tied to the momentum equation by f. all three equations (2.20) (2.21) and (2.22) must be solved.

A simple change of variables (Ref. 1) transforms the momentum equation into the familiar equation of Blasius. and existing numerical solu-tions of that equation with similar boundary condisolu-tions can be adapted to the present problem (see Table 1).

The energy and species conservation equations can be approached analytically. With the aid of Eq. (2. 20), the solution of the equation in g can be written ~

'L

P.,..

to yield

I j

M\

'ä(VL)

=

~(o)

'+

'3 (0)

L

{"(o)

1

d

1

c From Ref. 5.

J

oo

:f"(~Ü

1

P...

_

[f(O)]

d'l

0-47 RI'.>

c) (2. 25) (2.26)

Noting that g(oo)

=

1. this equation is substituted in Eq. (2. 25)

Using this and the numerical value f"(O)

=

0.47, we now

(2.28)

~~~ ~

A

1ä('1)

=

~(o)

-t

D-~(b)]

Fr

t

(OA7

rp..

S

[

-f"(

'1)

J

"017

è

Similarly, the solution of the equation in z is

. It

zh.) -

Sc.~

(O'41)I-SC

S

t

r(

,[)

J

s.c:

01

1

o

(2. 29)

The eJ.ectron-ion number density can now be obtained from equation (2.24), with zand g given by Eqs. (2. 29) and (2.28) respectively. This establishes n7:/ (n7:)e as a function of the coordinate

fl .

It is more convenient, however, to express n! / (n:!")e in term s of a dimensionless distance which can be directly related to the actual distance. In the stag-nation region, we set

r

=

x,

where a value for

~

:: ( * ) e has been given by Probstein (Ref. 6):

~

=

~~6 i~(2-~

(2.30)

Here V 1 represents the free stream velocity, D the body diameter, and the subscripts (1) and (2) refer to conditions upstream and downstream of the bow shock, respectively.

From the definitions of the coordinates

$

,V[

,we obtain

and

~

- p"rf

1

=

~2~:

The latter expression can be thrown into the form

'1.

~2)fe:

'j

~

j

(~d7

(2. 31)

(2.32)

We are now in a position to define a dimensionless distance

which is related to the coordinate

VL

s~

SCf-Jdl

~

o ~ by the equation

1

S

~

('I.)d1

o . (2. 34)

In summary, then, the relative electron-ion number density is obtained from (Eqs. 2.24,

2.

28, 2.29)

'!. 'Sc:

S:

(O.41')'-~ S

[f"(vt)

1

Jt

~

(0)

+

'[I -

~(o)J

Pr

-k

(641)'-P.

t[

-f"(.i)

JJ

1

This can be plotted against the dimensionless distance ~

CS

=

5

~(vij

d'1

o

(2.35)

(2. 36)

to yield profiles of electron-ion number density through the boundary layer which, for given values of Pr and Sc, have g(O) 0:: Tw/T e as a parameter.

The curves can be applied to specific problems by means of the equation relating the actual distance to the dimensionless one,

(2. 37)

Such curves are shown in Fig. 2 for the special case

Pr = Sc

=

1, for which the equations assume the particularly simple forms (2.38)

and

(2.39)

It is interesting to note that for T wiT e ~ g(O)

=

0, the charged particle number density remains constant throughout the boundary layer. This occurs whenever Pr "'" Sc and g( 0) = 0, for in such cases Eqs.

(2.21) and (2.22) are similar and possess identical boundary conditions. g and z are thus identical functions of

rt

and

ntl

(n±)e

=

z

I

g

=

1 for all values of

rt

Physically, the diffusion loss of charged particles is in

this case precisely balanced by the density increase associated with heat conduction, with the result that there is no net change in charged particle number density until a discontinuous drop occurs at the wall.

3. THE LANGMUIR PROBE

For a review of Langmuir probe theory, the reader is re-ferred to French (Ref. 2). Only a brief outline of the essentials will be dis-cussed here.

The Langmuir probe consists essentially of a metallic surface which collects charged particles from an ambient plasma. If the probe is small, the nature of its current-voltage characteristic can be directly

re-lated to some properties of the plasma in its vicinity.

Consider a probe with a plane circular collecting area in a stationary plasma. When the potentialof the surface is very much below that of the plasma, it collects ions and repels all electrons. A positive ion sheath forms on the collecting surface and restricts the penetration of the field of the negative probe into the plasma, thus mitigating the effect of the probe potential on ion collection. With the possible exception of the case when the electron temperature is smaller than the ion temperature, space-charge effects beyond the sheath proper make the ion current to the probe strongly dependent on the temperature of the electrons.

When the probe is only slightly below the plasma potential, both electrons and ions are able to reach the essentially sheath-free collect-ing surface . The electron current is made up of those electrons which have sufficient energy of random motion to penetrate the repulsive field of the

probe and strike its surface. Kinetic theory predicts that when the electrons

possess a Maxwellian distribution functionthe electron current density is given by

(3. 1)

Owing largly to the high ion-to-electron mass ratio, the electron current in this voltage region is much larger than the ion current, and the latter

may be neglected entirely in the analysis.

Equation (3. 1) implies that

1\ CoDO

,

\ - (3. 2)

Thus if the electrons have a Maxwellian distribution of velocities, the

log-arithm of the electron current varies linearly with the probe potential for slightly negative potentials, and the electron temperature can be obtained

For an ideal probe, the plasma potential is indicated by the

point at which the curve begins to deviate from the linear. The break occurs

as a result of the transition from essentially sheath-free electron collection at slightly negative probe potentials to sheathed collection at positive potent-ials; in the latter mode, the rate of increase of electron current with potential is much smaller than in the former. As a result of the gradual nature of the

changeover, the break is not infinitely sharp. It is debateable, however,

whether the rounding which is observed in practice is the result of the natural response of the probe or whether it is partly caused by external effects such

as, for example, the presence of a resistive coating on the probe or the probe

surface having a non-uniform work function (Ref. 7). Arguments in favour of the latter alternative have led many investigators to identify the plasma po-tential with that of the intersection point of the linear extrapolations of the slightly negative and slightly positive portions of the semi-log plot. This procedure was also adopted in the work reported here.

When the probe is at plasma potential (V = 0), Eq. (3. 1) yields

YÎ_

-=

e~) ~~2~_

(3.3) One can thus obtain, from the electron collecting portion of the current-voltage characteristic of a Langmuir probe, a measure of the local electron

temperature Eq. (3.2) and electron number density Eq. (3.3) . It is

possible that the method of determining the plasma potential which has been adopted here results in an overestimation of the absolute electron number

density (only if the break were. infinitely sharp would no question of

inter-pretation arise), but it should introduce little error into the relative readings

with which we shall be concerned here .

. ' The analysis up to this point has been based on kinetic theory

and is valid for small probes in a stationary plasma. The criterion for

"smallness" here is that the characteristic length of the probe (e. g. diameter)

should be smaller than the mean free paths of collisions of charged particles

with neutral atoms, so that the collected particles arrive at the probe on

effectively collision-free trajectories. In the opposite extreme, when these

mean free paths are much smaller than the probe diameter, the process

which governs the current collection of the probe is the diffusion of charged

particles through the neutral atoms, and in such cases an appropriate analysis must be used.

As to the question of whether the probe analysis given above retains its validity in a flowing plasma, we refer to French (Ref. 2), who has indicated that the interpretation should hold, provided that care is taken

to orient the probe so that the normal to its collecting surface is

trans-verse to the flow direction. When a probe is faced into the flow, an increase

of ion density tends to occur ahead of it when electrons are collected and ions

repelled; this in turn tends to raise the electron number density ahead of the probe (the Debye shielding distance being very much smaller than the probe),

which results in the electron density reading being thrown into error.

4. DESCRIPTION OF EXPERIMENTAL WORK

The experiments were carried out in the U. T. 1. A. Intermediate

Plasma Tunnel, described fully in Refs. 2 and 8. Argon gas was used

ex-clusively. Two mechanical Kinney pumps, each with 485 ft3/ min . capacity, maintained a pressure of 300-500 microns of mercury in the test section

when the plasma source was operating at a mass flow of about O. 1 gm / sec.

The static pressure in the test section was read with a McLeod gauge and independently with a three-tube butyl phtalate precision micromanometer,

which also registered the impact préssure and the nozzle exit pressure.

The nozzle (1. 5" exit diameter) was operated in an almost

balanced (slightly underexpanded) condition, and the model was located about

3 cm downstream of the nozzle exit. An approximate Mach number was

cal-culated from the local im pact pressure and the test section static pressure, ignoring the possibility of small error due to the slight unbalance. All ex-periments were done with the model at the centre-line of the stream.

The plasma souree is shown schematically on Fig. 3 (see also Ref. 8). The working gas enters tangentially at the periphery of the arc chamber, swirls through an arc struck between a button anode and a sleeve cathode (thoriated tungsten, water cooled) and exits into the nozzle. No

plenum chamber was used in these experiments. In typical running conditions,

the statie pressure at the periphery of the arc chamber was about 100 mrn Hg,

and the arc current and voltage were about 80 amps and 25 volts, respectively.

Power was provided by a welding generator.

Figure 4 shows the blunt-nosed model. Since the body must

"float" electrically in order to satisfy the condition of zero current, the

copper nose-cap was fitted onto an insulating teflon trunk. The Langmuir

probes (Fig. 4) were made of 0.004" dia. tungsten wire, over which was

drawn a sheath of nonex glass, about 0. 001 to 0.002" thick. A plane circular

collecting area, 0.004t1

in diameter, was formed by grinding one end of the

wire flush with the glass. The probe circuit diagram is shown on Fig. 5

(reproduced from Ref. 2).

Boundary layer traverses were made with both forward and

sideways-facing probes. For the former, an 0.010" dia. hole was drilled

into the centre of the copper nose-cap and an 0.008" O.D. probe was passed

through it into the boundary layer. The collecting area could be accurately

positioned by means of a micrometer screw mechanism at the downstream end of the model. In addition to the traversing probe, a fixed reference

probe was used in most experiments. The arrangement of model and probes

was mounted on a 6" arm and rotated through the plasma stream at a velocity of 1. 15 rn / sec. Mercury troughs were arranged so that sinusoidal voltages were applied to the probes roughly for the duration of their passage through

the stream, and a photocell system automatically triggered the oscilloscope to register one cycle of the probe characteristics when the body reached the centre-line. A dual beam osciUoscope (Tektronix type 555) with an inde-pendent drive for each beam simultaneously registered the current-voltage characteristics of the traversing and reference probes .

Since the traversing probe was positioned manually, it was necessary to re-start the· arc a number of times in the course of a traverse. To detect any changes in the free-stream electron density due to changes in the arc running conditions, the reference probe mentioned above wasplaced roughly between the shock and the boundary layer, and was left in that

position throughout the experiment. The reading of the traversing probe could then be normalized with respect to that of the reference probe in order to avoid errors due to changes in free-stream conditions. This procedure was of ten found to be unnecessary, however, for stream conditions could be closely reproduced.

To check that the probe did not disturb the flow, experiments were carried out with two probes

t

mm apart in the boundary layer, their collecting areas on the stagnation streamline. The current-voltage charact-eristic of the downstream probe was recorded for the case when both probes we re fired as weU as for the case when the upstream probe was removed. No ostensible difference was detected in the readings. This indicates that the effective sampling region of the probe, i. e. its resolution, is less than

.

t

mm. In the traversing expe.riments, the reference probe was actually slightly offset from the stagnation streamline to lessen the possibility.of disturbanc e.

5. RESULTS AND DISCUSSION 5. 1 Experim ental Results

Because the plasma source operated stably only in rather narrow ranges of mass flow and arc power, most experiments were con-ducted under roughly similar conditions . In a typical experiment, the arc was operated at a mass flow of about O. 1 grams/sec. and a power input of 2 kilowatts, most of which was lost to the water cooling the electrodes. Typical test section conditions wiU be quoted below; these correspond to ionization levels of the order of

10/0.

A photograph of the model in the plasma stream appears on Fig. 6 (the probe arrangement has been removed here). Note the luminous region directly upstream of the nose. Measurements along the stagnation streamline indicated that the outer bounda~y of this region coincided

approximately with the bow shock.

Figure 7 shows a probe trace corresponding to one reading in a traverse of the boundary layer. The corresponding plot of log J v.s. V,

i.

from .. which T _ and n_ were determined (Eqs. 3.2 and 3.3, respectively) is shown on Fig. 8. The readÏI!gs were interpreted as if the probes were in a stationary plasma. Note the linear semi-log plot, an indication of a

Maxwellian electron veloc ity distribution.

Out of a series of experiments, the results of three boundary layer traverses are shown for illustration (Figs. 9, 10, 11 and 12). They are identified here by the chronological experiment numbers 5, 8 and 12. The relevant data for the respective experiments is given in the figures.

Each point on the figures represents the ave rage of two to five readings, the scatter of which is indicated by a fl~g. The readings of electron number density are normalized with respect to the value measured at the outer edge of the boundary layer, which was identified as the point at which the profiles appeared to level off. Note the constant electron temperature across the boundary layer (Fig. 10), which indicates that thermal contact between electrons and Argon atoms is extremely poor under the conditions of these tests.

5. 2 Comparison with Theory

A comparison of experiment with theory requires a knowledge of P e, Tw/T e' ~ , Sc and Pro Assuming that the thermodynamic properties at the outer edge of the boundary layer are approximately equal to those be-hind the shock, P e can be obtained from the measured free stream pressure and the normal shock relations . T w can safely be set equal to room tempera-ture (3000_{K) since in the time the model spends in the hot plasma prior to}

measurement the wall temperature is increased by no more than a few degrees K. (The order of magnitude of the surface temperature rise was estimated by applying the methods of Ref. 9, p. 56, with the heat flux taken from Ref. 5). Since the free stream static temperature was not accurately known" Te is kept as a parameter. The parameter ~ is calculated by using Eq. (2.30) in conjunction with the measured free stream Mach number (from which

P, /

E'1.

is determined by means of the normal shock relations) and the assumed free stream tem perature.

For simplicity, let us first consider the case Sc = Pr = 1. To illustrate the method, let us consider Expt. 5 (Fig. 9) as an example:

M 1 =1.7

PI

=

350

r-

Hg

=

46.5 Newton/m2

T_w = 3000K

D

=

0.970 x 10-2 m

In the case Tw/T e

=

0.1, which implies that T 2

=

30000_{K and Tl}

_{= 1750oK,}

we have V 1 = _{M 1} V

1S

_{RT 1}

=

1. 32 x 10

3

m/sec (3 = '8V,

\1~/'2..-.e..\

= 1 01 x

105 sec- 1 \- 31rO ~ p~\ ~J . p

=

p,

=

2.49 x 10- 4 kg/m 3 \ e RT'2.

and from Ref. 10,

-6 3/2

2.07 x 10 T2 _{-4 2}

=

1. 07 x 10 Newton sec /m 170

+

_{T 2}

Using Eq. (2. 37), we now obtain the relation between the physical distance y and the dimensionless distance s,

'" - \ jfe.

s

=

1. 46 10-3 smeters

<l - 2

~r<e-n_ / (~r<e-n_)e ' known as a function of s from Eqs. (2.38), (2. 39) and the data in Table I (see,_{Fig. 2), can now be plotted against y. A profile for Tw/T e}

=

0.3 may be obtained in a similar manner. (These two values have been chosen because, as wiU be indicated below, the free stream temperatures which they imply should bracket the true free stream temperature. )

We observe first that the experimental results (Figs. 9, 11 and 12) compare weU with theory as regards the boundary layer thickness. In other words, the scaling factor

'fji:-/

.

~.2fSfl in Eq. (2.37) appears to be correct. Referring to the experimental points on Figs. 9 and 11, we observe further that the profiles obtained with sideways facing probes follow the theo-retical profile for Tw/T e = O. 3 fairly closely, although the points do not extend to the region very close to the wal!. A forward facing probe (Fig. 12), on the other hand, indicates a higher value of normalized electron density near the wall than that predicted by the theory. Now it may be ex-pected that a forward facing probe is more susceptible to being influenced by the flow than one which faces sideways. As was already mentioned~ all readings were normalized with respect to the apparent number density at the edge of the boundary layer, where the word "apparent" is used to indi-cate that the reading there, as those elsewhere, was interpreted as if the plasma were stationary. Assuming for the sake of argum ent that complete stagnation of ions occurs ahead of a forward facing probe, we may conclude that the reading of (nt)e in Expt. 12 can be an overestimate by at most 22%

(the ratio of the total to static density behind the shock being 1. 22 in this case). This is insufficient to account for the higher values of n±/ (nt)e near the wal! where, as a result of the very low velocities, the n:t- reading should be

practicaUy uninfluenced by stagnation effects . It should also be emphasized here that the electron density was obtained from a reading of the electron current when the probe was at plasma potential, in which case the coUecting

surface is not sheathed, and the higher readi~gs of the forward facin,s probe cannot therefore be ascribed to the effect of a sheath which .. - as the effective collecting surface - protrudes into the region of higher electron-ion concen-tration in the boundary layer (Ref. 1). It is more likely that the higher indi-cation is a. spurious one which may be attributed to an improper understanding of the probe's response in the immediaie vicinity of the wall.

For a better comparison with theory, it is necessary to con-sider more realistic values of the Prandtl and Schmidt numbers. In the Prandtl number the C ,as defined in Sec. 2, is very nearly equal to the Cp of the neutral gas in this case in which the degree of ionization was small. As regards the coefficient of viscosity we note that the transport of momentum by electrons is small compared to, and that of the ions no larger than, the transport of momentum by the neutral atoms. Hence the coefficient of vis-cosity may with good approximation be considered to be that of the neutral gas alone (this has indeed already been assumed above). The coefficient of thermal conductivity, on the other hand, depends on the transport of energy, in which the electrons are very efficient, and in this the electrons may, despite their small concentration, be expected to play an influential role if they are in good thermal contact with the other species. However, on the basis of experimental evidence (Fig. 10) that the electrons are in extremely poor therm al contact with the heavier particles and maintain a constant tem-perature across the boundary layer, we may assume that the electronic con-duction of heat does not play a significant role in determining the shape of the number density profile. We may thus consider the Prandtl number to be given by the kinetic theory value for the neutral atoms,

Pr~ ~

1S'

= 0.67

5

Unlike the Prandtl number, which is constant, the Schmidt number based on the ambipolar diffusion coefficient is temperature dependent. In Appendix A it is shown that it is reasonable to use Eq. (2. 1) with the ambi-polar ciiffus ion c oefficient taken as

Da

=

D+ (1

+

T - ) (5. 1) T+

where D+ is the diffusion coefficient for the ions. Now kinetic theory (Ref. 12) indicates that the diffusion coefficient for a minority species like the ions is very nearly proportional to

'fT/

f '

i. e.

D T3/2

+

Q( (5.2)

P

An empirical expression for the value of D+ may now be obtained by referring to existing experimental data. In slightly ionized Argon at p

= 0.41 mm Hg

and T = 3000K, Biondi (Ref. 11) found that

Using this in Eq. (5.2) we obtain 34 T 3/2 D+

=

P

(300) where p is in mm Hg. In MKS units, D+ P

=

0.87 x 10- 4 T3/2 m2/sec - .Newton/m2 If we write, in general, D+ P

=

CT3/2 AT3 / 2

~ =

B +T (5.4)

where A, Band Care constants, and use Eq. (5. 1) for Da, then we find that Sc

₌

_{D (:)} .;M.

₌

_C

AR _(B T

₊

_{T) (T} T

₊

_T)

a \

-(5. 5)

where R is the specific gas constant. For Argon we have (in MKS units), C

=

0.87 x 10- 4 , and from Ref. 10, A

=

2.07 x 10- 6, B

=

170. Thus for Argon,

Sc = (5. 6)

It is now apparent that if the electrons are in equilibrium with the ions and atoms, a constant value Sc -== 2.5 is a good approximation for Argon at high temperatures. In the present case, however, the electron temperature was approximately constant through the boundary layer, and hence a considerable variation of Sc may be expected. For example, in the case Tw/T e

=

0,1, with T _ = 4000oK, Sc has a value 2.0 at the outer edge of the boundary layer where T = 3000oK, and decreases to O. 22 at the wall, where T = 300oK. For

T wiT e = O. 1, on the other hand, Sc decreases from 0.85 at the boundary layer edge to O. 22 at the wall.

The assumption of constant Sc in the boundary layer equations is thus a poor one. A refinement of the theory to include a variabie Schmidt number will not be atternpted here, but for the sake of comparison, it may be of interest to use, in the simple theory, the average values Sc

=

1 for

Tw/T e = 0.1 and Sc = 0.5 _{for Tw/T e} = 0.3. Equation (2.35) was solved numerically with these Schmidt numbers and g(O) C& Tw/T e values, with Pr

=

0.67 in both cases. The resulting electron-ion number density profiles are shown on Fig. 13. A comparison with experiment (Fig. 14) indicates that, again, the experimental points fall close to the curve corresponding to the free stream temperature 5820K (Tw/T e

=

O. 3). In fact, th is profile is relatively insensitive to the value of Sc.

Unfortunately. the actual gas temperature in these experiments could only be roughly estimated. It is impossible to calculate a free stream temperature at the nozzle exit because the conditions in the arc chamber are inadequately known and the subsequent expansion of the plasma is in all pro-bability a non-isentropic and non-equilibrium process. lf. on the other hand. a plenum chamber were interposed between the arc chamber and the nozzle throat. one could obtain a rough estimate of the total temperature by compar-ing stagnation pressures. measured in the plenum chamber. with the arc on and off. keeping the mass flow constant (Ref. 2). Although no plenum chamber was used in these experiments. a very rough estimate of the total tempera-ture may be obtained by referring to the experiments inRef. 2, done with similar arc chamber conditions. in which such a chamber was used. These considerations imply that the statie temperature at the exit plane of the Mach

1. 7 nozzle should be somewhat below 1 x 103 OK in these tests. Although this is merely a rough estimate, it does indicate that the theoretical num-ber density profile appropriate to these experiments should lie somewhere intermediate to the two curves plotted. which correspond to the free stream temperatures 5820_{K and 1750}0_{K respectively.}

5.3 Discussion

The ass urn ption that the theory of Sec. 2 is applicable here must be given more concrete consideration. Let us consider the following typical conditions: Pe

=

1. 18 mm Hg

=

1. 56 x 102 Newton/m2 (PI

=

o.

350 mm Hg) Te

=

17200K (T 1

=

1000oK) T = 40000K (n_)e

= 10

20 jm3

For a continuum boundary layer theory to be applicable, the ratios of the mean free paths to the boundary layer thickness

6

must be small com-pared to unity.

In this case. taking

~

= 3 mm, and using the conventional kinetic theory definition of the mean free path for atom-atom collisions (Ref. 13. p. 147), the ratio of this mean free path to the boundary layer thickness decreases from the rather large value of

o.

13 at the outer edge of the boundary layer to 0.015 at the wall. Secondly. we recall also that in the theory of Sec. 2 it was implicitly assumed that the ratio of the boundary layer thickness to the local radius of curvature of the nose is small compared to unity. Taking the diameter of the body as the relevant dimension here, the ratio was about 0.3 in these tests. It is hoped that the choice of a practically flat nose renders this violation less serious in

the stagnation region. These shortcomings. associated with operating con-ditions which correspond to too low Reynolds numbers. could unfortunately not be improved up on significantly. for the choice of body diameter was re-stricted. by the 1. 5 inch nozzle exit diameter. and raising the Reynolds num-ber by other means (e. g. higher density) would have resulted in a reduction of the absolute thickness of the boundary layer. which in turn would have necessitated the use of smaller Langmuir probes. The latter alternative was impracticable because of difficulties of manufacture.

Turning next to the charged particles. we assume. for lack of collision cross section data at low energies. that the ion-atom mean free path is of the same magnitude as that for atorn-atom collisions. The remarks in the paragraph above therefore apply here also. The electron-atom mean free path can be estimated from the cross section data given by Kivel (Ref.

14). For an electron energy of about 0.5 eV. we find that the mean free path for collisions between electrons and Argon atoms is considerably larger than the 3 mm boundary layer thickness. (This is largely due to the Ramsauer-Townsend effect. which predicts a minimum in the cross section for elastic scattering of electrons by Argon atoms near this energy). The process can thus be envisaged as one in which the ions diffuse toward the wall through the neutral parent gas. while the motion of the electrons. which suffer few direct

collisions with the neutral atoms. is influenced mainly by the space charge field which is set up (see Appendix A). Since. in addition. the Debye shield-ing distance

(MKS) (5. 7)

has a value 4.4 x 10- 4 mrn (evaluated at the boundary layer edge) which is rnany orders of magnitude smaller than the boundary layer thickness. the situation is compatible with the assumption of ambipolar diffusion.

Lastly. since thermal ionization of Argon is negligible at the temperatures in question. the satisfaction of the frozen flow assumption de-pends entirely on the extent to which the electrons and ions recombine in their passage through the boundary layer. The transit time ' I for an electron-ion pair (Ref. 1) is given approximately by 1/

e .

where ~ is given by Eq. (2.30). Taking Expt. 5 as an example. and considering again the two limits of temperature T2

=

30000_{K and T2}

_{= 1000}

0_{K between which it}

is estimated that the true gas temperature lies. we obtain

o for T 2 = 3000 KJ o for T₂

= 1000

K.

rr

= 1. 0 x 10- 5 sec.

'i'

= 1. 7 x 10- 5 sec.

The rate of loss of electron-ion pairs through ideal electron-ion recombination is given by the equation

_ . _

-(5.8)

where

ex

is the recombination coefficient. Setting 'fα

=

(Y''l±)e..

at t = 0, this

integr~tes to

= \

-+

(5. 9)

In the light of the unreliability of theoretical expressions for D( , it is per-haps best to extrapolate from existing experimental data. The work of Sayers (see Ref. 15) is particularly useful here since it was conducted under much the same conditions as the present experiments. Using p'robes in a dying arc plasma, he obtained a value 0(

=

4.2 x 10- 10 cm 3 sec-I/ion in Argon at pressures of 0. 1 to 1. 0 mm Hg, with n_

>

10 11 cm- 3 and T _

=

1250oK,. and noted that

ex

decreases as T-:,3/2 but is constant with pressure. Extra-polating to T _ = 4000oK, this yields

0<.

=

0.74 x 10- 10 cm 3 -s.ec.-1 /ion

Substituting this in Eq. (5. 9) and using the calculated transit times for the time interval, we obtain for Expt. 5, in which (n_)e = 1. 9 x 10 14 / cm 3

n+/ (n±)

=

0.88

- e

0. 3,

=

0. 81

In other words, the charged particle number density in an otherwise undis-turbed plasma would be decreased through ideal recombination by 12 and 19% in the respective times the charged partic1es take to pass from the outer edge of the boundary layer to the wall. Note that the free stream electron number density - and hence the amount of recombination - was higher in this experi -ment than in the others shown here. In any case, it appears that the assump-tion of frozen flow is not seriously violated.

Finally, a few words may be said about the Langmuir probe readings. Returning to the typical conditions quoted at the beginning of this section, we observe that the at om -atom mean free path decreases from 0.38 mm at the boundarly layer edge to 0.046 mm at the wall. The O. 15 mm outer diameter of the Langmuir probes is smaller than this mean free path except very near the wall. Assuming as before that the ion-atom and atom-atom mean free paths are equal, we may thus conc1ude that the probe theory outlined in Sec. 3 is appropriate here.

As far as the interactions between the charged particles them-selves are concerned, however, the situation is. different. A "mean free path"

may be defined by multiplying. Spitzer's (Ref. 16) " se lf-collision time" with the meq.n thermaLspeed of the particles in question. This yields an ion-ion mean free path of 0.01 mm, and an electron-electron mean free path of O. 04 mm; the electron-ion mean free path is

21[2

times larger than the latter.

(All these are evaluated at the outer edge of the boundary layer.) The probe

may thus be said to be in "continuum flow" with respect to the charged particles. French (Ref. 2) has indicated, however, that even in these conditions the

kinetic probe theory outlined in Section 3 should be applicable, provided that the probe is oriented so that the normal to its collecting surface is perpen-dicular to the flow direction (Figs. 9, 10 q.nd 11). As was mentioned earlier,

a check was made to determ ine whether the probes themselves disturbed the flow, and the results showed that no disturbance could be detected within

i

mm.

In view of the above, we may conclude that the experimental profiles of normalized electron number density should correspond to the true ones where the measurements were done with sideways facing probes (It was pointed out in Sec. 3 that, as a result of the manner of interpretation adopted here, the absolute electron densities inferred from the probe characteristics may be overestimates). The error inherent in the probe readings is difficult to estimate, for it is primarily due to the manner of reading thé point of

plasma potentiaion the plot of log J v. s. V. The scattering of the experiment-al points, shown on the figures by flags, should be an indication of the error inherent in the measurements of relative number density.

6. CONCLUDING REMARKS

It was indicated in the previous section that the experimental profiles of normalized electron number density obtained with sideways fac-ing probes (Figs. 9 11 and 14) should be close to the true ones. The expèri-mental results are in good agreement with the simple theory as far as the boundary layer thickness is concerned, and in fair agreement as regards the shape of the profile provided one assumes T wiT e

=

0.3. This implies a free stream temperature of about 600oK. Unfortunately, the lack of areliabie

.method of temperature measurement allows no stronger a statement that the true gas temperature in these experiments should be bracketed by the value13 Tw/T e

=

O. 3 and

o.

1, which represent the free stream temperatures 5820K and 17500_{K respectively.}

A more conclusive comparison must await the application of a more accurate method of gas temperature measurement and, in addition .. it would be desirabie to have a sufficiently flexible test facility to be able to

isolate and establish the influences of the various parameters involved and thereby eliminate the possibility of spurious agreement.

APPENDIX A

A Note on Ambipolar Diffusion

The equations governing the diffusion of electrons and ions in the neutral parent gas of a weakly ionized mixture may be written .(Ref,. 18, p.

415) ~ ~

f'I-

V_

'VVL

kT -

e

YI_

E

(A. 1)

r-(A. 2)

~ ~

Where V _ and V

+

are the velocities of diffusion of the electrons and ions, respectively, relative to the neutral gas and )A- and

#-+

are interpreted as the respective coefficients of mobility. These two equations are supple-m ented by Pois son' s equation,

---\7.E (A.3)

-:...

which relates the space charge electric field E to the charge separation which gives rise to it.

When there is no net current, i. e.

(A. 4) and negligible charge separation, i. e.

(A. 5)

then the diffusion is termed ambipolar. This type of diffus~on is compatible with situations in which the Debye shielding distance is very much smaller than the characteristic length in the diffusion configuration. With these two assumptions, Eq. (A. 3) may be dispensed with and we obtain, by adding Eqs. (A. 1) and (A. 2),

(A.6)

Making use of the fact that

)A--

'>"">

JA-

+ ,

this reduces to

Y1±\J:.

=

-r~

VLVLtle\+(I+

~~J

(A.7)

Suppose now that, as in the experiments discussed in this paper, the electrons have a constant temperature and suffer few collisions, while the ions diffuse in the usual manner according to Eq. (A. 2). In this case, since the electron flux is still controlled by the ion flux and is therefore

much smaller than,the electron random flux, we may replace the equation of motion for the electrons by the statement that the electrons are in equilibrium in the potential field

cp

which results from charge separation, th at is,

e(4-4oj

IL \

'RT-Y't_

=-

{1-)o

e

(A.8)

Differentiation of this equation yields the result that

Ë

= - \7Á -=- -

~T-

\7V\-'f' ca v1- ' (A. 9)

Substituting this into the equation of motion for the ions, Eq. (A. 2), and making use of the assumption (A. 5), we obtain Eq. (A. 7) again. This is indeed not surprising, since in the previous approach the neglect of

ft+

in favour of

,.AA-

is equivalent to the neglect of the left hand side of Eq. (A. 1) in favour of the right hand side, and with this simplification Eq. (A. 1) is precisely equivalent to Eq. (A.8). The coUisions of electrons with neutral atoms are therefore of little consequence to the process of ambipolar

diffusion.

We note that Eq. (A. 7) wiU not reduce to precisely the same form as Eq. (2.1), which was used in the boundary layer equations, unless it is assumed that all species have a common temperature and that the total gas pressure is constant. In that case, p

=

nkT is constant, where

n

=

rin + n+ + n_ is the total number density, and the use of this reduces Eq. (A. 7) to the form

~

Vl

VI

±

V

-+

~

- V\

D~

V

±.

-

n

(A. 10)

where

o~= (\+~T')D-t

-=2D+

(A. 11)

is the ambipolar diffusion coefficient, expressed in terms of the diffusion coefficient of the ions,

(A. 12)

In Sec. 2 of this paper existing methods of diffusion boundary layer solution, which rely on a diffusion equation of the form of Eq. (2.1), were followed. Since in the actual experiments the electron temperature was not the same as th at of the other species, and equation of the form (A.7) should really be more appropriate. In view of the approximate nature of the theory, however, and the several other approximations made in its application to the experimental results, it was felt that the essential features of the diffu-sion process should be sufficiently weU retained with the use of Eq. (2. 1) when the ambipolar diffusion coefficient was taken to be

1. Talbo.t, L. 2. French, J. B. 3. Lees, L. 4. Fay, J. A. Riddell, F. R. 5. Go.ulard, R. 6. Pro.bstein, R. F. 7. Medicus, G. 8. French, J. B. Muntz, E. P. 9. Carslaw, H. S. Jaeger, J.C. 10. Kaye, G. W. C. Laby, T. H. 11. Bio.ncli, M. A. 12. Hirschfelder, J. O. Curtiss, C.F. Bird, R.B. REFERENGES

Theo.ry o.f the Stagnatio.n Po.int Langmuir Pro.be. Rep. ,No.. HE-150-168, Institute o.f Eng. Research, University o.f Califo.rnia (1959). '

Langmuir Pro.bes in a Flo.wing Lo.w Density Plasma. UTIA Rep. No.. 79, University o.f To.ro.nto. (1961).

Laminar Heat Transfer Over Blunt-No.sed Bo.dies at Hyperso.nic Flight Speeds, Jet Pro.pulsio.n, Vo.l. 26, p. '259 (1956).

Stagnatio.n Po.int Heat Transfer in Disso.ciated Air. AVCO Research Lab. R. N. 18 (1956)

On Catalytic R~co.mbinatio.n Rates in Hyperso.nic Stagnatio.n Heat Transfer, Jet Pro.pulsio.n, Vo.l. 28, p. 737 (1958).

!nviscid Flo.w in the Stagnatio.n Po.int Regio.n o.f Very Blunt-No.sed Bo.dies at Hyperso.nic Flight Speeds, WADC TN 56-356 (1956) Bro.wn Uni-versity.

In Io.nizatio.n Pheno.mena in Gases (H. Maecker,

ed. ), No.rth-Ho.lland Publishing Co.. , Amsterdam, (1961)

Design Study o.f the UTIA Lo.w Density Plasma Tunnel, UTIA TN 34, University o.f To.ro.nto. (1960)

Co.nductio.n o.f Heat in So.lids, (2nd ed. ) Oxfo.rd University Press (1950).

Physical and Chemical Co.nstants (10th ed. ) Lo.ngmans, Green and Co.. , Lo.ndo.n, 1949.

Diffusio.n Co.o.ling o.f Electro.ns in Ionized Gases, Phys. Rev. Vo.l. 93, 1954, p. 1136

Mo.lecular Theo.ry o.f Gases and Liquids, Jo.hn Wiley andSo.ns" !nc., New Yo.rk (1954)

13. Kennard, E. H. 14. Kivel, B. 15. Loeb, L. B. 16 . . Spitzer, L. 17. Schlichting. H. 18. Chapman, S. , Cowling. T.

G.

19. Dorrance. W. H.

'Kinetic Theory of Gases, McGraw-Hill Book Co., Inc., New York, (1938).

Electron Scattering by Noble Gases in the Limit of Zero Energy, Phys. Rev. Vol. 116, p. 1484

(1959).

Basic Processes of Gaseous Electronics, Univ. of California Press, Berkely & Los Angeles (1955).

Physics of Fully Ionized Gases, Interscience Publishers, Inc., New York (1956).

Boundary Layer Theory. McGraw-Hill Book Co. !nc .• New York (1955).

The mathematical Theory of Non- Uniform Gases (2nd ed. ) Cambridge Univ. Press (1960)

Viscous Hypersonic Flow, McGraw HilI Book Co. !nc .• New York (1962).

T[CH

':S(r:~

HOGESCHOOL

DelFT

VU:CïUICCOUVvi·:UND.e

m~UOTHfEK

TABLE I

Numerical solution of the equation f'" (~ )

+

f( ~ ) f" (

rt)

= 0, adapted from

Ref. 17, p. 107.

1

+(1)

t'(~)

+'I(~)

_~

_-f.

(1.)

-t

'(~)

-f''(Vl}

0 0 0 . 47·0 3.111 1.903 .976 .0551 3.394 2.181 .988 .0309 .141 .0047 .0664 .470 3.676 2.462 .994 .0160 .283 .0188 . 133 .470 3.959 2.743 .997 .0077 .424 .0422 . 199 .467 4.242 3.025 .999 .0034 .566 .0750 .265 .463 .707 .117 .330 .457 .848 .168 .394 .448 .990 .228 .456 .436 1. 131 .297 .517 .420 1. 273 .374 .575 .400 1. 414 . 460 .630 .377 . 1. 556 .552 .681 .351 1.697 .652 .729 .323 1. 838 .759 .772 .292 1. 980 .870 .812 .260 2. 121 .988 .846 .228 2.262 1. 109 .876 . 197 2.404 1. 235 .902 .167 2.545 1. 364 .923 .139 2.687 1. 496 .941 .113 2.828 1. 630 .956 .0909

r FIG. 1 , , "

. .

.

..

. .

.

,

BODY

..

,

.

.

..

.

.

.

COORDINATE SYSTEM

l.0 ~ E-< ... U) Z ri:l Ç)

c5

z

z

0 p:: [-4 U ri:l 0 .5 ..4 ri:l Ç) ri:l N H ..4 ~ :;E p:: 0

z

ï

,Cl CC 0 1+1+ _++R H+1 _{4+l 4+l +H H+}

-=

m : j : j : j : ++l- +1+1 T w _ 0 0 05 ' " ' " '"

~

,

'"

"

+t,

,

~

ff

,

I

,':

i'

~"

'.

TE7'7 Te , . , 0.1, 0.15, 0.2, 0 3 H+

i,

~

L!l-l

b:

I"

I

I'-j;

r

:'.;

~

:tl

~

!k~

~

'"

H+t _• +1+ _{TIJ hTl'-::J.i 1 ' 'TT .l14f-::} +11

r~;:Il+t:t

~t~

1,,)

~fnr.r.:

·:

o

l.0 !i ' ., ril It-fn-!tJ rl_" r ;-~t4.r+

..

...

"'"

.;;

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~

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..:b]:

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!

,:

I

'

,je j

~

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.

_{j,_t}

:t:tr:t~

,

IIH iHP' Il-:t "

, ct

~

~i

l±iJ

l:~

l

r:·:

h

.

~W.

!

:::r I J' i

tHi

.

t1rH : ;:D ITmruii

'

rr

~

'

u

i I

~Y

f~;T :l1!I, ':...,.

I, .,

~-, I ' ~: ; , ",,'~' ' k _ , . I, ' : 1 t1~ I I t . I'Jj I...r-~.. I I~, ~:r]~:-lf~ +î+AIT;!~i~

,

_ ' 1 rRif .:

-

fu

+!' I "

~

~

;_; Ü

Rf:

~

rJ~:

rt11

:

l

if~

;~

IJ .

lffi

Iti.

tr±jll:iil1iJll'~i~I

~';

j

Sc Pr

=

1

rtt++

-H4+I[ll..."I

1

iE.:

~:

'

:

~

rl

ij~

H

l

!~

'"

I"

rr

~

LiliU +: I

~'

, " " _ rl rr"

,,

_

"'l~J 1+I;'llttjl

ilIl.

:

l

.,tt

' '1 _~ I '

..

_{rl '} tI +1

IT

I~

;:

~

'nu

·-t'+

HrltLl!.

~

t:1 l:l

~

:

S

r;4· ij '

I

W

f.i:;'~

I

~TJ

R~~

w I,'

~

1 J:ill(' 1+1 L.,-l " -",I,r~ IH" t!'1 1, 'r',,' j t H f , II "'~', ' . l 1 I

;.

~~ ':t'~it

:FQl'::Jt#lb

l

ft'I' :j

L

i ti

~

+-,' i1 !l:tH-:"

t~Lfi-l-

~

t, lit~l _1 :!+!-'I'" 4 ·'4

tr

-

HnJ-I

ij1-j

1}t

~

F~

f-.-

,

-

··

tI . ' , ' 11 •• ,; T

_

.

-

~

~

m

'

',

1H!

FIrtl

UI

,;;~

1-

:

,il

i:

r:

~

tJl

t"

,~

l~

ï'

i

~

~

R

"lt

.~

It

1W

Il;il;

6~

w~

îil

~

11

!:t

Err

~f:

lil:

r

L!..,:T-:Tf ill ':J1~ [,jiii , I" ~c-f ,,·t,1 . n TîI

-

:n

.

Kr

R~

.,rlf ! .... tl 2.0

I-!::-L:i I:t: J -: t.t= I i! -4~1+1lL..l

. ru

1,""

ct l rl,;1 IT.r :-r.tt ~ ,1 rJt~

~, III ' t i L iLI-:f..tI

-"'~

l:t:i:t:l±ttWfElmlt!:illl:tF±!:!fEHm

;C

_u

8

_:

:T:

₁₁

ill

!

_i TÜ_tWl:tltitt[LfT u _l ,:tt-_'}iiJ '1~r -

[fu

-+tltr_'

tr

_{I r~}

i

l _{I i}

3.0

S ,

DIMENSIONLESS DIST ANCE FROM ST AGNA TION POINT

FIG. 2 THEORETICAL ELECTRON ION NUMBER DENSIT-{ rROF'._ f~ci

ARGON SUPPLY SYNCHRONOUS 72 RPM MOTOR FLOWRATORI ~ PLASMA JET + - - - ; 25 KW DC :.~'-:~:-~-:.:~:::::::::: .. :: .. :.:.

ROTATING DISC WITH RADIAL SLIT TO OSCILLOSCOPE TRIGGER AMPLI-FIER TO TEST SECTION (STATIC PRESSURE

f7Z77T/ZZZZZZZZZZTJ/Ji?"

BLUNT BODY NOZZLE

SLIP RINGS IN MERCURY TROUGHS

TEST SECTION

TEFLON TRUNK COPPER NOSE CAP

T

0.970cm

~RAVERSING

U

I

PROBE

---1

...

REFERENCE PROBE

(a) TRAVERSE WITH SIDEWAYS FACING PROBE

(b) TRAVERSE WITH FORWARD FACING PROBE

0.004 in. dia. TUNGSTEN WIRE

...

O. 008 in.

O. 006 in.

j

SHEATH

(c) LANGMUIR PROBE CROSS SECTION

+

25 KW DC

TO DUPLICATE PROBE CIRCUIT

PROBE

--

-VARIABLE RREQUENCY SINE WAVE GENERATOR

10 WATT AUDIO

AMPLIFIER

I50LA TING TRAN5FOER

300 High Mag. Low Mag. IX INPUT OF TEKTRONIX 555 OSCILL05COPE (PROBE VOLTAGE) YInputs (Probe Current)

FIG. 7

1 VOLT

12.1 V

PROBE VOLTAGE (VOLTS ABOVE GROUND )

CURRENT-VOLTAGE TRACE OF A LANGMUIR PROBE (READING TAKEN IN EXPT. 5 AT Y = 0.12 mm).

10 0·1 10 FIG. 8 (J -)0 = 9.5 MA ..,

-,,<ol / 0 /

-l

1-/0

I

V

~

I

T_ - 3800°1[ n_ - 0.77 x 1020/ m3 / 1. L L

!

)

I

!

I

11 12 13

PROBE VOLTAGE (VOLTS ABOVE GROUND)

1·0

T

_w

-

=

0.1 \

Te

_~

>-t E-t H Cl) Z ril Q

.

0 :zoo :zoo ~ E-t U ril H ril ~ 0·5 N H H

~

~

0 :zoo

-..

-I ·

_c::

, -

_I

_ E

o

o

. . ()

f

V--T

_w ~ - =

0.3

Te

Experiment 5

MI

=

1.7

PI • 0.350 mm

Hg '\.

T

w ..

300

0

_K

n_e - 1.9 x 10

20

/ m3

- - --- - --- - -- -1·0

2·0

3·0

y , DISTANCE FROM STAGNATION POINT (MM)

I

I

-5000

1

~

<t

<l> 4000

cl>

-

~ 'L. I E-< 3000 ~ ~ ~

_Experiment

₅

~ E-< ~

_{See data on Flg.9}

tx: ~ p.., :::E 2000 ~ E-< Z 0 tx: E-< U ~ ...:l ₁₀₀₀ ~

o

o

1 2 3

DIST ANCE FROM ST AGNA TION POINT, Y (MM)

:>ot 1.0 E-t ... rn Z ril Cl 0 Z Z 0 0:: E-t U ril ...:l ril Cl ril 0.5 N 0 ... ...:l ~ ~ 0:: 0 ( Z

Ill

w s:: s::

I

o

o

T_w Te = 0.1

'""

(I) (~ v

p--

(~

L>-'

/:

V

)

.

(~ "" TT = 0 . 3 w

Exl2eriment 12

e FIG. 12

M1

=

1.7

PI : 0.315 mm Hg

T

_w

=

300

0

K

n_e

=

0.52

X

10

20

/ m3

1.0 2.0 3.0

y, DISTANCE FROM STAGNATION POINT (mm )

RESULTS OF EXPT. 12 COMPARED TO THEORETICAL

CURVES FOR Sc

=

Pr = 1.

I

i

1

'·0

~ ~ H Cl) :zo ril ~ 0:: Pil

~

i-) ~ :zo 0 Il! ~ ( ) Pil ....:I Pil

~

0·5

N H ....:I

~

0 :zo j

_I

.-

C ~

.

-

-

~

o

o

T

_w

-

=

0.1,\

Te

1

₁

f"-

T

0

~

/2

T

_w

-

=

0.3

Te

Experiment 8

141 • 1.?

P1 • 0.4?5

mm Hg

T

_{w •}

300

0

X

20

3

n

_-8

- 1.1 x 10 / m

I

1·0

2·0

3·0

y,

DISTANCE FROM STAGNATION POINT (MM)

-1·0

nt(s)

(n:t)e

0·5

o

o

-Pr = 0·67 Sc = 1·00

_,

_{, Te}

.!w

= O' I

T

_w

Pr=0·67 Sc=0·50

_,

_{, Te}

-=0~3

1·0

2·0

5, DIMENSIONLESS DISTANCE

3·0