An atomic hydrogen supersonic nozzle beam source

•

Mareh, 1978

At! ATOtlIC IiYDROGEI'J SUPERSONIC fWZZLE BEAf,1 SOURCE

by

John Potj ewyd

"'

3

JULI

1979

1971

TECHNISCHE

HOGESCHOOL

DElFT

WCHTVAART· EN RUIMTEVAARITECHNIEK

BIBLlOTHEEf·

Kluyverweg

1 -

DELFT

UTIAS Report No. 220 CN ISSN 0082-5255

AN ATOMIC HYDROGEN SUPERSONIC NOZZLE BEAM SOURCE

Submitted December,

1975

March, 1978

by

John Potjewyd

UTIAS Report No. 220 CN ISSN 0082-5255

.

_•

Acknowledgements

This project was carried out tmder the supervJ.sl.on of Dr. J. B. French, whose advice and support are gratefully acknowledged. The helpful

and stimulating talks with Dr. N. M. Reid, Mr. W. R. Jones, MI'. C. K. Lam, Mr. G. V. L. Marson, Mr. D. M. Ca.meron, Mr. D. J. B. Venturi and Mr. J. Leffers in both the theoretical and experimental aspects of this work are greatly appreciated and acknowledged with the most sincere thanks.

Financial support was provided in part by the National Research COUIlCil of Canada under Operating Grant No. NRCA-2731

>.

Abstract

An atomic hydrogen supersonic nozz1e beam source has been designed, built and tested that meets its expected requirements . Operating up to 3000K, it produces a high degree of dissociation and high atomic hydrogen density at high velocity. The choice of operating modes, ei ther operation in an inverse

seeded mode with helium or as an atomic hydrogen beam depends on end use requirements of beam velocity, degree of dissociation, or maximization of atomic hydrogen flux.

A seeded beam velo city of 7.06 km/sec has been indicated at an experimental1y determined degree of dissociation of 0.75. At the mass spectrometer detector stat~on 50 centimeters downstream ofthe nozzle, this corresponds to 3.013 x 1014 hydrogen atoms/ cn?-/ sec flux. The source operated with pure hydrogen at higher pressures wi11 produce an estimated 9.1 km/sec beam. At much smaller source Knudsen numbers, an atomic hydrogen flux of 8.46 x 1015 atoms/cmF/sec is theoretical1y possib1e, at an estimated velocity of 10.4 km/sec.

1.

2.

3·

4.

5.

6.

7.

CONrENTS

Acknowledgements

Abs'tract

List of Synhols

INTRODUCTION

EXPERIMEMAL F ACILITY

THE MOMIC HYDROGEN SUPERSONIC NOZZLE BEAM SOURCE

3.1

Description

3.2

The OVen Hot Zone

3.3

Electrode Assembly

3.4

Capper Block Subassembly

3 .5

Gas Flow System

3.6

Tungsten Tubes

GAS DYNAMICS OF DISSOCIA!rING FLOWS

4.1

Nature of the Gas Flow

4.2

Equilibrium Flow

4.3

Nonequilibrium Flow

4.4

Heat Addition in the Nozzle

4.5

The Free Jet Freezing Phenomena

4.6

Sunnnary

HYDROGEN DISSOCIA!rION-RECOMBINMION

5.1

The Equilibrium Constant

5.2 Chemical Relaxation Times

5.3

Measurement of 'the Degree of Dissociation

EXPERIMEN.rAL DMA

AND

ANALYSIS

6.1

Introduc'tion

6.2

Pneumatic Calibration

6.3

Degree of Dissociation

6.4

Beam Flux Estimate

DISCUSSION OF RESULTS

AND

CONCLUSIONS

7.1

Lifetime

7.2

Impurity Levels

7.3

Degree of Dissociation versus Atomic Hydrogen

In-Flight Beam Density and Veloci'ty

7.4

Conclusions

REFERENCES

T.ABLES

FIGURES

iv

i i

iii

1 1 2 2 2

3

3

4

5 6 6

7

12

16

17

19

20 20 22

25

27

27

30

32

33

34

34

34

35

36

37

A

A

sk

af

a

B

b

C

..

'

CD

~

C(T)

Ce(T)

C

p

Cp

D

dq

e

e

trans

e

vib

, e

v

e

rot

e

diss

f e

1

f e

2

f(r)

h

h

Lis't of Symbols

Cross sectional area, or a constant

(in

Eq. 50)

Skimmer orifice area

Frozen speed of sound

Arbitrary constant

Constant used once in Eq. 50

Arbitrary constan't

A constant used in dissociational equilibrium constant

Discharge coefficient as defined in Eq. 71

Discharge coefficient at the state of the gas at temperature T

R

The collection efficiency defined in Eq. 77

The modified collection efficiency

Oh

Specific heat at constant pressure

=

df

I

p

Specific heat at constant density

=

~I

p

Damkohler number

Arbitrary energy change

Average internal energy per grrum of gas

Average 'translational energy per grrum of gas

Average vibrational energy per grrum of gas

Average rotational energy per grrum of gas

Average dissociational energy per grrum of gas

Electronic par'ti'tion function of atomic species

Electronic partition function of molecular species

Function of r as defined in Eq. 70

Average enthalpy per grrum of gas

Planck constant as used only in Eq. 17

K P k kd k r }; s M M f ~,

IrJI'

-m

.

m

IIJ.

IIfi2 ' (m)th (m)exp NA ~, n₂

.

n p Q.l' ~ R Re S

IrJIe

Equilibrium constant =

4J!::l

1 -

cl

Equilibrium constant based on molar concentration

Equilibrium constant in the 3-body reaction with atomic hydrogen as the partner

Boltzmann constant

Dissociation rate coefficient Recombination rate coefficient Scale length

Mach number

Frozen Mach nuIDber

Molecular and atomic weight per mole of gas

Gram molecular weights of atomic hydrogen, molecular hydrogen and helium

Mean molecular weight of a gas mixture Mass flow rate

Theoretical mass flow rate Experimental mass flow rate Avogadro's number

Number density of atomic and molecular species, respectively Number flow rate

Pressure

Skimmer chaIDber pressure

Total collision cross sections for electron impact ionization of the atomic and molecular hydrogen, respectively

Universal gas constant = 8.31441 x 10

7

erg mol-l

K-

l Flow Reynolds number

Spectra peak height

vi

s T t s u w

x

~e' XJI2 ~

x

opt ex ex e

f3

r rf € €

Molecular hydrogen peak height if' no dissociation were occurring

Observed atomic hydrogen spectra peak height Observed molecular hydrogen spectra peak height Observed helium spectra peak height

Average entropy per gram of gas Temperature

ChemicaJ. temperature

Temperature based on K , defined in Eq. 100 P

Temperature as measured wi th the thermocouple Characteristic relaxation time for dissociation

Characteristic relaxation time, or, if in the dissociation-recambination section, then the characteristic relaxation time for recombination

Time scale parameter Flow velocity

Characteristic constant

Axial coordinate along beam centreline Mole fractions of helium and hydrogen MaCh disc axiaJ. location from the nozzle Optimum nozzle-skimmer-axial separation Degree of dissociation

Equilibrium degree of dissociation Dissociation fraction

The local polytropi c exponent

Frozen value of the polytropic exponent

Average vibrational energy per gram of diatomic gas

Lennard-Jones (6-12) potentiaJ. model parameter, the character-istic energy of interaction between molecules

J..t

p

Temperature corresponding to the characteristic vibrational energy of the harmonie oscillator

Temperature corresponding to the characteristic rotational energy

Temperature corresponding to the characteristic dissociation . energy

Mean free path

Coefficient of viscosity Density

Function af

kT/E

used in the expression of the viscosity

coefficient based on Lennard-Jones (6-12) potential model parameters

Subscript-only Symbols

Denotes gas states '1' and '2', respectively Effecti ve value of parameters or variable Freezing point parameter or variable

rot Characteristics of rotational energy

vib Characteristics of vibrational energy

R Reference value

Superscript or Superscript

*

Throat variable or parameter

o Stagnation variable or parameter

Special Symbols

[A]

Concentration of species A in moles cm-

3

Reference Data:-Universal Constants

Planck constant, h

=

6.626176 x 10 -27 erg sec

Boltzmann constant, k = 1.380 2 x 10 66 -16 erg K -1 viii

Avogadro number, NA

=

6.022045 x 1023 mol-l

Universal gas constant, R

=

8.31441 x 107 erg mol-l K-l Hydrogen Constants

~

=

2.016 gm mol-l

~

= 1.008 gm mol"'l :f e

=

1 l :f 2 e

=

-2 9_r

= 84

.97~K 9

=

6324K v 9 D

=

51950K

cr

₌

2.9l5}l

elk

=

38.0K Helium Constants m rT = 4.003 gIn mol-l

=

2.5765(

elk

= 10.2K ix

1. INrRODUCTION

The aim of this work is the design and testing of an atomic hydrogen supersonic nozzle baam source. This source is ahle to produce a molecular beam, a directed stream of neutral atoms or molecules in a low pressure back-ground. On the average, few intermolecular collisions occur to disturb the nature of the beam. In order that the beam be cOllimated, the centre line, or core portion, of' the free jet is skimD:ed. The free jet is produced by a large pressure ratio expansion across the source orifice forcing a supersonic

expan-sion into the vacuum. system • . At large distances from thesource orifice, this axisyIlllOOtric free jet behaves very much like a three dimensional source flow into a vacuum.. It is accompanied by rapid transitions in flow regimes from continuum to free molecular • Atomic and molecular beam systems have been

extensively reviewed by numerous authors (see Refs. 1, 4, 10, 25, 40, and others), and several characteristic source designs have been tested (Refs.

1, 2,

4, 7, 8,

11,

15,25;

26,

29, 30, 34, 36, 41). Ding et al (Ref.

8)

present a good paper on their cooled microwave discharge source that can produce a hydrogen atom bèam.

Production of atomic hydrogen in this beam source is centred on the joule heating design philosophy. A tungsten tube is heated by the passage of high current. The flow of molecular hydrogen within this tungsten oven increases in enthalpy through wall and gas phase collisions until a state of equilibrium with the high temperature wall is reaehed. Supersonic expansion is directed through a small orifice in the tube wall located at the centre of the hot zone, perpendicular to the tungsten~tuPe centreline. Although other methods exist for dissociating molecular hydrogen, this means of atomie hydrogen production was pursued because i t appeared the simplest and the most promising approach towards high atomic f'lux rates. It was hoped that operation near 3000K was realistic for mechanical stahility of the souree • At these temperatures both the degree of dissociation and the velocity are high.

, I

The abili ty of this source to maintain the gas in an unvarying state while the source structure remains intact over the period of a test qualifies this source as a useful research tool. In order to measure temperature , pneu-matic thermometry (Ref. 14) was tested. This technique requires a knowledge of

sOurce nozzle discharge char~cteristics and corresponding accurate pressure measurements. In order to understand the nature of highly dissociated flows both the e.quilibrium and nonequilibrium. aspects of such flows must be known.

Then it is possible to state withsome assurance the relation between the measured degree of dissociation and the equ:i,.librium. state in the oven.

2 • EXPERIME:N.r.AL. FAC,ILITY

Beam formation from the supersonic free jet is accomplished in the minibeam f'acility (see Figs. 1 and 2), which consists of an initial expansion

chamber, a skinnner chamber, containing the source and skimmer i tself, followed by a collimator by which the final beam is defined. An operating pressure of no higher than 10-3 torr is maintained in the ·skimmer chaniber by a 4000 1i tre per second 2~!.i4Q cm ,diameter diffusion pump. A 15.24 cm 1500 li tre pgr second diffusion pump maintains the collimator background pressure below 10- torr. Both pumps operate with liquid nitrogen cooled 'baffles augmenting the system pumping speed and eliminating the backstreaming of diffusion pump oils.

An isolation valve separates theminibeam facility from the detector chamber. The beam passes completely through themass spectrometer without collision on surfaces so that the atomie species cannot recombine. Ions ·are produced by 70 eV electron impact and are analyzed in an Extranuclear Type II quadrupole mass spectrometer. Thesubsequent ion signal is amplified through an electron multiplier, the output of which is monitored on a Kiethley 6l0C solid state electrometer • Scans are made over ·a mass tO charge range· from zero to four. These recorded peak heights are used to determine the relative "in flight" densities ofthe helium and the atomie and molecular hydrogen in the beam and the background. Placing a flag, or obstruction, in front of the beam allows a measure of the thermalized background signal to be made. Beam peaks are found by the ·subtraction of these background peaks from the full beam plus background signal.

3 .

THE ATOMIC HYDROOEN SUPERSONIC NOZZLE BEAM SOURCE 3.1 Description

The design aim was toward the produetion of a high temperature oven capahle of operating wi th strongly reducing gases. This determined both the choice of material and the component layout in the remainder of. the assembly. Figure

3

is a photograph of the source while Fig. 4 presents a schematic layout of the arrangement of·water, gas, and electrical flows. Design finalization followed lengthy design iteration involving designer and technician, based on the supervisor' s initial suggestions concerning the configuration.

3.2 The Oven Hot Zone

The beam source itself is a tungsten tube clamped in water cooled c-opper bloeks si tuated at the end of the electrodes (Figs.

3

and 4) • As long a section of tube as was physically possible in the space availahle was used. It measured 7.2 cm of exposed length between the copper blocks wi th a nominal inner diameter of 0.208 cm and a 0 .. 010 cm ·wall thickness.

A heat shield 6.1 cm in length surrounds the oven, reducing the radiated power. The design is based on the· Crawford heat shield (Ref. 5). Since the radiated power for a n la,yer heat shield constructed of ma:terials -of emissivity Er and with an oven area of Ao is

A E cr(T

4 _

T

4)

o r o n

(2 -

€ )n

r .

p =

where Tn is the temperature of the last layer, then the power radiated is n times less than that of an oven radiating to the same ambient temperature , Tn . This holds ·if all materials have thesame emissivities and for n ~ 1. The present design involved forty-eight layers. The first twenty-one are 0.0254 mm thiek tungsten foil while the remainder are 0.0127 mm tantallum. Inden"bations are made in the foil surface to ~pace successive layers in the spiral construc-tion (Ref. 5) with negligible conducconstruc-tion losses. These losses are proporconstruc-tional to the full cross section area, which, being small, show insignificant loss of heat in conduct i on •

An _{outer layer temperature lower than To/nl/4 is possible. On the} basis of minimizing the hot volume, the first layer is 0.58 cm diameter or only 0.14 cm from the hot tube. A high area ratio reduction is achieved by an outer ·diameter of 2.0 cm, a ratio of 11.9 being estal>lished. Thus in the case of a 30.o0K tube this forty-eight layer heat shield has an outer layer temperature .less than 114oK. .

Copper melts at 1356K. To ensure radiative cobling of the end of the tube section from the hot zone to the copper bloek, a 0.5 cm space was provided between the· end of the .heat shield and the copper block. Thus the

copper was not expected to deform under radiati ve or conduct i ve heat loads. A forty-five degree half-angle hole is located in the centre of the wallof the heat shield to minimize perturbationsto the gas dynamic formation

of the free jet. Hole production proceeds in two stages. stage one involves sandblasting with aluminum oxide abrasive powder to form the gener al shape of the . hole. Subsequent cleaning wi th a tungsten carbide drill finishes the taper to form a hole of required inner diameter at the inner heat shield surface, allowing at least a forty degree half-angle free jet expansion from the nozzle exit plane.

3.3 Electrode AsseIDbly

Electrode design is based on bending flexibili ty and heat resistance. Each eiectrode is made of copper and mounted by two 'O'-ring compression

fittings held in place by teflon rings to provide electrical insulation from ground. The tube will contact the copper block over a surface 1.5 cm long, producing bending stresses as the tube expands upon heating. The use of the buna-N 'O'-rings allows a limited but sufficient degree of rotation about the

compression point. The prime feature is the symmetry 'O'-ring mounting, allowing matched flexing of the .support system, leaving the orifice aligned as the tube expands. It was felt th at providing a fixed mount increased the chances of compressi ve failure while risking distortions of the tube that would twist and bend the orifice plane out of alignment.

3.4 Copper Block SubasseIDbly

The copper bloek is secured by stainless steel bolts tothe main electrode body. Since the water coolingenters at the electrode base and must enter the block, a sili cone '0' - ring seal was provi ded at the .j oint • The water flows through twin loops around the clamping region, discharging at the base

. of the block. Through the use of copper, the block temperature i s close to the water temperature of 281K. Even the tungsten remains relatively cool as i t is dominated by the cold copper bath and the cross sectional area of the electrode in this region is such that ohmic heating of tungsten is made

difficult. A thermal gradient begins in the tungsten tube only from the point where it projects from the block. Copperwas chosen over a higher melting point material such as molybdenum as i ts superior thermal conducti vi ty enabled

larger heat loads to be more ·easily absorbed by the water ·flow. In this marmer the rest of the· electrode and the water and gas nylon tube connectors attached to the block would not be jeopardized.

Gas connection was provided by channeling through a thick copper-plate and turning theflow to a cylindrical depression into which the open end of the tungsten tube projected. Sealing was made by use of silicone 'O'-ring compression between 'the tungsten and the block. An adequately smooth compression surface at the bloek was provided byfi tting a -stainless steel foil between the

. gas connector and the block.

Tube clamping is effected through the action of a stainless steel bolt compressing the tungsten tube against the walls of a tubular cavi ty at the base of a slot extending perpendicularly out to the copper block surface . Since the blo ek is then of a single piece, the wallof the clamped tube is at a uniformly cold temperature • The tube itself is positioned within its own

copper bushing. Each bushing is also slotted to transmit the compressive

loads to clamp the tungsten. These are used in order to be able to accommodate different tungsten tube diameters. Although the compression is not excessive, there is sufficient electrical contact. The compressive 'Ol-ring fitting at each end of the tube is sufficient to prevent movement of the tube relative to the source assembly. Thus adjustments to the orifice alignment can more easily be effected at the gas connection assembly.

Also supported from the copper block is the heat shield mounting rings. Insulating bushings (Fig.

4)

are of boron nitride. Each ring is tapered down to a knife edge contact on the outer layer of the heat shield. This eliminates excessive heat conduction and allows the copper electrodes to rotate as the tube expands upon heating instead of transmitting stresses to the hot tube by a rigid heat shield support. Both rings are compressed to contact the shield with a stainless steel bolt. However, one ring is left loose enough

so"as to allow same free motion between ring and shield.

Only one heat shield used a thermocouple temperature indicator. A tungsten-rhenium alloy thermocouple located at the third layer of one heat shield is used as a model shield to monitor the tube tempera:ture. The alloys are one of

5%

rhenium, the other of

2&/0

rhenium, with a

0.25

mm diameter.

"Alumina insulation tubing covers the leads, up to the junction, preventing the formation of shunting thermocouple junctions through the tungsten body of the shield. Operation in this male severely limited the possible temperatures of the heat shield curtailing high temperature measurements for this model. Each shield operates under heat load so well that it was nat necessary to replace the unit even after eighty hours of high temperature operation.

All six water and gas lines into the source are

1/8"

nominal diameter nylon tubing. Connection to the bloek assembly is by way of

1/8"

copper tubing wi th brass swageloek fittings.

Two gas inlets were used to double the gasdynamic expansion ratio from throat to oven. This has the effect of lowering oven flow Mach numbers and the gas velocity in the oven, shortening the distance required for heat transfer.

3.5

Gas Flow System

A series of flowelements between the source and the gas reservoir bottle are used to monitor the gas flow to the source. Source pressure is read from a

100

torr Wallace and Tiernan gauge. A choked flow

0.08

mm. ruby control orifice provides a constant mass flow rate, provided the upstream pressure and temperature remain constant. Te:mperature remains within narrow limi ts as the room temperature is controlled by an air conditioning system. Since the gás regulator at the supply battle delivers a constant pressure demand flow rate, ~his condition is allowed to exist up to the control orifice. The flow rate

can be read from a Matheson 601 double float flowmeter tube, upstream of the control orifice. A

3750

torr Heise gauge monitors the" pressure at the outlet

\

of the flowmeter and upstre~ of the control orifice serving the dual purpose of monitoring the Choked flow condition and providing a calibration préssure so that the actual flow rate can be calculated from the flowmeter reading. Shut-off valves are located both upstre~ of' the control orifice and downstream of "the regulator •. Thus gas bottles can be isolated from the gas flow system when desired.

In one test, gas impurities evolving from cont~natedcopper tubing and sealant that penetrated the fittings were found to degrade the

source operation. Tbe centreline flux density sampled by the mass spectrom-eter showed ~ncreases as impurities deposited in the nozzle and on walls near the entrance to tbe nozzle. In the nozzle the obstruction which formed consisted of a crystalline structure with a microhole very close to the centreline.

By way of contrast, routine tests are run with the gas flow system cleanedwith alcohol and no such complications are observed. Instead a bright clean surface is maintained throughout the oven zone. Thus, by

operating the system with an eye for reutine cleanliness and proper main-tenance p~ocedures this deposi~ion phenomena can be avoided.

What, "then, is the process by which impurities beeome attached in the nozzle? It is suspected that hydrocarbon impurities aredissociated in the reducing atmosphere to react with tungsten vapour near the oven wall surface near and in the nozzle. The drop in wall telllPerature near the nozzle ex! t plane would favour depQsi"tion and crystal.lizaticm of tungstenie carbides and oxides. Tbe .presence of atomic hydrogen promo"tes reduction. Tbe ordi ... narily l~ar boundary la;yer becomes turbulent, enhancing deposition even more, when the first appearance of deposits triggers the destruction of l~arflQw. This effect should accelerate the rate of deposition until only a smal! ehannel is left near the nozzle centreline.

3.6

Tungsten Tubes

Tungsten's high melting point

(3655K)

makes it well suited as a refractory oven material. Its high strength and chemica! resistance are

compatible wi th the design

aims.

A ;reducing hydrogen environment or a vacuum is the best eondition under whieh to heat tungsten. Tungsten offers. less permeation to hydrogen than any other high temperature ma"terial in use today. For typica! source conditions of

100

torr and

3000K,

the loss due to permea-tion is less than

7

x

10-

4 torr

t

sec-l at flow rates near

2.0

torr

t

sec-l •

~see Ref. ~3). Thus this loss is less than

.04%.

Tubes can be specified in customer requireddimensions and our s~plier guarantees at least

99.99%

purity by virtue of its vapour d.eposition process. Thus even impurities in the tube are not more than

.01%.

Tbe tube size is selected for long operation lifetime where life-time is defined as .time taken to loose lCY'/o of the initia! mass. Operating temperature determines the evaporation rate (Refs.

20, 21, 22)

so·that mere massive tubes provide longer operation times. However, the ratio of length to cross sectiQnal area determ:l.nes the operating re si stance , whieh, for the tube size used in this report, is

17.4

milli-ohms at

3000K.

Note that at

2700K

ali>out

145

amperes were required at

2.10

volts indicating ali>out

14.5

milli-ohms resistance. The actual resistance will dep end strie"tly on the

temperature distribution so that this method could not be used to determine a source temperature . That is, integration of resistance over the tube length to determine aresistanee value to indicate the operating temperature requires a knowledge of the temperature distribution along the length of the tube.

Typical tube dimensions are a 2.08 ImIl inner diameter with a 1.0 ImIl wall thickness. Nozzle sizes used are 0.355 mm and 0.463 ImIl. Tube walls

were not ground down to provide a true orifice. Instead a nozzle was preferred so that long term use would not enlarge the nozzle diameter. Consequently the length to diameter ratio~ are from 2.7: 1 to 2.2: 1 resulting in some boundary layer growth and some beam attenuation.

A power supply was. built to deliver up to 1000 amperes current with a maximum power' rating of 1500 VA. The primary of the .control transformer was adjusted using a powerstat ." Output monitoring was done on built-in voltmeter and rummeter gauges.

4 •

GAS DYNAMICS OF DISSOCIATING FLOWS 4.1 Nature of the Gas Flow

Both 'the equilibrium and nonequilibrium nature of the gas flow are significant considerations in the determination of gas dynamic processes in the transition from the cold zone to the high telI!Perature oven and the nozzle throat region. The two zones requiring careful description are the oven entrance transition and the nozzle itself. The former involves conditions where the gas must come to a state of equilibrium at the oven wall temperature • However, the latter involves nonequilibrium processes and nozzle flow may

proceed under one or more frozen degrees of freedom due to the short expansion time scale.

For hydrogen diluted with a helium carrier gas, dissociation

equilibrium can be calculated if the assumption is made that heliUm. does not interfere with the forward and backward equilibrium reaction rates. Because of the large dilution used, namely ~ H2 in He, the transport properties are essentially those of monatomc helium. Due to the small hydrogen partial pressure, a high degree of dissociation can be achieved and the nonequilibrium features of reacting hydrogen nozzle flow can be arrested by the dominanee of the monatomic carrier gas. That is, the variable ex does net play an important role in the gasdynamic expansion. These points are elaborated in upcoming sections.

Nat urally , thenature ·of the gas used can be chosen to suit various requirements . For instanee , in order to achieve a high beam veloei ty, the highest source telI!Peratures and pressures are necessary at a low molecular weight. However, the maximum in-flight atomie hydrogen number density results under eonditions of 10CY'/o hydrogen source gas flow in which there is a low source pressure at the highest availa:ble temperature . But in order to satisfy both requirements of high beam velocity and high atomie hydrogen number density means that a compromise can be made. Usually, pressure does not influenee the terminal beam velocity to a great extent at very high telI!Peratures (near 3000K). Thus there exists a pressure for a given tempera-ture where both requirements are met. Note that the use of 100% hydrogen is

ideally suited for low molecular weight operation and thus high beam velocity with accompanying benefits of high in-flight atomie hydrogen number density.

In

the sourçe operation in this paper, lower partial pressures of hydrogen were used. A high degree of dissociation can be easily detected at a sacrificeto velocity.

To visualize the previous opening remarks, refer to Tables VI and VII which show how dilution affects both realistic and theoretical beam velo ei ties and how i t affects the atomie hydrogen nwnber densi ty . Brief mention will be made here of these calculations but they are of more signi-ficanee in the caming section on the discussion of results and investigation of the source potential. However, for the existing system, typical veloeities can be near

9.1

km sec-l at a corresponding in-flight atomie hydrogen number fraction of

.39

based on total density. Alternatively, with a

2!'/0

H2 dilution in He, at a velocity of

6.7

km sec-l the .dissociation partial fraction can be

.93

with respect to total hydrogen, corresponding to only

.019

of the total

densi ty. These conditions are valid for

3000K

and,.

80

torr.

4.2

E~uilibrium Flow

1

f

It is worthwhile to examine the nature of the equilibrium flow of a dissociating gas in order to develop relatiens useful in the prediction of beam veloeities and state in the free jet flow field.

Consider a pure diatamic homonuclear gas in one dimensional steady flow in the dissociation-recombination equilibri~ reaction:

(1)

without considering the rate processes leading to the equilibrium gas state. The assumption of isentropic flow can simplify the flowanalysis. Such isentropy is justified for several reasons that are explained now. Provision of an adequate length of heated oven ensures that heat addition to the gas is complete within a. very short distanee of the oven entrance .With the oven at an essentially uniform temperature , thermal gradients wi thin the flow are not problematic af ter heat addition is complete. Flow Reynolds nwnber is greater. than unity so that inertial farces dominate vis.cous forces. In order that the internal energy can be simply described, molecular vibra-tions are assumed harmonie. To summarize, an isentropic flow region exists following the heat addition region, where entropy is known to risee

A complete description of the state of the gas is provided by the

equation of state and t~e internal energy per gram of gas (Ref.

18):

;e

=

(1 +

ex)

RT

p ~

e=e +e +e +e

trans vib rot diss

7

(2)

where m is the gram molecular weightof: the diatomic gas. Since the average energies per gram of gas f:or each mode of energy storage are proportional to T and CX, then only T and

ex

are the independent variables for the state

description of the gas. The degree of dissociation,

ex,

is defined (Ref. 18) in a pure gas as the ratio of the mass of atomie species to that of the total

mass of the gas (Ref. 28). .

The individual energies can be written (Ref. 18):

e

=

J

RT (1 +

ex)

trans 2 ~

(4)

e = RT (1

~ ex)

rot ~ R e _diss

=

m.... _~ aD

ex

.

(6)

and e vib = (1 -

ex)e

where

e

is the vibrational energy per gram of the diatomic gas and aD is the characteristic dissociation temperature. For a system of harmonie

oscillators at a temperature T, in equilibrium with themse1ves and the trans1ational degree of freedom:

(8)

Therefore the average total internal energy per gram of gas is:

e =

~

(5

+ex

T a

)

+ ex

aD

+

(1 -

ex)

v 2 ajT e - 1

Total average entha1py per gram is h, where

hl::..e

+R

p

(10)

so that h =

~

(

7

+

:P

T

+ cx

a

+

(1 -

ex)

a

)

v ~ 2 D ajT e - 1 (11)

The flow is not completely described unless the gasdynamic conser-vation equations are satisfied. Theyare, assuming no heat addition:

Energy: u du + dh = 0

(12)

Momentum.: p u du + dp

=

0

(13)

Mass continuity: dep u A) = 0

(14)

The independent variables are p, u,

a

and T and there are three equations to describe the state, namely

(14), (13),

and

(12),

where

(e),

(9) and

(11)

define pep,

a,

T),

e(a,

T) and h(a, T). The statement of isentropic flow defines the entropy and provides the closure for the above set of equations. In equilibrium. flow, the differential entropy can be derived from. Heims

(Ref. 18): .

-ejT

ds

= én

{$n [ (1 -

Ct) c

Jr,y-

el] _

;n

}

(15)

so that, as a consequence of the isentropic assumption 2 -ejT -e

IT

.

l~a=c.JT(l-e

)e D /:).Kp(T) (16)

where the constant cis:

(17)

Substitute (2) and the regular form of the dissociational· equilibrium. equation is found:

4

2

4

I

-ejT -e

IT

p:x

= Re T

3

2 (1 _ e ) e D ~ TC (T) 1 _

a

2 ~ --p (18) so that (19)

Elementary statistical mechanics analysis also result in the same solution for this equilibrium. constant.

The local polytropic exponent, "I, relates pressure and density variations in the flow (see Ref. 18):

_ d

.en

p

"/ - d

in

p

(20)

Here, "I is equivalent to the variation of gas enthalpy wi th internal energy:

dh

" / = -_de

(21)

The equiv.alence is automatically trivial if one combines the energy and momentum equations

(12)

and

(13)

and applies the definition of enthalpy,

(10) .

An explicit form of "/ is formed by fin ding the differential of h(a, T) and e(a, T) in terms· of the independent variables:

Then R8 db

=

de +!L 7 +

3:x

dT+ J RT ctx +

2

ctx

v

~ 2 2~ ~

(7

+

3:X)

2~ de + OT + 28 ) ctx + _

-2.

D dT R dT

"/ =

---~--~--~~~--ctx 2~ dey

(5

+~) + (T + _{28D) dT}

+

R

dT

(22)

(24)

In a reacting, vibrationally excited gas in which both modes of erergy storage participate in the flow process, bath ctx/dT and dev/dT are nonzero. If, however, bath dissociation and vibration cannot achieve equilibrium, that is in a nonequilibrium situation, then these derivatives must be zero, and:

(25)

recovering the value of "/ which is based on the ratio of frezen specific heats. That is, the specific heats must remain constant in this si tuation and the gas must be non-reacting. Only then will

c

"/ =...R-

_c

p

(26)

This still holds even when chemistry does not change and vibration is in equilibrium with translation due to the nature of isentropic flow. It does not hold when the reaction is in equilibrium with translation.

,--- -

-The throat velocity of the nozzle is still in the usual form. Combining the enthalpy definition (10) plus the conservation equation (12),

(13)

and

(14)

with dA

=

0 at the throat easily yields:

( _p_* _ _ _ u* ) du

=

0 p* u* ï'

*

(27)

SQ that 2

n*

U

=

'V ~

*

I IJ*"

(28)

As

this is a consequence of the work done by the gas, this definition does not require equilibriUlll flow, but the flow must be isentropic.

The r"atio of specific heats for total equilibrium flow are found fram the definitions of specific heat per unit mass of gas:

c

=

~

I

and c

~

I

(29)

p p p p

Substitute h(a, T) and e(a, T) from (11) and

(9).

Thus:

Cp =

~

[

7

~

)Ct

+

(~T

+aD - € )

~Ip

+

(1 - a)Br/

C.~~T

1

feB/T]

(30)

cp

~ ~

[

5 ;

a

+

0

+ _{BD -} € )

~

lp + (1 -

a)

'1J

2

C.

~~T

_ 1

fe

BjT]

(31)

where

are found by partial differentiation of equations (16) and (19). Note that in

(29),

the ratio of specific heats cannot equal the local polytropie exponent since: d

Oe

..1m +

Oe

,;:vv e = ~ I..U.

di

l.6.N

:11

(32)

(33)

while

(29)

holds. We identif'y cp and cp only in a stricter sense with constant pressure and constant density processes.

It is in the nozzle where importantprocesses takes place. Here, the equilibrium Cp and cp are irrelevant, as nozzle flow is highly nonequi-librium in the instance of the scale of our nozzles.

4.3

Nonequilibrium Flow

The nonequilibrium processes occurring in the throat determine the nature of the gas~amic expansion characteristics into the skilllIOOr chamber, influencing the character of the free jet.

The basic postuiate is that of local thermo~a.mic equilibrium (see, for example, Tirumalesa, Ref.

37).

This way, the thermo~amic quant1ties of the nonequilibrium system are the identical functions 'of the local state variables as already exist for equilibrium flow. A consequence of the postulate is that within each degree of freedom a Boltzmann distribu-tion exists for each, and consequently a temperature and entropy as well as

all average energy is def1ned for each degree of freedom. In this manner,

all the local temperatures for vibration, Tv, rotation, Tr , chemistry, Tc, and translation, Tt, enter as independent variables which are useful in describ1ng the energy per grrum of gas for each degree of freedom. Suppose further that equilibrium between classes or degrees of freedam is controlled by rate processes stated in terms of the state variables characterizing the partial equilibrium of these classes (Ref.

37).

Since the translational

class is a single class and the internal classes are specified by their respective temperatures and composition variables alone, only translational temperature and pressure are to be used in the equation of state. Consider

only vibrational and dbemical nonequilibrium.

The fuil form of the differential entropy is gi ven by Heims as (Ref.

18):

ds = dey

(~y

-

~

)

+

~

0Ci { .en

l

(1 -

a)

c

.fi

~

-

e -

9v/

T

y) ] _

;n }

(34)

·

Note that for an arbi trary energy change dq dq ds =

T

This leads to the identification of the chemical temper.ature,

(35)

9? =

{..en [

(1 -

a)c

.JE

(1 -

e-

Sv/

T)

l _

aD }-l (36)

c

'

J

~

T

Four limiting cases of isentropic flow result fram the above entropy equation (see Refs. 18,

37).

Since

ev

as expressed in

(7)

leads to:

de

=

(1 - a)dE - Eda v

then both the total vibrational energy per gram of gas,

_{ev '}

or the vibrational energy per gram of diatomic gas, E, will vary. The systell).s are:

1. FuJ.ly Frozen Flow;

del

=

0, dey

=

0 so that bath Tv and a are constant. At low temperatures, or when vibrational relaxation is too slow to introduce any energy change, this wiil occur.

2. Partially Frozen Flows

(a) Tv

=

T and del

= o.

If the chemical relaxation time scale is too

long or where temperatures are too low, then chemistry is frozen. However, vibrational relaxation must always be able to occur.

(b) dey = 0. . Vibrational energy is constant, but the chemical reaction is in equilibrium. This is physically unrealistic since an increase in a wouJ.d imply an increase in E, while dissociation increases favour the

depletion of the average vibrational energy per gram of diatomic gas. (c) dE

=

0 with chemical equilibrium. Vibration temperature is constant but the average vibrational energy can change to acconnnodate a change in

degree of dissociation. The nonequilibrium entropy differential (34)

becomes, upon stibstitution of

(37)

and the condition dE

=

0: 2

pa

1 -

a

3.

Full Equilibrium Flow

_ (8n- EIIl2/R)

T

Relaxation times are short compared to the dominant physical scale time. Now the usual case resuJ.ts, where T

=

Tv and (16) hOlds, so that the analysis of Section 4.2 must be followed.

The Damköhler Number and Relaxation Times

In order to formalize the concept of relaxation times, the

Damköhler number, D, used by Zierep (Ref. 42) relates the time· SCale of the rate of energy transfer, t r , to that of the physical transit time, t s , of the physical dimension,

i.

_{s .}

where

D ;::

t /t

s r

t =

t

/u

s s (40)

and u is the bulk flow velocity. So if D » 1 , equilibrium is reached almost instantaneously, whereas if D

«

1 the gas cannot respond to the process that

is rela.xing ahd the flow ean be considered frozen in that energy mode. Dove and Teitelbaum (Ref. 9) have fitted experimental da'ta in Landau-Teller plots (Ref. 38). They used the laser schlieren 'technique in incident shock waves at l350K to 3000K to find vibrational relaxation times for H2 by H2' He, Ne, Ar and Kr. For 'this werk, the relaxation 'time will be that of infini'te hydrogen dilution in helium:

log(pt

r)

=

(41.35 ±

0.80)T-l_{/ 3 - (8.984 ± 0.63) (41)}

where p is 'the total pressure expressed in atmospheres, and tr is the relaxation 'time in seconds. The limits shown encompass experimental error and indica'te that the value of tr is uncertain within a factor of about five with their data.

Estimates of D for vibration ean be made assuming for now that the ehemistry and vi bra'tional energy are frozen in an isentropic flow. Then the ef'fective '1 is that of a monatomie gas. Leipmann and Roshko (Ref. 28) give the throat isentropie s'tate variable relation which for '1

=

5/S. beeomes:

T* 2

=

= 0.750 (42)

2

-~

= (

r

~

1 fl

=

0.487

(43) 1

P*=(

2 )'1-1 =0.650 Po '1 + 1 (44)

Consider one possible state of the gas in order to describe the relaxation proeesses in the oven and the throat. Take To

=

3000K and Po

=

100 torr. Then for vibrational relaxation, the Da.mköhler number can be

caleulated based on a throat diameter of 0.0355 cm. See the accompanying table for values. Throat velocity is caleula'ted from equation (28).

Vibrational Relaxation based on Da.mköhler Numbers Expansion ratio of 68.7, d*

=

0.03 cm OVen Throat t _r 5.81 x 10 -6 sec 2.32 x 10-5 sec

t

_s 3.5 cm .04 cm 't 1.32 x'10-3 sec 1.43 x 10-7 sec s D

₌

t /t 227

»

1 6.2 x 10- 3

«

1 s r

Calculations show vibrational "~nergy to be f'rozen in the throat. This is in agreement wi th Gallagher and

Fenn

(Ref's . 15, 16) who f'ound

analytically that "there was essentially no vibra"tional relaxa"tion in a case where To = 2000K and Po

=

2000 torr. Their parameters indicate a throat Damkohler number near 0.09. Thus, in" our typical case, vibra"tion must reinain f'rozen at the oven value even in the throat exitregion .

Note that equilibrium in the oven is aJ.most immediate f'or vibrational rela.xa"tion. A scale length of' 0.02 cm will give a D of' unity. This suggests that wall collisions introduce large vibrational energy rearrangements. Since the colli sion f'requency is near 105 sec-l and since the path length is .02 cm, then only 513 collisions correspond to D = 1. In the same time period, gas molecules experience 11080 wal1 collisions • Thus we can see that wal1

collisions channel oven temperature equilibrium into the gas phase very rapidly.

The question of' the extent of' the rotational and translational energies f'ram the throat and into the continuum portion of the f'ree jet expansion can be estimated assuming frozen chemistry. App1ying the same analysis as for vibrational energy, a Da.mlêóhler nUlIlber can be established for the typical expansion process. Jonkman and Ertas (Ref. 19) give trans-lational-rotational rela.xation times f'or gaseous mixtures of parahydrogen and noble gases at low temperatures. Based on the 1inearity of the inverse re1a.xation time with the mole fraction of' helium (Ref's. 9, 19, 23), trP at l70K f'or a

9&10

helium,

2!'/0

normal hydrogen mixture becomes:

so that and

-8

t~

=

1.815 x 10 atm sec 1.38 x 10-7 sec at p

=

100 torr t*

=

2.83 x 10-7 sec at p

=

48.7 torr r (45)

the former re1a.xation time representing the typical high tempera"ture process in the oven and the latter, that in the throat. Using the same values of t s as previ0us1y, then the respective Damkohler nuIDbers are

D~ot

=

9568» 1 in the oven and

D* = 0".51 ,...., 0(1) in the throat over one nozz1e dia.nieter 1èngth rot

ás indicated in the accompanying table. Since the nozz1e length to diameter ratio in the typical tungsten tube is 2.7:1, the result over a nozz1e 1ength is that D~ot ,., 1.0 and still it isof' order unity. However, it should be

noted that although trP is given at 170K, f'or higher temperatures trP decreases (Ref'. 19), and thus these values of' Drot' as stated, are 10wer limits.

Papers by Gallagher and Fen (Refs. 15, 16) on rotational re1axation of' molecular hydrogen are a source of higher temperature data on the rotational

Rotational Re1axation based on DaJDkb"hler NunDers

{same

typical gas state}

Oven Throat t 1.38 x 10-7 sec 2.83 x 10-7 sec r t 1.32 x 10- 3 sec 1.43 x 10-7 sec s D :;: t /t 9568

»

1 0.51 ..., 0(1) s r

temperature dependence on the Po<4 of the molecular beam source. In this instance, for Po~ _{of 30.5 torr-mm at a To of 1900K, a va1ue adjusted on the}

basis of constant mass flow rate equation (69), an extrapo1ation to this 10w value yie1ds Tr/To ~ 0.95. The values of Tr/To at the same Po

c4

increase

with

stagnation temperature increases despite the increase in Po for a constant flow rate, so that Tr must be very close to To at 3000K. In fact Gal1agher and Fenn (Ref. 15) in an earlier paper gave Runge-Kutta solutions for axisymmetric expansions of molecular hydrogen at room temperature. In the typical case, if tbe source stagnation teII!Perature were 300K, the pressure would be 31.6 torr and Po

c4 :;:

12.1 torr-mm. The authors I data gi ve Tr/To a value near 0.90 consistent with their experimental results . It is then safe to conc1ude that some rotational re1axation wi11 occur, but that the extent in the typical high temperature case is insignificant. To - Tr is on the order of 60K for To

=

3000K.

4.4 Heat Addition in the Nozz1e

The heat addition in the nozz1e, if severe, wi11 on1y effect the sonic point in the flow. Zierep (Ref. 42) studied f10ws in compressib1e media with heat addition, and in particular , in nozz1es. He indicates that a transition through sonic conditions without singularities is possib1e only if

!...dA=_l_~

A* dx C T* dx

P

(~6) ho1ds at M :;: 1. Thus sonic velocity can occur with heat addition, only in the divergent portion of the nozz1e, where dA/dx

>

O. Then, there are no prob1ems of a possib1e shock wave system being set up at the throat. The same conc1usion can be reached by rewriting the energy equation (12) as:

u du + dh :;: dq

(47)

This results in:

dA

=

E.

7 - 1 dq + du [~ _ 1]

. . . - - - _._--- - --

-where M is based on a frozen speed of sound:

where '1f is the effective frozen polytropic exponent. Therefore, from

(48),

at M = 1, with heat addition it follows that dA

>

0 in the region where the some point ean be found. At the throat the Mach number is subsonic . Thus in

(48),

for dA/A

=

0 at thLs point, heat can be added.

4.5

The Free Jet Freezing Phenomena

In order to have confidenee that the species measured in the mole-cu1.ar beam. are the same as those in the oven, the free jet freezing phenomena ,must be considered. As the flow expands from the nozzle, the Mach nUlnber

rises as the translational temperature drops. Energy, extracted from gas enthalpy is eonserved as an ordered translational motion • However, the expansion reaches a posi tion, for our low backg'round pressures, at whichthe mean free path becomes sufficiently large compared to the scale of the jet that collisional processes involving the translational degrees of freedom effectively cease. This approach of a eollisionless limit is imagined as oecurring in a translational freezing surface, at a definite distance from the throat related to the inverse source Knudsen nuni::>er. In fact, i t can be imagined that a similar process is undergone by the rotational and vibra-tional degrees of freedom at an earlier point in the expansion whose position is governed by the Da.mköhler number.

Within the continuum free jet, the Mach number variation is that established experimentally by Ashkenas and Sherman (Refs.

4,

10),

( )

'1-1 (

)1-'1

M=A

~*

+B

~*

'

(50)

where và.lues of A and B both depend on '1 as shown in the chart below:

'1 A B

5/3 3·31 -1.01

7/5

3.64

-1.21 9/7

4.00

-1.58

Previously, the Dämkohler nunil:>ers determined for the throat relaxation process indicated no vibrational relaxation or chemical recombination. Rotational relaxation was in doubt but data from Gallagher and Fenn had previously indicated that rotation is also frozen. The scale of the free jet is now determined by theposition of the translational freezing surface' at which a frozen Mach number is set by virtue of the frozen translational temperature and the existence of a frozen speed of sound. Based on the inverse source

-1

Knudsen number, Kno, there are two regimes of frozen Mach numbers. Scott (Ref'. 32) compares these terminal

Mr

relations for a monatomic gas. A two-thirds power variation is expected for Maxwell molecules. The Hamel and Willis formulation (Ref. 32):

(51)

holds for 10

<

Kn~l

<

50. A two fifths power variation was expected for hard sphere molecules by Anderson and Fenn (Ref'. 32)

(52)

f'or 50

<

Kn~l

<

2000. A criterion for freezing determines the constant. In any case using (50) and equating M to

M.f,

a value for (x/d*)f can be found which should indicate the location of the freezing surface.

Table V gives typical computation values for (x/d )1' using this criterion. Evan at best, f'or the gas used (based on

r

=

5/~),

high temper-ature values of' the position of' the freezing surface are less than 1.66 nozzle diameters. This value is the physical scale length

tso

The gas bulk velocity will be de:fined as the final velocity of the gas mixture at the freezing surf'ace. Ignoring rotational contributions from the 'ë!'/o hydrogen seed gas, an energy balance yields

1 2

'2

u e _trans

+E

_p

(53)

based on (4) and (10), since enthalpy must be conserved in isentropic flow (Ref. 28). Therefore ignoring the hydrogen contribution everywhere except in molecular weight: 2 R u

=

10 - (T - Tt) - 0 m

(54)

where To b,. Tk, and Tt is the translational temperature remaining with the gas translational degree of freedom. For an isentropic expansion, Tt is known from Mf (Ref. 28): T o T = ----.;;...".---=-t 1 +

r -

1 M 2 2 f'

(55)

Assume

r

=

5/3

f'or this process, since vibration and rotation do not relax in the nozzle, they should not be able to do so in 1.66 diameters or less. Take a typical value of high temperature velocity of 7.06 x 105 cm sec-l

or less. This is the worst case. Therefore, at the location of the freezing surf'ace,

t s

1.66

x

0.0355

-

---=~ =

7.06

x

105

-8

8.35

x

10

sec

The vibrational :r:elaxation time will be greater than in the throat due to the isentropic pressure drop, given as

In the best case

M.r

=

3.65

so that when Po =

107

torr, Pf =

1.45

torr. Based on this vaJ.ue of Pf and a corresponding temperature of Tt

=

536

PK

6

-2

(tr)vib

=

.7

x

10

sec from

(41)

{tr)rot

=

9.5

x

10-

6

sec from

(45)

This is tabulated below as a summary.

RotationaJ. Relaxation VibrationaJ. Relaxation

Freezing Surface Relaxation Characteristics Based on Damköhler Numbers

8

8.35

x

10-

sec t r

-6

9.5

x

10

sec

6

.7

x

10

-2 sec D

.0088

«

1

-6

1.2

x

10

«1

Consequently, rotation and vibration are frozen energy modes, and both rotation and vibration c:annot participate. In the next section, it will be shown that chemical relaxation times in a three body gas phase reaction greatly exceed vibrationaJ. relaxation times in the case of hydrogen. Thus we expect no ChemicaJ. recombination in th~ continuum portion of the free

jet.

4.6

SUIID1lary

For the typicaJ. high temperature case considered, it is possible to drawa. conclusion about the nature of the flow in the oven, and in the t~roat.

These conclusions can be extended into the initial c·ontinuum portion of the gas dynamic expansion into the vacuum if one assumes that a typical high temperature ·scale length is approximately less than 5 nozzle diameters.

Oven continuum gas flow is characterized by a highly equilibrium state of the vibrational, rotational and hence the translational energies. However, nozzle gas flow is distinctly nonequilibrium. Vibration and rotation remain frozen. On the as.s.umption that thespecies composition remainsfrozen, to be verified in the following section, then· the effecti ve polytropic exponent becomes equal to the ratio of specific heats and 7f

= 5/3. This justifies the original

assumptions of monatomiccompression for (42), (43) and (44). Assume as a typical scale length that ~s = 5 nozzle diameters. On this basis and on the assumption of a lower translational temperature coupled with a higher gas velocity , the Damkohler mmiber will still be less than uni ty. Thus hydrogen

should not undergo appreciable rotational relaxation. Vibrational relaxation cannot occur. Hence all that remains is to verify that the chemistry , in fact, remains frozen at the oven values.

5. HYDROOEN DISSOCIATION-RECOMBINATION

)

5.1 The Equilibrium Constant

Hydrogen dissociational equilibrium exists in the oven stagnation region. The ~low is definitely continuum so that intermolecular collisions are dominant. However, equilibrium with the wall temperature is rapid and each wall collision induces massive vibrational rearrangement (Ref. 33). Thus it is reasonable to expect from the former observations, that the experiment al individual rate constants for the forward and reverse processes forming the equilibrium may be instantaneously higher than the simple homogeneous equili-brium case. But no te that there is no reason to assume that this is the only

reason. Reddy (Ref. 31) notes th at the anharmonic specific heat with rotational-vibrational coupling exceeds that of a usual harmonic oscillator, serving to

drive the equilibrium constant higher by producing highly energetic vibrations which easily dissociate the molecule.

In Section 4.2 dissociational equilibrium is expressed through (18):

/

(18)

where C'

= 4c. The theoretical value of the constant C' for hydrogen is

C'

=

6.1052 atm K- 3/2 where e_v

₌

6324óK er 84.97loK e_D

=

519500K e _{f e}

See Table I for a comparison of the theoretical eq1..li.librium constant to those of'researchers. The equilibrium constant of Woolley (Ref.

39)

comes fram the National Bureau of Standards Series III of June

1948

and span a range from

10000K

to

5000

o

_K.

_{The theoretical constants exceed these values by almost}

a factor of

1.10

over most of the range. The equilibrium constant supplied by Wooliey fitted to the functional form: of equation

(18)

using the least squares error technique over the range of

24.00

o

_K

_{~ T}

=S

3400

0

K

yields C'

=

5.5389

atm K-

3/2

for the same values of

ev

and 9D as indicated above. The goodness of fit based on the correlation coefficient differed from unity only in the seventh

decimal place.

Lockwood, Helbig and Everhart (Ref. 27) xooasured the low pressure thermal dissociation of hydrogen in the presence of hot tungsten furnace walls. The authors suggest that the catalytic effect of tungsten on

dissocia-tion rate does not affect the recombinadissocia-tion rates as much. They showed an equilibrium constant valid over a temperature range from

1800

0

K

to

2200

o

K.

By their technique of angular scattering of kiloelectron volt protons from atomic and molecular hydrogen targets directly in the oven, equilibrium need not be established to determine the degree of dissociation. Nonetheless, their results do not markedly differ from those of Woolley (Ref.

39)

(see Table I).

The three body gas phase reaction was studied by Shui (Ref.

33)

by applying a generalized Monte Carlo technique. The reaction studied is:

kd,

H₂ +H F H+H+H

k

r

The rate coefficients are defined in the context of

.At equilibrium,

dl

dt 6. 0 and

(57)

(58)

(59)

Shui noted that the thermaldissociation and recombination of molecules proceeds via intermediate energy states in the three body gas phase reaction. This way, the distribution of vibrational energy states has significant effects on the kinetic process in the author 's simulation . However, the postulate of local equilibrium is assuxood. The author believes that the development of a gas phase equilibrium state proceeds in two stages. First, dissociation and recom-bination are negligible for a time on the order of the ~ibrational relaxation time. Then, in the second phase, most af the dissociation and recombination occurs (for temperatures less than 60000

K) over a much longer time periode

Consequently, dissociation depletes the population of excited vibrational states. Similarly, the contribution to the total rate of dissociation from low-lying vibrational states decreases as temperature is lowered.

Shui IS observation of the effect of different collision partners are noteworthy. For recombination, hydrogen atoms are much more efficient collision partners in promoting reaction than were inert gas atoms or hydrogen molecuJ.es. Compared to argon, the hydrogen atom recombination rate coefficient was some

4

to 15 times larger, whereas· hydrogen molecuJ.e rate coefficients were only 1.5 those of argon.

Shui quotes the data book of BauJ.ch, Drysdale, Horne and Lloyd (Ref.

33) who give evaluated rate coefficients over a temperature range of 3400 to 50000

K (modified to use moles instead of molecuJ.es):

(H)

4

30

-4

I

3

-1

-1

kd

=

.216 x 10 T

exp(

-51900 T) cm mole sec

(60)

valid to a factor of five uncertainty. Using these values in (59) resuJ.ts in the function: KC(H)(T)

=

3.2288 exp(-5l900/T) mole cm-

3

Since Kp(T)

=

RT Kc(T), then K

(H)(T)

=

264.94T exp(-5l900/T) atm p (61) (62)

Although the functional dependenee is different, the values of Kp (H) (T) do not differ much from those presented by Woolley and yet the Kp of the three body process is being used below i ts calcuJ.ated range of usef'ulness.

5.2 Chemical Relaxation Times

The extent of the reaction within the throat can be estimated using the rate coefficients presented in the paper by Shui (Ref. 33) mentioned in the previous section. Consider the reaction (57), then as concentration is proportional to number density and the number density can be written as

(Ref. 38): n =

E.

ex

1 m = pel -

ex)

n₂ 2m (63)

where m is the molecuJ.ar weight of the atomie species. Then (58) becomes: