High current discharges in a forced gas flow

High current discharges in a forced gas flow

Citation for published version (APA):

Andriessen, F. J. (1973). High current discharges in a forced gas flow. Technische Hogeschool Eindhoven.

https://doi.org/10.6100/IR67250

DOI:

10.6100/IR67250

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Published: 01/01/1973

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FORCED GAS FLOW

FORCED GAS FLOW

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE

TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE

HOGESCHOOL TE EINDHOVEN, OP GEZAG VAN DE

RECTOR MAGNIFICUS, PROF. DR. IR. G. VOSSERS,

VOOR EEN COMMISSIE AANGEWEZEN DOOR HET

COLLEGE VAN DEKANEN IN HET

OPENBAAR

TE

VERDEDIGEN OP DINSDAG 9 OKTOBER 1973 TE 16.00 UUR.

DOOR

FRANCISCUS JOZEF ANDRIESSEN

GEBOREN TE GOOR

The investigations reported in this thesis were carried out in the Labaratory for High Voltages and High Currents of the Eindhoven University of Technology, the Netherlands.

The author wishes to express his sineere gratitude to Prof.Dr. D.Th.J. ter Horst, head of this Laboratory, for his continuous encouragement, his many valuable suggestions and fruitful discusslons. He is furthermore mostly indebted to Mr. A.J. Aldenhoven for his assistance during the experiments, to

Mrs. M.E. Mulder - Woolcock for the translation of the manuscript, to Mr. F.M. van Gompel for the preparatien of the figures and to Mrs. L.F.P.A. Bleijerveld - van den Akker for her valuable assistance in the preparatien of the manuscript.

a a A A A , A T q B c

c

c

_p

cv

f e d D e e E ED E. I g G G( v) h h I V I V I p z Ig' I _m k k k, k,' k2 K

R:

velocity of sound

absorbed radiation energy cross sectien

transition probabi I ity turbulent mixing coefficients magnetic induction

velocity of 1 ight capacity

specific heat per unit mass at constant pressure specific heat per unit mass at constant volume mean thermal velocity of the electrens

diameter density

electronic charge emitted radiation energy electric field strength dissociation energy reduced ionization energy os c i 1 la tor strengt h statistica\ weight turbulent heat flux

normal ized Gaussian 1 ine shape function Planck's constant

enthalpy per unit mass

current

speetral radiation intens i ty

speetral blackbody radiation intensity

ionisation energy of a Z-fold ionized partiele excitation energies

Boltzmann's constant

normalized frequency deviation dimensionless numbers

proportional ity factor length

L L " L L(v} m

m

M M n n N p Pr p el p u p ->-conv q Q Q(v} r R R Re R _e T u

u

-+ V V V(v} va. V ac intensity

distance between the upstream electrode and the throat of the nozzle

relative intensity

normal ized lorentzian 1 ine shape function ma ss

mass flow rate

mass of the heavy particles Mach number

partiele density principal quanturn number integer

pressure Prandtl number electric power input

rad i a ted power

power loss by convection radiative energy flux

momenturn transfer cross-section normal ized 1 ine shape function radial variabie

optica! radius of the discharge

universa! gas constant Reyno 1 ds numbe r

radius of the outer free jet (exposure} time

absolute temperature radiative balance partition function velocity volume normal ized Voigt function vo 1 tage

x, y,

z

~·(

z

"

"

V B S

Bo

y Ye /!, E 0 E V T1

e

I( Kt À A ~0 V p Cl q, wee + Q z Cartesian coordinates eh a rge numbe r

effective nuclear charge

ratio of the half-half widths for Stark and Doppier broadening

speetral absarptien coefficient half-half width forStark broadening half-half width for Doppier broadening ratio of the specific heats

factor accounting for electron-electron interaction radius of the free jet

internal energy per unit mass vacuum permittivity

speetral emission coefficient dynamic viscosity

angle

molecular heat conductivity

turbulent heat conductivity wave length

Cou Iomb factor vacuum permeabi I ity frequency mass density electrical conductivity shear stress angle cyclotron frequency unit vector

Chapter 1-1 1-2 Chapte r I I I I -1 I 1-2 11-3 I 1-4 Chapter lil I I 1-1 I 11-2 I 11-3 111-4 I I 1-5 Chapter IV IV-1 IV-2 IV-3 IV-4 Chapter V V-1 V-2 Introduetion Genera 1

Purpose of the invest[gation

Plasma properties Introduetion Partiele densities

lnternal energy, enthalpy and specific neat Electrical conductivity 10 10 11 13 13 15 18 20

Emission and absorption of radlation in a high 25 pressure discharge

lntroduc~on 25

Free-free and free-bound emission and absarptien 27

The NI continuurn as a function of temperature 32 and pressure

Line absarptien and emission coefficients 37 NI and NI I 1 i nes 40

Experî.mental arrangements and techniques 44

The arcing device 44

lsentropic compressible flow through tubes with 46 variabie cross-sectien

Pressure measurements 49 Electric ei rcuit and are stability 55

Temperature measurements 64

Introduetion 64

Density measurements and calculation of the

65

relative radial distr[bution of the continuurn

inte

Chapter Vl-1 Vl-2 Vl-3 VIl Vll-1 VI 1-2

The energy equatlon The radiative balance

The effective heat conductivity

Turbulent heat transfer in forced blown dtscnarges Turbulent viscosîty and neat conductivity in an axi-symmetric compressióle free jet

Model for tne calculation of tne turbulent visco-77 82 88

90

90

9

4

sity and heat conduct i v i ty in a gas blast discharge VI 1-3 Numeri ca I ca I c u I at i on of

in gas blast discharges

Chapter VI 11 Summary and conclusions

List of references

Samenvatting en conclusies Curriculum vitae

temperature distributton

97

110

114 119 123

CHAPTER I.

I n trad uc ti on.

The proteetion of transmission and distribution systems for electrical energy from short-circuits and overloads, is usual ly obtained by means of electro-mechanical switching devices, commonly known as circuit breakers. Al 1 electro-mechanical switching devices contaJn a pair of electrical contacts, of which usually one is movable, the other being fixed. When the cantacts are in the closed position, two electrical circuits are connected tagether via a smal 1 contact resistance. lf, forsome reason, the current exceeds a given value, interruption is obtained after separating the contacts. During this interruption process, a current wil 1 be conducted through the medium between the two cantacts (gas, oi 1 or vacuum. dependent on the type) of the circuit breaker.

The current flows in the medium because, at the moment of separation of cantacts an electric discharge occurs with a certain electrical conductivity. The current in the circuit can be interrupted only by extinguishing the discharge in some way. The physical processes which take place during the interruption of the current should lead from a conducting medium to an insulating medium.

The interruption of an alternating current normally occurs in a very short period of time near a current zero. During this short peri ad only a smal 1 amount of energy per unit time is supplied to the discharge in the circuit breaker. During this short period, interruption of current can be achieved only when the energy dissipation of the discharge is so large that the electrical conductivity of the medium between the cantacts is reduced to such a level that the transient restriking voltage which appears between the cantacts as soon as the current is interrupted can be withstood. Under practical conditions, the whole interruption process takes a few hundreds of microseconds. lf one bears in mind that in modern electrical transmission systems voltages are applied of 50 to 80 kV r.m.s. per interruptor and that the r.m.s. short-circuit currents may reach values of tens of kA, and moreover, that the rate of rise of the restriking voltage

in these systems may be of the order of thousands of volts per microsecond, it wi 11 be apparant that very stringent requlrements must be satisfied by circuit breakers. The relevant physical processes occurring during the

interruption (referred to as current zero processes) are nat yet ful ly understood, despite intensive research and many theoretica! attempts to explain the interruption process.

This thesis describes a study of the behaviour of a discharge in a forced gas flow during the pre-current-zero process, that is to say, the investigation has been restricted to the period, the discharge shows a more or less stabie behaviour.

In practical designs of gas-blast circuit breakers, the forced gas-flow is aften obtained by allowing a gas to expand from a high toa low pressure through a single or double nozzle. The behaviour of a discharge during the pre-current-zero process is determined mainly by a number of physical processes which can be described by means of the energy balance. In discharges under the

influence of a forced gas flow energy is dissipated by: convection, molecular-and turbulent-conduction, expansion molecular-and radiation. In order to discover the

relative importance of the various dissipation mechanisms just mentfoned, one has to study theseparate mechanisms in detai 1. For thfs purpose, experi-mental conditions should be created in such a way that only une or two disst-pation mechanisms dominate. One of the dissidisst-pation mechanisms most intenstvely studied is molecular heat conduction. Most of the experfments were carrted out with long wali-stabil ized discharges with a very smalt gas flow. These discharges showthermal equilibrium at pressures of the order of one atmos-phere and higher (with the possible exception of the immediate vfcintty of the electrodes).

In these discharges the energy suppl ied per unit length is almast entirely dissipated radially to the wall by means of molecular heat conduction, with the exception of a smalt dissipation percentage as a result of radiation. lf the molecular heat conductivity of the gas is known as a function of temperature, then for given experimental conditions we are able to calculate the temperature distribution as a function of the radius of the dlscharge. Experimental verification can be carried out. Conversely, the experimental

temperature distributton can be used to calculate the molecular heat conduction as a functîon of temperature in a manner whîcn was probably used for the flrst time by Pflanz [1].

In the same way, an attempt has been made to study the losses resultlng from convection. Here, too, preferenee was gtven to long wall-stallîlîzed discharges with a lamlnar gas flow. The properties of discharges tn a lamlnar gas flow have been described at length in 1 iterature [2, 3, 4, S]. Furthermore, rrlnd et al. [6, 7, 8, 9] have studled the effect of turbulence of the heat transfer by conduction. Thelr experlments were carrled out on discharges in very long tubes (length up to 100 cm) and a diameter of 1 cm. The measurements were made on the fully developed flow region of the tube. Frind [6] showed that heat transfer by conductlon, either molecular or turbulent, was the only important mechanism. 1heir experiments showed that the heat transfer by turbulence may be ten times ~he transfer by molecular conduct ion.

However, when a discharge Is situated in a forcerl gas flo~, as is the case in many modern power circuit breakers, all the dissîpation mechantsms mentloned above, are active simultaneously, and it is diffîcult to dîvide the dlsslpation between the various mechantsms. One of the reasons ts the Interaction between the different mechanisms.

ln this thesis we have trled to throw some light on the quantitative effect of the varlous dissipation mechanlsms which occur in discharges tn a forced gas flow. The experlments were carried out with help of a model of a breaker consistlng of a single nozzle. The model of a breaker was destgned In such a way that ft could reptace the arc-exttnguisfling part of a commercial gas-blast circuit breaker fora rated voltage of 10 kV r.m.s.; a rated current of 400 Amp. r.m.s. and an interrupting current of about 23 kA r.m.s. per phase. The quenchlng medium used was dry air. The calculations of the varlous

thermodynamic and transport quantltles were carrled out for nitrogen. Since the differences between the· lonlzation energies and momenturn transfer cross-sectlons of oxygen and nitrogen [10] are very smalt, the simpt I ft cation from air to nitrogen Is justlfied toa flrst approxtmatlon.

CHAPTER I I .

Plasma Propertles.

The dlstribution of energy among the indivîdual particles of a plasma Is determined by the energy exchange wlth the surroundings - tncludtng the Interaction with external fields- and the energy transfer between the particles themselves.

A plasma is In local thermadynamie equilibrium (L.T.E.) îf:

a) Collision-induced transitlans and reactions are more frequent than radiative ones;

b) the relative changes in temperature and density are small wîthtn times and distances equal to the relaxation times and mean free paths of the particles under consideration;

c) external perturbing agencies, sufficiently streng to distart apprectably the velocity-distribution function of the charged particles (especially of the electrons), are net present.

In sufficlently dense plasmas (atmospheric pressure or htgherl the condl-tions mentioned in a) and b) are usually fulfilled Til, 12].

The condition stated In c) wilt be considered more in detatl for the case there is an externally applied electric field.

In ~n electric discharge the charged particles wilt suffer accelerations

+

due to the electric field, resulting in a drift velocity vd in the direction of the electric field, superimposed on theîr thermal (randoml velocity ~. Therefore the mean kinetic energy of the charged particles can be higher than the mean kinetic energy of the neutral parttcles.

In a discharge at a pressure of several atmospheres this effect w{ll ne

of minor importance as far as the ions are concerned, stnce the colltston frequency, colt isions between !ons and neutral particles, is high (of the

8

order of 10 per partiele per unit time) and the mass of the tons ts of the same order as that of the neutral particles. In first approximatton tne mean kinetic energy of the ions and the neutral particles wilt be equal.

However, electrens in elastic collislons with neutral particles wil\ transfer only a fraction of the energy gained from the electrtc field to

the neutral particles, due totheir smalt mass. This fraction is propor-tienat to 2me/M, me and M being the masses of electron and neutral particle, respectively.

As a result the mean kinetic energy of the electrens wilt approach toa higher value than that of the neutral partlcles, so that equipartition of energy is no langer guaranteed, teading to an electron temperatur~ Te different from the neutral and ion temperature.

Toa first approximation the discharge can be regardedas a mixture of two gases, each with its own temperature, conslsting of heavy particles (neutra! and charged) with a temperature Tg and electrens with a temperature Te. The temperature Te can be derived from the mean kinetic energy of the electrens even when they do not possess a Maxwel lian velocity distribut ion. Te is not equal to Tg and in general Te wilt be higher than Tg.

lf the energy gained by the electrens from the electric field is transferred by means of elastic collisions to the heavy particles, the relative temperature di fference (T _e - T ) I T wi 11 be gl ven by: [13]

g e T - I ~= T M(IeeE) 2 4me(3/2kTe)2 where: Ie e k E e

the mean free path of the electrens [m] -19

electronic charge (1 .602 10 C)

Boltzmann's constant (1 .38 10-23 J °K- 1) -1

electric field strength Ivm ]

( 11-l)

In a high current discharge (several hundred amperes), situated în a super-sonic air stream at a pressure of a few atmospheres, the electric field strength is of the order 10 4 V/m (see chapter IV). With these data, the factor I eE/(3/2 kT) In equation (I 1-1) for temperatures of about 10 4 to

4

5

-3 -3

2 x 10 K wilt be respectlvely 4.5 x 10 and 2 x 10 approximately. For 4

a mass ratio M/me ~ 2.5 x 10 , the relative temperature dtfference wil! be approximately 0.1 and 0.03 respectlvely, which is negligible.

Another reasen whlch possibly could lnduce a dlsturbance of L.T.E. is radlation which escapes from the discharge without being absorbed. Calculations of the total radlation losses which occur in electrlc arcs under the above-mentloned expertmental conditons have shown, however, t&at

these Jasses represent only a fractîon (10% ar less) of the electrical input (see chapter Vl).

The conclusion may be that a discharge at a pressure of a few atmospheres 4

and wtth an electrlc field strength notmore than 10 V/m îs in first approximation in L.T.E. (With the exception of the immedtate vlcîntty of the electrodes).

lt should be noted that as early as 1934 this was proved by Ornstein and Brinkman [12] for discharges under simtlar experimental conditions.

Fora gas at high temperature which is in L.T.E .• the densities of the molecules and their dissociation products and the densitles of the molecules and atoms and their tonization productscan be calculated wîth th·e help of the mass action and Saha-Eggert equations.

Conslder a gaseaus mixture composed of the following particles:

molecules; slngly ionized molecules; atoms; lans (singly and multîply charged)

+ + ++

and electrons, denoted by A2, A2, A, A , A and e respectlvely.

The total partiele density is determined by the sum of theseparate denstties of the above -mentioned parttcles.

( 11-2)

Assumlng charge neutral lty, lt fol lows that the electron density "e wil! be glven by the equation:

( 11-3)

Fora system conslstlng of the particles 1 lsted above, the mass action and Saha equatlons will be as follows:

(nA)2 3/2 (UA)2 ('ll'mAkT) ( -E0/kT) h3 - - exp nA UA 2 2 ( 11-4) 3/2

n nA+ (211'mekT) 2UA+

~-

2

(-EA+/kT)

hl

'""'([""' exp

"A _A2 2

( 11-6)

( 11-7)

where:

E0; EA+ ; EA+en EA++ are the dissociation and reduced ionization energies

2

resp.

UA ; UA; UA+ ; UA+ and UA++ are the partition_ functions of the particles

ln~icated

bg the subscript, and h is Planck's constant. The reduced lonlzation energy can be given by:

(11-8)

in which I2 Is the lonlzatlon energy of a Z-fold ionized partlcle. The lowering of the lonlzation energy àiz' which is caused by electric mlcro-fields, can be calculated by means of the Debye-Hückel approximation

[14, 15, 16, 17, 18]. e3 lT l I 2 hi = 2 (Z + 1) _.::__...,)3:-/2 (kT) Z (4lTE 0 ( 11-9) where: the . 100 -11 F m -1) E vacuum permrttlvlty

(3!rr

10 0

n _e the density of the electrans [m-3] n.

I the density of the particles i wi th electric charge

z

1e

(Z i = 0 for neutrals;

z.

= 1 for single ionized particles; etc.)

I

With the help of equations (I 1-2) to (I 1-9), the partiele densities have been

calculated in {19] for a number of gases as a function of the temperature with pressure as a parameter. The results of these calculatlons for nitrogen at pressures of 3 and 10 atm.abs. are reproduced in figures I 1-1 and I 1-2 respectively.

The corresponding mass densities, calculated from:

p I n.m.

I I

I"''"'

~~ iti"OQIH'o (prew_n·l~tm. ~trs.J rl' 1::-1--

...

I \ / \ H ...

·"

"

\

\ N I~

•1

/I\

"

\ rl'

.,

I

\

.,

_\

_I

u

\I\

•

~ ll"'l()

"

0 5

"

15 20

,...,.,

Fig. 11-1. 10'

=±:,-t",...,'

\

10 0

\

M>atm.abs. '1. \ \

'

latm.llbs:'

'

'

\

'

lfi

'

...

'

""

...

r-....

Tc•)(! l ~ 10 0 12 16 Fig. 11-3. ~ ).. Nitrogen l.,r ... ~n .. k)Mm.ab&J

J'l"'""

_0::,...

\

wl

1\

IN.

··~

I~ 10"

IV

I

_-

1--'--

7'

1

I

.,

\

•..

ul'

I

\

1'\.

.1'

..

l Tc•1o 0 5

"

,.

20 >!i.." Fig. 11-2. F Jg. 11-1.

Partiele densittes of Nltrogen as a funct Ion of temperature

( pressure · = 3 atm. abs.) Flg. 11-2.

Idem. (pressure • 10 atm.abs.)

Flg. 11-3 •

Mass denstty of Nltrogen

The internal energy Es per unit mass of a partiele of the type s, ~~ glven by [20, 21]: al nU _ _ s_+

ar

(11-11} where:

3/2 kT z the klnetic part of the lnternal energy

Us the partition function of the partiele s

Es the contribution of the dissociatlon and fontzation energy

to the lnternal energy, depending on the considered particles The internal energy per unit mass of a gaseaus mixture is obtalned from the

lnternal energy per unit mass per partiele Es by multiplication by the corresponding mass concentratien Cs of the s-particles and subsequent summatien over all particles:

E • L

c

E s s s ( 11-12) where ( 11-13} s

Then for a gaseaus mixture consisting of the general particles of sectien (11-2) the sum of the termsof table 11-1, taken from !16] , give the total l nterna I energy.

The internat energy per unit mass for nitrogen at a pressure of 3 and l [ atm.abs. has been calculated wit~ the ~elp of the partiele densities and partition functlons of nitrogen frorn [19], the dissociatlon and ionizatlon energies are taken from [22, 23].

The results are shown in figures I 1-4 and I 1-5.

The corresponding enthalpy, h, per unit mass has been calculated by replacing the factor 3/2 by 5/2 intheserles of equations in table I 1-1. These results are a lso given in figure I 1-4 and I 1-5.

~lï:gible

tn the case of Nitrogen

~ee

flg. ll-1 and 11-2. (See table 11-1.)

TABLE 11-1. General particles Csts A2 "A 3 ___1. _p [3/2 kT + kT 2 _1i'f' In UA ] 2 A+ _nA+ •l 2 3 A+

--!'-

[3;2 kT + kT2 ä'f In UA; + E1 2 l A _nA 3 ED p Ph kT + kT2 ä'f In UA • y-l A+ _{n +} 3 ED A+ _A_ p Ph kT + kT2 _ä'f In UA++2+EI J A++

~(3/

.~.

+ A++ 3 A p 2 kT + kT2 _äf In UA++ ₂ EI + El J ele,trons n ~ _{Ph kT]} p whe re:

E0 = the disscociation energy, Ei is given by equations (11-8) and (11-9).

,"

_lhfJ/~1 f--. [IJ/kg!. -~ -Hi1r09fn lpteuurr-l.lotm.X~Ll

rOllig

I E!J/kgl

Nilrog.nlorth..,, IOM"'»s1

\0

.~

p.-•

\0

k:

~ 1:::= ~

•

1:::( ; !!:' IV lU

A

7 10

~

\0

'

IJ

I. /, (/

.

u

T1.-.c:1 0

'

•

12 16

"'

"

"'

10

'I

·~

-·

I T1•1o 0

•

12 lli

"'

"

""

\0 Fig. 11-4. F lg. 11-5.

•

10 ~~jlkf"'IO

'

10 r~

I

I

I l -10 ~ 0 4

•

12 Fig. 11-6. Jatm.~l.

/

~

~·tm.abt:. f(•l(l

"

20dl' F lg. 11-4 •

Internat energy and enthalpy of NI tragen as a functlon of ternl)erature

(pressure a 3 atm.abs.) Flg. 11-5.

Idem. (pressure·· 10 atm.abs.)

Flg. 11-6.

Speciflc heat per unit mass as a functlon of temperature for Nltrogen (pressure 3 and 10 atm.abs.)

The specific heat at constant pressure per unit mass CP is given by:

( 11-14)

The results of the calculations of CP as a function of the temperature for nitrogeri at a pressure of 3 and 10 atm.abs. are shown in ft gure I t-6.

The electrical conductivlty of a partlally ionized gaseaus mixture in L.T.E. is glven toa flrst approxlmatlon by [lb, 18, 24, 25, 26]:

n e2

e

0

=--

_{m v}

e e

( 11-15)

where e is the elementary charge of the electron, ve the colliston frequency of the electrons. The latter Is given by:

V e ë' e t e I {Kfmk.Te ( "k

_V----;;;;:-

_..

_{nko_;} k. _{+ ; nz~} z) ( 11-16)

where Ce ~nd

1

are the mean thenual veloctty and mean free patn of the free e 1 ec trens· respect tve I '(•

nk c density of the neutral partlcles· per unit volume (molecules and atoms)

Qk the electron momenturn transfer cross sectton for the neutral parti-e

cl es [m2]

nz the density of the Z-fold charged ions (Z ~ 1)

Qz the electron momenturn transfer cross sectien for the Z-fold charged e

i ons •

. In the expression for the electrical conductivity according to equat!on (I 1-15), the following assumptions have been made: The electrical conduc-tion depends entirely upon the diffusion mechanism of charged particles (charge exchange due to coll isions has been neglected).

Because of the relatively high mobil ity of the electrens as compared wit ions, the contrlbutlon of the latter to the electrical conductivity has been neglected. The lnfluence of 'the self-magnetic field of the discharge upon the electrical conductivity can be neglected, provlded the cyclotron frequency- w - is small wlth respect to the col! ision-frequency v of the electronscf27, 24]. Th is condition is satisfied*). The electron-!omen-tum transfer cross-sections for molecular and atomie particles, the effect of which on the electrical conductivity is dominant at lower temperatures (see fig. 11-1) can be obtained from the literature {28, 29, 30, 31, 32). The electron momenturn transfer cross-sections for nitrogen molecules QN2

e taken from Brown [33) and Frost et al. [30] have been reproduced in fig. I 1-7 as a functlon of the electron energy. This il lustrates the good agreement between the calculated and experimentally determined values. For the electron-momenturn transfer cross-sections for nitrogen atoms QN,

e only theoretica! estimates are available in the

1

iterature

[31, 32, 33].

The calculated va lues from

[31, 32),

which showthebest agreement, have been reproduced in fig. l l-8 as a function of the electron energy.

~ _{The azimuthal magnetic induction B~ occurring here is of the order of}

0.1 Tesla (see chapter VIl). Wlth w =eB it fol lows that w

~

I .5 x 10 10

ce m ce

rad/sec. In a discharge In nitrogen at a ~ressure of several atmospheres at temperatures of 104 and 2 x 1c4 °K, the electron col! is!on frequency ve wil! be = 5 x 10 11 and 1012 respectively, so that wee << ve.

At sufficiently high temperatures the gas reaches the fully tonized state. (For nttrogen at a pressure of a few atmospheres, thts occurs at about 2 x 104 °K, see figures 11-1 and I 1-2).

This being the case the electrtcal conductivtty ts detenmtned solely óy

the electron-ion, electron-electron, and ion-ton interactions.

,..,

11\. 24 o';.J,I _I!\ 20

_/_

"

~\ 12

Jl

1\

'1

"

-

..

,

-8 _{, .} ."..

'

0 a-lectron tnt~l I

"

10 F lg. 11-7.

Electron momenturn transfer cross sectlons formolecular Nltrogen as a functlon of electron energy,

-taken from [33] (theory)

taken from [30] (theory) --- taken from (33] (e•perlment)

l~!,J) \ !-""--

_...

12 ~ electron •nergyceVJ o I ₁₀ FIg. 11-8.

Electron mome-nturn transfer cross sectlons for atomie: Nltrogen as a functlon of electron energy.

- taken from [31] -·- taken from [32]

10'

Spltzer and Härm [34] have derlved an expressJon for the electrical con-ductivlty of a fully lonized gas whlch has obtalned a ce~taln expertmental

justification [25, 351.

Fora fully slngly ionlzed gas (Z 1) this expression Is given by [34, 36]:

2 a =

-/iil

_e

( 11-17)

where lnA for singly charged lons (Z = 1) is given by [36]:

[ [kT 4 11_{e: ]}3_{/ 2 ]} lnA 2 In - 3- - - -0

IBïi"rï'

e2 e ( 11-18)

The factor ye in equation (I 1-17) takes into account the effect of the electron-electron interaction; For slngly charged ions Ye has the value 0. 581 6 [34] .

When the electrical conductivity is calculated according to equation (1 1-15) the electron-electron interaction is neglècted.

This is compensated by introductng an "effective" electron momenturn transfer

cross sectien for tne Z-fold charged ions Qz I16]. The· effective cross

sec-e

tion Q1 _{(Z = 1) is obtained for the case of a practically fully singly}

e

ionized gas, i.e. for negligîble densities of the neutral partic les, by

putting equation (11-15) equal to equatlon (11-17).

Th is glves for the effectlve cross sectien ~ (Z = 1):

3;

2 _{ln A lT} 2 I Ql ~ ( e )2

2

) e ~ _{2/3(kT)2 Ye} 0 ( 11-19)

Substituting the numerical values of the contstants e, E 0 , k and ye In

( I I -19) g i ves:

( 11-20)

Using equations (11-15) and (11-20) and the electron momenturn transfer cross sections for the neutral particles QN2 and QN from flgures I 1-7 and I 1-8,

e e

the electrical conductlvity of nitrogen as a function of the temperature wlth pressure as a parameter has been calculated, the results of which are shown

in fig. I 1-9. The partiele densities used in these calculations have been

taken from [19].

'

lCJ(Q,...)., 10 )

i//!

I I

.,

'

10 I 'I ·I

~

I

'·

11

•

I ~~

:-.:....:

~~

-

·

I",_,

""'

I 12

•

lO.ID'

--~ ~ Fl g. 11-9.

Electrlcal conductlvl ty of Nltrogen as a function of

temperature.

- p .. 1 atm.abs.;

..... p .. 3 atm.abs.; --- p = 10 atm.abs.;

- p • I atm.abs. taken from

For comparatlve purposes, fig. I 1-9 a lso includes the electr[cal conductivity of ni trogen at atmospfterlc press-ure as found experlmentally by Hermann and

Schade

[37]

by measurements on cascade arcs.

Thls shows a very good agreement wlth the values we have calculated for the electrical conductivity at temperatures higher than approxlmately 9000 °K.

CHAPTER I I I .

Emission and absarptien of radfation tn a high pressure d[scharge.

lnvestigations carried'out on wali-stabil ized e\ectric arcs in n[trogen and argon [38, 39, 40] have shown that radiative energy transfer is na langer negl igible when the central temperature rises above about 12,000 °K. In

particular, the energy transfer by means of ultra-violet radlation (À< 2000 ~). which is subject to reabsorption in the discharge, plays an important part

in the total energy balance of the discharge.

Ta what extent the radiative energy transfer in a cylindrical discharge of high current intensity (several hundreds of amperes) at high pressure (several atmospheres) situated in a streng, axially-directed gas stream, as described in chapter IV, influences the total energy balance deserves

closer consideration.

In the general case, where reabsorption of emitted radlation in a discharge which is in L.T.E. is nat negligible, the calculation of the radiatlve energy transfer is basedon the stationary radiative transfer equation whlch is given by [41, 42]:

...

n grad I" a I [I - 1\)]

\) "p ( 111-1)

in which the unit vector

n

indicates the propagation direction of the radiatlon, I" is the intensity of the radlatton with frequency v, l"p Is the intensity of the black body radlatton and a~ the (spectra\) absarptien coefficient,

corrected for the induced emission. (a~=

a"

{1 - exp (-hv/kT)}).

The contribut ion made by radlation from al 1 directlans to the total radiatlve energy balance at a certain point, is obtained by integrating equation

(I I 1-1) over the total solid angle 4w.

Wi th

n

grad I = div (lll ) , (diva

=

0) this gives:

\) \)

J

div 4w div

_J

e dn-

J

4w " 4w a' I dn \) \) ( 111-2)

...

E

\) a.' I \) l)p is the (spectra!) emission coefficlent.

The term

f

Q

411

+

lvdn represents the speetral flux vector qv.

The two terms on the right-hand si de of equation (I I 1-2) represent respec-tively the total emitted and the total absorbed radlation energy per unit volume, time and frequency; these are indicated by ev and av respectîvely. Obvlously (I I 1-2) can now be written as:

dlv

q

u

\) ( 111-3)

Here the term u is the balance between the emitted and absorbed radlation \)

energy per unit volume, time and frequency.

Assumlng that the coefficients E _{and "v are isotropic, it can be shown that}

\)

point r ; 0 the + and given by [39]:

at a terms q\), a e are \) \) + ( r 0)

_{f f f}

:n

E (-;)

r

1 ( 11 1-4) ql) exp (- a.' dp) -dV \) o v r2 a ( r 0) a.' ( r

_O)

f f f

E

m

exp (-

r

a' dp) .l_ dV ( 111-5) \) \) \) \) r2 0 0) 0) ( 111-6)

One can distlnguish two extremes:

a) Where the absarptien coefficient a~ is very large, sa that the mean free pathIv of the photons (Iv; 1/a.~) is so smal I that at a gîven point ( r ; 0) the only radlation arriving wil! be that from the immediate neighbourhood of that point, for which Ev is practically constant. Then from equations (111-4, 5, 6) with E _\) (-;) ; E _\) ( r ; 0) and a_\) ' (-;) ; a.' _\)

( r ; 0) fellows:

e -\) a \) "' 0 (equilibrium radiatlon) ( 111-7)

b) For very smal 1 values of a~ the situation may occur in which in a medium of limited extent the absarptien per unit volumeav is negligibly small with respect to the emission per unit volume ev.

In determining tne radtative energy transfer, a first approxrmation wi ll be given by:

e _\) u

\) (optically thin radiation)

( 111-8} lf the value of a'

V lies between the two extremes mentioned above, the

cal-culation of

q

and a at a given point wil! require integration

V V

total volume of the medium emittlng the radiation.

The calculation of the total radiatlve energy flux q(Jqvdv}; the emission

e

(f

ev

dv)

and absorption a (Javdv) of radiative energy volume and time wil l a lso require integration over the frequency the emitted radiation.

over the

total per unit range of

In the special case of cylinder symmetry, equations (I I 1-4) and (111-5)

can be simplifled considerably so that the calculation of

q

and a or

q

V V and a are rather less compl icated. (See chapter VI). However, it is

essen-tlal for the calculation of q and a that the coefficients e and a' from

V V V V

equation (I I 1-2) are known as a function of position in the medium. The following sections wi 11 deal with this in more detail.

In a partially ionized gas containing atoms, ions and free electrons, photons are emitted and absorbed during transitions of electrens (free and bound) from one energy state to another.

Emission of a photon occurs when an electron in a high energy state jumps toa lower state; in the reverse case, absorption of a photon occurs. A schematic diagram of the energy levels in an atom or ion is given in fig. 111-1.

The zero energy level is fixed at the ground level (lowest bound state) of the atom or ion. The higher bound levels have a positive energy with respect to the ground level. The energy I - in fig. I 11-1 - Is the tonizatton

energy which separates the free and bound statesof the electron.

Al 1 the pP.rmitted "electrontc transtHons" which are accompanted by ab-sorption and emission of photons, can óe suódtvtded into three groups, as indicated by arrows in fig. I I 1-1:

a) free-free absarptien c) bound-free aósorption e) bound-bound absarptien

1

l'

c d I e

-rt

0 b) free-free emission d) free-óound emission f) bound-bound emisslon

I

freestales

hv

9

hvn

j

boundstates

h

g roundlevel

Fig. 111-1. Schematic diagram of energy st:ates and transitlens

for atom, ion or electron.

As early as 1923 Kramers [43] dertved th.e follow-lng equati·on for tfie

free-free absarptien coefficient for one ion with charge Ze and one aósorlilng electron, with velocity v, per unit volume:

where: h e me c c 0 = 4n

313

z

2

hcm2 e Planck's constant elementary charge electron rest mass speed of 1 ight in vacuum permittivity (I I 1-9.)

-34

(6.6256 10 J sec) ( 1 .60210 10- 19 C) (9. 1091 1 o-31 kg} vacuum (2.9979 1

o

8 m se'c -1) (100/36n 10-11 F/m)

With n1 ionsper untt volume and dne electrans tn the velocity interval between v and v + dv, assumlng L.T.E., tntegration over the Maxwell îan velocity distributton of the electrans gives the following expression for

the free-free absorption coefficient

[44]:

a = g

"ff ff

3/3

(I 11-1 0)

where k is Boltzmann's constant and gff is the Gaunt factor.

An expression for this factor is given by Griem

[45) .

The value of gff is usual ly of the order of unity.

lf the distributton of atoms among the excited states is a Boltzmann distribution, then for hydrogen, the bound-free absorption coefficient is found as fo 11 ows [ 44) :

Kramers' formula (I I 1-9) is applied to all states with the same principal quanturn number n, and a summation over the lower excited levels and an

integration over the upper excited levels can be carried out. Unsöld

[46]

extended the expression which holds for hydragen to complex atoms. The structural peculiaritles of complex atoms were taken into account by intro-ducing an effective nuclear charge Z and a factor y/UA. y is the ratio of the number of sub-levels in a complex atom for the given principal and orbital quanturn numbers n and

L,

to the analogous quantity for the hydragen atom and UA Is the partltion function of the complex atom.

The quantity

z*

is given by Unsöld as:

( 111-11)

where In,L corresponds to the áctual energy of the level of the complex atom with the given quanturn numbers n and

L.

IA and IH are the tonization energies of the complex atom and hydragen atom respectively. Hence the fol lowing expresslons for the bound-free absorption coefficients were obtained

[44, 47]:

\) ~ \) g .~2 Z ykT nA UA h4cv3

exp (-IA/kT) [exp{hv/kT)-1) ( 111-12)

= 16112

\) > \)

g ( 111-13)

where nA is the partiele density of the complex atorns per unit volume and v is the frequency limit of the close lying terms given by

g

v = ( I - I ) I h (see fig. 111-1).

g g

The absorption coefficient for the whole continuurn is obtatned from the expresslons fora by the addition of the free-free absorption coefficient

\)bf

( l

vff found from equation (111-10). With help of the Saha equation (11-6), the p·oduct "i"e in equatlon (I I 1-1~) can be expressed in termsof the number uf atoms nA per unit volume, giving for the continuurn absarptien coefficient for complex atoms the following expresslons

[44, 47]:

\) ~ \) g 16112 e6 a \)

3/3

(41Te:0 )3 ( 111-14) ~t2 YZ kT nA ( 111-15) UA h4_{cv 3}

When L.T.E. applies, the relationship between theemission coefficient e:\1

and the absorption coefficient av is given by Kirchhoff's law:

a\1 I [1 - exp (- h\J/kT)]

\lp = a' \1 I \) (I 11-16)

p

in which the term {1 - exp (- hv/kT)} takes into account the effect of the induced emission; " is the intensity of the black body radlation as given by Planck's formula:p

exp(Etv/k.T)- ( 111-17)

Appl i cation of Kirchhoff's law results in the following expressions for the continuurn emission coefficient E :

\1 E \1 V ~ E V = 321f2

_{- - -}

é

3/3 (47rE0)3 V g = 321f2 eG 3/3 (47rE0 )3 V > V g y UA y UA ~·,2

z

( 111-18) [exp ({h(vg - v) - IA }/kT)] ( 111-19)

As can beseen from equation (111-18), the continuurn emission coefficient is independent of the frequency for v ~ v . For v > v , E decreases

proportien-g g V

al ly to exp(-hv/kT). Figure I I 1-2 illustrates the variatien of E₁₁ as a

function of the frequency according to the Kramers-Unsöld theory.

V Vg

Fig. I 11-2.

Varlation of t as a functlon of

V

frequency \)1 accordlng to theory, (Schematlc) Taken from [47].

Calculation of the bound-free absorptlon coefficient for photons whose energy is greater than the ionlzation energy of the complex atom (hv >IA)' making use of equation (I I 1-13) gives rlse to considerable deviations [44].

By employlng the fact that these photons are mainly absorbed by atoms in the ground level, the fol lowing approximation formula can be derived for complex atoms [44]: a \) 321!2 _e_G _ _ 3/3 ( 41fE ) 3 0 hv > IA ( 111-20)

*2

The value of Z , according to Unsöld

146]

and Vîtense

148],

ts of the

order of 4 to 7 for all levels whîch correspond to tne ground state of the

atoms. With the help of Kirchhoff's la1ot, we fînd for tfie emlsslon coefflcient:

641!2 e6

z

;,2

IA exp (-hv/kT) ( 111-21)

E _{\) nA}

3/3 (41fE)3 h3c3

hv > IA

lt should be noted that the ionizatton energy I ls decreased óy an amount

ói, as a result of electric micro-fields in the plasma generated óy charge

carriers (see chapter 11). This correction must be introduced when calculatlng

the coefficients av and Ev. An expression for ói, from the Debye-Hückel

approximation, is given in equation (11-9).

Flgure I I 1-3 reproduces part of the term diagram of NI which has been taken

from ( 49]. 15 11 2 1o1 ns 2s 1 1o1 " o 1 ~ J J p 2 " d ~,'P-

_r--

_'o{~~

=

..

~~·lp Jo(:=:~ h'"'-

r--

r

)d{:~· _~lr)s'--'o "••

.

'~; _'•

_,

_,

_'

_.

_~ hlp'- - " '

»f%=t::::=

10

leV

"\19) " 1191 "vil~ 5 lo 1rJI )pJDO 0 2p 4 S0 Fig. 11 1-3.

Part of the term diagram of Nl.

The horizontal I ine at

1•

.s3

eV denotes the ionization energy of

N. which Is the series limit of

terms beienging to the configuration (1s 2) 2s 22p 2ns, 2s 2 2p 2np and 2s 22p 2nd.

The terms go i ng to other llmi ts are

given at the rlght-hand side. For

notatien of the terms see [50].

I t can be seen from fig. 111-3, that the 4P level in the 1s2_2s2_2p2_{3s system} farms the lower limit of the group of strongly excited levels, lying close tagether.

The lowering of the ionization energy 6I, calculated by means of equation (11-9), is about 0.4 eV at a pressure of 3 atm.aós. and 0.5 eV at a pressure of 5 atm.abs. The series limit frequency v corresponding to tne 3s4P level

(see fig. 111-3) is then a5out 9.38 10 14

s~c-

1

(3200

R,).

The series ltmtt frequencies v , v and ~g corresponding to the levels 2p2_P0 , 2p2

_oa

_and

2p4 sO (ground9!tat;Y have t::e values- 2.54 10 15 sec-I (1180 R_), -2.82 10 15

sec- 1 (1060 R_) and -3.38 10 15

se~~

1 (885

R)

(principal series 1 îmit) respec-t i ve ly.

The effect of the different values of the lowering of the ionization ener9y

6I at the two pressures, mentioned above, has been neglected in calculatin9 the series 1 imit frequencies.

The factor y/UN has been calculated making use of the tables of Wiese et al. [50) (calculatlon of y) and the tables of Pflanz et al. [19] (calculation of UN),the resultin9 value of the factor y/UN is roughly approximate to unity. The effective nuclear charge

z*

for the frequency interval v $ v (9.38 10 14

-1 g

sec ) has been calculated from the 2 P and 4 P levels of the 1s22s22p2ns system, from the 2 sO, 2oo, 4 pO and 4o0 levels of the 1s22s22p2np system and the 2 P, 20, 2 F, 4P, 4D and 4 F levels of the 1s22s22p2nd system, employin9 equation (111-11) and is found to be about 1.4.

The effective nuclear charge Z for the frequency interval v₉ ~ v ~ v 9h has been calculated from the 3s4_{P and from the} 2P0 , 2 oO and 4 s 0 levels of the 1s22s22p22p system, the resulting value bein9 about 1

.7.

lntroducing the quant i ties calculated above in equations (I I 1-14) and (I I 1-15), the fol lowing expressions for the absorption coefficient av of the NI continuurn are obtained:

'

= 16rr2 e6 ( 1 .4 )2 kT nN

a exp [- (iN--~ 6I)/kT] exp(hv/kT)

V

3/3

(4rre)3 h4_cv3

V ~ \) ( 111-22)

9

=· 16rr2 e6 (1.7)2 kT nN

a exp [- (IN - 6I)/kT) exp(hv /kT}

V

3/3

(4rre0 )3 h4cv3 9

\) > V (I I 1-23)

9

In these equations IN is the ionization energy of the nitrogen atom, (14. 53 eV).

16rr2

The factor

0.87 1020 m2 sec-3 °K- 1.

in equat lans ( 111-22) and ( 111-23) equa 1 s:

The absarptien coeffident, corrected for tf\e induced emis:sion, a~,

(a~= av {1 - exp(- hv/kT)]) has been ca1cu1ated using av from equatîons (111-22) and (111-23) fora pressure of 5 atm.abs. with tfte temperature as a parameter. The resu1ts are shown as a function of the frequency in fig.

I I 1-4. The partiele densities required forthese ca1culations have been taken from {19).

The corresponding values of a~ at a pressure of 3 atm.abs. were obtained from the calculated va1ues for T = 20,000 °K and T = 12,000 °K at a pressure of 5 atm.abs. by multiplying by 0.396 and 0.581 respectively; which is the ratio of the densitles of the nitrogen atoms at 3 and 5 atm.abs. for

the temperatures mentioned above.

1a'v<m1> 1 1Ö 2 10 14 10

'

T =2

ooo•

'

'r-T· 0 0 ~

..

.

-::

/'

/

1

_I

--}

l

:

' I

'

'

i

l\

'

I

'

'

i '

\

_'

I ' I

'

' ' :

"

'

:

:

'

I\

I Vgl: I ! "v, :v~

1\.

'

'

.

' ' ' ~h

!

v

~

Fig. 111·4. The absorpt1on coefflclent of the NI contlnuum

(v ~ v_9h) as a functron of the frequency wi th temperature

A number of values of the absorption coefficient for the extreme ultra-violet part of the continuurn at a pressure of 2 atm.abs. and a temperature of 12,900 °K, which were obtained by experiments [51], have a lso been lnclud-ed in fig. I I 1-4. The experimental values of a~ are therefore of the same order as these calculated on the basis of the Kramers-Unsöld theory.

lt fellows from fig. 111-4 that the mean· free pathof the photons tv; 1/a~

is greater than about 5 10-2 m. (With the exception of that part of the spectrum where the frequency is v ~ v at temperatures around 12,000 °K.)

9h

When a high-pressure discharge has a diameterdof a few mi 11 imetres (see chapter V), this means that for the NI continuurn (v ~ v ) practical ly no

9h

reabsorption occurs: in ether words, the discharge is optically thin for the NI continuurn (v ~ v₉h).

The balance between the emitted and absorbed radiative energy per unit volume, time and frequency, u" is then given by:

u

\) e \) - a \) e \) 41Ts \) (I I 1-24)

Substitution of the expresslons for sv found in equations (111-18) and

(111-19) with the corresponding values for y/UN, / and fJ in equation (I I 1-24) and integration over the corresponding frequency interval, gives theemission per unit time and volume of radiative energy e whîch leaves the discharge

(e

~ u).

The results of these calculations, as a function of temperature wîth pressure as a parameter, are reproduced in figure I I 1-5.

The bound-free absorption coefflcient a" for high-energy photons (hv>

IN -tJ) is given, toa first approximatigl:, by equation (111-20).

Calculation of the effective charge

z'

1

, by means of equatlon (111-11) fór the ground level of the nitrogen atom, gives a value of about 2.

*

Substitution of the numerical values of the constants, e, s0 , h, c and Z

in equation (I I 1-20) results in the following expressionfora

11_bf

\) > \)

,~.uiW/m3_i 0 101

I

i//

9 10

h

:/ '

I

IJ

1

12 / poSatm.aos.

/V

J

)'"'

po3atm.abs.

1/

u.

16 18 20 T I'Kl Fig. 111-5.

Radiative energy per unit volume

and time of the "optlcally thin1' NI continuurn (v ~ v_9h) as a

funct ion of the temperature wl th

pressure as a parameter.

Multiplication of equation (111-25) by the term (1- exp{- hv/kT}) , which takes into account the effect of the induced emission, gives the expression for ct'

\)bf

Figure I I 1-6 reproduces the results of the calculations of ct' as a function

\)bf

of frequency at 5 atm.abs. pressure wi th temperature as a parameter. The corresponding values for ct' at

\)bf the calculated values for T

a pressure of 3 atm.abs. are obtained from

20,000 °K, T

=

16,000 °K and T

=

12,000 °K in figure 111-6 by multipi i cation by 0.396, 0.478 and 0.581 respectively.

lt can beseen from the calculated values of ct' that the mean free pathof

\)bf

the photons Iv is of the sameorder as or much smaller than the diameter d, (d is several mi I I i metres), of the discharge. For this part of the NI continuurn (v < v ) reabsorption of emitted radiation wi 11 take place.

9h

This implies that the absorbed radiative energy per unit volume, time and frequency av is nat equal to zero and therefore uv #ev.

Calculatlon of the absorbed radiative energy per unit volume and time, a, at a given point in the discharge, necessitates integration over the whole volume of the discharge, as is indicated in principle by equatlon (I I 1-5), and

integration over the frequency interval. In chapter VI these calculatlons wlll bedealt with more in detail.

10' Ja~(m-1₎ 10 2 1 10 15 10

.

_:\

:

1\

~\

\

'

1\

'\ ' ~ ' '

~

'

\

' ' ' ' ' ' Vgl'l:

1

\

1

\

\

1\

\

1\

T,12,000 °K

1\

I

\

I

T'16,000°K

'

I

\

1,20,000 °K

1\~<

s

e

c-

1

₁

Fig. 111-6.

The absarptien coefficient of the NI

continuurn (v > v_9h) as a function

of the frequency, wi th temperature as a

parameter. at a pressure of 5 atm.abs.

The dependenee as a function of the frequency, of the absorption coefficient a~ of a speetral 1 ine is given by the fol lowing relationship, derived from

the classica I theory [52]:

a'

V

m c

e

Q(v). (1 - exp(- hv/kT)) (I I 1-26)

where nj is the popuiatien density per unit volume of the energy level

j; fjm is the osci 1 later strength for the transition of the lower level to the higher level mand Q(v) is a normal ized 1 ine shape function

( JQ(v) dv 1).

The popuiatien nj of the energy level is given, in the case of L.T.E., by:

nj = n

~

_u exp (- I ./kT)

J ( 11 1-27)

Appl i cation of Kirchhoff's law (equation 11 1-16) to equation (I I 1-26) gives for theemission coefffcient Ev:

1Te2 -..c:..=.--n. f. Q(v) exp (- hv/kT) J Jm m c e ( 111-28)

The I ine shapes of speetral I i nes are almast never determined by natura! broadening only. Besides natura! broadening, Doppier broadening is always present and dominates the I ine shapes near the 1 ine centre at high temper-aturesar low densities.

However, in a high pressure discharge (pressure some atmospheres) such as dealt with here, the two above-mentioned universa! line broadening mechanisms are aften negl igible, because the line shapes are strongly influenced by the Interaction of the radlating atoms ar ions with surrounding particles.

This broadening mechanism is referred to as pressure broadening. Interaction with the radiating atoms ar ions can be achieved by either neutral ar charged particles. The effect of charged particles, however, is sa much greater than that of neutral particles that the interaction of the latter can be neglected as soon as there is any appreciable ionization [53]. (For nitrogen at a pressure of a few atmospheres, thls occurs when the temperature rises above 104 °K).

Hence there are two main broadening agents, ions and electrons. Because

electric fields are involved, this type of broadening is cal led Stark broadening. A fundamental study of pressure broadening has been made by

Baranger [53]. Basedon this study Griem [54] calculated the Stark

broade-ning of several elements and tabulated numerical results [55L The shape

of a 1 ine broadened by the Stark effect can be described, toa first approximatlon, by a Lorentz function [56], whlch is given in normal ized

farm by [52]:

L(v)

'

:,

[.1~}'

l

(I I 1-29)

where vmj is the central 1 ine frequency and 85 ts the half-·half wldth for Stark broadening.

of the normal ized frequency deviation (v - "mj)/Bs·

ID

Fig. 111-7.

·J ·2 lorentz func tI on.

The earl i er mentioned Doppier broadening results in a Gaussian 1 ine shape, which is given in normal ized farm by [52]:

G(v)

B

1 exp[

D

(111-30)

The half-half width of this function is given by

[57]:

( 111-31)

where M is the mass of the emitting atoms.

When the half-half width due to the Stark effect (at high densities the natura! broadening can be entirely neglected) is not appreciably greater than the half-half width due to the Doppier effect, the resultant 1 ine profile is obtained by fotding the two line shapes Ls(v) and GD(v), that is [58]:

V(v)

_J

G ( v 1 ) L ( v - v 1 ) dv 1

0 D s

( 11 1- 32)

Th is leads to Voigt profiles, which are avai lable in bath tabular and graphic farm for a large number of conditions 159, 60, 61].

q

normal ized frequency deviation k, with the ratio

a

₅

1

s

₀ = a as a parameter .

The frequency deviation k is deflned by: k = (v - v .)/w, where w Is the

mJ effective half-half widtn of tne Volgt profile.

Fora> 0.4 a first approximatîon for w is given by [60):

( 111-33)

lt Is apparent from figure I I 1-8 that the Volgt functions for large values of k, i.e. in the "wings", behave as a Lorentz functîon. Th i's Is a result

of the fact that the Gauss function at large values of k approaches zero

more rapidly than the Lorentz function. The influence of the Gauss (Doppler)

kernel on the Voigt functions becomes less as a increases.

Fora> 1, it appears that the Voigt profile, apart from a relatively smal I

kernel, approximates wel I to the Lorentz profile.

10° ~ ViaiJl

I

,

1Ö 2 1Ö 3 1Ö 4 \

\

~~ \. "~ ~'\

_~

::-....

\\

"'

~

~ ... ...

...

'

CLo0 ...

\

a•~

"

8 a·2

-

-

.._

a=

..._

_~-

_--.:::

a·0.5

---

-

---..:

_:::-=

a.0.1

...

-

-

-2..

-12 16 20 21.

Fig. 111-8. Values of the functlon V(a, kw)IV(a, 0) for some values of a.

Part of the NI and NI I term diagrams wttn a number of bound-óound transitoons taken from [49, 50, 62), are shown In figures 111-9 and 111-10 respectively.

~

I t is worthwhi le to di vide the Nl and Nll ltnes into two groups: a) one group for which the centrat wave tength ~ . < 2000 ~ and

mJ b) one group for whicn A . > 2000

~-mJ

The first group camprises nearty all the lines arising from transitlans to the terms 2p2_Po, 2p2DO and 2p4S0 in the case of the NI 1 ines and the terms 2pls, 2p!D and 2p3P in the case of NI I lines.

For the lines in group b)

(À,

2000 ~) we find that at a pressure of several atmospheres the half-half widths due to the Doppier effect, as calculated form equation (111-31) in a temperature range from about 104 to 2 x 104 °K,

are smal 1 with respect to the half-half widths due to the Stark effect as calculated by Griem [55], (S5/s0 ~ 10). 30 2 12lp .. , 2 ~2 2 p ~ d '•

{

'• 10 ""' 20

1:"

5 0

leV

10

Fig. 111-9. Part of the term diagram of NI Fig. 111-10. Part of the term diagram of NII

The bound-bound transitlens are designed by arrows and averaged wave lengths in Rngström units. The notatien and symbols are slmular as In fig. 111-3.

In other words, for this group of 1 i nes the Stark effect is by far the

,,..,

'"'

'o'

most important broadening mechanism with the result that the line shapes can bedescribed by a Lorentz function as goven by equation (I I 1-29). Calculation of the absorption coefffcfents for tne central I ine frequenc[es 6y means of equatlon (I I 1-26), the data on NI and NI I llnes from the tables of Wlese et al. [50], the half-half widtfis from Grtem's tables !55], produces values of