search.jsp?R=19690008459 2020-03-12T07:08:49+00:00Z · 2020. 3. 12. · Predictions of radiative hecting are givan 'n Section 4 fcr iso- thermal and nonisothennal slabs. These serve

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A e r o t h e r m R e p o r t NO. 68-38, P a r t I11 C o p y No. -

FINAL REPORT

FURTHER STUDIES O F THE COUPLED CHEMICALLY REACTING BOUNDARY LAYER AND CYARFUNG ABLATOR

A NONGREY RADIATION TRANSPORT W D E L SUITABLE FOR USE I N ABLATION-PPODUCT

CONTAMIE3TED BOUNDARY LAYERS

by K i l l i a m E. N i c o l e t

Prepared f o r

NATIONAL AERONAUTICS AND SPhCE ADMINISTRATION

O c t o b e r 15 , 196 8

CONTRACT hlAS!' p.6 7 1 9

T e c h n i c a l Manacrement NASA M a n n e d Spacec: a f t C e n t e r

H o u s t o n , Te : a s S t ruc tu res and Mecha: .ics D i v i s i o n

D.M. Curry Go S t r o u h a l

ABSTRACT

This r epor t descr ibes a predic t ion method which can be used t o obta in nongrey r a d i a t i v e f luxes o r i n t e n s i t i e s a t any po in t within a plane-paral le l s l a b ( f o r t h e f lux ca lcula t ion) o r a t any poin t on a r ay ( f o r t h e i n t e n s i t y ca lcu la t ion) . The method was developed f o r t h e study of r ad ia t ion hea t ing phenomena i n t h e mass in jec ted , hypersonic boundary l a y e r environment; however, it is no t limited t o such s tud ies , The raciiative p roper t i e s model assumes l o c a l thermodynamic e q u i l i b r i m and considers t h e continuum t r a n s i - t ions , molecular bands and atomic l i n e s of t h e spec ies of t h e C-H-0-N elemental system. The smeared l i n e model f o r t h e molecular bands is t h e only approximation which is an i n t e g r a l p a r t of t h e method. Any o f t h e o the r aspects of the proper t i e s model can be made t o include as m a c h (or l i t t l e ) detail as desired, allowing t rade-offs t o be made between accuracy and compu ta t iona l e f f o r t .

The p r e s e n t r e p o r t is one of a series of fou r r e p o r t s , publ ished s imul taneously , which desc r ibe ex tens ion and app l i - ca t i on of analyses and computational procedures f o r p r e d i c t i n g the in-depth response o f cha r r ing a b l a t i o n materials and nonsimilar chemically r e a c t i n g boundary l a y e r s which were generated under a previous c o n t r a c t (NAS9-4599). I n p a r t i c u l a r , t h e p r e s e n t r e p o r t s desc r ibe t h e ex tens ion o f a laminar multicompo- nen t chemical ly-react ing (equi l ibr ium) boundary-layer program t o inc lude nongrey r a d i a t i o n coupl ing, the ex tens ion of t h i s compu- t a t i o n a l 2rocedure t o t u r b u l e n t flow ( a t t h i s p o i n t f o r incompres- s i b l e flaws o n l y ) , the f u r t h e r checkout of a code which couples the laminar boundary l a y e r procedv-- t o a t r a n s i e n t cha r r ing a b l a t i o n code, and t h e appl icat ior . f t he se and o t h e r computa- t i o n a l procedures t o t h e Apollo h e a t s h i e l d m a t e r i a l and t y p i c a l Apollo missions. P a r t I se rves as a summary r e p o r t and descr ibes t h e p r e s e n t status o f and s o l u t i o n s ob ta ined wi th t h e var ious computational procedures. I n P a r t I1 a thermochemical a b l a t i o n program based on a t r a n s fer-coef f i c i e n t approach is u t i l i 9ed t o i n v e s t i g a t e a b l a t i o n mechanisms f o r t he Apollo h e a t s h i e l d material. The r a d i a t i o n t r a n s p o r t model which is u t i l i z e d is descr ibed i n P a r t 111, whereas t h e t u r b u l e n t boundary l a y e r code is d iscussed i n P a r t I V .

The t i t les i n t h e series are:

P a r t I: Summary Report: Fu r the r S tud ies of t h e Coupled Chemically Fteacting Boundary Layer and Charring Ablator , by E.P. B a r t l e t t , W.E. N ico le t , L.W. Anderson, and R.M. Kendall.

P a r t 11: An Evaluat ion of Surface Recession Models f o r the Apollo Heat S h i e l d Material, by E.P. B a r t l e t t , and L. W. Anderson.

P a r t I11 : A Nongrey Radiation Transport Model S u i t a b l e f o r U s e i n Ablation-Product Contaminated Boundary Layers, by W. E. Nico le t

Part IV: Nonsimilar So lu t ion of an Incompressible Turbulent ..- Boundary Layer by an I n t e g r a l Matrix Method, by L. W. Anderson and R. M. Kendall.

This e f f o r t w a s conducted f o r t h e S t ruc tu re s and Mechanics Division of the Manned Spacecraf t Center , Nat ional Aeronautics and Space Administrat ion under Contract NAS9-6719 with M r . Donald M. Curry a s t h e NASA Technical Monitor. Development of t h e t u rbu len t boundary l a y e r code w a s sponsored j o i n t l y by NASA/MSC and by the A i r Force Weapon8 Laboratory, Ki r t l and A i r Force Base,with L t . Ronald H. Aungier as Project Engineer. Extension of t h e t u rbu len t boundary l a y e r a n a l y s i s to compressible flows is cont inuing under AFWL spon- sorship . M r . Eugene P. B a r t l e t t of Aerotherm Corporation was Program Manager and P r i n c i p a l I n v e s t i g a t o r f o r t h e e f f o r t s repor ted here.

iii

TABLE OF CONldNTS

Title

ABSTRACT FOREWORD LIST OF TABLES LIST OF FIGURES LIST OF SYMBOLS

1 INTRODUCTION

2 ABSORPTION COEFFICIENTS OF THE BOUNDARY LAYER SPECIES 2.1 Important Absorption Mschanisms

2.1.1 Atomic Continuum Transitions 2.1.2 Atomic Line Transitions 2.1.3 Molecular Band Transitions 2.1.4 Other Transitions

2.2 The Radiation Properties Model 2.2.1 Molecular Band Model 2.2.2 Atomic and Ionic Line Model 2.2.3 Sources of Data

3 TRANSPORT MODEL 3.1 Formulation 3.2 Evaluation of Flux Integrals

3.2.1 Nodal Points in Frequency 3.2.2 Nodal Points in Space 3.2.3 Integration Scheme

3.3 The Radiation Transport Pi-ogram (RAD)

4 APPLICATION OF THE METHOD 4.1 Comparison with Other Predictions 4.2 Comparisons with Measured Data 4.3 Application to Uniform Slabs 4.4 Application of Nonuniform Slabs

5 CONCLUDING REMARKS

6 REFERENCES

ii iii v v

vii

LIST OF TABLES

Number Title Page

15

16

17

17

I Atomic and Ionic Continuum

11 Atomic and Ionic Lines

I11 Molecular Band Systems

IV Other Contributors

LIST OF FIGURES

Number

1

2

Title Page

Continuum Cross Sections of Nitrogen 5

Typical Absorption Coefficient Predictions Including Important Molecl?lar Contributions

Coordinate System 19

Spatial Grid Transformations 24

Comparison with the Predictions of Morris et.al. (Ref. 39) 2 7

Effect of N- on the Spectral Distribution of the Intensity of Nitrogen, 6 = 1 cm 28

Flux Emitted from a Uniform Air Slab, P = 1 atm 30

Flux Emitted from a Uniform Air Slab, P = 1 atm 31

Intensity Emitted from a Uniform Nitrogen Slab, 6 = 1 cm 33

Intensity Emitted from a Uniform Hydrogen Slab, T = 10,000°~ 34

Comparison with the Shock Tube Data of Nerem (Ref. 50), P1 = 1 mm Hg 35

Comparison with the Shock Tube Data of Nerem (Ref. SO), P1 = 2 mm Hg 36

Comparison with Shock Tube Data of Gruszynski and Warren (Ref, 51) 37

Comparison with Shock Tube Data of Wood (Ref. 53) and Wood et,al. (Ref. 52) 39

Comparison with Total Intensity Measurements of Wood (Ref. 53) and Wood et.al.. (Ref. 52) 40

Variation of Flux from a Uniform Air Slab, 6 = 1 cm 41

Number

17

18

19

20

21

22

23

LIST OF FIGURES (concluded)

Title

Variation of Flux from a Uniform Air Slab, 6 = 10 cm

Thermodynamic State Variation

Spatial Distribution of Radiating Species

Spectral Distribution of Continuum Fluxes

Spectral Variation of Positive Flux at Wall

Positive Flux Variation

Net Flux (Positive When Directed Toward the Wall)

Page

42

43

45

46

47

49

50

LIST OF SYMBOLS

GREEK SYMBOLS

Planck function

l i n e shape function

constant

speed of l i g h t

black body emissive power

charge on an e lec t ron

exponential i n t e g r a l of order n

one s ided r a d i a t i v e f lux

o s c i l l a t o r s t r eng th of a l i n e (f- numb,^)

growth parameter (Eqs. ( 2 5 ) and (37) )

a r b i t r a r y constants (Eqs. (34) and (35))

Gaunt f ac to r

Planck's constant

emission per u n i t volume

Boltzmann's constant

mass of an e l ec t ron

number of quan t i t i e s , Eq. (1)

number dens i ty

p r inc ipa l quantum number

p a r t i t i o n function

ne t f lux

Rydberg constant

l i n e s t rength

ioniza t ion p o t e n t i a l

s p a t i a l coordinate

a p a r t i c u l a r s p a t i a l poin t

defined a s h ~ y / k ~

p a r t i t i o n function r a t i o v i i

SUBSCRIPTS

SUPERSCRIPTS

LIST OF SYMBOLS (concluded)

l i n e h a l f width

path length

emiss iv i ty

absorption c o e f f i c i e n t

absorption c o e f f i c i e n t corrected f o r induced emission

photon frequency

Gaunt-like correc t ion fac to r f o r non-hydrogenic atoms

o p t i c a l depth

s t r e t c h i n g parameter (Eq. (38) )

bound f r e e t r a n s i t i o n

quant i ty evaluated a t t h e cen te r of a l i n e

f r e e f r e e t r a n s i t i o n

species i

abosr3ing l e v e l j

l i n e k

quant i ty evaluated a t t h e ou te r edge of the l aye r

s p e c t r a l quant i ty

continuum quant i ty

quant i ty due t o the Doppler e f f e c t

f r e e f r e e t r a n s i t i o n

l i n e property

photodetachment t r a n s i t i o n #

quant i ty due t o t h e resonance e f f e c t

quant i ty due t o t h e S ta rk e f f e c t

lowest l e v e l t o be included i n i n t e g r a l formulation

h ighes t l e v e l allowed under given plasma c o ~ ~ d i t i o n

averaged quant i ty v i i i

A NONGREY RADIATION TRANSPORT MODEL SUITABLE FOR USE I N ABLATION-PRODUCT CONTAMINATED BOUNDARY LAYERS

SECTION 1

INTRODUCTIOX

The a b i l i t y t o p red ic t r ad ian t energy tl-ansfer is required i n order t o

understand t h e heat ing phenomena associated with bodies en ter ing planetary

atmospheres a t high ve loc i t i e s . The rad ia t ion t r anspor t i s important both as

an energy source i n d i r e c t thermal contac t with t h e body and indi.. x t l y through

t h e coupling between it and the thermal boundary layer . Quan t i t a t ive predic-

t i o n s of r a d i a t i v e t r anspor t under these condi t ions requi re frequency depen-

dent p roper t i e s of t h e r ad ia t ing species and a frequency dependent t r anspor t

model. I n t h i s study a t t e n t i o n is d i rec ted toward a t r anspor t model which is

simple enough t o be used as p a r t of a coupled flow f i e l d ca lcula t ion , y e t is

s u f f i c i e n t l y d e t a i l e d t o be able t o p r e d i c t t h e more s i g n i f i c a n t fea tures of

measuxed s p e c t r a l data ,

Transport within t h e C-H-0-N elemental system is considered. This s y t m

is representa t ive of boundary layers adjacent t o most ab la t ing bodies. Local

thermodynam!.~ equilibrium is assumed t o e x i s t a t a l l times. Molecular, atomic

and ion ic species a r e a l l considered with those which appear i n t h e 3 , 0 0 0 ' ~ t o

1 5 , 0 0 0 ° ~ temperature range (0.1 t o 10 a tm~spheses pressure range) being given

primary consideration.

The proper t ies u t i l i z e d and the t r anspor t model a re presented i n Section

2 and 3, respect ively. The cont r ibut ions from atomic l i n e s a r e pa r : i cu la r ly

emphasized. Predict ions of r ad ia t ive hect ing a r e givan 'n Section 4 f c r iso-

thermal and nonisothennal slabs. These serve t o i l l u s t r a t e appl ica t ions o .

t h e method and, i n some cases , allow f o r comparisons with the predic t ions of

o the r inves t iga tors ,

SECTION 2

Ab- 3RPTION COEFFTC ' aNTS OF THE aOUNDARY LAYER SPECIES

2.1 IMPORTANT ABSORPTION MECHANISMS

The spectral absorption coefficient for a plasma consisting oE a mixture

of elements is in general

where the first term represents the continuum contribution with its summation

taken over all continuum transitions (Nl), and the second term represents line

contributions with the summation taken over all the lines (N ) . For the plas- L ma condi..ions of interest, the important continuxm transitions include atomic

photoionization, photodetachment, free-free transitions, photodissociation and

molecular photoionization in approximately a decreasing order of importance.

The atomic 'ine transitions are very important. The molecular band systems

can be important for some co-iditions,

2 Atomic Contin~um Transitions - In general, continuum contributions depend on the plasma state (to a satis-

factory approxivation) only through the populations of the absorbing levels,

v;?.

C where Nij is the number density of the absorbing level and oi j (v) is its cross-

section. The number densities must be obtained from thermodynamic stat2 cal-

culations and the cross sections either from quantum mechanical calculation or

experiment.

Hydrogen cross sections are well understood, exact values being available

(Ref, 1)

where t h e subsc r ip t s bf and f f r e f e r t o bound-free (photoionization) and free-

f r e e t r a n s i t i o n , respect ively. The quan t i ty 5 i s t h e p r i n c i p a l quantum num-

ber of t h e absorbing l eve l ; Ne is the number dens i ty of f r e e e lec t rons : qbf

and gff a r e Gaunt f a c t o r s which are ava i l ab le i n t a b u l a r form (Ref. 1) . and t h e

o t h e r symbols have t h e i r usual meanings. The Gaunt f a c t o r s a re o f t en taken t o

be u n i t y (a good approximation) o r expressed i n terms of s h e l l averages (qf f ) . This allows a c lose9 form expression t o be obtained f o r t h e free-free cont r i -

but ion f o r t h e z,z+l s t age of ioniza t ion (Ref . 2), viz .

where Q~ and QZ+' are t h e e l e c t r o n i c p a r t i t i o n funct ions of t h e atom and i ts

ion, respect ive ly , were xZ is t h e ion iza t ion energy of the atom, where z+l is

t h e charge on t h e r e s idua l ion. The photoionizat ion cont r ibut ion is a l s o s i m -

p l i f i e d , v iz ,

1 T exp -

v, <v "a ("I:.) where Vna is the threshold frequency c f t h e l e v e l na. A t s u f f i c i e n t l y high values of n,, t h e l e v e l s are s o c lose ly spaces t h a t t h e summation can be

approximated by an in teg ra t ion (see Ref, 3, f o r example)

PI V i (v) r. ~ly jexp (hRy/n2 kT) - exp (h~y,? k ~ ) 1

where n is t h e lowest l e v e l t o be included i n t h e i n t e g r a l formulation and 5 i s the highes t l e v e l allowed a t t h e given plasma condition,

I n t h e case of t h e hezvy atoms, no exact c ross sec t ions a r e avai lable .

The most widely accepted method of ca lcu la t ing them is the "quantum defec tw

approximation a s p u t f o r t h by Bur-jess and Seaton (Ref. 4) . This method has

been used by Armstrcng, Johnston and Kelly (Ref. 5, see a l s o Refs. 6, 7, and

8) t o obta in t h e l e v e l c ross sec t ions f o r t h e l e v e l s of N and 0. A compilation

-4-

of t h i s work is ava i l ab le (Ref. 2) i n which " e f f e c t i v e cross sec t ionsM ( ( v ) ) a r e tabulated.

A useful approximation f c r t h e heavy atom cont r ibut ions a t low frequen-

c i e s ha^ been proposed by Biberman and Norman [Ref. 6) . They a l s o use t h e

quantum defec t method b u t have obtained an approximate, closed form so lu t ion

f o r t h e t o t a l contr ibut ion. Their r e s u l t s are presented i n terms of two

cocrect ion f a c t o r s t o be applied t o t h e hydrogenic formula, v iz ,

A L A J hydrogenic

where r = 2Q z+l,*z . Tabulated values of 5 (hv,T) a r e ava i l ab le (Ref. 6) f o r

many d i f f e r e n t atoms. The absorption c o e f f i c i e n t s ca lcula ted from Equation

(8) show surp r i s ing ly good agreement w i t h t h e f u l l y d e t a i l e d ones of Armstrong

e t . a l . ( R e f . 5 ) . Tabulated values of t h e e f f e c t i v e c ross sec t ions of C and C+

a s obtained from ~ q u e t i o r . (3) are ava i l ab le (Ref. 2) .

Figure 1 shows ni t rogen continuum c ross sec t ions taken from References

5 and 6 and i l l u s t r a t e s t h e dominant importance of t h e photoionization edges.

2.1-2 Atomic Line Trans i t ions

The absorption c o e f f i c i e n t s of t h e atomic l i n c t r a n s i t i o n c depend on t h e

plasma condition both through t h e population of t h e absorbing l e v e l and a l s o

through t h e shapes of t h e l ines . Thus,

where f k ( j )

is t h e o s c i l l a t o r s t r eng th of t h e kth l i n e i n the jth s e r i e s of

l i n e s and b k ( j )

(v.T,P,xl,x2, ...) is the l i n e shape and is a function of f re -

quency and t h e plasma condition. The l i n e shape obeys the normalization con-

d i t i o n (omitt ing t h e e x p l i c i t l y wr i t t en funct ional dependence on t h e plasma

condit ion f o r brevi ty)

b u t

the

otherwise is f r e e t o

species involved and

take on a v a r i e t y of funct ional forms depending upon

t h e broadening mechanisms (o r combination of mechanisms).

jo LI -v r I !

h

I0 I-" n - w ~ n z~

U W

I Z L z Z

For t h e heavy atomic species , t h e dominant broadening mechanism i s Stark

broadening by e lec t ron impacts. Following Armstrong e t . a l . (Ref . 5) it i s

assumed t h a t each mul t ip le t can be t r e a t e d a s a l i n e with a Lorentz shape,viz,

where vk is t h e frequency of t h e l i n e center and is t h e Stark (ha l f ) ha l f - width which is a function of t h e plasma condition. Corrections t o Equation (10) due t o J - s p l i t t i n g and e f f e c t s due t o ion per turbers a r e of ten ignored.

The (ha l f ) half-widths, can be ca lcula ted from t h e e l ec t ron impact

approximation. Griem (Ref. 9) has worked out t h e proper formulation and

t abu la tes da ta fo r ma;ly t r a n s i t i o n s and f o r severa l elements. Wilson and

Nicolet (Ref. 2) have performed t h e calculat.ions* f o r a l l t h e important t ran-

s i t i o n s i n N, N+, 0, 0+, C and C+. A comparison given by Wilson and Greif

(Ref. 10) shows t h a t t h e d a t a of Reference 2 compares favorably with t h a t of

Reference 9. Page e t .al. (Ref. 11) repor t t h a t the (ha l f ) half-widths from

Reference 2 can be approximated a s

where n = 0.25 and 0.46 fcir nitrogen and oxygen, respect ively.

For many plasma conditions of present i n t e r e s t , t h e degree of ioniza t ion

is very low. Under such conditions, resonance broadening can be g rea te r than

Stark broadening, y ie ld ing l i n e s with Lorentz shapes having (ha l f ) half-widths

which can be ca lcula ted i n an impact approximation (Ref. 15) a s

Here fres, vres, g1 and gu a l l belong t o the resonance l i n e and are , respec-

t i v e l y , t h e absorption o s c i l l a t o r s t rength , t h e cen te r frequency, t h e lower

s t a t i s t i c a l weight and t h e upper s t a t e s t a t i s t i c a l weight. Na is the number

of per turbing atoms pe r u n i t volume. t

* I t should be noted t h a t u n t i l r ecen t ly an erronious approximation was usually made i n t h i s ca lcu la t ion (see Ref. 2 f o r a discussion of t h i s point) which resu l t ed i n half-widths which were much too large.

The broadening mechanisms f o r atomic hydrogen requi re except ional t r e a t -

ment. The S t s r k s p l i t t i n g of hydrogen l i n e s is much g r e a t e r than t h a t of

&her spec t ra (Ref. 1 2 ) . This e f f e c t is connected with t h e acc identa l degen-

eracy i n hydrogen whereby terms of t h e same p r i n c i p a l quantum number b u t d i f -

f e r e n t o r b i t a l quantum numbers have (very nearly) t h e same energy. Further,

it is known t h a t broadening caused by ion per turbers i s not neg l ig ib le com-

pared t o t h a t caused by e lec t ron impacts ( see Ref. 13) . Thus, t h e Lorentz

l i n e shapes cannot be used, The l i n e shape i n t h e core region of each of t h e important hydrogen l i n e s w a s obtained by G r i e m , Kolb and Shen (Ref. 13) and

is ava i l ab le i n numerical form. Asymptotic equations a r e given by G r i e m (Ref,

9) f o r t h e shape of t h e f a r wings, viz .

and

The parameter a (v) is defined as

and values of C and R a r e ava i l ab le i n Ref. 9. For t r a n s i t i o n s involving highly exci ted l eve l s , Griem (Ref. 14) has shown t h a t e lec t ron impact broaden-

ing i s dominant, Thus, t h e 1.orentz shape can be used with t h e (ha l f ) h a l f -

widths taken from O r i e m (Ref. 14) , viz.

where nL and nu a r e t h e p r i n c i p a l quantum numbers of t h e lower and upper l eve l s

of the t r a n s i t i o n , respect ively.

The l i n e s of a l l t h e spec ies can a l s o be broadened by t h e Doppler e f f e c t .

Hunt and Sibulkin (Ref. 15) have recent ly pointed out t h a t it is important t o

include t h i s mechanism fox some plasma condit ions of ixlterest . The Doppler

broadened l i n e acquires t h e shape

where is t h e Doppler (ha l f ) half-width and is given by

where mi is t h e mass of t h e r ad ia t ing p a r t i c l e . I n order for t h i s mechanism

t o p a r t i c i p a t e i n a meaningful way i n t r anspor t ca lcula t ions , Y: must be very m n 3 n ?uch 1 a r ~ 0 1 than yk + yk. This e f f e c t occurs because the Doppler wings Zecay

exponentially; whereas, t h e Lorentz wings decay quadri-al ly , and most of the

r ad ian t energy is (usually) t ransported i n t h e wings.

Some of t h e atomic l i n e s can be t r e a t e d l i k e a continuum r a t h e r than

individual ly. The high l i n e s * i n a cjlven s e r i e s always become overlapped as

they approach t h e i r s e r i e s l i m i t . Armstrong (Ref. 16) has shown t h a t t h e

photoionization threshold can be s h i f t e d t o lower frequencies t o account f o r

t h e contr ibut ions from these l i n e s , v iz .

where Ahv is the s h i f t i n terms of photon energy.

Lines belonging t o high series** can a l s o he t r ea ted as a continuum con-

t r ibu t ion . They become overlapped a t low frequencies; i n addi t ion, they a r e

usual ly weak. I n t e g r a l formulations f o r the cont r ibut ions of these l i n e s have

been obtained i n a va r i e ty of inves t iga t ions (Refs. 15 and 17, fo r exbwple) .

* The high l i n e s i n a s e r i e s a re associated with t r a n s i t i o n s which have upper l e v e l s with l a rge p r i n c i p a l quantum number.

** The high series a r e associated with t r a n s i t i o n s which have lower l e v e l s with l a rge p r inc ipa l quantum numbers.

A l l approximate t h e l i n e s t r eng ths by s h e l l averaged hydrogenic values. This is cons i s t en t with t h e f u l l y d e t a i l e d ca lcu la t ions (Ref. 5) . For a uniform

and o p t i c a l l y t h i n plasma, Vorobyov and Norman (Ref. 17) have obtained t h e 1 t o t a l emission per u n i t volume (j) of a l l t h e t r a n s i t i o n s o r ig ina t ing a t o r above t h e l e v e l k, viz .

j = const. ' exp (- 6 ) r [ t (ex. 48

- exp A) - ( n i a - n") exp 'iia Za

(19) where $ = h ~ y / k ~ , where z is t h e charge of t h e atom, X' i ts ioniza t ion poten- t i a l and the h ighes t l e v e l allowed a t t h e given plasma condition. vorcbyov

and Norman ( R e f . 17) do no t d iscuss t h e s p e c t r a l d i s t r i b u t i o n of t h e in tens i - f

ties thus obtained; howver, Hunt and Sibulkin (Ref. 15) have obtained a some- I i I

what similar equation and i n d i c a t e t h a t t h e d i s t r i b u t i o n should be t h e same as t h a t f o r f ree-free t r a n s i t i o n s . I f one i n t e r p r e t s high s e r i e s a s t r ans i -

t i o n s of nea r ly f r e e e l ec t rons , then t h e p l a u s i b i l i t y of such an argument is

seen.

Even f o r s t rong and/or i s o l a t e d l i n e s , a f u l l y d e t a i l e d evaluat ion of Equations (1) , (9) , and (10) is no t required; r a the r , t h e l i n e group approximation can be u t i l i z e d . I n i t i a l l y , t h e frequency range of i n t e r e s t is divided i n t o a number of increments. Each increment def ines as a l i n e group those l i n e s which are centered within it. The l i n e cont r ibut ions a t a frequency po in t within a frequency increment is obtained by summing over only those l i n e s within i t s l i n e group. This approximation usual ly y ie lds a s i g n i f i c a n t s impl i f i ca t ion with only a neg l ig ib le l o s s i n accuracy. 1 2.1.3 Molecular Band Trans i t ions

The r o t a t i o n a l l i n e s s i t u a t e d i n molecular band systems can a l s o be

t r e a t e d a s a pseudo continuum, r a t h e r than individual ly, fo r many plasma con- d i t i o n s of i n t e r e s t . The dens i ty of molecular l i n e s i s always very large, and it can be assumed t h a t they are f u l l y overlapped and/or weak. Frequency

averaged models are near ly always employed i n which only t h e bands within t h e system a r e considered indiv idual ly (smeared band model) o r only t h e gross 1; shape @f t h e band system i tself is re ta ined (bandlees model). Such da ta is I

adequate f o r s tud ies i n which high s p e c t r a l reso lu t ion is not required.

The low frequency band systems of t h e air species (N2 (I+) , N2 (2+) , N2+(1-) , O2 (S-R) , NO ($) , n ( y ) ) have been extensively studied. Frequency averaged con-

t r i b u t i c r : ~ ~ are presented by Aroente and Magee (Ref. 18) and Bibenaan and

Mnatsakanyan (Ref. 19) , among others. A representative comparison* of their

results is given in Figure 2. It is seen that noticeable differences exist in

some of the details, but the general levels are in agreement except in the

ultraviolet where Biberman and Mnatsakanyan (Ref. 19) included contributions

from more transitions. Biberman and Mnatsakanyan (Ref. 19) showed that their

data is also in good agreement with the data (viewed with low spectral resolu-

tion) of Churchill et.al. (Ref. 20) in which over 150,000 lines were considered

individually.

A large number of high frequency band systems have been identified for

the air species (see Ref. 21, for example). Sufficient data is not available

to allow accurate calculations of the contributions from each individual band

system; however, the total contribution from each molecular species can be

estimated. Gilmore (Ref. 22) investigated the relative importance of 02. No

and N2 in air (contributions from other spbcies being negligible). at hv = 9.77 ev and for 2.00O0K s T s 8,000~~. He found that 02, NO and N2 contribu-

tions dominate at low, medium and high temperatures, in that order. The o2 contribution was attributed to the usual photodissociation continuum. The sources of the NO and N2 contributions have not been positively identified.

The high frequency contributions of NO were investigated by Biberman and

Mnatsakanyan (Ref. 19) . They found that the NO(&) and NO(€) systems make sig- nificant contributions in the range 5.2 ev s hv 8.2 ev and for an extended range of temperature. At higher frequencies, the room temperature data of

Watanabe (Ref. 23) show that importurt contributions from NO exist above

hv w 9.3 ev. Biberman and Mnatsakanyan (Ref. 19) used the room temperature

cross sections to estimate the high temperature contribution in this frequency

range. The resulting total contribution for NO is high below 8.2 ev and above

9.3 ev with a pronounced minimum in between. This minimum is probably spuri-

ous, caused by a lack of data and the use of room temperature cross sections

rather than an absence of absorbing mechanisms. Accordingly, Biberman and

Mnatsakanyan (Ref. 19) recommend interpolating between the two contributions

and disregarding the ninimum. In view of the absence of more nearly complete

data, this approach seems superior to the usual one of ignoring these systems.

The total high frequency contribution of N2 has been investigated experi-

mentally by Appleton and Steinberg (Ref. 24) and theoretically by Allen (Ref.

25).** In the experimental study, the contribution in the frequency range

" It was necessary to interpolate between the data points presented in Ref. 19 to allw the two sets of data to be compared. ** The contributions of particular band rystemr have also been studied by Churchill et.al. (Ref. 26) and Gilmore (Ref. 22)

2 4 6 8

PFOTON ENERGY, hv ( E V )

Figure 2. Typical Absorption Coeff ic ient Predictions Including Important Molecular Contributions

9.27 ev r hv r 10.4 sv are measured, and a method is suggested by which they

can be obtained to 12.75 ev. In the theoretical study, Allen assumes that

the total contribution is dominated by the long wave length "tailw of the N, I

Birge-Hopfield systems b L.J ,J + - X 'x and b 'nu - X 'zg+ for the frequency range 7.0 ev s hv s 14.25 ev. For frequencies above about 10.5 ev,

the experimental data is substantially above the calculated data indicating

that contributions from sources other than the Birge-Hopfield systems are

important. However, there is no reason to believe that Allen's data is in

error at the lower frequencies. Thus, the experimental data can be used in

the frequency range 9.27 ev s hv s 12.75 ev and the theoretical data in the

range 7.0 ev s hv s 9.27 ev.

The contributions from the ablation product species are available from

several sources. The species associated with the C02 - N2 system have been studied by Arnold, Reis and Woodward (Ref. 27) and more recently by Woodward

(Ref. 28) . They considered the CN(V) , CN (R) , C2 (Swan) and C0 (4+) along with the usual air systems. The contribution from the first three band sys-

tems are considered to be reasonably well established. In the case of the

~0(4+) system, significantly lower estimates of the magnitude of its contri-

bution are also available (see Ref. 29, for example). The contributions of

the C2 (Freymark) , C2 (Fox-Herzberg) , C2 (Mulliken) , H2 (~yman) and H2 (l'erner) band systems are also available from Weisner (Ref. 30) . 2.1.4 Other Transitions

The O2 Schumann-Runge photodissociation continuum is known to be the

most important ultraviolet contributor for the air system at lower tempc--a-

tures (Ref. 22). Evans and Schexnayder (Ref. 31) have studied this transi-

tion experimentally and numerically. A coinparison between their results and

the approximate Sulzer-Wieland formula

indicates that it is accurate up to temperatures sufficiently high to cause

O2 disoociation, provided that a threshold frequency of 7.1 ev is imposed.

The photodetachment trarrsition of the negative ions of atomic oxygen and

hydrogen are known to exirrt and to i~ important for some plasma conditions.

i . . . . , . I.. . p - . . . .- '. li . * e . - L..

The c r o s s s e c t i o n s f o r 0- a r e a v a i l a b l e from t h e work o f Church i l l , Armstrong f and Mueller (Ref. 26). The c r o s s s e c t i o n s c?f: H- are very w e l l known and a r e 1 a v a i l a b l e from severir l sources (Refs. 32, 33.. and 34) , agreement be ing on t h e I 2

orde r of f i v e percent . The C- i on is a l s o known t o undergo a photodetachment t r a n s i t i o n , and c r o s s sect ior is have been obtained by Cooper and Martin (Ref.

3 5 ) . and are i n f a i r agreement with t h e measurements of Seman and Branscomb (Ref. 36) .

The c o n t r i b u t i o n of t h e N- ion is c u r r e n t l y being debated. Norman (Ref.

37) argues i n favor of an apprec iab le photodetachment con t r ibu t ion and p o i n t s o u t t h a t t h e u t i l i z a t i o n of a reasonable c r o s s s e c t i o n (- 8 x 1 0 - ~ ~ c m ' ) g r e a t l y enhances t h e agreement between c a l c u l a t e d (Ref. 6) and measured (Ref. 38) continuum emission. The r ecen t measurements o f Morris et.al. (Ref. 39) a l s o in- t

d i c a t e s a s u b s t a n t i a l N- con t r ibu t ion . However, t h e N- con t r ibu t ion is no t i l

taken i n t o account i n most o f t h e t h e o r e t i c a l papers on t h e emission of an a i r plasma, inc lud ing many of t h e most r e c a n t ones (Refs. 5, 10 and 1 5 ) . I t

fl ir

has no t been p o s i t i v e l y e s t a b l i s h e d t h a t t h e i on i s s t a b l e , e s p e c i a l l y i n i t s

ground s t a t e . Fur ther , i t s photodetac'hment c r o s s s e c t i o n i s completely un- '?

known. Thus, t h e N- photodetachment con t r ibu t ion is requi red t o make predic- !I - -

t i o n s agree wi th r ecen t experimental f ind ings , b u t t h e t r a n s i t i o n i s no t suf- f i c i e n t l y w e l l understood t o a l low p r e d i c t i o n s t o be made from first p r i n c i p l e s . i 1 - - A numerical i n v e s t i g a t i o n of t h e poss ib l e e f f e c t s of t h i s t r a n s i t i o n i s given i n Sec t ion 4.1.

The c o n t r i b u t i o n s from molecular photo ion iza t ion t r a n s i t i o n s a r e impor-

t a n t on occasion. Biberman and Mnatsakanyan (Ref. 19) found t h a t t h e NO ,. . photo ion iza t ion con t r ibu t ion makes an important .mt r ibu t ion t o t h e i r t o t a l

h igh frequency NO con t r ibu t ion . However, o t h e r molecules such as CO., li;. and o2 have photo ion iza t ion th re sho lds a t such h igh f requencies t h a t t hey can

u s u a l l y be neglected.

F ina l ly , t h e r e is reason t o be l i eve t h a t absorp t ion and s c a t t e r i n g by 1 ! p a r t i c u l a t e mat te r (carbon ~ r i m a r i l y ) can be i m p o r t m t f o r some cases ( s ee

7 - Ref. 29, f o r example). I t is f e l t that t h i s phenomena l r n o t s u f f i c i e n t l y f r w e l l understood t o be dsscr ibed q u a n t i t a t i v e l y . Consequently, t h e p re sen t

model is app l i cab ie t o those cases where on ly a neglicj ible i n t e r a c t i o n e x i s t s b t w e e n t h e r a d i a t i o n f ie ld and any p a r t i c u l a t e matter.

2.2 THE RADIATION PROPERTIES MODEL

The rad ia t ion p roper t i e s which are f e l t t o be most near ly cons i s t en t with

t h e objec t ives of t h e present s tudy are given i n Sect ions 2.2.1, 2-2.2, and

2.2.3. These are ul t imate ly used i n t h e numerical work presented i n Sect ion 4,

2.2.1 Molecular Band M o d e l

The bands within each band system are smeared according t o t h e scheme

where the Av a r e se lec ted such t h e < v a r i e s smoothly. The "bandless model" obtained i n t h i s fashion is e s s e n t i a l l y t h e same as t h a t proposed o r i g i n a l l y

by Meyerott e t - a l . (Re f . 40) and used more recent ly by Bibeman and matsakan-

yan (Ref. 19). It is f e l t t o be s a t i s f a c t o r y f o r r ad ia t ion hea t ing calcula-

t i o n s .*

2-2.2 A t o m i c and Ionic Line Model

The l i n e group approximation is employed t o s implify the ca lcula t ion . The frequency range of i n t e r e s t is divided i n t o a number of frequency increments - 15 t o 20 o r thereabout, These are no t necessar i ly connected. Each one def ines

as a l i n e grouF those l i n e s which are centered within it, The l i n e cc t r ibu-

t i o n s a t a frequency po in t within a frequency increment is obtained by summing

over only those l i n e s withln its group.

An addi t ional c l a s s i f i c a t i o n of the l i n e s is required, L i z e s having a

lower l e v e l with a p r inc ipa l quantum number (2) of four o r g r e a t e r are con-

s idered t o be high series l i n e s . Contributions from them a r e obtained from

i n t e g r a l formulas (Eq. (19)). Lines having an equal t o o r less than two

are t h e very s t rang l i n e s s i t u a t e d i n the u l t r a v i o l e t . The contr ibut ions from

them a r e ca lcula ted i n f u l l d e t a i l (within t h e l i n e group approxinstion). Lines

having an o f t h r e e appear i n the in f ra red and, t o a l e s s e r extent , t h e v i s i -

ble range of t h e spectrum. They are of intermediate s t r eng th - probably too

s t rong t o be ca lcula ted by an i n t e g r a l formula bu t not s t rong enough t o requi re

a f u l l y de ta i l ed ca lcula t ion . Contributions from these l i n e s a r e obtained by

considering t h e cont r ibut ions from an waveragen l i n e (defined by an averaged

l i n e width and f-number) and multiplying by t h e number of l ines . The averag-

ing process is l imi ted t o l i n e s within t h e same l i n e group and having lower

l eve l s with t h e same pr inc ipa l and o r b i t a l quantum numbers. This approach

neglects l i n e overlapping, an e f f e c t which should be of negl ig ib le importance f o r these l i n e s f o r m o s t cases of i n t e r e s t .

* A discussion is given i n k c t i o n 3.2 on tile e f f e c t this approximate model has on t h e ca lcula t ion of r a d i a t i v e f luxes ,

2.2.3 Sources of D a t a

The mole f r a c t i o n s of t h e r a d i a t i n g spec ies were ca lcu la ted w i n g a chem-

ical equilibrium computer program ( t h e ACE program - s e e Ref. 41). The bas ic d a t a ( s k z c i f i c hea ts , hea t s of formation, etc.) we^ . obtained from t h e compila-

t i o n of Hochstim (Ref, 42) with some minor modifications t o t h e negative ion

d a t a t o make them agree better with t h e r ecen t ca lcu la t ions of Gilmore (Ref.

43). When l e v e l populations a r e required (atomic l i n e and continuum calcula-

t i o n s ) , t h e appropriate Boltzmann f r a c t i o n s a r e evaluated as p a r t of t h e ab-

sorpt ion c o e f f i c i e n t ca lcula t ion . The required s t a t i s t i c a l weights, l e v e l

energies and p a r t i t i o n functions were taken from Gilmore (Refs . 43 and 44) .

The r a d i a t i v e p roper t i e s model adopted f o r t h e present ca lcu la t ions was

taken from t h e sources 0 5 d a t a given i n Tables 1-IV. The atomic and i on ic

continuum cont r ibut ions are given i n Table I, t h e atomic and ion ic l i n e s i n

T a b l e 11, t h e molecular bands i n Table 111, and t h e o ther c o ~ t r i b u t i o n s i n

Table IV.

TABLE I

ATOMIC AND IONIC CONTINUUM

Species Hydrogen

Oxygen and Nitrogen

Carbon

A l l atomic ions

Trans i t ions I Source High frequency photo- ion iza t ion edges

Low frequency photo- ioniza t ion

free- f ree

High frequency photo- ioniza t ion edges

Low frequency photo- ion iza t ion

free-free

High frequency photo- ion iza t ion edges

Low frequency photo- ioniza t ion

free-free

A l l t r a n s i t i o n s

Eq. (3) with Gaunt f a c t o r of u n i t y

Eq. (5) with Gaunt f a c t o r of un i ty

Numerical d a t a from Ref. 2

Eq, (8) modified empiri- c a l l y t o b e t t e r agree with d a t a from Ref. 2

Eq. (5) with Gaunt f a c t o r of un i ty

Num~ :ical da ta from Ref. 2

Eq. (5) with Gaunt f - l t o r of un i ty

Not included a s t h e i r continuum t r a n s i t i o n s are negl ig ib ly small

TABLE I1

ATOMIC AND IONIC LINES

- Species

Hydrogen

Oxygen Nitrogen Carbon (atoms and ions)

rans sit ions

Lyman a

B

B a l m e r a

B

Other l i n e s acccunted f o r indiv idual ly

High l i n e s

High s e r i e s

Sources of

f width

--

I

Eq. (16)

f-no.

Ref.45

Ref .45

I !

Data

Other

Eqs. (13-15) used where appl icable

Eq. (17) used when appl icable

Ioniza t ion threshold lowered (';i = 8)

p; taken equal

Ref. 2 and

Eq- (11)

Line Shape

Numer- i c a l da ta from Ref.13

Eq. (10)

t o j/~,, with j from Eq. (19)

Eq. (17) used when appl icable

- Ioniza t ion threshold lowered (z = 8)

taken equal k j / ~ with j from ~ 4 . (19)

4

Eg. (10) A l l l i n e s accounted f o r indiv idual iy

.--

H i ~ h l i n e s

.High s e r i e s

Ref. 2

TABLE I11

MOLECULAR BAND SYSTEMS

TABLE IV

OTHER CONTRIBUTORS

Source

Curve fits to numerical data from Ref. 19

Curve fits to numerical data from Ref. 25 for hv s 10.5 ev and from Ref. 24 for hv > 10.5



Curve fits to numerical data from Ref, 30

b

Species

N2

O2

NO

N2

CN

=2

H2

Transitions

1+

2+

Birge Hopfield

Schumann-Runge Band System

Y

8 ~~6,photoionization

1-

red

violet

Swan Freymark

Mulliken

Lyman

Werner

Species

O2

0- -

H-

N-

C-

N2' O2 CO8 H2 CN, C2

ALL

- Transitions

Photodissociation

Photodetachmnt

Photodetachment

Photodetachment

Photodetachment

Photoionization

Particulate absorption and Scattering

Source

Eq- (20)



It is not included in the numerical work unless it is called out. Then, the contribution is consistent with the data of Ref. 39

Numerical data from Ref. 36 - Neglected

Keglected

I

SECTION 3

TRANSPORT MODEL

3.1 FORMULATION

The bas ic equation governing t h e t r a n s f e r of r ad ia t ion through a medium

i n l o c a l thermodynamic equi l ibr ium can be w r i t t e n as

where Iv is t h e s p e c t r a l i n t e n s i t y , Bv is t h e Planck function, S i s the length

o f t h e ray and p ' is t h e absorption c o e f f i c i e n t cor rec ted f o r induced emission, v viz.

and wkere p is t h e ordinary absorption coe f f i c i en t . v

I n computing rad ia t ion f luxes across boundary and/or shock layexs, it is

convenient t o make t h e tangent s l a b approximation. Thus, the proper t i e s along

any ray can be r e l a t e d t o those along t h e normal coordinate ( y ) by applying a cosine transformation. The coordinate system is shown i n Figure 3. The re- s u l t i n g expressions f o r t h e o p t i c a l depth, s p e c t r a l f luxes and t o t a l f lux a r e

w e l l known and can be w r i t t e n i n t h e form

where f luxes en te r ing o r r e f l e c t e d a t y = 0 have been neglected ( a cold, black

wall is assumed). The quant i ty Ev i s t h e black body emissive power defined a s

E,, = nBv (27)

F ina l ly , Equations (23) and (24) u t i l i z e e rn iss iv i t ies a s independent va r i ab les

where

and t h e &(x) functions a r e exponential i n t e g r a l s of order n.

The exponential approximation can be used t o fu r the r s implify t h e equa-

t i o n s without an appreciable l o s s i n accuracy, It is known (see Ref. 15, f o r

example) t h a t when t h e approximation

2 8 (x) o exp (-2x) (30) 3

is made, exact t r anspor t s o l u t i o n s a r e obtained i n t h e o p t i c a l l y t h i n and op-

t i c a l l y t h i c k l i m i t s , and good approximations a r e obtained a t intermediate

o p t i c a l depths. Thus, t h e e m i s s i v i t i e s become

t + r v a 1 - exp 2 ( ~ ~ - tv) (31)

- E a 1 - exp 2 ( t Y - T ~ )

V (32)

which a r e more convenient t o work with than those given by Equations (28) and

( 2 9 ) . Emiss iv i t ies wr i t t en i n t h i s form have t h e addi t ional advantage t h a t

by suppressing the f a c t o r of 2 i n t h e exponential arguments and replacing Ey

by Bv i n Equations (23) and (24). t h e same formulation can be used t o ca lcu la te

i n t e n s i t i e s .

Before evaluat ing Equations (23)-(26) it is advantageous t o separa te t h e

i n t e g r a l s i n t o l i n e and continuum par t s . This allows optimum coordinates t o

be se lec ted i n frequency space. The continuum contr ibut ion t o the absorption

c o e f f i c i e n t (r p t ) is j u s t the f i r s t term i n Equation (1). Subs t i tu t ing U: C f o r pv i n Equations (23) - (26) y ie lds corresponding values f o r T~ and Fv, v

where the 2 s igns have been dropped from t h e l a t t e r f o r b rev i ty , The l i n e

cont r ibut ions t o t h e f lux a r e obtained by skfference

where the Fv values are evaluated using t h e t o t a l absorption c o e f f i c i e n t -. (Eq. 1 Thus, t h e l i n e cont r ibut ion is treatad as a correc t ion t o t h e con- :I

X

t inuum flux. *

A d i r e c t evaluat ion of Equations (23)-(26) requi res an i m p l i c i t assump- - t i o n - t h a t only a reasonable number of frequency po in t s need be s l eec ted t o

adequately descr ibe t h e v a r i a t i o n of t h e spectrum. I n t h e case of t h e con-

tinuum spectrum, t h e assumption is well s a t i s f i e d w i t h about 25-50 po in t s 3 : being s u f f i c i e n t , I n t h e case of atomic l i n e s , t h e number is approximately 3 1,000 frequency po in t s (100 l i n e s with 10 po in t s per l i n e ) , which i s g e t t i n g i

l a r g e b u t can s t i l l be handled f o r most cases. I n t h e case of molecular l i n e s

t h e requirement r i s e s t o approximately 1,500,000 frequency poin ts (150,000 3 1

l i n e s with 10 poin ts per l i n e ) , which is q u i t e impract ical fo r r ad ia t ion heat- I

i n g ca lcu la t ions ,

I n t h e present study, t h e problem of resolving t h e molecular zpectrum

is solved summarily. The u t i l i z a t i o n of t h e molecular band model ( t h e band- % : i

I l e s s model) converts the molecular spectrum i n t o an "equivalent" continuum

process. Such an approach ignores t h e d e t a i l s of the i n t e r n a l s t r u c t u r e of

t h e band system. The calcula ted s p e c t r a l f luxes y i e l d t h e co r rec t t o t a l f luxes

(when in tegra ted over a band system) only when the layer i s o p t i c a l l y th in ,

o p t i c a l l y thick, o r when t h e l i n e s a re spaced s u f f i c i e n t l y c lose together t o

* It should be noted t h a t the l i n e cont r ibut ions defined by Equation (33) can be negative f o r nonisothermal layers .

be s t rongly overlapped. Otherwise, only an approximation t o t h e t o t a l f lux

is obtained.*

No add i t iona l approximations a r e required. The continuum f luxes and t h e

approximation t o t h e molecular l i n e f luxes can be ca lcula ted i n a s t r a i g h t -

forward manner. The atomic and ion ic l i n e f luxes can a l s o be obtained d i r e c t l y . While t h i * ca lcu la t ion is lengthy, it is no t f e l t t o be excessively so. I n

a l l cases c a r e f u l a t t e n t i o n m ~ t s t be paid t o t h e s e l e c t i o n of nodal poin ts ( i n frequency and i n space) and in te rpo la t ion formulas t o insure t h a t high accuracy

i s maintained.

3.2.1 Nodal Points i n Frequency

For t h e continuum cont r ibut ion t o t h e f l u x i n t e g r a l s t h e frequency g r i d

can be spec i f i ed a p r i o r i and i s only s l i g h t l y dependent on t h e charac ter i s - t i c s of t h e layer . Basical ly , t h e va r in t ion of t h e Planck function and t h e

continuum absorption c o e f f i c i e n t s must be adequately described. The f i r s t requirement can be m e t by d i s t r i b u t i n g nodal po in t s across t h e frequency range hv FJ 0.25 ev t o hvm, RI 1 2 kTm ev. Usually about 30 po in t s spaced a t roughly equal i n t e r v a l s a re s u f f i c i e n t . This g r i d a l s o s a t i s f i e s t h e second require-

ment except i n the u l t r a v i o l e t . There t h e frequency g r i d must be c a r e f u l l y se lec ted t o resolve t h e photoionization thresholds.**

* The accuracy of t h e approach can be defended on the following grounds: (1) The molecular spec t ra f o r which good property da ta a r e avai lab le a r e a l l s i t u a t e d i:r t h e irifrared, v i s i b l e , o r near u l t r a v i o l e t . ~t is w e l l known t h a t t h i s region of t h e spectrum is o p t i c a l l y t h i n f o r typ ica l boundary or shock l aye r conditions and, hence, t h e bandless model is accurate f o r these data. (2) The molecular spec t ra s i t u a t e d i n t h e u l t r a v i o l e t and t h e vacuum u l t r a v i o l e t a r e o f t en s t rong enough t o absorb appreciably. However, good rad ia t ion property datin a r e not ava i l ab le i n t h i s region; consequently, t h e bandless model approach is cons i s t en t with t h e q u a l i t y of the data. (3) Final ly , and of primary importance, it i s known t h a t whenever r ad ia t ion heating is comparable t o convective heating, t h e molecular species w i l l be con- f ined t o a narrow l a y e r near the w a l l . The rad ia t ion w i l l o r i g i n a t e i n t h e ex te rna l region of t h e flow which w i l l be h o t enought t o be dissoc ia ted and a t l e a s t p a r t i a l l y ionized. Thus, i n any competition f o r accuracy and/or d e t a i l t h e atom and ion rad ia t ion must be given p r e f e r e n t i a l treatment. It is t o be noted t h a t more sophis t ica ted band models a r e used by some inves t iga tors . These extend t h e range of v a l i d i t y somewhat, but intraduce m a t is b a s i c a l l y t h e same type of approxin~ation and a r e accurate i n the same regimes.

** It should be noted t h a t t h e movement of the thresholds ( i n frequency) caused by plasma i n t e r a c t i o n e f f e c t s i s not considered. This is cons i s t en t with t h e approximation discussed i n Sect ion 2.

For the l i n e cont r ibut ion t o the t r anspor t i n t e g r a l s , a g r i d must be

se lec ted f o r each l i n e and should be dependent upon the c h a r a c t e r i s t i c s of

t h e l aye r a s w e l l a s the individual l i n e . This i s accomplished by in t rodu. ing a parameter cp which is c h a r a c t e r i s t i c of t h e width of t h e l i n e (and w i l l be discussed f u r t h e r i n a subsequent paragraph) t o s t r e t c h t h e coordinate system.

The smal les t increment and t h e d is tance t o t h e most remost nodal poin t a r e

defined by Equations (34) and (35) , respect ive ly , where ; = I v - V C I .

The q u a n t i t i e s f l and f 2 a r e se lec ted a r b i t r a r i l y and are usual ly taken t o be 0.5 and 10 ( re spec t ive ly ) , on t h i s order. The intermediate nodal poin ts

are es tabl i shed using a growth l a w , viz.

where t h e nodal spacing increases ( increasing subsc r ip t j) with increas ing d is tance from t h e cen te r of t h e l i n e . The growth f a c t o r f i s determined i m - p l i c i t l y from t h e r e l a t i o n

where N i s t h e number of increments t o be used. Usually, t h e center of the

l i n e sr.d a b u t 5 t o 7 addi t ional po in t s i n each d i r e c t i o n from it a r e s u f f i -

c i en t .

The s t r e t c h i n g parameter i s defined by Equation (381,

where t h e (half) h a l f width y(y) is evaluated a t a p a r t i c u l a r s p a t i a l loca t ion (usual ly a s an edge condition) and 7, is t h e o p t i c a l depth of the ' en t i r e l aye r

I at t h e frequency of t h e cen te r of t h e l i n e . I t can be shown t h a t cp has t h e

following propert ies :

That is, t h e q~ i s t h e d i s t m c e i n frequency space from tha cen te r of t h e l i n e

t o t h e h a l f i n t e n s i t y pos i t ion when t h e l i n e i s weak, and it .is approximatd y t h e d is tance t o t h e frequency a t which t h e l aye r has u n i t o p t i c a l depth when t h e l i n e is strong. A t intennediate values of T,, q~ should a l s o be a r e a r m - a b l e approximation t o t h e width of t h e l i n e .

3.2.2 Nodal Points i n Space

I n i t i a l l y a s p a t i a l g r i d is se lec ted s o t h a t it adequately descr ibes t h e v a r i a t i o n s of t h e thermodynamic p roper t i e s across t h e rxd ia t ing l aye r ( the c i r c l e s shown i n Figure 4, f o r example). The o p t i c a l depths a r e then calcula ted* and used t o evaluate t h e emiss iv i t i e s . Typical values a r e shown i n Figure 4 (b) where Curves 1, 2 and 3 i l l u s t r a t e t h e o p t i c a l l y t h i n , moderately

t h i c k and very th ick layers , respectively. Transforming i n t o t h e Ey VS. E v plane y ie lds t h e poin ts shown (as circles) i n Figure 4(c) . Curves 1 and 2

a r e well-behaved; however, Curve 3 is very ill behaved, showing e s s e n t i a l l y a

s i n g u l a r i t y a t the ou te r boundary. To circumvent t h e problems caused by t h i s

V

0 1.0 0 1 .0 0 1 .o ~ 1 5 Y I ~ € "

(a) (b) ( c )

Figure 4. S p a t i a l Grid Transformations

* The methods ueed t o evaluate t h e i n t e g r a l s appearing i n Zquations (23)-(26) w i l l be described subsequently.

I.

behavior, a spec ia l coordinate system is used f o r t h e o p t i c a l l y th ick l a y e r s

( Tv > 2 f o r present purposes). The l aye r i s divided i n t o equal increments of

e ~ ~ i s s i v i t i e s (crosses i n Figure 4 (c ) from which t b e new value:; of o p t i c a l

depth a r e calculated.* The new E values a r e then obtained b y cubic interpo-

1 v

l a t i o n i n t h e plane lnEv vs. lnTV.** The new curve i n t h e E vs. E plane v v (crosses i n Figure 4 ( c ) ) is s u i t a b l e f o r accurate evaluatlon by numerical

I methods.

3.2.3 In tegra t ion Scheme

With one exception, a l l t h e i n t e s r a l s a re evaluated using th ree term Taylor ' s s e r i e s expansions a s in te rpo la t ion formulas. The de r iva t ives re-

I quired i n t h e second two terms a r e obtained from Equation (41) , where x and y a r e t h e general independent and dependent var iables , respect ively. I

3 = YS ( $1)

The s values a r e taken equal t o Lhe s lopes i n the plane I n y vs. x and *re obtained from curve f i t s . I t has been found t h a t t h i s procedure minimizes

unwanted excursions i n the higher order terms.

The Taylor 's series expansions cannot be used f o r the frequency integra- t i o n of rhe continuum fluxes due t u t h e d i scon t inu i t i e s i n t h e in teg ra l s . These i n t e g r a l s were evaluated using l i n e a r in terpola t ion formulas, instead. This causes no ser ious l o s s i n accuracy as t h e i n t e g r a l s a r e slowly varyinq except a t t h e d i scon t inu i t i e s .

3.3 THE RADIATION TRANSPORT PROGRAM (RAD) ! The e n t i r e so lu t ion procedure \as been programmed fo r the Phi lco 2 1 2 and

Univac 1108 computers. A descr ip t ion of the code has been presented elsewhere (Ref. 46) . ~ y p i c a l l y , about 3C seconds of Univac 1108 computer time a re re-

quired t o ca lcu la te t h e f lux from t h e boundary of a uniform layer . The calcu- l a t i o n of t h e f lux d i s t r i b u t i o n across a l aye r containing a C-H-0-N elemental mixture requi res s l i g h t l y more than 1 minute.

"1n evaluat ing t h e o p t i c a l depths, t h e maximum allowable value of q,, is taken t o be 0.99.

** The o r i g i n of t h e o p t i c a l depth in teg ra t ion i s s h i f t e d and/or inverted i n space t o insure t h a t t h e lnEvvs. l n ~ curve is w e l l behaved f o r t h e interpo- l a t ion . v

SECTION 4

APPLICATION OF THE NETHOD

A matrix of t h e o r e t i c a l predic t ions has been obtained. Selected cases

are compared with predic t ions from o the r s tud ies i n Sect ion 4.1 and with ex-

perimental data i n Sect ion 4.2. Calculat ions of t h e r a d i a t i v e t r anspor t

through isothermal l aye r s are given i n Sect ion 4.3 and through a noniosthermal

layer i n Section 4.4, both a s examples of t h e use of the predic t ion method.

4.1 COMPARISON WITH OTHER PREDICTIONS

The f i r s t set of t h e o r e t i c a l predic t ions is given i n Figure 5 and com-

pared with the semi-empirical predic t ions of Morris e t . a l . (Ref. 39). Three

predict ions a r e presented using t h e present model - one without an N- cont r i -

bution, one with an N- contr ibut ion and a cross sec t ion of 8 x 1 0 - ~ ' c d as

suggested by Norman (Ref. 37), and one with an N- cont r ibut ion and a c ross

sec t ion of 1.6 x 10-16cma which makes t h e present predic t ions agree with the

predict ions of Morris e t - a l . (Ref. 39) . While a c ross sec t ion of 1.6 x 10-l6

c& is l a r g e r than t h a t of most continuum t r a n s i t i o n s , it is ne i the r SO l a rge

nor s o d i f f e r e n t from t h e one suggested by Norman (Ref. 37) a s t o appear t o

be unreasonable. However, based on this evidence alone, it cannot be con-

cluded t h a t the present predic t ions are cons i s t en t with those of Morris e t , a l .

(Ref, 39) .

Additional predic t ions of t h e ni t rogen continuum i n t e n s i t i e s a re given

i n Figure 6 and compared with t h e predic t ions of Biberman and Norman (Ref. 6)

and with the da ta of Boldt (Ref. 38) snd Morris e t . a l . (Ref. 39). It is seen

t h a t the present predic t ions (without N-) a r e g rea te r than those by Biberman

and Norman (Ref, 6) by W?lt a f a c t o r of two;* l i k e w i s e , t k e measurement5 of

Morris et.al ( Z e f . 39) are g rea te r than t h e measurements of Boldt e t . a l . (Ref,

38) by about a f ac to r of two. Thus, assigning t h e difference between measure-

ment and predict ion t o t h e N- contr ibut ion can y ie ld e i t h e r t h e rough agreement noted previously o r gross d i f ferences i n the N- c ross sect ion, depend-

ing on t h e combinations of measurements and predic t ions selected. It is a l s o

seen i n Figure 6 t h a t t h e present predic t ions (without N-) are i n f a i r agree-

ment with both sets o f measurements and t h a t t h e s lopes of a l l t h e predict ions

* I n t h i s s p e c t r a l region, t h e absorption c o e f f i c i e n t s used by Bibeman and Norman (Ref. 6) d i p below those used in t h e present study; otherwise, the agreement is general ly very good.

Figure 5. Comparison with the Predictions of Morris et .ale (Ref. 39)

I P R E S S U R E = 1 A T M

T E M P E R A T U R E = 1 3 , O O O ° K

P R E S E N T C A L C U L A T I O N W I T H N- ( a = 1 . 6 x

l o - ' ' C M 2 )

4 B I B E R M A N ? N D NORMAN ( R E F . 6 ) W I T H O U T N C O N T R I B U T I O N

P H O T O N E N E R G Y , hv ( e V ) /

Figure 6. Effect of N- on the Spectral Distribution of the Intensity of Nitrogen, 6 = 1 cm

(without N-) a r e i n good agreement. The agreement i n s lope is destroyed when

t h e N- cont r ibut ion i s included i n t h e present predic t ions .* Thus, forcing

agreement between t h e present predic t ions and t h e t o t a l i n t e n s i t i t e s predicted

by Morris e t . a l . (Ref. 39) decreasss t h e agreement between the predic t ions and

t h e i r s p e c t r a l data. Thus, it can be concluded t h a t the present predic t ions

(without N-) a r e i n f a i r agreement with t h e predic t ions and measurements of

Morris e t . a l . (Ref. 39) and t h a t t h e inc lus ion of an N- increases only c e r t a i n

aspects of t h i s agreement. The numerical work t o be presented subsequently

does not contaj m N- cont r ibut ion unless it i s s p e c i f i c a l l y c a l l e d out. - The t h e o r e t i c a l prodict ions f o r a i r a r e presented i n Figure 7 and com-

pared with t h e t h e o r e t i c a l p red ic t i cns of Wilson and Greif (Ref. 10). Their

p roper t i e s model f o r t h e atomic and (i30sitive) ionic spec ies is very s imi la r

t o t h a t used i n t h e present study (Ref. 2 ) . However, they d id not include

cont r ibut ions from t h e molecular species or t h e negative ions. I n addi t ion,

they employ t h e e f f e c t i v e width approach which approximates t h e cont r ibut ions

due t o t h e atomic and ion ic l i n e s . The r e s u l t i n g predic t ions (Fig, 7) a r e i n

exce l l en t agreement f o r t h e 6 = 1 cm case exf.!ept a t t h e lower temperatures

where t h e molecular cont r ibut ions a r e c l e a r l y important. For t h e 6 = 10 cm

case, t h e predic t ions d i f f e r by about 25 - 35 percent, even i n t h e temperature

range where no d i f ferences should e x i s t . This d i f ference cannot be a t t r i b u t e d

t o fundamental d i f ferences i n e i t h e r t h e p roper t i e s o r t h e t r anspor t models.

Molecular contr ibutrons a r e neg l ig ib le above about T % 1 2 , 0 0 0 ~ ~ . I n addi t ion , t h e d i f ferences i n ca lcu la t ing l i n e e f f e c t s cannot be responsible a s they

would be i n t h e opposite d i rec t ion . However, Wilson and Greif (Ref. 10) u t i -

l i z e Equation (8) a s a convenient approximation t o t h e continuum absorption

c o e f f i c i e n t s , r a t h e r than t h e numerical da ta presented i n Reference 2. While

t h e agreement between t h e two is genera l ly very good (see Fig. l), a?l e r r o r

e x i s t s i n t h e 1.5 - 4 ev range which is s u f f i c i e n t t o cause t h e d i f ferences

observed.

A comparison with t h e t h e o r e t i c a l predic t ions of a i r by Biberman e t . a l .

(Refs. 8 and 19) is given i n Figure 8. The d i f ferences between t h e i r property

d a t a and t h a t used i n t h e present study a r e not l a r g e f o r t h e l i n e radia t ion ,

both f - numbers and l i n e widths being i n good agreement. Further, t h e prop-

e r t i e s used f o r t h e molecular species cont r ibut ions a re from t h e same source

* I f t h e N- c ross sec t ion were assigned a va r i a t ion i n frequency, agreement with t h e s p e c t r a l da ta of Morris e t . a l . (Ref. 39) could be maintained; however, t h e r e present ly e x i s t s no b a s i s f o r such an assignment.

Figure 8 , Flux Emitted from a Uniform A i r Slab, P = 1 atm

(Ref. 19). The d i f ferences are s l i g h t l y more appreciable i n the case of t h e

continuum rad ia t ion proper t ies . They use Equation (8) and include a con t r i -

but ion from t h e N- ion. The addi t ion of t h e N- continuum i n t h e 1.5 ev t o

4 ev s p e c t r a l region more than compensates (usual ly) f o r the d i f ferences be-

tween Equation (8) and t h e d a t a given i n Reference 2. Thus, t h e f luxes Bibex-

man e t - a l . (Refs. 8 and 19) p r e d i c t s a r c cons i s t en t ly higher than those pre-

d ic t ed i n t h e present study. However, reference t o Figure 8 ind ica tes t h a t

t h e d i f ferences are not large.

A comparison with t h e t h e o r e t i c a l predic t ions of Hunt and Sibulkin (Ref. 15) f o r pure ni t rogen is given i n Figure 9. They obtained t h e i r molecular

spec ies cont r ibut ions from Allen (Ref. 47) , t h e i r continuum rad ia t ion from t h e

tabula t ions of Sherman and Kulander (Re :. 48) , t h e i r f-numbers from Kelly

(Ref. 49) and t h e i r h a l f widths from Griein (Ref. 9) . The tabula t ions of Sher-

man and Kulander (Ref. 48) a r e somewhat higher th.a,r. " :1:~3e used i n the present

study (Ref. 2). done of t h e da ta obtained from t h e o ther sources is s i g n i f i -

c a n t l y d i f f e r e n t from t h a t used i n t h e present study. Hunt and Sibulkin do

handle the t r anspor t s l i g h t l y d i f f e r e n t l y i n t h a t J - s p l i t t i n g is considered

and quas i - s t a t i c l i n e p r o f i l e s a r e used f o r some of t h e higher l ines . However,

reference t o ~ i g u r e 9 ind ica tes t h a t these d i f ferences a r e not s i g n i f i c a n t a s

good agreement e x i s t s between t h e two sets of predict ions.

A comparison with t h e t h e o r e t i c a l predic t ions of Lasher e t . a l . (Ref. 3)

f o r hydrogen is given i n Figure 10. The c l a s s i c a l formulas f o r f-numbers

and continuum absorption c o e f f i c i e n t s w e r e used i n both s tudies . However,

Lasher e t .a l . (Ref. 3) u t i l i z e somewhat d i f f e r e n t l i n e shapes (Eqs. (13) - (15) ,

only) , and they employ t h e equivalent width approximation. In addi t ion, Lasher

et.al. (Ref. 3) a r e concerned only with t h e atomic cont r ibut ions and do not

include cont r ibut ions from molecular o r ionized species . A s shown i n Figure

l G , t h e i r model i s not appl icable a t t e n atmospheres f o r t h e lower tempera&

tu res . However, t h e two s e t s of predic t ions are i n good agreement a t the low r

pressures.

4.2 COMPARISONS WITH MEASURED DATA

Predic t ions of t h e emission c o e f f i c i e n t s of shock heated eir a re given

i n Figures 11, 1 2 and 13 and compared with t h e measurements of Nerem (Ref. 5C)

and Gruszczynski and Warren ( R e f . 51). The predic t ions a r e i n excel len t agree-

ment with t h e da ta of Nerem (Ref. 50), b u t they are somewhat lower than t h e

d a t a of Gruszczynski and Warren (Ref. 51). The bas ic techniques used i n t h e

10 11 1 2 i 3

TEMPERATURE, T y ~ $ - ? ( O K )

Figure 9. Intensity Emitted from a Uniform Nitrogen Slab, 6 = 1 cm

0 *. + 1 LOGlO PATH LENGTH, 6 ( C M )

Figure 10. Intensity Emitted from a Uniform Hydrogen Slab, T = 1 0 , 0 0 0 ~ ~

Figure 11. Comparison with the Shock Tube Data o f Nerela (Ref. S O ) , PI = 1 nun Hg

Figure 12. Comparison with the Shock Tube Data of Nerem ( R e f . 5 0 ) . P1 = 2 nun Hg

A > 2 , 0 0 0 i

- P R E S E N T P R E D I C T I O N S

f \\\\\\ R E F . 5 0

-- . I l l

E X P E R I M E N T A L S E T U P

\

j7$'Jvs - E M I S S I O N M E A S U R E D , I N \ H I S R E j I O N

(3 r r 4

Figure 13. Comparison with Shock Tube Data of Ssuazynaki and Warren (Ref. 51)

two experimental s tud ies are very s imi la r , and t h e spne general range of con-

d i t i o n s were covered. It is unfortunate t h a t t h e two s e t s of r e s u l t s a r e not i n better agreement with each 0th3r. Nevertheless, t h e general agreement be-

tween predic t ion and experiment is f e l t t o be encouraging.

The predicted i n t e n s i t i e s of a i r behind a r e f l e c t e d shock wave a r e given i n Figures 14 and 15 and compared with t h e measurements of Wood e t . a l . (Ref. 52) a s revised and supplied by Wood (Ref, 53). The predic t ions a r e i n excel- l e n t agreement with t h e aeasurements f o r inc ident shock v e l o c i t i e s less than

7.5 m m / ~ set-and a r e above t h e measurements a t higher ve loc i t i e s . Golobic and N e r e m (Ref. 54) have aualyzed a r e f l e c t e d shock experiment s i m i l a r t o

t h a t performed by Wood et.al. ( R e f . 52) and have concluded t h a t t h e r ad ia t ion

losses s i g n i f i c a n t l y cool the gases f o r inc ident shock v e l o c i t i e s above 7.5

mm/p sec. They present ca lcula ted i n t e n s i t i e s f o r t h e low frequency spectrum 0

( A > 1,700 A) showing t h e effect of the losses . Obtaining t h e i r predicted r a t i o s of i n t e n s i t i e s with r ad ia t ion l o s s e s t o those without losses (as a

functioh of incident shock ve loc i ty) and using them t o (approximately) c o r r e c t

t h e predic t ion given i n Figure 14 y ie lds t h e dashed curve and a dramatical ly improved comparison between predic t ion and measurement. Such an e f f e c t would obviously a f f e c t t h e t o t a l i n t e n s i t i e s also. Thus, t h e present predic t ions appear t o agree w , . the data of Wood ( R e f . 53) .

This concludes the series of compzrisons with t h e war-: of o the r inves t i -

ga tors , It is felt chat t h e model has been proven t o be a v a l i d one. Sub- sequent numerical work w i l l be used t o i l l u s t r a t e interesting applicat ions.

4.3 .WPLSCATION TO UNIFORM SLABS

Theoretical predic t ions f o r uniform slabs a r e given i n Figures 16 znd 17.

They show t h e v a r i a t i o l of t h e i luxes e n i t t e d from uniform slabs of a i r . The ranges of temperature, pressure and path lengths se lec ted a r e t y p i c a l of tnose

which might be encountered i n t h e s tagnat ion region of a vehic le reenter ing from a lunar o r p lans tary mission.

4.4 APPLICATION OF NONUNIFORM SLABS

To fu r the r i l l u s t r a t e appl ica t ions of t h e method, a slab with nonuniform

properties was se lec ted and its d i s t r i b u t i o n s of f lux i n space and frequency obtained. The va r i a t ions of the thermodynamic state proper t ies across the

s l a b a r e given i n Figure 18 where the o v e r a l l length of the slab (y6) is 11.4 cm. This vaxiation .-6 t y p i c a l of t h a t which might be found i n t h e s tagnat ion region

6 . 5 7 7 . 5 8 8 . 5 9 9 . 5 10.0

INCIDENT S H O C K V E L O C i T Y V, (MM/pSEC)

Fi-re 14. Comparison with SIlock Tube Data of Wood (Ref. 53) and Wood et.al. (Ref. 52)

I I I I I I. 1: I I I I I 1 I I I I Figure 15. Comparison with Total Intensity Measurements

of Wood (Ref. 53) and Wood et.al. (Ref. 52)

I I

J -

7 - 1

6 . 5 7 7 . 5 8 8.5 9 9 .5 1 0 . 0

I N C I D E N T SHOCK V E L O C I T Y , V, ( M M l p S E C )

Figure 16. Variation of Flux from a Uniform Air Slab, 6 = 1 an

Figure 17. Variation of F l u from a Uniform Air S l a b , 6 = 10 CM

0

Figure 18. Thermodynamic State Variation

0 0 .1 0 . 2 2 . 3 0 .4 0 .5 0 .6 0 .7 0 . 8 0 . 9 1 .0

it; n a f I 4

T E M P E R A T U R E A N D P R E S S U R E R A I I O S , T / T t , P / P 6

T ~ ' = 1 0 , 0 5 0 ~ ~ P 6 = 0 . 4 A T M

I I

T / T 6 -. \

1

I I 1

! I

of a vehic le reenter ing from a lunar mission. Carbon and hydrogen gases were

included as they a r e usual ly present due t o mass i n j e c t i o n from t h e sur face

of t h e vehicle.

The s p a t i a l d i s t r i b u t i o n of t h e r ad ia t ion spec ies is given i n Figure 19.

The molecules r e s i d e i n a narrow region immediately adjacent t o t h e wal l

(y/y6 < 0.01). I n t h i s case t h e region war so narrow t h a t it had essential2.y

no i n t e r a c t i o n w i t h t h e r ad ia t ion f i e l d . A mixing region e x i s t s f u r t h e r ou t

where t r a n s i t i o n from mainly atomic t o mainly molecular spec ies occurs

(0.01 < y/y6 < 0.1) . The species i n t h i s region tend t o e m i t i n t h e v i s i b l e

and absorb i n t h e u l t r a v i o l e t . Note t h a t atomic carbon and hydrogen a r e pres-

e n t i n s i g n i f i c a n t q u a n t i t i e s . The e x t e r i o r f i e l d (y/y6 > 0.1) c o n s i s t s of pure a i r species. I n t h i s case, t h e atomic species dominate with t h e N con-

t r i b u t i o n s being of p a r t i c u l a r importance.

The s p e c t r a l d i s t r i b u t i o n of t h e p o s i t i v e continuum fluxes (d i rec ted

toward t h e wall) a r e shown i n Figure 20, both a t t h e wal l and a t the edge of

t h e boundary l aye r (which i s set a r b i t r a r i l y a t y/y6 = 0.211). The peaks i n

t h e wal l f lux a t hv = 1 ev and 3.3 ev are pr imar i ly due t o smission from t h e

red and v i o l e t CN bands i n the boundary layer . There i s a l s o a peak i n t f r e

boundary f lux a t hv = 3.3 ev, but t h i s must be a t t r i b u t e d t o %(I-) emission

from t h e e x t e r i o r flow region. The u l t r a v i o l e t cont r ibut ions t o t h e boundary

f lux above 10.8 ev a r e due t o N emission from i t s photoionization edges with

t h e f lux l e v e l s beyond 12 ev being near ly black body. Note that the wall f l u x

has undergone severe a t tenuat ion i n t h e region nv > 11.5 ev. This is due t o

absorption by t h e photo'onization edges of t h e carbon atoms i n the mixing re-

gion. Typically, l a r g e f r a c t i o n s of t h e inc ident u l t r a v i o l e t continuum w i l l

be absorbed i n the boundary layex.

The s p e c t r a l d i s t r i b u t i o n of the w a l l f lux is presented again i n Figure

21, b u t here tlie l i n e cont r ibut ions are a l s o included (and presented i n terms

of averages over a r b i t r a r i l y spec i f i ed frequency increments). The l i n e con-

t r i b u t i o n is extremely important (probably dominant) i n t h e v i s i b l e region.

I n t h e u l t * l o l e t region t h e lir!=.-3 cont r ibute a modest addi t ion t o t h e f lux

i n t h e 9-1b.d ev range. However, the . l i n e cont r ibut ion i n t h e 10.8-11.5 ev

range is negative (dark l ines ) and t h e t o t a l f1u.x t o t h e wall is s i g n i f i c a n t l y

less than it would have been i f l i n e contributions had not %en included. T h i s

is a l s o t y p i c a l as t h e u l t r a v i o l e t l i n e s can be s t rongly absorbed i n noniso-

thern.31 regions.

l o - '

N O R M A L I Z E D D I S T A N C E FROM W A L L , y / y g

Figure 19. Spatial Distribution of Radiating Spe~ies

T O T A L C O N T R I B U T I O N I C O N I ' I N U U M C O N T R I B U T I O N - - - - --

P O S I T I V E L I N E C O N T R I B U T I O N .-

2 4 6 8 10 1 2 14 P H O T O N E N E R G Y , h v ( E v )

Figure 21. Spectral Variation of Positive Flux at Wall

The energy exchange mechanism between t h e boundary l aye r and t h e e x t e r i o r flow is i l l u s t r a t e d fu r the r i n Figures 22 and 23. The s p ~ t i a l va r i a t ion of

t h e t ransmit ted f r a c t i o n of t h e p o s i t i v e l y d i rec ted t o t a l f l u x and t h e r e l a t i v e f r a c t i o n s of i ts cons t i tuen t s are shown i n Figure 22. Thus, t h e p o s i t i v e f lux undergoes a s teady a t tenuat ion from q w 6 t o q w 0.75 with t h e u l t r a v i o l e t continuum accounting f o r t h e bulk of t h e loss . For q < 0.75 t h e l o s s r a t e in- c reases not iceably due t o l i n e at tenuat ion. The e f f e c t i s fu r the r i l l u s t r a t e d i n Figure 23 where t h e s p a t i a l v a r i a t i o n of t h e n e t f lux is shown. The region 6 < q < 0.7 shows t h a t t h e en te r ing f l u x is always less than t h e depart ing f lux causing r a d i a t i v e cooling. Nearer t h e wal l (q < 0.7) the e f f e c t reverses causing r a d i a t i v e heating. Thus, the fullowing q u a l i t a t i v e e f f e c t s a r e present: (a) t h e gases near t h e w a l l are heated, (b) t h e gases i n t h e outer regions of t h e boundary l a y e r a r e cooled, (c) t h e emission from the ablated spec ies i n t h e boundary l a y e r is no t s u f f i c i e n t t o o f f s e t t h e l o s s e s from t h e boundary f lux by absorption and (d) the boundary f lux is at tenuated s i g n i f i - c a n t l y ( i n t h i s case by 25 percent) before it reaches the w a l l .

I

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SECTION 5

CONCLUDING REMARKS

A predic t ion method has been described which can be used t o obtain f luxes

or i n t e n s i t i e s a t any speci f ied point within a plane-paral lel s l a b ( f o r the

f lux ca lcula t ion) o r a t any point on s ray ( f o r the i n t e n s i t y ca lcu la t ion) .

While the method was developed f o r the study of radia t ion heat ing phenomena

within the mass in jec ted boundary l aye r environment, it is not l imited t o such

s tudies . Indeed, its primary v i r t u e is its v e r s a t i l i t y . The only approxima-

t i o n which is an i n t e g r a l p a r t the method is the smeared l i n e m ~ d e l f o r the

m i e c u l a r bands. Any of the o ther aspects of the proper t ies model can e a s i l y

be made t o include more (o r less) d e t a i l , a l lowins important t rade-offs t o be

made between accuracy and computational e f f o r t .

The comparisons presented i n Sections 4.1 and 4.2 showed t h a t the present

predict ions are i n generally good agreement with predict ions from other s tudies .

Comparison with avai lable experiments yielded no addi t ional information as the

inconsistencies between t h e d i f f e r e n t sets of experimental da ta are g rea te r

than a r e those between the sets of predict ions. However, it was possible t o

conclude t h a t N- t r a n s i t i o n does not necessari ly improve the prcper t ies model.

Several examples were given i n Sections 4.3 and 4.4 which serve t o fu r the r

i l l u s t r a t e the v e r s a t i l i t y of the method. Additional predict ions w e r e pre-

sented i n Pa r t 1 of t h i s s e r i e s of repor ts and include a predict ion of a radia-

t ion coupled, mass-injected boundary layer flow.

SECTION 6

REFERENCES

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G i l m o r e , F.S.: The Cont r ibu t ion of Generally-Neglected Band Systems and Continua t o t h e Absorption C o e f f i c i e n t o f Hiuh-Temperature A i r . JQSRT, 5 , 125, 1965.

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Appleton, J .P. and S t e i n b e r g , M.: Vacuum-Ultraviolet Absorpt ion o f Shock- Heated V i b r a t i o n a l l y E x c i t e d Ni t rogen . J.Chem, Phys., 46, 1521, 1967.

Al len , R.A.: Rad ia t ion Graphs, S p e c t r a l l y I n t e g r a t e d Fluxes Inc lud inq Line C o n t r i b u t i o n s and S e l f -Absorption. AXXO Resedrch Report 2 30, September 1965.

C h u r c h i l l , D.R., Armstrong, B.H. and Muel ler , K.G.: Absorption C o e f f i c i e n t s o f Heated A i r : A Compilation t o 24,COOO. Lockheed Report 4-77-65-1, 1965.

Arnold, J . O . , R e i s , V.H. , and Woadward, H.T.: S t u d i e s o f Shock-Layer Rad ia t ion of Bodies E n t e r i n g P l a n e t a r y Atmospherzs. AIAA 2 , 2029, 1965.

Woodward, H.T. : P r e d i c t i o n s of Shock-Layer Rad ia t ion from Molecular Band Systems i n Proposed P l a n e t a r y Atmospheres. NASA Technica l Note, NASA TN D-3850, Ames Research Cente r , Moffe t t F i e l d , C a l i f o r n i a , February 19 6 7.

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32. Chandrasekhar, S. and E l b e r t , D.D.: On t h e Continuous Absorption Coef- f i c i e n t o f t h e Negative Hydrogen Ion. V. Astrophys. J. , 128, 114, 1958. I

34. Ohmura, H. and Ohmura, T.: Astrophys. J . , 131, 8, 1960; Phys. Rev. 121, 513, 1961.

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37. Norman. G.E. : The Role of the Negat ive Ni t rogen Ion N- i n t h e Product ion of t h e Continuous Spectrum of Ni t rogen and A i r Plasmas. Opt. Spec t ry . , 17, 94 , 1964.

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40. Meyerott , R.E., Sokolof f , J . , and N i c h o l l s , R.A.: Absorpt ion C o e f f i c i e n t s o f Air. Report No. LMSC-288052, Lockheed M i s s i l e s & Space Co., Sunnyvale, C a l i f o r n i a , September 1359. A l s o , Geophysics Res. Di r . , GRO TR-60-277.

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43. G i l m o r e , F.R.: Thermal Rad ia t ive Phenomena, V o l . 1, The Equi l ibr ium Thermo- dynamic P r o p e r t i e s of Ai r . Lockheed Report 3-27-67-1, Vol. I , May 1967.

44. G i l m o r e , F.R.: Energy Leve l s , P a r t i t i o n Funct ions and F r a c t i o n a l Elec- t r o n i c Popula t ions f o r Ni t rogen and Oxygen Atoms and Ions t o ~5 ,0000K, RAND Corp. RM-3748-PR, 1963.

45. Wiese, W.L., Smith, M.W., and Glennon, B.M.: Atomic T r a n s i t i o n P r o b a b i l i t i e s . Na t iona l aureau o f S tandards Report NSRDS-4, 1966.

46. Nicolet, W.E.: User's Manual - Radia t ion T r a n s f e r Code (RAD). Report \ No. 68-39, Aerotherm Corporat ion, Mountain View, C a l i f o r n i a , October 15 , 1968.

47. Al len, R.A.: Air Radia t ion Tables: S p e c t r a l D i s t r i b u t i o n Funct ions f o r Molecular Band Systems. Research Report 236, AVCO-Everett Research Laboratory , E v e r e t t , Massachuset ts , A p r i l 1966.

48. Sherman, M.P. and Kulander, J.L.: Free-Bound Radia t ion from Nit rogen, Oxygen and Air. General E l e c t r i c Co. Report R65SI>15, May 1965.

49. Kel ly , P.S.: T r a n s i t i o n P r o b a b i l i t i e s i n Nitrogen and Oxygen from Hart ree-Fork-Sla ter Wave Funct ions . JQSRT, 4 , 117, 1964.

50. N e r e m , R.M.: S t a g n a t i o n P o i n t Heat T r a n s f e r i n High Enthalpy Flaws. Ohio S t a t e Univ. Report FDL-TDR-64-41, P a r t 11, March 1964.

51. Gruszczynski, J.S. and Warren, W.R.: Study o f Equi l ibr ium A i r T o t a l Radia t ion. A I M Paper No. 66-103, January 1966.

X 5 2 . Wood, A.D., Hoshizaki, H . , Andrews, J.C., and Wilson, K . H . : Measurements

o f the Total Radiant Intens i ty of Air. AIAA Paper No. 67-311, April 1967. P 53. Wood, A . D . : Private comnunication. i. 5 4 . Golobic, R . A . , and Nerem, R.M. : Shock-Tube Measurements o f End-Wall

Radiative Heat Transfer i n A i r . AIAA J . , 6 , 1741, 1968.

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