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''KtA

%r.

00

-- R62SD72

THE ABLATION OF GRAPHITEIN DISSOCIATED AIR

1. THEORY

S.M. SCALA >)1CA

c ***A SPACE SCIENCES, LABORATORtY

GENERAL (0, ELECTRIC

~~~SPAU DIVISION

SPACE SCI LABORATORY

AEROPHYSICS SECTION

THIE ABLATION OF GRAPHITE TN DISSOCIATED AIRPART I: THEORY.'

By

Sicar- 1 c l

Prese'nted at the IAS Natior'al. Summer Meeting,0,3 Ange-Ie',, California, June 19-22, 1962

R62SD72 -Class ISeptember, 196?

MISSILE AND SPACE DIVISION

GfNERAL ELECTRIC

%I

TABLE O FCON TENTPAGE

Abst rac~t

1. hIt roduct ion

IT. SyniboI1 d

111. The~ Re~action Rate Controll-ed Regime. 14

I V. DWhr.,io Cont rolled Re'gimne 24

V. Discussion of Results 32

Vt Cunclu.,ions 42

A ckn oxv vd c m vnts *45

bI v~ r vnc VS 46

List of Tables 52

List of Figures 59

ABSTBhACT

An analysis is presented of the combustion of graphite in a

high speed stream of dissociated air., Many features of the analysis

are qtlite gene-.l ard may be applied to the oxidation ot different

materials in arbitrary chemically reactive environments. However,.

because of the current interest in Lhe hyp .sonic leading edge

problem, numerical results are presented here which are directly

applicable to surface oxidation at the leading edge region of fins

and wings, and the nose cap of axially-symmetric hypersonic

vehicles.

*The reaction rate controlled- rgdme and "o. transition-regime;

are fi re considered at length, and it is shown how the grade of

graphite and its specific chemical properties influence the over-all

uxiaation rate.

It is t, ei shown how the governing equations ti change may

be reduced to a cocipled set of non-linear differential equations of

the fiftoon.r order with variable enefficients and split boundary

conditions. These differential equations are t.a* utilized in treating

the laminar, compressible, multicomponent, chemically reacting

boundary layer in the diffusion controlled regime, and solutions

a- obtained for both the equilibrium and frozen gas flow chemical

constraints. .m..... . r.. . -.P _ 0. W,-. IBM. 4 ,u t 1"

eomtptei, aft& correlated results are obtained for the heat transfer-1

/t

. . ,-. . - .;;.

p~~i,-&,mastransfer rate and the skin friction coefficient for the

complee rane of suborbital hypersonic flight coriinsfineet

1tn adttition ;--, azo- ~to etbfi sit & better, unde-ratanttng-&f

4.a-gomrplex ihy~t.it .. hemical processeg whic.L-cu.D many details

of the StVUCt'ire of the boundary layer, including the variation of

rnaercscopic 4a.- velocity, gas temp*.rature, chemical compositioa

aind chemical source~ turms through the boundary layer, are

presented.V1

2

I. INTRODUCTION

In considering the design of hypersonic lifting vehicles, special

attention must be given to the icading edge surfaces which are exposed

to sustained aerodynami. heating, and hence must function for long time

periods at leading edge temperatures in the vicinity of eCA0oR

(Refs. ! -6).

A class of superior carbonaceous materials known as graphite

immediately suggests itself becauee this form of carbon is a refractory

material having high thermal shock resistance, good high temperature

strength, excellent machinabil;y. high thermal conductivity, a high subli-

mation temperature and a relatively low oxidation rate (see Table 1).

The type of graphite which is in current use in industry is usually

manufactured from carbon base materials, rather than mined as the natural

substance, and hence is commonly called "artificial graphite'" The latter

is suFerior to either natural graphite or carbon, both of which have relatively

low mechanical strength. It is noted that manufactured Praphite is not one

specific material, but a family of materials which are all essentially pure

carbon. They differ from each in other in regard to the orientation of the

crystallites, !I grain sgze, the size and number of pore spaces. the degree

of graphitization, and the level cf impurities, which therefore lead to certain

differences in the physical and chemical properties. Thus, the wide variation

found in the properties of graphite can be attributed to the choice of starting

materials, Pnd to the degree of control during the manufacturing process. In

reference 7 the reader will find a concise review of the propertiee and

3

- -7_1

applications of different grades nf manufactured graphite.

Che rical reactions between carbon, coal, graphite and oxygcn

have been studied intensively for over one hundred years, and attention

has been given to the reaction rate controlled, transition, and diffusion

controlled oxidatiun regimes. Consequently, a voluminous literature exists,

and excellent reviews on the mechanism of carbon oxidation have been

written byGolcvina (Ref. m). Frank- Kamenetskii (Ref. 9), Arthur (Ref. 10),

Townsend (Ref. I). Strickland-Constable (Ref. UZ),von Loon and Smeets

(Ref. I i). Gerstein and Coffin (Ref. 14 , Khitrin (Ref. 15), and Blakeley

(Ref. Io). However, very little of this previous work appli,!s to the en-

vironmental cond'tions encountered during hypersonic flig.:r. .Specifi '

for the regime of greatest interest to the glide vehicle designer, little

information is available other than the theoretical work of Scala (Refs. 17,

16), Lues (Ref. i9, Denni.4on and Dooley (Ref. 20), and Moore and

Zlotnick (Ref. 21).

None of the previous work, either theoretical or experimental, considers

the problem of determining systematic 1 1 ¢ the relationship between mass loss,

heat transfer, and viscous skin friction, as a function of the significant

environmental parameters, such as the flight speed, the ambiett. pressure,

the surface temperature and the mcdel geometry.

In the study presented here, the hypersonic ablation of graphite is

considered, and the heat transfer and mass transfer processes, and viscous

4

. . .. .-. :

drag efficts, which are experienced by hypersonic vehicles flying in the

earth's atmosphere, are analyzed in detail. Although the-analysis developed

here is quite general, because of great current interest, numerical results

Lhave .een obtained which are applicable to tin ileading edges of fins and

w l ng.s and at the forward stagnation point of axially-symmetric vehicles.

Upon i "-4ducing available expe-imental data on heterogeneous

reaction kinetics (Reft'), 4.2-.6), the mass transfer and heat transfer rates

will be determined at low surface temperatures, fn" the reaction rate con-

trolled regime. At higher Lurface temperatures, the transition regime

behavior will be determined utilizing the results obtained in the rate con-

trolled and ditfusion controlled regimes, by applying the concept of resistances

to mass transfer ir, series. Since the heat transfer to the surface depends

on the ratio of carbon monoxide to carbon dioxide at the surface, the recent

data of Arthur (Ref. 27) and Bonnetain (Ref. 28) are also introduced.

At still higher surface temperatures, in the diffusion controlled

regime, Pva'.t solutions wil! be obtained for the laminar flow of a compressible

multicomponent chemically reacting gas over a reaLting solid. It will be

assumed that dissociated air produced bythe upstream shock waves can be

treated as. a four component gas, cc.nsisting of oxygen and nitrogen atoms,

and oxygen and nitrogen m'olecules. Since the primary combustion products

include carboi- monoxide and carbon dioxide, the total number of gaseous

species considered is six. Therefore, the analysis requires the solution of

a coupled set of no n-linear partial differential equations, (including the

conservation of mass, momentum, energy and chemical species) v. ich is

5j

of the t r, having . plI.t hot.lry conditions and variable

trancport and thermodynamic coefficients.

In the work presented herp0 as in earlier studies (Refs. Z9.30),

th. transport properties uf the individual atcmic and molecular specie.

will be calculated utilizing the rigid sphere and Lennard-Jone, n..iel,

respectively. The thermodynamic properties of the pure species will be

determined usilqi the formulae of statist.-al mechanics. The transport

and thermodynamic properties of the gaseous mixture will be evaluated

durinjg the solution of the problem, in trmq of the gas composition, the

pressire and the temperature.

Since the homogeneous rates of reaL Lion of the various species

preser. 4-1 the high temperature gas stream are not yet known precisely,

calculations will be performed for the two limiting cases of "frozen"flow

(infinitesimally slow Zas phase reaction rates), and "local equilibrium"

flow (infinit-ly fast gas phase reaction rates), which bracket the actual

situation. It will be shown that, as In the case of hlypersonic stagnation

point heat transfer (Refs. 30 to 33i, when the gas is in local equilibrium

at the surface, for arbitrary hypersonic free stream conditions, both the

overall rate of miss transfer and the net heat transfer rate are virtually

independent of the rates of gas phase reaction in the diffusion controlled

regime.' This precludes the necessity of having an exact knowledge of

gan phase kinetics.

In order to establish a better understanding of the physicochemical

processes, many details of the structure of the multicomponent boundary

6

' T•

layer will be'pre-sented, including the variation of velocity, temperature,

And gas composition as ;t function of distance from the surface. The zones

in witich chemical reactions occur in the gas phase and the magnitude of

the various chemical source distributions w. will be shown in detail.1

Calculations will also be made to determine the fraction nf the heat trans-

ported to the rearting, surface by the various fluid dynamic and molecular

interacticn processes. Finally, the magnitude of the viscous shear stress

will be evaluated and 'correlated in the form of a skin friction coefficient.

Utilizing the graphical results, and the correlation formulas which

will be presented here, one may predict the heat conducted into the solid,

the mass loss from tne leading edge and the skin friction coefficient, for

a wide range of hypersonic flight conditions. i. e. Mach numbers in a

range from 10 to 24, an altitude range from 10, 000 ft. to 240, 000 ft. , and

surface tenperaturcc from the threshold range through 6000 0 R, for vehicle?

of arLitrary nose radius or wing leading edge radiusi and angle of yaw.

7

if. S'i NwIol's

a, b coefficients in eqiiilibrium constant

Alt. attitude

2 sw kin friction coefficientli /Ow e

C. mass fraction of species.

C P specific heat'at constant pressure of species i

C P CCp1 . frozen specific heat of the mixturer

CS specific heat-of the solid

binary diffusion coefficient

self diffusion coefficient

Dij multicomponent diffusior coefficient

E activation energy

f similarity -stream function

f u/ue . dimensionless velocity

hi s~atic enthalpy of species i, including chemical

8

hf standard heat of formation of species i evaluated at Tref.

h Cih , static enthalpy of iixture

4 H heat of vaporizationyap.

/'-iVi, diffusion flux of species i

mechanical equivalent of heat

k specific reaction rate

k effective collision frequency0

K frozen thermal conductivity of the mixture

Kpi equilibrium constant of species i

P -DijL1 * frozen Lewis number

Ij K

LiT CpDiT

Li T frosen thermal Lewis numberK

2

;n w Vwvw = Pi d) w , interphase masi transfer

Mi molecular welght of species i

49

M XiM i. mean molecular weight of the mixture

N number of chemical species

n -hder of the reaction

n i number of moles of species i per unit volume

Pi" parti. pressure of species i

P Pi static pressure

Pr _. frozen Prandtl numberK

SQ heat of reaction

O energy transfer function KVT - Vi h i

Q heat transfer rate

effective heat ot ablation

universal gas constant

RB nose radius of body

Re x uOwUex/w. Reynolds uumber based on x

(s) solid state, condensed phase

t time

10

T temperature

u x component of velocity

v y compnnent of velocity

v macroscopic stream velocity

vi absolute velocity of species i

V diffusioi velocity of species i

Vag flight speed

W. chemical source term, mass rate of production of

species i by chemical reaction per unit volume per

unit time

X, mole fraction of species i

X. y, r bodyoriented coordinate system

linear rate of surface recession

.A yaw angle

Eemissivity, depth of potential well

similarity variables

Q T/T e . dimensionless temperature

11/

e characteristic vibrational temperature

1 viscosity coefficient of species i.

/44.viscositY of mixture

density

Stefan - Boltzmann constant

'1' viscaiin shear sitres

* collision diameter

3treamn function

Subsc ripts

AIR treated as if the gas is dissociated air

c condensed phase

cal. calorimeter

e outer edge of boundary layer

eq. equilibrium

5 gaseous species

I~ ith species

rad. radiation

12

s stagnation point

yap. vaporization '

w wall, interface

00 upstream of shock, edge of boundary 11.yer

denotes differentiation with respect to V

* 13

III. THE REACTION RATE CONTROLLED REGIME

a) Mechanism of Surface Degradation

In an investigation of the behavior of graphite -in dissociated air, one

requires data on the nature and extent of the chemical reactions betwee.n carbon

and the primary products of dissociated air, including atomic and molecular

oxygen and nit:ogen.

The recent work of Stieber (Ref. 14) indicates that nitrogen molecules

can be considered chemically inert on carbon surfaces at temperatures as high

as 5400 0 R. An early study by Strutt (Ref. 35) indicates that active nitrogen

(primarily atomic nitrogen in the ground state) does not react with carbon at ,'

room temperature. In addition. Zinman (Ref. 360 who studied the interaction

between atomic nitrogen a.ad carbon at 8000 C. did not detect measurable

amounts of either cyanogen or paracyanogen. Consequently, one may conclude

that at surface temperatures up to approximately Z000 0 R, no permanent

carbon-nitrogen compounds are formed at a carbon surface. While experimental

data is lacking at higher surface temperatures, it will be assumed that molecular

nitrogen is chemically inert, and that atomic nitrogen undergoes heterogeneous

recombination at a graphitic surface. Thus, if atomic nitrogen diffuses to the

surface without undergoing gaf phase recombination, then the graphite acts to

catalyse the recombination of the atoms at the surface. In this stady, therefore,

surface degradation will be assumed to be primarily a consequence of a surface

oxidation process. That is, chemical reactions between nitrogen and carbon, and

mechanical effects such as spallizag,will not be included in the theoretical calculation

nf the ablation rate, during hypersonic flight.

14

" " , .i f -.-- .... .. / . .... "- "-+ - "" ' .... .. 1 ' -t - . " * '/ :

The carbon-oxygen reaction has been studisO extensively for over

one hundred years and hence, fortunately, although the mechanism is still

not completely understood, sufficient experimental data exists upon which

reasonable theoretical calculations may be based. The fact that graphite burns

to form a mixture of carbon monoxide and carbon dioxide haq .-en discussed

in the literature, although there is some disagreement as to the sequence of

the steps in the chemical reactions. At this time, as wi:" 'e discussed, there

is also some disagreement as to whether the overall reaction is first order

with respect to the concentration of oxygen or of fractional order. It Is

interesting to note, however, that th widest differences in the oxidation

behavior of the various grades nf graphite are found at the lowest surface

temperatures where the process is rate controlled, and these individual

differences tend to disappear as the surface temperature rises.

Since the reaction between carbon and oxygen produces both CO and

CO2 , these Products can be the result of either parallel or consecutive reactions.

A number of different mnechanisms are possible and these include the following:

1. The formation of both CO and CO2 in a surface reaction between

C(s) and 02 or 0.

Z. The formation of C02 from C(s) and 02 or .) &t the surface, followed

by the dissociation of CO? to CO, O and 0 in the gas phase, or reduction of

CO2 to CO at the surface.

3. The formation of CO at the surface from C(s) and OZ or 0, or

C(s) and C0 2 , the CO beirg oxidized to CO2 in the gas phase.

/ ., 15

*i s

A,

*~\

Measurements of the gas composition in the vicinity of an oxidizing

carbor, surface have been made by a number of different investigators, including

Arthur (Ref. 271, Bonnetain (Ref. 28) and Snow et al (Ref. 57). They have

verified that both species are detected adjacent to the surface, even r.t low

surface temperatures, and that the ratio of the mass fraction of CO to CO 2

at the surface r-./: rapidly with increases in surface temperature. Their

data can be represented by an Arrhenius equation.

CCO/CCOZ w ke' E/A(1)

and are shown in Fig. i5.

It is noted that if the gas at the reacting surface had sufficient

time to achieve thermochenical equilibrium during the low temperature

oxidation process, then the componition of the gas could be determined

from the equilibrium constant for the reaction,

C(s) + CO? ZCO (2)

Since the equilibrium constant for this reaction may be written in the

form (see Table I),

(F CO) a-bK - - e T (3)

~CCZ

it is not surprising thaL A;ien the boundary la,.;r *olutions, which are

based on the assumption of local thermochemical equilibrium at the siurface.

are compared with the experimental data of Arthur, Bonnetain, and Snow

et al, see Fig. 15. the same general trend ;s exhibited. One may conclude

16

-/ ' -4..'

/

that either the gasecus CO- CO 2 system is actually never removed very far

from an equilibrium state during oxidation, or that the experimental technique

utilized by the investigators produces a shift toward the equilibrium composition.

Although these data cannot be utilized to rationalize the presence of

ary of the three suggested mechanisms, it is fortunate that the mass transfer

and heat traadfer at Oe surface can be predicted reasonably well without

specifying tne specific oxidation mechanism.

1j) Reaction Raxe Data

It is commonly accepted that the manner in which the oxidation of

graphite proceeds depends on the type of graphite, the environmental conditions

(e.g. the pressure, temperature, velocity and composition of the stream), the

surface temperature, and at high surface temperatures, on the geometry of

the model. The temperature at which a measurabe mass loss first occurs,

is called the threshold temperature (1000-1800°R) and is not a constant but

depends or. the partial pressure of the reacting gas at the surface.

At *,ww durface temperatures, the mass loss increases rapidly with

surface temperature, and the ablation rate is limited by the speed of the

chemical processes, including adsorption, reaction and desorption.

At somewnat higher temperatures (1400-32000 R), the speed of the

chemical processes is comparable to the rate at which fresh rcactant is brought

to the surface and the products of reaction are removed by convection and -,

diffusion. Therefore, the overall process is in a transition regime, where

the speed of the overall oxidationprocess is limited by the presence of two

resii-,L.sgc6 la series, onc chemical, and the second gaa dynamic.

17

At temperatures above 32000 R, the chemical oxidation processes

are overshadow'ed by the gas dynamic proces ses. In this difusion controlled

regime, the mass loss is relatively insensitive to the surface temperature.

This result has been found experimentally for subsonic flow (Ref. 37) and

will be demonstrated here for hypersonic flow.

Eventually, when the surface temperature is sufficiently high (5500-

80000 R). the sublimation rate of carbon atoms and-molecules can exceed the

surface oxidation rate, and these species are then present in the gans phase.

Different investigators have studied one or more of these oxidation

regimes. For example, the influence of environment uponi the combution

rate of carbon nas been studied by Hottel et &I (Refs. 22, 39, 40), Chukhanov

and Grozdovskii (Ref. 38), Guibransen (Ref. 424. and xr.ro

recently by Kuchta, Kant and Damron (Ref. 43).

The effect of the nature of the carbon on the oxidation rate has also

been~ subject to investigation. For example. Riley (Ref. 44). and Smith and

Polley (Ref. 45) have studied the effect of varying degrees of crystallinity of

-the carbon. Winslow et al (Refs. 46, 47) and Akaaaatsu et &l (Ref. 48),

investigated the relationship between the starting materials and the degree

of graphitization. Wicke and Hedden (Refs. 49. 5ib. bstvo posLQ...ted that for

porous types of carbon, the diffusion of oxygen into the pores can be the rate

controlling step in a transition regime between. the rate controlled a"d diffusion

controlled oxidation regimes; hence, the apparent activation energy is found

to be half the true activation energy. However, Blakeley (Ref. 16), who

investigated natural and artificial graphite under a variety of conditions. feeln

718

I.!

that t pore diff'sion mechanism is not required to explain his experimental

It has also beer, shown. e. g. Arthur (Refs. 51, 52), that impurities

;n the solid phase, such as sodium c..rbonate and zinc chloride, will augment

the rate of oxidation in the reaction rate controlled regime. These experiments..

/ udicate that at relatively Jow tem-eratures, impurities tend to .- t ,ts favorable

sites, ur catalytic agents, which promote the rate of reaction. Although little

positive experimental data exists, one may anticipate that trace amounts of

impurities 'n the gas phase will also influenc,, the oxidat.on rate in the rate

controlled regime.

With regard to the 6ependenc, of the rate of oxidation upon the pressure.

of oxvfen in the stream, there is some experimental evidence that the reaction

rate is first order with respect to oxygen pressure (Refs. '22, 49, 53). However,

Frank-Kamenetskii (Ref. 9) has re-examined the experimental data of Parker

and Hottel (Ref. 2-1 and has shown that the' reaction rate can be interpreted to

be of fractioi:al order. Further, Semechkova and Frank-Kamenetskii (Ref. 54)

have ehown that the rate of reaction between carbon and carbon dioxide in the

purely kinetic regime is lower than first order, while Klibanova and Frank-Kamenetska

(Ref. 55) have established that the reaction between carbon and oxygen is not

first order, but fractional, lying between 1/3 and 1/2.

Vulis (Ref. 26) tabulated a large amount of data on the kinetics of the

reaction between carbon and oxygen or carbon dioxide. Upon applying an

Arrhenius formula to the data, he found that the activation energy'E varied

between limits of 8 and 37 K cal/mole for the carbon-oxygen reaction, and was

approximately 2. 2 timea larger for the carbon-dioxide reaction. Vulis also

19

found that the logarithm cf the specific reaction rate wac a linear function

of ti-e activation energy, and hence.was ledtothe conclusion that the only

experimentally determinable characteristic of a given variety of carbon is

the activation energy of either j, these reactions. However, Vulis' treatment

of the data rests on the assumption that the true chemical kinetics at the

surface follnw * ".st order reaction, and it has been pointed ott by Frank-

Kamenetskii (Ref. ?) that this assumption lacks theoretical or experimental

substantiation.

Examination of a large mass d experimental data (e. g. . Refs. 8,

9, 10, 11, 12, 13i 14, 15, lb, 4, 22, 3, 25, 26. 27, 34, 39, 40, 41, 42, etc...

indicates tnat in the reaction rate controlled regime, the oxidation process

follows a rate law which may be written in the form:

m k(Po),n (4)WReact. 2

where P0. is the partial pressure of the element oxygen near the surface.

n is the order of the reaction, and k is the specific reaction rate.

As noted, there is some uncertainty about the precise value of n, as

values in a rang,-

0 'S n - 1.0 (5)

have been reported in the literature for different oxidation regimed. Also,

as discussed, the specific reaction rate k is an exponentially increasing

function of temperature whose precise magnitude is directly related to the

type of graphite and its treatment during manufacture. Custumarily, ".e

20

reaction rate constant is written in the standard Arrhenius form,

k = koe-E(6)

where the pre-exponential factor can vary over several orders of magnitude,

and the activation energy has been reported to fall Within the limits

:f- E -5 60 K cal. /mole (7)

For example, Gulbransen and Andrew (Ref. 42), and Blyholder and Eyring

(Ref. .A) report their dat. in the form of Eq. (6). Parker and Hottel (Ref. 22)

utilized the form:

k = ko T" /2 e -E/ T (8)

Vulis, (Ref. 26), obtained the empirical formula

k kT-l e E(T-T*)/ATT* )

and Frank-Kamenetskii (Ref. 9) has suggested,

k - k T •/2eE(TT,)/,T0 (10)

where the associated rate data for eqs. (6), (8), (9). and (10) appear in

Table Ill. These data also appear in Fig. 16, where it is seen that the data

of Parker and Hottel, and Vulis are based on a first order ".action, but the

data of Gulbransen and Andrew, Frank-Kamenetskii, and Blyholder and

Eyring, have been tp'cet as following a on--half ord.r reaction. This figure

shows clearly that one can expect different grades of graphite to exhibit large

differences in oxidatior behavkur itk the rate controlled regime, i. e.,

21

A

-\

1400 0 R Tw 1! 32000"R. I owever, fortunately at higher surface

temperatures, the proce~ss becomes diffusion controlled and these large

differences will no longer appear.

Although our subsequient results will usually be presented in a

general form. wherever representative calculations are required to clarify

the differences ;n . ;havior between grades of graphite having high dr~d low

specific reactivity, numerical calculations will be performed for "fast"

iind "slow" heterogeiieu reactions, reapectively. In particular, in the

illustrative examples, we will utilize eqs. (4) and (6), with a value of

n 1/2. The vaLes of the rate data will be arbitrarily taken as,

E=44. U k cal. /mole"fasttko 6. 729 . 108 lb. /ft. 3 / 2 sec. atm l/2. I

.Slow"S E =42. 3 k cal. /mole (12)

[,0 4. 47 3 x 104 lb. /ft. 3Zsec. aml

In the rate controlled regime, the rpte -if oxygen consumption

adj-acent to the suarface is so low that tne mass fraction of the element oxygen

will be essentially the stiit i. it thne ,.idisturbed streasts. Hnwev.er, the atomic

species will recombine in tne low gas phase temperature regime adjacent to

the surface and hence the mole fraction of molecular oxygen -! the surface

will be vcr,. nearly, equal to its value in undissociated air. Further, the

static pressure gradic"' in the reaction zone i?- :i direction normal to the

surface, is negli .ible. 1 hese conisiderztions tnerefore enable ais to write,

N P (13)

lOr it reaCtion ratt, controlled reginie only, and hence equation (4) becomes

(r*hw)R.a(-. ~ 1/2 _E/RT

where X 0. 21 for undisuociated air,W

It is note,!'.: t if oxygen does reach the surface in the atomic ;Ltate.

reactions will o'ccur, but at a somewhat different rate (Ref. 57), which will

promote a shift In the transition regime.

23

WV. DITFFUSION cw4NTROLLED REGIME "

.)Gove rning~ KItations.

1'h. noti- Ii va r partiadl di fferni iiqinitions of c Lansge for a

:tii( wil :, jt .1, :-.caLll a(.tn ~,;re d-rjvced, for teXAIIIiiAC. inl

'-c. ', trd incic~d, thy: conve rvatjon of mass~, chemrical species,.

tnorflc i .1 'n !I l ,' . bv low:

3P + (19)

at

%%hu-c i. tht- in.,, ro-t tpic - t ear velbc ity,

+_V_ (16)

hc : V i - thte h.tn ici ,oit rt u te .-:t

* (~~~r +('171Lit

%h--re Tr the- prmv.nure tcn~or. add

Je * I +(18)

vwh rt~e. P, 1i c iad the cheit Al vitergy of formation.

Upon ititrordicint, fl.- b)oundary laye~r approximation for

the horly-ori~nterl coordinaite systemn givejn in Figure 1, the conservation

of mass bet omnes:

* ~ ~ ~ ~ ~ ~(CrtA) 2 (g ) -Q(9

LA >2. 1 w .(20)

poen m!~l~~rL.rometitmi bLecomu s:

P ~ (22)

heciri~y equa.tion becomte,:

')T,

- (23)

b) Trk-.sport and Thermodynamic Properties

rh tran~port coefficient., required for the definition nf

the physical problem include the coefficient of viscosity for each 'of

the n pure speries, (11r./ symmetric binary diffusion coefficieam,

and n self diffusion voefficienth. These properties may be calculated

from the following equations (ref. 58 )

2'5

I(. 'rJ-(24)5 cU. Lj "IV rAU "l

._ _ 2...-.k3 ' j (2 i3,

00 , 1~ 1 GS a, I) 1P T(% Z i ' j!

h I'rt" ti .: r".. q - r it1ti tht.o yv I''rm lli. for the vikct'.ity and

th" !ifIf. On ,, :it rvt. rel-.p ,tive 'ly. Iln order to ev.aluLIat th±! t!

prop,.rti, on,. reqaiiires it knowicd-'t- oi th. , olli.sion diamrneter To,

and the" colli.sion inteiaral Ia L funt tion of the reduced

temprature, where T*= KTA/

In the above, the s.ymbol/i " is the reduced mass given by:

J~'j _ __ __ __ (Z6)

,and the collision diameter by: -

± (c4- ~~\ . (27)

Note further that the constants to be utilized in the above equations are

given in Table IV and the resulting transport pauperties of thc pure

species which are based on the Rigid Sphere and the Lenfnard Jones 6:12

.potenti-d respectively are shown in Figures 1,3 and 4.

26

-. L ..... . -...

PIP -

c) Similarity Transformation <

'1 f.....Sy (28)

f- ~ ~w&r 7 (29).

and assuming that local similarity holds, then equdtons (19) through

(Z3) may be reduced to a set of ordinary non-linear equations. The

diffusion equation becomes:

-~--L" (~j)1 ,~0 (30)

The c6..ervation of mome.tum becomes:

(%1)i f~ 1 ~(a 1 Le.$ (31)

while thi energy equation becomes:

T. AUg ~1 "

(32)27

f

d) Chemical Constraints and Boundary Conditions

It will be asumed here that the total nmber of dominant species

present ii, the gas phase is six, including atomic and molecular oxygen,

atomic and molecular nitrogen and the combustion products, carbon

monoxide and carbun dioxide. Thus, there are six unknown chemical

source ternis i and ,ix unknown concentrations X; at each point

within the boundary layer.

At the -,urfacv, there are six unkniown concentratiois which

must satisfy the chumical constraints imposed by the surface oxidation

proces3s.

Let us therefore consider the chemical constraints. The

conserv.'ation of "Pmical species in the gas phase requires that

v(33)

Since it has been -assumed that the nitrogen species do not react with

oxygen or carbon to form permanent compounds (i.e. the formation

of NO or CN is not considered explicitly), one may also write for the

case of thermochemical equilibrium

%N - (34)

28

II

%4-W 0 C (35)

w C 0 (36)

For the case of frozen reactions uste has

0 (37)

The surface ho-.ndary conditions on velocity and temperature may

then be written:

U" W (38)Ae

- -~ w ri~ (39)

__v (40)

The boundary conditions on the composition are obtained from

the simultaneous solution of the equilibrium constants:

29

| II

- £"

X0

X (OL43)Y" cow

It is also notud that in the theoretical models considered here, the

nitrogen atoms are permitted to recombine on the surface, one 1uiay

also write:

(rAN,, "" " , = (:4,')

t and hence ,since

one immediately obtains for the oxidation rate

yV'4 d . - ~ h(46)T (C + C0 , . C c+ CO. ,,,

The boundary rnnditions at the outer edge of the boundary layer

"or velocity and temperature are given by:

I'Mt W- 0 a 1 -0'

30

w A

g-ndiin addition, one Ila" n-l relatiurns Of the form:

= (48

th otlA number oz. boundary conditions therefore equals the

order of the mathematical system.

7a

31

NV

discussed in detail. As al rfady not# ci, a convenient coordinate sytem

for, the study of the hypersonic J-imitiar boundary layer is the bt A-

oriented coordindle ,ystem sho%.%n in Figure 1. In order to treat the

-problem in the .bsence of experimental data, the high temperature trans-

port coeffieients and thermodynamic properties of the reacting species

%ere determinvd theoretically utilizing kinetic theory, statistical thermo-

dynamics and the ioa, constants gien 'n Table The theoretical rul's

are shown in Figures to :. Here it is remarked tnat the properties of

the gas mixtje have been treated as variable and were computed through

the hnundary layer as a function of both the local gas CMoSiotr and the,

propertis of the pure specirts at the local gas temperature. Thus, since

the gas properties are computed as part of the solution, one o not require \

simpliiying assumptions such as constant Prandtl and Lewis niimbers, or

a constant product of density and viscosity in order to solve the problem.

That is, once an app -opriate intermolecular force law between .1 pair nf

molecules ',as been selected, uric may immediately cakl-uate uniquely the

properties .,t the pure species, and then one may proceed to utilize the-se

to determine the mixture properties as dictated by the chemitry of the

particular physical problem.

In Figure 7 are shown typical values of Che variation of the normalized

product of density and viscosity through the boundary layer, as a function of

* 32

*, ~ *- * * - dkproerIe. of th.-r pc~ ttelclgstmeaue hs i

the stretched norryfal coordinate. Qualitatively, the trends are the same,

namely that the quantity decreases with increasing gas tempera-

ture. This may readily be explained as follows. I. a dissociating gas

boundary layer, which is locally a constant pressure layer. the diensity

ratio is given by:

and hence, clearly, the decrease in molecular weight with increasing temp.

e rature produces a smaller density ratio /p (greater density change)

with increasing gas temperature thlan will occur in a non-dissociating gas.

This .s particularly true for the case of the mass transfer of foreign

£peics at the surfa ce having a higher molecular weight than the primary

gas species. Since the increase ir viscosity witn gas temperature is less

than line.er. the increase in the viscosity ratio 4 with increasing

temp#-rature does not compcnsate for the decrease in the density ratio

-Thus, the larger the ratio of the gas temperature at the edge of

the boundary layer to that at the %all, the smaller will be the value of the

normalized product at the outer edge of the bou.-dary layer, and the

poorer the assumption that (/J. is constant. This is shown in curves 1,

2, and 5 of Figure 7, which were calculated for the chemical constraint of

loc.,l equilibrium flow. By way of further comparison, curve 3 of Figure 7

was Lomputed for the case of equilibrium dissociated'air (no graphite com.

buscion), and curve 4 was calculated for tne chemical constraint of frozen

13

i " / ,

r eactions, all for the same flight conditions. It its seen that under certain

con'Iitions the dens ity-vi scos ity variation may have nearly the samne

beha±vior for both the frozen flow and local equilibrium constr;*.;-ts,

(compare curves 4 aind 5), although the other gas propertia-, such as, for

.!X:mplte, the thermal conductivity, do not.

When one defines the Pranriti number of the ga* in terms of the

arn. spcfchaoth mitutre.-~ the viscosity of the mixture/A

nthe frozen thermal conductivity ~,one obtains the results shown in

Fig~ure 1. H, rv, the value of the Prandtl ntimber of the giss at the surface

ye r. .s the stirface temperature over a -id range of pressures. It appears

from this fillUre that the presence of carbon dioxide at the surface at the

lowe.-r surface ttempkratures s,.ts to increase the Prandtl number. At a

gi'nvalue of the temperature, the effect of a press.Ire increase is to

decrease tht- dissociation of carbon dioxide and hence the shifting equil-

ibrium composition with increasing pressure results in an increase in th.%

Prandtl number. The'tutal effect, however, is less than four percent

3.ice th r x:mumn mole f raction oi carbon dioxide ijk less than 0. Z. Of

gred'. r Lnterebt tce the Lewis numbers which have be-ra defineci here in

terms of the multiromponent difiusion coefficiputs D... which, unlike the

binary dif~usion coefficient., are not wkt11"Atrc. Consequently.

Since' all othfer gas properties which appear In the niulticomponent Lewis

number :*-re not riependonut an 0-'r particular pair of particles involved in a

given binary encotinter, while the inulticomponeit diffusion coefficients

34

arc both temperature and composition dependent. It is clear that there

27are eenerally n -n values of the Lewis number; in this case there are 30,

of which 12 representative values are -.hown in Figure 9.

Waen the boundary layer equations arc integrated subject to the appropriate

I mdary conditions, one obtains the veiocity distribution, the temperature

distribution and the distribution ot species through the layer.

The are shown in Figure 10 and 11. Corresponding chemical source

terms are sho-vn in Figure 12.

Si, et preferentid, diffusion can occur, it is found that the ratio of the element

oxygent to the element nitrogen at the surface is not necessarily equal to its

value at the edge of the boundary layer, and is in fact an eigenvalue of the

mathematical system. This is shown in Figure 13 and 14.

In the rate controlled regime, it is found that the ratio of the mass fraction

of carbuin monoxide to the mass fraction of carbon dioxide is a sensitive function

of su:face temperature, which increases with increasing surface temperature.

This can be shown to be the case from both theoretical and experimental con-

siderations and is demonstrated in Figure 15.

Examination of the literature indicates that the rect.on rat, controlled

mass transfer is also a very sensitive function of surface temperature and in

fact as many different acti-ation energies and specitic reactivities have been

reported in the literature as there are combination& of investigators and grades

of graphite.

35

/+

The work of five different investigators is shown in Figure 16. The

expunential dtpenfi.cv of the oxidation rate ^n surface temperature is clearly

seen which leads to the usual Arrhenius representation.

Typical values of the reaction rate results therefore appear in Figure 17

for A representative flight condition.

In Figure 18 we have shown the correlated resul:s for the mass transfer

oi oxidation products for both the rate controlled-and diffusion controlled

r,.Limes. Wher the surface temperature exceeds 3000R the oxidation rate

lev.els off and becomes insensitive to the magnitude of surface temperature,

simply because the mass loss is controlled by the diffusion of uxygen-bearing

s peci,.s to the surface rather than the specific reactivity of graphite.

Upon making khe of the concept of th- sum of resistances in series,

one may write:

WDiff" React.

where the reaction term predominates at low surface temperatures and diffu-

siun predominates at high surface temperatures. This equation wan utilized

in calculating the transition regime in Figure 19.

In order to obtain an over-all feeling for the combined effect of pressure and

surface temperature on the oxidation rate, Figure ZO has been prepared. Herc

it is seen that th-!- are 4 different regimes for the oxidation of graphite. In

36

- p.+'f

this paper-we have considered the first three reuimes from the reaction rate

c'ontrolledlr~n,~tri~ the transition to diffusion controlled oxidation.

It is also of interest to. consider the heat transfer into the solid graphite.

- It is instructive tu compare the heat transfer to the solid graphite to the heat

transfer to an inert surface.

This is sitown in Figure 21. where it is seen that one may not insist that

the pr)cess.- remains diffusion controlled down to the lowest surface temperature,

which would result in an overestimate of the heat transfer. It is clear that

as surface temperature decreases, the heat transfer must reduce to the aero-

dynarnic heating, since the oxidation rate decreases exponentially with decreasing

.urtace t.-mperature.

Sinc-, tht- total 't-orvctivo' untery trani.ft-r including conduction, convec-

tiun diiffusioni and chtemical rvac! iinn is c ivcn by9:

rK a~ T~w L 3y -V w 51

it is instructive to examine Figure 22 which compares the results'for both

the equilibrium and frozen gas chemical constraints. V. is seen that although

the separate contributions differ from each other, the total heat transfer is in-

dependent of chemrical constioints provided only that the surface is purely catalytic.

In Figure 23, we have shown the correlated heat transfer for the heat con-

ducted into solid graphite for an axially bsitemnatlc object. however, numerical

results have also been obtained for a two-dimensional body and the heat transfer

37

SB3 -

P , ft-S - . s. a. *.

rate wa. correlated by means, of the folowing equation:

I.wR fA ) BTU

.. e / f(A ) I I 33.3 0O.0333(H e hwi)" ft. 3 1 2 sec:. atm. I/2 (2

Swhere .£(A ) represeat~. ae effects of yaw, £ is the kronecker 0 such "hat

this equation ;appiies to both axially symmetric ( S = 1), and two-dimensional

z 0) geometries. P. is the local static pressure at tha, outer cdgc of the

boi nvary layer, He is the stagnation enthalpy and hwAir is the static enthalpy

of the gas evalated at the surface temperature and treated as if it were. air,

expressed in BTU/Ib. Figures 23 and 24 represent different ways of depicting

the correlated heat transfer.

If one utilizes the horizontal portion of the graph shown in Figure 25

for r~a , one obtains the following result .for the diffusion controlled mass loss:

(r~w Diff. f(JA.)ILDif -- 6.2 x 10'3 .12 (53)

P e ) (!Z ' ft. 32sec. tr. . 1/

Note that equation ( 14), which represents the reaction rAte controlled

mass loss is independent of the geometric factors whereas the diffusion con-

trolled mass loss, equation (53), j.not.

It is noted that in bath equations (52) and 1531. the effects oi yaw, which

act to decreise the heat and mass transfer, appear in the function f(A). Although

more complicated correlations have been suggested, (Ref. 59), the auth'or

believes that the following approximation is satisfactory.

38

It is of some interest to calculate the so-called "effective heat of

ablation". This may be done readily for the diffusion controlled regime by

taking the ratio of Q to 41 and one obtains a result which is independentw

of the geometry of the body, that is:

Q 5370 + 5. 37 (14 - h .. ), BTU/lb. (54)

It should be noted that since the driving force for the mass transfer

proceb is not really the heat transfer but rather the oxidation process, one

should not be surprised that the effective heat of ablation is of the order of

70,000 BTU/lb. at orbital velucity. It must be understood that graphite is /actually an exceilent sink material which also undergoes surface oxidation.

In this paper we have also determined the surface shear stress Tw

defined by

Tw /w ("-) (51)

and a typical result is shown in Figure 26.

Upon defining a local skin friction coefficient cf

twK- -c (56)fw 7 e 2 1

39

-. -'". -._ - - .-

" :- "' "" " N -'

. . ..- " " . -' -""

and a Reynolds nimber based on x:

Re , (57)

onp also obtains the correlated results shown in Figure 27. For purpos-s

oi comparison, results are also given for the skin friction cuefficient in the

absence of s urface oxidation.'

Finally, it may be of some interest to be aole to rapidly estimate

the surface temperature as a function of environmental conditions such as

H. and Pe. In general, one has:

Qw 0 6T - ET 4) (K, S4T (58)

where Q,,, ii the total convective heat flux to the ;urface, C- T 4/ represents

radiaLoa iu tt: surface from the hot gas cap, Tw4 is the reradiation fromw w

the surface and (K 22T ) is the heat conducted into the interior of they solid

solid graphite.

If one assumes that radiation equilibrium obtains, the --- ht hand

side of equation (58) may be set equal to zero. In this case, one may write

that the radiation equilibrium temperature is:

• q. Qw + 6 Tg " /4 (59) .TWRad. Eq. d" . w

40

• . ,. "

In general, for suborbital flight, Cl-C T e4 Q and if this term is neglected

one obtainq the typical results shown in Figure 28 for a value of the radiation

parameter 'R~ 0. 85 ft. 1/2

41

VI. CONCLUSIONS

A study of :hc ablation of graphite indicates that one may recognize

at least four distinct acrothermochemical regimes. As the, surface temper-

atUre rises into the threshold range, the mass transfer process is initially

rate cnntr';led and "cilows a fractional (half order) reaction law. At

surface temperaturec of approximately 1400°R < Tv % 3300 0 R. a transition

regime exists and the overall rate of the process is controlled both by

c :e..ical and gas dynanic factors.' At higher surface temperatures extend-

ing tLiro,,gh 60000 R, the process is diffusion controlled and the rate of mass

loss froni the surface is lirnited by the rate at which oxidizing species

diffuse to the surface. Finally, there is a regime in which the surface

temperature is sufficiently high so that the sublimation rate exceeds the

surface oxidation rate, and carbon atorms and molecules are present in the

gas phase.

In the rate controlled and transition regimes, the ab'ation rate ot

graphite is a sensitive function of the surface temperature, and depends

critically on the specific reaction rate which has an exponential temper-

ature dependence. It is in the low surface temperature regime that one

can distinguish between different grades of graphite, since the activation

energy and the effective coll. @ion frequency depend on the molecular

structure of the particular carbonaceous material. Therefore, unless

one has an independent knowledge of the chemical kinetics of the comb" stion

process for the specific carbonaceous material, one cannot make a precise

41

prediction :f the oxidation rate for surface temperatures below 32000 R.

":e thermocheiical response of graphite in the transition regime

caui be easily synthesized from a knowledge of its behavior in the purely

rate cnntrolled ard diffusion controlled regimes. A more precise study

of the transition reglime itself does not appear to be warr "'-d at this time

beca use of anr incomplete knowledge of certain of the chemical factors, e. g.

the CO/CO2 ratio.'

In the diffision controlled regime, it is found that the ablation

rate is proportional to taie square root of the pressure and is essentially

independent of surface temperature. It is further noted that for surface

temperat ires above. 3200 0 R, barring spalling effects, th c mass loss

theoretically independent of the grade of the graphite. The rr 2a 4 incertainty

resides in the somewhat arbitrary. selection of the high temperature transport

properties due to the lack of experimental data for any of the six gaseous

species present at'the tern-voratures of interest. However, judging bya

comparison between theory and experiment for stagnation point heat transfer

in the absence of mass transfer, it is estimated that the uncertainty in the

transport properties introduces no more than approximately a fifteen percent

error in tne present final restilts.

A study of the calculated results indicates that in the diffusion

controlled regime, the he.at and mass transfer are independent of the rates

of the homogeneous reactions, provided that the heterogeneous reactions are

rapid. That is, when it was assumed that thermochemical equilibrium was

, III-15

obtaiiied at the surface, but not necessarily throughout the botindary layer,

b :)t6 tho -froz~en reactions" -and "local 'eq- ilibrium" chemical conxtraints

yielded es.-ei-i:mlly identical results.

In cunsidering the heat conducted irnto the solid, one notes that

the %eparate contri- icns of convection, diffuion. conduction and radiation

must be included. In this attidy, it was found that the net heat transfer,

including combustion effects,exceeds tne aerodynarruc heat transfer. One

may conclude therefore. that the exothermicity of the oxidation reactions

exceeds the decrease in heat transfer due to mass transfer cooling effects,

(thickeninR of the boundary layer), which results in a net increase ir. the

heat transfer to the solid. However, tiiis increase is usually less than

ton percent for the full range of interest.

With regard to viscous drag effects. it is concluded that for the

hypersonic flight regime the reduction in skin friction du&e to the mass

transfer of u~idation products does not appear to be simski.icant,, and is of

the order of seven percent or less.

441

..,.-...

AC KNOW LEDGEMENTS

The author wishes to acknowledge the fruitful discussions held

with his colleagues, Dr. Guido Vidale, Dr. Albert Myerson and Dr.

Joseph Farber, and the assistance of Charles Baulknight in'the

calculation of the transport properties. In addition, the author wishes

to thank Leon Gilbert for his valued assistance in the analysis, in the

preparation of the graphs and in the computations, and Messrs.

Frank Bosworth and Paul Gordon, who programmed the differential

equations for the IBM 704 digital computer.

This paper is based on work performed under the auspices of the

United States Air Force, Ballistic Mib=iles Division, Contract No.

AF04(647) -269.

AlI

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48. Akamatsa; H. , Inukuchi, H. , Tokahashi, H., and Matsnugu, V.,ui.Chc--' Soc. (Japan) Vol. 29, 1956, p. 1,744.

41s. %V'cke. i . * Ccnt ributions to the Combustion Mechanism ofCarbon", Fifth S~rmpcisium on Combustion, Reinhold PublishingCo., 1955, pp. 245-254.

50. Hudden, K. and Wicke, E. "About Some Influen,:es on theReactivity of Carbons", Proceedings of the Third Conferaence, onCarbon, Pergamon Press, 19-39, pp. 249-Z56.

51. Arth.!r, J.R-.. BCURA, Vol. Ii. 1944, p. 296.

52. Arthur. 3.11., 8CURA, Vol. Ii, 1949, p. 297.

53. Yagi. S. ind Kunji, D.,. "Studies on Combulition of CarbonParticles in Flames dnd rlidized Beds', Fifth Symposium onCombustion, Rheirlhold Publishing Co., 1955, pp. 231-244.

54. Semechkova, A. F., and Frank -Kame netskii. D.A., "Vosatanovionleuglekisloti uglem", 7.hur. Fkz. Khim., Voi. 14, 1940, pp. 291-304.

55. Klibanova, Z., and Frank -Kameiietskii, D.A.. "Ignition ofCarbon and Kinetics of its reaction wit)h Oxygen.l Acta Phys.,Vol. Ia, 1943, pp. 387-405.

so

IJ

56. Horton, W. S., "Oxidation Kinetics of Pyrolytic Graphite*,General Electric Go. , G. E.. L. R~eport No. bUGLII8, Jan. 1961.

51'. Streznewski, J. and Turkevich, J. "The Reaction of Carbon withl

Oxygen At omns " Proceedings of th6 Third Conference on Carbon,P-ergdmon P~ress, 1959, pp. 273-478.

58. Hirschfulder, J. 0. , Cartiss, C. F., and Bird, R. B., "MolecularTheory of Gats.-- and Liquids". John Wiley, 1954.

59. Egitt-rs, A. J. Jr. , Hansen, C. F. and Cunningham, B. E.,"St;Ignttiun - Point Ile-at Tranisfer to Blunt Shape., in HypersonicFlight Including Effects of Yaw". NACA TN 4229, April 1958.

Sl/

Ta l IL Eq i . .. '. o stns l

Table~ 11. Seii ecinRts fCroaeu aeil

Table V. Oxida Prapte of High Temperature Materials

.. .A•- - " •- - , ..1 1. / . °

iP * .• ... ... a 0 A ~ ~

0o x ..w 0

*n U

- 0 -

W\

0. to to

S 0 K00N 00

C of t.-

-o- - . - Iz -x x x x - 00 0 0 -

*- I O- - - .

0 -- - - - - E

j .. .

N /. o-

Re-erences - Table I

a) Hlandbook of Chemistry and Physics, 40th Ed., Chem. Rubber Co.,

Ohio, 1958

b) R. 1. Joiffe, "Refractur Metras", f-attelle Memorial Institute,

Presented at Symposium on High Temperature Technology. Stanford

Researcn Inst. , C-if. . Oct. 1959

c) W. G. Bradshaw and C. 0. Matthews. "Properties of Refractory

Materials: Collected Data and Refe..rnces". Lockheed Report

#LMSD-2466, Jan. 1959

d) R. G. Charles. S. Barnartt. E. A. Gulbransen. "Prolonged Oxidationof Zirconium at 350 and 450°C", Trans. of Metallurgical Soc. of

ATME. . Vol. 212, No. 1. p. 101. Feb. 1958

e) J. W. Evans and S. K. Chatterji, "Influence of Silicon on the High

Temperature olcidation of Copper and Iron", J. Electrochemical Soc.,Vol. 1Ob, No. In, Oct. 1959

f) G. E. Zima, "Some Ifign Temperature Oxidation Characteristics of

Nickel with Chromium Additions". Trans. of American Society for

Metals. Vol. 49, 1957, pp. 924-947

g E. A. GuiSransen and K. F. Andrew, J. Electrochem. Sac. 97,

363- 395, 1950

-

0 00 0 0 aP m M~- *

K ~. a. . .

.0 0 r- r- .0l

m 0.

IT ix

0 0o .. c .r~

.,. 'q.-\ ......

or , . In-.

I- '. -U

IN;IJ.114

- ; - *1 * '

-v o . N

00 It

o o jd ,

S --~t t....S..

U , - ,52 -

4,I J *

04 . , 16 No

-'

!E A. It,

-1 4

eq N! !: 6 0 0

3. '. - 0~-

.~a.

I~ 0 1 Cr

0 70 -

0.4 0,a IV a*

0 0 0 - . C

'0 < A -0 *'

N N - 0 0 0 a 0

u N *0

x4 1. 000

w I0 0 C

-'~, a 57

r4 c

-' 4

-II 4) -a

x -I'- -I';

w VL

-.0aN 0

-. w

a' -

to a f04cf

'N. - ...-

igtire 1. Coordinate Sytvrt

Fi,Ze .. V:sco.sity C ufficints of Pure Species

Fig,:rc J. Binary !I.'ffution Coefficients of Pure Species

Figure 4. Self-Diffusion Coefficients of Pure Species

Figure S. 5pecific Heat oi" Pure Species

Figure 6. Enthalpy of Pure Species

Figure 7. Derity-ViScosity Variation for Graphite Combustion

Fig,-e s. Variation rf Surface Prandtl Number with SurfaceTemperature

.Figure 9. Typical Surface Lewis Number%

Figure 10. Velocty and Temperature Profiles

Figure 11. Concentration Profiles for Graphite Combustion

Figure IZ. Reaction Rate Profiles

Figure 13. Typical Ei&;envalues for Graphie Cambustion

Figure 14. Ratio of Oxygen to Nit rogen tt the Surface of BtrningGraphite

;t.'

FiLUe 1L. Re.action Rate Controlled ,Mass rransfer

Figure 16. Mass Fraction Ratio of Combustion Products at .Graphite Surface

Figur-e i7. Ma.ss Tran~sfer Comparison fur a Typical Flight Condition

Figure 18. Mass Transfer for Graphite Combustion

Figure 19. Mass Transfer Transition Curves for GraphiteCombustion

Figure 20. Mass Transfer Regimes for Ablating.Graphite

Figure 21. Typicil Heat rransfer into Solid Graphite

Figure 22. Comparison of Modes of Energy Transfer to a BurningGraphite Surface

FiguLc Z3 Heat Transfer Correlaticn for Heat Conducted into SolidGr'aphite

Figure 24. Correlated Heat Transfer intc Solid Graphite versusSu.rface Ternrxratuzre

Figure 25. The Effective Heat of Ablation for Graphite in theDiffusion Controlled Oxidation Regit-

Figure 26. Shear Function for aTypical Flig',. Cusadition

Figure 27. Correlatcd Skin Friction Coefficient

Figure 28. Axially -Symmetric Stagnation Point Radiation EquilibriumrSurface Tlemperature

60

~cI

U, >

C'C-

(I 0 Z ~ 0 z O_

u0

c3m

0u I

5--

U)N

IL

U

0~ 00

N. 0~LUU

LJ

00

00

C) I

0--

LLJ _____

Zz

a'z

M.I

0

0 C4,

0~0C'C)

LU'I)

_2 0

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00

U-

z LUo(0) _(

IL.

I-LJ OD__ (0 C__

*:3/idZOIIIi

Li- Zc~ ZO __ o_ co

((D

I~~~ W_ _

(I)N

_ 10

LiZ c

00

tDCj O (Do 0

8o 4, m

10

r J

0 cc

/ ~~U-____N

11o 0 w -m cLLz-

(/)J ni __Ol 4

LiLrI

H 0 0

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0

00 0 L

0 z'00

0 N00

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U) G

z S

IL 0

LJI

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0 .C) 0

Z LLJ <:

<-t it a

a-L cr0 I-

0

(1)

0Iz01

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000 00031

> bw.

CL

TYPICAL SURFACELEWIS NUMBERS(GRAPHITE COMBUSTION)

Pe 5 .7 AT M.

2.0-

1.5

N2 +CO BCO+N 2

CO2 +N2 a C02+CO.5 L

1000 2000 3000 4000 5000

r~R FIGURE 9

VELOCITY AND TEMPERATUREPROFILES

(GRAPHITE COMBUSTION)-EQUILIBRIUM BOUNDARY-LAYER---- FROZEPN BOUNDARY LAYER1.0-

* / ALT=I10000 FT.u Vo0 20,000 FT/SEC.Ue .4 Tw 2000 0

0o 1.0 2.0 3.0

1.0-

.8-

*.6-T

0 1.0 2.0 3.0 17 FIGUyRE 10

o 0I 0I 0I

IL 2~ /9

0 L I -

on con

L. . 0 I I0cZ C*-c 9

0 0 Eo zo It/

08 Icr I 0/

CLJrr N O

4,:C~

EQUILIBRIUM BOUNDARY LAYERPROFILES

(GRAPHITE. COMBUSTION)

1 0 '0 0 TEMPERATURE PROFI LE8000k

~6000L0 I ALT.= 100,000 FT.F-4000 Va,:z 20,000 F T/EC.

2000T:200R

4000_GO 2

S3000 RE AC TIO0N Z~E

~-2000-

-1000 s

0 .2 .4 ..6 .8 1.0 1.277VIG1 RE. 12

TYPICAL EIGENVALUES FOR GRAPHITECOMBUSTION

ALT: -100,000 FT Va, 20,OOO FT /SEC.l.a 0 ~ I

EQUILIBRIUM BOUNDARY LAYER---- FROZEN BOUNDARY LAYER

w'V~

z

0.2

01000 2000 3000 4000 5000

FIGUtRE 13

LLI

Liz

DUl)

Z3LJ (D 0~

0 -z

0 0 CcIr a:

5- z

LU 0

z~ U) -o 3 \(NO0IL - L

___________________________________ ______________________________________

/I

~MASS FRACTION RATIO OF COMBUSTIONPRODUCTS AT GRAPHITE SURFACE

\! BOUNDARYLAE4.0' SOLUTIONS!R

10 O.01 A

j :5O ,.--- \ I --- THEORETICAL,0~~P =I00 AT, DFUSO ONRLE

u 2.0 V C IFSO~cMBUSTIONTRL~

I Y\ -- EXPERIMENTAL DATA,k \\ \ RATE CONTROLLED

00 COMBUSTION

BONNIMSCR A (Pe IONATM )

PODCT AT GRAPSNOW, ET AL

4(005 <Pe < 25 ASMT)-2.0_

0 5.0 3.0 2.0 1.5 1.0I w X 10-3 NR

F.IGURE. is

I.-. "" . / '

$ /

REACTION RATE CONTROLLED MASS TRANSFER1/2 ,z2 lZ

mw/ ,LEJ/FT SEC.ATM.r 1 2.008 rnw/ ,LB./FT SEC.ATM.

GULBRANSEN & ANDREW-007PO 010 ATM. I 11.007[ 2P * ----- 7

. BLYHOLUER S EYRING-- I006-(EXTRAPOLATED),Po2=26pLHg

PARKER & HOTTEL--\ -1--.005 1- - -

FRANK -KAMENETSKIIBASED ON: =IATM. I

.003 ! VULIS ..

.002 -_ _IINOTE:P ° 21P /

.001 j /Ii

500 1000 1500 2000 2500 3000TwFR

FIGURI 1b

MASS TRANSFER COMPARISON FOR ATYPICAL FLIGHT CONDITION

EXACT SOLUTIONS(DIFFUSION CONTROL' ED

-C COMBUSTION)

Ld

1.21 ~APPROXIMATE" SolUI TIONSLL (DIFFUSION CONTROLLED

COMBUSTION)

N . ALT z 100, 000 FT.Vco =20, 000 FT/SEC.

K.81

-o-o-oRECTION RATE CONTROLLED

0.~ TRANSITION CURVE

1000 2000 3000 4000 OW0TW,*R FIGURE 17

-~I rI-IC)) -l x

(I) (f) 0J _2)Lj 2fW W 2

0 0 n LA

sL...

U 4c~ - o CD~L~

O-0.~ Ur I

006 IL<"-<

Liu <

09 00iir')10

I'f

MASS.TRANSFER TRANSITION CURVES,7.0 GRAPHITE COMBUSTION

107

U,

o0 4.0

~ 3.0K0-0-o REACTION RATECC ONTROLLED

DIFFUSIONE CONTROLLED

2.06-t- TRANSfl ION CURVE

1000 1400 1800 2200 .2600OwR FIGURE 19

0

LLL. 0A 0 0 r

O () 1)*

zL LLLw -Q -J x70 e- 0e)M

cn Z L4r~m

< o 0V

<_ I

0 00u 0 70.

~a- z JUJLOF1 -- Z 0

0U 0 0L (fL)v) ~~ ~ ~ I 0 -5; L0

LU cr Wfor

00

0 ~ ~ 7 6 0L f

0 0 0 'u~00 0

0 0 0)CoQ

q0

COMPARISON OF MODES OF

ENERGY TRANSFER TO A BURNINGGRAPHITE SURFACE

A=O, 8=1 Ew/RB=0

ALT 100,000 FT. Vco = 20,000 FT/SEC.

10 0 0 - - EQUILIBRIUM GAS PHASE

--- FROZEN GAS PHASE

800-

600-F-;60 TOTALS,, -

4 0 0L -won&F-

200 - " m--o -Lo K - T CONDUCTION

0 0--. Pi v hi , DIFFUSION + CONVECTION

-200 LL I I1 2 3 4 5 6

Tw x I0 , "R FtrLRE 22

- 0

003r

0~~LL1 ~ 0

_ _ 0r~c)

LLO. 0d< +

0 '0

0 0

oz i

z

z 00

00Hol 0f 0 0

*I V -303S ?,i.L /11.. (sd /Sg8

0

0 -

0. 0 0(1) 0 g 's 00

u-i 0C LJLUj 0

O0 00

Lu z zLLJ- OZ)F- - 0y-)i

UjL- > 11UEr M

Ld <

Hq) z 3

QQ 0< 0)D 0< cr <0w1JV3 %i/fJ. X,(dA

Ld uLLJ

-< 1'Nj0

__ 0

0-

I j

< x I 1,)+ o-

0' C____-__ C) P 0

> z 0

cc 0F- 0 0

LL LL

Lii 01J MZ 0 0

0 0

81i/ flw8o

020

0car

N In

(~0) 0Cl ;ii Lfl

O - F-

0_F- ) - K'U (D

z _ _- _ _

IL.

LL U) H

UI zLL- <I

0 Z LJ 0< :r

Lj0I 0

K0 8L&JJ

Z0 000

"S.- toI

(D 0U) C'J(O0

cr-j

a1 00 0

Gui

j I

Lii,

LL 0

uI oI "

SPAC SCiNC5 LAO~&O~yGENERAL * ELECTRICM~ISSILE AN.D SPA~cE DlVISIoN

TECHNICAL INFORMATION SERIESI *U~~~k@ SBCI CLASS6ICATIOei 00

R62SD72S. '*V. Scakla Ablation, Hlypersonic Heat- ---rr;insft*.,, GraphiteSpti6

Tht, A W,tioli coi Crap~i.ite in Dissociated 4m CAl~~ir T 'tir 1 heory NN

00wff .& WN .T * ION" r..W.&.M.IC9C~*"

mI ~ os coa.' 2...U& 0

W.......

fit, k

we 06dd *04Of

Avg"" ~1 Matr&i r. figh Alt iturt,' Aroeynim icl