1490 J. SPACECRAFT VOL. 3, NO. 10

Mechanical and Chemical Contributions to the Erosion Ratesof Graphite Throats in Rocket Motor Nozzles

V. R. GOWARIKER*Imperial Metal Industries (Kynoch) Ltd., Summerfield Research Station,^ Kidderminster, England

The paper presents an analytical approach, substantiated by experimental results, whichpredicts throat erosion in a rocket motor nozzle with fair accuracy over a wide range of graphitegrades, pressures, and propellants. Both surface chemical reactions and mechanical re-moval are assumed to occur; the effectiveness of each depends on the composition of the react-ing products in the combustion gases, the temperature, the pressure, the quality of the graph-ite, and the geometry of the nozzle. A simplified diffusion equation has been solved for theturbulent boundary layer close to the throat surface, and a suitable expression for the masstransfer coefficient that considers the geometry of the convergent portion of the nozzle hasbeen employed to evaluate the chemical contribution. For the mechanical effect, a simplelogarithmic function depending on porosity of the carbon and characteristic velocity of thepropellant gases has been determined using dimensional analysis and experimental data.The results obtained under a nozzle material evaluation program, from both full-size andsmall-scale motors operating under different pressures and using various graphite grades andpropellants, show a fairly close agreement with the theory. A new dimensionless numberbased on physical constants of the graphite throat has been defined for the graphical repre-sentation of results.

a, ai, &2BCc*d, CsD

DRPDthe/K

kMmi, m2, nNRZn\, ri2, UsPr

rrcj rm

ReSctxXp, XR

XROz

Nomenclature

throat areas before and after firing, respec-tively, in.2

constantscr(log P)2/o-'total molar concentration, moles /ft3characteristic exhaust velocity of gases, ft/hrconstantsvarying diam along convergent portion of

nozzle, in.diffusion coefficient, ft2/hrthroat diam (2r) in.measure of roughnessfriction factorfirst-order rate constant, ft/hr (also tempera-

ture unit Kelvin)mass transfer coefficient, Ib/ft2-hrmass, Ibconstantsmolar flux of R in the z direction, Ib/ft2-hrconstantspercentage porosity of carbon chokethroat radius, in.dr/dt, over-all erosion rate, mils /secchemical and mechanical contributions, respec-

tively, to r, mils /secReynolds numberSchmidt numbertime, secdistance along convergent portion of nozzle, in.mole fractions of products arid reactants, respec-

tivelymole fraction of R in the main stream at throatdistance perpendicular to the surface

Presented as Preprint 65-351 at the AIAA Second AnnualMeeting, San Francisco, Calif., July 26-29, 1965; submittedAugust 16, 1965; revision received May 16, 1966. The authoris grateful to G. F. P. Trubridge arid II. M. Darwell for goingthrough the manuscript and making some useful suggestions; toA. R. Parkes for preparing Fig. 5; to G. P. Thorp for his help inthe numerical calculations; and M. J. Chase for some advicein the preparation of the appendixes.

* Senior Technical Officer, Ballistics and Mathematical Ser-vices Department.

t An agency establishment of the Ministry of Aviation.

8 = boundary-layer thickness, in.M = gas viscosity, Ib/ft-hrp = gas density, lb/ft3a = bulk density of carbon, lb/ft3a' = theoretical (maximum) density of carbon, lb/ft3TW = shear stress at wall, lb//ft2

Introduction

THROAT erosion can affect the ballistic performance ofrocket motors, and it is important to be able to predicterosion rates at the design stage for various nozzle materialsunder the attack of combustion gases from different pro-pellants. From among various materials, graphite is usedon a wide scale for rocket motor nozzle throat inserts, dueto its excellent shock resistance, high vaporization tempera-ture, and low density. The main disadvantage of the mate-rial is its poor erosion resistance, especially at high gas tem-peratures and long burning times. The complexity of theproblems associated with erosion and the lack of adequateknowledge as to the relative importance of different factorsthat influence the erosion mechanism renders a rigorousmathematical treatment beyond practical conception. Inthe state of the present art however, a simplified model canbe proposed which, although being essentially mathematical,would utilize the experimental evidence to further the modeltoward perfection and in turn help to smooth out the apparentcontradictions in the available data.

This paper considers erosion problems associated withgraphite inserts. The analysis assumes that the erosion isattributable to both surface chemical reactions and somekind of mechanical scrubbing at the carbon surface. Thedegree of effectiveness of each of these factors will depend onthe composition, temperature and pressure of the reactingproducts in the combustion gases, and the quality of carbon.Generally, as gas temperatures and C02 and water contentsin the exhaust products increase, chemical effect becomesmore significant, and a point is reached at which little erroris introduced by ignoring the mechanical contribution.An attempt has been made in this analysis to include thepertinent parameters so that, under a set of existing condi-tions in a rocket motor, the relative importance of chemical

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OCTOBER 1966 EROSION OF GRAPHITE NOZZLE THROATS 1491

and mechanical effects can be predicted and the calculationsthereof can be performed to give the over-all erosion rate.

The results from firings of small-scale motors at differentpressures using various cast-double-base propellants andemploying several grades of graphite for the throats checkclosely with the theory. No claim is made as to the universalapplicability of this treatment, which clearly warrants a moredetailed investigation over a wider range of pressures andpropellants, but it is believed that the paper suggests a gen-eral pattern of the future analytical work on the problem.

Model

General

Assuming that the total erosion rate can be expressed asthe sum of the chemical and mechanical contributions, then

r = fc + rm (1)It will be shown later that rc depends on the composition of

exhaust gases, the flame temperature, Reynolds number andSchmidt number of the flow, density of the graphite throat,and the geometry of the nozzle; fm is a function of the char-acteristic velocity of the gases and porosity of the throat.

Effect of Chemical Reactions

Out of the mixture of gases resulting from the burning ofthe solid propellant, the reactions of only three reactantswith carbon are considered to be significant1:

Solid Gas GasC + C02 -* 2COC + H20 -> CO+H2

2C -}- H2 > C2H2

(2)(3)(4)

The atmosphere in the motor being generally oxygen negative,reactions with 02 are omitted, but may be considered in highenergy propellants.

It is assumed of course that the reactants diffuse throughthe turbulent boundary layer to come into contact with thecarbon surface and reciprocate, and that the products of re-actions do not accumulate in the boundary layer but diffusethrough to mix with the gas mixture. These assumptionsappear to be reasonable considering the high gas speed andthe nature of flow at the throat. Because of different massfractions of the main chemical elements in the gas mixture,there may exist different shaped concentration profiles for eachof the products along the normal to the surface of the nozzlematerial; the state of chemical aggregation of the fluid doesnot need to be known in detail, however, if the laws of theconcentration of individual chemical elements are notviolated.

From Eqs. (2) and (3), it follows that each mole of the react-ing gas evolves two moles of gas. Therefore,

NP = -2NR (5)The equation for the mass transfer rate at a distance z fromthe free-stream-boundary layer interface, when diffusion ofone species toward the carbon surface and one productaway from the surface is taking place, may be written as

NRZ = -C DRP dXR/dz - XRNRZ (6)Without introducing significant error, Eq. (6) may be

applied to the multicomponent diffusion case. Applyingthe^boundary conditions

at z = 0

at z = d

XR = XRQ

XR = NBZ/CK

Eq. (6) can be solved1 to giveNRZ = k ln[(l + XRO)/(l + (8)

Forwhere k is the mass transfer coefficient C DRp/d.erosion calculations therefore, the expressions for masstransfer coefficient and first-order reaction rate constantsmust be known. For the former, a choice must be made ofa suitable relation out of the existing many. The followingrelation for the turbulent boundary layer2 is employed herefor k:

k = 0.0288 -e X

(9)

In the propellants under study the temperature at thethroat is in the region of 2500K. There seems to be someuncertainty as to the correct correlation of the first-orderreaction rate data at this temperature, when carbon reactswith C02 and H20. However, since this temperature is nearenough to the one for which Khitrin's data3 are reported,the same equation as was used by Jones et al.1 for theCC02, CH20 reactions is employed:

InK - 19.96 - 2.12 X 104/T (10)where T is the temperature in K.

Regarding the contribution due to reaction (4) to thechemical erosion as negligible at the temperature considered,the final expression for the molar flux of R in the z directionis obtained by combining Eqs. (8-10). This is

NRZ = 0.0288 -f- X/Ah

Re-8 c--667 In1 + X

C exp(19.96 - 2.12 X 104/T)_

D(ID

N RZ appears on both sides and the computation has to bedone in an iterative manner. The final rate of carbon lossdue to chemical reactions is obtained by adding the effectsof C02 and H20 reactions on carbon, which are obtained byapplying Eq. (11) for R = C02 and H20. The chemicalcontribution is then given by

fc = 2 NRZ/ a- (12)where or is the density of the graphite (see Appendix 1).

Mechanical Contribution

The experience with different grades of graphite used in6-in. motor firings at Summerfield indicates that the surfacereactions discussed previously account for only a part of thetotal erosion. At pressures and temperatures generally at-tained in rocket motors, the mechanical effect is also im-portant with a number of propellants. The graphite used fornozzle inserts of the same density seems to vary widely inporosity (apparent percentage porosity, which bears no simplerelation with bulk density; see Appendix 1) and it has beenfound that for a given propellant and operating conditions,the erosion rate varies with the throat porosity. A possibleexplanation for this phenomenon may be given by introducingan analogy with turbulent flow in a rough pipe. For in-stance, Hopf's equation4 relating the friction factor (/) andthe surface roughness is

where e is a dimensionless number based on the pipe radius

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1492 V. R. GOWARIKER J. SPACECRAFT

Table 1 Predicted and experimental0 erosion rates

Fig. 1 Fired choke of graphite.

and the average height of roughness projections and ci andc-2 are constants, von Karman5 gives an expression for f(e)that has been satisfactorily tested for a wide range of e.In their studies, the roughness was created artificially on thesurface in a predetermined manner so that it could be ac-curately measured and the friction factor would be expressedas a function of the roughness. If the porosity of the graph-ite is taken as a measure of natural intra-molecular rough-ness at the surface, the resistance to the gas flow will varydirectly with the porosity, and the erosion rate will vary too.Let the mass eroded off per unit area in time 8t be 5M. Thiscorresponds to an increase in throat radius from r to r +5r in time 8t, so that the mass eroded per unit area is a dr.

Therefore

dM = adr = M/a

The flow of gases creates shear stress at the wall TWJ and fromdimensional and experimental analyses one may show thelinear relationship6 between rm and the usual group in fluiddynamics, viz. rw/(pc*) where p is the average density ofgases and c* is their characteristic exhaust velocity. Ex-pressing the shear stress at the wall in terms of the frictionfactor/, one gets

fm = aic*f (13)where a\ is a constant of proportionality.

If the porosity (P) of carbon is taken as a measure ofstructural resistance, then within the limits of throat diam-eters used in rocket nozzles of this program, a relation similarto von Karman's may be taken as follows:

Material

MlM*Mz

M,M,M6M7M8M,M10MuM12M13MuM13M16M17M18M19-

lb/ft3

109.3113.0112.4115.5116.7110.5112.4111.7111.7111.7115.5106.8106.1116.1116.1116.1116.1109.9116.7

Porosity, %

13.17.99.36.85.5

13.811.211.310.49.7

10.113.313.53.57.54.27.39.67.1

f, mils/secPredicted Actual

2.92.12.41.91.63.02.62.72.52.42.52.92.91.22.01.32.02.42.0

3.42.72.71.61.53.32.43.22.62.02.32.62.81.31.71.51.62.02.4

= o,(logF)'- (14)

a The data are for a chamber pressure of 1600 psia and an average burn-

ing time of 20 sec. The "actual" value is the mean of two measurementsthat differ by around 15% in some cases. The typical irregularity in throatenlargement is shown in Fig. 1.

where a2 is a new constant. Combining (13) and (14),rm = ac*(logP)2 (15)

wiiere a is a final constant to be determined using a set ofexperimentally determined over-all erosion rates. It isfound that a = 3.2 X 10~8 is satisfactory for all the pro-pellants tested. The over-all erosion rate is obtained bycombining Eqs. (12) and (15).

Theoretical Predictions and Comparison withExperiment

The theoretical calculations may be performed throughthe following steps:

1) From the percentage composition of different gases inthe exhaust mixture, the molar concentration of an indi-vidual gas can be determined at the required pressure andtemperature. The results for each species are then added togetC.

2) For both CCO, and CH20 reactions, the value ofK is obtained at the flame temperature considered fromEq. (10). This, in conjunction with (1), gives CK.

3) From motor pressure, throat diameter, characteristicvelocity, and viscosity of gas mixture, Re can be determined;M and Sc for both CO-2 and H-20 are obtained from the standardtables.

4-O- 4 - O

3 - 0 -

2-0-

oo i - O r -

Fig. 2 Over-all ero-sion rate against

density.

2 - 0 -

O I OL

Fig. 3 Over-all ero-sion rate againstpercentage porosity.

IO5 IIO~ 115DENSITY, LB / FT 3

120 5 IO 15

PERCENTAGE POROSITY

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OCTOBER 1966 EROSION OF GRAPHITE NOZZLE THROATS 1493

4) X for both C02 and H20 are again obtained from theirpercentage compositions in the exhaust gases.

5) After evaluating the geometrical quadrature of theconvergent portion of the nozzle, with the knowledge of thenozzle semiangle, k for both CO-2 and H20 can be calculated,Re, Sc, ju, Dth, etc., all being known [Eq. (9)].

6) As a result of calculations in steps (1) to (5), the compli-cated expression (11) for NRZ reduces to the following simpleequation:

NRZ = n,\n[n,/(n, + NRZ)]where n^ n-2, and n$ are known numerical constants. NRZcan then be determined by iteration, separately, for R =C02 and R = H20.

7) NRZ being known for C02 and H-20, rc is obtained fromEq. (12) if the density of the choke is known.

8) With the knowledge of characteristic exhaust velocityof the gases and porosity of the throat, rm can be calculatedfromEq. (15).

Several 6-in. motors were fired using propellant A and vari-ous grades of graphite for the nozzle material and the actualerosion rates determined. (See also Appendix 2.) Thecalculations for the theoretical prediction for this propellantwere worked through the given steps, and gave the followingresults:

C = 0.0163 Ib moles/ft3K = 9.550 X 104 ft hr: CK = 1557 Ib

moles/ft2 hrfie0-8 = 9.032 X 104r-:>.667 for CQo = 1.028; Sc--667forH,0 = 1.406M =0.12 Ib/ft-hr for both CO-> and H2OX for C02 and H,0 = 0.0059 and 0.01020, respectivelyk for C02 and H20 - 8939 and 12204 Ib ft'2-hr, respectively2 NRZ = 21.91b/ft2-hr

losing c* = 4728 fps for this propellant, one obtains fromEqs. (1, 12, and 15)

I2O MO IOO_lb/cuft

r = 0.073/0- + 1.816 X 10-3(logP)2 (16)It may be noted that the coefficient of cr~l depends on

the characteristics of the combustion gases (composition,temperature, etc), motor pressure, and the geometry of thenozzle, whereas the coefficient of (logP)2 is a straight func-tion of the characteristic velocity of the gases.

Comparison with Experiment

Table 1 summarizes the erosion rates predicted by Eq.(16) and measured \vhen throats of different grades of graph-ite were used in 6-in. firings with propellant A.

The predicted erosion rates for some other propellantsburning at different pressures in both small and full-sizemotors also show a good agreement with the actual values.For instance, in the case of one oxygen-balanced propellantburning at a pressure of 800 psia in a 6-in. motor, the predicted

380: i34O; ~ --/

-fel^ 2601 -- - " ?b

' / -

I8O1 6 /-

1494 V. R. GOWARIKER J. SPACECRAFT

FORWARDENDCLOSURE

DURESTOS.

BODY

This is calculated asENDCLOSURE

PRIMED CAMBRIC/IGNITER

CHQKE

Fig. 6 Test motor assembly.

effects have to be separately considered for close agreementwith data. Under a set of existing conditions in a rocketmotor, the two effects can be calculated to give the over-allerosion rate.

2) With the same propellant and at the same pressure,graphite throats of the same density but different porositygive different erosion rates. The porosity mainly affects themechanical contribution, and the degree of its influence hasbeen determined. The chemical effect depends largely uponsuch characteristics of the exhaust gases as the composition,the temperature, and the conditions of the flow.

3) A dimensionless number based on the physical con-stants of the choke material is defined for the representationof theoretical and actual erosion rates.

Appendix 1: Porosity and Bulk Densityof Graphites

The refractory grades of graphite are made by mixingpetroleum cokes from various sources and binding with aresinous material. This mix is then carbonized and graphi-tized. Intermediate resin or gas impregnations may be usedto increase the density. As the material is porous to varyingdegrees, it is possible to obtain a denser graphite with ahigher proportion of pores that still has the same bulkdensity as another graphite that is less porous. The termdensity used in this paper refers to bulk density, and, becauseof methods of manufacture, it bears no simple relation toporosity. It is defined as the weight of the sample dividedby the volume of its external shape, and it is determined asfollows: The pores of the sample are filled with whitespirit by vacuum impregnation. Excess liquid is wiped offand the sample is weighed in air and also weighed immersedin white spirit. The difference between these weights is theweight of the liquid displaced. The volume displaced iscalculated knowing the density of the white spirit.

The "apparent solid density" is defined as____________weight of the sample____________(volume of external shape) - (volume of open-ended pores)

The difference in weight (between sample and sample withpores impregnated and weighed in white spirit) gives theweight of liquid displaced. The volume displaced is calcu-lated knowing the density of the liquid.

The apparent porosity is defined as

bulk densityapparent solid

ity__\density/ X 100

volume of open poresvolume of external shape X 100

The word "apparent" is used since some pores may betotally enclosed in the graphite.

Appendix 2: Nozzle Material EvaluationProgram (NMEP)

Potential nozzle throat materials were evaluated in theform of small inserts in a 6-in.-diam rocket motor incorporat-ing the most advanced propellant system available in quan-tity at the time. The components were examined beforeand after firing, and efforts were made to determine the reasonfor success or failure where appropriate.

Four typical propellants, one with low flame temperature,two with high temperatures (aluminized and nonaluminized)and the last oxygen-balanced, burning for 20 sec, 13-15 sec,26-30 sec, and 20 sec, respectively, have been employed withdifferent pressures; throat diameters ranged between 0.4and 0.575 in. From thermodynamic calculations, the per-centage composition of different products in the combustiongases is known; these propellants give a fair cross section ofthe products resulting from most existing propellants.

The experimental motor used for these trials consisted of a6-in.-diam, thick-walled steel tube fitted with a forward endclosure and containing the solid cylindrical charge. An aftend closure carried the test insert. Figure 6 shows the gen-eral assembly in diagrammatic form. Both the forwardand aft end closures were drilled and tapped to allow pressuretransducers to be fitted, enabling chamber pressures to berecorded continuously during firing. Each insert was in-spected before and after firing and its performance was as-sessed by:

1) Comparison of throat diameters before and after firing.2) Pressure variations during firing, which indicated the

course of erosion.3) General appearance, e.g., crack formation, evidence of

spalling or chemical attack, etc.4) Structural examination. Most components were ex-

amined metallographically and in many cases evidence offailure mechanisms was furnished.

References1 Jones, W. H. and Delaney, L. J., "An analysis of the material

problems for throat inserts of high energy solid propellantrockets/' Institute for Defence Analyses, Wash. D. C., TR 62^.19, UBG 62-559 (October 1962).

2 Ambrok, G. S., "Approximate solution of the equation for

the thermal boundary layer with variations in boundary layerstructure," Sov. Phys. Tech. Phys. 2, 1979 (1957).

3 Khitrin, L. N., "Fundamental problems of carbon com-

bustion and factors intensifying the burning of solid fuels,r'Sixth Intern. Symp. Combust. 565-573. (1956).

4 Hopf, L., "Die messung der hydraaulischen rauhigkeit," Seit,

Angew Math. U Mech. 3, 329 (1923); summarized in Knudsen,L. G. and Katz, D. L., "Fluid Dynamics and Heat Transfer,"Eng. Res. Bull. 37 (September 1953).

5 von Karman, T., "Mechanical similitude and turbulence,"

NACATM611 (1931).6 Garner, F. H., private communication, Univ. of Birmingham,.

England (1961).

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