Indian Journal of Engineering & Materials SciencesVol. 5, October 1998, pp. 271-277

Thermal stress analysis of a solid rocket motor nozzle throatinsert using finite element method

R C Mehta", K Surcsh " & R Narayana lyer"

"Vikram Sarabhai Space Centre, Trivandrum 695 022. India

'Department of Mechanical Engineering, College of Engineering. Trivandrum 695 016, India

Received 22 September 1997; accepted 20 April 1997

A finite element formulation is developed for the analysis of axisymmetric, transient, anisotropicheat conduction problem with the temperature dependant thermo-physical material properties of agraphite throat nozzle for the solid rocket motor. A standard Galerkin method using linear triangularelement is employed for the space discretization. The time integration is done using an implicit timemarching scheme of the first order differential equation. The convective heat transfer coefficient iscalculated using the Bartz correlation. A thermal stress analysis is also carried out on the graphite throatof the nozzle using finite element method with two degrees of freedom. The developed computer codesfor this purpose are validated with known analytical solution and available ANSYS code.

A convergent-divergent nozzle is used as apropulsive device in the launch vehicle. The nozzleexpands high pressure and temperature gases fromsubsonic to supersonic velocities. Modern highenergy solid propellant produces combustion gasesof high temperature in order to enhance the specificimpulse. The throat region of the convergent-divergent nozzle is therefore, exposed to hightemperature and pressure environment as comparedto other zone of the nozzle. The structural failure ofthe nozzle material may occur if the temperatureexceeds the permissible operating design limit. Athermo-structural analysis is required for satisfactoryperformance of the solid rocket motor.

Analytical and numerical methods are availableto obtain temperature distribution inside thematerial. Carslaw and Jaeger' have obtained solutionfor a simple geometrical configuration. A numericalalgorithm to solve heat conduction problememploying finite difference method is described indetail by Rosenberg.' The finite difference approachneeds jacobian transformation to changediscretization from physical to computationaldomain, or special mathematical treatment to bedone to take into an account irregular boundary ofthe nozzle wall.

Swaminathan and Rajagopalarr' have ana lysedthe nozzle heat transfer problem using finite

difference method. Lee4 has obtained temperaturedistribution inside a nozzle of a solid rocket motoremploying finite difference method. Henderson) hascomputed in-depth temperature response by solvingheat conduction equation with temperature-dependent thermal properties.

The finite element method can easily solve a heatconduction problem for a complex geometricalconfiguration. Various grid arrangement can be usedto accommodate a complex geometrical configura-tion and resulting simultaneous equations can besolved by using standard algorithm." The finiteelement technique to solve a heat transfer problem isreported elsewhere in detail. A two-dimensionalheat conduction equation with time dependantheating condition at one surface of the specimen anda radiation boundary condition at the other end issolved using finite element method by Mehta et a/.x

The stress analysis of axisymmetric configurationis described by Zienkiewicz. To the authors' bestknowledge, a coupled thermo-structure analysis of arocket nozzle is not available in open literature. Themain aim of the present paper is to investigatethermo-structural analysis of a graphite throat insertof a typical solid rocket motor nozzle. A finiteelement analysis is carried out to solve axisymmetrictransient anisotropic heat conduction equation withtemperature dependant thermal properties.

272 INDIAN J. ENG. MATER. sci., OCTOBER 1998

Temperature distributions are computed at eachnode point and at each time interval for a givenconvective heat transfer coefficient variation. Usingthe computed temperature profile at the end ofmotor bum-out time, the displacements and resultingthermal stresses are computed using finite elementmethod. This coupled thermo structural analysis isvalidated using ANSYS software."

Heat Transfer AnalysisThe governing axisymmetric transient anisotropic

heat conduction equation with temperaturedependant thermal properties can be written as

!~(rkr aT)+~(kz aT)= pCI' aT (I)r ar ar az az or

with the following initial and boundary conditions:T= To on region t = 0 . (2)and

DT-k-=h(T,,...-T) onsurfaceratt>O ... (3)

Dnwhere k. and kz are the thermal conductivity in radial andaxial directions, respectively. p is the density of thematerial, ep is the specific heat of the material, / is time,h is convective heat transfer, n is the outward normaland Taw is combustion gas temperature. The other sidesof the nozzle wall are insulated.

The spacewise discretization of Eq. (I) subjectedto the above boundary condition can beaccomplished using Galerkin method. Let theunknown function temperature T be approximatedthroughout the solution domain at any time t by therelationship

nT= I N, (r,z)T; (I) ... (4)

i=1where N, is usual shape function defined elementby element, T; being the nodal parameters. For atypical triangular element e with nodes numberedanticlockwise as i, j, k and placed at vertices of thetriangle, the shape function N, is

N;= a+br+cawhere

... (5)

2 A"(z-z.)b i = '--"..I----'~

2 A"(n -r ,)

c, = 2 A'"

and Ae is the area of the element, e.

... (6)

The general procedure for solving Eq. (I) is toevaluate the Galerkin residual integral with respectto space coordinate for a fixed instant of time. Thisyields a system of ordinary differential equationsolved to obtain temperature distribution. The nequations can be written in a matrix form as

dT[C]-+[K]T={j} .. (7)

dtwhere [C], [K] and {f} are capacitance, stiffnessand load vector matrix, and can be written as

[C] = Iv pCI' [N] r [N]dV (8)

[K] = Iv[kr 8[Nl' 8[N] +k, 8[Nl' 8[N]]dV (9)ar 8r az az

j = Ir [N]" ur.:-T)dr ...(10)

where V is the volume of the element that is equalto 27r rcAe about z axis, rc is the centroid of thetriangular element.

Time integrationThe set of ordinary differential Eq. (7) which

defines the discretized problem is solved usingimplicit time marching a scheme. The equation canbe written for time level n to n+ I as

[C] +[p]/r'+1 = [C](Tf + jn ... (11)~t ~r

where [P] is combination of [C] and [K] matrix.The algorithm is unconditionally stable andconvergent in the context of finite difference.Numerical experiments have been done at differentvalues of aspect ratio and time-step in order toverify the stability of the algorithm.

Grid generationThe finite element solver requires the element

coordinate and connectivity in the domain ofcomputation. An algebraic procedure is used togenerate a triangular element. A schematic sketch ofthe nozzle insert is displayed in Fig.l. The outerwall and inner wall of the nozzle insert arerepresented as rtop(z)and rw(z). The radial incrementis computed as difference of inner and an outer wallradius divided by number of sub division. The gridis structured. Therefore, the connectivity of thetriangular elements is obtained using recurrencerelation. Fig. 2 shows typical grid arrangement in thenozzle insert.

Numerical stress analysisIn the axisymmetric problem of a nozzle throat

insert, a cylindrical coordinate system is taken in

MEHTA et al.: THERMAL STRESS ANALYSIS OF SOLID ROCKET MOTOR 273

computation of thermal stress. The strain equationfor the axisymmetric configuration can be written as(l')' = (l'rr eflll ezz erz] (12)

(I:)' = [G'" G'HH l:zz l:rz] (13)

kr;' = [a()T asr aOTo] (14)

where {e} is the total strain, {s} is the elastic strainand {cr} is the thermal strain, a is the coefficientof thermal expansion and '&T is the change oftemperature. The above elastic and thermal strainvector can be written as

{ e} = { s}+ {G°r} . . . (15)the stress can be calculated using following Hooke'slaw:

d

E b[D]=-

1+ fJ b

b b

d b

b d

ooo · .. (17)

o 0 0 12

where d=( 1-J.1)/(1-2J.1) and b=J.1I( 1-2J.1), J.1 is thePoisson's ratio. The stress vector components foraxisymmetric case can be written as

{cr}T=[crrrcrOOcrzzcrrz] ... (18)

Displacement calculation

The general expression for displacement can bewritten as

{a}=[D]{E} ... (16) u=f(r,z), w=g(r,z) and v=0 · .. (19)

where [D] is the material stiffness matrix and canbe written as

boundo.ry 'r:C'.i: ~

NOZZLE AX1SInner boundo.r y (WQll)

Fig. 2-A typical grid system for the graphite throat nozzle

where u, v and ware displacement in r, e, and zdirections, respectively. The strain displacementrelationship is given as

au ue =-, eoo=-,

rr ar rau aw

er;:= az +7);' e:o=O

~he unknown displacements can be written interms of element nodal values

aw=:«· .. (20)

erO =0,

U!k·/

U}i-/

· .. (21)Ulj./

Fig. I-A schematic sketch of a nozzle throat

274 INDIAN J. ENG. MATER. SCt., OCTOBER 1998

It can be written in matrix form as

{u}=[N]{ue} ... (22)

By differentiating equation 21 using the straindisplacement relationship, we get

s.o ) i 0 bk 0

1(e)=-

2A

2A N I 0 2A Nj 0 2 AN k 0r r r

c, b, c , b, c, b,(23)

Table I gives the geometrical parameters of G Iand G2 for typical nozzle throat inserts of a solidrocket motor. The nozzle G 1 was modelled by 456nodes and 784 triangular elements while G2 wasmodelled by 648 nodes and 1166 triangularelements. The non-linear temperature dependentthermo-physical and other material properties ofgraphite were taken from Table 2. The convectiveheat transfer coefficient for the analysis is calculatedfrom Bartz correlation II

0' [ ]o.X[ ]0.'[ ]o.yh = 0.026 j..Jo·C''K p, D,h ~ aD~i/ r-" C' . R' A""

... (3\)and

It can be written in matrix form as Ia= ---------::-=------

{e}=[B] {ue} . (24) J~l~l+r-1 M1]+0.5}1.r..I[I+!-1 M"]0.'5

2rc 2 2The element stiffness matrix can be given as . . . (32)

[K]=[Bf[D][B]Jv dv

[K]=27tA r. [Bf [D] [B]

(25) where ,"4 is Mach number and Tg local gastemperature. The characteristic velocity C', the(26)

The column vector associated with the thermalchange isJv[BHD]{ET}dv=21t rcA[B]T[D]{ET} .. (27)

The integral involving the boundary condition for aload vector can be expressed as

U;l=JdNl'[;}r ... (28)

where pr and pz are the surface stresses in the randz directions.

The element stress components are calculatedafter finding out the nodal displacements by finiteelement method, then by Hooke's law

{cr}=D( {e}-{ET}) ... (29)

The stress components as a function of theelement a nodal displacements and thermal strainvector are

... (30)

Results and DiscussionUsing the above finite element formulation, the

thermostructural problem of a graphite throat nozzleis analysed. The computer programme developed isvalidated for heat transfer and thermal stressesseparately. The validation cases are described indetail elsewhere!".

Table I--Geometrical parameters of nozzle throat inserts

Type and "rop rexit r, r, a pIdentification ofNozzle

Graphite Throat, G I 247 217 165 176 40 15Graphite Throat, G2 238 208 153 150 25 15

Table 2-Properties of graphite':'Property Graphite

Xo +7.07390E+01XI -3.76325E-D2X2 +9.5842IE-D6XJ -8. I8337E-1O

Thermal conductivityK.(n

Thermal conductivityK,(D

-7.10309E+01-3.60873E-D2+7. I4803E-D6+ 1.69792E-1 0

Specific heatC,,(D

-5.991012E+02-4.76525E+00-3.08595E-D3+6.56786E-D7

1700.0487x10-6

DensityCoefficient of thermalexpansionModulus of ElasticityPoisson's ratio

9.20x 10Q0.11

Kz(7)=Xo+XI T+X2r+XJTJ W/mK(

A',(7)=}'o+ }'I T+ r27~+ rJT.l W/mK

Co(7)=Lo+LI T+L27-2+L3TJ .l/kg K

MEHT A et al.: THERMAL STRESS ANALYSIS OF SOLID ROCKET MOTOR 275

Table 3-Properties of combustion products'?

PropertiesChamber Pressure, P; (N/m2)Chamber Temperature, Tc(K)Specific heat Cpg(J/kgK)Ratio of specific heats yCharacteristic velocity C· (m/s)Molecular weight, W (kg/kmol)

HTPB+NH4CI04

4.415E+063410.01716.55

1.\51582.25

29.0

specific heat, Cpg and chamber temperature T; arecomputed from combustion gases in frozencomposition. Chamber pressure P; corresponds tothe test condition and the combustion gasproperties were used for the convective heat fluxcalculation are taken from Table 3.

The temperature distribution on the nozzle wall ofgraphite throat nozzle G I is shown in Fig. 3 and theisothermal contour plot at various locations areshown in Fig 4. The throat region of the nozzle isattained a maximum temperature of 2977 K after thespecified period of performance of the motor. Theconvergent region experiences a temperature of therange of 2825-2950 K and a divergent region ishaving a range of 2925-2700 K. The convectiveheating of the nozzle liner causes, the hightemperature at the inner wall and the maximumvalue of convective heat transfer coefficient is at thethroat region. The minimum temperature is at theouter wall and at the region corresponding to throatand nearby. The maximum throat material at thisregion is the reason behind this minimumtemperature. In a divergent region, this temperatureincreases due to less throat material. The maximumheat transfer coefficient of 14900 W1m K is at thethroat region and minimum at the exit region of thedivergent of the nozzle which are calculatedemploying the above mentioned Bartz's correlation.The heat transfer coefficient increases from theconvergent region up to throat and then decreases.

In nozzle G2, the temperature distribution of thewalls of a nozzle and isothermal contour plots are asin Figs 5 and 6. Here also, the maximumtemperature of 3014 K, which is more than the first

.case, is at the throat region. The minimumtemperature of 303 K is at the outer wall regioncorresponding to the throat area where maximumthroat material exists. The convective heat transfercoefficient plot is similar like G 1, nozzle. Bycomparing nozzle G I and G2, it reveals that the heattransfer behaviour of nozzles are similar.

The model taken for the structural analysis is the

'" 1750<II

~ 1500

i1250

'"' 1000

3000 r----::==.-:::=:::::::------------..,2750

2500

2250

2000

__ nozzle mer Slrface_._nozzle outer surface

750

500

250 -.--.---.-.--.~o ~ __ ~ __ ~ __ ~ __ -L__~ __~ __~~~

-so -100 -50 o 50 100 150 200 250Oistance.mm

Fig. 3- Temperature distribution on the nozzle wall ofgraphite throat nozzle G I

l\T=92

Tmax=2977K

Fig. 4;-Isothermal contour plot for graphite throat nozzle G 1

4000

3500

3000 - -2500

__ Nozzle mer surface

_._ Nozzle outer Slrface

1000

500 -.--.--_._.__._-_.-o ~ __ -L L-__-L L-__-L L-__~

-100 -50 0 50 100 W 200 250

Fig. 5- Temperature distribution on the nozzle wall ofgraphite throat nozzle G2

276 INDIAN J. ENG. MATER. scr., OCTOBER 1998

G I nozzle. The temperature distribution at all nodalpoints determined by thermal analysis is an input forthe structural analysis purpose along with thepressure applied inside the boundary of the nozzle asshown in Fig. 7. The zero displacement on thelateral boundary is considered assuming a rigidassembly of the nozzle insert with the back-upmaterial. The pressure distribution is assumedconstant along the inner wall of the nozzle.However, the calculation can be carried out with thevarying pressure distribution. The boundarycondition for structural analysis is also shown in thesame figure.

The nodal displacement at each node isdetermined in z and r directions. Displacement v istaken zero due to the axisyrnmetry condition.Maximum value of displacement is 0.040697 mm

~T=94

Tmax=3014K

Fig. 6--lsothennal contour plot for graphite throat nozzle G2

noticed at the node, which is the last comer node ofthe inner surface of the nozzle. At this node'-region, the boundary is free to expand. The radialdirection displacement is maximum at the nodalpoint near the throat region and value is 0.12236mm.

The distribution of stresses are obtained such as(Jz, o, and (Js For the sake of the brevity, we arepresenting the o, and (Jz in Figs 8 and 9. In all thethese cases, the compressive stresses are acting onthe inner surface of the nozzle and change to tensi Ietowards the outer wall. It is noticed that the throatregion and the inner surface are subjected to highcompressive stress. The inner surface is under thecompressive stresses and changes to tensile stressestoward the outer surface.

The stress values obtained by the finite element

B

bouncJary whtrt displ.lctmqnt IS as sumeo zero

A

Pr~ssur€ & Ibet ma t loa,<.J

c

Fig. 7-Schematic sketch of a nozzle throat with application ofpressure and thermal load with boundary condition

Fig. 8-Distribution of stress c,

Fig. 9-Distribution of stress cr,

re H.lS-~-Z'.7l.s.; 7.HZ-L- -10.Z3-"- -Z7.70~ -, s . 17~ -1l.64----L- -eO.lO_J._ -97.~7

f -r is .«

...!.2- lB."~ U.7!_8 _ _S.O&l_7 _ _2LH

_1._ _u.eo..5- -70."..k-- -92. S2_3_ -11'"-L- -1-:16.2

, -1~a.l,-

MEHTA et al.: THERMAL STRESS ANALYSIS OF SOLID ROCKET MOTOR 277

IIICSI'S 4.411JIll 9 199113:42:47POSTl STIOSST[P:lITU:1SlI (AU(;)S'LOIA!.1m ~.I22882SIIH :-115.158SIIII:-lSZ.388SIIX :42.177SlIXI:87 .592

ZV :1DlST:I98.776xr :2116tt :-51 :211o -11:1.158D -~7.676

l~ :~:m'0 --45.mI 0 -a7.7~10 -16.268I mJ 7.214;0 24.6~5~ 42.177

!i

y

+"Fig. IO--Distribution of stress SX corresponding to o,

method and the computer code is validated by asoftware ANSYS 4.4, which deals with the thermaland structural problems. The model selected for thevalidation is same as that of nozzle G I. Theboundary conditions applied are also the same. Thestress plots are shown in Fig. 10. Comparing theresults obtained by the computer code and ANSYSprogram, it is understood that they are in goodagreement with each other.

ConclusionsIn this thermostructural analysis of a rocket

nozzle, a numerical method has been developed forsolving an axisymmetric, transient, anisotropic heatconduction equation with temperature dependantthermo-physical properties. For the numericalsimulation finite element method is used due to itsability to handle the arbitrary geometry, initialconditions, boundary conditions and materialproperties. Linear triangular elements have beenused for the space discretization. An implicit timemarching scheme is used for time integration of the

system of first order differential equation. Thedisplacements and stresses formed inside thegraphite throat nozzle is calculated using finiteelement method in conjunction with temperaturedistribution obtained from the thermal analysis. Thecomputer program generated for the solution ofthermal analysis is validated against an analyticalsolution available. The numerical values are foundin good agreement with the analytical solutions. AnANSYS 4.4 software is used to validate thecomputer program for stress calculation. Thecomputed results are found in good agreement withthe software results.

ReferencesI Carslaw H S & Jaeger J C, Conduction of heat in solids

2nd Ed. (Oxford University Press, London), 1959.2 Von Rosenberg D U, Methods for the numerical solution

of partial differential equations (American ElsvierPublishing Co, New York), 1969.

3 Swaminathan V & Rajagopalan S, A mathematical modelfor the heat transfer of rocket motor nozzle walls, inMultiphase flow and heat transfer symposium workshop.(Hemispherical publishing corporation, Washington),1980,527-541.

4 Lee R E, Measurements and correlation of heat transfer ina solid propellant rocket nozzle, NOL TER-64-163, 1964.

5 Henderson H P, Correction of the measured and predictedin-depth thermal response of multidimensional wovencarbon/carbon ITE under static test firing conditions.AIAA paper No. 79-1218, 1979.

6 Segerlind L J, Applied finite element analysis (John Wileyand Sons, New York), 1982.

7 Zienkiewicz 0 C, The finite element methods inengineering sciences (McGraw-HilI. London), 1971.

8 Mehta, R C, Jayachandran, T & Sastri, V M K, Wdrme-und Stroffubertragung, 22 ( 1988) 227.

9 ANSYS 4.4. Software. developed by Swanson AnalysisCorporation, USA.

10 Suresh, K, Thermostructural analysis of a solid rocketmotor nozzle throat insert using finite element method.M.Tech.Thesis, University of Kerala, 1997.

II Bartz, D R, Jet Propulsion, 27 (1957), 49-51.12 Scout Nozzle data book. NASA CR-145136, 1976.13 Gordon, S & Mcbride B, Computer program for

calculation of complex chemical equilibriumcompositions. rocket performance. incident and reflectedshocks and Chapman-Jouget detonations, NASA SP-273.1971.