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Lorenzo ValdevitMechanical and Aerospace Engineering

Department,University of California,Irvine, CA 92697-3975

Natasha Vermaak

Frank W. Zok

Anthony G. Evans

Materials Department,University of California,

Santa Barbara, CA 93106-5050

A Materials Selection Protocol forLightweight Actively CooledPanelsThis article provides a materials selection methodology applicable to lightweight activelycooled panels, particularly suitable for the most demanding aerospace applications. Thekey ingredient is the development of a code that can be used to establish the capabilitiesand deficiencies of existing panel designs and direct the development of advanced mate-rials. The code is illustrated for a fuel-cooled combustor liner of a hypersonic vehicle,optimized for minimum weight subject to four primary design constraints (on stress,temperatures, and pressure drop). Failure maps are presented for a number of candidatehigh-temperature metallic alloys and ceramic composites, allowing direct comparison oftheir thermostructural performance. Results for a Mach 7 vehicle under steady-stateflight conditions and stoichiometric fuel combustion reveal that, while C–SiC satisfies thedesign requirements at minimum weight, the Nb alloy Cb752 and the Ni alloy InconelX-750 are also viable candidates, albeit at about twice the weight. Under the most severeheat loads (arising from heat spikes in the combustor), only Cb752 remains viable. Thisresult, combined with robustness benefits and fabrication facility, emphasizes the poten-tial of this alloy for scramjets. �DOI: 10.1115/1.2966270�

Keywords: active cooling, lightweight structures, sandwich panels, hypersonics, multi-functional optimization, thermal stresses, materials selection

IntroductionComponents that experience extreme heat flux, while simulta-

eously supporting pressure loads, are frequently encountered inerospace and power systems. In some cases, the challenge can beddressed by using an efficient means for spreading the heat andhen convecting or radiating to the environment from a large area.eat pipes are especially effective for this purpose �1�. This strat-

gy is not always viable, whereupon active cooling by a fluidumped through the structure is required. In such cases, beforembarking on materials development and fabrication, it would beost beneficial to have a procedure that simultaneously selects the

referred material and design, while also highlighting the inad-quacies of existing materials. The task is complicated by thentertwining of material properties and geometric parameters.amely, the optimal geometries depend on materials properties inhighly coupled way. The purpose of this article is to describe therinciples governing the development of a code that couplesaterial choices with design parameters and to present an

llustration.The procedure is illustrated for a fuel-cooled combustor liner ofhydrocarbon-powered hypersonic vehicle �Fig. 1� �2,3�. This

hoice is timely because, while the potential to achieve positivehrust from a scramjet has been recently demonstrated �2–4�, se-ecting materials and generating designs that resist the thermome-hanical loads for the duration of a typical mission have proved toe daunting. Some aspects of the design and performance of ac-ively cooled combustion systems have been explored �5–8�, in-luding geometry optimization �9–12�. However, a comprehensivereatment that accounts for the complete set of thermomechanicalonstraints is lacking.

The structure of this article is as follows. A synopsis of thenalysis and optimization protocol is outlined in Sec. 2. Analytical

Contributed by the Applied Mechanics Division of ASME for publication in theOURNAL OF APPLIED MECHANICS. Manuscript received February 24, 2008; final manu-cript received June 6, 2008; published online August 22, 2008. Review conducted
y Martin Ostoja-Starzewski.
ournal of Applied Mechanics Copyright © 20

om: https://appliedmechanics.asmedigitalcollection.asme.org/ on 02/22/20

models for temperature distributions and thermomechanicalstresses are presented in Secs. 3 and 4. Also included are theresults from computational fluid dynamics �CFD� and finite ele-ment �FE� calculations, designed to critically assess the accuracyof the model predictions and the key underlying assumptions. For-mulation of the optimization scheme and its application to a com-bustor liner of a notional Mach 7 scramjet vehicle are contained inSec. 5: inclusive of an assessment of the suitability of a widerange of candidate structural materials. The implications for ma-terials selection follow. For facility of presentation, the analyticdetails are presented in Appendixes.

2 Principles and ProceduresA prototypical combustor wall for an aerospace system �Fig. 1�

comprises a sandwich plate subject to three loading mechanisms:external pressure from the combustion gases, internal pressurefrom the coolant, and thermal loads due to the temperature differ-ences between the combustion side and the vehicle exterior. Inaddition to the obvious thermostructural requirements �no meltingand no yielding/fracture�, the design may be limited by fuel-specific constraints �e.g., avoiding coking while promoting crack-ing� and the need to limit pressure losses in the cooling system.

A variety of shapes can be envisioned for the cooling ducts.Rectangular, triangular, or rhombic cross section can be manufac-tured to ensure thin walls and are easiest to model analytically.The present study focuses on rectangular ducts. Extension to otherperiodic shapes is elementary and is not expected to modify themain conclusions.

The protocol employed for thermostructural analysis and designoptimization consists of the following steps �Fig. 2�. �i� A range isdefined for the expected heating loads �represented by the heattransfer coefficient hG of the hot gases� and the cooling efficiency

�represented by the coolant flow rate per unit width of panel,V̇eff�.�ii� A candidate material is selected and its physical and mechani-cal properties either measured or obtained from handbooks. �iii�At each point in �hG , V̇eff� space, the design parameters are sys-
tematically varied over a prescribed range and the temperatures
NOVEMBER 2008, Vol. 75 / 061022-108 by ASME

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nd stresses calculated for each combination. Upon comparisonith material and coolant properties, the viability of the design is

scertained. �iv� provided that solutions exist, the design is opti-ized for minimum mass, subject to a number of design con-

traints. Otherwise, if a solution is not found, the point �hG , V̇eff� iseemed external to the design space. �v� Once the entire designpace has been scanned for each candidate material, comparisonsre made of materials on the basis of structural robustness

namely, the extent of feasible solution area in �hG , V̇eff� space�nd weight efficiency.

Temperatures in the panel have been derived using a two-imensional resistance network model and the solutions verifiedy the CFD and FE calculations. The utility of the temperatureredictions is twofold. First, they are used to ensure that the con-itions remain within allowable limits for the material and theoolant. Second, they become input for calculation of thermaltresses. To permit formulation of the structural constraints, thesetresses are superimposed on those induced by the pressure loads,oth external to the liner �inside the combustion chamber� andithin the cooling channels. The thermomechanical stresses are

equired to remain below the local temperature-dependent mate-ial strength. A constraint on pressure drop is also imposed.

Front view

(b)

(a)

H pf0

tcL

w

pcomb

b

Fig. 1 „a… Artist rendition of a proto„b… Schematic of actively cooled pane

The assessment facilitates three goals. �i� It determines the rela-

61022-2 / Vol. 75, NOVEMBER 2008


tive merits of representative categories of high-temperature mate-rials �Tables 1 and 2�, inclusive of refractory alloys and ceramicmatrix composites �CMCs�. �ii� It provides a focus for the devel-opment of advanced materials that outperform existing options.�iii� It assesses the possible benefits of superposing a thermal bar-rier coating �TBC�, such as yttria-stabilized zirconia �YSZ�, mo-tivated by the extensive use of such coatings in aeroturbines �13�.

3 Temperature DistributionTo obtain analytic estimates of the temperatures, three simpli-

fications are invoked. �i� The top face of the panel is exposed tohot gases at a uniform adiabatic wall temperature Taw and constantheat transfer coefficient hG, whereas the bottom face and the sidesare thermally insulated. Consequently, all of the heat passedthrough the top face is carried away by the cooling fluid. �ii� Noheat is conducted along the length of the panel in either the struc-ture or the coolant. This assumption results in slightly conserva-tive temperature estimates. �iii� The coolant temperature is uni-form at each cross section, increasing monotonically with distancez along the panel length from an initial value Tf

o at the channelinlet to its maximum Tf

max at the outlet.The thermal resistance network is illustrated in Fig. 3 and the

Side view

Combustion chamber

tfx

y

z

Tf0

Tf(Z)

Taw

hG

Z

ical hypersonic air-breathing vehicle.ith thermostructural loads.

typl w

solutions for the key temperatures are detailed in Appendix A. The

Transactions of the ASME


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ain goals are to ascertain the maximum temperatures in the ma-erial Tm

max and in the fuel Tfmax. Additionally, for the ensuing de-

ermination of thermal stresses, four temperature differences areefined. The first two are those across the top face �in the-direction�: one directly above the core web, �Ttf

c , and the sec-nd at the midpoint between a pair of webs, �Ttf

w. The second two,Tpanel

c and �Tpanelw , are those across the entire panel �also in the

-direction�, measured from the middle of the top face to theottom face, directly above and between the core webs. Thenalysis indicates that all the preceding temperature differencesre greatest at the inlet �z=0�.

CFD calculations, performed using the commercial code,

able 1 Approximate chemical compositions of the candidateetallic alloys

aterial Approximate chemical composition �wt %�

nconel 625 Ni–20% Cr–10% Mo–5% Fe–3% Nbnconel X-750 Ni–15% Cr–7% Fe–2.5% Tii 6Al 4V Ti–6% Al–4% Vi � 21S Ti–15% Mo–2.7% Nb–3% Al–0.2% SiARloy-Z Cu–3% Ag–0.5% ZrRCop-84 Cu–6.5% Cr–5.8% Nbb-Cb752 Nb–10% W–2.5% Zr

Table 2 Thermal and mechanical

MaterialT*�K�

�Y �Tf0�

�MPa�d�Y /dT�MPa/K�

Inconel 625 1100 427 −0.31Inconel X-750 1100 795 −0.39Ti–6Al–4V 675 909 −0.83Ti-�-21S 815 1222 −1.46NARloy-Z 811 99 −0.009GRCop-84 973 205 −0.18Nb–Cb752 1470 382 −0.17SiC–SiC 1640 200 —C–SiC 1810 200 —TBC �ZrO2�

Define a range for:- thermal loads (hG)- cooling efficiency (Veff)

Choose a material

Choose (hG , Veff)

Calculate:- temperatures- stresses

- Verification (FE+CFD)- Validation (Experiments)

Optimize geometrysubject to design constraints

DESIGNMAPS

COMPAREMATERIALS

Fig. 2 Schematic of the materials selection procedure

ournal of Applied Mechanics


FLUENT©, were used to assess the uniformity of the heat transfercoefficient around the internal surface of the channel and the ef-fect of longitudinal conduction. The panel used for these calcula-tions has a near-optimal geometry �detailed in Sec. 5� and madefrom Inconel X-750 �a Ni-based superalloy�. Approximate chemi-cal compositions and pertinent material properties are summarizedin Tables 1 and 2. The coolant is taken to be JP-7 jet fuel �Table3�. Both the fuel and the solid are meshed using three-dimensionalelements. The fuel flow rate and hot gas temperature are selectedto be representative of a notional Mach 7 cruise vehicle �Sec. 5�.The results �Fig. 4�a�� affirm that hC is essentially uniform overthe interior surface of the top face, where the vast majority of theheat is transferred to the fuel. The variations around the cornersand along the core and bottom face are deemed unimportant, be-cause the heat transfer averaged over the channel perimeter con-forms to the value obtained from established correlations �14�,with an accuracy of about 10%. Additionally, the axial distributionof the section-averaged hC confirms that full thermal and kine-matical developments are attained after the fuel has traveled adistance of a few hydraulic diameters �Fig. 4�b��. Effects of lon-gitudinal conduction within the solid were ascertained by compar-ing CFD calculations with and without axial conduction.For the parameter values selected, the two sets of results areindistinguishable.

A further assessment of the predicted temperatures was madethrough FE calculations of the same panel, performed using theABAQUS© code. The mesh consists of quadratic generalized planestrain elements with reduced integration �CPEG8RHT�. Convec-tive boundary conditions are applied both to the top face �hG=445 W /m2 K, Taw=3050 K� and the internal channel surfaces�hC=2266 W /m2 K, Tf =653 K�. The fuel temperature corre-sponds to the predicted exit temperature for the relevant geometryand boundary conditions, assuming an entry fuel temperature Tf

o

=400 K. The remainder of the cell perimeter is thermally insu-lated.

The steady-state temperature distribution at the channel outletand the corresponding analytic predictions at eight critical loca-tions are shown in Fig. 5. The comparisons reveal that the maxi-mum temperature in the structure is captured to within 1% accu-racy. Moreover, the temperature differences that drive the thermalstresses ��Tpanel= ��Tpanel

c +�Tpanelw � /2 and �Ttf =�Ttf

c , see Appen-dix B for details� are also predicted adequately �within about 8%�.FE calculations for other panel designs and material propertiesyielded similar consistency between the numerical results andanalytic predictions.

4 Stress DistributionsStress estimates were obtained using standard concepts of plate

bending and stretching and assuming the materials to be linearelastic. Derivations and solutions are in Appendix B. In practice,some nonlinearity may occur in the most highly stressed locations,enabling stress redistribution and shakedown �15,16�. Conse-

perties of the candidate materials

E�GPa�

CTE�10−6 /K�

ks�W/m K�

�s�kg /m3�

164 14.0 20.0 8440128 16.0 23.0 827690 10.0 11.0 4430100 10.3 21.0 4940125 17.0 350.0 913090 19.0 285.0 8756110 7.4 50.0 9030240 4.1 25��, 20 �� 2900100 2.0 15��, 5 �� 2000

1.0 3000

pro

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uently, in the absence of high cycle fatigue, the ensuing resultsre conservative. A subsequent article will incorporate yieldingnd shakedown and provide an assessment of the extent of theonservatism. Although the present analysis is for a flat panel, itsxtension to cylindrical configurations is straightforward.

4.1 Boundary Conditions. Two idealized sets of boundary

Table 3 Physical pr

Fuel kf �W/m K� � f �Pa s� cp

JP-7 0.11 1.984�10−4

Rh

qw(z) qc(z)

qh(z)

Taw

Tf(z)

R1

R2w

Ttfw(z)

(a)

(b)

qc(z)qw(z)

qw(z)+2 qh(z) tf/w

qh(z)

Rh

RcoolR

planes of symmetry

R

RRTBC

RG

q(z) Taw

TBC

qc(z) - 2 qh(z) tf/tc

Tf(z) T

qc(z)

qh(z)

Rfin

RG

RTBC

Tf(z)

Ttfw(z)

R1

R2c

Ttfc(z)

Rh

qh(z)

qc(z)

Rface / 2

Rface / 2

qw(z)++2 qh(z) tf/w

qc(z)+-2 qh(z) tf/tc

qc(z)+-2 qh(z) tf/t

Rface / 2

Rface / 2 R

RTtfc(z)

Fig. 3 „a… Thermal resistance network used to dpressions for all relevant thermal resistances. „b

0 1.00.2 0.4 0.6 0.8

-6

-8

-4

-2

0

2

4

8

6

Non-Dimensional Perimeter Position, ξ

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Hea

tTr

ansf

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)

NumericAverag

Gnielinski

ξ=1

ξ=0

(a)

Fig. 4 Distribution of the heat transfer coefficiesimulation of a near-optimal Inconel X-750 panel.eter at z /Z=0.9. „b… Variation of cross-section avtracted from the Gnielinski correlation „Eq. „A1…
comparison.
61022-4 / Vol. 75, NOVEMBER 2008


conditions are considered.I. Linear frictionless supports along the edges in the z-direction

(Fig.6(a)). This constraint prevents bending in the z-direction,while allowing it in the x-direction �albeit with no rotation at theends�. Uniform thermal expansion is allowed along all directions.The analog for a cylinder would be the absence of constraint on

rties of JP-7 jet fuel

kg K� Prf � f �kg /m3� Tcoke �K�

75 4.64 800 975

1

2c

tfc(z)

R1= R

G+ R

TBC+ R

f/ 2

R2w= R

f/ 2+ R

cool

R2c= R

f/ 2+ R

fin

⎧

⎨⎪⎪

⎩⎪⎪

RG = 1 / hG

RTBC = tTBC / kTBC⊥

Rface = t f / ks⊥

Rh = w+ tc / 2( ) / 4 ks�

Rcool = 1 / hc

Rfin = tanh−12hcks�tcL2

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

2hcks�tcks�

⎡

⎣⎢⎢

⎤

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−1

2

2

rmine temperature distributions, along with ex-fective network.

(b)

Non-Dimensional Axial Position, z / Z0 0.2 0.4 0.6 0.8 1.0

4

3

5

6

2

Ave

rag

eIn

tern

alH

eat

Tran

sfer

Co

effic

ien

t(k

W/m

2 K)

GnielinskiNumerical Results

0

1

in the cooling channel extracted from the CFDVariation of pointwise hC around channel perim-ged hC along the axial direction. The value ex-

nd used in the analytical model is depicted for

ope

�J/

25

R

R

T

fin

G

TBC

f(z)

c

face /

face /

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hanges in both the diameter and axial length. This boundary con-ition can also represent multiple linear frictionless supports �Fig.�b�� by simply re-interpreting the panel width b as the supportpacing.

II. Two-dimensional continuous bed of rollers (Fig. 6(c)). Uni-orm thermal expansion is permitted in all directions. The externalressure does not cause panel-level bending but the internal pres-ure can bend individual face segments.

The use of rollers instead of frictional supports allows uniformhermal expansion of the panel �with no bending�. While the prac-ical implementation may be challenging, attaining these condi-ions is essential to viable solutions. Otherwise, if the plate islamped on all sides, the maximum temperature increase that cane sustained without yielding is only �Tmax= �1−��Y /E� �withbeing Young’s modulus, � the thermal expansion coefficient, �Y

he material yield strength, and � Poisson’s ratio� �17�, well belowhe upper use temperature of all of the materials �Fig. 6�d��.

Both the pressure drop and the temperature variation along theanel length have been neglected. This assumption, combinedith the imposed boundary conditions, ensures that generalizedlane strain conditions are attained along the z-direction.

4.2 Failure Locations. Although the temperature differences,nd hence the thermal stresses, are greatest at the channel inlet,he material strength is also greatest at this location. Typicaltrength reductions with increasing temperature suggest the possi-ility of preferential failure at the outlet, where the temperature ist its maximum. To ensure accurate prediction of failure initiation,

Temp (K)

650678705733761789816844872900927955983

Tfuel

= 653 K

(987)983

(972)968

(664)657

(664)657

(analytic)FE

Tpanel =(261)285Δ

Fig. 5 Comparison of analytical and numericInconel X-750 panel. All temperatures are in Kdictions. Both the maximum temperature in thdrive the thermal stresses are captured accur

hermal stresses should be ascertained at each cross section and



compared with material strength at the pertinent �local� tempera-ture. Additionally, the stresses due to pressure loads vary withlocation within the same cross section and can be of opposite signrelative to those caused by thermal loads. Thus, establishing apriori the failure location is not straightforward.

To address this problem, a set of 18 “critical points” has beenidentified for close scrutiny, clustered around two failure-susceptible channels: one at the periphery closest to the supportsand the other at the center �Fig. 7�. Failure of the structure isaverted provided that, for metallic alloys, the Mises stress at eachpoint remains below the elastic limit. The analog for CMCs isbased on a maximum or minimum principal stress criterion. Theinternal pressure in the core channels �which induces large tensilestresses in all members� combined with the relatively stubbyshape of the optimized members make it unnecessary to designagainst buckling �18–26�.

The accuracy of the stress predictions was verified by FE cal-culations. Illustrative results are presented for the optimized In-conel X-750 panel subject to the thermal loads described in thepreceding section. The calculations use type II boundary condi-tions and an internal fuel pressure of 4 MPa �The pressure in thecombustion chamber can be neglected since it has minimal effecton the stresses for the selected boundary condition.� The bottom isconstrained against translation in the y-direction and periodicboundary conditions are imposed on the vertical sides �one side isconstrained against translation in the x-direction, whereas allnodes on the other are required to displace equally in the

ΔTtf =(14)15

(881)924

(845)893

(664)660

(664)660

Normalized distance along fin

Tem

per

atu

re(K

)

600

700

800

900

1000

0 0.2 0.80.4 0.6 1

FE

analytic

„FE… temperature distributions for an optimalin. Values in parentheses are analytical pre-tructure and the temperature differences thatly.

alelv

e sate

x-direction�. The top is traction free.

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(a)

(b)Te

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erat

ure

diff

eren

ce,�

T(K

)

0

200

400

600

800

1000

1200

1400

1600

Alloy

Inconel625

MARM246

NbCb752

SiC-SiC C-SiC

�Tyield�Tmaterial

(d)

(c)

Fig. 6 Mechanical boundary conditions. †„a… and „b…‡ Linear rollers on twosides „Type I…. „b… Multiple linear rollers with regular spacing. „c… Uniform two-dimensional bed of rollers, with impeded rotation at the ends „Type II…. „d…Benchmark boundary condition: plate sitting on rigid foundation „inset…. Thechart compares the temperature increase from room temperature to the mate-rial upper use limit needed to cause yielding. Under this boundary condition,
the full high-temperature potential of the materials is not exploited.
61022-6 / Vol. 75, NOVEMBER 2008 Transactions of the ASME

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The resulting distribution in the Mises stress is plotted on Fig.. Also shown are comparisons between the numerical results andhe analytic predictions along the four critical trajectories, corre-ponding to the external and internal surfaces of both face sheets,or thermal, mechanical, and combined thermomechanical load-ngs. The analytic prediction is accurate to within 1% at the mostighly stressed location �Point 1� and within �10% at other loca-ions on the top face. The corners �Point 2�, where stress intensi-cation is evident, are exceptions. This discrepancy has not beenursued for several reasons. �i� For this particular simulation, be-

b

1

2

3

5

4

6

7

8

9

10

11

12

13

14

15

16

17

18

ig. 7 Unit cells susceptible to local yielding and the 18 criti-al points

Mises S

4912233445567

0 0.5 1 1.5 20100200300400500600700800

VonMise

sStress

(MPa)

Distance (mm)8 4

AnalyticalNumerical

5

6

7

8

0 0.5 1 1.5 20

100

200

300

400

500

600

700

800

total

VonMise

sStress

(MPa)

Distance (mm)5 1

thermalmechanical

total

thermal

mechanical

Fig. 8 Comparison of analytical and numerical „FE…X-750 panel. The plots show the results for thermal,along the four paths. With the exception of Points
agreement is satisfactory, particularly on the top face, w


cause of the relatively low temperature at the corners �Fig. 5�, theyield strength is high �Table 2� and, given the stress distribution�Fig. 8�, both corners remain elastic. We speculate that this con-cept generalizes to metallic systems, but a formal proof requiresfurther analysis. �ii� Even if localized plasticity at the cornerscould not be avoided, the metallics can be readily designed toassure shakedown, wherein local plasticity occurs only during thefirst few thermomechanical cycles. Such a solution may not bepossible for CMCs. �iii� The stress intensification can be reducedby increasing the fillet radius without imposing a significantweight penalty.

The agreement on the bottom face is somewhat worse ��20%at Point 3�. This result is implicit in the model, which underesti-mates the thermal stress in the bottom face to ensure conservativestress estimates on the top face. This choice was made because thecritical condition �yielding or fracture� typically occurs in the topface. Comparisons performed for other materials and geometriesshowed similar correlations.

The numerical calculations confirm that the combined thermo-mechanical stresses are not necessarily the most dangerous. AtPoint 5, for instance, the thermal stresses alone are greater thanthose under thermomechanical loading.

5 Materials Selection for Scramjet Combustor LinersThe materials selection procedure exposed in Sec. 2 is applied

to the combustor liner of a Mach 7 scramjet cruise vehicle oper-ating with JP-7 jet fuel. The choice is motivated by the realization

s (MPa)

0 0.5 1 1.5 20100200300400500600700800

VonMise

sStress

(MPa)

3Distance (mm)7

0 0.5 1 1.5 20100200300400500600700800

VonMise

sStress

(MPa)

Distance (mm)6 2

1

2

3

4

total

thermal

mechanical

total

thermalmechanical

n Mises stress distributions for an optimal Inconelchanical, and combined thermomechanical stressesnd 3, clearly affected by stress intensification, the

tres

165106611671268136914601

vome2 a

here the highest stresses generally occur.

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hat the design of vehicles in this velocity range is most maturend in urgent need of technological advancements in high-emperature materials and structures.

5.1 Thermomechanical Loads. The thermomechanical loadsn the combustor liners depend on nearly every aspect of theehicle design, including the size and shape of the compressionamp, the size of the combustion chamber, details of the injectionystem, and combustion efficiency. For the present illustration, theehicle is assumed to be 10 m long and 2.5 m tall, with a bilinearompression ramp that generates three oblique shocks. The perti-ent aerothermodynamic conditions for the prescribed vehicle ve-ocity and optimal flight altitude are detailed in Ref. �3�. Fuelnters the cooling channels with pressure pf

o=4MPa and initial

emperature Tfo=400 K. On the combustion side, the pressure

comb=0.16 MPa, the adiabatic wall temperature Taw=3050 K andhe heat transfer coefficient hG=445 W /m2 K. To assess the ef-ects of potential heat spikes in the combustion chamber, heatransfer coefficients up to 1800 W /m2 K are considered. For thetress analysis, Type I boundary conditions are used with an un-upported span, b=0.5 m.

For stoichiometric combustion, the fuel flow rate �per unit˙ 2

445

1600

1400

1000

800

600

1200

445

1600

1400

1000

800

600

1200

445

1600

1400

1000

800

600

1200

1800

2 3 4 5 6

Coolant flow rate

HeatTransferC

oefficien

t,h G

(W/m

2 K)

Uncoated

Coated

No feasiblesolution

No feasible solution

Uncoated

Coated


Uncoated

Coated

Equivalenc1.0 1.5 2.0

Inconel X-75

Nb-Cb7

NARloy

Fig. 9 Design maps for several matewithout a TBC. The normalizing paramand the heat transfer coefficient „hG /hMach 7 flight conditions.

idth of combustor� is Vst=0.008 m /s. Since the total perimeter

61022-8 / Vol. 75, NOVEMBER 2008


of the combustor liner is 2�b+Hcomb�, with Hcomb the height of the

chamber, the effective flow rate per unit width of liner is V̇steff

= V̇stb /2�b+Hcomb�. Upon specifying the dimensions �b=0.5 m

and Hcomb=15 cm�, then V̇steff=0.003 m2 /s. To address offstoichio-

metric combustion, an equivalence ratio is introduced, definedby = f / fst, where f is the actual fuel-to-air mass ratio and fst isthe corresponding stoichiometric value. The actual flow rate then

becomes V̇eff=V̇steff. The range 0.62.5 is used for subse-

quent calculations.

5.2 Design Constraints. A candidate design is deemed ac-ceptable provided it satisfies four principal constraints: �i� thestresses induced by the pressure and the thermal loads remainbelow representative levels of material strength, �Y; �ii� the maxi-mum material temperature Tm

max does not exceed the upper uselimit, T*; �iii� the fuel temperature remains below that for coking�Tcoke=975 K �27��; and �iv� the pressure drop through the chan-nels is acceptably low ��p�0.1 MPa�. Additionally, to ensurethat designs can be manufactured, secondary constraints are im-posed on some dimensions, notably: channel width w�2 mm,channel height L�5 mm, face and core wall thicknesses tc and

2 3 4 5 6 7

idth, Veff (10-3m2/s)


Coated


Uncoated

Coated


C-SiC

Relativ

eHeatTransferC

oefficien

t,h G

/hGno

m2

4

3

1

2

3

1

atio, φ = f / fst

2

1

3

Ti β 21 S

Uncoated

GRCop-84

1.0 1.5 2.0

ls considered in this study, with andrs for the equivalence ratio „�= f / fst…

… are those expected for steady-state

7

/w

e r

0

52

-Z

riaete

Gnom

tf �0.4 mm, and TBC thickness�0.3 mm.



lat

dittgIv

wpaaphc

r�efi

J

Downloaded Fr

For metallic alloys, the yield strength is assumed to decreaseinearly with temperature for T�T*. For the CMCs, the tensilend compressive strengths are assumed equal and independent ofemperature. Caveats on this choice are discussed later.

5.3 Optimization Scheme. Whenever a solution exists, theesign was optimized for minimum weight. The mass of the TBCs included to ensure that a finite layer emerges only if it reduceshe overall weight. Numerical optimizations were performed usinghe quadratic optimizer MINCON in MATLAB™. Several randomlyenerated initial guesses were used to escape from local minima.n some cases, a manual optimization scheme was employed toerify the accuracy of the numerical results.

5.4 Principal Results and Interpretation. The procedureas implemented for a suite of high-temperature materials. Ap-roximate chemical compositions of the candidate metallic alloysre listed in Table 1; relevant mechanical properties �for metallicnd ceramic systems� are summarized in Table 2. Design maps areresented in two formats. �i� In the first �Fig. 9�, the ordinate isG, motivated by the appreciation that shocks passing through theombustor can cause local elevations. The abscissa is the fuel flow

ate, V̇eff. The normalizing parameters for the equivalence ratio= f / fst� and the heat transfer coefficient �hG /hG

nom� are thosexpected for steady-state Mach 7 flight conditions. The map speci-

Weigh

t/Area(kg/m

2 )

0

20

40

60

80

Coolant flow r

2 3 4 5 6

0

20

40

60

80

Equiva

1.0 1.5 2.0100

NAR

GRCop-8

MAR-M246Nb-Cb752

C-SiC Ti β

Inco X-750

Nb-Cb752

GRC

(c)

(a) hGUncoated

hUncoated

Fig. 10 Minimum weight comparison at twand „c… and „d… 890 W/m2 K. †„a… and „c…‡coated materials whereas †„b… and „d…‡ tmaterials.

es domains within which the material can function, with and



without a TBC, as well as a domain of inadmissibility. In thisscheme, the weight is a function of the location on the map. �ii� Inthe second �Fig. 10�, hG is fixed �at either 445 W /m2 K or890 W /m2 K�, the ordinate is the weight of the optimized paneland the abscissa is again the fuel flow rate.

The overarching implications from Fig. 9 are as follows. �i� Inall cases where solutions are obtained, increasing the flow rate isbeneficial, indicating that the cooling efficiency limits the design�as opposed to the fuel pressure drop�. �ii� Among the selectedmaterials, most provide a solution for the nominal Mach 7 condi-tions �hG=445 W /m2 K�. The exceptions are Inconel 625, Ti–6Al–4V, and SiC–SiC, which are not viable anywhere within thedesign space. �iii� The outcome changes radically if the heat loadis doubled. Namely, for hG=890 W /m2 K, only the Nb alloyCb752 and the Cu alloy GRCop-84 are viable without a TBC�albeit an environmental barrier coating will be needed to avertoxidation �28��. Furthermore, Cb752 is the only material that cansurvive without a TBC at near-stoichiometric fuel flow rates �=1�. �iv� The operational design space of essentially all metallicscan be increased by using a TBC, although the benefit differsamong materials: Cb752, Inconel X-750, and GRCop-84 showingthe largest advantage.

Figure 10 compares the optimal panel weights for cases withand without a TBC. For hG=445 W /m2 K, C–SiC offers the light-

2 3 4 5 6 7

/Width, Veff (10-3m2/s)

7

ce ratio, φ = f / fst1.0 1.5 2.0

Z

C-SiC Ti β 215

Inco X-750

Narloy-Z

GRCop-84MAR-M246

Nb-Cb752

With a TBC layer (b)

Nb-Cb752

GRCop-84

Inco X-750

(d)

4

445W/m2K

With a TBC layer890W/m2K

levels of heat transfer: „a… and „b… hG=445solid lines represent the results for un-

dashed lines are those for TBC coated

ate

len

loy-

4

215

op8

=

G =

othehe

est structure �by a factor greater than 2 relative to most metallic

NOVEMBER 2008, Vol. 75 / 061022-9


aaprewpntccsoiwt

ntssoatay1fiOli

6

ccastTm

0

Downloaded Fr

lloys�, followed by Inconel X-750 and Cb752. Ti-�21S provideslightweight alternative but only when high fuel flow rates are

ermitted. For hG=890 W /m2 K, the only viable uncoated mate-ials are Cb752 and GRCop-84. However, these two materialsxhibit vastly different weight efficiencies: Cb752 is half theeight of GRCop-84. When a TBC is used, the selection is ex-anded to include Inconel X-750. �Perhaps surprisingly, C–SiC isot viable at this heat flux, with or without a TBC, because bothhe thermal and the pressure-induced stresses on the hot face ex-eed its compressive strength�. A caveat to this outcome is dis-ussed below. Although a TBC on the Cb752 would enable alightly lighter structure, its use would be predicated on the trade-ff between weight savings and added cost. Finally, while increas-ng the fuel flow rate generally results in lighter structures, theeight savings is unlikely to be significant enough to overcome

he weight penalty associated with the extra fuel.The design maps of Fig. 9 and the weight analysis of Fig. 10 do

ot divulge the significant amount of valuable information con-ained in the code about optimal geometries, temperatures,tresses, and the relative importance of the various design con-traints. Complete description of these results is beyond the scopef this paper but will be presented in subsequent more detailedssessments. One notable observation is that, for essentially all ofhe materials and design space, the thermomechanical constraint islways active. That is, the design is limited by the occurrence ofielding or fracture. This feature is illustrated for Cb752 in Fig.1, expressed in terms of constraint activity parameters, �de-ned in such a way that unity signals activation of the constraint�.ccasionally, other constraints are also active. For example, at

ow fuel flow rates and high heat flux, the maximum temperaturesn both the structure and the fluid reach their allowable limits.

ConclusionsA materials selection strategy has been presented for actively

ooled panels, with implications for aerospace structures. The pro-edure encompasses a geometry optimization tool coupled withnalytical models for temperatures and thermomechanicaltresses. A thermal network approach has been used to derive theemperature distribution, accounting for the possible presence of aBC. A sandwich panel analysis has been adopted for the thermo-

1.0

0.8

0.6

0.4

0.2

0

Yielding (mech. and combined loads)

Yielding (thermal loads)

Fuel coking

Metal softening

Excessive pressure drop

Constraint

activ

itypa

rameter,Π

hG = 445W/m2K

(a)

Coolant flow rate/width, Veff (m2/s)

0.002 0.003 0.004 0.005 0.006 0.007

Equivalence ratio, φ = f / fst1.0 1.5 2.0

Fig. 11 Constraint activity map for Cb752 forConstraint is active when its activity parametertherm-omechanical constraints are active „yieldinthermal load is doubled, at low flow rates, both thmum allowable value.

echanical stress calculations. The accuracy of the model predic-

61022-10 / Vol. 75, NOVEMBER 2008


tions and the underlying assumptions has been verified numeri-cally, employing a combination of FE and CFD calculations.

The methodology has been applied to the combustor liner of aMach 7 hydrocarbon-powered vehicle. Realistic operating condi-tions have been estimated based on established aerothermodynam-ics considerations �3�. Many of the candidate materials presentfeasible designs for at least a limited range of operating loads,representative of steady-state flight conditions and stoichiometricfuel combustion. However, the suite of material options is sensi-tive to the operating loads and the permitted fuel flow rate. In thepresent illustration, only Cb752 is viable at the highest heat loadand under stoichiometric fuel combustion. This result, combinedwith robustness benefits and fabrication facility, emphasizes itspotential for superior performance, subject to the proviso that oxi-dation is averted through the use of environmental barrier coat-ings. For higher fuel flow rates and/or the addition of a TBC,GRCop-84 and Inconel X-750 become viable, although Cb752remains the most weight efficient.

Finally, since the thermomechanical constraint is almost alwaysactive in the optimized designs, elevations in the high-temperaturestrength of the candidate alloys would yield direct benefits inweight efficiency. Furthermore, based on observations that C–SiCcan sustain larger temperature gradients than the present modelassumes �29�, it should re-emerge as a viable candidate for themore severe thermal environments once a revised local-basis fail-ure criterion has been established.

AcknowledgmentThis work was supported by the ONR through a MURI pro-

gram on Revolutionary Materials for Hypersonic Flight �ContractNo. N00014-05-1-0439�. The authors are thankful to David Mar-shall of Teledyne, and Thomas A. Jackson and William M. Ro-quemore of AFRL for insightful discussions.

NomenclatureAf � cross-sectional area of the face in a unit cell

�m2�Ac � cross-sectional area of the core web in a unit

cell �m2�b � combustor width �m�

1.0

0.8

0.6

0.4

0.2

0

Yielding (mech. andcombined loads)

Yielding (thermal loads)

Fuel coking

Metal softening

Excessive pressure drop

Constraint

activ

itypa

rameter,Π

hG = 890W/m2K

(b)

0.002 0.003 0.004 0.005 0.006 0.007

Coolant flow rate/width, Veff (m2/s)

Equivalence ratio, φ = f / fst1.0 1.5 2.0

values of „a… 445 W/m2 K and „b… 890 W/m2 K.ches unity. At the lower thermal load, only thender mechanical and combined loads…. When theolid and fuel temperatures approach their maxi-

hGreag ue s

cp,f � specific heat of the coolant �J/kg K�



J

Downloaded Fr

Dh � hydraulic diameter of the cooling ducts �m�E � Young’s modulus �Pa�f � friction factor

� fuel/air mass ratiofst � stoichiometric fuel/air mass ratioH � panel thickness �m�

Hcomb � combustion chamber height �m�hG � heat transfer coefficient on the combustor side

�W /m2 K�hG

nom � heat transfer coefficient on the combustor sidefor a notional Mach 7 vehicle �W /m2 K�

hC � heat transfer coefficient on the cooling channelwalls �W /m2 K�

ks� � through-thickness thermal conductivity of the

material �W/m K�ks

�� in-plane thermal conductivity of the material

�W/m K�kf � thermal conductivity of the coolant �W/m K�

kTBC � through-thickness thermal conductivity of theTBC �W/m K�

L � height of cooling channel �m�pf � pressure in the coolant �Pa�pf

0 � entry pressure in the coolant �Pa�pcomb � pressure in the combustion chamber �Pa�

Pr � Prandtl numberqw � heat flux into the web �W /m2�qc � heat flux convected from the top face into the

coolant �W /m2�qh � horizontal heat flux in the top face �W /m2�

RG � external convective thermal resistance�W /m2 K�−1

Rcool � internal convective thermal resistance�W /m2 K�−1

RTBC � conductive thermal resistance across the TBC�W /m2 K�−1

Rface � conductive thermal resistance across the topface �W /m2 K�−1

Rh � conductive thermal resistance along the topface �W /m2 K�−1

R1 ,R2w ,R2

c � combination of thermal resistances�W /m2 K�−1

Rw* ,R

c*,R

h* � non-dimensional combination of thermal

resistancesRe � Reynolds number

Taw � adiabatic wall temperature in the combustor�K�

Tf � coolant temperature �K�Tf

0 � entry coolant temperature �K�Tf

max � maximum coolant temperature �K�Tm

max � maximum temperature in the material �K�Ttf

w � temperature on the top side of the top face,over a web �K�

Ttfc � temperature on the top side of the top face

away from a web �K�T�i� � temperature at a location i in the material �K�T* � maximum allowable temperature in the mate-

rial �K�Tcoke � coking temperature of the coolant �K�

tf � face sheet thickness �m�tc � core web thickness �m�u � coolant velocity �m/s�

V̇eff � volumetric fuel flow rate per unit width of thepanel �m2 /s�

V̇st � stoichiometric volumetric fuel flow rate perunit width of the combustor �m2 /s�



V̇steff � stoichiometric volumetric fuel flow rate per

unit width of the panel �m2 /s�w � width of cooling channel �m�Z � panel length �m�� coefficient of thermal expansion of the material

�K−1��p � viscous pressure drop across the panel �Pa�

�Tpanelc � temperature difference across the panel away

from the core web �K��Tpanel

w � temperature difference across the panel abovea core web �K�

�Tpanel � relevant temperature difference across thepanel �K�

�Ttfc � temperature difference across the top face

away from the core web �K��Ttf

w � temperature difference across the top faceabove a core web �K�

�Ttf � relevant temperature difference across the topface �K�

x, y, z � geometric coordinates � equivalence ratio� f � kinematic viscosity of the coolant �m2 /s�v � Poisson’s ratio of the material

�Y � yield strength of a metallic material �Pa��ult � ultimate stress for a CMC �Pa�

�core,ypf � stress in the core web along the y-direction

due to the pressure pf �Pa��core,z

pf � stress in the core web along the z-direction dueto the pressure pf �Pa�

�face,xpf � stress in the face sheet along the x-direction

due to the pressure pf �Pa��face,z

pf � stress in the face sheet along the z-directiondue to the pressure pf �Pa�

�face,xpcomb � stress in the face sheet along the x-direction

due to the pressure pcomb �Pa��face,z

pcomb � stress in the face sheet along the z-directiondue to the pressure pcomb �Pa�

�face,x�Tpanel � stress in the face sheet along the x-direction

due to the temperature difference �Tpanel �Pa��face,z

�Tpanel � stress in the face sheet along the z-directiondue to the temperature difference �Tpanel �Pa�

�face,x�Ttf � stress in the face sheet along the x-direction

due to the temperature difference �Ttf �Pa��face,z

�Ttf � stress in the face sheet along the z-directiondue to the temperature difference �Ttf �Pa�

�m,x�i�

� mechanical stress at location i along thex-direction �Pa�

�m,z�i�

� mechanical stress at location i along thez-direction �Pa�

�T,z�i�

� thermal stress at location i along thex-direction �Pa�

�T,x�i�

� thermal stress at location i along thez-direction �Pa�

� f � mass density of the coolant �kg /m3�� nondimensional fin temperature:

�T�y�−Tfuel� / �T�0�−Tfuel�

Appendix A: Thermal Resistance Model

The ModelAmong the nine thermal resistances in the model network �Fig.

3�a��, six are independent:

NOVEMBER 2008, Vol. 75 / 061022-11


Hstdivp

d

wPR

wp

A

w�

oa

0

Downloaded Fr

• top face/hot gas boundary: RG=1 /hG• across TBC �when present�: RTBC= tTBC /kTBC

�

• across hot face �y-direction�: Rface= tf /ks�

• along hot face �x-direction�1: Rh= �w+ tc /2� /4ks�

• top face/coolant boundary: Rcool=1 /hc• core web �modeled as a 1D thermal fin �30,31��: Rfin

= �tanh−1��2hC /ks�tcL

2��2hC /ks�tcks

��−1.

ere, hG and hC are the heat transfer coefficients on the hot gaside and coolant sides, respectively; ks

� and ks� are the through-

hickness and in-plane thermal conductivities of the material �theistinction being important for CMCs, wherein ks

��ks��; and kTBC

s the through-thickness conductivity of the TBC �its in-planealue is taken as zero, due to the columnar structure of coatingsroduced by physical vapor deposition�.

Assuming flow in the cooling channels is turbulent and fullyeveloped, hC is given by the Gnielinski correlation �14,32�:

hC =kf

DhNu =

kf

Dh

�f/2��Re − 1000�Pr

1 + 12.7�f/2�Pr2/3 − 1��A1�

here kf is the thermal conductivity of the coolant, Pr is therandtl number, Dh=2wL / �w+L� is the hydraulic diameter, ande is the Reynolds number:

Re uDh

� f=

V̇effDhH�w + tc�LwH� f

�A2�

ith � f being the kinematic viscosity, V̇eff the volumetric flow rateer unit width of panel �see Sec. 5.2�, and f is the friction factor,

1The horizontal resistance is not properly conductive, as convection occurs alongne of the sides. FE calculations reveal that using an effective length equal to half thectual length yields accurate results �hence the factor of 4�.

A5� gives

61022-12 / Vol. 75, NOVEMBER 2008


given by f =0.046Re−1/5 in the domain 2�104Re106 �32,33�.The model can be simplified into the effective network of Fig.

3�b�, characterized by four resistances, R1 ,R2w ,R2

c ,Rh, where

R1 = RG + RTBC + Rf/2

R2w = Rf/2 + Rcool

R2c = Rf/2 + Rfin �A3�

�with Rh previously defined�. The temperatures of interest arethose on the top face, directly over �Ttf

c � and midway between�Ttf

w� the core members, at both the inlet �z=0� and the outlet �z=Z�. Since the three fluxes qw ,qc ,qh are also unknown, five equa-tions are needed to close the system:

Taw − Tfuel = qwR1 + �qw + 2qhtf/w�R2w

Taw − Tfuel = qcR1 + �qc − 2qhtf/tc�R2c

Taw − Ttfw = qwR1

Taw − Ttfc = qcR1

Ttfc − Ttf

w = qhRh �A4�The solution gives the heat fluxes:

qc =Taw − Tf

R1R

c*

qw =Taw − Tf

R1R

w*

qh =Taw − Tf

R1R

h* �A5�

where

Rw* =

R1�R2cRh + R1�Rh + 2R2

ctf/tc + 2R2wtf/w��

R2cR2

wRh + R12�Rh + 2R2

ctf/tc + 2R2wtf/w� + R1�R2

wRh + R2c�Rh + 2R2

w�tf/tc + tf/w��

Rc* =

R1�R2wRh + R1�Rh + 2R2

ctf/tc + 2R2wtf/w��

R2cR2

wRh + R12�Rh + 2R2



w�tf/tc + tf/w��

Rh* =

R12�R2

c − R2w�

R2cR2

wRh + R12�Rh + 2R2



w�tf/tc + tf/w��A6�

dditionally, at the channel inlet,

Taw − Ttfw

Taw − Tf= R

w*

Taw − Ttfc

Taw − Tf= R

c* �A7�

The coolant temperature is obtained via an energy balance

� fcp,fV̇effdTf

dz=

wqw�z� + tcqc�z�w + tc

�A8�

here � fcp,f is its volumetric specific heat. Combining with Eq.

d�Taw − Tf�dz

+1

R1� fcp,fV̇eff� w

w + tcR

w* +

tc

w + tcR

c*��Taw − Tf� = 0

�A9�

The solution to this differential equation yields the longitudinaldistribution of the coolant temperature:

Taw − Tf

Taw − Tf0 = exp�− �z� �A10�

where



Ts

w

w

gtb

ab

J

Downloaded Fr

� =1

R1� fcp,fV̇eff� w

w + tcR

w* +

tc

w + tcR

c*� �A11�

he temperature distribution on the hot face can be expressed inimilar form, via Eqs. �A7� and �A10�:

Taw − Ttfw

Taw − Tf0 = R

w* exp�− �z�

tc R1 �0



Taw − Ttfc

Taw − Tf0 = R

c* exp�− �z� �A12�

All temperatures achieve their maximum at the outlet �z=Z�.From the preceding analysis, the temperatures at the 18 points

in Fig. 5 are as follows:

T�i� = Taw − �Taw − Tf0� ·

�1 −1

2

Rf

R1�R

c* exp�− �z� at Points 1 and 10

�Rc* + �1

2R

c* − R

h* tf

tc�Rf

R1�exp�− �z� at Points 2 and 11

�1 −��L��0

Rfin

R1�R

c* − 2R

h* tf

tc��exp�− �z� at Points 3, 4, 7, 8, 12, 13, 16, and 17

�1 −1

2

Rf

R1�R

w* exp�− �z� at Points 5 and 14

�Rw* + �1

2R

w* + R

h* tf

w�Rf

R1�exp�− �z� at Points 6 and 15

�1 −��L/2�

�0

Rfin

R1�R

c* − 2R

h* tf

tc��exp�− �z� at Points 9 and 18

� �A13�

here ��y� /�0 is the nondimensional fin temperature:

��y�

�0

=T�y� − Tf

T�0� − Tf

=

cosh��2hC

ks�tc

�L − y��cosh��2hC

ks�tc

L� �A14�

ith y the coordinate oriented along the fin.Once all the temperatures in the system are known, simple al-

ebraic manipulation provides the temperature difference acrosshe top face �directly above and midway between the core mem-ers�:

�Ttfc �z�

Taw − Tf0 = �R

c* − R

h* tf

tc�Rf

R1exp�− �z�

�Ttfw�z�

Taw − Tf0 = �R

w* + R

h* tf

w�Rf

R1exp�− �z� �A15�

nd across the entire panel �above and between the core mem-ers�:

�Tpanelc

Taw − Tf0 = �R2

c

R1−

Rfin

R1

��L��0

��Rc* − 2R

h* tf

tc�exp�− �z�

�Tpanelw

Taw − Tf0 = ��R

w* + 2R

h* tf

w�R2

w

R1

− �Rc* − 2R

h* tf �Rfin ��L��exp�− �z� �A16�

Appendix B: Stress Analysis

Coolant PressureThe coolant pressure pf �assumed uniform along z and equal to

pf0, given that �p� pf

0� induces uniform tensile stresses in the coremembers �at Points 9 and 18�, given by

�core,ypf

pf=

w

tc,

�core,zpf

pf= �

�core,ypf

pf�B1�

with � the Poisson ratio of the material. It also induces combinedtension/bending in the face segments. For Boundary Condition I,these are

�face,xpf

pf=

L/2tf + �w/tf�2/2 at Points 2, 3, 11, and 12

L/2tf − �w/tf�2/2 at Points 1, 4, 10, and 13

L/2tf + �w/tf�2/4 at Points 5, 8, 14, and 17

L/2tf − �w/tf�2/4 at Points 6, 7, 15, and 16�

�face,zpf

pf= �

�face,xpf

pf�B2�

The same solutions apply to boundary condition Type II with theexception of those for the bottom face segments, which lack thebending component. Along this face �at Points 3, 4, 7, 8, 12, 13,16, and 17�, the stresses are simply

�face,xpf

pf=

L

2tf,

�face,zpf

pf= �

�face,xpf

pf�B3�

Combustor Gas PressureFor Boundary Condition I, the panel behaves globally as a

clamped-clamped plate under uniform pressure, pcomb. With theusual assumption that the shear force is supported by the core and
the moment by the face sheets �19�, the stresses in the faces are
NOVEMBER 2008, Vol. 75 / 061022-13


Ianm

wer

F

M

wgst

astt�

0

Downloaded Fr

�x,facepcomb

pcomb=

1

12

b2

�H − tf�tfat Points 1, 2, 5, and 6

−1

12

b2


1

24

b2


−1

24

b2


��z,face

pcomb

pcomb= �

�x,facepcomb

pcomb�B4�

n contrast, for boundary condition Type II, bending is prohibitednd, since pcomb� pf, the additional stress on the top face can beeglected. For similar reasons, the stresses exerted on the core
embers can also be neglected for both boundary conditions.
i=1–18 ult ult

61022-14 / Vol. 75, NOVEMBER 2008


Thermal LoadThe temperature difference across the top face causes compres-

sion along its top surface and tension along its bottom surface �atthe boundary with the coolant�. These stresses are

�face,x�Ttf = �face,z

�Ttf = −E��Ttf

2�1 − ��at Points 1, 5, 10, and 14

E��Ttf

2�1 − ��at Points 2, 6, 11, and 15�

�B5�

with E and � the Young modulus and the coefficient of thermalexpansion of the material, respectively. Additionally, the averagetemperature difference between the top and bottom faces,�Tpanel= ��Tpanel

w +�Tpanelc � /2, causes the panel to deform uni-

formly in each of the x- and z-directions, inducing compression inthe top face and tension in the bottom face �17�. Accounting forthe stretching stiffness of the core members along the z-directionand assuming that the temperatures of the core and the bottomface are the same at steady state, the resulting additional stresses
are
�face,x�Tpanel = −

E��Tpanel

2�1 − ��at Points 1, 2, 5, 6, 10, 11, 14, and 15

E��Tpanel

2�1 − ��at Points 3, 4, 7, 8, 12, 13, 16, and 17�

�face,z�Tpanel = −

E��Tpanel�Af + Ac��1 − ��2Af + Ac�

at Points 1, 2, 5, 6, 10, 11, 14, and 15

E��TpanelAf

�1 − ��2Af + Ac�at Points 3, 4, 7, 8, 12, 13, 16, and 17� �B6�

here Af = tf�w+ tc� and Ac= �H−2tf�tc are the cross-sectional ar-as of the face and the core in a unit cell, respectively. Theseesults apply to both boundary conditions.

ailure ConditionsFor metals, failure is defined as the onset of yielding. The vonises criterion is used, namely,

maxi=1–18

�� m,x�i�

�Y�T�i��+

�T,x�i�

�Y�T�i��−

�m,z�i�

�Y�T�i��−

�T,z�i�

�Y�T�i��2

+ � �m,x�i�

�Y�T�i��

+�T,x

�i�

�Y�T�i��2

+ � �m,z�i�

�Y�T�i��+

�T,z�i�

�Y�T�i��2� = 2 �B7�

ith the stress components and the temperature at each location iiven by Eqs. �B2�–�B6� and �A13�, respectively. The yieldtrength of the material �Y is assumed to linearly decrease withemperature �Table 2�.

Well-designed CMCs typically fail when the normal stresslong the primary fiber orientation attains either the ultimate ten-ile strength or the compressive strength. Assuming for simplicityhat the strengths in tension and compression are identical andemperature independent �reasonable for SiC /SiC and C /SiC34,35��, the ensuing condition is

max�max� ��m,x�i� + �T,x

�i� ��

,��m,z

�i� + �T,z�i� �

�� = 1 �B8�

Pressure DropThe pressure drop in the coolant due to viscous dissipation over

the length of the panel is �32�

�p =2� f fZ�V̇eff�2

H2�1 − �̄�2Dh

�B9�

with �̄=1−Lw / �H�w+ tc�� the relative density of the panel.

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