valdevit aiaa canberra

7/29/2019 Valdevit AIAA Canberra

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American Institute of Aeronautics and Astronautics

1

Design of actively cooled panels for scramjets

Lorenzo Valdevit1, Natasha Vermaak2, Kathryn Hsu3, Frank W. Zok4 and Anthony G. Evans5

University of California, Santa Barbara, CA 93106-5050

The operating conditions of scramjet engines demand designs that include active cooling

by the fuel and the use of lightweight materials that withstand extreme heat fluxes under

oxidizing conditions. The goal of this analysis is to provide an optimization tool that can be

used to direct the development of advanced materials that outperform existing high

temperature alloys and compete with ceramic matrix composites. For this purpose an

actively cooled plate has been optimized for minimum weight under three primary

constraints. (i) Resistance to pressure loads arising from fuel injection and combustion, as

well as thermal loads associated with the combustion temperature. (ii) A temperature

distribution in the structure during operation that does not exceed material limits, subject to

a reasonable pressure drop. (iii) A maximum temperature in the fuel (JP-7) low enough to

prohibit coking. It is shown that all design requirements typical of Mach 5-7 hypersonic

vehicles can be met by a small subset of material systems. Those made using C/SiCcomposites are the lightest. Others made using Nb alloys and (thermal barrier coated)

superalloys are somewhat heavier, but might prevail in a design selection because of their

structural robustness, facility of fabrication and cost-effectiveness.

Nomenclature

A = area

a = speed of sound

b = width of the actively cooled panel

pc = specific heat

D = diameter

E = Youngs modulus

f = friction factor

= fuel/air mass ratio

= generic function

g = generic function

H = thickness of the actively cooled panel

h = enthaply

= heat transfer coefficient

k = thermal conductivity

L = height of the cooling channel

= characteristic lengthM = mass of the actively cooled panel

m = mass flow rate

Ma = Mach number

1 Postdoctoral scholar, Materials Department. Member AIAA.2 Graduate student, Materials Department.3 Graduate student, Materials Department.4 Professor and Associate Chair, Materials Department.5 Professor, Materials Department.


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= pressure

Pr = Prandtl number

Q = heat flux

q = specific heat flux

R = universal gas constant

= thermal resistance per unit width

r = thermal resistance per unit areaRe = Reynolds number

St = Stanton number

T = temperature

t = thickness

u = velocity of the coolant

V = velocity of the air

V = volumetric flow rate of the coolant per unit width of the panel

w = width of the cooling channel

x = non-dimensional geometric variable

x,y,z = spatial coordinate

Z = length of the actively cooled panel

Greek symbols = thermal expansion coefficient (CTE)

= thermal diffusivity

= inverse length constant

= difference = mixture richness (actual fuel/air ratio divided by stoichiometric fuel/air ratio)

= dynamic viscosity

v = kinematic viscosity

= Poissons ratio

= non-dimensional objective function

* = dimensional objective function

= mass density = areal density of the cross-section of the panel

= normal stress

Subscripts and superscripts

0 = inlet conditions

= free-stream conditions

1,2 = generic symbols

3,4 = stations along the vehicle

air = relative to the air

aw = adiabatic wall conditions

c, core = relative to the core web

c, cool = relative to the cooling channels

coke = coking conditions in the fuelcomb = within the combustion chamber

f, face = relative to the face sheet

f, fuel = relative to the fuel (coolant)

fin = relative to conduction/convection in the core web

G = relative to the hot gases in the combustion chamber

h = hydraulic

= horizontal (x) direction

i = generic location in the cross-section

m = mechanical


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max = maximum

min = minimum

panel = relative to the whole cross-section of the panel

s = relative to the solid material the panel is made of

st = stoichiometric

T = thermal

= total

t = top face

top = top side

TBC = relative to the thermal barrier coating (TBC)

w = wall exposed to the combustion chamber

= between core webs (i.e. near the middle of the channel)

x,y,z = along the respective directions

Y = yielding

panelT = due to the temperature drop across the panel

tfT = due to the temperature drop across the top face sheet

* = Eckert reference

= non-dimensional

= generic symbol

I. IntroductionThe high specific impulse enabled by air-breathing hypersonic vehicles has motivated continued development of

key technologies. The aerothermodynamics of scramjet engines has been extensively analyzed, and the potential to

achieve positive thrust successfully demonstrated. However, the integration with vehicle design remains a challenge.

Selecting materials and generating designs that resist the associated thermal loads for the duration of the mission is

especially daunting. Active cooling by the fuel is a natural choice, because the proximity of the injection points

facilitates its circulation just before injection. Aspects of the design and performance of actively cooled combustion

systems have already been explored1-4, including geometry optimization5-8. But, to the best of our knowledge, a

comprehensive treatment that accounts for the thermo-mechanical constraints has yet to be devised. In this paper,

optimization results are presented for a prototypical planar combustor with rectangular channels (Fig. 1), for a

variety of materials. All the work refers to a hydrocarbon-powered scramjet operating at Mach 7. The goal is todirect the development of advanced materials that outperform existing high temperature alloys and compete with

ceramic matrix composites (CMCs). For options based on high temperature alloys, the possible benefits of

superposing a thermal barrier coating (TBC), such as yttria-stabilized zirconia (YSZ), are explored.

For optimization purposes, it is convenient to have analytic expressions for the temperatures and stresses in the

structure. The temperatures are derived using a two-dimensional resistance network, building upon a fin analogy 9,10

and verified using finite elements. These temperatures are used to provide a constraint on the maximum allowable

material temperature. They are also used as input for analytic calculation of the thermal stresses, again checked

using finite elements. These stresses are superimposed on those induced by the pressure loads inside the combustion

chamber and within the cooling channels, to permit formulation of the structural constraints. By requiring that the

total stresses (thermal plus pressure) remain below the failure strength of each material at its maximum operating

temperature, the optimization results allow the ranking of candidate materials and designs in terms of their weight-

efficiency. While it is preferred to operate scramjets at (or near) stoichiometric fuel/air ratios, in some instances

extra fuel is needed to provide adequate cooling and/or to overcome combustion inefficiencies. To allow for thispossibility, results are also presented for a range of fuel/air mixture richness.

The structure of the paper is as follows. In section II the primary aerothermodynamics concepts that provide the

boundary conditions for the thermo-mechanical problem are summarized. Section III presents the thermal network

used for temperature and heat flux calculations. The details of the stress analysis are presented in Section IV.

Section V illustrates the formulation of the optimization scheme in non-dimensional form and section VI presents its

results. Discussion and conclusions follow.


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II. Thermo-mechanical loads in a scramjet combustion chamberFor tractability of presentation, the ensuing analysis focuses on a specific hydrocarbon-powered hypersonic

vehicle operating at Mach 7 (near the ramjet to scramjet transition). However the procedure is general and can be

readily extended to other flight situations. The preferred fuel at this velocity is JP-7, which facilitates compact and

robust designs11. Furthermore, the design of vehicles in this velocity range is most mature and in urgent need of

technological advance. The thermo-mechanical loads on the combustor depend on nearly every aspect of the vehicle

design, including the size and shape of the compression ramp, the size of the combustion chamber, the details of the

injection system, the combustion efficiency, etc. Nevertheless, reasonable (albeit inevitably approximate) estimates

can be derived based on a small set of initial assumptions, namely: (a) the vehicle speed is Mach 7; (b) the vehicle

is 10 m long and 2.5 m tall; and (c) the compression ramp is bi-linear, generating three oblique shocks. The

aerothermodynamic considerations underlying these derivations are clearly exposed in Ref. 11; details governing the

application of these concepts to the present design are provided in Appendix I. The main results are summarized

below.Three thermo-mechanical loads act on the actively cooled wall of the combustion chamber:

(i) An external pressure load, evenly distributed on the face exposed to the combusting gases,

0.24 MPacombp = .(ii) A pressure load inside the cooling channels, 10 MPacoolp = .(iii) Thermal loads derived from exposure of one face to the combusting gases, with adiabatic wall

temperature, 3050 KawT = and heat transfer coefficient,2535 W/m K

Gh = .

The incoming heat is carried away by the fuel (Table 2) flowing in the cooling channels at a rate per unit width

of the vehicle, 20.0082 m /sV = ( represents the fuel/air mixture richness, defined in equation (47)). The fuel

enters the cooling channels at a temperature,0

400 Kf

T = .

III. Temperature distributionsTo obtain analytic estimates of the temperatures, three simplifications are invoked:

(i) The top face of the structure is exposed to hot gases at specified temperature, whereas the bottom face is

thermally insulated. Consequently, all the heat received through the top face is carried away by the

cooling fluid.

(ii) No heat is conducted along the length of the panel (z-direction), either in the structure or in the cooling

fluid. This assumption results in slightly conservative temperature estimates.

Figure 1. (a) Artist rendition of a hypersonic air-breathing vehicle. The proposed multifunctional actively

cooled plate is shown in the inset. (b) Schematic of the actively cooled plate with thermo-structural loads,

variables definition and coordinate system.


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(iii) The fuel temperature is uniform within the channels at any given cross-section: ( )fuel fuelT T z= only.Namely,

fuelT is the mixing-cup temperature.

The temperature at every cross-sectionzis calculated using an electrical analogy12 (Fig. 2a). The nine conductive

resistances (per unit width in the z-direction) needed to characterize the problem can be grouped into six categories

(see Fig. 1b for definition of the geometric variables):

On the hot side: 1/G G cR h t= , 1/w

G GR h w=

Across a TBC (when present): /TBC TBC TBC cR t k t= , /w

TBC TBC TBC R t k w=

Across the face (y-direction): /face f s cR t k t= , /w

face f sR t k w=

Figure 2. (a) Thermal resistance network used for the temperature calculations, together with the

expression for all the relevant thermal resistances. (b) Effective network.


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Along the face (x-direction): ( )/ 2 / 2h c s f R w t k t= +

Convective resistance on the coolant side (for the portion of the face in direct contact with the coolant):

1/cool cR h w= Combined conductive and convective resistance of the core web (modeled as a one-dimensional thermal

fin9,10):

1

1 22 21tanh c cfin s

s c c s c

h hR L k

k t t k t

=

.

Here,Gh is the heat transfer coefficient on the hot gas side, sk and TBCk are the thermal conductivities of the solid

and the TBC, respectively, andch is the heat transfer coefficient on the coolant side. Flow in the cooling ducts is

assumed to be turbulent and fully developed. Under these conditions,ch is given by the correlation

13,14:

( )2 / 3(Re 1000)Pr

2

1 12.7 / 2 Pr 1

f f

c

h h

fk k

h NuD D f

= =

+ (1)

wheref

k is the thermal conductivity of the coolant, Pr is the Prandtl number, 2 /( )hD wL w L= + is the hydraulic

diameter of the ducts, and Re is the Reynolds number in the ducts, defined as:

( )Re

1

h h

f f

u D DV

H =

(2)

with f being the kinematic viscosity of the fluid, V the prescribed volumetric flow rate of the fuel per unit width

(section II), the areal density of the cross-section (Eq. (29)) and fthe friction factor. For fully developed turbulent

flow, the following correlation can be used for the friction factor13,15:

1/ 40.0791Ref = (3)

Eqs. (1)-(3) allow calculation of the thermal resistances coolR and finR as a function of the geometric variables and

the fuel properties and flow rate.

The temperatures of the structure and the fluid increase continuously along the z-direction, becoming maximum

at the outlet. For convective boundary conditions (and uniform hot gas temperature) the solid and fluid temperature

are interrelated. A three-step approach is adopted:

(i) The thermal network (fig. 2a) is solved as a function of the unknown temperature ( )fuelT z .(ii) The principle of energy conservation is used to relate the fuel and the solid temperatures, thus closing the

system.

(iii) The resulting differential equation is solved for ( )fuelT z and subsequently all the other temperatures ofinterest are derived.

For step (i), the model can be simplified into the effective network of Fig. 2b, characterized by the five

resistances1 1 2 2

, , , ,w c w ch

R R R R R , where:

1

1

2

2

2

2

2

2

w w w w

G TBC face

c

G TBC face

w w

face cool

c

face fin

R R R R

R R R R

R R R

R R R

= + +

= + +

= +

= +

(4)


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The pertinent temperatures are the averages of the top face, both over ( ctfT ) and between (

w

tfT ) the core members.

Since the three fluxes , ,w ch

Q Q Q are also unknown, five equations are needed to close the system:

( )

( )

( )

1 2

1 2

1

1

( 2 )

( 2 )

/

/

/

w w

aw w w h fuel

c c

aw c c h fuel

w w

w aw tf

c c

c aw tf

c w

h tf tf h

T Q R Q Q R T

T Q R Q Q R T

Q T T R

Q T T R

Q T T R

+ = = =

=

=

(5)

By defining the heat fluxes per unit area as:

/

/

/

w w

c c c

h h f

q Q w

q Q t

q Q t

=

=

=

(6)

Eqs. (5) can be rewritten as:

1 2

1 2

1

1

( 2 / )

( 2 / )

w

aw fuel w w h f

c

aw fuel c c h f c

w

aw tf w

c

aw tf c

c w

tf tf h h

T T q r q q t w r

T T q r q q t t r

T T q r

T T q r

T T q r

= + +

= +

=

= =

(7)

where the thermal resistances per unit area have been defined as:

( )

1

2

1

1 2

2

1 1

2

1 1

2

2 21tanh

2

2 / 2

fTBC

G TBC s

fw

s c

fc c c

s

s s c s c

c

h

s

ttr

h k k

tr

k h

t h hr L k

k k t k t

w tr

k

= + +

= +

= + +

=

(8)

Solving the linear system in Eq. (7), gives:

*

1

*

1

*

1

aw fuel

c c

aw fuel

w w

aw fuel

h h

T Tq R

r

T Tq R

r

T Tq R

r

=

=

=

(9)


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where *w

R and *c

R are given by:

1 2 1 2 2*

2

2 2 1 2 2 1 2 2 2

1 2 1 2 2*

2

2 2 1 2 2 1 2 2

( 2 / 2 / )

( 2 / 2 / ) { [ 2 ( / / )]}

( 2 / 2 / )

( 2 / 2 / ) {

c c w

h h f c f

w c w c w w c w

h h f c f h h f c f

w c w

h h f c f

c c w c w w c

h h f c f h

r r r r r r t t r t wR

r r r r r r t t r t w r r r r r r t t t w

r r r r r r t t r t w

R r r r r r r t t r t w r r r r

+ + + =+ + + + + + +

+ + +

= + + + + +

( )2

2

1 2 2*

2

2 2 1 2 2 1 2 2 2

[ 2 ( / / )]}

( 2 / 2 / ) { [ 2 ( / / )]}

w

h f c f

c w

h c w c w w c w

h h f c f h h f c f

r r t t t w

r r rR

r r r r r r t t r t w r r r r r r t t t w

+ +

=

+ + + + + + +

(10)

Additionally:

*

*

w

aw tf

w

aw fuel

c

aw tf

c

aw fuel

T TR

T T

T TR

T T

=

=

(11)

For step (b), the temperature in the fluid is obtained via an energy balance:

( ) ( ),fuel

f p f c

dTc V w t Q z

dz + = (12)

where,f p f

c is the specific heat at constant pressure per unit volume of the fluid and

( ) ( ) ( )1 1

w c

aw tf aw tf

w c c

T T T T Q z Q z Q z w t

r r

= + = + (13)

is the total heat flux impinging on the top face over a unit cell of widthcw t+ (Fig. 2a). By using Eq. (11) and Eq.

(13), Eq. (12) can be rewritten as:

( )( )* *,

1

10

aw fuel c

f p f w c aw fuel

c c

d T T twc V R R T T

dz r w t w t

+ + =

+ + (14)

With the boundary condition ( ) 00uel f T z T= = , the solution for the spatial variation of the fluid temperature is:

( )0

expaw fuel

aw f

T Tz

T T

=

(15)

where

* *1

,

1/ cw c

c cf p f

tr wR R

w t w t c V

= +

+ + (16)

The temperatures at the middle of the top face can be expressed in similar form via eqs. (15) and (11):


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( )

( )

*

0

*

0

exp

exp

w

aw tf

w

aw f

c

aw tf

c

aw f

T TR z

T T

T TR z

T T

=

=

(17)

All temperatures achieve their maximum at the fuel outlet,z Z= . The relative magnitudes of the convectiveresistance at the inner wall and the fin resistance dictate whether the hot spot appears on the top face over or

between core webs; namely:

1

1 2

2 2

2 2 1tanh

c w c w c c

tf tf s

s c s c c

h hT T r r L k

k t k t h

> > > (18)

The temperature at the hot spot (at the face sheet/TBC interface) is then:

max

, *

in

0 1

11

2

aw tf top f

aw f

T T rR

T T r

=

(19)

where * * *min

min{ , }c w

R R R= .

For the calculation of the thermal stresses, two temperature differences are relevant:

I. The temperature drop across the top face (in the y-direction), both over and between the core webs:

( )

( )

* *

0 1

* *

0 1

exp

exp

c

tf f f

c h

aw f c

w

tf f f

w h

aw f

T t rR R z

T T t r

T t rR R z

T T w r

=

= +

(20)

II. The temperature drop across the entire panel cross section, measured from the middle of the top faceto the bottom face, both over and between the core webs:

( )

( )

* * 2

0 1

* * 2

0 1

2 exp

2 exp

w wpanel f

w h

aw f

c cpanel f

c h

aw f c

T t rR R z

T T w r

T t rR R z

T T t r

= +

=

(21)

These temperature differences are greatest at the inlet ( 0z= ).The accuracy of this model has been verified with select finite element calculations. With the exception of

structures made using highly conductive alloys (such as copper), the predicted temperatures were consistent and

remarkably accurate.

IV. Stress analysis

A. Problem statement and boundary conditionsThe thermo-mechanical stresses depend on the constraint exerted on the plate by the surrounding components of

the vehicle. Because the actual vehicle design is not known, and for the sake of generality, two idealized sets of

boundary conditions are considered (Fig. 3a-c):


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Case I: Linear frictionless supports along the edges in the z-direction (Fig. 3a). This constraint prevents

bending in the z-direction, while allowing it in the x-direction (albeit with no rotation at the ends).

Uniform thermal expansion is allowed along all directions, but panel-level thermal bending is

forbidden.

Case II: Two-dimensional continuous bed of rollers (Fig. 3c). Uniform thermal expansion (but no panel-level

bending) is permitted along either direction. The external pressure does not cause the panel to bend

globally, but the internal pressure can bend individual top face segments.

Note that in case I the distance b between the clamps

need not equal the width of the entire combustion chamber.

The same boundary conditions can be used to model

multiple linear, frictionless supports at the back of the plate

(Fig. 3b). In this case, the dimension b represents the

distance between two consecutive supports. Moreover,

case II can be interpreted as the limit of case I as the

spacing between consecutive supports approaches zero. In

both cases, the use of rollers instead of frictional supports

allows uniform thermal expansion of the panel (with no

bending). While the practical implementation can be

challenging, attaining these conditions is essential to viable

solutions. As a simple demonstration, consider the thermalstress in a uniform plate clamped on all sides (with

frictional supports) subject to a uniform temperature

increase T 16: /(1 )x y E T = = . The maximum

temperature increase that can be sustained by the plate

without yielding is,max (1 ) /YT E = . This quantity

is remarkably small for most structural materials of interest

(Fig. 3d). Note that, with the exception of C-SiC, most

materials would yield well below their limit temperature.

To further simplify the problem, both the pressure drop

and the temperature variation along the panel length are

neglected. This assumption, combined with the imposed

boundary conditions, ensure that generalized plane strainconditions are obtained along thezdirection6. Furthermore,

it requires that the calculations be performed on one cross-

section only. The inlet cross-section ( 0z= ) is selected forthis calculation, since the temperature differences in Eqs.

(20)-(21) and thus the thermal stresses are maximum at

that location.

The next step is the identification of the most failure-

susceptible locations in the structure at this cross-section.

The stresses due to both pressure and thermal loads vary

along the member lengths and thicknesses; those due to the

pressure in the combustion chamber depend on the location

within the entire plate (in the xy plane). Because the

thermal and pressure loads often induce stresses of

opposite sign, it is not straightforward to establish a priori

the location of first yield. For this reason, a set of 18

critical points has been identified, clustered around two

failure-susceptible cells: one at the periphery closest to the

supports and the other at the center (Fig. 4). The presence

6 Generalized plane strain refers to conditions wherein the cross-sections z= 0 and z=Zare not allowed to rotaterelative to each other (although they are free to translate).

Figure 3. Possible mechanical boundary

conditions. (a-b) Case I: linear rollers on two sides

(in (b), a mechanically equivalent situation is

shown, where multiple supports are used). (c) Case

II: uniform two-dimensional bed of rollers, with

impeded rotation at the ends. (d) Benchmark

boundary condition: plate sitting on rigid

foundation. The chart compares the temperature

increase from room temperature that would induce

melting (light grey) to the temperature increase that

would yield the structure (dark grey); several

materials of interest are compared. Note that under

this boundary conditions, the full high-temperature

potential of the material would not be exploited.


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11

Figure 4. Cross-sections that are likely to fail by yielding: the 18 critical points are depicted. The

two unit cells considered (close to the support, points 1-9, and around the middle of the panel, points

10-18) are loaded differently under boundary condition case I(due to the effect of the pressure in the

combustion chamber,combp ), whereas they are thermo-mechanically identical undercase II.

of internal pressure in the core channels (which imposes significant tensile stresses on all members) combined with

the relatively stubby shape of the optimized members (section VI and Fig. 6) makes it unnecessary to design against

buckling17-25. Consequently, failure of the structure is averted provided that the Mises stress in the most highly

stressed location remains in the elastic range.

The analytic methods used to determine the stresses in sandwich structures consider the face and core members

in each unit cell as independent beams; the connection between members is modeled using translational (and some

degree of rotational) constraints17-25. The accuracy of this approach is dependent upon the aspect ratio of each

element, and decreases as the elements become stubbier. When this approach is used, finite element analyses are

needed to ensure that predictions at the optimal geometries are sufficiently accurate.

B. Stresses due to coolant pressureCase I. The pressure

cool

inside the cooling channels subjects the core members to uniform tension and induces

a combination of tension and bending on the faces. Using the notation of Figs. 1b and 4, the resulting stresses in the

core members are:

,

,,

at pts 9,18cool

coolcool

p

cor e y

cool c

pp

core ycore z

cool cool

w

p t

p p

=

=

(22)

and in the face segments:

( )( )

( )

( )

2

2

,

2

2

, ,

/ 2 / / 2 at pts 2,3,11,12

/ 2 / / 2 at pts 1,4,10,13

/ 2 / / 4 at pts 5,8,14,17

/ 2 / / 4 at pts 6,7,15,16

cool

cool cool

f f

pf fface x

coolf f

f f

p p

face z face x

cool cool

L t w t

L t w t

p L t w t

L t w t

p p

+ =

+

=

(23)


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12

Consistent with beam analysis, the through-thickness stresses are neglected.

Case II. The stresses in the core members and the face segments exposed to the combustion chamber are the

same as those in Eq. (22) and (23), respectively, whereas those in the bottom face segments lack the bending

component. Consequently:

( )

( )

( )

( )

2

2

2,

2

, ,

/ 2 / / 2 at pts 2,11

/ 2 / / 2 at pts 1,10

/ 2 / / 4 at pts 5,14

/ 2 / / 4 at pts 6,15

/ 2 at pts 3,4,7,8,12,13,16,17

cool

cool cool

f f

f fp

face x

f f

cool

f f

f

p p

face z face x

cool cool

L t w t

L t w t

L t w tp

L t w t

L t

p p

+

= +

=

(24)

C. Stresses due to combustion chamber pressureCase I. The entire panel behaves as a clamped-clamped beam under uniform pressure, comb . With the usual

assumption that the shear force is supported by the core and the moment by the face sheets 18, the stresses in the faces

are:

( )

( )

( )

( )

2

2

,

2

2

,

1at pts 1,2,5,6

12

1at pts 3,4,7,8

12

1at pts 12,13,16,17

24

1 at pts 10,11,14,1524

comb

f f

pf fx face

comb

f f

f f

z fa

b

H t t

b

H t t

p b

H t t

bH t t

=

,comb combp p

ce x face

comb combp p

=

(25)

The stress exerted on the core members are of the order of the pressure itself, and can be safely neglected relative to

those induced by the pressure in the coolant (Eq. (22)).

Case II. The combustion pressure does not cause global panel bending. The pressure on the top face segments

counteracts that from the coolant; thus sincecomb cool p


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13

( )

( )

( )( )( )

,

,

at pts 1,2,5,6,10,11,14,152 1

at pts 3,4,7,8,12,13,16,172 1

at pts 11 2

panel

panel

panel

T

face x

panel

panel f c

f cT

face z

E T

E T

E T A A

A A

=

+ +

=

( )( )

,2,5,6,10,11,14,15

at pts 3,4,7,8,12,13,16,171 2

panel f

f c

E T A

A A

+

(26)

where ( )f cA t w t= + and ( )2c f cA H t t= are the cross-sectional areas of the face and the core in a unit cell,respectively. Note that neglecting the core stiffness (letting 0cA in Eq. (26)) results in a perfectly bi-axial stateof stress in both faces. Contribution 2. By ignoring x-variations, the temperature difference across the top face

thickness, wtf tf

T T = , causes the upper surface of the top face to experience compression and the lower surface to

be in tension, giving:

( )

( )

, ,

at pts 1,5,10,142 1

at pts 2,6,11,152 1

tf tf

tf

T T

face x face z

tf

E T

E T

= =

(27)

Case II. The distributed support plays no role in defining the thermal stresses, so the solution is identical to that

forcase I.

The accuracy of the stress distributions have been tested using selected finite element analyses for near-optimal

geometries. For most materials the agreement is within 10% . The exception was for structures made of highlyconductive material (such as Cu alloys) because the thermal network fails to capture the flow of heat into the bottom

face through the core. Consequently, the model gives overly conservative estimates of yielding. An extension of the

thermal network is underway to accurately model highly conductive materials.

V. Optimization

The objective is to optimize the geometry for minimum weight, subject to the constraints associated with the

structural and functional requirements. In order to fully exploit their potential, air-breathing hypersonic vehicles

should operate at a minimum mass flow rate of the fuel,st

V V (or equivalently 1 ). In practice, richer mixtures

might be necessary to compensate for combustion inefficiencies. In order to capture a spectrum of conditions of

interest, optimizations are performed for a range of mixture richness, 1 4 = . Several materials have beenexplored (Table 1) and the resulting structures separately optimized. This allows exploration of the feasible alloys as

well as ranking according to weight efficiency. The properties of JP-7 (Table 2) have been used for all calculations.

For generality, and in order to minimize the number of input parameters, the optimization is posed in non-

dimensional terms. Table 3 contains the numerical values of all such parameters. In the following sub-sections, theoptimization variables, the objective function and all constraints are defined.

A. Geometric variables

The geometric variables depicted in Fig. 1b have the non-dimensional forms:


14/27


14

1

2

3

4

5

[0.5 0.99]

[0.2 10]

2 [0.01 100]

[0.005 0.02]

[0 0.001]

c f

TBC

x L H

x W L

x t t

x H Z

x t Z

=

=

=

=

=

(28)

with reasonable bounds reported within square brackets. Forboundary condition 1, 0.3b Z= .

B. Objective function

The areal density of the cross section depicted in Fig. 2 is given by:

( )

2

1 2

1 2 3 1

11

x x

x x x x =

+ (29)

and the mass of the panel:

s TBC TBCH b Z t b Z = + (30)

The non-dimensional objective function (for each material) is:

2

TBC TBC

ss

tM H

Z Zb Z

= = + (31)

In order to compare the performance of different materials, the results will be presented in dimensional version,

where the merit function is *s

= . The mass of the thermal barrier layer is included in Eq. (30) to assure that alayer of finite thickness will only emerge if it actually reduces the overall weight of the structure.

C. Constraints

Failure by yielding. The stress along the length of a member at any of the critical points (1-18) due to

mechanical loading can be expressed in non-dimensional form as:

Table 1. Candidate materials and their properties ( Tf0 = 400 K ).

Material T*

[K] Y [MPa] E[GPa] CTE[10-6

/K] ks [W/m K] s [kg/m3]

E T* Tf0( )Y

Nb/Si 1470 370 140 10.5 50.0 6900 4.24

MAR-M246 1090 400 161 16.7 26.0 8440 4.63Inconel 625 1100 265 164 14.0 20.0 8190 6.06SiC-SiC 1640 200 240 4.1 17.3 2900 6.18C-SiC 1810 200 100 2.0 8.7 2000 1.41

TBC (ZrO2) 1.0 3000

Table 2. Thermo-physical properties of fuel at 450 K.

Fuel kf [W/m K] f [m2 /s] f [m

2 /s] Prf f [kg/m3 ] Tcoke [K]

JP-7 0.11 2.48 107 5.34 108 4.64 800 600


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15

( )

, ( ) ( )

( ) ( )

, ,

( ) ( )

i

m x i icool combm m

Y Y Y

i i

m z m x

Y Y

p pf geom g geom

= +

=

(32)

where the superscript (i) identifies the location (Fig. 4). The functions ( )imf and ( )img only depend on the geometric

variables1 5,...,x x . Similarly, the thermal stresses can be expressed as:

( )( )

( )( )

*( )0 0, ( )

,*

0

*( )0 0, ( )

,*

0

1( , )

2 1

1( , )

2 1

if aw fT x i

T x tf panel

Y Y f

if aw fT z i

T z tf panel

Y Y f

E T T T Tf T T

T T

E T T T Tf T T

T T

=

=

(33)

The functions ( )im

f , ( )im

g ,( )

,

i

T xf and

( )

,

i

T zf are summarized in Appendix II7. The Mises yield criterion for mechanical

loads only can be expressed as:

2 2 2

, , , ,

1 18max 2

m x m z m x m z

iY Y Y Y

=

+ +

(34)

The corresponding criterion for thermal loads only is:

2 2 2

, , , ,

1 18max 2

T x T z T x T z

iY Y Y Y

=

+ +

(35)

Table 3. Non-dimensional parameters used in the optimization.

Nb/Si MAR-M246 Inconel 625 SiC-SiC C-SiC

cool Y p 0.027 0.025 0.038 0.05 0.05f

V

415 10

comb Y p 0.00065 0.0006 0.0009 0.0012 0.0012 ( )

2

2

critcrit

f

p Zp

V

=

71.9 10

*

0

aw

aw f

T T

T T

0.59 0.74 0.73 0.53 0.47 Prf 4.6

( )* 0fY

E T T

4.24 4.63 6.06 6.18 1.41

0

aw coke

aw f

T T

T T

0.93

0.3 0.3 0.3 0.3 0.3

f sk k 0.0022 0.0042 0.0055 0.0064 0.013

TBC sk k 0.02 0.039 0.05

TBC s 0.43 0.36 0.37

GG

s

h ZBi

k= 10.7 20.6 26.7 30.9 61.8


16/27


16

and that for combined loads is:

2 2 2

, , , , , , , ,

1 18max 2

m x T x m z T z m x T x m z T z

iY Y Y Y Y Y Y Y

=

+ + + + +

(36)

To ensure that the structure does not yield, imposing combined loading only does not suffice since, in somelocations, the stresses due to thermal and mechanical loads have opposite sign.

Maximum material temperature. To ensure that the structure remains below its maximum allowable

temperature *T , the maximum material temperature, axtf

T (calculated using Eq. (19) at z Z= )), must satisfy the

condition

ax*

0 0

aw tf aw

aw f aw f

T TT T

T T T T

(37)

Maximum coolant temperature. Similarly, the maximum fuel temperature (Eq. (15) at z Z= ) must remain

below that for coking; notably:

ax

0 0

aw fuel aw coke

aw f aw f

T TT T

T T T T

(38)

Maximum pressure drop. The pressure drop is a result of viscous dissipation and other losses in the cooling

channels. To minimize requirements on pumping power, the maximum allowable pressure drop over the 1m length

is taken as 1 MPacritp . Pressure losses at the manifold/panel connections are neglected8. From the definition of

the friction factor,

2

( / )

2

h

f

Z D

f u

= (39)

and the introduction of the non-dimensional parameter

2

2

critcrit

f

p Zp

V

=

(40)

the pressure drop over the length of the structure can be expressed as:

( )

2

2

2

1 h

f Z Zp

H D

=

(41)

where the friction factorfis given by Eq. (3).Thus the constraint is:

critp (42)

7 Equations (32) and (33) also encompass the core member, although in that case (i=9,18) the stresses are in the y-

direction rather than the x-direction.8 This assumption simplifies the calculations, but can easily be relaxed, should there be a need to consider other

components as well.


17/27


17

Manufacturing. To ensure that the optimal designs are compatible with manufacturing constraints, the face and

core member thicknesses are required to be at least 0.4 mm. In non-dimensional form, these constraints are

expressed as:

( )

14

1 3 4

10.0004

2

1 0.0004

f

c

xt Z x

t Z x x x

=

=

(43)

D. Optimization scheme

The fuel flow rate was discretized into 100

intervals over the range 1 4st

V V = . At each

, the objective function (31) was minimized

subject to the constraints (34)-(43) using the

quadratic optimizer MINCON in the MATLAB

Optimization Toolbox. To avoid local minima, a set

of at least 20 randomly generated initial guesses was

combined with the solution at the previous value of

. The best solution among these was assumed to

be the global minimum. In some cases, a manualoptimization scheme was employed to verify the

accuracy of the numerical results.

VI. Principal results

The minimum mass of panels is plotted in Fig. 5

as a function of the mixture richness, : case I in

Fig. 5a and case II in Fig. 5b. When a TBC layer

enables lower weight, the result is plotted as a

dotted curve. The plots convey a wealth of

information, collected below in three highlights.

(i) For boundary condition I, which inducesthe more severe stresses (Fig. 5a), somematerials (notably Inconel and SiC/SiC)

are inadmissible. That is, they do not yield

viable solutions and are thus absent from

the figure. Others only give solutions

above a critical fuel flow rate,crit

. These

include Inconel with a TBC ( 3.6crit = )

and uncoated MAR-M246 ( 1.4crit = ).The remaining materials (the Nb alloy,

MAR-M246 with a TBC and C/SiC)

provide solutions for all flow rates. For the

less stringent boundary condition II,

solutions exist for all materials at all flowrates.

(ii) The two materials (Nb alloy and C/SiC)

that enable the lightest designs for both

boundary conditions have weight

independent of the fuel flow rate. That is,

they can be designed to operate at the

stoichiometric fuel/air ratio, so that extra fuel for cooling is not needed. For the other materials, lighter

designs are obtained at higher flow rates. However, the weight of the extra fuel may obviate the benefit.

Figure 5. Minimum weight of the optimized structures as

a function of the mixture richness, . Different materialsof interest offer solutions. (a) Boundary condition case I; (b)

Boundary condition caseII. The dashed curves refer to

designs that include a thermal barrier coating. Properties for

all the materials are reported in Table 1 .


18/27


18

(iii) The merits of using a TBC are evident from the dotted lines. For case I: (a) a TBC is essential if Inconel is

to be used, but only with 3.6crit = ; (b) the weight of the MAR-M246 design is reduced by a TBC and,more importantly, the coating allows a solution for stoichiometric fuel; and (c) no benefit is obtained for

the Nb alloy. Forcase II, the only benefit of the TBC is the weight reduction for the Inconel design for flow

rates close to stoichiometric.Examples of the optimal panel dimensions for the MAR-M246 are summarized on Figs. 6 and 7. Under case

I, the coated and uncoated geometries differ significantly (Fig. 6); conversely, undercase IIthey are nearly identical,

and only the uncoated geometry has been reported (Fig. 7). The optimal dimensions of C/SiC are independent on the

mixture richness and expressed in tabular form (Table 4).

A few remarks hold for both materials. Forcase II, the panel thickness,H/Z, always hits the lower bound (this is

not true forcase I, where the bending loads imposed by the combustion pressure force a thicker panel). Similarly,

the manufacturing constraints (Eq. (43)) are active for both face and core members, resulting in / 2 0.5c f

t t = at all

values of . Finally, / 0.8L H , implying that the face thickness is roughly 10% of the panel thickness; the

exception is MAR-M246 undercase Iat low .

For MAR-M246, the aspect ratio of the cooling channels, w/L, increases as the mixture richness increases; for

the coated case, an asymptotical value ( / 0.5w L ) is soon approached.Finally notice that, not surprisingly, the thickness of the coating layer decreases as increases; at 4 = almost

no TBC is necessary.

VII. Interpretation and implications

Figure 6. Optimal dimensions for MAR-M246 (boundary condition case I).


19/27


19

Designs tend to become lighter as the

mixture richness increases because the cooling

efficiency increases and hence the thermal

stresses decrease. At sufficiently large , the

thermal stresses become insignificant and the

optimal geometry is dictated solely by

mechanical considerations. The transitionbetween these two regimes occurs at lower

for materials more suited to high temperature

applications; in particular, C/SiC and Nb are

mechanically dominated at all .

Among the materials considered, C-SiC is

the most weight-efficient. The best metallic

system, Nb, is considerably heavier (by a factor

3); the best superalloy (MAR-M246) is even

heavier (by a factor of almost 5). In some

instances, this weight penalty may be obviated

by the structural robustness and ease of

fabrication of metallic systems.

TBCs play a less significant role forscramjet combustion chambers than for turbine

engines. The main benefit of interposing a

thermally insulating layer between the structure

and the hot gases is an increase in the

temperature of the wall exposed to the gases, which, in turn, reduces the

heat flux transmitted to the structure (Eq. (58)). Since the adiabatic wall

temperature is much higher than the maximum allowable material

temperature, an increase in the latter by 200-300 C does not reduce the

heat flux significantly.

VIII. Conclusions and future work

An analytical model has been developed for the prediction of

temperature distributions and thermo-mechanical stresses in an activelycooled multifunctional plate to be employed in the combustion chamber

of a Mach 7 hypersonic air-breathing vehicle. The thermo-mechanical

loads have been estimated based on aerothermodynamics considerations.

A thermal network approach has been used to derive the temperature

distribution, also accounting for the possible presence of a thermal barrier coating. A sandwich panel analysis has

been adopted for the thermo-mechanical stress calculations. The analytic model agrees well with finite element

calculations, with the exception of highly conductive materials. Work is under-way to address this deficiency.

Based on the temperature and stress calculations, the geometry of actively cooled plates has been optimized for

minimum weight under constraints representative of service conditions. The mechanical constraints exerted by the

vehicle have a substantial effect on the thermo-structural stresses and, consequently, on the optimal geometries. A

realistic model of the actual boundary conditions in service is necessary for accurate predictions.

Future work will employ computational fluid dynamics (CFD) calculations to verify the convective correlations

used in the thermal network, coupled with a set of active cooling experiments on near-optimal geometries at heatfluxes representative of hypersonic flight conditions (via a high-power CO2 laser). Also, alternative core topologies

will be studied within the framework presented in this paper, with the goal of ascertaining the optimal combinations

of materials and structural designs.

Figure 7. Optimal dimensions for MAR-M246 (boundary

condition case II).

Table 4. Optimal dimensions for

C/SiC (the results are independent

on in the range considered).

Boundary conditions

Dimension Case I Case II

/L H 0.77 0.84

/W L 0.43 0.56/ 2c ft t 0.16 0.5

/H Z 0.011 0.005


20/27


20

Appendix I

The aerothermodynamic conditions within the combustion chamber are essentially determined by the Mach

number and the shape of the forebody. In the present work, a medium-size vehicle (~10 m long, and ~2.5 m tall) was

assumed, with a compression ramp shaped to generate three oblique shock waves. Although a smooth compression

ramp shaped to offer isoentropic compression would be more efficient, its benefit over a three-wave design rarely

justifies the increased complexity11. All calculations assume a free stream Mach number of 7. Using a standard

thermodynamic approach, the transformation of the working fluid from the front of the vehicle to the outlet of thecombustion chamber (Fig. A1) can be followed.

Conditions at the front of the vehicle (station 0)9

Since both the aerodynamic loads on a vehicle and the mass flow rate of the air entering the engine scale with the

dynamic pressure, optimized hypersonic vehicles are designed to fly within a very narrow range of dynamic

pressures11; the dependence of the dynamic pressure on the atmospheric pressure and the flight velocity, and the

strong variation of atmospheric pressure with altitude dictate optimal trajectories where a specific Mach number is

associated with a specific altitude (Fig A2).The optimal altitude for Mach 7 flight is approximately 30 km. At this

altitude, the air properties can be obtained from the US Standard Atmosphere Chart:

0

0

3

0

0

0

4 2

0

1.197 kPa

227 K

0.0184 kg/m

302 m/s

2114 m/s

8.01 10 m /s

p

T

a

V

=

=

=

=

=

=

(44)

9 The station numbers are not consecutive, consistent with the standard engine thermodynamics notation.

Figure A1. Sketch of a hypersonic air-breathing vehicle with a forebody designed to compress the

air through three oblique shocks (geometry not to scale). The calculated properties of the air at three

locations (stations 0, 3 and 4) are reported (see Appendix 1 for details). Based on the conditions at stations

3 and 4, the properties of the air in the combustion chamber were assumed uniform and idealized as shown.


21/27


21

Here, 0p is the static pressure, 0T the static

temperature,0 the mass density, 0a the speed

of sound, 0V the velocity and 0 the kinematic

viscosity.

Conditions at the inlet of the combustion

chamber (station 3)

The compression system increases the

temperature and the static pressure of the air,

while slowing the flow significantly. Generally

speaking, the efficiency of the combustion

process increases with increasing air

temperature at the inlet of the combustion

chamber. There is, nonetheless, a limit above

which losses due to dissociation and other

effects prevail. These two considerations result

in an optimal temperature at the combustor

inlet, typically3 0 7T T = . This condition,

together with Eq. (44) and the shape and size of

the vehicle (which determines the number of

oblique shock waves), fully characterize the

compression process. The software HAP

(GasTables)11 was used to determine the

conditions of the air at the inlet (for simplicity

assumed to coincide with the end of the

compression stage).

For this design:

3 0

3 0

3 air

3

3

3

3 0

200 0.24 MPa

7 1590 K

1460 kJ/kg

1.5

770 m/s

1150 m/s

0.062

p p

T T

h

Ma

a

V

A A

=

=

=

==

=

=

(45)

whereA3 is the cross-sectional area of the combustion chamber and A0 is the capture area at the inlet of the vehicle

(i.e. the area defined by the streamlines that enter the combustion chamber). If the overall vertical size of the vehicle,

0 2.5 mH = , this results in a combustion chamber height, 3 15.5 cmH = .Conditions at the outlet of the combustion chamber (station 4)

The chemistry of the fuel dictates the stoichiometric fuel/air ratio,st

f , i.e. the proportion in which all reagents

are transformed into combustion products (assuming perfect combustion). The complex chemical composition of JP-

7 makes it extremely difficult to calculate this quantity from first principles. Nonetheless, a reasonable estimate canbe obtained by modeling JP-7 with

12.5 26C H (Ref. 11), which yields:

uel air 0.0675 kg /kgfuel

st

air

mf

m= =

(46)

In actual service, air-breathing engines often operate at fuel/air ratios that deviate considerably from stoichiometric.

By introducing thefuel/air mixture richness, , the actual fuel/air ratio can be expressed as:

Figure A2. Flight trajectories (altitude VS Mach number).

Adapted from Ref. 11. Contours of uniform dynamic pressure

( 20 0 0

2q V= ) and air mass flow rate entering the engine (0 0V )

are depicted. Dynamic pressures of ~ 48 kN/m2 and air mass flow

rates of ~ 50 kg/m s2 are reasonable, resulting in the correlation

~ 7 ~ 30 kma H .


22/27


22

stf f= (47)

This allows calculation of the enthalpy increase of the working fluid within the combustion chamber. At an injection

temperature of approximately 500 K, the increase associated with complete combustion of a unit mass of the fuel is

approximately:, 7 f

27500 kJ/kgPR JP

h (Ref. 11). Neglecting the mass increase of the working fluid due to fuel

injection (the combustion process only adds heat to the working fluid)10, and modeling the combustion process as

isobaric, yield the conditions at the exit of the combustion chamber (station 4):

4 3

4 3 air

0.24 MPa

3320 kJ/kgPR

p p

h h f h

= =

= + =(48)

All other properties of air at station 4 can be calculated from Eq. (48) via the Mollier diagram 11; in particular:

4

3

4

4

2800 K

0.295 kg/m

989 m/s

T

a

=

=

=

(49)

Since the velocity of air does not change significantly within the combustion chamber11

, the Mach number at the exitof the combustion chamber can be calculated as:

34

4

4 4

1.17VV

Maa a

= = (50)

As expected for a Mach 7 vehicle, the flow in the combustion chamber is barely supersonic (consistently with Mach

6-7 denoting the ramjet-scramjet transition).

To be conservative, the temperature in the combustion chamber is assumed to be uniform and equal to the exit

temperature. The chamber will therefore be characterized by the following conditions:

2800 K

3320 kJ/kg1.2

1150 m/s

0.24 MPa

comb

comb

comb

comb

comb

T

hMa

V

p

=

==

=

=

(51)

Adiabatic wall temperature and heat transfer coefficient

The heat flux from the combusting gases to the walls of the combustion chamber is typically calculated using the

following correlation1211:

( )* * comb aw wq St V h h= (52)

10 The validity of this assumption was checked against a more elaborate thermodynamic analysis of the combustion

process (via the software HAP(Combustion) (Ref. 11)): no significant differences were observed in the conditions of

air at station 4.11 As we are in a regime where non-linear effects (such as dissociation) can be dominant, cp is not constant,

requiring the use of the Mollier diagram in converting enthalpies to temperatures.12 This correlation was originally introduced to estimate heat flux to leading edges, but has been largely adopted for

combustion chambers as well.


23/27


23

where *St and * are the Stanton number and the density of the air, both calculated at the Eckert reference

temperature (see below), andcombV is the velocity of the air; awh and wh are the adiabatic and the actual wall

enthalpies, respectively. For turbulent flow, the following correlation is available for the Stanton number:

*

*2 / 5 *1/ 5

0.0287

Pr Rex

St = (53)

where *Pr is the Prandtl number of the air and

*

* *Re comb combx

p V L

RT = (54)

is the Reynolds number.L represents the length-scale of the problem, which in this case can be approximated as the

length of the forebody of the vehicle (~6 m).

The quantities marked with an asterisk need to be evaluated at the reference temperature, *T . The Eckert

reference enthalpy is defined as11:

2

* 0.222 2

comb w combh h Vh += + (55)

where a unitary recovery factor was assumed (equivalent to neglecting the thermal conductivity of the gas). The

reference temperature can be obtained from Eq. (55) via the Mollier diagram. Note that the enthalpy at the wall, wh ,

in Eq. (55) is unequivocally linked to the wall temperature, and therefore is an unknown. Eqs. (52)-(55) were solved

for variousw

T between 800 and 1000 K, and no appreciable difference was observed in *T (and even less on *St );

* 2050 KT = was used in all subsequent calculations. This implies:

* 5 2

* 3

* 7

* 4

6.27 10 N s/m

0.4065 kg/m

Re 4.47 108.47 10St

=

=

= =

(56)

with *Pr 1= and 288 J/kg KR = .The adiabatic wall enthalpy, or recovery enthalpy, is that attained by the stream-line at the wall if perfectly

insulated. With a unitary recovery factor, the adiabatic wall enthalpy is equal to the total enthalpy:

2

2

combaw T comb

Vh h h= = + (57)

Conversion of Eq. (52) into the more customary heat transfer coefficient expression assumes that a constant

specific heat could be used for both the adiabatic and the actual wall temperatures. Although this is not the case, the

error introduced with such an approximation would not be significant, relative to the degree of uncertainty in Eqs.(52)-(53). For 1.35 kJ/kg K

pc = , Eq. (52) can be rewritten as:

( )G aw wq h T T = (58)

where the heat transfer coefficient is

* * 2535 W/m KG comb ph St V c= = (59)


24/27


24

The adiabatic wall temperature is 3050 KawT = .

Static pressure inside the cooling channels,cool

The pressure inside the cooling channels is dictated by optimal combustion considerations, as it greatly affects

the mixing efficiency of air and fuel in the combustion chamber: the higher the Mach number, the lower the

residence time of air molecules in the combustion chamber, which in turn requires higher fuel exit velocities at the

injectors. This is achieved by pressurizing the fuel channels. An analytical expression forcool

p in terms of the Mach

number of the vehicle would require a detailed knowledge of the vehicle design. In the present work, a

representative value 10cool MPa= was used, which is realistic for hypersonic flight conditions in the Mach 7-9range13.

Volumetric flow rate of the fuel (coolant)

The volumetric flow rate of the fuel (per unit width of the combustion chamber) can be expressed as:

20 0 0 0.0082 m /sf air st

f

f f f

m f m f V HV

= = = =

(60)

where 3800 kg/mf = is the density of JP-7, 0 0,V are given by Eq. (44) and 0H is the height of the overall

vehicle, from the leading edge to the bottom of the combustion chamber (0 2.5 mH = ).

Appendix II

All the stress components in Eqs. (22)-(27) can be conveniently expressed in the form of Eqs. (32)-(33) via the

introduction of four non-dimensional functions (refer to Fig. 4 for the locations of the points):( ) ( ) ( ) ( )

, ,, , ,i i i i

m m T x T z f g f f .

They assume different values depending on the boundary condition.

Boundary condition 1

2

2

2

( )

2

2 1at pts 1,4,10,13

2 2

2 1at pts 2,3,11,12

2 2

2 1at pts 6,7,15,16

2 4

2 1at pts 5,8,14,

2 4

f

f f

f

f f

fi

m

f f

f

f f

H t w

t t

H t w

t t

H t wf

t t

H t w

t t

+

=

+

17

at pts 9,18c

w

t

(61)

13 D. Marshall and T. A. Jackson, personal communication.


25/27


25

( )

( )

( )

( )

2

2

2

( )

2

at pts 1,2,5,612

at pts 3,4,7,812

at pts 12,13,16,1724

at pts 10,11,14,1524

0 at pt

f f

f f

im

f f

f f

b

t H t

b

t H t

bgt H t

b

t H t

=

s 9,18

(62)

0 0

( )0 0,

0

at pts 1,5,10,14

at pts 2,6,11,15

at pts 3,4,7,8,12,13,16,17

0

panel tf

aw f aw f

panel tf

iaw f aw f T x

panel

aw f

T T

T T T T

T T

T T T T f

T

T T

+

=

at pts 9,18

(63)

( )

( )0 0

( )

,0 0

0

2at pts 1,5,10,14

2

2at pts 2,6,11,15

2

2 at pt2

f c panel tf

f c aw f aw f

f c panel tfi

T zf c aw f aw f

f panel

f c aw f

A A T T

A A T T T T

A A T T

f A A T T T T

A TA A T T

+

+

+ += +

+ s 3,4,7,8,12,13,16,17

0 at pts 9,18

(64)

Boundary condition II

The thermal functions ( ),

i

T xf and( )

,

i

T zf are the same as forcase I. The mechanical functions are:


26/27


26

2

2

2

( )

2

2 1at pts 1,10

2 2

2 1at pts 2,11

2 2

2 1at pts 6,15

2 4

2 1at pts 5,14

2 4

2

2

f

f f

f

f f

f

if f

m

f

f f

f

f

H t w

t t

H t w

t t

H t w

t tf

H t w

t t

H t

t

+

=

+

at pts 3,4,7,8,12,13,16,17

at pts 9,18c

w

t

(65)

( )

0 1:18i

mg i= = (66)

Acknowledgments

This work was supported by the ONR through a MURI program on Revolutionary Materials for Hypersonic

Flight (Contract No. N00014-05-1-0439).

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