Abstract
A scheme for identifying and visualizing the material properties that limit the performance of candidate materials for actively cooled aerospace propulsion components is presented and illustrated for combustor panels for Mach 7 hypersonic vehicles. The method provides a framework for exploring the nonlinear interactions between design and materials optimization. By probing the active constraints along the border of feasible design space, the limiting properties have been elucidated for a representative group of candidate materials. Property vectors that enhance design options have also been determined. For one of the promising candidate alloys (the Nibased superalloy, INCONEL X750), the possibilities of reclaiming design space and lowering optimal combustor panel weight by tailoring its strength properties are assessed.
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Notes
INCONEL is a trademark of The Special Metals Corporation Group of Companies, Huntington, WV.
Numerical optimizations were performed using the quadratic optimizer MINCON in MATLAB (The MathWorks, Inc., Natick, MA). Several randomly generated initial guesses were used to escape local minima. In some cases, a manual optimization scheme was employed to verify the accuracy of the numerical results.
A grid of 20 by 20 points was used to create these figures.
Abbreviations
 A ^{(i)}, B ^{(i)} :

nondimensional functions
 \( C_{x}^{(i)} ,C_{z}^{(i)} \) :

nondimensional functions
 c _{ p,f } :

specific heat of the coolant (J/kg K)
 \( D_{x}^{(i)} ,D_{z}^{(i)} \) :

nondimensional functions
 E :

Young’s modulus (Pa)
 f :

fuel/air mass ratio
 f _{ st } :

stoichiometric fuel/air mass ratio
 F ^{(i)} :

nondimensional function
 G _{1}, G _{2} :

nondimensional functions
 h _{ G } :

heattransfer coefficient on the combustor side (W/m^{2} K)
 \( h_{G}^{\text{nom}} \) :

nominal combustion heattransfer coefficient for a Mach 7 vehicle (W/m^{2} K)
 k _{ s } :

thermal conductivity of the material (W/m K)
 k _{ f } :

thermal conductivity of the coolant (W/m K)
 L :

height of cooling channel (m)
 p _{cool} :

pressure in the coolant (Pa)
 p _{comb} :

pressure in the combustion chamber (Pa)
 s :

trajectory denoting the border of feasible design space
 T _{ aw } :

adiabatic wall temperature in the combustion chamber
 T _{coke} :

coking temperature of the coolant (K)
 T _{ f } :

coolant temperature (K)
 \( T_{f}^{0} \) :

coolant entry temperature (K)
 \( T_{\text{fuel}}^{ \max } \) :

maximum coolant temperature (K)
 \( T_{\text{solid}}^{ \max } \) :

maximum temperature in the material (K)
 T ^{(i)} :

temperature at location (i) in the material (K)
 T ^{*} :

maximum allowable temperature in the material (K)
 t _{ c } :

core web thickness (m)
 t _{ f } :

face sheet thickness (m)
 t _{TBC} :

thickness of TBC (m)
 \( \dot{V}_{\text{eff}} \) :

volumetric fuel flow rate per unit width scaled by combustor area ratio (m^{2}/s)
 W :

width of cooling channel (m)
 Z :

panel length (m)
 Δp :

viscous pressure drop across the panel (Pa)
 Δp ^{c} :

critical pressure drop (Pa)
 α :

coefficient of thermal expansion of the material (K^{−1})
 β :

inverse length constant in the temperature distribution (1/m)
 χ :

nondimensional heattransfer coefficient for combustion (\( {{h_{G} } \mathord{\left/ {\vphantom {{h_{G} } {h_{G}^{\text{nom}} }}} \right. \kern\nulldelimiterspace} {h_{G}^{\text{nom}} }} \))
 ϕ :

equivalence ratio (f/f _{ st })
 v _{ f } :

kinematic viscosity of the coolant (m^{2}/s)
 v :

Poisson’s ratio of the material
 ∏:

nondimensional constraint activity index
 ρ _{ f } :

mass density of the coolant (kg/m^{3})
 σ _{ult} :

critical strength of a CMC material (MPa)
 σ _{yield} :

yield strength of a metallic material (MPa)
 \( \sigma_{{VM,{\text{comb}}}}^{(i)} \) :

von Mises stress due to combined thermal/mechanical loads at location (i) (MPa)
 \( \sigma_{{VM,{\text{mech}}}}^{(i)} \) :

von Mises stress due to mechanical loads at location (i) (MPa)
 \( \sigma_{{VM,{\text{therm}}}}^{(i)} \) :

von Mises stress due to thermal loads at location (i) (MPa)
 \( \sigma_{m,x}^{(i)} \) :

mechanical stress at location i along the x direction (Pa)
 \( \sigma_{m,z}^{(i)} \) :

mechanical stress at location i along the z direction (Pa)
 \( \sigma_{T,x}^{(i)} \) :

thermal stress at location i along the x direction (Pa)
 \( \sigma_{T,z}^{(i)} \) :

thermal stress at location i along the z direction (Pa)
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Acknowledgments
This work was supported by the ONR through a MURI program on Revolutionary Materials for Hypersonic Flight (Contract No. N000140510439). The authors are thankful to David Marshall, Teledyne, and Thomas A. Jackson, Daniel J. Risha, and William M. Roquemore, AFRL, for insightful discussions.
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Manuscript submitted July 22, 2008.
Appendices
Appendix I: Synopsis of the code
The protocol employed for thermostructural analysis and design optimization consists of the following steps (Figure A1). (1) A range is defined for the expected heating loads (represented by the heattransfer coefficient h _{ G } of the hot gases) and the cooling capability (represented by the coolant flow rate per unit width of panel \( \dot{V}^{\text{eff}} \)). (2) A material is selected with the property profile indicated in Table I. (3) At each point in \( \left( {h_{G} ,\,\dot{V}^{\text{eff}} } \right) \) space, the design parameters are systematically varied over a prescribed range and the temperatures and stresses calculated for each combination. Upon comparison with material and coolant properties, the viability of the design is ascertained. (4) If provided solutions exist, the design is optimized^{Footnote 2} for minimum mass, subject to a number of design constraints. Otherwise, if a solution is not found, the point \( \left( {h_{G} ,\;\dot{V}^{\text{eff}} } \right) \) is deemed external to the design space. (5) Once the entire design space has been scanned^{Footnote 3} for each candidate material, comparisons are made of materials on the basis of structural robustness (namely, the extent of feasible solution area in \( \left( {h_{G} ,\;\dot{V}^{\text{eff}} } \right) \) space) and weight efficiency.
Analytical Model for the Temperature Distributions
Analytical expressions for the temperatures at critical locations in the panel are obtained via a thermal network approach, subject to four simplifying assumptions: (1) the adiabatic wall temperature T _{ aw } and the heattransfer coefficient h _{ G } are uniform on the hot face; (2) the rest of the panel is thermally insulated, whereby all the heat is removed by forced convection in the cooling channels; (3) longitudinal conduction is negligible in the panel; and (4) the mixingcup temperature T _{ f } is used to model the coolant, hence neglecting crosssectional variations in fluid temperature.
Based on the thermal network in Figure A2, the temperature in the fluid is
and the temperature distributions at the 18 locations depicted in Figure A3 are
where F ^{(i)} and β depend on (Reference 1 for details): geometry of the panel (W, L, t _{ f }, t _{ c }); thermal conductivity of the material k _{ s }; thermal properties of the coolant (thermal conductivity k _{ f }; kinematic viscosity ν _{ f }; volumetric specific heat ρ _{ f } c _{ p,f }); prescribed heattransfer coefficient on the hot side h _{ G }; and prescribed volumetric flow rate \( \dot{V}_{\text{eff}} \). Importantly, these functional dependencies are intertwined, thus precluding straightforward interpretation of the effect of each quantity on the temperature distribution.
The thermal stresses in the panel depend on two temperature differences: (1) across the top face ΔT _{face} (z) and (2) between the two faces ΔT _{panel} (z). For the sake of simplicity, these differences are averaged in the x direction, and can be expressed as
where G _{1} and G _{2} depend on the same quantities as F ^{(i)}.
The accuracy of this analytical model was verified with a number of selected computational fluid dynamics and finite element simulations; the temperature distribution was captured by Eq. [2] to within ~1 pct and the temperature gradients were captured by Eq. [3] to within ~8 pct.[1]
Analytical Model for Stress Distributions in the Panel
Stresses are induced in the panel by mechanical phenomena (combustion chamber pressure p _{comb} and coolant pressure p _{cool}) and thermal phenomena (ΔT _{ tf } and ΔT _{panel}). Assuming generalized plane strain conditions (no rotation about the x and y axes), the mechanical stresses (membrane + bending) at any of the 18 locations depicted in Figure A3 are given by
where the functions A ^{(i)} and B ^{(i)} depend on the panel geometry and the design scenario (Figure 2, details in Reference 1) and ν is the Poisson’s ratio of the material.
Similarly, the thermal stresses can be expressed as
with E and α the Young’s modulus and the coefficient of thermal expansion of the material, respectively. The quantities \( C_{x}^{(i)} ,\;D_{x}^{(i)} ,\;C_{z}^{(i)} ,\;{\text{and}}\,D_{z}^{(i)} \) are a function of geometry only (different expressions pertain to the x and z stresses because of the anisotropy of the core). Reference 1 provides details.
The signs of the various stress contributions at the 18 critical locations are catalogued in Figure A3 for both design scenarios. The accuracy of this analytical model was verified with a number of selected finite element simulations; the stress distribution was captured to within ~10 pct at the top face and ~20 pct at the bottom face.[1] The internal corners exhibit stress intensifications and the agreement is somewhat worse. In reality, localized plastic deformation will ameliorate the stress concentration.
Appendix II: Activity indices
For any choice of input parameters (h _{ G } and \( \dot{V}_{\text{eff}} \)), a quadratic optimizer is invoked to select the geometry that minimizes the weight of the panel while meeting all the prescribed design constraints. Temperature and stress distributions are calculated with the methodology described in Appendix I and listed in Table III. This appendix describes the nature of the applied constraints, dividing them in three groups: group I includes constraints that enforce the resistance of the panel under the prescribed thermal and mechanical loads; group II includes systemimposed constraints (on coolant temperature and allowable pressure drop); and group III contains design and manufacturing constraints (minimum gages).
Group I Constraints
(1–3)Yielding or Fracture
For metals failure is defined as the onset of yielding. The Von Mises criterion is used. Under the simultaneous action of thermal and mechanical loads (Appendix I), the Von Mises stress at any critical location (i) is given by
with the stress components given by Eqs. [4] and [5].
The temperature of the solid at each of these 18 critical locations (Eq. [2]) is used to identify the associated (temperaturedependent) yield strength, modeled as linearly decreasing when T ^{(i)} < T ^{*}, with T ^{*} the softening temperature of the alloy (Table I).[1]
The constraint can then be expressed as: \( \mathop \text{max}\limits_{i = 1  18} \left\{ {\sigma_{{VM,{\text{comb}}}}^{(i)} /\sigma_{\text{yield}} (T^{(i)} )} \right\} \le 1 \). The constraint activity is monitored through the ratio
To avert yielding, it is insufficient to restrict only the combined stresses because the thermal and mechanical stresses may have opposite signs in some locations (Figure A3). Two more constraints must be imposed, one for mechanical stresses (\( \sigma_{{VM,{\text{mech}}}}^{(i)} \)) and the other for thermal stresses (\( \sigma_{{VM,{\text{therm}}}}^{(i)} \)). The resulting constraint activity indices are
Welldesigned CMCs typically fail when the normal stress along the primary fiber orientation attains either the ultimate tensile strength or the compressive strength. Assuming for simplicity that the strengths in tension and compression are identical and temperature independent (reasonable for SiC/SiC and C/SiC),[13,14] the ensuing constraint is
where the ultimate strength σ _{ult} is assumed temperature independent. Activity indices analogous to Eqs. [7] through [9] arise.
(4) Material softening
The maximum temperature in the panel, found at the exit (Eq. [2]), must remain below the material softening limit T ^{*}. Above this limiting temperature, the loadbearing capacity decreases dramatically. The constraint can be expressed as: \( T_{\text{solid}}^{ \max } \le T^{*} \). The constraint activity is monitored through the ratio
Group II Constraints
(5) Fuel coking
The maximum temperature in the fuel, taken at the exit (Eq. [1]), must remain below the coking temperature (Table II). At the coking temperature, the hydrocarbon fuel undergoes chemical reactions causing deposition of carbon and fouling of the heattransfer surfaces.[15] The constraint can be expressed as \( T_{\text{fuel}}^{ \max } = T_{\text{fuel}} (Z) \le T_{\text{coke}} \). The constraint activity is monitored through the ratio
(6) Pressure drop
A pressure drop Δp is caused by viscous dissipation and other losses in the cooling channels. The Δp is quantified through a correlation for the friction factor.[16] Pressure losses at the manifold/panel connections are neglected. To minimize requirements on pumping power, the maximum allowable pressure drop (over a 2m combustor length) is taken as Δp ^{c} ≈ 0.1 MPa. The constraint is then simply Δp ≤ Δp ^{c}. The constraint activity is monitored through the ratio
Group III Constraints
The remaining constraints refer to the allowable range of the geometric variables to be optimized, depicted in Figure 1. These are as follows.
(7) TBC thickness
The presence of a TBC can be enabling in some cases. It functions by decreasing the maximum temperature in the solid. To avoid unnecessary costs, the optimizer will only add a coating if the thermal benefit will allow a lowerweight solution. In practice, in high heat flux situations, TBCs can be deposited to a maximum thickness of ≈0.3 mm without spalling.[17] Additionally, the TBC must be significantly thinner than the substrate t _{TBC} ≤ 0.25t _{ f }. The ensuing dimensional constraint for the TBC layer is \( t_{\text{TBC}} \le t_{\text{TBC}}^{ \max } \), where \( t_{\text{TBC}}^{ \max } = { \min }\left\{ {0.25t_{f} ,0.3\,{\text{mm}}} \right\} \). The constraint activity is monitored through the ratio
(8–11) Additional constraints on dimensions
Additional constraints are imposed on the dimensions L, W, t _{ f }, and t _{ c } (Table III), resulting in four more activity indices.
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Vermaak, N., Valdevit, L. & Evans, A. Materials Property Profiles for Actively Cooled Panels: An Illustration for Scramjet Applications. Metall Mater Trans A 40, 877–890 (2009). https://doi.org/10.1007/s116610089768y
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DOI: https://doi.org/10.1007/s116610089768y
Keywords
 Thermal Barrier Coating
 Softening Temperature
 Coolant Flow
 High Heat Transfer
 Hypersonic Vehicle