Proeedings of IMECE2007 2007 ASME International Mehanial Engineering Congress and Exposition Novembe1-15, 2007, Seattle, Washington, USA IMECE2007-41348 THE DESIGN SPACE OF SUPERALLOY-BASED ACTIVELY COOLED COMBUSTOR WALLS FOR H 2 -POWERED HYPERSONIC VEHICLES Lorenzo Valdevit Mehanial and Aerospae Engineering Department University of California, Irvine Irvine, CA 92697-3975 valdevit@ui.edu Natasha Vermaak Materials Department University of California, Santa Barbara Santa Barbara, CA 93106-5050 natasha@engineering.usb.edu Frank W. Zok Materials Department University of California, Santa Barbara Santa Barbara, CA 93106-5050 zok@engineering.usb.edu A. G. Evans Materials Department University of California, Santa Barbara Santa Barbara, CA 93106-5050 agevans@engineering.usb.edu ABSTRACT The alls of ombustion hambers used for air-breathing hypersoni vehiles are subjet to substantial thermomehanial loads, and require ative ooling by the fuel in onjuntion ith advaned material systems. Solutions based on metallis are preferable to erami matrix omposites due to their loer ost and greater strutural robustness. Previous oruggested that a number of metalli materials (e.g. Nikel, Copper and Niobium alloys) ould be used to fabriate atively ooled sandih strutures that ithstand the thermomehanial loads for a Mah 7, hydroarbon-poered vehile (albeit ith different eight effiienies). Hoever, this onlusion hanges hen the Mah number is inreased. This ork explores the feasibility of the Nikel superalloy MAR- M246 for a ide range of Mah numbers (7-12). Sine hydroarbon fuels are limited to Mah 7-8, Hydrogen is used as the oolant of hoie. A previously derived analytial model (appropriately modified for gaseous oolant) is used to explore the design spae. The relative importane of eah design onstraint is assessed, resulting in the distillation of essential guidelines for optimal design. NOMENCLATURE A = area [m 2 ] a = speed of sound [m/s] b = idth of the atively ooled panel [m] p = speifi heat [J/kg K] C μ C k = oeffiient in the vis./temp. relation [K] = oeffiient in the ond./temp. relation [K] D = diameter [m] E = Young s modulus [Pa] f = frition fator H = thikness of the atively ooled panel [m] h = heat transfer oeffiient [W/m 2 K] k = thermal ondutivity [W/m K] L = height of the ooling hannel [m] m = mass flo rate [kg/s] Ma = Mah number Nu = Nusselt number p = pressure [Pa] Pr = Prandtl number q = speifi healux [W/m 2 ] R = universal gas onstant [J/kg K] = thermal resistane per unit idth [(W/m K) -1 ] r = thermal resistane per unit area [(W/m 2 K) -1 ] Re T t u x,y,z Z = Reynolds number = temperature [K] = thikness [m] = veloity of the oolant [m/s] = idth of the ooling hannel [m] = spatial oordinate [m] = length of the atively ooled panel [m] 1 Copyright 2007 by ASME Donloaded From: http://proeedings.asmedigitalolletion.asme.org/ on 02/26/2018 Terms of Use: http://.asme.org/about-asme/terms-of-use
Greeymbols = thermal expansion oeffiient (CTE) [K -1 ] = non-dimensional parameter = non-dimensional parameter = non-dimensional parameter = differene μ = dynami visosity [Pa s] v = Poisson s ratio = onstraint ativity parameter = mass density [kg/m 3 ] = areal density of the ross-setion of the panel = normal stress [Pa] = fin temperature: T ( y) T fuel [K] Subsripts and supersripts 0 = inlet onditions = referene value 1,2 = generi symbols, ore = relative to the ore eb, ool = relative to the ooling hannels f, fae = relative to the fae sheet f, fuel = relative to the fuel (oolant) fin = relative to ondution/onvetion in the ore eb G = relative to the hot gases in the ombustion hamber h = hydrauli = horizontal (x) diretion i = generi loation in the ross-setion m = mehanial max = maximum min = minimum panel = relative to the hole ross-setion of the panel s = relative to the solid material the panel is made of T = thermal ft = top side of the top fae fb = bottom side of the top fae = beteen ore ebs (i.e. near the middle of the hannel) x,y,z = along the respetive diretions Y = yielding = due to the temperature drop aross the panel T panel T tf = due to the temperature drop aross the top fae sheet * = generi symbol 1. INTRODUCTION The potential of air-breathing hypersoni vehiles for spae aess and military appliations has been learly demonstrated. Although the fundamental aerodynamis and the feasibility of stable supersoni ombustion have been studied in detail, the integration ith vehile design remains a hallenge. Among the most daunting tasks is the development of materials and strutures that ithstand the enormous thermo-mehanial loads in severe environments. The alls of the ombustion hamber and the leading edges of the vehile are the most ritial surfaes. This paper disusses metalli sandih panels, atively ooled by the fuel, as thermo-strutural solutions for ombustion hamber alls. An optimal design ode that enables ranking of materials on the basis of eight effiieny has already been presented, and results have been derived for Mah 7, hydroarbonpoered vehiles [1]. In this ork e plan to extend those results to higher Mah numbers. Sine hydroarbon is not realisti for Mah >7-8, hydrogen is assumed as the fuel (and hene the oolant) of hoie. The overarhing goal is the extration of design guidelines that allo the seletion of the optimal material and struture for a ide range of Mah numbers. A neessary step is the ability to predit ith onfidene the thermo-mehanial loads as a funtion of the Mah number of the vehile. Unfortunately, this is not trivial, beause of the omplex aerothermodynamis in the ombustion hamber, and the influene of the details of the vehile design (e.g. shape, size and numbers of injetors, possible by-pass solutions, et ). In this ork, e take a more fundamental approah. We investigate the design spae of ombustor liners made of Nikel superalloy (MAR-M246, see Table 1 for properties), for a ide range of thermo-mehanial loads and oolanlo rates. A previously derived analytial approah for the alulation of temperatures and stresses in the panel is modified to aount for the ompressibility of the oolant (setion 2). For every hoie of the external loads (here envisioned as input parameters), the geometry of the panel is optimized on the base of eight-effiieny, subjet to a number of design onstraints (setion 3). When the optimization ode fails to onverge to a solution, the ombination of input parameter is onsidered external to the design spae. This allos the onstrution of design maps (setion 4). Investigation of the optimal designs for a speifi value of the healux ( q G = 3MW / m 2 ) allos a lear understanding of the effet of oolanlo rate and internal pressure (setion 5). Conlusions and future ork follo. 2. THE THERMO-MECHANICAL MODEL This setion presents an analytial model for the alulation of temperature and stress distributions in an atively ooled panel (Fig. 1) under uniform flux boundary onditions. Assumptions leading to realisti estimates of healuxes and oolant pressures are presented elsehere [2]. The oolant is assumed gaseous (hydrogen), and the dependene of its physial properties on temperature and pressure is addressed. Setion A presents the thermal netork model used to alulate the temperatures in the solid at a number of disrete points, at any given ross setion. Stress alulation is presented in Setion B. A. Temperature distribution To failitate the derivation of analyti estimates for the temperatures, three simplifiations are invoked: 2 Copyright 2007 by ASME Donloaded From: http://proeedings.asmedigitalolletion.asme.org/ on 02/26/2018 Terms of Use: http://.asme.org/about-asme/terms-of-use
(i) The top fae of the struture is subjeted to a uniform healux, q G, hereas the bottom fae and the sides are thermally insulated. Consequently, all the heat reeived through the top fae is arried aay by the ooling fluid. (ii) No heat is onduted along the length of the panel (zdiretion), either in the struture or in the ooling fluid. This assumption results in slightly onservative temperature estimates. (iii) The fuel temperature is uniform ithin the hannels at any given ross-setion: T fuel = T fuel (z) only. Namely, T fuel is the mixing-up temperature. The temperature at every ross-setion z is alulated using an eletrial analogy (Fig. 2a). Five thermal resistanes (per unit length in the z-diretion) are needed to haraterize the problem (refer to Fig. 1b and the nomenlature for definition of the geometri variables): Condutive resistane aross the fae (y-diretion): R fae = / t, R fae = / Condutive resistane along the fae (x-diretion) 1 : R h = ( + t /2)/4 Convetive resistane on the oolant side (for the portion of the fae in diret ontat ith the oolant): R ool = 1/ h Combined ondutive and onvetive resistane of the ore eb (modeled as a one-dimensional thermal fin [3, 4]): 1 2h R fin = tanh 1 L 2 1 2h k t t t s. Here, h is the heat transfer oeffiients on the oolant side, and is the thermal ondutivities of the solid material. Flo in the ooling duts is assumed to be turbulent and fully developed. Under these onditions, h is given by the orrelation [5, 6]: h = k f D h Nu = k f D h ( f / 2)(Re 1000) Pr 1+ 12.7 f /2 Pr 2/3 1 ( ) here k f is the thermal ondutivity of the oolant, Pr is the Prandtl number, D h = 2L /( + L) is the hydrauli diameter of the duts, and Re is the Reynolds number in the duts, defined as: Re f ud h μ f = μ f m 1 ( ) D h bh (1) (2) ith μ f being the dynami visosity of the fluid, m the presribed mass flo rate of the fuel, = 1 L /(H(+ t )) the areal density of the ross-setion and f the frition fator. For fully developed turbulenlo in the range 210 4 < Re < 10 6, the folloing orrelation an be used for the frition fator [5, 7]: f = 0.046 Re 1/5 (3) Eqs. (1)-(3) allo alulation of the thermal resistanes R ool and R fin as a funtion of the geometri variables, the fuel properties, and flo rate. If e assume that the speifi heat of the oolant, p, f, is not a strong funtion of temperature (Table 2), then the temperatures of the fluid inreases linearly along the z- diretion, beoming maximum at the outlet. A simple energy balane yields: T fuel ( z)= T f 0 + q b G z (4) m p, f For reasonable values of the parameters, the oolant temperature varies onsiderably over the length of the panel. Hene, variations in density, visosity and ondutivity (and onsequently in frition fator (Eq. (3)) and heat transfer oeffiient (Eq. (1)) annot be negleted. The impliation is that the temperature in the solid is not linear in z. The model an be simplified into the effetive netork of Fig. 2b, haraterized by the five resistanes R 1, R 1, R 2, R 2, R h, here: R 1 = R fae 2 R 1 = R fae 2 R 2 = R fae R 2 = R fae 2 + R ool 2 + R fin The unknons are the temperatures at the top side of the top fae, above the eb and aay from it ( T ft and T ft, respetively), the orresponding temperatures at the middle of the top fae ( T fm and T fm, respetively) and the horizontal heat flux, q h. Five equations are needed to lose the system: (5) 1 The horizontal resistane is not properly ondutive, as onvetion ours along one of the sides: finite elements analyses revealed that using an effetive length equal to half the physial length yields aurate results (hene the fator 4). 3 Copyright 2007 by ASME Donloaded From: http://proeedings.asmedigitalolletion.asme.org/ on 02/26/2018 Terms of Use: http://.asme.org/about-asme/terms-of-use
T ft T fuel + (q G 2q h / t ) r 2 T ft T fuel + (q G + 2q h / ) r 2 T ft T fm T ft T fm T fm T fm = q h r h here the thermal resistanes per unit area have been defined as: = 1 2 r 2 = 1 + 1 2 h r 2 = 1 + r 2 k fin s 2h r fin = tanh 1 L 2 t ( r h = + t 2 )/2 2h k t s 1 (6) (7) temperature at every point in the solid does not have the same z-variation as the mixing-up temperature of the fuel. For the alulation of the thermal stresses, to temperature differenes are relevant: (i) the temperature drop aross the top fae (in the y- diretion), both over and beteen the ore ebs: T f = 2 q G t T f = 2 q G 1+ t f (ii) T panel T panel (10) the temperature drop aross the entire panel ross setion, measured from the middle of the top fae to the bottom fae (the bottom fae is assumed to be isothermal and at the same temperature as the bottom end of the ore eb), both over and beteen the ore ebs: ( ) 1 2 t f t 1+ r fin 1 L r 1 0 1+ 2 t f r 2 ( L ) 1 2 t f 0 t r fin (11) Solving the linear system in Eq. (6), gives: ( ) T ft T fuel 1+ r 2 ( ) T ft T fuel 1+ r 2 T fm T fuel r 2 T fm T fuel r 2 q h = q G (8) Here, ( y) 0 is the non-dimensional fin temperature, namely: ( y) 0 2h = T ( y osh ( L y) ) T fuel t = T ( 0) T fuel 2h osh L t (12) here, and are non-dimensional parameters given by: = r + 2r t h 2 ( f / t + / ) r h + 2 r 2 / t + 2 r 2 / r h + 2r 2 ( / t + / ) = r h + 2 r 2 / t + 2 r 2 / r = 2 r 2 r h + 2 r 2 / t + 2 r 2 / Notie that Eq. (8) holds at any ross-setion. Sine r 2 and r 2 depend on the oolant temperature, though, the (9) and y is the oordinate oriented along the fin. B. Stress Distributions Problem statement and boundary onditions The thermo-mehanial stresses depend on the onstraint exerted on the plate by the surrounding omponents of the vehile. We assume that the bending effets due to the pressure in the ombustion hambers an be eliminated by appropriately supporting the panel along the bak fae [1]. This results in the idealized boundary onditions of Fig. 3a. Uniform thermal expansion (but no panel-level bending) is permitted along either diretion. The external pressure does not ause the panel to bend globally, but the internal pressure an bend individual top fae segments. The use of rollers instead of fritional supports allos uniform thermal expansion of the panel (ith no bending). 4 Copyright 2007 by ASME Donloaded From: http://proeedings.asmedigitalolletion.asme.org/ on 02/26/2018 Terms of Use: http://.asme.org/about-asme/terms-of-use
To further simplify the problem, both the pressure drop and the temperature variation along the panel length are negleted. This assumption, ombined ith the imposed boundary onditions, ensure that generalized plane strain onditions are attained along the z diretion 2. Furthermore, it requires that the alulations be performed on one ross-setion only. Beause the properties of the oolant are affeted by temperature, the temperature differenes that drive the thermal stresses (Eqs. (10)-(11)) vary ith z. Conomitantly, the yield strength of the metal dereases ith inreasing temperature, indiating potential failure at the outlet. The impliation is that a number of ross-setions should be analyzed for potential failure to guarantee onservative results. The next step is the identifiation of the mosailuresuseptible loations in the struture at eah ross-setion. The stresses due to both pressure and thermal loads vary along the member lengths and thiknesses, but are periodi in the idth of the panel, b (i.e. they are the same at eah ell). Beause the thermal and pressure loads often indue stresses of opposite sign, it is not straightforard to establish a priori the loation of first yield. For this reason, a set of 9 ritial points has been identified (Fig. 3b). The presene of internal pressure in the ore hannels (hih imposes signifiant tensile stresses on all members) ombined ith the relatively stubby shape of the optimized members makes it unneessary to design against bukling [8-16]. Consequently, failure of the struture is averted provided that the Mises stress in the most ritial loation remains in the elasti range. The analyti methods used to determine the stresses in sandih strutures onsider the fae and ore members in eah unit ell as independent beams; the onnetion beteen members is modeled using translational (and some degree of rotational) onstraints [8-16]. The auray of this approah is dependent upon the aspet ratio of eah element, and dereases as the elements beome stubbier. When this approah is used, finite element analyses are needed to ensure that preditions at the optimal geometries are suffiiently aurate. Stresses due to oolant pressure The pressure p ool inside the ooling hannels subjets the ore members to uniform tension and indues a ombination of tension and bending on the faes. Using the notation of Figs. 1b and 3b, the resulting stresses in the ore members (point 9) are: p ool ore,y p ool p ool ore,z p ool = t = p ool ore,y p ool (13) 2 Generalized plane strain refers to onditions herein the ross-setions z = 0 and z = Z are not alloed to rotate relative to eah other (although they are free to translate). and in the fae segments: L /2 / L /2 + ( / ) 2 / 2 at pt 2 = L /2t p f + ( / ) 2 / 4 at pt 5 ool L /2 ( / ) 2 / 4 at pt 6 L /2 at pts 3,4,7,8 p ool fae,x p ool fae,z p ool = p ool fae,x p ool ( ) 2 / 2 at pt 1 (14) Consistent ith beam analysis, the through-thikness stresses are negleted. Thermal stresses At time zero the entire struture is at the same temperature as the fuel at the inlet, T f 0, and no thermal stresses are present. Upon exposure to the ombusting gases, the temperature distribution reahes a steady state. Beause the boundary onditions allo uniform expansion of the panel, the problem an be modeled by superimposing the stresses due to to temperature differenes. Contribution 1. By ignoring x- variations, the average temperature differene beteen the top and bottom faes, T panel = T panel ( + T panel ) 2 auses the panel to deform uniformly in both the x and z diretions, induing ompression in the top fae and tension in the bottom fae [17]. Note that it is not legitimate to ignore the ore strething stiffness in the z-diretion, here the ore behaves as a ontinuous plate. When the ore and the bottom fae are at the same temperature, the resulting stresses are: T panel fae,x = T panel fae,z = E T panel 21 ( ) E T panel 21 ( ) E T A panel f + A ( 1 ) ( 2A f + A ) E T panel A f ( 1 ) ( 2A f + A ) ( ) at pts 1,2,5,6 at pts 3,4,7,8 at pts 1,2,5,6 at pts 3,4,7,8 (15) here A f = (+ t ) and A = (H 2 )t are the rosssetional areas of the fae and the ore in a unit ell, respetively. Note that negleting the ore stiffness (letting A 0 in Eq. (15)) results in a perfetly bi-axial state of stress 5 Copyright 2007 by ASME Donloaded From: http://proeedings.asmedigitalolletion.asme.org/ on 02/26/2018 Terms of Use: http://.asme.org/about-asme/terms-of-use
in both faes. Contribution 2. By ignoring x-variations, the temperature differene aross the top fae thikness, T f = T f, auses the upper surfae of the top fae to experiene ompression and the loer surfae to be in tension, giving: T tf fae,x T tf = = fae,z E T tf 21 ( ) E T tf 21 ( ) at pts 1,5 at pts 2,6 (16) 3. CONSTRAINTS For every hoie of the input parameters (the oolant pressure, p ool, mass flo rate, m, inlet temperature, T f 0, and the inoming healux, q G ), designs are aeptable if three onditions are met: (i) the thermo-mehanial stresses do not exeed the (temperature dependent) yield strength of the material; (ii) the maximum temperature in the material is belo a ritial level (defined as the temperature at hih the yield strength drops signifiantly); (iii) the pressure differene neessary to overome fritional dissipation in the ooling hannels is ithin alloable limits. Condition (i) is expressed in terms of Mises stress: (i) max m,x i=1 9 Y (T (i) ) + (i) T,x Y (T (i) ) (i) m,z Y (T (i) ) 2 (i) T,z Y (T (i) ) + (17) (i) + m,x Y (T (i) ) + 2 (i) (i) T,x Y (T (i) ) + m,z Y (T (i) ) + 2 (i) T,z Y (T (i) ) 2 Eq. (17) guarantees that the panel survives the ombined appliation of thermal and mehanial stresses. It does not guarantee survivability after the appliation of thermal or mehanial stresses independently. To ensure onservativeness, to more onditions similar to Eq. (17) must also be imposed for these alternative senarios. Condition (ii) an be expressed as: max T { ft,t ft } T * (18) here T * is the temperature at hih the yield strength of the material drops unaeptably (Table 1). The pressure drop is not linear in z, sine the properties of hydrogen (and onsequently the oolant veloity) are a funtion of temperature (and hene of z). The total pressure drop over the length of the panel an be obtained by integration: p = = Z Z 0 0 f 2 f ()u z 2 ()f z () z D h dz = 2 m 2 f () z ()D z h H 2 b 2 ( 1 ) 2 dz (19) here f is the frition fator (Eq. (3)), f is the density of the oolant (Table 2), and Z the areal density and the length of the panel, respetively. 4. DESIGN MAPS We ant to explore the feasibility of atively ooled panels made of the Nikel superalloy MAR-M246 as ombustor alls for hypersoni vehiles. In order to enompass a large range of Mah numbers, e define a parameter spae for healux and oolanlo rate: q G = 1 6 MW/m 2 m = 0.05 25 kg/s (20) Additionally, three different oolant pressures are onsidered: p ool = 5, 10, 30 MPa. The inlet temperature of the oolant is T f 0 = 200 K. The ombustor (and hene the panel) is assumed to be 1m long (Z = 1 m) and 30m ide (b = 30 m). Notie that the results apply to any idth, provided that the flo rate be appropriately saled. Also note that m is the flo rate into one panel. If all the four alls of the ombustor are ooled simultaneously, this value ill be 2-3 times smaller than the overall flo rate at the injetors. Again, ith appropriate saling of the flo rate the results apply to any onfiguration. The geometry of the panel is required to lie ithin the folloing spae: L = 5 20 mm W = 1 50 mm = 0.4 5 mm t = 0.4 5 mm (21) For every hoie of the input parameters (q G, m, p ool ), the mass of the panel is minimized subjet to the onstraints of setion 3. The quadrati optimizer FMINCON, inluded in the pakage MATLAB, as used; multiple random guesses ere used to assure aurate solutions. When the optimizer failed to find a solution for a given hoie of the input parameters, it as assumed that no solution existed for that partiular parameter ombination, resulting in a point outside the design spae. The results an be summarized in the design map of Fig. 4. The lines separate parameter ombinations that allo a feasible solution (inside the bell-shaped urve) from unaeptable 6 Copyright 2007 by ASME Donloaded From: http://proeedings.asmedigitalolletion.asme.org/ on 02/26/2018 Terms of Use: http://.asme.org/about-asme/terms-of-use
hoies of parameters (outside the urve). For any value of oolanlo rate and internal pressure, Fig. 4 shos the maximum healux that an be imparted to the struture ithout violating any of the onstraints. As every point in the map ontains the result of an optimization proess, very large amount of information is ontained in this figure. Here e try to onvey the major onlusions: (i) At any given oolant pressure, there exists an optimal flo rate, m opt 3 kg/s, resulting in the maximum possible healux. This optimal flo rate is (ii) essentially independent on the oolant pressure. An inrease in oolant pressure is benefiial; that is, higher healuxes an be safely tolerated at larger oolant pressures. To rationalize these onlusions, it is neessary to analyze the optimization results in more detail. Every onstraint an be assoiated to an ativity parameter, : a value = 1 implies that the onstraint is ative (and thus deign-limiting). Fig. 5 shos the onstraint ativity parameters for the optimal geometries relative to a healux, q G = 3 MW/m 2. Notie that alo rates loer than m min 0.5 kg/s, no geometry an satisfy all the onstraints. Sine the only ative onstraint at m min is the resistane to the thermo-mehanial loads (for all the three values of p ool ), e an onlude thaor m < m min the ooling effet is not suffiient to maintain the stresses belo the yield point of the material. As the flo rate inreases, the pressure drop onstraint eventually beomes ative (and remains suh afterards). Notie that this event is retarded ith inreasing oolant pressure. This is easily explained using the equation of state (Eq. (22)). An inrease in the gas pressure implies a proportional inrease in its density; sine the mass flo rate is kept onstant, this must imply that the oolant slos don. The pressure drop has a quadrati dependene on the oolant veloity (Eq. (19)), resulting in a substantial advantage. This is responsible for the larger design spae obtained at higher oolant pressures (Fig. 4). It is interesting to look at the optimal dimensions of the strutures as a funtion of the flo rate (and the oolant pressure). Fig. 6 shos the variation of the optimal height (L) and idth () of the hannels. Notie that at lo flo rates, the height of the hannel assumes its minimum alloable value (5 mm). After the pressure drop onstraint beomes ative, L inreases to ompensate for the inrease in flo rate (Eq. (19)). The idth of the hannel,, also inreases, but a muh loer rate: this is due to the fat that the idth of the hannel largely ontrols the bending stress in the fae sheet members (Eq. (14) ). Fig. 6 learly shos the advantage of a large oolant pressure: sine the oolant is sloer at large p ool, the hydrauli diameter (and hene L and ) an be loer ithout violating the pressure drop onstraint, resulting in stress mitigation. The thiknesses of the fae sheet and ore members almost alays assume their minimum alloable value (0.4 mm), and hene are not plotted (the exeption is the fae sheet thikness for p ool = 30 MPa, hih slightly inreases at large flo rates). As the flo rate is inreased, eventually the opposing requirements on the panel geometry imposed by the pressure drop and the stress onstraints result in impossible solutions. In this study, it has been shon that an inrease in the oolant pressure up to 30 MPa is benefiial, both at lo and high flo rates. This trend is not expeted to hold at arbitrary large oolant pressures. As the oolant pressure inreases, the mehanial loads beome proportionally more severe (Fig. 6). At p ool = 30 MPa, the onstraint assoiated ith the resistane to the mehanial loads is ative for moslo rates. If the pressure is inreased ell over 30 MPa, the resistane of the struture is likely to beome a onern. Future ork is needed to explore this regime, and identify the optimal oolant pressure. 5. CONCLUSIONS AND FUTURE WORK Atively ooled metalli strutures made of the Nikel superalloy MAR-M246 are proven to be effetive as ombustor alls for hypersoni vehiles over a large range of thermomehanial loads ( q G = 1 6 MW/m 2 and p ool = 5 30 MPa ). Although the atual loads in servie onditions depend on a number of fators (e.g. the nature of the mission, the size and shape of the vehile, the dynami pressure), this range should enompass the lo-to-moderate Mah number flight onditions for H 2 -poered hypersoni vehiles (Mah 7-12) [18-22]. The design spae for the ombination of parameters (oolanlo rate, healux and oolant pressure) has been presented. An optimal flo rate, m 3 kg/s, has learly emerged, independent on the oolant pressure. Somehat surprisingly, the pressure in the ooling hannels has a benefiial effet on the design spae; that is, inreasing the pressure from 5 to 30 MPa results in (i) higher aeptable healux and (ii) larger range of alloable flo rates. This trend is expeted to reverse at higher pressures, hen the mehanial resistane of the struture beomes a onern. Future ork inludes the extension of these results to a larger range of parameters, the study of the effet of the oolant temperature at the inlet, and the exploration of different metalli materials. Also, a formal derivation of the thermomehanial loads as a funtion of the vehile Mah number is underay. ACKNOWLEDGMENTS This ork as supported by the ONR through a MURI program on Revolutionary Materials for Hypersoni Flight (Contrat No. N00014-05-1-0439). The authors are thankful to David Marshall of Teledyne, and Thomas A. Jakson and William M. Roquemore of AFRL for insightful disussions. REFERENCES 1. Valdevit, L., Vermaak, N., Zok, F. W., Evans, A. G., Optimal design of atively ooled panels for sramjets. AIAA Journal, Submitted, 2007. 7 Copyright 2007 by ASME Donloaded From: http://proeedings.asmedigitalolletion.asme.org/ on 02/26/2018 Terms of Use: http://.asme.org/about-asme/terms-of-use
2. Valdevit, L., Vermaak, N., Zok, F. W., Evans, A. G., The effet of Mah number on materials seletion for hypersoni vehile design. In preparation, 2007. 3. Lu, T.J., L. Valdevit, and A.G. Evans, Ative ooling by metalli sandih strutures ith periodi ores. Progress in Materials Siene, 2005. 50(7): p. 789-815. 4. Valdevit, L., et al., Optimal ative ooling performane of metalli sandih panels ith prismati ores. International Journal of Heat and Mass Transfer, 2006. 49(21-22): p. 3819-3830. 5. Bejan, A., Convetion heat transfer. 3rd ed. 2004, Hoboken, N.J.: Wiley. xxxi, 694. 6. Gnielinski, V., Ne Equations for Heat and Mass- Transfer in Turbulent Pipe and Channel Flo. International Chemial Engineering, 1976. 16(2): p. 359-368. 7. Moody, L.F., Frition fators for pipe flo. Trans. ASME, 1944. 66: p. 671-684. 8. Rathbun, H.J., F.W. Zok, and A.G. Evans, Strength optimization of metalli sandih panels subjet to bending. International Journal of Solids and Strutures, 2005. 42(26): p. 6643-6661. 9. Valdevit, L., J.W. Huthinson, and A.G. Evans, Struturally optimized sandih panels ith prismati ores. International Journal of Solids and Strutures, 2004. 41(18-19): p. 5105-5124. 10. Valdevit, L., et al., Strutural performane of nearoptimal sandih panels ith orrugated ores. International Journal of Solids and Strutures, 2006. 43(16): p. 4888-4905. 11. Wei, Z., F.W. Zok, and A.G. Evans, Design of sandih panels ith prismati ores. Journal of Engineering Materials and Tehnology - Trans. of the ASME, 2006. 128(2): p. 186-192. 12. Wiks, N. and J.W. Huthinson, Optimal truss plates. International Journal of Solids and Strutures, 2001. 38(30-31): p. 5165-5183. 13. Wiks, N. and J.W. Huthinson, Performane of sandih plates ith truss ores. Mehanis of Materials, 2004. 36(8): p. 739-751. 14. Zok, F.W., et al., Strutural performane of metalli sandih panels ith square honeyomb ores. Philosophial Magazine, 2005. 85(26-27): p. 3207-3234. 15. Zok, F.W., et al., Design of metalli textile ore sandih panels. International Journal of Solids and Strutures, 2003. 40(21): p. 5707-5722. 16. Zok, F.W., et al., A protool for haraterizing the strutural performane of metalli sandih panels: appliation to pyramidal truss ores. International Journal of Solids and Strutures, 2004. 41(22-23): p. 6249-6271. 17. Boley, B.A. and J.H. Weiner, Theory of thermal stresses. 1960, Ne York: Wiley. 18. Heiser, W.H. and D.T. Pratt, Hypersoni airbreathing propulsion.1994, Washington: AIAA. 19. Walters, F.M. and O.A. Buhmann, Heat transfer and fluid flo analysis of Hydrogen-ooled panels and manifold systems, NASA CR-66925. 20. Youn, B. and A. F. Mills, Cooling panel optimization for the ative ooling system of a hypersoni airraft, Journal of Thermophysis and Heat Transfer, 1995. 9(1): p.136-143. 21. Buhmann, O.A., Thermal-strutural design study of an airframe-integrated sramjet, 1979. NASA CR- 3141. 22. D. Marshall, personal ommuniations. APPENDIX 1 THE PHYSICAL PROPERTIES OF HYDROGEN Hydrogen is a highly ompressible gas, and as a result, many of the properties hange signifiantly over the relevant range of temperatures (and pressures). For large healuxes and lo flo rates, the temperature inrease in the oolant over a 1m-long panel an be as large as 600K (Eq.(4)). Over this range, e need to aounor properties variation. Sine the speifi heat of hydrogen doesn t vary onsiderably over the range T = 50-1000K, e assume it onstant and equal to p = 14, 600 J/kg K. Similarly, e ignore variations in the Prandtl number and assume Pr=0.68. Density is assumed to vary ith oolant pressure and temperature, hereas visosity and thermal ondutivity vary ith temperature only. Density For temperatures loer than 1000K (the range of interest for ative ooling, sine injetion typially ours at temperature loer than or roughly equal to 1000K), H 2 an be modeled as an ideal gas. Iollo that the density is a funtion of temperature and pressure aording to: = p ool (22) RT here R is the gas onstanor H 2 (Table 2). It is legitimate to ignore the pressure drop over the hannels, and treat the pressure as uniform. Consequently, the density hanges uniquely as a funtion of the temperature. As the fluid gets hotter, its density dereases. Sine the mass flo rate of H 2 is onstant (by ontinuity), iollos that the volumetri flo rate (and the veloity in the duts) inreases toards the end of the hannels. Dynami visosity The variation of the dynami visosity ith temperature an be modeled ith Sutherland s formula: 3/2 T μ = μ 0 + C μ T 0 T + C μ T 0 here the parameters for hydrogen are given in Table 2. The dynami visosity is not signifiantly affeted by pressure. 8 Copyright 2007 by ASME Donloaded From: http://proeedings.asmedigitalolletion.asme.org/ on 02/26/2018 Terms of Use: http://.asme.org/about-asme/terms-of-use
Thermal ondutivity The thermal ondutivity is fairly unaffeted by pressure, but shos some temperature sensitivity. A Sutherland temperature dependene an be assumed: 3/2 T k = k 0 + C k T 0 T + C k T 0 here the parameters for hydrogen are given in Table 2. Note that these formula are typially valid for temperatures loer than 600K; e speulate that the error on t be too large in the range 600K<T<1000K. TABLES Table 1 Materials properties for the nikel superalloy, MAR-M246 Property s [kg/m 3 ] [W/mK] E [GPa] s [10-6 /K] s Y 0 [MPa] d Y /dt [MPa/K] T * [K] Value 8440 25.94 161 16.7 0.3 800-0.564 1089 Table 2 Physial properties of hydrogen (speifi heat and Prandtl number are assumed temperatureindependent for simpliity) Property p [J/kgK] Pr R [J/kgK] μ 0 [Pa s] T 0 μ [K] C μ [K] k 0 [W/mK] T 0 k [K] C k [K] Value 14,600 0.68 4,016 0.876 10 5 293 27 0.162 250 110 9 Copyright 2007 by ASME Donloaded From: http://proeedings.asmedigitalolletion.asme.org/ on 02/26/2018 Terms of Use: http://.asme.org/about-asme/terms-of-use
FIGURES Figure 2 (a) Thermal resistane netork used for the temperature alulations. (b) Effetive netork. Figure 1 (a) Artist rendition of a hypersoni air-breathing vehile. The proposed multifuntional atively ooled plate is shon in the inset. (b) Shemati of the atively ooled plate ith thermo-mehanial loads, variables definition and oordinate system. Figure 3 (a) Boundary onditions used for the thermomehanial stress alulations. (b) The ritial points onsidered for failure at any given ross-setion. 10 Copyright 2007 by ASME Donloaded From: http://proeedings.asmedigitalolletion.asme.org/ on 02/26/2018 Terms of Use: http://.asme.org/about-asme/terms-of-use
Figure 4 Design map for the Nikel superalloy MAR-M216 for 3 different values of oolant pressure. Notie ho inreasing the oolant pressure signifiantly expands the design spae. The inset shos the most relevant portion of the map. Figure 5 Constraint ativity parameters, for optimal designs at a presribed healux, q G = 3 MW/m 2. A parameter =1 indiates that the onstraint is ative (and thus design-limiting). 11 Copyright 2007 by ASME Donloaded From: http://proeedings.asmedigitalolletion.asme.org/ on 02/26/2018 Terms of Use: http://.asme.org/about-asme/terms-of-use
Figure 6 Optimal dimensions for the panels optimized for a presribed healux, q G = 3 MW/m 2. The other to dimensions (fae sheet thikness,, and ore eb thiknesses, t, alays assume the minimum value of 0.4 mm (the exeption is for the ase p ool = 30 MPa, hih inreases slightly at large flo rates). 12 Copyright 2007 by ASME Donloaded From: http://proeedings.asmedigitalolletion.asme.org/ on 02/26/2018 Terms of Use: http://.asme.org/about-asme/terms-of-use