Statistical, Modular Systems Integration Using

48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition 4 - 7 January 2010, Orlando, Florida

AIAA 2010-278

Statistical, Modular Systems Integration Using Combined Energy & Exergy Concepts John H. Doty1 Engineering Management & Systems The University of Dayton Dayton, OH, 45469

José A. Camberos2 Multidisciplinary Technologies Center U. S. Air Force Research Laboratory Wright-Patterson AFB, OH 45433

Downloaded by John Doty on November 30, 2015 | http://arc.aiaa.org | DOI: 10.2514/6.2010-278

ABSTRACT

This paper details statistical concepts to systems-level applications with relevance to integration, operation, and optimization of engineering components and systems. A physicsbased exergy analysis is combined with system performance goals. Designed experiments are used to pre-determine relevant simulation points for the analysis in order to develop the statistical models most effectively and efficiently. The results of the simulation are then processed via advanced statistics to create a surrogate model that identifies the component and/or system within desired or anticipated operational ranges. These statistical surrogate models represent the system in a modular fashion. In this manner, a statistical module may be interfaced independently from its origination and a “system of systems” may be built from the surrogate modules that may be used to efficiently investigate engineering trades and perform preliminary design studies. Quantitative examples from aerospace components and systems are used to demonstrate the overall process.

I. Introduction

S

tatisitcal concepts have been applied in the aerospace engineering process for many years. Rheinfurth and Howell present an overview and summary of statistical engineering applications1. Physics-based designs, however, have only received more emphasis in the recent years2,3,4,5,6,7,8,9. A major issue with the implementation of both concepts simultaneously is that advanced inferential statistics are just that—advanced, and require requisite education and experience to actually implement on complex aerospace systems, which are in turn themselves complex. Couple the requirement for advanced knowledge with the capability to perform advanced physics-based modeling and simulation in computational fluid dynamics (CFD), and very few individuals possess the expertise to oversee and/or implement such a multi-disciplinary analysis and design program. As a result, we often perform such advanced studies in teams comprised of Subject Matter Experts (SMEs) that are individually knowledgeable but not necessarily systems-oriented. The overall product of the process is often a non-integrated and often sub-optimal system-of-systems. The methodology outlined herein applies a top-down systems approach to integration implemented using physics-based analyses that are modular and extensible. For example, a turbine module from one system may be used in another system operating under similar (but not necessarily the same) conditions in a different system. Any level of fidelity simulation model may be used to develop the statistical surrogate model, making the process scalable, realistic, and adaptable to a design process. A methodology for combined statistical and physics-based analysis and design is used to efficiently study advanced concepts where experimental data is not available. Herein, a physics-based approach is referred to as one that incorporates the conservation equations along with 2nd Law feasibility constraints as well as operational 1 2

AFRL Researcher, University of Dayton, Department of Engineering Mgt. & Systems, AIAA Senior Member Research Aerospace Engineer, Air Vehicles Directorate, AIAA Associate Fellow

This paper is declared a work of the U.S. Government and is not subject to copyright protection in the United States. The views expressed in this paper are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U. S. Government.

1 AMERICAN INSTITUTE OF AERONAUTICS & ASTRONAUTICS This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.


requirements for performance. While many newer models and methods are being deployed that used this physicsbased approach10,11,12,13,14,15,16,17, much more needs to be done in order to put the process in the hands of working engineers. The present focus is to extend these concepts in such a fashion that the engineer may benefit from the development of these advanced techniques while still maintaining relevance at the project or design level. The ultimate benefit is to leverage the SME (advanced statistical and work flow knowledge) in order to increase engineering productivity and raise the individual engineer’s awareness level to the systems level by being able to perform multiple interdependent system interactions efficiently in a modular fashion. The process methodology will be presented in a workflow manner such that the development of the modular system-level components is clear. The process is itself modular and various sets of computational and analytical tools may be used at any point in the process, depending upon the desired level of fidelity for the end product. This modular concept supports the typical “building block” approach employed in systems engineering (e.g. subcomponent, component, subsystem, system, and “system-of-systems). With a system identified in such a manner, a system-level assessment may be performed from the individual integrated parts. Note that the process is “open box” in nature, enabling the engineer to choose the requisite level of fidelity for each component and/or subsystem. The goal is to be able to identify the system statistically, use the surrogate model in lieu of the timeintensive computational tools, “assemble” the system, interface the components, prescribe the inputs and select the desired outputs, and “run” the assembled model. The value added in this methodology is that the engineer can focus on the interpretation of the results, perform impact assessments and trades easily, and help form the basis for learning the important parameters to guide preliminary design and optimization.

II. Concept Incorporating a physics-based approach into advanced analyses and designs offers several advantages compared to other methodologies7,8,11-17. In addition to satisfying the governing equations for a system, the physical feasibility criteria are automatically satisfied, ensuring that the process considered may actually occur. Figure 1 presents a conceptual and architectural illustration of the system concept detailed in this investigation. The left panel conceptually represents a generic system which may have multiple inputs, outputs, constraints, as well as performance goals for the system. The right panel of Figure 1 represents an actual system, that of a turbojet engine.

Illustration of an actual turbojet system http://commons.wikimedia.org/wiki/File:Jet_engine.svg

Conceptual sketch of a generic system

Figure 1: Illustration of generic system concept with many different inputs, outputs, constraints, and performance goals as an example model for an actual system (turbojet engine). For the system represented in the right panel of Figure 1, we may consider, for example, the input analysis or design parameters to be the flight Mach number (M), compressor pressure ratio (CPR), and turbine inlet temperature ). Similarly, constraints may be maximum revolutions per minute (RPM) of the spool, minimum and ( . Outputs might be thrust, specific maximum CPR, minimum and maximum M, minimum and maximum impulse, exit velocity, etc. Additionally, a performance goal (which may also be a combination of output responses), might be maximum thrust with minimum fuel flow rate. Varying levels of fidelity may be employed in the computational models to simulate the response of the system for a given set of inputs. A typical low-fidelity model might be a one-dimensional, steady flow, perfect gas air model cycle analysis code18. On the other hand, a much more complex, and therefore higher physical fidelity

2 AMERICAN INSTITUTE OF AERONAUTICS & ASTRONAUTICS


model, might focus on just the compressor and consider three-dimensional, unsteady, compressible flow with dynamic blade interactions19. Additionally, one may employ actual experimental data for the operation of the system20. Regardless of the source of the data, the conceptual model of the system illustrated in the left panel of Figure 1 remains valid. We seek to employ statistical design of experiments and advanced inferences to determine the surrogate system identification model(s) that emulate the system and its interaction with the surroundings. In general terms, system identification attempts to determine interactions that characterize the system’s response for given bounds21. There are three major areas of system identification that are typically used to characterize a system: the “white box” model, the “grey box” model, and the “black box” model. The white box model is the most fundamental and highest fidelity, of the system identification subset and assumes one knows sufficient mathematical physics to fully describe the behavior of the system. Therefore, the white box model is essential its own system identification and is usually the result of detailed experimental and/or validated computational models. A simple example of a white box system model is a fully-specified mass-spring-damper system with known properties, characteristics, and system responses (could be modeled or experimental data). The grey box model is a lower fidelity model of the system than the white box model. In the grey box model, one seeks reasonable approximations and physically consistent trends, rather than “truth” models. The grey box models are often empirical in nature and use extensive reduced models and correlations to characterize a system’s behavior. It is presumed one knows the general mathematical physics to partially describe the behavior of the system. A simple example of a grey box system model is a heat transfer model with known properties and empirical coefficients, such as Newton’s law of cooling. Most engineering models are classified as grey box. On the other hand, black box models are often quite general in nature and apply over a wide class of inputs and responses. The goal is to choose a basis function that is scaled and/or normalized such that the model provides an approximation for many varied classes of problem. An example of such a low fidelity black box model may be the probabilistic density function such as a sigmoid function (Gaussian “bell” distribution). Note: The generalized approach suggested herein may employ any level of system identification for each module for each of the “box models” in any combination, although it is typically best to be consistent in application of level of fidelity for the system-of-system integration process. Extending the concept from a simple process or subsystem to a more complicated system-of-systems is reasonably straightforward using statistically-realized system identification concepts. The consistency of system integration is aided by employing physics-based models and exergy accounting as a consistent measure of integrated performance7. Additionally, the physics-based approach also helps illuminate areas where loss mechanisms reduce system-level performance. The concept of combining single subsystems into an integrated system is illustrated in Figure 2, with the left panel representing the turbojet subsystem, the middle panel representing a wing subsystem, and the right panel as the combined turbojet/wing system.

Subsystem 1: Turbojet

Subsystem 2: Wing section

Combined system-of-systems

Figure 2: Illustration of two individual systems combined into a single, integrated system. Left panel: isolated turbojet; middle panel: isolated wing section; right panel: integrated wing/jet system. The two subsystems illustrated in Figure 2 were analyzed and optimized individually and then combined into a system-of-systems in order to demonstrate the surrogate methodology as it supports the system identification process. A supersonic turbojet was modeled using simplified thermodynamic cycle analyses employing perfect gas air computations and utilizing typical component efficiency parameters22. The station notation and illustration of the



turbojet model are illustrated in Figure 3. Note the inclusion of the reference dead state for exergy-based computations. Also, the normal shock wave standing off the inlet in supersonic flow is assumed to act as a “device” with requisite loss characteristics acting as part of the engine system. Each other device is clearly labeled in the figure in preparation for the modular development of the surrogate models for each subcomponent of the turbojet system.

Figure 3: Subsystem 1: Turbojet model illustrating station notation and device enumeration. The wing was modeled as a simple rectangular wing with diamond airfoil shape. The left panel of Figure presents a conceptual sketch of a generic wing section. For the rectangular wing section, the taper ratio (ratio of tipto-root chord) is equal to unity. In order to be consistent with the turbojet gas dynamic model, the perfect gas model was again employed along with shock-expansion theory for the analysis of the external flow field around the wing section23. The station notation and device enumeration is presented in the sketch on the right panel of Figure 4. The shocks and expansions were again treated as “devices” in preparation for the statistical surrogate models required in the system identification process.

Conceptual sketch of a generic wing section

Shock-expansion theory representation of diamond airfoil

Figure 4: Subsystem 2: Wing section model illustrating station notation and device enumeration. Often, each individual subsystem is analyzed/optimized separately and then combined into a total system. This “off-the-shelf” design concept often negatively impacts integrated system performance. This degradation of performance will be demonstrated via the statistically-based system identification process by first optimizing the turbojet and the wing as separate systems, then optimizing the combined system operating under the same conditions. Combined energy and exergy concepts will be compared and contrasted in the analyses and optimizations.


III. Methodology


The overall methodology required to combine individual systems into an integrated system will be detailed in this section. The overall goal is to characterize each subsystem statistically as a surrogate for the actual analysis which, itself, obeys the governing physical laws of the system subject to the appropriate constraints and boundary conditions. Any model and level of fidelity representation of the physical system may be employed in this process, making it extensible, general, and robust. The result of the overall methodology is a relatively simple set of equations that represent the individual subsystems and/or components in a modular fashion. As long as sufficiently broad and relevant range of operating conditions are utilized in the development of the models and modules—the combination of the modules may be performed after-the-fact (“posteriori”). This enables the working engineer to utilize the modules in any desired computational environment with ease and portability (e.g. spreadsheet, calculator, computer subprogram, etc.). One can effectively “drag & drop” a surrogate module from one system to another as long as the operating environments are similar to that in which the models were developed. Depending upon the fidelity of the computational or experimental data employed in the development of the surrogates, there is minimal loss of accuracy of the original calculation compared to the surrogate calculations. A work flow representation of the system identification process employed in this study is illustrated in Figure 3. The baseline methodology in the top block refers to the one-case-at-a-time analysis typically employed for a given set of inputs (see left panel of Figure 1). This methodology is amenable to “looping” or repetitive analysis in computational codes and forms the basis for direct optimizations. The bottom block of Figure 3 illustrates the more complex process path which involves advanced statistical design and evaluation of the results, formulation of the surrogate models that identify the system, and then porting these results to standalone modules that simulate the system under consideration. The results of both the baseline and surrogate methodology are then compared for overall computational accuracy and efficiency. Only the surrogate methodology will be described in detail.

Figure 5: Illustration of work flows for baseline and surrogate methodologies. Baseline methodology taken to be the development of the analysis as basis for optimization. The surrogate methodology uses the same analysis, but represents the results via surrogate equations which are then used for the optimization. Note: The keys in the system identification process is proper identification of operating ranges, correctly pre-identified analysis points, data sufficiency and quality, correct inferential statistical interpretation of results, and proper surrogate model development. A Subject Matter Expert (SME) sufficiently versed in Designed Experiments and advanced statistical inference should be employed during this portion of the process. Thereafter, using the models is relatively straightforward.



The surrogate methodology employed in this investigation requires several steps in order to properly represent the system a posteriori modular fashion. An understanding of the physical operation of the system to be modeled is coupled to an advanced statistical design matrix for evaluation and interpretation. The major steps in the surrogate methodology are listed below: 1) Decide upon model level of fidelity. E.g.: a. Experimental data, high-fidelity computational analysis (“White box”), b. Combined physics and empirical/phenomenological data (“Grey box”), c. Simplistic, generalized relationships (“Black box”), 2) Select relevant operating ranges for each subsystem. a. A wider range of operating conditions for each system makes each surrogate module more general and portable, but also requires more runs required to properly capture the physics of the modeled systems (particularly nonlinear systems), 3) Select operating points within the desired ranges to statistically represent the modeled system, a. This is best performed using Design of Experiments (DOE), 4) Perform the analyses using steps 1 to 3, 5) Use the results of the analyses from step 4 to determine statistically significant input terms, interactions, and the appropriate surrogate model for each subsystem, 6) Build the modules that identify the system, 7) Use the surrogate models as “stand-ins” for the original models from step 1, 8) Compare the results of the surrogate models from step 7 to those of the original models from step 1. As might be expected from a quick perusal of the above-outlined surrogate methodology, there is a noted overhead in process compared to the baseline “standard” methodology. As in all engineering efforts, there exists a tradeoff between alternative methodologies. The purpose and intent of the development of the surrogate models and system identification process is to enable repeated utilization of the modules in preliminary design. In such cases, the efficiencies gained upon utilization of the surrogate models more than makes up for the initial overhead for the development time. Qualitatively, a properly developed surrogate model may yield accurate results in microseconds compared to the hours/weeks that may be required for experimental and/or high-fidelity computations. Additionally, the portability and usability of the results in modular fashion by non-Subject Matter Experts more than justifies the development effort of the surrogate modules. Lastly, the system identification process readily provides a basis from which to study the influence of uncertainty and error propagation from a systems level interpretation. (This topic will be further addressed in upcoming papers).

IV. Theory As mentioned earlier, a lower-fidelity computational model was selected for demonstration purposes. Each model of the individual system has its own requisite theory and will be addressed separately. The systems considered in this investigation consist of a supersonic turbojet and supersonic diamond-shaped rectangular wing. Each individual system’s theory will be outlined first, and then the statistical design and evaluation process will be detailed. While engineering-level models (“grey box”) are suitable for preliminary trades, higher fidelity computational or experimental data may be easily substituted in the modular surrogate model development supporting the system identification process. The systems considered in this investigation consist of a supersonic turbojet and supersonic diamond-shaped rectangular wing. Each model of the individual system has its own requisite theory and will be addressed separately.

A. Turbojet Equations As mentioned earlier, a lower-fidelity model set was selected for demonstration and efficiency purposes. These engineering-level models (“grey box”) are suitable for preliminary trades, but higher fidelity computational or experimental data may be easily substituted in the modular surrogate model development supporting the system identification process. The following general assumptions are used for the turbojet model: 1) 2) 3) 4) 5)

Steady flow, Local macroscopic thermodynamic equilibrium, One-dimensional, uniform flow at each station, Air is treated as a thermally and calorically perfect gas with constant properties, Fuel addition influences mass and energy addition, but not mixture properties,


a. I.e. the combustion products are still treated as high temperature air, 6) Losses in each component (“device” as noted in Figure 3), are incorporated via standard engineering relationships 7) Kinetic energy of the flow is neglected except at stations 0, 1, and 6 (Figure 3) relative to other energy forms, 8) Turbulent, well-mixed flow, 9) Potential energy changes of the flow are neglected relative to other energy forms, 10) Body forces and viscous effects are negligible, With reference to Figure 3, the conservation equations are augmented with the appropriate equations of state and entropy/exergy equations in order to determine the flow path.


Dead State (Station 10): Given a flight altitude and the 1976 US Standard Atmosphere24, the values for density, pressure, temperature, and speed of sound are obtained. Additionally, reference properties of enthalpy, entropy, and exergy are specified at this altitude. Station 10 serves as the operational dead state for the turbojet system, viz: , , ,

,

(1) (2) (3) (4)

Free Stream (Station 0): The free stream conditions are the same as the dead state reference for the specified altitude, except that the velocity is not zero. , , ,

, , ,

(5) (6) (7) (8) (9) (10)

Device “A”, Normal Shock Wave (Station 0 to 1): The continuity, momentum, and energy equations are solved simultaneously along with the thermal and caloric equations of state22. Steady, uniform flow, with no heat transfer or work is assumed. Additionally, entropy generation and exergy change are calculated across the device.

(11) (12) (13)

: : :

(14) :

(15) :

(16) (17)

∆

(18) (19) (20)


Device “B”, Diffuser (Station 1 to 2): The continuity, momentum, and energy equations are solved simultaneously along with the thermal and caloric equations of state22. A diffuser total pressure ratio is used to account for irreversibilities, and is defined as: ,

⁄

(21)

,

Entropy generation and exergy change are calculated across the device. Kinetic energy at the exit of the diffuser is neglected relative to enthalpy and incoming kinetic energy. : :

,

,

:

(25) :


(22) (23) (24) (26)

:

(27)

~0

(28)

∆

(29) (30) (31)

Device “C ”, Compressor (Station 2 to 3): The continuity, momentum, and energy equations are solved simultaneously along with the thermal and caloric equations of state22. Kinetic and potential energies are neglected. A compressor pressure ratio (CPR) is assumed to be provided, and is given by: 3

CPR

(32)

2

Lastly, an isentropic efficiency is used to account for irreversibilities and is defined as: ,

,

,

,

(33)

,

Entropy generation and exergy change are calculated across the device. Kinetic energy at the exit of the diffuser is neglected relative to enthalpy and incoming kinetic energy.

(34) (35) (36)

: : :

,

,

,

:

(37) (38)

:

(39)

~0

(40)

∆

~0

~0

(41) (42) (43)

Device “D”, Combustor (Station 3 to 4):


The continuity, momentum, and energy equations are solved simultaneously along with the thermal and caloric equations of state22. Kinetic and potential energies are neglected relative to the other energy terms. Air properties are assumed unchanged after the heat addition process. A combustor exit temperature is presumed known (input) consistent with a maximum turbine inlet temperature. The energy content per mass of fuel is used to back-calculate the required fuel flow rate for the given temperature rise in the combustor. Entropy generation and exergy change are calculated across the device. A combustor pressure loss factor is assumed and is defined as: ⁄

1

,

(44) (45) (46)

: : :

/

1

(47) (48)

,

1 / HeatingValue/

/


:

∆

(49) :

(50)

~0

(51)

~0

~0

(52) (53) (54)

Device “E ”, Turbine (Station 4 to 5): The continuity, momentum, and energy equations are solved simultaneously along with the thermal and caloric equations of state22. Kinetic and potential energies are neglected. An isentropic efficiency is used to account for irreversibilities and is defined as: ,

,

, ,

(55)

,

The compressor is assumed to be work-matched to the turbine in that no additional turbine work is required for auxiliary devices, giving: |

|

|

|

(56)

Entropy generation and exergy change are calculated across the device.

(57) (58) (59)

: : :

,

, |

,

, | ,

:

∆

(60) (61)

:

(62)

~0

(63)

~0

~0

(64) (65) (66)

Device “F”, Nozzle (Station 5 to 6): The continuity, momentum, and energy equations are solved simultaneously along with the thermal and caloric equations of state22. Kinetic energy of the inlet stream (Station 5) is neglected relative to the exit stream (Station 6).


The nozzle is assumed to operate adiabatically, have no work interactions, and expand to atmospheric pressure. An isentropic efficiency is used to account for irreversibilities and is defined as: ,

,

, ,

,

(67)

Entropy generation and exergy change are calculated across the device. : : :

,

,

,

:

(68) (69) (70) (71) (72)

:

(73)


(74) ∆

~0

(75) (76) (77)

Turbojet Performance Parameters: With the flow path evaluation performed, the overall performance of the turbojet may be characterized. ), the thrust, propulsive power, and specific Assuming the exit flow equilibrates to atmospheric pressure ( thrust are, respectively:

(78) (79) 1

(80)

B. Wing Section Equations The external supersonic air flow over the wing section illustrated in Figure 4 is modeled using two-dimensional shock-expansion theory23,24. Depending upon the wedge angle ( ) and angle of attack ( ) for the wing section, each station flow path (or device) may be a shock wave or an expansion fan. Therefore, careful logic must be implemented to check for flow path turning before applying the appropriate equations for a shock wave or an expansion fan. Oblique Shock Wave Model: 1) of The oblique shock wave flow geometry is illustrated in Figure 6. Supersonic flow upstream ( a flow turn ( ) produces an oblique shock wave of angle ( ) with a downstream Mach number that may be supersonic or subsonic. The weak shock wave solution produces supersonic downstream Mach number as illustrated the lower branch of the so-called “ " diagram (Figure 7). The strong shock wave produces subsonic flow downstream of the flow turn and is typically not realized for external flows. There is a maximum amount of flow turning for which the shock wave remains “attached” near the surface. In typical applications, the flow turning is known, and the shock wave angle is iterated via the diagram, or the equation24: 2

(81)


With the shock wave angle known, the remaining downstream properties are calculable in terms of the ), specific heat ratio ( ), and upstream normal component of the upstream Mach number ( , 24 properties as :


Figure 6: Conceptual sketch of supersonic incoming flow turning through an angle ( ) producing an oblique shock wave of relative angle ( ).

Figure 7: The

diagram illustrating the relationship between flow turning ( ) and shock wave angle ( ) for a given upstream Mach number (M). ( 1.4 .

(82)

, / ,

,

(83)

/

, ,

(84)

,

1

,

1

(85) (86)

,


(87)

(88) (89) (90)

: :

(91) ∆

~0

(92) (93) (94)


Prandtl-Meyer Expansion Fan Model: The isentropic Prandtl-Meyer expansion fan geometry is illustrated in Figure 8. Supersonic flow upstream 1) turns through an angle ( ) and produces an expansion fan delineated by upstream Mach angle ( ( ) and downstream Mach angle of ( ). There is a maximum amount of flow turning for which the flow remains may turn.

Figure 8: Conceptual sketch of supersonic incoming flow turning through an angle ( ) producing an ) to Mach angle ( ). expansion fan from Mach angle (

In typical applications, the flow turning ( ) is known, and the expansion fan is determined along with the flow properties23,24.

(95) Where the Prandtl-Meyer function

is defined as: 1

√

1

(96)

) known, the remaining downstream With the flow turning angle and upstream Mach number ( , properties are calculable in terms of the isentropic relations for the perfect gas, specific heat ratio ( ), and upstream properties as24: /

(97)

/ ⁄

(98) :

(99) ,

:

(100) (101) (102)


(103) ∆

(104) 0 0

(105) (106)

Wing Section Model:


With reference to Figure 4, we make the following assumptions typical for supersonic shock-expansion theory: 1) 2) 3) 4) 5) 6) 7) 8) 9)

Steady flow, Local macroscopic thermodynamic equilibrium, Two-dimensional, uniform flow at each station, Air is treated as a thermally and calorically perfect gas with constant properties, Losses in each component (“device” as noted in Figure 4), are due to shock waves only, if present, Expansions are treated as reversible Prandtl-Meyer fans, Flow is assumed to be adiabatic with no work interations, Potential energy changes of the flow are neglected relative to other energy forms, Body forces and viscous effects are negligible,

The reformulated conservation equations (Equations 81 to 106) are augmented with the appropriate equations of state and entropy/exergy equations in order to determine the flow path. Each “device” for the wing section is either an oblique shock wave or an expansion fan, depending upon the relative flow turning. Note: There are two possible flow paths for the external flow over the wing section: over the upper wing or below the lower wing. For smaller angles of attack and thinner airfoils, the shock and expansion structures will be similar. Accordingly, only the upper flow path will be detailed. Dead State (Station 10): The same dead state is used for the wing section as was used for the turbojet, enabling consistent reference conditions for later integration of the individual systems. These conditions are specified in Equations 1 to 4 and are not repeated here.

Free Stream (Station 1): The same free stream state is used for the wing section as was used for the turbojet, enabling consistent reference conditions for later integration of the individual systems. These conditions are specified in Equations 5 to 10 and are not repeated here. Device “A”, Attached Oblique Shock Wave (Station 1 to 2): For smaller wedge angles and angles of attack, the air will turn “into” itself, and an attached oblique shock wave will be present. (For larger wedge angles and higher angles of attack, it must also be considered that the resulting flow may be an expansion.) 2

(107) (108)

, / ,

,

(109)

/

,

1

1

, , ,


(110) (111)

(112) ,2

(113) (114) (115) (116)

: :

(117) ∆

(118) (119)


(120) Device “B”, Prandtl-Meyer Expansion (Station 2 to 4): From station 2 to 4, the flow turns “away” from itself and an isentropic expansion fan occurs. For a given flow ) and upstream Mach number ( ), the downstream Mach number ( ) is calculated via: turning (

(121) Where the Prandtl-Meyer function

is defined as:

2

1

1

(122)

1

1

(123)

⁄

(124) /

(125)

/

: :

(126) (127) (128)

∆

(129) 0

0 Devices “E & F”, Oblique Shock Wave/Prandtl Meyer Expansion Fan (Station 4 to 6 & 5 to 7):

(130) (131)

In the wake region (stations 6 and 7) the flow leaving the upper surface from (station 4) interacts with the flow leaving the lower surface (station 5). This flow interaction is incorporated into the model by matching the pressures and velocity flow angles in regions 6 and 7 as: :

(132) (133)

: Depending upon the relative flow turning from station 4 to 6 and from station 5 to 7 as well as angle of attack, it is possible that devices “E” and “F” may be a shock wave or an expansion wave23. Accordingly, shock and/or expansion theory must be applied in the iterative solution for the wake region properties. The procedure for solution is outlined below: 1) Guess a flow angle for station 6 (same as station 4, Equation 133) 2) Determine flow turning from station 4 to 6 for upper surface, a. If flow turning indicates shock wave, use Equations 107 to 110 to solve for pressure ( ),


b. If flow turning indicates expansion wave, use Equations 121 to 124 to solve for pressure ( ), 3) Determine flow turning from station 5 to 7 for lower surface, a. If flow turning indicates shock wave, use Equations 107 to 110 to solve for pressure ( ), b. If flow turning indicates expansion wave, use Equations 121 to 124 to solve for pressure ( ), 4) Determine convergence of iterative scheme: ? a. Does i. Yes? Done ii. No? Repeat steps 1 to 4 until convergence achieved. Due to the highly nonlinear and interactive nature of the above iterative scheme, the standard root solvers in Matlab were not robust enough to produce consistent results. Accordingly, a modified Newton-Raphson/secant method was written to perform the numerical solution of the wake region flow matching.


Wing Section Performance Parameters: With the flow path evaluation performed, the overall performance of the wing section may be characterized. Assuming the flow properties are uniform downstream of each shock wave and expansion wave, the lift coefficient and drag coefficient are calculated based upon the pressure fields and projected geometries. For given chord length (c) and diamond wedge half angle ( ) the wing section lift (L) and lift coefficient ( ) are calculated, respectively, as23: ⁄

⁄

⁄

⁄

(134) (135)

Similarly, the wing section drag and drag coefficient are calculated, respectively, as23: ⁄

⁄

⁄

⁄

(136) (137)

With the models for both individual systems developed, the analysis of each system may be performed. Additionally, each system may be independently optimized for a given operating condition. This independent optimization then serves as a basis for comparison of the integrated system, which may also be optimized for the same conditions. The comparison of the individual system’s performance will be compared to the integrated performance in the next sections.

V. Operating Conditions, Parameters, and Design Variables During the course of analyses and optimization procedures, it is helpful to isolate the influences of operating conditions and independent variables that influence the output response of a system. Operating conditions typically are fixed for a given study. There are two major types of variables in design: parameters and design variables, which serve different roles in the surrogate model development process. Parameters often define the bounds within which the study is performed or are chosen as fixed at a particular value for the entire study. For example, flight altitude is known to be a key variable which influences turbojet and wing performance. It may be varied parametrically within a range to be studied but held fixed for each analysis point. Hence, flight altitude in this case may be considered as a fixed parameter. On the other hand, a design variable is allowed to vary as a “free” variable during the course of the analysis and/or optimization. For the turbojet, such typical design parameters are chosen that tend to dominate the performance of the engine and therefore characterize the system on an overall basis. These are the variables of interest in the surrogate model development. An example of a typical design variable for the turbojet is the compressor pressure ratio (CPR). Both the turbojet and wing section illustrated in Figure 2 have unique characteristics that define their operation and are discussed further in terms of fixed parameters and variable design variables (also known as factors in statistics). Each of the two systems selected for this study have different parameters and design variables of interest. Careful pre-screening (or prototyping) of the parameters of interest and the design variables must be performed in order preclude errant statistical design and analysis points from the surrogate model. This prototype testing ensures maximal utility and extensibility of the developed modules for later use. During this prototyping process, one must


also consider the integrability of the individual systems into a system-of-systems. The operating conditions, fixed parameters, and design variables for each individual system and combined system-of-systems are now detailed.

Operating Conditions The operating conditions for the development of the surrogate models for system identification were selected to represent supersonic flight and are presented in Table 1. Both the turbojet and wing section are assumed to operate under the same conditions which are held fixed for the study. (Note that the number of significant digits for specific heat is carried for validation purposes.)


Table 1. Operating conditions for turbojet and wing section systems. Operating Condition Flight Altititude (km) Working Fluid Thermodynamic Model Air Specific Heat Ratio (γ) Air Specific Gas Constant (Rair, kJ/kg/K) Air Specific Heat (Cpair, kJ/kg/K) Reference Temperature for enthalpy, entropy (Tref,, K) Free Stream Property Values Fuel Heating Value (kJ/kg) 24 +

Value 6 Air Perfect Gas 1.4 0.287 1.00349 298.15 1976 US Standard Atmosphere *

Comment Both turbojet and wing section Both turbojet and wing section Both turbojet and wing section Both turbojet and wing section Both turbojet and wing section Both turbojet and wing section Both turbojet and wing section

42,700

Turbojet only

+

Both turbojet and wing section

25

*[Reference ], [Reference ] Fixed Parameters In both the turbojet and wing section, there are certain variables that directly influence the performance of a system, but which are held fixed for a particular study. These parameters are detailed for the turbojet system26 in Table 2 and for the wing section in Table 3.

Table 2. Fixed parameters for wing system (Figure 3).

**[Used for validation of turbojet cycle performance27]

Table 3. Fixed parameters for wing section system (Figure 4). Wing Section Fixed Parameters Dead State Reference Entropy (kJ/kg/K)**

Value 6.92015

Comment Significant digits used for validation

Chord length (c, m)

1

Arbitrary reference value

Wing span (b, m)

10

Arbitrary reference value

**[Used for consistency of integration with turbojet]


The fixed parameter values for the turbojet (Table 2) and the wing section (Table 3) are first used for analysis and optimization of each system separately, then used collectively as an integrated system-of-systems. For the integration, one seeks a physically consistent and feasible “tie component” which is a design variable common to both systems. These design variables are outlined in the next section.

Design Variables


Key variables that dominate the performance of a system are designated as design variables. As such, they are useful in characterizing the entire system and are therefore of prime interest as surrogates for the actual system. We seek the following traits of design variables in the system identification process: 1) There exists a philosophy of integration before the design variables are selected, a. In order to leverage the individual systems in integration, there must be a means of integrating/relating the systems together that is both meaningful and feasible, 2) The design variables selected must be amenable to advanced statistical analyses, a. A valid relationship between input design variables and output responses must exist and must be quantifiable and statistically significant, 3) The range of interest for the design variables is useful for the engineer, a. The surrogate modules developed are relevant for a wide range of applications, 4) The range of applicability is feasible, a. The performance of each model system obeys the physics of the actual system, 5) The surrogate models, although developed independently, may be integrated posteriori, a. The engineer may use the models for each independent system in a modular fashion, In both the turbojet and wing section, the design variables take on a potentially infinite number of values, but are typically constrained to certain ranges of interest for a particular study. These design variables are detailed for the turbojet system in Table 4 and for the wing section in Table 5.

Table 4. Design variables for turbojet system (Figure 3). Turbojet Design Variables

Range

Comment

Mach Number (M )

1.5 < M < 2.5

Upper limit "aggressive" in terms of current technology limits

Compressor Pressur Ratio (CPR )

14 < CPR < 30

Turbine Inlet temperature (T turb , K)

1600 < T turb < 2000

Upper limit "aggressive" in terms of current technology limits Upper limit "aggressive" in terms of current technology limits

Table 5. Design variables for wing section system (Figure 4). Wing Section Design Variables Mach Number (M ) Angle of attack (α, deg) Wedge half-angle (ε, deg )

Range

Comment Used as integration variable for both 1.5 < M < 2.5 turbojet and wing section Small angles to maintain attached 1

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