EML 4905 Senior Design Project

A B.S. THESIS PREPARED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF BACHELOR OF SCIENCE IN MECHANICAL ENGINEERING

Design Optimization of a Supersonic Nozzle Final Report Marc Linares Alessandro Ciampitti Marco Robaina Advisor: Professor George S. Dulikravich Cover Page

November 22, 2015 This B.S. thesis is written in partial fulfillment of the requirements in EML 4905. The contents represent the opinion of the authors and not the Department of Mechanical and Materials Engineering.

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Table of Contents Chapter

Page

COVER PAGE................................................................................................................................. I TABLE OF CONTENTS .............................................................................................................. III LIST OF FIGURES ....................................................................................................................... V LIST OF TABLES ....................................................................................................................... VII ABSTRACT .................................................................................................................................... 1 1. INTRODUCTION ...................................................................................................................... 2 1.1 Problem Statement 1.2 Motivation 1.3 Literature Survey 1.4 Survey of Related Standards

2 3 4 6

2. PROJECT FORMULATION...................................................................................................... 6 2.1 Project Overview 2.2 Project Objectives 2.3 Design Specifications 2.4 Addressing Global Design 2.5 Constraints and Other Considerations

6 7 7 7 8

3. PROJECT MANAGEMENT ...................................................................................................... 8 3.1 Overview 3.2 Breakdown of Work and Responsibilities into Specific Tasks 3.3 Gantt Chart

8 9 9

4. DESIGN ALTERNATIVES ..................................................................................................... 10 4.1 Overview 4.2 Design Alternate 1 Conical Nozzle: 4.3 Design Alternate 2 Bell Nozzle 4.4 Design Alternate 3 Dual Bell Nozzle 4.6 Feasibility Assessment

10 10 11 13 14

5. ENGINEERING SIMULATION & OPTIMIZATION ............................................................ 15 5.1 Gas Dynamics Overview 5.2 ANSYS Simulation 5.3 Loci 5.3.1 Overview and Simulation 5.3.2 Boundary Conditions 5.3.2.1 Conical 5.3.2.2 Bell 5.4 Overview of Optimization Process 5.4.1 Evolutionary Method 5.4.2 Random Number Generator 5.4.3 Response Surface 5.4.4 Optimization Analysis 5.4.5 Pareto Fronts for First Optimization 5.4.6 Pareto Fronts for Second Optimization 5.4.7 Parallel Coordinate Chart for First Optimization 5.4.8 Parallel Coordinate Chart for Second Optimization 5.4.9 Response Surface Validation for First Optimization 5.4.10 Response Surface Validation for Second Optimization 5.4.11 Discussion 5.5 Optimized Simulation Results

15 20 25 25 27 27 29 30 30 31 32 34 35 40 45 46 46 46 47 47

6. PROTOTYPE CONSTRUCTION ........................................................................................... 61 iii

6.1 Overview 6.2 Description of Prototype 6.3 Prototype Design 6.4 Parts List 6.5 Prototype Cost Analysis

61 61 62 66 67

7. TESTING AND EVALUATION ............................................................................................ 68 7.1 Overview 7.2 Design of Experiment 7.3 Test Results and Data 7.4 Evaluation of Results 7.4.1 Physical Evaluation 7.4.2 Simulation Evaluation 7.5 Improvement of Design 7.6 Discussion

68 68 69 70 70 71 71 72

8. DESIGN CONSIDERATIONS ............................................................................................... 74 8.1 Health and Safety 8.2 Assembly and Disassembly 8.3 Manufacturability 8.4 Maintenance of the System 8.5 Economic Impact

74 74 74 75 75

CH.9 DESIGN EXPERIENCE ..................................................................................................... 76 9.1 Overview 9.2 Standards 9.3 Contemporary Issues 9.4 Impact of Design in a Global Context 9.5 Life Long Learning 9.6 Discussion

76 76 77 77 78 78

CH.10 CONCLUSION ................................................................................................................. 80 10.1 Conclusion/Discussion 10.2 Future Work 10.3 Evaluation of Intangible Experiences

80 80 81

APPENDICES .............................................................................................................................. 82 A. Detailed Engineering Drawings of All Parts, Subsystems and Assemblies 82 B. Multilingual User’s Manuals (in English, Spanish, and at least one more language as appropriate: German, French, Japanese, Chinese, Portuguese, Italian, etc.) 82 C. Excerpts of Guidelines Used in the Project: Standards, Codes, Specifications and Technical Regulations (Quotes with references, or scanned material as appropriate) 84

REFERENCES ........................................................................................................................... 100

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List of Figures Figure

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Figure 1 - Schematic of De Laval Nozzle [1] ................................................................................... Figure 2 – Cold Spray Process [3] .................................................................................................... Figure 3 – Schematic of Conical Shape [10] ................................................................................ 11 Figure 4 – Schematic of Divergent Section of Bell Shaped Nozzle [10] ..................................... 12 Figure 5 – Shape of Dual Bell Nozzle [9]..................................................................................... 13 Figure 6 – Mesh Model of 2-Dimensional Conical Nozzle (ANSYS) ......................................... 20 Figure 7 – Mach Number for Conical Nozzle (ANSYS) .............................................................. 21 Figure 8 – Density for Conical Nozzle (ANSYS) ........................................................................ 22 Figure 9 – Temperature of Conical Nozzle (ANSYS) .................................................................. 22 Figure 10 – Pressure Distribution of the Conical Nozzle (ANSYS) ............................................ 23 Figure 11 – Velocity Contour of Conical Nozzle (ANSYS) ........................................................ 23 Figure 12 – Velocity Streamlines of Conical Nozzle (ANSYS)................................................... 24 Figure 13 – Flow Separation Bell Nozzle ..................................................................................... 25 Figure 14 – Shockwaves Bell Nozzle ........................................................................................... 26 Figure 15 – Conical Nozzle Example ........................................................................................... 28 Figure 16 – Streamline Velocity on Conic ................................................................................... 29 Figure 17 – Bell Nozzle example.................................................................................................. 30 Figure 18 - Length vs Density STD .............................................................................................. 35 Figure 19 - Length vs Pressure STD ............................................................................................. 36 Figure 20 - Length vs Temperature STD ...................................................................................... 36 Figure 21 - Length vs Thrust ........................................................................................................ 37 Figure 22 - Pressure STD vs Density STD ................................................................................... 37 Figure 23 - Temperature STD vs Density STD ............................................................................ 38 Figure 24 - Temperature STD vs Pressure STD ........................................................................... 38 Figure 25 - Thrust vs Density STD ............................................................................................... 39 Figure 26 - Thrust vs Pressure STD .............................................................................................. 39 Figure 27 - Thrust vs Temperature STD ....................................................................................... 40 Figure 28 - Density STD vs Length .............................................................................................. 40 Figure 29 - Density STD vs Pressure STD ................................................................................... 41 Figure 30 - Pressure STD vs Length ............................................................................................. 41 Figure 31 - Thrust vs Length ........................................................................................................ 42 Figure 32 - Temperature STD vs Density STD ............................................................................ 42 Figure 33 - Temperature STD vs Length ...................................................................................... 43 Figure 34 - Temperature STD vs Pressure STD ........................................................................... 43 Figure 35 - Thrust vs Density STD ............................................................................................... 44 Figure 36 - Thrust vs Pressure STD .............................................................................................. 44 Figure 37 - Thrust vs Temperature STD ....................................................................................... 45 Figure 38 - Parallel coordinate chart - first optimization.............................................................. 45 v

Figure 39 - Parallel coordinate chart - second optimization ......................................................... 46 Figure 40 – Mach Distribution of First Optimized Nozzle ........................................................... 48 Figure 41 – Mach Distribution of Final Optimized Nozzle .......................................................... 49 Figure 42 – Speed of Sound (m/s) in First Optimized Nozzle ...................................................... 50 Figure 43 – Speed of Sound (m/s) in Final Optimized Nozzle ..................................................... 50 Figure 44 – Temperature Distribution (ᵒK) in First Optimized Nozzle ........................................ 51 Figure 45 – Temperature Distribution (ᵒK) in Final Optimized Nozzle ....................................... 52 Figure 46 – Optimized First Nozzle Axial Velocity (m/s) ........................................................... 53 Figure 47 – Optimized Final Nozzle Axial Velocity (m/s)........................................................... 53 Figure 48 – Optimized First Nozzle Radial Velocity (m/s) .............................................................. Figure 49 – Optimized Final Nozzle Radial Velocity (m/s) ......................................................... 55 Figure 50 – Pressure (Pa) Distribution in First Optimized Nozzle ............................................... 56 Figure 51– Pressure (Pa) Distribution in Final Optimized Nozzle ............................................... 57 Figure 52 – Pressure (Pa) Distribution in First Optimized Nozzle Scaled Down for Exit ........... 58 Figure 53 – Pressure (Pa) Distribution in Final Optimized Nozzle Scaled Down for Exit .......... 58 Figure 54 – Density (kg/m3) of Air in First Optimized Nozzle .................................................... 59 Figure 55 – Density (kg/m3) of Air in Final Optimized Nozzle................................................... 60 Figure 56 – Connections from Compressor to Expansion Chamber ............................................ 62 Figure 57 – PVC Coupling Connected at both ends of Expansion Chamber ............................... 63 Figure 58 – Connector of ¾” diameter decreased to ¼” diameter ................................................ 63 Figure 59 – Pressure Valve and Pressure Gauge .......................................................................... 64 Figure 60 – ½” Male to Male Connector ...................................................................................... 65 Figure 61 – Primer, Heavy Duty Glue, and Seal Tape ................................................................. 66 Figure 62 – Logarithmic Scale for Residuals of First Optimized Nozzle..................................... 69 Figure 63 – Logarithmic Scale for Residuals of Final Optimized Nozzle .................................... 70 Figure 64 - Expansion Chamber and Pressure Valve with Gauge ................................................ 74 Figure 65 – Optimized Nozzle adjusted for Manufacturing ......................................................... 82

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List of Tables Table

Page

Table 1 – Responsibilities and Project Executables ....................................................................... 9 Table 2 – Gantt Chart ...................................................................................................................... 9 Table 3 - Geometrical Configuration for Conical Nozzle ............................................................ 21 Table 4 - RSM validation for first optimization ........................................................................... 46 Table 5 - RSM validation for second optimization ....................................................................... 46 Table 6 – Geometrical Configuration for First Optimized Nozzle (Conical) ............................... 47 Table 7 – Geometrical Configuration for Final Optimized Nozzle .............................................. 48 Table 8 – Part list for Final Design Nozzle................................................................................... 66 Table 9 – Part List and Total Cost ................................................................................................ 67 Table 10 – Experimental Results .................................................................................................. 69 Table 11 - Simulated Thrust Results for Optimized Nozzles ....................................................... 69

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Abstract This project has been completed with the aim to introduce a more efficient and better performing nozzle for rocket propulsion applications. In an era of increasing space related exploration and industries the implementation of more efficient and higher performing rocket engines will be essential. This project’s objective is to bring these goals closer to reality.

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1. Introduction 1.1 Problem Statement The use of a better performing supersonic nozzle will help rocket propulsion and the space industry to reach further into the unknown. The following project presents this design team with a number of complex challenges including: design, calculations, and testing. Firstly, the design team must expand its knowledge on fluid mechanics namely in the area of supersonic flow [for this report Mach 1 through 4], de Laval nozzles [convergence-divergence], shock wave formation, and pressure distribution inside the nozzle. Additional considerations and solutions must be found with respect to the testing of this design which includes: supersonic wind tunnel testing, machining of nozzle [according to tolerances and standards consistent with aerospace applications], operation of various temperature and pressure probes [to obtain quantitative measurements to validate data], and finally scaling considerations and dimensional analysis to accurately translate scale model calculations to full scale testing results. Computational considerations such as appropriate software choice is imperative. Simulation software whose capabilities include high velocity flows [greater than Mach 1] and the ability to calculate high pressures and shock wave formation associated with such flows. Such capabilities must extend to handling complex geometries associated with differing initial designs and their optimized shape. LOCI, a family of codes specifically designed for this purpose, is used for computational modelling in this report. The second consideration, the machining of the nozzle, must be done according to tolerances and standards which are common to the aerospace industry, including AS9100 and ASME Y14.5.

2

1.2 Motivation With an increased emphasis on rocket propulsion including the founding of notable private space agencies and a noticeable increase in the number of launches of satellites and other devices into space; improvements in rocket propulsion are a necessary component in making the future of space exploration more efficient. Specifically in regards to the profile, geometry, and pressure distribution inside the nozzle, each application will have its own specific optimized shape. These improvements, namely how pressure is distributed throughout the nozzle will indicate whether the highest thrust is being achieved, minimal boundary layer separation, and best flow properties are being achieved for any given geometric shape which is being optimized for a given function. These enhancements in the pressure distribution conditions inside the nozzle will optimize the given propulsion system for whatever its given operating conditions will be. Simultaneously, greatest efficiency will be achieved while also saving on the cost from losses associated with previous less efficient designs. The outcome is an expected maximization in efficiency while minimizing resources, such a combination must be achieved if progression of private space enterprise is to be sustained and expanded through the approaching decades.

3

1.3 Literature Survey When nozzles were invented, their purpose were primarily to change the characteristic of the flow such as an increase in pressure or velocity. In 1890 Swedish engineer and inventor Karl Gustaf Patrik de Laval developed a convergent-divergent (condi) nozzle that had the capacity to increase a steam jet to a supersonic state [1]. This nozzle as shown in Figure 1, was termed a de Laval nozzle and later was used for rocket propulsion. An American engineer Robert Goddard would be the first to integrate a de Laval nozzle in connection with a combustion chamber, increasing efficiency and achieving supersonic velocities in the region of Mach 7 [1].

Figure 1 - Schematic of De Laval Nozzle [1]

The typical uses for a de Laval nozzle fall under the category of rocket propulsion; however, there has been an increase for the use of the supersonic nozzle in other areas. The American military has been using rocket nozzles to apply high velocity particles, which are a

4

combination of metals, ceramics, and polymeric materials, onto the surfaces of weapon systems by a process called cold spray as shown in Figure 2 [2].

Figure 2 – Cold Spray Process [3]

As the use of de Laval nozzles in rocket design have become prominent, so have the parameters of the nozzle. Several research papers and works have been done to optimize the nozzle to meet certain criteria more effectively. Due to the multi-objective optimization of this project, this survey is focused on de Laval nozzle simulation and optimization. In 2012 Karla Quintano published a master thesis that detailed work in adjusting the shape parameters of the de Laval nozzle in order to find an optimal setup for making the gas flow exiting the condi-nozzle more uniform. Several software programs were used for the work. A FORTRAN code was used to develop 40 different nozzle shapes. ANSYS and modeFrontier were used to optimize specified parameters of the shapes, and to run simulations on flow and heat transfer. The thesis results showed that the shape of the nozzle had a significant impact on exit flow formation [5]. Jean-Baptiste Mbuyamba published a dissertation regarding nozzle design for a cold gas dynamic spray. While not directly related to rocket nozzles, the dissertation considers de Laval nozzles for design. It also describes several theoretical elements regarding compressible gas flow in a convergent-divergent nozzle as well as methods to simulate and calculate specific parameters [4]. 5

1.4 Survey of Related Standards In order to ensure that the design choices conducted in this project are fit for their tasks certain standards are adhered to. It is expected that throughout the duration of this project, the following standards will be followed: 1. AS9100 – Standardized Quality Management for Aerospace Industry 2. AIA – Aerospace Industries Association 3. ASM – American Society for Metals 4. ASME Y14.5– American Society for Mechanical Engineers 5. ASTM – American Society of Testing and Materials 6. ANSI – American National Standards Institute

2. Project Formulation 2.1 Project Overview The process of optimization began with a detailed literary research in order to find all relevant information on supersonic nozzles. This project had a large emphasis on the behavior of flow consisting primarily of compressible fluids. Modeling the flow of air as an ideal cold gas was done using a variety of software platforms such as ANSYS fluent, and a LOCI FORTRAN program. Furthermore, a hot-gas application was modeled following the exact testing conditions as the cold gas while utilizing software programs to analyze different fluid gradients such as pressure, temperature, density, and velocity. A Computer Aided Engineering (CAE) program, modeFrontier, was run to optimize the supersonic nozzle using a variety of geometric configurations.

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2.2 Project Objectives The optimization of the de Laval nozzle for rocket applications can consist of many parameters. The proposed objectives for this project are to minimize the total length of the nozzle while increasing the magnitude and uniformity of the exit flow. Firstly, the minimization of the length serves a dual purpose; reduction of friction due to surfaces in contact, and reduction of materials used as well as weight. Second, maximization of the exit Mach number; to achieve greatest speed and mass flow rate. Finally, maximize the uniformity of the flow by reducing inconsistencies in temperature and pressure.

2.3 Design Specifications Design specifications taken into consideration for this project were based upon simulation, optimization and manufacturing results. Simulations were based upon the modeling performed on three different nozzles (conical, bell, and dual bell shaped) using Solidworks and an input program, LOCI. In order to accomplish the project objectives, the optimization goals are realized through calculations derived from modeFrontier and computation fluid dynamics (CFD) multi-platform programs. In order to validate simulated tests, considerations between the size of the theoretical optimized nozzle and prototype was strongly considered. Dimensional analysis was completed to obtain a prototype used to acquire testing results. A key specification for the supersonic nozzle was the ability to directly observe shock wave formation inside the divergent section.

2.4 Addressing Global Design A global approach was considered during the design phase of this project. As space programs advance in many countries around the world, rocket propulsion with the use of optimized nozzles is vital for any developing program. The use of supersonic nozzles are vastly used in 7

rockets and missiles. Due to the fact that a nozzle can be used for a military operation, a thoughtful approach was applied regarding the information developed during the project’s design.

2.5 Constraints and Other Considerations Certain constraints were to be upheld during the project’s duration. In a real-time rocket application, non-uniform flow from the combustion chamber deals with a chemical energy change; however, this project models a uniform flow from the chamber. Analysis and simulation was completed using hot gases and cold gases in a 2-dimenional and 3-dimensional flow pattern. One constraint that had a major implication for this project dealt with the validation stage. Testing facilities for hot gas flow through the supersonic nozzle were not available through the student’s university and validation of a hot gas was felt to be out of the scope of an undergraduate capstone project. To validate the simulated results, cold gas was considered as the primary flow. This type of flow changed the way this nozzle was manufactured as the material used for hot gas flow had to be made with thermal considerations as opposed to cold gas that could be made from a transparent material such as Plexiglas.

3. Project Management 3.1 Overview The project execution was completed by all three members following a specific guideline for an eight month curriculum. In order to evenly distribute the project objectives and meet each goal within the established time frame, the tasks were given to each individual that had an interest or strong aptitude in that topic. Research and development were conducted by all members during the initial stages in order to acquire a strong scientific background. A consensus was reached between all members for a concept design based 8

upon research and development. This project allocated significant amount of time in software methods. Two members of the group had primary focus on analysis and behavior of the system using LOCI software. One member utilized modeFrontier for the optimization stage. In the following sections, the key responsibilities and the overall timeline for this project are displayed.

3.2 Breakdown of Work and Responsibilities into Specific Tasks Table 1 – Responsibilities and Project Executables

3.3 Gantt Chart Table 2 – Gantt Chart

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4. Design Alternatives 4.1 Overview The design of convergent-divergent supersonic nozzles for rocket applications have evolved from simple conical designs to several more advanced schematics. Each of these nozzle designs hold several advantages and disadvantages over the other depending on what rocket is used and what its designated goal is. For this project, several nozzle designs are considered for optimization. The selection, however, is limited by the difficulty of manufacturing and testing certain nozzle designs as well as the feasibility of running these designs with computational fluid dynamics analysis and optimization software. From the selected designs, simulation work was conducted and the parameters of each designed were optimized based on the optimization goals discussed in chapter 2.3. The nozzle that achieved the optimization goals by the largest amount was considered for a final design for manufacturing and testing. The following nozzle designs have been selected for optimization: conical, bell, and dual bell nozzles.

4.2 Design Alternate 1 Conical Nozzle: The conical nozzle design is the predecessor to all other designs of convergent-divergent supersonic nozzles. While other designs hold more effective ways of dealing with the common problems of rocket gas flow, this design is the most simple. Figure 4 contains a schematic of the conical nozzle.

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Figure 3 – Schematic of Conical Shape [10]

One unique quality of this nozzle is the constant half angle (symbolized as 𝛼) of the cone in divergent section. The simplicity of this parameter allows for a cleaner optimization and a cheaper production. The nozzle, however, loses efficiency due to the non-axial velocity components of the exit flow velocity. This occurs because of the simple cone geometry [10].

4.3 Design Alternate 2 Bell Nozzle The most common nozzle design used is the bell nozzle. While not as simple as the conical nozzle, the nozzle geometry provides a more efficient design in terms of less length and non-axial flow reduction. The next figure shows the geometry of the divergent section of a bell shape nozzle.

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Figure 4 – Schematic of Divergent Section of Bell Shaped Nozzle [10]

A noticeable difference from the conical design is the changing half angle of the cone in the divergent section. The angle leads to a parabolic contour which is the main factor to the increase in efficiency. Having a varying angle, however, makes analysis, optimization and manufacturing more difficult compared to the conical design. One of the main drawbacks to the bell nozzle is the lack of altitude compensation. The exit flow from the nozzle can under expand or over expand depending on the difference in pressure between the exit and the ambient. Pressure changes at different altitudes making the bell nozzle have losses at the non-optimal altitudes. Despite these losses, the bell nozzle is widely used due to how feasible it is to manufacture and the efficiency of the design [9].

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4.4 Design Alternate 3 Dual Bell Nozzle As an attempt to improve the bell nozzle design described in the previous section, the dual bell nozzle is designed to compensate for the wide spectrum of pressures at different altitudes. The schematics of a dual bell nozzle can be found in the figure below.

Figure 5 – Shape of Dual Bell Nozzle [9]

The divergent section of this nozzle is divided into the following three parts: the first bell, wall inflection, and second bell. The first bell is similar to the divergent section of the bell nozzle. The wall inflection is in between the first and second bells. The second bell represents the larger sized bell. Due to this geometry, when the exit flow under expands due to higher ambient pressure, the nozzle behaves like a standard bell nozzle. The flow does not touch the second bell due to the wall inflection causing flow separation. If the nozzle is at a high altitude where pressure is lower than the flow pressure, the exit flow over expands at the first nozzle exit and continues onto the second bell. While the dual bell nozzle has a greater overall efficiency in flights that deal with continuously changing pressures, the bell nozzle at its designed optimal pressure is more efficient

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than either of the two nozzles of the dual bell at their optimal pressures. The dual bell nozzle is also the most difficult of the three selected designs to manufacture, simulate, and optimize [9].

4.6 Feasibility Assessment The three nozzle designs selected for this project are the simplest to manufacture of the current convergent-divergent supersonic nozzles designs. Testing an optimal nozzle was done through cold gas. Hot flow testing was beyond the reach of this project due to a great deal of requirements. A few of these requirements were combustion chamber analysis, specific testing facilities, and obtaining the required permission for testing. For simulation, a 2-dimensional flow was assumed and an attempt to analyze 3-dimensional flow was conducted as well. Due to time constraints, running the optimization software using 3-dimensional flow analysis might not have yield desired results. For choosing a final design, optimization goals achieved were prioritized, however, only if it was not difficult to manufacture based on available facilities [10].

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5. Engineering Simulation & Optimization 5.1 Gas Dynamics Overview Gas Dynamics is fundamental to the understanding of compressible flows and their behavior in conditions where the Mach number is subsonic, sonic, and supersonic. This entails principally the understanding of shockwave formation and how pressure, temperature, density, Mach number, entropy generated change over the mathematical discontinuity known as the shock wave. During the modeling of compressible flows (flows in which ∆𝜌 ≠ 0) there exist three principal concepts for their mathematical expression. These mathematical expressions are Quasi one-dimensional isentropic, Euler, and Navier-Stokes. The quasi one-dimensional flow is the least complex of the group. Its modeling only considers the velocity in the x-direction, adiabatic, isentropic, and no viscosity term considered. The Euler equation increases complexity to include two-dimensional turbulence and can include viscosity. Finally, the full Navier-Stokes equation includes three-dimensional turbulence, heat transfer, viscosity, and entropy changes. Due to the complexity of the fully defined Navier-Stokes equation it is simpler and less time consuming to use the quasi one-dimensional and Euler equations to obtain initial values. To begin the derivation of the equations we will begin by defining the properties of our working fluid, air. For thermally and calorically perfect gases (an assumption we are taking for air) the following equations are defined: 𝑐𝑝 = 𝑐𝑝 (𝑇) and 𝑐𝑣 = 𝑐𝑣 (𝑇). And where the ratio of these values is, 𝛾 =

𝑐𝑝 𝑐𝑣

where 𝛾 = 1.4 for diatomic gases such as air. The 𝑐𝑝

gas constant R, can be defines in the following manner: 𝑅 = 𝑐𝑝 − 𝑐𝑣 = 𝑐𝑣 𝑐 − 𝑐𝑣 = 𝑐𝑣 (𝛾 − 1) 𝑣

𝑐𝑣 =

𝑅 𝛾−1

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𝑐

1

Similarly, 𝑐𝑝 − 𝑐𝑣 = 𝑐𝑝 (1 − 𝑐𝑣 ) = 𝑐𝑝 (1 − 𝛾) yielding, 𝑝

𝑐𝑝 =

𝛾𝑅 𝛾−1

From the second law of thermodynamics, the previous definitions can be used to further develop our understanding of this flow. 𝑇 𝑑𝑠 = 𝑑𝑒 + 𝑝 𝑑𝑣

;

1

𝑣=𝜌

;

𝑑𝑒 = 𝑐𝑣 𝑑𝑇 −𝑑𝜌 𝑇 𝑑𝑠 = 𝑑𝑒 + 𝑝 ( 2 ) 𝜌

𝑇 𝑑𝑠 = 𝑐𝑣 𝑑𝑇 − 𝑅𝑇

𝑑𝜌 𝑑𝑇 𝑑𝜌 𝑑𝜌 → 𝑑𝑠 = 𝑐𝑣 −𝑅 → ∫ 𝑑𝑠 = ∫ 𝑐𝑣 𝑑𝑇 − ∫ 𝑅 𝜌 𝑇 𝜌 𝜌

After substituting, integrating, and taking the natural logarithm: 𝑇2 𝑐𝑣 ( 𝑇2 𝜌2 𝑇1 ) ) )) ) )) ∆𝑠 = 𝑐𝑣 ((ln(𝑇2 ln(𝑇1 − 𝑅(ln(𝜌2 − ln(𝜌1 → ln ( ) − ln ( ) = ln ( ) 𝜌 𝑅 𝑇1 𝜌1 (𝜌2 ) 𝑐𝑣

𝑅

1

1

𝜌2 𝑇2 𝛾−1 −∆𝑠 =( ) 𝑒 𝑅 𝜌1 𝑇1 Noting in the previous equation how variations in density are greater than variations in the temperature. Similarly, 𝛾

𝑃2 𝑇2 𝛾−1 −∆𝑠 =( ) 𝑒 𝑅 𝑃1 𝑇1 Therefore, it can be stated that for isentropic processes: 16

𝛾

𝑃2 𝜌2 𝛾 𝑇2 𝛾−1 =( ) =( ) 𝑃1 𝜌1 𝑇1 Moving on from our thermodynamic expressions, we can now look to sound waves moving through an air medium. By definition, sound waves are disturbances in the medium. Before this disturbance medium has certain properties: 𝜌, 𝑝, 𝑇, 𝑎, where 𝑎 is the acoustic speed. After the disturbance of the sound wave (assume a weak sound wave) the properties of the medium change: 𝜌 + 𝑑𝜌, 𝑝 + 𝑑𝑝, 𝑇 + 𝑑𝑇, 𝑎 + 𝑑𝑎. Continuing onto total change of mass: 𝐷𝑀 𝐷 𝐷 𝜕𝜌 | = ∫ 𝑑𝑚 = ∫ 𝜌 𝑑𝑉 = ∫ 𝑑𝑉 + ∫ ∇ ∙ (𝜌𝑣⃗) 𝑑𝑉 = 0 𝑑𝑡 𝑠𝑦𝑠𝑡𝑒𝑚 𝐷𝑡 𝐷𝑡 𝜕𝑡 𝑉(𝑡)

𝑉(𝑡)

𝑉(𝑡)

𝑉(𝑡)

𝐷𝑀 𝜕𝜌 = ∫ 𝑑𝑉 + ∫ (𝜌𝑣⃗) ∙ 𝑛̂ 𝑑𝐴 ≡ 0 𝐷𝑡 𝜕𝑡 𝑉(𝑡)

𝐴(𝑡)

For steady state conditions,

∫(𝜌𝑣⃗) ∙ 𝑛̂ 𝑑𝐴 = 0 𝐴

Taking the velocity conditions before (L-left side) and after (R-right side) the acoustic wave, 𝑣̂𝐿 = 𝑎𝑖̂

and 𝑑𝑎

∴ 𝑎 = −𝜌 𝑑𝜌

𝑣̂𝑅 = (𝑎 + 𝑑𝑎)𝑖̂ and

;

𝜌𝑎 = (𝜌 + 𝑑𝜌)(𝑎 + 𝑑𝑎)

𝑎2 = 𝛾𝑅𝑇

Definitions of the Mach number, as well as substituting previously derived equations will be presented as follows, 17

𝑀≡

𝑙𝑜𝑐𝑎𝑙 𝑠𝑝𝑒𝑒𝑑 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 𝑣 𝑣 = = 𝑙𝑜𝑐𝑎𝑙 𝑠𝑝𝑒𝑒𝑑 𝑜𝑓 𝑠𝑜𝑢𝑛𝑑 𝑎 √𝛾𝑅𝑇 𝑝2 2𝛾 =1+ (𝑀12 − 1) 𝑝1 𝛾+1

𝜌2 𝑢1 = = 𝜌1 𝑢2

(𝛾 + 1) 2 2 𝑀1 (𝛾 − 1) 2 1+ 2 𝑀1

𝛾−1 1 + 2 𝑀12 𝑐𝑝 𝑇2 ℎ2 𝑇2 2𝛾 2 = [1 + (𝑀1 − 1)] [ ]= = 𝛾+1 2 𝑇1 𝛾+1 𝑐𝑣 𝑇1 ℎ1 𝑀 1 2

𝑀12

𝛾−1 2 2 𝑀1 = 𝛾−1 𝛾𝑀12 − 2 1+

For the case of quasi one-dimensional isentropic flows, that is, flows where the areas along the sections of the flow do not deviate by a considerable amount, only the x-axis velocities are considered; no turbulence. This mathematical model is also inviscid, steady, and adiabatic. For steady state conditions, 𝜌1 𝑢1 𝐴1 = 𝜌2 𝑢2 𝐴2 . Balancing forces inside a flow, integrating, and solving results in, 𝑝2 + 𝜌2 𝑢22 = 𝑝1 + 𝜌1 𝑢12 When the previously derived equations are applied to converging-diverging (de Laval) nozzles, equations are obtained relating a given cross sectional area to the Mach number at that point. These following equations apply only to de Laval nozzles with quasi one-dimensional isentropic flows. It is important to keep in mind that the minimum cross-sectional area for these

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equations is equal to the cross-sectional area at the throat of the nozzle. For the cross-sectional area in the throat, the theoretical value of the Mach number is equal to 1. At this point in the flow the stream changes from subsonic to sonic. After the throat area, the flow is supersonic. 𝛾+1

(𝛾 − 1) 2 2(𝛾−1) 𝐴 1 2 ∴ = [( ) (1 + 𝑀 )] 𝐴∗ 𝑀 𝛾 + 1 2 Where, 𝐴 ≥ 𝐴∗ as alluded to previously. These equations are valid for all points along the length of the nozzle. The mass flow in nozzles can be similarly found. Understanding the mass flow rate in the nozzle is an important factor in determining the thrust for the entire system. 𝛾 𝑝𝑜1 𝑅 √𝑇𝑜1

𝑀

𝑚̇ = √

𝛾+1

(𝛾 − 1) 2 2(𝛾−1) [1 + 2 𝑀 ]

Finally, we can relate the mass flow rate for the cross sectional area of the throat of the nozzle, 𝛾+1

𝑚̇ 𝑝𝑜1 √ 𝛾 2 𝛾−1 = ( ) 𝐴∗ √𝑇𝑜1 𝑅 𝛾 + 1

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5.2 ANSYS Simulation ANSYS is a commercial software package that was used to model and simulate the two geometrical configurations for this project. One other reason this software was used in conjunction with the LOCI software was to validate the results that the LOCI program produced. As LOCI is a non-commercialized software, it was apparent to be able to mirror the results based on a simple geometrical configuration and symmetrical boundary conditions that the ANSYS software could handle.

Figure 6 – Mesh Model of 2-Dimensional Conical Nozzle (ANSYS)

The 2D conical nozzle shown above has a sweep method mesh and has proximity and curvature for sizing with a fine relevance center. There are 3082 nodes with 1452 elements on this mesh. The default geometrical configuration used for this conical nozzle mirrors the default conical nozzle used on the LOCI software.

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Table 3 - Geometrical Configuration for Conical Nozzle

Object Chamber Diameter Chamber Length Throat Diameter Inlet Radius of Curvature Throat Radius of Curvature Exit Diameter Converging Angle Diverging Angle

Symbol 𝐷𝑐 𝐿𝐶 𝐷𝑇 𝑟𝑐𝑢𝑟𝑣𝑖𝑛𝑙𝑒𝑡 𝑟𝑐𝑢𝑟𝑣𝑡ℎ𝑟𝑜𝑎𝑡 𝐷𝐸 𝜃 𝛼

Unit 0.1275 m 0.04572 m 0.0458 m 0.0361 m 0.04572 m 0.075 m 30° 15°

Figure 7 – Mach Number for Conical Nozzle (ANSYS)

The 2D conical nozzle above was simulated using ANSYS 14.5 CFX and displays the Mach number for this simulation. The maximum Mach number obtained was 1.927, which compared to the same simulation from LOCI (M= 2.573), is a 36% difference. One can observe a normal shockwave forming at the nozzle exit.

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Figure 8 – Density for Conical Nozzle (ANSYS)

Figure 9 – Temperature of Conical Nozzle (ANSYS)

As observe from the figure above, we have conditions at the inlet that resemble typical sea surface temperatures (T=300K) on average.

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Figure 10 – Pressure Distribution of the Conical Nozzle (ANSYS)

The pressure distribution shown in the above figure is a prime example of the flow’s performance based on the initial conditions set forth in ANSYS at 5 bar at the inlet and the outlet condition of an average static pressure of 1 bar. Again as demonstrated, a shockwave begins to form near the top of the outlet location.

Figure 11 – Velocity Contour of Conical Nozzle (ANSYS)

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Figure 12 – Velocity Streamlines of Conical Nozzle (ANSYS)

The two previous figures are important to understand since they help define how the Mach number changes and assist in the visualization of the particles that flow throughout the conical nozzle. Under these geometrical configurations, we do not see any flow separation on the wall of the nozzle shown by the figure with velocity streamlines. The divergent section, where the velocity reaches the maximum of 527 m/s, the diverging angle plays a vital role on flow separation. Since the diverging angle is constant in a conical nozzle, flow separation is rare but may happen.

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5.3 Loci 5.3.1 Overview and Simulation Loci is a non-commercial CFD program developed by Dr. Ed Luke from Mississippi State University. The purpose of using this software over commercial software such as ANSYS and COMSOL is to be able to capture flow separation and shockwaves accurately through simulation. The following figures show the program’s ability to capture these two elements.

Figure 13 – Flow Separation Bell Nozzle

The figure above displays the streamline velocity profile of the flow. In the divergent section past the red gradient, the streamline detaches from the wall of the nozzle. This is where the flow separation is occurring.

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Figure 14 – Shockwaves Bell Nozzle

The above figure displays the Mach gradients of the flow within the nozzle and the flow that has exited the nozzle. The splits between the red gradients of Mach, show that the shockwaves are occurring. These figures were plotted using the 3rd party software ParaView. All of the tools for this program were run on Ubuntu Linux for compatibility. Both figures have a bell shaped geometry and only show half of the nozzle. This is because it is assumed that both halves of the nozzle are symmetrical. While these figures show the capability of the software, having flow separation and shockwaves in the nozzle is not optimal. Dr. Luke provided the team with tools that were specifically for this program to develop conical and bell shaped nozzles for 2-Dimensional analysis. The nozzle creation tool allows change to the following parameters: chamber diameter, chamber length, throat diameter, inlet radius of curvature, throat radius of curvature, exit diameter, converging angle, diverging angle, and diverging length. Since the emphasis of this optimization is on the exit flow of the nozzle, 26

more focus was placed on the exit diameter, throat diameter, throat radius of curvature, and nozzle length parameters. For the simulation, temperature is not considered as a factor because the gas selected for the nozzle is cold air. With these inputs, the software analyzes the unstructured mesh and calculates the following variables at each node: velocity, Mach number, speed of sound, temperature, density, pressure. One hundred conical and one hundred bell nozzles were run by this program. Refer to the optimization section for the approach on how these nozzles were used in the optimization process. Due to time constraints and difficulties with handling the software, the dual bell nozzle shape referred to in the previous sections was not analyzed. With the elimination of the dual bell shape, the optimization of the nozzle no longer included altitude compensation as a parameter, meaning the nozzle is not optimized to be efficient in varying pressures.

5.3.2 Boundary Conditions 5.3.2.1 Conical The conical nozzle is defined by the following boundary conditions: inlet, outlet, nozzle wall, and symmetry. For the inlet boundary condition, pressure, temperature and Mach number are specified. The inlet is also defined as isentropic. For the outlet boundary condition, only pressure is specified. The nozzle wall boundary condition is defined as both viscous and adiabatic. The symmetry condition is applied to the entire nozzle geometry and mirrors the same results to the other half of the nozzle. Specifying three values at the inlet provides smoother convergence of results. The isentropic factor assumes that the inlet flow is at a constant temperature. The nozzle wall boundary

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condition allows the simulation to capture the flow behavior at the nozzle wall, such as flow separation. The adiabatic condition prevents heat transfer through the nozzle wall.

Figure 15 – Conical Nozzle Example

The above figure represents one of the hundred nozzles generated for optimization. The inlet boundary condition is represented by the left edge of the nozzle and the outlet is at the right end. The nozzle wall covers the top curve of the nozzle, while two symmetry boundary conditions extends the nozzle wall to the other directions.

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Figure 16 – Streamline Velocity on Conic

This figure represents the streamline velocity of the flow inside the frame of the nozzle. At the divergent section of the nozzle, the wall is slightly blue. This shows that the velocity at the wall is zero. This is the result of the viscous boundary condition.

5.3.2.2 Bell The bell nozzle is defined by inlet flow, outlet flow, nozzle wall, nozzle lip, and symmetry boundary conditions. The boundary conditions for the bell are the same as those of the conic shape except for the nozzle lip which defines the section where the curve of the bell shape ends. This is

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used with the far field boundary conditions which is used to define flow outside of the bell nozzle.

Figure 17 – Bell Nozzle example

The figure above represents one of the bell nozzles used for optimization. The same boundary conditions of the conic can be seen for the bell as well as the nozzle lip boundary condition. The lip begins at where the second block intersects the nozzle. This is used separate the nozzle as its own object when attached to the block. The blue block on the right represents an area behind the nozzle. This block is used to capture the flow as it passes the exit of the nozzle. A far field condition is place on this block so that it would not alter the data acquired from simulation.

5.4 Overview of Optimization Process 5.4.1 Evolutionary Method Particle Swarm, an evolutionary based optimization algorithm created to be faster than its counterpart the genetic algorithm. The theory behind PS models and mimics the behavior of flocking animals, namely birds. Most prominent is the example of a flock of birds looking for food in a given area. In any flocking group there is a constant struggle between the individuality of its members and the sociability cohesion of the group. Like other evolutionary optimization methods, PS is an iterative method which searches an n-dimensional hypercube for optimal solutions. In this individuality versus sociability factor, it is important to note that sociability 30

increases new ideas and alternate search areas decrease. A very high importance on sociability will lead to convergence on a local minimum instead of a global minimum. The importance of individuality is meant so that even if food is found by the group of birds, there are still members who will continue searching nearby areas in case a better solution is found. Although, an over reliance on individuality will lead to non-convergence of the system. For the PS method, there exists only one formula and it based on particle dynamics, 𝑥𝑖 𝑘+1 = 𝑥𝑖𝑘 + 𝑣𝑖𝑘+1 Where 𝑣 is the velocity vector defined as the following: 𝑣𝑖𝑘+1 = 𝛼𝑣𝑖𝑘 + 𝛽𝑟1𝑖 (𝑝𝑖 − 𝑥𝑖𝑘 ) + 𝛽𝑟2𝑖 (𝑝𝑠 − 𝑥𝑖𝑘 ) In the previous formula, the first term expresses the inertia of the group, which is set to decrease at subsequent iterations. The second term expresses the individuality of its members, while the last term indicates the sociability of the entire group [6].

Values for “r” are chosen by our random number generator. While, {

0 Dc Or (Dt - (1 / 1000)) > de Randomize Dt = (Int((10 - 3 + 1) * Rnd + 1)) / 1000 Randomize Dc = (Int((20 - 5 + 1) * Rnd + 1)) / 1000 Randomize de = (Int((20 - 5 + 1) * Rnd + 1)) / 1000 Loop rs.Cells(i + 2, 3) = Dt rs.Cells(i + 2, 4) = ra rs.Cells(i + 2, 5) = rt rs.Cells(i + 2, 6) = cangle rs.Cells(i + 2, 7) = dangle rs.Cells(i + 2, 8) = de

ss.Cells(i, 1) = Chr(34) & "./noz -Dc " & Dc & " -Lc " & Lc & " -Dt " & Dt & " -ra " & ra & " -rt " & rt & " -cangle " & cangle & " -dangle " & dangle & " -De " & de & Chr(34) &""

Next ss.Columns.AutoFit

End Sub --------------------------------------------------------------------------------------------------------------------Sub RandomBells() 'generate random bell shapes 'generate 500 shapes 'nozzledimensions 86

Dim Dc As Double Dim Lc As Double Dim Dt As Double Dim ra As Double Dim rt As Double Dim Ln As Double Dim cangle As Double Dim dangle As Double Dim de As Double

'randomizer worksheet Dim rs As Worksheet 'text sheet for code Dim ss As Worksheet

Set rs = Sheets("Randomizer") Set ss = Sheets("Text")

For i = 1 To 500 'find random values for bell Randomize Dc = (Int((20 - 7 + 1) * Rnd + 7)) / 1000 Randomize Lc = (Int((15 - 8 + 1) * Rnd + 8)) / 1000 Randomize Ln = (Int((40 - 10 + 1) * Rnd + 10)) / 1000 Randomize 87

Dt = (Int((15 - 5 + 1) * Rnd + 5)) / 1000 Randomize ra = (Int((10 - 4 + 1) * Rnd + 4)) / 1000 Randomize rt = (Int((6 - 1 + 1) * Rnd + 1)) / 1000 Randomize cangle = (Int((50 - 20 + 2) * Rnd + 20)) Randomize dangle = (Int((50 - 20 + 2) * Rnd + 20)) Randomize de = (Int((50 - 10 + 1) * Rnd + 10)) / 1000

'print random values for each nozzle geometry rs.Cells(i + 2, 1) = Dc rs.Cells(i + 2, 2) = Lc Do While (Dt + (2 / 1000)) > de Randomize Dt = (Int((20 - 5 + 1) * Rnd + 5)) / 1000 Randomize de = (Int((50 - 10 + 1) * Rnd + 10)) / 1000 Loop Dc = de rs.Cells(i + 2, 3) = Dt rs.Cells(i + 2, 4) = ra rs.Cells(i + 2, 5) = rt rs.Cells(i + 2, 6) = cangle rs.Cells(i + 2, 7) = dangle 88

rs.Cells(i + 2, 8) = de rs.Cells(i + 2, 9) = Ln

'print text format for Loci code ss.Cells(i, 1) = Chr(34) & "./bell -Dc " & Dc & " -Lc " & Lc & " -Dt " & Dt & " -ra " & ra & " -rt " & rt & " -cangle " & cangle & " -dangle " & dangle & " -De " & de & " -Ln " & Ln & Chr(34) & " " Next ss.Columns.AutoFit End Sub --------------------------------------------------------------------------------------------------------------------Codes for Convergence Checks

Sub Checker() Set ws = Sheets("residuals") Set gs = Sheets("GoodN") Set ts = Sheets("TextArr") For i = 1 To Rows.Count If IsNumeric(ws.Cells(i, 1)) = False Then l=i counter = 1 Do While IsNumeric(ws.Cells(l, 1)) = False ws.Cells(l, 1) = counter l=l+1 counter = counter + 1 Loop i=l End If Next 89

ncount = 0 gs.Cells(1, 2) = "Nozzle Number" gs.Cells(1, 3) = Residual gcount = 2 For i = 10 To ws.Range("A1", ws.Range("A1").End(xlDown)).Rows.Count Step (9) ncount = ncount + 1 If ws.Cells(i, 1) = 10001 Then If ws.Cells(i, 2) < 0.000001 Then gs.Cells(gcount, 1) = "Nozzle" gs.Cells(gcount, 2) = ncount gs.Cells(gcount, 3) = ws.Cells(i, 2) gcount = gcount + 1 End If End If i=i+1 Next Call DeleteRows(gs) For i = 2 To gs.Range("B2", gs.Range("B2").End(xlDown)).Rows.Count ts.Cells(i - 1, 1) = Chr(34) & gs.Cells(i, 2) & Chr(34) & " " ts.Cells(i - 1, 3) = gs.Cells(i, 2) Next End Sub --------------------------------------------------------------------------------------------------------------------Sub DeleteRows(ws) Dim LR As Long Dim i As Long LR = ws.Cells(Rows.Count, "C").End(xlUp).Row For i = LR To 3 Step -1 90

If WorksheetFunction.CountIf(ws.Range("C2:C" & i), ws.Range("C" & i).Value) > 1 Then ws.Rows(i).EntireRow.Delete End If Next i End Sub --------------------------------------------------------------------------------------------------------------------Sub goodparam()

Dim Dc As Double Dim Lc As Double Dim Dt As Double Dim ra As Double Dim rt As Double Dim cangle As Double Dim dangle As Double Dim de As Double

Set ns = Sheets("GoodN") Set rs = Sheets("randomizer") Set ps = Sheets("GoodP") Set ts = Sheets("GText")

ps.Cells(1, 1) = "nozzle" ps.Cells(1, 2) = "Dc" ps.Cells(1, 3) = "Lc" ps.Cells(1, 4) = "Dt" ps.Cells(1, 5) = "ra" ps.Cells(1, 6) = "rt" 91

ps.Cells(1, 7) = "cangle" ps.Cells(1, 8) = "dangle" ps.Cells(1, 9) = "De"

For i = 2 To ns.Range("B2", ns.Range("B2").End(xlDown)).Rows.Count noz = ns.Cells(i, 2) ps.Cells(i, 1) = "Nozzle " & noz ps.Cells(i, 2) = rs.Cells(noz + 2, 1) ps.Cells(i, 3) = rs.Cells(noz + 2, 2) ps.Cells(i, 4) = rs.Cells(noz + 2, 3) ps.Cells(i, 5) = rs.Cells(noz + 2, 4) ps.Cells(i, 6) = rs.Cells(noz + 2, 5) ps.Cells(i, 7) = rs.Cells(noz + 2, 6) ps.Cells(i, 8) = rs.Cells(noz + 2, 7) ps.Cells(i, 9) = rs.Cells(noz + 2, 8)

Dc = rs.Cells(noz + 2, 1) Lc = rs.Cells(noz + 2, 2) Dt = rs.Cells(noz + 2, 3) ra = rs.Cells(noz + 2, 4) rt = rs.Cells(noz + 2, 5) cangle = rs.Cells(noz + 2, 6) dangle = rs.Cells(noz + 2, 7) de = rs.Cells(noz + 2, 8)

ts.Cells(i - 1, 1) = Chr(34) & "./noz -Dc " & Dc & " -Lc " & Lc & " -Dt " & Dt & " -ra " & ra & " -rt " & rt & " -cangle " & cangle & " -dangle " & dangle & " -De " & de & Chr(34) & " " 92

Next End Sub --------------------------------------------------------------------------------------------------------------------Sub GoodBells() Set gs = Sheets("GBell") Set ts = Sheets("Text") For i = 1 To gs.Range("A1", gs.Range("A1").End(xlDown)).Rows.Count gs.Cells(i, 2) = ts.Cells(i, 1) Next End Sub Sub optimizecheck() 'grab bells from ModeFrontier excel worksheet and convert dimensions for Loci simulations Dim Dc As Double Dim Lc As Double Dim Dt As Double Dim ra As Double Dim rt As Double Dim Ln As Double Dim cangle As Double Dim dangle As Double Dim de As Double Dim numofBells As Double Set rs = Sheets("OptB") Set ss = Sheets("TextB") numofoptBells = 46 For i = 1 To numofoptBells Dc = rs.Cells(i + 1, 1) 93

Lc = rs.Cells(i + 1, 2) Dt = rs.Cells(i + 1, 3) ra = rs.Cells(i + 1, 4) rt = rs.Cells(i + 1, 5) cangle = rs.Cells(i + 1, 6) dangle = rs.Cells(i + 1, 7) de = rs.Cells(i + 1, 8) Ln = rs.Cells(i + 1, 9) ss.Cells(i, 1) = Chr(34) & "./bell -Dc " & Dc & " -Lc " & Lc & " -Dt " & Dt & " -ra " & ra & " -rt " & rt & " -cangle " & cangle & " -dangle " & dangle & " -De " & de & " -Ln " & Ln & Chr(34) & " "

Next ss.Columns.AutoFit End Sub --------------------------------------------------------------------------------------------------------------------Tec Plot Macro (Gather all Nozzle Results into text files)

$!VarSet |MFBD| = '/usr/local/tecplot360ex/bin' $!VarSet |Counter| = 0 $!Loop 100 $!VarSet |Counter| = (|Counter|+1) $!READDATASET '"STANDARDSYNTAX" "1.0" "FILENAME_CASEFILE" "/home/youngest/Desktop/GoodNozzles/gnozzle.|Counter|/n|Counter|.case" "LOADERVER" "141"' DATASETREADER = 'EnSight Loader' $!EXTENDEDCOMMAND COMMANDPROCESSORID = 'CFDAnalyzer4' COMMAND = 'SetFieldVariables ConvectionVarsAreMomentum=\'F\' UVar=11 VVar=12 WVar=13 ID1=\'Pressure\' Variable1=7 ID2=\'Density\' Variable2=10' 94

$!EXTENDEDCOMMAND COMMANDPROCESSORID = 'CFDAnalyzer4' COMMAND = 'SetGeometryAndBoundaries Axisymmetric=\'F\' SymmetryVar=\'Y\' SymmetryValue=0 ConnectZones=\'F\' NodeTolerance=1e-06 DefaultBC=\'Outflow\'' RAWDATA Inflow,[3] Outflow,[4] Wall,[5] Symmetry,[6-7] $!EXTENDEDCOMMAND COMMANDPROCESSORID = 'CFDAnalyzer4' COMMAND = 'Integrate [4] VariableOption=\'ForcesAndMoments\' XOrigin=0 YOrigin=0 ZOrigin=0 ScalarVar=1 Absolute=\'F\' ExcludeBlanked=\'F\' XVariable=1 YVariable=2 ZVariable=3 IntegrateOver=\'Cells\' IntegrateBy=\'Zones\' IRange={MIN =1 MAX = 0 SKIP = 1} JRange={MIN =1 MAX = 0 SKIP = 1} KRange={MIN =1 MAX = 0 SKIP = 1} PlotResults=\'F\' PlotAs=\'Result\' TimeMin=0 TimeMax=0' $!EXTENDEDCOMMAND COMMANDPROCESSORID = 'CFDAnalyzer4' COMMAND = 'SaveIntegrationResults FileName=\'/home/youngest/Desktop/gthrust2/thrustforce0|Counter|.txt\''

$!WRITEDATASET "/home/youngest/Desktop/gparam2/parameters0|Counter|.dat" INCLUDETEXT = YES INCLUDEGEOM = NO INCLUDEDATASHARELINKAGE = YES ZONELIST = [3-4] VARPOSITIONLIST = [1-3,5,7-8,10-13] BINARY = NO USEPOINTFORMAT = YES PRECISION = 9 95

TECPLOTVERSIONTOWRITE = TECPLOTCURRENT $!Endloop --------------------------------------------------------------------------------------------------------------------Code to Group Data for Optimization

Sub Group() Dim counter As Integer Dim counter1 As Integer Dim counter2 As Integer Dim os As Worksheet Dim ps As Worksheet Dim ns As Worksheet Dim ws As Worksheet

Set os = Sheets("OptSheet") Set ps = Sheets("param-File") Set ws = Sheets("thrust-file")

os.Cells(1, 5) = "XForce" os.Cells(1, 6) = "Vxi Avg" os.Cells(1, 7) = "Vxo Avg" os.Cells(1, 8) = "Vxo SD" os.Cells(1, 9) = "Vyo Avg" os.Cells(1, 10) = "Vyo SD" os.Cells(1, 11) = "M Avg" os.Cells(1, 12) = "M SD" os.Cells(1, 13) = "P Avg" os.Cells(1, 14) = "P SD" 96

os.Cells(1, 15) = "To Avg" os.Cells(1, 16) = "To SD" os.Cells(1, 17) = "Length"

os.Columns.AutoFit counter = 2 counting = 2 counter1 = 2 counter2 = 2

For i = 1 To Rows.Count If InStr(ps.Cells(i, 3), "nozzleInlet") > 0 Then os.Cells(counter1, 6) = Mean(i, ps, 9) os.Cells(counter1, 17) = ps.Cells(i + 6, 2) counter1 = counter1 + 1 End If

If InStr(ps.Cells(i, 3), "nozzleOutlet") > 0 Then os.Cells(counter2, 7) = Mean(i, ps, 9) os.Cells(counter2, 8) = StdDev(i, ps, 9) os.Cells(counter2, 9) = Mean(i, ps, 10) os.Cells(counter2, 10) = StdDev(i, ps, 10) os.Cells(counter2, 11) = Mean(i, ps, 5) os.Cells(counter2, 12) = StdDev(i, ps, 5) os.Cells(counter2, 13) = Mean(i, ps, 6) os.Cells(counter2, 14) = StdDev(i, ps, 6) os.Cells(counter2, 15) = Mean(i, ps, 7) 97

os.Cells(counter2, 16) = StdDev(i, ps, 7) os.Cells(counter2, 17) = Abs(os.Cells(counter2, 17) - ps.Cells(i + 6, 2)) counter2 = counter2 + 1 End If

If InStr(ws.Cells(i, 2), "X-Force:") > 0 Then os.Cells(counter, 5) = ws.Cells(i, 3) counting = counting + 1 If counting Mod 2 > 0 Then counter = counter + 1 End If End If Next End Sub --------------------------------------------------------------------------------------------------------------------Function Mean(ref, wksht, col) Dim Sum As Single Dim h As Integer Sum = 0 For h = 1 To 30 Sum = Sum + wksht.Cells(h + ref + 5, col) Next h Mean = Sum / 30 End Function Function StdDev(ref, wksht, col) Dim i As Integer Dim avg As Single, SumSq As Single 98

avg = Mean(ref, wksht, col) For l = 1 To 30 SumSq = SumSq + (wksht.Cells(l + ref + 5, col) - avg) ^ 2 Next l StdDev = Sqr(SumSq / (30 - 1)) End Function

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References 1. Neilson, Robert M. The Steam Turbine. 1903: Longmans, Green, and CO, 1903. 2. Comparison of Empirical and Theoretical Computations of Velocity for a Cold. Dinavahi, Surya, Champagne, Victor and Helfritch, Dennis. s.l.: IEEE, 2010. 3. http://www.tessonics.com/products-cold-spray.html. Cold Gas Spray Coatings. [Online] 2013. [Cited: March 17, 2015.] 4. Mbuyamba, Jean-Baptiste Mulumba, “Calculation and Design of Supersonic Nozzles for Cold Gas Dynamic Spraying using MATLAB and ANSYS Fluent” [2013] 5. Quintao, Karla K., "Design Optimization of Nozzle Shapes for Maximum Uniformity of Exit Flow" [2012]. FIU Electronic Theses and Dissertations. Paper 779. 6. Colaco, Colaco J, Helcio RB Orlande and George S Dulikravich. "Inverse and Optimization Problems in Heat Transfer." J. of the Braz. Soc. of Mech. Sci. & Eng. XXVIII.No.1 (2006). 7. Sobol,I.M. (1967), "Distribution of points in a cube and approximate evaluation of integrals". Zh. Vych. Mat. Mat. Fiz. 7: 784–802 (in Russian); U.S.S.R Comput. Maths. Math. Phys. 7: 86–112 (in English). 8. Colaço, Marcelo J., Dulikravich, George S. and Sahoo, Debasis(2008)'A response surface method-based hybrid optimizer',Inverse Problems in Science and Engineering,16:6,717 — 741 9. Hagemann, Gerald, Hans Immich, Thong Nguyen, and Gennady Dunmov. "Advanced Rocket Nozzles." Journal of Propulsion and Power 14.5 (1998). Print. 10. Sutton, George, and Oscar Biblarz. Rocket Propulsion Elements. 7th ed. John Wiley & Sons, 2001. Print.

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