CFD Modeling of an Over-Expanded Ejector Nozzle for a Gas Turbine Engine Application By Jonathan M. Jause A Project Submitted to the Graduate Faculty of Rensselaer Polytechnic Institute In Partial Fulfillment of the Requirements for the degree of MASTER OF ENGINEERING Major Subject: MECHANICAL ENGINEERING Approved: _________________________________________ Ernesto Gutierrez-Miravete, Project Adviser Rensselaer Polytechnic Institute Hartford, Connecticut December, 2014 (For Graduation May 2015) i © Copyright 2014 by Jonathan M. Jause All Rights Reserved ii CONTENTS LIST OF TABLES ............................................................................................................ iv LIST OF FIGURES ........................................................................................................... v LIST OF VARIABLES .................................................................................................... vi LIST OF ACRONYMS AND KEY WORDS ................................................................ viii ACKNOWLEDGMENT .................................................................................................. ix ABSTRACT ...................................................................................................................... x 1. INTRODUCTION ....................................................................................................... 1 1.1 COMPRESSIBLE FLOW SUMMARY............................................................. 2 1.2 EFFECT OF NPR ON FLOW PROPERTIES ................................................... 4 1.3 EJECTOR NOZZLES ........................................................................................ 7 1.4 PROBLEM STATEMENT ................................................................................. 8 2. METHODOLOGY .................................................................................................... 10 2.1 2.2 CFD Validation Methods .................................................................................. 11 2.1.1 Method for Estimating Quasi 1D Solutions ......................................... 11 2.1.2 Method for Calculating Thrust from CFD Outputs ............................. 15 2.1.3 Pratt & Whitney Empirical Data Tool – PPS ....................................... 16 Ejector Nozzle Design of Experiments (DOE) ................................................. 16 3. RESULTS AND DISCUSSION ................................................................................ 20 3.1 Establishing the Operating Envelope of the Nozzle ......................................... 20 3.2 CFD Validation ................................................................................................. 21 3.3 3.2.1 Quasi 1D and Inviscid Solutions .......................................................... 21 3.2.2 Viscous Solutions ................................................................................. 27 Ejector Nozzle Results ...................................................................................... 28 3.3.1 Statistical Results from the DOE ......................................................... 32 4. CONCLUSION.......................................................................................................... 39 BIBLIOGRAPHY............................................................................................................ 40 iii LIST OF TABLES Table 2.1 List of variables and levels used in DOE. ....................................................... 18 Table 2.2 List of models and parameters which were run in DOE. ................................. 18 Table 3.1 Quasi 1D solutions at Different Axial Positions in the Nozzle ....................... 21 Table 3.2 Quasi 1D Solutions for Ideal Thrust, Actual Thrust, and Cv. .......................... 25 Table 3.3 Calculated percent difference between inviscid CFD solution and quasi 1D and PPS................................................................................................................................... 26 Table 3.4 Calculated percent difference between viscous CFD and PPS........................ 27 Table 3.5 Terms and coefficients for ejector nozzle regression model, NPR = 5. .......... 36 iv LIST OF FIGURES Figure 1.1 Nozzle geometry, dimensions are in mm. ........................................................ 5 Figure 1.2 Nozzle operating diagram (Mattingly, 2006). .................................................. 6 Figure 1.3 Schematic of 2 types of ejector nozzles in which the secondary airflow is used to vary the expansion ratio of the nozzle, (Kerrebrock, 1992). ......................................... 8 Figure 1.4 Baseline geometry for the ejector nozzle with parameters. ............................. 9 Figure 2.1 Graphical representation of method used to estimate thrust. Each box represents a finite element with corresponding nodes for data points. ............................ 15 Figure 2.2 Models A thru H that were run in DOE. ........................................................ 19 Figure 3.1 NPR vs. Cv and the associated boundaries through the operating range of the given nozzle. .................................................................................................................... 20 Figure 3.2 Contour of axial velocity (m/s) through the conventional nozzle. ................. 22 Figure 3.3 Contour of static pressure (Pa) through the conventional nozzle................... 22 Figure 3.4 Contour of static temperature (K) through the conventional nozzle. ............. 22 Figure 3.5 Mach number quasi 1D comparison to CFD solutions for NPR = 5.............. 23 Figure 3.6 Static pressure quasi 1D comparison to CFD solutions for NPR = 5. ........... 24 Figure 3.7 Static temperature quasi 1D comparison to CFD solutions for NPR = 5. ...... 24 Figure 3.8 Cv comparison of different solution methods assuming inviscid flow. ......... 26 Figure 3.9 Cv comparison of different solution methods assuming viscous flow........... 27 Figure 3.10 Comparison of all solutions for a conventional nozzle. ............................... 28 Figure 3.11 Contour of axial velocity (m/s) through the ejector nozzle, NPR = 5. ......... 29 Figure 3.12 Contour of static pressure (Pa) through the ejector nozzle, NPR = 5. ......... 29 Figure 3.13 Contour of static temperature (K) through the ejector nozzle, NPR = 5. ..... 30 Figure 3.14 Cv comparison of various ejector configurations with Ptej = 24,538 Pa. ...... 31 Figure 3.15 Cv comparison of various ejector configurations with Ptej = 49,075 Pa. ...... 31 Figure 3.16 Actual thrust comparison for various nozzle configurations. ...................... 32 Figure 3.17 Statistical results from DOE for ejector configurations on a Cv basis. ........ 34 Figure 3.18 Statistical results from DOE for ejector configurations on an actual thrust basis. ................................................................................................................................ 35 Figure 3.19 Actual thrust comparison between conventional nozzle and ejector nozzle by using regression model. ................................................................................................... 38 v LIST OF VARIABLES Subscript or Superscript Description * Throat a Ambient A Actual e Exit ej Ejector I Ideal I8 Primary Nozzle Ideal t Total tej Ejector Total Symbol Unit Description A m² Area c - Number Of Levels Cv - Thrust Coefficient e J/kg Specific Internal Energy F N Thrust f N Body Force (bold indicates vector notation) h J/kg Specific Enthalpy h m Height k - Number Of Factors L m Distance To Inlet From Throat ̇ kg/s Mass flow P Pa Pressure p Pa Pressure ̇ J/(s*kg) R J/(kg*K) Volumetric Rate Of Heat Addition Per Unit Mass Specific Gas Constant S - Surface T K Temperature vi t s Time V m/s Velocity (bold indicates vector notation) γ - Heat Capacity Ratio θ ° Angle ρ kg/m³ Density m³ Volume vii LIST OF ACRONYMS AND KEY WORDS Acronym Definition NPR Nozzle Pressure Ratio CFD Computational Fluid Dynamics DOE Design Of Experiments CD Convergent/Divergent PPS Performance Prediction System Key Words Ejector nozzle, convergent-divergent nozzle, gas turbine engine, CFD, DOE, inviscid, K-ε turbulence model, NPR viii ACKNOWLEDGMENT I’d like to thank my advisors Ernesto and Brendan for guiding me and helping me complete this project. I’d like to thank my professional advisor Robert Bush for providing his expert advice on the subject and in helping me outline the goals of my research. I’d also like to thank my wife, Rebekah, and my son, Aaron, for holding the family together while I spent long nights and many weekends working towards my Master’s degree. I couldn’t have done it without them. And most importantly I’d like to thank God for giving me direction in life and giving me the strength and endurance to finish the race. ix ABSTRACT Convergent-divergent nozzles have been the topic of study and application ever since the first jet engine was created in 1939. Like all technology, the need arises to improve performance characteristics and this is the case for nozzles in the form of thrust generation. In this paper, an effort is made to study the performance impacts of incorporating a secondary stream in the divergent section of a convergent-divergent nozzle using computational fluid dynamics. The modeling and analysis is carried out in ANSYS Workbench 14.5 and Fluent 14.5.7 which is a CFD processor. The models employed for study are Inviscid, while the K-ε turbulence model was briefly exercised to validate the CFD solutions. In addition, Minitab 16 was used to run a design of experiments and evaluate the interactions between several key parameters, ejector total pressure, location, area, and angle. It is found that the inviscid solution produces a more conservative thrust estimate when compared to the turbulent case. This is because flow separation actually reduces the difference between ambient pressure and exit pressure of the nozzle in the region of separation. Hence, improved thrust efficiency is expected when considering turbulent flow against inviscid flow in the over-expanded region. In addition, it is found that the key parameters that influence thrust generation are ejector total pressure, angle, area, and location, listed in order of importance. While the key parameters that affect thrust generation also affect thrust efficiency, it was discovered that ejector angle affects thrust efficiency in opposing directions. Higher ejector angles produce less thrust while improving thrust efficiency. x 1. INTRODUCTION Since the introduction of jet engines in 1939, propulsion systems have been increasing in thrust capability. Ever since the first propulsion system, they have consisted of four main modules; a compressor, a combustor, a turbine, and a nozzle. The need for increased thrust capability has driven improvements and, therefore, complexity in all components of the engine. Specifically, nozzles have grown in complexity with the need for subsonic and supersonic aircraft. Nozzles can be divided into three main categories of which there are many subcategories contained within each. Depending on the requirements, an aircraft propulsion system may require the use of a specific nozzle. Some of these requirements may include whether or not an engine has an afterburner or whether or not it is required to produce supersonic speeds. These types of design considerations will dictate which nozzle is selected for an engine. One key parameter for nozzle design is the nozzle pressure ratio, NPR. This value is determined from the ratio of total pressure to ambient pressure and it is this pressure differential that drives the flow through a nozzle. The first type of nozzle is the Fixed Convergent Nozzle. These types of nozzles are well suited for subsonic applications because they do not require variation in throat area for optimum performance. They will typically operate at nozzle pressure ratios, or NPR, less than 2 so that any variation provides insignificant benefit. However, since these nozzles are fixed they cannot be used in applications that require afterburning. As such they are most often seen in early, non-afterburning jet engines as well as modern subsonic commercial aircraft. The second type of nozzle is the Variable Convergent Nozzle. These types of nozzles are generally used to allow the use of afterburning which allows the aircraft to fly at high subsonic and transonic Mach numbers. They will typically operate at an NPR less than 3, and will fluctuate the throat area depending on whether or not the afterburner is on. Most applications use a relatively simple actuation system consisting of only two area ratios. Variable area convergent nozzles will typically only operate at Mach numbers up 1 to 1.2~1.5. Beyond this point, these types of nozzles provide inadequate performance and the need for the final category of nozzle arises. The final type of nozzle is the Converging/Diverging, CD, Nozzle. Similar to the convergent nozzles above, the CD nozzle comes in two variants, fixed and variable, although the most used is the variable geometry. A fixed area ratio nozzle can attain ideal thrust at only one design point and this is why it is necessary to incorporate variable geometry to allow for more ideal expansion for more flight points. These types of nozzles generate supersonic Mach numbers and operate at large NPR’s in excess of 3. Typical examples of these types of nozzles are used on the F-14, F-15, F-16, F-18, F-22, and F-35. A further discussion on how these nozzles operate will be discussed in the following section. 1.1 COMPRESSIBLE FLOW SUMMARY Compressible flow theory is a topic covered in many introductory aerodynamics textbooks. Presented in this paper is a high level summary of equations and assumptions used for Quasi 1D theory for analysis of a fixed-area-ratio, CD nozzle (Anderson, 2003). The equations presented herein can be used to estimate the performance of a given CD nozzle. Quasi one-dimensional flow is a flow in which the area, pressure, density, and velocity for steady flow vary as a function of distance along the axis. This means that the properties in the flow are uniform at any given cross-section which is not entirely accurate but it is frequently sufficient for a wide variety of engineering problems with compressible flow. The governing equations for quasi-one-dimensional flow are derived from the full continuity (1-1), momentum (1-2), and energy (1-3) equations listed below taken from (Ward, 2010). Continuity Equation − ∯ ∙ = ∰ 2 (1-1) Momentum Equation ∰ − ∯ = ∰ () + ∯( ∙ ) (1-2) Energy Equation ∰ ̇ − ∯ ∙ + ∰ ( ∙ ) (1-3) 2 =∰ [( + ] 2 + ∯ ( + 2 ) ∙ 2 These equations are non-trivial and numerical methods and significant computational power is required to obtain solutions. By making several assumptions, simpler forms of these equations can be obtained. By assuming steady adiabatic flow and no body forces, the quasi-one-dimensional and isentropic equations can be derived. These equations are listed below. −1 2 =1+ 2 (1-4) − 1 2 ⁄(−1) = (1 + ) 2 (1-5) − 1 2 1⁄(−1) = (1 + ) 2 (1-6) 1 1 1 = 2 2 2 (1-7) 2 1 1 + 1 12 1 + ∫ = 2 2 + 2 22 2 1 3 (1-8) 12 22 (1-9) ℎ1 + = ℎ2 + 2 2 From these equations, the analysis of flow through variable area ducts can begin. By assuming a calorically perfect gas and that the flow is isentropic throughout, the following area relation can be derived as a function of Mach number. 2 1 2 − 1 2 (+1)⁄(−1) ( ∗) = 2 [ (1 + )] +1 2 (1-10) Now that the properties of the nozzle can be calculated throughout the nozzle, a free body diagram can be used to determine the thrust from a given nozzle based on the exit conditions. The equation for uninstalled engine thrust for a stationary nozzle derived from this method is, = ̇ + ( − ) (1-11) Through the use of these equations, one can now begin to see the effect nozzle pressure ratio, NPR, has on a nozzle in terms of the Mach number, pressure, temperature, and velocity at different axial stations along the nozzle. 1.2 EFFECT OF NPR ON FLOW PROPERTIES To begin to understand the effect NPR has on performance, a given nozzle geometry was chosen from previous research (Padmanathan, 2012), which analyzed low NPR and shock formation inside a nozzle. This geometry was the topic of subsequent experiments to be discussed further in Chapter 2. The nozzle geometry is depicted below in Figure 1.1 and has an area ratio of 2.56. 4 Figure 1.1 Nozzle geometry, dimensions are in mm. The method for calculating the thrust produced by this nozzle begins with the geometric features, most importantly the area ratio. Since the geometric area of the nozzle is given it is proper to start with equation (1-10) and solve for the exit Mach number. Since the square root is involved there will be two solutions, one for subsonic nozzles and the other for supersonic nozzles. Although this may seem counter-intuitive, supersonic flow is achievable in the divergent section because of conservation of mass flow. It can be shown that velocity increases in the divergent section as long as the throat is sonic, i.e. M* = 1, for any nozzle using equation (1-7). This is the intent of all CD nozzles; hence, the supersonic solution will be desired, i.e. Me > 1. Once the exit Mach number is calculated and inlet total properties are known, static properties can be calculated for the pressure and temperature along the axis of the nozzle. Running through this type of calculation will establish the ideal conditions for the nozzle. As can be seen in Figure 1.2, there is a single nozzle pressure ratio, NPR I, for which this ideal condition can be achieved. Any NPR beyond this ideal condition will result in inefficiencies for thrust generation. When the NPR is greater than NPRI, this is known as under-expanded flow in which the exit pressure, Pe, is greater than the ambient pressure, Pa. This will result in expansion 5 waves forming at the exit of the nozzle, which can cause increased drag for an aircraft. When the NPR is less than NPRI, this is known as over-expanded flow in which Pe < Pa. This will result in the formation of oblique shocks forming at the exit of the nozzle as well as an adverse pressure gradient along the nozzle exit. If this pressure gradient is large enough, it will cause separation on the nozzle wall that will reduce performance. Of the two conditions, over-expansion is considered the worst because it causes thrust losses in the form of increased drag. Although under-expansion has reduced thrust, it is considered lost thrust potential because by simply increasing the area ratio of the nozzle ideal thrust thrusts could be attained. Additionally, there is a point at which the NPR becomes so low and the difference between Pe and Pa is large enough that a normal shock will form at the exit plane of the nozzle. The NPR associated with this condition is NPRNS. It is important to understand the full operating envelope of the nozzle and as such a graphical summary of these different conditions has been created as a function of area ratio and presented in Elements of Propulsion (Mattingly, 2006). This is shown in Figure 1.2. Figure 1.2 Nozzle operating diagram (Mattingly, 2006). 6 NPR can be driven in two ways. The first way is with the total pressure entering the nozzle which can be determined from a parametric analysis of a given engine. The second is related to the ambient conditions and, therefore, the altitude at which the aircraft is flying. As altitude increases, ambient pressure reduces and, therefore, a nozzle will behave more ideally. Both of these combinations of changing altitude and changing engine conditions make controlling a CD nozzle very complex in scheduling movement and actuation in order to achieve ideal thrust. 1.3 EJECTOR NOZZLES As mentioned previously, there are many subcategories of CD nozzles in which any one may produce the desired thrust. One subcategory of CD nozzles, and the one which will be the topic of further discussion, is the ejector nozzle. An ejector nozzle is one that changes the effective nozzle exit area without repositioning the divergent flap by allowing air to fill the over-expanded portion of the divergent nozzle, thus reducing its effective expansion ratio. Ejector nozzles are used as a means to alter the area ratio of a nozzle without introducing significant complexity with actuation systems and monitoring for a fully variable CD nozzle. The most well-known application of this type of nozzle is Pratt & Whitney’s J58 engine which was used to power the SR-71 Blackbird. Ejector nozzles are used in many supersonic aircraft because they can vary the throat area by use a secondary stream. A typical ejector nozzle design is shown in Figure 1.3. Primary air from the core of the engine flows through a convergent nozzle while a secondary airflow of higher pressure flows over the primary stream. The pressure provided by this secondary air is what controls the mass flow through the secondary nozzle thus controlling the nozzle area available to the primary airflow and, therefore, the nozzle expansion ratio. 7 Figure 1.3 Schematic of 2 types of ejector nozzles in which the secondary airflow is used to vary the expansion ratio of the nozzle, (Kerrebrock, 1992). If this nozzle expansion ratio is not ideal, the ejector nozzle will face similar performance affects to that of the CD nozzle through over and under-expansion. 1.4 PROBLEM STATEMENT The problem for nozzle design becomes how to maintain ideal thrust for as wide an operating envelope as possible while still maintaining other design considerations such as cost and weight for the engine. As discussed in section 1.2, the conditions of over and under-expansion produce below ideal thrust. Under-expansion does not fully expand the flow which is seen as lost thrust potential. In other words, if the nozzle had a larger area ratio the thrust could be more ideal. On the opposite side, over-expansion expands the airflow too much and produces increased drag from the nozzle. The condition of overexpansion is more detrimental to nozzle performance and, as such, the question of how to improve thrust while operating in this over-expanded region will be answered through the addition of a ejector. In the following sections, the solutions to the performance of the nozzle given in Figure 1.1 will be presented and discussed in detail. Subsequent analyses have been conducted 8 examining the performance impact of the ejector shown in Figure 1.4. In addition, trade studies have been developed and analyzed on different geometry modifications to assess the performance impact. The variables include p110, p112, as well as inlet angle and total pressure feeding the ejector. Ejector Inlet Figure 1.4 Baseline geometry for the ejector nozzle with parameters. 9 2. METHODOLOGY When designing or developing a new configuration or analysis it is always important to have a baseline from which the tools and methodology can be validated prior to any experimental testing of such a new configuration. For this project the chosen tool of analysis was ANSYS Workbench 14.5 and Fluent. The validation for this analysis tool was conducted on the baseline geometry previously discussed in Figure 1.1 through the use of Quasi 1D theory as well as a Pratt & Whitney proprietary tool which predicts nozzle performance based on empirical data. Within Fluent there are many options for analysis models to choose from. In order to capture the physics of compressible flow through the nozzle the density based solver was chosen. Also, the energy equation must be turned on in order to capture the temperature effects and increase in energy. The final model that was used during the analysis was related to the viscosity of the fluid. For high NPR values, viscosity has a very minor effect on the results. However, when the NPR is reduced sufficiently viscosity begins to play a major role in separation along the nozzle wall. As was discussed in Chapter 1, the separation, although still below ideal, helps regain some thrust by reducing the effective area of the nozzle which prevents some over-expansion. Results were produced for both models and compared. The models chosen to adequately capture these viscous effects were inviscid and κ-ε. The boundary conditions and setup of the analysis is critical in ensuring the physics of the nozzle is captured. As such, the following boundary conditions were applied for the baseline nozzle analysis. Since the nozzle is driven by a pressure differential across the inlet and exit, pressure inlets and outlets were used to drive the flow through the nozzle. And in order to control the analysis via a single variable, the outlet pressure which would be considered the ambient pressure, Pa, was chosen to vary in order to obtain the desired NPR while the inlet total pressure, Pt, was set to 101.325 kPa. In addition to setting the total pressure for the inlet, the total temperature must be designated, this was set to 300 K. The value chosen for the total pressure and temperature is somewhat arbitrary because all comparisons to performance are based solely on NPR. Hence, the 10 expected outcomes of this analysis could be scaled based on actual total pressures and temperatures. The remaining boundary conditions are simply set to walls and axis boundary conditions for this 2D axis-symmetric model. For the analysis that looked into the addition of the ejector nozzle, a similar boundary condition was applied to the inlet as the nozzle inlet. It was set to a pressure inlet which sets the total pressure and temperature. This intuitively makes the most sense for a boundary condition because in a real application the supply air can either be taken from two different sources; the ambient air or air bled from a specific compressor stage. The source of this air does not need to be determined for the problem setup but it can be accounted for in the post-processing of the results which will be discussed in the subsequent results section. 2.1 CFD Validation Methods 2.1.1 Method for Estimating Quasi 1D Solutions To obtain the quasi-1D results for the baseline nozzle, the equations developed in Chapter 1 were used. A sample calculation process is given below which shows the methodology behind calculating the performance of the nozzle for a given NPR = 10. Since nozzle performance is always measured at the exit of the nozzle the subscript e will be used to denote the exit condition. For the given nozzle with Ae/A* = 2.56 and from Eq. (1-10) an iterative solution can be obtained to find Me. (2.56)2 2 − 1 2 (+1)⁄(−1) = [ (1 + )] 2 2 + 1 1 Me = 2.47 Assuming Pt and Tt are constant through the nozzle, the isentropic relations given by Eq. (1-4) and Eq. (1-5) can be used with the initial conditions to solve for Pe and Te. −1 2 =1+ = 2.22 2 11 1 = ( ) 300 = 135 2.22 − 1 2 ⁄(−1) = (1 + ) = 16.31 2 1 = ( ) 101,325 = 6,212 16.31 While these properties were calculated at the exit, this same methodology can be used for any point in the nozzle using A as the area at a certain location. This was calculated for the baseline nozzle in order to show how the pressures, temperatures, and Mach numbers vary with axial position through the nozzle in order to validate the CFD solutions. Furthermore, thrust performance can be estimated using the exit properties previously calculated and the following methodology. Assuming an ideal gas, the mass flow can be determined from the following equation ̇ = = √ ̇ = 973.5 / From Eq. (1-11), the actual thrust can be calculated. = ̇ + ( − ) = (973.5)(575.3) + (6,212 − 10,132.5)(10.55) = 518.7 Additionally, the ideal thrust must be calculated based on optimum expansion, i.e. Pe = Pa. The equation for ideal thrust has been defined in (Gamble, 2004) and is only based on NPR and throat area. 2 2 2 +1⁄−1 −1⁄ ( )( ) [1 − ( ) ] −1 +1 ∗√ 8 = Plugging in the initial values from the problem 8 = 524.9 12 (2-1) Therefore, the thrust performance in terms of a non-dimensional parameter called the thrust coefficient is given by the ratio of these two thrusts given by the following equation = = 8 = 0.988 8 13 (2-2) 2.1.2 Method for Calculating Thrust from CFD Outputs The method for calculating thrust from the CFD output is one that involved integration across the exit plane for each mesh element and taking the average parameter values from the two nodes bounding each elemental area. The output data from Fluent, pressure, temperature, and velocity, can be used to determine the direct properties for calculating thrust in Eq. (1-11). Fluent outputs this data in a notepad format, which can be imported into Excel and then manipulated to produce elemental thrusts through each area. Because the mesh is fine and the properties do not vary drastically from one node to the next it is a good assumption to take an average of the nodal values and assume these values act across the elemental area. Once the elemental thrusts are calculated, they can be added together to estimate the total thrust produced from the nozzle. A graphical schematic of how this is done is shown in Figure 2.2. The ideal thrust and thrust coefficient are still calculated using Eq. (2-1) and Eq. (2-2), respectively. Exit Plane 5 F5 r5 4 F4 r4 3 F3 r3 2 F2 r2 1 F1 r1 0 Figure 2.1 Graphical representation of method used to estimate thrust. Each box represents a finite element with corresponding nodes for data points. 14 2.1.3 Method for Calculating Thrust from CFD Outputs The method for calculating thrust from the CFD output is one that involved integration across the exit plane for each mesh element and taking the average parameter values from the two nodes bounding each elemental area. The output data from Fluent, pressure, temperature, and velocity, can be used to determine the direct properties for calculating thrust in Eq. (1-11). Fluent outputs this data in a notepad format, which can be imported into Excel and then manipulated to produce elemental thrusts through each area. Because the mesh is fine and the properties do not vary drastically from one node to the next it is a good assumption to take an average of the nodal values and assume these values act across the elemental area. Once the elemental thrusts are calculated, they can be added together to estimate the total thrust produced from the nozzle. A graphical schematic of how this is done is shown in Figure 2.2. The ideal thrust and thrust coefficient are still calculated using Eq. (2-1) and Eq. (2-2), respectively. Exit Plane 5 F5 r5 4 F4 r4 3 F3 r3 2 F2 r2 1 F1 r1 0 Figure 2.2 Graphical representation of method used to estimate thrust. Each box represents a finite element with corresponding nodes for data points. 15 2.1.4 Pratt & Whitney Empirical Data Tool – PPS In addition to the quasi-one-dimensional estimations, a proprietary analysis tool from Pratt & Whitney has been used to offer a more realistic comparison between the CFD results. This tool took key geometric features such as Ae and A* and runs through a specified range of NPR and plots thrust coefficient, Cv, as a function of NPR. Because of the proprietary nature of this tool only the numerical predictions will be shown and compared in the results section. 2.2 Ejector Nozzle Design of Experiments (DOE) Once the Fluent analysis was proven to produce reliable results based on comparisons to the two methods above for inviscid flow, a series of tests were run in a controlled manner to evaluate the effects of various changes and determine which input factors have the most impact on the ejector design. This is called a design of experiments, DOE. In a typical DOE, a series of tests are run in which purposeful changes are made to input variables for the system so that one may observe and identify the reasons for changes in the output response. There are three different types of experimental designs which are dependent upon the end use of the analysis. The first is a screening DOE which is typically used to determine which few of many factors impact the output response the most. The second type is a characterization DOE in which several factors are experimented with in order to better understand how each of the input variables interacts with each other and their influence on the output response. The third and final type is an optimization DOE where a few input variables are experimented with in detail in order to optimize the output response to a desired output. In a typical application, a successive approach would be used starting with the screening DOE and finishing with the optimization DOE. In a controlled experiment, each input variable is changed independently from all the other variables in order to see its effect on the output. Each variable also has at least 2 different levels in order to see the change in output related to that variable. If a large amount of variables are chosen at two levels each, the amount of experiments can 16 increase dramatically which can cost time and money. The method of experimentation in which all the variables are tested individually is called a full factorial design. The number of experiments to be run for a full factorial DOE is determined by equation (2-3) where c is the number of levels for each input variable and k is the number of input variables. = (2-3) As one can see, the number of levels for each input variable can increase the number of experiments exponentially. In the case of the optimization DOE, the number of levels per variable would likely need to be large in order to adequately capture all the details. However, the number of experiments that can be run is based on time and money. Therefore, a fractional factorial design can be used to establish trends between the variables and allow for time savings while still capturing most of the detail. For the ejector analysis, a full factorial, characterization DOE has been implemented since a basic understanding of nozzle physics can be used to focus on several important factors with 2 levels of each. In this case, the output variable is thrust. Mass flow and NPR are key drivers in the thrust produced by the nozzle as can be seen from equation (2-1). A form of these two drivers, ejector total pressure, Ptej, and ejector area, Aej, assuming the ejector is a convergent nozzle, can alter the ideal thrust. Since the nozzle has two inlets, the ideal thrust equation given in (2-1) must be altered to account for it. This can be accomplished by taking equation (2-1) and instead of using the main nozzle parameters for A and Pt, use Aej and Ptej, and combine this with the main nozzle prediction. The equation given below should be used to estimate the ideal thrust. −1⁄ 2 2 2 +1⁄−1 √ = ( )( ) [1 − ( ) −1 +1 = 8 + ] (2-4) (2-5) The ejector area was established by setting its height, hej, and assuming an annular cross-section. Additionally, Ptej and its effect on the thrust will be highly dependent on 17 the location of the ejector since the static pressure within the main nozzle is changing with axial location. Hence, location, as defined from the throat, was another variable, Lej. The final variable chosen for the DOE was the angle of incidence relative to engine centerline, θej. These variables and their associated levels are shown in Table 2.1. Table 2.1 List of variables and levels used in DOE. Input Variable Low-Level Mid-Level High-Level Ptej (Pa) 24538 N/A 49075 hej (m) 0.100 N/A 0.150 Lej (m) 0.700 1.050 1.400 Θej (°) 0 N/A 30 The total number of models developed for this analysis was 12 based on the fact that Ptej is an input parameter from Fluent and further refinement on Lej was desired. The models are documented below in Table 2.2 and Figure 2.3. Also, as seen from the conventional CD nozzle for the inviscid solution, the NPR does not need to be run for each value as long as the NPR does not encounter a shock inside the nozzle, i.e. separation will not occur since we are assuming inviscid flow. As such, only the results from an NPR=5 were used while computing the results for the remaining pressure ratios for each model. Table 2.2 List of models and parameters which were run in DOE. Model hej (m) Lej (m) Θej (°) A 0.100 1.050 0 B 0.100 1.050 30 C 0.150 1.050 30 D 0.150 1.050 0 E 0.100 1.400 30 F 0.100 1.400 0 G 0.150 1.400 0 H 0.150 1.400 30 I 0.100 0.700 0 J 0.100 0.700 30 K 0.150 0.700 30 L 0.150 0.700 0 18 Figure 2.3 Models A thru H that were run in DOE. Once the models were established and the Fluent solution obtained, Minitab 16, a statistical analysis software, was used to evaluate the interactions, dependencies, and effects of each different variable. As part of this DOE analysis in Minitab, a regression model is proposed which will predict the thrust coefficient for an ejector. This regression model is valid only for the parameters tested within this DOE. The results of the DOE are shown and discussed in Chapter 3.3. 19 3. RESULTS AND DISCUSSION The following chapter presents the results of the simulations conducted in this project as well as a brief discussion for each. 3.1 Establishing the Operating Envelope of the Nozzle In order to guide the CFD analysis in making sure the correct region is focused on for ejector optimization, the bounding NPR’s must be established as discussed in Chapter 1.2. To determine NPRI the methodology presented in Chapter 2.1.1 was used. Based on the quasi 1D analysis for a nozzle with an area ratio of 2.56, NPRI = 16.3. This is the pressure ratio at which ideal thrust will be achieved for this nozzle and is based solely on the area ratio. The other bounding NPR is NPRNS, the pressure ratio at which a normal shock will exist at the exit of the nozzle. This pressure ratio was calculated to be NPRNS = 2.25. A figure showing a graphical representation of these bounding planes is shown in Figure 3.1. Also shown here is an estimated transition region for which the nozzle will experience Mach Reflections based on Figure 1.2 for the given area ratio. 1.00 Normal Shocks in Divergent Section 0.90 0.80 Under-expansion 0.70 Over-expansion Mach Reflection Cv 0.60 0.50 Over-expansion Regular Reflection 0.40 0.30 0.20 0.10 0.00 1 6 11 16 21 NPR Figure 3.1 NPR vs. Cv and the associated boundaries through the operating range of the given nozzle. 20 Based on these preliminary hand calculations, the given nozzle should experience normal shocks within the divergent section when NPR < 2.25. Between an NPR value of 2.25 and 16.3, this nozzle will experience over-expansion which causes inefficiencies in the form of increased drag. At NPR = 16.3 the nozzle will behave ideally with a Cv = 1. When NPR > 16.3, the nozzle is under-expanded. This can be thought of as lost thrust potential since the nozzle flow could be expanded more if there were a larger A9. With the basic operating region known, the project was tailored to experiment on the over-expanded region since this is the region which causes the most inefficiency in nozzle performance. The specific region of NPR’s experimented with ranged from 5 to 16. 3.2 CFD Validation 3.2.1 Quasi 1D and Inviscid Solutions The results presented in this section have been used to validate the Fluent analysis tool as a viable option for conducting further ejector analysis. Following the method described in Chapter 2.1.1, the flow properties at different axial stations along the nozzle are tabulated in Table 3.1 for an NPR = 5. Table 3.1 Quasi 1D solutions at Different Axial Positions in the Nozzle X (m) M P (Pa) T (K) .900 1 53528.2 250.0 1.250 1.483 28287.7 208.4 1.600 1.759 18772.6 185.3 1.950 1.972 13525.8 168.8 2.300 2.153 10199.2 155.7 2.650 2.312 7952.5 145.0 3.034 2.47 6213.8 135.1 Figure 3.2, Figure 3.3, and Figure 3.4 show the results of the Fluent analysis in the form of contour plots for axial velocity, static pressure and static temperature for an NPR of 5. 21 Figure 3.2 Contour of axial velocity (m/s) through the conventional nozzle. Figure 3.3 Contour of static pressure (Pa) through the conventional nozzle. Figure 3.4 Contour of static temperature (K) through the conventional nozzle. A comparison of the Fluent results with the quasi 1D solution is shown in Figure 3.5, Figure 3.6, and Figure 3.7. These plots show Mach number, static pressure, and static temperature as a function of axial distance in the nozzle, respectively. There are two 22 lines plotted from the CFD solution to try and capture the extremes of the flow field. One line represents the solution along the axis of the nozzle while the other is the solution along the nozzle wall. As can be seen in Figure 3.2, Figure 3.3, and Figure 3.4, the properties are not constant at any given cross-section of the nozzle. This is one assumption from the quasi-one-dimensional analysis that proves to be incorrect. However, even with this incorrect assumption the 1D solution gives roughly an average value between the two extremes. 2.500 Mach Number 2.000 1.500 CFD Axis CFD Nozzle Wall 1.000 Quasi 1D 0.500 0.000 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 Distance Along Nozzle (m) Figure 3.5 Mach number quasi 1D comparison to CFD solutions for NPR = 5. 23 Statis Pressure (kPa) 100.0 CFD Axis 90.0 CFD Nozzle Wall 80.0 Quasi 1D 70.0 60.0 50.0 40.0 30.0 20.0 10.0 0.0 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 Axis Title Figure 3.6 Static pressure quasi 1D comparison to CFD solutions for NPR = 5. 300.0 CFD Axis 280.0 CFD Nozzle Wall Quasi 1D Static Temperature (K) 260.0 240.0 220.0 200.0 180.0 160.0 140.0 120.0 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 Figure 3.7 Static temperature quasi 1D comparison to CFD solutions for NPR = 5. As can be seen in these figures, the approximate 1D solution matches closely with the solutions resulting from the inviscid analysis. Specifically at the exit of the nozzle, the 24 percent error between the two methods is approximately 3.5%, 9.2%, and 3.1% for temperature, pressure, and Mach number, respectively. These errors are well within the expected order of accuracy for the methods presented herein. The completed analysis for thrust at various pressure ratios has been conducted and tabulated in Table 3.2. Table 3.2 Quasi 1D Solutions for Ideal Thrust, Actual Thrust, and Cv. NPR 16.00 14.00 12.00 10.00 8.00 6.00 4.00 3.00 Me 2.47 2.47 2.47 2.47 2.47 2.47 2.47 2.47 Pe 6213.8 6213.8 6213.8 6213.8 6213.8 6213.8 6213.8 6213.8 Pa 6332.8 7237.5 8443.7 10132. 12665. 16887. 25331. 33775. Te 135.10 135.10 135.10 135.10 135.10 135.10 135.10 135.10 Ve 575.50 575.50 575.50 575.50 575.50 575.50 575.50 575.50 ̇ 973.47 973.47 973.47 973.47 973.47 973.47 973.47 973.47 FI 559235 550161 539044 524920 506015 478559 432368 392414 FA 558977 549428 536696 518871 492134 447572 358449 269325 Cv 0.9995 0.9986 0.9956 0.9884 0.9725 0.9352 0.8290 0.6863 Similar to the results presented above, the inviscid 1D solution has been overlaid with the results from the conducted analysis in Fluent and results taken from the Pratt & Whitney proprietary tool, PPS. The thrust coefficient, Cv, as a function of NPR is plotted in Figure 3.8 assuming inviscid flow. 25 1.0000 Cv 0.8000 0.6000 PPS Inviscid CFD Inviscid 0.4000 Quasi 1D Inviscid 0.2000 0.0000 0.00 5.00 10.00 15.00 20.00 NPR Figure 3.8 Cv comparison of different solution methods assuming inviscid flow. As can be seen in Figure 3.8, the CFD results show good correlation to the predicted 1D solution as well as Pratt & Whitney’s predicted values. A summary of the percent differences between these sets of data is tabulated in Table 3.3. Table 3.3 Calculated percent difference between inviscid CFD solution and quasi 1D and PPS. NPR 3 4 6 8 10 12 14 16 Quasi 1D 0.686328 0.829037 0.93525 0.972569 0.988477 0.995644 0.998667 0.999538 PPS 0.642681 0.7927 0.9057 0.9463 0.9642 0.9726 0.9766 0.9782 CFD 0.607199 0.757228 0.870379 0.91122 0.92934 0.938058 0.942246 0.944033 % Diff to Quasi 1D -11.5 -8.7 -6.9 -6.3 -6.0 -5.8 -5.6 -5.6 % Diff to PPS -5.5 -4.5 -3.9 -3.7 -3.6 -3.6 -3.5 -3.5 The results of the Fluent analysis show good correlation to the PPS solution. There are two notable outcomes of this comparison. The first is that there is an increase in error as NPR is reduced. This is likely caused by the rapid decrease in Cv in this region for inviscid flow. Secondly, the CFD more closely matches the PPS solution. This could be 26 explained by the many simplifications that go into the 1D analysis including the assumption that properties are constant across the entire cross section. This is a gross assumption and one that should not be assumed when doing detailed estimates. However, it has been shown to provide an approximate solution for simple applications. 3.2.2 Viscous Solutions To further evaluate the potential of CFD, turbulent modeling was looked into which shows good correlation to predicted values using PPS. These results are shown in Figure 3.9 and compared in Table 3.4. 1.0000 Cv 0.8000 0.6000 PPS Inviscid PPS Turbulent 0.4000 CFD Turbulent 0.2000 0.0000 0.00 5.00 10.00 15.00 20.00 NPR Figure 3.9 Cv comparison of different solution methods assuming viscous flow. Table 3.4 Calculated percent difference between viscous CFD and PPS. NPR 3 5 7 9 11 13 15 PPS 0.9065 0.9335 0.9523 0.9643 0.9716 0.9757 0.9777 CFD 0.93016 0.869333 0.8926 0.919485 0.932068 0.9382 0.941052 27 % Diff to PPS 2.6 -6.9 -6.3 -4.6 -4.1 -3.8 -3.7 There are several findings that can be stated from this data. First, the data shows that turbulence actually improves nozzle efficiency at lower NPR’s. When compared to the inviscid solution which dropped down to below 20%, the viscous solution never goes below 80%. This shows that assuming inviscid flow at lower NPR’s will under-estimate the thrust coefficient. Similar to the inviscid analysis, Fluent is generally predicting a lower Cv when compared to PPS. Although turbulent modeling was not used for the ejector portion of this project, these results further validate the results obtained from Fluent. Figure 3.10 is another comparison plot that shows all solutions obtained for the conventional nozzle geometry, both inviscid and viscous flows. 1.0000 Cv 0.8000 PPS Inviscid 0.6000 PPS Turbulent CFD Inviscid 0.4000 CFD Turbulent Quasi 1D Inviscid 0.2000 0.0000 0.00 5.00 10.00 15.00 20.00 NPR Figure 3.10 Comparison of all solutions for a conventional nozzle. 3.3 Ejector Nozzle Results The results presented in this section provide the basis of understanding the key parameters that influence ejector nozzle efficiency. As stated previously, various models 28 were run using the same methodology presented above and results were compiled and summarized into the following tables and figures. Figure 3.11, Figure 3.12, and Figure 3.13 show typical results for one of the ejector models run in the DOE for axial velocity, static pressure, and static temperature. The results shown are for model D with Ptej = 49,075 Pa. Figure 3.11 Contour of axial velocity (m/s) through the ejector nozzle, NPR = 5. Figure 3.12 Contour of static pressure (Pa) through the ejector nozzle, NPR = 5. 29 Figure 3.13 Contour of static temperature (K) through the ejector nozzle, NPR = 5. While it’s difficult to interpret the performance impacts of the ejector nozzle from the contour plots, a couple observations can be made. In Figure 3.11, the axial velocity is decreased in the local region immediately downstream of the ejector. While this alone would decrease thrust, the increased mass flow in this region helps neutralize the effect. In Figure 3.12, the static pressure is increased which helps recover thrust in the overexpanded operating region. Figure 3.14 and Figure 3.15 show Cv as a function of NPR for the entire set of experiments. One important observation regarding these two plots is that Cv is improved in all models by increasing Ptej. This physically makes sense because the ejector flow penetrates more deeply into the main stream since mass flow is increasing with Ptej. Similar to Figure 3.12, the static pressure is increased in this region which will help improve thrust according to equation (1-11). 30 Figure 3.14 Cv comparison of various ejector configurations with Ptej = 24,538 Pa. Figure 3.15 Cv comparison of various ejector configurations with Ptej = 49,075 Pa. 31 While Cv should be compared amongst similar types of nozzles, it does not provide the best means of comparing between the conventional nozzle and the ejector nozzle. The ideal thrust between the two nozzle configurations is calculated differently using equations (2-5) and (2-1). Instead, actual thrust was used to show the performance impacts made by the ejector nozzle. There is marked improvement between the two configurations as can be seen in Figure 3.16. Every ejector nozzle that was run in the analysis showed improvement when compared to the baseline nozzle. The lowest percent increase on average for actual thrust was 2.16% for model B with Ptej = 24,538 Pa. The highest percent increase was 18.00% for model G with Ptej = 49,075 Pa. 700 Actual Thrust (kN) 650 600 550 500 450 400 350 300 0 5 10 15 20 NPR CD A2 A4 B2 B4 C2 C4 D2 D4 E2 E4 F2 F4 G2 G4 H2 H4 I2 I4 J2 J4 K2 K4 L2 L4 Figure 3.16 Actual thrust comparison for various nozzle configurations. 3.3.1 Statistical Results from the DOE Once all the solution data was compiled for each test, the values for Cv and each corresponding model with input parameters were copied into Minitab 16 for statistical analysis and evaluation. The results from the statistical analysis are shown in Figure 3.17 and Figure 3.18. 32 (a) (b) 33 (c) (d) Figure 3.17a-d Statistical results from DOE for ejector configurations on a Cv basis. 34 (a) (b) Figure 3.18a-b Statistical results from DOE for ejector configurations on an actual thrust basis. Several observations can be made from the data presented in Figure 3.17 but first a brief description of the graphs is necessary. First, Figure 3.17a shows the interactions plot for Cv, which displays how each factor interacts with each other and how they both effect the thrust coefficient. By following the columns and rows labeled, one can interpret the results and see the relationship. Figure 3.17b shows the main effects plot for Cv, which gives the overall trending of each individual variable. The steeper the line, the greater 35 the effect will be on the thrust coefficient. Figure 3.17c shows a Pareto of the different parameters and interactions. Finally, Figure 3.17d shows the residuals based upon the regression model developed by Minitab. It tests for normality and shows how the data points correspond to the predicted regression model. Observed in Figure 3.17, the top five important factors that influence Cv according to the Pareto chart are Ptej, Lej, Aej, the interaction between Ptej and θej, and, finally, θej. Also, according to the residuals plot, the regression model accurately captures the data with an R-sq value equal to 99.93%. This means that greater than 99% of the data is fitted according to the regression model. The regression model coefficients are tabulated below in Table 3.5. Regression models were developed for each integer NPR from 5 to 20 using the same methodology in order to provide a complete analysis for an ejector nozzle containing similar geometry. Table 3.5 Terms and coefficients for ejector nozzle regression model, NPR = 5. Term Constant Ejector Pt Coef 0.813726 Ejector Area Ejector Loc Ejector Angle Ejector Pt*Ejector Area Ejector Pt*Ejector Loc Ejector Pt*Ejector Angle Ejector Area*Ejector Loc Ejector Area*Ejector Angle Ejector Loc*Ejector Angle Ejector Pt*Ejector Area*Ejector Loc Ejector Pt*Ejector Area*Ejector Angle Ejector Pt*Ejector Loc*Ejector Angle Ejector Area*Ejector Loc*Ejector Angle Ejector Pt*Ejector Area*Ejector Loc*Ejector Angle 3.45E-07 -0.111283 0.0067438 0.000193723 4.58E-06 -5.78E-08 4.79E-09 0.164681 -0.00102484 -4.20E-04 1.03E-07 9.35E-10 1.34E-08 -0.00222246 3.56E-08 A similar statistical analysis was conducted on an actual thrust basis in order to capture the magnitude of thrust generated by the nozzle. Since no regression model was used and similar Pareto charts were developed, these results were not included in Figure 3.18a-b. 36 Interestingly, the main effects plot shows different results than the Cv analysis for trends. Ejector angle improved Cv with increasing angle. However, actual thrust is reversed in favor of a lower angle to produce more thrust. This physically makes sense because of the reduction of axial velocity as the ejector angle increases. Similarly, ejector location appears to decrease actual thrust when at a certain value in between the two tested extremes. From these results, it appears there are two opposing performance characteristics, thrust efficiency and actual thrust. The angle of the ejector changes depending upon which characteristic is optimized. For example, in order to achieve the highest efficiency, values for the input parameters are Ptej (Pa) 49,075 hej (m) 0.1500 Lej (m) 1.4000 Θej (°) 30.000 In order to achieve the highest thrust, values for the input parameters are Ptej (Pa) 49,075 hej (m) 0.1500 Lej (m) 1.4000 Θej (°) 0.0000 As previously stated, only the ejector angle changes between the two configurations depending on which performance characteristic is optimized. Figure 3.19 shows an estimated ejector design, using the regression mode developed compared to the conventional nozzle. 37 700 650 Actual Thrust (kN) 600 550 500 450 400 350 300 0 5 10 15 20 25 NPR CD Ejector Figure 3.19 Actual thrust comparison between conventional nozzle and ejector nozzle by using regression model. 38 4. CONCLUSION As part of this exploratory analysis, ejector nozzles have been shown to improve thrust beyond the conventional convergent-divergent nozzle. Amongst the inviscid simulations presented, thrust was increased by as much as 18% on average. It could be shown in subsequent testing that further thrust improvements could be attained via increasing the total pressure supplying the ejector, which was the primary contributor in ejector thrust generation. The other major contributors to thrust generation were ejector angle, area, and location, listed in order of importance. In order to improve ejector efficiency, listed in order of importance, ejector total pressure, location, area, and angle all have significant impacts. While both these performance characteristics are related, the ejector angle influences both differently. For optimal thrust generation, the ejector angle should be small. For optimal thrust efficiency, the ejector angle should be large. In most applications, thrust is the primary objective and as such efficiency will have to be sacrificed. ANSYS Fluent has been shown to produce valid results when compared to 1D theory as well as experimental data for conventional convergent-divergent nozzles. Both analyses for inviscid and viscous assumptions have been shown to adequately capture the physics involved in the nozzle. 39 BIBLIOGRAPHY (1) Anderson, J. D. (2003). Modern Compressible Flow. McGraw Hill. (2) Gamble, E. (2004). Nozzle Selection and Design Criteria. American Institute of Aeronautics and Astronautics. (3) Kerrebrock, J. (1992). Aircraft Engines and Gas Turbines. The MIT Press. (4) Mattingly, J. (2006). Elements of Propulsion Gas Turbines and Rockets. (5) Nadeem. (2013, August). Modeling & Simulation of Flow Separation & Shocks in a CD Nozzle. International Journal of Mechanical Engineering and Applications, 1(3), 14-21. (6) Padmanathan, P. (2012, Mar- Apr). Computational Analysis of Shockwave in Convergent Divergent Nozzle. International Journal of Engineering Research and Applications, 2(2), 1597-1605. (7) Rao, S. (2014). Novel Supersonic Nozzles For Mixing Enhancement in Supersonic Ejectors. Applied Thermal Engineering Journal, 62-71. (8) Ward, T. A. (2010). Aerospace Prpoulsion Systems. Wiley. 40

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