© 2018 david scott hanon -...
TRANSCRIPT
SCRAMJET ENERGETIC ANALYSIS CONCERNING UPSTREAM FUEL INJECTION
By
DAVID SCOTT HANON
A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE
UNIVERSITY OF FLORIDA
2018
© 2018 David Scott Hanon
To my Mom, Dad, Bear and Roo Hornady
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ACKNOWLEDGMENTS
I would like to thank my parents, whom have inspired and supported me
throughout my life, extending into my academic tenure and beyond. My godparents,
whom apart from my parents have been instrumental in my education. My siblings for
providing support throughout this whole academic process. My closest friends and
roommate’s for making my years throughout college enjoyable through participation in
gator events. My lab colleagues for providing inspiration, advice and knowledge as we
all progressed toward graduation together. My best friend and love of my life, whom has
been the most wonderful and stabilizing force in my life. Finally, my advisor, whom has
opened doors both in my academic and professional career.
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TABLE OF CONTENTS page
ACKNOWLEDGMENTS ..................................................................................................... 4
TABLE OF CONTENTS ..................................................................................................... 5
LIST OF TABLES ............................................................................................................... 7
LIST OF FIGURES ............................................................................................................. 8
NOMENCLATURE ............................................................................................................. 9
ABSTRACT ....................................................................................................................... 12
CHAPTER
1 INTRODUCTION ....................................................................................................... 14
1.1 Theory and Previous Work .................................................................................. 14 1.1.1 Hypersonics ............................................................................................ 14 1.1.2 Scramjet .................................................................................................. 15
1.1.2.1 Scramjet theory .............................................................................. 15 1.1.2.2 Scramjet design ............................................................................. 17 1.1.2.3 Performance parameters ............................................................... 18
1.1.3 Combustion ............................................................................................. 20 1.1.3.1 Fuel type ......................................................................................... 21 1.1.3.2 Upstream fuel injection .................................................................. 23 1.1.3.3 Heat transfer................................................................................... 25
1.2 Motivation ............................................................................................................. 26
2 SELECTED DESIGN CONFIGURATION ................................................................. 28
2.1 Background .......................................................................................................... 28 2.2 Base Design ......................................................................................................... 32
2.2.1 Design constraints ..................................................................................... 32 2.2.2 Forebody and inlet ..................................................................................... 34 2.2.3 Isolator ........................................................................................................ 37
3 RESULTS AND DISCUSSION .................................................................................. 44
3.1 Upstream Fuel Injection ....................................................................................... 44 3.1.1 Case 1 (𝝃𝒊 = 𝟎. 𝟏𝟎) ..................................................................................... 48 3.1.2 Case 2 (𝝃𝒊 = 𝟎. 𝟐𝟎) ..................................................................................... 48 3.1.3 Case 3 (𝝃𝒊 = 𝟎. 𝟒𝟎) ..................................................................................... 49 3.1.4 Summary .................................................................................................... 49
3.2 Inlet Heat Addition ............................................................................................... 55
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4 RECOMMENDED STUDIES ..................................................................................... 60
APPENDIX MATLAB: Script ............................................................................................ 61
LIST OF REFERENCES ................................................................................................ 122
BIOGRAPHICAL SKETCH ............................................................................................. 125
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LIST OF TABLES Table Page 1-1 Performance parameters ....................................................................................... 19
1-2 Efficiency parameters ............................................................................................. 20
3-1 Parameters at Combustor Entrance for Un-Reacting Fuel Injection .................... 49
3-2 Entropy change across the combustor for un-reacting fuel injection .................... 50
3-3 Performance parameters, upstream un-reacting fuel-injection ............................. 52
3-4 Parameters at combustor entrance for reacting fuel Injection .............................. 56
3-5 Entropy change across the combustor for un-reacting fuel injection .................... 56
3-6 Performance parameters, upstream reacting fuel-injection .................................. 57
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LIST OF FIGURES Figure Page 2-1 Flow stations in a typical scramjet engine ............................................................. 28
2.2 External compression inlet ..................................................................................... 39
2-3 Internal compression .............................................................................................. 39
2-4 Mixed external and internal compression inlet ...................................................... 40
2-5 Propulsion system options as a function of Mach ................................................. 40
2-6 Flow stations in a gas turbine engine .................................................................... 41
2-7 Comparison of airbreathing and non-airbreathing flight corridors ........................ 41
2-8 Isolator section ....................................................................................................... 42
2-9 Model of scramjet engine ....................................................................................... 42
2-10 Base design with shocks ........................................................................................ 43
2-11 Contraction ratio of base design ............................................................................ 43
3-1 Isolator for all injections for non-reacting cases .................................................... 53
3-2 Isolator shock train for non-reacting cases ............................................................ 54
3-3 Last shock intersections for all injections in isolator .............................................. 54
3-4 Isolator for all injections for reacting cases............................................................ 58
3-5 Isolator shock train for reacting cases ................................................................... 59
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NOMENCLATURE
Subscripts
0 Freestream
1 Engine inlet plane
2 Isolator entrance
3 Combustor entrance
4 Nozzle entrance
9 Nozzle exit
𝑠 Static
𝑡 Total
Acronyms
DARPA Defense Advanced Research Projects Agency
NASA National Aeronautics and Space Administration
CFD Computational Fluid Dynamics
MATLAB Programming language and numerical analysis environment
scramjet Supersonic combustion RAMJET
TSFC Thrust specific fuel consumption
MFR Mass flow ratio
List of symbols
ℑ𝑠 Specific thrust
ℑ𝑓 Thrust flux
ℑ Thrust
𝑀 Mach
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List of symbols (continued)
𝑃 Pressure
𝑇 Temperature
𝜌 Density
𝑎 Speed of sound
𝐴 Area
𝑉 Local velocity
�̇� Mass flow rate
𝑔𝑐 Newton’s constant
𝑓 Fuel-air ratio
𝑓𝑠𝑡 Stoichiometric fuel-air ratio
𝛾 Ratio of specific heats
𝑅 Unique gas constant
𝐶𝑝0 Specific heat at constant pressure
𝑞𝑅 Heating value
𝜙 Equivalence ratio
𝜙𝑖 Equivalence ratio at injection site in inlet
𝜙𝑐 Equivalence ratio in combustion chamber
𝜉𝑖 Percentage of total fuel added in the inlet
𝜂𝑇 Thermal efficiency
𝜂𝑃 Propulsive efficiency
𝜂𝑂 Overall efficiency
𝜏 Combustor total temperature ratio
𝜋 Total pressure recovery
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List of symbols (continued)
𝐶𝑓𝑙 Wall frictional coefficient
𝐷𝐻 Hydraulic diameter
𝑤 Depth
𝑑𝑒 Infinitesimal internal energy
𝑑𝑞 Infinitesimal heat transfer
𝑑𝑤 Infinitesimal work
𝑠 Entropy
𝜇 Elastic or dynamic viscosity
𝜆 2nd Coefficient of viscosity
�⃡� Stress tensor
�⃡� Strain rate tensor
�⃗� Body force
�⃡� Identity tensor
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Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science
SCRAMJET ENERGETIC ANLAYSIS CONCERNING UPSTREAM FUEL INJECTION
By
David Scott Hanon
May 2018
Chair: Corin Segal Major: Aerospace Engineering
Injecting fuel in the inlet of a scramjet is a technique aimed at elevating the
propulsive efficiency of this technology. This paper has sought to understand the
physics and influence of this act on the scramjet engine from a fluid dynamic and
thermodynamic perspective.
Four different cases were explored in this study, corresponding to zero, ten,
twenty, and forty percent of the overall fuel injected directly in the inlet. Using a robust
MATLAB generated algorithm, the state at the leading edge, engine inlet plane, isolator
entrance and combustor entrance are calculated and fully defined for each case tested.
In addition to these states being calculated, this MATLAB code calculates performance
parameters and generates the geometry of a 2-d scramjet engine from the leading edge
to the end of the isolator.
As the percentage of fuel was increased in the inlet where auto-ignition did not
occur, it was observed that entropy across the combustor was driven down and the total
pressure recovery, along with the overall efficiency decreased. However, the thrust
specific fuel consumption, overall thrust and propulsive efficiency were observed to
increase. In the situation where the fuel was forced to combust, the additional heat
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release introduced prior to the combustor dramatically decreased the specific thrust fuel
consumption and total pressure recovery. It was observed that the overall thrust did
increase. However, when the fuel auto-ignites in the inlet, the isolator becomes an
obsolete component and impacts the overall performance negatively as this component
now imposes an additional drag and weight penalty on the engine.
Injecting the fuel in the inlet will allow for a smaller isolator given a Mach
boundary condition bounding this study and increase the thrust the engine can deliver.
Through mass addition, the flow slows down so the boundary condition of the Mach
entering the combustion chamber is reached sooner, eliminating a finite portion of the
isolator. How much fuel can be injected depends on the pressure recovery demanded
by a thrust requirement.
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CHAPTER 1 INTRODUCTION
1.1 Theory and Previous Work
1.1.1 Hypersonics
Hypersonics has been around since September of 1963, when Robert White
piloted the X-15 at a Mach of 6.06 while at an altitude of 354,200 ft. The landscape of
aerospace engines is constantly expanding to yield new and innovative propulsion
technologies to meet the requirements motivated by national defense, transportation,
and access to space. Ever since the Wright brothers flew the first heavier than air flyer
in 1903 (Flyer I) there has existed a community that strives to develop a more efficient,
faster, and superior performing aircraft. Currently the focus of this community is on
hypersonics. Reaching this flight regime will be accomplished with the aid of supersonic
combustion ramjets, a new and emerging propulsion technology. There exists a robust
and detailed history concerning the challenging task of developing this new class of
hypersonic capable aircraft. Researchers such as Bertin and Cummings discussed early
flights of hypersonic systems and issues encountered with this emerging technology [1].
One those challenges being the viscous to inviscid interactions produced by
components and how the physics of these interactions in this new flight regime tend to
damage parts of the aircraft. Challenges such as this will reveal themselves as testing
of various components of this new type of aircraft are explored.
As new innovative aircraft are introduced, publications become available that
establish a timeline for these emerging technologies. Jenkins, Landis and Miller
complied a monograph on American X-Vehicles [2], which serves as a useful resource
for this thesis so that a timeline of this technology may be represented.
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1.1.2 Scramjet
1.1.2.1 Scramjet theory
The scramjet engine is similar to that of a ramjet engine, the major difference
being that the combustion takes place above Mach one. Major components of a
scramjet include the inlet, forebody, isolator, combustor and nozzle. Curran and Murphy
wrote three volumes concerned with the status review on hypersonic propulsion
systems that was published by the American Institute of Aeronautics and Astronautics in
1991, 1996 and 2000 respectively. The third installment of this mini-series is devoted
almost entirely to this new engine technology and international progress in developing
these technologies [3]. This volume includes a chapter form Ortwerth that discusses
hypersonic theory for the scramjet flow path integration that is referenced in this study to
justify particular component designs.
Smart, a notable researcher in the area of hypersonics, has presented a
description of these major components and analysis used to determine the performance
of the scramjet engine [4]. Smart’s analysis will aid in justification of a design Mach
chosen for this study. Some challenges associated with this technology include fuel
injection, ignition, and proper compression needed for combustion to take place, just to
name a few.
Active research is investigating each of these scramjet sections, with the goal to
understand the physics of this emerging technology across the whole engine. Smart has
also investigated various inlets, classified based on the compression type, and found
that the inlet itself is a dominating feature of the whole engine [5]. The use of Smart’s
work will be used to define an inlet type for this study.
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This paper seeks to investigate the physics and consequences of injecting a
percentage of fuel upstream of the combustor in the inlet. In order to understand why
the engine behaves as it does when fuel is introduced outside of the combustor and
why the particular scramjet geometric quantities were chosen for this study, previous
work will be used to validate these decisions, reported results, and conclusions.
A fluid is a complex deformable media, and as such must be treated as a
continuum in order to analyze the physics of this transient substance without tracing
every single fluid parcel. A fundamental understanding of fluid mechanics is necessary
when dealing with concepts of propulsion and Cengal and Cimbala do a great job of
covering the basic principles and equations of this subject [6]. Fluid mechanics is a very
broad topic, where aerodynamics is a specialized class in fluid mechanics and
necessary when dealing with propulsion concepts as well. Anderson wrote a book that
discusses inviscid, viscous, incompressible, and compressible flows that will aid in the
understanding of the aerodynamics of this study [7]. Another subject imperative to the
understanding of the full fluid path throughout a scramjet engine is combustion.
Glassman and Yetter provide an excellent resource for this subject and are referenced
in this paper. This is due to the fact that this study investigates the possibility of the
injected fuel combusting [8].
Farokhi discusses fundamental propulsion principles focused on turbojet engines
that operate over a wide range of Mach numbers [9]. In addition to applications suitable
to turbojet engines, Farokhi gets into more complex principles that can be applied to
engines that operate in a flight envelope beyond that of a turbojet engine. Making
Farokhi a good resource for understanding the physics of a complete flight envelope
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that encompasses take off conditions to a hypersonic cruising state. There is consensus
among professional that the scramjet engine will want to be integrated into a system
that has a secondary engine, possible a low bypass turbofan, that can take off from a
runway and then transition to scramjet operation once a certain Mach is reached.
Industry is currently looking at making this concept a reality and in fact Aerojet
Rocketdyne was recently awarded a contract from DARPA to develop a ground test for
this mode transition in order to demonstrate that this combined engine design is
feasible.
Since this paper focuses on the hypersonic flight regime, a Heiser and Pratt text
becomes another useful resource; providing a broad and basic introduction to the
elements needed to work in this emerging hypersonics field as it expands [10]. The
difference between a scramjet and a rocket engine is that a scramjet is an air breather,
not needing to carry oxidizer on board. Turner discusses the advantages of scramjet
technologies as it applies to spacecraft propulsion [11], and discussed in this paper.
Segal, a leading researcher in this hypersonic/scramjet field, produced a book
that provides an informative initial look at this engine and its applications [12]. This is yet
another resource that will provide some background as to the purpose of this paper.
1.1.2.2 Scramjet design
There have been many proposed scramjet designs’ recently, emerging from both
academia and industry. Lindsay et. al, from North Carolina A and T State University
used a Fortran code to evaluate the aerodynamic performance of a two dimensional
scramjet engine. An inlet was designed, and the analysis performed using this Fortran
code proved that the design choices for this inlet resulted in a valid design for a
scramjet application. This is based on the lift-to-drag vs Mach profile that the Fortran
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code converged to, which compared favorably with Kuchemann barriers [13]. The
geometry of the inlet used in Lindsay’s et. al academic work influenced the geometry of
the inlet chosen for this present study.
John Hicks from NASA wrote that advanced analysis with ground and flight tests
are needed to design and develop new aircraft with a scramjet engine [14]. This NASA
technical memorandum illustrated operational envelopes for a scramjet application
among limitations associated with ground testing of these engines. This information is
used to choose the particular altitude and further strengthen the case for the design
Mach chosen in this present study.
More extensive studies have been performed involving testing of physical
components. An example of this is Kanda et. al investigation and testing of a subscale
scramjet research engine model at Mach six [15]. This subscale model included an inlet,
isolator and combustor with the external nozzle removed for testing. This experiment
revealed that intensive combustion was achieved when the combustion efficiency
reached approximately 0.9 or greater. Assuming a similar injection scheme (beyond
scope of this study) to promote intensive combustion, the Mach entering the combustor
of this study is chosen to be similar to that of the experiment accomplished by Kanda et.
al.
1.1.2.3 Performance parameters
Specific performance parameters will be used to evaluate each condition
imposed on the scramjet geometry of this study. Roux and Tiruveedula published a
technical note in order to provide certain parameters that describe the ideal scramjet
engine cycle, which is the cycle this study is based on [16]. The most efficient cycle a
scramjet can follow is that of a Brayton cycle.
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Important performance parameters used in this study can be observed in Table
1-1. The mass flow ratio is also an important parameter as it describes how much mass
flow is spilled around the engine if an off-design condition were to ensue. Since the
design of the engine for this study will assume all the flow is captured without any
spillage, the MFR will be set to a constant of one. If the engine were to throttle down,
this parameter would become less than unity, leading to a loss in thrust and generating
additional drag on the cowl of the scramjet. Several useful efficiency parameters will be
used in this study and summarized in Table 1-2.
Table 1-1. Performance parameters
Parameter Definition
Specific thrust ℑ𝑠 ≝
ℑ
�̇�0=
𝑀0𝑎0
𝑔𝑐[𝑉9
𝑉0− 1]
Thrust specific fuel
Consumption
𝑇𝑆𝐹𝐶 ≝𝑓
(ℑ/�̇�0)
Thrust flux ℑ𝑓 ≝
ℑ
𝐴3
Mach at exit of combustor 𝑀4 =
𝑀3
√𝑇𝑚𝑎𝑥𝑇𝑡3
(1 + (𝛾 − 1
2 )𝑀32)
Total pressure recovery 𝜋 =
𝑃𝑡3
𝑃𝑡0
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Table 1-2. Efficiency parameters
1.1.3 Combustion
Scramjet engine literature is largely dominated by combustor studies [3]. Recall
one of the potential applications of scramjet technology is suborbital insertion. Achieving
reusable propulsive applications to insert payloads into an orbit is one of the
advantages of scramjets for access to space over of a conventionally staged rocket.
Preller and Smart discussed how scramjets could be integrated into a reusable small
satellite launch system [17] and the cost effectiveness of this method. Preller and
Smart’s analysis is used in this study to motivate design choices from a combustion
standpoint.
A high-level view of the one of the benefits of a scramjet application and how the
combustor affects the overall performance of this engine was just presented. Looking
specifically at the component design, Kay et. al tested a combustor from an
experimental aircraft program, which was designed to be powered by a hydrocarbon-
fueled scramjet engine. The combustor was tested using an ethylene pilot to initiate and
sustain supersonic combustion over a wide range of equivalence ratios [18]. Kay et. al
analysis will be used to motivate a fuel choice for this study and their research
demonstrates that there is a lot to consider when designing a combustor for a scramjet.
Parameter Definition
Propulsive efficiency 𝜂𝑃 =
2
(𝑉9𝑉0
+ 1)
Thermal efficiency 𝜂𝑇 = 1 −
1
𝜏𝑟 [(𝑇9/𝑇0) − 1
𝜏 − 1]
Overall efficiency 𝜂𝑂 = 𝜂𝑃𝜂𝑇
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A few of the main considerations for a combustor of this application are broken up and
discussed in the proceeding sections.
1.1.3.1 Fuel type
The fuel type chosen is essential in how the scramjet engine behaves and
operates. In most scramjet combustion studies currently being conducted, the fuel being
used is either liquid hydrogen or a hydrocarbon. Taha et al. used Fluent generated CFD
to investigate ethylene as a possible hydrocarbon fuel choice in a scramjet experiment
[19]. Inserting a rearward facing-step enhanced the fuel-air mixture and combustion was
achieved with a stable flame. Another researcher in the area of Combustion, Mateu,
carried out a study in which a quasi-one-dimensional model of a scramjet combustor
was investigated. This one-dimensional analysis is computationally less expensive than
performing analysis that attempts to solve the full three-dimensional Navier Stokes
equations coupled with a turbulent model [20].
Some of the main concerns with achieving combustion in a scramjet is being able
to successfully ignite a fuel-air mixture in a supersonic environment while maintaining a
stable flame and mixing the fuel and air sufficiently prior to combustion. Pandey et. al
performed a set of experiments where they took a double cavity combustor and injected
hydrogen while varying the injection pressure and freestream total temperature.
Implementing cavities appears to solve both of these problems since it was found that
increasing the pressure at which hydrogen is injected into the cavities promotes a larger
vortex structure near the cavity regions, enhancing the fuel-air mixing [21].
In order to have an aircraft capable of taking off from a runway and accelerating
to speeds that will be efficient for a scramjet engine to take over, a mode transition must
occur between a coupled engine configuration if a dual engine model were chosen. Cao
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et al. performed a numerical analysis in which they transitioned from ramjet-mode to
scramjet-mode and achieved this transition through changing the amount of fuel that is
injected in the combustor [22]. In this zero-dimensional analysis of Cao et al., hydrogen
was determined to be the optimal fuel for their application.
Another combustion study where a cavity was used to investigate supersonic
combustion was performed by Liu et al. Here they used a single cavity with two fuel
injection sites: one transverse injector just upstream of the cavity and the other directly
in the cavity. Liu et al. found that this fuel injection scheme, when both were operational,
provided an increased mixture distribution and achieved a successful ignition than if one
were used over the other [23].
Once a fuel is chosen for this study, properties will have to be reported and
discussed. Bing et. al performed a study in which they characterized a fuel type and
analyzed injector configurations, inflow total temperatures of the fuel, and other
quantities that are important when analyzing combustion for a given fuel [24]. Bing et. al
determined certain transition points given for both a hydrogen and ethylene case. The
properties measured in that study will be used in this current study for the fuel chosen,
whether that be hydrogen or ethylene.
Although the combustor component is removed from this study, the cumulative
work of Pandey et al., Liu et al., Taha et al., and Mateu demonstrate why certain fuels
are chosen for given scramjet applications. Their work is used to help make a decision
on what type of fuel will be used in this paper and because the maximum temperature of
a scramjet occurs in the combustion chamber, which directly affects the amount of
thrust possible, the reaction will be considered stoichiometric. If the fuel being
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considered is a hydrocarbon, all the hydrogen atoms form water vapor and all the
carbon atoms form carbon dioxide [10].
1.1.3.2 Upstream fuel injection
The purpose of this study is to investigate what happens in a particular design
when fuel is introduced in the inlet. Barth, Wheatley and Smart simulated a scenario
where they injected hydrogen into the inlet of a practical scramjet design. This study by
Barth et. al concluded that inlet injection, when above a particular equivalence ratio,
leads to enhanced combustion downstream without significantly increasing the drag on
the inlet [25]. The results of Barth et. al will be used to motivate the importance of this
study and how injecting fuel in the inlet affects parts of the scramjet engine relevant to
this study.
Effects of upstream injection using kerosene on local ignition in a dual cavity
experiment was reported in a study performed by Bao et. al. They found that within the
cavities the mixture was fuel rich and that the corner of the cavity nearest the
freestream performs a significant role in stabilization of the local flame [26]. Once a local
flame can be established it will spread to the primary flow outside the recirculation
zones if the pressure rise in the combustor is significant.
The Mach entering the combustion chamber is a critical design point and must be
chosen with care for supersonic combustion to take place for a given design. Rockwell
et. al designed a unique fuel injection approach, which produced a uniform fuel-air
mixture around Mach 0.7 leading into the combustor and resulting in premixed
combustion [27]. Fuel was injected in the isolator to achieve this. However, if the fuel
were to combust in the isolator it would have a negative impact on the performance of
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the engine. Some of these negative impacts are presented in Rockwell et. al and
discussed in this study.
Mahto et. al also looked at the Mach entering the combustion chamber and
investigated Mach on the performance of a double cavity scramjet combustor [28]. The
study revealed that when the Mach rose about a certain value, the combustion
efficiency measured was not consistent. Those results will help set a Mach boundary
condition for this study.
Qiuya Tu evaluated isolator to combustor interactions when injecting fuel directly
into the isolator [29]. Through experimental data, Tu found that the wall frictional force
has a profound impact on the shock stabilization. The wall frictional coefficient for this
study was chosen based off of Qiuya’s results and experiments performed.
Ogawa is another researcher who also investigated the effects of upstream fuel
injection in a scramjet engine. Ogawa developed some important insights into the
behavior of a supersonic flow as different injection schemes were implemented and
tested. It was found that the injection angle has significant influence on the mixing of
fuel and oxidizer [30]. Even though the method by which fuel is delivered to the system
is beyond the scope of this study, Ogawa stated the main objectives of what injection
should accomplish and is touched upon in this study.
Scramjet performance may be enhanced through thermal compression at high
Mach numbers. Fuel can be ignited in regions of high compression regions, and the
surrounding regions will be compressed as a result. This will drive the cycle efficiency
up because the required internal/external compression at the inlet will go down
increasing thrust potential due to the availability of new internal energy added to the
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system. Scramjet designs generally take advantage of this thermal compression effect
and Bricalli et al. performed an experiment using a scramjet engine designed to take
advantage of just this effect. Bricalli found that the combustion observed was not as
robust when a similar design was chosen that didn’t inject fuel in a region that would
have had the potential to add energy to the fluid through thermal compression [31].
Bricalli et al. efforts in understanding performance parameters of scramjet engines will
aide in determining a design Mach, given a particular fuel choice.
1.1.3.3 Heat transfer
Thermally choking the inlet will occur if too much heat is released in the inlet,
causing the Mach to drive toward unity and potentially unstarting the engine. Srinivasan
and Jayanti combined an algorithm with CFD to generate a procedure for optimizing
guide plate positions to achieve desired flow distributions though multiple channels [32].
This code can be used to optimize the fuel lines in a scramjet combustor, varying the
flow rates of the fuel in each injector to distribute the heat release and avoid thermally
choking the engine. Srinivasan and Jayanti’s work will be used in this study in the
pursuit of an optimally design Isolator.
In hypersonic airbreathing engines it is necessary to cool particular components
due to the significant heat release in combustion areas and downstream of those sites.
Traditionally this can be accomplished through conventional regenerative cooling.
However more exotic methods of cooling the engine’s components are being
investigated, such as a re-cooled cycle. A re-cooled cycle uses a turbine to pump the
fuel/coolant around the engine in a thermal jacket, containing cooling finned-channels.
This re-cooled cycle method was investigated by Bao et. al in which a new optimization
index is introduced with the important result that the individual duct sizes must be as
26
small as possible [33]. Bao et. al emphasis why cooling is a major design constraint and
discussed in detail later in this study.
1.2 Motivation
Experimental X-designated aircraft supported by the U.S. government have
sought to push the boundaries of both manned and unmanned aircraft performance
throughout the previous decades. Chuck Yeager’s piloting of Bell Aircraft Company’s X-
1 (first aircraft given the X-designation) on October 14, 1947 was the first time an
aircraft pushed through the speed of sound in controlled level flight [2]. This aircraft
attained a Mach of 1.45, which demonstrated that a supersonic combat aircraft was
feasible. On April 26, 1951 Lockheed Missiles and Space Co. flew the X-7, which was
an unmanned supersonic demonstrator that utilized a ramjet, attaining a Mach of 4.31
[2]. Then on June 8, 1959 North American Aviation flew the X-15, a vehicle developed
for studying a new flight regime, hypersonics. During this aircraft’s fasted flight, it
reached Mach 6.06 becoming the fastest flying manned fixed wing aircraft prior to the
advent of the Space Shuttle [2].
Bertin and Cummings proposed three mission goals that should be required of
vehicles capable of hypersonic flight, Including, delivery of decisive blows at the outset
of hostilities, delivering cost-effective weapons to defeat time-critical targets, and to
accomplish access to space [1]. All these objectives will benefit the military, which
primarily provide funding in seeking these technologic breakthroughs. This paper seeks
to add to the history of propulsion progression and become a new source of information
for a scramjet application, which lies at the forefront of propulsive engineering.
Hypersonic airbreathing propulsion offers an alternative to conventional rockets
that are staged for access to orbit and beyond. This class of propulsion offers a higher
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potential specific impulse over rockets between flight Mach number of 6-14, providing
cheaper way of delivering payloads into orbit [31]. Additionally, this technology has the
potential to strengthen military reconnaissance and defense, offering high speed
atmospheric flight. This is a promising technology, due to the fact that the exclusion of
on board oxidizer necessary for combustion in a rocket application, decreases weight of
aircraft, allows for greater performance and enables efficient and flexible transportation
systems [30].
Traditionally hypersonic demonstrators such as NASA’s X-43 and Aerojet
Rocketdyne’s X-51A have utilized fuel by injecting it directly into the combustion
chamber. However, there has yet to be an experimental aircraft with hypersonic abilities
that injects a portion of the fuel upstream of the combustion chamber. Barth et. al
suggested that Inlet injection increases fuel residence times in a scramjet and reduces
the combustor length required to contain complete fuel combustion whilst avoiding
engine unstart [25].
Fuel injection at a site upstream of the combustor for a supersonic combustion
applicator seeks to: (i) promote high mixing efficiency, (ii) avoid as much stagnation
pressure loss as possible, and (iii) proper penetration of the injected fuel into the
airstream for ignition purposes [30]. All three of these design objectives directly affects
the efficiency and effectiveness of this technology. This paper will deal exclusively with
(ii), seeing how various percentages of fuel injected upstream of the combustor affects
the performance of the engine.
28
CHAPTER 2 SELECTED DESIGN CONFIGURATION
2.1 Background
A scramjet engine operating above Mach five at an altitude greater than 36 km
requires significant compression and heating of the air entering the combustion
chamber for an efficient combustion process to take place [5]. In the hypersonic flight
regime, mechanical compression is unnecessary and complicates the system with
moving parts. Ram compression at the inlet into the engine compresses the freestream
enough to accomplish the compression necessary for efficient burning in the combustor.
This is accomplished using multiple compression ramps, which turns the flow through
strong compression waves while maintaining a stagnation pressure ratio across the
engine necessary for useful conversion of the energy inherent of the fluid into kinetic
energy downstream within the nozzle. This generates momentum and pressure thrust
operating on the nozzles walls prior to the products of combustion being ejected into the
ambient environment through which there is no pressure gradient between the expelled
gasses and the environment.
Figure 2-1. Flow stations in a typical scramjet engine
A conventional labeling system of an uninstalled scramjet engine can be seen in
Error! Reference source not found., which is a simple modification of the common
29
labeling system for a simple gas turbine engine that will indicate the stations of interest
for this study. Farokhi illustrated the labeling system for air breathing engines in Figure
2-. Modifying Figure 2- to reflect this scramjet application is simple, where the
compressor is replaced with the isolator and the turbine is simply taken out. Looking at
Error! Reference source not found. station zero indicates the freestream conditions
and represents the leading edge of the engine, where the incoming flow first encounters
the engine. Station one is the state of the flow that once compressed due to the flows
interaction with the forebody (external compression), then enters the inlet. Station two is
the state of the fluid, once further compression has taken place (internal compression),
that enters the isolator. Station three is the state of the fluid that exits the isolator and
enters the combustion chamber. Station four is the inlet to the nozzle (or combustor
exit) and station nine is the state of the flow as it the exits the nozzle of the scramjet.
Properties at each of the states just described are referenced using a subscript. For
example, M0 is referring to the Mach at the location/state of the leading edge of the
engine. Mach number, static pressure, stagnation pressure, static temperature, and
stagnation temperature are some of the properties that will be reported on at the
stations of interest just described.
For this study, the system will be considered on design, operating at a design
Mach and the flow fully expanded within the nozzle. Turner discussed that the back
pressure should be designed to match the exit pressure in order to maximize the
potential thrust [11]. Flow entering the isolator should be parallel to the surrounding
walls, since the high-speed flow will interact with the boundary layers that form on the
upper and lower surfaces of the walls, producing a shock train within this component. If
30
the flow were not parallel entering the isolator the shocks would be stronger in this
region, and the overall thrust would be reduced. As such, Heiser and Pratt investigated
and provide illustrations of three possible design configurations for this studies inlet
geometry, Error! Reference source not found., Figure 2-, and Figure 2-. All the
compression takes place external to the engine in Error! Reference source not
found.. all the compression take place inside the engine in Figure 2- and there is a mix
of both internal and external compression for the design depicted in Figure 2-. All three
are practical designs, however Error! Reference source not found. has large cowl
drag because the flow is approaching the combustor at an angle and designs of the
type depicted by Figure 2- are difficult to integrate into the engine [5]. As such Figure 2-
is the inlet design configuration chosen for this study in order to minimize the external
drag.
Once the desired compression is reached using ram-compression due to the
presence of the forebody and inlet, the flow passes into the isolator, which is an
important component consisting of a short duct that further compresses the flow and
serves to prevent engine unstart [12]. This isolator section of the engine is extremely
important, as it serves to isolate the shock system prior to the flow entering the
combustor. If this section were omitted from the engine design, normal shocks would
form within the engine and propagate upstream to just outside the engine leading to a
loss in mass flow entering the engine, spilling the flow around the cowl. This occurs
because there is an upper limit to the stagnation pressure ratio across the engine. If
exceeded the only way the pressure field can adjust to the back pressure of the engine
is to have a strong shock located at the inlet. It’s much easier and cheaper to introduce
31
a constant area isolator than to have a very complicated forebody with many
compression ramps to reduce the stagnation pressure ratio across the engine, resulting
in the prevention of an unstart condition. An unstart situation would drastically reduce
the amount of thrust produced in the nozzle, since thrust is directly proportional to the
amount of incoming mass flow of the environment ingested by the engine. Drinivasan
and Jayanti saw that this phenomenon will reduce the performance of that engine at
large Mach numbers [32], demonstrating why the isolator is an essential component to
the scramjet engine.
The flow then passes into a combustion chamber where the compressed air is
mixed with a given fuel and ignited. Cao et al. saw that hydrogen is the ideal choice to
fuel a hypersonic application due to the fuel’s high specific impulse [22]. Since it’s
cryogenic, the fuel may be used to actively cool the walls and other components. This
active cooling technique provides significant cooling to counter the extreme
aerodynamic heating resulting from the hypersonic/supersonic flow within and around
the engine [33]. The heating value of hydrogen when compared to a hydrocarbon is
about two and a half times greater, resulting in a greater energy release per unit mass
of fuel burned [20]. Hydrogen is an ideal choice not only for the energy potential, cooling
capabilities, but it’s also a clean fuel. When burned, hydrogen doesn’t produce any
harmful products of combustion. Actually, the only product of combustion of hydrogen is
water, which may be safety exhausted into the surrounding environment [20]. Another
reason why hydrogen fuel is chosen to run high speed scramjet applications (i.e. Mach
greater than eight) is that reaction mechanism for hydrogen fuel is about ten times
faster than that of any hydrocarbon fuel available [21]. This is important because as the
32
time scale of axial diffusion of incoming oxidizer increases, the reaction time scale of
combustion must increase to have an efficient combustion process.
It is largely agreed that there are currently two fuel options for scramjets:
hydrocarbons and hydrogen. For this study a hydrocarbon will be used in place of
hydrogen, since the operational Mach is less than eight. There is a general consensus
that if the Mach is less than eight, then a hydrocarbon fuel is advantageous over liquid
hydrogen [20]. Figure 2- reveals a qualitative chart of propulsion options based on the
operational flight Mach number.
2.2 Base Design
2.2.1 Design constraints
To come up with a design configuration for this analysis, which is referred to as
the base design, it is first assumed that the vehicle is operating at a design Mach of six,
(M0 = 6) at an altitude of 30 km. This falls within the operational flight envelope of a
scramjet application according to Hicks [14]; refer to Figure 2-. Smart agrees that for a
scramjet application, no net thrust is produced below a flight Mach number ranging from
3.5-5 and that a low speed propulsion system or combined cycle is required before a
scramjet mode can be initiated [4]. At this altitude, the properties of the air are as
follows: 𝜌0 = 0.01786 kg/m3, P0 = 1186 Pa, T0 = 231 oK, 𝛾0= 1.4, R0 = 287 J/kg-oK,
CP,o = 1005 J/kg-oK. These properties of air are taken at the geometric altitude of 30 km
as indicated by Anderson [7]. It is assumed that the primary flow path through the
engine will have a consistent ratio of specific heats, unique gas constant, and specific
heat at constant pressure. Meaning that once the fuel is injected at the inlet, the
properties of the fluid will not change downstream in terms of those three properties just
listed.
33
Since the operational flight Mach is less than eight, it is established that the fuel
of choice must be a hydrocarbon. In flight kerosene is often used due to its high
enthalpy, density, and easy storage under flight conditions [26]. In ground tests,
kerosene has been used as the hydrocarbon of choice for applications limited to the
operational flight Mach as previously discussed. Another hydrocarbon, ethylene, is
usually used in the experiments of supersonic combustion because its hydrogen to
carbon ratio is very close to that of kerosene as investigated by Liu et al. [23]. In
addition to this desirable ratio, Taha et al. published that ethylene is often chosen for
scramjet engine tests because it can be used as a surrogate test fuel for hydrocarbon
fuels [19]. This is significant because Kay et al. clearly demonstrated that supersonic
combustion of various hydrocarbons can be achieved with a unique piloting device that
uses hydrogen to stabilize the flame [18]. These piloted supersonic combustion
experiments performed by Kay et al. suggested that hydrogen piloting is very effective
for certain fuels, including kerosene, which conveniently can be tested with ethylene.
Throughout this study the properties of the air approaching the engine are
assumed to be constant and the fuel chosen for this application is the hydrocarbon
ethylene. Important properties for this hydrocarbon include a heating value of 45 MJ/kg,
and a stoichiometric fuel-to-air ratio of 0.0679 [24]. It should also be noted that the
maximum temperature in the system will occur in the combustion chamber during an
approximate constant pressure heat addition process and limited by the material
properties of the chosen combustion chamber, while insuring no dissociation takes
place during the combustion process. This maximum temperature will be set to 2500
Kelvin, assuming that the fuel injection and combustion scheme is in place to produce
34
this result (beyond scope of study). The maximum temperature will almost certainly be
less than 4000 K because nitrogen starts to dissociate around this temperature [20] and
most metals melt around 5000 K [13].
Properties of this fuel are also assumed to be constant throughout this study,
whether the fuel is injected into the inlet or the combustor. This is a zero-dimensional
analysis since we are interested in only the state of the fluid at various locations
throughout the engine and other performance parameters that are obtained for the
overall application. Although this is a zero-dimensional analysis in terms of the
aerodynamic-thermodynamic model, the length scales are specified for both for the x
and y directions. The focus of the analysis performed in this paper will be on the region
encompassing the leading edge of the scramjet up to the exit plane of the isolator.
However as previously mentioned, an assumption is made about the max temperature
of the system within the combustion chamber in order to calculate performance
parameters as seen in Table 1-1. The boundary conditions at both locations will be
specified for computational purposes and chosen based off of previous work in the field
of hypersonics.
This study adopts an inviscid assumption. Meaning, that the boundary layers that
would naturally form along any solid surface are neglected. However, a shock train is
induced in the isolator by introducing solid symmetric wedges at a constant angle of 2o,
as seen in Error! Reference source not found.. It is important to model the physics
within this component to give validity to the results of this study.
2.2.2 Forebody and inlet
The forebody and inlet geometry is based off of work performed by Lindsay et al.
[13], with the condition that both oblique shocks converge at the cowl lip and the
35
corresponding reflected shock serves to turn flow back parallel to the freestream
utilizing mixed compression.
Two compression ramps were used to achieve the necessary compression for
the external compression part of the mixed compression inlet, with the deflection angle
for each ramp being -8.3o. The initial shockwave angle is -16o, second shockwave angle
is -18.1o, and the reflected shockwave angle off the cowl is 28.5o to turn the flow back
parallel to the free stream flow. The transverse (y-direction) measurement of the first
compression ramp is 8.7 cm, the transverse measurement of the second compression
ramp is 16.7 cm and the height of the isolator is 3.3 cm. The total length of the external
inlet (or forebody) as defined by where compression takes place outside the internal
region of engine is 100 cm. The length of the internal inlet is 15.6 cm, which brings the
entire forebody/inlet to 115.6 cm from the leading edge of the aircraft to the entrance of
the isolator. The depth of the entire engine is assumed to be 20 cm.
At the Leading edge (station zero), the Mach is six, static pressure is 1.19 kPa,
stagnation pressure is 1.87 MPa, static temperature is 231 oK, and stagnation
temperature is 1894 oK. Ram-compression taking place due to the forebody/inlet
geometry changes the properties significantly leading into the isolator; heating the air
and increasing the pressure. Coming into the isolator component (station two), the
Mach is 2.86, the static pressure is 38.2 kPa, the stagnation pressure is 1.13 MPa, the
static temperature is 719 oK, and the stagnation temperature is 1894 oK. Using
conservation of mass, the mass flow rate throughout the engine is constant leading up
to the point when fuel mass is added. That mass flow rate exiting the isolator (station
three) is 1.87 kg/s.
36
For the design condition chosen, both oblique shocks converge at the cowl lip
and deflect back up to turn flow parallel to the freestream flow. If you were to draw a
vertical line that intersects where the cowl starts, this is where station one is located.
Further inspection of the base design reveals that there is no spillage drag because the
free stream capture area is fully being directed into the engine.
The working fluid of interest is mostly comprised of air; an ideal gas undergoing
an isentropic process between the abrupt discontinuous phenomena of strong
compression waves. In assuming an ideal gas undergoing an isentropic process, the
mass flow rate may be expressed in terms of total quantities of pressure and
temperature as well as a function of the Mach and specific heat of the fluid, Equation 2-
1. Mass, momentum and energy were used to develop the jump conditions across the
oblique shocks to obtain variations of Mach, total, and static quantities across those
shocks, Equation 2-1, Equation 2-2, Equation 2-3, Equation 2-4, Equation 2-5, Equation
2-6 and Equation 2-7.
�̇� = 𝑃𝑡𝐴𝑀√𝛾
𝑅𝑇𝑡[1 + (
𝛾 − 1
2)𝑀2]
(𝛾+1)2(1−𝛾)
(2-1)
𝑃𝑏
𝑃𝑎= 1 +
2𝛾
𝛾 + 1(𝑀𝑎
2 − 1) (2-2)
𝑃𝑡,𝑏
𝑃𝑡,𝑎=
𝑃𝑏
𝑃𝑎((𝛾 − 1
2 )𝑀𝑏2 + 1
(𝛾 − 1
2 )𝑀𝑎2 + 1
)
𝛾𝛾−1
(2-3)
𝜌𝑏
𝜌𝑎=
(𝛾 + 1)𝑀𝑎2
𝑀𝑎2(𝛾 − 1) + 2
(2-4)
𝑇𝑏
𝑇= (
𝜌𝑏
𝜌𝑎)−1 𝑃𝑏
𝑃𝑎
(2-5)
37
2.2.3 Isolator
All the fuel is injected into the combustion chamber for the base design. Since
the Mach of the flow exiting the isolator and entering the combustion chamber is a
boundary condition, this will set the length of the isolator for this this design. This
boundary condition will be set to a Mach of 2.1 and complete combustion is assumed
when fuel is injected into the combustor. According to Mahto, optimal combustion
efficiency occurs when the Mach leading into the combustor is in the range of 2-2.5 and
a further increase in Mach beyond this range will result in decreased efficiency [28]. To
reiterate, the fuel chosen for this study is ethylene.
One of the design constraints of this study is to hold the Mach entering the
combustion chamber to 2.1. This value is reasonable when looking at previous
experiments performed on Mach six scramjet engines. During Kanda et al. experiment,
the Mach of the combustion gas in the constant area section of the combustion
chamber was estimated to be 2.4 for a Mach six scramjet test [15]. Since this study’s
design includes two compression ramps at a slightly larger magnitude of deflection (-
8.3o) than the two compression ramps observed in Kanda (-6o), Figure 2-, it is
reasonable to assume that the Mach entering the combustion chamber will be slightly
less than the one found in the work of Kanda and taken to be 2.1. This is due to more
external compression taking place, which produces stronger shocks and slows down
𝑀𝑏2 =
1 + 𝑀𝑎2 (
𝛾 − 12 )
𝛾𝑀𝑎2 − (
𝛾 − 12
)
(2-6)
tan(𝛿) = 2 cot(𝛽) [𝑀𝑎
2 sin2(𝛽) − 1
𝑀𝑎2(𝛾 + cos(2𝛽)) + 2
] (2-7)
38
the flow more than the Mach at the same location reported by Kanda et al. prior to
entering the isolator. The Mach boundary condition for this base design entering the
combustion chamber will be set to M3 = 2.1.
With a Mach boundary condition of 2.1 at the exit of the isolator, and a Mach of
2.86 entering this component, the length of the isolator is found to be 25.1 cm. Since
this analysis is assumed to be inviscid throughout, the boundary layer is simulated using
constant area ramps of 2o to perturb a shock train within this component, Error!
Reference source not found.. Navigating across this shock train using equations
discussed in the previous section, the length of the isolator is determined when the
Mach reaches the boundary condition. That length is 25.1 cm. The resulting geometry of
the base design with a free-stream of Mach six and the exit boundary condition of Mach
2.1 can be observed in Figure 2-.
The contraction ratio as defined by the throat area with respect to the internal
inlet area, is plotted in figure 2-11. Here the point plotted is bounded by the minimum
area ratio that will allow the inlet to self-start for the Mach at that location called the
Kantrowitz contraction ratio, Equation 2-8 and the maximum area ratio (isentropic
contraction limit) Equation 2-9 [34]. The Area chosen falls nearly in the middle of this
plot, making the geometry of the inlet and isolator a valid one.
(𝐴3
𝐴2)𝐾𝑎𝑛𝑡𝑟𝑜𝑤𝑖𝑡𝑧
= 𝑀2
−(𝛾+1𝛾−1)
[2
𝛾 + 1 (1 +
𝛾 − 1
2𝑀2
2)]
12[1 +
2𝛾
𝛾 + 1(𝑀2
2 − 1)]
1𝛾−1
(2-8)
(𝐴3
𝐴2)𝐼𝑠𝑒𝑛𝑡𝑟𝑜𝑝𝑖𝑐
= 𝑀0 (𝛾 + 1
2)
𝛾+12(𝛾−1)
(1 + (𝛾 − 1
2)𝑀0
2 )−
𝛾+12(𝛾−1)
(2-9)
39
Figure 2.2. External compression inlet
Figure 2-3. Internal compression
40
Figure 2-4. Mixed external and internal compression inlet
Figure 2-5. Propulsion system options as a function of Mach
41
Figure 2-6. Flow stations in a gas turbine engine
Figure 2-7. Comparison of airbreathing and non-airbreathing flight corridors
42
Figure 2-8. Isolator section
Figure 2-9. Model of scramjet engine
43
Figure 2-10. Base design with shocks
Figure 2-11. Contraction ratio of base design
44
CHAPTER 3 RESULTS AND DISCUSSION
3.1 Upstream Fuel Injection
Having the geometry up to the combustion chamber set (base design), different
percentages of the available fuel will be injected into the inlet at station two. The
influence of this action on the flow field downstream of the injection site and the overall
performance of the Engine as a result of this new modification will be investigated. The
details of the plumbing delivering the fuel to both the combustion chamber and
secondary fuel injection site upstream of the combustion chamber are beyond the scope
of this study.
Three different cases were investigated. Case one corresponds to 10% of the
fuel being delivered to the inlet, specifically at a distance of 115.6 cm from the leading
edge of the engine. Case two increases the amount of fuel injected into this location to
20% of the available fuel for this application and case three further increases the
amount of fuel injected to 40%. From a continuity perspective, the amount of mass of
the fuel added to the primary core flow path is significantly less than that of the total
mass of air that the fuel is being injected into.
Ethylene, C2H4, is the fuel chosen for this study with a heating value of 45 MJ/kg,
and a stoichiometric fuel-to-air ratio of 0.0679 [24]. The chemically balanced equation
for a hydrocarbon mixing with air and resulting in complete combustion can be seen in
Equation 3-1 [10]. The stoichiometric fuel-to-air ratio for this chemical reaction is then
obtained from Equation 3-2 [10].
𝐶𝑥𝐻𝑦 + (𝑥 +𝑦
4) (𝑂2 +
79
21𝑁2) → 𝑥𝐶𝑂2 +
𝑦
2𝐻2𝑂 +
79
21(𝑥 +
𝑦
4)𝑁2
(3-1)
45
The equivalence ratio “𝜙” is a parameter necessary to evaluate whether or not a
combustion process is fuel-rich or fuel-lean. This parameter is defined as the fuel-to-air
ratio of the actual reaction with respect to the stoichiometric fuel-to-air ratio as seen in
Equation 3-3. If all the fuel is injected into the combustion chamber, then the
equivalence ratio is set to one (base design). According to Preller and Smart the most
efficient design resulting in a peak thrust will occur when the equivalence ratio
approaches one [17]. At this equivalence ratio, all the fuel that is mixing with the oxidizer
is being used to generate thrust and any more fuel added to the air would not be used,
which makes an engine design with an equivalence ratio of one the most efficient
choice.
The overall equivalence ratio for this engine is referred to the global equivalence
ratio throughout this study and appear in equations without any subscript “𝜙”. When
diverting a percentage of the fuel around to the inlet, this percentage of this total fuel will
be described by the variable “𝜉𝑖”. For instance, if ten percent of the total fuel is to be
injected in the inlet, this value would be set to 𝜉𝑖 = 0.10. When the fuel is broken up in
this manner there will be two equivalence ratios used in this study. Specifically, “𝜙𝑖” will
be the equivalence ratio at the injection site upstream of the combustion chamber and
“𝜙𝑐” will be the equivalence ratio used in the combustion chamber. Together these two
parameters sum to the global equivalence ratio “𝜙”. This notation is reflected in
Equation 3-4 and Equation 3-5. Using conservation of mass for a steady-state flow,
Equation 3-6 is then used to describe the relationship between all mass addition across
the engine.
𝑓𝑠𝑡 =36𝑥 + 3𝑦
103(4𝑥 + 𝑦)
(3-2)
46
Using the mathematical description of the flow-field across the engine as just
described, Equation 3-7 can be obtained and is subsequently used to calculate the
mass flow rate of the fuel being injected into the inlet. The condition when the fuel
happens to combust in the inlet will be investigated in section 3.2. At present, it’s
assumed that none of the fuel being injected into the inlet will ignite.
The length of the isolator for the base design was found to be 25.1 cm, obtained
from code written for this study as seen in the Appendix. This value was obtained by
having the Mach boundary condition at the exit of the isolator set to 2.1 and navigating
across an induced shock train till it hit this boundary condition. A close-up view of this
section for the base design can be seen in Figure 2-8.
To verify this result, a secondary method for obtaining an Isolator length was
sought and found in a publication by Ortwerth et al. Ortwerth et al. introduced Equation
3-8 that determines the interaction isolator length required for any pressure [3]. Looking
at Equation 3-8, the �̃� represents the static pressure ratio increase across the isolator
and found to be 3.27 for the base design. The f1 in Equation 3-8 is a parameter defined
to make Equation 3-8 more legible, and a function of the stream-thrust at the entrance
to the isolator. The stream thrust at this location for the base design is found to be 3132
𝜙 =𝑓
𝑓𝑠𝑡=
�̇�𝑓/�̇�𝑎𝑖𝑟
(�̇�𝑓/�̇�𝑎𝑖𝑟)𝑠𝑡𝑜𝑖𝑐ℎ
(3-3)
𝜙𝑖 ≝ 𝜙𝜉𝑖 (3-4)
𝜙𝑐 ≝ 𝜙(1 − 𝜉𝑖) (3-5)
𝜕
𝜕𝑡∰𝜌𝑑𝑉 + ∯𝜌(�⃗⃗� ∙ �̂�) 𝑑𝑆 = 0
�̇�0 + �̇�𝑓 = �̇�1
(3-6)
�̇�𝑓 = �̇�1𝑓𝑠𝑡𝜙𝜉𝑖 = �̇�1𝜙𝑖𝜉𝑖 (3-7)
47
N. Another parameter defined for Equation 3-8 is g1 and is a function of the mass-flow-
rate and the total enthalpy at the entrance of the isolator. The ratio of specific heats is
taken to be 1.4, and the initial friction coefficient, Cfl, is taken to be 0.0109. This friction
coefficient was obtained from Qiuya, in which this friction coefficient agreed well with
experimental data for an isolator with an entrance Mach of 1.9 [29]. Therefore, this is
friction coefficient is an approximation for this study and justified through Qiuya’s
results.
The hydraulic diameter in Equation 3-8 was shown by Çengel and Cimbala for a
rectangular duct to be Equation 3-9 [6]. Using Equation 3-9, the hydraulic diameter of
the isolator for the base design is found to be 5.66 cm, with the area at the entrance of
the isolator (A1) being 0.66 cm2 and with a uniform depth (w) of 20 cm.
The total length of this isolator using Ortwerth’s Isolator equation, also referenced
in Qiuya’s dissertation, was found to be 20.1 cm for this study’s base design. Equation
3-8 models the flow for separation in a constant cross-section isolator with the boundary
𝑙𝑖𝑠𝑜 =𝐷𝐻𝑔1
2
(44.5)𝐶𝑓𝑙𝛾𝑓1{
�̃� − 1
(𝑓1 − �̃�)(𝑓1 − 1)+
1
𝑓1ln (
�̃�(𝑓1 − 1)
𝑓1 − �̃�)} +
(𝛾 − 1)
2𝛾ln(�̃�)
(3-8)
𝑓1 ≝ℑ1
𝑃1𝐴1
ℑ1 = 𝑃1𝐴1 + 𝑢12𝜌1𝐴1
𝑔1 ≝𝑚1̇ 𝑎𝑇
𝑃1𝐴1
𝑎𝑇 = √(𝛾 − 1) (𝑢1
2
2+ 𝐶𝑃𝑇1)
𝐷𝐻 =2 (
𝐴1𝑤)𝑤
(𝐴1𝑤) + 𝑤
(3-9)
48
condition that the Mach is subsonic exiting that duct. The fact that these two different
calculation yield measurements on the same order of magnitude helps strengthen the
case that the isolator length chosen for this study is a reasonable one. The result being
only about a five cm difference between this papers method for obtaining the isolator
length and Ortwerth’s et al. isolator length calculation. Now that the geometry for this
study, from the leading edge to the entrance of the combustion chamber (total length of
140.6 cm) is defined, the inlet fuel addition cases can be investigated.
3.1.1 Case 1 (𝝃𝒊 = 𝟎. 𝟏𝟎)
Case one corresponds to dumping only ten percent of the total fuel into the inlet,
just prior to entering the isolator. The mass flow rate of the fuel at this percentage is
0.0127 kg/s, using Equation 3-7. The ethylene being injected is not combusting and
changes the Mach exiting the combustion chamber from the design constraint of 2.1 to
2.086. It is also observed that the not only does the Mach change, but the static and
stagnation pressures decrease. This makes sense because mass addition brings with it
momentum that reacts with the incoming air to slow down the flow, so the shocks that
are generated in the isolator are weaker than the base design and compress the flow
less. The overall mass flow rate increases, the static temperature increases, and the
stagnation temperature is assumed to stay constant throughout the isolator, since there
is no observable heat transfer.
3.1.2 Case 2 (𝝃𝒊 = 𝟎. 𝟐𝟎)
Case two corresponds to dumping 20 percent of the total fuel into the inlet, just
prior to entering the isolator. The mass flow rate of the fuel at this percentage is 0.0255
kg/s, using Equation 3-7. This fuel being injected is not combusting and changes the
49
Mach exiting the combustion chamber from the design constraint of 2.1 to 2.081. The
parameters change similarly as in case one.
3.1.3 Case 3 (𝝃𝒊 = 𝟎. 𝟒𝟎)
Case three corresponds to dumping nearly half, 40 percent, of the total fuel into
the inlet just prior to entering the isolator. The mass flow rate of the fuel at this
percentage is 0.051 kg/s, using Equation 3-7. This fuel being injected is not combusting
and changes the Mach exiting the combustion chamber from the design constraint of
2.1 to 2.07. The parameters change similarly as in case one as well.
3.1.4 Summary
Table 3-1 is a summary of parameters at the exit of isolator as the independent
of this system is modified. The primary independent of this engine that is being changed
is the inlet fuel percentage added and the dependents are reported as a result. As more
fuel is introduced upstream of the combustor, the Mach at the exit of the isolator, static
pressure, stagnation pressure all decrease. The static temperature, total temperature
and mass flow rate all increase.
Table 3-1. Parameters at Combustor Entrance for Un-Reacting Fuel Injection
𝑃𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝜉𝑖 = 0.0 𝜉𝑖 = 0.10 𝜉𝑖 = 0.20 𝜉𝑖 = 0.40
M3 2.10 2.09 2.09 2.07
P3 [kPa] 125.06 124.69 124.40 123.83
Pt3 [kPa] 1130.88 1115.88 1104.09 1080.83
T3 [K] 1009.70 1012.70 1015.10 1020.00
Tt3 [K] 1894.20 1894.20 1894.20 1894.20
�̇�3 [kg/s] 1.87 1.8875 1.90 1.93
50
The base design along with the upstream fuel runs are considered to follow the
Brayton cycle. Using the 2nd law of thermodynamics, Equation 3-10, an equation for
entropy can be derived. Equation 3-11 is derived assuming the fluid behaves as an
ideal gas, pressure work being the dominant work term and is heat added to the air in
the combustion chamber reversibly. Using Equation 3-11, the entropy rise may be
computed across the combustor and represented in Table 3-2.
From Table 3-2 it is seen as more fuel is added in the inlet, the less entropy is produced
across this component. This is a desirable effect, because in order to have an efficient
energy conversion during the combustion process, the entropy rise needs to be limited
as much as possible.
Table 3-2. Entropy change across the combustor for un-reacting fuel injection
Figure 3- shows the shock trains that develop throughout the scramjet design for
this study as a result from the different percentages of fuel added in the inlet. The shock
trains describing case 1, case 2, case 3 and the base design with no fuel injection in the
inlet are superimposed on top of each other in this figure for analysis. Figure 3- zooms
𝑑𝑒 = 𝑑𝑞 − 𝑑𝑤 (3-10)
𝑠4 − 𝑠3 = 𝐶𝑃,0 ln(𝑇4/𝑇3) − 𝑅𝑙𝑛(𝑃4/𝑃3) (3-11)
𝐹𝑢𝑒𝑙 𝑎𝑑𝑑𝑒𝑑 (𝑠4 − 𝑠3) [J/kg-K]
𝜉𝑖 = 0.00 475.79
𝜉𝑖 = 0.10 474.66
𝜉𝑖 = 0.20 473.75
𝜉𝑖 = 0.40 471.96
51
in on the isolator from Figure 3-1, so that the interactions of the shock waves can more
clearly be observed. Since it’s still somewhat hard to see the effect of different fuel
percentages added in the inlet at this scale, Figure 3-3 zooms in on the last point in the
isolator where the reflected shocks from the upper and lower surfaces of the ramps
intersect. The green lines in these three figures represent the shock train for the base
design when zero fuel was injected. The blue lines represent the shock train when ten
percent of the total fuel is added. The red lines illustrate the shock train when 20
percent of the fuel is injected. And the cyan lines are the oblique waves when 40
percent of the fuel is supplied at the inlet. Looking at these figures, as more fuel is
added the shocks become stronger, becoming less oblique and pushing upstream. This
effect cascades as more reflections occur, so when viewing Figure 3-, the distances
between the shockwave intersections are significantly greater than the relative
distances between the same shocks at the first intersection.
The performance parameters as a result of upstream fuel injection are reported
below in Table 3-3. These performance parameters were obtained by assuming this is
an ideal scramjet engine that operates on a Brayton cycle. The flow initially goes
through an isentropic compression process, both externally then internally. The flow
then is assumed to undergo a constant static pressure combustion process where the
velocity remains constant. The conservative form of Navier Stokes can be seen in
Equation 3-12. Expanding out the stress tensor in Equation 3-13, assuming that there
are negligible body forces, negligible viscous forces, and steady flow then Navier
Stokes reduces to Euler’s Equation in 3-14. If the pressure within the combustion
52
chamber remains constant, the flow will stay at a constant velocity. Mathematically this
described by Euler’s equation.
Although only the geometry up until the combustor is set, it is assumed that the
flow in the nozzle is isentropically expanded and that a constant pressure heat rejection
process occurs when the flow exits the engine. Understanding the cycle of this engine
allows the following parameters in Table 3-3 to be obtained. Equation 3-15, Equation 3-
16, Equation 3-17 Equation 3-18, Equation 3-19, and Equation 3-20 describe these
parameters and are derived using detailed knowledge of this cycle [16].
Table 3-3. Performance parameters, upstream un-reacting fuel-injection
𝜌𝐷�⃗⃗�
𝐷𝑡= ∇ ∙ �⃡� + 𝜌�⃗�
(3-12)
�⃡� = [−𝑃𝑠 + 𝜆(∇ ∙ �⃗⃗� )]�⃡� + 2𝜇�⃡� (3-13)
𝜌(�⃗⃗� ∙ ∇�⃗⃗� ) = −∇(𝑃𝑠) (3-14)
Parameter Base design Case 1 Case 2 Case 3
ℑ [N] 339.2 398.14 397.29 395.58
ℑ𝑠 [m/s] 212.87 212.30 211.85 210.94
ℑ𝑓 [kN/m2] 129.27 60.41 60.28 60.02
TSFC [s/kg-N] 63.9 x 10-6 64.0 x 10-6 64.2 x 10-6 64.5 x 10-6
𝜂𝑃 0.94498 0.94512 0.94523 0.94545
𝜂𝑇 0.6767 0.6748 0.6732 0.6702
𝜂𝑂 0.6394 0.6377 0.6364 0.6336
M4 1.33 1.328 1.326 1.322
𝜋 0.6039 0.5959 0.5896 0.5772
53
Figure 3-1. Isolator for all injections for non-reacting cases
𝑉9
𝑉0=
𝑀9
𝑀0(𝑎9
𝑎0) =
𝑀9
𝑀0√
𝑇9
𝑇0
(3-15)
𝑀9 = √[(𝑃𝑡4
𝑃0)(𝛾−1)/𝛾
− 1]2
(𝛾 − 1)
(3-16)
𝑃𝑡4 =𝑃𝑡3
[(1 + 𝑀32 (
𝛾 − 12 )) −
1𝜏 (𝑀3
2 (𝛾 − 1
2 ))]
𝛾/(𝛾−1)
(3-17)
𝑇𝑚𝑎𝑥 = 𝑇𝑡4 = 𝑇𝑡3 +�̇�𝑓
�̇�0(
𝑞𝑅
𝐶𝑝,𝑜)
(3-18)
𝑇9 =𝜏(𝑇𝑡3)
1 + 𝑀92 (
𝛾 − 12 )
(3-19)
𝑓 = 𝑇𝑡3 (𝑇𝑡4
𝑇𝑡3− 1) (
𝐶𝑝,𝑜
𝑞𝑅)
(3-20)
54
Figure 3-2. Isolator shock train for non-reacting cases
Figure 3-3. Last shock intersections for all injections in isolator
55
3.2 Inlet Heat Addition
If hydrogen or a heavier hydrocarbon were the fuel of choice, it would be used to
actively cool walls around engine before being injected into either the combustion
chamber or inlet. If this were case, at high enough Mach numbers, this might heat the
fuel enough such that when injected into the primary flow path in the inlet combustion
would immediately occur without the aid of a burning element. Some heavier
hydrocarbons such as JP-7 are also used in active cooling techniques and can become
partially cracked when injected into a compressed environment. If this were to occur,
come concentrations of ethylene could be produced and then combust at the injection
site. So, although Ethylene is not a cryogenic fuel and not typically used in active
cooling schemes, it’s useful to investigate the situation where ethylene happens to
combust in the inlet will and will be investigated in this section. If the fuel being injected
into the inlet then happens to combust, this will lead to a drastic loss in performance as
flashback will occur [27] and would result in addition drag on the inlet.
When combustion occurs, it raises the stagnation temperature of the fluid.
Conservation of Energy was used, Equation 3-21 to calculate this increase in stagnation
temperature when the fuel combusted in the inlet, Equation 3-22. When the ethylene is
forced to combust, it increases the Mach exiting the combustion chamber as well. This
contrasts with the non-combusting cases where the Mach was observed to slightly
decrease across the combustion chamber. This is due to the parameter, Tt3. Since this
total temperature is increasing dramatically when fuel is combusting in the inlet it’s
driving M4 to increase as seen in Table 1-1. Figure 3-4 and Figure 3-5 provide a look at
the shock trains on the overall engine and within the isolator respectively. These are
56
developed similarly to those of Figure 3-1 and Figure 3-2, just with the fuel combusting
in the inlet.
It is also observed for all combusting injection cases that when the fuel combusts
in the inlet the static pressure fluctuates. Not only does the Mach decrease, but so does
the stagnation pressure. The amount of fuel injected and combusting never thermally
chokes the engine and the overall mass flow rate increases because of fuel addition.
Parameters at the exit of the isolator can be seen in Table 3-4, entropy change across
the combustor in Table 3-5, and performance parameters for theses combusting cases
represented in Table 3-5.
Table 3-4. Parameters at combustor entrance for reacting fuel Injection
𝑃𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝜉𝑖 = 0.0 𝜉𝑖 = 0.10 𝜉𝑖 = 0.20 𝜉𝑖 = 0.40
M3 2.10 1.97 1.92 1.77
P3 [kPa] 125.06 136.92 133.97 143.53
Pt3 [kPa] 1130.88 1019.87 929.03 790.35
T3 [K] 1009.7 1195.59 1349.5 1712.04
Tt3 [K] 1894.2 2122.02 2346.78 2787.38
�̇�3 [kg/s] 1.8747 1.8875 1.9002 1.9257
Table 3-5. Entropy change across the combustor for un-reacting fuel injection
𝐹𝑢𝑒𝑙 𝑎𝑑𝑑𝑒𝑑 (𝑠4 − 𝑠3) [J/kg-K]
𝜉𝑖 = 0.0 475.79
𝜉𝑖 = 0.10 278.68
𝜉𝑖 = 0.20 110.05
𝜉𝑖 = 0.40 n/a
57
Table 3-6. Performance parameters, upstream reacting fuel-injection
�̇� + �̇� =𝜕
𝜕𝑡∰𝑒𝑡𝑑𝑉 + ∯𝜌𝑒𝑡(�⃗⃗� ∙ �̂�) 𝑑𝑆
(3-21)
𝑒𝑡 ≝𝑉2
2+ 𝑒
Parameter Base design
(𝜉𝑖 = 0.0)
Case 1
(𝜉𝑖 = 0.10)
Case 2
(𝜉𝑖 = 0.20)
Case 3
(𝜉𝑖 = 0.40)
ℑ [N] 339.20 419.29 433.69 458.70
ℑ𝑠 [m/s] 212.87 223.53 231.26 244.59
ℑ𝑓 [kN/m2] 129.27 63.60 67.80 69.60
TSFC [s/kg-N] 63.87 x 10-6 37.95 x 10-6 14.87 x 10-6 n/a
𝜂𝑝 0.9450 0.9424 0.9405 0.9373
𝜂𝑇 0.6767 0.4838 n/a n/a
𝜂𝑂 0.6394 0.4559 n/a n/a
M4 1.33 1.36 1.41 1.47
𝜋 0.6039 0.5446 0.4961 0.4221
58
→ �̇� = ∯𝜌𝑒𝑡(�⃗⃗� ∙ �̂�) 𝑑𝑆
→ �̇� = �̇�2ℎ𝑡2 + �̇�𝑓ℎ𝑡𝑓 − �̇�1ℎ𝑡1
→ 𝑞𝑅,𝐼�̇�𝑓𝜂𝑏 = �̇�2ℎ𝑡2 − �̇�1ℎ𝑡1
→ 𝑞𝑅,𝐼�̇�𝑓𝜂𝑏 = �̇�2𝑇𝑡2𝐶𝑃2 − �̇�1𝑇𝑡1𝐶𝑃1
→ 𝑇𝑡2 =𝑞𝑅,𝐼�̇�𝑓𝜂𝑏 + �̇�1𝑇𝑡1𝐶𝑃1
�̇�2𝐶𝑃2(�̇�1
�̇�1)
→ 𝑇𝑡2 =𝑞𝑅,𝐼 (
�̇�𝑓
�̇�1)𝜂𝑏 + (
�̇�1�̇�1
)𝑇𝑡1𝐶𝑃1
(�̇�2�̇�1
)𝐶𝑃2
𝑇𝑡2 =𝑞𝑅,𝐼𝜉𝑖𝑓𝑠𝑡𝜂𝑏 + 𝑇𝑡1𝐶𝑃1
(1 + 𝑓𝑠𝑡𝜉𝑖)𝐶𝑃2
(3-22)
Figure 3-4. Isolator for all injections for reacting cases
59
Figure 3-5. Isolator shock train for reacting cases
60
CHAPTER 4 RECOMMENDED STUDIES
This study has investigated a situation where fuel is injected upstream in the
inlet. It was mentioned that the injection scheme was beyond the scope of this text, and
as such future work could focus on this aspect for this study’s particular design. This
could involve a physical experiment where cavities would be designed with an injection
scheme in mind for the design outlined in this study.
Only one condition was tested for this study at Mach six. More cases can be
investigated where the Mach was not six, causing the engine to operate off design.
Throttling an inlet and isolator combination up or down would be useful as more data
can be generated and analyzed. If this study were to be expanded to include additional
flight conditions, the code included in the appendix will have to become more robust and
expanded upon. CFD could even be utilized to get a 2-d or 3-d picture of this engines
design. This would make a zero-dimensional analysis more accurate as the planes
where states were taken can now be a result of an averaging technique brought on by
CFD.
61
APPENDIX MATLAB: SCRIPT
A description of the MATLAB functions used to develop the geometry and
perform analysis throughout this study are presented along with the codes themselves.
▪ Assumptions_given_info.m: This function defines the constants and assumptions
used throughout this study. Here the user is asked to input the freestream Mach,
deflection angle of the isolator ramps, design Mach at the exit of the isolator, and an
array of the percentages of the fuel to be tested in the inlet. All the parameters that
are defined here are then saved at the end, which are then loaded into the
proceeding MATLAB functions.
▪ Energy.m: This is the top-level function that is called in MATLAB’s command
window and where each case is looped through so that the Scramjet function below
may initiate the analysis. The results of each case tested are directed here to be
used for plotting purposes.
▪ Scramjet.m: This function walks through the scramjet’s individual components to
calculates the states described in this study for a specified percentage of fuel added
in the inlet. Geometry subroutines and performance functions are also called within
this function. There are two Inputs into this function; a counter to indicate the case
that is being analyzed (ten percent, twenty percent of fuel added in the inlet, etc.),
and the length of the isolator. At first this is initialized with a dummy variable, but
once the length is found it is passed into the code analyzing each case studied.
What is called from this function are the state arrays for station one, two and the
isolator length. These state arrays contain the Mach, static pressure, stagnation
62
pressure, static temperature, stagnation temperature, mass flow rate, and area at a
particular state. This is where the parameters in Table 3-1 came from.
▪ External_Inlet.m: This function marches across the inlet where external
compression takes place and defines the states between the oblique shocks. The
state array at the leading edge (freestream) and a counter are passed into this
function. The state array of station one and the points matrix are returned by this
function. The points matrix is two-dimensional giving the x and y location of pts of
interest along the flow path (shock intersections, intersection of shocks with solid
boundaries, etc.)
▪ Isolator.m: This function marches across the isolator where internal compression
occurs and defines the states between the oblique shocks up to the combustor. In
addition to sweeping out the states across this component, this function serves to
calculate the length of the base design based on the boundary condition at the
entrance to the combustor. The state array at station one, points matrix, and a
counter are passed into this function. It then returns the state array of station two, an
updated points matrix and an isolator length.
▪ Isolator2.m: This function also marches across the isolator where internal
compression occurs and defines the states between the oblique shocks up to the
combustor. However, in this function, the isolator length is now defined and the
boundary condition for this component is that length. This second isolator function is
called for the cases where fuel is injected in the inlet. The state array at station one,
points matrix, isolator length, a flag for combustion to occur in the inlet and a counter
are passed into this function. It then returns the state array of station two.
63
▪ Isolator_Length_Verification.m: This is where the isolator length was obtained,
from [29], and used to compare with this study’s calculation for that dimension. The
state array at station one, station two and isolator length are passed into this
function. It then returns no information.
▪ Forebody_Geometry.m: The shockwaves and boundaries of the forebody leading
up to the cowl for the base design are plotted in this function. The length of the
forebody, the location of where the shocks hit the cowl, depth of the scramjet, state
array for the freestream, and shockwave parameters are passed into this function.
An updated points matrix, the area at the entrance to the isolator, and lengths of
both compression ramps are found and produced by function when called.
▪ Isolator_Geometry.m: Plots the shocks within the isolator. Passes the current
points matrix, shockwave angle information, length of the forebody and a few
counters pass into this function. Some geometry information is called using this
function.
▪ Miscel_Geometry.m: Plots the upper surface of the engine, the cowl, the ramps in
the isolator and the centerline for the flow path inside the engine. Passes the points
matrix and geometric information about shocks occurring. This function then
produces the exit height of the isolator when called.
▪ Isen_Prop.m: Performs local isentropic stagnation calculations for an ideal gas.
This function is called often in this study. Requires the ratio of specific heats and
Mach to calculate and return these results.
▪ Mflow.m: Calculates the mass flow rate for an ideal gas given a static pressure,
area, gas constant, total pressure, ratio of specific heats and Mach
64
▪ Oblique_Shock.m: Calculates the shockwave angle given deflection angle and
vice-versa. Also calculates the total pressure ratio and static temperature, pressure
and density ratios across an oblique shock using Rankine-Hugoniot relations. Inputs
into this function are an upstream Mach, ratio of specific heats, either the shockwave
or turning angle and a counter value to keep track of the current state.
▪ Performance_Parameters.m: Calculates the performance parameters for each
case in this study, obtained from Roux and Tiruveedula [16]. Some results of this
function can be seen in Table 3-3. All the states for each case run in this study are
kept track of and passed into this function.
65
Assumptions_given_info.m function Assumptions_given_info %{ FUNCTION NOTES -------------- This function defines the constants and assumptions used throughout this study: 1. Specific heat, ratio of specific heats and gas constant at each component 2. Universal gas constant 3. Drag on the Aircraft 4. Percentages of fuel to be added in inlet ('I') for each case 5. Free stream conditions based on altitude 6. Shock compression geometry 7. Combustion Efficency 8. Heating value of fuel 9. Stochiometric fuel-air-ratio 10. Freestream velocity 11. Freestream dynamic pressure 12. Mass flow rate injested by engine 13. Fully define state 0: M0,P0,Po0,T0,To0,mdot0,A0 14. Equivalence ratio 15. Actual fuel-air-ratio Requests (user input): 1. The Mach of the freestream 2. If it's on design, underexpanded or overexpanded 3. The boundary condition of the isoloator (Mach at the exit of this study) 4. Isolator ramps slope (deg) 5. Fuel chosen, this code is set up for 3 different fuel options, we use ethylene fuel for this study 6. Boundary Condition (Mach at the Exit of Isolator) ---------------------------------------------------------------- FUNCTION VARIABLES ------------------ M0[ndim] - Freestream Mach, this is user defined
66
exit_flow_cond - Allow user to specify if flow is underexpanded, overexpanded, or on-design. Currently this code can handle an on-design condition, however I built this option into the code so that at a later time if you wanted to run off-design the structure is in place to support that case. Drag[N] - Specifying the drag for the overall engine in this function Thrust[N] - Propulsion force on nozzle walls and pressure force I - Array, whose components give percentage of the fuel added in inlet in order to be analyzed. cpA[J/kgK] - Specific heat at constant pressure for inlet gammaA[ndim] - Ratio of specific heats for inlet RA[J/kgK] - Specific gas constant in inlet cpB[J/kgK] - Specific heat at constant pressure for isolator gammaB[ndim] - Ratio of specific heats for isolator RB[J/kgK] - Specific gas constant in isolator cpC[J/kgK] - Specific heat at constant pressure for combustion chamber gammaC[ndim] - Ratio of specific heats for combustion chamber RC[J/kgK] - Specific gas constant in combustion chamber cpD[J/kgK] - Specific heat at constant pressure for a section prior to nozzle gammaD[ndim] - Ratio of specific heats for upstream of nozzle RD[J/kgK] - Specific gas constant in sec upstream of nozzl cpE[J/kgK] - Specific heat at constant pressure for nozzle gammaE [ndim] - Ratio of specific heats for nozzle RE[J/kgK] - Specific gas constant in nozzle Ru [J/kmol-K] - Universal Gas constant h [km] - Altitude at cruise M0[ndim] - Freestream Mach, this is user defined T0[K] - Static Temperature of freestream
67
P0[Pa] - Static Temperature of freestream den0[kg/m^3] - Static Density of freesteram Po0[Pa] - Stagnation Pressure in freestream To0[K] - Stagnation Temperature in freestream A0[m^2] - Capture area of incoming Air z0[m] - capture height (y-measurement from cowl to top of engine) u0[m/s] - Freestream velocity q0[N/m^2] - Freestream dynamic velocity mdot0[kg/s] - mass-flow-rate of incoming air Po_P_0[ndim] - Stagnation prssure wrt static pressure of freestream (lives in 'Isen_Prop.m') To_T_0[ndim] - Stagnation temperature wrt static temperature of freestream (lives in 'Isen_Prop.m') X0 - [M0 P0 Po0 T0 To0 mdot0 A0], state array for Leading edge of the engine. N - Number of compression ramps in inlet/forebody w[m] - Depth of Scramjet (made uniform for study) L_fb[m] - Length of the forbody/inlet deltaB[deg] - User defined slope of isolator ramps, to induce shock-train betaA[deg] - shockwave angle, of a shockwave in inlet ex) betaA(1) is the shockwave angle of the 1st shockwave to appear in inlet from leading edge nb[ndim] - Combustion Efficiency fuel - user defined fuel type, built into the code 3 choices, can add more if needed qr[J/kg] - Heating value of fuel x - carbon subscript y - hydrogen subscript fst[ndim] - Stoichiometric fuel-air ratio phi[ndim] - equivalvence ratio f[ndim] - actual fuel-air-ratio P5[Pa] - Pressure at exit of engine (nozzle exit) M2[ndim] - User defined Mach at exit of isolator %} %=============================================================== % USER INPUT % ----------
68
M0 = 6; % Design condition, Freestream Mach number deltaB = 2; % deflection of isolator ramps CombIn = 'Y'; % Does fuel combust in Inlet? Y/N M2 = 2.1; % Mach at exit of Isolator for design condition I = [0 0.10 0.20 0.40]; % Where I'm varying the fuel added in % inlet %=============================================================== % GAS CONSTANTS % --------------- cpA = 1010; gammaA = 1.4; RA = 287; % Forebody-Inlet cpB = 1010; gammaB = 1.4; RB = 287; % Isolator cpC = 1010; gammaC = 1.4; RC = 287; % Combustion Chamber cpD = 1010; gammaD = 1.4; RD = 287; % Pre-Nozzle cpE = 1010; gammaE = 1.4; RE = 287; % Nozzle % [J/kg-K] for cp % [J/kg-K] for R Ru = 8314; % [J/kmol-K] %--------------------------------------------------------------- % FREE STREAM CONDITIONS BASED ON ALTITUDE % ------------------------------------------ h = 30; % [km] T0 = 231; % [K] P0 = 1186; % [Pa] den0 = 0.01786; % [kg/m^3] [Po_P_0,~,To_T_0,~] = Isen_Prop(M0,gammaA); Po0 = Po_P_0 * P0; % [Pa] To0 = To_T_0 * T0; % [K] Drag = 321; % [N] (Fred) %--------------------------------------------------------------- % SHOCK COMPRESSION GEOMETRY
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% -------------------------- % Given: betaA(1) = -16; % 1st oblique shockwave angle N = 2; % 2 compression ramps w = 0.20; % [m] L_fb = 1; % [m] % Calculated: z0 = tand(betaA)*L_fb; % Capture height A0 = abs(z0*w); % The distance is taken to be % in negative direction, area % is a magnitude [m^2] %--------------------------------------------------------------- % CHEMISTRY % --------- nb = 0.80; % Ethylene (C2H4) qR = 45e6; % Heating value of Ethylene x = 2; % Carbon subscript y = 4; % Hydrogen subscript %CxHy+(x+y/4)(O2+(79/21)N2) --> (x)CO2+(y/2)H20 + 79/21(x+y/4)N2 % fst = (36x+3y) / 103(4x+y) - Stoichiometric fuel-air-ratio fst = (36*x+3*y) / (103*(4*x+y)); Tmax = 2500; % [Kelvin] - maximum temperature of system %--------------------------------------------------------------- % MORE INLET & EXIT CONDITIONS % ---------------------------- u0 = M0*sqrt(gammaA*RA*T0); % [m/s] q0 = 1/2*den0*u0^2; % [N/m^2] mdot0 = mflow(Po0,A0,RA,To0,gammaA,M0);
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X0 = [M0 P0 Po0 T0 To0 mdot0 A0]; % On Design (for this study) P5 = P0; % Static pressure is equal to that of the % ambient at exit plane of nozzle Thrust = Drag; % [N] phi = 1; f = phi*fst; %--------------------------------------------------------------- % Note: Found out that 34 variables is the max that can be % stored in a .mat file % % Saving these variables to a '.mat' file, allows us to % load these variables into other parts of the code save prob_constraints1.mat M0 Drag I cpA cpB... cpC cpD cpE gammaA gammaB gammaC gammaD gammaE RA RB RC... RD RE T0 To0 P0 Po0 betaA N w L_fb A0 qR nb fst M2 save prob_constraints2.mat P5 q0 mdot0 phi X0 f z0 Thrust... deltaB Ru u0 Tmax CombIn end
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Energy.m function Energy %{ FUNCTION NOTES -------------- Unit System: SI This design is using a mixed compression inlet The compression is performed by shocks both external and internal to the engine When the oblique shock hits the lip of the cowl, the reflected shock is angled in such a way that when it hits the upper inlet, the flow is turned parallel to freestream Labeling System: --------------- Station 0: Leading Edge of Engine Station 1: Isolator Inlet (in thesis this is station 2) Station 2: Combustor Inlet(in thesis this is station 3) Component A: Forebody/Inlet Component B: Isolator Component C: Combustor Component D: Pre-Nozzle Component E: Nozzle ---------------------------------------------------------------- FUNCTION VARIABLES ------------------ L_Isolator - The length of the isolator, we initialize it in the the beginning of this code, so that we may manipulate it in this parent function. During case 1, we deal with the base design and calculate this value. It is then used as a constant in the rest of the study. XX - This array contains how many stations there are in this study. For this study stations 0,1,2 are
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relevant: Leading edge, entrance of isolator, exit of isolator respectively. s1 - String, greek symbol 'xi' with 'inlet' as a subscript. This string will be used for plotting purposes. Telling how much fuel is added in inlet s2 - String, static pressure and units. This string will be used for plotting purposes. s3 - String, stagnation pressure and units. This string will be used for plotting purposes. s4 - String, static temperature and units. This string will be used for plotting purposes. s5 - String, stagnation Temperature and units. This string will be used for plotting purposes. s6 - String, '%', used for plotting purposes. Doing this so we can use 'strcat()' command for legends,titles , etc. This combines multiple strings. s7 - String, 'Mach' with subcript 'infinity symbol'. Used for labeling purposes. L1 - String, this writes fuel percentage at inlet, but does in such a way that if we vary percentage of fuel for engine, it will change the string. This creates a string for first entry in fuel percentage to be added, 1st entry in variable 'I' defined in Assumption.m L2 - This creates a string for second entry in fuel percentage to be added, 2nd entry in variable 'I' defined in 'Assumption_given_info.m' L3 - This creates a string for third entry in fuel percentage to be added, 3rd entry in variable 'I' defined in Assumption.m title_graph - Creates a string to be used as a title for all graphs. We utilizes 's7', and whatever the user defines the freestream Mach to be will be added to the string. loops/iterative subcode variables: j - This variable is used to keep track of each case of this study. Ranges form 1-4 (j=1 base design, j=2 1st case of percentage of fuel added to inlet, corresponding to 1st entry in 'I' variable, etc.) i - This variable is used to refer to entry in state
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array. ex) X1 = [M1 P1 Po1 T1 To1 mdot1 A1], this is the state array for a pt at the entrance to the isolator. X1(i) is used to call specific properties at a given state from the state array. k - Used for plotting, accessing information X0 - [M0 P0 Po0 T0 To0 mdot0 A0], state array for Leading edge of the engine. X1 - [M1 P1 Po1 T1 To1 mdot1 A1], state array for Isolator inlet X2 - [M2 P2 Po2 T2 To2 mdot2 A2], state array for Isolator exit of the engine. I - Array, whose components give percentage of fuel added in inlet in sequence. Defined in 'Assumptions_given_info.m' ex) I(1) is for the base design, with 0% fuel added ininlet M0 - Freestream Mach, this is user defined, lives in 'Assumptions_given_info.m' function N1 - 7x3 matrix, stores the state arrays (X0,X1,X2) for case 1. This case defined in 'Assumptions_given_info.m', is for the base design. N2 - 7x3 matrix, stores the state arrays (X0,X1,X2) for case 2. This case defined in 'Assumptions_given_info.m', is for the case where 10% of fuel is added in inlet (corresponds to 2nd entry in the 'I' variable). N3 - 7x3 matrix, stores the state arrays (X0,X1,X2) for case 3. This case defined in 'Assumptions_given_info.m', is for the case where 20% of fuel is added in inlet (corresponds to 3rd entry in the 'I' variable). N4 - 7x3 matrix, stores the state arrays (X0,X1,X2) for case 4. This case defined in 'Assumptions_given_info.m', is for the case where 40% of fuel is added in inlet (corresponds to 4th entry in the 'I' variable). YY - Just orgainizing the state variables for each case. After each case is executed, this takes all the states for a given case and groups each parameter. ex) YY(i=1,:) = [X0(i=1) X1(i=1) X2(i=1)]
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YY(1,:) = Mach at each station YY(2,:) = Pressure at each station [Pa] YY(3,:) = Stagnation Presssure at each station [Pa] YY(4,:) = Temperature at each station [K] YY(5,:) = Stagnation Temperature at each station [K] YY(6,:) = mass flow at each station [kg/s] YY(7,:) = Area at each station [m] check - string that if 'yes' will exit code and disconnect all plots (hold off command issued for each plot). If 'no' then the code will re-run %} %--------------------------------------------------------------- tic L_Isolator = 0.001; % initializing % The On-Design Condition requires that the first Oblique shock % doesn't enter the engine, and that the 2nd oblique shock hits % the cowl and is angled back to where the 2nd compression ramp % meets the internal comonents of the engine. XX = [0 1 2]; % Note: this study is from station 0 to 2 Assumptions_given_info % Running this function to define all the % the contraints for this problem while (1) load prob_constraints1 % Loading Assumptions/variables for load prob_constraints2 % this study, that were defined in % 'Assumptions_given_info' s11 = '\xi_i_n_l_e_t'; s22 = 'P_3 [Pa]'; s33 = 'P_t_3 [Pa]'; s44 = 'T_3 [\circK]'; s66 = ' %'; s77 = 'M_\infty = '; s88 = ' of fuel added in inlet'; l1 = sprintf('%3d',I(1)*100); % Legend entry 1 L1 = strcat(l1,s66,s88); l2 = sprintf('%3d',I(2)*100); % Legend entry 2
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L2 = strcat(l2,s66,s88); l3 = sprintf('%3d',I(3)*100); % Legend entry 3 L3 = strcat(l3,s66,s88); l4 = sprintf('%3d',I(4)*100); % Legend entry 4 L4 = strcat(l4,s66,s88); title_graph = sprintf(' %2.1f',M0); tg = strcat(s77,title_graph); for j = 1: 4 switch j case 1 [X1,X2,L_Isolator] = ScramJet(j,L_Isolator); N1 = [X0;X1;X2]; % for I(1) case 2 [X1,X2,~] = ScramJet(j,L_Isolator); N2 = [X0;X1;X2]; % for I(2) case 3 [X1,X2,~] = ScramJet(j,L_Isolator); N3 = [X0;X1;X2]; % for I(3) case 4 [X1,X2,~] = ScramJet(j,L_Isolator); N4 = [X0;X1;X2]; % for I(4) end for i = 1: length(X0) % seven parameters solving for, % Xi = [Mi Pi Poi Ti Toi mdoti Ai] YY(i,:) = [X0(i) X1(i) X2(i)];
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%YY(1,:) = Mach at each station %YY(2,:) = Static Pressure at each station [Pa] %YY(3,:) = Stagnation Presssure at each station [Pa] %YY(4,:) = Static Temperature at each station [K] %YY(5,:) = Stagnation Temperature at each station [K] %YY(6,:) = mass flow at each station [kg/s] %YY(7,:) = Area at each station [m] end switch j case 1 figure(5) % Plotting stagnation Pressure at the % exit of the isolator vs. fuel inject plot(YY(3,3),I(j),'g o','MarkerFaceColor','g') hold on figure(6) % Plotting static Pressure at the exit % of the isolator vs. fuel injection plot(YY(2,3),I(j),'g s','MarkerFaceColor','g') hold on figure(9) % Plotting static Temprature at the % exit of the isolator vs. fuel inject plot(YY(4,3),I(j),'g ^','MarkerFaceColor','g') hold on case 2 figure(5) plot(YY(3,3),I(j),'b o','MarkerFaceColor','b') hold on figure(6) plot(YY(2,3),I(j),'b s','MarkerFaceColor','b') hold on figure(9) plot(YY(4,3),I(j),'b ^','MarkerFaceColor','b') hold on case 3
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figure(5) plot(YY(3,3),I(j),'r o','MarkerFaceColor','r') hold on figure(6) plot(YY(2,3),I(j),'r s','MarkerFaceColor','r') hold on figure(9) plot(YY(4,3),I(j),'r ^','MarkerFaceColor','r') hold on case 4 figure(5) plot(YY(3,3),I(j),'c o','MarkerFaceColor','c') hold on xlabel(s33); ylabel('\xi'); title(tg); legend(L1,L2,L3,L4); grid figure(6) plot(YY(2,3),I(j),'c s','MarkerFaceColor','c') hold on xlabel(s22); ylabel('\xi'); title(tg); legend(L1,L2,L3,L4); grid figure(8) xlabel(s11); ylabel('M_3'); title('Fuel Injection'); legend(L1,L2,L3,L4); figure(9) plot(YY(4,3),I(j),'c ^','MarkerFaceColor','c') hold on xlabel(s44); ylabel('\xi'); title('Fuel Injection'); legend(L1,L2,L3,L4);
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grid end end %--------------------------------------------------------------- % Plotting Performance Parameters: Performance_Parameters(N1,N2,N3,N4) %--------------------------------------------------------------- toc % Time of code without any user inputs check = input('quit [yes/no]: ','s'); fprintf('\n') if strcmp(check,'yes') figure(7) load('p1'); load('p2'); load('p3'); load('p4'); PP = [P1 P2 P3 P4]; title('Shockwave Comparison') legend(PP,L1,L2,L3,L4) figure(1) title(strcat(tg,', ',L1)); figure(2) title(strcat(tg,', ',L2)); figure(3) title(strcat(tg,', ',L3)); figure(4) title(strcat(tg,', ',L4)); %axis([1.10 1.45 -0.30 -0.22]) % Want to have same scale %axis([1.385 1.415 -0.274 -0.265]); % Last intersection break end end end
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Scramjet.m function [X1,X2,L_Isolator] = ScramJet(j,L_Isolator) %{ FUNCTION VARIABLES ------------------ X0 - [M0 P0 Po0 T0 To0 mdot0 A0], state array for Leading edge of the engine. X1 - [M1 P1 Po1 T1 To1 mdot1 A1], state array for Isolator inlet X2 - [M2 P2 Po2 T2 To2 mdot2 A2], state array for Isolator exit of the engine. Pts - [x1 y1; x2 y2; x3 y3; ....] just stores the points of interest throughout the code. The 1st point is where the two compression ramps meet, the 2nd pt is where the 2 obli shock intersect, pt 3 is where the reflected shock interse the upper geometry of the engine, pt 4 is where the 1st shock in the isolator intersects the centerline, pt 5 is where the shock intersects hits the ramp in the isolator, all pts after this are defined in the same way, the units are all in [m] s - string that tells information about each case, ex) for case 1, j = 1, s = 0 percent of Total Fuel added in Isolator j - integer that is passed into the function that indicates the case that is currently being processed (j = 1 design case, j = 2 10% of fuel added, etc.) I - Array, whose components give percentage of fuel added in inlet in sequence. Defined in 'Assumptions_given_info.m' CombIn - 'Y' or 'N', user inputs this string after being asked, if 'Y' then the fuel combust in the inlet. z2 - [m] exit height of isolator A2 - [m^2] cross sectional area of isolator mdot2 - [kg/s] mass flow rate at the exit of the isol gammaB [ndim] - Ratio of specific heats for isolator RB[J/kgK] - Specific gas constant in isolator L_Isolator - [m] The length of the isolator
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%} %--------------------------------------------------------------- load prob_constraints1 load prob_constraints2 [X1,Pts] = External_Inlet(X0,j); if j == 1 [X2,Pts,L_Isolator] = Isolator(X1,Pts,j); figure(8) plot(X2(1),I(j)*100,'g d','MarkerFaceColor','g') xlabel('Mach'); ylabel('\xi'); grid figure (j) hold on else % Isolator2 is now solving for the Mach given Length of % Isolator X2 = Isolator2(X1,Pts,L_Isolator,j,CombIn); if j == 2 figure(8) hold on plot(X2(1),I(j)*100,'b d','MarkerFaceColor','b') figure (j) hold on elseif j == 3 figure(8) plot(X2(1),I(j)*100,'r d','MarkerFaceColor','r') figure (j) hold on elseif j == 4
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figure(8) plot(X2(1),I(j)*100,'c d','MarkerFaceColor','c') hold off figure (j) hold on end end z2 = Miscel_Geometry(Pts,deltaB,j); A2 = w*z2; % mdot2 = mflow(X2(3),A2,RB,X2(5),gammaB,X2(1)); % X2(6) = mdot2; X2(7) = A2; if j==1 Isolator_Length_Verification(X1,X2,L_Isolator); end end
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External_Inlet.m function [X1,Pts] = External_Inlet(X0,j) % 2 Wedge Compression Ramps, Characterized by 2 oblique shocks % % X0 = [M0 P0 Po0 T0 To0 mdot0 A0] {nond Pa Pa K K kg/s m} load prob_constraints1 load prob_constraints2 % 1st oblique shock % ----------------- [deltaA(1),Mob1_II,Po_rat_ob1,~,P_rat_ob1,~] = Oblique_Shock... (X0(1),gammaA,betaA(1),'beta given'); P_ob1_II = P_rat_ob1*X0(2); deltaA(2) = deltaA(1); % 2nd oblique shock % ----------------- [betaA(2),Mob2_II,Po_rat_ob2,~,~,~]= Oblique_Shock(Mob1_II... ,gammaA,deltaA(2),'delta given'); Po_ob2_II = Po_rat_ob2*Po_rat_ob1*X0(3); [Po_P_ob2_II,~,~,~] = Isen_Prop(Mob2_II,gammaA); P_ob2_II = Po_P_ob2_II*Po_rat_ob2*1*Po_rat_ob1*X0(3); To_ob2_II = X0(5); P = [P_ob1_II P_ob2_II]; %--------------------------------------------------------------- % Here there is a reflected shock on the cowl, and after this % first shock the fuel is added in this isolator. % 3rd oblique shock % ----------------- deltaA(3) = -1*(deltaA(1)+deltaA(2));
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[betaA(3),Mob3_II,Po_rat_ob3,~,~,~] = Oblique_Shock(Mob2_II,... gammaA,deltaA(3),'delta given'); M1 = Mob3_II; Po1 = Po_rat_ob3*Po_ob2_II; To1 = To_ob2_II; % From energy, stagnation pressure across % oblique shock is same [Po_P_ob3_II,~,To_T_ob3_II,~] = Isen_Prop(M1,gammaA); P1 = Po_P_ob3_II^-1*Po1; T1 = To_T_ob3_II^-1*To1; thetaA = [betaA(1) betaA(2)+deltaA(1) betaA(3)-deltaA(3)]; [A1,Pts,b1,b2] = Forebody_Geometry(L_fb,z0,w,X0,betaA,... deltaA,thetaA,j); mdot1 = mflow(Po1,A1,RA,To1,gammaA,M1); X1 = [M1 P1 Po1 T1 To1 mdot1 A1]; end
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Isolator.m function [X2,Pts,L_isolator] = Isolator(X1,Pts,k) %{ X1 = [M1 P1[Pa] Po1[Pa] T1[K] To1[K] mdot1[kg/s] A1[m]] A_Is = [m] Constant Area of the Isolator (not including the inserted Ramp) deltaB = [deg] deflection angle, in the isolator betaB = [deg] shockwave angle of each shockwave in isolator, starting with the 4th shock in the system %} n = k; load prob_constraints1 load prob_constraints2 j = 1; % initializing variable % 4th shock in system, "delta 4" Assume for now, however the % ramp we are using is added both to the upper part of the % isolator and the lower part of the isolator at the same angle. % Solve for half the flow field, then mirror across the jet % centerline. A_Is = X1(7); % Isolator Area mdot_fuel = X1(6)*I(k)*phi*fst; % [kg/s] Mass flow rate of the % fuel To_Isol = X1(5); % Constant total temperature across shocks %--------------------------------------------------------------- % Shock-Train Analysis % ------------------------- even = 2:2:1000; odd = 1:2:1000; while(1)
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% Don't know how many shocks there will be, so make generic, % unlike the the Forebody-Geometry analysis where I labeled each % of the 3 principle shocks. if j == 1 % Continuity, mass addition prior to 1st shock % in isolator T_I_ob(j) = X1(4); P_I_ob(j) = X1(2); M(j) = (X0(6)+mdot_fuel)/(P_I_ob(j)*A_Is)*sqrt(RB*... T_I_ob(j)/gammaB); [Po_P_ob(j),~,To_T_ob(j),~] = Isen_Prop(M(j),gammaB); Po_ob(j) = Po_P_ob(j)*P_I_ob(j); [betaB(j),M(j+1),PoII_PoI_ob(j),TII_TI_ob(j),... PII_PI_ob(j),denII_denI_ob(j)] = ... Oblique_Shock(M(j),gammaB,-deltaB,'delta given'); [Po_P_ob(j+1),deno_den_ob(j+1),To_T_ob(j+1),~] = ... Isen_Prop(M(j+1),gammaB); P_II_ob(j) = PII_PI_ob(j)*P_I_ob(j); T_II_ob(j) = TII_TI_ob(j)* T_I_ob(j); Po_ob(j+1) = PoII_PoI_ob(j)*Po_ob(j); % To is constant throughout, % Po is constant b/w shocks denI_ob(j) = P_I_ob(j)/(RB*T_I_ob(j)); den_II_ob(j) = denII_denI_ob(j)*denI_ob(j); deno_ob(j+1) = deno_den_ob(j+1)*den_II_ob(j); % static density is constant between c = 'odd'; [xp,yp] = Isolator_Geometry(-deltaB,Pts,betaB(j)... ,L_fb,c,j,n); Pts(4,:) = [xp yp]; elseif j~=1 % These are all now reflected shocks
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% delta is defined as being positive initially [Po_P_ob(j),~,To_T_ob(j),~] = Isen_Prop(M(j)... ,gammaB); Po_ob(j) = Po_P_ob(j)*P_II_ob(j-1); for k = 1: numel(even) if j == even(k) % want a positive deflection [betaB(j),M(j+1),PoII_PoI_ob(j),TII_TI_ob(j)... ,PII_PI_ob(j)] = ... Oblique_Shock(M(j),gammaB,deltaB,'delta given'); c = 'even'; [xp,yp] = Isolator_Geometry(deltaB,Pts,... betaB(j),L_fb,c,j,n); Pts(3+j,:) = [xp yp]; % saving all the points to automate isolator % geometry break elseif j == odd(k) % want a negative deflection [betaB(j),M(j+1),PoII_PoI_ob(j),... TII_TI_ob(j),PII_PI_ob(j)] = ... Oblique_Shock(M(j),gammaB,-deltaB,... 'delta given'); c = 'odd'; [xp,yp] = Isolator_Geometry(deltaB,Pts,... betaB(j),L_fb,c,j,n); Pts(3+j,:) = [xp yp]; break end end P_II_ob(j) = PII_PI_ob(j)*1/Po_P_ob(j)*Po_ob(j);
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T_II_ob(j) = TII_TI_ob(j)*1/To_T_ob(j)*To_Isol; Po_ob(j+1) = PoII_PoI_ob(j)*Po_ob(j); end err(j) = abs(M(j+1)-M2)/M2; if err(j) < 0.01 M2 = M(j+1); [Po_P_2,~,To_T_2,~] = Isen_Prop(M(j+1),gammaB); Po2 = Po_ob(j+1); P2 = 1/Po_P_2*Po2; To2 = To_Isol; T2 = 1/To_T_2*To2; mdot2 = X1(6)+mdot_fuel; X2 = [M2 P2 Po2 T2 To2 mdot2]; % last entries are % calclulated in Miscel_Geom % Calculating some values to be used in Qiuya function % ---------------------------------------------------- i = floor(j/2)+1; % just want an integer, trying to find % midpoint values of shock train Mn = M(i); Tn = T_II_ob(i-1); % TI_bo3 = TII_ob2 Pn = P_II_ob(i-1); denn = Pn/(Tn*RB); [a,~] = size(Pts); L_isolator = abs(Pts(3,1)-Pts(a,1)); break end j = j+1; end end
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Isolator2.m function X2 = Isolator2(X1,Pts,L_Isolator,k,CombIn) load prob_constraints1 load prob_constraints2 j = 1; % initializing variable n = k; A_Is = X1(7); % Isolator Area mdot_fuel = X1(6)*I(k)*phi*fst; To_Isol = X1(5); if strcmp(CombIn,'Y') % Note: This is when fuel combust Prior to the isolator Tt1 = X0(5); Tt2 = (qR*nb*phi*I(k)*fst+Tt1*cpA) / ((1+fst*phi*... I(k))*cpB); X1(5) = Tt2; % replacing Stagnation temperature after combustion else Tt2 = X1(5); end %--------------------------------------------------------------- % Shock-Train Analysis % ------------------------- even = 2:2:1000; odd = 1:2:1000; while(1) % Don't know how many shocks there will be, so make generic, % unlike the the Forebody-Geometry analysis where I labeled each % of the 3 principle shocks.
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if j == 1 % Continuity, mass addition prior to 1st shock % in isolator T_I_ob(j) = X1(4); P_I_ob(j) = X1(2); Ma = 1; mdot = mdot_fuel + X1(6); while(1) A = X1(7); Pt = X1(3); Tt = X1(5); RHS = A*Pt/sqrt(RB*Tt) * sqrt(gammaB) * Ma * (Ma^2*... (gammaB-1)/2 +1)^((gammaB+1)/(2*(1-gammaB))); error = abs(mdot-RHS)/RHS; if error < 0.001 M(j) = Ma; break end Ma = Ma + 0.001; end [Po_P_ob(j),~,To_T_ob(j),~] = Isen_Prop(M(j),gammaB); Po_ob(j) = Po_P_ob(j)*P_I_ob(j); [betaB(j),M(j+1),PoII_PoI_ob(j),TII_TI_ob(j),... PII_PI_ob(j),denII_denI_ob(j)] = ... Oblique_Shock(M(j),gammaB,-deltaB,'delta given'); [Po_P_ob(j+1),deno_den_ob(j+1),To_T_ob(j+1),~] = ... Isen_Prop(M(j+1),gammaB); P_II_ob(j) = PII_PI_ob(j)*P_I_ob(j); T_II_ob(j) = TII_TI_ob(j)* T_I_ob(j); Po_ob(j+1) = PoII_PoI_ob(j)*Po_ob(j); % To is constant throughout,
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% Po is constant b/w shocks denI_ob(j) = P_I_ob(j)/(RB*T_I_ob(j)); den_II_ob(j) = denII_denI_ob(j)*denI_ob(j); deno_ob(j+1) = deno_den_ob(j+1)*den_II_ob(j); % static density is constant between c = 'odd'; [xp,yp] = Isolator_Geometry(-deltaB,Pts,betaB(j),... L_fb,c,j,n); Pts(4,:) = [xp yp]; else % These are all now reflected shocks % delta is defined as being positive initially [Po_P_ob(j),~,To_T_ob(j),~] = Isen_Prop(M(j),gammaB); Po_ob(j) = Po_P_ob(j)*P_II_ob(j-1); for k = 1: numel(even) if j == even(k) % want a positive deflection [betaB(j),M(j+1),PoII_PoI_ob(j),TII_TI_ob(j),... PII_PI_ob(j)] = Oblique_Shock(M(j),gammaB,deltaB,... 'delta given'); c = 'even'; [xp,yp] = Isolator_Geometry(deltaB,Pts,betaB(j),... L_fb,c,j,n); Pts(3+j,:) = [xp yp]; % saving all the points to automate isolator geometry break elseif j == odd(k) % want a negative deflection [betaB(j),M(j+1),PoII_PoI_ob(j),TII_TI_ob(j),... PII_PI_ob(j)] = Oblique_Shock(M(j),gammaB,-deltaB,... 'delta given');
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c = 'odd'; [xp,yp] = Isolator_Geometry(deltaB,Pts,betaB(j),... L_fb,c,j,n); Pts(3+j,:) = [xp yp]; break end end P_II_ob(j) = PII_PI_ob(j)*1/Po_P_ob(j)*Po_ob(j); T_II_ob(j) = TII_TI_ob(j)*1/To_T_ob(j)*To_Isol; Po_ob(j+1) = PoII_PoI_ob(j)*Po_ob(j); end [a,~] = size(Pts); L = Pts(3,1)-Pts(a,1); err(j) = (abs(L_Isolator)-abs(L))/abs(L); if err(j) < 0.01 % Changing iterative constraint to Be Lentgth of Isolator % calulated form the 1st equivalance M2 = M(j+1); [Po_P_2,~,To_T_2,~] = Isen_Prop(M(j+1),gammaB); Po2 = Po_ob(j+1); P2 = 1/Po_P_2*Po2; T2 = 1/To_T_2*Tt2; mdot2 = X1(6)+mdot_fuel; X2 = [M2 P2 Po2 T2 Tt2 mdot2]; % last entries are calclulated in Miscel_Geom % Calculating some values to be used in Qiuya function % ----------------------------------------------------
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i = floor(j/2)+1; % just want an integer, trying to % find midpoint values of shock train Mn = M(i); Tn = T_II_ob(i-1); % TI_bo3 = TII_ob2 Pn = P_II_ob(i-1); denn = Pn/(Tn*RB); break end j = j+1; end end
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Isolator_Length_Verification.m function Isolator_Length_Verification(X1,X2,L_Isolator) load prob_constraints1 load prob_constraints2 A1 = X1(7); M1 = X1(1); P1 = X1(2); P2 = X2(2); T1 = X1(4); mdot = X2(6); u1 = M1*sqrt(gammaB*RB*T1); gamma = gammaB; rho1 = P1/(RB*T1); % Ideal Gas a = A1/w; b = w; Dh = 2*a*b/(a+b); % [Eq. 8-4] Cengel Cimbala % (Rectangular duct) %--------------------------------------------------------------- at = sqrt((gamma-1)*(u1^2/2+cpB*T1)); F1 = P1*A1+u1^2.... *rho1*A1; g1 = mdot*at/(P1*A1); f1 = F1/(P1*A1); prat = P2/P1; Cfl = 0.0109; % Qiuya l = Dh/(44.5*Cfl)*g1^2/(gamma*f1)*((prat-1)/((f1-prat)*... (f1-1))+1/f1*log(abs((prat*(f1-1))/(f1-prat))))+... (gamma-1)/(2*gamma)*log(prat); fprintf('\nLength of Design Isolator = %4.3f m',L_Isolator) fprintf('\nLength of Isolator [Qiuya] = %4.3f m\n\n',l) end
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Forebody_Geometry.m function [A1,Pts,b1,b2] = Forebody_Geometry(L_fb,z0,w,X0,... beta,delta,theta,k) % Solving for intersection of the 2nd oblique shock and the % start of the 2nd compression ramp. This will determine the % length (b1) of the first compression ramp x = linspace(0,L_fb,10000); % Test "x" yob1_fun = @(x) tand(beta(1))*x; xp2 = L_fb; yp2 = z0; % known (point 2 is intersection of % shocks) m1 = tand(delta(1)); % slope of 1st compression ramp y1_fun = @(x) m1*x; % Equation of 1st compression ramp % (pt at origin in pt-slope) mob2 = tand(theta(2)); % slope of 2nd oblique shock yob2_fun = @(x) mob2*(x-xp2) + yp2; for i = 1: numel(x) err(i) = abs((y1_fun(x(i)) - yob2_fun(x(i))) /... yob2_fun(x(i))); if err(i) < 0.01 % <1 percent error yp1 = y1_fun(x(i)); xp1 = x(i); b1 = xp1; break end end %--------------------------------------------------------------- % Solving for intersection of the 3rd oblique shock and the end % of the 2nd compression ramp. This will determine the length % (b2) of the second compression ramp clear xb yb x i err
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x = linspace(b1,L_fb+L_fb*2,10000); % Test "x" m2 = tand(delta(1)+delta(2)); % slope of 2nd compression ramp y2_fun = @(x) m2*(x-xp1) + yp1; % anonoymous equation for % y-component of the 2nd % comression ramp m3 = tand(theta(3)); yob3_fun = @(x) m3*(x-xp2) + yp2; for i = 1: numel(x) err(i) = abs((y2_fun(x(i)) - yob3_fun(x(i))) /... yob3_fun(x(i))); if err(i) < 0.001 % <1 percent error yp3 = y2_fun(x(i)); xp3 = x(i); b2 = xp3-b1; % pt 3 is where 1st reflected shock % hits the upper geometry break end end %--------------------------------------------------------------- % Plotting the forebody-Inlet with shockwaves % Body % ---- switch k case 1 x1 = linspace(0,b1,10000); y1 = y1_fun(x1); x2 = linspace(b1,b1+b2,10000); y2 = y2_fun(x2); c1 = 'b'; % color of body % Shockwaves
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% ---------- xob1 = linspace(0,L_fb,10000); yob1 = yob1_fun(xob1); xob2 = linspace(xp1,xp2,10000); yob2 = yob2_fun(xob2); xob3 = linspace(xp2,xp3,10000); yob3 = yob3_fun(xob3); c2 = 'g'; % color of shocks if (k==1) for n = 1:4 % Default increment is one switch n case 1 figure(7) plot(x1,y1,c1,x2,y2,c1,xob1,yob1,c2,xob2,yob2,c2,... xob3,yob3,c2) xlabel('x [m]') ylabel('y [m]') hold on figure(1) hold on case 2 figure(2) hold on case 3 figure(3) hold on case 4 figure(4) hold on end plot(x1,y1,c1,x2,y2,c1,xob1,yob1,c2,xob2,yob2,c2,... xob3,yob3,c2) % plotting shocks and Inlet compression ramps
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xlabel('x [m]') ylabel('y [m]') hold on end end end A1 = abs(z0-yp3)*w; % again area is a magnitude Pts = [xp1 yp1; xp2 yp2; xp3 yp3];
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Isolator_Geometry.m function [xp,yp] = Isolator_Geometry(deltaB,Pts,beta,L_fb,c,j,n) %{ Plots the shocks in the Isolator Pts = [xp1 yp1; xp2 yp2; xp3 yp3;.....] point1: where 1st two compression ramps meet point2: where 1st 2 oblique shocks intersect point3: where 1st reflected shock intersect upper geometry of engine z0: caputure length of the Engine %} for Count=1:2 if Count == 1 figure (n) hold on else figure (7) hold on end break2 = 'no'; % Initializing to break out of nested for % loop below %--------------------------------------------------------------- switch c case{'even'} % positive deflections slope & condition for intersecting % upper ramp theta = beta-deltaB; feven = @(x) tand(-deltaB)*(x-Pts(3,1))+Pts(3,2); % pt slope equation for upper ramp
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case{'odd'} % negative deflections slope & condition for % intersecting centerline theta = beta; x_cl = linspace(Pts(3,1),L_fb*2,1e5); fodd(1,1:length(x_cl)) = Pts(3,2)+(Pts(2,2)-... Pts(3,2))/2; % y - Centerline end m = tand(theta); x = linspace(Pts(j+2,1),L_fb*2,1e5);% Test "x" y_fun = @(x) m*(x-Pts(j+2,1)) + Pts(j+2,2); % anonymous function for shock for k = 1: numel(x) if strcmp(break2,'yes') % breaks out of nexted for loop break end for i = 1: numel(x) switch c case{'even'} err = abs((y_fun(x(i)) - feven(x(k))) / ... feven(x(k))); case{'odd'} err = abs((y_fun(x(i)) - fodd(1,k)) / fodd(1,k)); end if err < 1e-5 % < 1 percent error yp = y_fun(x(i)); xp = x(i);
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% where shock hits upper wall or centerline xob = linspace(Pts(j+2,1),xp,1e5); yob = y_fun(xob); if Count == 1 c2 = 'g'; % color of shocks plot(xob,yob,c2) plot(xob,-yob+2*(Pts(3,2)+(Pts(2,2)-Pts(3,2))... /2),'g'); % mirror across Centerline else switch n case 1 c2 = 'g'; % color of shocks P1 = plot(xob,yob,c2); save('p1','P1'); % saving variable to % call for last case plot(xob,-yob+2*(Pts(3,2)+(Pts(2,2)-... Pts(3,2))/2),'g'); case 2 c2 = 'b'; % color of shocks P2 = plot(xob,yob,c2); save('p2','P2'); plot(xob,-yob+2*(Pts(3,2)+(Pts(2,2)-... Pts(3,2))/2),'b'); case 3 c2 = 'r'; % color of shocks P3 = plot(xob,yob,c2); save('p3','P3'); plot(xob,-yob+2*(Pts(3,2)+(Pts(2,2)-... Pts(3,2))/2),'r'); case 4 c2 = 'c'; % color of shocks P4 = plot(xob,yob,c2); save('p4','P4'); plot(xob,-yob+2*(Pts(3,2)+(Pts(2,2)-... Pts(3,2))/2),'c'); end
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end break2 = 'yes'; break % "Break" command breaks only the loop where you use it end end end end end
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Miscel_Geometry.m function z2 = Miscel_Geometry(Pts,deltaB,k) %{ Plotting: (a) upper surface (x-axis of engine) & cowl (b) placing the two ramps in the isolator (c) centerline %} for m=1:5 if k == 1 switch m case 1 figure (1); hold on case 2 figure (2); hold on case 3 figure (3); hold on case 4 figure (4); hold on case 5 figure (7) hold on end end c1 = 'b'; % color of body Q = length(Pts); % finds the number of elements along the % largest dimension %---------------------------------------------------------------
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% (a) Upper geometry and cowl % ------------------------ xb = linspace(0,Pts(Q,1),1e5); % upper geometry yb = zeros(1,1e5); xcowl = linspace(Pts(2,1),Pts(Q,1),1e5); % cowl ycowl(1,1:length(xcowl)) = Pts(2,2); xI_x = linspace(Pts(3,1),Pts(Q,1),1e6); % upper isolator % geometry if k == 1 plot(xb,yb,c1,xcowl,ycowl,c1) % coloring in upper geometry % (vertices starting at origin and moving ccw) v_interior = [0 0; Pts(1,1) Pts(1,2); Pts(3,1) Pts(3,2);... Pts(Q,1) Pts(3,2); Pts(Q,1) 0]; patch(v_interior(:,1),v_interior(:,2),'b','EdgeColor',... 'blue','FaceAlpha',0.1) end %--------------------------------------------------------------- % (b) Ramps % ------ m1 = tand(-deltaB); m2 = tand(deltaB); f_ramp1 = @(x) m1*(x-Pts(3,1))+Pts(3,2); f_ramp2 = @(x) m2*(x-Pts(3,1))+Pts(2,2); if k == 1 v1 = [Pts(3,1) Pts(3,2); max(xcowl) Pts(3,2); max(xcowl)... f_ramp1(max(xcowl))]; patch('Vertices',v1,'EdgeColor','blue','FaceColor','blue',... 'FaceAlpha',0.1) v2 = [Pts(3,1) Pts(2,2); max(xcowl) Pts(2,2); max(xcowl)... f_ramp2(max(xcowl))]; % verticies of lower ramp patch('Vertices',v2,'EdgeColor','blue','FaceColor','blue',... 'FaceAlpha',0.1)
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end %--------------------------------------------------------------- % (c) Jet Centerline % --------------- x_cl = linspace(Pts(3,1),Pts(Q,1),1e5); y_cl(1,1:length(x_cl)) = Pts(3,2)+(Pts(2,2)-Pts(3,2))/2; if m ==1 for M = 1:4 figure(M) plot(x_cl,y_cl,'k :') hold on end figure(7) plot(x_cl,y_cl,'k :') hold on end %----------------------------------------------------------- % Calculate the exit height of the isolator z2 = abs(2*(y_cl(1,1)-f_ramp1(Pts(Q,1)))); end end
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Isen_Prop.m function [p_rat,d_rat,T_rat,A_rat] = Isen_Prop(M,gamma) % Isentropic flow of an Ideal Gas p_rat = (M^2 *(gamma-1)/2 ... % Total Pressure wrt Static +1)^(gamma/(gamma-1)); % Pressure d_rat = (M^2 *(gamma-1)/... 2+1)^(1/(gamma-1)); % Total Density wrt Static % Density T_rat = (M^2 *(gamma-1)/... % Total Temperature Static 2+1); % Temperature A_rat = 1/M * ((1+(gamma-1)... % Ficticious Area at Mach 1 wrt /2 * M^2)/((gamma+1)/2))^...% Real Area (A/A*) ((gamma+1)/(2*(gamma-1))); end
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mflow.m function massflowrate = mflow(Po,A,R,To,gamma,M) %{ Ideal gas Assumed Equations used for derivation: 1. Isentropic relations 2. Definition of mflow 3. Definition of Mach M = F_inertial ------------------ F_compressibility %} massflowrate = M*Po*A*sqrt(gamma/(R*To))*(1+(gamma-1)/2*M^2)^... ((gamma+1)/(2*(1-gamma))); end
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Oblique_shock.m function [Z,M2,Po2_Po1,T2_T1,P2_P1,den2_den1] =... Oblique_Shock(M1,gamma,P,count) %{ Iterative Procedure for finding shockwave angle or deflection angle (P determines if beta or delta is given) beta: shockwave angle delta: deflection angle M1: Mach just upstream of shockwave M2: Mach just downstream of shockwave "P" is either shockwave or turning (input into this code) "Z" is either shockwave or turning angle(code produces this) Note: The shockwave angle that the code is solving for, "Beta", is measured clockwise "-" from a datum defined by a line that is parallel to the upstream Mach. Positive direction is ccw from that plane %} switch count case {'beta given'} beta = P; deltafn = @(gamma,M1,beta) atand(cotd(beta) *((M1^2 ... *sind(beta)^2 - 1)/((gamma+1) /2 *... M1^2 - (M1^2 *sind(beta)^2 -1)))); delta = deltafn(gamma,M1,beta); Z = delta; case {'delta given'} delta = P;
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if delta < 0 delta = delta*-1; end A = M1^2 - 1; B = ((gamma+1) / 2) * M1^4 * tand(delta); C = (1 + (gamma+1)/2 * M1^2) *tand(delta); xn = sqrt(A); while(1) xnew = sqrt( A - (B / (xn + C))); er = abs((xnew-xn) / xnew); xn = xnew; if er < 0.00001 beta = acotd(xnew); % arccot(x) in Degrees, changing % the datum defined in Keith to % be in the cw direction break end end Z = beta; end %--------------------------------------------------------------- Mt = M1*cosd(beta); % Tangential component of velocity same on % both sides of oblique shock (momentum and % energy) Mn1 = M1*sind(beta); Mt1 = M1*cosd(beta); M2 = sqrt(((M1^2*sind(beta)^2+2/(gamma-1)) / (2*gamma/(gamma... -1)*M1^2*sind(beta)^2-1))/(sind(beta-delta)^2));
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Mn2 = M2*sind(beta-delta); Mt2 = M2*cosd(beta-delta); Po2_Po1 = (((gamma+1) / 2 * M1^2 * sind(beta)^2) / (1 + (... gamma-1) /2 * M1^2 * sind(beta)^2))^(gamma/(gamma... -1)) * (1 / ((2*gamma)/(gamma+1)* M1^2 * sind... (beta)^2 - (gamma-1)/(gamma+1)))^(1/(gamma-1)); T2_T1 = (1+(gamma-1)/2*M1^2*sind(beta)^2)*(2*gamma/(gamma-... 1)*M1^2*sind(beta)^2-1)/((M1^2*sind(beta)^2*(gamma... +1)^2)/(2*(gamma-1))); P2_P1 = 2*gamma /(gamma+1) * M1^2 *sind(beta)^2-(gamma-1)... /(gamma+1); den2_den1 = (gamma+1)*M1^2*sin(beta)^2/((gamma-1)*M1^2*sin... (beta)^2+2); if P < 0; switch count case{'beta given'} Z = delta; % sign takes care of itself case{'delta given'} Z = -beta; % changing the sign notation end end end
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Performance_Parameters.m function Performance_Parameters(N1,N2,N3,N4) load prob_constraints1 % loading assumptions load prob_constraints2 % loading assumptions %{ Notes: ----- subscripts in paper: Station 0 - Freestream Station 1 - Engine Inlet Station 2 - Isolator Inlet Station 3 - Isolator Exit, Combustor Inlet Station 4 - Combustor Exit code: X0 - Freestream X1 - Engine Inlet X2 - Isolator Exit Ideal gas undergoing an isentropic process I - Array, whose components give percentage of fuel added in inlet in sequence. Defined in 'Assumptions_given_info.m' ex) I(1) is for the base design, with 0% fuel added in inlet N1 - States across inlet/forebody for 0 fuel added in the inlet, I(1) N2 - States across inlet/forebody for 10 fuel added in the inlet, I(2) N3 - States across inlet/forebody for 20 fuel added in the inlet, I(3) N4 - States across inlet/forebody for 40 fuel added in the inlet, I(4) N1 = [X0; X1; X2] = [M0 P0 Pt0 T0 Tt0 mdot0 A0]
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|M1 P1 Pt1 T1 Tt1 mdot1 A1| [M2 P2 Pt2 T2 Tt2 mdot2 A2] ex) N1(3,2) = P2 gc - Newton's constant, unitless in SI units, if not its set to 32.174, this is a conversion factor M0 - Freestream Mach number M3 - Mach leading into the combustion chamber M9 - Mach at exit of nozzle V9[m/s] - Velocity at exit of nozzle V0[m/s] - Velocity of the freestream a0[m/s] - freestream speed of sound sos = sqrt(gamma*R*Ts) f - fuel to air ratio gamma - ratio of specific heats Cp0[J/kg-K] - Specific heat at constant pressure R[J/kg-K] - Unique gas constant Th[N] - Thrust A3[m^2] - Area at isolator exit Tmax = Tt4[K] - Maximum Temperature in Combustion Chamber (material limit) Tt3[K] - Temerature at exit of isolator Pt3[Pa] - Total pressure at exit of isolator Pt0[Pa] - Total pressure of the freestream %} %--------------------------------------------------------------- % Setup Variables % --------------- % equations 1-1 through 1-6 in paper from Roux & Tiruveedula % resource are where these setup variables come from for i=1:numel(I) % default increment is one (loop will execute % four times, assuming I has 4 elements) % can dynamically create variable in this way:
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eval(['N' strcat('=','N',num2str(i))]); % N = N1 for i = 1, etc. gc = 1; gamma = gammaA; Cp0 = cpC; R = RC; hc = qR; % heat of combustion, defined in assumptions M0 = N(1,1); a0 = sqrt(gammaA*RA*N(1,4)); V0 = M0*a0; mdot0 = N(1,6); Pt0(i) = N(1,3); P0 = N(1,2); Tt0(i) = N(1,5); T0(i) = N(1,4); T2(i) = N(2,4); Pt2(i) = N(2,3); Tt3(i) = N(3,5); Pt3(i) = N(3,3); P3(i) = N(3,2); T3(i) = N(3,4); M3(i) = N(3,1); A3 = N(3,7); Tt4(i) = Tmax; %defined in assumptions, can also find using Eq. 1-4 Pt4(i) = Pt3(i)/((1+M3(i)^2*((gamma-1)/2))-((1/(Tt4(i)/Tt3(i)))... *(M3(i)^2*(gamma-1)/2)))^(gamma/(gamma-1)); M9(i) = sqrt(( (Pt4(i)/P0)^((gamma-1)/gamma)-1)*(2/(gamma-1))); T9(i) = Tt4(i)/(1+M9(i)^2*(gamma-1)/2); V9(i) = V0*M9(i)/M0*sqrt(T9(i)/T0(i)); f = Tt3(i)*(Tt4(i)/Tt3(i)-1)*Cp0/hc; % fuel-air ratio, verify this % with value I used %--------------------------------------------------------------- % Performance Parameter Claculations % ---------------------------------- % Specific Thrust: ST(i) = M0*a0/gc *(V9(i)/V0 -1);
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% Thrust Specific Fuel Consumption: S(i) = f/(ST(i)); % Propulsive Efficiency: np(i) = 2/(V9(i)/V0+1); % Thrust Flux: Th(i) = ST(i)*mdot0; % Thrust, Verify that this is thrust approx TF(i) = Th(i)/A3; % Thrust Flux % Mach at Exit of Combustor: M4(i) = M3(i)/sqrt(Tmax/Tt3(i)*(1+((gamma-1)/2)*M3(i)^2)); P4(i) = Pt4(i)*(M4(i)*(gamma-1)/2 + 1)^ (gamma/(1-gamma)); T4(i) = Tt4(i)*(M4(i)*(gamma-1)/2 + 1)^-1; % Total Temperature Recovery: TR(i) = Tt4(i)/Tt3(i); % Thermal efficiency: TR0(i) = Tt0(i)/T0(i); % Freestream temperature ratio nt(i) = 1-1/TR0(i)*((T9(i)/T0(i)-1)/(TR(i)-1)); % Overall efficiency: no(i) = np(i)*nt(i); % Overall Pressure Recovery: PR(i) = Pt3(i)/Pt0(i); % Total Pressure Recovery: PR2(i) = Pt2(i)/Pt0(i); % Entropy increase Across the Combustor:
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del_s(i) = Cp0*log(T4(i)/T3(i))-R*log(P4(i)/P3(i)); % Kinetic energy Efficiency nke(i) = 2/((gamma-1)*M0^2)*(Tt3(i)/Tt0(i))*... (1+(gamma-1)/2*M0^2-(Pt0(i)/Pt3(i))^(gamma/(gamma-1))); end %--------------------------------------------------------------- % Plotting the Kantrowitz contraction ratio, Isentropic % contraction limit on same graph then plotting point of % design area ratio across the isolator Arat_desi = N1(3,7)/N1(2,7); % A3/A2 M = linspace(1.01,4,10000); for i = 1:numel(M) Arat_kant(i) = M(i)^(-1*((gamma+1)/(gamma-1)))*(2/... (gamma+1)*(1+(gamma-1)/2*M(i)^2))^(1/2)*(1+2*gamma/... (gamma+1)*(M(i)^2-1))^(1/(gamma-1)); end Arat_isen = M.*((gamma+1)./2).^((gamma+1)./(2.*(gamma-1))).*... (1+((gamma-1)./2).*M.^2).^(-1.*(gamma+1)./(2.*(gamma-1))); % Somewhere in code, Area at entrance to isolator is overiding % Area at exit of isolator for fuel injection cases. Doesn't % make a difference, corrected in Energy element. %--------------------------------------------------------------- % Plotting % ========= % Contraction Ratio if(strcmp(CombIn,'Y'))
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s3= 'Injected Fuel Auto Ignites'; else s3 = 'Injected Fuel Does not Ignite'; end figure(10) hold on plot(N1(2,1),Arat_desi,'c d','MarkerFaceColor','c'); plot(M,Arat_kant,'g'); plot(M,Arat_isen,'b'); xlabel('Contraction Ratio'); title(s3); legend('(A_3/A_2)_B_a_s_e_d_e_s_i_g_n',... '(A_3/A_2)_K_a_n_t_r_o_w_i_t_z',... '(A_3/A_2)_I_s_e_n_t_r_o_p_i_c'); grid %-------------------------------- % Kinetic energy Efficiency figure(11) hold on plot(nke(1),I(1),'g o','MarkerFaceColor','g') plot(nke(2),I(2),'b o','MarkerFaceColor','b') plot(nke(3),I(3),'r o','MarkerFaceColor','r') plot(nke(4),I(4),'c o','MarkerFaceColor','c') s1 = ' %'; s2 = ' of fuel added in inlet'; l1 = sprintf('%3d',I(1)*100); % Legend entry 1 L1 = strcat(l1,s1,s2); l2 = sprintf('%3d',I(2)*100); % Legend entry 2 L2 = strcat(l2,s1,s2); l3 = sprintf('%3d',I(3)*100); % Legend entry 3 L3 = strcat(l3,s1,s2);
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l4 = sprintf('%3d',I(4)*100); % Legend entry 4 L4 = strcat(l4,s1,s2); xlabel('\eta_K_E'); ylabel('\xi'); legend(L1,L2,L3,L4); title(s3); grid %-------------------------------- % Overall Efficiency figure(12) hold on plot(no(1),I(1),'g o','MarkerFaceColor','g') plot(no(2),I(2),'b o','MarkerFaceColor','b') plot(no(3),I(3),'r o','MarkerFaceColor','r') plot(no(4),I(4),'c o','MarkerFaceColor','c') xlabel('\eta_O'); ylabel('\xi'); legend(L1,L2,L3,L4); title(s3); grid %-------------------------------- % Propulsive Efficiency figure(13) hold on plot(np(1),I(1),'g o','MarkerFaceColor','g') plot(np(2),I(2),'b o','MarkerFaceColor','b') plot(np(3),I(3),'r o','MarkerFaceColor','r') plot(np(4),I(4),'c o','MarkerFaceColor','c') xlabel('\eta_P'); ylabel('\xi'); legend(L1,L2,L3,L4); title(s3); grid %--------------------------------
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% Overall Pressure Recovery (Pt3/Pt0) figure(14) hold on plot(PR(1),I(1),'g o','MarkerFaceColor','g') plot(PR(2),I(2),'b o','MarkerFaceColor','b') plot(PR(3),I(3),'r o','MarkerFaceColor','r') plot(PR(4),I(4),'c o','MarkerFaceColor','c') xlabel('P_t_3/P_t_0'); ylabel('\xi'); legend(L1,L2,L3,L4); title(s3); grid %-------------------------------- % Total Pressure Ratio (Pt2/Pt0) figure(15) hold on plot(PR2(1),I(1),'g o','MarkerFaceColor','g') plot(PR2(2),I(2),'b o','MarkerFaceColor','b') plot(PR2(3),I(3),'r o','MarkerFaceColor','r') plot(PR2(4),I(4),'c o','MarkerFaceColor','c') xlabel('P_t_2/P_t_0'); ylabel('\xi'); legend(L1,L2,L3,L4); title(s3); grid %-------------------------------- % Total Temperature figure(16) hold on plot(Tt3(1),I(1),'g o','MarkerFaceColor','g') plot(Tt3(2),I(2),'b o','MarkerFaceColor','b') plot(Tt3(3),I(3),'r o','MarkerFaceColor','r') plot(Tt3(4),I(4),'c o','MarkerFaceColor','c') xlabel('T_t_3 [K]'); ylabel('\xi');
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legend(L1,L2,L3,L4); title(s3); grid %-------------------------------- % Specific Thrust figure(17) hold on plot(Tt3(1),I(1),'g o','MarkerFaceColor','g') plot(Tt3(2),I(2),'b o','MarkerFaceColor','b') plot(Tt3(3),I(3),'r o','MarkerFaceColor','r') plot(Tt3(4),I(4),'c o','MarkerFaceColor','c') xlabel('T_t_3 [K]'); ylabel('\xi'); legend(L1,L2,L3,L4); title(s3); grid %-------------------------------- % Thrust Flux figure(18) hold on plot(TF(1),I(1),'g o','MarkerFaceColor','g') plot(TF(2),I(2),'b o','MarkerFaceColor','b') plot(TF(3),I(3),'r o','MarkerFaceColor','r') plot(TF(4),I(4),'c o','MarkerFaceColor','c') xlabel('Thrust Flux [N]'); ylabel('\xi'); legend(L1,L2,L3,L4); title(s3); grid %-------------------------------- % TSFC figure(19) hold on plot(S(1),I(1),'g o','MarkerFaceColor','g') plot(S(2),I(2),'b o','MarkerFaceColor','b') plot(S(3),I(3),'r o','MarkerFaceColor','r') plot(S(4),I(4),'c o','MarkerFaceColor','c')
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grid xlabel('TSFC [kg/N-s]'); ylabel('\xi'); legend(L1,L2,L3,L4); title(s3); %-------------------------------- % Specific Thrust figure(20) hold on grid plot(ST(1),I(1),'g o','MarkerFaceColor','g') plot(ST(2),I(2),'b o','MarkerFaceColor','b') plot(ST(3),I(3),'r o','MarkerFaceColor','r') plot(ST(4),I(4),'c o','MarkerFaceColor','c') xlabel('Specific Thrust [m/s]'); ylabel('\xi'); legend(L1,L2,L3,L4); title(s3); %--------------------------------------------------------------- % Uncontrolling Figures figure(10) hold off figure(11) hold off figure(12) hold off figure(13) hold off figure(14) hold off figure(15) hold off figure(16) hold off figure(17) hold off
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figure(18) hold off figure(19) hold off figure(20) hold off
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BIOGRAPHICAL SKETCH
David Scott Hanon was born in Winter Park, Florida. The majority of early life
was spent in the Longwood Florida where he attended Woodlands Elementary School,
Rock Lake Middle School, Lake Mary High School and Seminole Community College.
Accepting enrollment to University of Florida in the pursuit of an engineering degree in
fall of 2010, he was awarded two Bachelor of Science degrees in aerospace and
mechanical engineering in the spring of 2014 from the University of Florida. At that point
he worked in a lab for a brief time under Professor Fitzcoy while he contemplated on
working in industry or perusing an advanced degree. Deciding on the latter, he was
accepted into University of Florida’s graduate program and began perusing a Master of
Science in aerospace engineering with a thesis option under Professor Segal in the
spring of 2015. Completing all classes by fall of 2016, he worked as a teaching assistant
to Professor Jackson during the spring of 2017. He then accepted an engineering
position with Aerojet Rocketdyne in West Palm Beach, where he continued to
communicate with Professor Segal, developing the thesis to completion. In Fall 2018,
he was awarded a Master of Science.