University of Calgary

PRISM: University of Calgary's Digital Repository

Graduate Studies The Vault: Electronic Theses and Dissertations

2017

Assessment and Analysis of a Novel Intake for a

Ramjet Engine

Wilson, Steven James

Wilson, S. J. (2017). Assessment and Analysis of a Novel Intake for a Ramjet Engine (Unpublished

master's thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/25025

http://hdl.handle.net/11023/4160

master thesis

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UNIVERSITY OF CALGARY

Assessment and Analysis of a Novel Intake for a Ramjet Engine

by

Steven Wilson

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF MASTER OF SCIENCE

GRADUATE PROGRAM IN MECHANICAL AND MANUFACTURING ENGINEERING

CALGARY, ALBERTA

September, 2017

c© Steven Wilson 2017

Abstract

The Atlantis Intake System (AIS) is a novel intake design intended to supply a combustible

fuel/air mixture to a ramjet engine without the use of any moving parts. The operation of

the AIS is similar to an ejector pump, it operates via the continuous release of a gaseous fuel

jet into a system of inlet stages open to the surrounding ambient air. Interactions between

the fuel jet, the intake geometry and the surrounding air result in a relatively high velocity,

high pressure, and high temperature combustible mixture entering the intake of a ramjet

engine coupled with the intake.

A control-volume analysis is used to develop a means of predicting the performance of an AIS

coupled ramjet engine given a set of input conditions, to better understand the influence of

the controlled variables. This control volume (CV) analysis is expanded to include a method

for predicting the ratio of air entrained by the AIS based on the characteristics of the fuel

inlet jet and the geometry of the AIS. This model is compared to a series of computational

fluid dynamic (CFD) simulations, and shows strong agreement in terms of the ratio of air

entrained. The potential for the use of the models as a tool for rapid assessment of multiple

AIS designs is discussed. An exergetic analysis tool is developed and utilized on the results

of the CFD simulations to quantify the dominant sources of exergy destruction. The results

are to be used as a guide to better optimize the AIS.

i

Acknowledgements

I would like to thank my supervisor Dr. Craig Johansen for being supportive throughout my

extended program. I would also like to thank my good friend Schuyler Hinman for always

being there to provide a sanity check and intelligent insight whenever I have been staring at

the same set of equations for far too long. And, I am very grateful for the support of my

wife Elise, and my parents Brian and Lorrie, for always providing me with all of the family

support that I could ask for.

ii

Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Background and Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Jet Mixing and Entrainment . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Exergy Analysis of Aerospace Systems . . . . . . . . . . . . . . . . . . . . . 103.1 Review of Modern Applications of Exergy-Based Analysis . . . . . . . . . . . 113.2 Local Volumetric Entropy Generation . . . . . . . . . . . . . . . . . . . . . . 12

3.2.1 Turbulent Entropy Generation . . . . . . . . . . . . . . . . . . . . . . 193.3 Exergy Destruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 AIS Performance Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.1 Static Operation Control Volume Analysis . . . . . . . . . . . . . . . . . . . 22

4.1.1 Implications of Multiple Solutions for Um . . . . . . . . . . . . . . . . 284.1.2 CV Analysis Validation . . . . . . . . . . . . . . . . . . . . . . . . . . 304.1.3 Ideal Jet Propulsion Cycle . . . . . . . . . . . . . . . . . . . . . . . . 30

5 AIS Entrainment Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.2 Modified AIS Mixture Properties Model . . . . . . . . . . . . . . . . . . . . 356 CFD Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.1 Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6.1.1 Turbulence Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.2 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6.2.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.2.2 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457.1 CFD Simulation Results & Verification . . . . . . . . . . . . . . . . . . . . . 45

7.1.1 Flow Visualization Results . . . . . . . . . . . . . . . . . . . . . . . . 457.1.2 Grid Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

7.2 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487.2.1 Entrainment Model Comparison . . . . . . . . . . . . . . . . . . . . . 48

7.3 Exergy Destruction Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 517.4 Parametric Analysis Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 547.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598.1 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61A CFD Simulation Flow Visualization Results . . . . . . . . . . . . . . . . . . 66B Exergy Destruction Post-Processing Utility Source Code . . . . . . . . . . . 72C CFD Simulation Exergy Destruction Visualization Results . . . . . . . . . . 84

iii

D Additional Parametric Analysis Contour Plots . . . . . . . . . . . . . . . . . 89E Copyright Permission Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

iv

List of Tables

6.1 Standard model coefficients (OpenCFD, 2017). . . . . . . . . . . . . . . . . . 406.2 Fuel jet inlet conditions of CFD simulations conducted. . . . . . . . . . . . . 42

7.1 Grid properties and results for Case 2. . . . . . . . . . . . . . . . . . . . . . 487.2 Grid convergence index results for Case 2. . . . . . . . . . . . . . . . . . . . 487.3 Mean outlet properties for CFD simulations conducted. . . . . . . . . . . . . 497.4 Exergy destruction rates for CFD simulations conducted. . . . . . . . . . . . 53

v

List of Figures and Illustrations

1.1 Simplified schematic of an ideal engine utilizing the AIS. Not to Scale. . . . . 3

2.1 Maximum entrainment ratio vs stagnation pressure ratio for a constant areaejector (AR=1.51, M1=1.4) (Rao & Jagadeesh, 2015). . . . . . . . . . . . . . 7

2.2 The thrust augmenting ejector design (Whitley, Krothapalli, & VanDomme-len, 1996). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.1 The control volume around the three stages of the Atlantis Intake System.Image is not to scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2 A comparison of exiting mixture stagnation pressure predicted with the methodoutlined and CFD simulations using the software OpenFOAM for differentfuel jet Mach numbers, all values are normalized by the ambient air pressure(Wilson, Johansen, & Mravcak, 2015). . . . . . . . . . . . . . . . . . . . . . 31

4.3 A comparison of exiting mixture stagnation temperature predicted with themethod outlined and CFD simulations using the software OpenFOAM fordifferent fuel jet Mach numbers, all values are normalized by the ambient airtemperature (Wilson et al., 2015). . . . . . . . . . . . . . . . . . . . . . . . . 32

5.1 Flowchart of the developed method for predicting AIS entrainment. . . . . . 37

6.1 A schematic of the computational domain used for the AIS simulations. Initialboundary conditions and points of measurement are shown. Figure is not toscale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6.2 Example of simulation coarse mesh. . . . . . . . . . . . . . . . . . . . . . . . 446.3 Example of simulation coarse mesh near the fuel jet outlet and first AIS stage. 44

7.1 A contour of normalized static pressure, (p − patm)/patm, for an AIS withJTR=2.0, JPR=5.0, and a fuel jet Mach number of 1.0 from one of the CFDsimulations conducted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

7.2 A contour of normalized static temperature for an AIS with JTR=2.0, JPR=5.0,and a fuel jet Mach number of 1.0 from one of the CFD simulations conducted.The yellow frame indicates the core region of the fuel jet for which a syntheticSchlieren image is presented in Figure 7.4. . . . . . . . . . . . . . . . . . . . 46

7.3 A contour of velocity magnitude for an AIS with JTR=2.0, JPR=5.0, and afuel jet Mach number of 1.0 from one of the CFD simulations conducted. . . 47

7.4 A synthetic Schlieren image of the core region of the fuel jet for an AIS withJTR=2.0, JPR=5.0, and a fuel jet Mach number of 1.0 from one of the CFDsimulations conducted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7.5 Area averaged outlet static pressure normalized by ambient static air pressurefor all CFD cases conducted. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

7.6 Mass air-to-fuel ratio vs. JPR for an AIS with a fuel jet Mach number of 1and JTR of 2. Comparison of entrainment model to CFD results. . . . . . . 50

vi

7.7 Mass air-to-fuel ratio vs. JPR for an AIS with a fuel jet Mach number of 2and JTR of 2. Comparison of entrainment model to CFD results. . . . . . . 50

7.8 Mass air-to-fuel ratio vs. JPR for an AIS with a fuel jet Mach number of 1and JTR of 3. Comparison of entrainment model to CFD results. . . . . . . 51

7.9 Mass air-to-fuel ratio vs. JPR for an AIS with a fuel jet Mach number of 1and multiple JTR’s. Comparison of entrainment model to CFD results. . . . 52

7.10 Mass air-to-fuel ratio vs. JPR for an AIS with multiple fuel jet Mach numbersand a JTR of 2. Comparison of entrainment model to CFD results. . . . . . 52

7.11 Contours of the local rate of exergy destruction near the fuel jet exit for anAIS with JTR=2.0, JPR=5.0, and a fuel jet Mach number of 1.0 from one ofthe CFD simulations conducted. . . . . . . . . . . . . . . . . . . . . . . . . . 57

7.12 Specific impulse as a function of JPR and JTR for an AIS with a fuel jet Machnumber of 2 at zero forward velocity. . . . . . . . . . . . . . . . . . . . . . . 58

A.1 A contour of the mass fraction of air for an AIS with area ratio of 300,JTR=2.0, JPR=5.0, and a methane fuel jet Mach number of 1.0 from oneof the CFD simulations conducted. . . . . . . . . . . . . . . . . . . . . . . . 67

A.2 A contour of the mass fraction of methane for an AIS with area ratio of 300,JTR=2.0, JPR=5.0, and a methane fuel jet Mach number of 1.0 from one ofthe CFD simulations conducted. . . . . . . . . . . . . . . . . . . . . . . . . . 68

A.3 A contour of static pressure for an AIS with area ratio of 300, JTR=2.0,JPR=5.0, and a methane fuel jet Mach number of 1.0 from one of the CFDsimulations conducted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

A.4 A contour of static temperature for an AIS with area ratio of 300, JTR=2.0,JPR=5.0, and a methane fuel jet Mach number of 1.0 from one of the CFDsimulations conducted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

A.5 A contour of velocity magnitude for an AIS with area ratio of 300, JTR=2.0,JPR=5.0, and a methane fuel jet Mach number of 1.0 from one of the CFDsimulations conducted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

C.1 A contour of the local rate of exergy destruction due to thermal effects for anAIS with an area ratio of 300, JTR=2.0, JPR=5.0, and a methane fuel jetMach number of 1.0 from one of the CFD simulations conducted. . . . . . . 85

C.2 A contour of the local rate of exergy destruction due to viscous effects for anAIS with an area ratio of 300, JTR=2.0, JPR=5.0, and a methane fuel jetMach number of 1.0 from one of the CFD simulations conducted. . . . . . . 86

C.3 A contour of the local rate of exergy destruction due to species diffusion foran AIS with an area ratio of 300, JTR=2.0, JPR=5.0, and a methane fuel jetMach number of 1.0 from one of the CFD simulations conducted. . . . . . . 87

C.4 A contour of the total local rate of exergy destruction for an AIS with an arearatio of 300, JTR=2.0, JPR=5.0, and a methane fuel jet Mach number of 1.0from one of the CFD simulations conducted. . . . . . . . . . . . . . . . . . . 88

D.1 Specific impulse as a function of JPR and JTR for an AIS with a fuel jet Machnumber of 1 and area ratio of 300 at zero forward velocity. . . . . . . . . . . 89

vii

D.2 Specific impulse as a function of JPR and JTR for an AIS with a fuel jet Machnumber of 1.25 and area ratio of 300 at zero forward velocity. . . . . . . . . . 90

D.3 Specific impulse as a function of JPR and JTR for an AIS with a fuel jet Machnumber of 1.5 and area ratio of 300 at zero forward velocity. . . . . . . . . . 91

D.4 Specific impulse as a function of JPR and JTR for an AIS with a fuel jet Machnumber of 1.75 and area ratio of 300 at zero forward velocity. . . . . . . . . . 92

viii

List of Symbols, Abbreviations and Nomenclature

Symbol Definition

U of C University of Calgary

A Area, m2

AR AIS area ratio (Outlet/Inlet)

CP Specific heat capacity, J/g K

e Specific internal energy, J/K

E Energy flow rate, J/s

ER Equivalence ratio

FT Thrust, N

Isp Specfic impulse, s

J Diffusion flux, mol/m2 s

JPR Jet pressure Ratio

JTR Jet temperature Ratio

M Molar mass, mol/g

Ma Mach number

m Mass flow rate, kg/s

P Momentum flow rate, N

p Pressure, Pa

q Energy flux, W

Q Heat source, W

R Gas constant, J/g K

Ru Univesal gas constant, J/mol K

s Specific entropy, J/K

t Time, s

ix

T Temperature, K

Tmax Maximum temperature reached in combustion chamber, K

U, u Velocity, m/s

x Position, m

Y Species mass fraction

γ Ratio of specific heats

µ Dynamic viscosity, Pa s

ν Specific weight, m3/kg

ρ Density, kg/m3

σij Cauchy stress tensor

τij Viscous stress tensor

ψ Air-to-Fuel mass flow ratio

ψstoichiometric Stoichiometric air-to-fuel mass flow ratio

ω Molar rate of production/destruction of species, mol/s

Subscript

a Air property

CV Control Volume

f Fuel property

i Directional index

m Mixture property

0 Stagnation property

α Species index

x

Chapter 1

Background and Theory

The Atlantis Intake System (AIS) is a novel intake design intended to supply a combustible

fuel/air mixture to a ramjet engine without the use of any moving parts. The operation

of the AIS is similar to an ejector pump in that it uses a primary stream (gaseous fuel) to

entrain a secondary stream (ambient air). Entrainment in this context represents the mech-

anism by which the irrotational secondary stream acquires vorticity from, and becomes part

of, the turbulent primary stream (Carlos, Rodrigo, & Guillem, 2014). Entrainment is the

result outward spreading small scale vortices along the entire interface between the primary

and secondary stream (Westerweel, Fukushima, Pedersen, & Hunt, 2005).

An ejector traditionally consists of a primary stream tube encompassed by a secondary

stream tube. At the point where the primary tube ends, the primary stream introduces a

source of momentum to the surrounding stream, usually resulting in a mixture with a final

velocity somewhere between that of the primary and secondary streams. The AIS operates

via the continuous release of a fuel jet into a system of inlet stages open to the surrounding

environment. The fuel, traditionally propane due to its high vapor pressure at room tem-

perature, is preheated through a heat exchanger in the combustion chamber. Interactions

between the fuel jet, the intake geometry and the surrounding air result in a high velocity

(relative to the air) stream entering the intake of a ramjet engine coupled with the intake. A

simplified schematic of the intended engine design in use with the AIS is provided in Fig. 1.1.

The simplified engine consists of a fuel tank, a heating coil used to preheat the fuel, the AIS,

and an ideal engine consisting of a diffuser, combustion chamber and exit nozzle. Atlantis

Research Labs based the design of its AIS on the Gluhareff pressure jet engine, which was

1

designed and built by Eugene M. Gluhareff in the 1950s for tip propulsion on rotary wing

aircraft (Robert Q. Riley Enterprises, 1998). In the original Gluhareff design, the propane

fuel entered the heating coil as a liquid, where it was vaporized prior to injection by the heat

of combustion (Barrett & Gluhareff, 2008).

The depiction of the AIS in Fig. 1.1 provides a simplified example of a cross section of

the intake system. The arrow on the left represents the fuel jet, injected through a series

of 3 concentric, axi-symmetric ducts or intake stages. The last stage is depicted as being

coupled to the engine diffuser. The preliminary design used as the basis for the current study

measures approximately 0.54 m from the exit of the fuel jet, to the exit of the 3rd intake

stage. The fuel jet has a diameter of 0.003175 m, and the area ratio from the outlet of the

last stage to the fuel jet is 300.

The ramjet is one of the simplest jet engine designs as it has no moving parts and can

be efficient when operating at supersonic flight speeds. However, its inability to operate

standing still or at low subsonic speeds makes its applicability limited. Often a more tradi-

tional engine, such as a turbojet, is required to accelerate the vehicle until a speed is reached

where the inlet can provide the aerodynamic compression necessary to produce sufficient

thrust to maintain speed (Heiser H. H. & Pratt, 1994; Fry, 2004). As a result, an aircraft

must carry multiple engines, which do not operate simultaneously, increasing cost, weight,

and complexity. The AIS provides a means for introducing the airflow necessary to operate

the ramjet statically without the introduction of any moving parts. While the current study

focuses on the operation of the AIS on a vehicle with zero forward velocity, is it assumed that

at supersonic vehicle speeds the AIS could be retracted so as not to cause any interference

with the coupled ramjets intake.

2

While some experimental studies of the AIS and Gluhareff systems have been conducted

analyzing performance in terms of total thrust (Barrett & Gluhareff, 2008; Brilliant, Fortson,

Hess, & Torosian, 1980), the theoretical gains that may be realized by this system are still

largely unstudied. There are many input variables in terms of fuel jet properties and AIS

geometry that will impact the performance of and AIS coupled engine. Understanding the

impact of each of these parameters on performance is of great interest for design purposes.

However, experimental data for the AIS is very limited, and conducting CFD simulations for

all of the potential combinations of input variables is time consuming and computationally

expensive. Additionally, very little is known about the efficiency of the AIS. While the use of

a fuel jet to entrain air for combustion has been demonstrated by the Gluhareff jet (Barrett

& Gluhareff, 2008), the losses associated with this method are not thoroughly studied.

CombustionChamber

Heating Coil

Atlantis Intake System

Fuel Tank

Diffuser

Fuel Line

Engine

Control Volume

Exit Nozzle

Figure 1.1: Simplified schematic of an ideal engine utilizing the AIS. Not to Scale.

The research objectives for this work are to develop a method of rapidly predicting

the overall engine performance of a given AIS configuration, and to identify the significant

sources of losses within the AIS to better understand where improvements can be made.

3

Previous work by the current author demonstrated the feasibility of the AIS as a means of

introducing air-flow into an engine, and the potential performance increases that may be

realized (Wilson et al., 2015). A control volume analysis that allowed the prediction of the

outlet mixture properties based on the entrainment rate was also presented. The current work

outlines a method for rapidly predicting the entrainment rate of a given AIS configuration

based on the properties of the fuel injected into the AIS (pf , Tf , and Uf ) and those of the

ambient surroundings. This will allow the prediction of the mixture properties exiting the

AIS and entering the ramjet diffuser. The results of this method will be compared to some

simplified CFD simulations as a means of validation. If these properties entering the diffuser

can be determined, an ideal jet-propulsion cycle can be used to predict the overall engine

performance. This will provide a computationally inexpensive means of rapidly assessing the

performance of multiple AIS configurations which may be utilized as a tool for preliminary

design. The development of a post-processing utility to allow the visualization of the different

sources of exergy destruction based on the results of the CFD simulations is also presented.

This visualization will allow for identification of the dominant sources of losses in the AIS.

The relationship between the different sources of exergy destruction, and AIS performance

will be discussed.

4

Chapter 2

Jet Mixing and Entrainment

The most novel aspect of the AIS is that it utilizes the energy of the fuel gas to entrain

the air used for combustion at low speeds where the natural airflow isn’t sufficient for ram

compression. However the amount of entrainment expected during operation of the AIS and

the properties that have the greatest impact on performance are not explicitly known. A

significant part of this study was dedicated to developing a simplified analytical/numerical

method to predict the amount of air entrained given an AIS geometry, fuel jet properties,

ambient air conditions and vehicle speed. As experiments and CFD simulations can be

costly and time consuming, a simple method that will allow the analysis of a large range of

all of these input properties could allow for parametric analysis of the system to identify an

optimal design. While the AIS concept is novel, the general entrainment mechanism that

uses a high velocity primary stream of fluid to entrain a secondary stream has been studied

in the form of a traditional ejector pump. In this chapter, a literature review of ejector

performance models is presented. Special attention is given to models that can accurately

predict entrainment at a low computational cost. This information was utilized to develop

the entrainment model explored in Chapter 4.

The AIS is unique in that it is open to the atmosphere, though it shares many similarities

with the traditional ejector pump. Control volume analyses of constant area ejector pumps

or other mixing stream scenarios are available in the literature (Zhu, Cai, Wen, & Li, 2008;

Huang, Chang, Wang, & Petrenko, 1999; Liao, 2008; John & Keith, 2006). However, the

operating conditions and primary goals of those studies are typically different than those

of the AIS. Bernstein (1953) presented a one-dimensional analysis on parallel jet mixing

5

encountered in the testing of engines in a supersonic wind tunnel. A model was defined for

the mixing of two streams of the same gas that provided a means of predicting the down

stream Mach number of the uniform mixture. Relationships were also developed for the

total downstream pressure, though this was limited to cases of equal inlet Mach number

or equal inlet static temperature of the two streams. Huang et al. (1999) presented a 1-

D analysis using empirical coefficients to accurately predict ejector performance limited to

critical mode operation, where the downstream Mach number is equal to one. John and

Keith (2006) presented a simple control volume analysis for a jet ejector, though it requires

that the initial pressure, temperature, and velocity of the secondary stream are known. This

is not the case for the AIS. Liao (2008) produced a 1-D analytical model for the design of

gas ejectors, though it assumes that the Mach number of the secondary stream entering the

mixing region is known. This model is not suitable for the AIS as the velocity of the ambient

air as it is drawn into the mixing region is unknown.

Rao and Jagadeesh (2015) provided a thorough experimental, analytical and numerical

study on the performance of a supersonic ejector using different gases for the secondary

stream. Again, the assumption of operation in the critical mode, where the secondary flow

is choked, was applied allowing an analytical solution to be determined. However some valu-

able insight can be drawn from their results. The study focused on designing an optimized

purge ejector for the dilution of exhausted hydrogen from a fuel cell. It was assumed that

the stagnation properties of the secondary stream were known. Based on the desired purge

gas and dilution ratio an optimized area ratio (AR) and jet Mach number (M1) could be

determined analytically. It was found that the ratio of molecular weights between the gases

significantly impacts the resulting entrainment rate for similar inlet conditions. Based on

their analytical solution it was determined that mixtures with a higher primary to secondary

weight ratio produced higher entrainment rates, especially at low stagnation pressure ratios

as seen in Figure 2.1. Subscripts p and s denote the primary and secondary streams, respec-

6

tively. It was also found that the effective compression of the secondary stream was nearly

independent of the molecular weight ratio and almost entirely dependent on the stagnation

pressure ratio of the streams.

Stagnation Pressure Ratio (Pop/Pos)

Entr

ainm

ent

Rat

io(m

p/m

s)

Figure 2.1: Maximum entrainment ratio vs stagnation pressure ratio for a constant areaejector (AR=1.51, M1=1.4) (Rao & Jagadeesh, 2015).

There is a subsection of ejectors known as thrust augmenting ejectors; these were designed

and intended for use to entrain air at the exit of a rocket engine to increase the total thrust

produced by the engine. The mechanisms behind thrust augmenting ejectors are similar

to those experienced by the AIS. Ambient surrounding air is entrained by directing a high

pressure, high temperature stream of gas through a duct as seen in Fig. 2.2. Whitley et al.

(1996) use a control volume analysis to determine the performance of a thrust augmenting

ejector given a set of primary jet conditions and surrounding stagnation properties. Two

tangential discontinuities, similar to slip lines, are assumed in their case. One discontinuity

occurs at the exit of the primary jet, and one at the exit of the mixed flow. At each of

7

these locations two parallel streams meet and the discontinuity is present between the two

streams. It is stated that the pressure is required to be continuous across this discontinuity,

but other flow properties (e.g. velocity, temperature, density) are not necessarily so. This

assumption is only valid if both flows involved are subsonic, or ideally expanded supersonic.

This assumption provides a sufficient number of equations for a solution to be determined

given the variables provided. The assumption of a tangential discontinuity of the system

provides a means of closing the problem for the AIS control volume analysis as seen in

Chapter 4.

Figure 2.2: The thrust augmenting ejector design (Whitley et al., 1996).

Emanuel (1982) compared one-dimensional solutions for ejector performance to the in-

viscid theory of Fabri and Paulon (1958). It was found that the inviscid approach of Fabri

and Paulon is limited to certain ejector configurations. These limitations include the as-

sumptions of constant area mixing, and subsonic secondary inlet flow. It was determined

that secondary flow rates with a slightly supersonic velocity performed well, though other

restrictions imposed by this are discussed. High supersonic values resulted in poor ejector

performance, which is undesirable for a thrust augmenting ejector. Poor performance at

high-speeds is not detrimental to the AIS as its primary purpose is to support and improve

subsonic performance. At high speeds, the AIS deactivates and the engine operates as a

pure ramjet. The one dimensional constant area method discussed assumes the mass flow

rate of the secondary stream is known allowing for a solution to be obtained explicitly.

8

Kremar, Peddieson, and Han (2007) present a comparison study of three similar control-

volume models for subsonic/supersonic ejector systems. They found results of the three

methods to be qualitatively similar. The method discussed above by Emanuel (1982) is found

to provide the best combination of speed and accuracy. As previously mentioned this method

employs the assumption that the secondary mass flow rate is known. Similarly the method

outlined in Heiser H. H. and Pratt (1994) assumes the secondary stream Mach number is

known and the mixed outlet flow is sonic. As none of these assumptions are necessarily

true for the AIS application, the assumption outlined in Whitley et al. (1996), that the

outlet flow must have the same static pressure as long as both streams are subsonic, due to

tangential continuity is further explored. Most other analyses express solutions in terms of

secondary stream Mach number. In the current work, predicting the outlet conditions are

a required model output. So other assumptions need to be utilized to produce a solution

without knowing the secondary stream Mach number ahead of time.

9

Chapter 3

Exergy Analysis of Aerospace Systems

In most any thermodynamic system it is of interest to identify sources of losses in order

to optimize its design. An exergetic analysis of a system provides both a means of iden-

tifying the specific interactions and locations contributing to losses, as well as a means of

quantifying the overall efficiency of the system for comparison. Exergy-based analysis of

aerospace systems is a common method for performance analysis as exergy is considered a

useful tool in the assessment of machines that operate on the principals of thermodynam-

ics (Sohret, Ekici, Altunta, Hepbasli, & Karakoc, 2016; X. Zhao, Thulin, & Gronstedt, 2016).

The modern definition of exergy is the maximum shaft work that can be done by the

composite of the system and a specified reference environment that is assumed to be infinite,

in equilibrium, and ultimately to enclose all other systems (Sohret et al., 2016; Lucia, 2013).

Any irreversibilities in the operation of a system therefore decrease this maximum amount

of work that can be done, which is known as exergy destruction. The exergy form of the

Gouy-Stodola theorem shows that exergy destruction or lost work in a system is directly

proportional to the entropy generation (Sgen) due to irreversibilities within the system, where

the proportionality constant is equal to the absolute temperature (T0) of the environment

(Cengel & Boles, 2002).

Xdestroyed = T0Sgen ≥ 0 (3.1)

In order to quantify the sources of exergy destruction or lost work potential in a spa-

10

tial domain, the local rates of entropy generation must be determined. A review of some

examples of modern applications of entropy generation or exergy destruction analysis on

aerospace or related systems is provided in Section 3.1. A derivation of the terms frequently

used to quantify local entropy production can be found in Section 3.2.

3.1 Review of Modern Applications of Exergy-Based Analysis

Stanciu, Isvoranu, Marinescu, and Gogus (2001) investigated the volumetric generation of

entropy in both laminar and turbulent diffusion flames. The local magnitude of volumetric

irreversibilities due to viscous, thermal, diffusion, and chemical effects were quantified and

compared to determine the dominant factors. Independent terms for both the mean and

fluctuating exergy dissipation terms were developed to analyze the effects of turbulence on

the rate of entropy generation.

Yapici, Kayatas, Albayrak, and Basturk (2005) provided a numerical calculation of local

entropy generation in a methane-air burner based on the combustion results of a series of

CFD simulations. Part of their study investigated the impacts of varying simulation param-

eters on the rates of entropy generation. It is found that the irreversibilities due to heat

transfer are dominant in all cases. Spatial location of maximum entropy generation are dis-

cussed. In cases of high volumetric fuel flow rates, it is found that entropy generation rates

decrease exponentially with increasing equivalence ratios.

A review of the use of exergy as a performance assessment tool for aircraft gas turbine

engines was conducted by Sohret et al. (2016). A comparison of exergy consumption from

previous studies for the different components of a gas turbine engine wss presented. The

magnitude of irreversibilities for each component is quantified, and potential economic and

11

environmental costs are discussed based on these results. The value of exergy-based analysis

as a tool for economic, environmental, and sustainability assessments, in addition to perfor-

mance evaluation is emphasized.

An exergy-based analysis was conducted on the CFD simulation results of a supersonic

steam ejector by Boulenouar and Ouadha (2015). This study is of particular note as the pre-

vious chapter emphasized the similarities of the entrainment mechanisms used by ejectors,

and the AIS. In this study, exergy calculations are performed using average return steam

pressure and temperature from the CFD analysis. The relative magnitude of exergy losses

for each section of the ejector are compared. The use of exergy efficiency as a measure of

performance for ejectors, rather than entrainment ratio, is advocated.

3.2 Local Volumetric Entropy Generation

The volumetric entropy generation terms frequently utilized in exergy-based analysis are

fairly well agreed upon (Safari, Sheikhi, Janbozorgi, & Metghalchi, 2010; Stanciu et al., 2001;

Briones, Mukhopadhyay, & Aggarwal, 2009; L. Zhao & Liu, 2010; Sierra-Pallares, Garcıa

Del Valle, Garcıa Carrascal, & Castro Ruiz, 2016; Yapici, Basturk, Kayatas, & Albayrak,

2006; Gazzah & Belmabrouk, 2014). However, it is difficult to find a thorough derivation

of these terms in the literature. Therefore, a derivation of the entropy generation terms are

provided here. A good starting point is the general entropy transport equation Safari et al.

(2010); Hirschfelder, Curtis, and Bird (1954):

TρDs

Dt= ρ

De

Dt+ pρ

Dν

Dt−

Ns∑α=1

µαρDYαDt

(3.2)

T , ρ, s, e, p, ν, µα, Yα, are the temperature, density, specific entropy, specific internal energy,

12

pressure, specific volume, species chemical potential, and species mass fraction, respectively.

Expanding the material derivative of the term on the left hand side of Eq. 3.2 yields:

ρDs

Dt= ρ

[∂s

∂t+ ui

∂s

∂xi

](3.3)

=∂ρs

∂t+∂ρuis

∂xi(3.4)

In Eq. 3.4, ui is the vector component of velocity. This is valid if the region is considered

continuous, as expanding the right hand side of Eq. 3.4 is using product rule, and collecting

like terms results in the following:

∂ρs

∂t+∂ρuis

∂xi= ρ

∂s

∂t+ s

∂ρ

∂t+ ρui

∂s

∂xi+ s

∂ρui∂xi

(3.5)

= ρ

[∂s

∂t+ ui

∂s

∂xi

]+ s

[∂ρ

∂t+∂ρui∂xi

](3.6)

The last term on the right hand side of Eq. 3.6 will be equal to zero due to conservation of

mass:

∂ρ

∂t+∂ρui∂xi

= 0 (3.7)

From the conservation of energy, the first term on the right hand side of Eq. 3.2 becomes

(Kuo, 1986):

ρDe

Dt= − ∂qi

∂xi+ σji

∂ui∂xj

+ Q+ ρ

Ns∑α=1

Yαfα,iVα,i (3.8)

qi is the energy flux term, σji is the Cauchy stress tensor, Q is a heat source term (e.g.

ignition spark) which is assumed to be zero in the current work, and ρNs∑α=1

Yαfk,iVk,i is the

power produced by volume forces fα and diffusion velocity Vα on species α, which will also

be neglected. The Cauchy stress tensor can be further expanded as the difference between

the viscous stress tensor and the static pressure (Mei, 2007):

σji = τij − pδij (3.9)

13

So, the second term on the right hand side of Eq. 3.8 then becomes:

σji∂ui∂xj

= τij∂ui∂xj− p∂ui

∂xi(3.10)

As ρν = 1, the second term on the right hand side of Eq. 3.2 is equal to:

p

[ρDν

Dt

]= p

[∂ρν

∂t+∂ρνui∂xi

](3.11)

= p∂ui∂xi

(3.12)

From the conservation equation of chemical species mass fraction, the last term on the right

hand side of Eq. 3.2 becomes (Poinsot & Veynante, 2005):

Ns∑α=1

µαρDYαDt

=Ns∑α=1

µα

[∂ρYα∂t

+∂ρuiYα∂xi

](3.13)

=Ns∑α=1

µα

[∂

∂xi

(ρDα

∂Yα∂xi

)+ ωα

](3.14)

Dα and ωα are the species diffusivity, and molar rate of production/destruction of species

due to chemical reaction, respectively. The first term on the right hand side of Eq. 3.14, the

scalar flux, can be reduced by Fick’s law of diffusion (Safari et al., 2010):

Jαi = −ρDα∂Yα∂xi

(3.15)

Where Jαi is the diffusion flux. Substituting the results of Eq. 3.4, 3.8, 3.10, 3.11, and 3.13;

the transport of entropy, Eq. 3.2, can now be represented as:

∂ρs

∂t+∂ρuis

∂xi=

1

T

[τij∂ui∂xj− ∂q

∂xi+

Ns∑α=1

µα∂Jαi∂xi−

Ns∑α=1

µαωα

](3.16)

Using the relations provided for the chemical potential, µα, and the energy flux, q, from

Safari et al. (2010):

µα = hα − Ts0α (3.17)

14

qi = −λ ∂T∂xi

+Ns∑α=1

hαJαi (3.18)

The terms hα, s0α, and λ, represent the species specific enthalpy, partial entropy, and thermal

conductivity, respectively. From Eq. 3.17 and 3.18, the second term on the right hand side

of Eq. 3.16 can be expressed as:

∂qi∂xi

=∂

∂xi

[−λ ∂T

∂xi

]+

Ns∑α=1

Jαi∂hα∂xi

+Ns∑α=1

hα∂Jαi∂xi

=∂

∂xi

[−λ ∂T

∂xi

]+

Ns∑α=1

TJαi∂s0α∂xi

+Ns∑α=1

Jαi s0α

∂T

∂xi+

Ns∑α=1

Jαi∂µα∂xi

+Ns∑α=1

hα∂Jαi∂xi

(3.19)

The second two terms on the right hand side of Eq. 3.16 can then be expressed as:

− ∂qi∂xi

+Ns∑α=1

µα∂Jαi∂xi

=∂

∂xi

[λ∂T

∂xi

]−

Ns∑α=1

TJαi∂s0α∂xi−

Ns∑α=1

Jαi s0α

∂T

∂xi...

−Ns∑α=1

Jαi∂µα∂xi−

Ns∑α=1

Ts0α∂Jαi∂xi

=∂

∂xi

[λ∂T

∂xi

]−

Ns∑α=1

Jαi

[∂µα∂xi

+ s0α∂T

∂xi

]− T ∂

∂xi

Ns∑α=1

s0αJαi

(3.20)

From the product rule it is true that:

∂

∂xi

[1

Tλ∂T

∂xi

]=

1

T

∂

∂xi

[λ∂T

∂xi

]+

1

T 2

[λ∂T

∂xi

∂T

∂xi

](3.21)

Therefore, the first term on the right hand side of Eq. 3.20 may be expressed as:

∂

∂xi

[λ∂T

∂xi

]= T

∂

∂xi

[1

Tλ∂T

∂xi

]− 1

T

[λ∂T

∂xi

∂T

∂xi

](3.22)

15

Equation 3.2 may now be expressed as (Safari et al., 2010):

∂ρs

∂t+∂ρuis

∂xi=

1

T

τij ∂ui∂xj︸ ︷︷ ︸I

− λ

T

∂T

∂xi

∂T

∂xi︸ ︷︷ ︸II

−Ns∑α=1

µαωα︸ ︷︷ ︸III

−Ns∑α=1

Jαi

(∂µα∂xi

+ s0α∂T

∂xi

)︸ ︷︷ ︸

IV

− ∂

∂xi

[λ

T

∂T

∂xi+

Ns∑α=1

s0αJαi

]︸ ︷︷ ︸

V

(3.23)

Where the terms I, II, III, IV , and V represent the local rate of entropy generations due

to viscosity, heat transfer, chemical reaction, species diffusion, and the diffusion of entropy.

The term IV may be further reduced based on Eq. 3.17 and the enthalpy of a species

(Safari et al., 2010):

hα = h0α +

∫ T

Tr

CPdT (3.24)

CP is the mixture specific heat capacity, and Tr is the reference temperature. The term IV

in Eq. 3.23 can then be expressed as:

−Ns∑α=1

JαiT

(∂µα∂xi

+ s0α∂T

∂xi

)= −

Ns∑α=1

JαiT

(CP

∂T

∂xi− T ∂s

0α

∂xi

)

= −Ns∑α=1

Jαi

(CPT

∂T

∂xi− ∂s0α∂xi

) (3.25)

The partial entropy can be expressed in terms of the entropy of a pure substance as (Safari

et al., 2010):

s0α = sα −Rα lnXα (3.26)

Rα and Xα are the specific gas constant and mole fraction of the species, respectively.

16

Assuming an ideal gas, gradients of entropy can be computed from:

∂s0α∂xi

=CPT

∂T

∂xi− Rα

Yα

∂Yα∂xi

+Rα

n

∂n

∂xi− Rα

p

∂p

∂xi(3.27)

The variable n is the total number of moles. Eq. 3.25 can then be expressed as:

−Ns∑α=1

JαiT

(∂µα∂xi

+ s0α∂T

∂xi

)= −

Ns∑α=1

RαJαi

(1

Yα

∂Yα∂xi− 1

n

∂n

∂xi+

1

p

∂p

∂xi

)(3.28)

Similarly, the term V in Eq. 3.23 may be simplified based on Fick’s law of diffusion (Eq.

3.15), resulting in:

− ∂

∂xi

[λ

T

∂T

∂xi+

Ns∑α=1

s0αJαi

]=

∂

∂xi

[−λT

∂T

∂xi+

Ns∑α=1

s0αρDα∂Yα∂xi

](3.29)

As shown in Safari et al. (2010), if equal mass diffusivity for all species (Dα = D,α =

1, ..., Ns) and unity Lewis number (Le = λ/(DCP ) = 1) are assumed, the term V further

reduces to the diffusion of entropy:

− ∂

∂xi

[λ

T

∂T

∂xi+

Ns∑α=1

s0αJαi

]=

∂

∂xi

(ρDα

∂s

∂xi

)(3.30)

Substituting Eq. 3.28 and 3.30 into Eq. 3.23, the transport of entropy becomes:

∂ρs

∂t+∂ρuis

∂xi=

∂

∂xi

(ρDα

∂s

∂xi

)+

1

Tτij∂ui∂xj− 1

T

Ns∑α=1

µαωα +λ

T 2

∂T

∂xi

∂T

∂xi

+Ns∑α=1

ρDαRα

(1

Yα

∂Yα∂xi

∂Yα∂xi− 1

n

∂Yα∂xi

∂n

∂xi+

1

p

∂Yα∂xi

∂p

∂xi

)(3.31)

The two terms on the left hand side and the first term on the right represent the transport

17

of entropy, and the rest of the terms represent the volumetric rate of entropy generation:

S ′′′gen =1

Tτij∂ui∂xj− 1

T

Ns∑α=1

µαωα +λ

T 2

∂T

∂xi

∂T

∂xi

+Ns∑α=1

ρDαRα

(1

Yα

∂Yα∂xi

∂Yα∂xi− 1

n

∂Yα∂xi

∂n

∂xi+

1

p

∂Yα∂xi

∂p

∂xi

)(3.32)

Equation 3.32 is the equivalent of what is presented as Eq. 24 in Stanciu et al. (2001), and

Eq. 21 in Safari et al. (2010). The four generation terms represent entropy generation due

to viscosity, chemical reaction, heat transfer, and diffusion of species respectively:

S ′′′gen,V =1

Tτij∂ui∂xj

(3.33)

S ′′′gen,C = − 1

T

Ns∑α=1

µαωα (3.34)

S ′′′gen,Q =λ

T 2

∂T

∂xi

∂T

∂xi(3.35)

S ′′′gen,D =Ns∑α=1

ρDαRα

(1

Yα

∂Yα∂xi

∂Yα∂xi− 1

n

∂Yα∂xi

∂n

∂xi+

1

p

∂Yα∂xi

∂p

∂xi

)(3.36)

In the case of the AIS, it is assumed that no chemical reactions are taking place within the

intake, so S ′′′gen,C = 0. It should also be noted that these terms represent volumetric entropy

generation. In order to calculate the total rate of entropy generation within the system, the

sum of the generation terms must be integrated across the entire domain.

Sgen =

∫∫∫V

(S ′′′gen,V + S ′′′gen,Q + S ′′′gen,D

)dxdydz (3.37)

18

3.2.1 Turbulent Entropy Generation

Stanciu et al. (2001) provides additional generations terms for the application of the fluc-

tuating properties of the Reynolds Average Navier Stokes (RANS) equations. As chemical

reactions are being neglected for the current study, the turbulent generations terms are not

presented. Using the Reynolds average procedure, the instantaneous entropy generation

rates of Eq. 3.37 would now be expressed as:

Sgen =

∫∫∫V

(˙S′′′gen,V +

˙S′′′gen,Q +

˙S′′′gen,D

)dxdydz (3.38)

The terms for the averaged volumetric entropy generation can be determined by separating

the properties into their mean and fluctuating parts, and terminating their serial decompo-

sition after the first term (Stanciu et al., 2001). Resulting in the following:

˙S′′′gen,V ≈

˙S′′′gen,V M +

˙S′′′gen,V T

≈ 1

Tτij∂ui∂xj

+ρεK

T

(3.39)

εK is the dissipation of the turbulent kinetic energy.

˙S′′′gen,Q ≈

˙S′′′gen,QM +

˙S′′′gen,QT

≈ λ

T 2

∂T

∂xi

∂T

∂xi+ρCP

T 2εθ

(3.40)

εθ is the is the dissipation of fluctuating temperature variance.

˙S′′′gen,D ≈

˙S′′′gen,DM +

˙S′′′gen,DT

≈Ns∑α=1

ρDαRα

(1

Yα

∂Yα∂xi

∂Yα∂xi− 1

n

∂Yα∂xi

∂n

∂xi+

1

p

∂Yα∂xi

∂p

∂xi

)+

Ns∑α=1

Rα

Yαρεψ

(3.41)

εψ is the dissipation rate of fluctuating α-component mass fraction variance.

19

As the terms εθ and εψ in Eq. 3.40 and 3.41 are not typically represented in a RANS

simulation, an alternate formulation based on the equilibrium turbulence feature is presented

by Stanciu et al. (2001). This imposes the condition that the production and dissipation

terms are equal in both the temperature variance, and α-component mass fraction variance,

resulting in (Stanciu et al., 2001):

˙S′′′gen,QT ≈

ρCP

T 2εθ

≈ λT

T 2

∂T

∂xi

∂T

∂xi

(3.42)

˙S′′′gen,DT ≈

Ns∑α=1

Rα

Yαρεψ

≈ ρDT

Ns∑α=1

Rα1

Yα

∂Yα∂xi

∂Yα∂xi

(3.43)

Where λT and DT represent the turbulent contributions of thermal conductivity and molec-

ular diffusivity respectively.

3.3 Exergy Destruction

As the rate of exergy destruction is the quantity that is of interest to be extracted from the

CFD simulations conducted, equations for the different components of exergy destruction

must be established. As shown in Eq. 3.1, the local rate of exergy destruction is directly

proportional to the local rate of entropy generation. The equations for the local rate of

entropy generation are laid out in Section 3.2, applying the relation identified in Eq. 3.1

results in the following:

˙X′′′destroyed,V =

T0

T

[τij∂ui∂xj

+ ρεK

](3.44)

20

˙X′′′destroyed,Q =

T0

T 2

[(λ+ λT )

∂T

∂xi

∂T

∂xi

](3.45)

˙X′′′destroyed,D = T0ρ(D +DT )

Ns∑α=1

Rα

(1

Yα

∂Yα∂xi

∂Yα∂xi− 1

n

∂Yα∂xi

∂n

∂xi+

1

p

∂Yα∂xi

∂p

∂xi

)(3.46)

Equations 3.44, 3.45, and 3.46 are the equations utilized to visualize the exergy destruction

based on the results in the CFD simulations presented in Chapter 7, Section 7.3. The source

code for the post-processing utility is presented in Appendix B.

21

Chapter 4

AIS Performance Model

One of the primary goals of this research project was to determine a means of quickly and

accurately predicting the performance of the AIS based on a few key variables. These key

variables are the conditions of the fuel jet being injected into the AIS, the geometry of the

AIS, and the conditions of the air being entrained by the AIS. All of these factors will impact

the amount of air drawn into the AIS, and the conditions of the mixture entering the ramjet

engine coupled to the intake. As there are fairly well established means for predicting the

performance of a ramjet engine based in its inlet conditions, it is possible to predict the

overall performance indicators for the engine, such as specific impulse.

4.1 Static Operation Control Volume Analysis

Primarily, this work is focused on the static performance of the AIS. At zero velocity, there

will be no air induced into the engine by the movement of the vehicle. In order to supply

air to burn the fuel injected, all airflow must be supplied by entrainment through the use of

the AIS. Assuming that the properties of the fuel jet entering the AIS and the surrounding

air, as well as the relative physical dimensions of the inlet and outlet of the AIS, are known,

the goal is to predict the properties of the fuel-air mixture exiting the control volume into

the diffuser of the coupled ramjet.

A control volume is drawn encompassing the inlet and outlet of the AIS (see Fig. 4.1).

It is assumed that only fuel and air enter the left side of the domain, and a mixture of the

two is all that exits the domain through the outlet of the AIS. All inlet and outlet streams

are at uniform properties along the boundary of the control volume, and only normal com-

22

Ain Aout

CV

pf , Tf , Uf pm, Tm, Um

pa, Ta, Ua = 0

X

Figure 4.1: The control volume around the three stages of the Atlantis Intake System. Imageis not to scale.

ponents of velocity to the boundaries occur. Therefore it is assumed we have two uniform

inlet streams of fuel and air, and one uniform outlet stream, a mixture of fuel and air. It

is assumed that all gases and mixtures behave ideally. Knowing that the flux of mass and

energy entering the domain must be equal to those of the outlet stream leaving the domain,

and that the flux in linear momentum in the X direction is equal to the sum of external

forces, we have four equations, including the ideal gas law, which may be solved to determine

our three downstream properties pm, Tm, and Um.

Applying a conservation of mass to the CV produces:

m = min (4.1)

=pf

RfTfAinUf +

paRaTa

(Aout − Ain)Ua (4.2)

= mout (4.3)

=pm

RmTmAoutUm (4.4)

The velocity of the air entering the domain on the right hand side of Eq. 4.2, and therefore

the mass flow rate of air entering the domain is unknown. Due to this we implement the

23

relation ψ = ma/mf , resulting in the following:

m =pf

RfTfAinUf (1 + ψ) (4.5)

Expressing the mass flow rate in terms of outlet properties, Eq. (4.4) can be rearranged for

Um as:

Um =mRmTmAoutpm

(4.6)

Imposing the condition that the change in X-Momentum is equal to the sum of external

forces results in:

∑FX = FX,in − FX,out (4.7)

= pfAin + pa(Aout − Ain)− pmAout (4.8)

= mmUm − (mfUf + maUa) (4.9)

Combining Eq. 4.8 and 4.9, and collecting in terms of inlet and outlet properties results in

the following:

P = Pin (4.10)

= mfUf + maUa + pfAin + pa(Aout − Ain) (4.11)

= Pout (4.12)

= mmUm + pmAout (4.13)

The variable P is used to express the sum of the time rate of change of linear momentum

and external forces, at either the inlet or outlet. Frictional losses are being neglected in this

24

model for simplification, and are therefore not included in Eq. 4.7. It is assumed that were

frictional losses included in this analysis that an additional positive term for drag would be

added to Eq. 4.9. This would result in a lower value of mmUm, and less entrainment. It is

assumed that the pressure forces due to the air on either side of the CV beyond the outlet

area of the AIS are equal and opposite, resulting in the simplified term pa(Aout − Ain). It

is assumed that air entrained by the AIS enters the control volume upstream and radially

from the fuel jet. And, that as the control volume is extended in either direction, the area

through which air is entering the control volume will increase. As the area through which

air is entering the control volume increases, for a finite mass flow rate and fixed density,

the velocity of the air entering the control volume will decrease. For this analysis, it is

assumed that the control volume is sufficiently large that the magnitude of the term maUa

is significantly less than the term mfUf . Or that the fuel jet is the only significant source of

momentum entering the control volume, and the momentum contribution of the air entering

the domain may be neglected. Based on this assumptions, Eq. 4.11 can then be expressed

entirely in terms of known inlet properties.

Expressing pm in terms of mixture properties from Eq. (4.13) produces:

pm =P − mUmAout

(4.14)

Applying a conservation of energy to the control volume yields the following:

25

E = Ein (4.15)

= mfCPfT0f + maCPaT0a (4.16)

= Eout (4.17)

= mCPm

(Tm +

U2m

2CPm

)(4.18)

Solving Eq. (4.15) for Tm:

Tm =E

mCPm− U2

m

2CPm(4.19)

In the case of static operation, as the velocity of the air entering the control volume is as-

sumed to be negligible, the static and stagnation temperature can be assumed to be equal

(T0a = Ta).

Substituting Eq. 4.14 and 4.19 into Eq. 4.6 and collecting the like terms of Um results in

the following, which can be seen is in the form of a quadratic equation.

[m

(1− Rm

2CPm

)]U2m +

[−P]Um +

[RmE

CPm

]= 0 (4.20)

Eq. (4.20) can then easily be solved by putting it in the form of the quadratic formula:

Um =

P ±√P 2 − 4

[m(

1− Rm2CPm

)] [RmECPm

]2[m(

1− Rm2CPm

)] (4.21)

As the properties of the downstream mixture are unknown, the variables m, P , and E

26

in Eq. 4.21 should be expressed in terms of input parameters as seen in Eq. 4.5, 4.11,

and 4.16. Applying the relations AR = Ain/Aout, JPR = pf/pa, Ma = U/√γRT , and

R/CP = (γ − 1)/γ results in the following:

Um =

[1 + AR−1

JPR+ γfMa2f

]±√[

1 + AR−1JPR

+ γfMa2f]2 − 2

[γfMa2fRfTf

(1 + ψ)(γ2m−1γ2m

)][CPfT0f + CPaTa][√

γfRfTf

Maf (1 + ψ)(γm+1γm

)](4.22)

The gas properties of the mixture can be calculated based on the gas properties of the air

and fuel, and the value of ψ. The mass averaged specific heat capacity of the mixture is

(Cengel & Boles, 2002):

CPm =CPf + ψCPa

1 + ψ(4.23)

The molar mass of the mixture is:

Mm =1 + ψ1Mf

+ ψMa

(4.24)

The specific gas constant of the mixture is:

Rm =Ru

Mm

(4.25)

The ratio of specific heats of the mixture is:

γm =CPm

CPm −Rm

(4.26)

From Eq. 4.22 it can be seen that Um = f(AR, JPR, γf , γm, Rf , CPf , CPa,Maf , Tf , T0f , Ta, ψ),

though the term T0f may be expressed in terms of γf and Maf . All other variables are a

27

function of either the physical design of the AIS or its input operation, with the exception

of the mixture gas properties and the air-to-fuel mass flow ratio. The mixture gas properties

are a function of known inlet properties based on the air-to-fuel mass flow ratio as seen in

Eq. 4.23 to 4.26. Thus, the air-to-fuel mass flow ratio remains the only unknown variable.

If some value of ψ is assumed, Um can be calculated for a given array of input parameters

using Eq. 4.21 in conjunction with Eq. 4.23, 4.24, 4.25, and 4.26. It should be noted that as

Eq. 4.22 is in the form of a quadratic equation, there are multiple valid mathematical solu-

tions. Subsection 4.1.1 defines a means for determining which solution is physically valid, or

most likely. Once a solution to Um is determined, pm and Tm can be determined based on Eq.

4.14 and 4.19, where P , m, and E can be determined based on the given inlet parameters.

For static operation, the value of ψ achieved by the AIS will be dependent exclusively on

the amount of entrainment. Chapter 5 provides a means of predicting the value of ψ for a

system given the referenced input parameters.

4.1.1 Implications of Multiple Solutions for Um

It can be seen from Eq. (4.21) that the solution for the downstream mixture velocity is in the

form of a quadratic equation. There are two important observations: first, there will always

be two possible solutions for the velocity for a given array of input parameters, and second,

if the term under the square root of Eq. (4.21) is less than 0 the solution for the mixture

velocity will have a non-real component. So, the foloowing inequality must be satisfied for a

real solution:

P 2 − 4

[m

(1− Rm

2CPm

)][RmE

CPm

]≥ 0 (4.27)

28

If Eq. (4.27) is expressed in terms of the downstream properties it can be simplified into the

following equation, which has only one unique solution (Mam = 1).

Ma4m − 2Ma2m + 1 = 0 (4.28)

As the limit of the real solution represents the point where Mam = 1, it is deduced

from Eq. (4.22) that the double solution of Um must represent a supersonic and subsonic

downstream velocity. This implies that for any valid combination of input parameters there

are two possible mathematical solutions. The solution that is more physically accurate given

the inlet conditions must then be determined.

A similar analysis on the mixing of two gas streams was presented by Bernstein (1953)

where a double valued solution for downstream Mach number was found. In order to de-

termine which of the two solutions is correct for any case at hand, three distinct cases were

presented (Bernstein, 1953):

(a) In the case where both mixing streams are subsonic, obviously the subsonic solution is

the only valid case as the supersonic solutions solution would represent a net decrease

in entropy.

(b) In the case where both mixing streams are supersonic both solutions are physically

possible. The supersonic solution will occur when the back pressure (static pressure

after mixing) is low. The subsonic solution will occur when the back pressure is high,

this indicates shocks will exist in the mixing region.

(c) In the case where one stream is supersonic, and one subsonic, the subsonic solution is

always possible. The supersonic solution is only valid if the inlet area of the supersonic

stream is significantly greater than that of the subsonic region.

The current study is primarily focused on the stationary operation of the AIS. This will be

29

represented by case (c) as the air contributing to the mixture is initially at rest, and the area

of the air entering the AIS is significantly greater than that of the fuel. It is assumed that

the subsonic solution is the only valid solution.

4.1.2 CV Analysis Validation

Previous work by the current author compared the results of the above control volume anal-

ysis to several CFD simulations (Wilson et al., 2015). It was shown that given the resulting

equivalence ratio (φ = ψstoichiometric/ψ) from each simulation, the ambient air conditions, the

geometry of the AIS, and the fuel jet conditions; values of p0m, and T0m could be accurately

predicted. Figures 4.2, and 4.3 display some of the results. The cases simulated all represent

a methane fuel jet with a JPR of 5.92 injected through an AIS with in outlet-to-inlet area

ratio of 300. The temperature of the jet was calculated based on 10% of the heating value

of the fuel used to pre-heat the fuel from ambient temperature prior to injection through

the AIS. It was found that the mixture stagnation temperature is independent of the fuel

jet Mach number. This is shown in Figure 4.3, as all three series follow the same path. The

setup of the CFD simulations used included the same solver, mesh and boundary conditions

as those discussed in Chapter 6. Inlet conditions of the fuel jet are outlined in Wilson et al.

(2015).

4.1.3 Ideal Jet Propulsion Cycle

Once the properties of the mixture downstream of the AIS have been determined, the overall

performance of the engine may be evaluated. First, the rest of the properties through the

engine must be calculated. Following the AIS, it is assumed that the mixture is adiabatically

and isentropically decelerated and compressed to near stagnation properties. Losses due to

skin friction and shock waves are being neglected, so it can be assumed that static pressure

and temperature of the mixture then increase to their stagnation properties. The mixture

then passes through the combustion chamber, where it is burned at a constant static pressure.

30

Equivalence Ratio

0.5 1 1.5 2 2.5 3 3.5 4

p0m

/pa

1

1.05

1.1

1.15

1.2

Prediction Maf=0.5

Simulation Maf=0.5

Prediction Maf=1.0

Simulation Maf=1.0

Prediction Maf=2.0

Simulation Maf=2.0

Figure 4.2: A comparison of exiting mixture stagnation pressure predicted with the methodoutlined and CFD simulations using the software OpenFOAM for different fuel jet Machnumbers, all values are normalized by the ambient air pressure (Wilson et al., 2015).

This combustion is assumed to increase the stagnation temperature of the mixture, while

impacts on the velocity and static pressure are negligible (Heiser H. H. & Pratt, 1994).

It is assumed that there is a maximum temperature reached in the combustion chamber

independent of input conditions due to dissociation (Spakovsky, 1999). Heiser H. H. and

Pratt (1994) state that the maximum allowable compression temperature, Tmax, is almost

always found to be in the range of 1440-1670 K, and use 1560 K as a representative estimate.

The same value is utilized in the current study. As described in Chapter 1, the heat of

combustion is also used to pre-heat the fuel prior to injection through the AIS. It was

ensured that the heat of combustion of the fuel was sufficient to pre-heat the fuel jet to the

specified temperature and reach the maximum allowable compression temperature specified.

If the heat of combustion of the fuel is insufficient, the maximum temperature that could

ideally be achieved was calculated and used. Any losses due to the heat exchanger were not

accounted for in this analysis. Finally, the exhaust gases are assumed to be isentropically

and adiabatically expanded until the static pressure of the exhaust matches that of the

31

Equivalence Ratio

0.5 1 1.5 2 2.5 3 3.5 4

T0m

/Ta

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4

Prediction Maf=0.5

Simulation Maf=0.5

Prediction Maf=1.0

Simulation Maf=1.0

Prediction Maf=2.0

Simulation Maf=2.0

Figure 4.3: A comparison of exiting mixture stagnation temperature predicted with themethod outlined and CFD simulations using the software OpenFOAM for different fuel jetMach numbers, all values are normalized by the ambient air temperature (Wilson et al.,2015).

atmosphere. In this analysis standard temperature and pressure (T = 298.15 K, P =

101325 Pa) were assumed for the environment.

As the properties determined up to this point were shown to be independent of the inlet

area, and therefore the magnitude of the mass flow rate of fuel, it is useful to quantify this

performance independent of these variables too. A commonly used measurement for per-

formance of high speed air-breathing engines is specific impulse. Specific impulse quantifies

thrust as a function of the amount of mass flow rate of fuel used by the engine. In the case

of static operation the resulting equation is:

Isp =FTmfg

=(1 + ψ)Uexit

g(4.29)

The value of Uexit may be determined based on the stagnation temperature of the exhaust

gas, and the pressure ratio between the combustion chamber and the exit nozzle. As it

has been assumed that the flow through the exit nozzle is isotropically and adiabatically

32

accelerated from zero velocity relative to the vehicle, the stagnation temperature of the

exhaust gas is assumed to be equal to the temperature of the mixture following combustion.

The specific impulse of the engine can be computed from:

Isp =

(1 + ψ)

√2CPmTmax

(1−

(PaP0m

) γm−1γm

)g

(4.30)

It can be seen that the specific impulse is a function of ψ and the post AIS stagnation

pressure of the mixture. In the case of a propulsion system, it is usually desirable to keep

the equivalence ratio close to unity. The impacts of varying the jet inlet parameters on the

values of ψ and Isp are discussed in Section 7.4.

33

Chapter 5

AIS Entrainment Model

5.1 Assumptions

In Chapter 4 it was shown that given a value of ψ for an AIS with known input parameters,

the properties of the mixture exiting the system can be predicted quite accurately, and

the resulting overall performance can be quantified. It is then of interest to develop a

means of predicting the rate of entrainment of a system given these same input parameters.

As shown in the previous chapter, three conservation equations are insufficient to explicitly

calculate the outlet stream properties. It is then necessary to impose a value on one unknown

parameters of the control volume analysis in order for the problem to be solved.

This issue is explored in a similar application of a 2D control volume analysis predicting

air-to-exhaust ratios at the outlet for a thrust augmenting ejector is presented by Whitley,

Krothapalli, and Dommelen (1996). In the method presented, an assumption is made that

the static pressure at the exit of the mixture is equal to the ambient pressure of the sur-

rounding air. Given this assumption they were able to accurately predict the amount of air

entrained by the thrust augmenting ejector (Whitley et al., 1996). The same assumption

will be imposed on the problem of the AIS to provide a means of predicting rate of air

entrainment. Although the static pressure is assumed to be equal to the atmosphere, the

stagnation pressure can be much larger due to the high mixture velocity (Um). The engine

requires stagnation pressures above ambient to produce thrust. Section 7.2.1 will compare

the results of the entrainment model developed below. As well, the assumption of the static

pressure of the mixture matching that of the ambient air will also be assessed.

34

5.2 Modified AIS Mixture Properties Model

The assumption of an infinitely large control volume from the initial control volume analysis

is still considered for this case. The conservation of X-momentum and conservation of energy

equations will remain the same. The assumption of the outlet static pressure being equal to

the ambient pressure, pm = pa = pamb, will provide a means of validating the correct entrain-

ment rate has been found, though an explicit solution is not possible. An iterative approach

will be presented based on assuming an initial value for ψ. Based on the assumption for

the static pressure at the outlet, a modified version of the conservation of momentum, Eq.

4.7, can be used to find the outlet mixture velocity that will satisfy the initial value for ψ.

Again, the term maUa is assumed to be negligible, resulting in:

Um,j =mfUf + pfAin + pamb (Aout − Ain)− pambAout

(1 + ψj) mf

=mfUf + pfAin − pambAin

(1 + ψj) mf

(5.1)

Subscript j indicates the iteration index.

Mixture properties can be calculated using Equations 4.23) to (4.26 and the assumed

value of ψj. Using the velocity of the outlet mixture calculated, and a modified version of

the conservation of energy equation, the static temperature of the mixture at the outlet can

be determined:

Tm,j =CPfT0f + ψjCPaT0a

(1 + ψj)CPm,j−

U2m,j

2CPm,j(5.2)

All of the properties of the outlet mixture have been calculated or assumed, and a modified

version of the conservation of mass equation can be used to caculate a new value for the

35

mass air-to-fuel ratio, see Eq. 5.3.

ψj+1 =pambAoutUm,jRm,jTm,jmf

− 1 (5.3)

This value is compared to the original estimate, and if not within the desired tolerance

(0.01%) for a solution the process is repeated using this new value for ψj+1, until a solution

within the desired tolerance is found. This method ensures that all of the conservation

equations identified in Chapter 4 are satisfied, and allows a rapid method of predicting the

overall performance of an AIS with few assumptions being imposed. A flowchart of this

process is outlined in Figure 5.1. Results of this entrainment prediction model are compared

to several CFD simulation results in Chapter 7.

36

Figure 5.1: Flowchart of the developed method for predicting AIS entrainment.

37

Chapter 6

CFD Simulations

As there is limited experimental data available for either the AIS or the Gluhareff pressure

jet, some simplified CFD simulations were conducted to provide an application for visualizing

the exergy destruction equations outlined in Chapter 3, and a means for comparison for the

entrainment model outlined in Chapter 5. Due to costs associated with developing a model

of the AIS for experiments, and limits of the range of results that could be obtained it was

determined to be preferential to conduct CFD simulations for investigation and comparison.

10 cases of the static operation of the AIS using different inlet parameters were conducted

using the open source software OpenFOAM.

6.1 Solver

A modified version of the native OpenFOAM solver rhoCentralFoam was used for the turbu-

lent, multi-species, compressible simulations of a methane fuel jet injected through the AIS

into ambient air. rhoCentralFoam is a decoupled, explicit solver based on the conservative

form of the compressible Navier-Stokes equations. Interpolation schemes utilized for convec-

tive terms are both second order and total variation diminishing (TVD). The OpenFOAM

vanAlbada flux limiter was used, as experience has shown that it provides superior solution

quality near flow discontinuities (e.g. shocks or expansion waves). A modified version of this

solver, rhoCentralReactingFoam, was used in this study as it supports multi-species flow.

Both rhoCentralReactingFoam and rhoCentralFoam have shown good agreement with exper-

imental results and other commercial solvers (C. Arisman, Johansen, Galuppo, & McPhail,

2012; Arisman, 2014; C. Arisman, Johansen, Bathel, & Danehy, 2015; Hinman & Johansen,

2016a, 2016b; Teh & Johansen, 2016). rhoCentralReactingFoam was presented by C. Aris-

38

man et al. (2012), for the simulation of gas seeding into a hypersonic boundary layer flow.

The governing equations of rhoCentralReactingFoam are presented in the form:

∂ρ

∂t+∇ · (ρu) = 0 (6.1)

∂(ρu)

∂t+∇ · (ρu2) = −∇p+∇ · τ (6.2)

∂(ρhS)

∂t+∇ · (ρuhS) = ∇ ·

[αT∇hS +

Ns∑α=1

hαJαi

]+Dp

Dt+∇ · (τ · u) + Sh (6.3)

hS, αT , and Sh, are the sensible enthalpy, thermal diffusivity, and enthalpy source term

respectively. And the viscous stress tensor, τ , is defined as:

τ = µ

(∇u+ (∇u)T − 2

3∇ · uI

)(6.4)

µ and I are the viscosity and unit tensor respectively. The transport of multi-species mass

fraction is defined as:

∂ρYα∂t

+∇ · (ρuYα) = −∇ · Jαi +Rα (6.5)

A thorough discussion of the implementation of the multi-species mass fraction transport

model is provided in C. Arisman et al. (2012). The OpenFOAM native Sutherland transport

model was used to determine viscosity variations with temperature based on Sutherlands law.

The Janaf thermo model was used for the calculation of specific heat capacity and enthalpy

at different temperatures based on the 7 coefficient JANAF polynomials (OpenCFD, 2017).

6.1.1 Turbulence Model

The use of some variation of the k-ε turbulence model was common throughout the liter-

ature for both ejector pumps and exergy analysis (Rao & Jagadeesh, 2015), (Yapici et al.,

2005), (Stanciu et al., 2001), (Sierra-Pallares et al., 2016). So, the standard OpenFOAM

k-ε turbulence model was used for the CFD simulations conducted. As slip conditions were

39

Table 6.1: Standard model coefficients (OpenCFD, 2017).

Cµ C1 C2 C3,RDT σk σε

0.09 1.44 1.92 -0.33 1 1.3

imposed on all walls, the use of improved turbulence models that include wall factors was

not required. The OpenFOAM equations of the k-ε model vary slightly from the standard

model:

D

Dtρk = ∇ · (ρDk∇k) +Gk −

2

3ρ (∇ · u) k − ρε+ Sk (6.6)

D

Dtρε = ∇ · (ρDε∇ε) +

C1Gkε

k−(

2

3C1 + C3,RDT

)ρ (∇ · u) k − C2ρ

ε2

k+ Sε (6.7)

Buoyancy contributions are not included, and the third term on the right hand side of Eq.

6.7 includes the rapid distortion theory (RDT) contribution (OpenCFD, 2017). The default

model coefficients used are outlined in Table 6.1.

6.2 Simulation Setup

6.2.1 Boundary Conditions

Figure 6.1 shows a schematic of the computational domain and the boundary conditions.

The fuel jet nozzle is on the left of the domain, and three AIS stages are downstream (right)

of the jet outlet. Based on the control volume analysis presented in Chapter 4.1, it was

shown that the four primary design variables considered for the AIS are the fuel jet prop-

erties (temperature, pressure and velocity), and the outlet ti inlet area ratio of the AIS. As

the models developed do not take into consideration losses due to friction, the impacts of

changing axial dimensions are not explored. Due to the large variety of possible combina-

tions of these design parameters, the CFD simulations conducted used a fixed area ratio.

The dimensions used represent a simplified geometry of a preliminary design of the AIS. The

influence of changing jet properties on performance is explored.

40

While the diameter of each individual AIS stage is constant in the simulations conducted,

their respective diameter to each other increases further downstream. The diameter of the

fuel jet for all simulations is fixed at 0.003175 m. The diameter of each AIS stage is set such

that the area ratio between the fuel jet and the first, second, and third stage, are 90, 210,

and 300 respectively. The length of the first, second, and third stages are 0.12 m, 0.27 m,

and 0.12 m, respectively. A 0.01 m gap is left axially between the jet and first stage, and

between each of the other stages.

The static pressure, temperature, and velocity of the fuel inlet jet were specified in each

case as shown in Table 6.2. The static properties of the fuel jet are specified relative to the

standard ambient conditions by the Jet Pressure Ratio (JPR), and Jet Temperature Ratio

(JTR). The lower limit on the value of JPR was chosen as a perfectly expanded jet. The

upper limit was chose as 10, as this represents the static pressure following acceleration from

zero velocity, containment and handling at any higher ratios may be physically unrealistic.

The lower limits of JTR were restricted by the valid range of temperatures allowed by the

thermophysical models used. The upper limit is restricted by the maximum fuel pre-heating

temperature before expansion and injection through the AIS, which is limited by the maxi-

mum combustion temperature. The velocity of the fuel jet was specified in terms of jet Mach

number. The lower limit was chosen as 1, a sonic jet. Due to limited resources, only one

other fuel jet Mach number (Majet = 2) was explored in this study.

The boundary conditions at edges of the computational domain radially beyond the fuel

inlet and at the field opposite the axis of symmetry were set as fixed stagnation pressure and

temperature. At the radial field downstream of the measurement plane, as fields were speci-

fied as zero-gradient. In order to simplify the cases and reduce computational requirements,

41

the surfaces of the AIS intake stages and the jet were specified as a slip boundary condition

in terms of velocity. As mentioned in Chapter 4, if the no-slip condition were imposed it is

assumed that a decrease in entrainment would be observed.

ComputationalDomain

Fuel Inletpf , Tf , and Uf Set

Measurement Planepm, Tm, and Um Evaluated

Axis of Symmetry

Flow Direction

IntakeSystem

p0a and T0a Set

Figure 6.1: A schematic of the computational domain used for the AIS simulations. Initialboundary conditions and points of measurement are shown. Figure is not to scale.

Table 6.2: Fuel jet inlet conditions of CFD simulations conducted.

Case Number Fuel JPR JTR Jet Mach(pf/pamb) (Tf/Tamb) Number

1 Methane 2 2 12 Methane 5 2 13 Methane 10 2 14 Methane 1 2 25 Methane 2 2 26 Methane 5 2 27 Methane 10 2 28 Methane 2 3 19 Methane 5 3 110 Methane 10 3 1

Values for k and ε were specified for the inlet of the fuel jet based on Eq. 6.8 and 6.9.

k =3

2(I|uref |)2 (6.8)

42

ε =C0.75µ k1.5

L(6.9)

I, uref , and L are the intensity, reference velocity, and reference length scale (OpenCFD,

2017). And the value for I for a fully developed duct flow can be estimated by the empirical

correlation (Russo & Basse, 2016).

I = 0.16(ReDH )−18 (6.10)

ReDH is the Reynolds number based on hydraulic diameter. The internal field of the domain

initially has zero velocity, but a finite value for k is desired. So, the internal field is initialized

with a value of 1, and all external boundaries are set as zeroGradient. The internal field of

ε was similarly initialized based on Eq. 6.9, assuming k = 1 and using the jet diameter as

the characteristic length.

6.2.2 Mesh

The mesh is 2-D and axi-symmetric and has been created in the native OpenFOAM meshing

utility blockMesh. A grid convergence study was completed, and the results are discussed

in Section 7.1.2. Figures 6.2 and 6.3 show an example of the most coarse mesh used for the

simulations. A uniform grid was used for the primary region of interest, axially from the fuel

jet outlet to the outlet of the third AIS duct, and radially from the axis of symmetry to the

outermost AIS duct. Beyond this primary region of interest, the computational domain has

been extended 0.05 m radially, upstream, and downstream of this primary region of interest

to avoid potential negative interference of the with the boundary conditions. In these regions

the a cell inflation rate of 2.0 was used to retain a smaller mesh size to reduce computational

requirements.

43

Figure 6.2: Example of simulation coarse mesh.

Figure 6.3: Example of simulation coarse mesh near the fuel jet outlet and first AIS stage.

44

Chapter 7

Results

7.1 CFD Simulation Results & Verification

7.1.1 Flow Visualization Results

Figures 7.1, 7.2, and 7.3 display contours of some of the results from the CFD simulation

(Case 2) for an AIS with JTR=2.0, JPR=5.0, and a methane fuel jet Mach number of 1.0.

It can be seen that spatial variations in pressure are primarily contained to the core region

of the jet, while variation in temperature and velocity are observed in the mixing region of

the two streams. Figure 7.3 shows that the regions beyond the radius of the AIS have no

substantial velocity magnitude, this support the assumption made in Chapter 4 that the

velocity of the air entering the control volume may be neglected. In this case, the mixture

velocity at the end of the AIS was predicted to be 66.5 m/s. Plots of all of the properties

extracted from the flow visualization can be found in Appendix A. A synthetic Schlieren

image of the core jet is shown in Figure 7.4. The flow features associated with an under-

expanded jet can be seen in this image.

7.1.2 Grid Independence

A grid convergence study was performed on to ensure that the results obtained were inde-

pendent of the grid resolution. The parameter used for assessing grid convergence was the

mass air-to-fuel ratio, ψ, as this is the variable used for validation of the entrainment model

developed in Chapter 5. The grid convergence index (GCI) was calculated across two mesh

refinements for the results of the case used for the flow visualization, based on the method

outlined in Oberkampf and Trucano (2002). The final mesh (> 666000 nodes) showed only

45

Figure 7.1: A contour of normalized static pressure, (p − patm)/patm, for an AIS withJTR=2.0, JPR=5.0, and a fuel jet Mach number of 1.0 from one of the CFD simulationsconducted.

Figure 7.2: A contour of normalized static temperature for an AIS with JTR=2.0, JPR=5.0,and a fuel jet Mach number of 1.0 from one of the CFD simulations conducted. The yel-low frame indicates the core region of the fuel jet for which a synthetic Schlieren image ispresented in Figure 7.4.

small changes in ψ (< 1.1%) compared to the medium mesh. The properties of the three

grids are shown in Table 7.2, and the results of the GCI study are shown in Table 7.2.

To verify that the solution is withing the asymptotic range of convergence, the following

condition should be met (NASA, 2008):

GCI23rpGCI12

≈ 1 (7.1)

Where GCI23 and GCI12 are the grid convergence indices reported in Table 7.2 between

46

Figure 7.3: A contour of velocity magnitude for an AIS with JTR=2.0, JPR=5.0, and a fueljet Mach number of 1.0 from one of the CFD simulations conducted.

Figure 7.4: A synthetic Schlieren image of the core region of the fuel jet for an AIS withJTR=2.0, JPR=5.0, and a fuel jet Mach number of 1.0 from one of the CFD simulationsconducted.

meshes 2 to 3, and 1 to 2 respectively. r is the refinement ratio, and P is the order of

convergence calculated as:

P = ln

(f3 − f2f2 − f1

)/ ln(r) (7.2)

Where f is value of the property for which the convergence is being calculated, in this case

ψ. Equation 7.3 shows that for the case under consideration the relation shown in Eq. 7.1 is

approximately 1, indicating the solutions are within the asymptotic range of convergence.

0.0393

20.590.0259= 1.01 (7.3)

47

Table 7.1: Grid properties and results for Case 2.

Mesh Number of Mass-Air-to-FuelCells Ratio (ma/mf )

1 666084 14.902 166521 14.743 44715 14.51

Table 7.2: Grid convergence index results for Case 2.

Refinement Coarse Mesh Fine Mesh Refinement Grid ConvergenceNumber Number Number Ratio Index

1 3 2 1.92978 3.93%2 2 1 2.00000 2.59%

7.2 Model Validation

7.2.1 Entrainment Model Comparison

In Chapter 5, the assumption that the static pressure of the mixture exiting the AIS will be

equal to the ambient air static pressure is imposed in order to allow the prediction of the

AIS entrainment rate. Table 7.3 and Figure 7.5 show the area averaged static pressure at

the outlet plane for each of the CFD simulations conducted normalized by the ambient air

static pressure. It can be seen that all of the results of the simulations fall within 1.2% of

the ambient static pressure. This validates the assumption imposed previously.

For each CFD simulation conducted of the static operation of the AIS, the mass air-to-

fuel ratio was determined at the outlet plane. These results are compared to the predicted

entrainment from the model outlined in Chapter 5 in Figures 7.6 to 7.8. Table 7.3 provides

a summary of the mean outlet properties determined from each simulation. The values pre-

sented represent the average over the last 5 printed time steps.

The resulting mass air-to-fuel ratio for each simulation is plotted against the jet pressure

ratio specified for each unique combination of jet Mach number and jet temperature ratio. It

48

1 2 3 4 5 6 7 8 9 100.9

0.925

0.95

0.975

1

1.025

1.05

1.075

1.1

Case

Ou

tlet

Mix

ture

Sta

tic

Pre

ssu

re /

Am

bie

nt

Sta

tic

Pre

ssure

Ambient Static Pressure

Outlet Area Averaged Static Pressure

Figure 7.5: Area averaged outlet static pressure normalized by ambient static air pressurefor all CFD cases conducted.

Table 7.3: Mean outlet properties for CFD simulations conducted.

Case Number Outlet Static Outlet Static Mass Air-to-Fuel EquivalencePressure Ratio Temperature Ratio Ratio (ψ) Ratio (φ)

1 0.9996 1.15 18.5 0.932 0.9998 1.17 14.9 1.153 0.9976 1.23 10.51 1.644 0.9953 1.15 33.05 0.525 0.9987 1.23 19.78 0.876 0.9952 1.35 11.98 1.447 0.9902 1.5 7.62 2.268 0.9968 1.17 30.98 0.569 0.9884 1.24 21.07 0.8210 1.0027 1.42 10.85 1.59

can be seen in each case that the CV-based model developed in Chapter 5 accurately predicts

the amount of air entrained by the AIS in comparison to the RANS simulation results. The

results of the RANS simulations accurately follow the trend of decreasing mass air-to-fuel

49

ratio with increasing JPR as predicted by the entrainment model. The entrainment rates

achieved in each case fall into the range in which the heat of combustion of the fuel is

sufficient to reach the maximum combustion temperature as mentioned in Section 4.1.3.

1 2 3 4 5 6 7 8 9 10

0

5

10

15

20

25

30

35

40

Jet Pressure Ratio

Mass A

ir-t

o-F

uel

Rati

o

Control Volume Model

RANS Simulation

Figure 7.6: Mass air-to-fuel ratio vs. JPR for an AIS with a fuel jet Mach number of 1 andJTR of 2. Comparison of entrainment model to CFD results.

1 2 3 4 5 6 7 8 9 10

0

5

10

15

20

25

30

35

40

Jet Pressure Ratio

Mass A

ir-t

o-F

uel

Rati

o

Control Volume Model

RANS Simulation

Figure 7.7: Mass air-to-fuel ratio vs. JPR for an AIS with a fuel jet Mach number of 2 andJTR of 2. Comparison of entrainment model to CFD results.

50

1 2 3 4 5 6 7 8 9 10

0

5

10

15

20

25

30

35

40

Jet Pressure Ratio

Mass A

ir-t

o-F

uel

Rati

o

Control Volume Model

RANS Simulation

Figure 7.8: Mass air-to-fuel ratio vs. JPR for an AIS with a fuel jet Mach number of 1 andJTR of 3. Comparison of entrainment model to CFD results.

Figures 7.9 and 7.10 show a comparison between the model and simulation results for

multiple JTR and multiple jet Mach numbers plotted together. It can be seen in Fig. 7.9

that a lower ψ is associated with a lower JTR in both the simulation and entrainment model

results. As well, the converging of the solutions for the two JTR with increasing JPR is

reflected in both the model and the simulation results. Figure 7.10 show that both the

model and the simulations display a higher ψ associated with a lower Mach number, as well

as divergence of the two with increasing JPR.

7.3 Exergy Destruction Analysis

Based on the equations for the volumetric rate of entropy generation developed in Chapter 3,

a post-processing utility was developed to review the sources of exergy destruction present in

the AIS. The source code for this utility is available in Appendix B. The entropy generation

post-processing utility was applied to several of the AIS cases to identify prominent sources

of exergy destruction. Contours of the magnitude of local volumetric exergy destruction

51

1 2 3 4 5 6 7 8 9 10

0

5

10

15

20

25

30

35

40

Jet Pressure Ratio

Mass A

ir-t

o-F

uel

Rati

o

Control Volume Model JTR=2.0

RANS Simulation JTR=2.0

Control Volume Model JTR=3.0

RANS Simulation JTR=3.0

Entrainment Rate vs. Jet Pressure Ratio

Figure 7.9: Mass air-to-fuel ratio vs. JPR for an AIS with a fuel jet Mach number of 1 andmultiple JTR’s. Comparison of entrainment model to CFD results.

1 2 3 4 5 6 7 8 9 10

0

5

10

15

20

25

30

35

40

Jet Pressure Ratio

Mass A

ir-t

o-F

uel

Rati

o

Control Volume Model Maf=1.0

RANS Simulation Maf=1.0

Control Volume Model Maf=2.0

RANS Simulation Maf=2.0

Entrainment Rate vs. Jet Pressure Ratio

Figure 7.10: Mass air-to-fuel ratio vs. JPR for an AIS with multiple fuel jet Mach numbersand a JTR of 2. Comparison of entrainment model to CFD results.

from the identified sources, as well as relevant simulation properties, for one of the cases are

presented in Figures 7.11a to 7.11d.

It can be seen that the peak magnitudes for the volumetric rate of exergy destruction are

similar for each of the source terms. Additionally, the regions displaying the greatest rate

of exergy destruction in all three cases are quite similar. The mixing region between the

52

under-expanded jet and the ambient air appears to have the greatest amplitude of exergy

destruction. This is consistent with what is expected based on the equations for exergy de-

struction described in Chapter 3, as this region represents the steepest gradients in velocity,

temperature, and species mass fraction.

Table 7.4 shows the normalized magnitude of the exergy destruction rates due to vis-

cosity, heat transfer, and diffusion of species calculated. The rate of exergy destruction for

each case is normalized by its predicted thrust power (PT = FTUexit). Where FT and Uexit

are the force of thrust and exit velocity of the AIS coupled engine based on the simulation

value of ψ, and the ideal jet propulsion cycle outlined in Section 4.1.3. Comparing with

Table 6.2, it can be seen that the total normalized exergy destruction rates tend to decrease

with increasing JTR or Mach number. However, in at least the cases with Maf = 1 and

JTR = 3 there appears to be a local minima in terms of total normalized exergy destruction

with respect to JPR. Analyzing the impacts of varying the input variables of the fuel jet on

the individual sources of exergy generation can be done by comparing 7.4 to Table 6.2.

Table 7.4: Exergy destruction rates for CFD simulations conducted.

Case Number Xdest,V /PT Xdest,Q/PT Xdest,D/PT Xdest,T/PT

1 12.16 4.48 5.74 22.382 9.28 4.61 3.03 16.923 6.08 3.51 1.78 11.384 6.65 2.91 2.00 11.565 7.09 4.69 2.10 13.876 4.36 3.77 1.20 9.327 3.49 3.56 0.89 7.948 6.97 1.98 2.18 11.139 3.69 1.41 1.03 6.1210 5.37 3.38 1.43 10.18

It can be seen that increasing the JTR will actually reduce the normalized exergy destruc-

tion rates due to each of the three sources considered. This is as expected when reviewing

53

Eq. 3.44 to 3.46. It is shown that exergy destruction due to viscous effects is inversely pro-

portional to T . Similarly, exergy destruction due to heat transfer is inversely proportional

to T 2, and exergy destruction due to species diffusion is directly proportional to ρ, which for

an ideal gas is inversely proportional to T . It can also be observed that this effect is more

significant at lower JPR and less significant at higher JPR.

Similarly, it is seen that an increase in fuel jet Mach number results in an decrease in

normalized exergy destruction rates due to viscous effects and species diffusion. Though,

the normalized exergy destruction due to heat transfer shows relatively little response to a

change in Mach number, compared to a change in JTR. While normalized exergy generation

rates decrease with increasing Mach number, the effect is opposite when analyzing absolute

entropy generation rates. This indicates that the increase in fuel jet Mach number must

also be associated with an increase in engine performance. This will be further explored in

Section 7.4.

Increasing the JPR resulted in an increase in the overall rate of exergy destruction for all

three factors, but has mixed results when analyzing the normalized values. This may be due

to the fact that the engine performance appears to have competing factors with changing

JPR as will be shown in Section 7.4. It is interesting to note that the highest rate of viscous

exergy destruction occurs in the region of the Mach disk of the under-expanded fuel jet.

7.4 Parametric Analysis Results

Together, the entrainment model outlined in Chapter 5, and the performance model out-

lined in Chapter 4 provide tools for the rapid analysis and of potential AIS configurations to

identify optimal design parameter values. While CFD simulations provide a great amount

of information, they are time consuming and computationally expensive. In order to find

a design that produces a specific desired result, or maximizes a specific parameter, a wide

54

variety of simulations are required. This will allow for determination of the influence of input

parameters in relation to one another. A tool for the rapid analysis of a design for a set of

given input parameters allows a greater number of options to be tested without significant

input of time or resources. Additionally, the ability to test a greater number of options

allows for more refinement near a potential solution.

A brief study of the possible application of the tools developed in Chapter 4 and 5 for an

AIS with an outlet-to-inlet Area Ratio of 300 is presented. Figure 7.12 provides a contour

of predicted specific impulse vs. JPR and JTR for a jet Mach number of 2 for a vehicle with

zero forward velocity. Contours of specific impulse vs. JPR and JTR for several different

jet Mach numbers are presented in Figures D.1 to D.4 in Appendix D. The limits of jet

Mach number, temperature and pressure ratio are matched to those of the CFD simulations

conducted in Chapter 7.1.

It can be seen that the specific impulse increases with increasing JTR, but there is a

local maxima in terms of JPR. The JPR at which this local maxima occurs decreases with

increasing Mach Number. Additionally, as Mach number increases the amplitude of the local

maxima increases. Based on these results, for the given AIS configuration, the maximum

specific impulse would be achieved by a fuel jet Mach number of 2, a JTR of 3, and a JPR

of approximately 2.5.

7.5 Discussion

It is clear from the CFD, exergy, and parametric analyses results that increasing the JTR in-

creases entrainment, increases Isp, and decreases exergy destruction. Since the local speed of

sound increases with temperature, high jet velocities can be produced with a combination of

high jet temperature and low JPR. Decreasing the JPR was found to maximize the entrain-

55

ment rate. However, the exergy destruction showed mixed results and parametric analysis

results indicate that Isp is maximized when the JPR reaches approximately 2.5. Therefore,

clearly there are competing mechanisms to consider with regards to the selection of the JPR.

While the minimization of exergy destruction is desirable to improve efficiency, entrain-

ment through the jet shear layer and fuel-air mixing are necessary for engine operation. At

high JPR values, shocks form in the underexpanded jet region, which contribute to exergy

destruction, but to not contribute to entrainment. However, some JPR is required in order

to drive the jet to high velocities such that entrainment can occur.

Exergy destruction via heat transfer along the jet shear layer is not required for AIS

operation and can be minimized. It is not clear if the AIS efficiency can be improved by

better matching the temperatures between the jet and the secondary stream using a diverging

nozzle. In this scenario, exergy destruction through shocks would also be minimized and the

jet velocity, which drives entrainment could be maximized. Future work is required to look

at this configuration.

56

(a) Volumetric rate of exergy destruction due to viscous effects.

(b) Volumetric rate of exergy destruction due to heat transfer.

(c) Volumetric rate of exergy destruction due to species diffusion.

(d) Total volumetric rate of exergy destruction.

Figure 7.11: Contours of the local rate of exergy destruction near the fuel jet exit for an AISwith JTR=2.0, JPR=5.0, and a fuel jet Mach number of 1.0 from one of the CFD simulationsconducted.

57

2 4 6 8 102

2.2

2.4

2.6

2.8

3

Jet Pressure Ratio

Jet T

em

pera

ture

Ratio

320

325

330

335

340

Specific

Im

puls

e

Figure 7.12: Specific impulse as a function of JPR and JTR for an AIS with a fuel jet Machnumber of 2 at zero forward velocity.

58

Chapter 8

Conclusion

A control-volume analysis was used to develop a means of predicting the performance of an

AIS coupled ramjet engine given a set of input conditions. A method for predicting the ratio

of air entrained by the AIS based on the characteristics of the fuel inlet jet and the geometry

of the AIS was presented. An assumption that the static pressure of the mixture exiting the

AIS is equal to the ambient air static pressure is imposed to allow the iterative solution of ψ.

This assumption is is validated based on good agreement with CFD. The predicted values

of ψ also show good agreement with CFD.

It is shown that the specific impulse of an AIS with an outlet-to-inlet area ratio of 300

increases with increasing jet Mach number and increasing JTR, but a local maxima occurs at

a certain value for JPR. It was also shown that the value of JPR at which this local maxima

occurs decreases with increasing jet Mach number. At a jet Mach number of 2.0 and JTR

of 3.0, a local maxima for specific impulse is identified at a JPR of 2.5.

Equations for the calculation of local volumetric entropy generation due to viscous, heat

transfer, and dissipation effects are derived. These equations are implemented in a post-

processing utility and applied to the CFD simulations discussed to quantify exergy destruc-

tion. It is found that the rate of exergy destruction due to all three factors decreases with

increasing jet Mach number and JTR, but mixed results were observed with increasing JTR.

In the simulations reviewed it is shown that exergy destruction due to viscous effects is the

greatest contributor in nearly all cases. Contours of the local rates of exergy destruction

indicate the destruction due to all three factors primarily takes place at the boundaries of

59

the jet where the mixing of the streams occurs.

8.1 Recommendations

Performing experiments to provide physical data for validation is recommended for any future

work on this project. Exploration of the impacts of different turbulence models, including

large-eddy simulation to look at larger scale turbulent structures within the AIS is also

recommended. Further investigation into the competing forces observed with changing JPR

in terms of exergy generation and engine performance is needed.

60

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65

Appendix A

CFD Simulation Flow Visualization Results

66

Fig

ure

A.1

:A

conto

ur

ofth

em

ass

frac

tion

ofai

rfo

ran

AIS

wit

har

eara

tio

of30

0,JT

R=

2.0,

JP

R=

5.0,

and

am

ethan

efu

elje

tM

ach

num

ber

of1.

0fr

omon

eof

the

CF

Dsi

mula

tion

sco

nduct

ed.

67

Fig

ure

A.2

:A

conto

ur

ofth

em

ass

frac

tion

ofm

ethan

efo

ran

AIS

wit

har

eara

tio

of30

0,JT

R=

2.0,

JP

R=

5.0,

and

am

ethan

efu

elje

tM

ach

num

ber

of1.

0fr

omon

eof

the

CF

Dsi

mula

tion

sco

nduct

ed.

68

Fig

ure

A.3

:A

conto

ur

ofst

atic

pre

ssure

for

anA

ISw

ith

area

rati

oof

300,

JT

R=

2.0,

JP

R=

5.0,

and

am

ethan

efu

elje

tM

ach

num

ber

of1.

0fr

omon

eof

the

CF

Dsi

mula

tion

sco

nduct

ed.

69

Fig

ure

A.4

:A

conto

ur

ofst

atic

tem

per

ature

for

anA

ISw

ith

area

rati

oof

300,

JT

R=

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71

Appendix B

Exergy Destruction Post-Processing Utility Source

Code

1 /∗−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−∗\

2 ========= |

3 \\ / F i e l d | OpenFOAM: The Open Source CFD Toolbox

4 \\ / O perat i on |

5 \\ / A nd | Copyright (C) 2012 OpenFOAM Foundation

6 \\/ M an ipu l a t i on |

7 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

8 License

9 This f i l e i s part o f OpenFOAM.

10

11 OpenFOAM i s f r e e so f tware : you can r e d i s t r i b u t e i t and/ or modify i t

12 under the terms o f the GNU General Publ ic L icense as publ i shed by

13 the Free Software Foundation , e i t h e r version 3 o f the License , or

14 ( at your opt ion ) any l a t e r version .

15

16 OpenFOAM i s d i s t r i bu t ed in the hope that i t w i l l be use fu l , but WITHOUT

17 ANY WARRANTY; without even the impl i ed warranty o f MERCHANTABILITY or

18 FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Publ ic L icense

19 for more d e t a i l s .

20

21 You should have r e c e i v ed a copy o f the GNU General Publ ic L icense

22 along with OpenFOAM. I f not , s ee <http ://www. gnu . org / l i c e n s e s />.

23

24 Appl i cat ion

25 Mach

26

27 Desc r ip t i on

28 Ca l cu l a t e s and wr i t e s a f i e l d for each type o f l o c a l entropy generat i on ra t e

29

30 \∗−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−∗/

31

32 #inc lude ” ca l c .H”

33 #inc lude ” fluidThermo .H”

34 #inc lude ”fvCFD .H”

35 #inc lude ”OFstream .H”

36 #inc lude ”psiCombustionModel .H”

37 #inc lude ” turbulenceModel .H” //Arv

38

39 #inc lude ” zeroGradientFvPatchFie lds .H”

40 #inc lude ”mult ivar iateScheme .H” // nakul

41

42 //INCLUDE STANDARD C++ f e a t u r e s :

43 #inc lude <iostream>

44 #inc lude <fstream>

72

45 #inc lude <iomanip>

46

47

48 // ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ //

49

50 void Foam : : c a l c ( const a rgL i s t& args , const Time& runTime , const fvMesh& mesh)

51 {

52

53

54 bool wr i t eResu l t s = ! args . optionFound (” noWrite ”) ;

55

56 IOobject Uheader

57 (

58 ”U” ,

59 runTime . timeName ( ) ,

60 mesh ,

61 IOobject : :MUST READ

62 ) ;

63

64 IOobject pheader

65 (

66 ”p” ,

67 runTime . timeName ( ) ,

68 mesh ,

69 IOobject : :MUST READ

70 ) ;

71

72 IOobject rhoheader

73 (

74 ” rho ” ,

75 runTime . timeName ( ) ,

76 mesh ,

77 IOobject : :MUST READ

78 ) ;

79

80 IOobject N2header

81 (

82 ”N2” ,

83 runTime . timeName ( ) ,

84 mesh ,

85 IOobject : :MUST READ

86 ) ;

87

88 IOobject CH4header

89 (

90 ”CH4” ,

91 runTime . timeName ( ) ,

92 mesh ,

93 IOobject : :MUST READ

94 ) ;

95

96 IOobject rhoDi jheader

97 (

98 ” rhoDij ” ,

73

99 runTime . timeName ( ) ,

100 mesh ,

101 IOobject : :MUST READ

102 ) ;

103

104 IOobject muheader

105 (

106 ”mu” ,

107 runTime . timeName ( ) ,

108 mesh ,

109 IOobject : :MUST READ

110 ) ;

111

112 IOobject mutheader

113 (

114 ”mut” ,

115 runTime . timeName ( ) ,

116 mesh ,

117 IOobject : :MUST READ

118 ) ;

119

120 IOobject alphaheader

121 (

122 ” alpha ” ,

123 runTime . timeName ( ) ,

124 mesh ,

125 IOobject : :MUST READ

126 ) ;

127

128 IOobject phiheader

129 (

130 ”phi ” ,

131 runTime . timeName ( ) ,

132 mesh ,

133 IOobject : :MUST READ

134 ) ;

135

136 IOobject ep s i l onheade r

137 (

138 ” ep s i l o n ” ,

139 runTime . timeName ( ) ,

140 mesh ,

141 IOobject : :MUST READ

142 ) ;

143

144

145

146 Info<< ”Reading the rmophys i ca lProper t i e s \n” << endl ;

147

148 // Check headers exist

149 i f ( Uheader . headerOk ( ) && pheader . headerOk ( ) && rhoheader . headerOk ( ) )

150 {

151 autoPtr<vo lSca l a rF i e ld> sGenVPtr ;

152 autoPtr<vo lSca l a rF i e ld> sGenHPtr ;

74

153 autoPtr<vo lSca l a rF i e ld> sGenDPtr ;

154 autoPtr<vo lSca l a rF i e ld> sGenTPtr ;

155 autoPtr<vo lSca l a rF i e ld> mDotnormalPtr ;

156 autoPtr<vo lSca l a rF i e ld> kappaPtr ;

157 autoPtr<vo lVectorFie ld> gradTPtr ;

158

159 autoPtr<combustionModels : : psiCombustionModel> r e a c t i on

160 (

161 combustionModels : : psiCombustionModel : : New(mesh)

162 ) ;

163

164

165

166

167 //Molar mass o f s p e c i e s (CH4 and Air )

168 const s c a l a r& M1 = 16 .0425 ;

169 const s c a l a r& M2 = 28 . 966 ;

170

171 psiReactionThermo& thermo = react ion−>thermo ( ) ;

172 thermo . va l i d a t e ( args . executab le ( ) , ”h” , ”e ”) ;

173

174 IOdic t ionary the rmophys i ca lProper t i e s

175 (

176 IOobject

177 (

178 ” the rmophys i ca lProper t i e s ” ,

179 runTime . constant ( ) ,

180 mesh ,

181 IOobject : :MUST READ,

182 IOobject : :NO WRITE

183 )

184 ) ;

185 // Load data from time d i r e c t o r y

186 vo lVecto rF i e ld U(Uheader , mesh) ;

187 vo l S c a l a rF i e l d p( pheader , mesh) ;

188 vo l S c a l a rF i e l d N2(N2header , mesh) ;

189 vo l S c a l a rF i e l d CH4(CH4header , mesh) ;

190 vo l S c a l a rF i e l d rhoDij ( rhoDijheader , mesh) ;

191 vo l S c a l a rF i e l d mu(muheader , mesh) ;

192 vo l S c a l a rF i e l d mut(mutheader , mesh) ;

193 vo l S c a l a rF i e l d alpha ( alphaheader , mesh) ;

194 s u r f a c eS c a l a rF i e l d phi ( phiheader , mesh) ;

195 vo l S c a l a rF i e l d ep s i l o n ( eps i l onheader , mesh) ;

196

197 i f

198 (

199 IOobject

200 (

201 ” the rmophys i ca lProper t i e s ” ,

202 runTime . constant ( ) ,

203 mesh

204 ) . headerOk ( )

205 )

206 {

75

207

208 Info<< ”Reading the rmophys i ca lProper t i e s ” << endl ;

209

210 // Reference temperature and pre s su r e for s0

211 s c a l a r T0Air = 298 . 0 ;

212 s c a l a r p0Air = 101325 .0 ;

213 s c a l a r T0CH4 = 298 . 0 ;

214 s c a l a r p0CH4 = 101325 .0 ;

215

216 dimens ionedSca lar Cp0Air

217 (

218 ”Cp0Air ” ,

219 dimensionSet (0 ,2 ,−2 ,−1 ,0 ,0 ,0) ,

220 s c a l a r ( 1004 . 5 )

221 ) ;

222

223 dimens ionedSca lar R0Air

224 (

225 ”R0Air ” ,

226 dimensionSet (0 ,2 ,−2 ,−1 ,0 ,0 ,0) ,

227 s c a l a r (287)

228 ) ;

229

230 s c a l a r s0Air = 5754 .153832 ; // http ://www. n i s t . gov/data/PDFfi les / jpcrd581 . pdf pg 367

231 s c a l a r RAir = 287 . 0 ;

232

233 dimens ionedSca lar Cp0CH4

234 (

235 ”Cp0CH4” ,

236 dimensionSet (0 ,2 ,−2 ,−1 ,0 ,0 ,0) ,

237 s c a l a r (2223 .234)

238 ) ;

239

240 dimens ionedSca lar R0CH4

241 (

242 ”R0CH4” ,

243 dimensionSet (0 ,2 ,−2 ,−1 ,0 ,0 ,0) ,

244 s c a l a r ( 518 . 3 )

245 ) ;

246

247 s c a l a r s0CH4 = 6673 . 3 ; // NIST Thermo Lookup

248 s c a l a r RCH4 = 518 . 3 ;

249

250 // Janaf Polynomials for Air and CH4 from thermoPhys i ca lProper t i e s

251 // I f us ing constant Cp, only uses A0 , put Cp value for each sp e c i e in p lace o f A0 and a l l

other va lues to 0

252

253 s c a l a r A0AirLow = 9.47E+02; //1004 .5 ; //

254 s c a l a r A1AirLow = 4.04E−01;

255 s c a l a r A2AirLow = −1.14E−03;

256 s c a l a r A3AirLow = 1.62E−06;

257 s c a l a r A4AirLow = −7.02E−10;

258 s c a l a r A5AirLow = 0.00E+00;

259

76

260 s c a l a r A0AirHigh = 8.40E+02; //1004 .5 ; //

261 s c a l a r A1AirHigh = 4.27E−01;

262 s c a l a r A2AirHigh = −1.63E−04;

263 s c a l a r A3AirHigh = 2.90E−08;

264 s c a l a r A4AirHigh = −1.94E−12;

265 s c a l a r A5AirHigh = 0.00E+00;

266

267 s c a l a r A0CH4Low = 1.07E+03;

268 s c a l a r A1CH4Low = −7.18E−01;

269 s c a l a r A2CH4Low = 5.46E−03;

270 s c a l a r A3CH4Low = −4.22E−06;

271 s c a l a r A4CH4Low = 9.87E−10;

272 s c a l a r A5CH4Low = 0.00E+00;

273

274 s c a l a r A0CH4High = 1.18E+03;

275 s c a l a r A1CH4High = 2.14E+00;

276 s c a l a r A2CH4High = −7.57E−04;

277 s c a l a r A3CH4High = 1.20E−07;

278 s c a l a r A4CH4High = −7.10E−12;

279 s c a l a r A5CH4High = 0.00E+00;

280

281 s c a l a r Tcommon = 1283;

282 s c a l a r Ymin = 0 ;

283

284 Info<< ”Ca l cu la t ing F i e l d s ” << endl ;

285

286 vo l S c a l a rF i e l d Cp( thermo .Cp( ) ) ;

287 vo l S c a l a rF i e l d Cv( thermo .Cv( ) ) ;

288 vo l S c a l a rF i e l d T( thermo .T( ) ) ;

289 vo l S c a l a rF i e l d rho ( thermo . rho ( ) ) ;

290

291 Info<< ”Creat ing turbu lence model\n” << endl ;

292 autoPtr<compres s ib l e : : turbulenceModel> turbu lence

293 (

294 compres s ib l e : : turbulenceModel : : New

295 (

296 rho ,

297 U,

298 phi ,

299 thermo

300 )

301 ) ;

302

303 vo l S c a l a rF i e l d kappa ( turbulence−>kappaEff ( ) ) ;

304 vo l S c a l a rF i e l d Le (0 . 7∗ kappa /(Cp∗mu) ) ;

305

306 // I n i t i a l i z e f i e l d s for de l t a s

307 vo l S c a l a rF i e l d de l t a sA i r (Cp∗0) ;

308 vo l S c a l a rF i e l d deltasCH4 (Cp∗0) ;

309

310 // Ca lcu la te mole f r a c t i o n for each c e l l / f a c e

311 vo l S c a l a rF i e l d yAir (N2∗(N2∗M2 + CH4∗M1)/M2) ;

312 vo l S c a l a rF i e l d yCH4(CH4∗(N2∗M2 + CH4∗M1)/M1) ;

313

77

314 // Def ine temperature g rad i en t s

315 vo lVecto rF i e ld gradT = fvc : : grad (T) ;

316 vo l S c a l a rF i e l d dTdx = gradT . component (0) ;

317 vo l S c a l a rF i e l d dTdz = gradT . component (2) ;

318

319 // Def ine v e l o c i t y Gradients

320 vo lVecto rF i e ld graduX = fvc : : grad (U. component (0) ) ;

321 vo lVecto rF i e ld graduY = fvc : : grad (U. component (1) ) ;

322 vo lVecto rF i e ld graduZ = fvc : : grad (U. component (2) ) ;

323

324 // Def ine s p e c i e s Gradients

325 vo lVecto rF i e ld gradN2 = fvc : : grad (N2) ;

326 vo l S c a l a rF i e l d dN2dx = gradN2 . component (0) ;

327 vo l S c a l a rF i e l d dN2dz = gradN2 . component (2) ;

328

329 vo lVecto rF i e ld gradCH4 = fvc : : grad (CH4) ;

330 vo l S c a l a rF i e l d dCH4dx = gradCH4 . component (0) ;

331 vo l S c a l a rF i e l d dCH4dz = gradCH4 . component (2) ;

332

333 // Ca lcu la te entropy generat i on due to heat t r a n s f e r sgen = k/Tˆ2 ∗ (dTdxˆ2 + dTdyˆ2 + dTdzˆ2)

334 Info<< ”Ca l cu la t ing SGenH” << endl ;

335 vo l S c a l a rF i e l d SgenH = ( kappa /(pow(T, 2 ) ) ) ∗(pow( gradT . component (0) ,2 )+pow( gradT . component (1) ,2 )

+pow( gradT . component (2) ,2 ) ) ;

336

337 // Ca lcu la te entropy generat i on due to v i s c o s i t y sgen = mu/T ∗ (2∗( dudxˆ2 + dvdyˆ2 + dwdzˆ2) +

dudyˆ2 + dudzˆ2 + dvdxˆ2 + dvdzˆ2 + dwdxˆ2 + dwdyˆ2 + 2∗(dudy∗dvdx + dudz∗dwdx + dvdz∗

dwdy)

338 Info<< ”Ca l cu la t ing SGenV” << endl ;

339 vo l S c a l a rF i e l d SgenV = ((mu+mut) /T) ∗ ( (pow( graduX . component (0) ,2 )+pow( graduY . component (1) ,2 )+

pow( graduZ . component (2) ,2 ) ) ∗2 + (pow( graduX . component (1) ,2 )+pow( graduX . component (2) ,2 )+pow

( graduY . component (0) ,2 )+pow( graduY . component (2) ,2 )+pow( graduZ . component (0) ,2 )+pow( graduZ .

component (1) ,2 ) ) + 2∗( graduX . component (1) ∗graduY . component (0)+graduX . component (2) ∗graduZ .

component (0)+graduY . component (2) ∗graduZ . component (1) ) )+rho∗ ep s i l o n /T;

340

341 // I n i t i a l i z e entropy generat i on due to d i f f u s i o n f i e l d s

342 vo l S c a l a rF i e l d SgenDN2(SgenV) ;

343 vo l S c a l a rF i e l d SgenDCH4(SgenV) ;

344

345 // Ca lcu la te entropy generat i on due to d i f f u s i o n sgen = rho∗Dij∗R/ yi ∗( dyidx∗dyidx + dyidz∗

dyidz )

346 Info<< ”Ca l cu la t ing SGenD” << endl ;

347

348 f o rA l l (N2 , i )

349 {

350 i f (N2 [ i ] <= Ymin)

351 {

352 SgenDN2 [ i ] = 0 ;

353 }

354 else

355 {

356 SgenDN2 [ i ] = rhoDij [ i ] ∗ ( ( RAir/N2 [ i ] ) ∗(dN2dx [ i ]∗dN2dx [ i ]+dN2dz [ i ]∗dN2dz [ i ] ) ) ;

357 }

358

359 i f (CH4[ i ] <= Ymin)

78

360 {

361 SgenDCH4 [ i ] = 0 ;

362 }

363 else

364 {

365 SgenDCH4 [ i ] = rhoDij [ i ] ∗ ( (RCH4/CH4[ i ] ) ∗(dCH4dx [ i ]∗dCH4dx [ i ]+dCH4dz [ i ]∗dCH4dz [ i ] ) ) ;

366 }

367 }

368

369 vo l S c a l a rF i e l d SgenD(SgenDN2+SgenDCH4) ;

370

371 // Total vo lumetr i c entropy generat i on equa l s some o f each term

372 vo l S c a l a rF i e l d SgenT(SgenD+SgenV+SgenH) ;

373

374 // Total a c t rua l entropy genera t i on equa l s vo lumetr i c entropy generat i on times volume

375 vo l S c a l a rF i e l d sGenHV(SgenH∗0) ;

376 vo l S c a l a rF i e l d sGenVV(SgenV∗0) ;

377 vo l S c a l a rF i e l d sGenTV(SgenT∗0) ;

378 s c a l a r SGENH = 0 ;

379 s c a l a r SGENV = 0 ;

380 s c a l a r SGEN = 0 ;

381 s c a l a r massCV = 0 ;

382 s c a l a r energyCV = 0 ;

383 s c a l a r Volume = 0 ;

384

385 f o rA l l (SgenT , i )

386 {

387 sGenHV [ i ] = SgenH [ i ]∗mesh .V( ) [ i ] ;

388 sGenVV [ i ] = SgenV [ i ]∗mesh .V( ) [ i ] ;

389 sGenTV [ i ] = SgenT [ i ]∗mesh .V( ) [ i ] ;

390 SGENH = SGENH + sGenHV [ i ] ;

391 SGENV = SGENV + sGenVV [ i ] ;

392 SGEN = SGEN + sGenTV [ i ] ;

393 massCV = massCV+rho [ i ]∗mesh .V( ) [ i ] ;

394 energyCV = energyCV+rho [ i ]∗mesh .V( ) [ i ]∗Cp[ i ]∗T[ i ] ;

395 Volume = Volume + mesh .V( ) [ i ] ;

396 }

397

398

399

400 Info<< ”Ca l cu la t ing Entropy at each f a c e . . . ” << endl ;

401

402

403 vo l S c a l a rF i e l d p a r t i a l sA i r (Cp) ;

404 vo l S c a l a rF i e l d part ia lsCH4 (Cp) ;

405 vo l S c a l a rF i e l d s (Cp∗0) ;

406 vo l S c a l a rF i e l d mDotnormal (U. component (1) ∗0) ;

407 s c a l a r sTrans = 0 ;

408 s c a l a r massTrans = 0 ;

409 s c a l a r energyTrans = 0 ;

410

411 const fvPatchLi s t& patches = mesh . boundary ( ) ;

412 f o rA l l ( patches , patch i )

413 {

79

414

415 f o rA l l (mesh . boundaryMesh ( ) [ patch i ] , f a c e i )

416 {

417 l a b e l Tface = T. boundaryField ( ) [ patch i ] [ f a c e i ] ;

418 l a b e l p face = p . boundaryField ( ) [ patch i ] [ f a c e i ] ;

419

420 i f ( Tface > Tcommon)

421 {

422 de l t a sA i r . boundaryField ( ) [ patch i ] [ f a c e i ] = A0AirHigh∗ log ( Tface /T0Air )−RAir∗ log (

p face /p0Air ) +(((( A5AirHigh /5∗( Tface ) + A4AirHigh /4) ∗( Tface ) + A3AirHigh /3) ∗(

Tface ) + A2AirHigh /2) ∗( Tface ) + A1AirHigh ) ∗( Tface ) −((((A5AirHigh /5∗(T0Air ) +

A4AirHigh /4) ∗(T0Air ) + A3AirHigh /3) ∗(T0Air ) + A2AirHigh /2) ∗(T0Air ) + A1AirHigh

) ∗(T0Air ) ;

423 deltasCH4 . boundaryField ( ) [ patch i ] [ f a c e i ] = A0CH4High∗ log ( Tface /T0CH4)−RCH4∗ log (

p face /p0CH4) +((((A5CH4High/5∗( Tface ) + A4CH4High/4) ∗( Tface ) + A3CH4High/3) ∗(

Tface ) + A2CH4High/2) ∗( Tface ) + A1CH4High) ∗( Tface ) −((((A5CH4High/5∗(T0CH4) +

A4CH4High/4) ∗(T0CH4) + A3CH4High/3) ∗(T0CH4) + A2CH4High/2) ∗(T0CH4) + A1CH4High

) ∗(T0CH4) ;

424 }

425 else

426 {

427 de l t a sA i r . boundaryField ( ) [ patch i ] [ f a c e i ] = A0AirLow∗ log ( Tface /T0Air )−RAir∗ log (

p face /p0Air ) +((((A5AirLow/5∗( Tface ) + A4AirLow/4) ∗( Tface ) + A3AirLow/3) ∗( Tface

) + A2AirLow/2) ∗( Tface ) + A1AirLow) ∗( Tface ) −((((A5AirLow/5∗(T0Air ) + A4AirLow

/4) ∗(T0Air ) + A3AirLow/3) ∗(T0Air ) + A2AirLow/2) ∗(T0Air ) + A1AirLow) ∗(T0Air ) ;

428 deltasCH4 . boundaryField ( ) [ patch i ] [ f a c e i ] = A0CH4Low∗ log ( Tface /T0CH4)−RCH4∗ log (

p face /p0CH4) +((((A5CH4Low/5∗( Tface ) + A4CH4Low/4) ∗( Tface ) + A3CH4Low/3) ∗( Tface

) + A2CH4Low/2) ∗( Tface ) + A1CH4Low) ∗( Tface ) −((((A5CH4Low/5∗(T0CH4) + A4CH4Low

/4) ∗(T0CH4) + A3CH4Low/3) ∗(T0CH4) + A2CH4Low/2) ∗(T0CH4) + A1CH4Low) ∗(T0CH4) ;

429 }

430

431 i f (N2 . boundaryField ( ) [ patch i ] [ f a c e i ] <= Ymin)

432 {

433 p a r t i a l sA i r . boundaryField ( ) [ patch i ] [ f a c e i ] = 0 ;

434 }

435 else

436 {

437 p a r t i a l sA i r . boundaryField ( ) [ patch i ] [ f a c e i ] = de l t a sA i r . boundaryField ( ) [ patch i ] [

f a c e i ]+ s0Air−RAir∗ log ( yAir . boundaryField ( ) [ patch i ] [ f a c e i ] ) ;

438 }

439

440 i f (CH4. boundaryField ( ) [ patch i ] [ f a c e i ] <= Ymin)

441 {

442 part ia lsCH4 . boundaryField ( ) [ patch i ] [ f a c e i ] = 0 ;

443 }

444 else

445 {

446 part ia lsCH4 . boundaryField ( ) [ patch i ] [ f a c e i ] = deltasCH4 . boundaryField ( ) [ patch i ] [

f a c e i ]+s0CH4−RCH4∗ log (yCH4 . boundaryField ( ) [ patch i ] [ f a c e i ] ) ;

447 }

448

449 s . boundaryField ( ) [ patch i ] [ f a c e i ] = N2 . boundaryField ( ) [ patch i ] [ f a c e i ]∗ pa r t i a l sA i r .

boundaryField ( ) [ patch i ] [ f a c e i ]+CH4. boundaryField ( ) [ patch i ] [ f a c e i ]∗ part ia lsCH4 .

boundaryField ( ) [ patch i ] [ f a c e i ] ;

80

450

451 mDotnormal . boundaryField ( ) [ patch i ] [ f a c e i ] = phi . boundaryField ( ) [ patch i ] [ f a c e i ] ;

452

453 sTrans = sTrans + mDotnormal . boundaryField ( ) [ patch i ] [ f a c e i ]∗ s . boundaryField ( ) [ patch i ] [

f a c e i ] ;

454 massTrans = massTrans + mDotnormal . boundaryField ( ) [ patch i ] [ f a c e i ] ;

455 energyTrans = energyTrans + mDotnormal . boundaryField ( ) [ patch i ] [ f a c e i ]∗Cp. boundaryField

( ) [ patch i ] [ f a c e i ]∗T. boundaryField ( ) [ patch i ] [ f a c e i ] ;

456 }

457 }

458

459 Info<< ”Asss ign ing F i e l d s ” << endl ;

460

461

462 sGenVPtr . set

463 (

464 new vo l S c a l a rF i e l d

465 (

466 IOobject

467 (

468 ”sGenV” ,

469 runTime . timeName ( ) ,

470 mesh

471 ) ,

472 SgenV

473 )

474 ) ;

475

476 sGenHPtr . set

477 (

478 new vo l S c a l a rF i e l d

479 (

480 IOobject

481 (

482 ”sGenH” ,

483 runTime . timeName ( ) ,

484 mesh

485 ) ,

486 SgenH

487 )

488 ) ;

489

490 sGenDPtr . set

491 (

492 new vo l S c a l a rF i e l d

493 (

494 IOobject

495 (

496 ”sGenD” ,

497 runTime . timeName ( ) ,

498 mesh

499 ) ,

500 SgenD

501 )

81

502 ) ;

503

504 sGenTPtr . set

505 (

506 new vo l S c a l a rF i e l d

507 (

508 IOobject

509 (

510 ”sGenT” ,

511 runTime . timeName ( ) ,

512 mesh

513 ) ,

514 SgenT

515 )

516 ) ;

517

518 mDotnormalPtr . set

519 (

520 new vo l S c a l a rF i e l d

521 (

522 IOobject

523 (

524 ”mDotnormal ” ,

525 runTime . timeName ( ) ,

526 mesh

527 ) ,

528 mDotnormal

529 )

530 ) ;

531

532 kappaPtr . set

533 (

534 new vo l S c a l a rF i e l d

535 (

536 IOobject

537 (

538 ”kappaEff ” ,

539 runTime . timeName ( ) ,

540 mesh

541 ) ,

542 kappa

543 )

544 ) ;

545

546

547 gradTPtr . set

548 (

549 new vo lVecto rF i e ld

550 (

551 IOobject

552 (

553 ”gradT” ,

554 runTime . timeName ( ) ,

555 mesh

82

556 ) ,

557 gradT

558 )

559 ) ;

560

561 In fo << SGEN << tab << sTrans << tab << massTrans << tab << massCV << endl ;

562 In fo << SGENH << tab << SGENV << endl ;

563 In fo << energyTrans << tab << energyCV << endl ;

564 In fo << Volume << endl ;

565

566 ofstream myf i l e ;

567 myf i l e . open (” exergyDest . txt ” , std : : i o s : : app ) ;

568 myf i l e << ”\n” << std : : s e t p r e c i s i o n (6) << SGEN << tab << std : : s e t p r e c i s i o n (6) << sTrans << tab

<< std : : s e t p r e c i s i o n (6) << massTrans << tab << std : : s e t p r e c i s i o n (6) << massCV << tab ;

569 myf i l e . close ( ) ;

570

571 }

572 else

573 {

574

575 }

576

577 i f ( wr i t eResu l t s )

578 {

579 sGenVPtr ( ) . wr i t e ( ) ;

580 sGenHPtr ( ) . wr i t e ( ) ;

581 sGenDPtr ( ) . wr i t e ( ) ;

582 sGenTPtr ( ) . wr i t e ( ) ;

583 mDotnormalPtr ( ) . wr i t e ( ) ;

584 kappaPtr ( ) . wr i t e ( ) ;

585 gradTPtr ( ) . wr i t e ( ) ;

586 }

587 }

588 else

589 {

590 Info<< ” Miss ing U or T or P or rho” << endl ;

591 }

592

593 Info<< ”\nEnd\n” << endl ;

594 }

595

596

597 // ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ //

83

Appendix C

CFD Simulation Exergy Destruction Visualization

Results

84

Fig

ure

C.1

:A

conto

ur

ofth

elo

cal

rate

ofex

ergy

des

truct

ion

due

toth

erm

aleff

ects

for

anA

ISw

ith

anar

eara

tio

of30

0,JT

R=

2.0,

JP

R=

5.0,

and

am

ethan

efu

elje

tM

ach

num

ber

of1.

0fr

omon

eof

the

CF

Dsi

mula

tion

sco

nduct

ed.

85

Fig

ure

C.2

:A

conto

ur

ofth

elo

calra

teof

exer

gydes

truct

ion

due

tovis

cous

effec

tsfo

ran

AIS

wit

han

area

rati

oof

300,

JT

R=

2.0,

JP

R=

5.0,

and

am

ethan

efu

elje

tM

ach

num

ber

of1.

0fr

omon

eof

the

CF

Dsi

mula

tion

sco

nduct

ed.

86

Fig

ure

C.3

:A

conto

ur

ofth

elo

cal

rate

ofex

ergy

des

truct

ion

due

tosp

ecie

sdiff

usi

onfo

ran

AIS

wit

han

area

rati

oof

300,

JT

R=

2.0,

JP

R=

5.0,

and

am

ethan

efu

elje

tM

ach

num

ber

of1.

0fr

omon

eof

the

CF

Dsi

mula

tion

sco

nduct

ed.

87

Fig

ure

C.4

:A

conto

ur

ofth

eto

tal

loca

lra

teof

exer

gydes

truct

ion

for

anA

ISw

ith

anar

eara

tio

of30

0,JT

R=

2.0,

JP

R=

5.0,

and

am

ethan

efu

elje

tM

ach

num

ber

of1.

0fr

omon

eof

the

CF

Dsi

mula

tion

sco

nduct

ed.

88

Appendix D

Additional Parametric Analysis Contour Plots

2 4 6 8 102

2.2

2.4

2.6

2.8

3

Jet Pressure Ratio

Jet T

em

pera

ture

Ratio

180

200

220

240

260

Specific

Im

puls

e

Figure D.1: Specific impulse as a function of JPR and JTR for an AIS with a fuel jet Machnumber of 1 and area ratio of 300 at zero forward velocity.

89

2 4 6 8 102

2.2

2.4

2.6

2.8

3

Jet Pressure Ratio

Jet T

em

pera

ture

Ratio

210

220

230

240

250

260

270

Specific

Im

puls

e

Figure D.2: Specific impulse as a function of JPR and JTR for an AIS with a fuel jet Machnumber of 1.25 and area ratio of 300 at zero forward velocity.

90

2 4 6 8 102

2.2

2.4

2.6

2.8

3

Jet Pressure Ratio

Jet T

em

pera

ture

Ratio

250

260

270

280

290

Specific

Im

puls

e

Figure D.3: Specific impulse as a function of JPR and JTR for an AIS with a fuel jet Machnumber of 1.5 and area ratio of 300 at zero forward velocity.

91

2 4 6 8 102

2.2

2.4

2.6

2.8

3

Jet Pressure Ratio

Jet T

em

pera

ture

Ratio

290

295

300

305

310

315

Specific

Im

puls

e

Figure D.4: Specific impulse as a function of JPR and JTR for an AIS with a fuel jet Machnumber of 1.75 and area ratio of 300 at zero forward velocity.

92

Appendix E

Copyright Permission Forms

93

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