AEROMECHANICAL CONTROL OF HIGH-SPEED AXIAL COMPRESSOR STALL

AEROMECHANICAL CONTROL OF HIGH-SPEED AXIAL COMPRESSOR STALL

An Undergraduate Honors Thesis

Presented in Partial Fulfillment of the Requirements for the

Bachelor of Science of Civil Engineering with Distinction in the

College of Engineering at The Ohio State University

By

Keith LaMar Coleman

* * * * *

The Ohio State University 2006

Examination Committee: Approved By Dr. Oliver G. McGee III, Advisor Dr. Patrick J. Fox

Advisor

Department of Civil & Environmental Engineering & Geodetic Science

First and foremost, I would like to dedicate this work to my Lord and Savior Jesus Christ. With Him all things are possible. This is also for my family for their continual support.

Thanks Mom and Jess.

i

ABSTRACT

General methodologies will be developed in this work for the evaluation of passive high-

speed compressor stabilization strategies using tailored structural design and aeromechanical

feedback control. These passive stabilization strategies will be compared in their

performance of several aeromechanical stabilization approaches which could potentially be

implemented in high-speed axial compressors used by industry. The stability of

aeromechanically-compensated high-speed compressors will be determined from linearized,

compressible structural-hydrodynamic equations of stall inception developed in this study.

This work offers a systematic study of the influence of ten aeromechanical feedback

controller schemes to increase the range of stable operation of two high-speed laboratory

compressors, using static pressure sensing and local structural actuation to postpone modal

(long wave) stall inception. The maximum operating range for each scheme is determined for

optimized structural parameters, and the various schemes are compared. Ten passive

stabilization schemes that could potentially be used by industry were discussed and examined

in a high-speed compressible flow environment. The concept of elasticity was introduced

and implemented to examine the effects of flow non-uniformity, entropic loss, and

unsteadiness on thermodynamic state changes within the compression system. Finally,

pumping and aeroelastic characteristics of these laboratory compressors both with and

without feedback were analyzed.

ii

ACKNOWLEDGEMENTS

I would like to first thank my advisor and friend, Dr. Oliver G. McGee III, for inspiring me

and also believing in my potential. I appreciate you for spending those longs nights in your

office with me to make sure that I understood the material. Thanks for ‘hurtin’ my head’ so

that I may become a better thinker. I truly believe that you have prepared me to be successful

in the next stage of my academic career. I look forward to the times ahead. Here’s to a long

lasting friendship.

Secondly, I thank Dr. Chih Fang, for his advice and support. I wish you and your family well.

I am grateful for the support of The Ohio State University Summer Research Opportunities

Program, The OSU Civil and Environment Engineering & Geodetic Science Department,

and The OSU Engineering Experiment Station for deeming my work as “worthy of research”.

Last and certainly not least, I must thank my lab partners and friends, Celeste Chavis and

Dan Work, for making my undergraduate research experience worthwhile. You two will truly

be missed. Best wishes in your future endeavors. Maybe someday we may work together as

colleagues. Only time will tell.

Thank you all!

iii

TABLE OF CONTENTS

Abstract ………………………………………………………………………………….. . i

Acknowledgements ………………………………………………………………….. …... ii

Table of Contents …………………………………………………………………………. iii

List of Figures …………………………………………………………………………….. iv

List of Tables ……………………………………………………………………………... vi

Chapter 1 ………………………………………………………………………………….. 7

Chapter 2 ………………………………………………………………………………….. 14

Chapter 3 ………………………………………………………………………………….. 21

Chapter 4 ………………………………………………………………………………….. 27

Chapter 5 ………………………………………………………………………………….. 42

Chapter 6 ………………………………………………………………………………….. 64

References ………………………………………………………………………………… 71

Appendix ………………………………………………………………………………….. 72

iv

List of Figures

Figure 1.1a……………………………………………………………………………….. 8 Figure 1.1b……………………………………………………………………………….. 9Figure 2.1a……………………………………………………………………………….. 14 Figure 3.2a……………………………………………………………………………….. 22 Figure 3.4a……………………………………………………………………………….. 24 Figure 4.2a……………………………………………………………………………….. 29 Figure 4.2b……………………………………………………………………………….. 29 Figure 4.2c……………………………………………………………………………….. 30 Figure 4.2d……………………………………………………………………………….. 30 Figure 4.3a……………………………………………………………………………….. 31 Figure 4.3b……………………………………………………………………………….. 32 Figure 4.4.1a……………………………………………………………………………... 33 Figure 4.4.1b…………………………………………………………………………….. 33 Figure 4.4.2a……………………………………………………………………………... 34 Figure 4.4.3a……………………………………………………………………………... 35 Figure 4.4.3b……………………………………………………………………………... 36 Figure 4.4.4a……………………………………………………………………………... 36 Figure 4.4.4b……………………………………………………………………………... 37 Figure 4.5.1a……………………………………………………………………………... 38 Figure 4.5.1b……………………………………………………………………………... 38 Figure 4.5.2a……………………………………………………………………………... 39 Figure 4.5.3a……………………………………………………………………………... 40 Figure 4.5.3b……………………………………………………………………………... 40 Figure 4.5.4a……………………………………………………………………………... 41 Figure 4.5.4b……………………………………………………………………………... 41 Figure 5.2a……………………………………………………………………………….. 44 Figure 5.2b……………………………………………………………………………….. 44 Figure 5.3a……………………………………………………………………………….. 45 Figure 5.4a……………………………………………………………………………….. 46 Figure 5.4b……………………………………………………………………………….. 47 Figure 5.4c……………………………………………………………………………….. 47 Figure 5.5a……………………………………………………………………………….. 48 Figure 5.5b……………………………………………………………………………….. 48 Figure 5.5c……………………………………………………………………………….. 49 Figure 5.6a ………………………………………………………………………………. 50 Figure 5.6b……………………………………………………………………………….. 51 Figure 5.6c……………………………………………………………………………….. 51 Figure 5.6d……………………………………………………………………………….. 52 Figure 5.6e……………………………………………………………………………….. 52 Figure 5.7a ………………………………………………………………………………. 53 Figure 5.7b……………………………………………………………………………….. 54 Figure 5.7c……………………………………………………………………………….. 54 Figure 5.7d……………………………………………………………………………….. 55 Figure 5.7e……………………………………………………………………………….. 55 Figure 5.8a……………………………………………………………………………….. 56

v

List of Figures (cont’d)

Figure 5.8b……………………………………………………………………………….. 56 Figure 5.8c……………………………………………………………………………….. 57 Figure 5.9a……………………………………………………………………………….. 58 Figure 5.9b……………………………………………………………………………….. 59 Figure 5.9c……………………………………………………………………………….. 59 Figure 5.9d……………………………………………………………………………….. 60 Figure 5.9e……………………………………………………………………………….. 60 Figure 5.10.1a……………………………………………………………………………. 61 Figure 5.10.1b……………………………………………………………………………. 62 Figure 5.10.2a……………………………………………………………………………. 63 Figure 5.10.2b……………………………………………………………………………. 63

vi

List of Tables

Table 1…………………………………………………………………………………….24 Table 2………………………………………………………………………………….... 26

7

CHAPTER 1: Introduction

1.1 Introduction

The operating range of aeroengine compression systems is limited by two classes of

aerodynamic instabilities known as rotating stall and surge (Emmons et al, 1955). Rotating

stall is a multi-dimensional instability in which regions of low or reversed mass flow (i.e.,

stall cells) propagate around the compressor annulus due to incidence variations on adjacent

airfoils (Greitzer, 1976, 1980, 1981). Surge is primarily a one-dimensional instability of the

entire pumping system (compressor, ducts, combustion chamber, and turbine). It is

characterized by axial pulsations in annulus-averaged mass flow, including periods of flow

reversal through the machine. In high-speed compressor hydrodynamics (Fréchette, 1997),

rotating stall is generally encountered first, which then (loosely) “triggers” surge (often after

a few rotor revolutions, Greitzer, 1976). Therefore, the focus of this work will be on rotating

stall. With either instability, the compression system experiences a substantial loss in

performance and operability, which sometimes result in mechanical failure.

8

Fig. 1.1a Illustration of rotating stall and surge. A sketch of the transient signatures that would be given by high response pressure probes in the compressor (for rotating stall) or in the combustor, or other volume downstream of the compressor (for surge). (Fréchette 1997)

An experience-based approach for avoiding such performance loss is to operate the

compressor at a safe range from the point of instability onset (i.e., with stall margin). The

stall margin ensures that the engine can endure momentary off-design operation. The margin

also reduces the available pressure rise and efficiency of the machine (see Fig 1.1b). It is

proposed here that addition of tailored structural dynamic components or aeromechanical

feedback controllers, locally sensed by unstable perturbations in annulus pressure and

actuated by non-uniformities in the high-speed flow distribution around the annulus, can be

shown to inhibit the inception of rotating stall of high-speed compressor devices. As a result,

the stable operating range will be effectively extended, allowing higher performance

operating conditions.

9

Fig. 1.1b Compressor map illustrating the surge margin (from Fréchette 1997).

Aeromechanical feedback can be loosely defined as the dynamic interaction between

flexible structures and fluid dynamics of the compression system, without external

electromechanical input. When approaching the stall line, flow disturbances induce local

pressures on the structures. When tailored with the appropriate dynamic characteristics, the

structure deforms to counteract flow disturbances, either directly or by modifying the local

unsteady pressure rise of the compressor. High-speed compressor stabilization using

aeromechanical feedback control is investigated here as a passive means of improving

stability so that the compressor can operate safely at lower mass flows. This approach

incorporates tailored structural feedback control within the machine that can alter the fluid

dynamic behavior, so that the performance of the compressor can be extrapolated to

operating ranges outside the empirical database generated by years of experience. Passive

approaches to high-speed compressor stability have received no previous attention in the

open literature. However, some important fundamentals in aeromechanical control of low-

speed devices have been achieved (Gysling and Greitzer, 1995; McGee et al, 2004; Fréchette

et al, 2004).

10

1.2 Objectives and Scope of Study

This work presents a systematic development and evaluation of tailored structural

design and aeromechanical feedback stabilization of rotating stall in high-speed axial

compressors. The focus of this research is to evaluate aeromechanical feedback stabilization

strategies of long wave- length, high-speed compressible, modal stall (Fréchette, 1997) that

employ static pressure sensing and structural actuation for dynamic compensation. Ten

aeromechanical feedback stabilization strategies are considered, which generally include: (1)

aeromechanically incorporating variable duct geometries for dynamic impedance control, (2)

dynamically restaggered inlet guide vanes and rotor blades for diffusing or contracting blade

passage control, (3) movable casing walls for dynamic tip clearance flow control, and (4)

dynamic fluid injection for unsteady circumferential flow and pressure rise control across the

compressor. To quantify the effectiveness of the various schemes, the analysis is applied to

two laboratory compressors (ideally modified for high speeds) for which the empirical

modeling inputs have been previously determined: (i) the MIT single-stage compressor

(Gsyling and Greitzer, 1995) and (ii) the MIT three-stage compressor (Haynes et al, 1994).

The proof-of-concept studies of Gysling and Greitzer (1995), McGee et al (2004), and

Fréchette et al (2004) demonstrated the feasibility of aeromechanical control of low-speed

stalled compressors. However, some additional questions remain for high-speed devices. The

motivation of the present modeling development and evaluation is to address three

overarching questions: (1) What is the high-speed stall control capability of other

aeromechanical feedback stabilization strategies, and how do they compare to that modeled

and demonstrated by Gysling and Greitzer (1995), McGee et al (2004), and Frechette et al

(2004)? (2) Are there destabilizing high-speed, compressible fluid-structural interactions

which should be avoided through tailored structural design? (3) How do the aeromechanical

11

feedback dynamics couple with the pre-stall compressible fluid dynamics (Fréchette, 1997)

to postpone or induce the inception of high-speed rotating stall?

The significance of the problem posed here is extremely unique and timely. Fluid-

structural interaction effects are not only essential in devising high-speed compressor

stabilization strategies, but also useful in establishing constraints on the structural design of

compressors used by industry. As lightweight, less rigid structures are incorporated into new

high-speed compressor designs, the level of fluid-structure interactions is likely to increase

and result in reduced stall margin if the structures are not properly tailored. The practical

insight and motivation here is to achieve light-weight, more efficient compressor builds using

tailored, less rigid structures, while preventing potential stall margin reductions. There is

therefore a need for broader study evaluating the potential of various passive control schemes

to better assess the effect of aeromechanical interaction on high-speed compressor stability.

The overarching goals of this study are to evaluate the role of flexible structures on

compressor stability, and to elucidate that a proper choice of local structural dynamic

compensation close-coupled to the compressor affects (either beneficially or detrimentally)

the stability of the system. The paper presented here re- introduces the ten aeromechanical

feedback schemes developed by McGee et al (2004) and the nonlinear measured high-speed

compression system dynamics calculated from a MIT single-stage and a MIT three-stage

compressor characteristics employed therein (Fréchette et al, 2004; Gysling and Greitzer,

1995 (MIT single-stage); Haynes et al, 1994 (MIT three-stage)).

1.3 Methodology

An extensive literature review of low-speed and high-speed compressor system

hydrodynamic stability models and aeromechanical feedback approaches for passive control

of such devices was conducted. Specifically, a state-of-the-art high-speed, compressible

12

hydrodynamic model (Fréchette 1997) was examined and extending to a linearized,

compressible structural-hydrodynamic model of stall inception examined herein.

Subsequently, proof-of-concept schemes of aeromechanical control technologies were

developed in order to describe how such feedback can be utilized to stabilize high-speed

compressor stall of axial compressors, and how different tailored structural designs impact

high-speed compression system stability. Optimal structural parameters for aeromechanical

compensators were determined to maximize the stable operating range of the high-speed

compression system. The use of optimized aeromechanical feedback control to stabilize the

system and extend the operating range will be discussed later. The theoretical basis of ten

aeromechanical control schemes examined here was evaluated under a compressible flow

environment. These schemes are broadly classified as: (1) dynamic fluid injection upstream

of the compressor for control of inlet flow non-uniformities, (2) variable compressor inlet

and exit duct geometries for impedance control inside the ducts, (3) flexible compressor

casing wall providing control of tip clearance flow processes, and (4) dynamically re-

staggered inlet guide vanes and rotor blades for control of deviation and dissipation loss

mechanisms.

Completing the evaluation of the aforementioned aeromechanical control

methodologies developed, the low-speed computer codes of McGee et al (2004) was

modified and extended to high-speed, compressible flow regimes. Off-design operation of an

aeromechanically-controlled (or dynamically-compensated) high-speed compressor can

dramatically affect the performance characteristic curve shape of the device. Any change in

inlet conditions can change the discharge pressure and gas horsepower. Besides changing the

characteristic pressure and horsepower curves, the characteristic head curve and the head

curve sensitivity (associated with the stability of the high-speed compressor) also changes.

This phenomenon is due to specific volume or gas density ratio effects and equivalent speed

13

effects on the compressor. Since the performance map curves change with speed (higher

losses at higher speeds), the overall shape of these curves change, which can be compounded

by compressibility (specific volume or gas density ratio) effects closely-coupled with

dynamic compensation associated with aeromechanical feedback controls.

The modified computer codes were utilized in order to construct the essential

performance maps, discharge pressure vs. flow (bringing forth a measure of gas density and

speed changes affecting a dynamically-compensated compressor’s stability).

14

Fig. 2.1a Compressor characteristics used in this study: (a) MIT single stage (Gysling and Greitzer 1995), (b) MIT three-stage (Haynes et al, 1994)

CHAPTER 2: Thermodynamics and the Compressible Rotating Stall Inception Model

2.1 Compressor Characteristic

As previously mentioned, empirical compressor

characteristics associated two low-speed laboratory

compressors were used in this study by way of

simplicity and illustrative purpose of the concepts

proposed herein. Characteristic curves and pressure

loss buckets for the MIT single-stage compressor

(Gysling and Greitzer, 1995) and the MIT three-stage

compressor (Haynes et al, 1994) are shown in Figure

2.1a. The polynomial expressions for the MIT single-

stage compressor are as follows:

94.14.104.1475.5 23 −+−=Ψ φφφts (2-1)

1039.1 +−=Ψ φisen (2-2)

198.0 +−=Ψ φideal (2-3)

The MIT three-stage compressor expressions are:

85.1430.94.10 2 −+−=Ψ φφts (2-4)

15

φφφ

011.1374.3995.8499.7 2 +−+−=Ψisen (2-5) 3559.2547.0 2 +−−=Ψ φφideal (2-6)

φγ

02.1−=∂

Ψ∂ ideal (2-7)

Where, Ψ ,φ , and γ are the pressure rise coefficient, flow coefficient, and stagger angle

respectively.

The thermal loss due to viscous dissipation is estimated by:

tsisenL Ψ−Ψ=φ (2-8)

and the propulsive loss due to blade flow incidence deviation changes was estimated by :

isenidealdL Ψ−Ψ= (2-9)

These compressor characteristics were used to create critical performance maps, which will

be discussed in the proceeding sections. The compressor input parameters are assumed as

follows:

o design shaft speed - 10,000 rpm

o axial velocity ratio - 0.5

o density of atmospheric air - 1.229 kg/m3

o specific heat ratio - 1.4

o atmospheric pressure - 101.3 kPa

Through manipulation of the total- to-static pressure rise coefficient equation shown above

and assuming the given inlet pressure, P1, inlet density, ?o and wheel speed, U, the

performance map for a MIT single-stage and MIT three-stage compressor can be developed.

This map can then be used to evaluate essential thermodynamic properties (Gresh 2001).

First, the exit to inlet pressure ratio must be obtained for the compressor. The total-to-static

pressure rise coefficient equation can be defined as:

16

2

12

21 U

PP

o

ts

ρ

−=Ψ (2-10a)

Solving for P2,

Ψ+= 2

12 21

UPP ots ρ (2-10b)

Finally, dividing by the inlet pressure, yields the pressure ratio

1

2

1

2 21

1P

U

PP ots

Ψ

+=ρ

where P2= pressure at the compressor exit (2-10c)

Now that we have attained the exit-to-inlet pressure ratio, we can now develop the

exit-to- inlet temperature ratio using the laws of thermodynamics for a polytropic process

using the following relation (Gresh 2001):

( ) nn

PP

TT

/1

1

2

1

2

−

= where n=polytropic exponent (2-11)

Finally, the exit-to-inlet density ratio can be found by dividing the pressure ratio by

the temperature ratio (Gresh 2001):

=

1

2

1

2

1

2

TT

PP

ρρ

(2-12)

2.2 Losses and Efficiency

The propulsive pressure loss due to deviation (Ld) was estimated by taking the

difference between the ideal pressure rise characteristic (? ideal) and the isentropic

characteristic (? isen). Similarly, the thermal pressure loss due to viscous dissipation (Lf ) was

estimated from the difference between the isentropic pressure characteristic (? isen) and the

measured total-to-static characteristic (? ts). To obtain a reasonable measure of loss at each

17

speed, both deviation and viscous loss were proportioned appropriately. For example, if the

engine operates at 70 percent design speed, loss is then reduced by 30 percent. Accordingly,

if the engine is sped up to 110 percent design, loss is then increased by 10 percent.

Propulsive, thermal, and overall efficiency can be found by considering the losses.

Thermal efficiency is:

ϕη Lt −= 1 (2-13)

Propulsive efficiency is defined as:

dp L−= 1η (2-14)

The total loss experienced by the compressor is:

ϕLLL dtot += (2-15)

Therefore, the overall efficiency of the compressor can now be simply described as

follows:

tottot L−=1η (2-16)

Now that all efficiencies of the machine have been defined, the work and head can

now be considered (Gresh, 2001). Specifically, propulsive head, thermal head, propulsive

work, and thermal work will be evaluated.

Propulsive head is:

−

−=

−

11

/)1(

1

2

1

1

nn

p PPP

nn

Hρ

(2-17)

Thermal head is:

−

−=

−

11

/)1(

1

2

1

1

kk

t PPP

kk

Hρ

(2-18)

where k=specific heat ratio

Propulsive work is:

18

p

p

HW

=η

(2-19)

Thermal work is:

t

tHW

=

η (2-20)

2 .3 Compressible Rotating Stall Inception Model

The stability model of McGee et al (2004) was similar to the model of Moore and

Greitzer (1986), which was an extension of Emmons’ original work on stall inception in

1955. Emmons theorized that the stall cell starts as a flow separation in a single blade

passage, which causes blockage in the passage. As a result, approaching flow is diverted to

the adjacent blade passage causing the stalled cell to propagate. Circumferentially, the length

of this type of disturbance was short scale, meaning that it was limited to a small number of

blade passages. The more inclusive model of Moore and Greitzer (1986) idealized a multi-

stage compressor mathematically as a two-dimensional, incompressible flow machine of

large hub-to-tip ratio, with three-dimensional unsteady effects at the casing walls. Their

model assumes that the flow and its perturbations are radially uniform. This model also

assumes that the initial harmonic wave-like disturbance is circumferentially longer than that

of the Emmons-type disturbance. As the point of instability is approached, this disturbance

grows in intensity until it becomes a fully developed stall cell. The low-speed model of

McGee et al (2004) captures the same essential physics and mathematics as that of Moore

and Greitzer (1986) with the addition of several aeromechanical feedback stabilization

schemes. McGee’s low-speed stability model, however, made a few simplifying assumptions

such as, constant shaft speed, constant axial velocity, constant density ratio, and constant

19

temperature ratio suggesting that the fluid is incompressible. This notion is fairly valid at

relatively low speeds. However, as the compressor approaches higher speeds this theory

collapses and the fluid is, in fact, compressible. Since the current study involves high-speed

compressible fluid, the low-speed stability model of McGee et al (2004) and Frechette et al

(2004) had to be modified to account for variable shaft speed, variable axial velocity,

variable density ratio, and variable temperature ratio to incorporate compressibility affects.

To achieve this, Fréchette’s (1997) fluidic compressibility parameters were integrated into

the model. These parameters are (Fréchette, 1997):

o axial velocity-density ratio,

1,1

2,2

x

x

VV

AVDRρρ

= (2-21)

o blade row continuity parameter,

21 cI br

+= where 1,

2,

x

x

VV

c = (2-23)

These two compressibility parameters modify the fluidic blade row inertias of McGee

et al (2004) in the rotors, ?, and also in the rotors + stators, µ. The blade row inertia

parameters are now defined as:

AVDRIrcbr

rotors

ox

= ∑ γ

λ2cos/

(2-24)

AVDRIrcbr

statorsrotors

ox

= ∑

+ γµ

2cos/

(2-25)

with cx , ro , and γ representing the axial chord, mean radius, and blade row stagger angle,

respectively.

For low-speed incompressible flow assumptions, the velocity and density are held

constant, thus, the AVDR and Ibr is unity. In addition, the fluidic blade row inertia parameters

(equations 2-24 and 2-25) are identical to the low-speed model of McGee et al (2004) and

20

held constant. In the present study, the Ibr is constant at 0.75 and AVDR is variable due to the

changes in the density ratio.

21

CHAPTER 3: Aeromechanical Feedback Control

3.1 Static Pressure Sensing, H(s)

The structural feedback responds to fluctuations in static pressure in the ducts either

upstream, downstream, or within the compressor depending on the scheme. These unsteady

pressure disturbances are then used to serve as input parameters to the structural controller. A

transfer function for the sensor was developed by McGee et al. (2004) relating the upstream

or downstream velocity with the pressure disturbance. The transfer function is defined as

follows:

Φ+−= um

ssH 2)( where s=iω (3-1)

with m and Fu representing the mth spatial harmonic mode and steady sate upstream axial

flow, respectively. In the present analysis m was set to unity, assuming the instability of the

first harmonic as the initial inception of stalling condition, and Fu was variable. By definition,

the disturbance rotational frequency is:

n

mµ

λω −= where µµ +=mn

4 (3-2)

3.2 Structural Controller, C(s)

22

McGee et al (2004) also developed a transfer function for the structural controller.

This structural controller provides the feedback to the compression system. This controller is

defined as:

22 2

)(QsQs

WsC

++=

ξ (3-3)

with W , Q , and ξ as the mass ratio, frequency ratio, and critical damping ratio, respectively.

In McGee et al (2004) these parameters were restricted to constructible sizes for low-speed

compressor builds using materials that were readily obtainable. In the current high-speed

compressor study, these parameters carry the same restrictions. The constraints restricted the

mass ratio to 583.0=W , the frequency ratio within 9.23.0 ≤≤ Q , and the critical damping

ratio within 9.23.0 ≤≤ Q (Gysling & Greitzer, 1995). These ranges of values for the

frequency and critical damping ratio were chosen to maximize stability in the compression

system and are shown in Figure 3.2a below (McGee et al, 2004; Frechette et al, 2004).

Fig. 3.2a Structural control parameters for maximum stable extension. Optimal structural frequency, Q, and damping ratio, ξ are shown for the various aeromechanical

schemes. (McGee et al, 2004) Note: Time lags are not considered in the present study and the frequency needed to be increased from 0.3 to 1.1 for the 3-stage scheme # 10 in order to achieve reasonable stability.

23

3.3 Effective Slope

In order to delay the inception of stall in the machine, the compression system was

aeromechanically dampened instantly. To achieve this, McGee et al (2004) defined the

“ideal” effective growth rate of the compression system at neutral stability as:

βφφ

cosHCbrts

eff

ts +∂Ψ∂

=

∂Ψ

(3-4a)

with φ∂

Ψ∂ ts representing the slope of the steady-state compressor characteristic and rb

denoting the essential feedback control parameter for this study. The β term measures the

degree to which the fluid is in phase with the structural response of the controller. In this

study we ideally assume that the fluid is 180 degrees out of phase with the structural response

thus reducing the equation to:

HCbrts

eff

ts −∂Ψ∂

=

∂Ψ

φφ (3-4b)

with the term HCbr defined as the ideal control authority. The rb parameter is defined

differently for each scheme. These definitions are presented in Table 2 of the next section.

The equation of the best fit line through the points that create the effective slope versus

corrected flow scatter plot is integrated with a certain initial condition to produce an effective

total-to-static characteristic curve ( )efftsΨ . The initial condition simply implies that ( )efftsΨ

must initially equal ( )tsΨ at stall.

24

Fig. 3.4a Illustration of the ten aeromechanical feedback schemes (McGee et al, 2004)

3.4 Description of Aeromechanical Feedback Stabilization Schemes

The aeromechanical feedback

strategies studied are shown in Fig. 3.4a

and listed in Table 1. They are

categorized as (i) dynamic fluid injection

(with and without exit flow recirculation)

to supplement the axial momentum

entering the compressor, implemented

with a circumferential array of reed valve inj ectors that react to local static pressure (Gysling

and Greitzer, 1995) (Schemes

#1-#4); (ii) movable compressor

inlet and exit duct walls for flow

field impedance control,

potentially implemented as

flexible wall liners or as a

structurally tuned case that

resonates with the pre-stall, local

static pressure fluctuations

(Schemes #5-#7); (iii) flexible

compressor casing wall to

provide dynamic control of rotor tip clearance flow processes, implemented through

structurally- tuned casing or flexible casing treatment (Scheme #8); and (iv) dynamically

restaggered inlet guide vanes and rotor blades, possibly implemented through flexible root

attachments or structurally tuned blades (Schemes #9 & #10). (McGee et al, 2004)

25

The broad range of control schemes are considered without limiting the study to the

most feasible schemes to implement. This provides further insight on the impact of the

interactions between the fluid and structure during the inception of stall. It also indicates

which phenomena are most beneficial or detrimental in stabilizing the compression system.

Different combinations of actuation principles and sensing locations (upstream, downstream,

and average) are investigated. Further design work would be necessary to construct a

practical configuration that implements the most promising approaches.

A schematic of the system considered is shown in Fig. 3.4a. Flow enters through an

inlet duct, a compressor then pumps the flow through an exit duct to a plenum, which then

exhausts through a throttle (not shown) and an exit nozzle (not shown). The aeromechanical

components are integrated at one of the various locations shown. The table shown below

defines the essential feedback control parameter for the current study, br. As mentioned in the

previous section, this br factor is an essential component of the ideal control authority of

McGee et al (2004).

26

*Note that br is divided by µn for all schemes. For schemes #2 and #4 64.0=∂Ψ∂

i

ideal

α, for scheme

#8 8.0=∂

Ψε

ideal , and for scheme #10 ϕγ

02.1−=∂

Ψ∂ ideal (McGee et al, 2004).

Table 2 Essential Feedback Control Parameter, br*

# 1

Φ−Φ+

∂Ψ∂

Φ )(2 uits

i φ

#2 ( )2/ uii

ideal ΦΦ

∂Ψ∂

α

#3

Φ+

∂Ψ∂

Φ )2 iideal

i φ

#4 ( ) uiuii

ideal ΦΦ+ΦΦ

∂Ψ∂

2/ 2

α

#5

Φ−

∂Ψ∂

Φ )2 uideal

u φ

#6 22 uts Φ−Ψ

#7 uts

uts Φ

∂Ψ∂

−Φ−Ψφ

22

#8 ε∂

Ψideal

#9 igvuideal n µλγ

Φ−∂

Ψ 2

#10 γ∂

Ψideal

27

CHAPTER 4: Nonlinear Measured High-Speed Compression System Dynamics

4.1 Introduction

The following data has been produced for the MIT single-stage compressor using the

compressor characteristic of Gysling and Greitzer (1995). Subsequent data was produced for

the MIT three-stage compressor using the characteristic of Haynes et al (1994). The curves

were then manipulated to include aeromechanical feedback for each scheme. A detailed

explanation of findings will be presented using scheme #1 (radially mixed-out injection at the

compressor face) as the basis for discussion. The results for the remaining schemes can be

found in the Appendix section. A new metric called elasticity, which is a non-dimensional

measure of relativeness, will be discussed in the next chapter.

4.2 Corrected Pressure Ratio (Density Ratio)

Figure 4.2a depicts the Corrected Pressure Ratio vs. the Corrected Flow at variable

speed. In order to obtain these “corrected” quantities the pressure was divided by the

temperature ratio (T2/T1) and the flow was divided by the pressure ratio (P2/P1). The

corrected pressure ratio can now be defined as the density ratio. See Section 2.1 Compressor

Characteristic for the development of these quantities. The corrected T2/T1 values on the

opposite side of the y-axis depict lines of constant temperature ratios between the compressor

inlet (T1) and the compressor exit /combustor inlet (T2). Figure 4.2b also represents the

28

Corrected Pressure Ratio vs. the Corrected Flow; however, the values on the opposite side of

the y-axis depict lines of constant corrected T3/T1 ratios between the compressor inlet (T1)

and the combustor exit/turbine inlet (T3). The critical T3/T1 ratio was found to be 1.126 for

the MIT single-stage, as shown in the diagram, because it is the highest T3/T1 temperature

ratio that can be achieved at 70 percent design before the machine begins to stall. This value

was estimated by obtaining the slope from the highest point on the 70 percent corrected

pressure ratio curve to the origin at one. The remaining T3/T1 lines were then proportioned

accordingly. Notice that the safe operating line is not located along the maximum efficiency

line; instead, it is located a safe distance away from the surge line called the stall margin

(recall the discussion in Chapter 1).

The effective surge line with scheme #1 aeromechanical feedback is found in Figure

4.2c and Figure 4.2d. With the new surge line in place, the compressor will now be capable

of operating closer to the original maximum efficiency line. Also, notice how the T2/T1

(compressor exit to compressor inlet) temperature ratio lines shift as a result of the increased

corrected pressure due to the feedback scheme. The T3/T1 (combustor exit to compressor

inlet), however, remains unchanged; but, Figure 4.2d gives a new critical T3/T1 ratio of 1.24,

which is about a 10% increase in temperature.

29

Corrected Pressure Ratio (Density Ratio)

94.7%

95.2%

95.7%

96.2%

96.7%

0.99

1

1.01

1.02

1.03

1.04

1.05

1.06

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Co

rrec

ted

Pre

ssu

re R

atio

(Den

sity

Rat

io)

110% 100% 90% 80% 70% Surge Line

Maximum Efficiency Line

70 80% 90% 100% 110%

102.0E-2

101.5E-2

101.0E-2

100.5E-2

Corrected T2/T1

100.75E-2

101.25E-2

101.75E-2

102.2E-2

Safe Operating Line

Fig. 4.2a Corrected Pressure Ratio & Density ratio versus Corrected Flow of a MIT single stage compressor with lines of constant T2/ T1 temperature ratio.


94.7%

95.2%

95.7%

96.2%

96.7%

0.99

1

1.01

1.02

1.03

1.04

1.05

1.06

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Co

rrec

ted

Pre

ssu

re R

atio

(Den

sity

Rat

io)

110% 100% 90% 80% 70% Surge Line


70 80% 90% 100% 110%

112.6E-2114.0E-2116.0E-2118.0E-2

110.0E-2

108.0E-2

106.0E-2

104.0E-2

102.0E-2

Corrected T3/T1

Safe Operating Line

Fig. 4.2b Corrected Pressure Ratio & Density ratio versus Corrected Flow of a MIT single stage compressor with lines of constant T3/ T1 temperature ratio.

30

Corrected Pressure Ratio (Density Ratio)with Scheme 1 Feedback

94.7%

95.2%

95.7%

96.2%

96.7%

92.6%

93.3%

94.0%

94.6%

95.3%

94.4%

93.6%

92.8%

92.0%

91.2%

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

0.1 0.15 0.2 0.25 0.3 0.35 0.4

Corrected Flow

Co

rrec

ted

Pre

ssu

re R

atio

(Den

sity

Rat

io)

110% 100% 90% 80% 70% Surge Line Effective Surge Line

Safe Operating Line


101.75E-2

101.5E-2

101.25E-2

101.0E-2

100.75E-2

100.5E-2

102.0E-2

102.2E-2

Corrected T2/T1

102.6E-2

102.3E-2

102.0E-2

101.7E-2

101.4E-2

101.1E-2

Fig. 4.2c Corrected Pressure Ratio & Density ratio versus Corrected Flow of a MIT single stage compressor with lines of constant T2/ T1 temperature ratio and aeromechanical feedback.


96.7%

96.2%

95.7%

95.2%

94.7%

95.3%

94.6%

94.0%

93.3%

92.6%

91.2%

92.0%

92.8%

93.6%

94.4%

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

0.1 0.15 0.2 0.25 0.3 0.35 0.4

Corrected Flow

Co

rrec

ted

Pre

ssu

re R

atio

(D

ensi

ty R

atio

)


Safe Operating Line


112.0E-2

110.0E-2

108.0E-2

106.0E-2

104.0E-2

102.0E-2

114.0E-2

116.0E-2

118.0E-2Corrected T3/T1

120.0E-2122.0E-2124.0E-2126.0E-2

Fig. 4.2 d Corrected Pressure Ratio & Density ratio versus Corrected Flow of a MIT single stage compressor with lines of constant T3/ T1 temperature ratio and aeromechanical feedback.

31

4.3 Compressor Stability

Shown below is the stability of the compression system. It measures the slope of the

MIT single-stage compressor at various speeds. The surge line extends across the horizontal

axis at a value equal to zero. A negative slope means that the compression system is stable

and corresponds to the values below the surge line on Figure 4.3a. Note that the effective

surge line extends to the left of the original surge, thus enabling a dynamically-compensated

compressor using Scheme #1 aeromechanical feedback to reach stability at lower flow as

shown in Figure 4.3b.

Slope of the Pressure Rise Characteristic vs. Corrected Flow

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Slo

pe

110% 100% 90% 80% 70% Surge Line

UNSTABLE

STABLE

Fig. 4.3 a Corrected Compressor Stability versus Corrected Flow of a MIT single-stage compressor.

32

Slope of the Pressure Rise Characteristic vs. Corrected Flowwith Scheme 1 Feedback

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Slo

pe

of

Pre

ssu

re R

ise

Ch

arac

teri

stic

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Eff

ecti

ve S

lop

e o

f Pre

ssu

re R

ise

Ch

arac

teri

stic


UNSTABLE

STABLE

Fig. 4.3 b Corrected Compressor Stability versus Corrected Flow of a MIT single-stage compressor

with aeromechanical feedback.

4.4 Compressor Efficiency

The efficiency of the compression system can be divided into three categories-

thermal efficiency, propulsive efficiency, and overall efficiency. It should be duly noted,

however, that thermal efficiency is a more useful measure in analyzing high-speed

compressor performance. The efficiency was calculated and plotted using the compressor

characteristic equations. See the Section 2.2 Losses and Efficiency for a description of the

efficienc ies. Aeromechanical feedback causes maximum thermal and overall efficiency at

design speed to be regained by 95.794% and 96.660%, respectively (see Figs. 4.4.1b and

4.4.3b). The propulsive efficiency, however, is shown in Figure 4.4.2 exhibit negligibly small

change with shaft speed. This is because we are assuming the propulsive efficiency is not

affected by changes in the total- to-static pressure rise characteristic, although propulsive

efficiency is assumed to be affected by empirically-measured affected aerodynamic flow

losses due to blade flow incidence deviation changes, whose overall aggregated effects are

considerably smaller by comparison to viscous dissipation losses attributing to the

33

considerably larger thermal efficiency of the Scheme #1 dynamically-compensated

compression system.

4.4.1 Thermal Efficiency (refer to equation 2-13)

Thermal (Dissipation) Efficiency

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ther

mal

Effi

cien

cy

110% 100% 90% 80% 70% Surge Line

Fig. 4.4.1a Thermal Efficiency versus Corrected Flow of a MIT single-stage compressor at

various speeds.

34

Thermal (Dissipation) Efficiency with Scheme 1 Feedback

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ther

mal

Effi

cien

cy


Fig. 4.4.1 b Thermal Efficiency versus Corrected Flow of a MIT single-stage compressor with

aeromechanical feedback at various speeds.

4.4.2 Propulsive Efficiency (refer to equation 2-14)

Propulsive (Deviation) Efficiency

0.94

0.945

0.95

0.955

0.96

0.965

0.97

0.975

0.98

0.985

0.99

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Pro

pulsive

Efficien

cy

110% 100% 90% 80% 70% Surge Line

Fig. 4.4.2a Propulsive efficiency versus Corrected Flow of a MIT single-stage compressor at various

speeds . Note: Propulsive efficiency is not affected by changes in the total-to-static characteristic according

to its definition. Hence, aeromechanical feedback will not affect the propulsive efficiency of the machine.

Refer to section on Losses and Efficiency.

35

4.4.3 Overall Efficiency (refer to equation 2-16)

Overall (Propulsive & Thermal) Efficiency

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ove

rall

Eff

icie

ncy

110% 100% 90% 80% 70% Surge Line

Fig.4.4.3a Overall efficiency versus Corrected Flow of a MIT single-stage compressor at various

speeds .

36

Overall (Propulsive & Thermal) Efficiency with Scheme 1 Feedback

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ove

rall

Effi

cien

cy


Fig.4.4.3 b Overall efficiency versus Corrected Flow of a MIT single-stage compressor with

aeromechanical feedback at various speeds.

4.4.4 Efficiency Ratio (Propulsive/Thermal)

Efficiency (Propulsive/Thermal)

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Effic

iency

Rat

io

110% 100% 90% 80% 70% Surge Line

Fig. 4.4.4a Efficiency Ratio (Propulsive/Thermal) versus Corrected Flow of a MIT single-stage

compressor at various speeds.

37

Efficiency (Propulsive/Thermal) Ratiowith Scheme 1 Feedback

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Eff

icie

ncy

Rat

io


Fig.4.4.4b Efficiency Ratio (Propulsive/Thermal) versus Corrected Flow of a MIT single-stage

compressor with aeromechanical feedback at various speeds.

4.5 Compressor Losses

The loss of the compression system can be divided into the same three categories as

efficiency- thermal loss, propulsive loss, and overall loss. The loss was calculated and

plotted using the compressor characteristic equations. See the Section 2.2 Losses and

Efficiency for a description of the losses. In accordance with compressor efficiency, notice

that aeromechanical feedback causes loss at design speed to be reduced to 95.794% and

96.660%, respectively (see Figs. 4.5.1b and 4.5.3b). The propulsive loss, however, remains

unchanged because of how it was defined in Section 2.2 Losses and Efficiency. We are

assuming the propulsive loss is not affected by changes in the total-to-static pressure rise

characteristic.

38

4.5.1 Thermal Loss (refer to equation 2-8)

Thermal (Dissipation) Loss

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Th

erm

al L

oss

110% 100% 90% 80% 70% Surge Line

Fig. 4.5.1a Thermal Loss versus Corrected Flow of a MIT single-stage compressor at various speeds.

Thermal (Dissipation) Losswith Scheme 1 Feedback

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ther

mal

(Dis

sipat

ion)

Loss


Fig. 4.5.1 b Thermal Loss versus Corrected Flow of a MIT single-stage compressor with aeromechanical

feedback at various speeds.

39

4.5.2 Propulsive Loss (refer to equation 2-9)

Propulsive (Deviation) Loss

0

0.01

0.02

0.03

0.04

0.05

0.06

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Pro

pu

lsiv

e L

oss

110% 100% 90% 80% 70% Surge Line

Fig. 4.5.2a Propulsive Loss versus Corrected Flow of a MIT single-stage compressor at various speeds.

Note: Propulsive loss is not affected by changes in the total-to-static characteristic according to its definition.

Hence, aeromechanical feedback will not affect the propulsive loss of the machine. Refer to section on

Losses and Efficiency.

40

4.5.3 Overall Loss (refer to equation 2-15)

Overall (Propulsive & Thermal) Loss

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ove

rall

Lo

ss

110% 100% 90% 80% 70% Surge Line

Fig. 4.5.3a Overall Loss versus Corrected Flow of a MIT single-stage compressor at various speeds.

Overall (Propulsive & Thermal) Losswith Scheme 1 Feedback

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ove

rall

Effic

iency


Fig. 4.5.3b Overall Loss versus Corrected Flow of a MIT single-stage compressor with aeromechanical

feedback at various speeds.

41

4.5.4 Loss Ratio (Propulsive/Thermal)

Loss Ratio (Propulsive/Thermal)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Lo

ss R

atio

110% 100% 90% 80% 70% Surge Line

Fig. 4.5.4a Loss Ratio (Propulsive/Thermal) versus Corrected Flow of a MIT single-stage compressor

at various speeds.

Loss (Propulsive/Thermal) Ratiowith Scheme 1 Feedback

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Loss

Rat

io


Fig. 4.5.4 b Loss Ratio (Propulsive/Thermal) versus Corrected Flow of a MIT single-stage compressor

at various speeds.

42

CHAPTER 5: Compressor Elasticity

5.1 Introduction

Compressor elasticity is a measure of the unsteadiness in the compression system.

This chapter will explain how to develop an elasticity quantity. Further explanation is

discussed in the Appendix. This chapter will also demonstrate the affect of aeromechanical

feedback on two elasticity measures (corrected pressure-flow and corrected pressure-density).

The effect of aeromechanical feedback on the other elasticity measures will be le ft for future

study. However, the open- loop (without aeromechanical feedback) results are presented. In

the subsequent sections, findings shown illustrate the unsteadiness of the compression system

with respect to several different thermodynamic properties. Also introduced in this chapter is

the concept of relative non-dimensional flow, which will be briefly explained. Finally, the

chapter ends with presenting the results of the pumping and aeroelastic characteristic for a

gas generator based on the MIT single-stage compressor with and without aeromechanical

feedback. The aeroelastic characteristic was only measured for the corrected pressure-flow

and corrected pressure-density elasticities, the measurement of the other aeroelastic

characteristics are left for future study. The results of other elasticities, pumping, and

aeroelastic characteristics for the MIT single-stage and MIT three-stage compressor for the

remaining schemes can be found in the Appendix.

43

5.2 Corrected Pressure -Flow (Pf ) Elasticity

In order to create an elasticity, or unsteadiness, measure you first need to develop a

relationship between two quantities- in this case corrected pressure ratio and corrected flow.

In Section 4.2 this relationship was formed and plotted (see Fig. 4.2a). Once the relationship

of the corrected pressure ratio and corrected flow has been obtained, the relative change in

the corrected pressure ratio to the relative change in the flow (or the slope of the corrected

pressure ratio vs. corrected flow curve) is measured. This quantity is then multiplied by the

original corrected flow to corrected pressure ratio. The result is a corrected pressure-flow

elasticity quantity. Mathematically, it is:

∆∆

=o

oelasticity P

PP

ϕϕ

ϕ

All other elasticities (other than the corrected pressure-density elasticity) were derived in a

similar manner. See the Appendix for a more general formulation.

Figures 5.2a and 5.2b show the corrected pressure-flow elasticity of a MIT single-

stage compressor at various speeds with and without feedback, respectively. Note the

similarity of the relationship between the slope of the pressure rise characteristic versus

corrected flow (Fig. 4.3b) and the corrected pressure-flow elasticity versus corrected flow

(Fig. 5.2b), which both show scheme #1 feedback enabling stability at lower flow; however,

the slope of the latter is flatter than the slope of the former, which suggests that the corrected

pressure-flow elasticity is less prone to fluctuate when the corrected flow changes.

44

Corrected Pressure- Flow (Pf) Elasticity vs. Corrected Flow

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Co

rrec

ted

Pre

ssu

re-

Flo

w (

Pf)

Ela

stic

ity

110% 100% 90% 80% 70% Surge Line

STABLE

UNSTABLE

Fig. 5.2a Corrected Pressure-Flow Elasticity versus Corrected Flow of a MIT single-stage compressor

at various speeds.

Corrected Pressure-Flow Elasticity (Pf ) with Scheme 1 Feedback

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Co

rrec

ted

Pre

ssu

re-F

low

(Pf)

Ela

stic

ity


UNSTABLE

STABLE

Fig. 5.2 b Corrected Pressure-Flow Elasticity versus Corrected Flow of a MIT single-stage compressor

with aeromechanical feedback at various speeds.

45

5.3 Corrected Pressure -Density (P?) Elasticity

The corrected pressure-density elasticity is the polytropic exponent, n, and was

derived from the propulsive efficiency, pη , and specific heat ratio, k. It is described as

follows (Gresh 2001):

( )

( ) 11

1

−==

−

−

kk

p

kk

pelasticity nP

η

ηρ

Note that it is assumed that aeromechanical feedback doesn’t affect the corrected

pressure-density elasticity because neither propulsive efficiency nor the specific heat ratio is

affected by changes in the total-to-static pressure characteristic.

Corrected Pressure-Density Elasticity (P?) vs. Corrected Flow

1.405

1.41

1.415

1.42

1.425

1.43

1.435

1.44

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Co

rrec

ted

Pre

ssu

re-D

ensi

ty (P

?) E

last

icit

y

110% 100% 90% 80% 70% Surge Line

Fig. 5.3a Corrected Pressure-Density Elasticity versus Corrected Flow of a MIT single-stage

compressor with aeromechanical feedback at various speeds. Note: Corrected Pressure-Density

Elasticity is not affected by changes in the total-to-static characteristic but rather changes in the polytropic

exponent.

46

5.4 Corrected Pressure-Specific Volume Elasticity (Pv) (see Appendix for basic elasticity derivation)

Since this is the first section where it is derived, it is fitting to discuss the relative

non-dimensional corrected flow. The relative non-dimensional corrected flow is simply a

quantity that measures the relative change in the corrected flow with respect to the original

flow. It is defined as follows:

oϕϕϕ ∆=′

Altering the x-axis from corrected flow to relative non-dimensional corrected flow has

significant consequences; particularly, the surge line becomes almost vertical, which means

that the relative non-dimensional corrected flow at surge is constant (see Fig. 5.4c).

Corrected Pressure Ratio vs. Corrected Specific Volume Ratio(WORK)

0.99

1

1.01

1.02

1.03

1.04

1.05

1.06

0.94 0.95 0.96 0.97 0.98 0.99 1 1.01

Corrected Specific Volume Ratio v2/v1

Co

rrec

ted

Pre

ssu

re R

atio

P2/

P1

100% 110% 90% 80% 70%

Fig. 5.4 a Corrected Pressure Ratio versus Corrected Specific Volume Ratio of a MIT single-stage


47

Corrected Pressure-Specific Volume (Pv) Elasiticity vs. Corrected Flow

-1.04

-1.02

-1

-0.98

-0.96

-0.94

-0.920 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Corr

ecte

d P

ress

ure

-Spec

ific

Volu

me

(Pv)

Ela

stic

ity

110% 100% 90% 80% 70% Surge Line

Fig. 5.4 b Corrected Pressure-Specific Volume Elasticity versus Corrected Flow of a MIT single-stage


Corrected Pressure-Specific Volume (Pv) Elasiticity vs. Relative Non-Dimensional Corrected Flow

-1.04

-1.02

-1

-0.98

-0.96

-0.94

-0.920.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024

Relative Non-Dimensional Corrected Flow

Corr

ecte

d P

ress

ure

-Spec

ific

Volu

me

(Pv)

E

lasi

ticity

110% 100% 90% 80% 70% Surge Line

Fig. 5.4c Corrected Pressure-Specific Volume Elasticity versus Relative Non-Dimensional Corrected

Flow of a MIT single-stage compressor at various speeds.

48

5.5 Corrected Pressure-Temperature (PT) Elasticity (see Appendix for basic elasticity derivation)

Corrected Pressure Ratio vs. Temperature Ratio (T2/T1)

0.99

1

1.01

1.02

1.03

1.04

1.05

1.06

0.995 1 1.005 1.01 1.015 1.02 1.025

Temperature Ratio (T2/T1)

Corr

ecte

d P

ress

ure

Rat

io

110% 100% 90% 80% 70%

Fig. 5.5a Corrected Pressure Ratio versus Corrected Temperature Ratio (T2/T1) of a MIT single-stage


Corrected Pressure- Temperature (PT) Elasticity vs. Corrected Flow

2.46

2.47

2.48

2.49

2.5

2.51

2.52

2.53

2.54

2.55

2.56

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

PT

Ela

stic

ity

110% 100% 90% 80% 70% Surge Line

Fig. 5.5b Corrected Pressure-Temperature Elasticity versus Corrected Flow of a MIT single-stage

compressor at various speeds .

49

Corrected Pressure- Temperature (PT) Elasticity vs. Relative Non-Dimensional Corrected Flow

2.46

2.47

2.48

2.49

2.5

2.51

2.52

2.53

2.54

2.55

2.56

0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024


PT

Ela

stic

ity

110% 100% 90% 80% 70% Surge Line

Fig. 5.5c Corrected Pressure-Temperature Elasticity versus Relative Non-Dimensional Corrected Flow

of a MIT single-stage compressor at various speeds.

50

5.6 Corrected Temperature-Loss (Ts) Elasticity (see Appendix for basic elasticity derivation)

The elasticities that contain losses (thermal & propulsive) also have rather interesting

effects- most likely because of the strange relationships that it has with the other

thermodynamic properties in this study. In this section, the corrected temperature ratio (T2/T1)

and losses are compared (see Fig. 5.6a). Note the considerable amount of thermal loss in

comparison to propulsive loss during operation. As a result, it is apparent that when

considering the overall efficiency of the compressor, it is more appropriate to consider the

thermal efficiency versus the propuls ive efficiency of the machine. The peculiar shape of the

temperature ratio versus thermal loss curve leads to asymptotic elasticity curves around the

x-axis (Fig. 5.6b). In addition, because the surge line is also along the x-axis ranging from

about 0.18 to 0.27 in Fig. 5.6b, it becomes a point at about 0.0186 when this axis is altered in

Fig. 5.6c. This phenomenon occurs in a similar manner in all the elasticities that involve loss.

(see Sections 5.7 and 5.9)

Temperature Ratio (T2/T1) vs. Thermal & Propulsive Loss

0.995

1

1.005

1.01

1.015

1.02

1.025

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Loss

Tem

per

atu

re R

atio

110% 100% 90% 80% 70%

(solid) Denotes thermal Loss (dotted) Denotes propulsive Loss

Fig. 5.6a Corrected Temperature Ratio (T2/T1) versus Thermal & Propulsive Loss of a MIT single-

stage compressor at various speeds.

51

Temperature- Thermal Loss (Ts_o) Elasticity vs. Corrected Flow

-0.5

0

0.5

1

1.5

2

2.5

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ts_

o E

last

icity

110% 100% 90% 80% 70% Surge Line

Fig. 5.6b Corrected Temperature-Thermal Loss Elasticity versus Corrected Flow of a MIT single-stage


Temperature- Thermal Loss (Ts_o) Elasticity vs. Relative Non-Dimensional Corrected Flow

-0.5

0

0.5

1

1.5

2

2.5

0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024


Ts_

o E

last

icity

110% 100% 90% 80% 70% Surge Line

Fig. 5.6c Corrected Temperature-Thermal Loss Elasticity versus Relative Non-dimensional Corrected


52

Temperature-Propulsive Loss (T_sd) Elasticity vs. Corrected Flow

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

T_s

d E

last

icity

110% 100% 90% 80% 70% Surge Line

Fig. 5.6d Corrected Temperature-Propulsive Loss Elasticity versus Corrected Flow of a MIT single-


Temperature-Propulsive Loss (T_sd) Elasticity vs. Relative Non-Dimensional Corrected Flow

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024


T_sd

Ela

stic

ity

110% 100% 90% 80% 70% Surge Line

Fig. 5.6e Corrected Temperature-Propulsive Loss Elasticity versus Relative Non-Dimensional

Corrected Flow of a MIT single-stage compressor at various speeds.

53

5.7 Corrected Pressure -Loss (Ps) Elasticity Elasticity (see Appendix for basic elasticity derivation)

Corrected Pressure Ratio vs. Thermal & Propusive Loss

0.99

1

1.01

1.02

1.03

1.04

1.05

1.06

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Loss

Co

rrec

ted

Pre

ssu

re R

atio

110% 100% 90% 80% 70%


Fig. 5.7a Corrected Pressure Ratio versus Thermal & Propulsive Loss of a MIT single-stage


54

Pressure-Thermal Loss (Ps_o) Elasticity vs Corrected Flow

-1

0

1

2

3

4

5

6

0 0.1 0.2 0.3 0.4 0.5

Corrected Flow

Ps_

o E

last

icity

110% 100% 90% 80% 70% Surge Line

Fig. 5.7b Corrected Pressure-Thermal Loss Elasticity versus Corrected Flow of a MIT single-stage


Pressure-Thermal Loss (Ps_o) Elasticity vs Relative Non-Dimensional Corrected Flow

-1

0

1

2

3

4

5

6

0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024


Ps_

o E

last

icity

110% 100% 90% 80% 70% Surge Line

Fig. 5.7c Corrected Pressure-Thermal Loss Elasticity versus Relative Non-dimensional Corrected Flow

of a MIT single-stage compressor at various speeds.

55

Corrected Pressure-Propulsive Loss (Ps_d) Elasticity vs. Corrected Flow

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ps_

d E

last

icity

110% 100% 90% 80% 70% Surge Line

Fig. 5.7d Corrected Pressure-Propulsive Loss Elasticity versus Corrected Flow of a MIT single-stage


Corrected Pressure-Propulsive Loss (Ps_d) Elasticity with respect to Propulsive Loss vs. Relative Non-Dimensional Corrected Flow

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024


Ps_

d E

last

icity

110% 100% 90% 80% 70% Surge Line

Fig. 5.7e Corrected Pressure-Propulsive Loss Elasticity versus Relative Non-Dimensional Corrected


56

5.8 Corrected Specific Volume-Temperature (vT) Elasticity Elasticity (see Appendix for basic elasticity derivation)

Corrected Specific Volume (v2/v1) vs. Temperature Ratio (T2/T1)

0.94

0.95

0.96

0.97

0.98

0.99

1

1.01

0.995 1 1.005 1.01 1.015 1.02 1.025

Temperature Ratio

Cor

rect

ed S

peci

fic V

olum

e (v

2/v1

)

110% 100% 90% 80% 70%

Fig. 5.8a Corrected Specifc Volume (v2/v1) versus Temperature Ratio (T2/T1) of a MIT single-stage


Corrected Specific Volume-Temperature (vT) Elasticity vs. Corrected Flow

-2.65

-2.6

-2.55

-2.5

-2.45

-2.40.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

vT E

last

icity

110% 100% 90% 80% 70% Surge Line

Fig. 5.8b Corrected Specifc Volume-Temperature Elasticity versus Corrected Flow of a MIT single-


57

Corrected Specific Volume-Temperature (vT) Elasticity vs. Relative Non-Dimensional Corrected Flow

-2.65

-2.6

-2.55

-2.5

-2.45

-2.4

0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024


vT E

last

icity

110% 100% 90% 80% 70% Surge Line

Fig. 5.8c Corrected Specifc Volume -Temperature Elasticity versus Non-dimensional Corrected Flow of

a MIT single-stage compressor at various speeds.

58

5.9 Corrected Specific Volume-Loss (vs) Elasticity Elasticity (see Appendix for basic elasticity derivation)

Corrected Specific Volume Ratio (v2/v1) vs. Thermal & Propulsive Loss

0.94

0.95

0.96

0.97

0.98

0.99

1

1.01

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Loss

Cor

rect

ed S

peci

fic

Vol

ume

Rat

io (

v2/v

1)

110% 100% 90% 80% 70%


Fig. 5.9a Corrected Specific Volume Ratio (v2/v1) versus Thermal & Propulsive Loss of a MIT single-


59

Corrected Specific Volume-Thermal Loss (vs_o) Elasticity vs Corrected Flow

-6

-5

-4

-3

-2

-1

0

1

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

vs_o

110% 100% 90% 80% 70% Surge Line

Fig. 5.9b Corrected Specific Volume–Thermal Loss Elasticity versus Corrected Flow of a MIT single-


Corrected Specific Volume-Thermal Loss (vs_o) Elasticity vs Relative Non-Dimensional Corrected Flow

-6

-5

-4

-3

-2

-1

0

1

0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024


vs_o

110% 100% 90% 80% 70% Surge Line

Fig. 5.9c Corrected Specific Volume–Thermal Loss Elasticity versus Relative Non-dimensional


60

Corrected Specific Volume-Propulsive Loss (vs_d) Elasticity vs. Corrected Flow

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

vs_d

Ela

stic

ity

110% 100% 90% 80% 70% Surge Line

Fig. 5.9d Corrected Specific Volume–Propulsive Loss Elasticity versus Corrected Flow of a MIT

single-stage compressor at various speeds.

Corrected Specific Volume-Propulsive Loss (vs_d) Elasticity vs. Relative Non-Dimensional Corrected Flow

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024


vs_d

Ela

stic

ity

110% 100% 90% 80% 70% Surge Line

Fig. 5.9e Corrected Specific Volume–Propulsive Loss Elasticity versus Relative Non-dimensional


61

5.10 Pumping and Aeroelastic Characteristics. Using the compressor map in Section 5.1

the pumping characteristic of a gas generator can be determined. In a similar manner the

aeroelastic characteristic with respect to density and flow can also be determined. The

purpose of these characteristics is to reveal the changes that occur to certain quantities when

the engine changes speed. The changes in corrected pressure ratio, corrected temperature

ratio, and corrected flow were measured at stall. The changes in these quantities at maximum

efficiency before and after stall were also compared.

5.10.1 Pumping Characteristic.

Compression System Pumping Characteristicat Maximum Efficiency & Stall

1

1.01

1.02

1.03

1.04

1.05

1.06

70% 80% 90% 100% 110%

Corrected Speed

Co

rrec

ted

Pre

ssu

re &

T

emp

erat

ure

Rat

ios

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Co

rrec

ted

Flo

w

Pressure Ratio @ Maximum Efficiency Temperature Ratio @ Maximum Efficiency

Pressure Ratio @ Surge Temperature Ratio @ Surge

Flow @ Maximum Efficiency Flow @ Surge

Design Line

Corrected T3/T1= 1.126

Fig. 5.10.1a Pumping Characteristic for a gas generator based on a MIT single-stage compressor.

62

Compression System Pumping Characteristicat Maximum Efficiency & Stall with Scheme 1 Feedback

1

1.01

1.02

1.03

1.04

1.05

1.06

70% 80% 90% 100% 110%

Corrected Speed

Co

rrec

ted

Pre

ssu

re &

T

emp

erat

ure

Rat

ios

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Co

rrec

ted

Flo

w




Design Line

X Denotes Effective Surge Line - - Denotes Effective Maximum Efficiency


Fig. 5.10.1b Pumping Characteristic for a gas generator based on a MIT single-stage compressor with

aeromechanical feedback .

63

5.10.2 Aeroelastic Characteristic.

Compression System Aeroelastic Characteristicsat Maximum Efficiency & Stall

1.41

1.412

1.414

1.416

1.418

1.42

1.422

1.424

1.426

70% 75% 80% 85% 90% 95% 100% 105% 110%

Corrected Speed

Co

rrec

ted

Pre

ssu

re-D

ensi

ty (

P?

) A

ero

elas

tici

ty

-0.45

-0.35

-0.25

-0.15

-0.05

0.05

0.15

0.25

0.35

0.45

Co

rrec

ted

Pre

ssu

re-F

low

(P

f)

Aer

oel

asti

city

P? Elasticity @ Stall P? Elasticity @ Maximum EfficiencyPf Elasticity @ Stall Design LinePf Elasticity @ Maximum Efficiency



Fig. 5.10.2a Aeroelastic Characteristic for a gas generator based on a MIT single-stage compressor.

Compression System Aeroelastic Characteristicsat Maximum Efficiency & Stall with Scheme 1 Feedback

1.41

1.412

1.414

1.416

1.418

1.42

1.422

1.424

1.426

70% 75% 80% 85% 90% 95% 100% 105% 110%

Corrected Speed

Corr

ecte

d P

ress

ure

-Den

sity

(P?) A

eroel

astic

ity

-0.45

-0.35

-0.25

-0.15

-0.05

0.05

0.15

0.25

0.35

0.45

Corr

ecte

d P

ress

ure

-Flo

w (P

f) A

eroel

astic

ity

P? Elasticity @ Stall P? Elasticity @ Maximum EfficiencyPf Elasticity & Effective Pf Elasticity @ Stall Design LinePf Elasticity @ Maximum Efficiency Effective Pf @ Max Efficiency

Note: Effective E_density= E_density



Fig. 5.10.2 b Aeroelastic Characteristic for a gas generator based on a MIT single-stage compressor

with aeromechanical feedback.

64

CHAPTER 6: Summary and Future Work

6.1 Summary

General methods were developed in this work to evaluate passive high-speed

stabilization of two laboratory gas generator devices - the MIT single-stage and MIT three-

stage compressors. A high-speed compressible rotating stall inception model was developed

and compressible flow was analyzed along with the effects of changes in its thermodynamic

properties. Ten passive stabilization schemes that could potentially be used by industry were

discussed and examined in a high-speed compressible flow environment. The concept of

elasticity was introduced and implemented to examine the effects of flow non-uniformity,

entropic loss, and unsteadiness on thermodynamic state changes within the compression

system. Finally, pumping and aeroelastic characteristics of these laboratory compressors

both with and without feedback were analyzed.

65

6.1 Future Work

In the present research, we were able to answer two questions that lingered for high-

speed aeroengine devices: (1) Are there high speed fluid-structural interactions associated

flow non-uniformity, entropic loss, and unsteadiness, which can be avoided using tailored

structural design to dynamically compensate compression systems through aeromechanical

feedback passive control? (2) How do aeromechanical feedback dynamics combine with the

pre-stall compressible fluid dynamics to postpone or induce the inception of high-speed

rotating stall of aeroengine compressors? However, two additional questions still remain: (1)

What are other stabilization schemes associated with geometry, steady flow, fluidic

compressibility, and operating performance of high-speed (compressible flow) compressors

and engines in contrast to those fundamental to low-speed (incompressible flow) devices? (2)

How do changes in the parameters that govern the pre-stall dynamics affect the stability of

not only the compression system, but also the matching components (compressor, combustion,

turbine) of the entire pumping system of the engine?

It is proposed here that in addition to the ten aeromechanical schemes presented in the

current research, other innovative aeromechanical schemes exist, through a careful sensitivity

analysis on geometric, steady flow, fluidic compressibility, and operating performance

parameters that can indeed inhibit the inception of rotating stall in high-speed engine devices.

As a result, the stable operating range will be effectively extended, allowing higher

performance operating conditions.

In order to achieve the goals presented in this proposal, a reduced-order model of

fluid-structure interaction inside an axial compressor system used in modern aircraft engines

has been devised. First, proof-of-concept schemes of innovative aeromechanical control

technologies to describe how such feedback can be utilized to stabilize high-speed

compressor stall of axial compressors, and how different tailored structural designs impact

66

high-speed compression system stability will be developed. Optimal structural parameters for

these unique aeromechanical compensators will be determined to maximize the stable

operating range of the high-speed compression system. The use of optimized aeromechanical

feedback control to stabilize the system and extend the operating range will be discussed.

The theoretical basis of additional control methods examined here will be evaluated under a

compressible flow environment. To complete the evaluation of the aeromechanical control

methodologies developed, a sensitivity analysis of the parameters that govern the pre-stall

dynamics will be conducted. Thirteen input parameters are required in a high-speed

compressor stall inception model, which include: geometry, steady flow values, fluidic

compressibility, and overall performance slopes of the compressor device. Specifically these

parameters are (Fréchette, 1997):

1. Geometry

o axial chords (cx/ro)

o stagger angles ( γ )

2. Steady flow values

o inlet axial flow coeffient (ϕ )

o inlet swirl angle ( inα )

o axial velocity ratios (1

2

ϕϕ )

o density ratios (1

2ρρ )

3. Overall compressor performance slopes

o overall pressure rise sensitivity to inlet flow coefficient ( ϕ∂Ψ∂ ts )

o overall pressure rise sensitivity to inlet swirl angle (in

tsαtan∂

Ψ∂ )

o last blade row deviation sensitivity to inlet flow coefficient ( ϕα

∂∂ extan )

67

o last blade row deviation sensitivity to inlet swirl angle (in

exαα

tantan

∂∂ )

4. Fluidic Compressibility Parameters

o axial velocity-density ratio (AVDR)

o axial momentum ratio (Y)

o blade row continuity parameter (Ibr)

These parameters are all part of Fréchette’s extended stability model which is:

+

+

−=

µ

ϕσ

nAVDR

dYnTS exIR

1,

2

where:

in

tsintsRTS

αϕα

ϕ tantan

∂Ψ∂

∂−

∂Ψ∂=

in

exexI n

dαα

ϕ tantan1

, ∂∂

=

AVDRIrcbr

statorsrotors

ox

= ∑

+ γµ

2cos/

1,1

2,2

x

x

VV

AVDRρρ

=

2

11,

2,

x

x

br

VV

I+

=

2

1,

2,

1

2

=

x

x

VV

Yρρ

Differentiating this extended stability equation with respect to each of these

parameters results in the following:

1. Geometry

68

o axial chords (cx/ro)

( )

++

−−=

∂∂

γµ

ϕσ22

,2

cos1

AVDR

nAVDR

dYnTS exIR

rc

ox

o stagger angles ( γ )

( ) AVDRIr

c

nAVDR

dYnTSbr

o

x

exIR

++

−=

∂∂

γµ

ϕ

γσ

22

,2

cos1

2. Steady flow values

o inlet axial flow coefficient (ϕ )

( ) 2

,2

1

12tan

tan

++

++

−

∂Ψ∂

=∂∂

µ

µϕαϕ

α

ϕσ

nAVDR

nAVDRYdn exI

in

tsin

o inlet swirl angle ( inα )

( )

++

∂

Ψ∂

−=∂

∂

µ

αϕα

σ

nAVDR

in

ts

in 1

tan1

tan

o axial velocity ratios (1

2

ϕϕ )

( )

[ ]

2

1

22

1

2,

2,

1

22

1

21

cos112

12

++

++

−−

++

−

=∂

∂

µ

ϕϕ

γρρ

ϕµϕϕ

ϕ

σ

ϕϕ

nAVDR

rc

ndYnTS

nAVDRdn o

x

exIRexI

69

o density ratios (1

2ρρ )

( )

[ ]

2

21

2,

2,

2

1

2

1

cos11

12

+

+

+

−−

+

+

−

=∂

∂

µ

γϕϕ

ϕµϕϕ

σ

ρρ

nAVDR

Ir

c

ndYnTS

nAVDR

dn bro

x

exIRexI

3. Overall performance slopes

o overall pressure rise sensitivity to inlet flow coefficient ( ϕ∂Ψ∂ ts )

( ) 21

1

++

=∂∂

∂Ψ∂

µ

σ

ϕ

nAVDR

ts

o overall pressure rise sens itivity to inlet swirl angle (in

ts

αtan∂Ψ∂ )

( )

++

−=∂

∂

∂Ψ∂

µ

ϕα

σ

α

nAVDR

in

in

ts 1

tan

tan

o last blade row deviation sensitivity to inlet flow coefficient ( ϕα

∂∂ extan )

( ) 0tan

=∂

∂

∂∂

ϕα

σex

o last blade row deviation sensitivity to inlet swirl angle (in

ex

αα

tantan

∂∂ )

( )

++

−=∂

∂

∂∂

µ

ϕσ

αα

nAVDR

Y

in

ex 1tantan

4. Fluidic Compressibility Parameters

o axial velocity-density ratio (AVDR)

70

( )

[ ]

2

2,2

1

cos1

++

+−−

=∂

∂

µ

γϕ

σ

nAVDR

Ir

c

ndYnTS

AVDR

bro

x

exIR

o axial momentum ratio (Y)

( )[ ]

++

−=∂∂

µ

ϕσ

nAVDR

dnY

exI

1,

2

o blade row continuity parameter (Ibr)

( )

[ ]

2

2,2

1

cos

++

−−

=∂∂

µ

γϕ

σ

nAVDR

AVDRr

c

dYnTS

I

o

x

exIR

br

which is the basis for the sensitivity analysis posed.

71

References Emmons, H.W., Pearson, C.E., and Grant, H.P., 1955, “Compressor Surge and Stall Propagation,” Trans. ASME, 77, pp. 455-469. Frechette, L.G., 1997, “Implications of Stability Modeling for High-Speed Axial Compressor Design,” Master Thesis, Department of Aeronautics and Astronautics, M.I.T., Cambridge, MA. Frechette, L.G., McGee, O.G., and Graf, M.B., 2004, “Tailored Structural Design and Aeromechanical Control of Axial Compressor Stall – Part II: Evaluation of Approaches,” ASME J. Turbomachinery, 126, pp. 63-72. Greitzer, E.M., 1976, “Surge and Rotating Stall in Axial Flow Compressors, Part I & II,” ASME J. Eng. Power, 99, pp. 190-217. Greitzer, E.M., 1980, “Review: Axial Compressor Stall Phenomenon,” ASME J. Fluids Eng., 102, pp. 134-151. Greitzer, E.M., 1981, “The Stability of Pumping Systems, The 1980 Freeman Scholar Lecture,” ASME J. Fluids Eng., 103, pp. 193-242. Gresh, M. Theodore, 2001, “Compressor Performance: Aerodynamics for the User,” Butterworth & Heinemann, Boston. Gysling, D.L., and Greitzer, E.M., 1995, “Dynamic Control of Rotating Stall in Axial Flow Compressors Using Aeromechanical Feedback,” ASME J. Turbomachinery, 117, pp. 307-319. Haynes, J.M., Hendricks, G.J., and Epstein, A.H., 1994, “Active Stabilization of Rotating Stall in a Three-Stage Axial Compressor, “ ASME J. Turbomachinery, 116, pp. 226-239. McGee, O.G., Graf, M.B., and Frechette, L.G., 2004, “Tailored Structural Design and Aeromechanical Control of Axia l Compressor Stall – Part I: Development of Models and Metrics, ASME J. Turbomachinery, 126, pp. 63-72. McGee, O.G. and Coleman, K.L., 2006a, “Aeromechanical Control of High-Speed Axial Compressor Stall and Engine Performance– Part I: Development of Models, to be submitted, ASME J. Turbomachinery. Coleman, K.L. and McGee, O.G., 2006b, “Aeromechanical Control of High-Speed Axial Compressor Stall and Engine Performance– Part II: Evaluation of Approaches, to be submitted, ASME J. Turbomachinery.

72

APPENDIX

A.1 Introduction This appendix contains the formulation of the elasticity concept and the results of the

nonlinear measured high-speed compression sys tem dynamics for the MIT single-stage and

MIT three-stage compressor for schemes #1-#8 and schemes #1-#10 (including the open-

loop problem), respectively.

73

A.2 Basic Elasticity Formulation

Step 1: Begin with a relationship between two quantities, X and Y.

X vs. Y

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

X

Y

Fig. A.2 Plot of the relationship between X and Y.

Step 2: Find the slope, m, of the XY relationship.

XYm

∆∆=

Step 3: Multiply the slope at point (X,Y) by the X to Y ratio at that point to obtain a

cross-elasticity measure .

=

YX

mYX elasticity

74

A.3 Additional Nonlinear Measured High Speed Compression System Dynamics

MIT 1 Scheme 2


94.7%

95.2%

95.7%

96.2%

96.7%

97.0%

96.6%

96.2%

95.7%

95.3%

92.6%

93.3%

94.0%

94.6%

95.3%

1

1.02

1.04

1.06

1.08

1.1

1.12

0.08 0.13 0.18 0.23 0.28 0.33 0.38

Corrected Flow

Co

rrec

ted

Pre

ssu

re R

atio

(Den

sity

Rat

io)


Safe Operating Line


Corrected T2/T1

10.75E-2

101.25E-2

101.75E-2

102.2E-2

104.1E-2

103.8E-2

103.5E-2103.2E-2

102.9E-2

102.6E-2


with Scheme 2 Feedback

94.7%

95.2%

95.7%

96.2%

96.7%

97.0%

96.6%

96.2%

95.7%

95.3%

92.6%

93.3%94.0%

94.6%95.3%

1

1.02

1.04

1.06

1.08

1.1

1.12

0.08 0.13 0.18 0.23 0.28 0.33 0.38

Corrected Flow

Co

rrec

ted

Pre

ssu

re R

atio

(Den

sity

Rat

io)


Safe Operating Line

Maximum Efficiency Line116.0E-2

112.0E-2

108.0E-2

104.0E-2

Corrected T3/T1

120.0E-2

124.0E-2

128.0E-2

132.0E-2140.0E-2166.0E-2 148.0E-2156.0E-2

Efficiency (Propulsive/Thermal) Ratio


0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Eff

icie

ncy

Rat

io

110% 100% 90% 80% 70% Surge Line Effective Surge Line Loss (Propulsive/Thermal) Ratio


0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Loss

Rat

io


Work (Propulsive/Thermal) Ratiowith Scheme 2 Feedback

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

0 0.05 0.1 0.15 0 . 2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Wor

k R

atio

110% 100% 90% 80% 70% Surge Line Effective Surge Line Head (Propulsive/Thermal) Ratio


0.9999

1

1.0001

1.0002

1.0003

1.0004

1.0005

1.0006

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Hea

d R

atio

110% 100% 90% 80% 70% Surge Line Effective Surge Line Pressure-Flow and Pressure-Density Cross-Elasticities


1.405

1.41

1.415

1.42

1.425

1.43

1.435

1.44

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Co

rrec

ted

Pre

ssu

re-D

ensi

ty E

last

icity

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Co

rrec

ted

Pre

ssu

re-F

low

Ela

stic

ity


Pressure-Flow

Pressure-Density

Propulsive (Deviation) Efficiency and Loss


0.94

0.95

0.96

0.97

0.98

0.99

1

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Pro

pu

lsiv

e E

ffic

ien

cy

0

0.01

0.02

0.03

0.04

0.05

0.06P

ropu

lsiv

e Lo

ss

110% 100% 90% 80% 70% Surge Line

Propulsive Efficiency

Propulsive Loss

75

Thermal (Dissipation) Efficiency and Losswith Scheme 2 Feedback

0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Th

erm

al E

ffic

ien

cy

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35


Thermal Efficiency

Thermal Loss

Overall (Propulsive & Thermal) Efficiency and Loss


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ove

rall

Eff

icie

ncy

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Ove

rall

Lo

ss


Overall Efficiency

Overall Loss

Compression System Aeroelastic Characteristics

at Maximum Efficiency & Stall

1.41

1.412

1.414

1.416

1.418

1.42

1.422

1.424

1.426

0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re-D

ensi

ty A

ero

elas

tici

ty

-0.45

-0.35

-0.25

-0.15

-0.05

0.05

0.15

0.25

0.35

0.45

Co

rrec

ted

Pre

ssu

re-F

low

Aer

oel

astic

ity

E_Density @ Stall E_Density @ Maximum EfficiencyE_Flow & Effective E_Flow @ Stall Design Line

E_Flow @ Maximum Efficiency Effective E_Flow @ Maximum Efficiency




Compression System Pumping Characteristic


1

1.02

1.04

1.06

1.08

1.1

1.12

0.7 0.8 0.9 1 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re &

T

emp

erat

ure

Rat

ios

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Co

rrec

ted

Flo

w




Design Line

X Denotes Effective Surge Line - - Denotes Effective Maximum Efficiency Line


Dynamically Compensated Compression System Stability


-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Slo

pe o

f P

ress

ure

Ris

e C

hara

cter

istic

s

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Eff

ectiv

e S

lop

e o

f Pre

ssu

re R

ise

Ch

arac

teri

stic

s


UNSTABLE

STABLE

76

MIT 1 Scheme 3


96.7%

96.2%

95.7%

95.2%

94.7%

94.9%

95.4%

95.8%

96.3%

96.8%

95.3%

94.6%

94.0%

93.3%

92.6%

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

0.1 0.15 0.2 0.25 0.3 0.35 0.4

Corrected Flow

Co

rrec

ted

Pre

ssu

re R

atio

(D

ensi

ty R

atio

)


Safe Operating Line


Corrected T2/T1

101.75E-2

101.25E-2

101.75E-2

102.2E-2

103.0E-2

102.5E-2

102.0E-2

101.5E-2



96.7%

96.2%

95.7%

95.2%

94.7%

94.9%

95.4%

95.8%

96.3%

96.8%

95.3%

94.6%

94.0%

93.3%

92.6%

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

0.1 0.15 0.2 0.25 0.3 0.35 0.4

Corrected Flow

Co

rrec

ted

Pre

ssu

re R

atio

(Den

sity

Rat

io)


Safe Operating Line


118.0E-2

116.0E-2

114.0E-2

112.0E-2

110.0E-2

108.0E-2

106.0E-2

104.0E-2

102.0E-2

Corrected T3/T1

133.0E-2 128.0E-2 124.0E-2 120.0E-2



0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Eff

icie

ncy

Rat

io



0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Loss

Rat

io



0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

0 0.05 0.1 0.15 0 . 2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Wor

k R

atio



0.99995

1

1.00005

1.0001

1.00015

1.0002

1.00025

1.0003

1.00035

1.0004

1.00045

1.0005

0 0.05 0.1 0.15 0.2 0.25 0 . 3 0.35 0.4 0.45 0.5

Corrected Flow

Hea

d R

atio



1.405

1.41

1.415

1.42

1.425

1.43

1.435

1.44

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Co

rrec

ted

Pre

ssu

re-D

ensi

ty E

last

icity

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Co

rrec

ted

Pre

ssu

re-F

low

Ela

stic

ity


Pressure-Flow

Pressure-Density



0.94

0.95

0.96

0.97

0.98

0.99

1

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Pro

pu

lsiv

e E

ffic

ien

cy

0

0.01

0.02

0.03

0.04

0.05

0.06

Pro

puls

ive

Loss

110% 100% 90% 80% 70% Surge Line


Propulsive Loss

77


0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Th

erm

al E

ffic

ien

cy

0

0.1

0.2

0.3

0.4

0.5

0.6


Thermal Efficiency

Thermal Loss



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ove

rall

Eff

icie

ncy

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Ove

rall

Lo

ss


Overall Efficiency

Overall Loss



1.41

1.412

1.414

1.416

1.418

1.42

1.422

1.424

1.426

0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re-D

ensi

ty A

ero

elas

tici

ty

-0.45

-0.35

-0.25

-0.15

-0.05

0.05

0.15

0.25

0.35

0.45

Co

rrec

ted

Pre

ssu

re-F

low

Aer

oel

astic

ity








1

1.01

1.02

1.03

1.04

1.05

1.06

0.7 0.8 0.9 1 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re &

T

emp

erat

ure

Rat

ios

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Co

rrec

ted

Flo

w




Design Line





-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Slo

pe o

f P

ress

ure

Ris

e C

hara

cter

istic

s

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Eff

ectiv

e S

lop

e o

f Pre

ssu

re R

ise

Ch

arac

teri

stic

s


UNSTABLE

STABLE

78

MIT 1 Scheme 4


94.7%

95.2%

95.7%

96.2%

96.7%

95.6%

95.0%

94.3%

93.7%

93.1%

92.6%

93.3%94.0%

94.6%95.3%

1

1.02

1.04

1.06

1.08

1.1

1.12

0.1 0.15 0.2 0.25 0.3 0.35 0.4

Corrected Flow

Cor

rect

ed P

ress

ure

Rat

io (

Den

sity

Rat

io)


Safe Operating Line


Corrected T2/T1

100.75E-2

101.25E-2

101.75E-2

102.2E-2

143.0E-2

140.0E-2

136.0E-2

132.0E-2

128.0E-2



96.7%

96.2%

95.7%

95.2%

94.7%

93.1%

93.7%

94.3%

95.0%

95.6%

95.3%94.6%

94.0%

93.3%

92.6%

1

1.02

1.04

1.06

1.08

1.1

1.12

0.1 0.15 0.2 0.25 0.3 0.35 0.4

Corrected Flow

Co

rrec

ted

Pre

ssu

re R

atio

(Den

sity

Rat

io)



120.0E-2

116.0E-2

112.0E-2

108.0E-2

104.0E-2

Corrected T3/T1

Safe Operating Line

124.0E-2

128.0E-2

132.0E-2140.0E-2148.0E-2166.0E-2 156.0E-2



0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Eff

icie

ncy

Rat

io



0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Loss

Rat

io



0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

0 0.05 0.1 0.15 0 . 2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Wor

k R

atio



0.9999

1

1.0001

1.0002

1.0003

1.0004

1.0005

1.0006

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Hea

d R

atio



1.405

1.41

1.415

1.42

1.425

1.43

1.435

1.44

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Co

rrec

ted

Pre

ssu

re-D

ensi

ty E

last

icity

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Co

rrec

ted

Pre

ssu

re-F

low

Ela

stic

ity


Pressure-Flow

Pressure-



0.94

0.95

0.96

0.97

0.98

0.99

1

0 0.05 0 . 1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Pro

pu

lsiv

e E

ffic

ien

cy

0

0.01

0.02

0.03

0.04

0.05

0.06

Pro

puls

ive

Loss

110% 100% 90% 80% 70% Surge Line


Propulsive Loss

79


0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Th

erm

al E

ffic

ien

cy

0

0.1

0.2

0.3

0.4

0.5

0.6

Th

erm

al L

oss


Thermal Efficiency

Thermal Loss



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ove

rall

Eff

icie

ncy

0

0.1

0.2

0.3

0.4

0.5

0.6

Ove

rall

Lo

ss


Overall Efficiency

Overall Loss



1.41

1.412

1.414

1.416

1.418

1.42

1.422

1.424

1.426

0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re-D

ensi

ty A

ero

elas

tici

ty

-0.45

-0.35

-0.25

-0.15

-0.05

0.05

0.15

0.25

0.35

0.45

Co

rrec

ted

Pre

ssu

re-F

low

Aer

oel

astic

ity








1

1.02

1.04

1.06

1.08

1.1

1.12

0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re &

T

emp

erat

ure

Rat

ios

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Co

rrec

ted

Flo

w




Design Line





- 2

-1.5

- 1

-0.5

0

0.5

1

1.5

2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Slo

pe

of

Pre

ssu

re R

ise

Ch

arac

teri

stic

s

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Eff

ectiv

e S

lop

e o

f Pre

ssu

re R

ise

Ch

arac

teri

stic

s


UNSTABLE

STABLE

80

MIT 1 Scheme 5


96.7%

96.2%

95.7%

95.2%

94.7%

95.3%

94.6%

94.0%

93.3%

92.6%

1

1.01

1.02

1.03

1.04

1.05

1.06

0.1 0.15 0.2 0.25 0.3 0.35 0.4

Corrected Flow

Co

rrec

ted

Pre

ssu

re R

atio

(D

ensi

ty R

atio

)


Safe Operating Line


Corrected T2/T1

102.0E-2

101.5E-2

101.0E-2

100.5E-2

100.75E-2

101.25E-2

101.75E-2

102.2E-2



94.7%

95.2%

95.7%

96.2%

96.7%

92.6%

93.3%

94.0%

94.6%

95.3%

1

1.01

1.02

1.03

1.04

1.05

1.06

0.1 0.15 0.2 0.25 0.3 0.35 0.4

Corrected Flow

Co

rrec

ted

Pre

ssu

re R

atio

(Den

sity

Rat

io)


Safe Operating Line


118.0E-2 116.0E-2

114.0E-2

112.3E-2

110.0E-2

108.0E-2

106.0E-2

104.0E-2

102.0E-2

Corrected T3/T1



0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Eff

icie

ncy

Rat

io



0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Loss

Rat

io



0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

0 0.05 0.1 0.15 0 . 2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Wor

k R

atio



0.99995

1

1.00005

1.0001

1.00015

1.0002

1.00025

1.0003

1.00035

1.0004

1.00045

0 0.05 0.1 0.15 0.2 0.25 0 . 3 0.35 0.4 0.45 0.5

Corrected Flow

Hea

d R

atio



1.405

1.41

1.415

1.42

1.425

1.43

1.435

1.44

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Co

rrec

ted

Pre

ssu

re-D

ensi

ty E

last

icity

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Co

rrec

ted

Pre

ssu

re-F

low

Ela

stic

ity


Pressure-Flow

Pressure-Density



0.94

0.95

0.96

0.97

0.98

0.99

1

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Pro

pu

lsiv

e E

ffic

ien

cy

0

0.01

0.02

0.03

0.04

0.05

0.06

Pro

puls

ive

Loss

110% 100% 90% 80% 70% Surge Line


Propulsive Loss

81


0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Th

erm

al E

ffic

ien

cy

0

0.1

0.2

0.3

0.4

0.5

0.6


Thermal Efficiency

Thermal Loss



0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ove

rall

Eff

icie

ncy

0

0.1

0.2

0.3

0.4

0.5

0.6

Ove

rall

Lo

ss


Overall Efficiency

Overall Loss



1.41

1.412

1.414

1.416

1.418

1.42

1.422

1.424

1.426

0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re-D

ensi

ty A

ero

elas

tici

ty

-0.45

-0.35

-0.25

-0.15

-0.05

0.05

0.15

0.25

0.35

0.45

Co

rrec

ted

Pre

ssu

re-F

low

Aer

oel

astic

ity


E_Flow @ Maximum Efficiency Effective E_Flow @ Maximum Efficiciency






1

1.01

1.02

1.03

1.04

1.05

1.06

0.7 0.8 0.9 1 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re &

T

emp

erat

ure

Rat

ios

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Co

rrec

ted

Flo

w




Design Line





-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Slo

pe o

f P

ress

ure

Ris

e C

hara

cter

istic

s

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Eff

ectiv

e S

lop

e o

f Pre

ssu

re R

ise

Ch

arac

teri

stic

s


UNSTABLE

STABLE

82

MIT 1 Scheme 6


96.7%

96.2%

95.7%

95.2%

94.7%

95.3%

94.6%

94.0%

93.3%

92.6%

1

1.01

1.02

1.03

1.04

1.05

1.06

0.1 0.15 0.2 0.25 0.3 0.35 0.4

Corrected Flow

Co

rrec

ted

Pre

ssu

re R

atio

(Den

sity

Rat

io)


Safe Operating Line


Corrected T3/T1

118.0E-2 116.0E-2

114.0E-2

112.3E-2

110.0E-2

108.0E-2

106.0E-2

104.0E-2

102.0E-2



0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Eff

icie

ncy

Rat

io



0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Loss

Rat

io

110% 100% 90% 80% 70% Surge Line Effective Surge Line Work (Propulsive/Thermal) Ratio


0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

0 0.05 0.1 0.15 0 . 2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Wor

k R

atio


Head (Propulsive/Thermal) Ratiowith Scheme 6 Feedback

0.99995

1

1.00005

1.0001

1.00015

1.0002

1.00025

1.0003

1.00035

1.0004

1.00045

0 0.05 0.1 0.15 0.2 0.25 0 . 3 0.35 0.4 0.45 0.5

Corrected Flow

Hea

d R

atio



1.405

1.41

1.415

1.42

1.425

1.43

1.435

1.44

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Co

rrec

ted

Pre

ssu

re-D

ensi

ty E

last

icity

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Co

rrec

ted

Pre

ssu

re-F

low

Ela

stic

ity


Pressure-Flow

Pressure-Density



0.94

0.95

0.96

0.97

0.98

0.99

1

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Pro

pu

lsiv

e E

ffic

ien

cy

0

0.01

0.02

0.03

0.04

0.05

0.06

Pro

puls

ive

Loss

110% 100% 90% 80% 70% Surge Line


Propulsive Loss

Thermal (Dissipation) Efficiency and Loss


0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Th

erm

al E

ffic

ien

cy

0

0.1

0.2

0.3

0.4

0.5

0.6


Thermal Efficiency

Thermal Loss

83

Overall (Propulsive & Thermal) Efficiency and Losswith Scheme 6 Feedback

0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ove

rall

Eff

icie

ncy

0

0.1

0.2

0.3

0.4

0.5

0.6

Ove

rall

Lo

ss


Overall Efficiency

Overall Loss



1.41

1.412

1.414

1.416

1.418

1.42

1.422

1.424

1.426

0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re-D

ensi

ty A

ero

elas

tici

ty

-0.45

-0.35

-0.25

-0.15

-0.05

0.05

0.15

0.25

0.35

0.45

Co

rrec

ted

Pre

ssu

re-F

low

Aer

oel

astic

ity








1

1.01

1.02

1.03

1.04

1.05

1.06

0.7 0.8 0.9 1 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re &

T

emp

erat

ure

Rat

ios

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Co

rrec

ted

Flo

w




Design Line





-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Slo

pe o

f P

ress

ure

Ris

e C

hara

cter

istic

s

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Eff

ectiv

e S

lop

e o

f Pre

ssu

re R

ise

Ch

arac

teri

stic

s


UNSTABLE

STABLE

84

MIT 1 Scheme 7


96.7%

96.2%

95.7%

95.2%

94.7%

95.3%

94.6%

94.0%

93.3%

92.6%

1

1.01

1.02

1.03

1.04

1.05

1.06

0.1 0.15 0.2 0.25 0.3 0.35 0 . 4

Corrected Flow

Co

rrec

ted

Pre

ssu

re R

atio

(Den

sity

Rat

io)


Safe Operating Line


Corrected T2/T1

102.0E-2

101.5E-2

101.0E-2

100.5E-2

100.75E-2

101.25E-2

101.75E-2

102.2E-2



96.7%

96.2%

95.7%

95.2%

94.7%

95.3%

94.6%

94.0%

93.3%

92.6%

1

1.01

1.02

1.03

1.04

1.05

1.06

0.1 0.15 0.2 0.25 0.3 0.35 0.4

Corrected Flow

Co

rrec

ted

Pre

ssu

re R

atio

(Den

sity

Rat

io)


Safe Operating Line


118.0E-2 116.0E-2

114.0E-2

112.3E-2

110.0E-2

108.0E-2

106.0E-2

104.0E-2

102.0E-2

Corrected T3/T1



0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Eff

icie

ncy

Rat

io



0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Loss

Rat

io



0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

0 0.05 0.1 0.15 0 . 2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Wor

k R

atio



0.99995

1

1.00005

1.0001

1.00015

1.0002

1.00025

1.0003

1.00035

1.0004

1.00045

0 0.05 0.1 0.15 0.2 0.25 0 . 3 0.35 0.4 0.45 0.5

Corrected Flow

Hea

d R

atio



1.405

1.41

1.415

1.42

1.425

1.43

1.435

1.44

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Co

rrec

ted

Pre

ssu

re-D

ensi

ty E

last

icity

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Co

rrec

ted

Pre

ssu

re-F

low

Ela

stic

ity


Pressure-Flow

Pressure-Density



0.94

0.95

0.96

0.97

0.98

0.99

1

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Pro

pu

lsiv

e E

ffic

ien

cy

0

0.01

0.02

0.03

0.04

0.05

0.06

Pro

puls

ive

Loss

110% 100% 90% 80% 70% Surge Line


Propulsive Loss

85


0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Th

erm

al E

ffic

ien

cy

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7


Thermal Efficiency

Thermal Loss



0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ove

rall

Eff

icie

ncy

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Ove

rall

Lo

ss


Overall Efficiency

Overall Loss



1.41

1.412

1.414

1.416

1.418

1.42

1.422

1.424

1.426

0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re-D

ensi

ty A

ero

elas

tici

ty

-0.45

-0.35

-0.25

-0.15

-0.05

0.05

0.15

0.25

0.35

0.45

Co

rrec

ted

Pre

ssu

re-F

low

Aer

oel

astic

ity








1

1.01

1.02

1.03

1.04

1.05

1.06

0.7 0.8 0.9 1 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re &

T

emp

erat

ure

Rat

ios

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Co

rrec

ted

Flo

w




Design Line





-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Slo

pe o

f P

ress

ure

Ris

e C

hara

cter

istic

s

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Eff

ectiv

e S

lop

e o

f Pre

ssu

re R

ise

Ch

arac

teri

stic

s


UNSTABLE

STABLE

86

MIT 1 Scheme 8


94.7%

95.2%

95.7%

96.2%

96.7%

92.6%

93.3%

94.0%

94.6%

95.3%

1

1.01

1.02

1.03

1.04

1.05

1.06

0.1 0.15 0 . 2 0.25 0.3 0.35 0.4

Corrected Flow

Co

rrec

ted

Pre

ssu

re R

atio

(Den

sity

Rat

io)


Safe Operating Line


Corrected T2/T1

102.0E-2

101.5E-2

101.0E-2

100.5E-2

100.75E-2

101.25E-2

101.75E-2

102.2E-2



94.7%

95.2%

95.7%

96.2%

96.7%

92.6%

93.3%

94.0%

94.6%

95.3%

1

1.01

1.02

1.03

1.04

1.05

1.06

0.1 0.15 0.2 0.25 0.3 0.35 0.4

Corrected Flow

Co

rrec

ted

Pre

ssu

re R

atio

(Den

sity

Rat

io)


Safe Operating Line


118.0E-2 116.0E-2

114.0E-2

112.3E-2

110.0E-2

108.0E-2

106.0E-2

104.0E-2

102.0E-2

Corrected T3/T1



0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Eff

icie

ncy

Rat

io



0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Loss

Rat

io



0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

0 0.05 0.1 0.15 0 . 2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Wor

k R

atio



0.99995

1

1.00005

1.0001

1.00015

1.0002

1.00025

1.0003

1.00035

1.0004

1.00045

0 0.05 0.1 0.15 0.2 0.25 0 . 3 0.35 0.4 0.45 0.5

Corrected Flow

Hea

d R

atio



1.405

1.41

1.415

1.42

1.425

1.43

1.435

1.44

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Co

rrec

ted

Pre

ssu

re-D

ensi

ty E

last

icity

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Co

rrec

ted

Pre

ssu

re-F

low

Ela

stic

ity


Pressure-Flow

Pressure-



0.94

0.95

0.96

0.97

0.98

0.99

1

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Pro

pu

lsiv

e E

ffic

ien

cy

0

0.01

0.02

0.03

0.04

0.05

0.06

Pro

puls

ive

Loss

110% 100% 90% 80% 70% Surge Line


Propulsive Loss

87


0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Th

erm

al E

ffic

ien

cy

0

0.1

0.2

0.3

0.4

0.5

0.6


Thermal Efficiency

Thermal Loss



0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ove

rall

Eff

icie

ncy

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Ove

rall

Lo

ss


Overall Efficiency

Overall Loss



1.41

1.412

1.414

1.416

1.418

1.42

1.422

1.424

1.426

0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re-D

ensi

ty A

ero

elas

tici

ty

-0.45

-0.35

-0.25

-0.15

-0.05

0.05

0.15

0.25

0.35

0.45

Co

rrec

ted

Pre

ssu

re-F

low

Aer

oel

astic

ity








1

1.01

1.02

1.03

1.04

1.05

1.06

0.7 0.8 0.9 1 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re &

T

emp

erat

ure

Rat

ios

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Co

rrec

ted

Flo

w




Design Line





-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Slo

pe o

f P

ress

ure

Ris

e C

hara

cter

istic

s

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Eff

ectiv

e S

lop

e o

f Pre

ssu

re R

ise

Ch

arac

teri

stic

s


UNSTABLE

STABLE

88

MIT 3 Open Loop


59%

55%

63%

67%

71%

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Corrected Flow

Cor

rect

ed P

ress

ure

Rat

io (

Den

sity

Rat

io)

110% 100% 90% 80% 70% Surge Line


70 80% 90% 100% 110%

105.0E-2

104.0E-2

103.0E-2

102.0E-2

101.0E-2

Corrected T2/T1


59%

55%

63%

67%

71%

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Corrected Flow

Co

rrec

ted

Pre

ssu

re R

atio

(Den

sity

Rat

io)

110% 100% 90% 80% 70% Surge Line


70%

80% 90% 100% 110%

130.0E-2

135.0E-2139.4E-2145.0E-2

125.0E-2

120.0E-2

115.0E-2

110.0E-2

105.0E-2

Corrected T3/T1


0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Eff

icie

ncy

Rat

io

110% 100% 90% 80% 70% Surge Line

Loss (Propulsive/Thermal) Ratio

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Loss

Rat

io

110% 100% 90% 80% 70% Surge Line

Work (Propulsive/Thermal) Ratio

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Wor

k R

atio

110% 100% 90% 80% 70% Surge Line

Head (Propulsive/Thermal) Ratio

0.998

1

1.002

1.004

1.006

1.008

1.01

1.012

1.014

1.016

1.018

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Hea

d R

atio

110% 100% 90% 80% 70% Surge Line Pressure-Density Elasticity vs. Corrected Flow

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Pre

ssu

re-D

ensi

ty E

last

icity

110% 100% 90% 80% 70% Surge Line Pressure-Flow Elasticity vs. Corrected Flow

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Pre

ssu

re-F

low

Ela

stic

ity

110% 100% 90% 80% 70%

89


0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Pro

pu

lsiv

e E

ffic

ien

cy

110% 100% 90% 80% 70% Surge Line Propulsive (Deviation) Loss

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Pro

pu

lsiv

e L

oss

110% 100% 90% 80% 70% Surge Line Thermal Efficiency

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ther

mal

Eff

icie

ncy

110% 100% 90% 80% 70% Surge Line Thermal (Viscous) Loss

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ther

mal

Los

s

110% 100% 90% 80% 70% Surge Line

Overall (Propulsive & Thermal) Efficiency

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ove

rall

Eff

icie

ncy

110% 100% 90% 80% 70% Surge Line Overall (Viscous+Dissipation) Loss

0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ove

rall

Loss

110% 100% 90% 80% 70% Surge Line Compression System Aeroelastic Characteristics


1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

70% 75% 80% 85% 90% 95% 100% 105% 110%

Corrected Speed

Co

rrec

ted

Pre

ssu

re-D

ensi

ty A

ero

elas

tici

ty

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Co

rrec

ted

Pre

ssu

re-F

low

Aer

oel

asti

city

E_Density @ Maximum Efficiency E_Density @ Stall E_Flow @ StallDesign Line E_Flow @ Maximum Efficiency



Compressor System Pumping Characteristic


1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

70% 75% 80% 85% 90% 95% 100% 105% 110%Corrected Speed

Co

rrec

ted

Pre

ssu

re &

Tem

per

atu

re

Rat

ios

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Co

rrec

ted

Flo

w

Temperature Ratio @ Maximum Efficiency Temperature Ratio @ SurgePressure Ratio @ Maximum Efficiency Pressure Ratio @ SurgeFlow @ Maximum Efficiency Flow @ SurgeDesign Line


90

Temperature Ratio vs. Entropy

0.99

1

1.01

1.02

1.03

1.04

1.05

1.06

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Entropy

Tem

pera

ture

Rat

io

110% 100% 90% 80% 70% Temperature- Propulsive Loss (Ts_d) Elasticity with respect to Propulsive Loss vs.

Corrected Flow

-4-3.5

-3

-2.5

-2

-1.5

-1

-0.50

0.5

1

1.5

0 0.1 0.2 0.3 0.4 0.5

Corrected Flow

Ts_d

Ela

stic

ity

110% 100% 90% 80% 70% Surge Line Temperature- Propulsive Loss (Ts_d) Elasticity with respect to Propulsive Loss vs. Relative Non-Dimensional Corrected Flow

-4-3.5

-3-2.5

-2-1.5

-1-0.5

00.5

11.5

0 0.1 0.2 0.3 0.4 0.5


Ts_

d E

last

icity

110% 100% 90% 80% 70% Surge Line Temperature-Viscous Loss (Ts_o) Elasticity with respect to Viscous Loss vs.

Corrected Flow

-2

0

2

4

6

8

10

12

14

16

18

20

0 0.1 0.2 0.3 0.4 0.5

Corrected Flow

Ts_

o E

last

icity

110% 100% 90% 80% 70% Surge Line

Temperature-Viscous Loss (Ts_o) Elasticity with respect to Viscous Loss vs. Relative Non-Dimensional Corrected Flow

-2

0

2

4

6

8

10

12

14

16

18

20

0 0.005 0.01 0.015 0.02 0.025 0.03


Ts_o

Ela

stic

ity

110% 100% 90% 80% 70% Surge Line Corrected Specific Volume Ratio (v2/v1) vs. Entropy

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Entropy

Co

rrec

ted

Sp

ecif

ic V

olu

me

Rat

io (v

2/v1

)

110% 100% 90% 80% 70% Corrected Specific Volume-Propulsive Loss (vs_d) Elasticity with respect to Propulsive Loss vs.

Corrected Flow

-4

-2

0

2

4

6

8

10

0 0.1 0.2 0.3 0.4 0.5

Corrected Flow

vs_d

Ela

stic

ity

110% 100% 90% 80% 70% Surge Line Corrected Specific Volume-Propulsive Loss (vs_d) Elasticity with respect to

Propulsive Loss vs. Relative Non-Dimensional Corrected Flow

-4

-2

0

2

4

6

8

10

0 0.005 0.01 0.015 0.02 0.025 0.03


vs_d

Ela

stic

ity

110% 100% 90% 80% 70% Surge Line

91

Corrected Specific Volume-Viscous Loss (vs_o) Elasticity with respect to Viscous Loss vs. Corrected Flow

-50

-40

-30

-20

-10

0

10

0 0.1 0.2 0.3 0.4 0.5

Corrected Flow

vs_o

Ela

stic

ity

110% 100% 90% 80% 70% Surge Line Corrected Specific Volume-Viscous Loss (vs_o) Elasticity with respect to

Viscous Loss vs. Relative Non-Dimensional Corrected Flow

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

0 0.1 0.2 0.3 0.4 0.5


vs_o

Ela

stic

ity

110% 100% 90% 80% 70% Surge Line Corrected Specific Volume Ratio (v2/v1) vs. Temperature Ratio (T2/T1)

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

0.99 1 1.01 1.02 1.03 1.04 1.05 1.06

Temperature Ratio

Co

rrec

ted

Sp

ecif

ic V

olu

me

Rat

io (v

2/v1

)

110% 100% 90% 80% 70% Corrected Specific Volume-Temperature (vT) Elasticity with respect to

Temperature vs. Corrected Flow

-2.9

-2.8

-2.7

-2.6

-2.5

-2.4

-2.3

-2.2

-2.1

-2

0 0.1 0.2 0.3 0.4 0.5

Corrected Flow

vT E

last

icity

110% 100% 90% 80% 70% Surge Line

Corrected Specific Volume-Temperature (vT) Elasticity with respect to Temperature vs. Relative Non-Dimensional Corrected Flow

-2.9

-2.8

-2.7

-2.6

-2.5

-2.4

-2.3

-2.2

-2.1

-20 0.005 0.01 0.015 0.02 0.025 0.03


vT E

last

icity

110% 100% 90% 80% 70% Surge Line Corrected Pressure Ratio vs. Entropy

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Entropy

Co

rrec

ted

Pre

ssu

re R

atio

110% 100% 90% 80% 70%


Pressure-Propusive Loss (Ps_d) Elasticity with respect to Propulsive Loss

vs. Corrected Flow

-10

-8

-6

-4

-2

0

2

4

0 0.1 0.2 0.3 0.4 0.5

Corrected Flow

Ps_

d E

last

icity

110% 100% 90% 80% 70% Surge Line Pressure-Propusive Loss (Ps_d) Elasticity with respect to Propulsive Loss vs.


-10

-8

-6

-4

-2

0

2

4

0 0.005 0.01 0.015 0.02 0.025 0.03


Ps_

d E

last

icity

110% 100% 90% 80% 70% Surge Line

92

Pressure-Viscous Loss (Ps_o) Elasticity with respect to Viscous Loss vs. Corrected Flow

-5

0

5

10

15

20

25

30

35

40

45

50

0 0.1 0.2 0.3 0.4 0.5

Corrected Flow

Ps_

o E

last

icity

110% 100% 90% 80% 70% Surge Line Pressure-Viscous Loss (Ps_o) Elasticity with respect to Viscous Loss vs.


-5

0

5

10

15

20

25

30

35

40

45

50

0 0.005 0.01 0.015 0.02 0.025 0.03


Ps_

o E

last

icity

110% 100% 90% 80% 70% Surge Line Corrected Pressure vs. Temperature Ratio

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

0.99 1 1.01 1.02 1.03 1.04 1.05 1.06

Temperature Ratio

Co

rrec

ted

Pre

ssu

re R

atio

110% 100% 90% 80% 70% Corrected Pressure- Temperature (PT) Elasticity with respect to Temperature vs.

Corrected Flow

2.3

2.35

2.4

2.45

2.5

2.55

2.6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

PT

Ela

stic

ity

110% 100% 90% 80% 70% Surge Line Corrected Pressure- Temperature (PT) Elasticity with respect to Temperature vs.


2.3

2.35

2.4

2.45

2.5

2.55

2.6

0 0.005 0.01 0.015 0.02 0.025 0.03


PT

Ela

stic

ity

110% 100% 90% 80% 70% Surge Line

Corrected Pressure Ratio vs. Corrected Specific Volume Ratio (v2/v1)

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02

Corrected Specific Volume Ratio (v2/v1)

Co

rrec

ted

Pre

ssu

re R

atio

110% 100% 90% 80% 70% Corrected Pressure-Corrected Specific Volume (Pv) Elasiticity with respect to Corrected Specific Volume vs.

Corrected Flow

-1.15

-1.1

-1.05

-1

-0.95

-0.9

-0.85

-0.8

-0.75

-0.70 0.1 0.2 0.3 0.4 0.5

Corrected Flow

Pv

Ela

stic

ity

110% 100% 90% 80% 70% Surge Line Corrected Pressure-Corrected Specific Volume (Pv) Elasiticity with respect to Corrected Specific Volume vs. Relative

Non-Dimensional Corrected Flow

-1.15

-1.1

-1.05

-1

-0.95

-0.9

-0.85

-0.8

-0.75

-0.7

0 0.005 0.01 0.015 0.02 0.025 0.03


Pv

Ela

stic

ity

110% 100% 90% 80% 70% Surge Line

93

MIT 3 Scheme 1


59%

55%

63%

67%

71%

50%

55%

59%

64%

68%

53%

46%

39%

32%

25%

1

1.02

1.04

1.06

1.08

1 . 1

1.12

1.14

1.16

0 . 1 0.15 0 . 2 0.25 0.3 0.35 0 . 4 0 . 4 5

Corrected Flow

Corr

ecte

d Pr

essu

re R

atio

(De

nsit

y Ra

tio)

1 1 0 % 100% 90% 8 0 % 70% Surge Line Effective Surge Line

Safe Operating Line


Corrected T2/T1

105.0E-2

104.0E-2

103.0E-2

102.0E-2

101.0E-2

105.7E-2

104.8E-2

103.9E-2

103.2E-2

102.5E-2



59%

55%

63%

67%

71%

50%

55%

59%

64%

68%

53%

46%

39%

32%

25%

1

1.02

1.04

1.06

1.08

1 . 1

1.12

1.14

1.16

0 . 1 0.15 0 . 2 0.25 0.3 0.35 0 . 4 0 . 4 5

Corrected Flow

Corr

ecte

d Pr

essu

re R

atio

(Den

sity

Rat

io)

110% 100% 9 0 % 80% 70% Surge Line Effective Surge Line

Safe Operating Line


130.0E-2

135.0E-2140.0E-2145.0E-2

125.0E-2

120.0E-2

115.0E-2

110.0E-2

105.0E-2

Corrected T3/T1150.0E-2155.0E-2160.0E-2165.0E-2



0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Eff

icie

ncy

Rat

io



0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Loss

Rat

io



0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Wor

k R

atio



0.998

1

1.002

1.004

1.006

1.008

1.01

1.012

1.014

1.016

1.018

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Hea

d R

atio



1.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

1.9

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Co

rrec

ted

Pre

ssu

re-D

ensi

ty E

last

icit

y

Co

rrec

ted

Pre

ssu

re-F

low

Ela

stic

ity


Pressure-Density

Pressure-Flow



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Pro

pu

lsiv

e E

ffic

ien

cy

110% 100% 90% 80% 70% Surge Line


Propulsive Loss

94


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Th

erm

al E

ffic

ien

cy


Thermal Efficiency

Thermal Loss



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ove

rall

Eff

icie

ncy

Ove

rall

Lo

ss


Overall Efficiency

Overall Loss



1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re-D

ensi

ty A

ero

elas

tici

ty

- 1

-0.8

-0.6

-0.4

-0.2

0

0 . 2

0 . 4

0 . 6

Co

rrec

ted

Pre

ssu

re-F

low

Aer

oel

astic

ity

E_Density @ Maximum Efficiency E_Density @ Stall E_Flow & Effective E_Flow @ StallDesign Line E_Flow @ Maximum Efficiency Effective E_Flow @ Maximum Efficiency






1

1.05

1.1

1.15

1.2

1.25

0.7 0.8 0.9 1 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re &

Tem

per

atu

re

Rat

ios

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Co

rrec

ted

Flo

w

Temperature Ratio @ Maximum Efficiency Temperature Ratio @ SurgePressure Ratio @ Maximum Efficiency Pressure Ratio @ Surge


Design Line





-7

-6

-5

-4

-3

-2

-1

0

1

2

3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Slo

pe o

f P

ress

ure

Ris

e C

hara

cter

istic

s

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

Eff

ectiv

e S

lop

e o

f P

ress

ure

Ris

e C

har

acte

rist

ic


95

MIT 3 Scheme 2


59%

55%

63%

67%

71%

50%

55%

59%

64%

68%

54.1%

47.5%

40.9%

34.4%

27.8%

1

1.02

1.04

1.06

1.08

1 . 1

1.12

1.14

1.16

0.1 0.15 0 . 2 0.25 0 . 3 0.35 0.4 0.45

Corrected Flow

Corr

ecte

d Pr

essu

re R

atio

(Den

sity

Rat

io)

110% 1 0 0 % 9 0 % 8 0 % 7 0 % Surge Line Effective Surge Line

Safe Operating Line


Corrected T2/T1

105.0E-2

104.0E-2

103.0E-2

102.0E-2

101.0E-2

106.2E-2

105.3E-2

104.4E-2

103.7E-2

103.0E-2



59%

55%

63%

67%

71%

50%

55%

59%

64%

68%

54.1%

47.5%

40.9%

34.4%

27.8%

1

1.02

1.04

1.06

1.08

1 . 1

1.12

1.14

1.16

0.1 0.15 0 . 2 0.25 0 . 3 0.35 0.4 0.45Corrected Flow

Corr

ecte

d Pr

essu

re R

atio

(Den

sity

Rat

io)

110% 100% 90% 80% 7 0 % Surge Line Effective Surge Line

Safe Operating Line


130.0E-2

135.0E-2

140.0E-2145.0E-2

125.0E-2

120.0E-2

115.0E-2

110.0E-2

105.0E-2

Corrected T3/T1150.0E-2155.0E-2160.0E-2165.0E-2170.0E-2



0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Eff

icie

ncy

Rat

io



0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Loss

Rat

io



0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Wor

k R

atio



0.998

1

1.002

1.004

1.006

1.008

1.01

1.012

1.014

1.016

1.018

1.02

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Hea

d R

atio



1.4

1.5

1.6

1.7

1.8

1.9

2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Co

rrec

ted

Pre

ssu

re-D

ensi

ty E

last

icit

y

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Co

rrec

ted

Pre

ssu

re-F

low

Ela

stic

ity


Pressure-Density

Pressure-Flow



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Pro

pu

lsiv

e E

ffic

ien

cy

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

110% 100% 90% 80% 70% Surge Line


Propulsive Loss

96


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Th

erm

al E

ffic

ien

cy

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Thermal Efficiency

Thermal Loss



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ove

rall

Eff

icie

ncy

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ove

rall

Lo

ss


Overall Efficiency

Overall Loss



1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re-D

ensi

ty A

ero

elas

tici

ty

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Co

rrec

ted

Pre

ssu

re-F

low

Aer

oel

astic

ity







1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18

0.7 0.8 0.9 1 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re &

Tem

per

atu

re

Rat

ios

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Co

rrec

ted

Flo

w



Design Line





-7

-6

-5

-4

-3

-2

-1

0

1

2

3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Slo

pe o

f P

ress

ure

Ris

e C

hara

cter

istic

s

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

Eff

ectiv

e S

lop

e o

f P

ress

ure

Ris

e C

har

acte

rist

ic


97

MIT 3 Scheme 3


59%

55%

63%

67%

71%

50%

55%

59%

64%

68%

53%

47%

3 9 %

3 4 %

27%

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

0.1 0.15 0.2 0.25 0.3 0.35 0 . 4 0 . 4 5

Corrected Flow

Corr

ecte

d Pr

essu

re R

atio

(De

nsit

y Ra

tio)


Safe Operating Line


Corrected T2/T1

105.0E-2

104.0E-2

103.0E-2

102.0E-2

101.0E-2

105.9E-2

105.0E-2

104.1E-2

103.4E-2

102.7E-2



71%

67%

63%

55%

59%

68%

64%

59%

55%

50%

27%

3 4 %

3 9 %

47%

53%

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

0.1 0.15 0.2 0.25 0.3 0.35 0 . 4 0 . 4 5

Corrected Flow

Corr

ecte

d Pr

essu

re R

atio

(Den

sity

Rat

io)


Safe Operating Line


130.0E-2

135.0E-2139.4E-2145.0E-2

125.0E-2

120.0E-2

115.0E-2

110.0E-2

105.0E-2

Corrected T3/T1

151.0E-2155.0E-2160.0E-2



0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Eff

icie

ncy

Rat

io



0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Loss

Rat

io



0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Wor

k R

atio



0.998

1

1.002

1.004

1.006

1.008

1.01

1.012

1.014

1.016

1.018

1.02

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Hea

d R

atio



1.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

1.9

0 0.05 0.1 0.15 0 . 2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Co

rrec

ted

Pre

ssu

re-D

ensi

ty E

last

icit

y

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Co

rrec

ted

Pre

ssu

re-F

low

Ela

stic

ity


Pressure-Density

Pressure-Flow



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Pro

pu

lsiv

e E

ffic

ien

cy

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

110% 100% 90% 80% 70% Surge Line


Propulsive Loss

98


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Th

erm

al E

ffic

ien

cy

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Thermal Efficiency

Thermal Loss



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ove

rall

Eff

icie

ncy

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ove

rall

Lo

ss


Overall Efficiency

Overall Loss



1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re-D

ensi

ty A

ero

elas

tici

ty

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Co

rrec

ted

Pre

ssu

re-F

low

Aer

oel

astic

ity







1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

0.7 0.8 0.9 1 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re &

Tem

per

atu

re

Rat

ios

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Co

rrec

ted

Flo

w



Design Line





-7

-6

-5

-4

-3

-2

-1

0

1

2

3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Slo

pe o

f P

ress

ure

Ris

e C

hara

cter

istic

s

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

Eff

ectiv

e S

lop

e o

f P

ress

ure

Ris

e C

har

acte

rist

ic


99

MIT 3 Scheme 4


71%

67%

63%

55%

59%

6 8 %

6 4 %

5 9 %

5 5 %

5 0 %

2 9 %

36%

42%

49%

55%

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18

0.08 0.13 0.18 0 . 2 3 0.28 0.33 0.38 0.43

Corrected Flow

Corr

ecte

d Pr

essu

re R

atio

(Den

sity

Rat

io)

110% 1 0 0 % 90% 80% 70% Surge Line Effective Surge Line

Safe Operating Line


Corrected T2/T1

105.0E-2

104.0E-2

103.0E-2

102.0E-2

101.0E-2

106.4E-2

105.5E-2

104.7E-2

103.9E-2

103.3E-2



71%

67%

63%

55%

59%

68%

64%

59%

55%

50%

29%

36%

42%

49%

55%

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18

0.1 0.15 0.2 0.25 0 . 3 0.35 0 . 4 0.45

Corrected Flow

Corr

ecte

d Pr

essu

re R

atio

(Den

sity

Rat

io)


Safe Operating Line


130.0E-2

135.0E-2

140.0E-2145.0E-2

125.0E-2

120.0E-2

115.0E-2

110.0E-2

105.0E-2

Corrected T3/T1

170.0E-2 155.0E-2185.0E-2 160.0E-2



0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Eff

icie

ncy

Rat

io



0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Loss

Rat

io



0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Wor

k R

atio



1

1.002

1.004

1.006

1.008

1.01

1.012

1.014

1.016

1.018

1.02

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Hea

d R

atio



1.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

1.9

0 0.05 0.1 0.15 0 . 2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Co

rrec

ted

Pre

ssu

re-D

ensi

ty E

last

icit

y

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Co

rrec

ted

Pre

ssu

re-F

low

Ela

stic

ity


Pressure-Density

Pressure-Flow



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Pro

pu

lsiv

e E

ffic

ien

cy

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

110% 100% 90% 80% 70% Surge Line


Propulsive Loss

100


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0 . 3 0.35 0.4 0.45 0.5

Corrected Flow

Th

erm

al E

ffic

ien

cy

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Thermal Efficiency

Thermal Loss



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ove

rall

Eff

icie

ncy

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ove

rall

Lo

ss


Overall Efficiency

Overall Loss



1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re-D

ensi

ty A

ero

elas

tici

ty

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Co

rrec

ted

Pre

ssu

re-F

low

Aer

oel

astic

ity







1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18

0.7 0.8 0.9 1 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re &

Tem

per

atu

re

Rat

ios

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Co

rrec

ted

Flo

w



Design Line





-7

-6

-5

-4

-3

-2

-1

0

1

2

3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Slo

pe o

f P

ress

ure

Ris

e C

hara

cter

istic

s

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

Eff

ectiv

e S

lop

e o

f P

ress

ure

Ris

e C

har

acte

rist

ic


101

MIT 3 Scheme 5


71%

67%

63%

55%

59%

6 8 %

64%

5 9 %

5 5 %

5 0 %

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

0.1 0.15 0.2 0.25 0 . 3 0.35 0 . 4

Corrected Flow

Corr

ecte

d Pr

essu

re R

atio

(De

nsit

y Ra

tio)


Safe Operating Line


Corrected T2/T1

105.0E-2

104.0E-2

103.0E-2

102.0E-2

101.0E-2



71%

67%

63%

55%

59%

6 8 %

64%

5 9 %

5 5 %

5 0 %

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

0.1 0.15 0.2 0.25 0 . 3 0.35 0 . 4

Corrected Flow

Corr

ecte

d Pr

essu

re R

atio

(Den

sity

Rat

io)


Safe Operating Line


130.0E-2

135.0E-2139.4E-2145.0E-2

125.0E-2

120.0E-2

115.0E-2

110.0E-2

105.0E-2

Corrected T3/T1



0.8

1.3

1.8

2.3

2.8

3.3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Eff

icie

ncy

Rat

io



0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Loss

Rat

io



0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Wor

k R

atio



1

1.002

1.004

1.006

1.008

1.01

1.012

1.014

1.016

1.018

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Hea

d R

atio



1.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

1.9

0 0.1 0.2 0.3 0.4 0.5

Corrected Flow

Co

rrec

ted

Pre

ssu

re-D

ensi

ty E

last

icit

y

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Co

rrec

ted

Pre

ssu

re-F

low

Ela

stic

ity


Pressure-Density

Pressure-Flow



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Pro

pu

lsiv

e E

ffic

ien

cy

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

110% 100% 90% 80% 70% Surge Line


Propulsive Loss

102


0

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

0 . 6

0 . 7

0 . 8

0 . 9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Th

erm

al E

ffic

ien

cy

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Thermal Efficiency

Thermal Loss



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ove

rall

Eff

icie

ncy

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ove

rall

Lo

ss

110% 100% 90% 80% 70% Surge Line

Overall Efficiency

Overall Loss



1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re-D

ensi

ty A

ero

elas

tici

ty

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Co

rrec

ted

Pre

ssu

re-F

low

Aer

oel

astic

ity

E_Density @ Maximum Efficiency E_Density @ Stall E_Flow @ StallDesign Line E_Flow @ Maximum Efficiency






1

1.05

1.1

1.15

1.2

1.25

0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re &

Tem

per

atu

re

Ra

tio

s

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Co

rrec

ted

Flo

w



Design Line





-7

-6

-5

-4

-3

-2

-1

0

1

2

3

0 0 . 2 0.4 0.6 0.8 1 1.2

Corrected Flow

Slo

pe o

f P

ress

ure

Ris

e C

hara

cter

istic

s

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

Eff

ectiv

e S

lop

e o

f P

ress

ure

Ris

e C

har

acte

rist

ic


103

MIT 3 Scheme 6


59%

55%

63%

67%

71%

50%

55%

59%

64%

68%

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Corrected Flow

Corr

ecte

d Pr

essu

re R

atio

(De

nsit

y Ra

tio)


Safe Operating Line


Corrected T2/T1

105.0E-2

104.0E-2

103.0E-2

102.0E-2

101.0E-2



59%

55%

63%

67%

71%

50%

55%

59%

64%

68%

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

0.1 0.15 0.2 0.25 0 . 3 0.35 0 . 4 0.45

Corrected Flow

Corr

ecte

d Pr

essu

re R

atio

(Den

sity

Rat

io)

110% 100% 90% 80% 70% Surge Line

Safe Operating Line


130.0E-2

135.0E-2139.4E-2145.0E-2

125.0E-2

120.0E-2

115.0E-2

110.0E-2

105.0E-2

Corrected T3/T1



0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Eff

icie

ncy

Rat

io



0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Loss

Rat

io



0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Wor

k R

atio



1

1.002

1.004

1.006

1.008

1.01

1.012

1.014

1.016

1.018

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Hea

d R

atio



1.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

1.9

0 0.05 0.1 0.15 0 . 2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Co

rrec

ted

Pre

ssu

re-D

ensi

ty E

last

icit

y

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Co

rrec

ted

Pre

ssu

re-F

low

Ela

stic

ity


Pressure-Density

Pressure-Flow



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Pro

pu

lsiv

e E

ffic

ien

cy

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

110% 100% 90% 80% 70% Surge Line


Propulsive Loss

104


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Th

erm

al E

ffic

ien

cy

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Thermal Efficiency

Thermal Loss



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ove

rall

Eff

icie

ncy

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ove

rall

Lo

ss


Overall Efficiency

Overall Loss



1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

70% 75% 80% 85% 90% 95% 100% 105% 110%

Corrected Speed

Co

rrec

ted

Pre

ssu

re-D

ensi

ty A

ero

elas

tici

ty

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Co

rrec

ted

Pre

ssu

re-F

low

Aer

oel

astic

ity







1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

70% 80% 90% 100% 110%

Corrected Speed

Co

rrec

ted

Pre

ssu

re &

Tem

per

atu

re

Rat

ios

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Co

rrec

ted

Flo

w



Design Line





-7

-6

-5

-4

-3

-2

-1

0

1

2

3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Slo

pe o

f P

ress

ure

Ris

e C

hara

cter

istic

s

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

Eff

ectiv

e S

lop

e o

f P

ress

ure

Ris

e C

har

acte

rist

ic


105

MIT 3 Scheme 7


71%

67%

63%

55%

59%

68%

64%

59%

55%

50%

1

1.02

1.04

1.06

1.08

1 . 1

1.12

1.14

1.16

0.1 0.15 0.2 0.25 0.3 0.35 0 . 4 0.45

Corrected Flow

Corr

ecte

d Pr

essu

re R

atio

(De

nsit

y Ra

tio)


Safe Operating Line


Corrected T2/T1105.0E-2

104.0E-2

103.0E-2

102.0E-2

101.0E-2



71%

67%

63%

55%

59%

68%

64%

59%

55%

50%

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

0.1 0.15 0 . 2 0.25 0 . 3 0.35 0 . 4 0.45

Corrected Flow

Corr

ecte

d Pr

essu

re R

atio

(Den

sity

Rat

io)

1 1 0 % 100% 9 0 % 8 0 % 70% Surge Line Effective Surge Line

Safe Operating Line


130.0E-2

135.0E-2139.4E-2145.0E-2

125.0E-2

120.0E-2

115.0E-2

110.0E-2

105.0E-2

Corrected T3/T1



0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Eff

icie

ncy

Rat

io



0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Loss

Rat

io



0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Wor

k R

atio



1

1.002

1.004

1.006

1.008

1.01

1.012

1.014

1.016

1.018

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Hea

d R

atio


Pressure-Flow and Pressure-Density Cross-Elasticitieswith Scheme 7 Feedback

1.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

1.9

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Co

rrec

ted

Pre

ssu

re-D

ensi

ty E

last

icit

y

Co

rrec

ted

Pre

ssu

re-F

low

Ela

stic

ity


Pressure-Density

Pressure-Flow



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Pro

pu

lsiv

e E

ffic

ien

cy

110% 100% 90% 80% 70% Surge Line


Propulsive Loss

106


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Th

erm

al E

ffic

ien

cy


Thermal Efficiency

Thermal Loss



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ove

rall

Eff

icie

ncy

Ove

rall

Lo

ss


Overall Efficiency

Overall Loss



1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re-D

ensi

ty

Aer

oel

asti

city

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Co

rrec

ted

Pre

ssu

re-F

low

Aer

oel

astic

ity

E_Density @ Maximum Efficiency E_Density @ Stall

E_Flow & Effective E_Flow @ Stall Design LineE_Flow & Effective E_Flow @ Maximum Efficiency






1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re &

Tem

per

atu

re

Rat

ios

Co

rrec

ted

Flo

w



Design Line





-7

-6

-5

-4

-3

-2

-1

0

1

2

3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Slo

pe o

f P

ress

ure

Ris

e C

hara

cter

istic

s

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

Eff

ectiv

e S

lop

e o

f P

ress

ure

Ris

e C

har

acte

rist

ic


107

MIT 3 Scheme 8


71%

67%

63%

55%

59%

68%

6 4 %

5 9 %

5 5 %

50%

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

0.1 0.15 0.2 0.25 0 . 3 0.35 0 . 4 0.45

Corrected Flow

Cor

rect

ed P

ress

ure

Rat

io (D

ensi

ty R

atio

)


Safe Operating Line


Corrected T2/T1105.0E-2

104.0E-2

103.0E-2

102.0E-2

101.0E-2



71%

67%

63%

55%

59%

68%

6 4 %

5 9 %

55%

50%

1

1.02

1.04

1.06

1.08

1 . 1

1.12

1.14

1.16

0.1 0.15 0 . 2 0.25 0 . 3 0.35 0 . 4 0.45

Corrected Flow

Corr

ecte

d Pr

essu

re R

atio

(Den

sity

Rat

io)


Safe Operating Line


130.0E-2

135.0E-2

139.4E-2145.0E-2

125.0E-2

120.0E-2

115.0E-2

110.0E-2

105.0E-2

Corrected T3/T1



0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Eff

icie

ncy

Rat

io



0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Loss

Rat

io



0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Wor

k R

atio



1

1.002

1.004

1.006

1.008

1.01

1.012

1.014

1.016

1.018

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Hea

d R

atio



1.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

1.9

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Co

rrec

ted

Pre

ssu

re-D

ensi

ty E

last

icit

y

Co

rrec

ted

Pre

ssu

re-F

low

Ela

stic

ity


Pressure-Density

Pressure-Flow



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Pro

pu

lsiv

e E

ffic

ien

cy

110% 100% 90% 80% 70% Surge Line


Propulsive Loss

108


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Th

erm

al E

ffic

ien

cy


Thermal Efficiency

Thermal Loss



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ove

rall

Eff

icie

ncy

Ove

rall

Lo

ss


Overall Efficiency

Overall Loss



1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

70% 75% 80% 85% 90% 95% 100% 105% 110%

Corrected Speed

Co

rrec

ted

Pre

ssu

re-D

ensi

ty

Aer

oel

asti

city

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Co

rrec

ted

Pre

ssu

re-F

low

Aer

oel

astic

ity








1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

70% 75% 80% 85% 90% 95% 100% 105% 110%

Corrected Speed

Co

rrec

ted

Pre

ssu

re &

Tem

per

atu

re

Rat

ios

Co

rrec

ted

Flo

w



Design Line





-7

-6

-5

-4

-3

-2

-1

0

1

2

3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Slo

pe o

f P

ress

ure

Ris

e C

hara

cter

istic

s

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

Eff

ectiv

e S

lop

e o

f P

ress

ure

Ris

e C

har

acte

rist

ic


109

MIT 3 Scheme 9


59%

55%

63%

67%

71%

50%

55%

59%

64%

68%

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

0 . 1 0.15 0 . 2 0.25 0.3 0.35 0.4 0 . 4 5

Corrected Flow

Corr

ecte

d Pr

essu

re R

atio

(Den

sity

Rat

io)


Safe Operating Line


Corrected T2/T1

105.0E-2

104.0E-2

103.0E-2

102.0E-2

101.0E-2



59%

55%

63%

67%

71%

50%

55%

59%

64%

68%

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

0 . 1 0.15 0 . 2 0.25 0.3 0.35 0.4 0 . 4 5Corrected Flow

Corr

ecte

d Pr

essu

re R

atio

(Den

sity

Rat

io)


Safe Operating Line


130.0E-2

135.0E-2

139.0E-2145.0E-2

125.0E-2

120.0E-2

115.0E-2

110.0E-2

105.0E-2

Corrected T3/T1



0.8

1

1.2

1.4

1.6

1.8

2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Eff

icie

ncy

Rat

io



0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Loss

Rat

io



0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Wor

k R

atio



0.998

1

1.002

1.004

1.006

1.008

1.01

1.012

1.014

1.016

1.018

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Hea

d R

atio



1.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

1.9

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Co

rrec

ted

Pre

ssu

re-D

ensi

ty E

last

icit

y

Co

rrec

ted

Pre

ssu

re-F

low

Ela

stic

ity


Pressure-Density

Pressure-Flow



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Pro

pu

lsiv

e E

ffic

ien

cy

110% 100% 90% 80% 70% Surge Line


Propulsive Loss

110


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Th

erm

al E

ffic

ien

cy


Thermal Efficiency

Thermal Loss



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ove

rall

Eff

icie

ncy

Ove

rall

Lo

ss


Overall Efficiency

Overall Loss



1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re-D

ensi

ty A

ero

elas

tici

ty

- 1

-0.8

-0.6

-0.4

-0.2

0

0 . 2

0 . 4

0 . 6

Co

rrec

ted

Pre

ssu

re-F

low

Aer

oel

astic

ity







1

1.05

1.1

1.15

1.2

1.25

0.7 0.8 0.9 1 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re &

Tem

per

atu

re

Rat

ios

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Co

rrec

ted

Flo

w



Design Line





-7

-6

-5

-4

-3

-2

-1

0

1

2

3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Slo

pe o

f P

ress

ure

Ris

e C

hara

cter

istic

s

Eff

ectiv

e S

lop

e o

f P

ress

ure

Ris

e C

har

acte

rist

ic


111

MIT 3 Scheme 10


71%

67%

63%

55%

59%

68%

64%

59%

55%

50%

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

0.1 0.15 0.2 0.25 0 . 3 0.35 0 . 4 0.45

Corrected Flow

Corr

ecte

d Pr

essu

re R

atio

(Den

sity

Rat

io)


Safe Operating Line


Corrected T2/T1

105.0E-2

104.0E-2

103.0E-2

102.0E-2

101.0E-2



71%

67%

63%

55%

59%

68%

64%

59%

55%

50%

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

0.1 0.15 0.2 0.25 0 . 3 0.35 0 . 4 0.45

Corrected Flow

Corr

ecte

d Pr

essu

re R

atio

(Den

sity

Rat

io)


Safe Operating Line


130.0E-2

135.0E-2139.4E-2145.0E-2

125.0E-2

120.0E-2

115.0E-2

110.0E-2

105.0E-2

Corrected T3/T1



0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Eff

icie

ncy

Rat

io



0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Loss

Rat

io



0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Wor

k R

atio



1

1.002

1.004

1.006

1.008

1.01

1.012

1.014

1.016

1.018

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Hea

d R

atio



1.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

1.9

0 0.05 0.1 0.15 0 . 2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Co

rrec

ted

Pre

ssu

re-D

ensi

ty E

last

icit

y

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Co

rrec

ted

Pre

ssu

re-F

low

Ela

stic

ity


Pressure-Density

Pressure-Flow



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Pro

pu

lsiv

e E

ffic

ien

cy

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

110% 100% 90% 80% 70% Surge Line


Propulsive Loss

112


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Th

erm

al E

ffic

ien

cy

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Thermal Efficiency

Thermal Loss



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Ove

rall

Eff

icie

ncy

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ove

rall

Lo

ss


Overall Efficiency

Overall Loss



1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re-D

ensi

ty

Aer

oel

asti

city

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Co

rrec

ted

Pre

ssu

re-F

low

Aer

oel

astic

ity








1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

0.7 0.8 0.9 1 1.1

Corrected Speed

Co

rrec

ted

Pre

ssu

re &

Tem

per

atu

re

Rat

ios

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Co

rrec

ted

Flo

w



Design Line





-7

-6

-5

-4

-3

-2

-1

0

1

2

3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Corrected Flow

Slo

pe o

f P

ress

ure

Ris

e C

hara

cter

istic

s

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

Eff

ectiv

e S

lop

e o

f P

ress

ure

Ris

e C

har

acte

rist

ic


AEROMECHANICAL CONTROL OF HIGH-SPEED AXIAL COMPRESSOR STALL

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Transcript of AEROMECHANICAL CONTROL OF HIGH-SPEED AXIAL COMPRESSOR STALL