research article modeling techniques for a computational...

Research ArticleModeling Techniques for a Computational Efficient DynamicTurbofan Engine Model

Rory A Roberts and Scott M Eastbourn

Wright State University Dayton OH 45435 USA

Correspondence should be addressed to Rory A Roberts roryrobertswrightedu

Received 29 April 2014 Revised 15 August 2014 Accepted 9 September 2014 Published 12 October 2014

Academic Editor Hyochoong Bang

Copyright copy 2014 R A Roberts and S M Eastbourn This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A transient two-stream engine model has been developed Individual component models developed exclusively in MAT-LABSimulink including the fan high pressure compressor combustor high pressure turbine low pressure turbine plenumvolumes and exit nozzle have been combined to investigate the behavior of a turbofan two-stream engine Special attention hasbeen paid to the development of transient capabilities throughout the model increasing physics model eliminating algebraicconstraints and reducing simulation time through enabling the use of advanced numerical solvers The lessening of computationtime is paramount for conducting future aircraft system-level design trade studies and optimization The new engine model issimulated for a fuel perturbation and a specified mission while tracking critical parameters These results as well as the simulationtimes are presented The new approach significantly reduces the simulation time

1 Introduction

Conceptual design groups have traditionally designed aircraftfrom a subsystem-level viewpoint Consequently subsystemssuch as the propulsion electrical and thermal managementsystems are often optimizedwithout consideration of vehicle-level interactions resulting in a final aircraft design that isnot truly optimized It is believed however that vehicle-levelanalysis of subsystem interactions could result in significantperformance gains across the aircraft potentially improvingthe overall effectiveness of future platformsThe developmentof a vehicle level tip-to-tail (T2T) modeling and simulationtool would allow these performance gains to be quantified ina cost effective manner [1 2]

Recent work completed by the authors focused on thedevelopment of a nonproprietary thermal T2T aircraftmodelin Simulink [3] At least some of the motivation for buildinga T2T model was to perform design trade studies In orderto run these trade studies effectively and efficiently compu-tation times should be no slower than real time The T2Tmodel is intended to stimulate the optimization of individualsubsystems for improving overall vehicle-level performanceand mitigating the thermal and power challenges of future

aircraft platforms [4] In addition the nonproprietary natureof the model allows the tool to be distributed to variousconceptual design groups and researchers Specifically itis foreseen that conceptual designers will use the modelto conduct design trade studies allowing the analysis ofmultiple design configurations and the resulting subsysteminteractions in short time periods [5ndash7] In order for effectivetrade studies to be conducted the model must have relativelyfast computation times Previous work has demonstratedthat while effective and accurate the developed T2T modelhas extremely large simulation times of half real-time As aresult the tool would fail to meet a major requirement forconducting valuable design trade studies and optimization ina practical time frame Through further investigation it wasidentified that the turbofan engine model was contributingto slow simulation times The T2T aircraft model is a stiffsystem with time constants varying between milliseconds todecaseconds The turbofan engine has many interfaces withother subsystems and must be compatible with the varioustime constants captured in the T2T model

Gas turbine engine modeling tools such as NumericalPropulsion System Simulation (NPSS) tool provide a top-down propulsion system approach to provide designers with

Hindawi Publishing CorporationInternational Journal of Aerospace EngineeringVolume 2014 Article ID 283479 11 pageshttpdxdoiorg1011552014283479

2 International Journal of Aerospace Engineering

a tool to incorporate the relevant factors which affect propul-sion performance early in the design and analysis process [8]NPSS was primarily a steady-state tool that has been widelyadopted by government and industry over the past decadesbut has expanded its capabilities in dynamic simulationNPSS can be expanded to incorporate systems beyond themain engine The NPSS based models were inefficient whenutilized inMatlab-Simulink based T2Tmodel simulations [5]with simulation times 16 of real time Surrogate models wereused in the T2T model to reduce computational times [6]Dynamicmodels have been developed for real time operationwith unsteady effects due to mass accumulation consideredby adding a plenum between each compressor and turbinestage and shaft dynamics [9ndash11] These models capture someof the dynamics of the engine but assume mass flows fromprevious time-steps in different sections of the engine that isthe compressor mass flow Other work in dynamic modelingcaptured the plenum volume dynamics needed to predict thesecondary flows within the engine for blade cooling [12]

As previously mentioned using NPSS models with Sim-ulink is less efficient The computational inefficiencies residein the compiling of source code within the Matlab-Simulinkframework These subtle inefficiencies do not pose issueswhen executing a limited amount of trade studies or simu-lations but do pose a problem when performing thousandsof simulations for an optimization routine In this workan engine model developed entirely in Matlab-Simulinkenvironment is presented Several techniques were attemptedto increase the simulation speed of the turbofan enginemodel The techniques adopted to achieve a computationalefficient turbofan engine model are presented in this workThe turbofan engine model is developed without iterationloops (algebraic constraints) and all states are continuousThis approach is very important for complex system levelsimulations of stiff dynamic systems By modeling all thesignificant states as continuous states and not steady-stateapproximations with discontinuities advanced numericalstiff solvers for stiff systems may be used Numerical stiffsolvers rely on the Jacobian matrix and thus require accurateapproximations for gradients of all continuous states Stiffsolvers dramatically reduced the computational time forsimulating stiff systems Also by having the engine modelconstructed completely in Simulink a complete T2T modelcan be compiled then executed or exported as an efficientexecutable limiting unnecessary callbacks

The following sections provide a detailed description ofthe modeling approach for the transient engine model Sim-ulation results are presented along with the comparisonin simulation times Key approaches in performance mapinterpretation and plenum volume dynamics enable thedevelopment of a computationally efficient engine model

2 Fundamental Equation Development

The engine consists of several key component models pre-sented in Figure 1 These models include Fan High Pressure(HP) Compressor Combustor High Pressure (HP) TurbineLow Pressure (LP) Turbine Bypass Plenum Volume Nozzle

Bypass

Bypass

Fan

LP tu

rbin

e

Noz

zle

Heat exchanger

Heat exchanger

LP shaft

Combustor

HP shaft

HP

com

pres

sor

HP

turb

ine

Combustor

Figure 1 Two-stream turbofan engine diagram of major compo-nents

Speed 1

Speed 2

Speed 3

Speed 4

Corrected mass flow

Pres

sure

ratio

R-line 1R-line 2

R-line 3R-line 4

Figure 2 Compressor map with 119877-lines and approximations nearthe surge line

High Pressure (HP) Shaft and Low Pressure (LP) ShaftEach of the turbo-machinery components in the engineshare several common equations defining the physics of themodel Equations common to the fan HP compressor HPturbine and LP turbine are outlined in the following sectionincluding mass flow temperature and work terms

21 Mass Flow Each turbo-machinery model contains aperformance map that determines a corrected mass flowfor a given shaft speed and pressure ratio The traditionalapproach of interpreting the performancemaps is to create anadditional independent variable known as the 119877-line [13 14]Figure 2 shows the compressor map with 119877-lines includedTypically the first 119877-line is given a value of 10 that coincideswith the surge lineThe remaining119877-lines are roughly parallelto the surge line with increasing 119877-line value correspondingto a higher surge margin The 119877-lines have no physicalmeaning but are simply mathematical constructs orthogonalto corrected speed so that any value of 119877-line and correctedspeed will have a unique compressor operating point Theengine model did not adopt the 119877-line methodology dueto the need for additional iterations (algebraic constraint)within the model

The adopted approach adjusts the original compressormaps to eliminate the curl back of the speed lines near

International Journal of Aerospace Engineering 3

the surge line A minimum slope is assumed for the pressureratio versus corrected mass flow in order to accommodatetwo-dimensional interpolation in lookup tables as shown inFigure 2This approach introduces some error near the surgeline but it is assumed that the engine will not operate inthis region for all valid designs If operations near surge werepart of the study then the approximations near the surge linewould introduce excessive uncertainty

The maps are represented by two-dimensional lookuptables that contain a predetermined matrix for the specificturbo-machine being used Row and column vectors arealso defined within the map allowing interpolation withinthe matrix based on the input signals to the lookup tableThese input signals are normalized speeds and pressure ratioshown below by (1) and (2) respectively Using these twonormalized signals the performancemap interpolates withinthe predefined matrix an output of a normalized mass flowrate This normalized mass flow rate is used to calculate anactual mass flow rate using (3)

119875119903normalized =

119875out119875in119875119903design

(1)

119873normalized = (119873

radic119879in)(

radic119879indesign

119873design) (2)

= normalized (

designradic119879indesign

119875indesign)(

119875in

radic119879in) (3)

22 Temperature Each turbo-machinery model contains aperformance map that determines an efficiency for a givenshaft speed and pressure ratio In a manner similar to themass flow rate performance map the efficiency performancemap contains matrix defining efficiencies for predeterminedshaft speeds and pressure ratios The normalized signals forpressure ratio and shaft speed are shown by (1) and (2)respectively The efficiency term yielded from the perfor-mance map is then used to calculate the outlet temperaturefor the compressor and turbine models shown by

119879out119862 = 119879in (1 +1

120578119862

((119875out119875in

)

((119896minus1)119896)

minus 1)) (4)

119879out119879 = 119879in (1 + 120578119879((

119875out119875in

)

((119896minus1)119896)

minus 1)) (5)

23 Work The power (work rate) absorbed or produced isbased on the outlet mass flow rates as well as the inlet andoutlet temperatures for each of the turbo-machine modelsThe fan and compressor models consume power (negative)while the HP and LP turbines produce power (positive)The inlet and outlet temperatures of each model are used tocalculate an enthalpy value using (6) These inlet and outletenthalpies are combined with the outlet mass flow rate to

calculate the power for the compressor and turbine modelsas shown by (7) and (8) respectively

ℎ = int

119879

0

119862119901(119879) 119889119879 (6)

119862

= 119862(ℎin minus ℎout) (7)

119879


24 Plenum Volume Dynamics Another important tech-nique in increasing the simulation speed of the engine modelwas the implementation of an isentropic plenum volumedynamics using (9) Plenum volume dynamics provide acontinuous solution for the pressures within the enginemodel [15 16] For simplicity and efficiency typically plenumvolumes are assumed to be steady-state with the flow rateentering the volume equal to the flow rate leaving the volumeA steady-state assumption for the plenum volumes is validin the sense the time constants of the plenum volumes arenegligible when compared to the shaft and thermal timeconstants On the contrary the engine model has higher sim-ulation speeds with dynamic plenum volumes while addingmore physics The increase in simulation speed is attributedto the numerical methods employed within the advancedstiff solvers in Simulink Stiff solvers utilize the Jacobianmatrix which consists of the partial derivatives of all thestates The Jacobian matrix is used to estimate the new statesfor the next time step [17] If a very sensitive parametersuch as operating pressures within the engine is assumedsteady-state then the pressures are not states and excludedfrom the Jacobian matrix The numerical solver is unable toaccount for the changes in pressure and the gradients of allthe states with respect to pressure The numerical solver willhave a reduced time step increasing computational time andnumerical round off errors during changes in pressure

119875out = int(in minus out) 119877out119879

119881119889119905 (9)

3 Overview of Model Components

The one-dimensional turbofan engine model has incorpo-rated a lot of detailThemodel includes detailed performancemaps pressure drop plenum volume dynamics thermaltransients chemical reactions parallel flow paths with massflows dependent on local densities and pressure loses andshaft dynamics

Detailed descriptions of each of the componentmodels aswell as the unique equations used to model the appropriatephysics are covered in the following sections

31 Compressor Located at the front of the engine in Figure 1the fan is responsible for drawing air into the engine TheLP compressor also known as the fan is driven by the LPshaft and compresses the air entering the engine Some ofthis compressed air then enters the HP compressor (corestream) where it will be compressed even further but themajority of the fan air enters the bypass plenum volume


(bypass stream) Within the fan model several key equationsare modeled to describe the relevant physics In addition tothe common equations described in the previous sectionthe fan has a unique inlet pressure For subsonic conditionsthe inlet pressure is found by calculating the total pressureat the front of the aircraft as shown by (10) Equation (10)assumes 100pressure recovery for the inlet diffuser It is alsoworth mentioning that the outlet pressure term for the fan isrepresented by the bypass plenum volume pressure which isoutlined in the bypass component section

119875in = 119875ambient +1

2120588ambient(119872radic119896ambient119877ambient119879ambient)

2

(10)

Air from the fan that does not enter the bypass plenumvolume is sent to the HP compressor in Figure 1 The HPcompressor increases the core air pressure to its largest valuebefore it enters the combustor The HP compressor is drivenby the HP shaft which is powered by the HP turbine Theoutlet pressure is provided by the combustor and will bediscussed in the combustor section The inlet pressure isequivalent to the bypass plenum volume pressure and willbe discussed in the bypass component section The HPcompressor has bleed air extracted at the exit After the actualoutlet mass flow rate is calculated based on Section 21 bleedair is removed to cool turbine blades (secondary flow) andpower additional systems within the aircraft

32 Combustor The combustor in the center of Figure 1receives an air stream from the HP compressor as well as afuel streamof JP-8 Energy balances are used to determine thetemperature and composition of the outgoing air streamThismixture is sent to the HP turbine It is assumed that completecombustion of the JP-8 fuel occurs yielding CO

2 H2O and

N2as the sole products of the reaction The JP-8 combustion

equation is expressed by (11) [18]The heat of reaction and theenthalpy flow are calculated using (12) and (13) respectivelyThe temperature of the combustor outlet stream can be foundusing (14) Equation (15) yields the molar flow rate and themolar composition by (16) for combustion of the air and fuelstreams Equation (17) provides the reaction vector derivedfrom (11)

C103

H205

+ 15425 [O2+ 376N

2]

997888rarr 103CO2+ 1025H

2O + 57998N

2

(11)

119877119909

= sum119877(ℎ119891119877

) minus sum119875(ℎ119891119875

) (12)

ℎout =in +

119877119909

out (13)

119879outlet =ℎout119862119901out

(14)

out = in + sumR (15)

119881119888V119862

119889Xout119889119905

= in (Xin minus Xout) minus Xout sumR + R (16)

R = JP-8 [119903JP-8 119903CO 119903CO2

119903H2

119903H2O 119903N

2

119903O2]

= JP-8 [minus1 0 103 0 1025 0 minus15425]

(17)

33 Turbine The high pressure (HP) turbine receives thecombustor outletmixture shown in Figure 1 Power generatedby the turbine is used to apply a torque to the HP shaftwhich then drives the HP compressor The plenum volumelocated between the HP compressor and the HP turbine inletis modeled within the HP turbine to derive the HP turbineinlet pressureThe plenum volume primarily accounts for thevolume of the combustor The mass flow rate entering thisplenum volume is known from the HP compressor modelThe outlet mass flow rate of the HP turbine is specified bythe performance mapWith the incoming and outgoing massflows of the plenum volume known the dynamic pressure ofthe plenum volume can be calculated via (9)

As air enters the HP turbine a secondary air stream forblade cooling is added thus reducing the temperature of thecore airThis secondary stream is fed by the bleed air removedat the HP compressor exit Within the HP turbine model asubsystem exists to calculate the flow rate of bleed air whichcools the HP turbine inlet as well as the flow rate of airthat continues on to the LP turbine The bleed mass flowrate calculations are shown by (18) and (19) respectively Aspreviously mentioned the HP turbine bleed flow is mixedwith core air from the combustor outlet before it enters theHP turbine in order to provide cooling Two calculations arerequired to determine the resulting mass flow rate as well asthe temperature of the newly formed mixture that enters theHP turbine The required calculations for the mass flow rateand temperature signals entering the HP turbine are shownby (20) and (21) respectively

HPTbleed = 119909HPTbleed (18)

LPTbleed = (1 minus 119909HPT) bleed (19)

inHPT = HPTbleed + combustor (20)

119879inHPT = 119879HPCbleedHPTbleed

inHPT+ 119879combustor

combustorinHPT

(21)

After core air exits the HP turbine it enters the lowpressure (LP) turbine as shown in Figure 1 The LP turbineproduces power that drives the LP shaft which in turn drivesthe fan The inlet pressure is found in a similar fashion tothe inlet pressure of the HP turbine as shown by (9) The LPturbine bleed air mass flow rate is already known from (19)The inlet mass flow rate which includes the core air from theHP turbine outlet as well as the LP turbine bleed air and theinlet temperature are found using amethod equivalent to (20)and (21) respectively

34 Bypass Plenum Volume The bypass model determinesbypass flow rate and pressure The bypass labeled in Figure 1is comprised of the void space around the HP compressor


combustor HP turbine LP turbine and shafts The air thatbypasses the HP compressor combustor HP turbine andLP turbine travels through a bypass duct and enters a mixerplenum volume at the nozzle inlet The majority of the fanmass flow enters the bypass rather than the HP compressorThe pressure drop across the bypass plenum volume drivesthe amount of mass flow that bypasses the core of the engineThis mass flow rate is represented by (22) [19] Equation (22)assumes low Mach number flow incompressible flow Thedynamic pressure of the plenum volume is found using (9)

bypass = 119862119889119860bypassradic2120588 (119875in minus 119875out) (22)

35 Nozzle The nozzle is the final component in a turbofanengine flow path shown on the right of Figure 1 A con-verging-diverging nozzle creates the thrust needed to propelthe aircraft forward Air from the LP turbine outlet and thebypass plenum volume are combined in the mixer volumebefore entering the nozzle The temperature of the mixedstream is shown by (23) The pressure of the mixer volume isfound using (9)Within the actual nozzle two cases can existChoked Flow orNon-Choked Flow [20] To determine whichcase is occurring at a given time a critical pressure ratio isfound using (24) through (30) To determine the thrust exitvelocity and mass flow rate must be calculated

119879mixer = int119876net

119898119881119862119901out

119889119905 (23)

(119875out119875in

)

critical= (

2

119896 + 1)

119896(119896minus1)

(24)

351 Choked Flow Choked flow occurs when the actualnozzle pressure ratio is less than the critical pressure ratiowhile nonchoked flow occurs for pressure ratios larger thanthe critical value When the nozzle model has determinedthat the flow is choked the exit mass flow rate is shown by(25) The exit temperature is calculated using (26) The speedof sound and exit velocity are found using (27) and (28)respectively

out = 119875mixer119860 throatradic119896

119877119879mixer(

2

119896 + 1)

((119896+1)2(119896minus1))

(25)

119879out =119879mixer

1 + ((119896 minus 1) 2) (26)

119888out = radic119896119877119879out (27)

Vout = 119888out (28)

352 Non-Choked Flow When the flow is not choked sev-eral terms must first be derived including the exit Machnumber the exit temperature and the speed of sound atthe nozzle exit The nozzle exit Mach number is found by(29) Using the exit Mach number the exit temperature iscalculated using (30) With the exit temperature known thespeed of sound at the nozzle exit is calculated as in the choked

case with (27) The velocity of air exiting the nozzle can befound using the speed of sound and Mach number results asshown by (31) The density of air exiting the nozzle is foundusing (32) Finally exit mass flow of the nozzle is found using(33)

119872out = radic(2

119896 minus 1)[(

119875mixer119875out

)

((119896minus1)119896)

minus 1] (29)

119879out =119879mixer

1 + 119872out2

((119896 minus 1) 2) (30)

Vout = 119872out119888out (31)

120588out =119875out

119877out119879out (32)

out = 119872out119888out120588out119860out (33)

353Thrust Once the nozzlemass flow rate and exit velocityhave been established for choked or nonchoked flow thethrust is calculated The thrust produced by the engine isbased on themass flowentering and exiting the engine aswellas the pressure difference between the nozzle and ambient airThe inlet mass flow rate which is equivalent to the fan massflow rate is already known The inlet velocity however mustbe calculated using (34) Using the inlet velocity the totalengine thrust is represented by (35) for a convergent nozzle

Vinlet = 119872aircraftradic119896ambient119877ambient119879ambient (34)

Thrust = (outVout minus inVin) + 119860 in (119875out minus 119875ambient) (35)

36 Shaft The HP shaft connects the HP Turbine and theHP Compressor Power from the HP turbine is transferredby the HP shaft to drive the HP compressor The HP turbinework signal represents a positive load and theHP compressorwork signal represents a negative load The HP shaft speed isthe only calculation performed within the shaft model and isrepresented by (36) Additional auxiliary loads may be addedto the shaft (not shown)

119873HP shaft =30

120587int

119862

+ HPT119869HPshaft120596HPshaft

(36)

The LP shaft connects the LP Turbine and the fan Powerfrom the LP turbine is transferred by the LP shaft to drive thefanThe LP turbine work signal represents a positive load andthe fan work signal represents a negative load The LP shaftspeed is the only calculation performedwithin themodel andis represented by (37) The LP shaft has additional auxiliaryloads pumps and generator which are inputs to the enginemodel

119873LPshaft =30

120587int

Fan + LPT + Pumps + Generator

119869LPshaft120596LPshaft (37)


0 2 4 6 8 10 12 14 16 18 20

Thrust

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

4

45

5Fuel

08

11

14

SFC(lb

mh

rlb

f)

17

18

19

20Th

rust

(klb

f)

138

14

142

NH

P(k

RPM

)

NLP

(kRP

M)

76

78

8Shaft speeds

HPLP

SFCm

Fuel

(lbm

s)

mFuel

Figure 3 Engine fuel flow input specific fuel consumption thrust and shaft speeds responses to a step in fuel flow

4 Simulation Results

Two sets of simulations were performed using the enginemodel First a fuel perturbation was performed to demon-strate and isolate the transient response behavior of the tur-bofan engine Second a full mission was simulated to com-pare the simulation times of previous approaches

41 Engine Fuel Perturbation The turbofan engine modelwas simulated at an altitude of 20000 ft (6096m) and a speedof 06Mach numberThe fuel flow rate was stepped from 441to 48 lbms (2 to 22 kgs) Figure 3 presents the step in fuelflow at time equal to 1 second on the top graphwith the thrustresponse in the middle and shaft speeds at the bottomThereare essentially three time constants affecting the response ofthe engine fuel step approximately 0 seconds temperatureand pressure approximately 005 seconds and shaft speedapproximately 10 seconds The step increase in fuel flowincreases the thrust output of the engine from 156 to 183 klbfThere is a rapid increase (first 005 second) in thrust initiallyfollowing the step up in fuel which is a result in the rapidtemperature increase in the components downstream of thecombustor as shown in the HP and LP turbine temperaturespresented in Figure 4 In addition the pressures throughoutthe engine also experience a rapid increase following the stepup in fuel as shown for the turbines in Figure 4 and the fanand compressor in Figure 5 The increase in pressure alsocontributes to the increase in thrust

Initially there is sharp increase in SFC in Figure 3 due tothe sudden (step change) increase in fuel and delayed increasein thrust over a 005 seconds The SFC decreases sharplyduring the first 005 seconds following the perturbation as thethrust increases On a larger time scale of 10 seconds there isgradual increase in thrust due to the response in shaft speedswhich in turn increases the flow rate through the engine andcontinues to decrease the SFC

Figure 4 presents the turbine pressures temperatures andmass flows as well as bypass mass flow The effects of thepressure and temperature on thrust and SFC were previouslydiscussed The total temperature of the turbines in Figure 4increases rapidly due to the increase in fuel and temperaturein the combustor but begin to decline as the air mass flowrate through the combustor and turbines increase as the shaftspeeds increaseThe flow rates in Figure 4 present interestingbehavior to the fuel perturbation The turbine mass flowrates decrease (very slightly) and the bypass mass flow ratedecreases noticeably before returning to the previous initialvalueThe same trendsmay be seen in the fan and compressormass flows as for the turbine mass flows in Figure 5 Themass flow rates initially drop (very slightly) in the fan andcompressor with the step increase in fuel The initial drop inmass flow rate is due to the pressures increasing as shown inFigure 5 for the fan and compressor The drop in mass flowreverses with the increase in shaft speed shown in Figure 3The fan and compressor maps in Figure 6 support the massflowpressure interactions Figure 6 illustrates the initial


0 5 10 15 2038

40

42

44

46

PH

P(p

sia)

Total pressure

Total temperature

21

22

23

24

25

PLP

(psia

)

1300

1400

1500

1600

TH

P(∘

F)

1400

1300

1200

1100

TLP

(∘F)

240

250

260

270

280

mH

Pan

dm

LP(lb

ms

)

Time (s)

0 5 10 15 20

Time (s)

0 5 10 15 20

Time (s)

100

102

104

106

108

mby

pass

(lbm

s)Mass flow

HPTLPT

HPTLPT

HPTLPTBypass

Figure 4 Pressures temperatures and mass flows for the HP turbine and LP turbine including mass flow of the 2nd stream bypass duct

decrease in normalized mass flow with increased pressureratio (moving up and to the left) before the normalized massflow increases with shaft speed (moving right)

The bypassmass flow rate noticeably decreases in Figure 4as the flow rates rebalance between the HP compressor andbypass in the two-stream engine Some of the decline inbypass flow is due to the initial drop in fan mass flowAdditional decline inmass flow is due to the quicker responseof the compressor and HP turbine to the fuel perturbationThe HP turbine is effected initially by the fuel increase sinceits first downstreamof the combustor In turn the compressorspeed is increased since it is connected to the HP shaftwith the HP turbine The compressor speed increases thecompressor mass flow more rapidly before the fan mass flowincreases resulting in less flow in the bypass Once the fanspeed catches up with the compressor speed and fan massflow increases the bypass flow returns to its initial valueending with a slight increase in overall flow

If the plenum volumes were assumed to be steady-statethe initial mass flow responses to the pressure perturbationswould not be capturedThe engine model was able to captureboth the plenum volume dynamics and the shaft dynamics

42 Computational Time The engine model is assessed forcomputational efficiency in this section First the engine

model is simulated standalone for a specified mission pre-sented in Figure 7 The mission is 7700 seconds with varyingaltitudes and Mach number During a mission the enginemodel has 11 inputs The inputs are

(1) fuel flow(2) fuel temperature(3) air inlet temperature(4) air inlet pressure(5) air inlet composition(6) bleed flow demand from the fan(7) bleed flow demand from the compressor(8) shaft power extraction from the LP shaft(9) shaft power extraction from the HP shaft(10) pressure drop in the bypass heat exchangers(11) heat flux in the bypass heat exchangers

Throughout the mission all of the inputs are continuouslychanging All of the inputs are treated as a disturbanceexcept for the fuel flow The inputs described as disturbancesare provided by lookup tables utilizing previous data froma previous full mission T2T simulation The fuel flow is


0 2 4 6 8 10 12 14 16 18 20

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

Total pressure

Total temperature

Mass flow

FanCompressor

FanCompressor

FanCompressor

23

24

25

26

Pfa

n(p

sia)

215

220

225

230

235

Tfa

n(∘

F)

350

360

370

380

240

260

280

300

Pco

mp

(psia

)

1000

1020

1040

1060

1080

Tco

mp

(∘F)

240

250

260

270

mco

mp

(lbm

s)

mfa

n(lb

ms

)

Figure 5 Pressures temperatures and mass flows for the fan and compressor

0 02 04 06 08 1 12 1402

04

06

08

1

12Fan map (circle indicates end point)

Nor

mal

ized

pre

ssur

e rat

ioN

orm

aliz

ed p

ress

ure r

atio

Normalized mass flow

0 02 04 06 08 1 12 14


0

05

1

15Compressor map (circle indicates end point)

Figure 6 Fan and compressor map for a step in fuel flow to theengine

controlled to achieve a demanded thrust from the enginevia a feedback controller The thrust demand is also in theform of a lookup table utilizing previous data This approach

0 20 40 60 80 100 1200

5

10

15

20

25

30

35

40

Alti

tude

(tho

usan

d ft)

Time (min)

Vehicle mission

01

02

03

04

05

06

07

08

09

Mac

hnu

mbe

r

AltitudeMach number

Figure 7 Altitude and Mach Profile for a 21 hour mission

isolates the computational efficiency of the engine and itscontroller The data for the inputs contained in the lookuptables are from previous simulations of a T2T model Theengine model from previous work is also used to simulate


the same mission with the same inputs for comparison [3]The new engine model took 15 seconds to complete theentire mission simulation The engine model from previouswork took 29304 seconds to simulate the same missionwith less detail The new turbofan engine model with theplenum volume dynamics and approximated performancemaps near the surge line reduced the simulation time by9995 while providing more detail and capturing highfrequency dynamics

The engine model was then integrated with a T2T modelThe T2T model is exercised to simulate the same predeter-mined mission presented in Figure 7 Throughout the mis-sion demands on shaft power extractions bleed flows fuelflow inlet conditions and the bypass heat exchanger arecontinuously changing but are not implemented via lookuptables The T2T model contains dynamic models for theair vehicle thermal management system electrical systemand fuel system The T2T model with the old engine modelwithout the techniques presented completed the missionsimulation in 72650 seconds The T2T model with the newturbofan engine model completed the mission simulation in186 seconds The new engine model reduced the simulationtime for the T2T model by 9975 All simulations werecarried out on the same hardware The hardware used wasan Intel Core 2 Duo Processor E8600 with 4G of RAM Theincrease in computational efficiency is significantly openingthe opportunity for design optimization

5 Discussion of Results

The results for the turbofan engine model built in Matlab-Simulink were presented The engine model is capable ofcapturing the shaft dynamics thermal dynamics and theunsteady mass flowplenum volume dynamics By capturingthe unsteadymass flowplenumvolume dynamics two thingswere achieved First unsteady mass flow and pressure aresimulated providing more detail in engine behavior Theseperturbations propagate throughout the engine and inte-grated subsystems that utilize compressor bleed air Secondthe engine model reduces the simulation time of a vehiclelevel T2Tmodel tool making it feasible to simulate thousandsof simulations needed for a complex optimization routineof a large system The ability to perform simulations in atimely manner for system level optimization including tran-sient behavior and control architectures provides advancedcapability to design highly integrated dynamic systems

The modeling techniques shown here introduce lim-itations For example simulations with the compressorsoperating near the surge line of the map should be abortedor considered with caution due to the approximations madein the maps to remove iteration loops The unsteady massflowplenum volume dynamics captured by the model areapproximated by using large control volumes within theengine components For more detailed results the controlvolumes need to be discreetly divided into smaller controlvolumes creating a one-dimensional distribution ofmass flowand pressure for each of the engine components This wouldincrease the complexity and simulation time depending

on the resolution of the one-dimensional grid of controlvolumes

6 Conclusion

A dynamic turbofan engine model was developed in themodeling and simulation environment of Matlab-SimulinkThe model has been built without the aid of proprietary dataallowing the tool to be made available to multiple designand research groups Special attention was also paid to thecapturing of dynamic behavior These dynamics not onlyincrease the physics being captured but even more impor-tantly they reduce the occurrence of algebraic constraintsleading to increased simulation speed The new techniquesof approximating the performance maps near the surge linewithout the use of 119877-lines and implementing the plenumvolume dynamics have proven very efficient as the currentengine model completes the 7700 second mission in just 15seconds When the engine model was integrated with the fullT2Tmodel significant gains in computational efficiency weremaintained The new modeling techniques of incorporatingplenumvolume dynamics and approximating the compressormaps near the surge line result in a more efficient T2Tmodel The T2T model with the new engine model will becapable of conducting trade studies and vehicle level designoptimization

Nomenclature

119860 Area (m2)AVS Air vehicle system119862 Concentration (kmolem3)119862119889 Discharge coefficient

119888 Speed of sound (ms)119862119901 Specific heat at constant pressure (kJkmolelowastK)

119889119905 Differential time (s)ℎ119894 Enthalpy 119894 (kJkg)

119867 Enthalpy (kJ)HP High pressure119869 Polar moment of inertia (kglowastm4)119896119894 Heat capacity ratio 119894

LP Low pressure119872 Mach number119898 Mass (kg)119894 Mass flow rate (kgs)

119873119894 Shaft speed 119894 (RPM)

Molar flow rate (kmols)120578119894 Efficiency for component 119894

120588 Density (kgm3)119875119894 Pressure 119894 (kPa)

119875119903 Pressure ratio

119894 Thermal energy rate 119894 (kW)

119877119894 Ideal gas constant 119894 (kJkgK)


R Reaction vector (kmolkmolfuel)RPM Shaft speed (revmin)119879119894 Temperature 119894 (K)

TMS Thermal management system


T2T Tip-to-Tail119881 Volume (m3)V Velocity (ms)119894 Power or work rate of component 119894 (kW)X Mole fraction vector (mol frac)119909 Flow fraction120596 Angular velocity (rads)

Subscripts

aircraft Aircraft parametersambient Ambient parametersnormalized Normalized parameterbleed Bleed flow parameters119862 CompressorCombustor Combustorcritical Critical condition for choked flowdesign Design point parametergenerator Generator parameterin In to control volumeHP High pressureHPT High pressure turbineJP-8 Jet propellant 8 jet fuel specified by

MIL-DTL-83133LP Low pressureLPT Low pressure turbinemixer Mixer parametersnet Net heat rateout Exit to control volumepump Pump parametershaft Shaft parameter119877119909 Net reaction heat rate of combustion119879 Turbinethroat Parameters at the throat of nozzle

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank AFOSR ASEE and AFRLin all of the support provided for this research The authorswould also like acknowledge the funding provided byWrightState University Research Sponsored Program

References

[1] M Amrhein J R Wells E A Walters et al ldquoIntegrated elec-trical system model of a more electric aircraft architecturerdquo inPower Systems Conference 2008

[2] J R Wells M Amrhein E Walters et al ldquoElectrical accumu-lator unit for the energy optimized aircraftrdquo SAE InternationalJournal of Aerospace vol 1 no 1 pp 1071ndash1077 2009

[3] R A Roberts S M Eastbourn and A C Maser ldquoGenericaircraft thermal tip-to-tail modeling and simulationrdquo in Pro-ceedings of the 47th AIAAASMESAEASEE Joint Propulsion

Conference and Exhibit pp 2011ndash5971 San Diego Calif USAAugust 2011

[4] E A Walters M Amrhein T OrsquoConnell et al ldquoINVENTmod-eling simulation analysis and optimizationrdquo in Proceedings ofthe 48th AIAA Aerospace Sciences Meeting Including the NewHorizons Forum and Aerospace Exposition Orlando Fla USAJanuary 2010

[5] ACMaser EGarcia andDNMavris ldquoFacilitating the energyoptimization of aircraft propulsion and thermal managementsystems through integrated modeling and simulationrdquo in Pro-ceedings of the 2010 Power Systems Conference January 2010

[6] C J Miller A C Maser E Garcia and D N Mavris ldquoINVENTsurrogate modeling and optimization of transient thermalresponsesrdquo in Proceedings of the 50th AIAA Aerospace SciencesMeeting Including the New Horizons Forum and AerospaceExposition AIAA 2012-1123 January 2012

[7] A C Maser E Garcia and D N Mavris ldquoThermal man-agement modeling for integrated power systems in a tran-sient multidisciplinary environmentrdquo in Proceedings of the45th AIAAASMESAEASEE Joint Propulsion Conference andExhibit August 2009

[8] R W Claus A L Evans J K Lylte and L D Nichols ldquoNumer-ical propulsion system simulationrdquo Computing Systems in Engi-neering vol 2 no 4 pp 357ndash364 1991

[9] SM Camporeale B Fortunato andMMastrovito ldquoAmodularcode for real time dynamic simulation of gas turbines insimulinkrdquo Journal of Engineering for Gas Turbines and Powervol 128 no 3 pp 506ndash517 2006

[10] J H Kim T W Song T S Kim and S T Ro ldquoModel devel-opment and simulation of tansient behavoir of heavy duty gasturbinesrdquo Journal of Engineering for Gas Turbines and Powervol 123 no 3 pp 589ndash594 2001

[11] R Chacartegui D Sanchez A Munoz and T Sanchez ldquoRealtime simulation of medium size gas turbinesrdquo Energy Conver-sion and Management vol 52 no 1 pp 713ndash724 2011

[12] L Chuankai Q Tian and D Shuiting ldquoTransient analysis ofvolumepacking effects on turbofan enginerdquo inProceedings of the2nd International Symposium on Aircraft Airworthiness (ISAArsquo11) pp 549ndash558 October 2011

[13] S M Jones An Introduction to Thermodynamic PerformanceAnalysis of Aircraft Gas Turbine Engine Cycles Using the Numer-ical Propulsion System Simulation Code 2007

[14] G C Oates Aircraft Propulsion Systems Technology and DesignAIAA Washington DC USA 1st edition 1989

[15] N U Rahman and J F Whidborne ldquoReal-time transient threespool turbofan engine simulation a hybrid approachrdquo Journalof Engineering for Gas Turbines and Power vol 131 no 5 ArticleID 051602 pp 29ndash36 2009

[16] S Kim P Pilidis and J Yin ldquoGas turbine dynamic simulationusing simulinkrdquo in Proceedings of the Power Systems ConferenceJanuary 2000

[17] Y Yu Stiff Problems in Numerical Simulation of Biochemical andGene Regulatory Networks The University of Georgia AthensGa USA 2004

[18] J A Cooke M Bellucci M D Smooke et al ldquoComputationaland experimental study of JP-8 a surrogate and its componentsin counterflow diffusion flamesrdquo Proceedings of the CombustionInstitute vol 30 pp 439ndash446 2005


[19] B Munson D Young T Okiishi and W Huebsch Fundamen-tals of FluidMechanics JohnWiley amp Sons NewYork NY USA3rd edition 1998

[20] S Yarlagadda Performance Analysis of J85 Turbojet EngineMatching Thrust with Reduced Inlet Pressure to the CompressorUniversity of Toledo 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



a tool to incorporate the relevant factors which affect propul-sion performance early in the design and analysis process [8]NPSS was primarily a steady-state tool that has been widelyadopted by government and industry over the past decadesbut has expanded its capabilities in dynamic simulationNPSS can be expanded to incorporate systems beyond themain engine The NPSS based models were inefficient whenutilized inMatlab-Simulink based T2Tmodel simulations [5]with simulation times 16 of real time Surrogate models wereused in the T2T model to reduce computational times [6]Dynamicmodels have been developed for real time operationwith unsteady effects due to mass accumulation consideredby adding a plenum between each compressor and turbinestage and shaft dynamics [9ndash11] These models capture someof the dynamics of the engine but assume mass flows fromprevious time-steps in different sections of the engine that isthe compressor mass flow Other work in dynamic modelingcaptured the plenum volume dynamics needed to predict thesecondary flows within the engine for blade cooling [12]

As previously mentioned using NPSS models with Sim-ulink is less efficient The computational inefficiencies residein the compiling of source code within the Matlab-Simulinkframework These subtle inefficiencies do not pose issueswhen executing a limited amount of trade studies or simu-lations but do pose a problem when performing thousandsof simulations for an optimization routine In this workan engine model developed entirely in Matlab-Simulinkenvironment is presented Several techniques were attemptedto increase the simulation speed of the turbofan enginemodel The techniques adopted to achieve a computationalefficient turbofan engine model are presented in this workThe turbofan engine model is developed without iterationloops (algebraic constraints) and all states are continuousThis approach is very important for complex system levelsimulations of stiff dynamic systems By modeling all thesignificant states as continuous states and not steady-stateapproximations with discontinuities advanced numericalstiff solvers for stiff systems may be used Numerical stiffsolvers rely on the Jacobian matrix and thus require accurateapproximations for gradients of all continuous states Stiffsolvers dramatically reduced the computational time forsimulating stiff systems Also by having the engine modelconstructed completely in Simulink a complete T2T modelcan be compiled then executed or exported as an efficientexecutable limiting unnecessary callbacks

The following sections provide a detailed description ofthe modeling approach for the transient engine model Sim-ulation results are presented along with the comparisonin simulation times Key approaches in performance mapinterpretation and plenum volume dynamics enable thedevelopment of a computationally efficient engine model

2 Fundamental Equation Development

The engine consists of several key component models pre-sented in Figure 1 These models include Fan High Pressure(HP) Compressor Combustor High Pressure (HP) TurbineLow Pressure (LP) Turbine Bypass Plenum Volume Nozzle

Bypass

Bypass

Fan

LP tu

rbin

e

Noz

zle

Heat exchanger

Heat exchanger

LP shaft

Combustor

HP shaft

HP

com

pres

sor

HP

turb

ine

Combustor

Figure 1 Two-stream turbofan engine diagram of major compo-nents

Speed 1

Speed 2

Speed 3

Speed 4

Corrected mass flow

Pres

sure

ratio

R-line 1R-line 2

R-line 3R-line 4

Figure 2 Compressor map with 119877-lines and approximations nearthe surge line

High Pressure (HP) Shaft and Low Pressure (LP) ShaftEach of the turbo-machinery components in the engineshare several common equations defining the physics of themodel Equations common to the fan HP compressor HPturbine and LP turbine are outlined in the following sectionincluding mass flow temperature and work terms

21 Mass Flow Each turbo-machinery model contains aperformance map that determines a corrected mass flowfor a given shaft speed and pressure ratio The traditionalapproach of interpreting the performancemaps is to create anadditional independent variable known as the 119877-line [13 14]Figure 2 shows the compressor map with 119877-lines includedTypically the first 119877-line is given a value of 10 that coincideswith the surge lineThe remaining119877-lines are roughly parallelto the surge line with increasing 119877-line value correspondingto a higher surge margin The 119877-lines have no physicalmeaning but are simply mathematical constructs orthogonalto corrected speed so that any value of 119877-line and correctedspeed will have a unique compressor operating point Theengine model did not adopt the 119877-line methodology dueto the need for additional iterations (algebraic constraint)within the model

The adopted approach adjusts the original compressormaps to eliminate the curl back of the speed lines near





119875out119875in119875119903design

(1)

119873normalized = (119873

radic119879in)(

radic119879indesign

119873design) (2)

= normalized (


119875indesign)(

119875in

radic119879in) (3)


119879out119862 = 119879in (1 +1

120578119862

((119875out119875in

)

((119896minus1)119896)

minus 1)) (4)

119879out119879 = 119879in (1 + 120578119879((

119875out119875in

)

((119896minus1)119896)

minus 1)) (5)



ℎ = int

119879

0

119862119901(119879) 119889119879 (6)

119862


119879




119881119889119905 (9)







119875in = 119875ambient +1


2

(10)



2 H2O and



C103

H205

+ 15425 [O2+ 376N

2]

997888rarr 103CO2+ 1025H

2O + 57998N

2

(11)

119877119909

= sum119877(ℎ119891119877

) minus sum119875(ℎ119891119875

) (12)

ℎout =in +

119877119909

out (13)

119879outlet =ℎout119862119901out

(14)


119881119888V119862

119889Xout119889119905


R = JP-8 [119903JP-8 119903CO 119903CO2

119903H2

119903H2O 119903N

2

119903O2]

= JP-8 [minus1 0 103 0 1025 0 minus15425]

(17)








combustorinHPT

(21)








119898119881119862119901out

119889119905 (23)

(119875out119875in

)

critical= (

2

119896 + 1)

119896(119896minus1)

(24)



119877119879mixer(

2

119896 + 1)

((119896+1)2(119896minus1))

(25)

119879out =119879mixer

1 + ((119896 minus 1) 2) (26)

119888out = radic119896119877119879out (27)

Vout = 119888out (28)



119872out = radic(2

119896 minus 1)[(


)

((119896minus1)119896)

minus 1] (29)

119879out =119879mixer

1 + 119872out2

((119896 minus 1) 2) (30)

Vout = 119872out119888out (31)

120588out =119875out

119877out119879out (32)






119873HP shaft =30

120587int

119862


(36)


119873LPshaft =30

120587int




0 2 4 6 8 10 12 14 16 18 20

Thrust

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

4

45

5Fuel

08

11

14

SFC(lb

mh

rlb

f)

17

18

19

20Th

rust

(klb

f)

138

14

142

NH

P(k

RPM

)

NLP

(kRP

M)

76

78

8Shaft speeds

HPLP

SFCm

Fuel

(lbm

s)

mFuel








0 5 10 15 2038

40

42

44

46

PH

P(p

sia)

Total pressure

Total temperature

21

22

23

24

25

PLP

(psia

)

1300

1400

1500

1600

TH

P(∘

F)

1400

1300

1200

1100

TLP

(∘F)

240

250

260

270

280

mH

Pan

dm

LP(lb

ms

)

Time (s)

0 5 10 15 20

Time (s)

0 5 10 15 20

Time (s)

100

102

104

106

108

mby

pass

(lbm

s)Mass flow

HPTLPT

HPTLPT

HPTLPTBypass










0 2 4 6 8 10 12 14 16 18 20

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

Total pressure

Total temperature

Mass flow

FanCompressor

FanCompressor

FanCompressor

23

24

25

26

Pfa

n(p

sia)

215

220

225

230

235

Tfa

n(∘

F)

350

360

370

380

240

260

280

300

Pco

mp

(psia

)

1000

1020

1040

1060

1080

Tco

mp

(∘F)

240

250

260

270

mco

mp

(lbm

s)

mfa

n(lb

ms

)


0 02 04 06 08 1 12 1402

04

06

08

1


Nor

mal

ized

pre

ssur

e rat

ioN

orm

aliz

ed p

ress

ure r

atio


0 02 04 06 08 1 12 14


0

05

1




0 20 40 60 80 100 1200

5

10

15

20

25

30

35

40

Alti

tude

(tho

usan

d ft)

Time (min)

Vehicle mission

01

02

03

04

05

06

07

08

09

Mac

hnu

mbe

r

AltitudeMach number










6 Conclusion


Nomenclature






119873119894 Shaft speed 119894 (RPM)











Subscripts





Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119875out119875in119875119903design

(1)

119873normalized = (119873

radic119879in)(

radic119879indesign

119873design) (2)

= normalized (


119875indesign)(

119875in

radic119879in) (3)


119879out119862 = 119879in (1 +1

120578119862

((119875out119875in

)

((119896minus1)119896)

minus 1)) (4)

119879out119879 = 119879in (1 + 120578119879((

119875out119875in

)

((119896minus1)119896)

minus 1)) (5)



ℎ = int

119879

0

119862119901(119879) 119889119879 (6)

119862


119879




119881119889119905 (9)







119875in = 119875ambient +1


2

(10)



2 H2O and



C103

H205

+ 15425 [O2+ 376N

2]

997888rarr 103CO2+ 1025H

2O + 57998N

2

(11)

119877119909

= sum119877(ℎ119891119877

) minus sum119875(ℎ119891119875

) (12)

ℎout =in +

119877119909

out (13)

119879outlet =ℎout119862119901out

(14)


119881119888V119862

119889Xout119889119905


R = JP-8 [119903JP-8 119903CO 119903CO2

119903H2

119903H2O 119903N

2

119903O2]

= JP-8 [minus1 0 103 0 1025 0 minus15425]

(17)








combustorinHPT

(21)








119898119881119862119901out

119889119905 (23)

(119875out119875in

)

critical= (

2

119896 + 1)

119896(119896minus1)

(24)



119877119879mixer(

2

119896 + 1)

((119896+1)2(119896minus1))

(25)

119879out =119879mixer

1 + ((119896 minus 1) 2) (26)

119888out = radic119896119877119879out (27)

Vout = 119888out (28)



119872out = radic(2

119896 minus 1)[(


)

((119896minus1)119896)

minus 1] (29)

119879out =119879mixer

1 + 119872out2

((119896 minus 1) 2) (30)

Vout = 119872out119888out (31)

120588out =119875out

119877out119879out (32)






119873HP shaft =30

120587int

119862


(36)


119873LPshaft =30

120587int




0 2 4 6 8 10 12 14 16 18 20

Thrust

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

4

45

5Fuel

08

11

14

SFC(lb

mh

rlb

f)

17

18

19

20Th

rust

(klb

f)

138

14

142

NH

P(k

RPM

)

NLP

(kRP

M)

76

78

8Shaft speeds

HPLP

SFCm

Fuel

(lbm

s)

mFuel








0 5 10 15 2038

40

42

44

46

PH

P(p

sia)

Total pressure

Total temperature

21

22

23

24

25

PLP

(psia

)

1300

1400

1500

1600

TH

P(∘

F)

1400

1300

1200

1100

TLP

(∘F)

240

250

260

270

280

mH

Pan

dm

LP(lb

ms

)

Time (s)

0 5 10 15 20

Time (s)

0 5 10 15 20

Time (s)

100

102

104

106

108

mby

pass

(lbm

s)Mass flow

HPTLPT

HPTLPT

HPTLPTBypass










0 2 4 6 8 10 12 14 16 18 20

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

Total pressure

Total temperature

Mass flow

FanCompressor

FanCompressor

FanCompressor

23

24

25

26

Pfa

n(p

sia)

215

220

225

230

235

Tfa

n(∘

F)

350

360

370

380

240

260

280

300

Pco

mp

(psia

)

1000

1020

1040

1060

1080

Tco

mp

(∘F)

240

250

260

270

mco

mp

(lbm

s)

mfa

n(lb

ms

)


0 02 04 06 08 1 12 1402

04

06

08

1


Nor

mal

ized

pre

ssur

e rat

ioN

orm

aliz

ed p

ress

ure r

atio


0 02 04 06 08 1 12 14


0

05

1




0 20 40 60 80 100 1200

5

10

15

20

25

30

35

40

Alti

tude

(tho

usan

d ft)

Time (min)

Vehicle mission

01

02

03

04

05

06

07

08

09

Mac

hnu

mbe

r

AltitudeMach number










6 Conclusion


Nomenclature






119873119894 Shaft speed 119894 (RPM)











Subscripts





Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119875in = 119875ambient +1


2

(10)



2 H2O and



C103

H205

+ 15425 [O2+ 376N

2]

997888rarr 103CO2+ 1025H

2O + 57998N

2

(11)

119877119909

= sum119877(ℎ119891119877

) minus sum119875(ℎ119891119875

) (12)

ℎout =in +

119877119909

out (13)

119879outlet =ℎout119862119901out

(14)


119881119888V119862

119889Xout119889119905


R = JP-8 [119903JP-8 119903CO 119903CO2

119903H2

119903H2O 119903N

2

119903O2]

= JP-8 [minus1 0 103 0 1025 0 minus15425]

(17)








combustorinHPT

(21)








119898119881119862119901out

119889119905 (23)

(119875out119875in

)

critical= (

2

119896 + 1)

119896(119896minus1)

(24)



119877119879mixer(

2

119896 + 1)

((119896+1)2(119896minus1))

(25)

119879out =119879mixer

1 + ((119896 minus 1) 2) (26)

119888out = radic119896119877119879out (27)

Vout = 119888out (28)



119872out = radic(2

119896 minus 1)[(


)

((119896minus1)119896)

minus 1] (29)

119879out =119879mixer

1 + 119872out2

((119896 minus 1) 2) (30)

Vout = 119872out119888out (31)

120588out =119875out

119877out119879out (32)






119873HP shaft =30

120587int

119862


(36)


119873LPshaft =30

120587int




0 2 4 6 8 10 12 14 16 18 20

Thrust

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

4

45

5Fuel

08

11

14

SFC(lb

mh

rlb

f)

17

18

19

20Th

rust

(klb

f)

138

14

142

NH

P(k

RPM

)

NLP

(kRP

M)

76

78

8Shaft speeds

HPLP

SFCm

Fuel

(lbm

s)

mFuel








0 5 10 15 2038

40

42

44

46

PH

P(p

sia)

Total pressure

Total temperature

21

22

23

24

25

PLP

(psia

)

1300

1400

1500

1600

TH

P(∘

F)

1400

1300

1200

1100

TLP

(∘F)

240

250

260

270

280

mH

Pan

dm

LP(lb

ms

)

Time (s)

0 5 10 15 20

Time (s)

0 5 10 15 20

Time (s)

100

102

104

106

108

mby

pass

(lbm

s)Mass flow

HPTLPT

HPTLPT

HPTLPTBypass










0 2 4 6 8 10 12 14 16 18 20

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

Total pressure

Total temperature

Mass flow

FanCompressor

FanCompressor

FanCompressor

23

24

25

26

Pfa

n(p

sia)

215

220

225

230

235

Tfa

n(∘

F)

350

360

370

380

240

260

280

300

Pco

mp

(psia

)

1000

1020

1040

1060

1080

Tco

mp

(∘F)

240

250

260

270

mco

mp

(lbm

s)

mfa

n(lb

ms

)


0 02 04 06 08 1 12 1402

04

06

08

1


Nor

mal

ized

pre

ssur

e rat

ioN

orm

aliz

ed p

ress

ure r

atio


0 02 04 06 08 1 12 14


0

05

1




0 20 40 60 80 100 1200

5

10

15

20

25

30

35

40

Alti

tude

(tho

usan

d ft)

Time (min)

Vehicle mission

01

02

03

04

05

06

07

08

09

Mac

hnu

mbe

r

AltitudeMach number










6 Conclusion


Nomenclature






119873119894 Shaft speed 119894 (RPM)











Subscripts





Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














119898119881119862119901out

119889119905 (23)

(119875out119875in

)

critical= (

2

119896 + 1)

119896(119896minus1)

(24)



119877119879mixer(

2

119896 + 1)

((119896+1)2(119896minus1))

(25)

119879out =119879mixer

1 + ((119896 minus 1) 2) (26)

119888out = radic119896119877119879out (27)

Vout = 119888out (28)



119872out = radic(2

119896 minus 1)[(


)

((119896minus1)119896)

minus 1] (29)

119879out =119879mixer

1 + 119872out2

((119896 minus 1) 2) (30)

Vout = 119872out119888out (31)

120588out =119875out

119877out119879out (32)






119873HP shaft =30

120587int

119862


(36)


119873LPshaft =30

120587int




0 2 4 6 8 10 12 14 16 18 20

Thrust

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

4

45

5Fuel

08

11

14

SFC(lb

mh

rlb

f)

17

18

19

20Th

rust

(klb

f)

138

14

142

NH

P(k

RPM

)

NLP

(kRP

M)

76

78

8Shaft speeds

HPLP

SFCm

Fuel

(lbm

s)

mFuel








0 5 10 15 2038

40

42

44

46

PH

P(p

sia)

Total pressure

Total temperature

21

22

23

24

25

PLP

(psia

)

1300

1400

1500

1600

TH

P(∘

F)

1400

1300

1200

1100

TLP

(∘F)

240

250

260

270

280

mH

Pan

dm

LP(lb

ms

)

Time (s)

0 5 10 15 20

Time (s)

0 5 10 15 20

Time (s)

100

102

104

106

108

mby

pass

(lbm

s)Mass flow

HPTLPT

HPTLPT

HPTLPTBypass










0 2 4 6 8 10 12 14 16 18 20

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

Total pressure

Total temperature

Mass flow

FanCompressor

FanCompressor

FanCompressor

23

24

25

26

Pfa

n(p

sia)

215

220

225

230

235

Tfa

n(∘

F)

350

360

370

380

240

260

280

300

Pco

mp

(psia

)

1000

1020

1040

1060

1080

Tco

mp

(∘F)

240

250

260

270

mco

mp

(lbm

s)

mfa

n(lb

ms

)


0 02 04 06 08 1 12 1402

04

06

08

1


Nor

mal

ized

pre

ssur

e rat

ioN

orm

aliz

ed p

ress

ure r

atio


0 02 04 06 08 1 12 14


0

05

1




0 20 40 60 80 100 1200

5

10

15

20

25

30

35

40

Alti

tude

(tho

usan

d ft)

Time (min)

Vehicle mission

01

02

03

04

05

06

07

08

09

Mac

hnu

mbe

r

AltitudeMach number










6 Conclusion


Nomenclature






119873119894 Shaft speed 119894 (RPM)











Subscripts





Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 2 4 6 8 10 12 14 16 18 20

Thrust

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

4

45

5Fuel

08

11

14

SFC(lb

mh

rlb

f)

17

18

19

20Th

rust

(klb

f)

138

14

142

NH

P(k

RPM

)

NLP

(kRP

M)

76

78

8Shaft speeds

HPLP

SFCm

Fuel

(lbm

s)

mFuel








0 5 10 15 2038

40

42

44

46

PH

P(p

sia)

Total pressure

Total temperature

21

22

23

24

25

PLP

(psia

)

1300

1400

1500

1600

TH

P(∘

F)

1400

1300

1200

1100

TLP

(∘F)

240

250

260

270

280

mH

Pan

dm

LP(lb

ms

)

Time (s)

0 5 10 15 20

Time (s)

0 5 10 15 20

Time (s)

100

102

104

106

108

mby

pass

(lbm

s)Mass flow

HPTLPT

HPTLPT

HPTLPTBypass










0 2 4 6 8 10 12 14 16 18 20

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

Total pressure

Total temperature

Mass flow

FanCompressor

FanCompressor

FanCompressor

23

24

25

26

Pfa

n(p

sia)

215

220

225

230

235

Tfa

n(∘

F)

350

360

370

380

240

260

280

300

Pco

mp

(psia

)

1000

1020

1040

1060

1080

Tco

mp

(∘F)

240

250

260

270

mco

mp

(lbm

s)

mfa

n(lb

ms

)


0 02 04 06 08 1 12 1402

04

06

08

1


Nor

mal

ized

pre

ssur

e rat

ioN

orm

aliz

ed p

ress

ure r

atio


0 02 04 06 08 1 12 14


0

05

1




0 20 40 60 80 100 1200

5

10

15

20

25

30

35

40

Alti

tude

(tho

usan

d ft)

Time (min)

Vehicle mission

01

02

03

04

05

06

07

08

09

Mac

hnu

mbe

r

AltitudeMach number










6 Conclusion


Nomenclature






119873119894 Shaft speed 119894 (RPM)











Subscripts





Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 5 10 15 2038

40

42

44

46

PH

P(p

sia)

Total pressure

Total temperature

21

22

23

24

25

PLP

(psia

)

1300

1400

1500

1600

TH

P(∘

F)

1400

1300

1200

1100

TLP

(∘F)

240

250

260

270

280

mH

Pan

dm

LP(lb

ms

)

Time (s)

0 5 10 15 20

Time (s)

0 5 10 15 20

Time (s)

100

102

104

106

108

mby

pass

(lbm

s)Mass flow

HPTLPT

HPTLPT

HPTLPTBypass










0 2 4 6 8 10 12 14 16 18 20

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

Total pressure

Total temperature

Mass flow

FanCompressor

FanCompressor

FanCompressor

23

24

25

26

Pfa

n(p

sia)

215

220

225

230

235

Tfa

n(∘

F)

350

360

370

380

240

260

280

300

Pco

mp

(psia

)

1000

1020

1040

1060

1080

Tco

mp

(∘F)

240

250

260

270

mco

mp

(lbm

s)

mfa

n(lb

ms

)


0 02 04 06 08 1 12 1402

04

06

08

1


Nor

mal

ized

pre

ssur

e rat

ioN

orm

aliz

ed p

ress

ure r

atio


0 02 04 06 08 1 12 14


0

05

1




0 20 40 60 80 100 1200

5

10

15

20

25

30

35

40

Alti

tude

(tho

usan

d ft)

Time (min)

Vehicle mission

01

02

03

04

05

06

07

08

09

Mac

hnu

mbe

r

AltitudeMach number










6 Conclusion


Nomenclature






119873119894 Shaft speed 119894 (RPM)











Subscripts





Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 2 4 6 8 10 12 14 16 18 20

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

0 2 4 6 8 10 12 14 16 18 20

Time (s)

Total pressure

Total temperature

Mass flow

FanCompressor

FanCompressor

FanCompressor

23

24

25

26

Pfa

n(p

sia)

215

220

225

230

235

Tfa

n(∘

F)

350

360

370

380

240

260

280

300

Pco

mp

(psia

)

1000

1020

1040

1060

1080

Tco

mp

(∘F)

240

250

260

270

mco

mp

(lbm

s)

mfa

n(lb

ms

)


0 02 04 06 08 1 12 1402

04

06

08

1


Nor

mal

ized

pre

ssur

e rat

ioN

orm

aliz

ed p

ress

ure r

atio


0 02 04 06 08 1 12 14


0

05

1




0 20 40 60 80 100 1200

5

10

15

20

25

30

35

40

Alti

tude

(tho

usan

d ft)

Time (min)

Vehicle mission

01

02

03

04

05

06

07

08

09

Mac

hnu

mbe

r

AltitudeMach number










6 Conclusion


Nomenclature






119873119894 Shaft speed 119894 (RPM)











Subscripts





Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















6 Conclusion


Nomenclature






119873119894 Shaft speed 119894 (RPM)











Subscripts





Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Subscripts





Acknowledgments


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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