SIMULATION of ISOTHERMAL COMBUSTION in GAS TURBINES By

SIMULATION OF ISOTHERMAL COMBUSTION IN GAS TURBINES by

Matthew Rice

Thesis submitted to the faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of

Masters of Sciences in Mechanical Engineering

Dr. Peter King, Committee Chairman Dr. Clint Dancey, Committee Member Dr. Uri Vandsburger, Committee Member

February 12th, 2004 Blacksburg, Virginia

Keywords: Turbines, Isothermal, Combustion, High-efficiency, Turbine Simulations

Abstract

SIMULATION OF ISOTHERMAL COMBUSTION IN GAS

TURBINES

Matthew J Rice

Committee Chair Dr Peter King

Dr ClintDancey

Dr Uri Vandsburger

Current improvements in gas turbine engine p erformance have arisen primarily due to increases

in turbine inlet temp erature and compressor pressure ratios However a maximum p ossible tur

bine inlet temp erature exits in the form of the adiabatic combustion temp erature of the fuel In

addition thermal limits of turbine blade materials also places an upp er bound on turbine inlet

temp eratures Thus the current strategy for improving gas turbine eciency is inherently lim

ited Intro duction of a new gas turbine based on an alternativework cycle utilizing isothermal

combustion ie combustion within the turbine aords signicant opp ortunities for impro ving

engine output andor eciency However implementation of such a scheme presents a number

of technological challenges such as holding a ame in highsp eed ow The currect research is

aimed at determining whether such a combustion scheme is feasible using computational metho ds

The geometry a simple D cascade utilizes surface injection within the stator or rotor b oundary

layers including the rotor pressure side recirculation zone a natural ame holder Computa

tional metho ds utilized b oth steady and time accurate calculations with transitional owaswell

as laminar and turbulentcombustion and sp ecies transp ort It has b een determined that burning

within a turbine is p ossible given a variety of injection schemes using typical foil geometries

under typical op erating conditions Sp ecically results indicate that combustion is selfigniting

and hence selfsustaining given the high temp eratures and pressures within a high pressure tur

bine passage Deterioration of aero dynamic p erformance is not pronounced regardless of injection

scheme However increased thermal loading in the form of higher adiabatic surface temp eratures

or heat transfer is signicantgiven the injection and burning of the fuel within the b oundary layer

This increase in thermal loading is however minimized when injection takes place in or near a

recirculation zone The eect of injection lo cation on pattern factors indicates that suction side

injection minimizes temp erature variation downstream of the injection surface for rotor injection

only In addition the most uniform temp erature prole in the ow direction is achieved by

injection of fuel and combustion nearest to the source of work extraction Namely injection at the

rotor pro duces the most isothermal temp erature distribution Finally a pseudo direct simula

tion of an isothermal machine is conducted bycombining simulation data and assumed pro cesses

The results indicate that isothermal combustion results in an increase in turbine sp ecic work and

eciency over the equivalent Brayton cycle

iii

Acknowledgements

Iwould like to express graditude to my graduate committee Dr Peter King Dr Clint

Dancey and Dr Uri Vandsburger for their feedback I also wish to thank Steve

Greeneld and Dr Jan Bhn for supplying the computing facilities and computing

hardware The Virginia Tech community is also deserving of my gratitude for providing

the opp ortunity to continue my education after a previously turbulent graduate scho ol

exp erience Much appreciation also go es to those down in the turb olab for their steady

presence Finally I wish to express my deep est love for those back home inlittle ole

Portland whose supp ort made my success in these last few years p ossible

Published in glorious L T X E

Contents

INTRODUCTION AND MOTIVATION

INTRODUCTION AND HISTORICAL BACKGROUND

IDEAL BRAYTON CYCLE

IDEAL ISOTHERMAL CYCLE

PERFORMANCE COMPARISONS FOR THE ISOTHERMAL AND BRAY

TON CYCLES

HYBRID BRAYTONISOTHERMAL CYCLE

TOPICS OF INQUIRY AND THESIS FORMAT

TOPICS OF INQUIRY

THESIS FORMAT

REVIEW OF THE LITERATURE

ISOTHERMAL COMBUSTION

COMBUSTION IN HIGH SPEED FLOW

APPLICABLE EXPERIMENTAL RESULTS

APPLICABLE COMPUTATIONAL RESULTS

OBSERVATIONS AND OPPORTUNITIES FOR INQUIRY

THERMODYNAMIC MODELING OF THE FLOW

INTRODUCTION

MASS MOMENTUM ENERGY AND SPECIES CONSERVATION

TURBULENCE MODELING

CHEMICAL REACTIONS

ARRHENIUS KINETICS

EDDY DISSIPATION MODEL

FLUID PROPERTIES

BOUNDARY CONDITIONS

TRANSITION MODELING

CONSERVED SCALAR APPROACH iv

CONTENTS v

CONSERVATION EQUATIONS FOR NONPREMIXED COMBUS

TION

NUMERICAL MODEL

APPROACH

THE GEOMETRY

BOUNDARY CONDITIONS

FLUID PROPERTIES

REACTION MECHANISM

SIMULATIONS

SIMULATION PROCEDURE

NONREACTING FLOW

REACTING FLOW ARRHENIUS KINETICSEDDYDISSIPATION

TURBULENT COMBUSTION CONSERVED SCALAR SIMULATIONS

STATOR SIMULATIONS

RESULTS AND DISCUSSION

PRESSURE DISTRIBUTIONS

HEAT TRANSFER

PATTERN FACTORS

ISOTHERMAL COMBUSTION

CYCLE ENHANCEMENTS

VERALL PRESSURE RATIOS IDENTICAL O

NONCONSTANT OVERALL PRESSURE RATIO

CONCLUSION

SUMMARY

RECOMMENDATIONS FOR FUTURE WORK

A HYBRID CYCLE PARAMETERS

B DIFFUSION VIA THE CHAPMANENSKOG CORR

C CYCLE COMPARISONS

C CONSTANT OVERALL PRESSURE RATIO

C NONCONSTANT PRESSURE RATIO

D PATTERN FACTORS

E BOUNDARY CONDITIONS

vi CONTENTS

F SIMULATION DATA

F BULK FLOW QUANTITIES

F ROTOR INJECTION

F STATOR INJECTION

F OTHER QUANTITIES OF INTEREST

F ROTOR

F STATOR

F REDUCED INJECTION RATE CASE

F INCREASED PRESSURE RATIO CASE

List of Figures

Lo ckheed Tristar L

RollRoyce RB Turb ofan Engine

Concord at takeo

RollRoyce Olympus

Temp erature vs year

Thrust to weight vs inlet temp erature

Thrust sp ecic fuel consumption vs pro duction year

T s diagram for ideal Brayton cycle

T s diagram for ideal isothermal cycle

vs for Isothermal and Brayton cycle

Normalized work vs for Isothermal and Brayton cycle

Ts diagram for hybrid isothermalBrayton cycle

vs for Hybrid combined and Brayton cycle

Normalized work vs for Hybrid combined and Brayton cycle

Normalize work and vs for Hybrid and Brayton cycle

Near wall ow regimes

Wall b oundary conditions

Flame sheet comp ositions

Mass fractions vs mixture fraction f

Linear cascade geometry see Table for dimensions as lab eled in paren

thesis

Stator geometry and grid

Rotor geometry and grid

Domain b oundary surfaces

Total pressure contours noninjection simulation

Total temp erature for noninjection simulation

FlowMachnumber vii

viii LIST OF FIGURES

Pathlines indicating pressure side rotor blade separation

Contours of y for no injection

y for stator

T proles for rotor injection

T for stator injection

y for midrotor injection

H O

Ma for midrotor injection

y for mid rotor injection

C H

Conserved scalar f for mid rotor injection adiabatic foil

for mid rotor injection adiabatic foil

H O

Temp erature for mid rotor injection adiabatic foil

Pathlines for mid rotor injection PDF adiabatic foil

Temp erature contours for laminar leading edge pressure side injection leps

adiabatic foil

Temp erature contours for eddydissipation trailing edge suction side injec

tion nonadiabatic foil

Rotor pressure distribution for various injection lo cations laminary com

bustion

Rotor pressure variation for various injection lo cations

Heat transfer q for various injection schemes

Crossstream temp erature distibution cord downstream of stator adi

abatic noncombustion

Temp erature contours ab out rotor for noninjection case

Axial lo cal temp erature distribution for various injection schemes

Ts diagram for cycle comparisons

Ts diagram for Hybrid yellow and Brayton cycle black for equal peak

temp eratures and isothermal cutoBrayton turbine inlet pressures

A Ts diagram for hybrid isothermalBrayton cycle

C Ts diagram for HybridnonisothermalBrayton cycle comparison

C Ts diagram for hybrid and Brayton cycle for equal p eak temp eratures and

isothermal cutoturbine inlet pressures

List of Tables

Stage geometry summary

Injector area

Stator geometry summary

Rotor geometry summary

Inlet b oundary conditions

Calculated constant binary diusivities for a temp erature of K and

pressure of Pa

Calculated constant binary diusivities for a temp erature of K and

pressure of Pa using ChapmanEnskog correlation

Inlet and exit ow prop erties for nonreacting owcase

Rotor foil velo city force and work extraction for nonreaction ow

Inlet and exit ow prop erties for reacting ow case rotor mid injection

rotor mid injection Fuel mass ow and p olytropic exp onent n

fuel

Net pressure force on rotor blade for dierent injection metho ds laminar

combustion

Calculated pattern factors laminar rotor adiabatic wall injection

Calculated pattern factors laminar stator adiabatic wall injection

Values for and w

net

simul ation theor etical

B Required data for calculation of binary diusion co ecients D

C Assumed state temp eratures C and

C Assumed pro cess averaged C and

C Simulation C and

C Assumed pro cess averaged sp ecic heats and

D Required data for calculation of pattern factor laminar rotor injection

D Calculated pattern factors laminar rotor injection ix

x LIST OF TABLES

D Required date for calculation of pattern factor laminar stator injection

D Calculated pattern factors laminar stator adiabatic wall injection

E Simulation b oundary conditions

F Inlet and exit ow prop erties for reacting ow case rotor less injection

F Inlet and exit ow prop erties for reacting ow case rotor mid injection

F Inlet and exit ow prop erties for reacting ow case rotor leps injection

F Inlet and exit ow prop erties for reacting ow case stator less injection

F Inlet and exit ow prop erties for reacting ow case stator leps injection

F Inlet and exit ow prop erties for reacting ow case stator tess injection

F Inlet and exit ow prop erties for reacting ow case stator teps injection

F Injection rates p eak temp eratures etc rotor less injection

F Injection rates p eak temp eratures etc rotor mid injection

F Injection rates p eak temp eratures etc rotor leps injection

F Injection rates p eak temp eratures etc stator less injection

F Injection rates p eak temp eratures etc stator leps injection

F Injection rates p eak temp eratures etc stator tess injection

F Injection rates p eak temp eratures etc stator teps injection

F Inlet and exit ow prop erties for reacting ow case reduced midinjection

F Injection rates p eak temp eratures etc reduced midinjection

F Inlet and exit ow prop erties for reacting ow case high pressure mid

injection

F Injection rates p eak temp eratures etc high pressure midinjection

Chapter

INTRODUCTION AND

MOTIVATION

INTRODUCTION AND HISTORICAL BACKGROUND

The need for continual improvements in gas turbine eciency and power is the driving

force b ehind current engine development Indeed the evolution of the gas turbine engine

Figure Lo ckheed Tristar L

used in aerospace applications can be describ ed as a sucession of technological break

throughs resulting in enhanced thermal eciency and power The principle historical

advances in gas turbine technology were the development of highbypass ratio turb ofan

engines in the s and externalinternal turbine blade co oling in the s and

s Figure shows one of the rst civilian aircraft to use highbypass engines supplied

by RollsRoyce Figure In addition to improvements in eciency of concern to

thrust engine manufacturers is the power pro duced in the form of shaft power andor

in comparison to engine size While this comparison is esp ecially imp ortant for military

CHAPTER INTRODUCTION AND MOTIVATION

Figure RollRoyce RB Turb ofan Engine

applications which require signicant thrust for rapid acceleration along with low engine

weight civilian applications also require in some cases signicant pro duction of thrust

if only for brief p erio ds A sp ecic technology used to enhance thrust pro duction is the

Figure Concord at takeo

technique known as afterburning used extensively for military aircraft and to a much

lesser extent in civil aviation The use of this technology has signicant drawbacks in

terms of reduced eciency and an increase in thrust sp ecic fuel consumption TSFC

Figure shows the Concord at takeo note the use of afterburners which utilizes the

Olympus turb o jet supplied by RollsRoyce Figure

With the exception of p erformance improvements arising from the use of turb ofan

as opp osed to turb o jet conguations the recent history of jet engine development

has b een dictated bymuc h more fundamental and rudimentry cycle analysis Sp ecically

enhanced p ower and engine eciency can and has b een brought ab out by raising turbine

inlet temp eratures The end result historically is a constant upward trend in turbine

inlet temp eratures and compressor pressure ratios in order to enhance the cycle eciency

and power of mo dern Brayton cycle gas turbine engines This trend is readily observed

High fuel consumption was the primary reason for the Concords lack of commercial viability

INTRODUCTION AND HISTORICAL BACKGROUND

Figure RollRoyce Olympus

Figure Temp erature vs year

in Figure and Figure which display turbine inlet temp eratures vs pro duction date

for various RollsRoyce engines and cruise sp ecic thrusttoweight ratio vs turbine

inlet temp erature for various turb o jet engines As Figure indicates the upward trend

in p eak engine temp eratures is gradual throughout the s whereup on the intro duction

of turbine blade co oling and advances in materials technology allowed for a more rapid

increase in turbine inlet temp erature Finally Figure illustrates the historical trend

in cruise thrust sp ecic fuel consumption as a result of engine improvements including

amongst others an increase in p eak engine temp erature ie turbine inlet temp erature

It should b e b orne in mind that turbine inlet temp eratures cannot b e raised indenitely

Apart from the issue of turbine comp onent durability there exists a temp erature limit in

the form of the adiabatic ame temp erature for a given combustor inlet temp erature

which the turbine inlet gases cannot exceed F or current aviation fuels this limit is in

excess of K Hence as p eak engine temp eratures approach this limit an alternative

CHAPTER INTRODUCTION AND MOTIVATION

Figure Thrust to weightvs inlet temp erature

strategy must b e found for increasing engine eciency without reducing p ower Suchan

alternative potentially exits with the intro duction of a new cycle and its hybrid counter

part in the form of isothermal heat addition within the turbine itself as opp osed to the

combustor alone which is the case for all mo dern Brayton cycle engines currently used

to day To see this we rst take a brief detour into the basic thermo dynamics governing

the Brayton work cycle

IDEAL BRAYTON CYCLE

The idealized Brayton cycle for which all mo dern gas turbine engines are based is

illustrated in Figure and comp osed of the following internally reversible pro cesses

Isentropic compression in compressor

Isobaric heat addition in combustor

Isentropic expansion in turbine

t Isobaric heat rejection to the ambien

From a rather straightforward cycle analysis assuming idealized pro cesses for a p erfect

gas wehave the following expression for the thermal eciency of the Brayton work cycle

B r ay ton

The following development in section resembles that in Ramohalli AIAA

IDEAL BRAYTON CYCLE

Figure Thrust sp ecic fuel consumption vs pro duction year

where and are the compressor temp erature and pressure ratios resp ectively In ad

dition sp ecic work scaled or dimensionless work p er unit mass of the working uid is

also given by

B r ay ton

C T

p c

ambient

The maximum normalized sp ecic work for a xed overall temp erature ratio

T T T T is also given by

tur bineinl et ambient

B r ay ton

p p

max

maxB r ay ton

C T

ambient

where we nd that

w or k max

This yields an expression for the maximum work thermal eciency

w or k max

B r ay ton

Thus from and we can see that in order to maximize output and reduce fuel

consumption ie increase thermal eciency while pro ducing the most work p ossible p er

unit mass ow turbine inlet temp eratures must b e as high as p ossible hence maximiz

ing the value of Unfortunately as stated previously physical limitations for such a

strategy which cannot b e avoided are the maximum combustion temp erature for the fuels

currently used and the thermal limits of turbine and combustor materials In addition

CHAPTER INTRODUCTION AND MOTIVATION

Figure T s diagram for ideal Brayton cycle

while increasing compressor pressure ratios results in an increase in eciency sp ecic

output or work per unit mass ultimately falls as a result Thus to a signicant ex

tent a tradeo exists b etween engine p ower and eciency for currentmachines based on

the Brayton cycle It is in the hop e of partially overcoming this tradeo that a new

isothermal work cycle could b e implemented for use in gas turbines

IDEAL ISOTHERMAL CYCLE

The Carnot cycle comp osed of constant temp erature or isothermal heat addition and

rejection pro cesses is the most ecientwork cycle Hence it makes sense to apply sucha

pro cess where p ossible to gas turbines Sp ecicallya newwork cycle might b e comp osed

of the following pro cesses as illustrated in Figure

tropic compression in compressor Isen

Isothermal heat addition combustion in turbine

Isentropic expansion work extraction in turbine

Isobaric heat rejection to ambient

For the maximum eciency op erating p oint and W

IDEAL ISOTHERMAL CYCLE

Figure T s diagram for ideal isothermal cycle

Note that a distinct combustor as used in the case of a Brayton cycle machine is absent

from the isothermal cycle Instead fuel is injected and combustion o ccurs at the lo cation

of work extraction ie in the turbine thus maintaining a constantgasow temp erature

Assuming an ideal isothermal pro cess as shown in Figure we nd the eciency for our

new cycle to b e

isother mal

T according to Figure Now assuming typical temp eratures asso where T

ciated with conventional Brayton cycle engines we have T T Thus the

isothermal thermal eciency can b e approximated as

B r ay ton

isother mal

Hence for typical jet engine op erating conditions in fact for all conditions conforming to

the cycle in Figure the isothermal thermal eciency is higher that that of the Brayton

cycle In addition we can express the normalized work again scaled work p er unit mass

ow for the isothermal cycle as

isother mal

W ln

c c

isother mal

C T

ambient

Similarly normalized work for the isothermal pro cess is maximized when

max

T T W

isother mal

CHAPTER INTRODUCTION AND MOTIVATION

which is hardly surprising given that net work per unit mass ow is the area bound by

the Ts curve In contrast for maximum eciency wehave

T T

which from the gure minimizes heat rejection p er unit heat addition Note that likethe

Figure vs for Isothermal and Brayton cycle

conventional Brayton cycle output is zero for the maximum eciency op erating p oint

PERFORMANCE COMPARISONS FOR THE ISOTHER

MAL AND BRAYTON CYCLES

Comparing cycle eciency and sp ecic output for the maximum work Brayton as a func

tion of p eak or turbine inlet temp erature we nd that the isothermal cycle p erformance

is sup erior for almost all op erating points Sp ecically Figure displays isothermal

T and Brayton cycle eciency for various values of the temp erature ratio T

in parenthesis including the maximum work condition T T as a function

improvement in cycle eciency over the maxi of Note that Figure indicates an

mum work Brayton cycle except in the case of the op erating p oints corresp onding to the

maximum work isothermal cycle This eciency p enalty is rather small and ranges from

to dep ending on turbine inlet temp erature If however we compare the slight re

duction in cycle eciency to the increase in work output for the maximum work Brayton

and isothermal cycle see Figure we see an increase in sp ecic output of to

HYBRID BRAYTONISOTHERMAL CYCLE

Figure Normalized work vs for Isothermal and Brayton cycle

for the isothermal cycle dep ending on turbine inlet temp erature prop ortional to

Hence we conclude that at the least isothermal combustion results in a signicant

increase in engine power with only a mo dest decrease in eciency Conversely if only

a small increase in power is desired over that of the conventional Brayton cycle then

isothermal combustion can result in a signicant improvement in cycle eciency

HYBRID BRAYTONISOTHERMAL CYCLE

Unfortunatelyto achieve the high output and eciency asso ciated with isothermal com

note that bustion very high compressor pressure ratios are required see Figure and

the temp erature rise T T is achieved solely through compression within the compres

sor as opp osed to compression followed by isobaric heat addition within a combustor as

is the case for the Brayton cycle Such pressure ratios on the order of to

are not technologically feasible at present The solution to this diculty is to utilize a

combustor with subsequent isothermal heat addition burning within the turbine yielding

ahybrid isothermalBrayton cycle Such a cycle depicted in Figure is comp osed of

the following pro cesses

Isentropic compression

Isobaric heat addition in combustor

Isothermal heat addition heat addition in turbine

CHAPTER INTRODUCTION AND MOTIVATION

Figure Ts diagram for hybrid isothermalBrayton cycle

Isentropic expansion work extraction in turbine no combustion

Isobaric heat rejection to the ambient

This hybrid Braytonisothermal cycle while not as ecient as the pure isothermal cycle

still provides many advantages over the traditional Brayton machine in terms of increased

output andor enhanced thermal eciency over a variety of op erating conditions The

expression for the thermal eciency for this hybrid cycle is

Hybrid

c c

again where and are dened as T T T T and T T As in the case of the

ideal Brayton and Isothermal cycle a normalized work can also b e calculated

W ln

Hybrid

Hence a comparison can now be made between the Brayton and Hybrid cycle Figures

and compare thermal eciency and normalized work for the Hybrid and Brayton

cycle assuming reasonable values for and Sp ecically from Figure we see

that the Hybrid or combined cycle in blue presents an eciency advantage over the

Brayton cycle in yellow over al l temp erature ratios Figure also illustrates the

See App endix A for details

Assuming a conventional Brayton machine overall pressure ratio of and a Hybrid device pressure

and are and resp ectively ratio of we nd for isentropic compression

c c

HYBRID BRAYTONISOTHERMAL CYCLE

Figure vs for Hybrid combined and Brayton cycle

hybrid cycle advantage in net sp ecic work over the Brayton cycle It should b e noted

that these results are only valid for the temp erature ratios assumed and in general for an

identical compressor and p eak engine temp eratures the Hybrid cycle must pro duce more

work but may not b e more ecient Indeed the latter p oint is illustrated by comparing

the combined cycle with the same compressor pressure ratio as the Brayton cycle This

combined cycle has a cuto temp erature ratio of and while pro ducing more work

than the Brayton cycle exhibits a lower eciency

Instead of plotting sp ecic work and eciency as a function of we might instead

be interested in holding the p eak engine temp erature compressor pressure ratio for the

constant hine and the cuto pressure constant ie holding T and T Brayton mac

These results are given in Figure and indicate that the Hybrid cycle always has sup e

rior p erformance in terms of enhanced output and eciency In conclusion we must b e

careful howwecharacterize the p otential b enets of isothermal combustion Sp ecically

isothermal combustion as used in the Hybrid cycle is only guaranteed to yield improved

eciencyand output for compressor pressure ratios in excess of the equivalent Brayton

machine with the same p eak engine temp erature

Note the following notation Conbined cycle refers to a compressor pressure ratio of

and value of for

These plots assume reasonable values for and of and resp ectively but this result is

c general

CHAPTER INTRODUCTION AND MOTIVATION

Figure Normalized work vs for Hybrid combined and Brayton cycle

In subsequentchapters we shall demonstrate some of these results using a pseudo

direct simulation of a Hybrid and Brayton machine

TOPICS OF INQUIRY AND THESIS FORMAT

TOPICS OF INQUIRY

Thus far no information has b een presented which relates to actual combustion inside a

turbine nor the real b enets asso ciated thereof As indicated in the next chapter research

into this topic has b een limited and conned to several areas

Simple cycle analysis comparing Brayton and Isothermal eciency and power no

treatment has b een found of a Hybrid cycle as such

Exp erimental results for p orous plate injection none of which were conducted at

pressures and temp eratures asso ciated with the interior of a gas turbine

Computational exp eriments for combustors and stabilized ames but few results for

yer injection of jet fuels under conditions similar to those existing in gas b oundary la

turbines

Hence this thesis attempts to extend using available computational to ols the range

of inquiry into the feasibility of b oundary layer and recirculation zone combustion for

TOPICS OF INQUIRY AND THESIS FORMAT

Figure Normalize work and vs for Hybrid and Brayton cycle

conditions sp ecic to gas turbines Sp ecicallyow inside a turbine with rst b e mo deled

afterwhich the following questions will b e addressed

Using all computational to ols at our disp osal via the computational package FLU

ENT can we demonstrate the feasiblityofcombustion within a gas turbine passage

under typical conditions

If combustion app ears to b e p ossible whichcombustion scheme is the most desirable

from an engineering standp oint In other words

Do es injection lo cation aect aero dynamic p erformance of the turbine airfoils

Do es injection lo cation aect blade heat transfer in magnitude and distribution

How do es injection lo cation aect pattern factors transverse temp erature vari

ations just downstream of the foil trailing edge

How do es injection lo cation rotor or stator injection aect the engine temp er

ature distribution ie whichscheme pro duces the most isothermal temp erature

distribution

Do es isothermal combustion indeed result in any of the p erformance improve

ments promised

CHAPTER INTRODUCTION AND MOTIVATION

THESIS FORMAT

Given that this research is computatonal in nature a heavy emphasis will be placed on

providing an in depth review of the theoretical underpinnings as well as detailed de

scriptions of the solution pro cedure Sp ecically organization of the material will be as

follows

Presentbackground into the desirability and motivation for investigation of isother

mal combustion

Review the progress previously made and justify the need for further inquiry

Carefully illustrate the development of a computational mo del for the simulated

phenomena reacting ow inside a gas turbinelikeenvironment

Review simulation results and answer the questions stated previously

In addition several App endices are provided for conciseness and eciency where needed

Chapter

REVIEW OF THE LITERATURE

ISOTHERMAL COMBUSTION

The p otential b enets of isothermal combustion are illustrated in the works of Ramohalli

Liu and Sirignano and Sirignano Delplanque and Liu Ramohalli the rst of

the ab ove to illustrate the b enets of isothermal combustion compared thermal ecien

cies and sp ecic or normalized work for both the idealized and real isothermal and

Brayton cycles Assuming typical compressor exit temp eratures Ramohalli showed that

the isothermal eciency advantage over the Brayton cycle is prop ortional to the ambient

temp erature and inversely prop ortional to the peak engine temp erature In addition he

found that the isothermal cycle maximum sp ecic work was substantially higher than that

of the maximum work Brayton cycle with only a mo dest decrease in eciency Sp ecically

for the maximum sp ecic w ork op erating conditions for b oth cycles assuming an overall

temp erature ratio of the ideal isothermal engine will pro duce threetimesthe normalized

work of the equivalent Brayton machine op erating b etween the same temp erature limits

while suering a reduction in thermal eciency of approximately Ramohalli p oints

out however that to achieve the necessary high turbine inlet temp eratures compressor

pressure ratios on the order of to would b e required which is unattainable using cur

renttechnology The Author also notes an inherentadvantage of the isothermal cycle for

high ightMachnumbers Finally Ramohalli also nds using order of magnitude anal

ysis that the chemical and ow residence times are of the same maginitude for droplet

combustion in a turbine like environment Thus yielding inconclusive results as to the

vestigated the p erformance p ossibility of burning within a turbine Liu and Sirignano in

Indeed under such conditions the lo cal compressor inlet temp erature can b e quite high hence for a

constant compressor pressure ratio increasing combustor inlet temp eratures for the traditional Brayton

cycle This given a maximum combustor outlet temp erature necessitates a reduction the fuel burn rate

and hence work extraction

CHAPTER REVIEW OF THE LITERATURE

b enets of interturbine burning as an alternative to afterburners and notes the enhanced

eciency asso ciated with the former The Authors also nd that the isothermal device

achieves a peak thrust eciency at a higher bypass ratio than the conventional Brayton

machine

As in the case of Ramohalli Liu and Sirignano nd that turbine interburning results

in improvements in eciency andor output manifested in lower TSFC thrustsp ecc

fuelconsumption over the conventional Brayton cycle Sp ecically for pressure ratios

in the range p erformance of the conventional Brayton cycle machine deterio

rates in terms of ST sp ecic thrust while the isothermal device exhibits p erformance

improvements for pressure ratios in excess of Sirignano Delplanque and Liu rep eat

the analysis of the two previous studies but also extends it to landbased regenerative

gas turbines and nds that isothermal combustion yields sup erior sp ecic power andor

thermal eciency over the conventional Brayton cycle utilizing reheat Finally Andriani

using a simple one dimensional co de solving the governing bulk ow equations nds

that a two stage turbine utilizing interstage reheating can pro duce sp ecic thrust com

parable to a conventional Brayton cycle engine with afterburner while at the same time

exhibiting thrust sp ecic fuel consumption equivalent to the nonafterburning engine

COMBUSTION IN HIGH SPEED FLOW

APPLICABLE EXPERIMENTAL RESULTS

Success of the isothermal scheme dep ends on the ability to light and sustain a ame in

high sp eed ows present in turbine passages Attemps have b een made b oth exp erimen

tally and computationally to simulate combustion in a turbine passagelikeenvironment

These attemps range from p orous at plate injection as in the case of Nasir Agrawal

and Ramachandra to blu body and step ame holders in high sp eed ow as

in the case of Owens and Schefer Sp ecically Nasir exp erimentally simulated

through which combustion within turbine passages using two parallel p orous at plates

passed a heated air stream and found that combustion of Methane could b e maintained

for freestream velo cities over ms The airstream exp erienced precombustion corre

sp onding to an equivalance ratio of at atmospheric pressure yielding a freestream

temp erature of approximately K Ramachandrea investigated the stabilization of a

ame in a freestream using p orous plate injection of Pentane However unlike the pre

vious attempt freestream temp eratures were limited to K Ramachandra found an

extinction freestream velo city of ms and that blowout injection velo city increased

with freestream oxygen content and decreased with fuel dilution Rohmat on the other

hand exp erimented with blo ck and step ame holders placed upstream of a p orous at

plate injection of methane and found the unsurprising result that ame holders blo ckor

step signicantly increase ame stability Like all previous exp eriments this was also

COMBUSTION IN HIGH SPEED FLOW

conducted at atmospheric pressure and a temp erature of approximately K in the

freestream In addition Lewis found that the laminar ame sp eed is prop ortional to

g where g refers to the gforce exp erienced bytheow In other words the presence of

swirl in the ow eld has the eect of increasing the reaction rate The presumed physical

mechanism resp onsible for this eect is the migration of unburned fuel of higher density

than the surrounding pro ducts due to centrifugal eects thus exp osing additional quan

titiesofunburned fuel to the surrounding oxidizer resulting in a faster burn rate Zelina

conducted exp eriments using a high swirl combustor and found that the centrifugal

force applied to a burning element of fuel indeed had the eect of increasing the eective

reaction rate These results although not utilized in our study are imp ortantgiven that

the turbine environmentischaracterized by high swirl

APPLICABLE COMPUTATIONAL RESULTS

Computational simulation of reacting highsp eed swirling ows was conducted byRoy

Sp ecicallyRoy utilized an Eddy Dissipation mo del and PDF approach in temp erature

to simulated reacting ows past a blu b o dy The turbulence mo del used was a is

dened as the mean vorticityvariance and the use of global as well as a fourstep reaction

mechanism yielded accurate results unlikethe turbulence mo del which is known to

and did p erform p o orly for separated ows In addition numerical simulations of more

complex geometries such as full combustors have b een conducted by Lee and Xia

Xia sp ecically compares the p erformance of two turbulence mo dels with a Reynolds

stress formulation and nds the familiar deciency in p eformance of the former compared

with exp erimental results The Author notes that manyoftheinteresting features of the

ow are not even qualitatively repro duced by the mo del Lee on the other hand

attempted to mo del combustion within a Pratt Whitney gas turbine combustor again

using a turbulence mo del with reaction rates based on eddydissipation Results were

less than satisfactory with some of the overall qualitative characteristics of the ow not

b eing predicted Peak combustion temp eratures were also incorrect error on the order

of however some uncertainty as to the geometric conguration of the device may

have b een a factor A computational study was also conducted by Zelina using the

enhanced reaction rate of Lewis utilizing a turbulence mo del with PDF sp ecies

transp ort and found the results to be in go o d agreement with exp eriment However it

must b e noted that the mo del used was sp ecically tuned to achieve this result

It should b e stated that Xias results are for cold incompressible ow ie nonreacting

These results along with the known p erformance advantage p osed bythe turbulence mo del for

separated ows led to our choice of this mo del for the turbulence formulation

CHAPTER REVIEW OF THE LITERATURE

OBSERVATIONS AND OPPORTUNITIES FOR IN

QUIRY

The previous studies all of which approach indirectly the issue of combustion within

a turbine do not attempt to fully simulate conditions thereof A simple cycle analysis

demonstrating the theoretical improvements in engine output or eciency due to isother

mal combustion may not b e realizable in real gas turbines Likewise order of magnitude

analysis on chemical and ow times in a gas turbinelikeenvironmentlack sp ecicityand

yield inconclusive results as to the p ossibilityofcombustion The exp eriments supp osedly

aimed directly at combustion in gas turbines are only lo osely applicable given the fuels

and op erating conditions used The same can also b e said for the computational studies

Hence the opp ortunity exists for conducting computational exp eriments to deter

mine whether isothermal combustion is clearly imp ossible in a realistic gas turbine

Sp ecicallywe wish to address the following shortcomings in the literature

Only simple cycle analysis comparing the Brayton and Isothermal cycle has b een

found no treatment of the previously describ ed Hybrid cycle has b een presented to

this Authors knowledge

The exp erimental results for p orous plate injection although of interest were not

conducted at pressures and temp eratures asso ciated with the interior of a gas turbine

Computational exp eriments for combustors and stabilized ames present few results

for b oundary layer injection of jet fuels under conditions similar to those existing in

a gas turbine

Indeed in reference to reaction rates and ame sp eeds are strong functions of the

ambient temp erature and relatively weak functions of pressure Sp ecically for most

hydro carb on fuels

laminar ame sp eed P T

Hence if p ossible exp eriments should b e conducted at the correct op erating conditions

It should however be stated that metho ds based on CFD are only as go o d as the

mo deling Thus strictly sp eaking we seek only to ascertain whether combustion is possible

according to various models for the owas opp osed to arriving at a denitive conclusion

as to the p ossibility of b oundary layer injection and combustion in a turbine

See Metghalchi and Keck Burning Velocities of Mixtures of Air with Methanol Isooctane and Indolene

at High Pressures and Temperatures Combustion and Flame

Chapter

THERMODYNAMIC

MODELING OF THE FLOW

INTRODUCTION

The combustion mo dels used for the simulations were based on Arrhenius Kinetics Eddy

Dissipation or Turbulent length scales and the conserved scalar or PDF approach all of

which utilize diering forms of the basic conservation equations for the ow eld mass

momentum energy sp ecies etc Hence in the interest of clarity the governing equations

of the ow for the PDF metho d will b e treated separately from the traditional Arrhenius

kinetics or reaction rate based approach and the Eddy Dissipation mo del

MASS MOMENTUM ENERGY AND SPECIES CON

SERVATION

The b ehavior of single comp onent uids ie uid containing a single sp ecies can b e de

scrib ed and predicted by application of the principles of conservation mass pro ducing the

continuity equation conservation of linear momentum or Newtons second law yielding

the NavierStokes equations and conservation of energy or the energy balance giving the

energy equation all for a dierential control volume Sp ecicallyinthe absense of mass

creation or destruction eg no nuclear pro cesses the application of mass conservation

on a dierential basis yields

where is the uid density and V is the bulk ow velo city With the application of

Newtons second law for uid motion for an op en system wehave the uid equations of

CHAPTER THERMODYNAMIC MODELING OF THE FLOW

motion neglecting gravitational forces

V r V rP r

where P is the lo cal or static pressure and is the shear stress tensor which includes

the viscous as well as dilation compressible eects Note that since we are dealing

with conservation of a vector quantity force the second term on the left has a sp ecial

interpretation Sp ecically

V j V V

i i i

V V V V

i i i j

x x x x

i j i j

r V V

for the twodimensional case where i and j refer to the x and y co ordinate directions The

stress tensor is taken to mean the following

h i

rV rV r V I

where is the absolute uid viscosity and I is the identity matrix In the case of turbulent

ow the typical approach is to replace absolute viscosity a prop erty of the uid with

an eective viscosity reecting the enhanced diusion due to uctuating or oscillatory

motion of the ow ie Finally conservation of energy can b e applied for

ef f

a dierential control volume for a multicomp onent mixture while neglecting radiative

energy transfer to yield

er V e P r k rT h J V S

j j

where k is the mixture thermal conductivity of the uid replaced by k k k in the

ef f

case of turbulentow J is the diusion ux of sp ecies j to b e discussed subsequently

S is an energy source term due for example to chemical reactions and e is dened as

the total energy of the mixture p er unit mass neglecting changes in gravitational p otential

energy e uT jV j where uT is the mixture sp ecic internal energy Note that

if weintro duce a turbulent Prandlt number

t p

we can express the eective thermal conductivityas

t p

k k k k

ef f

Pr t

MASS MOMENTUM ENERGY AND SPECIES CONSERVATION

Terms in whichmay b e unfamiliar to us include the diusive term second in brackets

on the right hand side resulting from sp ecies transp ort

Inthecaseofmulticomp onent uids the enthalpy h of the mixture is calculated via

the fact that enthalpy for an ideal mixture is an additive prop erty

h h

j j

where

ref ref

h h h h

j j

and is the mass fraction of the j sp ecies Also note that for an ideal gas enthalpyand

energy are related via the following

eT V uT jV j hT P jV j hT R T jV j

where R is the mixture gas constant Sp ecies conservation is imp osed via

r V r J R

j j j j

where R is the net rate of pro duction of sp ecies j The second term on the left refers

to the net convective ux of sp ecies j while the rst term on the right represents the net

diusiveux Note that conservation of mass implies and is enforced via the requirement

In the case of laminar ow J is calculated using Ficks law for a dilute mixture ie we

assume that each of the mixture comp onents behave as if they are dilutants within an

parent uid in our case air Or

J J D r

j ji ji j

where D is the assumed constant diusivity of sp ecies j into the surrounding uid i

air In the case of turbulentow mass transfer is enhanced and we can replace diusion

ef f

with an eective diusion D D as in the case with energy k and momentum

diusivity Finally the equation of state used previously is simply

Pv R T

where the mixture gas constant R is given by

R R

m u

j j

CHAPTER THERMODYNAMIC MODELING OF THE FLOW

TURBULENCE MODELING

While conservation of mass momentum and energy are relatively transparent concepts the

same cannot be said for turbulence quantities and mo deling in general The phenomena

of turbulence and the asso ciated diculty of mo deling arises out of the fact that not

all solutions to the NavierStokes continuity and energy equations are stable For some

ows slight p erturbations in velo city pressure temp erature etc within the uid stream

are damp ed out while for others these p ertubations which can b e microscopic or macro

scopic growuntil the overall ow eld is altered This purturbation or disturbance of the

ow eld is characterized by time and spatially dep endent uctuations in all prop erties

Sp ecically in the case of steady ow in the mean all prop erties can b e represented as

the sum of the mean prop ertyvalue R and a uctuating comp onent r t Hence Pressure

R r t Now given Temp erature and momentum are generally expressed as R t

that most problems of practical interest are concerned with quantities in the mean eg

average shear stress or total drag on a surface average heat transfer etc it is natural to

attempt to solve the governing equations of motion for the average values of the quantities

of interest Sp ecicallywe desire expressions for all timeaveraged quantities R which can

b e found byintegrating the governing expressions for the owover some time p erio d much

greater than the p erio d of turbulence This approach advo cated and implemented by Sir

Reynolds is refered to as Reynolds averaging

The next prudent question to ask is what physical situations in terms of geometry

we wish to mo del After all substitution of our timedep endent prop erties V t T t etc

would even after Reynolds averaging yield an intractable and complex set of governing

equations for the ow with dep endent variables V and v t in the case of the Navier

Stokes equations alone In fact in terms of velo cities we have as set of three equations

enforcing momentum conservation and six unknowns the three comp onents of the mean

and uctuating velo city This predicament is termed the closure problem of turbulence

and in a purely mathematical sense renders the ow solution unattainable However if

one assumes that the turbulence is isotropic in other words

u tv tw t

where again the mean is taken over some p erio d greater that the p erio d of the turbulent

uctuation then after making b oundary layer assumptions the number of unknowns is

reduced resulting in a set of timeaveraged NavierStokes equations where the molecular

Note that strictly sp eaking pressure has a uctuating comp onentaswell but we will b e neglecting this

MASS MOMENTUM ENERGY AND SPECIES CONSERVATION

viscosity is replaced by the eective viscosity

ef f

However strictly sp eaking one would app ear to b e none the wiser given that wehave just

given another name turbulent viscosity to an unknown quantity namely u v At

tempting another route we p erform the following op eration where NS is the j direction

NavierStokes equation

NS V v t

After Reynolds averaging the result and making b oundary layer assumptions we have a

conservation equation for the average turbulent kinetic energy p er unit mass v u

of the form

u G D

t x x x

i j j

where Y G and represent the dissipation pro duction and diusion of notice the

gradient transp ort formulation as with ow shear stress or heat transfer The generation

of turbulent kinetic energy is related to the vorticityoftheow

u u S G

i j

where S is the square of the mo dulus of the mean rateorstrain tensor In addition

dissipation or destruction and the diusivityof are given by

D f

where is the turbulent Prandlt numb er based on is a constant f Vorticityand

the new quantity is the dissipation p er unit turbulent kinetic energy of the oworif

you like the turbulence frequency To arrive at the conservation equation for turbulent

frequency we p ostulate that like is sub ject to convection diusion generation

and dissipation Thus wehave an analogous expression to that of

u G D

t x x x

i j j

The turbulent viscosity replaces the remaining after making BL assumptions timeaveraged uctu

ating velo city comp onents via

u v

according to the Boussinesq formulation

CHAPTER THERMODYNAMIC MODELING OF THE FLOW

where the generation and dissipation of are mo deled via

G G

Y d

where d f vorticity The diusion of is also given by

where is the Prandtl number based on Finally the set of momentum related

equations are closed by noting that on dimensional grounds

where the signicance of the dimensionless quantity will b ecome apparent with the

intro duction of transition mo deling Complete closure of the system of equations the

energy and sp ecies turbulent diusion have not b een dealt with can b e achieved by noting

that physically sp eaking all conservation equations except mass include all the same

basic pro cesses diusion and convection and within the b oundarylayer in the absense

of any pressure gradient take on identical mathematical forms Hence one might presume

k D Or to b e more rigorous

t t t

t p

where Pr is the turbulent Prandlt numb er assumed to b e a constant And in the case

of sp ecies diusion

S ch

where the turbulentSchmidt Sch numb er is also taken as a constant The values for the

turbulent Prandlt and Schmidt numb ers are and resp ectively

CHEMICAL REACTIONS

ARRHENIUS KINETICS

The eects of chemical reactions enter into the computation of the ow eld by way of

the energy equation and indirectly by way of the dep endence of various uid not ow

prop erties on comp osition Sp ecicallycombustion enters into the energy equation in the

CHEMICAL REACTIONS

form of the dep endence of sp ecic enthalpy h on mixture comp osition and the energy source

term S Mixture sp ecic enthalpy is given by assuming ideal gas mixture b ehavior

X X

ref

h h T h C dT

j j j pj

ref

j j

and the energy source term is

X X

ref

C dT MW S h R

pj j jr

ref

where R is the volumetric rate of pro duction of sp ecies j due to reaction r The reaction

corresp onds to a forward reaction of the form

X X X X

X X

jr jr

r r

j j

where the summation is over all reaction mechanisms r with forward reaction rate k and

In addition the and reactant and pro duct stoichiometric co eents are lab eled

jr jr

X s are simply lab els for the relevant sp ecies Note that global as opp osed to reversible

elemental reactions are utilized in this study Assuming that the overall reaction rate is

a function of reactant sp ecies concentrations C and temp erature the form for the rate

of sp ecies pro duction for sp ecies j is

X Y

R C k

j j

jr jr

where is the corresp onding reaction exp onen t for sp ecies j in reaction r The forward

reaction rate constant k f T isgiven by

R T

k A T e

where E and A are emp erical constants

r r r

EDDY DISSIPATION MODEL

Now the previous expression for the rate of sp ecies pro duction do es not in any way di

rectly take into account the eects of turbulence and turbulentmixing In fact the only

path of inuence for turbulence is via the rate of sp ecies and energy transp ort ie the

eect of turbulent mixing on mean concentrations C and uid temp erature T We can

Thus the simple stoichiometric equation CO C O would involve the sp ecies X s C O

and CO with stoichiometric co ecients for and and for the s

CHAPTER THERMODYNAMIC MODELING OF THE FLOW

b egin to appreciate the inuence which turbulence and turbulent structures have on the

combustion pro cess by intro ducing the concept of a ratelimiting pro cess An example

of this is would b e the assumptions made in b oundary layer analysis as to the governing or

ratelimiting pro cess for momentum transfer ie the comparative imp ortance of convec

tive vs diusive viscous transfer of momentum The same is true for turbulence and

analysis of turbulentow where the b oundary layer is partitioned into regions dominated

by certain momentum transfer pro cesses viscous eects in the laminar sublayer and tur

bulent diusion in the log and wake region Hence it should not come as a surprise

that there are limiting pro cesses asso ciated with chemical reactions as well Sp ecically

one could imagine the follo wing p ossible extremes Combustion o ccurs in regions of much

smaller scale than the smallest turbulent eddies Conversely the combustion region could

b e large compared to the mixing length etc Note that in the rst case this must imply

that on a dierential control volume basis the reaction is governed or if you like rate

limited by chemical kinetics that are on the scale of the ame laminar in character

Hence we can p ostulate that under the rst extreme the reaction is laminar and kinet

ically limited Another way of say this is that in the rst case turbulent uctuations

due to the motion of turbulent eddies app ears relatively sp eaking macroscopic when

compared with the scale of chemical reaction

This is in con trast to the latter extreme where we p ostulate that turbulent struc

tures p opulated the combustion region Hence in this case we can say that turbulence and

turbulent mixing play an imp ortant if not dominantroleingoverning the reaction rate

Sp ecically in the latter regime we can think of combustion o ccuring when clumps of

unburned fuel are broken down into smaller clumps of suciently small volume com

pared to surface area to allow for complete combustion We could op erationalize this

approachby surmising that as the eddies rotate there exist some likeliho o d of breakup

into smaller eddies per revolution Note that this implies the greater the number of

revolutions p er unit time ie the higher the turbulence frequency the higher the rate

and combustion In addition one would assume that as the amount of of eddy breakup

fuel present in the uid increases p er unit volume the greater the consumption rate for

the fuel Hence we can render these observations in mathematical form via the following

fuel

where is the fuel consumption p er unit volume p er unit time

fuel

Applying the ab ovetoasinglestep reaction mechanism wehave the slightly mo died

expressions for the reaction rate as the minimum of either of the two expressions

R MW A min

j j

Note that this combustion regime is refered to as the ameletsineddies regime

FLUID PROPERTIES

R AB MW

j j

where and are the mass fractions of particular pro duct P and reactant R sp ecies

P R

A and B are empirically derived constants Note that the ab ove formulation is required

since the overall reaction is limited by the breakup and combustion of the slowest reacting

comp onent sp ecies

FLUID PROPERTIES

Mixture uid prop erties such as the viscosity thermal conductivity k sp ecic heat C

were all calculated using the ideal gas mixing law based on indivudual comp onent mixture

prop erties Thus given individual uid comp onent prop erties on a p er mole basis we

can express the mixture prop erty as

y T T

where y is the mixture mole fraction of sp ecies j

The individual comp onent uid viscosities are given by Sutherland curve ts of

the following form

T T S

j j

T T S

j j

where T and S are empirical constants The individual uid comp onent molar

j j j

sp ecic heats C are given by the p olynomial curvet

C T T T T T

pj j j j j j

Likewise the individual comp onent uid thermal conductivities k T are calculated using

kinetic theory

R C MW

j pj j

k T k C T T

j j pj j

MW R

j u

where R is the universal gas constant

Mo del based on the work of Magnussen and Hjertager utilizing the eddybreakup mo del of Spalding

CHAPTER THERMODYNAMIC MODELING OF THE FLOW

BOUNDARY CONDITIONS

The b oundary conditions imp osed for momentum energy and sp ecies conservation should

be familiar eg no slip adiabaticconstant wall temp erature etc However b oundary

conditions which are most interesting and least obvious are those used in the conservation

equations for sp ecic turbulent kinetic energy and frequency The b oundaries of

whichwe sp eak are for any computational domain the inlets outlets and surfaces The

simulations to be conducted here sp ecify the values of and at inlets and outlets in

the case of backow But on surfaces ie blade surfaces the only b oundary condition

relevant to the momentum equation is no slip The metho d used for implementing

b oundary conditions for the turbulence quantities and near a surface ie the rst

grid point o the wall is as follows The region adjacent to a no slip surface a wall

Figure Near wall ow regimes

is in turbulent ow characterized by a sucession of ow regimes where certain physical

eects dominate Sp ecically at the wall turbulent velo city oscillations are damp ed in

fact turbulent kinetic energy vanishes very near the wall hence the owislocal ly laminar

and referred to as the viscous or laminar sublayer Further from the wall the eects of

turbulent transp ort dominate diusion It is this region which sp ecies the log layer and

where the law of the wal l applies note the buer region bridging the laminar and log

layers see Figure Thus wall b oundary conditions dep end on which region the rst

o the wal l grid point resides If the rst grid p oint is in the viscous sublayer which

extends approximately over the interval y we have for all intents and

y is dened as y y u y where is the wall shear stress

w w

BOUNDARY CONDITIONS

purp oses resolved the ow eld to the wall Sp ecically for a smo oth wall no surface

Figure Wall b oundary conditions

roughness the b oundary conditions for and are

wall

Thus our approach here is to resolve the ow all the way through into the viscous

sublayer

Note that the previous discussion only refers to surface b oundary conditions for

and In the case of inlet b oundary conditions for turbulent quantities we instead sp ecify

the values of turbulentintensity Tu and the hydraulic diametery H If one assumes the

inlet ow is turbulent and fully develop ed then a unique value for the length scale exists

at the inlet presumably some fraction of the total inlet diameter since the turbulent

mixing length cannot b e greater than the ows connement distance Sp ecically

l H

Note that this is in contrast to the wall function approach which assumes the near wall or rst grid

point resides in the loglayer and bridges the spatial gap via wall functions see Figure

This assumes a value of y less than where u is the friction velo city w

CHAPTER THERMODYNAMIC MODELING OF THE FLOW

where can b e related to the length scale l and turbulent sp ecic kinetic energy via

k k

l H

on the other hand is related to turbulentintensity Tu via

Tu TuU

ref

where the reference velo city U is the inlet bulk owvelo city magnitude jV j

ref

Finallya word must b e said ab out the metho dology employed by an elliptic solver

ie the generation of a solution for a set of eliptic equations where apriori the values of

all quantities are not known on all b oundaries Sp ecically while inlet and surface ow

conditions are known ie sp ecied by the user in the form of b oundary conditions the

outlet ow prop erties except lo cal pressure are unknowns part of our purp ose is to solve

for exit ow quantities as well as all prop erties in between The solution is to sp ecify

gradients of prop erties at the exit consistant with the fullydevelop ed solution no changes

in o w prop erties after the uid exits the computational domain In other words we

implicitly apply a b oundary condition at the ow exit in the form of vanishing prop erty

gradients Unfortunately this requires that we iterate the solution while up dating exit

ow prop erties based on the previous estimates just downstream of the computational

domain outlet

TRANSITION MODELING

The previous treatment do es not address transition even high Reynolds number ow

contains an initial laminar region Clearly the means of mo deling transition is to make

the turbulent viscosity of the following functional form

where f and small quantity or in the laminar typied by low

values for k or turbulent region typied by high values for resp ectively Bearing this

in mind is mo deled via the following

where Re

Hence the solver employs an iterative approach as opp osed to a marching solution as is the case for

sup ersonic where the governing equations are parab olic in nature

Note that the asymptotic b ehavior of is as desired ie for high values of or

CONSERVED SCALAR APPROACH

The approach previously taken attempts to incorp orate the eects of turbulence by cal

culating an eective viscosity thermal conductivity and sp ecies diusion However the

intro duction of these eective diusivities has no direct eect on rate of sp ecies creation

via the general expression for the reaction rate

X Y

R k C

j j

jr jr

where

R T

e k A T

Sp ecically small uctuations in temp erature ab out the mean indicative of turbulent

ow can have a signicant eect on the reaction rate due to the nonlinear behavior of

the exp onential The same is true for uctuating values of concentration C Another

way of stating this is that inserting the time mean values for T and C into the reaction

rate expression eectively neglects the uctuating nature of these values and renders the

reaction as essential ly laminar

CONSERVATION EQUATIONS FOR NONPREMIXED COM

BUSTION

Given that the eects of turbulence not captured by the previous laminar approachare

due to neglecting temp erature and concentration uctuations wemust restate sp ecies and

energy conservation if we are to takeinto account these eects of turbulence Sp ecically

we intend to neglect the reaction rate and its calculation entirely by assuming the reac

tion takes place instantaneously once the mixture has reached stoichiometric prop ortions

Thus the following approach will concentrate more on the transp ort mechanisms o ccur

ing within the mixture instead of the reaction chemistry in other words assume sp ecies

transp ort is the ratelimiting pro cess

To reduce computational complexity sp ecies conservation is enforced using the con

served scalar approach which is formulated as follows Note that the previous approach

for laminary combustion utilized m ultiple sp ecies in our case Oxygen Nitrogen water

vap or Carb on Dioxide and fuel Kerosene However it is p ossible to describ e the mix

ture comp osition in terms of material which has its origin either within the fuel or oxidizer

stream Sp ecicallywe can express the total mass of the mixture anywhere in the com

putational domain as

m m m m

mixtur e

fuel oxidiz er pr oducts

fuel pr od

CHAPTER THERMODYNAMIC MODELING OF THE FLOW

f f

fuel

where f and f are the fraction of material having its origin in the fuel and oxidizer

fuel

stream resp ectively this is of course assuming a single inlet fuel stream But in terms

of a quantity f dened as the ratio of mass have is origin in fuel stream and the mixture

mass wehave

f mass of material with origin in fuel streammass of mixture

m m m m m

fs f uel fs pr oducts fs

oxidiz er

m m m m m m

mixtur e mixtur e mixtur e

fuel pr oducts oxidiz er

ox stoc

fuel pr od f uel pr od

stoc

where m is the mass of fuel stu and

f AF

stoc stoc

This denition of the mixture fraction is highly useful if we make several assumptions

regarding the combustion pro cess some of whichhave already b een stated

Chemistry is fast compared to convection and diusion

Combustion o ccurs instantaneously and completely when mixture is stoichiometric

ame sheet assumption

Oxidant and fuel are segregated ie no unburned fuel in freestream and no oxidizer

inside ame

Diusion is binary and follows Ficks Law

Radiative energy transfer is neglected

Sp ecically item implies the partition of the oweldinto three zones the internal no

oxidizer present ame sheet innitely thin region where the AF ratio is stoichiometric

and external zone no unburned fuel present The assumptions regarding comp osition are

illustrated in Figure

Note that the previous assumptions regarding internalamesheetexternal to ame

imp ose the following conditions for mass conservation

inside ame internal zone

fuel pr od

at ame sheet

pr od

outside ame external zone

ox pr od

CONSERVED SCALAR APPROACH

Figure Flame sheet comp ositions

This leads to the following expressions for fuel and pro duct mass fraction inside the ame

sheet via

inter nal

f f f

stoc stoc

fuel pr od fuel fuel

f f

stoc

inter nal

fuel

stoc

f f f

stoc stoc

fuel pr od pr od pr od

inter nal

pr od

stoc

And for the region external to the ame sheet wehave

fuel

f f f

stoc stoc

fuel pr od pr od

exter nal

pr od

f stoc

CHAPTER THERMODYNAMIC MODELING OF THE FLOW

f f f f

stoc stoc stoc

fuel pr od pr od

exter nal

stoc

Note that all expressions are linear in f Thus the relationship b etween mixture fraction

and comp osition is a simple one as displayed in Figure Now for this approach to

Figure Mass fractions vs mixture fraction f

be useful not only must chemical comp osition be related to f but prop erties such as

density and temp erature must also be related to mixture fraction f and enthalpy where

the relevantformulation of the energy equation in terms of enthalpy is

hrVhr rhS

t C

where all quantities are as dened previously Note that the mixture fraction since we

are intending to mo del turbulent combustion must be a uctuating quantitiy p osessing

a mean f and uctuating comp onent f with variance f If we also assume equal

diusivities D for all sp ecies then sp ecies conservation can be expressed in terms of a

f and its variance f conservation equation involving the mean mixture fraction

r f rV f r f

t t

CONSERVED SCALAR APPROACH

f r V f

f C r f C f r r

g t

Note that the co ecients C and C are empirical constants Sp ecically the ab ove

t g

expression is arrived at via combining the sp ecies conservation equations and using the

denition of mixture fraction f The result is a conservation equation whose rst

right hand side term represents sp ecies diusion and rst and second terms on the left

represent unsteady eects and convection of sp ecies via bulk motion resp ectively Thus

we have b een able to reduce the n umber of sp ecies conservation equations from in our

case four to two Of course the direct incorp oration of turbulent uctuations into the

equation for sp ecies transp ort and energy has not yet b een p erformed This incorp oration

is done by again noting that f is a uctuating quantity Hence we can evaluate the mean

value of any dep endentow prop erty f according according to the weighting scheme

f pf f hdf

i i

where

pf df

Thus pf is termed the probabilitydensity function and the metho d used here is termed

a p df combustion calculation The functional dep endence of pf onthemuxture fraction

f has b een approximated using semiemp erical metho ds and is describ ed by the following

expression

f f

where

f f df f

f f

f f f

and

f f

f f f

Finally b oundary conditions for f and f are given at the computational b oundaries such

as inlets outlets and surfaces gradients in f and f vanish at solid surfaces

Chapter

NUMERICAL MODEL

APPROACH

The presentendeavor attemps to simulate the imp ortant geometric and uid mechanical

conditions within a gas turbine These conditions in particalar include ow around foillike

ob jects which can b e moving or stationary corresp onding to rotor or stator blades com

pressibility eects work extraction via rotor blades due to moving b oundaries rep eating

b oundary conditions to simulate multiple blades within stages as well as appropriate tur

bulence mo deling and chemical reactions for vap or combustion The result is a D linear

cascadetyp e geometry in the xy plane utilizing the low Reynolds number ie transi

tional turbulence mo del along with Arrhenius chemical kinetics eddydissipation

or PDF sp ecies conservation and transp ort as describ ed in the previous chapter

THE GEOMETRY

The chosen geometry is a simple D cascade of generic dimensions and shap e see Figure

Sp ecicallya stage consisting of a stator and matched rotor are used to simulate

ow inside a gas turbine alb eit in a dimensional sense The overall dimensions of the

computation space as illustrated in Figure and given in Table

Table Stage geometry summary

Domain heightm Itslem S tetr l em Rtetom

The geometry used was generatured using the FLUENT prepro cessor GAMBIT

THE GEOMETRY

Figure Linear cascade geometry see Table for dimensions as lab eled in paren

thesis

In addition injector surface areas are given in

Table Injector area

I nj ector l ocation Ar eam

LESS Rotor

LEPS Rotor

MID Rotor

LESS Rotor

LEPS Rotor

TESS Rotor

TEPS Rotor

The grid utilized is comp osed of a nonstructured triangular mesh with quadrihedral

elements within the b oundary layer and contains approximately grid p oints

CHAPTER NUMERICAL MODEL

Figure displays the stator used having overall turning in the relative and absolute

frame of a cord length c of m a cord to pitch ratio cs of and total

uncovered turning of see Table Figure displays the matched rotor used

Figure Stator geometry and grid

having overall turning in the relativeframe of a cord length c of m a cord to

pitchratio cs of and total uncovered turning of Note that these sp ecications

are summarized in Table

Table Stator geometry summary

Cordc m Pitchs m CS

Table Rotor geometry summary

cordc m pitchs m cs

BOUNDARY CONDITIONS

Figure Rotor geometry and grid

BOUNDARY CONDITIONS

Boundary surface conditions for the geometry are lab eled in Figure Sp ecicallythe

domain inlet utilizes a pressure b oundary condition for subsonic ow which resembles

the conditions found in the combustor exit of a gas turbine ie ow entering the high

pressure turbine of which we are mo deling In particular a total pressure of atm is

used with an inlet total temp erature of K In addition the sp ecies comp osition

reects a stoichiometric combustion in the combustor followed by dilution with air

These parameters are summarized in Table

Table Inlet b oundary conditions

P MP a T K Tu s

C H O CO H O

The outlet b oundary condition corresp onding to subsonic ow is simply an exit

lo cal pressure of MP a Boundary conditions at the fuel injection points on the

foils four on the stator and three on the rotor are of the mass ow typ e with a fuel

temp erature of K conforming to the pyrolosis limit for JP In contrast fuel

injection was varied from injection surface to surface to achieveagiven approximate total

injection rate This tailoring of bulk ow injection rates to yield a certain total mass

ow was necessarily conducted b y trial and error and thus resulted in slightly dierent

injection rates for the various injectors

CHAPTER NUMERICAL MODEL

Figure Domain b oundary surfaces

In addition foil surface conditions b oth included adiabatic and isothermal b ound

aries with a constant wall temp erature of K Finally the interface b oundary

conditions see Figure are simple grid point to grid point mappings from the top

of the domain to the b ottom and interp olative mappings for the statortorotor interface

since grid points in the sliding mesh will not in general remain in alignment Sp ecic

b oundary conditions are given in App endix E

FLUID PROPERTIES

Given that we have assumed ideal gas behavior for the uid the question arises as to

whether air and fuel can b e mo deled as ideal gases throughout the oweld In the case

air air

of the inlet ow whichisessentially air diluted with pro ducts of com bustion P and T

cr it cr it

are MP a and K This yields an approximate reduced pressure P and temp erature

T of and resp ectively and a compresibility factor Z of ab out one Hence treatment

of the freestream as an ideal gas is deemed acceptable In the case of the unburned fuel

Kerosene we are on much more contentious grounds Sp ecically the critical pressure

FLUID PROPERTIES

P and temp erature T for jet fuel JP are MP a and K resp ectively Hence

cr it cr it

given that fuel will be entering the computational domain at approximately freestream

pressures MP a and with an injection temp erature of K the reduced pressure

and temp erature are on the order of one Thus the assumption of ideal gas b ehavior will

b e a p o or one near the p oint of fuel injection While this would result in highly inaccurate

ideal gas b ehavior based calculations of density as a function of temp erature and pressure

calculations of physical prop erties suchas sp ecic heat C and thus changes in enthalpy

entropy etc may also suer in terms of accuracy From the denition of Z

ideal

h h Z R T

u cr it

Now we would exp ect fuel temp erature to vary more then fuel pressure up to the point

of combustion Thus taking the partial derivative of the ab ove expression with resp ect to

T holding pressure constantgives

ideal

cr it

h h Z R Z R

u u

h h

T T T T

r cr it r

Noting the denition of C the ab ove becomes for evaluation at the point of injection

where T

ideal

C C

ideal

Z T P Z T P C C R

r r r r u

ideal ideal

T T

r r

P P P

C C

r r r

Z Z

ie an error in the estimate for sp ecic heat on the order of Thus estimates

for changes in sp ecic enthalpyentropy etc for the fuel near the p oint of fuel injection via

the ideal gas b ehavior assumption will likely result in an error of ab out However

it should be noted that rapid heating of the fuel wil l occur thus rendering the behavior of

the fuel as approximately ideal everywhere but very near the point of injection

Note that the reduced pressure for the fuel must exceed the value of one everywhere

in the domain This implies that the fuel must also b e at a reduced pressure in excess of one

approximately in fact anywhere in the injection system and hence within the internal

blade passages themselves This has rep ercussions in terms of using the fuel as a co olant

hange can o ccur as the fuel is heated say from ro om temp erature given that no phase c

within the blade to a temp erature near the coking limit approximately K at the

p oint of injection into the freestream Hence heat transfer will result only in a sensible

Sp ecically the compressibility factor takesonavalue of around

See generalized enthaloychart for enthalpy deviation Z h

CHAPTER NUMERICAL MODEL

enthalpy change in the fuel thus reducing its eectiveness as a co olant Having justied

the assumption of ideal gas behavior all uid prop erties are derived using FLUENTs

thermo dynamic database according to the correlations given in the previous chapter

The only remaining prop erties to determine are the binary diusivities D for

an assumed dilute mixture Sp ecicallyfor an ideal mixture assuming dilute mixture

b ehavior ie the multicomp onent mixture b ehaves as a binary mixture with fuel oxidizer

and pro ducts diusing into air wehave the correlation

D T P

Hence

T P

ref

D D

ref ref

T P

Table Calculated constant binary diusivities for a tem

p erature of K and pressure of Pa

ref

ref ref

m s Sp ecies i Sp ecies j T P MP a D D m s K

CO Air

H O Air

n D odecane N

Thus given data for the reference diusivity temp eratures and pressures a new

approximate diusivity can b e calculated via Table gives the reference as well as

newly calculated values for sp ecies diusivityofinterest at a pressure of Pa and a

temp erature of K

For the remaining diusion co ecients N O into air where reference values

are not available we can use the ChapmanEnskog correlation The calculated diusion

co ecients are given in Table for completeness and one should refer to App endix B

for details

constant binary diusivities for a Table Calculated

temp erature of K and pressure of Pa using

ChapmanEnskog correlation

Sp ecies i Sp ecies j D m s

O Air

Air N

REACTION MECHANISM

Finally the singlestep global reaction mechanism used describ es the combustion of Kerosene

in the presence of oxygen and corresp onds to the following reaction

C H O CO H O

The reaction rate or change in fuel molar concentration per unit time kmolm s is

given by

dC H

k T C H O

where the global rate co ecient k T is of the Arrhenius form

R T

k T e

Chapter

SIMULATIONS

SIMULATION PROCEDURE

The simulations in this study were conducted by slowly building from simple to more

complex mo dels Sp ecically the ow solution was iterated via the following steps

Inviscidincompressible ow solution steady ow

Laminar incompressible ow solution steady ow

Laminar compressible ow solution steady ow

Turbulent compressible ow solution steady ow

Turbulent compressible with sp ecies transp ort no injectionnonreacting steady

Turbulent compressible with sp ecies transp ort no injectionnonreacting unsteady

Turbulent compressible with sp ecies transp ort with injectioncombustion steady

Turbulent compressible with sp ecies transp ort with injectioncombustion un

steady ow

The simulations where p erformed in this order not only to increased stability running the

full combustion simulation even with very low relaxation always resulted in a diverging

solution but also to decrease the time to convergence and check basic ow conservation

and work extraction Finally the noninjection cases provided a common starting point

for all injection simulations with and without adiabatic foil surfaces In addition all

NONREACTING FLOW

reacting ow simulations were subsequently p erformed using time accurate unsteady

computations to gauge convergence The extent of convergence was determined by not

only examining residuals but by rep eatedly iterating the ow solution until exit ow

prop erties such as lo cal temp erature turbulence intensity and total pressure converged

to essentially constant quantities For the unsteady calculation the time step t was

chosen so as to provide at least steps p er blade pass Sp ecically the time required for

the freestream ow to transverse the domain is given by

L L

domain domain

t s

tr as

V V

mer idinal

where V is approximated by the inlet x velo city comp onent The time required per

blade pass on the other hand is given by

domain

t s

pass

r otor

Thus a time step t was chosen as s and the numb er of minimum computational

time steps were chosen so as to satisfy

NONREACTING FLOW

The rst results of interest are the nonreacting ow simulations used to test the accuracy

of the sp ecies transp ort solution presence of work extraction via the rotor and correct

minimum y values at foil surfaces to reduce computational eort a higher y ie a

coarser grid was used for the noninjection surface eg the stator in the case of rotor

injection Figure and Figure display total pressure and temp erature contours

Note that one in teresting anomaly is the lo calized region of high pressure and temp erature

near the leading edge of the rotor Sp ecically total pressure and temp erature exceed

freestream values by a maximum of approximately at several gid points near the

blade surface However at midstage total temp erature and pressure vary from their

theoretically constantvalue by approximately and resp ectively

Note that decreasing duration of the time step increases the numb er of steps required for the freestream

owtotravel the length of the computational domain

The values for P and T at midstage are MPa and K This result was robust in that

the region of high total temp erature and pressure p ersisted even for alternative grid densities

CHAPTER SIMULATIONS

Figure Total pressure contours noninjection simulation

Figure Total temp erature for noninjection simulation

NONREACTING FLOW

FlowMach number is given in Figure and varies in the freestream from at

the inlet to in excess of within the cascade itself In addition the ow includes a

Figure FlowMachnumber

pressure side separation on the rotor as indicated by the pathlines given in Figure

In terms of sp ecies conservation we see that the O mole fraction is conserved see Figure

Finally to ensure adequate grid quality we note that y values are on the order

of one over the foil surface the stator is displayed for illustrative purp oses and the rotor

exhibits similar values as displayed in Figure Inlet and outlet ow prop erties are

given in Table

Table Inlet and exit ow prop erties for nonreacting ow

case

Sim typ e v m v m P MPa P MPa T K T K kgm

inl et outl et inl et outl et

inl et outl et

Unstad

Unstnonad

The rst item to note is the reduction in lo cal temp erature at the outlet due primarily

to work extraction total temp erature also decreased by approximately K see Figure

In the case of fully laminar ow b oth the rotor and stator exhibited gross separation

The nomenclature used here is st steadyad adiabatic wall and nonad isothermal wall

CHAPTER SIMULATIONS

Figure Pathlines indicating pressure side rotor blade separation

In addition the nonadiabatic case ie isothermal foil surfaces at K exhibits

an increase in ow rate due presumablyto the decrease in average viscosityofthe uid

near the wall Indeed for air absolute viscosity increases with temp erature

We can describ e the overall pro cess by calculating the p olytropic exp onent n ac

cording to

n n

P v P v

inl et outl et

and in the adiabatic case a p olytropic eciency via the following

is the average of the ow inlet and outlet values where

Table Inlet and exit ow prop erties for nonreacting ow

case

Simulation typ e n

pol y

Unsteadyad

Unsteadynonad

The p olytropic exp onent is useful in that it may be sp ecied in a cycle deck to

describ e the work extraction pro cess taking place for a complete stage

is dened at the ratio of sp ecic heats C C

p v

NONREACTING FLOW

Figure Contours of y for no injection

Of interest is also the rate of work extraction which can b e directly calculated using

the expression

F V F kV k

bl ade bl ade y bl ade bl ade

The required quantities for calculating work extraction are giveinTable

Table Rotor foil velo city force and work extraction for

nonreaction ow

Simulation typ e jV j ms F kNm MWm

y r otor

bl ade

Unsteadyad

Unsteadynonad

Again note that due to the increased mass ow rate for the nonadiabatic foils there

is an increase in force applied to the blades in the y direction and hence an increase in

power extracted by the rotor

Finallyofinterest here is arriving at an estimate of the required heat addition and

hence fuel injection to yield an isothermal pro cess

The units given for force and p ower are per unit width since the simulation is dimensional

CHAPTER SIMULATIONS

Figure y for stator

Sp ecically for the nonreacting case we have from the energy balance neglecting

shear work heat transfer and elevation change

j q w h jw j jh j jhT

Thus in order to pro duce zero net change in lo cal temp erature the required heat addition

and fuel mass ow can b e expressed as

q jhT j

r eq uir ed

m m m

hT q

fuel required

HHV HHV

fuel fuel

kg h h

inl et outl et

HHV s

f uel

where HHV is the higher heating value of the fuel Note that this result is a minimum

given heat addition will accelerate the ow and result in a lower lo cal temp erature via

an increase in the Mach numb er It should also be restated that we are mo deling the

injection of Kerosene vap or as opp osed to liquid or droplet injection

Note that work calculated via the energy balance was in excellent agreement with the metho d employed

in the previous expression

The heating value used is for that of Do decane C H vap or

REACTING FLOW ARRHENIUS KINETICSEDDYDISSIPATION

REACTING FLOW ARRHENIUS KINETICSEDDY

DISSIPATION

Figure T proles for rotor injection

Fuel was injected from seven lo cations two at the leading and trailing edge of the

stator as well as two leading edge and one midspan injection p oint on the rotor The re

sults from the laminar combustion case as well as the eddydissipation mo del indicate that

combustion is selfigniting and selfsustaining for all injection p oints Figure illustrates

temp erature contours for the rotor injection p oints leading edge suction and pressure side

injection along with pressure side recirculation zone injection lower image Figure

displays the same for the stator injection points For the purp ose of demonstrating that

fuel is indeed b eing consumed the mole fraction contours for water vap or y are given

h O

in Figure and clearly showthevap or comp onentoftheow is minimized at the p oint

foil of injection approximately in the middle of the gure on the pressure side of the

whereup on y increases to a maximum value within the recirculaton zone and wake

h O

Diusion into the freestream subsequently o ccurs thereafter

CHAPTER SIMULATIONS

Figure T for stator injection

Figure y for midrotor injection

H O

REACTING FLOW ARRHENIUS KINETICSEDDYDISSIPATION

Figure Ma for midrotor injection

The Machnumb er proles are given in Figure Two items of interest are the fact

that the intro duction of cold fuel at a temp erature of K compared with the main ow

which is at approximately K pro duces a high lo cal Mach number about the point

of injection This is due simply to the fact that the lo cal sp eed of sound is prop ortional to

T In addition the combustion pro cess tends to drivethe Mach number downstream

of the injection point towards the sonicp oint This should not be surprising since any

heat addition pro cess will in general drive the Machnumber towards unity Sp ecically

the exit Ma for injection and noninjection cases are and Finallycombustion is

also demonstrated by examining the fuel concentrations over the computational domain

Figure illustrates the mole fraction of fuel y From the gure we Sp ecically

C H

can see that the fuel after being injected travels upstream of the main ow the ow

is lo cally moving clo ckwise in the recirculation zone and subsequently disp ersed and

consumed within the recirculation zone Various stage inlet and outlet ow quantities are

given as in the nonreaction case

Table Inlet and exit ow prop erties for reacting owcase

rotor mid injection

Sim typ e v m v m P MPa P MPa T T kgm K K

inl et outl et inl et outl et

inl et outl et

Unstad

Unstnonad

CHAPTER SIMULATIONS

Figure y for mid rotor injection

C H

As we can see the pro cess is now essentially isothermal for the complete stage In

addition the mass ow rate of fuel can b e calculated via

f uel

N MW m

f uel fuel fuel

m m m m

CO CO CO

fuel

m N MW

CO CO CO

where the s refer to the dierence b etween stage inlet and outlet From the previous we

can calculate a pro cess p olytropic exp onent n as in the nonreaction case as well as the

fuel injection rate

Table Fuel mass ow and p olytropic exp onent n

fuel

rotor mid injection

kgs Sim typ e n

fuel

Unstad

Unstnonad

EddyDissipation results were nearly identical to the laminar combustion cases in

terms of peak ame temp erature and ame structure see App endix F Leading edge

pressure and suction side injection bulk ow results were also similar to the midrotor

pressure side injection and are given in App endix F

TURBULENT COMBUSTION CONSERVED SCALAR SIMULATIONS

TURBULENT COMBUSTION CONSERVED SCALAR

SIMULATIONS

Figure Conserved scalar f for mid rotor injection adiabatic foil

The results for the turbulent combustion simulations based on a conserved scalar

approach are given as follows A contour plot showing the distribution for the scalar f

is given in Figure indicating for mid pressure side injection the intro duction of fuel

material on the pressure side of the foil Conversely Figure shows the increase in

H O

in the freestream Temp erature proles are given in Figure and show a substantially

reduced p eak ame temp erature of ab out K as opp osed to a value of approximately

K for the laminar and eddyviscosity case In fact all rotor simulations yielded

substantially lower p eak combustion temp eratures see App endix F An additional p oint

of interest to make is that the PDF calculation did not yield a pressure side separation as

indicated by the pathlines shown in Figure A p ossible explanation for this may be

that the separation is weak thus slight variations in prop erty calculation metho ds may

result in reattachment of the ow

Fluid prop erties suchas and k are assumed for the PDF metho d thus the values of viscosity and

thermal conductivitywere set to their approximate values at midstage and wm K

resp ectively via the previous laminar combustion case

Laminar combustion simulations are known to chronically overpredict peak ame temp eratures in

turbulentow Thus this result comes as no surprise

The PDF metho d assumes constant uid prop erties k etc

CHAPTER SIMULATIONS

Figure for mid rotor injection adiabatic foil

H O

STATOR SIMULATIONS

Given the similarity of the stator and rotor injection simulations only a few illustrative

gures are given to demonstrate the presence of combustion Sp ecically Figures

and display the static temp erature rise asso ciated with combustion on the stator foil

region It should be stated that diuculties were encountered in arriving at consistent

predictions b etween the stator and rotor PDF simulations Sp ecically the stator injec

tion cases yielded a much higher p eak combustion temp erature similar to the laminar and

eddydissipation mo del than those of the rotor Recall that the results for rotor combus

tion yielded p eak temp eratures of approximately K while stator PDF combustion

results gives a p eak temp erature of close to K In addition the stage outlet temp er

atures were not consistent between stator and rotor injection nor was there a reasonable

correlation b etween fuel injection and the overall stage temp erature rise

STATOR SIMULATIONS

Figure Temp erature for mid rotor injection adiabatic foil

Figure Pathlines for mid rotor injection PDF adiabatic foil

CHAPTER SIMULATIONS

Figure Temp erature contours for laminar leading edge pressure side injection leps

adiabatic foil

Figure Temp erature contours for eddydissipation trailing edge suction side injection nonadiabatic foil

Chapter

RESULTS AND DISCUSSION

Given that the results do not exclude the p ossibilityof combustion in a turbine we can

now turn our attention to some of the other questions at hand Namely what are the

benets and drawbacks of the dierent combustioninjection schemes in terms of

Deterioration of aero dynamic p erformance

Blade heat transfer

Transverse temp erature variations pattern factors

Axial temp erature variation ie how isothermal is the combustion pro cess

PRESSURE DISTRIBUTIONS

Table Net pressure force on rotor blade for dierent in

jection metho ds laminar combustion

Simulation typ e F kN

pr essur e

No Injection

Leading edge suct side

Leading edge pres side

Midspan pres side

Given the various lo cations for fuel injection it is of interest as to which fuel injection

scheme results in the least disruption of the ow in an aero dynamic sense A rst

indication of the eect of injection lo cation on aero dynamic p erformance is the net force on

CHAPTER RESULTS AND DISCUSSION

Figure Rotor pressure distribution for various injection lo cations laminary combus

tion

the blade Table From the table we can see that the choice of injection conguation

has very little eect on blade force and hence work extraction these force calculations were

made over one full blade pass although there would app ear to b e some minor advantage

to pressure side injection as indicated by the slight higher blade force for pressure side

fuel injection This relativeinvariance at least for the temp erature rises presentinthese

simulations can be seen by sup erimp osing blade pressure distribution for the various

injection schemes Figure From the gure we can see that indeed the pressure

distributions are largely identical However a more illustrative plot is of the p ercentage

noninjection case as shown in Figure From the deviation in pressure from the

gure we see that the injection of fuel has the eect of reducing foil surface pressure

via the addition of heat accelerating the ow Sp ecically surface pressure for the

combustion cases match those for noncombustion at two spatial lo cations the leading

edge at the stagnation p oint to the left and at the trailing edge where the ow ceases

to turn and lo cal static pressure is determined by the exit b oundary condition the same

for all cases However immediately downstream of the leading edge stagnation point

pressure is less than the noncombustioning case again due to the expansion of the hot

These results were from the laminar combustion simulations

AFORTAN program was written to read extract and combine pressure plot information from multiple

les

Note that as in the blade force calculations these proles are timeaverages over one blade pass

HEAT TRANSFER

Figure Rotor pressure variation for various injection lo cations

gases and acceleration of the ow Of interest is the qualitative dierence between the

maximum pressure deviation in that the leading edge suction side injection case has the

greatest negative deviation indicativeof strong acceleration near the point of injection

compared to pressure side injection where separationinduced mixing tends to mo derate

the temp erature rise of the ow Hence mixing due to the pressure side separation

tends to mo derate any dierences between surface pressure distributions for the leading

edge pressure side and midinjection cases b oth utilizing injection of the pressure side

of the blade This can be seen by noting that the leading edge and midinjection cases

exhibit almost identical pressure deviations from the noncombustion case

HEAT TRANSFER

In assessing the resp ective merits and deciencies of various injection schemes we are also

interested in the p otential impact of heat transfer on blade durability Sp ecically one

would exp ect that given the enhanced mixing asso ciated with the pressure side injection

due to the recirculation zone heat transfer would be less pronounced on the pressure

than on the suction side This is indeed the case as illustrated via the plot of surface

The gure can b e viewed as follows The leading edge on the left b egins at an x distance of approx

imately m The curve of minimum absolute pressure deviation pro ceeding downstream represents

the pressure side while at the p oint of inection the suction side exhibits the minimum absolute pressure deviation

CHAPTER RESULTS AND DISCUSSION

heat ux Figure Sp ecically while the pressure side injection cases exhibit their

maximum heat transfer at the trailing edge resulting presumably from enhanced mixing

at the trailing edge separation point due to the thickness of the blade suction side

combustion yields a much higher heat transfer rate forward of midspan This is due to

the fact that on the pressure side gradients are muchlower than on the suction side which

do es not contain a separation bubble Note it is not approriate to refer to a b oundary

layerassuch given that near the p ointofcombustion heat is not b eing transferred from

the freestream or the blade to the blade or freestream Sp ecically the maximum

gas temp erature must be attained within the b oundary layer not at the surface or the

freestream Also the acceleration of the ow on the suction side increases the convective

transp ort of energy thus yielding a higher heat transfer rate on the suction side the

analogy for b oundary layers here would b e a relatively high heat transfer co ecient h on

the suction side of the blade in comparison to the pressure side

Figure Heat transfer q for various injection schemes

PATTERN FACTORS

Of interest is not only heat transfer to the blades but the eect of combustion on the

foil wake temp erature distibution Sp eccally Figure illustrates the crossstream

Temp erature nonuniformity due to foil b oundarylayer combustion will aect comp onent durability

downstream of the combustion zone eg blades in the following stage

PATTERN FACTORS

temp erature distribution just downstream of the stator trailing edge Hence we dene

Figure Crossstream temp erature distibution cord downstream of stator adia

batic noncombustion

two pattern factors as follows

T T

max mean

Pattern Factor

foil

T T

max min

Pattern Factor

foil

where T is the temp erature prole mean value at the rotor exit plane or stator

mean

exit if we are interested in pattern factors for the stator approximately a quarter cord

length downstream of the blade trailing edge Note that temp erature dierence in the

numerator is normalized via the quantityT whichistaken to be the rotor or stator

foil lo cal temp erature drop from leading to trailing edge for the nonreacting case

CHAPTER RESULTS AND DISCUSSION

The calculated pattern factors are given in Table

Table Calculated pattern factors laminar rotor adia

batic wall injection

Injection lo cation PF PF

No Injection

LEPS Injection

Mid Injection

LESS Injection

These results indicate that not only do es combustion on the foil increase the pattern

factor in comparison to the case without fuel injection but combustion on the pressure

side of the foil lends itself to higher pattern factors than combustion on the suction side

The reason for this can be readily seen by viewing a contour plot of static temp erature

ab out the rotor for the injectionless case Figure Sp ecically we see that the ex

Figure Temp erature contours ab out rotor for noninjection case

pansion pro cess for the ow takes place on the suction side resulting in relatively high

temp eratures on the pressure side As the ow pro ceeds downstream towards the trailing

edge and b eyond there is recompression on the suction side heat transfer and uid mixing

between the relatively hot pressure side ow and the co ol suction side uid Thus heat

ISOTHERMAL COMBUSTION

addition on the pressure side would accentuate the temp erature dierence between the

suction side and pressure side uid hence increasing the value of the pattern factor

Similarly for the stator injection cases we have the result that injection further

upstream ie at the leading edge of the blade is b est for minimizing the pattern factor

This is reasonable since overall diusion of the hot pro duct gases into the freestream is

enhanced by the longer travel distance These results can b e seen from Table

Table Calculated pattern factors laminar stator adia

batic wall injection

Injection lo cation PF PF

non injection

less

leps

tess

teps

Unlike in the case of rotor injection where work extraction is taking place by expan

sion on the suction side of the foil we see no disadvantage in burning fuel on the pressure

side in terms of lowering the pattern factor

ISOTHERMAL COMBUSTION

Given that the goal is to determine the most advantageous scheme for instigating isother

mal combustion wecontinue by lo oking at the axial xdirection temp erature distribution

for the various injection schemes Sp ecicallyone would presume that given low enough

Mach numb ers it would make sense to burn fuel at or near the pointof work extraction

ie on the rotor This apparently is the case as displayed in Figure Sp ecically

we see that stator injection results in a mass ow averaged peak temp erature of ap

proximately K at midstage resulting in a temp erature departure of ab out K

trast rotor injection results in a minimum temp erature of approximately K In con

also at midstage and a temp erature departure from the mean inlet of only ab out

K a signicant improvement over the stator injection results However it should b e

noted that for high Mach number ows where velo city is high and lo cal temp eratures

are low at midstage stator combustion may provide a more uniform axial temp erature

distribution

Note that the normalization temp erature T is much smaller for the stator case Hence direct

foil

comparisons b etween rotor and stator pattern factors are meaningless

CHAPTER RESULTS AND DISCUSSION

Figure Axial lo cal temp erature distribution for various injection schemes

CYCLE ENHANCEMENTS

IDENTICAL OVERALL PRESSURE RATIOS

A suitable comparison can b e made see Figure between the Hybrid cycle op erating

over the closed path an intermediate cycle op erating over

and the equivalent Brayton cycle

where the pro cesses and are assumed isen tropic In

addition simulation data exists for the paths and corresp onding

to the combustion and noncombustion cases resp ectively Hence the expressions for

net sp ecic turbine work for the Brayton cycle is assuming wecan neglect compressible

eects

F jV j

bl ade

av er ag e B r ay ton

w T C T w

comp

net

the hybrid cycle

F jV j

bl ade

Hybrid av er ag e

w C T T w

comp

net

A reduced fuel injection simulation was conducted to give results for an intermediate nonisothermal heat addition case

CYCLE ENHANCEMENTS

and for an intermediate heat addition cycle

F jV j

bl ade

av er ag e

I nter mediate

w C T T w

comp

net

where w is the isentropic compressor sp ecic work w In addition thermal e

comp

Figure Ts diagram for cycle comparisons

ciency for the Brayton Hybrid and Intermediate cycle can b e expresses as see App endix

w w

net net

B r ay ton

q q

w w

net net

I nter mediate

q q q

w w

net net

Hybrid

q q q

is the combustor heat addition for pro cess q is the approximately where q

isothermal heat addition asso ciated with the simulated pro cess and q is the

heat addition for a nonisothermal combustion pro cess reduced fuel injection

Calculating the ab ove quantities assuming pseudo constant sp ecic heats while neglecting

Machnumb er eects yields the following net work and thermal eciencies Table

A separate simulation was conducted for this reduced fuel injection nonisothermal case data is

presented in App endix F

CHAPTER RESULTS AND DISCUSSION

Table Values for and w

net

simul ation theor etical

Simulation w kJkg

net

simul ation theor etical

Brayton

Hybrid

Intermediate

Note as exp ected a machine utilizing isothermal combustion when compared with

a Brayton machine having identical overall pressure and temp erature ratios pro duces

more work

NONCONSTANT OVERALL PRESSURE RATIO

If we instead not only hold the overall temp erature ratio constant but set the Hybrid cuto

pressure P equal to that of the turbine inlet temp erature of a Brayton cycle we see the

enharentadvantage of the Hybrid cycle Sp ecicallythe psuedo simulated Brayton cycle

Figure Ts diagram for Hybrid yellow and Brayton cycle black for equal peak

temp eratures and isothermal cutoBrayton turbine inlet pressures

can be compared with the Hybrid equivalent

The calculated eciency and sp ecic net work for this high pressure

Where the simulation results apply to the subpro cesses and

CYCLE ENHANCEMENTS

Hybird cycle again based on pseudo simulation data is and kJkg resp ectively

Recall that the eciency and sp ecic net output for the equivalentBrayton cycle case are

and kJkg Hence the Hybrid cycle indeed presents under certain conditions

an absolute advantage in terms of eciency and p ower over a Brayton machine Thus in

the case of xed overall temp erature ratios the Hybrid cycle can pro duce more work at a

higher eciency than the corresp onding equivalent Brayton cycle as depicted in Figure

Indeed via the expression for Brayton cycle sp ecic output we nd that takes on a

value of for values of and corresp onding to a hyp othetical highpressure Brayton cycle with

pressure ratio and a p eak temp erature of and K resp ectively Hence a highpressure Brayton

cycle will pro duce less work but exhibit a higher thermal eciency found to b e via than the

comparable Hybrid or lowpressure Brayton machine Thus the advantage of the Hybrid cycle in pro ducing

more work as wel l as exhibiting a higher eciency for increasing pressure ratios is demonstrated

Chapter

CONCLUSION

SUMMARY

A number of simulations using a generic turbinelike geometry under turbinelike condi

tions has b een p erformed using a singlestep global reaction mechanism The simulation

results indicate that b oundary layer combustion is selfigniting and selfsustaining under

the representativeow conditions The simulations indicate that heat transfer to the foil

surfaces is mimimized when injection takes place in a recirculation zone Pattern factors

also app ear to be minimized when combustion o ccurs on the suction side of the foil for

rotor injection only In addition pattern factors are minimized when injection takes

place near the leading edge of the foil The ow itself remains the most isothermal if

combustion takes place at the surface of work extraction on the rotor foil Also there

app ears to be little aero dynamic advantage or p enalty asso ciated with the dierent in

jection schemes at least for the rates of heat addition exhibited by the simulations used

in this study Finally after conducting a pseudosimulation of a cycle incorp orating

isothermal combustion and comparing to the equivalent Brayton machine we nd that

isothermal combustion yields at the least an increase in engine sp ecic work and hence

power Sp ecicallygiven a xed overall temp erature ratio isothermal combustion even

cycle aords the opp ortunity to enhance output and eciency in the form of a Hybrid

over the traditional Brayton cycle

RECOMMENDATIONS FOR FUTURE WORK

Although this study has answered some of the fundamental questions regarding the feasi

bility of isothermal combustion in turbines a numb er of opp ortunities can and should b e

pursued These include but are not limited to the following

RECOMMENDATIONS FOR FUTURE WORK

Foil optimization study to assess with improved accuracy the extent to which

b oundarylayer combustion results in airfoil aero dynamic deterioration

Three dimensional simulations investigating the eect of swirl induced reaction rate

enhancement in the case of isothermal combustion

Assesment of other reaction mechanisms suchasmultistep laminarEddyDissipation

global reactions

Assesment of the eect of radiative energy transfer on the previous results

Full turbine multistage simulations to verify p erformance improvements asso ciated

with isothermal combustion

App endix A

HYBRID CYCLE

PARAMETERS

To derive the Hybrid cycle eciency and sp ecic output we start by describing the new

cycle as depicted in Figure A Sp ecically this cycle is comp osed of the following

pro cesses

Isentropic compression

Isobaric heat addition in combustor

Isothermal heat addition heat addition in turbine

Isentropic expansion work extraction in turbine no combustion

Isobaric heat rejection to the ambient

The expression for thermal eciency for this new hybrid cycle can b e found b eginning

with the general expression

w q

net out

Hybrid

q q

in in

where noting the sequence of heat additions and rejections wehave

Hybrid

q q

where q q and q are comp osed of internally reversible isobaric isothermal and

isobaric pro cesses resp ectively Thus we arriveat

C T

H y br id

C T T s

Figure A Ts diagram for hybrid isothermalBrayton cycle

For the isothermal heat addition pro cess wenote

P T

Rl n s C ln

T P

P P

P P P

Rl n Rl n A Rl n

P P P P P

T Furthermore P T where for the isentropic pro cess we have P

we can rewrite P P via the following

P P T

where we are now reintro ducing the notation T T T T and T T

and represent the isobaric heat addition for the Hence the isobar connecting state

equivalent Brayton cycle with peak temp erature ratio and compressor temp erature

ratio and also serves to dene an isothermal cuto temp erature T Combining the

previous developments gives

A s Rl nT T Rl n

Making substitutions and dividing by C yields

Hybrid

T ln

p c

APPENDIX A HYBRID CYCLE PARAMETERS

T T

ln T T

p c

which can b e almost completely expressed in terms of temp erature ratios already dened

Hybrid

T T

c c

Finally the temp erature ratio T T can b e found via the observationthats s

whichgives noting that the two paths and are isobaric

T T T T T T

C ln C ln A

p p

T T T T T T

This yields our nal general expression for the Hybrid cycle eciency

Hybrid

c c

Note in the limit as werecover the cycle eciency for the ideal Brayton cycle

c c

Hybrid

c c

B r ay ton c

Likewise in the limit as our combined cycle eciency reduces to that of the

isothermal cycle

Hybrid

c c

I sother mal

ln ln

c c

where is equivalent to As in the case of the ideal Brayton and Isothermal cycle a

normalized work can also b e calculated

q q q

Hybrid

C T C T

p p

C T T T s C T T

p p

C T

Using the previous expression for s A and substituting our previously dened

temp erature ratios wend

T T T T

W ln

Hybrid

T T T T

ln A

A W ln

H y br id

Taking the limit as we nd

ln W

Hybrid

W ln

I sother mal

we also nd that as exp ected And taking the limit as

Norm

W W

H y br id B r ay ton

App endix B

DIFFUSION VIA THE

CHAPMANENSKOG CORR

The co ecient estimates to b e used for D in equation are not readily available in

tabular form for Oxygen or Nitrogen However data do es exist for using the Chapman

Enskog correlation derived via psuedo emp erical metho ds and consists of the following

relationship

D B

PMW

ij ij

where

i j

B MW MW MW

j i

T T T

e e e

B T

k k

i B j B

where and are constants The required physical constants are given in Table

i j i j

B along with intermediate and nal calculated values relevanttoD

Svehla RA Estimated viscosities and thermal conductivities of gases at high temp eratures th

Ed McGrawHill New York

Table B Required data for calculation of binary diusion

co ecients D

Sp ecies i sp ecies j k k MW MW D m s

i B j B i j i j ij

O Air

N Air

App endix C

CYCLE COMPARISONS

C CONSTANT OVERALL PRESSURE RATIO

In order to utilize the results derived from the ow simulation for the purp ose of comparing

the isothermal to the Brayton cycle work output etc we pro ceed as follows Restating the

Brayton combined Hybrid and an intermediate nonisothermal cycle as displayed in the

Figure C a comparison can b e made b etween the Hybrid cycle op erating over the closed

path the equivalent Brayton cycle

and an intermediate cycle Note that the pro cesses

and are assumed isentropic ie rev ersible and adiabatic Also

computational data exists for the paths and corresp onding to

the combustion and noncombustion cases resp ectively To compare cycle output and

thermal eciency we rst characterize the Brayton cycle turbine work as the sum of the

subpro cess works for the turbine Or

F jV j

bl ade

B r ay ton

reversible

w w w C w

tur bine

where the rst quantity on the right is calculated via simulation data and the last term is

simply reversible b oundary work for an assumed internally reversible expansion pro cess

Likewise an expression for the turbine work for the Hybrid cycle is given by

F jV j

bl ade

Hybrid

reversible

w w w w C

tur bine

Also the expression for the intermediate injection case yields an expression for sp ecic

turbine work

F jV j

bl ade

reversible I nter mediate

C w w w w

tur bine

C CONSTANT OVERALL PRESSURE RATIO

Figure C Ts diagram for HybridnonisothermalBrayton cycle comparison

Next an expression for reversible adiabatic b oundary work is arrived at via the

following development For an op en system undergoing an internally reversible pro cess we

have the following dierential expressions for the energy and entropy balances b oth on a

p er unit mass

q w dh C

reversible reversible

q q

ds s C

ir r

T T

Combining these two expressions wehave the following relationship

w dh C

reversible

Thus the reversible adiabatic work can b e expressed as

av er ag e

w h C T C

reversible

If we assume low Mach numbers ie T T then the resulting expressions for total

sp ecic turbine work for the intermediate isothermal and Brayton cycle are

F jV j

bl ade

B r ay ton av er ag e

w C T C T

tur bine

F jV j

bl ade

av er ag e Hybrid

T T C C w

tur bine m

APPENDIX C CYCLE COMPARISONS

F jV j

bl ade

av er ag e

I nter mediate

w C T T C

tur bine

T for an isentropic pro cess can b e calculated by noting the following relationship b etween

temp erature and pressure total or lo cal assuming constant sp ecic heats

P T

isentr opic

T P

where is some suitably chosen average value over the pro cess

In addition to sp ecic work we may also be interested in comparing thermal e

ciency We can begin by noting the denition of cycle eciency where net work is the

dierence b etween turbine and compressor work for say a turb oshaft engine

w w w

net comp

tur b

q q

in in

Or more sp ecically

w w w w

comp comp

tur b tur b

B r ay ton

q q

w w w w

comp comp

tur b tur b

H y br id

q q q

w w w w

comp comp

tur b tur b

I nter mediate

q q q

where

C T dT C q h h

Note that C in the preceding cycle analysis can be approximated by that of air the

combustion pro cess results in a change in sp ecic heat due to changes in chemical comp o

sition but wecho ose to neglect this Sp ecic heat for air as a function of temp erature

is given by

C T T T T C

where the units are k J k g K Thus

q C T dT

T T T T

T T T C T

T T T T

As a check the simulation mixture sp ecic heat and calculated sp ecic heat for air were compared

and had the values and k J k g K Thus with an error of ab out we are justied in

approximating the mixture by a single comp onent uid air

C CONSTANT OVERALL PRESSURE RATIO

Now the heat addition for the isothermal pro cess can be approximated by using

the upp er heating value HHV for Kerosene Hence

fuel

q HHV C

Finally the compressor work can b e estimated via the previous reversible work expression

av er ag e

T T C w C

r ev er sibl e

or assuming low compressor inlet and outlet Mach numbers

av er ag e

w C T T C

reversible

where the inlet pressure and temp erature are that of the ambient MP a and K

Now for computational purp oses quantities of interest to us are and hence C

evaluated at T T T T and T th T T us let us assume values for the previous

temp eratures and iterate the solutions as needed see Table C for all assumed relevant

state temp eratures

Table C Assumed state temp eratures C and

State T C

Thus we can assign an average value for sp ecic heat and over the required

pro cesses see Table C for all assumed path averaged relevant prop erties

Table C Assumed pro cess averaged C and

Pro cess C

Again we are assuming vap or injection as opp osed to droplet where the upp er heating value for

Do decane was used whichis k J k g f uel

APPENDIX C CYCLE COMPARISONS

Note that for certain states namely and we have simulated data for

sp ecic heat and This information is presented in Table C

Table C Simulation C and

State C

Thus as in the previous case we can calculate an average value for sp ecic heat and

for relevant the pro cesses and given in Table C

Table C Assumed pro cess averaged sp ecic heats and

Pro cess C

For reversible adiabatic compression wehave

T T K C

whichisvery close to our initial guess of K Thus the compressor work via C

is given by

w C T T k J k g C

compr essor P

Next we determine the heat addition in the combustor pro cess via C which

gives a value for q of k J k g In addition wealsohave

fuel

q HHV kJ kg C

and

fuel

q HHV k J k g C

The ow rates were selected from the midrotor injection simulaton

C NONCONSTANT PRESSURE RATIO

Finallywemust determine the turbine exit temp erature for the simulated Brayton Hybrid

and Intermediate cycle via C whichgives

T P

B r ay ton

K C T

T P

P T

Hybrid

K C T

T P

P T

I nter mediate

K C T

T P

Note that these results are close to the assumed values in Table C Thus from these

results and utilizing C C and C wehave the following

B r ay ton

w kJ kg C

tur bine

Hybrid

w kJ kg C

tur bine

I nter mediate

kJ kg C w

tur bine

Finally using the general expression for thermal eciency wehave

w w w w

comp comp

tur b tur b

B r ay ton

q q

w w w w

comp comp

tur b tur b

Hybrid

q q q

w w w w

comp comp

tur b tur b

I nter mediate

q q q

C NONCONSTANT PRESSURE RATIO

Simulation date exists for the paths and in the form a high inlet pressure

simulation with an exit pressure and temp erature comforming to that of state and the

noncombustion simulations with an inlet state Note that all quantities relevant to

See App endix E for inlet and exit states

APPENDIX C CYCLE COMPARISONS

Figure C Ts diagram for hybrid and Brayton cycle for equal p eak temp eratures and

isothermal cutoturbine inlet pressures

pro cess and have previously been calculated Hence the only

q and w Sp ecically assuming remaining quantities to calculate are w q

identical average pro cess values for sp ecic heat etc for wehave

T T K C

w C T T k J k g C

compr essor P

Using the integrated expression for heat addition C wendavalue of kJ kgfor

q The heat addition q is also given by

fuel

HHV k J k g C q

Hence combining simulation date with the assumed isentropic expansion pro cess

wehave the expression for turbine sp ecic work for the high pressure Hybrid case

F jV j F jV j

y y

bl ade bl ade

HP Hybrid av er ag e

T T w C

T ur bine

m m

k J k g C

C NONCONSTANT PRESSURE RATIO

or in terms of net work

HP Hybrid

w w w k J k g C

C omp

Turbine

net

The eciency for the high pressrue Hybrid cycle is given by

w w

net net

HP Hybrid

q q q

Thus the high pressure hybird cycle has an eciency based on pseudo simulation data

of and sp ecic net work of kJkg

App endix D

PATTERN FACTORS

Of interest to us is the extenttowhich the ow is nonuniform in temp erature just down

stream of combustion Indeed for the case of rotor injection an accompanying foil such

as a stator blade may reside just downstream Thus wedenetwo pattern factors as

T T

max mean

PF D

stag e

T T

max min

D PF

stag e

where T is the temp erature change for the noninjection case

stag e

Table D Required data for calculation of pattern factor

laminar rotor injection

K K K Injection lo cation T T K T T

max mean stag e

min

No Injection

LEPS Injection

Mid Injection

LESS Injection

The chosen distance from the trailing edge in the meridonial direction is approx

imately one quarter of a cord length The required quantites are given in Table D

Note that a pattern factor for a turbine is traditionally dened where T is over a stage but in our case

change in lo cal temp erature is ideally zero given that our aim is to achieve isothermal combustion Hence

we use the noncombustion stage temp erature drop to normalize the nonuniformityinow temp erature

Note that the average temp erature used for T was not the prole mean but the stage outlet mean

mean

Finally the calculated pattern factors are given in Table D

Table D Calculated pattern factors laminar rotor injec

tion

Injection lo cation PF PF

No Injection

LEPS Injection

Mid Injection

LESS Injection

Similarlywehave the required data for the stator injection cases

Table D Required date for calculation of pattern factor

laminar stator injection

Injection lo cation T T K T T K K K

max mean stag e

min

No Injection

LESS Injection

LEPS Injection

LESS Injection

LEPS Injection

Table D Calculated pattern factors laminar stator adia

batic wall injection

Injection lo cation PF PF

non injection

less

leps

tess

teps

ow temp erature Given that no turning or work extraction is present after the ow pro ceeds downstream

of the rotor average lo cal temp erature should remain essentually constant

App endix E

BOUNDARY CONDITIONS

The b oundary conditions for all simulations are provided in Table E

Table E Simulation b oundary conditions

Quantity Boundary value

P MPa MPa for high pressure case

inl et

MPa MPa for high pressure case P

outl et

T K

inl et

T backow only K

outl et

T K

inj ection

inl et

Tu backow only

outl et

inj ection

H m

inj ection

m H

inl et

H m

outl et

T nonaddiabatic surface K

sur f ace

variable kgs target injection normal to b oundary

f uel

rotor blade

V ms

fuel

inj ection

inl et

H O

inl et

App endix F

SIMULATION DATA

F BULK FLOW QUANTITIES

F ROTOR INJECTION

Table F Inlet and exit ow prop erties for reacting owcase

rotor less injection

Sim typ e m m P MPa P MPa T K T K m kgm

inl et outl et inl et outl et

inl et outl et

Unstad LAM

Unstad ED

Unstad PDF

UnstnonadLAM

UnstnonadED

Unstnonad PDF

APPENDIX F SIMULATION DATA

Table F Inlet and exit ow prop erties for reacting owcase

rotor mid injection

Sim typ e m m P MPa P MPa T K T K m kgm

inl et outl et inl et outl et

inl et outl et

Unstad LAM

Unstad ED

Unstad PDF

UnstnonadLAM

UnstnonadED

Unstnonad PDF

Table F Inlet and exit ow prop erties for reacting owcase

rotor leps injection

Sim typ e m m P MPa P MPa T K T K m kgm

inl et outl et inl et outl et

inl et outl et

Unstad LAM

Unstad ED

Unstad PDF

UnstnonadLAM

UnstnonadED

Unstnonad PDF

F STATOR INJECTION

Table F Inlet and exit ow prop erties for reacting owcase

stator less injection

m kgm Sim typ e m m P MPa P MPa T K T K

inl et outl et inl et outl et

inl et outl et

Unstad LAM

Unstad ED

Unstad PDF

UnstnonadLAM

UnstnonadED

Unstnonad PDF

F STATOR INJECTION

Table F Inlet and exit ow prop erties for reacting owcase

stator leps injection

Sim typ e m m P MPa P MPa T K T K m kgm

inl et outl et inl et outl et

inl et outl et

Unstad LAM

Unstad ED

Unstad PDF

UnstnonadLAM

UnstnonadED

Unstnonad PDF

Table F Inlet and exit ow prop erties for reacting owcase

stator tess injection

Sim typ e m m P MPa P MPa T K T K m kgm

inl et outl et inl et outl et

inl et outl et

Unstad LAM

Unstad ED

Unstad PDF

UnstnonadLAM

UnstnonadED

Unstnonad PDF

Table F Inlet and exit ow prop erties for reacting owcase

stator teps injection

Sim typ e m m P MPa P MPa T T K K m kgm

inl et outl et inl et outl et

inl et outl et

Unstad LAM

Unstad ED

Unstad PDF

UnstnonadLAM

UnstnonadED

Unstnonad PDF

APPENDIX F SIMULATION DATA

F OTHER QUANTITIES OF INTEREST

F ROTOR

Table F Injection rates p eak temp eratures etc rotor less

injection

Sim typ e T K

fuel peak

Unstad LAM

Unstad ED

Unstad PDF

UnstnonadLAM

UnstnonadED

Unstnonad PDF

Table F Injection rates p eak temp eratures etc rotor mid

injection

Sim typ e T K

peak fuel

Unstad LAM

Unstad ED

Unstad PDF

UnstnonadLAM

UnstnonadED

Unstnonad PDF

Table F Injection rates peak temp eratures etc rotor

leps injection

Sim typ e T K

fuel peak

Unstad LAM

Unstad ED

Unstad PDF

UnstnonadLAM

UnstnonadED

Unstnonad PDF

F OTHER QUANTITIES OF INTEREST

F STATOR

Table F Injection rates peak temp eratures etc stator

less injection

Sim typ e T K

fuel peak

Unstad LAM

Unstad ED

Unstad PDF

UnstnonadLAM

UnstnonadED

Unstnonad PDF

Table F Injection rates peak temp eratures etc stator

leps injection

Sim typ e T K

fuel peak

Unstad LAM

Unstad ED

Unstad PDF

UnstnonadLAM

UnstnonadED

Unstnonad PDF

Table F Injection rates peak temp eratures etc stator

tess injection

Sim typ e T K

fuel peak

Unstad LAM

Unstad ED

Unstad PDF

UnstnonadLAM

UnstnonadED

Unstnonad PDF

APPENDIX F SIMULATION DATA

Table F Injection rates peak temp eratures etc stator

teps injection

Sim typ e T K

peak fuel

Unstad LAM

Unstad ED

Unstad PDF

UnstnonadLAM

UnstnonadED

Unstnonad PDF

F REDUCED INJECTION RATE CASE

Table F Inlet and exit ow prop erties for reacting ow

case reduced midinjection

K K Sim typ e m m P MPa P MPa T T m kgm

inl et outl et inl et outl et

inl et outl et

Unstad LAM

Unstnonad LAM

Table F Injection rates p eak temp eratures etc reduced

midinjection

Sim typ e

fuel

Unstad LAM

Unstonoad LAM

F INCREASED PRESSURE RATIO CASE

Table F Inlet and exit ow prop erties for reacting ow

case high pressure midinjection

Sim typ e m m P MPa P MPa T K T K m kgm

inl et outl et inl et outl et

inl et outl et

Unstad LAM

F INCREASED PRESSURE RATIO CASE

Table F Injection rates peak temp eratures etc high

pressure midinjection

Sim typ e F kN

CO y

fuel

Unstad LAM

Bibliography

SP Lukachko DR Kirk IA Wailt Turbine Durability Impact of High FuelAir

Ratio Combustors Part IPotential for IntraTurbine Oxidation of Partial ly Reacted

Fuel ASMETurb o Exp o GT

R Andriani F Gamma U Ghezzi and E Infante Design Proposals for Constant

Temperature Turbine Engine for Propulsion System AIAA

KNR Ramohalli Isothermal Combustion for Improved Eciencies AIAA

F Liu and WA Sirignano Turbojet and Turbofan Engine Performance Incre ases

Through Turbine Burners AIAA pages

WA Sirignano JP Delplanque and FLiu Selected Chal lenges in Jet and Rocket

Engine Combustion Research AIAA Pages

S Nasir AK Agrawal I McGregor and L Tuchinskiy Characteristics of Conned

Diusion Flames Stabilization over Porous Plates in a Flow Channel

AK Agrawal and Shakeel Nasir Noncatalytic Porous Combustion for Turbine

Burner Applications

TA Rohmat H Katoh T Obara T Yoshihashi and S Ohyagi Diusion Flame

Stabilized on a Porous Plate in a Paral lel Airstream AIAA Vol No Nov

Pages

A Ramachandra BN Raghunandan Investigation on the Stability and Extinction of

a Laminar Diusion Flame Over a Porous Flat Plate

MG Owens S Tehranian C Segal and VA Vinogradov FlameHolding Congu

rations for Kerosene Combustion in a Mach Airow Journal of Propulsion and

Power Vol No JulyAugust Pages

BIBLIOGRAPHY

RW Schefer N Namazian and J Kelly Velocity Measurements in a Turbulent Non

premixed BluBody Stabilized Flame AIAA

JR Roy BluBody Stabilized Flame Calculation with the Two Equation Tur

bulenceModel AIAA

MS Anand SB Pop e and HC Mongia PDF Calculations for Swirling FlowsAIAA

CL Hackert JL Ellzey OA Ezekoye Combustion and Heat Transfer in Model

TwoDimensional Porous BurnersCombustion and Flame

J Lee and D Fricher A Numerical Study of Reacting ow inside Combustors using

a Twoequation model of Turbulence and an Eddy dissipation model for Turbulent

Chemistry AIAA

K Chen and J Shuen ACoupled MultiBlock Solution Procedure for Spray Combus

tion in Complex Geometries AIAA

JL Xia G Yadigaroglu YS Liu J Schmidli and BL Smith Numerical and Ex

perimental Study of Swirling Flow in a Model CombustorInt J Heat Mass Transfer

Vol No Pages

J Zelina J Ehret RD Hancak UltraCompact Combustor Technology Using High

Swirl for Enhanced Burning Rate AIAA

GD Lewis CentrifugalForceEects on Combustion Symp osium International

on Combustion The Combustion Institute pp

Metghalchi and Keck Burning Velocities of Mixtures of Air with Methanol Isooctane

and Indolene at High Pressures Temperatures Combustion and Flame

VITA

Matthew Rice b orn in Portland Oregon to Thomas and Phyllis Rice received his BA

in Economics from Reed College Sp ending a brief time at Portland State as a physics

undergraduate and then at Columbia UniversityMr Rice joined the Virgina Tech

community in His activites during that time included internships at NASA Glenn

Research Center and Pratt Whitney as well as an instructorship at Virginia Tech

After completing an internship at Pratt Whitney Mr Rice plans to b egin a PhD

program at Kings College London at the Exp erimental and Computational Lab oratory

for the Analysis of Turbulence in the fall of