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Heat Transfer Analysis for Preliminary Design of GasTurbine Combustion Chamber Liners

by

Nikhil Sharma

A thesis submitted in conformity with the requirementsfor the degree of Masters of Applied Sciences

Graduate Department of Aerospace EngineeringUniversity of Toronto

c© Copyright 2015 by Nikhil Sharma

Abstract

Heat Transfer Analysis for Preliminary Design of Gas Turbine Combustion Chamber

Liners

Nikhil Sharma

Masters of Applied Sciences

Graduate Department of Aerospace Engineering

University of Toronto

2015

The objective of this thesis was to assess numerical techniques that can be utilized to

predict gas turbine combustion chamber liner temperature in preliminary design. There

are three main aspects of liner temperature prediction that were explored: (1) hot gas

temperature prediction; (2) radiation modelling; and (3) cooling technology modelling.

Reactor networks, zonal method along with simple one/two-dimensional models were

picked for the these three factors. Preliminary tests for zonal method very low run time

with results showing accurate trends. Reactor Networks provided valid trends for com-

bustor outlet temperatures; further validation would be required to assess its capability

to predict local temperature. One/two-dimensional models for cooling technologies were

tested potential benefits over empirical correlations were discussed. Additional validation

would be required for all the sub modules to be integrated into a larger Preliminary Mul-

tidisciplinary Design Optimization (PMDO) tool for gas turbine combustion chambers.

ii

Dedication

To my family and friends

iii

Acknowledgements

I am thankful to my supervisor Professor Sam Sampath for giving me the opportunity to

work on this project and for his guidance and support throughout its execution. I am also

thankful to Professor Clinton Groth, Professor Adam Steinberg, Professor Omer Gulder,

Professor Gottleib, Sri Sreekanth (PWC) and Haley Ozem (PWC) for their comments

and input in the project.

I am thankful to my colleagues at UTIAS, and to my friends and family for their

constant moral support.

Nikhil Sharma

University of Toronto Institute for Aerospace Studies

September 30, 2015

iv

Contents

List of Figures viii

List of Symbols xii

1 Introduction 1

1.1 Gas Turbine Combustor Design . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Combustor Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Thesis Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Background 6

2.1 Cooling Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Film Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2 Double Walled Cooling . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.3 Effusion and Transpiration . . . . . . . . . . . . . . . . . . . . . . 10

2.1.4 Thermal Barrier Coatings . . . . . . . . . . . . . . . . . . . . . . 12

2.1.5 Liner Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Combustor Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Heat Transfer Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Liner Durability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Literature Review and Modelling Strategy 20

4 Radiation Modelling 24

4.1 Zonal Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2 Radiative Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3.1 Direct Exchange Area Calculation . . . . . . . . . . . . . . . . . . 28

4.3.2 Grid Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

v

4.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.4.1 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.4.2 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.4.3 Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5 Combustion Modelling 40

5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2 Cantera Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6 Cooling Technology Modelling 47

6.1 Film Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.2 Double Walled Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.3 Effusion and Transpiration Cooling . . . . . . . . . . . . . . . . . . . . . 53

7 Conclusions 57

7.1 Thesis Accomplishments . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Bibliography 63

vi

List of Figures

1.1 Pressure ratio and outlet temperature trend for large gas turbine combustors 2

1.2 Inlet temperature and aspect ratio trend for gas turbine combustors . . . 3

1.3 Size comparison of an older and a modern combustor . . . . . . . . . . . 4

2.1 Combustor configurations . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Axial, reverse flow combustor layout and general airflow in combustor

primary zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Traditional cooling technologies . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Tiled cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Transpiration and effusion cooling . . . . . . . . . . . . . . . . . . . . . . 11

2.6 Combustor wall temperature effect through variation in inlet pressure,

inlet temperature, and air mass flow rate . . . . . . . . . . . . . . . . . . 14

2.7 One dimensional heat transfer on combustor liners . . . . . . . . . . . . . 15

4.1 Surface to surface interaction . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2 Approximation used to calculate transmissivity between major grid elements 29

4.3 Effect of integration step size on transmissivity calculation . . . . . . . . 30

4.4 Effect of grid density on calculation accuracy of gas to gas and surface to

surface direct exchange area . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.5 Geometry layout for direction vector used in direct exchange area verification 32

4.6 Comparison of discretization schemes that can be used for a cylindrical

geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.7 Cylindrical geometry to represent a combustor [46]. . . . . . . . . . . . . 33

4.8 Effect of zonal parameters on the net radiative heat flux . . . . . . . . . 34

4.9 Axial temperature, soot profiles and normalized radial profiles used to

predict . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.10 Effect of zonal parameters on the net radiative heat flux . . . . . . . . . 36

4.11 Effect of radial soot profile variation and constant radial temperature and

soot profile on net radiative heat flux on the wall . . . . . . . . . . . . . 37

vii

4.12 Measured vs predicted results for incident radiative heat flux . . . . . . . 38

4.13 A course discretization grid of primary and secondary zone of a reverse

flow combustor for zonal method implementation . . . . . . . . . . . . . 39

4.14 Combustor cross section schematic with eight sectors . . . . . . . . . . . 39

5.1 Engine A and Engine B profile schematics for comparison . . . . . . . . . 43

5.2 Engine A combustor outlet temperature; measured and predicted results 44

5.3 Engine A reactor network . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.4 Effect of mechanisms and reactor network setup on combustor outlet tem-

perature for Engine B . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.5 Network configurations for Engine B . . . . . . . . . . . . . . . . . . . . 46

5.6 Effect of reactor network setup on combustor NOx emission index (EINOx) 46

6.1 Schematic representing Counter Flow Film Cooling (CFFC) . . . . . . . 49

6.2 Temperature profiles predicted for panels of various lengths for (a) CFFC

and (b) PFFC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.3 Effect of number of nodes used in conduction solver on the overall tem-

perature profile for CFFC . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6.5 Predicted results from the effusion model for the wall temperature and the

effect of blowing angle on cooling effectiveness . . . . . . . . . . . . . . . 55

6.6 Parametric study for effusion cooling. . . . . . . . . . . . . . . . . . . . . 56

6.7 Predicted results from transpiration model for wall temperature distribu-

tion with and without a ceramic coating. . . . . . . . . . . . . . . . . . . 56

viii

List of Symbols

Variables

A Area (m2)

Cp Heat capacity (constant pressure) (J/(m ∗K))

D Diameter (m)

Eb Black body radiation emission (W/m2)

F View factor

G Incident radiation on a gaseous element (W )

gg Gas to gas direct exchange area (m2)

H Incident radiation on a surface element (W )

h Convective heat transfer coefficient (W/(m2 ∗K))

i Enthalpy (J)

J Radiosity (W )

j Chilton-Colburn j factor

L Luminosity factor

Lmb Mean beam length (m)

k Thermal conductivity (W/(m ∗K))

M Mass flux ratio

m Mass flow rate (Kg/s)

m mass (Kg)

Nu Nusselt number

P Pressure (Pa)

Pr Prandtl number

Q Heat transfer rate (W )

q Heat flux rate (W/(m2))

Ru Universal gas constant (J/(mol ∗K))

Re Reynolds number

S Path length used in Zonal method (m)

St Stanton number

ix

ss Surface to surface direct exchange area (m2)

sg Surface to gas direct exchange area (m2)

s Cooling slot height (m)

T Temperature (K)

t Lip thickness (m)

tr Residence time (s)

V Volume (m3)

x Axial distance on liner wall (m)

Y Mass fraction

z Normal distance through liner wall (m)

Greek

α Absorptivity

ε Emissivity

κ Absorption Coefficient

ρ Density (Kg/(m3))

σ Stefan Boltzmann constant (W/(m2K4))

τ Transmissivity

η Cooling film effectiveness

ν Velocity (m/s)

µ Dynamic Viscosity (Kg/(m ∗ s))ω Chemical production rate (mol/s)

Subscripts

a Air

an Combustor annulus

c Coolant

ca Combustor Casing

h1 Effusion hole exit

h2 Effusion hole entrance

hg Hot gas

i i species

in Inlet condition

l Liner

out Outlet condition

p Potential core

x

s Slot

w Wall

wad Adiabatic wall condition

w1 Hot side wall

w2 Cool side wall

Acronyms

AFR Air Fuel Ratio

CFD Computational Fluid Dynamics

CFFC Counter Flow Film Cooling

CO Carbon Monoxide

DEA Direct Exchange Area

DOM Discrete Ordinates Method

DTM Discrete Transfer Method

FAR Fuel Air Ratio

LCF Low Cycle Fatigue

MDO Multidisciplinary Design

MW Molar Weight

NOx Nitrogen Oxides

PFFC Parallel Flow Film Cooling

PFR Plug Flow Reactor

PMDO Preliminary Multidisciplinary Design

PSR Perfectly Stirred Reactor

TBC Thermal Barrier Coating

UHC Unburned Hydrocarbons

WSR Well Stirred Reactor

xi

Chapter 1

Introduction

1.1 Gas Turbine Combustor Design

Gas turbine engine design involves multiple disciplines, is highly complex and is iterative

in nature. Gas turbine design involves three main stages which are: conceptual design,

preliminary design, and detailed design stage. Success of an engine can be defined by the

conceptual design since decisions made early on in the project have a significant impact

on the project. Traditionally, an engineering group utilizes empirical correlations or low

fidelity one dimensional models to complete the conceptual design phase, followed by

specialized advanced design groups responsible for bringing the concept to production.

This results in end-loaded designs where information and effort is added towards the end

of the design cycle; changes made at the end of the design cycle are time consuming

and expensive [54]. In order to make the design process more efficient modern practices

involve more information being added early on in the conceptual stage itself.

Using empirical correlations in the conceptual stage can be disadvantageous as it may

lead to designs that are conservative in nature and not optimal or competitive. In order

to add information in the conceptual design stage, advanced tools are required that can

either provide higher accuracy or greater exploration of the design space. However, since

the design is in flux in the conceptual design stage the design optimization and decision

making have to have a quick turnaround time. Major work has been done in aerospace

industry to utilize automated computer frameworks that can incorporate design tools

from various disciplines and provide an optimized solution based on the object. Such

Multi-Disciplinary Optimization Tools (MDO) are discussed by Panchenko et al. [54].

Multiple engineering disciplines govern the design of each turbo-machinery compo-

nents in a gas turbine engine. These disciplines include but are not limited to: aero-

dynamics, dynamics, structures, heat transfer, materials, manufacturing, and acoustics.

1

2 Chapter 1. Introduction

1940 1950 1960 1970 1980 1990 2000 2010Year

0

10

20

30

40

50

60

Pre

ssu

reR

atio

(a)

1940 1950 1960 1970 1980 1990 2000 2010Year

1.00

1.25

1.50

1.75

2.00

Nor

mal

ized

Ou

tlet

Tem

per

atu

re

(b)

Figure 1.1: (a) Pressure ratio and (b) outlet temperature trend for large gas turbinecombustors [20]

The combustion chamber of the engine might not have any moving components but the

design and modelling is nonetheless complicated due to the combustion process. Fur-

thermore, there are five major conflicting design requirements that are critical for the

combustion chamber; these are: (1) emission control; (2) liner durability; (3) flame sta-

bility; (4) altitude relight capability; and (5) exit temperature profile requirement. To

prevent combustion liner from reaching critical temperature cooling air is usually added

along the liner wall as insulation. This cooling air quenches the combustion products,

prevents complete combustion resulting in higher carbon m onoxide (CO) emissions and

smoke emissions. Higher cooling flows also reduce the air temperature near the wall,

increasing the difference in peak and average exit temperature and making the exit tem-

perature profile worse. Hence, significant effort is required to make the combustor more

durable while meeting all other operational requirements [39].

1.2 Combustor Cooling

The problem of durability (and consequently of cooling) is even greater for small gas

turbine combustion chambers. Combustor volume scales with total flow level due to

engine starting requirements, whereas, combustor cooling requirement would scale with

total surface area (to maintain the same temperature the cooling air used per area must

remain the same). If the combustor size is scaled, keeping the shape proportionately the

same, then the combustor volume would scale to the cube of characteristic length and

the surface area to the square of the characteristic length. This leads to a higher surface

1.2. Combustor Cooling 3

1960 1970 1980 1990Year

1.0

1.1

1.2

1.3

1.4

1.5

1.6

Nor

mal

ized

Inle

tT

emp

erat

ure

All SubsonicNon-RecuperatedEngines

Up to 5000 Hp7000 lb ThrustEngines

(a)

1960 1970 1980 1990Year

0

1

2

3

4

5

6

Com

bu

stor

Len

gth

To

Inje

ctor

Sp

acin

gR

atio

(L/P

) All Engines

Up to 5000 Hp7000 lb ThrustEngines

(b)

Figure 1.2: (a) Inlet temperature and (b) aspect ratio trend for gas turbine combustors[9]

area to volume ratio for smaller combustor. The surface area to volume ratio for large

combustors may range from 5 to 10 ft−1 meanwhile it may range from 1 to 5 ft−1 for

small combustors [39]. This in turn requires that a larger portion of total air has to be

used to maintain certain liner temperature. Although for smaller gas turbine engines the

combustor operating point temperature is lower as compared to larger engines, this might

not be enough to offset higher percentage of cooling that is required [9]. Furthermore,

lower cooling flows would require smaller orifices or cooling holes in the liner. However,

the tolerances of machining do not scale down with size. Therefore, the tolerance levels

would cause greater temperature variation in a smaller combustor than a larger combustor

[9].

Higher customer requirements for engine durability and power output also require

the combustor engineer to further optimize the cooling flow. Gas turbine manufacturers

have been increasing pressure ratio for the past 50 years to increase the power output of

the engine. Increase in pressure ratio is accompanied by an increase in the combustor

inlet air temperature (usually termed T3) which lowers the heat sink capability of the

air and increases the combustion temperature. Both leading to higher heat load on the

combustor liner. The general industry trends for pressure ratio and combustor outlet

temperatures (usually termed T4) are shown in Figs. 1.1(a) and 1.1(b) respectively [20].

The data presented in Fig. 1.1 is for large gas turbines but the overall trend is followed

by smaller engines as well. Manufacturers have also reduced the combustor size to lower

the emission output and to shorten the overall engine. The general industry trends for

combustor inlet temperature and combustor size in shown in Figs. 1.2(a) and 1.2(b).

4 Chapter 1. Introduction

Introduced in

1970's

Modern

combustor

Figure 1.3: Size comparison of an older and a modern combustor [39]

The data presented in Fig. 1.2 is for General Electric combustors but these trends are

representative of the entire industry. The drastic change in combustor size can also be

seen by an overlay of an older combustor introduced in 1970s and a modern combustor

shown in Fig. 1.3. Furthermore, typical combustor life requirements before 1990s would

be about 5000 mission hours, whereas, for recent applications the requirement for mission

hours per overhaul is in the range of 6000 to 12,000 hours [7, 39].

1.3 Thesis Objective

In light of higher customer requirements and shorter design cycles, use of numerical tools

for analysis and MDO becomes essential in the design of gas turbine engine. Therefore,

the objective of this project is to explore heat transfer analysis tools that can predict

gas turbine liner temperature. A liner temperature prediction tool can be sub-divided

into three modules: (1) hot gas radiation prediction; (2) hot gas temperature prediction;

and (3) cooling technology modelling. Integration of these modules in larger PMDO tool

could be possible, hence, factors effecting this integration would be taken into account

in this thesis.

1.4 Thesis Outline

Following this introductory chapter, Chapter 2 of the thesis describes general gas turbine

combustor geometrical features and various layouts used in the industry. It also describes

common cooling techniques used to protect the liner, factors effecting the liner temper-

1.4. Thesis Outline 5

ature, and a general strategy used to predict the liner temperature. The chapter ends

with a brief discussion of durability issues with aviation gas turbine combustors. Chap-

ter 3 focuses on previous work that has been done to model a combustor using high and

low fidelity tools and the various levels of success they have achieved. Most of the work

discussed focuses on liner temperature prediction or combustor life prediction, however,

emission prediction techniques are also discussed as these strategies can be used in a tool

to predict liner temperature. Liner temperature prediction is dependent on modelling

of (1) combustion (for hot gas temperature prediction); (2) radiation; and (3) cooling

technology; Chapter 3 outlines the reasoning behind specific strategies that were selected

for each of these sub modules in the context of the literature. Details of these models

are discussed in Chapters 5, 4, and 6, respectively. Chapter 7 discuss the conclusions

drawn based on the results and future work that can be done to improve the predictions

by these proposed sub modules.

Chapter 2

Background

Overall combustor layout is largely dependent on the engine type and performance re-

quirements. In terms of the cross section of the combustor three main configuration

types have been used in the industry as shown in Fig. 2.1. ‘Can’ combustors consist

of cylindrical liners concentrically placed inside a cylindrical casing. They are relatively

inexpensive to develop and maintain; however, they generally weigh more than annular

counterparts and are seldom used in modern aviation engines [20]. An annular com-

bustor consists of an annular liner located inside an annular casing. These combustors

tend to weigh less and are aerodynamically clear resulting in less pressure losses [20].

Tuboannular configuration consists of combustor cans with an annular outer casing, hav-

ing some advantages from both configurations discussed above. The general industry

trend for aircraft engines after 1960s has been to use annular combustors. In terms of

heat transfer there are two major differences in cylindrical and annular liners. First, a

cylindrical liner allows for simple cylindrical correlations to be used for convective heat

transfer calculations. Second, in cylindrical liners radiation in a sector is contained and

does not propagate to adjacent sectors; this is not true for annular combustors. Details

of combustor heat transfer and a common technique to model is discussed in Sections 2.2

and 2.3 respectively.

There are two main axial combustor layouts used in the industry, straight through and

reverse flow combustors. These configuration layouts are shown in Fig. 2.2. Advantages

of straight through annular combustor are that it provides a compact combustor geom-

etry and low surface area to volume ratio which is beneficial for cooling requirements;

disadvantages include longer engine length and higher sensitivity to diffuser operation.

On smaller engines higher shaft speed requires close coupling of the compressor and the

turbines due to whirling issues [20]. Reverse flow combustors allow for shorter shaft

lengths, are compact for engines with centrifugal compressors and reduce overall engine

6

2.1. Cooling Technology 7

Figure 2.1: Combustor configurations [20]

length. Hence, this configuration is heavily used in smaller aviation engines. In terms

of heat transfer however, reverse flow configuration requires larger cooling flow due to

larger surface area when compared to straight-through configuration.

2.1 Cooling Technology

There are various configurations that are used in the industry to cool a gas turbine

combustor liner. Here a brief description is provided for three main categories of these

configuration, these are (1) film cooling; (2) double walled cooling; and (3) effusion/

transpiration cooling. Film cooling has been traditionally employed on combustors of

various sizes due to their manufacturing simplicity. Double walled cooling requiring

additional tiles, hence, are more suitable for larger combustors. Effusion cooling requires

large number of holes to be drilled on the liner, and has been made possible due to

modern manufacturing techniques that lower the cost of such an operation.

2.1.1 Film Cooling

Film cooling refers to cooling concepts that employ a cold film of air to insulate the

liner from hot gases. The film gets destroyed downstream of the slot due to turbulence,

therefore, usually the slots are provided in 40-80 mm intervals. Some commonly used

film cooling devices are wiggle strips, stacked rings, splash-cooling, and machined rings

[20].

8 Chapter 2. Background

(a) (b)

(c)

Figure 2.2: (a) Axial combustor layout, (b) reverse flow combustor layout and (b) generalairflow in combustor primary zone [20]

Wiggle strip configuration utilizes the total pressure available in the cooling air (rather

than only the static pressure in the case of splash cooling). The combustor liner consists

of separate sections with annular height difference between them. A corrugated metal

strip is used to connect the various sections, as can be seen in Fig. 2.3(b). Wiggle

strip configuration can be very sensitive to small manufacturing variances in the strip’s

thickness [20].

Stacked ring shown in Fig. 2.3(c), also uses the total coolant pressure available to

form a film. In this configuration, consecutive liner sections which vary in annular height

are joined via a metal plate. The metal plate contains holes that are drilled with high

precision. The plenum to the aft of the strip is required for the individual jets to coalesce

into a sheet. Although being less structurally sound than wiggle strip configuration,

the dimensional accuracy of the holes is higher in stacked ring configuration, therefore,

providing less variation in cooling air flow rate [20].

Splash cooling only utilizes the static pressure drop across the liner to form the cooling

film; the configuration is shown in Fig. 2.3(a). Holes on the combustor liner bleed air

and a lip redirects the cooling air in the required direction. Typical lip length is about

3-4 times the slot height which is on the order of 1.5-3 mm [20].


(a) (b)

(c) (d)

Figure 2.3: Traditional cooling technologies are: (a) splash cooling ring; (b) wiggle strip;(c) stacked ring; and (d) machined ring [20]

The machined ring configuration avoids the braze joint required in stacked ring and

therefore, the entire step section is machined from solid metal. This configuration is

shown in Fig. 2.3(d). Splash cooling (Louvre cooling) or any other cooling configuration

requiring a lip is most likely to crack at the tip of the lip. This is due to high temperature

gradients at this location; the film provided by previous slot deteriorates at this point

while incoming cooling air is almost at combustor inlet temperature. To avoid this

drawback, the holes can be drilled with smaller diameter, resulting in thinner cooling

jets which require less distance to coalesce into a sheet. A configuration that utilizes a

number of small holes without any lip is the Z ring configuration [20].


CoolAir

HotGas

Tile

(a)

SlotCoolingFins

Coolantfilm

HotTile

CoolantDirection

(b)

Figure 2.4: Tiled cooling: (a) Impingement with tiles and (b) Counter Flow Film Cooling(CFFC)

2.1.2 Double Walled Cooling

Double walled cooling refers to a class of cooling schemes where cooling air flows through

a passage between two surfaces before exiting and forming a film on the surface of the

liner. Various configurations are possible depending on the direction of the cooling air

in relation to the direction of the film that is formed on the top surface. In Counter

Flow Film Cooling (CFFC), the coolant air flows in the opposite direction of the film as

shown in Fig. 2.4(b), whereas in Parallel Flow Film Cooling (PFFC) it flows in the same

direction. If impingement is used to further enhance the heat transfer to the coolant,

then the configuration shown in Fig. 2.4(a) is achieved. These configurations allow for

the coolant to act as a heat sink and pick up heat before it forms a layer on top, hence,

improving the overall cooling effectiveness and reducing the total amount of cooling air

used as compared to traditional cooling louvres. To further enhance the convective heat

transfer to the coolant, various finned configurations can be placed in the passage between

the two panels as shown in Fig. 2.4(b).

2.1.3 Effusion and Transpiration

With Effusion cooling large number of small discrete holes are drilled directly on the

liner as shown in Fig. 2.5(b). Furthermore, these holes are drilled at an angle to provide

a twofold advantage in terms of cooling: (1) angled holes provide higher surface area

for heat transfer to occur between the liner and the coolant and (2) shallow hole angles

result in jets that are less likely to penetrate the hot gas and hence, are better at forming

a film downstream from the hole. Advancements in laser drilling have made effusion


(a)

CoolantDirection

(b)

Figure 2.5: (a)Transpiration and (b) effusion cooling [20]

cooling a viable technology for aircraft combustors. Hole diameters can be upwards

of 0.4 mm whereas the lowest angle that is attainable is around 20◦ [20]. Drawbacks of

effusion cooling include the increase weight of the combustor, this is mainly from increased

thickness that is required to provide buckling strength and higher cost associated with

drilling of high number of holes. The cooling effectiveness of this configuration can be

improved with holes drilled with diffuser shaped expansion at the exit portion, however,

this would further increase the cost of manufacturing.

Several geometrical factors effect the cooling effectiveness of effusion cooling; these

are briefly discussed here. Cooling effectiveness increases with increased hole size, mainly

due to the decrease in coolant velocity [17]. Lower coolant velocity would lead to lower

turbulence and hence, decrease the amount of mixing of hot gases with the coolant.

Effectiveness can also be increased by using a diffuser shaped cooling hole outlet. This

would increase the lateral spreading of the coolant and prevent coolant jet blow-off [17,

53].

The result of increasing span-wise or stream-wise distance between the holes is that

the overall effectiveness goes down [17]. Higher span-wise pitch prevents the coolant jets

to coalesce and higher stream-wise pitch increases the area that a single film has to cool.

Finally, increasing the blowing/ inclination angle results in lowering the overall ef-

fectiveness. With lower inclination angle the coolant passage in the liner increases in

length resulting in higher convective cooling and it also results in better attachment of

coolant film to the hot surface [17]. However, with increase in blowing ratio the effect of

inclination angle is decreased [1]. Transpiration cooling is defined as a process whereby

a fluid transpires through a porous medium and hence, the temperature of the medium

and the coolant is equal at the exit. Large internal area of a porous wall would facilitate

large amount of heat removal required for this configuration. Emerging jets would coa-


lesce to form a protective film over the surface similar to effusion cooling configuration.

Effusion cooling would approach the behaviour of transpiration cooling as the hole sizes

reduce. Transpiration cooling ideally enables a uniform temperature to be maintained

on the entire surface where the control of the temperature relies on cooling air flow rate.

This method would be very efficient in terms of cooling air requirements, however,

extremely small hole sizes or air passages in the porous wall increase the chances of

blockage due to external debris and oxidation. Due to these issues with porous materials

manufacturers have opted to use multi-laminated sheets that provide quasi-transpiration

configuration. High manufacturing cost and lack of mechanical strength make this con-

figuration impractical for application in current combustors.

2.1.4 Thermal Barrier Coatings

Thermal Barrier Coatings (TBC) are ceramic materials that can be applied to the inside

of the combustor liner. TBC material is usually low in emissivity and thermal conductiv-

ity, thus, reflecting large portion of the incident radiation and forming a thermal insulator

between the liner and hot combustion gases. A typical TBC has a metallic base coat and

one or two layers of ceramic coatings. Oxidation resistant base coat can also be applied

to prevent oxidation/ corrosion damage of the liner. Overall the coating thickness can be

around 0.4-0.5 mm which can give temperature reduction on the order of 40-70 K [20].

2.1.5 Liner Materials

New super-alloys are being developed for modern combustor liners, however, metals in-

herently have limitations for extremely high temperature applications. Monolithic ce-

ramic materials on the other hand, can operate on much higher temperature and provide

substantial weight benefits. Some disadvantages of ceramic materials are that they are

brittle, prone to attack by hot combustion gases and expensive to manufacture. One

method of avoiding catastrophic failure mode due to brittle nature of monolithic ceram-

ics is to introduce particles or whiskers that deflect and arrest cracks.

2.2 Combustor Heat Transfer

Common gas turbine combustor features and the complex airflow that can be expected

in the combustion chamber is shown in Fig. 2.2. In modern gas turbine combustors gases

can have temperatures peaking over 2100 ◦C, while nickel and cobalt based alloys that

are used for combustor liner material cannot operate above 1373 ◦C [20]. As a result,

2.2. Combustor Heat Transfer 13

various methods are employed to prevent the transfer of heat to the liner and effective

removal of heat from the liner as discussed in Section 2.1.

In general, the liner is heated by radiation and convection from hot gases inside

it and is cooled by radiation to outer casing and convective heat transfer to annulus

air. Thermal gradients produced by hot gases are an order of magnitude larger than

the thermal gradients in the liner due to conduction, therefore, a common practice in

conducting heat transfer calculation is to ignore liner conduction in the axial direction.

Internal radiation forms a major portion of total heat transferred from combustion

gases to the liner. Areas where cooling films form an effective barrier between the com-

bustion gases and the liner, radiation is the only means by which heat can be transferred.

Radiation from gas turbine fuels consists of luminous and non-luminous component. Non-

luminous radiation emanates from heteropolar gases such as carbon dioxide and water

vapour, while luminous radiation generally depends on solid particles in the flame. The

production of soot particles increases at elevated pressures resulting in high amount of

heat transfer through luminous radiation [20].

Outside the liner, majority of the heat is transferred through convection to the annulus

air. Due to convective transfer, the temperature of air outside the liner progressively gets

higher as it traverses the combustion chamber. Radiation from the liner to the engine

casing is less prominent as compared to external convection and it depends on surface

temperatures and material properties of the liner and the engine casing.

Internal convective heat transfer in gas turbine combustion chambers is extremely dif-

ficult to predict since the hot gases are going under rapid physical and chemical changes.

Various local factors such as boundary layer formation or destruction can have significant

impact on convective heat transfer. Any mathematical model used to predict internal

convective heat transfer must take into account the combustor aerodynamics, which is

heavily influenced by geometric features such as inlet swirler, diffuser and cooling louvres.

The liner material in the primary zone (specifically the dome) is subjected to mixture

that has high concentration of fuel as it is in close proximity to the fuel nozzle. In the

case where the fuel sprayed by the nozzle impinges on the liner, local hot spot can be

created. Therefore, the dome section of the primary zone requires special attention by

the designer.

Fig. 2.6 shows experimental results obtained for liner temperature with variation in

three parameters that change with the operating conditions: (1) pressure; (2) inlet air

temperature; and (3) air mass flow rate [21]. These results were part of a paper written

by Lefebvre and Herbert [21] which employed a methodology that relied on empirical

correlations (this is discussed in detail in Section 2.3). The parameters shown were varied


0.80 1.12 1.44 1.76 2.08 2.40Inlet Pressure, ( P3

P3min)

0.8

1.0

1.2

1.4

1.6

Nor

mal

ized

Lin

erT

emp

erat

ure

(T

L

Tm

eas,

min)

Measured

Predicted

(a)

0.8 1.0 1.2 1.4 1.6 1.8 2.0Inlet Temperature, ( T3

T3min)

0.8

1.0

1.2

1.4

1.6

Nor

mal

ized

Lin

erT

emp

erat

ure

(T

L

Tm

eas,

min)

Measured

Predicted

(b)

0.5 1.5 2.5 3.5Inlet Air Mass Flow Rate, ( W3

W3min)

0.8

1.0

1.2

1.4

1.6

Nor

mal

ized

Lin

erT

emp

erat

ure

(T

L

Tm

eas,

min)

Measured

Predicted

(c)

Figure 2.6: Combustor wall temperature effect through variation in: (a) inlet pressure;(b) inlet temperature; and (c) air mass flow rate [21]

individually on a test combustor in a lab, whereas, during a flight these parameters would

change simultaneously.

Fig. 2.6(a) shows the effect of pressure on liner temperature. An increase in pressure

increases the emissivity in the flame and in turn increases the radiation heat transfer to

the liner, it also suppresses chemical dissociation, increasing the total heat release. Due to

both of these reasons liner temperature increases, following an exponential law, whereby

at high pressures an increase in pressure would not result in similar wall temperature rise

as compared to low pressures [21].

Fig. 2.6(b) shows the effect of inlet air temperature on liner temperature. An increase

in inlet temperature increases the flame temperature and total heat transferred to the

liner, while it also decreases the effectiveness of cooling air and reducing the amount of

heat removed from the liner. Due to these reasons the wall liner temperature increases

2.3. Heat Transfer Modelling 15

Figure 2.7: One dimensional heat transfer on combustor liners [20]

with an increase in inlet temperature [21].

Fig. 2.6(c) shows the effect of air mass flow rate on liner temperature. External

convection removes more heat from the liner when compared to external radiation, on

the contrary, internal convection and internal radiation both transport heat to the liner

from hot gases. As a result, increase in mass flow increases effectiveness of convective

heat transfer, but overall it increases the amount of heat removed from the liner lowering

the temperature [21].

2.3 Heat Transfer Modelling

A number of researchers [21, 36, 16, 35, 37, 45, 38, 3, 48] have used the empirical strat-

egy discussed in this section (or a variation that involves improved correlations). This

methodology also becomes a reference to compare the new proposed methodology that

would be discussed in later sections. Details of the methodology were taken from a paper

by Herbert and Lefebvre [21], which was the oldest source the author could find on the

issue.

If it is assumed that heat flux in constant in the axial and circumferential direction

of the liner, then the balance of the heat flux in the normal direction is given by

qr1 + qc1 = qr2 + qc2 = qk1−2. (2.1)

This balance is also shown schematically in Fig. 2.7. Here qc1 and qc2 refer to internal

and external convective heat flux respectively, while qr1 and qr2 refer to internal and

external radiative heat flux respectively. Eq. (2.1) shows this by equating the radiation

and convective heat flux inside the liner to the radiation and convective heat flux outside

the line and to the conduction through the liner. Convection heat flux is predicted


through Newton’s law of cooling as described by

qc1 = hhg(Thg − Tw1). (2.2)

Hot gas temperature (Thg) can be obtained from combustion rise charts assuming near

100% combustion efficiency in the gas turbine combustion chamber. Lefebvre assumed

flow inside the combustor to be highly turbulent and to be similar to flow inside a straight

circular duct [21]. Stanton number relation for this assumption is given by

St = 0.0283 Re−0.2. (2.3)

Using relationship between Stanton number, Reynolds number and Prandtl number the

following is obtained:

qc1 = 0.02khg

D0.2l

(mhg

Alµhg

)0.8

(Thg − Tw1) , (2.4)

where Dl is the liner diameter and Al is the liner cross sectional area.

To adjust this methodology for cooled liner the concept of film effectiveness was

introduced; film effectiveness is defined as

η =Thg − Twad

Thg − Tc

. (2.5)

The combsutor liner temperature (Tw1) in Eq. (2.4) is replaced with the adiabatic tem-

perature (Twad) obtained from the cooling effectiveness correlations. Hence, the problem

of convective heat flux prediction is broken into two components: (1) of estimating the

heat transfer coefficient; and (2) of estimating adiabatic wall temperature. The heat

transfer coefficient can be obtained from Nusselt number correlation given below:

Nu = 0.069(

Res

(xs

))0.7

. (2.6)

Eq. (2.6) is valid for a blowing ratio (defined as density, velocity ratio of coolant gas to

hot gas m = ρUa

ρUg) ranging from 0.5 to 1.3. Conceptually, the film issuing from a slot

is assumed to have three main regions: (1) Potential core; (2) Transition zone; and (3)

Fully turbulent region [20]. In case of a cooling louvre, important variables that the film

effectiveness depends on are slot thickness, slot depth, distance downstream of the slot,

blowing ratio and the slot Reynolds number.

For the calculation of external convection (liner to casing) similar analysis is done as

2.3. Heat Transfer Modelling 17

described above without considering cooling geometry. However, hydraulic mean diam-

eter is used instead of cross sectional diameter to take into account the annulus area.

Further assumption is made that the temperature of the cooling gas does not change

axially and is the same as inlet temperature (T3) to give

qc2 = 0.02kc

D0.2an

(mc

Aanµc

)0.8

(Tw2 − T3) , (2.7)

where all the gas dependent variables in the equation correspond to the cooling gas in

the annulus.

The Stefan Boltzmann law is used along with emissivity and absorptivity adjustments

to predict radiative heat flux from hot gases to the combustor liner through the following:

qr1 = σ(εhgT4hg − αhgT

4w1) (2.8)

Empirical correlation was employed to relate the hot gas absorptivity and emissivity

(αg

εg= ( Tg

Tw1)1.5), while a correction factor (0.5(1 + εw)) is added to correct for non black

emission from the liner surface resulting in

qr1 = 0.5σ(1 + εw)εhgT1.5hg (T 2.5

hg − T 2.5w1 ) (2.9)

Original equation describing hot gas emissivity for non luminous gases by Reeves was

adjusted to take luminosity into account to form

εhg = 1 − e(−290P L(FAR Lmb)0.5T−1.5hg ), (2.10)

here luminosity factor (L) which depends on fuel hydrocarbon content is defined as

L = 7.53(C/H − 5.5)0.84 (2.11)

and mean beam length can be obtained from

Lmb = 3.4Volume

Surface Area(2.12)

For the calculation of external radiation (liner to the casing) view factor analysis for

two surface enclosure was employed [47] to obtain

qr2Aw =σ(T 4

w2 − T 43 )

1−εwεwAw

+ 1AwF.wc

+ 1−εcεcAc

. (2.13)


Using a view factor of unity for long annular space, estimated emissivity values for steel,

and diameter ratio the following equation was obtained

qr2 = 0.6σ(T 4w2 − T 4

3 ) (2.14)

Since this technique does not require large input data it can be useful in preliminary

design. Good results for liner temperature prediction using this technique were shown

in Fig. 2.6 and discussed in the previous section; although the operating conditions for

these results were not as extreme as in modern combustors.

2.4 Liner Durability

The thrust and power settings of an aircraft engine varies during aircraft flight depending

on the aircraft mission, which heavily influences the internal loads that the engine is put

under. The gas turbine combustor is put under high dynamic loads due to combustion

processes. Some desirable properties for the materials used are high strength, creep

resistance, oxidation resistance, low thermal expansion, high thermal conductivity and

ease of welding. Generally, super-alloy metals are used with high contents of nickel and

chromium such as Hastelloy X and Inconel 617 [42].

The major failure modes of the combustor are creep, Low Cycle Fatigue (LCF) and

oxidation which either directly result in surface cracking or exacerbate the issue by weak-

ening the structure. The interaction between each of these failure modes can be complex

which makes life prediction or failure modelling very difficult [42].

Creep can be defined as permanent deformation of metal at elevated temperature

for extended periods of time. This can happen during aircraft cruise phase and with

stress levels well below the yield strength of the material. Creep can become an issue for

hot gas turbine components when homologous temperature (ratio of actual to melting

temperature) exceeds 0.5. During cruise phase of an aircraft hot components can be well

above this temperature. Two methods commonly used in the industry to predict creep

are Larson-Miller parameter and Theta-projection method [42].

Low cycle fatigue is associated with a low number of cycles to failure (between 10,000

and 100,000 cycles). Each flight of an aircraft can be considered a single LCF cycle; for

a combustor, start up and shut-down cause temperature changes resulting in thermal

stresses. LCF can be correlated with the inelastic strain range, however, inelastic strain

range can be hard to determine, therefore, LCF is also correlated with total strain range

[42].

2.4. Liner Durability 19

Creep and LCF interaction is specially important where the damage by both the

mechanisms is almost equal. Synergistic damage process caused by cyclic thermal and

mechanical loading is called Thermo-mechanical fatigue. Life time under such loadings

can be very different from what is obtained with isothermal LCF tests conducted at

maximum temperature where creep damage would be the maximum. Simple analytical

methods apply a linear accumulation law to life due to LCF damage in absence of creep

and life due to creep damage in absence of LCF to predict overall life [42].

Due to the oxygen content and high temperature of combustion gases, hot sections

of the engine are susceptible to oxidation and corrosion. Contaminants in the fuel and

air such as Sulphur and Vanadium can cause corrosion attack on high surface materials.

Alloys with higher content of chromium and aluminium tend to be more resistant to

oxidation and can be used as surface coatings to further protect the combustor liner

fromn oxidation[42, 30].

Chapter 3

Literature Review and Modelling

Strategy

Hot gas temperature prediction can be considered a preliminary step for combsutor liner

temperature prediction. The designer may have this information from previous designs

or can estimate this using combustion temperature rise charts (assuming certain com-

bustion efficiency as shown in Section 2.3. This technique was also used by Odgers and

Kretschmer [16], however, instead of using a single efficiency factor throughout the com-

bustor they divided the combustor in three main zones (primary, secondary and dilution

zones) and varied the efficiency and temperature in these zones. Odgers et al. [37] went

over the limitation of the liner temperature calculation approach outlined in Section 2.3

for smaller combustors and Gosselin et al. [38] improved the correlations for better re-

sults with small combustors. However, it is difficult to extrapolate the results form these

correlations and they might not be applicable to new combustor designs.

Another method to predict temperature, along with flow properties is to divide the

combustor in smaller sub flows, and model these with empirical pressure-drop/ flow rate

equations. The PhD thesis by Stuttaford [45] used such a method for modelling gas tur-

bine combustors for preliminary design analysis. The network of sub flows also employed

momentum and recirculation effects through empirical factors. Terms relating heat flow

between nodes were modelled as sources in the flow equations and the conjugate heat

transfer method utilizing equations discussed in Section 2.3 was used for liner tempera-

ture prediction. Results of the study showed agreement with one-dimensional industry

code used for combustor design and also with experimental data. A drawback of using

such a technique is the extensive use of empirical correlations for flow modelling. As

discussed earlier accurate data might not be available for newer combustors.

A methodology that can relate hot gas residence time (which is related to combustor

20

21

volume and mass flow rate of fuel and air), temperature and pressure to hot gas tem-

perature is reactor network analysis. Reactor networks make the assumption that the

turbulence is infinitely high and therefore, the temperature and species concentrations

are limited by the chemical reaction rates. The equations for mass, energy and species

concentration reduce to non linear algebraic equations which can be solved numerically

[49]. The flow is usually represented through interconnectedness of multiple reactors,

where each reactor solves the equations aforementioned.

Hammond and Mellor [14], and Rizk et al. [32] used reactor networks for emission

prediction, while Swithenbank et al. [13] and Sturgess and Shouse [44] used reactor

networks to predict flame stability (blow off limits). More recently, Marchand [27] and

Lanewala [18] used reactor networks to predict CO and NOx emissions respectively;

these projects used CFD flow information that was available to direct the layout of the

networks. Although liner temperature was not verified in these models, they demonstrate

the ability of reactor networks to predict species concentration (CO and NOx) with

accuracy. Bradshaw [3], in his study focused on impact of manufacturing variability

on combustor liner durability. Simplified models were used to link combustor life, liner

temperature variability and effects of manufacturing variability; probabilistic analysis

was then applied on these models to asses combustor life. A simple reactor network

with one reactor to model the entire combustor was used to obtain bulk gas temperature

based on various air fuel ratios and inlet temperatures. Heat transfer analysis done in

this study was similar to what has been discussed in Section 2.3.

Although none of the literature discussed above compared liner temperature with

predicted results from reactor networks, the technique has significant advantages for

preliminary design temperature prediction. Reactor networks do not require large number

of inputs, are scalable (as the networks can be made as complex as required) and can

potentially model emissions and blow out. Emission prediction and blow out are not the

focus of the current project, however, these capabilities might be used in a larger PMDO

tool in the future. Chapter 5 goes over the details of the implementation of reactor

networks for the current project.

The importance of radiation modelling for liner temperature prediction is highlighted

by Lefebvre [20]. Viskanta and Menguc show that radiation flux can be 30-50% of the

total heat flux on the liner [26]. Carvalho and Coelho [6] conducted high fidelity simula-

tion of a representative gas turbine combustor sector; radiative exchange was calculated

through Discrete Transfer Method (DTM). Percentage of radiative flux to total heat

flux for 5, 15, and 25 bar operating points was calculated to be 48%, 43%, and 44%

respectively. Although this work was based on numerical simulations, it highlighted the

22 Chapter 3. Literature Review and Modelling Strategy

importance of modelling radiation for combustor heat transfer. The results also revealed

higher radiative flux in the primary zone of the combustor.

Lefebvre [19], in his review on radiation in gas turbine combustors, discusses the

effect of spray atomization in axial radiative flux variation in a combustor. Radiation

in gas turbine combustors is found to be dominated by luminous radiation, which is

radiation due to soot particles. Atomizers control the spray characteristics and soot

forming regions. Since, majority of the fuel is concentrated near the atomizer (in the

primary zone), radiation is higher in the primary zone and drops towards the end of the

combustor. Axial variation in radiation heat flux can have an impact on the final cooling

configuration picked by the combustor designer. Empirical correlations that produce

axially average radiation flux might not provide enough detail required to select cooling

schemes at various axial positions of the combustor.

There are a number of papers in literature that discuss implementation of high fi-

delity tools for accurate radiation modelling. Details of these implementation, are not

discussed, since high fidelity modelling would be out of the scope of this thesis. However,

Stuttaford [46] in his paper, assesses the viability of using Discrete Ordinates Method

(DOM) for preliminary combustor design. Stuttaford shows validation of the method

through multiple test cases where comparison was done between measured thermal paint

data for a combustor and predicted results from DOM. However, the computational time

was not reported for these cases.

On the other hand, the zonal method is a widely used method to solve radiative

transfer for practical engineering problems and the computational time required for this

method is usually smaller than the time required by the alternative methods [50]. Disad-

vantages of the zonal method include: (1) high computational time if it is coupled with

high fidelity finite difference solvers that contain fine grids; (2) high computational time

if the geometry is complex; and (3) inability to treat non-gray, temperature dependent

radiative properties [50]. For preliminary design, an assumption was made that impact

of minute geometrical detail is not required for radiation prediction. Also, that hot gases

within the combustor are gray and the combustor liner is a gray emitter as well (this

is true if the liner is covered with soot deposit). Under these assumptions, the zonal

method is a viable tool for preliminary design and the details are discussed in Chapter 4.

Although, development of a complete PMDO optimization tool is outside the scope

of this thesis, various strategies to take heat transfer into account in such tools were

explored. Tietz and Behnredt [48] developed a software tool for preliminary design of

a gas turbine combustor. The objective of the tool was to optimize the amount of

cooling air used for a given metal temperature, minimize NOx emissions and maintain

23

combustor volume for stable combustion. Reactor modelling was used to predict hot gas

temperature in the combustor using a simple network while a more dense reactor network

was proposed to predict UHC and CO emissions in the future. In order to calculate heat

transfer through the combustor liner, a methodology similar to the one described in

Section 2.3 was used. The tool focused on modelling effusion cooling with further option

of expansion to other configurations in the future. Major advantage of the tool is the

detailed modelling of effusion cooling as it takes into account conduction through the

liner surface (normal to hole face) and radially outward from the hole, thus providing

a two-dimensional analysis. Since heat transfer modelling of the cooling configuration

and emission modelling both depend on local fuel to air ratio and air flow split, the

optimization was done on specific modules (contour design, cooling air and emissions)

and iterated over one by one until the solution converged.

Pegemanyfar and Pfitzner [40] developed a software tool for preliminary design of

gas turbine combustors as well, however, the emphasis was not on optimization but on

automated design process. Operating requirements such as flame stability (relight and

weak extinction limit) determined the volume of the combustion chamber and the local

fuel to air ratio. After the initial layout of the combustor was picked, the designer

would be able to pick a cooling configuration and iterate the design until local air to fuel

targets were met. Results of preliminary design were used to develop a parametric CAD

geometry. The CAD model was converted to a CFD grid using automatic grid generation

software, and a final CFD simulation was run.

From the two tools discussed above, it can be inferred that any optimization tool

would require an iteration between local temperature prediction, fuel and air mass flow

rate adjustment and cooling performance prediction. It is possible to achieve this by

using the Reactor network and zonal method. The details of modelling various cooling

strategies is discussed in Chapter 6

Chapter 4

Radiation Modelling

4.1 Zonal Method

Zonal method was initially formulated by Hottel and Cohen in 1958 to be used for gas

furnaces but it lends itself well to preliminary design analysis for gas turbine combustion

chambers. In this method a radiative enclosure is divided in several isothermal surface

and volume zones. Radiative exchange between any two zones is dependent on radiative

exchange areas. An energy balance on each zone leads to a system of equations equalling

the number of zones and can be solved for unknown temperatures or heat fluxes.

The equations presented here are taken by a textbook on radiation by Modest [31].

The formulation would only be briefly discussed and the reader can refer to the book

for further details. Modest [31] discusses equations for gray, absorbing, emitting, non-

scattering medium with constant absorption coefficient, enclosed within a gray, diffusely

emitting and reflecting surface.

The most significant step in zonal method is the formulation of Direct Exchange Areas

(DEA). DEA can be described as the fraction of radiative heat flux arriving at receiving

element over the total flux leaving the emitting element; given by

Qi→j = sisjJi. (4.1)

Here the transfer is taking place between two surface components as shown in Fig. 4.1.

Note that the total flux leaving the element is its radiosity (sum of emitted black body

radiation and reflection of all incident radiation), defined by

Jj = εjEbj + ρjHj. (4.2)

24

4.1. Zonal Method 25

Aj, Tj

Ai, Ti

S

dAi

dAjnj

ni

θi

θj

(a)

dAi

dAj cos θj

ni

dωj−i

(b)

Figure 4.1: Surface to surface interaction

Considering the geometrical location of the two surface, radiation transfer from surface

to surface can be described by

dQi→j =

(Ji

π

)(dAi cos θi)

(dAj cos θj

S2

)τij. (4.3)

The first term on the right side of the equation is radiation that leaves the first element

(per solid angle), the second term is the projected area of the first element in the direction

of the second element, the third term is the solid angle subtended by the two elements

and the last term is transmissivity (i.e. radiation transferred after attenuation due to

absorption on path length S).

The Beer-Lambert law defines transmissivity for the case with no scattering as

τ = e−∫κ(S) dS. (4.4)

Scattering behaviour of particulate matter is governed by Mie theory; in cases where

the particles are small relative to the irradiated wavelength such as soot particles in

gas turbine combustors, scattering can be neglected [19]. For the special case where

absorption coefficient is constant over the path length transmissivity reduces to

τ = e−κS (4.5)

Comparing the definition of a DEA and the formulation in Eq. (4.3), we get the

26 Chapter 4. Radiation Modelling

following expression for surface to surface DEA:

sisj =

∫Ai

∫Aj

τijcos θi cos θj

πS2dAj dAi. (4.6)

For interaction of gas to gas elements, DEA definition, radiative heat transfer and DEA

formulation is given respectively by the following:

Qi→j = gigjEbi (4.7)

dQi→j = 4κiEbi dVidAj

4πS2τijκj dSj (4.8)

gigj =

∫Vi

∫Vj

τijκiκj

πS2dVj dVi. (4.9)

For interaction of gas to surface elements, DEA definition, radiative heat transfer and

DEA formulation is given respectively by the following:

Qi→j = gisjEbi (4.10)

dQi→j = 4κiEbi dVidAj cos θj

4πS2τij (4.11)

gisj =

∫Vi

∫Aj

τijcos θj

πS2κi dAj dVi. (4.12)

Note that with these definitions all DEA’s follow the rule of reciprocity given by

sisj = sjsi, gisj = sjgi, gigj = gjgi. (4.13)

The total radiation reaching a surface is the sum of radiation from all other surfaces

(given by the surface to surface DEA and multiplied by emitting surface’s radiosity) and

radiation from all the volumes (given by gas to surface DEA and multiplied by the black

body radiation for the gas volume). This formulation is given by

AiHsi =N∑

j=1

sisjJj +K∑

k=1

sigkEbgk, i = 1, 2, . . . , N. (4.14)

Similarly, summing the radiation on a gaseous element results in

κViGi =N∑

j=1

gisjJj +K∑

k=1

gigkEbgk, i = 1, 2, . . . , K. (4.15)

4.2. Radiative Properties 27

Radiosity term in Eqs. (4.14) and (4.15) can be eliminated with Eq. (4.2) and the final

form of equations is obtained where the unknowns are surface incident radiation and

gas incident radiation. The resultant is a linear system of equations with number of

equations (N + K) equalling the number of unknowns (N surface radiation and K gaseous

radiation).

4.2 Radiative Properties

For gas turbine combustion application radiation can be divided in two major modes,

luminous and non luminous. Non luminous radiation is from water vapour and carbon

dioxide gases present in combustion mixture. Since these are molecular gases their radi-

ation is in specific spectral bands. Luminous radiation is emitted by soot and at higher

pressures can contribute much more than non luminous radiation [20].

Accurate prediction of radiative heat flux requires accurate knowledge of radiative

properties. This can be a challenge since radiative properties depend on species concen-

tration, temperature, pressure, wavelength and path length [31]. For zonal method the

only radiative property required for interacting gas is the absorption coefficient.

Extensive review of methods to predict absorption coefficient is out of the scope of

this thesis, however, general techniques applicable for preliminary design are discussed

below.

One strategy to predict average absorption coefficient is to use a relevant correlation

based on local temperature. Correlation shown below was produced by Gibb and Joyner

based on data obtained from combustor testing:

κhg = 0.32 + 0.28e−Thg1135 . (4.16)

A drawback of this correlation can be that it does not scale directly with pressure.

Although the temperature would rise with higher pressure resulting in lower absorption

coefficient (for temperatures ranging from 700 to 2100 K the absorption coefficient from

this correlation varies from 0.47 to 0.36 m−1), the full impact of pressure might not be

taken into account.

Another strategy that is employed is computational radiation codes and can be em-

ployed with zonal method is to obtain average gray absorption coefficient from total gas

emissivity. Equation shown below describes this relationship:

κhg = − 1

Lmbln(1 − εhg). (4.17)


The Kayakol et al. [33], Viskanta and Menguc [50], and Lallemant et al. [34] all

mention this with Kayakol using it with Discrete Ordinates Method (DOM) to obtain

absorption coefficient for each cell. In absence of detailed information an average mean

beam length for an arbitrary shape (in this case the combustor’s shape)can be obtained

by the Eq. (2.12). Using this formulation with non homogenous gray medium is not

mathematically sound, however, due to prohibitive computational effort required for

detailed formulation this formulation is used in several high fidelity codes [34]. If water

and carbon dioxide concentrations are assumed constant throughout the combustor, total

emissivity correlations can be used to obtain an average absorption coefficient. To predict

total emissivity for a luminous flames , the following equation can be used [19]:

εtot = εgas + εsoot − εgasεsoot. (4.18)

4.3 Implementation

4.3.1 Direct Exchange Area Calculation

As mentioned earlier calculating DEA’s is the most significant step in zonal method; the

rest of the calculations involve simple linear algebra and would not be discussed in detail

here. A strategy that was adopted to estimate DEA was to conduct direct numerical

integration; this technique is described by Mechi’s [41] in detail.

A given geometry is divided into surface and gaseous elements. The elements are con-

sidered isothermal zones where properties such as emissivity, temperature and absorption

coefficient are assumed to be constant. To calculate DEA with direct integration each

element is further subdivided into a finer grid. If a surface element is subdivided in a

grid of ‘k’ by ‘k’ sub elements then Eq. (4.6) is transformed into

sisj =k∑

m=1

k∑n=1

τijcos θi cos θj

πS2dAj dAi. (4.19)

With this definition of DEA, finer resolution of discretization should result in higher

accuracy of total DEA calculated.

In the formulation discussed by Mechi [41], transmissivity is calculated for non homo-

geneous medium between each sub element which can be computationally very expensive,

therefore, for this project an average absorption coefficient was calculated between given

two elements.

It was assumed that the average coefficient did not vary with the location on each

4.3. Implementation 29

A1 A2

A3 A4

B1

B3

B2

B4

(a)

A1 A2

A3 A4

B1

B3

B2

B4

(b)

Figure 4.2: Approximation used to calculate transmissivity between major grid elements

element, and it was calculated on path length joining the center of the two elements as

shown in Fig. 4.2(b). Fig. 4.2(a) shows the original case where absorption coefficient

could vary from one sub element to the other.

To calculate the total attenuation due to non homogeneous absorption coefficient the

integral in Eq. (4.4) is calculated through direct numerical summation as well. The path

length ray S is traced from the emitting to the receiving element and further divided

into smaller elements. At each element the distance from the current node’s center point

and each adjacent node’s center point is calculated. The smallest distance indicates the

node that current step lies in. Absorption coefficient for that node is added to a running

total which is averaged based on ray divisions in the end. This method of calculating

inhomogeneous absorption coefficient requires that information regarding adjacent ele-

ment nodes be available at each step of the calculation. This method allows for accurate

the integration of transmissivity term given that the ray is subdivided into fine enough

elements. Effect of a coarse and fine path length step size is shown in Fig. 4.3. A finer

step size result in more information being added in the formulation (violet elements being

added )

Ideally, the ray subdivisions should be an order of magnitude smaller than the smallest

edges of the volume or surface zones, enabling representative traversal of these zones

and better weighting for the average coefficient. A major disadvantage of using direct

numerical integration for absorption coefficient estimation is that as the size of isothermal

zones becomes smaller and smaller step size is required, computational time can become

high.

To verify the implementation for DEA described above, results were compared to

the ones reported by Yuen et al. [52] for normalized exchange factors. Yuen et al. [52]


A

B

A

B

Figure 4.3: Effect of integration step size on transmissivity calculation. Reduced step sizeadds information from new cells (shown in purple).

provided normalized exchange factors for various combination of direction vectors and

optical depth (κD). Comparison of calculated and reported results for gas to gas and

surface to surface interactions are shown in Figs. 4.4(a) and 4.4(b) respectively. Fig. 4.5

shows the position used for direction vector corresponding to (x=5, y=5, z=5). Fig. 4.5

showed expected effect of number of sub elements on accuracy of DEA calculation; it also

served as a verification test for the current implementation for DEA calculation (with

the exception of transmissivity calculation).

4.3.2 Grid Generation

Implementation of a rectangular prism grid is the most straightforward. A small rectan-

gular prism is created to represent an isothermal zone; this is then copied and shifted in

three major axis for the corresponding required lengths to form the final geometry. For

a cylindrical geometry, small volumetric divisions are created for each radial position.

This arc is then copied and rotated circumferentially and then copied and shifted axially

to form a cylinder.

If the three major geometrical lengths (axial, radial, and circumferential for a cylin-

der) are evenly divided for refinement, the method described above creates a cylindrical

geometry that is extremely fine in the center, however, may still be coarse on the surface

of the cylinder. In order to refine a cylinder more evenly a specific scheme was adopted

whereby the cross sectional area, and by extension the volume (since axially it is evenly

divided) of each sub-volume is kept constant. This requires that for each radial position

the number of circumferential divisions varies and it increases as the radial position moves

from the inside of the cylinder to the outside. For preliminary studies an initial division

4.4. Results and Discussion 31

100 101 102 103

Grid Density (#)

0

2

4

6

8

10

12

Err

or(%

)Dir (1,1,5), κD =0.1

Dir (3,3,3), κD =0.1

Dir (3,3,5), κD =0.1

Dir (1,1,5), κD =2.0

Dir (3,3,3), κD =2.0

Dir (3,3,5), κD =2.0

(a)

100 101 102

Grid Density (#)

0

1

2

3

4

5

6

Err

or(%

)

Dir (1,1,5), κD =0.1

Dir (3,3,3), κD =0.1

Dir (3,3,5), κD =0.1

Dir (1,1,5), κD =2.0

Dir (3,3,3), κD =2.0

Dir (3,3,5), κD =2.0

(b)

Figure 4.4: Effect of grid density on calculation accuracy of (a) gas to gas and (b) surfaceto surface direct exchange area

of three was picked which meant the circumferential divisions varied from 3,5,7,9 for

radial positions 1,2,3,4 respectively. Fig. 4.6 shows the comparison of a regular circular

grid and the new scheme, both with equal refinement.

A note should be made here about the axisymmetric nature of the problem. Ax-

isymmetric assumption allows for significant time reduction in the computation of all

radiative matrices. For example, in the case of an axisymmetric cylinder with 3 sub-grid

model shown in Fig. 4.6, radiative interaction between all elements within sub grid A

are equal to radiative interaction between all elements within B and C. Similarly only

radiative interaction between AB is equal to BC and CA. It should be noted that all

execution times reported here take this implementation into account.

The direct exchange area is assumed to be the sum of exchange areas from all the fine

subdivision of the emitting element to all the fine subdivision of the receiving element.

The principal of reciprocity is held for DEA which means that the DEA1→2 is the same as

DEA2→1. Hence, to minimize computation time the DEA is calculated only once for each

pair of zones. Direct integration technique requires that if an accurate DEA is required

then the second sub grid level should be finer.

4.4 Results and Discussion

Implementation of zonal method as described in the previous section was tested against

data from three sources. In all cases the radiative heat flux over a cylindrical geometry


X axis

01

23

Yax

is

0

1

2

3

Zax

is

0

1

2

3

4

5

(a)

X axis

01

23

Yax

is

0

1

2

3

Zax

is

0

1

2

3

4

5

(b)

Figure 4.5: Geometry layout for direction vector used in direct exchange area verification.Direction (X5,Y5,Z5) is shown here

with axisymmetric input data was calculated; the details of the first geometry are given

in Fig. 4.7.

4.4.1 Case 1

Stuttaford [46] reported Monte Carlo results for net radiative heat flux along the axial

length of the cylindrical combustor (Case 1). Since the input data is axisymmetric the

resultant radiative heat flux is also axisymmetric.

There are three main parameters in the current implementation of zonal method: (1)

major grid panels which determine the number of isothermal zones to represent input data

and hence, overall accuracy of the solution; (2) path interval length which determines

the accuracy of transmissivity factor calculation from one panel to the other; and (3) the

grid density (divisions) of each panel which determine the accuracy of the DEA integral

calculated. These parameters were varied and solved for the geometry shown in Fig. 4.7;

the results are shown in Fig. 4.8.

For the results obtained in Fig. 4.8(a), the path length was kept constant at 0.005 m

and 2 axial, radial and circumferential divisions per panel; the grid number shown in the

figure corresponds to axial and radial divisions respectively. The execution time varied

from 36 seconds for the (10,2) grid to 3 hours for the (20,8) grid. For the results obtained

in Fig. 4.8(b), the major grid was kept constant with 20 axial and 4 radial divisions and


AB

C

AB

C

Figure 4.6: Comparison of discretization schemes that can be used for a cylindrical ge-ometry. On the left circumferential divisions vary with radial position, on the right theyare constant

2.5 m

5 m

0.5 m1 m

Tgas = 1700 K

κ = 0.6 m−1

Tgas = 1100 K

κ = 0.05 m−1

Twall = 500 K, ε = 0.8

Figure 4.7: Cylindrical geometry to represent a combustor [46].

each panel is resolved with 2 points in each direction. The execution time varied from 7

minutes for the path length of 0.01 m to 54 minutes for the path length of 0.001 m. For

the results obtained in Fig. 4.8(c), the major grid was kept constant with 10 axial and 2

divisions with the path length of 0.05 m. The execution time varied from 1 minute for 4

divisions to 6 minutes for 6 divisions per panel.

Overall, the predicted solutions follow the general trend of Monte Carlo solution,

however, results under predicts radiative flux by 20% at peak conditions. Overall impact

of parameter variation on the final results is minor; out of the three parameters, path

interval length had the most effect, followed by major grid divisions and grid density

for each panel. While the offset of the predicted and measured results cannot fully be

explained and might be a limitation of the zonal method, the small variation in results

with variation in input parameters can be explained due to the nature of the input

data. In the geometry under consideration, there are only two distinct zones where the

temperatures and absorption coefficient vary while the temperature and emissivity of the


0.0 0.2 0.4 0.6 0.8 1.0Axial Location ( llo)

0

20

40

60

80

100

120N

etR

adia

tive

Hea

tF

lux

%(

qq p

eak,

MC)

Monte Carlo Case 1

Zonal (10,2)

Zonal (20,4)

Zonal (20,8)

(a)


0

20

40

60

80

100

120

Net

Rad

iati

veH

eat

Flu

x%

(q

q pea

k,

MC)

Monte Carlo

Zonal ∆S =0.01

Zonal ∆S =0.005

Zonal ∆S =0.001

(b)


0

20

40

60

80

100

120

Net

Rad

iati

veH

eat

Flu

x%

(q

q pea

k,

MC)

Monte Carlo Case 1

Zonal 4 divs

Zonal 5 divs

Zonal 6 divs

(c)

Figure 4.8: Effect of (a) grid density; (b) path length divisions; and (c) divisions perpanel on net radiative heat flux on the wall

wall do not vary. Therefore, discretization of the geometry does not have a significant

impact on the final solution.

4.4.2 Case 2

Menguc et al. [26] presented numerical solution using P3 approximation technique for

radiative heat flux on a cylindrical combustor (Case 2). The input data presented in

the paper was used as a second test case and the predicted profile from P3 approxima-

tion solution has been presented in subsequent results for this case as reference. The

axial temperature and soot profiles that were used for all the conditions are shown in

Fig. 4.9(a). Normalized radial profiles that were used for the four subsequent cases are

shown in Fig. 4.9(b). The axial profiles were based on experimental data that was gath-


0 1 2 3 4 5 6 7z/ro

400

800

1200

1600

2000

2400

T(K

)

0

5

10

15

20

25

Cs

g m3

(a)

0.0 0.2 0.4 0.6 0.8 1.0r/ro

0.0

0.5

1.0

1.5

2.0

2.5

3.0Case 2.1 T (r)−Tw

Tm−Tw

Case 2.2 Cs(r)Cs,avg

Case 2.3 Cs(r)Cs,avg

Case 2.4 T (r)−TwTm−Tw &Cs(r)

Cs,avg

(b)

Figure 4.9: (a) Axial temperature and soot profiles and (b) normalized radial profiles usedby Menguc et al. [26]

ered previously, while the radial profiles were selected by the authors to represent various

conditions that might occur in an actual combustor. A correlation was used to obtain

average absorption coefficient based on temperature and local soot concentration.

Fig. 4.10 shows results for Case 2.1 where a constant radial soot profile but varying

radial temperature profile were used. The three major parameters for zonal method code

were varied similar to Case 1. For the results obtained in Fig. 4.10(a), the path length

was kept constant at 0.005 m and 2 axial, radial and circumferential divisions per panel.

The execution time varied from 29 seconds for (7,3) grid to 1 hour for (22,10) grid. For

the results obtained in Fig. 4.10(b), the major grid was kept constant with 15 axial and 7

radial divisions and each panel is resolved with 3 points in each direction. The execution

time varied from 27 minutes for path length of 0.01 m to 49 minutes for path length of

0.001 m. For the results obtained in Fig. 4.10(c), the major grid was kept constant with

15 axial and 7 divisions with path length of 0.005 m. The execution time varied from 9

minutes for 2 divisions to 6 minutes for 5 divisions per panel.

Similar to Case 1 results, variation of path length interval has the greatest effect on

the net radiative heat flux. This effect is greater than Case 1 since there is much more

radial and axial variation in absorption coefficient throughout the combustor. The effect

of major grid divisions and grid density on net radiative heat flux is extremely low. If the

major grid divisions are high in number, which is the case for Fig. 4.10(c), then changing

grid density only has a minor effect on DEA accuracy and hence on the final radiative

heat flux.

Grid density has an effect on DEA accuracy when there is large variation in calculation


0 1 2 3 4 5 6 7Axial Location ( lr0

)

0

24

48

72

96

120N

etR

adia

tive

Hea

tF

lux

%(

qq p

eak,

P3) P3 Approx. Case 2.1

Zonal (7,3)

Zonal (8,4)

Zonal (22,10)

(a)


)

0

24

48

72

96

120

Net

Rad

iati

veH

eat

Flu

x%

(q

q pea

k,


Zonal ∆S =0.01

Zonal ∆S =0.005

Zonal ∆S =0.001

(b)


)

0

24

48

72

96

120

Net

Rad

iati

veH

eat

Flu

x%

(q

q pea

k,


Zonal 2 divs

Zonal 3 divs

Zonal 4 divs

Zonal 5 divs

(c)

Figure 4.10: Effect of (a) grid density; (b) path length divisions; and (c) divisions perpanel on net radiative heat flux on the wall

of relative angles (i.e. θi and θj in Eqs. (4.6) and (4.12)) or path length over the elements

(S in Eqs. (4.6), (4.9), and (4.12)). For small panels that are relatively far away change

in angle and path length for points within the panel is negligible. Furthermore, for

cases where large amount of absorbing medium separates the panels the DEA integral is

dominated by transmissivity term (τ). As a result, variation in grid density has almost

no effect, while variation in major grid panels has very low effect on the net radiative

heat flux on the wall.

Effect of varying the soot profile on net radiative heat flux is shown in Fig. 4.11.

The trend of axial variation in net radiative flux matches what was obtained with P3

approximation, however, the magnitude of the flux varies significantly. As the soot

profile was changed for a higher peak in the center (Case 2.3), Menguc et al. reported an



)

0

24

48

72

96

120

Net

Rad

iati

veH

eat

Flu

x%

(q

Cas

e2.2q p

eak,

P3)

P3 Approx. Case 2.2

P3 Approx. Case 2.3

Zonal Case 2.2

Zonal Case 2.3

(a)


)

0

90

180

270

360

450

Net

Rad

iati

veH

eat

Flu

x%

(q

Cas

e2.1q p

eak,

P3)

P3 Approx. Case 2.1

P3 Approx. Case 2.4

Zonal Case 2.1

Zonal Case 2.4

(b)

Figure 4.11: Effect of (a) radial soot profile variation and (b) constant radial temperatureand soot profile on net radiative heat flux on the wall

increase in overall radiative flux and the reason for this change is explained to be lower

absorption of radiation near the wall due to lower soot. Results from the zonal method

show the opposite trend where the radiative heat flux decreased as the amount of soot

near the wall is lowered; shown in Fig. 4.11(a).

For the final Case 2.4 the radial soot and temperature profile were kept at a constant

average value; results are shown in Fig. 4.11(b). Results obtained by the authors from P3

approximation show an increase in net radiative heat flux as the temperature near the

wall is increased from Case 2.1 to Case 2.4 (80% increase in peak flux) and that trend

is matched by results obtained from zonal method although the increase in peak flux is

much higher at 300%.

By comparing results from the two methods for all four cases it can be seen that the

current implementation of zonal method is much more sensitive to absorption coefficient

near the wall.

4.4.3 Case 3

Kayakol et al. [33] compared measured results for incident radiative heat flux on a repre-

sentative gas turbine combustor with predicted results from DOM simulation. Combustor

length to radius ratio was approximately 8.1 therefore, it is comparable to the geometry

used by Menguc et al. Other details regarding the geometry of the combustor and input

parameters are described in detail by Kayakol et al. [33]. Absorption coefficient for thir-

teen radial and fourteen axial positions were measured, therefore, the major grid used


0.0 0.2 0.4 0.6 0.8 1.0Axial Location (l/l0)

0

24

48

72

96

120

Inci

den

tR

adia

tive

Hea

tF

lux

%(

qq p

eak,

mea

s)

Measured

Zonal Method

Figure 4.12: Measured vs predicted results for incident radiative heat flux

for zonal method matched that configuration with 3 divisions per panel (each direction)

and path length interval equal to 0.005 m. Since the major grid was extremely dense the

case finished execution in 5 hours.

Figure 4.12 shows a comparison of measured and predicted result from zonal method.

Results show a trend that matches measured results, with heat flux under predicted in

the beginning of the combustor but the location and magnitude of the peak matching

measured results.

So far the discussion has focused on radiation in cylindrical enclosures, however,

majority of the implementation remains the same for implementation on a geometry

representative of a gas turbine combustor. An example geometry to model a gas turbine

combustor is shown in Figure 4.13, where the primary and secondary zones for one sector

are modelled ignoring rest of the combustor geometry.

Curvature of the combustor requires that surface and gaseous elements that cannot

receive any radiation from other zones (due to the combustor core being in the way) be

recognized for the computation, these elements were ignored in the shown geometry. Since

the combustor is the hottest in the primary zone, this assumption might still produce

results that are representative of actual radiation flux.

Another geometry feature that is being ignored in Figure 4.13 is various adjacent

combustor sectors that would increase the radiative flux on the current sector. If it is

assumed that all nozzles in the combustor produce the same pattern and by extension the

same temperature profile, the radiative flux would be axisymmetric in nature, however,


Xaxis

− 0.2− 0.10.00.1

0.2 Y axis0.0 0.1 0.2 0.3 0.4

Zaxis

− 0.2

− 0.1

0.0

0.1

0.2

Figure 4.13: A course discretization grid of primary andsecondary zone of a reverse flow combustor for zonalmethod implementation

A

B

C

D

EF

G

H

Figure 4.14: Combustorcross section schematic witheight sectors

the interaction of various sectors would need to be taken into account.

Figure 4.14 shows a combustor cross section with eight sectors. Application of ax-

isymmetric condition was discussed in the context of cylindrical geometry with three

major sub grids; this can be further extended to any number of subgrids. Therefore,

for eight sectors the computation time would be eight times the time required for only

a single sector. This is the worst case scenario as in reality the combustor core blocks

a significant number of sectors. In case this strategy is computationally expensive, a

scaling factor could be applied to the results of a single sector as the trend for incident

radiative flux in axial direction should not vary significantly from single sector results.

Chapter 5

Combustion Modelling

5.1 Background

There are three main characteristic times that describe combustion within the combus-

tor, these are, chemical, evaporation and mixing characteristic times. In simulations

employing CFD an assumption that is usually made is that the mixing or evaporation

rate is the limiting factor while the chemical rate is infinitely fast. In reactor modelling

the assumption is made that chemical rate is the limiting factor while high turbulence

causes mixing to be instantaneous. Conceptual reactors make broad assumptions about

the flow and if used individually can seldom model real flows, but in combination might

be able to model complex flows.

Several theoretical reactors are described by Turns [49] but in the context of combus-

tion chambers that operate on almost constant pressure two are the most relevant which

are: (1) Perfectly Stirred Reactor (PSR); and (2) Plug Flow Reactor (PFR). A PSR is an

ideal reactor in which perfect, infinitely fast mixing is achieved inside a control volume

[49]. Mass conservation is given by

dmi,cv

dt= m′′′V + mi,in − mout. (5.1)

For a non compressible system the left hand side of the equation reduces to zero. Mass

generation can be given by each reactants chemical production rate changing the conser-

vation equation to

ωMWiV + m(Yi,in − Yi,out) = 0, (5.2)

where Y defines the mass fraction of each species. Energy balance can be done based on

40

5.2. Cantera Model 41

internal enthalpy of incoming reactants and outgoing products to give

Q = m(iout − iin). (5.3)

To close the system of equations equation of state is used to relate density, pressure and

temperature as follows:

ρ =PMWmix

RuTr

(5.4)

Finally, residence time of the reactor can be defined as

tR =ρV

m(5.5)

Regions in the combustor where the turbulence is high, for example the primary zone,

might approach the idealisation of a PSR reactor.

A PFR is an ideal reactor which is one dimensional in nature (compared to zero

dimensional for PSR) for which velocity, concentration, temperature and other properties

describing the state might vary axially. It also has the following attributes: (1) steady

state, steady flow, (2) no mixing in axial direction, (3) uniform properties in perpendicular

direction to the flow, (4) ideal frictionless flow and (5) ideal gas behaviour [49]. Regions

in the combustor where the flow is one dimensional and has relatively low turbulence,

for example the dilution zone, can be modelled with a PFR.

These conceptual reactors might be linked in a network with output from one being

fed as the input for the other to resemble a complex combustor flow. Ideally, the closer

the reactor would be to actual flow features, the better the accuracy of species production

and temperature data.

5.2 Cantera Model

For the current project an open source software, CANTERA [5] was used to solve chem-

ical equilibrium equations and model reactor network. CANTERA numerically solves

differential and algebraic equations discussed earlier; the software has the capability to

model various forms of reactors, namely, a general reactor, ideal gas reactor, constant

pressure reactor, ideal gas constant pressure reactor, and flow reactor. Differences in var-

ious models lie in various assumptions being made to solve the mass, momentum (in the

case of flow reactor i.e. a PFR) species conservation and energy equations being solved.

To match the equations discussed in previous section ideal gas constant pressure reactor

and flow reactor were selected for the current study. CANTERA inherently solves the

42 Chapter 5. Combustion Modelling

time dependent versions of equations discussed previously, as a result, to obtain steady

state results all cantera simulations were run until temperature had converged.

5.3 Implementation

Since the reactor networks would eventually be part of a larger combustor design system,

the creation of the network was generalized. The generalization would allow for a new

reactor network to be created based on the combustor geometry. The user can specify

the interconnectedness of the reactors via an input text file. Flow splits from one reactor

to every other reactor in the network are stored in a matrix structure (this is similar to

an adjacency matrix for a graph which would store a 1 or 0 but instead stores the split

value). Details of the amount of fuel and air to each reactor are stored and read as a

table in another input text file.

Flow splits are initially defined as percentage split of the input flow; this was done

to make the creation of input files easier. To get the actual mass flow rate from the

percentage is trivial if the network does not contain any recirculation zones or any part

of the flow coming back to the input. Since recirculation flow are to be expected an

iterative algorithm was used to find the absolute flow rate.

First, each reactor’s output is calculated based on the flow split and the input while

ignoring any recirculation. In second iteration previous results are used to calculate new

outputs while taking recirculation into account. This is continued until the mass flow

rates in the entire network converge.

Recirculation also poses a problem when it comes to solving the networks in CAN-

TERA directly. In cases where flow from downstream reactors is connected to upstream

reactors the CANTERA numerical solver fails to converge. If the evaporation and heat

loss are being solved iteratively it implies that the entire network would be solved a large

number of times. Since CANTERA inherently solves for time dependent system, a highly

stiff system would be computationally costly to solve in an iterative scheme. In order to

achieve convergence faster each reactor was ignited and solved separately, and then the

network solved iteratively similar to the approach used when calculating mass flow rates

from flow splits.

CANTERA utilizes mechanism files that contain list of elementary reactions along

with reaction rate data and thermodynamic data to predict chemical compositions. Marc-

hand [27] and Lanewala [18] had success in estimating emissions with CANTERA and

utilizing GRI3.0 mechanism.

GRI3.0 mechanism is designed for detailed combustion analysis of propane and methane,


(a) (b)

Figure 5.1: Engine A and Engine B profile schematics for comparison

contains 53 species and 325 reactions [8]. Since propane has higher heat value than jet

fuel the reactor network calculations were adjusted by decreasing the total amount of

fuel entering each reactor in proportion to the fraction of higher heating value of the two

fuels [27, 18] . This method ensures that the total energy entering the system is similar

to what it would be with the jet fuel.

Jet fuel mechanism by Westbrook was tested by Marchand but not used for the final

calculation as it was found significantly more computationally expensive (it contains 1421

species and 7851 reactions) and not suitable for preliminary design calculations [27].

In order to model temperature more accurately two additional reduced mechanisms

were tested; these were Kollrack’s jet fuel mechanism [11, 28] and University of California

San Diego’s (UCSD) JP 10 mechanism [29].

To ignite each reactor in the reactor network, a short Gaussian pulse of hot gas is

initially injected into each reactor. The pulse consists of combustion products of the

specified fuel with air at 1000K. This ensures that no concentration of species introduced

is somewhat comparable to what is expected at the end from the reactor.

5.4 Results and Discussion

Figs. 5.1(a) and 5.1(b) shows profiles for Engine A and Engine B that were used as test

cases discussed in this section. Fig. 5.2 shows results that were obtained for Engine A

using a simple linear reactor network of five reactors with GRI 3.0 mechanism and gaseous

propane as fuel. The reactor network configuration shown in Fig. 5.3 was run for idle,

approach, climb and take off power settings. Operating conditions and the combustor

outlet measured temperature were obtained from a technical report by Gratton et al.


Idle Approach Climb Take Off0.0

0.5

1.0

1.5

2.0

2.5

Com

bu

stor

Tem

per

atu

reR

ise,

T4

T3

Measured

Predicted

Figure 5.2: Engine A combustor outlet temperature; measured and predicted results

1 432 5

Figure 5.3: Engine A reactor network

[24] . Since the simulation was run for propane it is surprising that the results are within

6% of experimental data. This might be due to higher residence time and lower fuel to

air ratio that is used in relatively older combustors. If the residence time is higher then

the temperature prediction would be controlled by elementary reactions that take place

later in the combustion process, which are similar for hydrocarbon fuels.

In Figure 5.4 combustor outlet temperature (T4) is plotted against air to fuel ratio

(AFR) for Engine B. As compared to Engine A, Engine B is of a modern design with the

overall residence time being much lower and the fuel to air ratio being higher. Operating

conditions and combustor outlet measured temperature for engine B was obtained from

a book by Schumann [43]. Fig. 5.4(a) shows the effect of various reduced mechanisms on

outlet temperature. Out of the three mechanisms tested GRI 3.0 and UCSD mechanisms

follow the overall trend with UCSD being closer to experimental results even though it

models JP 10 fuel and not Jet A while Kollrack mechanism severely under-predicts T4.

Fig. 5.4(b) shows the effect of network configuration on temperature prediction. Con-

figuration 1 shown in the figure consists of 3 linear reactors similar to the linear reactor

configuration of Engine A in Fig. 5.3. Configuration 2 was a network that attempted to

improve the modelling of recirculation zone by sending a percentage output of reactor

2 to reactor 3 and then to 1 as shown in Fig. 5.5(a). The new model did not have any

impact on the final outlet temperature prediction. However, network configuration does


35 45 55 65AFR

1.50

1.75

2.00

2.25

2.50

Com

bu

stor

Tem

per

atu

reR

ise,

T4

T3

Measured

GRI 3.0

Kollrack

UCSD

(a)

35 45 55 65AFR

1.50

1.75

2.00

2.25

2.50

Com

bu

stor

Tem

per

atu

reR

ise,

T4

T3

GRI 3.0 Config 1

GRI 3.0 Config 2

UCSD Config 1

UCSD Config 2

(b)

Figure 5.4: Effect of (a) mechanisms and (b) reactor network setup on combustor outlettemperature for Engine B

have an impact on the local temperature in distinct combustor zones.

Local temperature is also an important factor for radiation prediction and for emis-

sion prediction (especially NOx). The effect of network configuration on NOx prediction

is shown in Figure 5.6. Here the third configuration of reactor networks tested is shown

in Fig. 5.5(b). Detailed analysis on emission prediction with network reactors was con-

ducted by Marchand and Lanewala and is not the focus of current project, however, since

heat transfer prediction and emission prediction would be part of a single PMDO design

tool it is important to note that the current tool gives the user the ability to predict emis-

sions while the configuration can be set based on a strategy outlined in aforementioned

projects.


1 42

3(a)

1 42

3(b)

Figure 5.5: Network configurations for Engine B: (a) configuration 2 and (b) configura-tion 3

1.8 2.0 2.2 2.4 2.6Combustor Temperature Rise, T4

T3

0.0

0.2

0.4

0.6

0.8

1.0

EINOx

%(

EINO

x

EINO

x,

Con

fig

1m

ax)

Config 1

Config 2

Config 3

Figure 5.6: Effect of reactor network setup on combustor NOx emission index (EINOx)

Chapter 6

Cooling Technology Modelling

6.1 Film Cooling

Film cooling regardless of the geometry being used has mainly been modelled using

correlations. The definition of cooling film effectiveness (Eq. (2.5)) and the consequent

calculations for liner temperature are described in detail in Section 2.3. Correlations for

louvres fall in two broad categories as described by Lefebvre [20]. There are correlations

based on turbulent boundary-layer model and wall-jet model; Lefebvre’s correlations for

these models are:

η = 0.6( x

Ms

)0.3(

ReMµc

µhg

)−0.15

(6.1)

and

η = 1.28

(µc

µhg

)0.15 (xs

)−0.2(t

s

)−0.2

. (6.2)

Boundary-layer models are derived based on idealized turbulent boundary layer down-

stream of the slot and might not represent conditions close to the slot; Eq. (6.1) was

based on measured skin friction coefficient to avoid this drawback.

Lefebvre used the following correlation for machined ring cooling slot:

η = 1.0 − 0.12S0.65N , (6.3)

and the following correlation for stacked ring cooling slots:

η = 1.0 − 0.094S0.65N , (6.4)

47

48 Chapter 6. Cooling Technology Modelling

where

SN =x− xp

Ms

(Re

µc

µhg

)−0.15Ao

Aeff

.

Li and Mongia [22] employed mass transfer analogy to measure film cooling effective-

ness in machined ring liners. Computational results were gathered for certain design of

experiment configurations and on the basis of experimental and numerical results corre-

lation with the functional form shown below was suggested:

η = e−aζb

(6.5)

where ζ is defined as

ζ =x

Ms=

(µinf

µ2

)Rex

Re2

(6.6)

and a and b are functions of blowing ratio (m), slot lip thickness to slot height ratio

(s/t), coolant injection angle, lip taper angle and starting edge angle on nugget-slot exit.

Lefebvre’s Eqs. (6.2) and (6.3) were also compared with the collected data. Equation

(6.2) was found not be a good match while equation (6.3) matched data to within 15%

accuracy. However, the new correlation provides predictive capabilities with greater

number of geometrical parameters.

Some authors have found film mass flow rate and hot stream turbulence level to be

of greater importance than slot geometry; based on this Juhasz and Marek [15] provided

a correlation of the form given by

η =1

1 + CmxMs

(6.7)

and for the case where the cold and hot air differ significantly in composition the corre-

lation included heat capacity terms determined using

η =1

1 + CmxMs

Cph

Cps

. (6.8)

The proposed correlation was able to predict test results within 20%.

Other correlations for various geometries take into account different geometric factors;

However, blowing ratio, local Reynolds number, and downstream distance are common

correlating parameters. Various empirical correlations have been derived for different flow

conditions (Nusselt number formulas) and cooling geometry (film effectiveness formulas).

These parameters are the key in order to optimize for minimum required flux. Various

conditions under which these equations are valid must be taken into account in order to

6.2. Double Walled Cooling 49

L

Tw,2Tc,2

Tc,i

1

2 4

3

Tw,1

Figure 6.1: Schematic representing Counter Flow Film Cooling (CFFC)

arrive at a realizable and valid solution.

6.2 Double Walled Cooling

The analytical model to solve for Counter Flow Film Cooling (CFFC) and Parallel Flow

Film Cooling (PFFC) was obtained from the NASA technical report by Colladay [4].

Fig. 6.1 shows the geometry setup used for the CFFC solver. The main CFFC tile of

length L is shown in striped pattern while the bottom surface shown in gray does not

take part in heat transfer.

Heat gained through convection by the coolant as it passes by the underside of the

tile (labelled ‘1’ in the figure) is given by

dQc = hcf(Tw2 − Tc) dAeff . (6.9)

A Coulbourn j factor correlation appropriate for fin geometry was used to obtain heat

transfer coefficient for the cool side. This correlation was taken from a paper by London

[23] and is given by

hcf =mc

Ap

Cp,cj Pr−2/3. (6.10)

The total area for the transfer is also increased and reflected by the effective area term

(Aeff) in Eq. (6.9). Energy transfer to the coolant is also limited by the heat capacity of

the coolant given by the following equation:

dQc = mcCp,c dTc. (6.11)

Similarly, on the hot side of the tile (labelled ‘3’) the convective heat transfer is described


by

dQhg = hhg(Taw − Tw1) dA. (6.12)

The value for hot side heat transfer coefficient is given by the following Nusselt number

correlation:

Nu =hhgx

k= 0.0296 Re0.8

x Pr1/3 (6.13)

To obtain the adiabatic wall temperature (Taw) in Eq. (6.12), the following correlation

for film effectiveness was used:

η = 16.9( x

Ms

)−8/10

(6.14)

Note that film effectiveness was defined in equation Eq. (2.5), here Tc,2 is taken as the

coolant temperature (Tc in the original definition). This equation was obtained from [12].

A two-dimensional conduction code was written for the tile itself using finite difference

technique. The heat equation for two-dimensional, steady state, with no generation and

constant thermal conductivity that was used for the code is given by

∂2T

∂x2+∂2T

∂y2= 0. (6.15)

The finite difference scheme was taken from textbook by Bergman et al. [47]. The

boundary condition for the finite difference code were heat flux for the top and the

bottom sides of the tile (labelled ‘3’ and ‘1’ respectively) while the other edges were

assumed to be at constant temperature, in this case being Tc,2.

As the resulting system of equations is coupled it was solved for iteratively; the pro-

gram logic that was used can be found in the original report and will be described here

briefly by the following steps: (1) take an initial guess for coolant temperature profile;

(2) calculate cold side heat transfer coefficient and fin effectiveness; (3) calculate film

effectiveness and hot side heat transfer coefficient; (4) run the two-dimensional conduc-

tion code for the tile; and (5) repeat steps (1) - (4) until the solution converges. To

obtain a stable solution the coolant temperature profile was calculated until it had con-

verged before the conduction code was run, this also ensured that less time was spent on

conduction code which relatively takes longer to execute.

Fig. 6.2 shows the results for temperature profiles that can be expected on the top

surfaces of tiles used in CFFC and PFFC. The coolant mass flow rate was adjusted

by Colladay [4] to achieve a maximum surface temperature of 1255 K. In the results

presented here maximum temperatures for all the lengths except 2.5 and 5 cm are higher

6.2. Double Walled Cooling 51

0.00 0.05 0.10 0.15 0.20 0.25Distance(m)

1.1

1.2

1.3

1.4

1.5

1.6

Tem

per

atu

re(T

w2

Tc,

i)

25 cm

20 cm

15 cm

10 cm

5 cm

2.5 cm

(a)

0.00 0.05 0.10 0.15 0.20 0.25Distance(m)

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Tem

per

atu

re(T

w2

Tc,

i)

25 cm

20 cm

15 cm

10 cm

5 cm

2.5 cm

(b)

Figure 6.2: Temperature profiles predicted for panels of various lengths for (a) CFFC and(b) PFFC.

than 1255. This discrepancy is due to ambiguity in certain input parameters that could

not be clarified. However, overall the trends shown here match what was presented in

the original report.

As discussed earlier determining and minimizing the coolant required for a combustor

is an important task for the designers. Colladay found that for tiles with longer length

(larger area) the total coolant required to maintain a certain temperature was larger,

however, the coolant required per area was lower [4]. Therefore, longer tiles were more

efficient in terms of coolant requirements. Increasing the length of the tiles increases the

time that the tile has to transfer energy to the coolant before it exits on top to form the

film layer. This also results in the temperature of the tile near the slot being higher for

longer section lengths.

A note should be made here that in his analysis Colladay predicted the required

pressure for the required mass flow rate, which increased for larger tiles [4]. In reality

this maximum pressure drop across the combustor cannot be changed by the combustor

designer, however, this analysis suggests that the longest tile that can sustain the mass

flow rate according to a given pressure should be picked.

Effect of wall thickness and operating pressure on overall heat transfer was also re-

ported by Colladay. A thin tile wall was reported to have higher temperature on its

coolant side which allow for higher heat flux transfer to the coolant, resulting in more

efficient cooling. Higher operating pressure had the effect of increasing coolant density

and hence, the total coolant mass flow rate while the heat flux increase to the tile was

not as high. As a result, the overall convective effectiveness (defined by η =Tc,2−Tc,i

Tw,2max−Tc,i )


0.00 0.05 0.10 0.15 0.20 0.25Distance(m)

1.0

1.2

1.4

1.6

1.8T

emp

erat

ure

(Tw

2,

pea

k

Tc,

i)

100

300

500

1000

2000

3000

(a)

102 103 104

Number of Nodes(#)

1.0

1.2

1.4

1.6

1.8

Tem

per

atu

re(T

w2,

pea

k

Tc,

i)

Tpeak

Tavg

(b)

0.000 0.005 0.010 0.015 0.020 0.025 0.030Distance (m)

1.30

1.35

1.40

1.45

1.50

1.55

1.60

Tem

per

atu

re(T

w2,

pea

k

Tc,

i)

2

4

8

12

20

100

(c)

100 101 102

Number of Nodes (#)

1.0

1.2

1.4

1.6

1.8

Tem

per

atu

re(T

w2,

pea

k

Tc,

i)

Tpeak

Tavg

(d)

Figure 6.3: Effect of number of axial nodes on a 25 cm section length CFFC’s temperatureprofile is shown in (a) while the change in peak and average temperature is shown in (b).Effect of number of normal nodes on a 2.5 cm section length CFFC’s temperature profileis shown in (c) while the change in peak and average temperature is shown in (d).

decreased with an increase in pressure [4].

In order to determine the density of the grid used for the conduction solver a number

of cases were run varying axial and normal nodes; the results for these cases are shown

in Fig. 6.3. Axial nodes were varied from 100 to 3000 for a CFFC panel with length of

25 cm. The peak of the temperature profile decreased with an increase in axial nodes

with not much improvement for higher than 1000 nodes as shown in Fig. 6.3(a) and

Fig. 6.3(b). Since computational cost was not a major issue, highest density of axial

nodes was selected for all the analysis shown here (120 nodes per cm). Number of

normal nodes did not have a significant impact on the temperature profile.

6.3. Effusion and Transpiration Cooling 53

(a)

Coolant DirectionTwm

Twc

Hot Gas, Tm

Coolant Gas, Tc

Tent

Text

(b)

Figure 6.4: Schematic representing effusion geometry

6.3 Effusion and Transpiration Cooling

Fig. 6.4 shows a schematic representation for the geometry that was solved for by Martiny

et al. [25]. The analytical model consists of one cooling hole (from an array of holes

that might be used to cover a large area shown in Fig. 6.4(a)). It is assumed that

property variation over the stream-wise and pitch-wise direction is negligible and due to

symmetry in both directions all planes can be considered adiabatic. Therefore, under

this assumption the heat transfer is only in the normal direction.

For an infinitesimal wall and air element shown in Fig. 6.4(b), equation representing

the heat balance is given bydQw

dz+

dQa

dz= 0, (6.16)

where dQw

dzrepresents heat lost by the wall and dQa

dzrepresents heat gained by the air.

Heat lost by the wall is due to conduction and given by

dQw

dz= −kwA

d2Tw

dz2, (6.17)

while the heat gained by the air is controlled by the total heat capacity and convective


heat transfer. These relationships are given by the following equations:

dQa

dz∗= maCp

dTa

dz∗(6.18)

dTa

dz∗=

hhπd

maCp

(Tw − Ta) (6.19)

This is a system of coupled equation and can be solved analytically to give equations for

temperature profile for the wall and for temperature profile for the air. The solution to

the resulting two coupled ODE’s defining the cooling in the effusion hole are

Tw = sinαmaCp

kwA

(C1

β1

eβ1z∗

+C2

β2

eβ2z∗)

+ C4 (6.20)

and

Ta = C1eβ1z∗ + C2e

β2z∗ − γ1C3

sinαγ2

, (6.21)

where,

γ1 =hhπd

maCp

,

γ2 =hhπd

kwA,

β1,2 = −γ1

2±√γ2

1

4+ sinαγ2.

The four constants of the two differential equations can be found with equations that

describe the boundary conditions of the system. The total enthalpy rise of the coolant

has to equal to the convective heat flux due to hot gases; this relationship is given by the

following equation:

hhgA(Thg − Tw1) − maCp(Th1 − Tc) = 0 (6.22)

Convective heat flux of the coolant on the cold side is equal to the enthalpy rise in the

coolant before it enters the hole; this relationship is given by the following equation:

hcA(Tw2 − Th2) − maCp(Th2 − Tc) = 0 (6.23)

The convective heat flux of the coolant equals the conductive heat flux on the cold side;

this relationship is given by the following equation:

hcA(Tw2 − Tent) − kwAdTw

dz

∣∣∣∣z=0

= 0 (6.24)

6.3. Effusion and Transpiration Cooling 55

0.0 0.2 0.4 0.6 0.8 1.0Normal Location (z/t)

0.265

0.270

0.275

0.280

0.285

0.290

Non

Dim

ensi

onal

Wal

lT

emp

erat

ure

,Θw

(Tw−T

c

Tm−T

c)

Predicted

(a)

0 20 40 60 80 100Incident Angle (◦)

0.60

0.65

0.70

0.75

0.80

0.85

Coo

ling

Eff

ecti

ven

ess,η

Predicted

(b)

Figure 6.5: Predicted results from the effusion model for (a) wall temperature distributionand (b) effect of blowing angle on cooling effectiveness

Eq. (6.19) can be evaluated for z = 0 to get a relation between Twc and Tent. With these

four equations the formulation of the analytical model is complete.

Martiny et al. presented numerous trends with the analytical model and these were

all matched to verify the implementation. Two of these predicted results are shown in

Fig. 6.5. Fig. 6.5(a) shows the trend for non dimensional wall temperature; one of the

benefits of the model is that it is able to capture the non linear behaviour of this profile due

to coupling of conduction and convection. Fig. 6.5(b) shows the effect of blowing angle

on cooling effectiveness; this matches the expected trend discussed in Section 2.1 where

with shallower angles the length of the hole increases allowing for greater heat transfer

to take place from the wall to the coolant and increasing the cooling effectiveness.

To further demonstrate the capabilities of this model for a designer the formulation of

the problem was changed to produce the required mass flow rate per area for a specified

hot side temperature. Implementation of this model as a module gives the designer the

opportunity to optimize the cooling scheme geometry for a given objective function (in

this case mass flow rate per area). DAKOTA [2] is one such open source software that is

available for optimization and parametric analysis (amongst other features).

Parametric study was done on the effusion cooling model using data presented in the

original paper, the required hot side temperature for the cooling surface was set at 600K.

Further assumption was made the the pitch in stream-wise and pitch-wise direction are

the same. With this assumption there are three main geometrical parameters that can

be varied with this model: (1) hole diameter; (2) blowing angle; and (3) liner thickness.

During combustor design phase liner thickness is most likely to be fixed so it was not


10 30 50 70 90Incident angle (◦)

0.20

0.66

1.12

1.58

2.04

2.50D

iam

eter

(mm

)

1.000

2.000

3.000

4.000

5.000

6.000

7.000

8.000

Figure 6.6: Parametric study for effusioncooling.

0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007Normal Location, X/m

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Non

Dim

ensi

onal

Wal

lT

emp

erat

ure

,Θ,Θ

Θs

Θf

Θs, Θs

Θf , Θf

Figure 6.7: Predicted results from tran-spiration model for wall temperature dis-tribution with and without a ceramic coat-ing.

considered in this analysis.

The contour plot from the parametric study based on variation of hole diameter and

blowing angle is shown in Fig. 6.6. The effect of variation in blowing angle is more

pronounced for larger hole diameters than smaller ones. The accuracy of the predictions

cannot be evaluated without experimental data relevant to actual gas turbine combustor

conditions, however, this demonstrates that models such as the one presented here can

provide more physical insight than a semi-empirical correlation while allowing for fast

execution time, hence being relevant in preliminary design environment. Furthermore,

the effect of manufacturing imperfections on cooling performance can be tested through

coupling of cooling modules with software such as DAKOTA that allow uncertainty

analysis.

Wolfersdof [51] compared effect of various cold side boundary conditions on an analyt-

ical model for transpiration cooling. One-dimensional analytical model for transpiration

cooling presented in the paper and a number of other studies [10] is very similar to the

one presented for effusion cooling, hence, the details would not be presented here. Cou-

pled relationship between conduction of the wall and convection of the air remains the

same, however, the convection relationship was modelled using volumetric heat transfer

coefficient rather than a heat transfer coefficient for a discrete hole. Results for non-

dimensional wall and coolant temperature can be seen in Fig. 6.7 along with a case

where a ceramic coating is added.

Chapter 7

Conclusions

7.1 Thesis Accomplishments

The objective of this thesis, as stated in Chapter 1, was to assess heat transfer analysis

tools that can be utilized to predict gas turbine combustion chamber liner temperature

in the context of preliminary design. Liner temperature prediction calculation was sub-

divided into three steps (or modules): (1) hot gas radiation prediction; (2) hot gas

temperature prediction; and (3) cooling technology modelling. Although currently the

modules operate independently, their integration in larger PMDO tool could be possible,

hence, factors effecting this integration were also taken into account.

In this thesis, a zonal method was assessed as a radiation prediction tool for gas

turbine combustors. Three cases were tested for cylindrical geometry with given radiative

properties. Trends produced by the zonal method for variation in net radiative flux on

the wall were similar to the ones obtained through higher fidelity numerical techniques

in all cases. The zonal method under-predicted radiative flux for the cases where the

absorption coefficient and temperature near the wall were low (case 1, 2.1, 2.2 and 2.3)

and over-predicted for the case where these properties were high near the wall (case 2.4).

The method also demonstrated to be insensitive to various program parameters that were

varied. This can be useful as accurate results can be achieved with low execution time.

Reactor network analysis was assessed as a combustor hot gas temperature prediction

tool. Comparison was made between predicted and measured combustor outlet temper-

atures for two combustors, with accurate predictions for the combustor with higher res-

idence time and valid trends for the combustor with lower residence time. The models

were able to predict the general trend of outlet temperature variation with air to fuel

ratio and the drop in local temperature with axial distance. Hence, reactor networks

can be used as a tool to assess the difference in average temperature in local sections

57

58 Chapter 7. Conclusions

of the combustor such as primary and secondary zones based on the amount of cooling

flow that is provided. Furthermore, a programming environment was created to give the

designer the ability to rapidly create reactor networks with various structures; this can

be a valuable tool when assessing the configuration of a network for new combustors.

Various methods that are available in literature for cooling technology modelling

were discussed in this study. Models for effusion/ transpiration cooling have the ability

to provide temperature profiles for the wall and coolant air in normal direction while

the model for CFFC/PFFC has the ability to predict axial and normal temperature

variation on a cooling tile. Both of the aforementioned models make broad assumptions,

however, they can provide more physical insight as compared to empirical correlations. It

is also possible for the designer to asses the impact of geometrical parameters on cooling

performance which might not be possible with correlations due to complicated geometry

(for example CFFC) or such detailed correlations being unavailable in literature.

7.2 Future Work

Although preliminary tests were conducted on each sub module as discussed in the main

body of the thesis, comparison should be made between relevant combustor geometry

and operating conditions for further validation. Integration of the three sub modules

would also be a future objective following this validation.

The combustion modelling can also be improved through implementation of evapora-

tion models and detailed network configurations that allow for emission prediction and

flame stability. This would allow the temperature prediction tool to be integrated with

a higher level preliminary design tool.

The zonal method should be implemented on a full annular combustor geometry to

assess the effect of adjacent sectors on the radiation prediction. Third party software can

be used for generation of parametric geometry which can be linked with numerical solver

for zonal method, reducing the programming overhead required for each new geometry.

The models for cooling technology lack the capability to predict mass flow rate based

on pressure drop and this relationship should be added for each model for greater accuracy

and proper integration.

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