Master Thesis

Analysis of Swirling ReactiveFlow in Gas Turbines and

Industrial BurnersAuthor:

Philip Brown (0108784b)

Supervisor:

Dr Nader Karimi

A thesis submitted in partial fulfilment of the requirements

for the degree of Master of Science

in

Aeronautical Engineering

August 31, 2015

Abstract

Power generation using gas turbines in the 21st Century is an increasingly

complex business. Globally, there are more and more pieces of enivironmental

legislation being passed that seek to restrict production of CO2 (in order to

alleviate climate change) as well as oxides of Nitrogen(NOx)(which contribute

to ’smog’ and acid rain). This leads to a need for gas turbine manufacturers to

develop turbines that produce less of these pollutants.The solutions, however,

do not come without a cost. Multi-fuel systems and systems using synthetic

gas (syngas) or hydrogen-enriched lean mix fuels can cause problems with

flame stability, either through thermoacoustic instabilities or flame flashback.

The aim of this paper is to create a framework for investigating the phe-

nomenon of flame flashback, in particular the theory of back-pressure drive

flame propagation mechanism, using a simplified model of the vortex pro-

duced in an experimental combustor (similar to the can combustor in a

gas turbine). A system of equations is solved using Mathematica and the

Burgers-Rott vortex model.

The result is a series of equations that determine values for previously

unknown properties of the flow with a view to further supporting the back-

pressure mechanism. If the back-pressure mechanism is accepted, this could

provide manufacturers with greater understanding of ways to reduce flame

instability when designing multi-fuel/syngas/hydrogen-enriched fuel systems,

thus improving the chances of gas turbines providing a greater contribution

to the transition to greener energy production.

2

Contents

1 Nomenclature 4

1.1 Latin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Greek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Subscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Superscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Introduction 6

2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 The Convergence of Flame Flashback and Vortex Break-

down . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.2 Flame Flashback . . . . . . . . . . . . . . . . . . . . . 8

2.2.3 Improvements to Current Back Pressure Drive Flame

Propagation Mechanism Theory . . . . . . . . . . . . . 8

2.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Theory 10

3.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1

3.1.1 Knowns and Unknowns . . . . . . . . . . . . . . . . . . 10

3.1.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1.3 Known Variables . . . . . . . . . . . . . . . . . . . . . 11

3.1.4 Determination of Unknown Variables . . . . . . . . . . 11

3.2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2.1 Burgers-Rott Vortex Model . . . . . . . . . . . . . . . 12

3.2.2 Core Momentum Equation . . . . . . . . . . . . . . . . 12

3.2.3 Conservation of Momentum . . . . . . . . . . . . . . . 12

3.2.4 Relationship between Angular Velocities and Dimen-

sions of the Vortex’s Cores . . . . . . . . . . . . . . . . 13

3.2.5 Conservation of Angular Momentum . . . . . . . . . . 13

3.2.6 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Experimental Techniques 16

4.1 Determining Velocities . . . . . . . . . . . . . . . . . . . . . . 17

4.1.1 Axial Velocities . . . . . . . . . . . . . . . . . . . . . . 17

4.1.2 Angular Velocities . . . . . . . . . . . . . . . . . . . . 20

4.1.3 Unknown Variables . . . . . . . . . . . . . . . . . . . . 24

5 Results 26

5.1 Non-Dimensional Units . . . . . . . . . . . . . . . . . . . . . . 26

5.1.1 Su . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.1.2 Vf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.1.3 ∆P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6 Discussion 29

2

6.1 Improvements to the Mathematica Code . . . . . . . . . . . . 29

6.1.1 User Input . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.1.2 Optimisation of Numerical Solving . . . . . . . . . . . 29

6.2 Continuation of the Project . . . . . . . . . . . . . . . . . . . 30

6.2.1 Computational Fluid Dynamics . . . . . . . . . . . . . 30

6.2.2 Improvements to experimental apparatus . . . . . . . . 30

6.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6.3.1 Conclusions of the Research . . . . . . . . . . . . . . . 32

6.3.2 Learning Outcomes . . . . . . . . . . . . . . . . . . . . 32

6.3.3 Thanks . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Appendices 34

A Derivation of Flow in Karimi et Al 2015 35

A.1 Conservation of Momentum . . . . . . . . . . . . . . . . . . . 35

A.1.1 Flame Volume . . . . . . . . . . . . . . . . . . . . . . . 36

A.1.2 Forced Vortex Volume . . . . . . . . . . . . . . . . . . 36

A.1.3 Unforced Vortex Volume . . . . . . . . . . . . . . . . . 37

A.2 Conservation of Angular Momentum . . . . . . . . . . . . . . 37

A.2.1 Flame Volume . . . . . . . . . . . . . . . . . . . . . . . 37

A.2.2 Forced Vortex Volume . . . . . . . . . . . . . . . . . . 38

A.2.3 Unforced Vortex Volume . . . . . . . . . . . . . . . . . 38

A.3 Radial Momentum Equation . . . . . . . . . . . . . . . . . . . 39

A.4 Determination of Velocities . . . . . . . . . . . . . . . . . . . 39

A.4.1 Axial Velocities . . . . . . . . . . . . . . . . . . . . . . 39

3

B Derivation of Flow using Burgers-Rott Vortex 42

B.1 Conservation of Momentum . . . . . . . . . . . . . . . . . . . 42

B.1.1 Flame Volume . . . . . . . . . . . . . . . . . . . . . . . 42

B.1.2 Forced Vortex Volume . . . . . . . . . . . . . . . . . . 43

B.1.3 Unforced Vortex Volume . . . . . . . . . . . . . . . . . 43

B.2 Conservation of Angular Momentum . . . . . . . . . . . . . . 44

B.2.1 Flame Volume . . . . . . . . . . . . . . . . . . . . . . . 44

B.2.2 Forced Vortex Volume . . . . . . . . . . . . . . . . . . 44

B.2.3 Unforced Vortex Volume . . . . . . . . . . . . . . . . . 44

B.3 Radial Momentum . . . . . . . . . . . . . . . . . . . . . . . . 45

B.3.1 Pressure along the bluff-body . . . . . . . . . . . . . . 45

B.3.2 Pressure along the flame tube . . . . . . . . . . . . . . 45

B.4 Bernoulli Equation . . . . . . . . . . . . . . . . . . . . . . . . 46

B.4.1 Pressure along the flame tube . . . . . . . . . . . . . . 46

4

Chapter 1

Nomenclature

1.1 Latin

Symbol Description Dimensions Units

A Surface Area L2 m2

n Normal Vector - -

r Radius L m

R Radius of Flame Tube L m

V Velocity LT-1 ms-1

z ( rη/2)2 - -

5

1.2 Greek

Symbol Description Dimensions Units

∆V Relative Axial Velocity - -

Γ∞ Circulation L2T-1 m2s-1

εr εr = rb/ru - -

η Diameter of forced vortex L m

ρ Density ML-3 Kgm-3

Ω Angular Velocity L2T-1 m2s-1

1.3 Subscripts

Symbol Description

b Value in the burning region

c Value at the face of the bluff-body

f Value in the flame

r Radial value

u Value in the unburned region

z Axial value

θ Angular value

1.4 Superscripts

Symbol Description

’ Relating to the forced vortex volume

” Relating to the unforced vortex volume∗ Non-dimensionalised value

6

Chapter 2

Introduction

2.1 Background

With an increase in stringent power generation emissions control, particularly

through the European Union’s Industrial Emissions Directive [16], manufac-

turers and energy producers are becoming more aware of the need for multi-

fuel and lean-mix options in their gas turbines/ combustors which would

produce lower temperature combustion products. This is because nitrogen

oxide (NOx) production is proportional to combustion temperature.

Figure 2.1: NOx Production against Flame Temperature [8]

7

There is also interest in using hydrogen as a ’carbon-neutral’ fuel source

to meet carbon dioxide targets and to help renewable energy sources, working

in tandem with conventional power generation solutions, to act as baseload

power stations [6]. However, this causes problems with phenomena linked

to Combustion Induced Vortex Breakdown (CIVB), a critical mechanism for

flame stabilisation. These are thermoacoustic instabilities, flame flashback

and blow-off.

Vortex breakdown allows engineers to accurately estimate and sustain

the position of flames within a carefully designed combustion chamber. By

achieving this, these machines can provide continuous hot gases to a turbine

or boiler.

Due to the complexity of vortex breakdown, i.e. there are chemical re-

actions, heat transfer as well as three-dimensional unstable fluid mechanics

involved, these phenomena cannot easily be investigated using Direct Numer-

ical Solution (DNS) simulations. It is therefore more efficient to attempt to

model these phenomena in isolation using simplifications of the flow. Some

work has been done in the past to understand flame flashback in particu-

lar.This paper will only seek to bring more understanding to flame flashback.1

1For further information on blow-off, see Santosh J. et al (2008) Lean Blowoff of BluffBody Stabilized Flames: Scaling and Dynamics

8

2.2 Literature Review

2.2.1 The Convergence of Flame Flashback and Vortex

Breakdown

Vortex breakdown was first described in detail in 1957 by Peckham and

Atkinson [1]; however, this was in relation to vortices shed from the tips of

delta wings. Given that this is well after the beginning of the ’Jet Age’ [12],

it could be assumed that either vortex breakdown had not been observed

within the gas turbines of the era or that combustion proceeded in such

a way as to not require flame stabilisation by vortex breakdown. Most of

the literature continues to focus on breakdown from delta wings; however,

Harvey, J.K (1960) begins to explore vortex breakdown in a tube, opening

the way for analysing combustion induced vortex breakdown(CIVB).

Ishizuka et al(2002) [2] mentions that investigation into flame flashback

seems to have started in 1953 by Martin, N.P.W and Moore, D.G [15]. How-

ever , vortex breakdown and flame flashback don’t seem to be linked until

1977 when Chomaik,J. [11] coins the term ’vortex bursting’, which he relates

to vortex breakdown.

2.2.2 Flame Flashback

Regarding flame flashback, Ishizuka et al (1998) [3] discusses two different

mechanisms for the propagation of flame along the vortex line, baroclinic

torque and vortex bursting. Using a simple model of the flow of a Rankine

Vortex (axial velocity is constant) for an unconfined rotating pre-mixed flow,

9

Ishizuka provides compelling evidence for the vortex bursting mechanism.

Experimental and analytical data matches relatively well given the simplicity

of the model.

Karimi et al (2015) [5] expands on this work, showing that Ishizuka’s

assertions still hold when the flow is confined and a bluff body is introduced

along the axis of the flow (effectively simulating a can-type combustor) using

an experimental setup first detailed in Nauert et al (2007) [7]. Equally,

Karimi et al (in press) [4] investigates the presence of an adverse pressure

gradient with a circumferential propagation of the flame. However, the flow

is still modelled as a Rankine vortex.

2.2.3 Improvements to Current Back Pressure Drive

Flame Propagation Mechanism Theory

Whilst Ishizuka’s flow may well be similar to that used in a Rankine vor-

tex, the introduction of a bluff body and enclosure will most likely change

the velocity profile (due to the need to analytically represent the effects of

boundary layers on the edges of the bluff body and enclosure). This is where

this report aims to improve on the model used in Karimi et al (2015). Us-

ing the Burgers-Rott vortex model, detailed in Escudier (1988) [14], rather

than a Rankine vortex, the author aims to more closely model the flow in a

confined flame with a bluff body.

10

2.3 Objectives

• Create a series of equations, solely in known values, for flow in a flame

tube using the Burgers-Rott vortex model .

• Use the known and previously unknown values to generate graphs of

non-dimensionalised pressure along the bluff-body against non-dimensionalised

laminar burning velocity, and non-dimensionalised flame speed against

non-dimensionalised laminar burning velocity.

11

Chapter 3

Theory

3.1 Outline

3.1.1 Knowns and Unknowns

As there are unknowns, equations are needed.

3.1.2 Assumptions

- The flow is laminar, incompressible, inviscid and axisymmetrical.

- The velocities of the flow are consistent with a Burgers-Rott Vortex model.

Knowns Unknownsρu ηu Ωu ∆Vuρb R Ωb ∆Vbrb rc Ω′u ∆V ′uru Su η′u ∆V ′′uηu Y VfAf

Table 3.1: List of known and unknown variables in the fluid flow

12

- The flame is laminar and premixed.

- Flame speed remains constant.

- Momentum is conserved during combustion.

- Burning gases are restricted to the forced vortex volume.

3.1.3 Known Variables

δ = ρuρb

(3.1)

εr = rbru

(3.2)

ε′r = η′uηu

(3.3)

k = ruηu/2

(3.4)

K = ηu/2R

(3.5)

K ′ = rcηu/2

(3.6)

Y = Afπr2

u

(3.7)

Y ′ = k2Y

k2 −K ′2= Afπr2

c − πr2u

(3.8)

(3.9)

3.1.4 Determination of Unknown Variables

There are nine unknowns, requiring nine equations:

- The core momentum can be assumed to be constant (ρuSuAf ) (1 Equa-

tion)

13

- The complete control volume can be split into 3 distinct volumes (burning

region, unburned forced vortex region and unforced vortex region) so each

one will produce an equation in momentum equations (conservation of mo-

mentum and angular momentum). (6 Equations)

- The flow is inviscid, which allows a relationship between angular velocities

and radii of their respective vortex cores to be used (Ω′uΩu = η2

u

η′2u= 1

ε′2r) (1

Equation)

- The radial momentum equation along the bluff body can be used to com-

plete a system of equations for solving all unknown variables (1 Equation)

From the three equations from the conservation of momentum, as well as

the core momentum equation, we can determine ∆Vu, ∆Vb, ∆V ′u and ∆V ′′u

in terms of known variables and ε′r and Vf .

From conservation of angular momentum and the relation between Ωu

and Ω′u for all inviscid flows, we can determine ε′r and Vf in terms of known

variables as well as Ωb in terms of Ωu and known variables.

From the radial momentum equation for a Burgers-Rott vortex in Vortic-

ity and Vortex Dynamics and the Bernoulli equation for the edge of the flame

tube, a value for Ωu can be determined. However, due to the complexity of

the analytical answer for Ωu, a numerical solution must be sought.

14

3.2 Equations

3.2.1 Burgers-Rott Vortex Model

Vz = ∆Vze−(r/η)2 (3.10)

Vr = 0 (3.11)

Vθ = q|Vz|η/r(1− e−(r/η)2) (3.12)

where q = Ωcη

Vz(3.13)

3.2.2 Core Momentum Equation

ρuSuAf = 0 (3.14)

3.2.3 Conservation of Momentum

The generic formula for conservation of momentum is:

∫∫ρ(Vr.n)dA = 0 (3.15)

However, with a cylindrical control volume dA can be replaced and Vr.n

= Vz − Vf .

A = πr2 (3.16)dA

dr= 2πr (3.17)

dA = 2πrdr (3.18)

15

∫ρ(Vz − Vf )2πrdr = 0 (3.19)

3.2.4 Relationship between Angular Velocities and Di-

mensions of the Vortex’s CoresΩ′uΩu

= η2u

η′2u= 1ε′2r

(3.20)

3.2.5 Conservation of Angular Momentum∫∫(V× r)ρ(Vr.n)dA = 0 (3.21)

As with the conservation of momentum, the generic formula can be altered

for a cylindrical control volume.

∫(V× r)ρ(Vr.n)2πrdr = 0 (3.22)

16

For the vector terms

(V× r) = Vθr = Ωcη2(1− e−(r/η)2

) (3.23)

(Vr.n) = Vz − Vf (3.24)

The altered formula is

∫Ωcη

2(1− e−(r/η)2)ρ(Vz − Vf )2πrdr = 0 (3.25)

3.2.6 Pressure

From vorticity and vortex dynamics [9], the radial momentum equation for

a Burgers-Rott vortex is

∂P

∂r= Vθ

r− Vz

∂Vz∂r

(3.26)

Vθ2

r= Ωc

2

zr(1− e−z)2where z = (r

η)2 (3.27)

Vz∂Vz∂r

= −2∆Vz2e−2zz

r(3.28)

∂P

∂r= Ωc

2

zr(1− e−z)2 − 2∆Vz2e−2zz

r(3.29)

(3.30)

The standard form of the Bernouilli equation is as below

P

ρ+ 1

2V2 − Φ = constant (3.31)

17

Rearranging for two different pressures along a streamline gives

P1 + 12ρ1V

21 − ρ1Φ = P2 + 1

2ρ2V2

2 − ρ2Φ (3.32)

In all our equations Φ is assumed to be zero.

P1 − P2 = 12ρ2V

22 −

12ρ1V

21 (3.33)

18

Chapter 4

Experimental Techniques

As in previous papers, the aim was to solve most of the unknowns using

analytical methods; however, due to the complexity of the model, Mathe-

matica [17] was used to improve the speed of simplifying the equations after

substitutions are performed. Mathematica was also able to provide numerical

analysis of variables when required.

19

Figure 4.1: Description of model from Karimi et al (2015) [5]

4.1 Determining Velocities

4.1.1 Axial Velocities

∆Vu

Taking the conservation of momentum equation for the core, the term for

the velocity in the burned region can be eliminated.

π∆Vu(ηu2 )2ρu(e

−( ruηu2

)2

− e−( rcηu

2)2

)− πVfρu(r2u − r2

c ) = AfSuρu (4.1)

Rearranging, reducing and using equations for variables (k, K’, Y and Y’)

20

gives ∆Vu in terms of variables, Vf and Su only.

∆Vu =(k2Y )(Vf

Y ′+ Su)

e−k2 − e−K′2(4.2)

∆ Vb

Again, the conservation equation for the core can be used, this time with the

term for velocity in the unburned region eliminated.

π∆Vbρb(η′u2 )2(e

−( rbη′u2

)2

− e−( rc

η′u2

)2

)− πρbVf (r2b − r2

c ) = AfSuρu (4.3)

Rearrangment, reduction and the substitution of variables (k,K’,δ, Y,εr)

gives ∆Vb in terms of variables, Vf and Su and ε′r only.

∆Vb = Vf (k2ε2r +K ′2) + δk2Y Su

ε′2r (e−k2ε2

rε′2r − e−

K′2ε′r2 )

(4.4)

21

∆V′u

∆V ′u was found using the second conservation of momentum equation.

14π∆Wuη

2uρu(e

− 4r2u

η2u − 1

e)− πVfρu(r2

u − η2u) = πVfρu(r2

b −η′2u4 ) + 1

4πρu∆W′uη′2u (e−4r2b

η′2u − 1e

) (4.5)

This time, previously calculated velocities were substituted into the rearranged equation to give ∆V ′u in terms of

variables and Vf only.

∆V′u = ek2ε2

rε′2r (Vf (eK

′(e(−k2ε2r − (K ′)2 + (ε′)2

r + 4) + ek2((K ′)2 − k2)) + ek

2+1(k2ε2r + k2 − (ε′)2

r − 4))− (ek2 − e)k2Y e(K′)2Su)

(ek2 − eK′)ε′2r (ek2ε2

rε′2r − e)

(4.6)

22

∆V′′u

∆V ′′u can be determined by using the third conservation of momentum equation.

π∆Wu(ηu2 )2ρu(e

−( ruηu2

)2

− 1e

)− πVfρu(r2u − η2

u) = πVfρu(r2b − (η

′u2 )2) + πρu∆W′u(η

′u2 )2(e

−( rbη′u2

)2

− 1e

) (4.7)

Again, rearranging and substituting in ∆Vu gives:

∆V′′u =e

1ε′2r−1

K2 (k2(e

1K2 −e)K2Y (

VfY ′+Su)

e−k2−e−(K′)2 − 4e1K2 +1Vf (K2ε′2r +K2 − 2))

K2ε′2r (e1

K2ε′2r − e)(4.8)23

Vf

Using the third conservation angular momentum equation, the relationship between Ωu and Ω′u, the equations for

∆Vu and ∆V ′′u and variables gives Vf.

π(ηu2 )4ρuΩu(Vf (1− (Rηu2

)2) + Vf (1e− e

−( Rηu2

)2

) + 12∆Vu(e

−2( Rηu2

)2

− 1e2 ) + ∆Vu(

1e− e

−( Rηu2

)2

))

= πρu(η′u2 )4Ω′u(Vf (1− (R

η′u2

)2) + Vf (1e− e

−( Rη′u2

)2

) + 12∆W′′u(e

−2( Rη′u2

)2

− 1e2 ) + ∆W′′u(

1e− e

−( Rη′u2

)2

))(4.9)

Vf = 2(e1K2 − e)Y Su(e

1K2ε′2r − e

1K2 )

(e−K2ε′2r − e−k2)( e1K2 (e

1K2 +1

(− 16K2 +ε′2r +2)+e

1K2ε′2r (e

1K2 (−3(e−1)ε′2r −5e+1)+ 16(2e−1)e

1K2

K2 +e))k2 + 2(e−e

1K2 )(K2−(K′)2)(e

1K2 −e

1K2ε′2r )ek2+K2ε′2r

K2(eK2ε′2r −ek2 ))

(4.10)

24

4.1.2 Angular Velocities

Ωu

Taking the radial momentum and Bernoulli equation along the flame tube, the only way to determine Ωu is by

putting numerical values for ∆Vu, ∆V ′′u ,Vf , the variables, and replacing Ω′u with Ωuε′2r

. The equation can then be used

with the Solve function in Mathematica to give a numerical value for Ωu.

12((Vf − e−

1K2 ∆Vu)2 − (Vf −∆V′ue

− 1K2ε′2r )2)

= (Vf (∆V′2u e− 1K2ε′2r − e−

1K2 ∆V2

u)−12(∆V′2u e

− 2K2ε′2r − e−

2K2 ∆V2

u))

−(K2Ω2u(−

2Ei(− 1K2ε′2r

)K2ε′2r

+2Ei(− 2

K2ε′2r)

K2ε′2r+ e

− 2K2ε′2r (e

1K2ε′2r − 1)2)−K2(

2Ei(− 2K2 )

K2 −2Ei(− 1

K2 )K2 + e−

2K2 (e

1K2 − 1)2)Ω2

u)

(4.11)

25

Ωb

Taking the first equation for conservation of angular momentum and rearranging gives Ωb in terms of Ωu.

π(ηu2 )4ρuΩu(Vf ((rcηu2

)2 − ( ruηu2

)2) + Vf (e−( rcηu

2)2

− e−( ruηu

2)2

) + 12∆Vu(e

−2( ruηu2

)2

− e−2( rcηu

2)2

) + ∆Vu(e−( rcηu

2)2

− e−( ruηu

2)2

))

= πρbΩb(η′u2 )4(Vf ((

rcη′u2

)2 − ( rbη′u2

)2) + Vf (e−( rc

η′u2

)2

− e−( rb

η′u2

)2

) + ∆Vb(e−( rc

η′u2

)2

− e−( rb

η′u2

)2

) + 12∆Vb(e

−2( rbη′u2

)2

− e−2( rc

η′u2

)2

))

(4.12)

26

Ωb = −AB

(4.13)

where A =e−k2−(K′)2δΩu(2e3k2+2(K′)2

k2 − 6eε2rk

2

(ε′)2r

+k2+1Ck2 + 2e

ε2rk

2

(ε′)2r

+2k2+1 + 2e2(k2+(K′)2) + 2e2k2+(K′)2+1 − 2e3k2+(K′)2

−2eε2rk

2

(ε′)2r

+k2+(K′)2+1 + 2eε2rk

2

(ε′)2r

+2k2+(K′)2

− 4eε2rk

2

(ε′)2r

+2k2+(K′)2+1 − 2ek2+2(K′)2(−e+ ek2)(ek2 − e

k2ε2r

(ε′)2r )(K ′)2

−4e3k2+1(k2 − 1) + 10eε2rk

2

(ε′)2r

+3k2+1(k2 − 1) + 8e3k2+(K′)2+1(k2 − 1)− 20eε2rk

2

(ε′)2r

+3k2+(K′)2+1(k2 − 1) + 4eε2rk

2

(ε′)2r

+k2+2(K′)2

(k2 − 1)

+20eε2rk

2

(ε′)2r

+2k2+2(K′)2+1(k2 − 1) + ek2+2(K′)2+1(4k2 − 6) + e

k2( ε2r

(ε′)2r

+1)C(4k2 − 3) + ek

2+1C(4k2 + 1)

+eε2rk

2

(ε′)2r

+3k2+(K′)2

(8k2 − 4) + ek2( ε2

r(ε′)2

r+3)(2− 4k2) + e

ε2rk

2

(ε′)2r

+k2+2(K′)2+1(14− 8k2) + ek2ε2

r(ε′)2

r+2(k2+(K′)2)(4− 10k2) + e2k2+2(K′)2+1(8− 10k2))Y ′

(4.14)

27

and B =(ε′)2r(2e(K′)2(−e+ ek

2)(ek2 − ek2ε2

r(ε′)2

r )(−e−k2ε2

r(ε′)2

r + e− (K′)2

(ε′)2r )Y ′(ε′)2

r + 2e2(K′)2

(ε′)2r (−e−

k2ε2r

(ε′)2r + e

− (K′)2

(ε′)2r )

(δ + e(K′)2(−e+ ek2)(ek2 − e

k2ε2r

(ε′)2r )((K ′)2 − k2ε2

r)Y ′D

)h+ e2(K′)2

(ε′)2r (e−

2k2ε2r

(ε′)2r − e−

2(K′)2

(ε′)2r )

(δ + e(K′)2(−e+ ek2)(ek2 − e

k2ε2r

(ε′)2r )((K ′)2 − k2ε2

r)Y ′D

)D + 2e(K′)2(−e+ ek2)(ek2 − e

k2ε2r

(ε′)2r )((K ′)2 − k2ε2

r)Y ′)

(4.15)

C = (2e2k2+(K′)2 − 2ek2+2(K′)2 − e2k2 + e2(K′)2)(ε′)2r (4.16)

D = Y ′(ek2 − e(K′)2)(2ek2k2ε2

r(3ek2ε2

r(ε′)2

r+1 − 2e

k2ε2r

(ε′)2r − 2e) + ek

2(ε′)2r(3e

k2ε2r

(ε′)2r − e) + 2(5(k2 − 1)e

k2ε2r

(ε′)2r

+k2+1 + ek2ε2

r(ε′)2

r+1

+(1− 2k2)ek2( ε2

r(ε′)2

r+1) − 2ek2+1(k2 − 1))) + (ek2 − e)k2Y e(K′)2(ek2 − e

k2ε2r

(ε′)2r )

(4.17)

28

4.1.3 Unknown Variables

ε′r

Taking the third equation for conservation of angular momentum and replacing Ω′u, the equation can be rerarranged

to give Ω′uΩu . If this is replaced with 1

ε′2r, the resulting equation can be solved with numerical values for variables.

1(ε′r)2 =

(e1K2 − e)ε4

r((−e−

2k2ε2r

(ε′r)2 +2e−k2ε2

r(ε′r)2 + 1

e2−2e

)e

1(ε′)2

r−1

K2 (k2K2(F (k2−(K′)2)

E+1)

e−k2−e−(K′)2 + 8e1K2 +1

(e1K2 −e

1K2(ε′)2

r )(K2(ε′)2r+K2−2)

E)

K2(ε′)2r(e

1K2(ε′)2

r −e)+

4( k2ε2r

(ε′r)2 +e−k2ε2

r(ε′r)2 − 1

e−1)(e

1K2(ε′)2

r −e1K2 )

E)

(−e−2k2+2e−k2+ 1e2−

2e

)k2(F (k2−(K′)2)E

+1)e−k2−e−(K′)2 + 2F (k2+e−k2− 1

e−1)

E

(4.18)

where E = (e−K2(ε′)2r − e−k2)(

e1K2 (e

1K2 +1(− 16

K2 + (ε′)2r + 2) + e

1K2(ε′)2

r (e1K2 (−3(e− 1)(ε′)2

r − 5e+ 1) + 16(2e−1)e1K2

K2 + e))k2

+2(e− e1K2 )(K2 − (K ′)2)(e

1K2 − e

1K2(ε′)2

r )ek2+K2(ε′)2r

K2(eK2(ε′)2r − ek2) )

(4.19)

29

and F = 2(e− e1K2 )(e

1K2 − e

1K2(ε′)2

r )ek2+K2(ε′)2r (4.20)

30

Chapter 5

Results

With values for all variables and velocities, graphs that investigate the back-

pressure drive flame propagation theory can be created. The two most im-

portant values of interest are flame speed(Vf) and the pressure along the

bluff-body (∆Pr=rc). As there is a particular interest in whether laminar

burning velocity (Su) affects this, they were plotted against it. To remove

the chance of incorrect dimension use, these are non-dimensionalised.

5.1 Non-Dimensional Units

For the graphs, only Su needs to be altered in the equations, due to it varying

in the graphs.

5.1.1 Su

S∗u = [ ρuPu

]× S2u (5.1)

where Pu= 105Pa and ρu = 1.225kgm−3

31

5.1.2 Vf

V ∗f = 1Su(CH4)

× Vf (5.2)

where Su(CH4) = 0.4ms−1

5.1.3 ∆P

∆P ∗ = ∆PPu

(5.3)

where Pu = 105Pa

∆P ∗ and V*f can be evaluated against a different variable (k, K, K’ or

εr) to give graphs similar to those below.

The author did not have enough time to complete a set of graphs for a

Burgers-Rott vortex.

Figure 5.1: Example of Graph of V ∗f vs S∗u

32

Figure 5.2: Example of Graph of ∆P ∗ vs S∗u

33

Chapter 6

Discussion

6.1 Improvements to the Mathematica Code

6.1.1 User Input

Currently, the variables must be changed by hand. With some understanding

of the Wolfram language, the author believes the code could be altered to

make these so the user can input them.

It may also be possible for the user to indicate which variable they wish to

use as the key variable. This would require extensive automation. However,

for multiple variables, the time investment may be worth it.

6.1.2 Optimisation of Numerical Solving

The time taken to solve ε′r is quite long at the moment, through some opti-

misation of the equation used (perhaps through use of FullSimplify), less

time can be spent determining a value that affects most other properties of

34

the flame tube flow.

6.2 Continuation of the Project

6.2.1 Computational Fluid Dynamics

As can be seen from the complexity of the analysis of the flame tube using

a Burgers-Rott vortex model, further analytical investigation of the flame

flashback mechanism would seem to be increasingly difficult. It may therefore

be preferable to attempt to model the flame tube using a Computational

Fluid Dynamics (CFD) package.

Reynolds-Averaged Navier Stokes

The first option for CFD would be a Reynolds-Averaged Navier Stokes (RANS)

simulation. This would add another layer of complexity to the model, in the

form of allowing turbulence. However, this would not include any chemi-

cal reactions or heat transfer between the volumes of the flame tube. The

advantage of this approach would be to test different combustion chamber

geometries without creating experimental apparatus.

Direct Numerical Simulation

By far the most computer-intensive but also most accurate way of checking

the validity of the back-pressure theory would be to use Direct Numerical

Simulation (DNS). With a carefully constructed control volume and an ac-

curate representation of initial conditions from the experiment, this would

35

provide very accurate results.

6.2.2 Improvements to experimental apparatus

There are several improvements that the experimental apparatus could un-

dergo to test other types of combustion chamber setup and to more closely

match a combustion chamber.

Dilution Holes

Unlike a genuine combustion chamber, the experimental apparatus ends

straight onto an opening to the atmosphere. A more accurate representa-

tion would be to add dilution holes prior to the nozzle. Although this may

not provide any more insight from a purely fluid mechanical point of view,

any thermal contributors involved in the back-pressure mechanism would be

affected by a lower temperature created by this common feature of combus-

tion chambers.

Figure 6.1: Diagram of dilution holes [18]

36

Cannular Setup

Rather than having a true cannular setup, it may be interesting to see the

effect of ’communication’ between can-like experimental apparatus. This

could be done in a similar system to some cannular setups (using ducts

which allow the interaction between flame tubes).

Figure 6.2: Diagram of a Can-Annular(Cannular) Combustor [19]

This should add another layer of authenticity as several manufacturers use

cannular combustion chambers as a means of saving on weight, by reducing

the need for each can to act as a pressure vessel.

6.3 Conclusions

6.3.1 Conclusions of the Research

Although the author was not able to create graphs to verify the back-pressure

drive flame propagation mechanism, he did manage to produce a tool for

future students to evaluate the theory using a more complex vortex model.

37

6.3.2 Learning Outcomes

Regarding learning outcomes, the author feels he has learned a great deal

from this research project, not least two more programming language (Math-

ematica and LaTeX). The assignment has been very mentally challenging,

taking on an open research question as well as verifying the results of three

papers( [3] [5] [4]).

6.3.3 Thanks

The author would like to thank Dr Nader Karimi for the opportunity of

investigating the phenomenon of flame flashback. The author would also like

to thank his course colleagues (Alberto Estellar Torres, Rémi Borfigat and

Hjalmar Gretarsson) and fiancée for their support during the year.

38

Appendices

39

Appendix A

Derivation of Flow in Karimi et

Al 2015

The first thing to notice is that the control volume is split into 3 distinct

sub-volumes. These are: the flame and control volume directly before it; the

forced vortex area and the free vortex area. The Rankine vortex model is

used to determine axial, radial and angular velocities.

Vz = constant for rc ≤ r ≤ R (A.1)

Vr = 0 (A.2)

Vθ =

Ωr for rc ≤ r ≤ ηu/2

Ωη2u

4r for ηu/2 ≤ r ≤ R

(A.3)

40

A.1 Conservation of Momentum

We start with the equation for the conservation of momentum

∫∫ρ(Vr.n)dA = 0 (A.4)

In all three sections of the control volume, the equation below can be de-

termined by an alteration to the above equation. A full derivation is provided

for the flame volume only due to similarities in the derivations.

∫ρ(Vr.n)2πrdr = 0 (A.5)

A.1.1 Flame Volume∫ ru

rcρu(Vu − Vf )2πrdr =

∫ rb

rcρb(Vb − Vf )2πrdr (A.6)

[ρu(Vu − Vf )πr2]rurc = [ρb(Vb − Vf )πr2]rbrc (A.7)

41

ρu(Vu − Vf )π(r2u − r2

c ) = ρb(Vb − Vf )π(r2b − r2

c ) (A.8)

A.1.2 Forced Vortex Volume

ρu(Vu − Vf )π((ηu/2)2 − r2u) = [ρu(V ′u − Vf )π((η′u/2)2 − r2

b ) (A.9)

A.1.3 Unforced Vortex Volume

ρu(Vu − Vf )π(R2 − (ηu/2)2) = ρu(V ′′u − Vf )π(R2 − (η′u/2)2) (A.10)

A.2 Conservation of Angular Momentum

Again, we start from the general equation.

∫∫(r× v)ρ(v.n)dA = 0 (A.11)

42

Equally, each volume is cylindrical in nature, as with the conservation of

momentum.

∫(r× v)ρ(v.n)2πrdr = 0 (A.12)

A.2.1 Flame Volume∫ ru

rcρuΩu(Vu − Vf )2πr3dr =

∫ rb

rcρbΩ′u(Vb − Vf )2πr3dr (A.13)

ρuΩu(Vu − Vf )[r4/4]rurc = ρbΩ′u(Vb − Vf )[r4/4]rbrc (A.14)

ρuΩu(Vu − Vf )(r4u/4− r4

c/4) = ρbΩ′u(Vb − Vf )(r4b/4− r4

c/4) (A.15)

43

A.2.2 Forced Vortex Volume

ρuΩu(Vu−Vf )((ηu/2)4/4−r4u/4) = ρuΩ′u(V ′u−Vf )((η′u/2)4/4−r4

b/4) (A.16)

A.2.3 Unforced Vortex Volume∫ R

ηu/2ρuΩu(η/u2)2(Vu − Vf )2πrdr

=∫ R

η′u/2ρuΩ′u(η′u/2)2(V ′′u − Vf )2πrdr

(A.17)

ρuΩu(ηu/2)2(Vu − Vf )2π[r2/2]Rηu/2

= ρuΩ′u(η′u/2)2(V ′′u − Vf )2π[r2/2]Rη′u/2(A.18)

ρuΩu(ηu/2)2(Vu − Vf )π[R2 − (ηu/2)2]

= ρuΩ′u(η′u/2)2(V ′′u − Vf )π[R2 − (η′u/2)2](A.19)

44

A.3 Radial Momentum Equation

There is only a need for one further equation, that is the radial momentum

equation.

∂P

∂r= ρ

V 2θ

r(A.20)

(A.21)

Integrating over the upstream and downstream values provides another equa-

tion for obtaining unknown properties of the flame tube.

(∆P )R − (∆P )rc =∫ρV 2θ

rdr (A.22)

=∫ρΩ2r for sections where rc ≤ r ≤ ηu/2 (A.23)

and∫ρ

Ω2(ηu/2)4

r3 for sections where ηu/2 ≤ r ≤ R (A.24)

45

A.4 Determination of Velocities

A.4.1 Axial Velocities

Vu

Using the core momentum equation and the first section of the flame volume

conservation of momentum equation, Vu can be determined.

ρu(Vu − Vf )π(r2u − r2

c ) = AfSuρu (A.25)

rearranging to Vu − Vf gives

Vu − Vf = − AfSuπ(rc2 − ru2) (A.26)

However, Y ′ = − Afπ(rc2−ru2) , so

Vu = Y ′Su + Vf (A.27)

Vb

Similar to Vu, Vb can be determined from the core momentum equation and

the second section of the flame volume conservation of momentum equation.

ρb(Vb − Vf )π(r2b − r2

c ) = AfSuρu (A.28)

46

Rearranged to Vb − Vf gives

Vb − Vf = − ρuAfSuπρb(rb2 − rc2) (A.29)

Replacing − Afπ(rb2−rc2) with Y ′

ε′′rgives

Vb = ρuρb

Y ′Suε′′r

+ Vf (A.30)

V′′u

Taking the unforced vortex volume’s equation for conservation of momentum,V′′u

can be determined .

ρuΩu(ηu/2)2(Vu − Vf )π[R2 − (ηu/2)2]

= ρuΩ′u(η′u/2)2(V ′′u − Vf )π[R2 − (η′u/2)2](A.31)

V ′′u − Vf = (Ωu

Ω′u)(ηu/2)2

(η′u/2)2[R2 − (ηu/2)2][R2 − (η′u/2)2] (Vu − Vf ) (A.32)

Using the relationship between Ωu and Ω′u and the previously determined

value for Vu, the equation reduces somewhat.

Ωu

Ω′u= (η′u)2

ηuand Vu − Vf = Y ′Su (A.33)

V ′′u − Vf = [R2 − (ηu/2)2][R2 − (η′u/2)2]Y

′Su (A.34)

47

The relationships η′u/2 = εrηu/2 and K = ηu/2R

complete the reduction.

V ′′u − Vf = [R2 − (ηu/2)2][R2 − (εrηu/2)2]Y

′Su (A.35)

V ′′u = [1−K2][1− (εrK)2]Y

′Su + Vf (A.36)

48

Appendix B

Derivation of Flow using

Burgers-Rott Vortex

As with Appendix A, the control volume can be split up into three sections.

Vz = ∆Vzer/η2

(B.1)

Vr = 0 (B.2)

Vθ = q|ηVz|η/r(1− e−r/η2) (B.3)

where q = Ωη/ηVz (B.4)

49

B.1 Conservation of Momentum

B.1.1 Flame Volume

πρu(∆Vuηu/22[e−(ru/(ηu/2))2 − e−(rc/(ηu/2))2 ]− Vf (ru2 − rc2))

= πρb(∆Vbηb/22[e−(rb/(ηb/2))2 − e−(rc/(ηb/2))2 ]− Vf (rb2 − rc2)) = ρSuAf

(B.5)

B.1.2 Forced Vortex Volume

πρu(∆Vuηu/22[e−(ru/(ηu/2))2 − e−(rc/(ηu/2))2 ]− Vf (ru2 − rc2))

= πρ(∆V ′uηb/22[e−(rb/(ηb/2))2 − e−(rc/(ηb/2))2 ]− Vf (rb2 − rc2))

(B.6)

B.1.3 Unforced Vortex Volume

ρu∆Wuηu2[e−1 − e−(R/ηu)2 ]− ρuVf (ηu2 −R2)

= ρu∆W ′′u ηb

2[e−1 − e−(R/ηb)2 ]− ρuVf (ηb2 −R2)(B.7)

50

B.2 Conservation of Angular Momentum

B.2.1 Flame Volume

π(ηu2 )4ρuΩu(Vf ((rcηu2

)2 − ( ruηu2

)2) + Vf (e−( rcηu

2)2

− e−( ruηu

2)2

) + 12∆Wu(e

−2( ruηu2

)2

− e−2( rcηu

2)2

) + ∆Wu(e−( rcηu

2)2

− e−( ruηu

2)2

)

= πρbΩb(η′u2 )4(Vf ((

rcη′u2

)2 − ( rbη′u2

)2) + Vf (e−( rc

η′u2

)2

− e−( rb

η′u2

)2

) + ∆Wb(e−( rc

η′u2

)2

− e−( rb

η′u2

)2

) + 12∆Wb(e

−2( rbη′u

2)2

− e−2( rc

η′u2

)2

)

(B.8)

B.2.2 Forced Vortex Volume

π(ηu2 )4ρuΩu(Vf ((ruηu2

)2 − 1) + Vf (e−( ruηu

2)2

− 1e

) + ∆Vu(e−( ruηu

2)2

− 1e

) + 12∆Vu(

1e2 − e

−2( ruηu2

)2

))

= πρu(η′u2 )4Ω′u(Vf ((

rbη′u2

)2 − 1) + Vf (e−( rb

η′u2

)2

− 1e

) + ∆V′u(e−( rb

η′u2

)2

− 1e

) + 12∆V′u(

1e2 − e

−2( rbη′u2

)2

))(B.9)

B.2.3 Unforced Vortex Volume

π(ηu2 )4ρuΩu(Vf (1− (Rηu2

)2) + Vf (1e− e

−( Rηu2

)2

) + 12∆Vu(e

−2( Rηu2

)2

− 1e2 ) + ∆Vu(

1e− e

−( Rηu2

)2

))

= πρu(η′u2 )4Ω′u(Vf (1− (R

η′u2

)2) + Vf (1e− e

−( Rη′u2

)2

) + 12∆V′′u(e

−2( Rη′u2

)2

− 1e2 ) + ∆V′′u(

1e− e

−( Rη′u2

)2

))(B.10)

51

B.3 Radial Momentum

B.3.1 Pressure along the bluff-body

Pur=rc − Pbr=rc = −Ω2b(−

2(K′)2Ei(− (K′)2

(ε′)2r

)

(ε′)2r

+2(K′)2Ei(− 2(K′)2

(ε′)2r

)

(ε′)2r

+ e− 2(K′)2

(ε′)2r (e

(K′)2

(ε′)2r − 1)2)

(K′)2

(ε′)2r

+ ∆VbVfe− (K′)2

(ε′)2r − 1

2∆V2be− 2(K′)2

(ε′)2r

+Ω2u(2(K ′)2Ei(−(K ′)2)− 2(K ′)2Ei(−(K ′)2) + e−2(K′)2(e(K′)2 − 1)2)

(K ′)2 − Vfe−(K′)2∆Vu + 12e−2(K′)2∆V2

u

(B.11)

B.3.2 Pressure along the flame tube

Pur=R − Pbr=R = ρu(Vf (∆V2ute− 4R2

(η′)2u −∆V2

ue− 4R2

η2u )− 1

2(∆V2ute− 8R2

(η′)2u −∆V2

ue− 8R2

η2u ))

−(Ω2u(−

(8R2)Ei(− 4R2(η′)2

u)

(η′)2u

+2(4R2)Ei(− 8R2

(η′)2u

)

(η′)2u

+ e− 8R2

(η′)2u (e

4R2(η′)2

u − 1)2)(4R2)(ε′)2

r

(η′)2u

−Ω2u(−

(8R2)Ei(− 4R2η2u

)

η2u

+2(4R2)Ei(− 8R2

η2u

)

η2u

+ e− 8R2

η2u (e

4R2η2u − 1)2)

4R2

η2u

)

(B.12)

52

B.4 Bernoulli Equation

B.4.1 Pressure along the flame tube

Pur=R − Pbr=R = 12ρu((Vf −∆Vue

− 4R2η2u )2 − (Vf − e

− 4R2η′2u ∆V′u)2) (B.13)

Further derivation is available in the Mathematica file(Project.nb) - https:

//www.dropbox.com/s/on5uwx9vodzwpca/Project.nb?dl=0

53

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