This thesis examines swirling reactive flow in gas turbines and industrial burners. The author develops equations to model flame flashback using the Burgers-Rott vortex model. Experimental data on vortex velocities is used to determine unknown properties of the flow. This improves understanding of the back-pressure flame propagation mechanism and how to reduce flame instability in multi-fuel systems, supporting greener energy production.
14 روحًا قُلبت حياتهم رأسًا على عقب جرّاء اعتقالهم قهرًا وظُلمًا في وطنِ انعدمت فيهٍ الحُريّات وانتُهكت القوانين، ليَعقبه حُكمًا قاسيًا بإعدامهم شنقًا دون أن يُكلف القضاء أنفسهم عناء البحثٍ عن الحقيقة.
فمن هُم؟ وماهي قصتهم؟ وما الذي أدى إلى الحُكم عليهم بالإعدام؟
#الحياة_حق | أوقفوا أحكام الاعدام بحق 14 بريء بالدقهلية.
14 روحًا قُلبت حياتهم رأسًا على عقب جرّاء اعتقالهم قهرًا وظُلمًا في وطنِ انعدمت فيهٍ الحُريّات وانتُهكت القوانين، ليَعقبه حُكمًا قاسيًا بإعدامهم شنقًا دون أن يُكلف القضاء أنفسهم عناء البحثٍ عن الحقيقة.
فمن هُم؟ وماهي قصتهم؟ وما الذي أدى إلى الحُكم عليهم بالإعدام؟
Internship Report: Interaction of two particles in a pipe flowPau Molas Roca
The present document sums up the development and results of the research internship carried out at LEGI Laboratory. The study aimed to understand the hydrodynamic forces involvement in the interaction between two red blood cells located in a capillary (pipe flow). The problem regarding Red Blood Cells (RBCs) moving through a capillary has been tackled from a two-dimensional point of view and has been both analytically and numerically outlined. Finite elements have been used to discretize the geometries considered. Several boundary conditions and geometries were simulated and deeply examined aiming to understand the mechanism governing hydrodynamic attraction and repulsion between red blood cells. The consequent results are analyzed in this report.
1d 2d heat transfer lumped capacitance model giesler chart, forced and free convection heat transfer, radiation, heat exchangers, boiling and condensation for mechanical engineering students
Definition, modes of heat transfer, critical radius of cooling, conduction from extended and finned surfaces, 1d and 2d steady heat conduction, unsteady head conduction with and without convective boundary layer, convection heat transfer, radiation heat transfer, heat exchangers, boiling and condensation
Fluid properties statics control volume integral and differential relations dimensional analysis pipe losses compressible and inexpressible boundary layers
Short notes on fluid properties, fluid statics, integral and differential relations of control volume, dimensional analysis and similitude, boundary layer theory, compressible and in compressible boundaries, measure of boundary layer thickness and few worked examples along with review questions in each section
1. Master Thesis
Analysis of Swirling Reactive
Flow in Gas Turbines and
Industrial Burners
Author:
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
2. 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.
8. 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 ( r
η/2
)2
- -
5
9. 1.2 Greek
Symbol Description Dimensions Units
∆V Relative Axial Velocity - -
Γ∞ Circulation L2
T-1
m2
s-1
εr εr = rb/ru - -
η Diameter of forced vortex L m
ρ Density ML-3
Kgm-3
Ω Angular Velocity L2
T-1
m2
s-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
10. 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
11. 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
1
For further information on blow-off, see Santosh J. et al (2008) Lean Blowoff of Bluff
Body Stabilized Flames: Scaling and Dynamics
8
12. 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
13. 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
14. 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
15. 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 Unknowns
ρu ηu Ωu ∆Vu
ρb R Ωb ∆Vb
rb rc Ωu ∆Vu
ru Su ηu ∆Vu
ηu Y Vf
Af
Table 3.1: List of known and unknown variables in the fluid flow
12
16. - 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 =
ρu
ρb
(3.1)
εr =
rb
ru
(3.2)
εr =
ηu
ηu
(3.3)
k =
ru
ηu/2
(3.4)
K =
ηu/2
R
(3.5)
K =
rc
ηu/2
(3.6)
Y =
Af
πr2
u
(3.7)
Y =
k2
Y
k2 − K 2
=
Af
πr2
c − πr2
u
(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
17. - 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 (Ωu
Ωu
= η2
u
η 2
u
= 1
ε 2
r
) (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, ∆Vu and ∆Vu
in terms of known variables and εr and Vf .
From conservation of angular momentum and the relation between Ωu
and Ωu for all inviscid flows, we can determine εr 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
18. 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
19. ρ(Vz − Vf )2πrdr = 0 (3.19)
3.2.4 Relationship between Angular Velocities and Di-
mensions of the Vortex’s Cores
Ωu
Ωu
=
η2
u
η 2
u
=
1
ε 2
r
(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
20. 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
Ωcη2
(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)
Vθ
2
r
=
Ωc
2
zr
(1 − e−z
)2
where z = (
r
η
)2
(3.27)
Vz
∂Vz
∂r
= −
2∆Vz
2
e−2z
z
r
(3.28)
∂P
∂r
=
Ωc
2
zr
(1 − e−z
)2
−
2∆Vz
2
e−2z
z
r
(3.29)
(3.30)
The standard form of the Bernouilli equation is as below
P
ρ
+
1
2
V 2
− Φ = constant (3.31)
17
21. Rearranging for two different pressures along a streamline gives
P1 +
1
2
ρ1V 2
1 − ρ1Φ = P2 +
1
2
ρ2V 2
2 − ρ2Φ (3.32)
In all our equations Φ is assumed to be zero.
P1 − P2 =
1
2
ρ2V 2
2 −
1
2
ρ1V 2
1 (3.33)
18
22. 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
23. 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(
ηu
2
)2
ρu(e
−( ru
ηu
2
)2
− e
−( rc
ηu
2
)2
) − πVf ρu(r2
u − r2
c ) = Af Suρu (4.1)
Rearranging, reducing and using equations for variables (k, K’, Y and Y’)
20
24. gives ∆Vu in terms of variables, Vf and Su only.
∆Vu =
(k2
Y )(
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(
ηu
2
)2
(e
−(
rb
ηu
2
)2
− e
−( rc
ηu
2
)2
) − πρbVf (r2
b − r2
c ) = Af Suρu (4.3)
Rearrangment, reduction and the substitution of variables (k,K’,δ, Y,εr)
gives ∆Vb in terms of variables, Vf and Su and εr only.
∆Vb =
Vf (k2
ε2
r + K 2
) + δk2
Y Su
ε 2
r (e
−
k2ε2
r
ε 2
r − e
−K 2
εr2
)
(4.4)
21
25. ∆V u
∆Vu was found using the second conservation of momentum equation.
1
4
π∆Wuη2
uρu(e
−
4r2
u
η2
u −
1
e
) − πVf ρu(r2
u − η2
u) = πVf ρu(r2
b −
η 2
u
4
) +
1
4
πρu∆Wuη 2
u (e
−
4r2
b
η 2
u −
1
e
) (4.5)
This time, previously calculated velocities were substituted into the rearranged equation to give ∆Vu in terms of
variables and Vf only.
∆Vu =
e
k2ε2
r
ε 2
r (Vf (eK
(e(−k2
ε2
r − (K )2
+ (ε )2
r + 4) + ek2
((K )2
− k2
)) + ek2+1
(k2
ε2
r + k2
− (ε )2
r − 4)) − (ek2
− e)k2
Y e(K )2
Su)
(ek2
− eK )ε 2
r (e
k2ε2
r
ε 2
r − e)
(4.6)
22
26. ∆V u
∆Vu can be determined by using the third conservation of momentum equation.
π∆Wu(
ηu
2
)2
ρu(e
−( ru
ηu
2
)2
−
1
e
) − πVf ρu(r2
u − η2
u) = πVf ρu(r2
b − (
ηu
2
)2
) + πρu∆Wu(
ηu
2
)2
(e
−(
rb
ηu
2
)2
−
1
e
) (4.7)
Again, rearranging and substituting in ∆Vu gives:
∆Vu =
e
1
ε 2
r
−1
K2
(
k2(e
1
K2 −e)K2Y (
Vf
Y
+Su)
e−k2
−e−(K )2 − 4e
1
K2 +1
Vf (K2
ε 2
r + K2
− 2))
K2ε 2
r (e
1
K2ε 2
r − e)
(4.8)
23
27. Vf
Using the third conservation angular momentum equation, the relationship between Ωu and Ωu, the equations for
∆Vu and ∆Vu and variables gives Vf.
π(
ηu
2
)4
ρuΩu(Vf (1 − (
R
ηu
2
)2
) + Vf (
1
e
− e
−( R
ηu
2
)2
) +
1
2
∆Vu(e
−2( R
ηu
2
)2
−
1
e2
) + ∆Vu(
1
e
− e
−( R
ηu
2
)2
))
= πρu(
ηu
2
)4
Ωu(Vf (1 − (
R
ηu
2
)2
) + Vf (
1
e
− e
−( R
ηu
2
)2
) +
1
2
∆Wu(e
−2( R
ηu
2
)2
−
1
e2
) + ∆Wu(
1
e
− e
−( R
ηu
2
)2
))
(4.9)
Vf =
2(e
1
K2
− e)Y Su(e
1
K2ε 2
r − e
1
K2
)
(e−K2ε 2
r − e−k2
)(
e
1
K2 (e
1
K2 +1
(− 16
K2 +ε 2
r +2)+e
1
K2ε 2
r (e
1
K2 (−3(e−1)ε 2
r −5e+1)+
16(2e−1)e
1
K2
K2 +e))
k2 + 2(e−e
1
K2 )(K2−(K )2)(e
1
K2 −e
1
K2ε 2
r )ek2+K2ε 2
r
K2(eK2ε 2
r −ek2
)
)
(4.10)
24
28. 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, ∆Vu ,Vf , the variables, and replacing Ωu with Ωu
ε 2
r
. The equation can then be used
with the Solve function in Mathematica to give a numerical value for Ωu.
1
2
((Vf − e− 1
K2
∆Vu)2
− (Vf − ∆Vue
− 1
K2ε 2
r )2
)
= (Vf (∆V 2
u e
− 1
K2ε 2
r − e− 1
K2
∆V2
u) −
1
2
(∆V 2
u e
− 2
K2ε 2
r − e− 2
K2
∆V2
u))
−(K2
Ω2
u(−
2Ei(− 1
K2ε 2
r
)
K2ε 2
r
+
2Ei(− 2
K2ε 2
r
)
K2ε 2
r
+ e
− 2
K2ε 2
r (e
1
K2ε 2
r − 1)2
) − K2
(
2Ei(− 2
K2 )
K2
−
2Ei(− 1
K2 )
K2
+ e− 2
K2
(e
1
K2
− 1)2
)Ω2
u)
(4.11)
25
30. Ωb = −
A
B
(4.13)
where A =e−k2−(K )2
δΩu(2e3k2+2(K )2
k2
− 6e
ε2
rk2
(ε )2
r
+k2+1
Ck2
+ 2e
ε2
rk2
(ε )2
r
+2k2+1
+ 2e2(k2+(K )2)
+ 2e2k2+(K )2+1
− 2e3k2+(K )2
−2e
ε2
rk2
(ε )2
r
+k2+(K )2+1
+ 2e
ε2
rk2
(ε )2
r
+2k2+(K )2
− 4e
ε2
rk2
(ε )2
r
+2k2+(K )2+1
− 2ek2+2(K )2
(−e + ek2
)(ek2
− e
k2ε2
r
(ε )2
r )(K )2
−4e3k2+1
(k2
− 1) + 10e
ε2
rk2
(ε )2
r
+3k2+1
(k2
− 1) + 8e3k2+(K )2+1
(k2
− 1) − 20e
ε2
rk2
(ε )2
r
+3k2+(K )2+1
(k2
− 1) + 4e
ε2
rk2
(ε )2
r
+k2+2(K )2
(k2
− 1)
+20e
ε2
rk2
(ε )2
r
+2k2+2(K )2+1
(k2
− 1) + ek2+2(K )2+1
(4k2
− 6) + e
k2(
ε2
r
(ε )2
r
+1)
C(4k2
− 3) + ek2+1
C(4k2
+ 1)
+e
ε2
rk2
(ε )2
r
+3k2+(K )2
(8k2
− 4) + e
k2(
ε2
r
(ε )2
r
+3)
(2 − 4k2
) + e
ε2
rk2
(ε )2
r
+k2+2(K )2+1
(14 − 8k2
) + e
k2ε2
r
(ε )2
r
+2(k2+(K )2)
(4 − 10k2
) + e2k2+2(K )2+1
(8 − 10k2
))Y
(4.14)
27
31. and B =(ε )2
r(2e(K )2
(−e + ek2
)(ek2
− e
k2ε2
r
(ε )2
r )(−e
−
k2ε2
r
(ε )2
r + e
−
(K )2
(ε )2
r )Y (ε )2
r + 2e
2(K )2
(ε )2
r (−e
−
k2ε2
r
(ε )2
r + e
−
(K )2
(ε )2
r )
(δ +
e(K )2
(−e + ek2
)(ek2
− e
k2ε2
r
(ε )2
r )((K )2
− k2
ε2
r)Y
D
)h + e
2(K )2
(ε )2
r (e
−
2k2ε2
r
(ε )2
r − e
−
2(K )2
(ε )2
r )
(δ +
e(K )2
(−e + ek2
)(ek2
− e
k2ε2
r
(ε )2
r )((K )2
− k2
ε2
r)Y
D
)D + 2e(K )2
(−e + ek2
)(ek2
− e
k2ε2
r
(ε )2
r )((K )2
− k2
ε2
r)Y )
(4.15)
C = (2e2k2+(K )2
− 2ek2+2(K )2
− e2k2
+ e2(K )2
)(ε )2
r (4.16)
D = Y (ek2
− e(K )2
)(2ek2
k2
ε2
r(3e
k2ε2
r
(ε )2
r
+1
− 2e
k2ε2
r
(ε )2
r − 2e) + ek2
(ε )2
r(3e
k2ε2
r
(ε )2
r − e) + 2(5(k2
− 1)e
k2ε2
r
(ε )2
r
+k2+1
+ e
k2ε2
r
(ε )2
r
+1
+(1 − 2k2
)e
k2(
ε2
r
(ε )2
r
+1)
− 2ek2+1
(k2
− 1))) + (ek2
− e)k2
Y e(K )2
(ek2
− e
k2ε2
r
(ε )2
r )
(4.17)
28
32. 4.1.3 Unknown Variables
εr
Taking the third equation for conservation of angular momentum and replacing Ωu, the equation can be rerarranged
to give Ωu
Ωu
. If this is replaced with 1
ε 2
r
, the resulting equation can be solved with numerical values for variables.
1
(εr)2
=
(e
1
K2
− e)ε4
r(
(−e
−
2k2ε2
r
(εr)2
+2e
−
k2ε2
r
(εr)2
+ 1
e2 −2
e
)e
1
(ε )2
r
−1
K2 (
k2K2(
F (k2−(K )2)
E
+1)
e−k2
−e−(K )2 +
8e
1
K2 +1
(e
1
K2 −e
1
K2(ε )2
r )(K2(ε )2
r+K2−2)
E
)
K2(ε )2
r(e
1
K2(ε )2
r −e)
+
4(
k2ε2
r
(εr)2 +e
−
k2ε2
r
(εr)2
−1
e
−1)(e
1
K2(ε )2
r −e
1
K2 )
E
)
(−e−2k2
+2e−k2
+ 1
e2 − 2
e
)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 )2
r − e−k2
)(
e
1
K2
(e
1
K2 +1
(− 16
K2 + (ε )2
r + 2) + e
1
K2(ε )2
r (e
1
K2
(−3(e − 1)(ε )2
r − 5e + 1) + 16(2e−1)e
1
K2
K2 + e))
k2
+
2(e − e
1
K2
)(K2
− (K )2
)(e
1
K2
− e
1
K2(ε )2
r )ek2+K2(ε )2
r
K2(eK2(ε )2
r − ek2
)
)
(4.19)
29
33. and F = 2(e − e
1
K2
)(e
1
K2
− e
1
K2(ε )2
r )ek2+K2(ε )2
r (4.20)
30
34. 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 = [
ρu
Pu
] × S2
u (5.1)
where Pu= 105
Pa and ρu = 1.225kgm−3
31
35. 5.1.2 Vf
V ∗
f =
1
Su(CH4)
× Vf (5.2)
where Su(CH4) = 0.4ms−1
5.1.3 ∆P
∆P∗
=
∆P
Pu
(5.3)
where Pu = 105
Pa
∆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
37. 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 εr 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
38. 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
39. 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
40. 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
41. 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
43. 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
Ωη2
u
4r
for ηu/2 ≤ r ≤ R
(A.3)
40
44. 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
rc
ρu(Vu − Vf )2πrdr =
rb
rc
ρb(Vb − Vf )2πrdr (A.6)
[ρu(Vu − Vf )πr2
]ru
rc
= [ρb(Vb − Vf )πr2
]rb
rc
(A.7)
41
45. ρu(Vu − Vf )π(r2
u − r2
c ) = ρb(Vb − Vf )π(r2
b − r2
c ) (A.8)
A.1.2 Forced Vortex Volume
ρu(Vu − Vf )π((ηu/2)2
− r2
u) = [ρu(Vu − Vf )π((ηu/2)2
− r2
b ) (A.9)
A.1.3 Unforced Vortex Volume
ρu(Vu − Vf )π(R2
− (ηu/2)2
) = ρu(Vu − Vf )π(R2
− (ηu/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
46. 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
rc
ρuΩu(Vu − Vf )2πr3
dr =
rb
rc
ρbΩu(Vb − Vf )2πr3
dr (A.13)
ρuΩu(Vu − Vf )[r4
/4]ru
rc
= ρbΩu(Vb − Vf )[r4
/4]rb
rc
(A.14)
ρuΩu(Vu − Vf )(r4
u/4 − r4
c /4) = ρbΩu(Vb − Vf )(r4
b /4 − r4
c /4) (A.15)
43
48. A.3 Radial Momentum Equation
There is only a need for one further equation, that is the radial momentum
equation.
∂P
∂r
= ρ
V 2
θ
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 = ρ
V 2
θ
r
dr (A.22)
= ρΩ2
r 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
49. 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(r2
u − r2
c ) = Af Suρu (A.25)
rearranging to Vu − Vf gives
Vu − Vf = −
Af Su
π(rc
2 − ru
2)
(A.26)
However, Y = −
Af
π(rc
2−ru
2)
, 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(r2
b − r2
c ) = Af Suρu (A.28)
46
50. Rearranged to Vb − Vf gives
Vb − Vf = −
ρuAf Su
πρb(rb
2 − rc
2)
(A.29)
Replacing −
Af
π(rb
2−rc
2)
with Y
εr
gives
Vb =
ρu
ρb
Y Su
εr
+ Vf (A.30)
Vu
Taking the unforced vortex volume’s equation for conservation of momentum,Vu
can be determined .
ρuΩu(ηu/2)2
(Vu − Vf )π[R2
− (ηu/2)2
]
= ρuΩu(ηu/2)2
(Vu − Vf )π[R2
− (ηu/2)2
]
(A.31)
Vu − Vf = (
Ωu
Ωu
)
(ηu/2)2
(ηu/2)2
[R2
− (ηu/2)2
]
[R2 − (ηu/2)2]
(Vu − Vf ) (A.32)
Using the relationship between Ωu and Ωu and the previously determined
value for Vu, the equation reduces somewhat.
Ωu
Ωu
=
(ηu)2
ηu
and Vu − Vf = Y Su (A.33)
Vu − Vf =
[R2
− (ηu/2)2
]
[R2 − (ηu/2)2]
Y Su (A.34)
47
51. The relationships ηu/2 = εrηu/2 and K = ηu/2
R
complete the reduction.
Vu − Vf =
[R2
− (ηu/2)2
]
[R2 − (εrηu/2)2]
Y Su (A.35)
Vu =
[1 − K2
]
[1 − (εrK)2]
Y Su + Vf (A.36)
48
52. 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
54. B.2 Conservation of Angular Momentum
B.2.1 Flame Volume
π(
ηu
2
)4
ρuΩu(Vf ((
rc
ηu
2
)2
− (
ru
ηu
2
)2
) + Vf (e
−( rc
ηu
2
)2
− e
−( ru
ηu
2
)2
) +
1
2
∆Wu(e
−2( ru
ηu
2
)2
− e
−2( rc
ηu
2
)2
) + ∆Wu(e
−( rc
ηu
2
)2
− e
−( ru
ηu
2
)2
)
= πρbΩb(
η u
2
)4
(Vf ((
rc
η u
2
)2
− (
rb
η u
2
)2
) + Vf (e
−( rc
η u
2
)2
− e
−(
rb
η u
2
)2
) + ∆Wb(e
−( rc
η u
2
)2
− e
−(
rb
η u
2
)2
) +
1
2
∆Wb(e
−2(
rb
η u
2
)2
− e
−2( rc
η u
2
)2
)
(B.8)
B.2.2 Forced Vortex Volume
π(
ηu
2
)4
ρuΩu(Vf ((
ru
ηu
2
)2
− 1) + Vf (e
−( ru
ηu
2
)2
−
1
e
) + ∆Vu(e
−( ru
ηu
2
)2
−
1
e
) +
1
2
∆Vu(
1
e2
− e
−2( ru
ηu
2
)2
))
= πρu(
ηu
2
)4
Ωu(Vf ((
rb
ηu
2
)2
− 1) + Vf (e
−(
rb
ηu
2
)2
−
1
e
) + ∆Vu(e
−(
rb
ηu
2
)2
−
1
e
) +
1
2
∆Vu(
1
e2
− e
−2(
rb
ηu
2
)2
))
(B.9)
B.2.3 Unforced Vortex Volume
π(
ηu
2
)4
ρuΩu(Vf (1 − (
R
ηu
2
)2
) + Vf (
1
e
− e
−( R
ηu
2
)2
) +
1
2
∆Vu(e
−2( R
ηu
2
)2
−
1
e2
) + ∆Vu(
1
e
− e
−( R
ηu
2
)2
))
= πρu(
ηu
2
)4
Ωu(Vf (1 − (
R
ηu
2
)2
) + Vf (
1
e
− e
−( R
ηu
2
)2
) +
1
2
∆Vu(e
−2( R
ηu
2
)2
−
1
e2
) + ∆Vu(
1
e
− e
−( R
ηu
2
)2
))
(B.10)
51
55. B.3 Radial Momentum
B.3.1 Pressure along the bluff-body
Pur=rc
− Pbr=rc
= −
Ω2
b(−
2(K )2Ei(−
(K )2
(ε )2
r
)
(ε )2
r
+
2(K )2Ei(−
2(K )2
(ε )2
r
)
(ε )2
r
+ e
−
2(K )2
(ε )2
r (e
(K )2
(ε )2
r − 1)2
)
(K )2
(ε )2
r
+ ∆VbVf e
−
(K )2
(ε )2
r −
1
2
∆V2
be
−
2(K )2
(ε )2
r
+
Ω2
u(2(K )2
Ei(−(K )2
) − 2(K )2
Ei(−(K )2
) + e−2(K )2
(e(K )2
− 1)2
)
(K )2
− Vf e−(K )2
∆Vu +
1
2
e−2(K )2
∆V2
u
(B.11)
B.3.2 Pressure along the flame tube
Pur=R
− Pbr=R
= ρu(Vf (∆V2
ute
− 4R2
(η )2
u − ∆V2
ue
−4R2
η2
u ) −
1
2
(∆V2
ute
− 8R2
(η )2
u − ∆V2
ue
−8R2
η2
u ))
−(
Ω2
u(−
(8R2)Ei(− 4R2
(η )2
u
)
(η )2
u
+
2(4R2)Ei(− 8R2
(η )2
u
)
(η )2
u
+ e
− 8R2
(η )2
u (e
4R2
(η )2
u − 1)2
)
(4R2)(ε )2
r
(η )2
u
−
Ω2
u(−
(8R2)Ei(−4R2
η2
u
)
η2
u
+
2(4R2)Ei(−8R2
η2
u
)
η2
u
+ e
−8R2
η2
u (e
4R2
η2
u − 1)2
)
4R2
η2
u
)
(B.12)
52
56. B.4 Bernoulli Equation
B.4.1 Pressure along the flame tube
Pur=R
− Pbr=R
=
1
2
ρu((Vf − ∆Vue
−4R2
η2
u )2
− (Vf − e
−4R2
η 2
u ∆Vu)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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