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Determination of laminar flame speeds under engine relevant conditions
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Determination of laminar flame speeds under engine relevant conditions
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Content
Determination of Laminar Flame Speeds
under Engine Relevant Conditions
by
Tailai Ye
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements of the Degree
DOCTOR OF PHILOSOPHY
(MECHANICAL ENGINEERING)
May 2017
2
Acknowledgements
I would like to thank Professor Fokion N. Egolfopoulos for giving me this opportunity of great
scientific adventure, and for consistently offering me insightful guidance and kind support throughout
this doctoral work.
I would also like to thank Christodoulos Xiouris and Dr. Jagannath Jayachandran for great
collaborative work, Dr. Kian Eisazadeh-Far for assisting with the original design and methodology of
the experiments.
I am grateful to my parents and paternal grandparents for their unconditional love and support. I am
sad that my mother cannot see me graduate. Her memory will be with me always.
3
Table of Contents
Acknowledgements .......................................................................................................... 2
Table of Contents ............................................................................................................. 3
Abstract ........................................................................................................................... 5
Chapter 1 Introduction ..................................................................................................... 6
1.1 Introduction ......................................................................................................... 6
1.2 Literature Review ................................................................................................ 8
Chapter 2 Methodology ..................................................................................................11
2.1 Flame fundamentals ...........................................................................................11
2.1.1 Laminar flame speed .................................................................................11
2.1.2 Flame stretch ............................................................................................11
2.1.3 Karlovitz number ......................................................................................12
2.1.4 Flame speed calculation ............................................................................12
2.2 Experimental approach .......................................................................................13
2.3 Direct numerical simulations of freely propagating flames .................................16
2.4 Direct numerical simulations of spherically expanding flames ...........................16
2.5 Hybrid thermodynamic-radiation model of spherically expanding flames ..........18
Chapter 3 Uncertainty Quantification..............................................................................21
3.1 Uncertainty propagation .....................................................................................21
3.2 Mixture preparation ............................................................................................21
4
3.3 Data acquisition ..................................................................................................25
3.4 Data processing ..................................................................................................25
Chapter 4 Results and Discussions ..................................................................................31
4.1 Hybrid thermodynamic-radiation model validation ............................................31
4.2 Parametric study on radiation model effect .........................................................35
4.3 Evaluation of stretch effect .................................................................................42
4.4 Laminar flame speed determination for H
2
-CO flames .......................................45
Chapter 5 Summary ........................................................................................................48
5.1 Conclusions ........................................................................................................48
5.2 Future directions of research ..............................................................................49
References ......................................................................................................................50
5
Abstract
The spherically expanding flame method is the only approach for measuring laminar flame speeds at
thermodynamic states that are relevant to engines. In the present study, a comprehensive evaluation of
data obtained under constant volume conditions was carried out through experiments, development of a
mathematically rigorous method for uncertainty quantification and propagation, and advancement of
numerical models that describe the experiments accurately. The proposed uncertainty characterization
approach accounts for parameters related to all measurements, data processing, and finally data
interpretation.
The propagation of spherical flames under constant volume conditions was investigated through
experiments carried out in an entirely spherical chamber and the use of two numerical models. The first
involves the solution of the fully compressible one-dimensional conservation equations of mass, species,
and energy. The second model was developed based on thermodynamics similarly to existing literature,
but radiation loss was introduced at the optically thin limit and approximations were made to account for
re-absorption. It was shown that neglecting radiation in constant volume experiments could introduce
significant error.
Incorporating the aforementioned techniques, laminar flame speeds were measured and reported
with properly quantified uncertainties for flames of synthesis gas for pressures ranging from 9 to 28 atm,
and unburned mixture temperatures ranging from 440 to 520 K. The approaches introduced in this study
allow for the determination of laminar flame speeds with notably reduced uncertainties under conditions
of relevance to engines, which has major implications for the validation of kinetic models of surrogate
and real fuels.
6
Chapter 1 Introduction
1.1 Introduction
Modern civilization depends on a continuous and abundant energy supply. Including transport,
industry, agriculture, commercial and public service, all aspects of modern civilization involves energy
consumption. Based on the Key World Energy Statistics 2015 by International Energy Agency (Figure
1.1), more than 80% of the world total energy supply comes from fossil fuel during 1973 to 2013.
Despite the recent growth in nuclear and hydro energy, conventional fossil fuel is and will continue
being the largest energy source in the foreseeable future.
Although fossil fuel is heavily used, it is both limited in supply and non-renewable. Inevitably, the
use of fossil fuel also produces greenhouse gases, which are believed to be closely connected to global
warming. These facts motivate the research of more efficient and environmental friendly combustion.
7
Figure 1.1 World total primary energy supply
Numerical simulation is a very important tool to design combustors with higher energy efficiency
and lower emissions. In order to perform such numerical simulation accurately, sophisticated chemical
kinetic models are required, which have been investigated extensively by numerous researchers.
Laminar flames in canonical configurations are widely utilized to study flame characteristics and
develop chemical kinetic models. With different configurations, various measurements can be
performed under pressures from 0.05 to 60.0 bar for laminar flame studies, as shown in Figure 1.2.
8
Figure 1.2 Experimental approaches and measurements applicable for laminar flame studies at
various pressures (bar)
1.2 Literature Review
The laminar flame speed,
o
u
S , is defined as the propagation speed of a steady, laminar, one-
dimensional, planar, stretch-free, and adiabatic flame, hereafter referred to as freely propagating flame.
o
u
S is an important fundamental property of a combustible mixture, being a measure of its reactivity,
diffusivity, and exothermicity (e.g., [1][2]), and an essential validation target for kinetic models. There
is extensive literature on the topic of laminar flame propagation (e.g., [4][5][6][7][8][9][10]).
The first measurements of flame propagation speeds were carried out in 1867 with the experiment of
Bunsen [11]. Since then, a larger number of data have been reported in the literature derived from
Bunsen flames (e.g., [12][13][14]), Spherically Expanding Flames (SEF) under constant pressure (e.g.,
9
[15][16][17][18][19]) and volume (e.g., [20][21][22][23][24][25][26]) conditions, Counter-Flow Flames
(CFF) (e.g., [27][28][29]), and recently the heat flux method (e.g., [30][31][32]).
With few exceptions,
o
u
S data are scarce for conditions encountered in piston and jet engines, that is
for liquid fuel flames at P = 20-50 atm and T
u
= 700-800 K, given that measurements can be rather
challenging and highly uncertain.
Steady-state burner-type laminar flame experiments can be carried out up to P ≈ 10 atm or so to keep
the Reynolds number below its transition value, while experiments at notably higher pressures can be
carried out only using SEFs [10][33].
Achieving T
u
up to 700-800 K for liquid fuels remains a challenge as well. In order for liquid fuels
to exist in the vapor phase at engine-like pressures, the required T
u
must be high enough and could result
in fuel decomposition both in burner type steady state and static experiments. The vast majority of the
flame data reported in the literature during the last 10 years or so for liquid fuels have been measured at
T
u
< 500 K to avoid decomposition in CFFs (e.g., [29][35]) and SEFs (e.g., [36]).
Burner-type experiments such as Bunsen, counterflow, and burner-stabilized flames (e.g.,
[13][29][31]) are operating under steady state conditions. On the other hand, in order for the reactants to
reach a quiescent state in static SEF experiments, several minutes are required upon the completion of
the reactant filling process. T
u
higher than 500 K in constant pressure SEF experiments could result in
fuel decomposition within the several minutes period between filling the chamber and ignition.
Additionally, high T
u
could reduce the effectiveness of the sealing materials and compromise the overall
integrity of the apparatus. Thus, in constant pressure SEF experiments (e.g., [16][18][19]) in which
measurements are obtained before the compression stage, only moderate T
u
values can be tolerated.
The need to measure
o
u
S at engine-relevant conditions has been recognized long time ago. Lewis
and von Elbe [20] were the first to propose that using the pressure data during the isentropic
10
compression stage in SEF experiments the flame propagation speed could be computed by invoking a
number of assumptions and a detailed thermodynamic analysis. This proposition was a pioneering one
as during the compression stage notably high pressures and T
u
could be achieved within 20 milliseconds
or less, a time that is short enough to result in any measurable fuel decomposition and/or to compromise
the integrity of the chamber. Bradley and Mitcheson [22] and subsequently Metghalchi and Keck
[23][24] adopted the approach of Lewis and von Elbe [20] and developed also thermodynamic models
that allowed for the measurement of
o
u
S for flames of gaseous and liquid fuels.
While the approach of Lewis and von Elbe [20] was a meritorious one, concerns have been raised
about the potential formation of cells that could be unaccounted for in cases that optical access was not
possible as well as about potential stretch effects. Metghalchi and co-workers [25][26] resolved the
issue of cell formation, by performing measurements first in a cylindrical chamber with optical access
and identify thus reactant compositions for which thermal-diffusive and/or hydrodynamic instabilities do
not develop during the compression stage when the flame radius is large and the stretch is small. Then,
identical initial conditions were established in a perfectly spherical chamber for which the assumptions
of the thermodynamic model are applicable. Regarding stretch effects, Metghalchi and co-workers
[25][26] argued that they should be small at large flame radii and supported this argument by a series of
carefully executed experiments.
11
Chapter 2 Methodology
2.1 Flame fundamentals
A flame is a thin non-equilibrium region where large gradients of species concentrations and
temperature exist. In laminar flames, the effects of kinetics, fluid mechanics, and molecular transport
can be simultaneously assessed and thus kinetic models can be validate for a wide range of temperatures
and species concentrations.
2.1.1 Laminar flame speed
The laminar flame speed,
o
S
u
, defined as the propagation speed of a steady, laminar, one-dimensional,
planar, adiabatic flame is a fundamental property of any combustible mixture. It is a measure of the
mixture’s reactivity, diffusivity, and exothermicity and depends primarily on ϕ, the temperature of the
unburned mixture, and the pressure.
2.1.2 Flame stretch
Flame stretch, K, at any point on the flame surface is defined as the time derivative of the logarithm
of the area, A, of an infinitesimal element of the surface [25]:
𝐾 =
1
𝐴 𝑑𝐴 𝑑𝑡
K can be also expressed in terms of flow velocity [26]:
𝐾 = ∇
𝑡 ∙ 𝑣 𝑡 + (𝑉 ∙ 𝑛 )(∇
𝑡 ∙ 𝑛 )
12
Where ∇
𝑡 and 𝑣 𝑡 are the tangential components of ∇ and 𝑣 evaluated at the surface, and 𝑛 is the unit
normal vector of the surface, pointed in the direction of the unburned gas. 𝑉 is flame surface velocity,
while the flow has a velocity 𝑣 . The first term represents the rate of change of the tangential velocity
along the flame surface. The second term represents the stretch by the movement of a curved flame.
The stretch for a spherically expanding flame is :
𝐾 =
2
𝑅 𝑑𝑅 𝑑𝑡
where R is the radius of the spherical flame. It can be easily derived from the time derivative of the
logarithm of flame area.
2.1.3 Karlovitz number
The Karlovitz number, Ka, is a measure of the flame time, 𝑡 𝑓 , in terms of aerodynamic time, as:
𝐾𝑎 =
𝑙 𝑓 𝑆 𝑢 𝐾 ~
𝛼 (𝑆 𝑢 )
2
𝐾
Large Ka means the flame experience stronger stretch effect. Weak flames may have same stretch
rate as strong flames, but their Ka are certainly different, where weakly burning flames have lower flame
speed and thus higher Ka.
2.1.4 Flame speed calculation
The definition of flame speed, 𝑆 𝑢 , gives:
𝑑 𝑚 𝑢 𝑑𝑡 = −4π𝑅 𝑓 2
𝜌 𝑢 𝑆 𝑢
13
where 𝑚 𝑢 is the mass of the unburnt gas, 𝑅 𝑓 is the flame radius, and 𝜌 𝑢 is the density of the unburnt
gas.
From spherical geometry:
𝑚 𝑢 =
4
3
π(𝑅 𝑤 3
− 𝑅 𝑓 3
)𝜌 𝑢
where 𝑅 𝑤 is the vessel radius.
It might be assumed that the unburnt gas is compressed isentropically:
𝑃 𝜌 𝑢 −𝛾 𝑢 = constant
where 𝛾 𝑢 is the specific heats ratio of unburnt gas.
The three equations above yield:
𝑆 𝑢 =
𝑑 𝑅 𝑓 𝑑𝑡 −
(𝑅 𝑤 3
− 𝑅 𝑓 3
)
3𝑅 𝑓 2
𝛾 𝑢 𝑃 ∙
𝑑𝑃 𝑑𝑡 (2.1)
a result derived by Fiock and Marvin [21]. Thus 𝑆 𝑢 can then be obtained using Eqn. 2.1, provided
the variation of 𝑅 𝑓 , with pressure, henceforth abbreviated [𝑅 𝑓 , P], is known.
2.2 Experimental approach
In order to study laminar flame propagation at variable pressures and T
u
, two facilities were used,
one cylindrical allowing for full optical access and one spherical without optical access to preserve the
sphericity of the apparatus.
Fig. 2.1 depicts the schematic of the cylindrical chamber facility. The chamber is constructed from
316-type stainless steel, measures 270 mm in diameter and 220 mm in length, and can operate up to
90 atm post-combustion pressure. It is fitted with 76 mm thick and 152 mm diameter fused quartz
14
windows at both ends, which are sealed to the chamber with heavy duty O-rings. The initial pressure, P
o
,
cannot exceed 8 atm in order to avoid failure of the quartz windows during the compression stage of the
experiment. The chamber is fitted with two opposing stainless steel electrodes that allow for central
ignition. An ignition system has been designed to offer accurate control over the energy that is being
discharged to the two electrodes.
The propagating flame was imaged by using a CMOS Phantom v710 monochrome high-speed
camera, capable of achieving 25,000 fps in a 512x512 pixel window. A 601B1 Kistler dynamic pressure
sensor was used to record the pressure trace during the experiment. Ignition and data acquisition were
synchronized and controlled by a LabView program.
The filling of the chamber was done using the partial pressure method. The partial pressures were
measured using high accuracy Omega PX409-A5V pressure transducers with the appropriate full-scale
range to minimize uncertainties in the equivalence ratio, .
Figure 2.1 Schematic of the cylindrical chamber configuration.
15
In order to achieve stable flames for the duration of the experiment, two conditions need to be met.
First the effective Lewis number, Le
eff
, of the mixture has to be greater than unity to suppress thermal-
diffusive instabilities. Second, the flame thickness needs to be relatively large at high pressures in order
to suppress hydrodynamic instabilities. Both of these requirements were met through the use of helium
as inert.
The experiments that were carried out in the cylindrical chamber to identify initial thermodynamic
conditions that result in flames free of thermal-diffusive and hydrodynamic instabilities during the entire
duration of the experiment including the compression stage. Then, use those conditions to measure
o
u
S
in the spherical chamber, as originally proposed by Metghalchi and co-workers [25][26].
The constant volume experiments that result in high pressures and T
u
were carried out in an entirely
spherical chamber with an internal diameter of 203.2 mm. The chamber is made of stainless steel and
can withstand post-combustion pressures up to 200 atm. Fig. 2.2 depicts the schematic of the
experimental configuration. Mixture preparation and dynamic pressure measurements were performed
using identical equipment and methodology as for the constant pressure experiments.
Figure 2.2 Schematic of the spherical chamber configuration.
16
Since there is no optical access, the only observable is the pressure time history, which was then
used to derive values for flame propagation speeds similarly to previous studies (e.g.,
[22][23][24][25][26]), and details will follow.
2.3 Direct numerical simulations of freely propagating flames
o
u
S was computed using the PREMIX code [44][45], integrated with CHEMKIN [46] and the
Sandia transport [47] subroutine libraries. The H and H
2
diffusion coefficients of several key pairs are
based on the recently updated set [48]. The code has been modified to account for thermal radiation of
CH
4
, CO, CO
2
, and H
2
O at the Optically Thin Limit (OTL) [49].
The computed
o
u
S value constitutes the “known” answer of the speed of a freely propagating flame
based on the kinetic and transport models used in DNS. These “baseline” values can be used then to
quantify non-ideal effects like stretch, heat loss, etc., present in flames by analyzing deviations from the
known
o
u
S value.
All the flame simulations in this study used the kinetic model of Davis and co-workers [50].
2.4 Direct numerical simulations of spherically expanding flames
In order to model SEFs under constant pressure and constant volume conditions, a Transient One-
dimensional Reacting flow Code (TORC) was employed. TORC was originally developed [37] to
model the evolution of SEFs under constant pressure. The code has been modified subsequently in
order to model the evolution of SEFs in a confined constant volume spherical domain thereby
accounting for the compression stage during which pressure is assumed to vary in time but not in space,
given the small Mach numbers associated with SEFs. The one-dimensional reacting flow conservation
17
equations of mass, species and energy were integrated numerically in spherical coordinates [53] with the
following Boundary Conditions (BC) for constant volume flame propagation:
(
𝑑𝑇
𝑑𝑅
)
𝑅 =0
= 0, 𝑢 𝑅 =0
= 0, (𝑉 𝑘 )
𝑅 =0
= 0 (BC1)
(
𝑑𝑇
𝑑𝑅
)
𝑅 =𝑅 𝑤 = 0 or 𝑇 𝑅 =𝑅 𝑤 = 𝑇 𝑤𝑎𝑙𝑙 , 𝑢 𝑅 =𝑅 𝑤 = 0, (𝑉 𝐾 )
𝑅 =𝑅 𝑤 = 0 (BC2)
where T represents the temperature, R the spatial coordinate, u the gas velocity, R
w
the chamber radius,
and V
k
the diffusion velocity of species k. Note that the chamber wall can be treated as either adiabatic
or isothermal by specifying its temperature T
wall
.
Second order upwind and central finite difference schemes, both derived for second order accuracy
on a non-uniform grid [55], were used to discretize the convective and diffusive terms respectively. A
Differential Algebraic Equation (DAE) system [56] was obtained upon discretization of the spatial
derivatives. High fidelity, fully implicit time integration of this DAE system was performed using the
DASPK [57] solver, which implements a backward-difference formula with adaptive time step and order
control.
An adaptive grid algorithm was utilized to improve computational efficiency. The flame structure
was resolved using at least 50 points at all times. This condition ensured grid independent solutions.
The code was integrated with the CHEMKIN [46] and Sandia transport subroutine libraries [47][48] and
is also capable of performing calculations involving OTL-based radiation heat loss [49]. Details of the
code relating to determination of a consistent initial condition, re-gridding algorithm, and BCs for
constant pressure SEFs can be found in [37].
18
2.5 Hybrid thermodynamic-radiation model of spherically expanding flames
The temporal variation of pressure, P, constitutes the only diagnostic in constant volume SEF
experiments. The flame speed, S
u
, can then be obtained using Eqn. 2.1 provided [R
f
, P] is known. The
underlying assumptions of Eqn. 2.1 include isentropic compression of the unburned gas, and an
infinitely thin, smooth, spherical flame. In all previous studies, [R
f
, P] was obtained using a
thermodynamic model based on several assumptions (e.g., [22],[24]). A simplified Hybrid
ThermoDynamic-Radiation (HTDR) model was developed based on similar assumptions but is
additionally capable of performing radiation heat loss calculations. The computational cost of HTDR is
notably lower compared to TORC, and is the only viable way that radiation heat loss can be modeled
with reasonable computational cost when interpreting experimental data, as will be discussed below.
The HTDR model computes R
f
and the thermodynamic states of the burned and unburned gases as a
function of P based on the following assumptions:
The flame front is smooth and spherical.
The flame is infinitely thin.
The unburned gas has uniform temperature and composition.
P is uniform throughout the vessel at any instant.
The burned gas is in chemical equilibrium at any instant, thus accounting for dissociation at
high temperatures attained during compression.
No chemical reactions occur in the unburned gas.
All calculations, except for radiation heat loss, were based on equilibrium thermodynamics utilizing
STANJAN-based subroutines [58]. The gas in the spherical vessel was treated as an assembly of a
number of very thin spherical shells. The innermost shell was treated in the first step. Each
computational step was associated with combustion of one shell, which involved three successive sub-
steps:
19
1. First, the reactants in a particular shell that undergoes combustion are allowed to attain
chemical equilibrium under constraints of constant pressure and enthalpy, which results in an
increase of its volume and temperature. All inner shells, which correspond to the burned gas,
are also allowed to attain chemical equilibrium to account for the shifting in equilibrium
during compression.
2. Second, each burned shell is allowed to lose heat through radiation at constant pressure,
lowering its temperature and volume.
3. Third, all shells are simultaneously compressed isentropically such that the total volume
equals that of the spherical chamber, thus obtaining P and R
f
.
The aforementioned three sub-step algorithm is repeated for the next outer shell until the reactants in all
shells are consumed.
For an adiabatic system, there is no need to determine a characteristic time scale in order to derive
[R
f
, P]. However, in the presence of radiation heat loss, a characteristic time scale for cooling, ∆t, which
corresponds to the complete burning of each shell and the attendant pressure rise in the chamber, needs
to be evaluated. ∆t was derived as ∆P/(dP/dt), where dP/dt is the experimentally obtained slope and ∆P
represents the pressure rise obtained in sub-step 3 (resulting from the combustion of the current shell
followed by radiation loss from the burned gas). However, ∆P itself depends on the magnitude of
radiation heat loss and hence ∆t. Thus, an iterative process was adopted that first uses ∆P resulting from
an “adiabatic” step to obtain ∆t. This iteration scheme results in a new ∆P and ∆t, and after a sufficient
number of iterations, converged values for ∆P and ∆t and hence [R
f
, P] are obtained.
The radiation heat flux was computed using two approaches. The first approach adopts the OTL
assumption [60], which over-estimates the heat loss by not accounting for re-absorption. The Planck
mean absorption coefficients used in this approach is provided by Ju et al. [51]. The second approach
involves RADCAL [59], a radiation subroutine with a narrow-band database for combustion gas
20
properties. In this approach, the radiation heat flux term is calculated more accurately by considering
the spectrally dependent re-absorption (REAB). However, the radiation calculation is performed in 1-D
slab geometry [60] instead of spherical geometry. The radiation heat flux is multiplied by 1.2 in order to
account for the difference between spherical and planar geometry, which is an approximation based on
the results reported by Chen et al. [61]. The approximation is justified for the following two reasons.
First, the temperature and species concentrations do not vary substantially inside the burned gas region.
Second, the uncertainty in radiative properties of gases at such high temperatures is substantial and
notably higher than those introduced by REAB.
21
Chapter 3 Uncertainty Quantification
3.1 Uncertainty propagation
The experiment procedure can be split in three stages, namely mixture preparation, data acquisition,
and data processing. First, the uncertainty of each stage will be evaluated separately, and subsequently
will be combined to provide a realistic estimate of the total uncertainty in the reported
o
u
S .
The general equations [65][66][67] for uncertainty propagation from all parameters Q to the function
F=F(Q
1
,Q
2
, ...,Q
M
), with M being the number of parameters involved, is:
2
22
1 1 1,
M M M
F i ij
i i j j i
i i j
F F F
a a a
Q Q Q
(3.1)
with
i
a
being the uncertainty of each parameter Q
i
,
ij
a the correlation coefficient between parameters Q
i
and Q
j
, and
F
a
the final propagated uncertainty of F
; the notation a will be used henceforth to
represent uncertainties.
In the case of uncorrelated parameters, Eqn. 3.1 can be simplified to:
2
22
1
M
Fi
i
i
F
aa
Q
(3.2)
It should be noted though, that the validity of the aforementioned formulas is limited only to small
perturbations around the mean value.
3.2 Mixture preparation
22
Experience shows that the sensitivity of
o
u
S on ϕ could be significant. It is also expected that errors
in the concentration of inert gases can also be of significance through its effect on T
ad
and thus kinetics.
Information of all the species involved is necessary to define a mixture and thus evaluate the sensitivity
on the propagation speed. Moreover there is an uncertainty in the reported values of initial temperature
and pressure whose effect can also be of significance. Typically the mixture preparation is performed
via successively inserting each mixture component following Dalton’s law of partial pressures, and
measuring the static pressure at each step via transducers.
The filling of the chamber was done using the partial pressure method. The partial pressures were
measured using high accuracy Omega PX409-A5V pressure transducers. Their uncertainty in the
pressure indication is 0.08% of the full range of the device. Thus, in order to improve accuracy the
experimentalist needs to use a variety of transducers with different ranges and operate as close to full
scale as possible.
For combustible mixtures involving a single fuel component, the absolute pressure indications are
chosen as independent variables, since they are non-correlated. Thus assuming that fuel and oxygen
were the first two components inserted in the chamber, Eqn. 3.2 can be used with
oxygen oxygen
fuel 1
12
fuel oxygen fuel 2 1
( , , )
v
v
st st
nn
p P P
F P P P
n p n P P
where (n
oxygen
/n
fuel
)
st
represents the stoichiometric molar oxygen to fuel ratio, and p
fuel
and p
oxygen
the
partial pressure of fuel and oxygen respectively; the pressure indications constitute the parameters Q. P
v
is the initial vacuum pressure indication, and P
1
and P
2
the absolute pressure indications of each of the
two successive steps.
The case involving a single fuel component in air is the simplest one, but not the most common
though in high-pressure SEF experiments, as helium is commonly used to assure flame stability. When
oxygen and inert gases are introduced separately and/or multi-component fuels are investigated, the
23
composition uncertainty cannot be expressed by ϕ alone. In this case, the ratios of partial pressures of
the successively inserted species during the filling procedure are chosen as parameters:
( 1)
12
23
12
, ,...,
N
N
S
Ss
N
S s S
p
pp
R R R
p p p
with
i
S
p
being the partial pressure of each component. First their uncertainties
Ri
a are calculated using
Eqn. 3.2, and subsequently Eqn. 3.1 is implemented, with Q
i
= R
i
and F =
o
u
S this time, to finally
propagate their uncertainties to
o
u
S . It should be noted that all ratios R
i
are coupled with each other due
to the successive nature of the filling process. A calculation of the cross-correlation uncertainty terms
ij
a in Eqn. 3.1 is not a straightforward task. Using, however, the Schwarz inequality,
ij i j
a a a
[67] the
maximum possible uncertainty can be calculated without determining
ij
a , as follows:
2
22
1 1 1,
M M M
F i i j
i i j j i
i i j
F F F
a a a a
Q Q Q
(3.3)
Simulations are necessary in order to obtain estimates for the partial derivatives needed above. To
that end, freely propagating flames were computed and brute-force variations of each Q
i
= R
i
parameter
were performed while keeping the rest of the R
j
ratios constant.
Similar approach was taken to evaluate the effects of T
u,o
and P
o
. In the present study all
experiments were conducted at T
u,o
= 298 K, and ± 1 K uncertainty was chosen as a realistic estimate.
Fig. 3.1 (a) depicts the computed Logarithmic Sensitivity Coefficients, (ln ) / (ln )
o
u
LSC d S d for
C
3
H
8
/air mixtures at T
u,o =
298 K and at different P
o
values. As expected, off-stoichiometric mixtures
exhibit larger LSC for all pressures. Fig. 3.1 (b) depicts the LSCs of
o
u
S to T
u,o
, P
o
, and all the R
i
component ratios as earlier described, for the case of Mixture 4 at P
o
= 5 atm. The sensitivities on ϕ,
N
2
/O
2
ratio, and initial temperature are high due to their effects on T
ad
while the sensitivity on He/N
2
ratio and pressure are minor.
24
Figure 3.1 Logarithmic sensitivity coefficients of
o
u
S to: (a) ϕ for a C
3
H
8
/air mixtures with T
u,o
= 298 K
at 1 (green), 5 (blue), and 10 atm (red), and (b) to all the pertinent parameters for a flame of Mixture 4 at
T
u,o
= 298 K and P
o
= 5 atm.
Note that a perturbation in initial condition will change the isentrope along which P and T
u
evolve
during the experiment, thus imparting an uncertainty in the state of unburned gas at each instant. To
calculate this uncertainty, 1-D planar flame computations were performed to obtain partial derivatives at
25
various points along the isentrope. It should be noted however, that this uncertainty remained relatively
constant along the isentropes for all mixture conditions adopted in the current study.
Finally using Eqn. 3.3, the uncertainties of all factors related to mixture preparation,
,
o
u
S MP
a , that
affect
o
u
S can be now propagated as follows:
,
2
22
2 2 2 2
,
1 1,
,
o
i u o o
u
o o o o o MM
u u u u u
R Ri Ri T P
S MP
i j j i
i i j u o o
S S S S S
a a a a a a
R R R T P
3.3 Data acquisition
Pressure measurements constitute the only uncertainty stemming from data acquisition. As
described in chapter 2.2, Kistler dynamic pressure sensor model 601B1 is used to record the pressure
trace during the experiment. It is calibrated for multiple ranges, such as 0-750 psi, 0-1500 psi, 0-15000
psi, etc. For each calibrated range, a corresponding linearity is provided. Among those ranges, 0-1500
psi is chosen for this study as it covers all typical measurement needs with also the smallest uncertainty.
Its corresponding linearity is 0.19% and thus its uncertainty is 0.19% * 1500 psi = ± 2.85 psi.
3.4 Data processing
The data processing of constant volume spherically expanding flame is obtaining laminar flame
speed from the temporal variation of pressure. The HTDR model was used to perform such task. The
uncertainty in data acquisition results in an uncertainty in
o
u
S , as the following quantities in Eqn. 2.1 are
affected: P, dP/dt, R
f
(calculated from the HTDR model), and dR
f
/dt (calculated as a product of dR
f
/dP
and dP/dt).
26
Figure 3.2 Pressure residual distributions of the locally fitted polynomial P(t) for Mixture 1 at
P
o
= 6 atm and T
u,o
= 298 K: (a) the first fit (at t = 95 ms). (b) the last local fit (at t = 110 ms).
A local 4
th
-order polynomial fit of the neighborhood points, P(t), was performed on the P vs. t data
in order to calculate dP/dt. The structure of the residuals is the proper indicator of the quality of the fit,
and should be randomly distributed around zero, not showing any specific structure. Fig. 3.2 (a) and (b)
show the residuals of polynomial fit for Mixture 1 (See Table 4.1) at P
o
= 6 atm and T
u,o
= 298 K. It is
27
clear that the polynomial fit residuals are small and structureless. Fig. 3.3 shows that all locally fitted
pressure values have very small differences with the pressure measurements, and Fig. 3.4 further shows
that the calculated flame speed values is insensitive to the order of the polynomial and the number of
neighborhood points used in the local fits. Note as each local polynomial fit needs data points before
and after the fitting location, some pressure measurements at the beginning and at the end does not have
a fit, and will not be used in the flame speed calculation. The weighted design matrix is calculated for
every fit and used for the calculation of the covariance matrix [65]. Thus, the uncertainty of polynomial
fit is calculated using Eqn. 3.1 with F = dP/dt, and having as Q
i
’s
the coefficients of the fitted
polynomial P(t).
Figure 3.3 Pressure calculated from polynomial fit P(t) comparing against experimental pressure
measurement for Mixture 1 at P
o
= 6 atm and T
u,o
= 298 K.
28
Figure 3.4 Flame speeds calculated with different local polynomial fit P(t) for Mixture 1 at P
o
= 6 atm
and T
u,o
= 298 K.
The HTDR model provides [R
f
, P], and thus [dR
f
/dP, P] in the form of many discrete pairs of R
f
and
P. They don’t necessarily fall right on top of the pressure measurement. As a result, in order to calculate
the flame speed with Eqn. 2.1, [R
f
, P] is interpolated to the pressure measurement values with a global
8
th
order polynomial fit. As shown in Fig. 3.5 (a) and (b), such polynomial fit is highly consistent to the
numerical result as well as has a very smooth first derivative. [R
f
, P] relation is considered error-free
when calculating the data processing uncertainty. Note that the assumptions used in the HTDR model
will introduced error in [R
f
, P]. However, such error is not caused by experimental or data processing
procedure, and will be addressed separately in chapter 4.2.
The HTDR model provides [ γ, P] for flame speed calculation with Eqn. 2.1. Similarly, a global 7
th
order polynomial fit is used for interpolation. Flame speed result is not sensitive to this fit, because Eqn.
2.1 only need the value instead of derivative of γ.
29
Figure 3.5 [R
f
, P] relation for Mixture 1 at P
o
= 6 atm and T
u,o
= 298 K: (a) [R
f
, P] from HTDR
simulation and its global 8
th
order polynomial fit. (b) [dR
f
/dP, P] calculated from the polynomial fit of
[R
f
, P].
Given error-free [R
f
, P] and [ γ, P], a perturbation in P will result in a change in R
f
and dR
f
/dP. Thus,
by perturbing the P values by the uncertainty of the experimental pressure measurement and looking up
30
the corresponding changes in R
f
and dR
f
/dP in the HTDM model result, the uncertainties in R
f
and
dR
f
/dP were calculated.
Finally the combined uncertainties resulting from mixture preparation and data processing can be
calculated as [65]:
2
22
1
()
oo
step i
uu
SS
i
aa
(11)
Note that the combined uncertainty a is a representation of the 68% confidence interval of the
parameter used as subscript.
31
Chapter 4 Results and Discussions
4.1 Hybrid thermodynamic-radiation model validation
All the following experiments in this section were carried out for syngas flames at an initial
temperature T
u,o
= 298 K. Measurements at P
o
= 3 and 6 atm were carried out using the two mixtures
reported in Table 4.1 in which X
i
represents the mole fraction of species i and T
ad
the adiabatic flame
temperature. He is used to suppress thermal-diffusive and hydrodynamic instabilities.
Table 4.1 Mixtures used in experimental measurements.
Mixture T
ad
(K) P
o
(atm) X
H2
X
CO
X
O2
X
HE
1 0.8 1600 3 and 6 0.0563 0.0563 0.0704 0.8169
2 0.8 1800 6 0.0257 0.1028 0.0803 0.7912
As discussed before, the constant volume SEF experiment is the only viable approach to measure
o
u
S at high pressures and T
u
simultaneously during the compression stage.
In order to determine S
u
from the experimental pressure trace, [R
f
, P] is obtained using the simplified
HTDR model. First, DNS results are used to assess the accuracy of HTDR model as well as the validity
of assumptions employed in HTDR model to determine S
u
. Subsequently, the experimental results along
with the method of interpretation of data are presented.
Fig. 4.1 depicts [R
f
, P] results obtained for Case A under constant volume adiabatic (ADB)
conditions, i.e. no radiative and conductive heat loss to the wall, using TORC and HTDR; Case A
corresponds to Mixture 2 at P
o
= 6 atm and T
u,o
= 298 K in a domain of size R
w
= 10.16 cm. The flame
32
front in TORC simulation results was defined as the location of maximum heat release rate. The
consistency of the two codes is confirmed as under the same conditions the results of Fig. 4.1 are
indistinguishable. Chen et al. [33] made similar observations.
Figure 4.1 Comparison of R
f
as a function of P obtained from TORC (o) and HTDR model ( ─) both
utilizing the ADB assumption for Mixture 2 at P
o
= 6 atm and T
u,o
= 298 K in a domain of size
R
w
= 10.16 cm (Case A).
In order to assess the consistency of TORC and HTDR model invoking the OTL assumption that is
manageable by both codes, simulations were performed for Case A and dP/dt obtained from TORC was
used as input to the HTDR model to obtain S
u
, and in Fig. 4.2 the results are compared. The agreement
confirms the ability of the HTDR model to infer burning/cooling time scales accurately and thus
perform radiation calculations with confidence.
33
Figure 4.2 Flame speed computed using HTDR model ( ─) with input dP/dt from TORC simulation
compared against that computed using TORC results alone (o). Both simulations were performed for
Case A utilizing the OTL assumption for the radiation heat loss.
Most previous constant volume SEF studies invoke the ADB assumption. To evaluate its validity,
computations were performed using TORC including OTL-based radiation heat loss and conduction to
the wall. The heat conduction was modeled using an isothermal wall boundary condition, T
wall
= T
u,o
,
which is an accurate and conservative estimate given the large heat capacity of the stainless steel vessel.
34
Figure 4.3 (a) and (b) Variation of R
f
and dP/dt with P for constant volume SEF computations
performed using TORC for Case A employing the ADB assumption ( ─); with conductive heat loss (---);
and with OTL-based radiation heat loss (-
.
-). (c) and (d) Percentage difference of R
f
and dP/dt
computed for the case with conductive heat loss (---); and OTL-based radiation heat loss (-
.
-) from that
of the ADB case, obtained using results shown in (a) and (b).
Fig. 4.3 (a) and (b) depict respectively the variation of R
f
and dP/dt with P for Case A. P, the sole
experimental observable, is chosen as the independent variable as it uniquely identifies the
thermodynamic condition of the unburned gas, excluding the thermal boundary layer at the wall. The
35
results obtained with conductive heat loss overlaps with those of the ADB case. Also, the effect of
radiation heat loss has a more profound effect on dP/dt compared to R
f
. Fig. 4.3 (c) and (d) illustrate the
extent of each heat loss effect by showing the percentage difference reflected in R
f
and dP/dt when
compared to the ADB case. The difference in R
f
is less than 1.5% in the case of the OTL-based
radiative loss, which is consistent with the estimates of Metghalchi and co-workers [24][70]. However,
the radiation heat loss can result in up to 13% difference in dP/dt, which has been overlooked in
previous studies. Also note that the effect of conduction heat loss on R
f
and dP/dt is less than 0.5% even
in the presence of the highly conductive helium in the mixture, and thus can be considered as minor.
4.2 Parametric study on radiation model effect
The radiation effect on laminar flame speed determination for constant pressure spherically
expanding flame has been studied by Chen et al. extensively [61][73]. However, no previous study
addressed its effect for constant volume spherically expanding flame. Constant volume spherically
expanding flame relies on an accurate [R
f
, P] relation to determine the flame speed. In order to evaluate
the effect of radiation heat loss on the flame speed determination, S
u
was calculated using TORC with
the ADB and OTL assumptions and Eqn. 2.1 using the DNS results as input. It is evident from Fig. 4.4,
that the S
u
as computed by Eqn. 2.1 and using the dP/dt and [R
f
, P] variation either from the ADB or
OTL computations, is nearly indistinguishable which is consistent with the well-established result that
radiation affects flame propagation only for near-limit mixtures [49]. However, in most previous studies,
the [R
f
, P] variation obtained using a thermodynamic model employing the ADB assumption was
utilized to extract S
u
from the experimental pressure measurements, which are inherently affected by
radiation heat loss. To assess the consequence of employing the ADB assumption, dP/dt from the OTL
TORC simulation result is treated as “experimental data” and S
u
is then calculated using [R
f
, P] from the
ADB TORC simulation result for the same conditions and the result is shown in Fig. 4.4 as well. It is of
36
interest to observe that a difference of 1.5% in [R
f
, P], as evident from Fig. 4.3 (c), can result in a
difference of up to 15% in S
u
values, calculated using the same dP/dt values. Evaluating the individual
terms of Eqn. 2.1 reveal that the factor that multiplies dP/dt is the most sensitive to a perturbation of [R
f
,
P] and causes the notable difference in S
u
.
Figure 4.4 Variation of S
u
with P computed using SEF computations performed using TORC with the
ADB (---) and OTL (-
.
-) assumptions and Eqn. 2.1 for Case A. The solid line ( ─) corresponds to S
u
calculated using Eqn. 2.1 with dP/dt from the OTL-based TORC simulation and [R
f
, P] from the ADB
TORC simulation.
As shown above, an inaccurate radiation model can introduce non trivial error in S
u
result, which
will be referred to as radiation model effect. A study was performed to further reveal how each
parameter affects the radiation model effect. First, a pressure trace was generated by HDTR based on
OTL assumption with an arbitrary S
u
, which is referred to as S
u0
. Then the pressure trace was treated as
experimental data, and used as input for HTDR to calculate S
u
with another radiation model. As a result,
37
the radiation model effect can be quantified by the percentage difference between the calculated S
u
and
S
u0
.
Instead of TORC, HTDR is used to generate the pressure trace. This choice serves two purposes.
First, as S
u
, as both one of the parameters and the benchmark of the study, will both influence and be
influenced by the radiation model effect. The use of a thermodynamics model like HTDR makes it
possible to fix S
u
throughout the simulation. As a result, the feedback from radiation effect to S
u
is
eliminated and the meaning of the difference between S
u0
and S
u
for each parameter is clear. Second, S
u
will also be influenced by kinetics, the changing temperature and pressure during the compression stage.
A fixed S
u
in HTDR also eliminates those effects and the pure effect of each parameter can be shown in
the result.
The parameter space considered in listed in Table 4.2. S
u0
is fixed to be 40 cm/s in all cases except
Low S
u0
, where S
u0
is reduced to 20 cm/s. Chamber radius is 10.16 cm in all cases except Small
chamber, where chamber radius is reduced to 5 cm.
Table 4.2 Cases considered in the parametric study of radiation model effect. T
ad
is the adiabatic
flame temperature at T
u
= 298 K and P
o
= 1 atm.
Case T
u0
(K) T
ad
(K) P
o
(atm) X
CH4
X
O2
X
N2
X
CO2
X
HE
Baseline 1.0 298 1800 1 0.0662 0.1324 0.8015 0 0
High T
ad
1.0 298 2400 1 0.1119 0.2238 0.6642 0 0
High T
u0
1.0 498 1800 1 0.0662 0.1324 0.8015 0 0
High P
0
1.0 298 1800 2 0.0662 0.1324 0.8015 0 0
CO
2
addition 1.0 298 1800 1 0.0691 0.1382 0.7235 0.0691 0
He dilution 1.0 298 1800 1 0.0515 0.1031 0 0 0.8454
38
Figure 4.5 Radiation model effect of OTL(Tien), REAB, and ADB model with respect to OTL model,
for Baseline, High T
ad
, and CO
2
addition cases.
Fig. 4.5 shows the radiation model effect between different radiation models, which can be
quantified with (1 - S
u
/S
u0
). OTL(Tien) refers to the optically thin limit model that adopts the Planck
mean absorption coefficients provided by Tien [52], while OTL uses the Planck mean absorption
coefficients provided by Ju et al. [51], as described in chapter 2.5. All OTL is the one label for all cases
where OTL assumption is used both for pressure trace generation and S
u
calculation. In those cases, S
u
always equals to S
u0
, as denoted by the green line. Several observations can be made from Fig. 4.5:
1. The radiation model effect of ADB model is very significant: more than 5% for Baseline, and
even more for High T
ad
and CO
2
addition cases. It means the S
u
result calculated with OTL model and
ADB model will have a very significant difference, and consequently an accurate radiation model is
essential to calculate accurate S
u0
.
39
2. In OTL(Tien) model, the S
u
results of are close to S
u0
, but they are not consistently greater than or
less than S
u0
. This is because the Planck mean absorption coefficients in OTL(Tien) are close to those of
OTL, but their relative magnitudes shift for major radiation species (CO
2
, H
2
O) across the temperature
range in the study.
3. In REAB and ADB model, the radiation model effect for High T
ad
case is always greater than the
Baseline case, as radiation heat loss increases rapidly with temperature.
4. In ADB model, the radiation model effect for CO
2
addition case is both greater than the Baseline
and High T
ad
case. However, in REAB model, this effect is less than High T
ad
case and close to the
Baseline case. This is because CO
2
is both important radiation emitter and absorber, but only REAB
model can take the latter role into account. For mixture with CO
2
dilution, it is very desirable to use a
radiation model considering reabsorption.
Keep using OTL model as the model to generate pressure traces, more parameters are studied for
ADB model, as shown in Fig. 4.6 (a) and (b). Considering OTL as an approximation to the real world
radiation mechanism, the radiation model effect of ADB model, or (1 - S
u
/S
u0
) in Fig. 4.6 (a) and (b),
indicates the minimum error in S
u
calculated with any thermodynamic model that fails to take radiation
into account. Several observations can be made from Fig. 4.6 (a) and (b):
1. Smaller chamber size helps reduce radiation model effect, as it takes less time for flame to
propagate to chamber wall, and thus less time for radiation cooling. Similarly, Low S
u0
has the opposite
effect.
2. High T
ad
and High T
u0
increase radiation model effect, as radiation heat loss increases rapidly
with temperature.
40
Figure 4.6 Radiation model effect of ADB model with respect to OTL model. (a) Baseline, Small
chamber, High T
ad
, High T
u0
, and Low S
u0
cases. (b) Baseline, High P
0
, High T
u0
, He dilution, and CO
2
addition cases.
41
3. Pressure has no effect on radiation model effect when adopting optically thin limit assumption.
However, in practice high pressure will usually decrease flame speed and thus increase radiation model
effect.
4. Replacing N
2
dilution with He will increase radiation model effect, as He has higher heat capacity
ratio, and thus He case has a higher T
u
and flame temperature.
5. Adding CO2 into dilution will increase radiation model effect, as CO
2
is a strong radiation
emitter, which has a high Planck mean absorption coefficients.
The above study only covers limited number of cases, while the radiation model effect is a nonlinear
function of various parameters. It is impossible to quantify radiation model effect for all combinations
of those parameters, which is also not the purpose of this study. The conclusion is that radiation model
effect is significant, even for a flame with low T
ad
and no CO
2
or He dilution, and an accurate radiation
model is crucial to get accurate S
u
data from pressure traces in constant volume spherically expanding
flame experiment.
To assess the extent to which dP/dt is affected by radiation heat loss for different chamber sizes,
constant volume SEF simulations were performed for mixture 2 at P
o
= 6 atm and T
u,o
= 298 K for three
domain sizes, R
w
= 3, 5 and 10.16 cm. The percentage difference of dP/dt results from simulations
including radiation loss from that of adiabatic ones are shown as a function of P in Fig. 4.7. Results
indicate that the effect of radiation heat loss on dP/dt is attenuated for smaller chamber sizes. This result
is consistent with the following simple analysis. Ratio of OTL-based radiation heat loss to heat release
rate of flame ~ (volume of burned gas ball/area of flame) ~ R
f
. For similar thermodynamic conditions of
the unburned mixture, the flame in the larger chamber will have a larger radius and hence suffer an
increased effect in pressure rise due to increased heat loss from the burned gas.
42
Figure 4.7 Percentage difference reflected in dP/dt for simulations with OTL radiation heat loss when
compared to the corresponding ADB case for different chamber sizes (R
w
).
4.3 Evaluation of stretch effect
The effect of stretch on S
u
was assessed similarly to Chen et al. [33]. The presence of stretch can
suppress instabilities, which is desirable, but will affect S
u
, which is undesirable as corrections will be
needed to derive
o
u
S . Simulations were performed for Mixture 2 at P
o
= 3 atm and T
u,o
= 298 K for
three chamber sizes of R
w
= 3, 5, and 10.16 cm. The initial thermodynamic condition was chosen such
that the mixture demonstrated substantial non-equidiffusion effects in the case of the constant pressure
SEF simulation.
Fig. 4.8 (a) and (b) depict the change of Karlovitz number Ka ≡ K/(α/S
u
2
) and S
u
respectively during
the course of flame propagation for the three chamber sizes, where α is the thermal diffusivity of the
mixture at unburned conditions.
o
u
S s computed at the same unburned gas conditions are also shown in
Fig. 4.8 (b) to discern the extent to which S
u
is affected by stretch. Results indicate that for values of
43
(P/P
o
) greater than 2.5, Ka reduces to very small values and, S
u
differs from
o
u
S
by less than 1.5% even
for the smaller chamber with R
w
= 3 cm.
Figure 4.8 Variation of (a) Ka and (b) S
u
as a function of P/P
o
for constant volume SEF computations
performed using TORC for Mixture 2 at T
u,o =
298 K and
P
o
= 3 atm for three chamber sizes, R
w
= 3 cm
(-
.
-), R
w
= 5 cm (
...
), and R
w
= 10.16 cm (---). The solid line ( ─) in (b) represents the stretch free
o
u
S
values computed for unburned gas conditions extracted from the constant volume SEF results.
44
To elucidate the reason why stretch decreases rapidly with increasing pressure, the variations of
dR
f
/dt
with R
f
obtained from constant volume (R
w
= 3 cm) simulations and constant pressure simulations
are shown in Fig. 4.9; note that dR
f
/dt ≠ S
b
because the burned gas is not stationary during the
compression stage. dR
f
/dt, which determines the magnitude of the stretch rate for a fixed R
f
, decreases
rapidly in the constant volume case with the pressure rise when compared to the constant pressure case.
Note that the decrease of dR
f
/dt happens despite the increase in
o
u
S (or S
b
), which is shown in Fig. 4.8
(b). The compression-induced inward flow [33] is the cause for this significant reduction in dR
f
/dt. This
results in near stretch-free flames for high values of (P/P
o
), which fortuitously corresponds to the range
where experimental pressure measurements are of high fidelity due to the large values of dP/dt. It is
also of interest to note that an additional contribution of (1/𝝆 )(d𝝆 /dt) to stretch rate exists as a result of
temporal variation of pressure [71][72]. However, this contribution is negligible, as the deviation of S
u
from
o
u
S
seems to be insignificant as evident from Fig. 4.8 (b).
Figure 4.9 Variation of the computed dR
f
/dt with R
f
using TORC for Mixture 2 at T
u,o
=
298 K and
P
o
= 3 atm; dashed line (---): constant pressure; solid line ( ─) constant volume.
45
4.4 Laminar flame speed determination for H
2
-CO flames
Constant volume SEF experiments were performed for Mixture 1 at P
o
= 3, 6 atm and Mixture 2 at
P
o
= 6 atm, both at T
u,o
= 298 K. The experimental pressure traces were post-processed using the HTDR
model employing the ADB, REAB, and OTL assumptions. As it has been shown already, for P/P
o
> 2.5
nearly stretch-free flames are obtained and S
u
calculated using Eqn. 2.1 is considered to be
o
u
S , and the
results are shown in Figs. 4.10-4.12. Note that T
u
increases non-linearly with P due to the isentropic
compression.
Figure 4.10 Experimentally measured laminar flame speeds of Mixture 1 for T
u,o =
298 K and
P
o
= 6 atm
obtained using the HTDR model with OTL (△), REAB (
…
), and ADB ( ▽) assumptions; green line ( ─)
represents 1-D planar flame simulation results obtained using the Davis et al. [50] kinetic model.
46
Figure 4.11 Experimentally measured laminar flame speeds of Mixture 1 for T
u,o =
298 K and
P
o
= 3 atm
obtained using the HTDR model with OTL (△), REAB (
…
), and ADB ( ▽) assumptions; green line ( ─)
represents 1-D planar flame simulation results obtained using the Davis et al. [50] kinetic model.
o
u
S s determined from the experimental data using the HTDR model employing the ADB, REAB,
and OTL assumptions are different as a result of different [R
f
, P] predictions. A counter-intuitive fact is
that the
o
u
S determined using the ADB assumption is lower than that obtained using the OTL
assumption. This is a result of using inconsistent [R
f
, P], and this trend is evident also in the DNS
results shown in Fig. 4.5. It can be seen also that over-estimating radiation heat loss leads to over-
prediction of the flame speed for a given pressure trace. This uncertainty in flame speed caused by
uncertainty in the radiation model is bounded by OTL and ADB calculations as they assume the
maximum and minimum possible radiation heat loss respectively.
47
Figure 4.12 Experimentally measured laminar flame speeds of Mixture 2 for T
u,o =
298 K and
P
o
= 6 atm
obtained using the HTDR model with OTL (△), REAB (
…
), and ADB ( ▽) assumptions; green line ( ─)
represents 1-D planar flame simulation results obtained using the Davis et al. [50] kinetic model.
It is worth noticing that for all cases, the
o
u
S uncertainty reduces when pressure increases. As an
example, for Mixture 1 at P
o
= 6 atm, the uncertainty decreases from 4.67 cm/s (12.4%) to 3.07 cm/s
(6.7%) when pressure increases from 17 atm to 28 atm. The uncertainty in P is constant and depends on
the transducer and its calibrated range. Also, the sensitivity of R
f
to P decreases with increasing P, as
evident in Fig. 4.1. Thus, for the same perturbation in P, the corresponding change in both R
f
and
dR
f
/dP is also smaller at higher pressure, making measured
o
u
S values more reliable during the late part
of the compression stage.
The fact that the
o
u
S uncertainty decreases with pressure makes the constant volume technique a
robust and reliable technique to measure flame speeds at conditions of high P and T
u
.
48
Chapter 5 Summary
5.1 Conclusions
Experimental data obtained from spherically expanding flames under constant volume conditions,
were evaluated using a detailed uncertainty quantification procedure and direct numerical simulations to
quantify and minimize uncertainties arising from various factors. Contribution of uncertainties resulting
from initial conditions, data acquisition, and data post-processing were first determined and then
propagated to the derived laminar flame speeds.
Direct numerical simulations were first carried out to assess the validity of the adiabatic assumption
used in all previous methods to extract flame speed from the experimental pressure trace in constant
volume spherically expanding flame experiments. It was found that neglecting radiation heat loss when
interpreting experimental data could lead to significant uncertainty as large as 15%. To account for
radiation heat loss accurately, a new hybrid thermodynamic-radiation model was developed. The model
uses the experimental pressure data to derive a time scale for radiative cooling of the burned gas. Thus,
radiation heat loss calculations can be performed accurately accounting for re-absorption. Direct
numerical simulations also show no stretch corrections are needed when pressure has risen substantially
(~ 2.5 times the initial value).
Flame speeds were measured using the constant volume approach for synthesis gas flames for 9-28
atm pressure and 440-520 K unburned mixture temperature. An uncertainty quantification analysis was
performed and the results revealed that the flame speed uncertainty decreases during the late parts of the
compression stage.
The present work on uncertainty quantification and the development of realistic models for
spherically expanding flames under constant volume conditions constitutes a major advancement, as it
49
allows for uncertainty minimization and proper accounting of radiation effects. But most importantly,
the proposed approach allows for the determination of laminar flame speeds with improved accuracy
under thermodynamic conditions that are relevant to piston and jet engines, and such data are critical to
the development of reliable kinetic models of surrogate or real fuels.
5.2 Future directions of research
Currently, no theoretical or experimental work exists in the literature regarding the onset of cells and
the degree of flame acceleration during the compression stage of the constant volume SEF experiment.
This is the stage during which experimental data are collected, in order to derive laminar flame speed.
Thus, gaining knowledge of the physics of instability growth during this stage is invaluable from an
experimental point of view. Experiments on carefully chosen fuels and conditions, as well as numerical
simulations, can be used to study this underlying physics.
During the course of the constant volume SEF experiment, the unburned gas temperature and
pressure is increasing monotonically. If the unburned temperature, pressure or both reach sufficiently
high values such that the time scales of chemical reaction become comparable to that of flame
propagation, a variety of phenomena which can include auto-ignition [74] or a transition from
deflagration to detonation [75] can occur. These conditions are similar to that encountered in spark
ignition (SI) internal combustion (IC) engines. The physics of the experimentally observed pressure
oscillations [75], in engines as well as constant volume experiments, is currently a matter of debate.
Experimental investigation of the problem can help understand the processes that lead to the
generation of pressure waves. Combined with numerical tools, further study can be conducted to reveal
its physics.
50
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Abstract (if available)
Abstract
The spherically expanding flame method is the only approach for measuring laminar flame speeds at thermodynamic states that are relevant to engines. In the present study, a comprehensive evaluation of data obtained under constant volume conditions was carried out through experiments, development of a mathematically rigorous method for uncertainty quantification and propagation, and advancement of numerical models that describe the experiments accurately. The proposed uncertainty characterization approach accounts for parameters related to all measurements, data processing, and finally data interpretation. ❧ The propagation of spherical flames under constant volume conditions was investigated through experiments carried out in an entirely spherical chamber and the use of two numerical models. The first involves the solution of the fully compressible one-dimensional conservation equations of mass, species, and energy. The second model was developed based on thermodynamics similarly to existing literature, but radiation loss was introduced at the optically thin limit and approximations were made to account for re-absorption. It was shown that neglecting radiation in constant volume experiments could introduce significant error. ❧ Incorporating the aforementioned techniques, laminar flame speeds were measured and reported with properly quantified uncertainties for flames of synthesis gas for pressures ranging from 9 to 28 atm, and unburned mixture temperatures ranging from 440 to 520 K. The approaches introduced in this study allow for the determination of laminar flame speeds with notably reduced uncertainties under conditions of relevance to engines, which has major implications for the validation of kinetic models of surrogate and real fuels.
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Determination of laminar flame speeds under engine relevant conditions
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