=Paper=
{{Paper
|id=Vol-1638/Paper59
|storemode=property
|title=Critical conditions of ignition of fuel spray containing liquid fuel droplets
|pdfUrl=https://ceur-ws.org/Vol-1638/Paper59.pdf
|volume=Vol-1638
|authors=Aina Zh. Agataeva,Elena A. Shchepakina
}}
==Critical conditions of ignition of fuel spray containing liquid fuel droplets ==
https://ceur-ws.org/Vol-1638/Paper59.pdf
Mathematical Modelling
CRITICAL CONDITIONS OF IGNITION OF FUEL
SPRAY CONTAINING LIQUID FUEL DROPLETS
A.Zh. Agataeva, E.A. Shchepakina
1
Samara National Research University, Samara, Russia
Abstract. The ignition of combustible gas containing liquid fuel droplets is in-
vestigated. The analysis is based on the theory of integral manifolds of singular-
ly perturbed systems. This approach allows us to define different types of chem-
ical regimes including the critical mode. The relation between the critical re-
gime and the phenomenon of delayed loss of stability in the dynamical model is
shown.
Keywords: integral manifold, singular perturbations, delayed loss of stability,
spray, combustion theory, critical behavior, thermal explosion.
Citation: Agataeva AZh, Shchepakina EA. Critical conditions of ignition of
fuel spray containing liquid fuel droplets. CEUR Workshop Proceedings 2016;
1638: 484-492. DOI: 10.18287/1613-0073-2016-1638-484-492
Introduction
In this paper we present a qualitative investigation of combustion dynamics in a mul-
tiphase medium. The investigation of the ignition process in a gas medium was car-
ried out by many authors, see, for instance, [1–16] and references therein. However,
the influence of liquid droplets within such a context is less studied [17]. The dynam-
ics of such systems is determined by two processes: heat loss due to the evaporation
of the flammable liquid medium (droplets) and heat release associated with an exo-
thermic oxidation reaction in the gas phase [17, 18]. Competition between these pro-
cesses determines the main dynamical features of the systems.
Model
A concise physical model for the thermal explosion in a two-phase medium (combus-
tible gas mixture - combustible liquid drops) is suggested using an adiabatic approach.
The main physical assumptions of the model are as follows. The combustible liquid
droplets are considered to comprise a monodisperse spray, whose effect on the blow-
up process is to be investigated qualitatively. As usual [2] for thermal explosion pro-
cesses, we neglect the pressure change in the reaction volume and its influence on the
combustion process. We assume that the thermal conductivity of the liquid phase is
much greater than that of the gas phase. Thus, the heat transfer coefficient in the liq-
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Mathematical Modelling Agataeva AZh, Shchepakina EA. Critical…
uid gas mixture is supposed to be defined by the thermal properties of the gas phase.
It is assumed that the quasi-steady state approximation is valid for the vaporizing
droplets [18]. The drop boundary is assumed to be on a saturation line (i.e., the liquid
temperature is constant and is equal to the liquid saturation temperature). The com-
bustion reaction is modeled as a first order, highly exothermic chemical reaction. The
exothermic oxidation reaction is usually modelled as a single step reaction obeying an
Arrhenius temperature dependence. Heat losses were assumed proportional to the
temperature excess over the ambient temperature (Newtonian cooling) [1, 4].
Under these assumptions a mathematical model of the process has a form [18]:
dTg E
c pg g g C f Cox Q f g Z f exp 4 Rd g nd Tg Td ,
RT
(1)
dt g
d Rd3 3g
dt
T T ,
L Rd L d g
(2)
dC f E 4 Rd g nd
dt
C f Cox Z exp
RT
L f g
Tg Td , (3)
g
dCox E
C f Cox Z exp (4)
RT
,
dt g
g Tg g 0Tg 0 , (5)
where T is a temperature; R d is a radius of the drops; C is a concentration; Z is pre-
exponential factor; E is an activation energy; Q is a energy of the combustion; L is a
liquid evaporation energy; R is the universal gas constant; c is a thermal capacity; n d
is a number of drops per unit volume; t is a time; is a volumetric phase content;
is a thermal conductivity; is a density; is a molar mass. The subscripts here
denote: d - liquid fuel droplets; f - combustible gas component of the mixture; g - gas
mixture; L - liquid; p - under constant pressure; ox - oxidizer; 0 - initial state.
Suppose that the fuel is a deficient reagent in a large amount of oxidant, thus equation
(4) can be omitted.
Following to the classical theory Semenov [1] we define the dimensionless variables
C E Tg Tg 0 R
f , , r d ,
Cf 0 RTg 0 Tg 0 Rd 0
t E
, treact A1 exp
RT
,
treact g0
RTg 0 c pg Tg 0 g 0 4 Rd 0 g Tg 0 nd E
, , 1 exp
RT
,
E C f 0Q f f AC f 0Q f g f g0
Q f C f 0 g f Qf
2 , ,
L L L L
and the system (1)-( 3) can be rewritten in dimensionless form:
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Mathematical Modelling Agataeva AZh, Shchepakina EA. Critical…
d
exp 1r 1 , (6)
d 1
d r3
1 2 r , (7)
d
d 1
exp 1 r , (8)
d 1 1
where is the dimensionless fuel gas temperature; r is the dimensionless radius of
the drops; is the dimensionless concentration of flammable gas; is the dimen-
sionless time; is the dimensionless parameter equal to the final dimensionless adia-
batic temperature thermally isolated system after the explosion; gives the initial
temperature; 1 , 2 characterize the interaction between the gas and liquid phases;
is a parameter characterizing the ratio of the energy of combustion gas mixture to
the liquid evaporation energy.
The initial conditions for the equations (6)-(8) are:
(0) 0, (0) 1, r (0) 1.
It should be noted that the system (6)-(8) has the energy integral:
1
ln 1
1 3
2
r 1 0. (9)
Taking into account (9) we can reduced the system (6)-(8) [19-23]:
d 1 3
1 ln 1
d 2
r 1 exp 1 1r 1 , (10)
d r3
1 2 r . (11)
d
Thus, the dynamic of the system depends on five dimensionless parameters:
𝛽 ≪ 1, 𝛾 ≪ 1, 𝜀1 , 𝜀2 , 𝛹. The property values of γ and β are small compared to unity
for most gas mixtures because of the high activation energy and the exothermic chem-
ical reaction [3, 4, 6, 17]. The smallness of the parameter γ implies that (10), (11) is a
singularly perturbed system, which allows us to apply the geometric methods of the
singular perturbation theory for its analysis [19-23].
Analysis
The degenerate equation [21, 23]
( , r ) 1
ln 1
11 r 3 exp r 1 0 (12)
2 1 1
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Mathematical Modelling Agataeva AZh, Shchepakina EA. Critical…
describes the zero-approximation of slow integral manifold of the system, a slow
curve. The analysis of the slow curve allows us to determine the basic types of chemi-
cal reaction regimes depending on the values of the additional parameters of (10),
(11).
For 0 the slow curve has a shape as shown on Figure 1. The parts PT and QS of
the slow curve are stable while the part TQ is unstable. Depending on the ratio of
other parameters following cases are possible: thermal explosion with delay or typical
thermal explosion. In the first case a trajectory of the system, starting from the initial
point A in the basin of attraction of the stable part PT, will be attracted to it with the
velocity of the fast variable (see path AB) and then follows along it with the velocity
of the slow variable (see path BT). After this moment the trajectory will jump from
the slow integral manifold (see path TC). The thermal explosion will have happened
long before the point C is reached because the temperature’ value at the point C ex-
tremely high.
The typical thermal explosion occurs if the initial point is located below the basin of
attraction of the stable part PT (see the trajectory DE on Figure 1).
Fig. 1. Slow curve and the trajectories of (10), (11) with 0
Figure 2 demonstrate the trajectory of the reduced system (10), (11) and the solutions
of the full system (6)-(8) in the case of thermal explosion with delay for
0.05, 0.01, 1 2.0, 2 0.8, 0.19.
Figure 3 shows the slow curve and a trajectory (AB) of the system (10), (11) in the
case ψ>1. This case also corresponds to the typical thermal explosion and the behav-
iors of the solutions of (6)-(8) are the similar as in previous case.
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Fig. 2. Thermal explosion with delay: (top) the dimensionless radius r vs. and
(bottom) the solutions of the system (6)-(8) vs. τ
Fig. 3. Slow curve and a trajectory of the system (10), (11) with ψ > 1
For 0 1 2 the slow curve is shown on Figure 4. In this case three regimes are possible
depending on the ratio of the parameters: the typical thermal explosion (the trajectory CD), the
slow combustion regime (the trajectory CTP), and the critical regime (the trajectory CTQ).
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Mathematical Modelling Agataeva AZh, Shchepakina EA. Critical…
Fig. 4. Slow curve and the trajectories of the system (10), (11) with
0 < ψ < 1 − ε2
Figure 5 demonstrate the trajectory of the system (10)-(11) and the solutions of the
system (6)-(8) in the case of slow combustion regime. Without loss of generality the
parameters of the system are chosen to be 0.05, 0.01, 0.19, 1 3.5, and
2 0.8.
Fig. 5. Slow combustion regime: (top) the dimensionless radius r vs. and
(bottom) the solutions of the system (6)-(8) vs. τ
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The critical regime separates an explosive regime from a nonexplosive one. The cru-
cial result is that the unstable slow manifold may be used to construct the separating
regime between the safe regimes and explosive ones.
Figure 6 shows the trajectory of the system (10), (11) and the solutions of the system
(6)-(8) in the case of critical regime ( 1 2.2, other parameters are the same as in
Figure 5). The critical trajectory is characterized by a comparatively rapid (but not
explosive) flow of the reaction till the essential degree of conversion takes place and
then a jump slow-down and a transition to the slow flow of the reaction to near the
origin.
Fig. 6. Critical regime: (top) the dimensionless radius r vs. and
(bottom) the solutions of the system (6)-(8) vs. τ
The critical value of a control parameter, say 1 , corresponding to the critical trajecto-
ry may be found in the form of the asymptotic expansion [24]:
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Mathematical Modelling Agataeva AZh, Shchepakina EA. Critical…
1
1 0 1 2/3 2 ln O( ).
Such approach to modeling of the critical phenomena in combustion problems has
been applied in [7, 12, 21, 23, 25].
Conclusion
The dynamical model of the ignition in the two-phase medium was considered. The
study of the zeroth order approximation of the slow integral manifold of the system
(the slow curve) has allowed to define the three basic types of chemical reaction
modes: safe slow combustion mode, typical thermal explosion regime and thermal
explosion regime with a delay. It was shown that the realizability conditions for these
regimes depend on the values of the additional system parameters. It was shown that
there is also the critical regime which divides the area of safe reactions and the area of
dangerous, explosive regimes. The conditions of the existence of the critical regime
have been obtained via the geometric theory of singular perturbation. The crucial
result is that the unstable slow manifold may be used to construct the separating re-
gime between the safe regimes and explosive ones.
Acknowledgment
This work is supported in part by the Russian Foundation for Basic Research (grant
14-01-97018-p) and the Ministry of Education and Science of the Russian Federation
under the Competitiveness Enhancement Program of Samara University (2013–2020).
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