INSTITUTE OF PHYSICS PUBLISHING
COMBUSTION THEORY AND MODELLING
Combust. Theory Modelling 6 (2002) 339–359
PII: S1364-7830(02)29799-5
Thermal explosion in a hot gas mixture with fuel
droplets: a two reactant model
Viatcheslav Bykov1 , Igor Goldfarb1 , Vladimir Gol’dshtein1 and
J Barry Greenberg2
1 Department of Mathematics and Computer Sciences, Ben-Gurion University of the Negev,
PO Box 653, Beer-Sheva 84105, Israel
2 Faculty of Aerospace Engineering, Technion—Israel Institute of Technology, Haifa 32000,
Israel
Received 11 October 2001, in final form 26 March 2002
Published 27 May 2002
Online at stacks.iop.org/CTM/6/339
Abstract
We extend previous analyses of thermal explosion in a gas–droplets mixture
to permit a more complete description of the chemistry via a single-step
two-reactant model of general order, rather than the prior deficient reactant
model. A detailed mathematical analysis has been carried out of this new
physical model that encompasses oxidizer effects (in both fuel rich and fuel lean
situations) on the thermal explosion of a hot combustible mixture of gases and
cool evaporating fuel droplets. The closed mathematical formulation involves
a singularly perturbed system of four highly non-linear ordinary differential
equations. The entire dynamical picture of the system is qualitatively exposed
by exploiting the geometrical version of the powerful asymptotic approach
known as the method of integral manifolds (MIM).
It was found that the system’s behaviour can be classified according to
the values of nine dimensionless parameters. All possible types of dynamical
behaviour of the system were studied and the parametric regions of their
existence were delineated, with emphasis on the underlying physico-chemical
processes at play. Both conventional explosive and delayed regimes were found
to occur, including the freeze delay regime. Whereas this latter important
regime had been associated with physically unviable operating conditions in
previous deficient reactant models, it was found that the current use of a
single-step two-reactant chemical kinetic model renders the freeze delay regime
physically plausible. Due to its practical importance the delayed regimes were
analysed in detail and explicit analytical formulae for delay and evaporation
times were extracted. The predictions were found to agree rather well with the
results of direct numerical simulations.
It was also found that the stoichiometry of the initial mixture per se does
not lead to a natural classification of different sorts of regimes. Rather, the
ratio of two key parameters plays the dominant role in defining the relevant
fast variables and their associated dynamical regimes, irrespective of the initial
mixture stoichiometry.
1364-7830/02/020339+21$30.00
© 2002 IOP Publishing Ltd
Printed in the UK
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1. Introduction
In a previous publication, the dynamical behaviour of thermal explosion in a droplet–gas
mixture was examined in depth (Goldfarb et al 2000). The term ‘thermal explosion’ is taken
to refer to the initial stages of the behaviour of a combustible mixture as its temperature begins
to rise and various competing physical and chemical processes are called into play. In the case
of a droplet–gas mixture, it is mainly the endothermic versus exothermic competition, arising
from heat loss due to droplet evaporation in opposition to heat gain due to intensive chemical
reaction, that determines the multifaceted nature of the system. A full picture of the various
possible dynamical regimes and their dependence on the physico-chemical parameters of the
system was derived. The analysis was carried out by exploiting the geometrical version of
the integral manifold method which leads to a simple, yet extremely insightful, geometrical
description of the system’s dynamics. However, a drawback of the physical model considered
was that only a situation in which the fuel is the deficient reactant was considered. This is
somewhat restrictive if practical ramifications of the theory to situations such as jet engine
re-light are to be relevant. A complete description of the dynamical behaviour for the entire
gamut of initial mixture stoichiometry is therefore in place.
In this paper, we extend the previous analysis (Goldfarb et al 2000) and present a qualitative
investigation of thermal explosion dynamics in a fuel droplet–gas cloud in which a more
complete description of the chemistry via a single-step two-reactant model of general order is
permitted. This enables oxidizer effects to be treated via the whole spectrum of initial mixture
stoichiometries.
The structure of the paper is as follows. Following a description of the governing equations,
we expose the possible multiple scales necessary for the analysis. The integral manifold
method is then briefly sketched and applied to a reduced set of governing equations. The
different possible dynamical regimes are revealed, the influence of the mixture stoichiometry
is discussed and delay time estimates are developed. Finally, we present calculated results
in which a comparison is made between the predictions of the theory and direct numerical
simulation.
2. Problem statement
We investigate the mutual effect of oxidizer and fuel concentrations on the thermal explosion
of a flammable spray in a pre-heated combustible gas mixture using a simplified physical
model. The following main physical assumptions apply. We consider an infinite medium
filled with a combustible gas mixture (oxidant and gaseous fuel) and liquid fuel droplets. The
combustible liquid droplets are distributed uniformly throughout space as a mono-disperse
spray undergoing evaporation.
An adiabatic approach is adopted for the analysis. As is appropriate (Semenov 1928,
Frank-Kamenetskii 1969) for thermal explosion processes of the type we are studying, the
pressure change in the reaction volume is negligible as is its influence on the combustion
process. All droplets are at the same constant temperature (on the saturation line), which
differs from that of the hot gas. Heat flux from the burning gas to the droplets is supposed
to be proportional to the temperature difference between the liquid and gas phases. The
chemistry is modelled as a one-step highly exothermic chemical reaction. The order of the
chemical reaction is quite general; the reaction rate contains the multiple of non-integer powers
of the fuel and oxidizer concentrations in addition to the usual Arrhenius exponential term.
The mathematical model consists of four highly non-linear ordinary differential equations: an
energy equation (1), the mass equation for a single droplet (2) and concentration equations
Thermal explosion in a hot gas mixture
341
for the fuel and oxidizer (3) and (4). Under the aforementioned assumptions and after
reformulation in terms of dimensionless parameters, the system of governing equations reads
as follows
u
du
a b
= η ξ exp
− ε1 r(u + θ0 )
(1)
γ
dτ
1 + βu
1 d(r 3 )
= −ε1 r(u + θ0 )
ε2 dτ
u
dη
ν̃f
= −ηa ξ b exp
+ ν̃f ψε1 r(u + θ0 )
dτ
1 + βu
dξ
u
ν̃0
= −ηa ξ b exp
.
dτ
1 + βu
In deriving these equations, the following definitions and notations have been used:
t
1
E
τ=
;
treact =
exp
b−0.5
treact
RTg0
ACffa−0.5 Cox0
Cox
Cf
E Tg − T d
Rd
ξ=
;
η=
;
θ=
;
r=
Cox0
Cff
RTg0 Tg0
Rd0
u = θ − θ0 ;
θ0 =
E Tg0 − Td
RTg0 Tg0
(2)
(3)
(4)
(5)
(6)
in which treact —characteristic reaction time (s), Cff —overall fuel concentration (kmol m−3 ),
τ —time (dimensionless), ξ —oxidizer concentration (dimensionless), η—fuel vapour
concentration (dimensionless), θ—temperature (dimensionless), r—radius of the drops
(dimensionless), A—pre-exponential factor (m3(a+b−1) s−1 kmol−(a+b−1) ), C–concentration
(kmol m−3 ), E—activation energy (J kmol−1 ), L—latent heat (J kg−1 ), Q—combustion energy
(J kg−1 ), R—universal gas constant, Rd —radius of the drops (m), T –temperature (K),
nd —number of droplets per unit volume, α—volumetric phase content, λ—thermal
conductivity, ν—stoichiometric coefficient (dimensionless), ρ—density (kg m−3 ), t—time (s),
µ—molar mass (kg kmol−1 ), subscripts are d—liquid droplets, f—combustible component
(fuel), g—gas mixture, ox—oxidizer, ff—overall concentration, p—under constant pressure,
0—undisturbed (initial) state. In addition, the following dimensionless parameters appear
cpg ρg Tg0
RTg0
γ =
β
(7)
β=
E
(Cff Cox0 )0.5 Qf µf
4πRd0 λg βTg0 nd
1
ρL LαL
1
ε1 =
exp
=
;
(8)
;
b
β
ε2
(Cff Cox0 )0.5 Qf αg µf
Acox0
Cffa Qf αg µf
cff 0.5
1
1 Cox0 0.5
Qf Cox0 0.5
ν̃f =
;
ν̃0 =
;
ψ=
.
(9)
νf Cox0
ν0 Cff
L
Cff
The initial conditions for the system (1)–(4) are
Cf0
.
(10)
Cff
The dynamical behaviour of the system depends on nine dimensionless parameters: β, γ ,
ε1 , ε2 , ψ, ν̃f , ν̃0 , a, b and two initial conditions η0 , θ0 . β and γ are commonly used in
the context of the thermal explosion problems and their physical meanings are well known.
β is the reciprocal of the activation energy at the reference temperature, and γ represents the
u0 = 0;
r0 = 1;
ξ0 = 1;
η0 =
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V Bykov et al
reciprocal of the final dimensionless adiabatic temperature of the thermally insulated system
after the explosion is completed. Characteristic values of β and γ are small compared with unity
for most gaseous mixtures due to the high exothermicity and activation energy of the chemical
reaction. Parameters similar to ε1 , ε2 , were introduced in previous works relevant to the
self-ignition phenomenon in multiphase media (Goldfarb et al 1996). They describe relations
between the thermo-physical properties of the gaseous and liquid phases. The parameter
ψ represents the ratio of the heat of reaction to the latent heat of evaporation of the fuel.
Characteristic values of this parameter generally lie in the range 10–50 (heat released as a
result of combustion of a unit mass of fuel is much greater than that needed to evaporate the
same mass of fuel). The parameters ν̃f , ν̃0 describe the stoichiometric properties of the initial
combustible mixture and a, b relate to the way in which the reactants are involved in the
one-step chemistry.
The system of governing equations (1)–(4) together with the initial conditions (10) describe
the phenomenon of the ignition of the cold spray by hot gas which we will attempt to analyse.
A note is in order concerning the spatial homogeneity assumption. What is implied by
this assumption is that the time scale associated with spatial redistribution of concentration
and thermal energy should be smaller than the time scale of chemical reaction. Suppose we
take n-decane as the fuel; its diffusion coefficient is of the order of 10−5 m2 s−1 . Now the
typical length scale for diffusion in the current context can be associated with the inter-droplet
spacing. Taking a unit volume and 108 droplets distributed homogeneously therein, it is not
hard to estimate the aforementioned length scale to be of the order of 10−2 m. This gives
−1
tdiff
≈ 10−1 . As discussed later the concentration and reaction data we use for computing
−1
yield treact
≈ 10−5 . Thus, during the ignition episode of the mixture under analysis, it can be
shown that tdiff (see equation (5)) is indeed much less than treact thereby justifying the spatial
homogeneity assumption.
3. Preliminary analysis
Rather than numerically tackling the aforedescribed problem, we seek to make the equations
tractable analytically so as to gain direct insight into the possible dynamical regimes, which
dictate the subsequent evolution of the system.
Appropriate combinations of the equations can be taken and, after integration, the
following results are readily obtained:
1
1
γ u + ν̃f η + (ν̃f ψ − 1)r 3 = A1 = ν̃f η0 = (ν̃f ψ − 1)
(11)
ε2
ε2
γ u + ν̃0 ξ −
1 3
1
r = A2 = ν̃0 −
ε2
ε2
(12)
where A1 and A2 are constants. These relationships enable the fuel and oxidizer concentrations
to be expressed as functions of u and r, whereby we obtain
η(u, r) =
(A1 − γ u)ε2 − (ν̃f ψ − 1)r 3
γ uε2 − (ν̃f ψ − 1)(1 − r 3 )
= η0 −
ν̃f ε2
ν̃f ε2
ξ(u, r) =
γ uε2 + (1 − r 3 )
(A2 − γ u)ε2 + r 3
=1−
.
ν̃0 ε2
ν̃0 ε2
(13)
(14)
For further simplification, we note that, as we are concerned with the initial stages of the
behaviour of the combustible medium as its temperature begins to rise, it is reasonable to make
use of the inequalities βθ 1 and γ θ 1 (see Frank-Kamenetskii 1969). Thus, applying
Thermal explosion in a hot gas mixture
343
the approximations implied by these inequalities and armed with equations (11) and (12),
equations (1)–(4) reduce to the pair of ordinary differential equations (ODEs)
du
γ
= η(u, r)a ξ(u, r)b exp(u) − ε1 r(u + θ0 )
(15)
dτ
1 d(r 3 )
= −ε1 r(u + θ0 ).
ε2 dτ
(16)
These two equations are characterized by the presence of the parameters γ and 1/ε2 on the LHS.
Both parameters can be small: γ 1 due to the assumed high exothermicity of the chemical
reaction, whereas 1/ε2 can be less than unity as a result of the appropriate combination of
physical properties of the system. This implies that, at least initially, equations (15) and (16)
can represent a singularly perturbed system of ordinary differential equations with appreciably
different rates of change of the dependent variables. The type of singularity depends on the
relative sizes of the two parameters. The following possibilities exist.
(a) If γ 1 and γ 1/ε2 , then the temperature changes much faster than the droplet radius.
Equation (15) describes a fast heat release process, whereas equation (16) describes the
slow process of the reduction in the droplets’ radius. Thus, the temperature θ is the fast
variable, and the droplet radius r is the slow variable.
(b) If 1/ε2 γ 1 then the radius of the droplets changes much faster than the gas
temperature. In this limiting case, equation (15) describes a slow heat release process,
whereas equation (16) describes the fast process of the reduction in the droplets’ radius
due to intensive evaporation. The temperature θ is the slow variable, and the radius r is
the fast one.
(c) If 1/ε2 ∼ γ 1, the rates of change of the droplets radius and the gas temperature
are approximately the same. In this intermediate case, the equations cannot be separated
according to their rates of change.
Generally speaking, the qualitative analysis of the behaviour of possible solutions of
equations (15) and (16) using conventional phase-space analysis is rendered extremely difficult
due to the nature of the right hand sides of the equations, and approximate numerical procedures
must be resorted to. Alternatively, the presence of the small parameters γ and 1/ε2 such
that equations (15) and (16) may form a singular perturbed system, raises the possibility of
using some sort of asymptotic treatment for developing solutions. In this paper, we exploit
a powerful technique, the geometrical version of the integral manifolds method, through
which the multiple-scale system under consideration is decomposed into separate studies
of its component fast and slow subsystems. The advantage of this decomposition is that
the subsystems have lower dimensions than the original problem. Their analysis permits
a compact, clear geometrical/analytical rendition and interpretation of all possible dynamical
scenarios associated with the governing equations, in terms of the physico-chemical parameters
of the system. Although numerical solution of equations (15) and (16) is straightforward,
general analytical parametric demarcation of the system’s dynamical behaviour such as will
be presented here, is unattainable by numerical means.
3.1. Methodology (brief description of the MIM)
We give a brief outline of the method we shall use. Every solution of equations (15)–(16)
can be represented by a trajectory in the u–r plane. Use of the method of integral manifolds
(MIM) (Strygin and Sobolev 1988, Gol’dshtein and Sobolev 1992) exploits the splitting of an
arbitrary trajectory into fast and slow parts. The fast part is characterized by a constant value of
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V Bykov et al
the slow variables. The slow part is quasi-stationary for the fast variable and is located close to
the integral manifold. The exact location of the integral manifold of the system (15) and (16)
is unknown, and its definition represents a separate complicated problem, which is beyond the
scope of the present paper. However, we can determine the zeroth approximation (with respect
to an appropriate small parameter) of the unknown integral manifold (this approximation of the
real manifold is known as the slow surface) in accordance with the general theory of integral
manifolds (Fenichel 1979, Strygin and Sobolev 1988). This slow surface lies within a close
neighbourhood of the integral manifold’s exact position. The size of the neighbourhood and
the proximity of the slow surface (a curve in the two-dimensional case) to the exact manifold is
determined by the appropriate small parameter. The entire picture of the dynamical behaviour
of the system is then compactly encapsulated within the geometrical description of the fast
part of the trajectory and the slow curve. The interested reader unfamiliar with the details of
this method should refer to Gol’dshtein and Sobolev (1992) and other references (Babushok
and Gol’dshtein 1988, Strygin and Sobolev 1988, Goldfarb et al 1998, Fenichel 1979) for a
full description of the pertinent mathematical ideas.
3.2. Slow curves equation
The slow curve of the set of equations ((15) and (16)) is derived by equating the RHS of the
equation for the fast variable to zero. Hence, for situation (a) when the temperature is the fast
variable the slow curve u for the current system is given by the equation
u (u, r) = η(u, r)a ξ(u, r)b exp(u) − ε1 r(u + θ0 ) = 0.
(17)
Similarly, for case (b) when the radius is the fast variable, the slow curve r is
r (u, r) = −ε1 r(u + θ0 ) = 0 ⇒ r = 0.
(18)
As we are concerned with the effects of mixture stoichiometry on the thermal explosion
behaviour of the mixture, it is of benefit at this stage to consider the fuel rich and lean limits of
the initial system. In referring to the mixture stoichiometry, we mean that both the fuel liquid
and vapour are accounted for in its determination. We make use of the stoichiometric ratio
ϕ = (ν0 Cff )/(Vf Cox0 ) describing the relation between initial amounts of fuel and oxidizer.
ϕ > 1 and ϕ < 1 correspond to the fuel rich and lean limits, respectively. It is not hard to
show from equations (11) and (12) that A1 > A2 for the fuel rich case and A1 < A2 for the
fuel lean case.
The location of the adiabatic point of the system (i.e. its final state after chemical reaction
and heat transfer processes are completed) depends on the initial stoichiometry. In the fuel
lean case (ϕ < 1) the fuel is the deficient reactant, and once evaporation of the droplets is
concluded all fuel vapour is consumed and the final parameters of the system can be simply
determined:
A1
ν̃f (η0 ε2 + ψ) − 1
lean
;
rad
= 0;
=
γ
γ ε2
1
ν˜f
lean
= 0;
ξad
= (A2 − A1 ) = 1 −
(η0 ε2 + ψ).
ν̃0
ν̃0 ε2
ulean
ad =
lean
ηad
(19)
When ϕ > 1 a number of scenarios are possible. The oxidizer is the deficient reactant.
The simplest situation is realized when the heat release is large enough to evaporate all the
droplets. However, chemical reaction can terminate (due to insufficient oxidizer) when droplets
are still present. The remaining liquid droplets continue to evaporate and cause the gas phase
temperature to decrease. The evaporation will take place while the temperature of the gas
Thermal explosion in a hot gas mixture
345
phase is larger than its initial value or until all droplets disappear. In this case, we have
ν̃0 ε2 − 1
A2
1
ψ
ν̃0
rich
=
;
ηad
= (A2 − A1 ) = η0 +
− ;
urich
ad =
γ
γ ε2
ν̃f
ε2
ν̃f
rich
= 0;
rad
rich
ξad
= 0.
(20)
4. Analysis and results
We now turn to the analysis of the possible dynamical scenarios. We catalogue the different
possibilities according to the three sets of inequalities mentioned before in section 3.
4.1. Temperature as the fast variable
(a) γ 1, γ 1/ε2 . The equation of the slow curve u is given by equation (17), which
can be rewritten in the form containing the two variables u and r as
a b
(1 − r 3 )
(ν̃f ψ − 1)(1 − r 3 )
1−
exp(u) − ε1 r(u + θ0 ) = 0. (21)
u (u, r) = η0 +
ν̃f ε2
ν̃0 ε2
As the location of the initial point of any trajectory relative to the slow curve is of vital
importance for the system dynamics, the shape and position of the slow curve u in the
u–r plane must be first determined.
4.1.1. Shape and position of the slow curve. The function u (u, r) is defined in the domain
× , where is the axis of real numbers, and is determined as follows:
ν̃f η0 ε2
3
3
χru
;
χrd
=1+
= 1 − ν̃0 ε2
(22)
r ∈ ;
= [χrd , χru ];
ν̃f ψ − 1
where χrd , χru are lower and upper possible values for the variable r. A negative value of
the droplet radius has no physical significance so we focus attention on positive values of r
only. This is important since the description of the dynamics of the system requires knowledge
of the position of that part of the slow curve belonging to the physically accessible region of
parameters.
We now seek the turning points of the slow curve. The turning points T are defined as
points where the slow curve has a horizontal tangent ((u, r) = ∂(u, r)/∂u = 0). The
distinguishing feature of the turning points is that they divide the slow curve into stable and
unstable parts. The stable parts attract trajectories. Conversely, the unstable ones repel them.
Upon approaching a stable part, a trajectory begins to move along the slow curve within its close
neighbourhood. The proximity of this part of the trajectory to the slow curve is determined
by the value of the appropriate small parameter (O(γ ) or O(1/ε2 )). In essence, the trajectory
adheres to the stable part of the slow curve. The movement along the attractive (stable) part
continues until the trajectory reaches the unstable part or the stationary point of the system.
To determine the number of turning points we need to solve equation (21) together with
a b
(1 − r 3 )
(ν̃f ψ − 1)(1 − r 3 )
∂u (u, r)
1−
= η0 +
exp(u) − ε1 r = 0
(23)
∂u
ε2 ν̃f
ε2 ν̃0
from which we conclude that the slow curve u has a single turning point T, with u coordinate
uT = 1 − θ0 . The r coordinate rT of the point T is found by substituting value uT in
equation (21)
a b
(1 − r 3 )
(ν̃f ψ − 1)(1 − r 3 )
1−
exp(uT ) − ε1 r = 0.
(24)
u (uT,r ) = η0 +
ν̃f ε2
ν̃0 ε2
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V Bykov et al
(a)
(b)
N
C
P2
P3
D
P1
Pc
M
E
A
T
P4
Q
J1
C
P2
J3 F
P1
D
B
Pc
–θ0
M
J2
P3
Dimensionless radius r
Dimensionless radius r
Q
New variable u
P5
E
N
J
4
T
P4
B
A
New variable u
Figure 1. Shapes of slow curves (QTNM and QJ1 TJ4 J2 M) and possible trajectories (P1 DTA,
P2 CDTA, P3 E, P4 B, PCTA); temperature as the fast variable.
Similarly, the partial derivative ∂u (u, r)/∂r enables further details of the slow curve’s
structure to be extracted. Solving the algebraic equations u = 0 and ∂u /∂r = 0 we can
find points where the tangent to the slow curve is vertical. The equation for the r coordinates
reads
P (r) = C1 r 6 + C2 r 3 + C3 = 0
3
3
C1 = 1 − 3(a + b);
C2 = χrd
(3b − 1) + χru
(3a − 1);
3 3
C3 = χrd
χru .
(25)
The quadratic equation (25) in r 3 can be easily solved. It is readily concluded that there are
two real roots only if the discriminant is positive and no real roots if it is negative.
The general shape of the slow curves is presented in figure 1: in figure 1(a) the curve
QCTNM has no vertical tangent, whereas in figure 1(b) the curve QJ1 CTNJ2 M has two pairs
of vertical tangents (points J1 , J2 and J3 , J4 ).
For the sign of the partial derivatives (∂u (u, r)/∂r, ∂u (u, r)/∂u) on the slow curve we
have the following expressions:
∂u (u, r)
Sign
= Sign(C1 r 6 + C2 r 3 + C3 )
r∈
(26)
∂r
∂u (u, r)
(27)
= Sign(u − (1 − θ0 )).
Sign
∂u
4.1.2. Classification of possible regimes. Having determined possible shapes of the slow
curve we are in a position to classify the dynamical regimes by examining possible trajectories
depending on the location of the initial point relative to the slow curves. In the general case
(when the initial point does not belong to the slow curve) an arbitrary trajectory begins with the
so-called ‘fast motion’ from the initial point (u = 0, r = 1) and moves parallel to the u-axis (in
the zeroth approximation). A trajectory may impinge on the slow curve or pass above or below
it. The parameter values of the specific system under consideration determine the location of
the initial point relative to the slow curve and, accordingly, the detailed dynamical picture.
We begin by finding the critical trajectory, which is the one that touches the slow curve at
the turning point (Pc TA in figure 1(a)). The solution corresponding to this critical trajectory
describes the critical dynamical regime. Substituting r = 1 and u = (1−θ0 ) into equation (21)
for the slow curve yields
ε1c = (η0 )a exp(1 − θ0 ).
(28)
Thermal explosion in a hot gas mixture
347
This critical value ε1c collapses to that of Semenov’s classical theory of thermal explosion
under the simplifying conditions η0 = 1 and θ0 = 0. If ε1 < ε1c only conventional thermal
explosion can be expected—the chemical reaction dominates the evaporation process, the
system explodes and the gas temperature rises sharply (P4 B in figures 1(a) and (b)).
If ε1 > ε1c the initial point lies above the horizontal tangent of the slow curve QTNM at
the turning point T and a more complicated set of scenarios results. Firstly, straightforward
explosive behaviour occurs when the initial point P3 is located just right of the branch TNM
of the slow curves (P3 E in figures 1(a) and (b)). The corresponding mathematical condition
guaranteeing the existence of this position of the initial point can be easily found. We are thus
able to write the following conditions for conventional thermal explosion to occur:
(29a)
ε1 < ε1c = (η0 )a exp(1 − θ0 )
a
(η0 )
> ε1 > ε1c = (η0 )a exp(1 − θ0 );
θ0 > 1.
(29b)
θ0
Further dynamical behaviour is also possible. Referring to figures 1(a) and (b) it can be seen
that the fast parts of the trajectories P2 CTA and P1 DTA reach the stable branch QT of the slow
curve QTNM. The next part of these trajectories (CT and DT, correspondingly) adheres to the
slow curve and describes the decreasing droplets’ radii, until the turning point T is reached.
Essentially what happens is that following an initial rapid temperature increase/decrease (P2 C
and P1 D, respectively), the two processes of chemical reaction and droplet evaporation become
balanced and the droplets’ radii begin to decrease (CT and DT, respectively). This fine balance
is responsible for a delay effect, which is accompanied by an increase in the fuel vapour
concentration. Finally, beyond the turning point T, explosion occurs. It is not difficult to
derive conditions dictating the various relevant locations of the initial points. The initial point
lies within the slow curve (P2 ) if the inequality (30a) is valid:
(η0 )a
(30a)
ε1 >
θ0
and is located to the left of the branch QT of the slow curve (above the critical trajectory Pc TA)
if the double inequality (30b) is valid
(η0 )a
> ε1 > ε1c = (η0 )a exp(1 − θ0 );
θ0 < 1.
(30b)
θ0
The results of this analysis are succinctly summarized as a map in the θ0 −ε1 /η0a parametric
plane (see figure 2).
ε1/η0a
3
B
2.5
2
B
1.5
1
A
A
0.5
1
2
3
4 θ
0
Figure 2. Classification of dynamical regimes; temperature as the fast variable. Axes: abscissa
(θ0 )—dimensionless initial gas temperature, ordinate (ε1 /η0a )—critical parameter. A denotes the
region of conventional fast explosion, dictated by the conditions of equations (29a) and (29b); B
denotes the region of delayed explosion, dictated by the conditions of equations (30a) and (30b).
348
V Bykov et al
Dimensionless radius r
–θ0
P1
P3
P2
V
A
B
0
New variable u
C
S
W
Figure 3. Shapes of slow curves (VW) and possible trajectories (P1 AS, P2 BS, P3 CS); radius as
the fast variable.
4.2. Radius as the fast variable
(b) 1/ε2 γ 1. The equation of the slow curve r is simply given by r = 0. This means
that the u-axis (VW in figure 3) is the slow curve in the framework of this approximation. The
slow curve consists of a single attractive branch and all possible trajectories are similar to each
other (i.e. there is no dependence on the location of the initial point). Figure 3 depicts some
trajectories (P1 AS, P2 BS and P3 CS—the dashed graph is the slow curve from the previous
case shown here for comparison only). The underlying physics concerns a rapid decrease
of the droplets’ radius, while the temperature is essentially ‘frozen’. At the point where a
trajectory reaches the slow curve VW (A, B or C, figure 3), the droplets’ radius vanishes and
the model being used loses its validity (as the droplets no longer exist). A conventional final
explosion begins in a purely gaseous atmosphere and behaves in accordance with Semenov’s
theory (Semenov 1928).
4.3. Radius and temperature are fast and of the same rate
(c) O(ε2−1 /γ ) ∝ 1, γ 1. When the original system contains two fast variables of the same
rate of change the situation is much more complex. The previous natural occurrence of a
delay effect characterized by a fine balance between evaporative heat loss and reaction heat
release does not formally arise here, so that we can only surmise as to the possible dynamical
scenarios that may arise. (Despite this we are still able to make estimates of the time lapse
before explosion occurs, see later.)
Consider figure 4 in which some possible trajectories are sketched. Let us begin at an
initial point P1 , where the radius of the droplets equals unity. The location of the initial point
dictates the direction of the vector field at this point. Now consider trajectories P2 DS, P2 ES,
P1 CS, which take place when ε2−1 < γ θ0 . The first stage of the trajectory P1 CS (P1 C) is
characterized by a moderate decrease in temperature and a simultaneous diminishing of the
droplets’ radius, whereas the stages P2 D and P2 E are characterized by a moderate increase in
temperature. Evaporation continues until the liquid phase disappears. At this point (C, D or E)
conventional explosion begins in a pure gaseous atmosphere, as per Semenov.
A more complex picture takes place when the first part of the trajectory reaches the vertical
line with coordinate u = −θ0 (P1 A, figure 4, this scenario occurs when ε2−1 > γ θ0 ). This
line serves as a left border of available points for the system’s trajectory in that no trajectory
can pass beyond this line due to the assumed adiabatic condition. Physically what happens
Dimensionless radius r
Thermal explosion in a hot gas mixture
349
P1
P2
A
B
–θ0
C
0
new variable u
D
E
S
Figure 4. Possible trajectories (P1 ABS, P1 CS, P2 DS, P2 ES); both temperature and radius as fast
variables. The dashed line (the slow curve for the case of temperature as the fast variable) is shown
here for comparison only.
is that the system is cooled and its temperature drops to its initial value as a result of droplet
evaporation. Further movement of the trajectory then occurs along this line (stage AB, figure 4)
down to the point where the liquid fuel vanishes and conventional thermal explosion sets in
(stage BS, figure 4).
4.4. Influence of stoichiometry and relative concentrations of liquid and gaseous fuel on
dynamical regimes
The next question we address is the way in which the mixture stoichiometry and the relative
concentrations of liquid and gaseous fuel can influence the possible dynamical regimes
we have discussed. To estimate this impact, let us consider the relation between the key
parameters—1/ε2 and γ . We express their ratio in the following form:
ε2−1
E
ρL αL L
.
=
cpg ρg0 αg0 Tg0 RTg0
γ
(31)
The numerator of the first fraction on the RHS of equation (31) (ρL αL L) represents the amount
of energy required to evaporate all the liquid fuel, whereas the denominator (cpg ρg0 αg0 Tg0 ) is
the thermal capacity of the gas phase at the initial temperature. The second factor on the
RHS is the reciprocal of the small parameter β. Now, for a given total fuel load the relative
contributions of the vapour and liquid portions can be controlled by changing the droplets’
radius, for example. When most of the initial fuel is in the form of vapour (i.e. αL → 0)
the effect of the droplets will be rather weak and it is expected that the delay time before
explosion will be close to zero. This correlates well with the fact that the ratio (31) is much
less than unity (i.e. 1/ε2 γ ) in this case. Thus, the system exhibits dynamic scenarios that
are characterized by fast radius and slow temperature (see section 4.2). The greater the fraction
of liquid fuel in the fixed initial fuel load becomes the greater the impact of the droplets. Under
such circumstances, it can be envisaged that the ratio (31) can actually become greater than
unity (i.e. 1/ε2 γ ), leading to a radical change in the system’s hierarchy whereby the
temperature becomes the fast variable (see section 4.1). Following this logic, it can be seen
that the maximum influence of the droplets will occur when the entire initial fuel content is
in the form of droplets. This case corresponds to the greatest possible value of the volumetric
liquid content, αL,max , which can be related to the total initial amount of fuel Cff in the mixture:
αL,max = Cff µf /ρL . However, it is important to point out that, even for this limiting value of
the volumetric moisture content, αL,max , it is still plausible for the ratio (31) to be less than
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V Bykov et al
unity, depending on the values of the other parameters that play a role, e.g. Cpg0 , Tg0 etc. In
such a case, dynamic regimes associated with temperature as a fast variable are precluded,
irrespective of the initial stoichiometry of the mixture. Thus, we conclude that in the current
context no single, dramatic change in dynamic behaviour is implied by a shift from rich to lean
in the initial stoichiometry.
5. The delay time
The delay time is defined here as the time from τ = 0 until the final explosion. The system’s
dynamics have been shown to depend critically on the various parameters arising in the analysis.
In this section, we show how estimates of the delay times can be deduced analytically for the
different regimes considered previously.
5.1. Temperature as the fast variable
Referring to our analysis (section 4.1) it is seen that the system’s history prior to explosion can
be subdivided into two stages. At the very initial stage (P1 D or P2 C, figures 1(a) and (b)) the
trajectory rapidly moves to the slow curve parallel to the u-axis. The second stage consists of
motion along the slow curve (DT or CT, figures 1(a) and (b)). Asymptotically, the time of the
first stage is much smaller than that of the second one. Therefore, the delay time in this case
can be defined as the period when the trajectory moves along the slow curve until the final
explosion occurs, i.e. the time between the intersection of the trajectory with the slow curve
(points C, D) and the turning point T.
To estimate the delay time defined in this way, we simply integrate equation (16),
integration with respect to r being carried out from r = 1 to its value at the turning point
rT , the latter being determined from equation (24). Since the time of the initial stage is
negligible, the lower limit of integraton is set equal to zero and we obtain
τdelay
1 rT
dτ = −
[ε1 r(u + θ0 )]−1 d(r 3 ).
(32)
ε2 1
0
Now, substituting the expression for ε1 r(u + θ0 ) from equation (21) into the integral permits
the latter to be rewritten as
−a −b
(1 − r 3 )
1 rT
(ν̃f − 1)(1 − r 3 )
τdelay = −
η0 +
1−
exp(−u(r)) d(r 3 ).
ε2 1
ν̃f ε2
ν̃0 ε2
(33)
Clearly, this integration can only be performed analytically if an explicit expression for u(r)
is known, which is generally not the case. Nevertheless, an accurate approximation for the
unknown function u(r) can be found whereby the delay time can be estimated. We note that
the u coordinate of the turning point T was found to be 1 − θ0 . Therefore, on the slow part of
the trajectory (i.e. during the delay period) the following double inequality is valid
exp(−θ0 ) exp(u) exp(1 − θ0 ).
(34)
Exploiting these inequalities (34) it is not hard to produce upper and lower bounds for the
delay time
I1 exp(θ0 − 1) τdelay I1 exp(θ0 )
(35)
where the upper and lower estimates differ by a factor of e only (see equation (34)). The
constant value of the integral I1 can be obtained from the following expressions:
−a −b
(1 − r 3 )
1 rT
(ν̃f − 1)(1 − r 3 )
I1 =
η0 +
1−
d(r 3 ). (36)
ε2 1
ν̃f ε2
ν̃0 ε2
Thermal explosion in a hot gas mixture
351
The integral on the RHS of equation (36) can be calculated analytically. We find
η0−a rT
(1 + L(1 − r 3 ))−a (1 + M(1 − r 3 ))−b d(r 3 )
I1 =
ε2 1
a
1
M
3 (1−b)
=
(1 + M(1 − r ))
(b − 1)M
M −L
rT
L(1 + M(1 − r 3 )) ×2 1 − b, a, 2 − b,
L−M
(37)
1
where 2 (a, b, c, z) is the hyper-geometric function of the second kind and
L=
(ν̃f ψ − 1)
;
ν̃f ε2 η0
M=−
1
.
ν̃0 ε2
(38)
5.2. Radius as the fast variable
When the radius is the fast variable the delay time must be viewed differently since the system
degenerates (droplets disappear) when the trajectory reaches the slow curve (VW, figure 3)
and there is no motion along a slow curve because conventional explosion begins. The time
before the final thermal explosion is therefore the time during which fast evaporation occurs
(whilst the variable u is constant). Hence, taking u constant and equal to its initial zero value,
equation (16) can be readily integrated and the delay time before explosion can be determined
0
τdelay
d(r 3 )
3
dτ = τdelay = −
.
(39)
=
2ε1 ε2 θ0
0
1 ε1 ε2 r(u + θ0 )
5.3. Radius and temperature are fast and of the same rate
The intermediate situation, when the original system contains two fast variables having the
same rate of change, is much more complex than the previous ones we have discussed. As
mentioned previously (and as is the case when the radius is the fast variable) the natural
occurrence of a delay effect characterized by a fine balance between evaporative heat loss
and reaction heat release does not formally arise here. Nevertheless, we are able to calculate
the time which the system needs to reach to explosion conditions for both cases presented in
figure 4 and analysed in section 4. The time of evaporation for the trajectory P1 CS (ε2−1 < γ θ0 )
can be calculated in the following manner. u(r) can be extracted from equation (12) (we cannot
put u = 0 as in equation (39), because of the close rates of the two variables). Then, after
exploiting equation (2) and noting that ξ = 1 to a first approximation (equation (14)) we get
τdelay
1
γ
d(r 3 )
dτ = τdelay =
ε1 0 r(r 3 − 1 + γ ε2 θ0 )
0
√
γ
2−p
π
1 − p + p2
(40)
=
+ ln
√ + 2 3 arctan √
2ε1 p
(1 + p)2
3
3p
p = (γ θ0 ε2 − 1)1/3 .
For the more complex case (trajectory P1 ABS, ε2−1 > γ θ0 ), we adopt an approach that
was suggested elsewhere (Goldfarb et al 1997, 2000). The total time of the trajectory from
the initial point P1 to the point where thermal explosion begins (B) can be found as the sum of
the times of movements along P1 A and then AB.
τdelay = τ1 + τ2 .
(41)
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V Bykov et al
Determination of these two times yields
3
τ1 =
{1 − (1 − γ θ0 ε2 )2/3 }
2ε1 ε2 θ0
ξ∗
η(r(ξ ), ξ )−a ξ −b exp
τ2 = −ν̃0
1
where
1
ξ =1−
ν̃0
∗
1
+ γ θ0
ε2
(42)
θ0
1 − βθ0
dξ
(43)
(44)
is the solution of r(ξ ) = 0 and r(ξ ) satisfies u(r(ξ ), ξ ) + θ0 = 0, with
r 3 (ξ ) = 1 − ε2 (ν̃0 (1 − ξ ) + γ θ0 ).
(45)
The integral (43) is of the same nature as the one in the equation (37) and can be obtained in
analytical form.
6. Results and discussion
We have solved the governing equations numerically in order to confirm the integral manifold
analysis of the dynamical regimes. The computed results for the temperature and fuel
concentration are presented in figure 5 and should be read together with the analytical
predictions of figures 1(a) and (b). In figure 5(a) conventional thermal explosion occurs
(corresponding to trajectories P4 B, P3 E in figures 1(a) and (b)). There is no slow regime due
to the particular data chosen for this case. In figure 5(b) a delayed regime of type P2 CTA
(figure 1(a)) is shown. The temperature decreases due to rapid heat absorption by the droplets,
followed by a moderate temperature rise (corresponding to (CT)), after which explosion occurs
once the chemical reaction becomes predominant. Another type of delayed regime is illustrated
in figure 5(c) (corresponding to P1 DTA, figures 1(a) and (b)). It is seen that initially there is
a small yet sharp increase (P1 D), followed by a moderate rise (DT) of the temperature until
explosion occurs (at point T). In this case, it is observed that the initial fuel concentration
actually decreases slightly for a short period of time before increasing until explosion occurs.
At the very beginning of this process, the chemical kinetic effect dominates the evaporation
effect, producing the drop in the fuel concentration, before the latter effect takes over.
Finally, in figure 5(d) a third type of delayed regime is shown corresponding to the
trajectory P2 CTA (in figure 1(b)). In this case, after the initial rapid drop in temperature,
a delay stage (CT) sets in, during which a mild temperature drop occurs leading up to the final
explosion at the turning point T. This type of dynamics is referred to as a ‘freeze delay’ (Goldfarb
et al 1996, 2000). The behaviour of the droplet radius and the oxygen concentration are not
illustrated here since they are qualitatively similar for the four regimes illustrated in figure 5,
with the radius decreasing to zero and the oxygen concentration decreasing monotonically until
explosion occurs. Although the fuel vapour concentration generally increases monotonically
until explosion, there is an exception to this rule shown in the lower graph of the figure 5(c)
(fuel concentration versus time). It is worth noting that the delay phenomenon before the
onset of explosion is of vital importance from a practical point of view. As we have shown the
whole system can ultimately explode despite a temporary temperature drop during the process.
Interestingly, this freeze delay was previously uncovered and analysed in detail in the context
of a single reactant reaction and for unrealistic values of the parameter ψ (less than unity)
(Goldfarb et al 1996, 2000). The novelty of the occurrence of freeze delay here lies in the
fact that we have used a two-reactant single-step reaction, with the reaction rate depending
Thermal explosion in a hot gas mixture
353
(b)
Temperature (θ)
2.6
2.4
2.2
0.002
0.004
0.006
2
Temperature (θ)
(a)
0.008
1.5
1
0.5
1.8
1.6
0.1
0.2
0.1
0.2
Time (T)
0.3
Time (T)
0.4
0.5
0.3
0.4
0.5
Fuel concentration (η)
Fuel concentration (η)
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0.002
0.004
0.006
0.8
0.6
0.4
0.2
0.008
Time (T)
(c)
Time (T)
(d)
0.025
Temperature (θ)
Temperature (θ)
1.5
1.25
1
0.75
0.5
0.25
0.0225
0.02
0.0175
0.015
0.0125
5
0.02 0.04 0.06 0.08 0.1 0.12 0.14
Time (T)
0.154
0.152
0.02 0.04 0.06 0.08
0.1 0.12 0.14
0.148
0.146
20
25
30
25
30
0.25
0.2
0.15
0.1
0.05
5
Time (T)
15
Time (T)
Fuel concentration (η)
Fuel concentration (η)
0.156
10
10
15
20
Time (T)
Figure 5. Temperature and fuel concentration evolution (common data a = 0.25, b = 0.25,
β = 0.04). (a) Conventional explosion: γ = 0.01, ε1 = 0.1, ε2 = 0.001, θ0 = 1.5, ν̃f = 0.1,
ν̃0 = 0.9, η0 = 0.15, ψ = 15, (b) Explosion with delay: γ = 0.001, ε1 = 1, ε2 = 0.1, θ0 = 1.5,
ν̃f = 0.127, ν̃0 = 0.327, η0 = 0.25, ψ = 15, (c) Explosion with delay: γ = 0.001, ε1 = 5,
ε2 = 0.1, θ0 = 0.05, ν̃f = 0.1, ν̃0 = 0.9, η0 = 0.15, ψ = 10.1, (d) Explosion with freeze delay:
γ = 0.001, ε1 = 1.5, ε2 = 0.333, θ0 = 3, ν̃f = 0.4, ν̃0 = 0.5, η0 = 0.1, ψ = 3.
much more strongly on the oxidizer concentration than on the fuel concentration (a = 0.25,
b = 1.5 for n-decane fuel, Westbrook and Dryer 1981). Thus, the weak dependence of the
reaction rate on the fuel vapour concentration and the presence of the additional reactant render
freeze delay a distinct possibility under a realistic combination of the system’s parameters, with
354
V Bykov et al
subsequent practical ramifications. The delay time’s dependence on the physical and chemical
characteristics of the liquid fuel and oxidizer under consideration can be critical in assessing
safety features of a given system.
For the numerical simulations to be discussed, the following thermophysical properties
of n-decane were used: heat capacity cp , 1050 (J kg−1 K−1 ); latent heat L, 321 kJ kg−1 ;
combustion energy Q, 44.2 MJ kg−1 ; liquid fuel density ρf , 730 kg m−3 ; thermal
conductivity λ, 0.0193 W m−1 K−1 ; molar mass µf , 142 kg kmol−1 ; pre-exponential factor A,
1.9 × 107 s−1 .
Figure 6(a) illustrates how the ratio 1/(ε2 γ ) changes as the initial droplet radius increases
for a initially lean mixture and for three different initial temperatures of the gas under a fixed
overall (i.e. vapour + liquid) fuel loading. The rightmost points of the curves correspond to
the situation when all the fuel initially present is in liquid form. It is clear that 1/(ε2 γ ) is
(a)
1
Ratio 1/(ε2γ)
0.04
2
0.03
3
0.02
0.01
0.01
0.02
0.03
0.04
0.05
0.04
0.05
radius (mm)
0.25
(b)
1
Delay time (sec)
0.2
2
0.15
3
0.1
0.05
0.01
0.02
0.03
radius (mm)
Figure 6. Effect of initial droplets radius on explosive behaviour—fuel lean mixture; (a) variation of
the parameter 1/(ε2 γ ) with initial droplet radius, for different initial gas temperature; (b) variation
of delay time with initial droplet radius, for different initial gas temperature. Digits designate:
1 − Tg0 = 550 K; 2 − Tg0 = 600 K; 3 − Tg0 = 650 K. System parameters: n-decane, nd0 = 107
(m−3 ); ϕ = 0.1.
Thermal explosion in a hot gas mixture
355
much smaller than unity, thus placing the system under the radius-as-fast-variable regime, for
the conditions under consideration. Figure 6(b) shows plots of the delay time as a function of
the droplet initial radius. All three curves increase monotonically with initial droplet radius.
This is not unexpected: an absence of droplets (zero radius) produces an immediate explosion
without delay (see section 4.4). As the initial droplet radius grows (the number of droplets
remains fixed) a redistribution of liquid and fuel vapour in the initial mixture occurs. With
more fuel concentrated initially in the droplets and less vapour in the gas phase, more time is
needed to evaporate a sufficient amount of liquid fuel in order to bring the system to explosive
conditions (at the end of the delay period). A comparison of our analytical predictions of
the delay time (equation (39)) with data obtained from a numerical solution of the governing
equations reveals excellent agreement, with a relative error that does not exceed 1.5%.
Consider, now, the case when the initial overall mixture is fuel rich. In figure 7(a) the
parameter 1/(ε2 γ ) is plotted as a function of the initial droplet radius (for different initial
gas temperatures) and is found to be less than unity implying, once again, the fast radius
(a)
1
0.06
Ratio 1/(ε2γ)
0.05
2
0.04
3
0.03
0.02
0.01
0.01
0.015
0.02
0.025
0.02
0.025
Radius (mm)
(b)
0.06
1
Delay time (sec)
0.05
2
0.04
3
0.03
0.02
0.01
0.01
0.015
Radius (mm)
Figure 7. Effect of initial droplet radius on explosive behaviour—fuel rich mixture. (a) Variation of
the parameter 1/(ε2 γ ) with initial droplet radius, for different initial gas temperature; (b) Variation
of delay time with initial droplet radius, for different initial gas temperature. Digits designate:
1 − Tg0 = 550 K, 2 − Tg0 = 600 K, 3 − Tg0 = 650 K. System parameters: n-decane, nd0 = 108
(m−3 ), ϕ = 20.
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V Bykov et al
regime. The corresponding delay times are plotted in figure 7(b) and qualitatively exhibit
similar behaviour to the results obtained in the fuel lean case.
The horizontal axes in figure 8(a) begins at the point where the axes in figure 7 terminated.
For this range of droplet radii the key ratio 1/(ε2 γ ) is of the order of unity implying a regime
of radius and temperature as fast variables. For completeness, the delay times are shown in
figure 8(b), although the relative error in their estimate reaches as much as 7%.
Figure 9 continues from figure 8. Now 1/(ε2 γ ) is very much greater than unity
(see figure 9(a)) so that the fast temperature regime is implied by our analysis. In figure 9(b)
the upper and lower bounds of the delay time (equation (35)) are plotted as a function of the
droplet radius. The final figure 9(c) compares these estimates (for an initial temperature of
Tg0 = 550 K) with the results of direct numerical calculations. One can readily see that the
numerical points lie close to the upper estimate. A similar effect was found and a possible
explanation was suggested in a previous work (Goldfarb et al 1997).
1
(a)
2
Ratio (ε2γ)
2
1.5
3
1
0.5
0.04
0.05
0.06
0.07
Radius (mm)
(b)
1
1
Delay time (sec)
0.8
2
0.6
3
0.4
0.2
0.04
0.05
Radius (mm)
0.06
0.07
Figure 8. Continuation of figure 7—effect of initial droplet radius on explosive behaviour—fuel
rich mixture: (a) Variation of the parameter 1/(ε2 γ ) with initial droplet radius, for different initial
gas temperature; (b) Variation of delay time with initial droplet radius, for different initial gas
temperature.
Thermal explosion in a hot gas mixture
357
(a)
1
Ratio 1/(ε2γ)
18
16
2
14
3
12
10
8
0.11
(b)
0.14
1
6000
Delay time estimation (sec)
0.12
0.13
Radius (mm)
5000
2
4000
3
3000
2000
1000
0.11
0.12
0.13
Radius (mm)
0.14
(c)
Delay time (sec)
6000
5000
4000
3000
2000
1000
0.11
0.12
0.13
Radius (mm)
0.14
Figure 9. Continuation of figure 8—effect of initial droplet radius on explosive behaviour—fuel
rich mixture: (a) Variation of the parameter 1/(ε2 γ ) with initial droplet radius, for different initial
gas temperature; (b) Variation of delay time with initial droplet radius, for different initial gas
temperature; (c) Comparison of estimated delay times with computed results.
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V Bykov et al
7. Conclusions
Previous analyses of thermal explosion in a gas-droplets mixture have been extended to permit
a more complete description of the chemistry via a single-step two-reactant model of general
order, rather than the prior deficient reactant model. A detailed mathematical analysis has been
carried out of this new physical model that encompasses oxidizer effects (in both fuel rich and
fuel lean situations) on the thermal explosion of a hot combustible mixture of gases and cool
evaporating fuel droplets. The closed mathematical formulation involves a singularly perturbed
system of four highly non-linear ordinary differential equations. The entire dynamical picture
of the system is qualitatively uncovered by exploiting the geometrical version of the powerful
asymptotic approach known as the MIM.
It was found that the system’s behaviour can be classified according to the values of nine
key dimensionless parameters (β, γ , ε1 , ε2 , a, b, ψ, ν0 , νf ). All possible types of dynamical
behaviour of the system were studied and the parametric regions of their existence were
delineated, with emphasis being on the underlying physico-chemical processes at play. Both
conventional explosive and delayed regimes were found to occur, including the freeze delay
regime. While this latter important regime had been associated with physically unviable
operating conditions in previous deficient reactant models, it was found that the current use of
a single-step two-reactant chemical kinetic model definitely renders the freeze delay regime
physically plausible. Due to its practical importance, the delayed regimes were analysed
in detail and explicit analytical formulae for the delay times and evaporation times were
extracted. Their predictions were found to agree rather well with the results of direct numerical
simulations.
It was also found that the stoichiometry of the initial mixture per se does not lead to a
natural classification of different sorts of regimes. Rather the ratio of the parameters 1/ε2 and
γ plays the dominant role in defining the relevant fast variables and their associated dynamical
regimes, irrespective of the initial mixture stoichiometry.
Finally, we note that although the current use of a global two-reactant single-step chemical
model does provide a broader more accurate perspective than that previously obtained with
a single-reactant model, it is not without its own deficiencies. The data generally used for
the exponents of the fuel and oxygen, the pre-exponential factor and the activation energy are
derived from correlations with experimental measurements or detailed computational results
for physical situations that are not necessarily identical to those under consideration here.
Nevertheless, it can be argued that the exponents must somehow express some driving chain
branching mechanism that is surely called into play in the ignition process. However, it
is indisputable that a more realistic description should ideally include more details of the
chemistry such that initiation of the chemistry via a radical pool can be accounted for. This
direction of further improvement of our model is currently under investigation.
Acknowledgments
This paper was partly presented at the 18th International Colloquium on the Dynamics
of Explosions and Reactive Systems, Seattle, Washington, August, 2001. JBG gratefully
acknowledges the partial support of the Lady Davis Chair in Aerospace Engineering and the
Technion Fund for the Promotion of Research.
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