=Paper=
{{Paper
|id=Vol-1490/paper21
|storemode=property
|title=Critical phenomena in a model of fuel's heating in a porous medium
|pdfUrl=https://ceur-ws.org/Vol-1490/paper21.pdf
|volume=Vol-1490
}}
==Critical phenomena in a model of fuel's heating in a porous medium==
https://ceur-ws.org/Vol-1490/paper21.pdf
Mathematical Modeling
Critical phenomena in a model
of fuel's heating in a porous medium
Shchepakina E.A.
Samara State Airspace University
Abstract. The autoignition of flammable liquid in an inert porous medium are
studied. In this paper we concentrated on the critical case which is concerned
with the phenomenon of delayed loss of stability in the dynamical model. The
realizability conditions for the critical regime are obtained. It is shown that
critical regime is modelled by a canard − a trajectory of slow–fast system,
which first move near the stable part of the slow invariant manifold, then move
near the unstable part of it.
Keywords: ignition, critical phenomenon, canard, invariant manifold, delayed
loss of stability.
Citation: Shchepakina E.A. Critical phenomena in a model of fuel's heating in
a porous medium. Proceedings of Information Technology and Nanotechnology
(ITNT-2015), CEUR Workshop Proceedings, 2015; 1490: 179-189. DOI:
10.18287/1613-0073-2015-1490-179-189
Introduction
This paper deals with the investigation of the critical conditions for autoignition of
combustible fluids in porous insulation materials [1, 2]. This phenomenon is usually
caused by a leaking of a combustible liquid into lagging material surrounding a hot
pipework. Due to highly insulation environment heat losses are remarkable low and
autoignition may occur as a result of exothermic oxidation reaction.
The investigation of the autoignition process has been carried out by many authors,
see for instance [3–12] and references therein.
We shall study a process which may be defined as the autoignition in two-phase
medium (combustible liquid and inert porous matrix). The possible depletion of oxygen or
its diffusion into porous structure and transport of the liquid or its vapour within the
insulation are all ignored, in order to focus attention on the competitive effects of the
reactive term of the dispersed liquid and evaporative heat loss. The dimensionless model
in this case has the form [1]
du
QK1 xe1/ u (u ua ) Qc K 2 xe βe / u ,
dt
(1)
dx
K 2 xeβe / u K1 xe 1/ u .
dt
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Here, u is a dimensionless temperature of the reactant phase; the dimensionless
concentration x represents the mass fraction of combustible liquid present in the porous
material; the dimensionless parameters Q and K1 characterize the heat of reaction and the
reaction frequency, respectively, for the exothermic oxidation reaction, while Qc and K 2
are the similar terms for the endothermic evaporation reaction; β e is the ratio of the
enthalpy of vaporization to the activation energy of the oxidation reaction; ua is ambient
temperature.
Introducing the new variables θ and τ by
u θ β2 , t τ exp(1/ β), β u(0),
and taking into account
1 1 1
exp exp exp ,
β(θβ 1) 1+βθ β
leads (1) to the form
QK1 x θ ua β 1
θ exp θ 2 exp
β2 1+βθ β β
Qc K 2 x 1-β βθ
exp e exp e , (2)
β2 β 1+βθ
1-β βθ θ
x K 2 x exp e exp e K1 x exp .
β 1+βθ 1+βθ
We introduce the new parameters
1 1-β u β
ε= exp , a K 2 exp e , θ a a 2 ,
β β β
QK 1 QK -β
μ 2 1 exp , ν= c 2 2 exp e .
β β β β
Due to the smallness of the parameter ε for typical combustible liquids, system (2) can
be rewritten in the singularly perturbed form (see, for instance, [13−17]):
θ βe θ
εθ μx exp θ θ a νx exp , (3)
1+βθ 1+βθ
βθ θ
x ax exp e K1 x exp . (4)
1+βθ 1+βθ
The chemically relevant phase space of system (3), (4) is defined by
x 0, θ 1/ .
In [2] system (1) was investigated numerically under quasi-steady-state assumption that
corresponds to the assumption ε 0 for system (3), (4). This approach allows
determining the main types of chemical regimes of the investigated process. The quasi-
steady-state assumption is widely used in the theory of combustion and gives good results
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to draw conclusions about the qualitative behavior of the full system for sufficiently small
ε . However, the critical phenomena are highly sensitive with respect to the parameters.
Hence, this fact implies the considerable difficulties under the numerical calculations.
Thus, detailed study of this mathematical object is possible with taking into account the
small perturbations and using of asymptotic methods, for example, methods of the integral
manifolds theory for singularly perturbed systems.
Criteria for the critical regimes
The trivial solution is the final steady state of the system. The degenerate equation
θ βe θ
0 μx exp θ θ a νx exp F ( x,θ) describes the slow curve S of
1+βθ 1+βθ
(3), (4) (see, for example, [17, 18]). The subset S s ( S u ) of S with
F ( x,θ)
0 ( 0)
θ
is called the stable or attractive (unstable or repulsive) part of S. A point A on S in
which F / θ=0 is called the jump or turning point. Stable and unstable parts of the
slow curve are zeroth order approximations of corresponding stable and unstable slow
invariant manifolds. The invariant manifolds lie in an ε-neighborhood of the slow
curve, except near jump or turning points (see [17] and references therein).
The slow curve has an asymptote θ θ , where
1
θ ln μ / ν ,
βe 1 β ln μ / ν
and intersects the axis Oθ in the point with θ θ0 . The shape of the curve S varies
with the relation between values of the parameters, which leads to a change in
qualitative behavior of the system. So, if we change the value of one parameter, with
fixed values of the other parameters, we can change the type of chemical reaction.
Following [2], we consider β e as a control parameter. For θ θ0 we have βe b0
where
1 βθ
b0 1 ln μ / ν .
θa
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a)
b)
c)
d)
Fig. 1. – The slow curve of (3), (4) for b0 1 , β>ua , and a) βe bT b0 ; b) bT βe b0 ; c)
βe b0 ; d) βe b0
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a)
b)
Fig. 2. – a) The trajectory (the solid line) and the slow curve (the dashed line) of (3), (4), and
b) the x− and θ −components of the solution in the case of the slow regime: b=0.3685
Consider the case β>ua and b0 1 . For βe bT the lower branch of S can consist
of two stable parts ( S1s ) and one unstable part ( S1u ), see Fig. 1a. These parts are
divided by two turning points, which merge with one another and disappear at a value
βe bT [2, 11], see Fig. 1b. In both cases the trajectories of system (3), (4) move
along the stable part of the slow curve to the final steady state. These trajectories
correspond to the slow regimes, which are safe, see Fig. 2.
For βe b0 the upper and lower branches of S consist of stable ( S1s and S 2s ) and
unstable ( S1u and S 2u ) parts, which are divided by the turning points A1 and A2 , see
Fig. 1d. And the system's trajectories starting at any point of the basin of attraction of
S 2s correspond to the slow regimes, see Fig. 3.
In other case, when the initial point is out of the basin of attraction of S 2s , we can
observe the thermal explosion (Fig. 4) or thermal explosion with delay [8, 18]. The
thermal explosion with delay occurs when the initial point belongs to the basin of
attraction of S1s and the system's trajectories having reached the jump point A1 along
S1s at the tempo of the slow variable jump into the explosive regime.
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a)
b)
Fig. 3. – The case of the slow regime: b=1; the initial point belongs to the basin of attraction of
S 2s
For βe b0 the point A1 merges with A2 to give one self–intersection point A of
the slow curve, see Fig. 1c. As it was noted above, in ε-neighborhood of the subset
s u
S1s ( S 2u ) there exists a stable (unstable) slow invariant manifold S1,ε ( S2,ε ). For some
value βe b* b0 O(ε), (ε 0) [11], the stable and unstable slow invariant
s
manifolds S1,ε u
and S2,ε are glued at the point A . As a result for βe b* system (3),
(4) has a canard trajectory [17-22] which, at first, follows an attractive invariant
manifold, and then a repulsive one. In both cases the distances travelled are O(1) as
ε 0 , see Fig. 5.
This canard simulates the critical regime, separating slow chemical regimes from
regimes with a self–acceleration in the case b0 1 .
If β>ua , one can observe the similar transformation of the slow curve (and the
qualitative behavior of the system) as shown in Fig. 1 but with a decreasing value of
the parameter β e ( bT b0 in this case).
For b0 1 plots of the slow curve are mirror images with respect to the vertical
axis of the graphs shown in Fig. 1, and for nonsignificant values of the initial
concentration of a combustible liquid the thermal behavior of the chemical system is
safe. Otherwise, the value βe bT determines the boundary of the safe region [2].
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a)
b)
Fig. 4. – The case of thermal explosion: b=1; the initial point is out of the basin of attraction of
S 2s
Our goal is to reveal the sufficient conditions for realization of the critical regime
for the case b0 1 . As it has been noted above, the main feature here consists in fact
that during the critical regime the temperature attains a high value but without
explosion. The interest in critical phenomena is occasioned by not only for reasons of
safety, but in many cases the critical regime is the most effective in technological
processes [6-9, 18, 20-22].
Realizability conditions for the critical regime
Using the method of integral manifolds and the canard techniques in [17,18] it is
possible to find the critical value of the parameter βe b* and corresponding
trajectory in the form of the asymptotic representation
θ φ( x,ε) φ0 ( x) εφ1 ( x) o(ε), (5)
βe b* b0 εb1 o(ε). (6)
We write (3), (4) as
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βθ θ
εθ ax exp e K1 x exp
1+βθ 1+βθ
βθ θ
θ θ a +νx exp e μx exp ,
1+βθ 1+βθ
or, taking into account (5), (6),
φ0 φ1
x εφ0 ε 2 φ1 K1 exp 1 ε
1+βφ0
2
1+βφ 0
b φ 1+βφ0 b1φ0 b0 φ1
a exp 0 0 1 ε
1+βφ0 1+βφ0
2
b φ 1+βφ0 b1φ0 b0 φ1
φ0 εφ1 θ a +νx exp 0 0 1 ε
1+βφ0 1+βφ0
2
(7)
φ0 φ1
μx exp 1 ε o(ε).
1+βφ0
2
1+βφ0
a)
b)
Fig. 5. – a) The trajectory (the solid line) and the slow curve (the dashed line) of (3), (4), and
(b) the x− and θ −components of the solution in the case of critical regime: b b* 0.56827
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Setting ε=0 in (7) we obtain the slow curve equation:
φ0
F ( x, φ0 ) μx exp φ0
1+βφ0 (8)
bφ
θ a νx exp 0 0 0.
1+βφ0
Conditions for self-intersection of the slow curve in point A xs ,φ0 ( xs )
F F
0 (9)
x xs ,φ0 ( xs ) φ0 x ,φ ( x )
s 0 s
give us the coordinates of the self-intersection point and the zeroth-order
approximations for critical value b* . Indeed, from (8), (9) we obtain
1 βθa θa
xs , φ0 ( xs ) θ a (10)
μ exp θ a / 1 βθ a ln ν / μ
and
1 βθ a
b0 1 ln μ / ν . (11)
θa
Equating the coefficients with ε1 in (7) we get
bφ φ0
xφ0 a exp 0 0 K1 exp
1+βφ 0 1+βφ 0
x bφ φ0
φ1 1+ b ν exp 0 0 μ exp
2 0 (12)
1+βφ0 1+βφ 0 1+βφ 0
b φ bφ
νx exp 0 0 1 0 .
1+βφ 0 1+βφ 0
From (9) we note that the expression in brackets in r.h.s. of (12) is equal to zero at point
A . To avoid a discontinuity in function φ1 ( x) at xs we put
b φ (x ) φ0 ( xs )
φ0 ( xs ) a exp 0 0 s K1 exp
1+βφ0 ( xs ) 1+βφ 0 ( xs )
b φ (x ) b φ (x )
ν exp 0 0 s 1 0 s .
1+βφ0 ( xs ) 1+βφ 0 ( xs )
From this and (10) we get
φ0 ( xs ) 1 βθ a (1- b0 )θ a
b1 = K1 exp a . (13)
νθ a 1 βθ a
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Note, that the value φ0 ( xs ) can be found from equation (8) after double
differentiation with respect to x, taking into account (9):
μ 1 βθ a ν
3
θ
φ0 ( xs ) 2 ln exp a
θa μ 1+βθ a (14)
1 βθ a μ
2β 1 βθ a 2 ln .
θa ν
Expressions (13) and (14) give us
2μ 3 1 βθ a
4
ν 3θ a K1 a
b1 = ln 3 exp
θa 4
μ 1+βθ a μ ν (15)
1 βθ a ν
2β 1 βθ a 2 ln .
θa μ
Thus, the expressions (8), (11), (12), and (15) determine the first−order
approximation for canard and corresponding critical value βe b* .
It should be noted that it is not possible to explicitly solve equation (8) with respect
to φ 0 , while the critical value b* has been found in the explicit form. However, one
can use the implicit or parametric representation for slow invariant manifold [17] to
obtain an approximation of the canard.
Conclusion
In this paper the model of autoignition of combustible fluids in an inert porous
medium has been studied. The realizability conditions for the critical regime have
been obtained as the explicit asymptotic expression for the control parameter. It was
shown that the critical regime is modelled by the canard. This regime plays the role of
a watershed between the safe processes and regimes with self-acceleration that leads
to the explosion.
It should be noted that the critical regime is not a slow regime, since the
temperature may attain a high value, and is not explosive, as the temperature increases
at the tempo of the slow variable. Thus, for the examined model the new type of the
safe regime has been revealed.
Acknowledgements
This work is supported in part by the Russian Foundation for Basic Research
(grants 13-01-97002-p, 14-01-97018-p) and the Ministry of education and science of
the Russian Federation in the framework of the implementation of Program of
increasing the competitiveness of SSAU for 2013−2020 years.
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