Critical ignition conditions in exothermically reacting systems for

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Research
Cite this article: Filimonov VY, Koshelev KB.
2016 Critical ignition conditions in
exothermically reacting systems for arbitrary
reaction kinetics. Proc. R. Soc. A 472: 20160529.
http://dx.doi.org/10.1098/rspa.2016.0529
Received: 29 June 2016
Accepted: 27 July 2016
Subject Areas:
chemical physics
Keywords:
thermal explosion, reactant consumption,
critical conditions, bifurcation, bistability,
degeneration conditions
Author for correspondence:
Valeriy Yu. Filimonov
e-mail: vyfilimonov@rambler.ru
Critical ignition conditions
in exothermically reacting
systems for arbitrary
reaction kinetics
Valeriy Yu. Filimonov1 and Konstantin B. Koshelev2
1 Altai State Technical University, Lenina strausse 46,
Barnaul 656038, Russia
2 Institute for Water and Environmental Problems SB RAS,
Molodyoznaya strausse 1, Barnaul 656038, Russia
VYF, 0000-0003-0229-7058
In this work, a universal method for determination
of the critical ignition conditions taking into account
the reactant consumption is proposed. Based on
the analysis of the phase trajectories equation, the
equation for maximal temperatures of exothermic
reactions was obtained. In this case, the asymptotic
criterion of ignition is determined by the impossibility
of slow reaction mode realization with low value
of maximum temperature. The method allows
demarcating the regions of low- and high-temperature
modes of exothermic reactions and to establish the
criteria of transition to the region of high-temperature
modes. The corresponding parametric diagrams can
be characterized as the bifurcation ones (bistability).
It was found that the region of thermal explosion (TE)
existence is bounded by the classical TE conditions
from below and by the degeneration conditions
from above. The comparison of analytical calculation
results with the results of numerical calculation gives
a satisfactory agreement.
1. Introduction
The mathematical theory of thermal explosion (TE) in
homogeneous systems was developed in the second
half of the twentieth century [1–3]. The most successful
approximation of the theory is the neglect of reactant
consumption during preheating that allows analysing
the stationary modes of reactions. However, the stationary
modes are impossible in the case when the system is not
a flow reactor with a continuous supply of reactants [3].
2016 The Author(s) Published by the Royal Society. All rights reserved.
The generalized set of self-heating dynamics and the product formation kinetics equations can be
written in the dimensionless form as follows [1–3]:
⎫
θ
dθ
⎪
= w + ϕ(y) exp
− δθ ⎪
⎪
⎪
dτ
1 + βθ
⎬
(2.1)
⎪
⎪
dy
θ
⎪
⎪
⎭
and
= γ ϕ(y) exp
.
dτ
1 + βθ
The initial conditions are τ = 0, θ = 0, y = y0 . Here, θ = (E/RT02 )(T − T0 ) is the dimensionless
temperature, 0 ≤ y ≤ 1 is the conversion depth (fraction reacted), ϕ(y) is the kinetic function, γ
is the Todes parameter (the ratio of the characteristic time of heat release to the characteristic
time of chemical reaction), δ is the Semenov parameter (the ratio of the characteristic time of heat
release to the characteristic time of heat removal), T0 is the initial temperature, E is the activation
energy. w = W/W0 is the ratio of the external heating power to the heating power of chemical
reaction at T = T0 , β = RT0 /E is the Arrhenius parameter. The form of γ , δ parameters depends on
the specificity of the exothermic reaction.
It should be noted that the analysis of the set (2.1) is primarily important in terms of
the critical ignition conditions determination. Consequently, the set of equation (2.1) must be
supplemented by the corresponding criterion of ignition. Shouman & El-Sayed [8] and Filimonov
[10] suggested that the simultaneous appearance of two inflection points on the thermogram
(time–temperature profile) can be considered as a criterion of TE. However, this does not allow
solving this problem mathematically and this criterion can be used in numerical simulation of
the set (2.1). Even in this case, the serious problems connected with TE identification appear at
relatively high values of γ parameter [11]. With an increase of this parameter, the transition to
...................................................
2. Kinetics of exothermic reactions: heuristic considerations
2
rspa.royalsocietypublishing.org Proc. R. Soc. A 472: 20160529
Indeed, the continuous decrease of reaction rate occurs during the reactant consumption.
Consequently, the reacting system tends to the stationary state which is determined by the
completion of reaction. This brings up the questions: What is the criterion of ignition in the case
when we cannot neglect the reactants consumption? What is the TE in this situation? Indeed,
the TE becomes less pronounced with an increase of reactants consumption influence. Thus, the
demarcation of explosive and non-explosive modes of exothermic reactions is an important and
non-trivial problem.
One of the best known methods which allow the influence of reactants consumption on the
critical conditions is the method of integral manifolds [4–7]. This method gives an opportunity
to define the critical autoignition conditions close to the classical conditions of TE at small
reactant consumption during preheating. However, as it was shown by Shouman & El-Sayed
[8], this model becomes not applicable with distance from the classical TE conditions. Thus, this
method gives an opportunity to determine only the small corrections to the classical conditions.
At the same time, the necessity of critical conditions determination arises for explosives which
are diluted by an inert component or reaction product (smouldering modes of combustion or
mild combustion [9]) or for the systems with low heat effect. In this situation, the regime of
exothermic reaction is an ‘intermediate’ one which is located between the classical (asymptotical)
conditions and degeneration conditions. The currently existing asymptotic methods are not
applicable for such systems. Therefore, the problem of critical TE conditions determination (up to
TE degeneration) for any kind of exothermic reaction remains unsolved. It is worth noting that the
main sign of TE is the high sensitivity of temperature to the change of parameters. Consequently,
the TE phenomenon can be considered as a fast increase of maximal temperature at small change
of parameters. This point should be given more attention.
The purpose of this work is the determination of ignition criteria in terms of maximum
temperatures of heating and definition of the connection between the kinetic parameters at the
limit of ignition.
From equation (2.1), the phase trajectory equation for any exothermic reaction has the form
γ
(w − δθ ) exp [ − θ/(1 + βθ)]
dθ
=1+
,
dy
ϕ(y)
with initial conditions θ = 0, y = y0 .
From equation (3.1), the maximum temperature condition can be presented as
−θm
,
ϕ(ym ) = (δθm − w) exp
1 + βθm
(3.1)
(3.2)
where θm is the maximal temperature of reaction, ym is the corresponding conversion depth:
−θm
,
(3.3)
ym (θm , δ) = ϕ −1 (δθm − w) exp
1 + βθm
where ϕ −1 is the inverse function.
Let us integrate equation (3.1) with respect to y between the limits y = y0 and y = ym :
ym
ym
θm
dy
−θ
.
dθ =
dy +
(w − δθ ) exp
γ
1 + βθ
ϕ(y)
0
y0
y0
(3.4)
Let us introduce the function
F(y) =
dy
.
ϕ(y)
Then, equation (3.4) can be written as follows:
F(ym )
−θ
dF.
γ θm = ym − y0 +
(w − δθ ) exp
1 + βθ
F(y0 )
After integrating equation (3.6) by the parts, we obtain
−θm
γ θm = ym − y0 + F(ym )(w − δθm ) exp
− F(y0 )w
1 + βθm
θm
−θ
.
−
F(y) d (w − δθ ) exp
1 + βθ
0
(3.5)
(3.6)
(3.7)
It is reasonable to assume that there are two characteristic modes of reaction: (i)
low-temperature mode ym ≈ y0 1; θm ≈ 1 and (ii) high-temperature mode ym ≈ 1; θm 1.
The transition between these modes must be realized by small change of parameters. In terms of
...................................................
3. The basic equations and transformations: equation of maximal temperatures
3
high-temperature modes is becoming increasingly blurred and the problem of fixing the critical
parameters becomes complicated. The well-known method of integral manifolds [5,6] does not
determine any novel criterion of ignition. This model allows analysing the problem by the method
of small perturbation using the small parameter γ 1. Thus, the small correction for reactants
consumption during preheating can be obtained in this case. However, it is unclear how small
should be the value of parameter γ in order to apply this method. Indeed, it should be understood
that the range of variation of kinetic parameters γ , δ, β is quite large for various exothermic
reactions. Thus, the problem of ignition criterion determination for the entire spectrum of
exothermic reactions has not been solved to date. Within the framework of the classical theory
[2], this criterion is formulated in an asymptotic approximation γ = 0 (excluding the reactants
consumption influence on the critical conditions). In this case, the TE critical conditions can
be determined as an impossibility of thermal equilibrium between the reacting system and its
environment. At the same time, the exact solution of the set (2.1) (analytical or numerical) does
not give an opportunity to obtain clear and unambiguous criterion of low- and high-temperature
modes demarcation. It is, therefore, necessary to find a reasonable approximation which would
allow finding this criterion at any value of γ parameter.
It is worth noting that the heating is small (for the low-temperature mode): θ ≤ θm ≈ 1 and the
condition γ 1 is the necessary condition for TE realization [2]. Consequently, we can suggest
that the following approximate condition takes place:
F(y) ≈ F(y0 ).
(3.9)
It is necessary to note that the classical theory of TE uses the simpler condition
ϕ(y) ≈ ϕ(y0 ).
(3.10)
The condition (3.10) can be considered as a zero approximation in the classical theory at γ = 0.
In this case, it is necessary to analyse only the first equation of the set (2.1) in order to obtain the
critical TE conditions. However, the condition (3.9) should be regarded as the first approximation
which takes into account the reactants consumption and dependence ϕ(y) at γ = 0. From the point
of view of the first approximation, the ignition condition means the impossibility of condition (3.9)
realization (the impossibility of low-temperature mode realization).
Thus, we obtain the solution of equation (3.7) in a first approximation (low-temperature mode)
−θm
.
(3.11)
y0 + γ θm = ym (θm ) + [F(ym ) − F(y0 )](w − δθm ) exp
1 + βθm
Equations (3.2) and (3.11) can be represented in the form
y0 + γ θm = ym (θm ) − [F(ym ) − F(y0 )]ϕ(ym ).
(3.12)
4. Analysis of maximal temperature equation
It is extremely important that equation (3.11) is valid for any form of kinetic function ϕ(y). As an
example, we can consider the following kinetic laws:
ϕ(y) = (1 − y)n
(4.1)
ϕ(y) = ( 23 )[(1 − y)−1/3 − 1]−1 .
(4.2)
and
Equation (4.1) determines the n-order homogeneous reactions; equation (4.2) defines the
heterogeneous reactions during diffusion of components through the reaction product layer
(Ginstling–Brounstein equation).
The dependences (3.11) for kinetic laws (4.1), (4.2) are presented in figure 1a. From figure 1a, the
decrease of γ parameter leads to the slow increase of maximal temperature up to the minimum
point.
Then, one can observe the abrupt increase of maximal temperature. It is necessary to note that
there is no any jump of maximal temperature in the real situation (figure 1b). With a decrease of γ
parameter, the maximal temperature increases monotonically. However, there is the region of high
parametric sensitivity when a small change of γ parameter leads to dramatic change of maximal
temperature (dγ /dθm 1). The real dependence γ (θm ) does not have any characteristic points
and this fact does not allow defining the strict boundaries of high parametric sensitivity area. At
the same time, the approximate solution (3.11) gives an opportunity to reveal the boundaries of
this region and to divide the areas of stable low- and high-temperature modes due to bistability.
...................................................
y0
4
the first approximation, we can assume that the function F[y(θ)] weekly depends on temperature
in the range 0 − θm in the low-temperature mode. Indeed, we can consider the series expansion
of the function F[y(θ )] about the point θ = 0, y = y0 . Taking into account equation (3.1), we obtain
dF
1 d2 F
1
γ θ + O(γ 2 θ 2 ) + · · · . (3.8)
θ+
θ 2 + · · · = F(y0 ) +
F(θ ) ≈ F(y0 ) +
dθ y0
2 dθ 2
w + ϕ(y0 )
(a) 0.40
(b)
stable low
temperature
modes
1- Eq. (3.12), n = 2, d = 1, 5
2- Eq. (3.12), n = 1, d = 1, 5
0.35
stable high
temperature
modes
1
0.25
g
critical
point
0.20
0.15
0.10
2
1
0.05
1
2
3
dg /dqm = 0
presented
model
3
4
5
qm
6
7
8
9
real
dependence
dg /dqm1
I
III
II
qm
10
Figure 1. The dependences γ (θm ) in accordance with equation (3.11) at w = 0, β = 0, y0 = 0: (a) the kinetic functions (4.1)
and (4.2); (b) the schematic of dependences: 1—the real dependence, 2—the asymptotic dependence (3.11).
(a) 0.30
1–d = 2, 5;
2–d = 2.0;
3–d = 1, 5;
4–d = 1.0.
b = 0, y0 = 0
0.25
0.20
g
b = 0, y0 = 0
1–g = 0, 1;
2–g = 0, 13;
3–g = 0, 16;
4–g = 0, 19.
1
3.0
2.5
d
3
0.10
0
3.5
4
0.15
0.05
(b) 4.0
2.0
1.5
2
1.0
2
3
0.5
1
1
2
3
4
5
6
7
0
4
2
4
qm
6
qm
8
10
12
Figure 2. The families of parametric dependences (4.5) for the first-order reactions at w = 0, β = 0, y0 = 0.
The zone of unstable modes is located between these boundaries (figure 1b). In this case, the
condition of high parametric sensitivity dγ /dθm 1 can be changed by asymptotic condition
dγ
= 0.
dθm
(4.3)
The condition (4.3) means the impossibility of condition (3.9) realization and it can be
considered as the condition of transition to high-temperature mode.
Let us analyse the dependence (3.11) for the first-order reactions in details (at β = 0). Taking
into consideration equations (3.2) and (3.5), we obtain
ϕ(ym ) = 1 − ym = δθm exp ( − θm ),
F(ym ) = −ln(1 − ym ).
(4.4)
The substitution of equation (4.4) in equation (3.11) gives
γ (θm , δ) =
1
− δ[1 + θm − ln δθm ] exp( − θm ).
θm
(4.5)
Figure 2 shows the families of parametric dependences for the first-order reactions at the
various values of parameters. From the figure, the decrease of δ parameter leads to increase of
the critical values of maximum temperature and γ parameter (figure 2a) and the increase of γ
parameter leads to decrease of critical value of δ parameter. With the change of parameters, the
regions I and II approach each other (figure 1b) and the area of unstable modes III disappears.
...................................................
2
0.30
5
3- Eq. (4.1), d = 30
region of high parametric sensitivity
(unstable modes)
The application of condition (4.3) to equation (3.11) gives (after transformation) the generalized
criterion for the critical ignition condition
w
ym − y0
θm
=
+
.
ϕ(ym )[F(ym ) − F(y0 )] w − δθm
(1 + βθm )2
(5.1)
Equations (5.1) and (3.11) and equation of connection (3.3) represent the complete system
of equations for critical conditions analysis. This algebraic system determines the relationship
between the two variables θm , ym and five parameters γ , δ, w, β, y0 . It is extremely difficult
to analyse this relationship with the use of numerical simulation methods because of the
large dimension of the parametric space. The most important dependences are θc (γ , δ, w, β, y0 ),
yc (γ , δ, w, β, y0 ) and parametric dependence γc (δ, w, β, y0 ) at the limit of ignition (hereinafter, the
index ‘c’ means the critical value). In this work, we consider the partial case of condition (5.1) for
the second-order reactions (the most common type of reaction) at w = 0 (self-ignition).
6. Second-order reactions: analysis of the critical conditions
(a) The basic equations
For the second-order reactions, the dimensionless parameters have the form [2]
γ=
cRT02
,
QE[A01 ]
δ=
αSRT02
E
exp
,
QVEk0 [A01 ][A02 ]
RT0
(6.1)
where [A01 ], [A02 ] are the molar concentrations of initial reactants under normal conditions, c
is the heat capacity of the system per unit volume, k0 is the pre-exponential factor, Q is the heat
effect of reaction, α is the heat transfer coefficient, V is the volume of reaction which is bounded by
surface area S. Therefore, these parameters are not independent. Excluding the initial temperature
T0 from consideration, we obtain (in accordance with [10])
b
,
(6.2)
δ = γ z exp √
γ
where
b2 =
cE
QRA01
and z =
αS
.
cVk0 A02
Taking into account equation (3.5), the functions ϕ(y) and
ϕ(y) = (1 − y)2
(6.3)
F(y) can be written as
and F(y) = (1 − y)−1 .
(6.4)
Then, equations (5.1), (3.11) and (3.3) have the form (after transformations)
(1 + βθc )2
1 − yc
=
,
1 − y0
θc
(yc − y0 )2
1 − y0
−θc
.
(1 − yc )2 = δθc exp
1 + βθc
γ θc =
and
(6.5)
(6.6)
(6.7)
...................................................
5. Generalized criteria of ignition
6
Thus, the TE is bounded by the classic conditions from below and by degeneration conditions
from above. In this connection, the problem of determination of the parametric boundaries of TE
existence reduces to the problem of the existence of extremum points.
θc
,
= (1 + βθc ) exp
1 + βθc
γ̃ =
γc
1 − y0
where
4
and δ̃c =
δc
(1 − y0 )2
.
(6.9)
(6.10)
It is known that the condition βθc 1 is valid for the systems with high activation energy [2].
With the use of this condition, it is important to obtain the relation between the parameters δ̃c , γ̃c
in an explicit form. One can introduce the deviation from the classical conditions
θ = θc − 1.
We can assume that this deviation is small. The product βθ we can consider as a
of the second order of smallness. The substitution of equation (6.11) in equation (6.9)
the expression for critical temperatures (with an accuracy up to the terms of the first
of smallness)
3
e
1+ β .
θc ≈
2
δ̃c
The substitution of equation (6.12) in equation (6.8) gives
⎤2
⎡
δ̃
δ̃ ⎦
3
1
1 − β ⎣1 − 1 + β
.
γ̃c ≈
e
2
2
e
(6.11)
value
gives
order
(6.12)
(6.13)
When considering the set of equations (6.8) and (6.9), one can analyse two problems:
1. Direct problem. It is necessary to define the conditions of ignition in the case when the
initial temperature T0 is known. This problem can be solved easily. For example, we
know the characteristic values of parameters: γ̃ = 0.03 and β = 0.01. The solution of
equation (6.8) gives: θc ≈ 1.27. Taking into account equation (6.5), we obtain yc ≈ 0.19.
The substitution of the values calculated in equation (6.9) gives δ̃c ≈ 1.8. Using equations
(6.2) and (6.3), we obtain the critical value z = αS/cVk0 A02 ≈ 2.7 × 10−3 .
2. Inverse problem. It is necessary to determine the critical temperature Tig and
corresponding preheating θc in the case when the heat removal conditions (z parameter)
and kinetics parameters of reacting system (b parameter) are known.
First of all, it is very important to consider the characteristic limiting case β = 0 which determines
the systems with high activation energy [2].
(b) Critical conditions: Frank-Kamenetsky problem
The set of equations (6.8) and (6.9) is significantly simplified in the case of the classical asymptotic
approximation [2] β = 0 (Frank-Kamenetsky problem):
⎫
1 2⎪
⎪
γ̃ θc = 1 −
⎬
θc
(6.14)
⎪
⎪
⎭
3
and
δ̃θc = exp θc .
The dependences (6.14) are presented in figure 3.
...................................................
δ̃c θc3
and
7
The algebraic set (6.5)–(6.7) gives an opportunity to consider the analytic dependences of the
critical temperature on parameters δ, γ :
2
(1 + βθc )2
(6.8)
γ̃c θc = 1 −
θc
(a) 6
(b)
8
qc 3
a
a
qc
2
1
0
1
2
3
d˜c
4
5
6
0
0.04
0.06
0.08
0.12
0.15
g˜c
Figure 3. The dependences (6.14) for the critical values. (a) The dependence of critical temperature on parameter δ̃c ; (b) the
dependence of critical temperature on parameter γ̃c . The point ‘a’ corresponds to degeneration condition.
Taking into account equations (6.1) and (6.10), the increase of parameter γ̃c (figure 3b) can be
connected with a decrease of heat effect or with an increase of dilution y0 . In both cases, this leads
to the necessity of additional heating of the system which can be achieved by reducing the rate of
heat removal δ̃c (figure 3a). However, there is the limiting value of γ̃ parameter when TE becomes
impossible (point a). The analysis of the set (6.14) allows determining the degeneration conditions:
θd = 3; γd =
4
27
and δd =
e3
.
27
(6.15)
With the use of equation (6.5), one can estimate the limiting conversion depth:
yd =
γd θd =
2
≈ 0.67.
3
(6.15’)
Further, it is necessary to compare the results obtained with the results of numerical calculation
of the set (2.1). The parametric diagrams of critical conditions are presented in figure 4. These
diagrams were obtained with the use of numerical calculation of the set (6.14) and with the use of
analytic dependence (6.13) at β = 0.
Figure 5 shows the thermograms which correspond to transitions 1–4 (figure 4).
From figure 5, the critical transitions are the transitions 2 → 3. Obviously, the TE is strictly
pronounced in the cases γ = 0.02 and γ = 0.04. With the increase of γ parameter, the gap
between the low- and high-temperature modes narrows (figures 1 and 2). The gap disappears
at γ ≈ 0.1 and dependence Todes parameter—maximal temperature becomes close to linear.
With a decrease of δ̃ parameter, the maximal temperature changes smoothly until the adiabatic
heating regime. The dependence (6.13) corresponds to the results of numerical calculation more
accurately. From the figure, the results obtained by Kassoy & Linan [6] (the method of integral
manifolds) give the overly underestimated values of critical parameters (the plot c).
In this case, the degeneration phenomenon is determined by the small value of heat effect,
whereas the value of activation energy is significant.
(c) Critical conditions: the general analysis
In order to calculate the critical conditions for a given system, the parameters γ and β cannot
be considered as independent values. Taking into account the form of parameter β = RT0 /E and
...................................................
4
5
0.16
a
9
2 - the critical points
b=0
A
...................................................
0.14
B
0.12
0.10
4 3 2 1
d
g~c 0.08
0.06
4 3 21
C
c
0.04
4 3 2 1b
0.02
4
3
2 1a
2
0
0.5
1.0
1.5
~
dc
2.0
2.5
3.0
Figure 4. Results of critical conditions calculation at β = 0. A—the numerical calculation of algebraic set of equations (6.14),
B—the dependence (6.13) at β = 0, C—the dependence obtained by Kassoy & Linan [6], 1–4 the trajectories of imaging
points which are defined by decrease of δ̃c parameter in the parametric space. The point a corresponds to degeneration
conditions (6.14).
equations (6.1) and (6.3), we obtain
γ̃ = b2 β 2 .
(6.16)
From equation (6.3), b parameter characterizes only the reacting system. The dependences (6.8)
and (6.9) (equation (6.16)) are presented in figure 6.
For a given system (with the fixed value of b parameter), the increase of γ̃ parameter is
connected with an increase of initial temperature T0 . The increase of this value leads to the
increase of reactant consumption influence during preheating. In its turn, this leads to the
necessity of heating of the system θc . At the points of degeneration, the influence of reactants
consumption dominates and TE becomes impossible. However, with an increase of b2 (transition
to another reacting system), the activation energy increases. This broadens the regions of
TE existence. The condition b2 → ∞ is equivalent to the Frank-Kamenetsky problem (β = 0).
Therefore, the dependences which are presented in figure 6 tend asymptotically to the limiting
dependences which are presented in figure 3.
Let us analyse the degeneration conditions which are determined by the conditions dγ̃c /dθc =
dδ̃c /dθc = 0. These conditions lead to equation
β 2 θc2 + (1 − 2β)θc − 3 = 0.
(6.17)
From the analysis of equation (6.17), TE cannot be realized under condition β > 0.25. The joint
consideration of equations (6.8), (6.9), (6.17) leads to the dependences γd (b2 ) and γd (δ) which are
presented in figure 7.
It is necessary to note that the dependences presented characterize unambiguously the all
exothermic reactions of second order. In the limiting case b2 → ∞, the degeneration conditions
tend asymptotically to conditions (6.15) for the Frank-Kamenetsky problem. Thus, we can
estimate the degeneration conditions for any given system which is characterized by the given
value of b parameter with the use of figure 7 and equations (6.2) and (6.3).
(a)
(b) 40
40
4
25
3
20
20
3
15
10
10
1
0
2
4
6
2
1
5
2
8
10
(c)
0
2
8
4
4
6
8
10
6
8
10
(d)
4
12.5
7
10.0
6
5
3
q 7.5
3
4
5.0
3
2
2.5
2
2
1
1
1
2
0
4
t
6
8
0
10
2
4
t
Figure 5. The time-temperature profiles corresponding to trajectories 1–4 (figure 4). The transitions from low- to hightemperature modes are determined by the transitions from the mode 2 to the mode 3.
(a)
(b)
4
1–b2 = 10
2–b2 = 10
1
3
3
2
3–b2 = 70
4–b2 = •
4
3.8
3.4
b2 = 10
3.0
b2 = 40
2.6
qc
b2 = 70
b2 = •
2.2
2
1.8
Frank-Kamenetsky
problem
1
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
~
dc
Frank-Kamenetsky
problem
1.4
1.0
0
0.02 0.04 0.06 0.08 0.10 0.12 0.14
g~c
Figure 6. The dependences of critical temperature on parameters: (a) the dependence (6.9) at various values of parameter b;
(b) the dependence (6.8) at various values of parameter b. The points indicate the degeneration conditions. The arrows indicate
the directions of the initial temperature T0 increase.
Figure 8 shows the dependences of critical parameters at the limit of ignition which were
obtained with the use of equations (6.13) and (6.16). From the figure, the results of analytical
calculation show the good agreement with the results of numerical calculation.
The deviations (about 10%) are observed only for relatively large values of γ̃ parameter. In
order to obtain the critical temperatures, it is necessary to take into account the dependence (6.2)
at the given value of b parameter.
...................................................
30
4
30
q
10
35
(a)
0.12
0.12
0.09
0.09
0.06
0.06
0.03
0.03
0
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
11
10
0
20
30
b2
d˜d
40
50
60
Figure 7. The dependences of the Todes criterion on parameters at the limit of degeneration.
1 – b2 = 10
0.14
2 – b2 = 70
0.12
3 – b2 = •
3
- numerical
calculation eq. (2.1)
0.10
g˜c
0.08
2
0.06
0.04
1
0.02
0
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
d˜c
Figure 8. The dependences of critical parameters at the limit of ignition which were calculated with the use of equations (6.13)
and (6.16) at various values of parameter b2 .
(d) An example of critical parameters calculation
For the analysis of critical conditions, it is practically important to consider the relationship
between the parameters of reacting mixture at the limit of ignition. In this case, the dependence of
critical temperature on the pressure of gas mixture is most often considered [2]. In order to obtain
this dependence, equation (6.13) (at β = 0) should be represented in the form
⎞
⎛
2
4 eγ̃
⎠.
(6.18)
δ̃ = e ⎝1 −
δ̃
From equation (6.1), the parameter γ̃ 2 /δ̃ does not depend on reactants concentration or gas
pressure. Then, the dependence of critical pressure on temperature (using equation (6.1)) can be
represented as follows:
hT02 exp(E/2RT0 )
,
(6.19)
pc =
√
[1 − H T0 exp( − E/4RT0 )]
where
h(Pa K−2 ) =
αSR3 /VQk0 Ee
and H =
4
c2 eVk0 R/QEαS.
(6.20)
...................................................
0.15
g˜d
(b)
0.15
2.5
12
pressure (atm)
1
1¢
a
b
c
d
1.0
e
0.5
2¢
0
1000
1050
f
g
h
2
A
B
1100
1150
1200
temperature (K)
1250
1300
1350
Figure 9. The dependence of critical pressure on temperature for reaction (6.21). 1, 1 —αS/V = 103 W m−3 K−1 ; 2,
2 —αS/V = 102 W m−3 K−1 . 1, 2—the calculation with the use of equation (6.19) at H = 0 (considering the reactant
consumption), 1 ,2 —the calculation with the use of equation (6.19) at H = 0 (the classical theory). A and B are the points
of degeneration. The squares indicate the results of experimental investigation [15].
The numerator of equation (6.19) defines the classical TE condition at γ̃ = 0, δ̃ = e excluding
the reactant consumption. Thus, the reactant consumption influence is determined by the
denominator of equation (6.19) at H = 0. The conditions E/4RT0 1, H 1 determine the
possibility of the classical theory application. Hence, for critical parameters calculation, it is
interesting to analyse the case when the difference between the classical condition and condition
(6.19) is significant (the upper part of the diagram presented in figure 8) and the denominator of
equation (6.19) is of primary importance. This may correspond to low pressures or concentrations
of the gas mixture and correspondingly to high values of critical temperatures. Moreover, the heat
removal rate (the value αS/V) should be significant. Let us consider the second-order exothermic
reaction of nitrous oxide decomposition:
2N2 O → 2N2 + O2 .
(6.21)
The kinetics of this reaction is well studied in the high-temperature region [12–14]. In
accordance with the results of the study [14], let us introduce the values of kinetic parameters:
Q = 163 kJ mol−1 , E = 240 kJ mol−1 , k0 = 5.1 × 108 m3 mol−1 s−1 and c = 2.18kJ m−3 K−1 in the
temperature range 1000–1300 K.
The dependence (6.19) (reaction (6.21)) is presented in figure 9 for two values of parameter
αS/V. From the figure, the results of critical parameters calculations obtained with the use of
classical theory (curves 1 , 2 ) are significantly different from the results obtained with the use
of equation (6.19) (curves 1, 2). With the decrease of pressure, the gap between the curves 1, 1
and 2, 2 increases. The discrepancy between the critical pressures is about 25% at T0 = 1050 K
and it reaches 100% at T0 = 1200 K. Consequently, the classical theory is inapplicable in terms
of the critical pressures calculation. With an increase of the pressure, the gap between the curves
narrows asymptotically. With a decrease of heat removal rate (the plots 2, 2 ) the curves are located
closer to each other and the dependence of critical pressure on temperature becomes weaker. From
calculation, the experimental results [15] correspond to theoretical ones with good accuracy at the
value αS/V ≈ 800 W m−3 K−1 . Theoretically, all the signs of TE disappear at the points A, B.
Figure 10 shows the results of numerical calculation of the equations set (2.1) for reaction
pathways d → a and h → e (figure 9) which correspond to the change of pressure at the
given temperatures.
...................................................
1.5
2.0
(a)
30
a
4
e
3
15
b
2
f
10
1
5
0
d
1
2
g
c
h
3
4
5
6
7
0
1
t
2
3
4
5
6
7
t
Figure 10. The self-heating thermograms which correspond to reaction pathways presented in figure 9.
From figure 10a, the critical transition is the transition c → b at T0 = 1050 K which is
accompanied by the abrupt increase in temperature. The corresponding change of maximum
temperature is T = 908 K. At the temperature T0 = 1200 K, the critical transition is the transition
f → e which is accompanied by the change of maximum temperature: T = 138 K. In the transition
h → g → f → e, the maximum temperature increases smoothly (without any jumps) and the mode
of self-ignition is close to a degenerate one.
It is useful to estimate the corresponding conversion depth. The joint consideration of
equations (6.5) and (6.6) at y0 = 0 gives
γ (1 − yc ) y2c = (γ + βy2c )2 .
(6.22)
From the calculation, the value of conversion depth is yc ≈ 0.21 at T0 = 1200 K. Therefore,
the conversion depth during preheating is significant (it cannot be ignored). At T0 = 1200 K, the
corresponding conversion depth is yc ≈ 0.6 that is close to the estimate (6.15 ).
7. Discussion
Realization of exothermic reactions in controlled mode is the extremely difficult problem in the
case when reaction rate strongly depends on temperature and the value of the heat effect is
significant. The basic methods which allow controlling the rate of fast exothermic reactions are the
methods of pressure reducing and dilution of gas mixture by an inert component. In this situation,
it is important to be able to predict the critical TE conditions until its complete degeneration. This
may be of practical interest in terms of the problems of so-called mild combustion [9], explosion
safety, heat and power engineering, chemical kinetics, etc. The currently existing methods of
critical conditions calculation [5–8] are developed for homogeneous reactions and they are based
on the technique of perturbation theory. Actually, these methods allow calculating the correction
to the classical theory taking into account the reactant consumption. As it was shown in this
work (figure 9), these corrections give the significant deviations from the results of numerical
calculation at relatively high values of Todes parameter (figure 4) for the second-order reactions.
In this study, the calculating formulae (6.13), (6.19) which can be easily verified experimentally
were suggested. The formula (6.19) allows to define directly the deviation from the classical TE
theory at H = 0. The model presented allows estimating the low limit of TE degeneration (β = 0)
which has the form (equations (6.15) and (6.13))
e3
exp(−2.6b).
(7.1)
zd =
4
This equation establishes a relationship between the heat removal conditions (αS/V) and
reactants concentration A0 (or pressure) when the reaction can be realized in a controlled manner.
...................................................
20
13
maximum
temperatures
Tm (e) = 1439K
Tm ( f ) = 1308 K
Tm (g) = 1249 K
Tm (h) = 1230 K
T0 = 1200 K
25
q
(b) 5
maximum
temperatures
Tm (a) = 2195 K
Tm (b) = 2004 K
Tm (c) = 1096 K
Tm (d) = 1088 K
T0 = 1050 K
In this study, the method of critical ignition conditions calculation for arbitrary exothermic
reactions has been proposed. The method is based on the approximate solution of integral
equation which was obtained for the maximal temperatures of exothermic reactions with
reactant consumption. In this case, the critical ignition conditions should be regarded as the
impossibility of slow reaction mode realization. The method allows revealing the regions of
low- and high-temperature modes on the parametric plane (bistability) and to formulate the
generalized mathematical criterion of transition of reacting system to high-temperature mode.
It was established that the region of critical conditions existence is bounded by the classical
conditions from below and by the degeneration conditions from above. The application of
the method to the particular case of the second-order reactions allowed establishing that the
calculated critical values correlate well with the results of numerical calculations. The method
gives an opportunity to analyse the critical conditions in a much broader range of parameters than
the method of integral manifolds. Therefore, it becomes possible to analyse the critical conditions
for reactions with low heat release or diluted mixtures when the method of integral manifolds is
not applicable. The method suggested may be of interest from the point of view of the problems
of explosion safety, heat power engineering and chemical kinetics.
Data accessibility. This work contains the experimental data presented in the reference list.
Authors’ contributions. V.Yu.F. carried out the analytical investigations, developed the mathematical model,
participated in the design of the study and drafted the manuscript. K.B.K. carried out the numerical
calculation of the basic equations of the mathematical model and helped draft the manuscript. All authors
gave the final approval for publication.
Competing interests. We declare we have no competing interests.
Funding. This work is not supported by any grant or programme presently.
References
1. Todes OM, Melent’ev PV. 1939 The theory of thermal explosion. J. Phys. Chem. 13, 52–58.
2. Frank-Kamenetsky DA. 2015 Diffusion and heat exchange in chemical kinetics, p. 382. Princeton,
NJ: Princeton University Press.
3. Bebernes J, Eberly D. 1989 Mathematical problems from combustion theory. Appl. Math. Sci.
83, 1–178. (doi:10.1007/978-1-4612-4546-9)
4. Gorelov GN, Sobolev VA. 1992 Duck-trajectories in a thermal explosion problem. Appl. Math.
Lett. 5, 3–6. (doi:10.1016/0893-9659(92)90002-Q)
5. Gray P, Lee PR. 1965 Thermal explosion and the effect of reactant consumption on critical
conditions. Combust. Flame 9, 201–203. (doi:10.1016/0010-2180(65)90068-4)
...................................................
8. Conclusion
14
Indeed, the smouldering reaction modes (or mild combustion) can be realized under condition
z > zd and this gives an opportunity to control the rate of reaction by the change of concentration
or (and) pressure in the parametric region z ≈ zd (figure 10b). The reaction is uncontrollable at
z zd . For a more accurate calculation(β = 0), one can use the dependences presented in figure 7.
Thus, the model suggested gives an opportunity to estimate the values of critical parameters in
the upper part of the critical diagram (figure 8) where the classical theory gives the incorrect
results. The equation (6.22) gives the possibility to estimate the value of conversion depth
during preheating.
In conclusion, it should be noted that the conditions (3.12) and (5.1) are applicable not
only to the calculation of TE critical conditions for homogeneous mixtures (4.1) which were
predominantly analysed with the use of integral manifolds methods. The mathematical theory
of critical phenomena for heterogeneous systems (taking into account the reactant consumption)
is poorly developed. Unlike the homogeneous reactions (law of mass action (4.1)), there are many
different kinds of kinetic laws ϕ(y) which describe the heterogeneous interaction [16]. This allows
using the conditions (3.12) and (5.1) in a wide range of applications (including forced ignition).
This problem requires additional investigations.
15
...................................................
6. Kassoy DR, Linan A. 1978 The influence of reactant consumption on the critical conditions
for homogeneous thermal explosion. J. Mech. Appl. Math. 31, 99–112. (doi:10.1093/gjmam/
31.1.99)
7. Gray BF. 1973 Critical behaviour in chemical reacting systems: an exactly soluble model.
Combust. Flame 21, 317–325. (doi:10.1016/0010-2180(73)90023-0)
8. Shouman AR, El-Sayed SA. 1997 Accounting for reactant consumption in the thermal
explosion problem. Part II: a direct solution with application to the Frank-Kamenetskii
problem. Combust. Flame 108, 361–386. (doi:10.1016/S0010-2180(96)00121-6)
9. Coelho PJ, Peters N. 2001 Numerical simulation of a mild combustion burner. Combust. Flame
124, 503–518. (doi:10.1016/S0010-2180(00)00206-6)
10. Filimonov VY. 2013 Thermal modes of monomolecular exothermic reactions: twodimensional model. Combust. Flame 160, 539–545. (doi:10.1016/j.combustflame.2012.11.016)
11. Shouman AR, El-Sayed SA. 1998 Accounting for reactant consumption in the thermal
explosion problem III. Criticality conditions for the Arrhenius problem. Combust. Flame 113,
212–223. (doi:10.1016/S0010-2180(97)00168-5)
12. Ross SK, Sutherland JW, Kuo S-C, Klemm RB. 1997 Rate constants for the thermal dissociation
of N2 O and the O(3 P) + N2 O reaction. J. Phys. Chem. A 101, 1104–1116. (doi:10.1021/jp96
2416y)
13. Allen MT, Yetter RA, Dryer FL. 1995 The decomposition of nitrous oxide at 1.5 < P < 10.5 atm.
and 1103 < T < 1173 K. Int. J. Chem. Kinet. 27, 883–909. (doi:10.1002/kin.550270906)
14. Johnsson JE, Glarborg P, Dam-Johansen K. 1992 Thermal dissociation of nitrous oxide
at medium temperatures. Symp. Int. Combust. Proc. 24, 917–923. (doi:10.1016/S0082-0784
(06)80109-8)
15. Kondrat’ev VN. 1964 Chemical kinetics of gas reactions, 810. Oxford, UK: Pergamon Press.
16. Sestac J. 2013 Thermal analysis of micro, nano- and non-crystalline materials, p. 481. Dordrecht,
The Netherlands: Springer.

Critical ignition conditions in exothermically reacting systems for

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