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Thermal modes of monomolecular exothermic reactions: Two-dimensional model
Combustion and Flame 160 (2013) 539–545
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Combustion and Flame j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m b u s t fl a m e
Thermal modes of monomolecular exothermic reactions: Two-dimensional model Valeriy Yu. Filimonov ⇑ Altay Technical State University, Barnaul, Lenina str. 46, 656038 Barnaul, Russia
a r t i c l e
i n f o
Article history: Received 16 June 2012 Received in revised form 25 November 2012 Accepted 26 November 2012 Available online 17 January 2013 Keywords: Heat release Heat removal Burn-up Thermal explosion Critical conditions Degeneration conditions
a b s t r a c t A new method for theoretical analysis of the self-heating kinetics for monomolecular exothermic reactions has been proposed. It has been shown, that consideration of the phase trajectories of reaction in the phase plane: the heating rate – temperature enables detection of the qualitative changes of the phase portrait under the changes of the Todes or Semenov criteria. This gives a possibility to analyze the diversity of various reaction modes for any reaction order. From this point of view, the classical Semenov theory of thermal explosion developed for zero-order reactions is a particular case. As an illustration of this method, the detailed phase trajectories analysis has been performed for the case of first-order reactions. The regions of the thermal explosion degeneration, fastest reaction mode region and transition regions have been established. The necessary and sufficient conditions for the thermal explosion have been formulated. Examples of the application of the method for calculation of specific reactions are presented; the comparison with the classical theory is performed. Ó 2012 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
1. Introduction Determination of the conditions of an exothermic reaction realization in the progressive self-heating mode (thermal explosion mode, TE) is one of the most important problems of the current combustion theory. In accordance with the classical theory developed by Semenov [1–3], the condition of TE means impossibility of thermal equilibrium between the exothermically reacting system and its environment. During TE, the burn-up of reactants leads to the decrease in the heating rate due to the kinetic inhibition by the reaction products. Nevertheless, in terms of the classical theory, realization of TE is possible in the case of sufficiently small burn-up of the initial reactants during preheating. This is the reason why the specific features of the products formation kinetics (the kinetic function structure) do not matter during preheating. This is determined by the small value of the Todes criterion [4]. Smallness of the Todes criterion is a basic assumption of the classical TE theory. In this case, the critical conditions for TE can be found from the condition of balance between the rate of heat release and rate of heat removal without taking into account a burn-up. This approximation allows obtaining the analytic expressions for the TE critical conditions, calculation of the critical value of the ambient temperature and corresponding critical heating of the reacting system [1–4]. However, we must bear in mind that ⇑ Address: St. Severo-Zapadnaya 173/31, 656052 Barnaul, Russia. Fax: +7 (3853)290930. E-mail address: vyfi[email protected]
the steady-state heat balance mentioned above is impossible with consideration process of the burn-up. Indeed, the rate of a homogeneous chemical reaction is determined by the law of mass action. Therefore, it is necessary to take into consideration both the heat balance equation and the equation of chemical kinetics in general case. Thus, the critical value of Semenov criterion must depend on the value of Todes criterion. However, the analytical solution of this problem is connected with some mathematical difficulties. For this reason, the thermal explosion theory was developed with the help of approximate analytical [5–8] and numerical methods [9–12]. At the same time, a qualitative analysis of these critical phenomenon is possible in terms of the dynamic system theory with considering the phase trajectories of self-heating process on the corresponding phase planes (the phase portrait reconstruction) [13]. In this paper, we suggest a method for analysis of self-heating modes of exothermically reacting systems in terms of the non-stationary thermal explosion theory by the studying of the structures of the phase trajectories. 2. Basic equations and parameters The classic system of the self-heating dynamics and the product formation kinetics equations for homogeneous n-th order reactions can be written in the dimensionless form as follows [13]: dH ¼ ds dy ¼ ds
ð1 yÞn exp H dH
cð1 yÞn exp H
s ¼ 0; H ¼ y ¼ 0
0010-2180/$ - see front matter Ó 2012 The Combustion Institute. Published by Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.combustflame.2012.11.016
ð1Þ
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V.Yu. Filimonov / Combustion and Flame 160 (2013) 539–545
where s = t/tad is the dimensionless time; tad ¼ cRT 20 exp ðE=RT 0 Þ=QEk0 qn1 is the adiabatic reaction time; H ¼ EðT T 0 Þ=RT 20 is the dimensionless temperature; T0 is the initial (ambient) temperature; y is the conversion depth (or fraction reacted); d ¼ tad =t ; t ¼ cqV=aS is the characteristic heat removal time; V is the volume of the reacting system; S is the area of the sample surface; c, q are the specific heat capacity and density of the reactant, respectively, a is the heat transfer coefficient; c Td ¼ cRT 20 =QE is the Todes criterion [4]; E is the activation energy; Q is the heat of reaction; k0 is the pre-exponential factor. It was assumed that the condition RT0/E 1 was holds. It is well known, that the classical theory gives exact values of the critical parameters for the zero-order reactions at n = 0 [1–4]:
H ¼ 1;
d¼e
ð2Þ
However, results Eq. (2) are approximately valid under condition c 1, (because the burn-up of a reagent has no significant effect on the heat balance on this condition) and hence y 1 during preheating. This condition, however, is not clearly defined. Therefore, it is necessary to investigate the kinetics of self -heating process at any value of c and to establish the relationship between critical value of Todes criterion and Semenov criterion. This problem was not considered in Semenov theory and is analyzed below. Let us find y from the first equation of the system (1) and substitute into the second equation; simultaneously, let a new variable u = dH/ds (heating rate) be introduced. As a result, we obtain:
u
du ¼ u2 þ duðH 1Þ cnðu þ dHÞ21=n expðH=nÞ dH
ð3Þ
For the further analysis, it is convenient to transform Eq. (3) to a reduced one-parameter form. After the introduction of new variables n = u/d, and g = cnd1/n, we obtain the equation:
n
dn ¼ n2 þ nðH 1Þ gðn þ HÞ21=n expðH=nÞ; dH
with the initial condition: H ¼ 0;
ð4Þ
n ¼ 1=d.
3. Phase trajectories analysis 3.1. Overall analysis Eq. (4) determines the dependence of the heating rate on temperature, or the phase trajectories in the plane n–H, and describes all monomolecular reactions of any order (including fractional
orders). This is the Abel equation of the second kind, which cannot be integrated by quadratures. However, the equality to zero of the right hand side of Eq. (4) determines a family of inflection isoclines on thermograms (dependence H(s)), or extrema of the phase trajectory n(g, H):
n2 þ nðH 1Þ gðn þ HÞ21=n expðH=nÞ ¼ 0
ð5Þ
The solution of algebraic Eq. (5) gives an expression for these isoclines ni(H, g, n), as illustrated by Fig. 1. Obviously, qualitative features of the heating process are determined by the intersection of the phase trajectory n(H) and inflection isocline ni(H). Let us consider some particular cases: 1. n = 0 (Semenov theory). The proposed method of analysis allows us to consider the problem in the plane n–H. If n = 0 and g = 0, the solution of Eq. (5) takes the simplest form
ni ¼ 1 H
ð6Þ
As it is seen in Fig. 1, the phase trajectory has the minimum point (point a). This point appears due to the influence of the heat removal. Obviously, the minimum point appears under conditions: n ¼ 0; H ¼ 1 and with the use of the first equation of the set (1) at y = 0 the well-known critical conditions (2) can be obtained. 2. n > 0, The extreme points can be observed on the phase trajectory, Fig. 1a. The appearance of the high temperature maximum point c is associated with the influence of the kinetic inhibition which prevents the unlimited growth of the temperature. The appearance of the minimum point b is associated with the influence of the heat removal. However, the point b does not always exist on the phase trajectory. Therefore, it is necessary to investigate the Eq. (5) for the presence of a minimum. By differentiating the expression (2) with respect to temperature H under condition dni/ds = 0 we can obtain:
ni ¼ gðni þ HÞ11=n ½2 1=n þ ðni þ HÞ=n expðH=nÞ
ð7Þ
Eqs. (5) and (7) determine the conditions for the existence of a minimum on the inflection isoclines. Combined consideration of Eqs. (5) and (7) gives:
nmin ¼ f ðnÞ H;
ð8Þ
where nmin is the ordinate of the minimum point on the isoclines (Fig. 1), f(n) is the function which depends only on the order of the reaction:
f ðnÞ ¼ 1 þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2 þ 4n n =2
ð9Þ
The substitution Eq. (8) into Eq. (5) gives:
f H ¼ gf 21=n expðH=nÞ=ðf 1Þ
ð10Þ
The graphical solution of this transcendental equation is presented in Fig. 2, where:
uðg; nÞ ¼ gf 21=n =ðf 1Þ
ð11Þ
As it is seen in Fig. 2, only one minimum point is present on the isocline when f1 > /. The minimum point disappears when H = 0 and f = /. Returning to the variables c, d, the condition for the existence of a minimum (f1 > /) can be written as follows:
c < d1=n f 1=n1 ðf 1Þ=n ¼ d1=n kðnÞ
ð12Þ
where: Fig. 1. Inflection isoclines of phase trajectory (4) for monomolecular reactions of various order (solid lines) at g = 0, 1. 1, 2 – Schematic representation of corresponding phase trajectories. (a–c) The extremum points of the phase trajectory (inflection points on the thermogram).
kðnÞ ¼ f 1=n1 ðf 1Þ=n
ð13Þ
From Eq. (13), we have k ¼ 0; 62 for the first order reactions and k ¼ 0; 24 for the second order reactions.
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The transition from the mode 1 to the mode 2 occurs under the condition of equivalent horizontal tangency of the phase trajectory and isocline at the point m. The transition from the mode 2 to the mode 3 is determined by the disappearance of low-temperature inflection point a at H = 0. This transition is possible under the condition: n0 = 1/d. The substitution this value in Eq. (14) gives:
c ¼ ð1 dÞ=n
ð15Þ
Consequently, we can observe the only one high-temperature inflection point on the thermogram under the following condition:
c < ð1 dÞ=n
Fig. 2. Graphical solution of Eq. (9). 1 – The left-hand side of Eq. (9), 2 – its righthand side. If f1 > u the only solution exists, if f2 > u solution of Eq. (9) does not exist.
Let us consider the characteristic structures of phase trajectories and the corresponding thermograms. The following typical situations are possible:
ð16Þ
II. Inequality (12) is not satisfied. Thus, two characteristic types of the relative position of the phase trajectories and inflection isoclines exist in this case, Fig. 4. 1. Monotonic decrease of the heating rate with temperature. 2. Progressive self-heating with the one inflection point on the thermogram. Obviously, this mode of self-heating takes place if inequality (16) is satisfied. Therefore, the possible reaction modes are determined by the various combinations of the signs in inequalities (12), (16). This allows us to generalize possible types of phase portraits n(H) considered above and the corresponding thermograms in a compact and visual form which is presented in Fig. 5. Curve (12) and straight line (16) divide the parametric plane
I. Inequality (12) is satisfied. Three characteristic types of the relative position of the phase trajectories and inflection isoclines exist in this case (Fig. 3a). 1. Monotonic decrease of the heating rate with temperature. The absence of inflection points (before the maximum temperature is attained). 2. Non-monotonic change of the heating rate and the presence of two extreme points on the phase trajectory or corresponding inflection points on the thermogram. It is worth noting that the appearance of the second (high temperature) inflection point is impossible in terms of the Semenov theory. 3. Fast heating rate and temperature increase from the very beginning of the heating process. There is the only one high temperature inflection point on the thermogram (this point is not shown in the figure) in this situation. The value n0 corresponds to initial point on the inflection isocline and can be found from Eq. (5) at H = 0:
n0 gn011=n ¼ 1
ð14Þ
c – d into four regions with the various kinetics of heating. The curve Ob (shown schematically) divides the modes with monotonic and non-monotonic behavior of the phase trajectory or determines the appearance of two inflection points on the thermogram. The point b of the diagram corresponds to the critical condition of the thermal explosion in the Semenov theory: d = e at c = 0. So, this diagram is an extension of the Semenov theory onto the plane c > 0. The O point is a characteristic point of the convergence of all the modes. Let us analyze each characteristic region. The limiting modes are the following: 1. A monotonic decrease in the heating rate with temperature; the influence of kinetic inhibition and heat removal is substantial (trajectory 1, Figs. 3 and 4). 3. The region of the fastest reactions modes (FRM region) which is characterized by a weak influence of heat removal and kinetic inhibition on the heating process. There is the only one hightemperature inflection point on the thermogram in this case (trajectory 2, Fig. 4).
Fig. 3. (a) Schematic representation of phase trajectories 1–3 when d parameter changes in vicinity of the isocline of extrema (4) and (b) the corresponding thermograms: d1 > d2 > d3.
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Fig. 4. Schematic representation of the characteristic reaction modes in the absence of a minimum on extrema isocline: (a) – Phase trajectories 1, 2; 3 – extrema isocline and (b) – corresponding thermograms.
The other parametric regions can be characterized as the transitional ones. 2. This region is characterized by the low heat removal rate and substantial influence of kinetic inhibition on the heating process. At the initial heating stage, the fast reaction takes place, but the heating rate starts to decrease during the later stages because of substantial burn-up. 5. This region is characterized by significant influence of the heat removal on the heating process. Heating rate decreases monotonically. 4. The critical region. The critical conditions are determined by horizontal tangency between inflection isocline and phase trajectory at point m (Fig. 3a). Two inflection points appear on the thermogram when the transition into this region occurs. The rates of the kinetic inhibition and heat removal are comparable. Thus, we can conclude that there is a region with an optimal combination of the c and d parameters in terms of the fast reaction (thermal explosion) realization (FRM-region 3). Next, let us consider the first-order reactions as a partial case of the presented model. 3.2. First – order reactions 3.2.1. Influence of the heat removal conditions For the first-order reactions (n = 1), we can obtain: g ¼ c=d; k ¼ 0; 62 and separating lines (12), (15) have the form: c ¼ 0; 62d; c ¼ 1 d in this case. Thus, regions 1–5 are formed by the intersection of the two straight lines (Fig. 6). Point O has the coordinates: d0 ¼ 0; 62; c0 ¼ 0; 38. The separating curve II is calculated using the numerical analysis of Eq. (1). It is worth noting, that the calculation of this dependence must be made between two fixed points O and b. Due to the fact that this dependence is the unambiguous one, we can find some approximating dependence between the points O and b which is closest to the calculated curve II, and assuming that this dependence is exponential, the approximation formula can be proposed as follows:
cðdÞ ¼ c0
1 exp½1; 5ðe dÞ 1 exp½1; 5ðe d0 Þ
ð17Þ
or:
c ¼ 0; 017fexp½1; 5ðe dÞ 1g
Fig. 5. Schematic parametric diagram of the heating modes: (1) and (3) are the limiting modes. Semenov theory is the projection of two-dimension area on the axis c = 0 Ob is the separating line.
Let us consider the all possible transitions between the characteristic parametric regions (Fig. 6). Based on the general form of the d, c parameters, we have:
c¼
cRT 20 RT 20 aS E ;d ¼ exp QEk0 V RT 0 QE
Therefore, we can conclude that the transitions in the vertical direction ðd ¼ const; c ¼ varÞ for a given reacting system are possible theoretically under the constant heat removal conditions, i.e., at a constant ambient temperature and constant thermokinetic parameters. Therefore, it is possible only due to the change of the heat capacity c of the reacting system. Transitions in horizontal directions (A, B) for a given reacting system can occur at a constant initial temperature (c = const) and they are possible with the change of the heat removal conditions (d = var) i.e. with the change of the relation aS/V. The necessary condition of the transition in the triangular FRM region 3 is as follows: c < 0, 38. Next, we turn to the analysis of specific systems. Let us consider the exothermic first-order reaction of iodomethane decomposition:
CH3 I ! CH3 þ I ð18Þ
This analytical dependence is presented in Fig. 6 by the curve I. The difference between the numerical calculation II and dependence (18) does not exceed a few percent.
ð19Þ
ð20Þ
The heat of reaction is: Q = 234 kJ/mole [14], the activation energy is: E = 231 kJ/mole. Thus, we can estimate c 3, 1 103 at the temperature 400 °C. As follows from the diagram presented in Fig. 6, the critical condition of ignition agrees well with the classical
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Fig. 6. Parametric diagram for first order reactions. II – Separating curve obtained by numerical calculation of the system (1). I – The approximating dependence (17). B – c = 0,05; A – c = 0,1. Points a, b, c correspond to the critical conditions.
Fig. 7. Thermogram of the heating process under the change of parameter d in A and B direction (Fig. 6). The thermograms correspond to the values of parameters which are represented by the points in Fig. 6.
Semenov theory and d e. However, in the case of the less value of the heat of reaction Q = 33 kJ/mole (ethyl chloride decomposition):
C2 H5 Cl ! C2 H4 þ HCl
ð21Þ
with the activation energy E = 228 kJ/mole [15] we obtain c 0, 05 at the same initial temperature. As follows from the diagram: d 1, 85. Consequently, the burn-up has a significant influence on the critical conditions in spite of the small value of the Todes criterion c 1.
Figure 7 shows the thermograms of the self-heating process obtained by the numerical integration of the system (1). We can observe, that the transitions 1 ? 2 and 4 ? 5 (Fig. 6) in the critical region 4 are characterized by the high sensitivity of the thermograms structure to the change of parameter d. But with an increase of parameter c, this sensitivity is reduced until the complete degeneration of the thermal explosion: c0 = 0, 38. Thus, the model presented is characterized by the continuous transitions from the slow reaction modes to the fast modes of self-heating in contrast to the approximate Semenov theory when
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V.Yu. Filimonov / Combustion and Flame 160 (2013) 539–545
Fig. 8. Phase trajectoris which are determined by Eq. (22) on the plane c–d at b = 4 (ammonium perchlorate) under the various values of parameter z. The arrows indicate the direction of the initial temperature increase.
the transition to a mode of unlimited temperature growth occurs discontinuously for an arbitrarily small change of the parameter d. 3.2.2. Influence of the ambient temperature As follows from Eq. (19), the c, d parameters are not independent and excluding the initial temperature T0 we obtain:
pffiffiffi d ¼ cz exp ðb= cÞ
ð22Þ
where:
z ¼ aS=cqk0 V pffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ¼ cE=QR
ð23Þ
z 1013 aS=V
ð24Þ
Taking into account the actual interval of changing of the value aS/ V:aS/V 10 102 Wm3 K1 [19], we can estimate that the ignition temperature is in the range of temperatures: T0 = 217 320 °C. This is in good agreement with data [17], where T0 = 200–300 °C.
The z parameter depends on the geometric scale and heat removal conditions. The b parameter depends on the kind of reagent. We can consider the z parameter as the following relation: z ¼ t r =t where tr = 1/k0 is the characteristic time of reaction (product formation), t ¼ cqV=aS is the characteristic time of heat removal. Thus, Eq. (22) determines the phase trajectory on the plane c–d under the change of the ambient temperature. As an example, let us consider the well-known exothermic firstorder reaction of ammonium perchlorate decomposition:
2NH4 ClO4 ! Cl2 þ 2NO þ O2 þ 4H2 O
Using the data of [16,17], we can calculate: b 4. Figure 8 illustrates dependence (22) at the various values of z parameter. The transition in the critical region 4 determines the critical conditions of thermal explosion when the ambient temperature increases. As follows from Fig. 8, the critical values of parameter c are as follows: ca ¼ 0; 015; cb ¼ 0; 022; cc ¼ 0; 031. With the use of Eq. (19) and reference data [18], we can calculate the critical values of the ambient temperatures: T 0a ¼ 2170 C; T 0b ¼ 3200 C; T 0c ¼ 4310 C: Based on the data in [16–18], we can obtain an estimate for z:
ð25Þ
ð26Þ
3.2.3. Degeneration conditions. Necessary and sufficient conditions for a thermal explosion Figure 9 shows the phase trajectories (22) at z = 1011 and various values of b. Increasing the b parameter leads to the increase of the c parameter at the constant value of parameter d and hence to the increase of the burn-up influence. The increase of parameter b may be associated with transition to a less exothermic reaction or
Fig. 9. Phase trajectories which are determined by Eq. (22) at z = 1011 under various values of parameter b.
V.Yu. Filimonov / Combustion and Flame 160 (2013) 539–545
to a reaction with a higher value of the activation energy. In this case, the characteristic temperature interval of FRM (between the points a and b) narrows and the degeneration of the thermal explosion occurs at the point O (in the case of b = 16). We can obtain that the phase trajectory passing through the point O is determined by the relation between z and bparameters:
pffiffiffiffiffi z ¼ ðd0 =c0 Þ exp b= c0 ¼ 1; 63 expð1; 62bÞ
ð27Þ
545
trajectory on the parametric plane can be changed. By varying these parameters, the critical temperatures, the maximum temperatures and the heating rates can be varied, and, therefore, the kinetics of the self-heating process can be controlled. Moreover, the characteristic temperatures of the self-heating process can be determined if the values of these parameters are known exactly. 5. Conclusion
⁄
Thus, the inequality z < z can be regarded as a necessary condition for the thermal explosion in the case of monomolecular first-order reactions. So, if z > z⁄ the thermal explosion is impossible at any value of ambient temperature. Nevertheless, inequality z < z⁄ is not sufficient. Indeed, the sufficient condition is determined by the intersection of the phase trajectory with the boundary of the region 4 i.e. the initial critical temperature must be determined by the point c (Fig. 9). The thermal explosion is impossible at a lower value of the temperature. This value of the temperature can be found from the set of Eqs. (18) and (22). 4. Discussion The model presented reveals the existence of the ‘‘optimal’’ values of the Todes and Semenov criteria from the point of view of the fastest reaction realization. This parametric region has the triangular form on the corresponding parametric plane for the first order reactions. Within this region, the combination of d parameter and c parameter is optimal when the rate of heat removal is low enough (d < 1 c) and simultaneously, the influence of burn-up is insignificant (c < 0, 62d). As follows from Figs. 8 and 9, a characteristic pla0 teau is present on the phase trajectories (to the right of point a , Fig. 8). In the plateau region, a small increase in the ambient temperature leads to a significant decrease of the parameter d and to a very small changes in parameter c. It means that the effect of burnup is insignificant in this temperature range. The approximations of the classical theory are applicable only within this range. At the same time, as follows from Fig. 6, the classical theory is valid quantitatively only when c < 102(d e). However, the c parameter becomes sensitive to the change of ambient temperature to 0 the left of point a (Fig. 8) in the region 2 of the high ambient temperatures. It means that the large fraction of the initial reactant has reacted during the heating to the ambient temperature T0. Thus, the critical conditions of the thermal explosion are determined by the value of parameter c in this case. These critical values ccr (point a0 , Fig. 8) are determined by the intersection of phase trajectory (Eq. (19)) and straight line c = 0, 62d. The classical theory is not applicable to this case. The heat release decrease (at increasing b) leads to a greater influence of the kinetic inhibition (parameter c increases) on the heating process and to an increase of the critical ambient temperatures. The critical phenomena disappear at the point O. The model presented determines three characteristic temperatures: (i) the temperature at point c which determines the critical conditions of the appearance of two inflection points on the thermogram (Fig. 9); (ii) the temperature at point b which determines the transition in FRM triangular region 3 where the only one inflection point appears on the thermogram; (iii) the temperature at point a which determines the transition in the high temperature region 2 where the influence of the kinetic inhibition is significant. The diagram presented in Fig. 6, gives an opportunity to determine these values of the characteristic temperatures. In order to make these calculations, we need to know the parameters b, z as accurately as possible. Parameters z, b, T0; can be considered as the governing parameters, through which the position of the phase
Under consideration of the monomolecular exothermic reactions taking into account the kinetic inhibition by the reaction product, it is established that the thermal modes of reactions may be qualitatively different depending on the ratio of Todes criterion – Semenov criterion. The variety of heating modes is determined by the characteristic regions on the corresponding parametric diagram. Five characteristic regions which converge at a single point are defined on the parametric plane. Under consideration of the partial case of the first-order reactions, it is found that the fastest reaction modes region (FRM region) has the triangular form and it is the optimal one in terms of the simultaneously small rates of the kinetic inhibition and heat removal. It is revealed that the critical conditions of the thermal explosion are characterized by the appearance of two inflection points on the thermogram and these conditions are determined by the functional connection between Todes and Semenov criteria unlike the classical theory when Semenov criterion equals to the constant value (base of natural logarithm). On the example of the calculation of the specific monomolecular reactions, it is shown that taking into account the kinetic inhibition by the reaction product leads to a substantial deviation from the existing classical theory in determination of the critical temperature and the critical heat removal conditions. Under conditions established in the work, the critical conditions disappear that corresponds to the impossibility of an explosive reaction to be realized at any value of the initial temperature. This defines the necessary condition for a thermal explosion. The sufficient condition is determined by the intersection of the phase trajectory and the boundary of the critical region on the parametric diagram. The model proposed allows us to analyze the thermal modes for monomolecular exothermic reactions of any order. References [1] N.N. Semenov, Some Problems in Chemical Kinetics and Reactivity. Parts 1 and 2, Pergamon Press, London, 1959. [2] J. Bebernes, D. Eberly, Mathematical Problems from Combustion Theory, Springer-Verlag, New York, 1989. [3] B.F. Gray, Combust. Flame 21 (1973) 317–325. [4] O.M. Todes, P.V. Melent’ev, J. Phys. Chem. 13 (1939) 52–58. [5] P. Gray, M.J. Harper, Trans. Faraday Soc. 55 (1959) 581–590. [6] J. Adler, Math. Phys. Sc. 433 (1991) 329–335. [7] K.S. Adegbie, A.I. Alao, Am. J. Appl. Sci. 4 (2007) 53–55. [8] J. Adler, Combust. Flame 8 (1964) 97–103. [9] S.O. Ajadi, O. Nave, J. Mater. Chem. 47 (2010) 790–807. [10] V.T. Gontkovskaya, A.V. Gorodetskov, A.N. Peregudov, V.V. Barzykin, Comb., Explos. Shock Waves 32 (1996) 424–426. [11] P.O. Olanrewaju, Int. J. Num. Math. 5 (2010) 447–455. [12] O.D. Makinde, Mech. Res. Commun. 32 (2005) 191–195. [13] A.P. Aldushin, Comb. Explos. Shock Waves 23 (1987) 338–341. [14] G.A. Skorobogatov, B.P. Dymov, I.V. Nedozrelova, Zh. Org. Khim. 64 (1991) 956–965. [15] K.A. Holbrook, Trans. Faraday Soc. 63 (1967) 643–645. [16] A.S. Shteinberg, Fast Reactions in Energetic Materials, Springer-Verlag, Berlin, 2009. [17] A.G. Kennan, F. Siegmund, The Thermal Decomposition of Ammonium Perchlorate, Miami University, Special Report No. 6, Contract Nonr-4008(07), 1968. [18] M.W. Chase, J. Phys. Chem. Ref. Data. 9 (1998). [19] A.G. Merzhanov, F.I. Dubovitskii, Zh. Fiz. Khim. 35 (1961) 2083–2089.
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Combustion and Flame j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m b u s t fl a m e
Thermal modes of monomolecular exothermic reactions: Two-dimensional model Valeriy Yu. Filimonov ⇑ Altay Technical State University, Barnaul, Lenina str. 46, 656038 Barnaul, Russia
a r t i c l e
i n f o
Article history: Received 16 June 2012 Received in revised form 25 November 2012 Accepted 26 November 2012 Available online 17 January 2013 Keywords: Heat release Heat removal Burn-up Thermal explosion Critical conditions Degeneration conditions
a b s t r a c t A new method for theoretical analysis of the self-heating kinetics for monomolecular exothermic reactions has been proposed. It has been shown, that consideration of the phase trajectories of reaction in the phase plane: the heating rate – temperature enables detection of the qualitative changes of the phase portrait under the changes of the Todes or Semenov criteria. This gives a possibility to analyze the diversity of various reaction modes for any reaction order. From this point of view, the classical Semenov theory of thermal explosion developed for zero-order reactions is a particular case. As an illustration of this method, the detailed phase trajectories analysis has been performed for the case of first-order reactions. The regions of the thermal explosion degeneration, fastest reaction mode region and transition regions have been established. The necessary and sufficient conditions for the thermal explosion have been formulated. Examples of the application of the method for calculation of specific reactions are presented; the comparison with the classical theory is performed. Ó 2012 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
1. Introduction Determination of the conditions of an exothermic reaction realization in the progressive self-heating mode (thermal explosion mode, TE) is one of the most important problems of the current combustion theory. In accordance with the classical theory developed by Semenov [1–3], the condition of TE means impossibility of thermal equilibrium between the exothermically reacting system and its environment. During TE, the burn-up of reactants leads to the decrease in the heating rate due to the kinetic inhibition by the reaction products. Nevertheless, in terms of the classical theory, realization of TE is possible in the case of sufficiently small burn-up of the initial reactants during preheating. This is the reason why the specific features of the products formation kinetics (the kinetic function structure) do not matter during preheating. This is determined by the small value of the Todes criterion [4]. Smallness of the Todes criterion is a basic assumption of the classical TE theory. In this case, the critical conditions for TE can be found from the condition of balance between the rate of heat release and rate of heat removal without taking into account a burn-up. This approximation allows obtaining the analytic expressions for the TE critical conditions, calculation of the critical value of the ambient temperature and corresponding critical heating of the reacting system [1–4]. However, we must bear in mind that ⇑ Address: St. Severo-Zapadnaya 173/31, 656052 Barnaul, Russia. Fax: +7 (3853)290930. E-mail address: vyfi[email protected]
the steady-state heat balance mentioned above is impossible with consideration process of the burn-up. Indeed, the rate of a homogeneous chemical reaction is determined by the law of mass action. Therefore, it is necessary to take into consideration both the heat balance equation and the equation of chemical kinetics in general case. Thus, the critical value of Semenov criterion must depend on the value of Todes criterion. However, the analytical solution of this problem is connected with some mathematical difficulties. For this reason, the thermal explosion theory was developed with the help of approximate analytical [5–8] and numerical methods [9–12]. At the same time, a qualitative analysis of these critical phenomenon is possible in terms of the dynamic system theory with considering the phase trajectories of self-heating process on the corresponding phase planes (the phase portrait reconstruction) [13]. In this paper, we suggest a method for analysis of self-heating modes of exothermically reacting systems in terms of the non-stationary thermal explosion theory by the studying of the structures of the phase trajectories. 2. Basic equations and parameters The classic system of the self-heating dynamics and the product formation kinetics equations for homogeneous n-th order reactions can be written in the dimensionless form as follows [13]: dH ¼ ds dy ¼ ds
ð1 yÞn exp H dH
cð1 yÞn exp H
s ¼ 0; H ¼ y ¼ 0
0010-2180/$ - see front matter Ó 2012 The Combustion Institute. Published by Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.combustflame.2012.11.016
ð1Þ
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where s = t/tad is the dimensionless time; tad ¼ cRT 20 exp ðE=RT 0 Þ=QEk0 qn1 is the adiabatic reaction time; H ¼ EðT T 0 Þ=RT 20 is the dimensionless temperature; T0 is the initial (ambient) temperature; y is the conversion depth (or fraction reacted); d ¼ tad =t ; t ¼ cqV=aS is the characteristic heat removal time; V is the volume of the reacting system; S is the area of the sample surface; c, q are the specific heat capacity and density of the reactant, respectively, a is the heat transfer coefficient; c Td ¼ cRT 20 =QE is the Todes criterion [4]; E is the activation energy; Q is the heat of reaction; k0 is the pre-exponential factor. It was assumed that the condition RT0/E 1 was holds. It is well known, that the classical theory gives exact values of the critical parameters for the zero-order reactions at n = 0 [1–4]:
H ¼ 1;
d¼e
ð2Þ
However, results Eq. (2) are approximately valid under condition c 1, (because the burn-up of a reagent has no significant effect on the heat balance on this condition) and hence y 1 during preheating. This condition, however, is not clearly defined. Therefore, it is necessary to investigate the kinetics of self -heating process at any value of c and to establish the relationship between critical value of Todes criterion and Semenov criterion. This problem was not considered in Semenov theory and is analyzed below. Let us find y from the first equation of the system (1) and substitute into the second equation; simultaneously, let a new variable u = dH/ds (heating rate) be introduced. As a result, we obtain:
u
du ¼ u2 þ duðH 1Þ cnðu þ dHÞ21=n expðH=nÞ dH
ð3Þ
For the further analysis, it is convenient to transform Eq. (3) to a reduced one-parameter form. After the introduction of new variables n = u/d, and g = cnd1/n, we obtain the equation:
n
dn ¼ n2 þ nðH 1Þ gðn þ HÞ21=n expðH=nÞ; dH
with the initial condition: H ¼ 0;
ð4Þ
n ¼ 1=d.
3. Phase trajectories analysis 3.1. Overall analysis Eq. (4) determines the dependence of the heating rate on temperature, or the phase trajectories in the plane n–H, and describes all monomolecular reactions of any order (including fractional
orders). This is the Abel equation of the second kind, which cannot be integrated by quadratures. However, the equality to zero of the right hand side of Eq. (4) determines a family of inflection isoclines on thermograms (dependence H(s)), or extrema of the phase trajectory n(g, H):
n2 þ nðH 1Þ gðn þ HÞ21=n expðH=nÞ ¼ 0
ð5Þ
The solution of algebraic Eq. (5) gives an expression for these isoclines ni(H, g, n), as illustrated by Fig. 1. Obviously, qualitative features of the heating process are determined by the intersection of the phase trajectory n(H) and inflection isocline ni(H). Let us consider some particular cases: 1. n = 0 (Semenov theory). The proposed method of analysis allows us to consider the problem in the plane n–H. If n = 0 and g = 0, the solution of Eq. (5) takes the simplest form
ni ¼ 1 H
ð6Þ
As it is seen in Fig. 1, the phase trajectory has the minimum point (point a). This point appears due to the influence of the heat removal. Obviously, the minimum point appears under conditions: n ¼ 0; H ¼ 1 and with the use of the first equation of the set (1) at y = 0 the well-known critical conditions (2) can be obtained. 2. n > 0, The extreme points can be observed on the phase trajectory, Fig. 1a. The appearance of the high temperature maximum point c is associated with the influence of the kinetic inhibition which prevents the unlimited growth of the temperature. The appearance of the minimum point b is associated with the influence of the heat removal. However, the point b does not always exist on the phase trajectory. Therefore, it is necessary to investigate the Eq. (5) for the presence of a minimum. By differentiating the expression (2) with respect to temperature H under condition dni/ds = 0 we can obtain:
ni ¼ gðni þ HÞ11=n ½2 1=n þ ðni þ HÞ=n expðH=nÞ
ð7Þ
Eqs. (5) and (7) determine the conditions for the existence of a minimum on the inflection isoclines. Combined consideration of Eqs. (5) and (7) gives:
nmin ¼ f ðnÞ H;
ð8Þ
where nmin is the ordinate of the minimum point on the isoclines (Fig. 1), f(n) is the function which depends only on the order of the reaction:
f ðnÞ ¼ 1 þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2 þ 4n n =2
ð9Þ
The substitution Eq. (8) into Eq. (5) gives:
f H ¼ gf 21=n expðH=nÞ=ðf 1Þ
ð10Þ
The graphical solution of this transcendental equation is presented in Fig. 2, where:
uðg; nÞ ¼ gf 21=n =ðf 1Þ
ð11Þ
As it is seen in Fig. 2, only one minimum point is present on the isocline when f1 > /. The minimum point disappears when H = 0 and f = /. Returning to the variables c, d, the condition for the existence of a minimum (f1 > /) can be written as follows:
c < d1=n f 1=n1 ðf 1Þ=n ¼ d1=n kðnÞ
ð12Þ
where: Fig. 1. Inflection isoclines of phase trajectory (4) for monomolecular reactions of various order (solid lines) at g = 0, 1. 1, 2 – Schematic representation of corresponding phase trajectories. (a–c) The extremum points of the phase trajectory (inflection points on the thermogram).
kðnÞ ¼ f 1=n1 ðf 1Þ=n
ð13Þ
From Eq. (13), we have k ¼ 0; 62 for the first order reactions and k ¼ 0; 24 for the second order reactions.
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The transition from the mode 1 to the mode 2 occurs under the condition of equivalent horizontal tangency of the phase trajectory and isocline at the point m. The transition from the mode 2 to the mode 3 is determined by the disappearance of low-temperature inflection point a at H = 0. This transition is possible under the condition: n0 = 1/d. The substitution this value in Eq. (14) gives:
c ¼ ð1 dÞ=n
ð15Þ
Consequently, we can observe the only one high-temperature inflection point on the thermogram under the following condition:
c < ð1 dÞ=n
Fig. 2. Graphical solution of Eq. (9). 1 – The left-hand side of Eq. (9), 2 – its righthand side. If f1 > u the only solution exists, if f2 > u solution of Eq. (9) does not exist.
Let us consider the characteristic structures of phase trajectories and the corresponding thermograms. The following typical situations are possible:
ð16Þ
II. Inequality (12) is not satisfied. Thus, two characteristic types of the relative position of the phase trajectories and inflection isoclines exist in this case, Fig. 4. 1. Monotonic decrease of the heating rate with temperature. 2. Progressive self-heating with the one inflection point on the thermogram. Obviously, this mode of self-heating takes place if inequality (16) is satisfied. Therefore, the possible reaction modes are determined by the various combinations of the signs in inequalities (12), (16). This allows us to generalize possible types of phase portraits n(H) considered above and the corresponding thermograms in a compact and visual form which is presented in Fig. 5. Curve (12) and straight line (16) divide the parametric plane
I. Inequality (12) is satisfied. Three characteristic types of the relative position of the phase trajectories and inflection isoclines exist in this case (Fig. 3a). 1. Monotonic decrease of the heating rate with temperature. The absence of inflection points (before the maximum temperature is attained). 2. Non-monotonic change of the heating rate and the presence of two extreme points on the phase trajectory or corresponding inflection points on the thermogram. It is worth noting that the appearance of the second (high temperature) inflection point is impossible in terms of the Semenov theory. 3. Fast heating rate and temperature increase from the very beginning of the heating process. There is the only one high temperature inflection point on the thermogram (this point is not shown in the figure) in this situation. The value n0 corresponds to initial point on the inflection isocline and can be found from Eq. (5) at H = 0:
n0 gn011=n ¼ 1
ð14Þ
c – d into four regions with the various kinetics of heating. The curve Ob (shown schematically) divides the modes with monotonic and non-monotonic behavior of the phase trajectory or determines the appearance of two inflection points on the thermogram. The point b of the diagram corresponds to the critical condition of the thermal explosion in the Semenov theory: d = e at c = 0. So, this diagram is an extension of the Semenov theory onto the plane c > 0. The O point is a characteristic point of the convergence of all the modes. Let us analyze each characteristic region. The limiting modes are the following: 1. A monotonic decrease in the heating rate with temperature; the influence of kinetic inhibition and heat removal is substantial (trajectory 1, Figs. 3 and 4). 3. The region of the fastest reactions modes (FRM region) which is characterized by a weak influence of heat removal and kinetic inhibition on the heating process. There is the only one hightemperature inflection point on the thermogram in this case (trajectory 2, Fig. 4).
Fig. 3. (a) Schematic representation of phase trajectories 1–3 when d parameter changes in vicinity of the isocline of extrema (4) and (b) the corresponding thermograms: d1 > d2 > d3.
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Fig. 4. Schematic representation of the characteristic reaction modes in the absence of a minimum on extrema isocline: (a) – Phase trajectories 1, 2; 3 – extrema isocline and (b) – corresponding thermograms.
The other parametric regions can be characterized as the transitional ones. 2. This region is characterized by the low heat removal rate and substantial influence of kinetic inhibition on the heating process. At the initial heating stage, the fast reaction takes place, but the heating rate starts to decrease during the later stages because of substantial burn-up. 5. This region is characterized by significant influence of the heat removal on the heating process. Heating rate decreases monotonically. 4. The critical region. The critical conditions are determined by horizontal tangency between inflection isocline and phase trajectory at point m (Fig. 3a). Two inflection points appear on the thermogram when the transition into this region occurs. The rates of the kinetic inhibition and heat removal are comparable. Thus, we can conclude that there is a region with an optimal combination of the c and d parameters in terms of the fast reaction (thermal explosion) realization (FRM-region 3). Next, let us consider the first-order reactions as a partial case of the presented model. 3.2. First – order reactions 3.2.1. Influence of the heat removal conditions For the first-order reactions (n = 1), we can obtain: g ¼ c=d; k ¼ 0; 62 and separating lines (12), (15) have the form: c ¼ 0; 62d; c ¼ 1 d in this case. Thus, regions 1–5 are formed by the intersection of the two straight lines (Fig. 6). Point O has the coordinates: d0 ¼ 0; 62; c0 ¼ 0; 38. The separating curve II is calculated using the numerical analysis of Eq. (1). It is worth noting, that the calculation of this dependence must be made between two fixed points O and b. Due to the fact that this dependence is the unambiguous one, we can find some approximating dependence between the points O and b which is closest to the calculated curve II, and assuming that this dependence is exponential, the approximation formula can be proposed as follows:
cðdÞ ¼ c0
1 exp½1; 5ðe dÞ 1 exp½1; 5ðe d0 Þ
ð17Þ
or:
c ¼ 0; 017fexp½1; 5ðe dÞ 1g
Fig. 5. Schematic parametric diagram of the heating modes: (1) and (3) are the limiting modes. Semenov theory is the projection of two-dimension area on the axis c = 0 Ob is the separating line.
Let us consider the all possible transitions between the characteristic parametric regions (Fig. 6). Based on the general form of the d, c parameters, we have:
c¼
cRT 20 RT 20 aS E ;d ¼ exp QEk0 V RT 0 QE
Therefore, we can conclude that the transitions in the vertical direction ðd ¼ const; c ¼ varÞ for a given reacting system are possible theoretically under the constant heat removal conditions, i.e., at a constant ambient temperature and constant thermokinetic parameters. Therefore, it is possible only due to the change of the heat capacity c of the reacting system. Transitions in horizontal directions (A, B) for a given reacting system can occur at a constant initial temperature (c = const) and they are possible with the change of the heat removal conditions (d = var) i.e. with the change of the relation aS/V. The necessary condition of the transition in the triangular FRM region 3 is as follows: c < 0, 38. Next, we turn to the analysis of specific systems. Let us consider the exothermic first-order reaction of iodomethane decomposition:
CH3 I ! CH3 þ I ð18Þ
This analytical dependence is presented in Fig. 6 by the curve I. The difference between the numerical calculation II and dependence (18) does not exceed a few percent.
ð19Þ
ð20Þ
The heat of reaction is: Q = 234 kJ/mole [14], the activation energy is: E = 231 kJ/mole. Thus, we can estimate c 3, 1 103 at the temperature 400 °C. As follows from the diagram presented in Fig. 6, the critical condition of ignition agrees well with the classical
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Fig. 6. Parametric diagram for first order reactions. II – Separating curve obtained by numerical calculation of the system (1). I – The approximating dependence (17). B – c = 0,05; A – c = 0,1. Points a, b, c correspond to the critical conditions.
Fig. 7. Thermogram of the heating process under the change of parameter d in A and B direction (Fig. 6). The thermograms correspond to the values of parameters which are represented by the points in Fig. 6.
Semenov theory and d e. However, in the case of the less value of the heat of reaction Q = 33 kJ/mole (ethyl chloride decomposition):
C2 H5 Cl ! C2 H4 þ HCl
ð21Þ
with the activation energy E = 228 kJ/mole [15] we obtain c 0, 05 at the same initial temperature. As follows from the diagram: d 1, 85. Consequently, the burn-up has a significant influence on the critical conditions in spite of the small value of the Todes criterion c 1.
Figure 7 shows the thermograms of the self-heating process obtained by the numerical integration of the system (1). We can observe, that the transitions 1 ? 2 and 4 ? 5 (Fig. 6) in the critical region 4 are characterized by the high sensitivity of the thermograms structure to the change of parameter d. But with an increase of parameter c, this sensitivity is reduced until the complete degeneration of the thermal explosion: c0 = 0, 38. Thus, the model presented is characterized by the continuous transitions from the slow reaction modes to the fast modes of self-heating in contrast to the approximate Semenov theory when
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Fig. 8. Phase trajectoris which are determined by Eq. (22) on the plane c–d at b = 4 (ammonium perchlorate) under the various values of parameter z. The arrows indicate the direction of the initial temperature increase.
the transition to a mode of unlimited temperature growth occurs discontinuously for an arbitrarily small change of the parameter d. 3.2.2. Influence of the ambient temperature As follows from Eq. (19), the c, d parameters are not independent and excluding the initial temperature T0 we obtain:
pffiffiffi d ¼ cz exp ðb= cÞ
ð22Þ
where:
z ¼ aS=cqk0 V pffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ¼ cE=QR
ð23Þ
z 1013 aS=V
ð24Þ
Taking into account the actual interval of changing of the value aS/ V:aS/V 10 102 Wm3 K1 [19], we can estimate that the ignition temperature is in the range of temperatures: T0 = 217 320 °C. This is in good agreement with data [17], where T0 = 200–300 °C.
The z parameter depends on the geometric scale and heat removal conditions. The b parameter depends on the kind of reagent. We can consider the z parameter as the following relation: z ¼ t r =t where tr = 1/k0 is the characteristic time of reaction (product formation), t ¼ cqV=aS is the characteristic time of heat removal. Thus, Eq. (22) determines the phase trajectory on the plane c–d under the change of the ambient temperature. As an example, let us consider the well-known exothermic firstorder reaction of ammonium perchlorate decomposition:
2NH4 ClO4 ! Cl2 þ 2NO þ O2 þ 4H2 O
Using the data of [16,17], we can calculate: b 4. Figure 8 illustrates dependence (22) at the various values of z parameter. The transition in the critical region 4 determines the critical conditions of thermal explosion when the ambient temperature increases. As follows from Fig. 8, the critical values of parameter c are as follows: ca ¼ 0; 015; cb ¼ 0; 022; cc ¼ 0; 031. With the use of Eq. (19) and reference data [18], we can calculate the critical values of the ambient temperatures: T 0a ¼ 2170 C; T 0b ¼ 3200 C; T 0c ¼ 4310 C: Based on the data in [16–18], we can obtain an estimate for z:
ð25Þ
ð26Þ
3.2.3. Degeneration conditions. Necessary and sufficient conditions for a thermal explosion Figure 9 shows the phase trajectories (22) at z = 1011 and various values of b. Increasing the b parameter leads to the increase of the c parameter at the constant value of parameter d and hence to the increase of the burn-up influence. The increase of parameter b may be associated with transition to a less exothermic reaction or
Fig. 9. Phase trajectories which are determined by Eq. (22) at z = 1011 under various values of parameter b.
V.Yu. Filimonov / Combustion and Flame 160 (2013) 539–545
to a reaction with a higher value of the activation energy. In this case, the characteristic temperature interval of FRM (between the points a and b) narrows and the degeneration of the thermal explosion occurs at the point O (in the case of b = 16). We can obtain that the phase trajectory passing through the point O is determined by the relation between z and bparameters:
pffiffiffiffiffi z ¼ ðd0 =c0 Þ exp b= c0 ¼ 1; 63 expð1; 62bÞ
ð27Þ
545
trajectory on the parametric plane can be changed. By varying these parameters, the critical temperatures, the maximum temperatures and the heating rates can be varied, and, therefore, the kinetics of the self-heating process can be controlled. Moreover, the characteristic temperatures of the self-heating process can be determined if the values of these parameters are known exactly. 5. Conclusion
⁄
Thus, the inequality z < z can be regarded as a necessary condition for the thermal explosion in the case of monomolecular first-order reactions. So, if z > z⁄ the thermal explosion is impossible at any value of ambient temperature. Nevertheless, inequality z < z⁄ is not sufficient. Indeed, the sufficient condition is determined by the intersection of the phase trajectory with the boundary of the region 4 i.e. the initial critical temperature must be determined by the point c (Fig. 9). The thermal explosion is impossible at a lower value of the temperature. This value of the temperature can be found from the set of Eqs. (18) and (22). 4. Discussion The model presented reveals the existence of the ‘‘optimal’’ values of the Todes and Semenov criteria from the point of view of the fastest reaction realization. This parametric region has the triangular form on the corresponding parametric plane for the first order reactions. Within this region, the combination of d parameter and c parameter is optimal when the rate of heat removal is low enough (d < 1 c) and simultaneously, the influence of burn-up is insignificant (c < 0, 62d). As follows from Figs. 8 and 9, a characteristic pla0 teau is present on the phase trajectories (to the right of point a , Fig. 8). In the plateau region, a small increase in the ambient temperature leads to a significant decrease of the parameter d and to a very small changes in parameter c. It means that the effect of burnup is insignificant in this temperature range. The approximations of the classical theory are applicable only within this range. At the same time, as follows from Fig. 6, the classical theory is valid quantitatively only when c < 102(d e). However, the c parameter becomes sensitive to the change of ambient temperature to 0 the left of point a (Fig. 8) in the region 2 of the high ambient temperatures. It means that the large fraction of the initial reactant has reacted during the heating to the ambient temperature T0. Thus, the critical conditions of the thermal explosion are determined by the value of parameter c in this case. These critical values ccr (point a0 , Fig. 8) are determined by the intersection of phase trajectory (Eq. (19)) and straight line c = 0, 62d. The classical theory is not applicable to this case. The heat release decrease (at increasing b) leads to a greater influence of the kinetic inhibition (parameter c increases) on the heating process and to an increase of the critical ambient temperatures. The critical phenomena disappear at the point O. The model presented determines three characteristic temperatures: (i) the temperature at point c which determines the critical conditions of the appearance of two inflection points on the thermogram (Fig. 9); (ii) the temperature at point b which determines the transition in FRM triangular region 3 where the only one inflection point appears on the thermogram; (iii) the temperature at point a which determines the transition in the high temperature region 2 where the influence of the kinetic inhibition is significant. The diagram presented in Fig. 6, gives an opportunity to determine these values of the characteristic temperatures. In order to make these calculations, we need to know the parameters b, z as accurately as possible. Parameters z, b, T0; can be considered as the governing parameters, through which the position of the phase
Under consideration of the monomolecular exothermic reactions taking into account the kinetic inhibition by the reaction product, it is established that the thermal modes of reactions may be qualitatively different depending on the ratio of Todes criterion – Semenov criterion. The variety of heating modes is determined by the characteristic regions on the corresponding parametric diagram. Five characteristic regions which converge at a single point are defined on the parametric plane. Under consideration of the partial case of the first-order reactions, it is found that the fastest reaction modes region (FRM region) has the triangular form and it is the optimal one in terms of the simultaneously small rates of the kinetic inhibition and heat removal. It is revealed that the critical conditions of the thermal explosion are characterized by the appearance of two inflection points on the thermogram and these conditions are determined by the functional connection between Todes and Semenov criteria unlike the classical theory when Semenov criterion equals to the constant value (base of natural logarithm). On the example of the calculation of the specific monomolecular reactions, it is shown that taking into account the kinetic inhibition by the reaction product leads to a substantial deviation from the existing classical theory in determination of the critical temperature and the critical heat removal conditions. Under conditions established in the work, the critical conditions disappear that corresponds to the impossibility of an explosive reaction to be realized at any value of the initial temperature. This defines the necessary condition for a thermal explosion. The sufficient condition is determined by the intersection of the phase trajectory and the boundary of the critical region on the parametric diagram. The model proposed allows us to analyze the thermal modes for monomolecular exothermic reactions of any order. References [1] N.N. Semenov, Some Problems in Chemical Kinetics and Reactivity. Parts 1 and 2, Pergamon Press, London, 1959. [2] J. Bebernes, D. Eberly, Mathematical Problems from Combustion Theory, Springer-Verlag, New York, 1989. [3] B.F. Gray, Combust. Flame 21 (1973) 317–325. [4] O.M. Todes, P.V. Melent’ev, J. Phys. Chem. 13 (1939) 52–58. [5] P. Gray, M.J. Harper, Trans. Faraday Soc. 55 (1959) 581–590. [6] J. Adler, Math. Phys. Sc. 433 (1991) 329–335. [7] K.S. Adegbie, A.I. Alao, Am. J. Appl. Sci. 4 (2007) 53–55. [8] J. Adler, Combust. Flame 8 (1964) 97–103. [9] S.O. Ajadi, O. Nave, J. Mater. Chem. 47 (2010) 790–807. [10] V.T. Gontkovskaya, A.V. Gorodetskov, A.N. Peregudov, V.V. Barzykin, Comb., Explos. Shock Waves 32 (1996) 424–426. [11] P.O. Olanrewaju, Int. J. Num. Math. 5 (2010) 447–455. [12] O.D. Makinde, Mech. Res. Commun. 32 (2005) 191–195. [13] A.P. Aldushin, Comb. Explos. Shock Waves 23 (1987) 338–341. [14] G.A. Skorobogatov, B.P. Dymov, I.V. Nedozrelova, Zh. Org. Khim. 64 (1991) 956–965. [15] K.A. Holbrook, Trans. Faraday Soc. 63 (1967) 643–645. [16] A.S. Shteinberg, Fast Reactions in Energetic Materials, Springer-Verlag, Berlin, 2009. [17] A.G. Kennan, F. Siegmund, The Thermal Decomposition of Ammonium Perchlorate, Miami University, Special Report No. 6, Contract Nonr-4008(07), 1968. [18] M.W. Chase, J. Phys. Chem. Ref. Data. 9 (1998). [19] A.G. Merzhanov, F.I. Dubovitskii, Zh. Fiz. Khim. 35 (1961) 2083–2089.