Combustion, Explosion, and Shock Waves, Vol. 48, No. 4, pp. 432–439, 2012. c V.Yu. Filimonov. Original Russian Text
Thermal Heating Regimes During First-Order Homogeneous Reactions V. Yu. Filimonova
UDC 536.46
Translated from Fizika Goreniya i Vzryva, Vol. 48, No. 4, pp. 68–75, July–August, 2012. Original article submitted May 11, 2011; revision submitted November 22, 2011.
Abstract: Qualitative features of self-heating regimes for first-order homogeneous reactions are considered by analyzing the structure of the phase trajectories. Consideration of the temperature dependence of the heating rate makes it possible to extend the main results of Semenov’s theory of thermal explosion and pass to a two-dimensional parameter diagram, which allows the determination of the critical ignition conditions taking into account kinetic inhibition. It is found that the diagram of the Todes criterion versus the reciprocal of the Semenov criterion has four characteristic regions with different kinetics, two of which are limiting and two transitional ones. The boundaries of the regions are calculated, and the classical theory of thermal explosion is shown to be a special case of the projections of these boundaries on the range of the Semenov criterion. Keywords: reaction kinetics, thermal explosion, phase trajectory, parameter diagram, heating regimes. DOI: 10.1134/S0010508212040090 INTRODUCTION Issues related to the self-heating of exothermically reacting mixtures (thermal explosion) are of considerable practical interest to many fields of science and technology, especially power engineering, machine engineering, pyrotechnics, fire and explosion safety, production of new materials. The theory of thermal explosion focuses on calculating the critical self-heating conditions, establishing degeneration conditions, and determining the induction period for different kinetics of formation of the reaction product [1, 2] under different conditions of thermal contact of the reacting system with the environment. The basic propositions of the theory were developed by Semenov [3]. With the development of methods and capabilities of numerical modeling, numerous studies of thermal explosion have been conducted and the key characteristics and practically significant parameters of the self-heating of reacting mixtures have been determined [4–7]. Thus, the theory of thermal explosion is now a well developed chemical direction of the macrokinetics of nonisothermal processes. a
Polzunov Altai State Technical University, Barnaul, 656038 Russia; vyfi
[email protected].
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At the same time, numerical solutions often doe not provide a clear description of the totality of possible regimes and the variety of qualitative features of the processes, especially in the case where the system is nonlinear and multiparameter and does not admit steady-state solutions. Note that the steady-state Semenov theory is only a very successful and physically justified approximation which admits an analytical solution (it is accurate only for zero-order reactions). The basic approximation of the theory involves a neglect of the burnout of the components of the mixture in the preflame period. In this case, this approximation is self-consistent because the possibility of thermal explosion disappears with the significant burnout of the components. For example, the theory fairly accurately describes the ignition of the quite broad class of hydrocarbon systems [8] characterized by high values of the activation energy and thermal effect (which is responsible for the negligible burnout in the preflame period). However, a consideration of even first-order reactions (exothermic decomposition of solid and gaseous substances, isomerization reactions) shows that the class of such reactions is characterized by a very wide range of thermokinetic parameters [9, 10]. In this connection,
c 2012 by Pleiades Publishing, Ltd. 0010-5082/12/4804-0432
Thermal Heating Regimes During First-Order Homogeneous Reactions an analysis of the effect of the burnout dynamics on the critical ignition conditions is practically important for establishing the parameter boundaries of the control of the reaction and predicting the heating kinetics in unexplored exothermic reactions (for example, a recent study found [11] a first-order reaction in nanostructured heterogeneous systems). The main difficulties which make impossible an analytical treatment of problems of nonisothermal synthesis even in a thermally nongradient formulation are, first, the nonlinear temperature dependence of the heat release rate, and second, the unsteady nature of the selfheating and product formation processes. In this connection, it is very useful to perform a qualitative analysis of the behavior of complex nonlinear systems based on an analysis of phase trajectories [12], which provides information on the variety of possible regimes, their stability, and the boundaries of the transition from one regions of states to another. In this sense, even for a system such as a first-order exothermic reaction, there are still many unclear issues regarding its unsteady regimes. In the work presented here, we performed a qualitative analysis of the possible unsteady regimes of firstorder exothermic reactions in the phase plane “selfheating rate–temperature,” approximately calculated the boundaries of the transition between the regimes, and analyzed and systematized all types of phase trajectories in the parameter diagram.
QUALITATIVE ANALYSIS The classical system of equations describing the self-heating dynamics and kinetics of product formation for first-order monomolecular reactions is written in dimensionless form as follows [9]: dΘ = (1 − y) exp Θ − δΘ, (1) dτ dy = γ(1 − y) exp Θ, τ = 0; Θ = y = 0. dτ Here τ = t/tad is the time normalized to the adiabatic heating time scale: tad = cρRT02 exp(E/RT0 )/QEk0 , Θ = E(T − T0 )RT02 is the dimensionless temperature, y is the depth of transformation, δ = tad /t− is the ratio of the adiabatic heating time scale to the characteristic time of heat removal t− = cρV /αS (in modern terminology, the reciprocal of the Semenov criterion: δ = 1/Se), γ ≡ Td = cρRT02 /QE is the Todes criterion, T0 is the initial temperature of the mixture, c and ρ are its specific heat and density, respectively, R is the universal gas constant, E is the activation energy of the reaction, Q is the thermal effect, k0 is the pre-exponential factor, S is the heat-removal surface area which bounds
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the reaction volume V , and α is the heat-release coefficient. In writing (1), we assumed that the condition RT0 /E 1 is satisfied. Determination of qualitatively different reaction regimes involves the following difficulty, noted in [13]. Whereas Semenov’s theory clearly distinguishes between the possibility and impossibility of steady reaction regimes (constant temperature—unlimited heating), the overall analysis of system (1) does not allow this to be done because with the burnout rigorously taken into account, there is no mathematical jump that separates the stable steady heating regime and the regime with an unlimited increase in temperature (the temperature in all cases increases to the maximum value and then decreases). For a detailed analysis of the problem, we proceed as follows. We express the variable y from the first equation of (1) and substitute it into the second equation. We obtain a second-order equation for the time dependence of the temperature: d2 Θ dΘ dΘ dΘ + δΘ − −δ dτ dτ dτ 2 dτ =γ
dΘ + δΘ exp Θ. dτ
(2)
Further, introducing the heating rate of the system u = dΘ/dτ , we pass to the equation u + δΘ du = u + δ(Θ − 1) − γ exp Θ (3) dΘ u with the initial condition Θ = 0, u = 1. Thus, the system of first-order equations (1) reduces to one firstorder equation (3), represented in different variables. In this case, the self-heating regimes of system (1) will be determined by the behavior of the phase trajectory in the plane u–Θ. Equation (3) can be brought to a reduced form by introducing the variables ξ = u/δ and η = γ/δ. Then, we finally obtain the first-order oneparameter equation dξ = ξ 2 + ξ(Θ − 1 − η exp Θ) − ηΘ exp Θ (4) dΘ with the initial condition Θ = 0, ξ = 1/δ, which is an Abel equation of the second kind and cannot be integrated in quadratures. However, in terms of the qualitative theory of differential equations, this, generally speaking, is not required. We first consider the difference between the representations of the classical steadystate theory of thermal explosion and the unsteady system (1) in the variables ξ and Θ. The corresponding trajectories 1 and 2 for the nonexplosive (subcritical) reaction regime are schematically shown in Fig. 1. ξ
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Fig. 1. Schematic of the phase trajectories (a) and their corresponding heating thermograms (b): (1) trajectory for the unsteady regime, in accordance with (4); (2) trajectory of the transition to the steady state in Semenov’s theory; (3, 4) supercritical unsteady trajectories in Semenov’s theory; δ4 < δ3 < δ2 .
From the standpoint of Semenov’s theory, the phase trajectory is characterized by a monotonic decrease in the heating rate ξ to zero; in this case, the temperature rises to the steady-state value Θn , which is an asymptotically stable steady-state solution of (1) for γ = 0 (path 2 in Fig. 1a). In the unsteady problem (4), the phase trajectory must necessarily cross the maximum point Θm and the inflection point a, which has an extremum dξ/dΘ = 0 in Eq. (4). According to the steady theory of thermal explosion, the critical condition is given by dξ = 0 if ξ = 0, η = 0, (5) dΘ which, as follow from (4), corresponds to Θn = 1 and is equivalent to the appearance of a minimum in the phase trajectory 2 (Fig. 1a) or an inflection point in the thermogram (Fig. 1b). For supercritical conditions, the isocline of minima is given by the equation ξb = 1 − Θb, which follows from (5) and (4). In the terminology proposed in [13], the disappearance of the inflection point is a condition for the transition to the substantially unsteady regime Θb = 0, ξb = 1/δ = 1. Consequently, the parameter region of existence of this regime is defined by the inequality δ 1. Note that in this case, both the curve of the phase trajectory and the thermogram are characterized by a strictly positive value of the second derivative (upward convexity of the curves). We now consider unsteady regimes, i.e., Eq. (4). Since the thermograms have the maximum point and the inflection point a (Fig. 1a), regardless of the reaction regime, qualitative changes in the phase portrait occur in the first quadrant of the plane ξ–Θ. We first analyze the characteristics of the family of isoclines of
Fig. 2. Schematic of the phase trajectory for the subcritical heating regime (curve 3) and isoclines of extrema (curves 1 and 2).
extrema in the phase trajectory, which determine the corresponding inflections in the heating thermograms: ξ 2 + ξ(Θ − 1 − η exp Θ) − ηΘ exp Θ = 0.
(6)
The family of isoclines of extrema is shown schematically in Fig. 2. It is evident from the figure that the qualitative heating regimes depend on the relative position of the isoclines and phase trajectories, and the isoclines of extrema can both have and not have a minimum point. The condition of a minimum in the isocline (6) takes the form
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1+Θ . (7) exp(−Θ) − η From Eqs. (6) and (7), it follows that the minimum in the first quadrant disappears if ξmin = η
η 2 + η − 1 = 0, η = 0.62.
(8)
Consequently, for η > 0.62 in the first quadrant of the plane, the isocline of extrema is a monotonically increasing function of temperature. As follows from the figure, the qualitative features of the temperature dependence of the heating rate are determined by the ratio of the ordinates of the isoclines and phase trajectories for Θ = 0. A possible transition to progressive self-heating is shown schematically in Fig. 3. Passage through the point d is responsible for the nonmonotonic change in ξ(Θ) and the occurrence of two inflections in the thermogram, unlike in Semenov’s problem (see Fig. 1b), where the critical condition is determined by the appearance of a single point of inflection. Naturally, the appearance of two inflections in the unsteady problem is related to the presence of burnout in system. Thus, the self-acceleration of the reaction in the region ab (Fig. 3b) is replaced by the kinetic self-inhibition at the point b. Regime 3 is identical to the substantially unsteady regime 4 (see Fig. 1), which is the most rapid of all the possible regimes, and in this case, the inhibition of the reaction is due only to the kinetic inhibition in the developed stages of the process. For the regime considered, the critical condition is the passage through the point d with subsequent transition to the significantly unsteady mode. In this case, the transformation of the phase trajectory proceeds continuously; however, with the disappearance of the extremum at the point a (Fig. 3a), the structure of the trajectory changes qualitatively. In other words, in a neighborhood of the point A, the system is sensitive to changes in the parameters. We shall call this transition a hard transition. Conditions for the existence of the hard transition are the following inequalities (see Fig. 2): η < 0.62, 1/δ < 1 + η
(9)
γ < 0.62δ, γ + δ > 1.
(10)
or The first of these defines the presence of a minimum in the isocline of extrema 2 and the second implies that the ordinate of the phase trajectory Θ = 0 is below the corresponding ordinate of the isocline. However, in addition to the transition considered, there may be a different transition to the substantially unsteady regime in the case where curves 1 and 3 (see Fig. 2) intersect each other at Θ = 0 in the absence of an extremum in the isocline and the monotonic decrease in the phase trajectory is replaced by its growth
Fig. 3. Types of phase trajectories (curves 1–3) plotted by Eq. (4) with a change in the parameters of the problem (a) and the corresponding thermograms (b) [curve 4 is the isocline of extrema (6)].
at the beginning of the process, which corresponds to the reversal of the signs in inequalities (10) when crossing the merger of the points a and b. This transition will be called soft. We note that this situation has some singularities associated with the presence or absence of inflection points in the phase trajectory. The equation of the isocline of inflections of the phase trajectory is obtained by differentiation of (4) and is given by ξ(ξ − η exp Θ)2 = η exp Θ[(1 − Θ)ξ + η(ξ + Θ) exp Θ].
(11)
The relative position of the isoclines of inflections and extrema is shown in Fig. 4. Isoclines 1 and 2 have a single point of intersection corresponding to the minimum of the isocline of inlfections, which disappear with the disappearance of the minimum in the isocline of extrema, i.e., for η = 0.62. For η > 0.62 the isocline of inflections 2 is above the isoclines of extrema 1 . Further, one can rigorously show that the slope of the isocline of inflections for Θ = 0 and η > 0.62 is greater than the corresponding slope of the phase trajectory, and, therefore, the first point of intersection of the isocline and the phase trajectory occurs for Θ = 0 (the point a in Fig. 4).
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Fig. 4. Relative position of the isoclines of the extrema of the phase trajectory (6) (curves 1 and 1 ) and isoclines of inflections (curves 2 and 2 ) for η = 0.05 (curves 1 and 2) and η = 0.8 (curves 1 and 2 ). Schematic of the phase trajectories (curves 3 and 4): the point a corresponds to the transition to a substantially unsteady heating regime.
Fig. 5. Parameter diagram of ignition regimes: regions 1 and 4 refer to the limiting regimes; soft transition between the two limiting regimes occurs in a counterclockwise direction, and a hard transition in a clockwise direction; ao and ob are the demarcation lines determined by the critical conditions.
It should be noted that different combinations of signs in inequalities (10) determine four basic heating regimes in the plane γ–δ. This allows us to make a compact and visual generalization of the possible types of the above phase portraits u(Θ) in the form of the diagram shown in Fig. 5. The straight lines γ = 0.62δ and γ = 1 − δ divided the parameter plane into four regions 1, 2–3, 4, and 5–6 which are bounded in the figure by solid lines. The point b of the diagram corre-
Filimonov sponds to the critical condition in Semenov’s problem: δ = e for γ = 0, and passage through the point δ = 1 (with a decrease in δ) corresponds to transition to region 4 of substantially unsteady heating regimes. The demarcation lines ao and ob bound regimes 2 and 5 and represent the critical conditions for the soft and hard transitions 2 → 3 and 6 → 5, respectively. This diagram is an extension of Semenov’s problem to the plane γ > 0; we can also say that Semenov’s problem is a projection of the regions shown in Fig. 5 on the δ axis. Let us analyze each of the characteristic regions (the limiting regimes): In region 1, a monotonic decrease in heating rate with increasing temperature. This region is the most distant from the explosive reaction regime. The effect of kinetic inhibition and heat removal is substantial. In region 4, the fastest reaction regime. The effect of heat removal and kinetic inhibition in the preheating is small, i.e., the regime is quasiadiabatic or substantially unsteady. The remaining regions can be considered transitional, with the transition from one limiting regime to another proceeding in two directions. These are the following regions: Region 2 is characterized by an insignificant heat removal and a significant effect of kinetic inhibition on heating. The initial stage is characterized by a slow (nonexponential) increase in the heating rate, but due to significant burnout in the stage of development of the reaction, the heating rate then begins to decrease. Region 3 is the first critical region. It is characterized by fast heating due to a decrease in the influence of burnout. The critical conditions are determined by the demarcation line ao, i.e., the condition of passage through the point a (see Fig. 4). Region 6 is characterized by a significant effect of heat removal and a monotonic decrease in the heating rate. Region 5 is the second critical region. A reduction in heat removal leads to the appearance of a region of increasing heating rate, and then, under the influence of kinetic inhibition, the heating rate begins to decrease. The critical conditions are determined by the horizontal contact between the isocline of extrema 4 and the phase trajectory 1 at the point d (see Fig. 3a). In this case, the transition of the system through the point of convergence of the regimes o(0.62; 0.38) is very interesting is. Indeed, small changes in the parameter γ in a neighborhood of this point can cause an abrupt transition from the region of reactions at low rate to the region of rapid increase in the rate of heat release, bypassing intermediate regimes, i.e., from region 1
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to region 4 and vice versa. Thus, there are three pathways for the transition from region 1 to region 4; the corresponding routes are indicated by arrows in Fig. 5. The state of the system at the point o is unstable. Indeed, any small change in the parameters can lead to the occurrence of any of the regimes.
QUANTITATIVE ANALYSIS Quantitative analysis reduces to determining the separating dependences γ(δ) in the segments ao and ob (see Fig. 5). Segment ao. In this case, it is easy to obtain the exact solution for the dependence γ(δ). The transition is determined by the position of the point a in Fig. 4, which, based on (11), satisfies the condition (ξ − η)2 = η(1 + η) when ξ = 1/δ, which leads to the relation γ = 1/(2 + δ).
(12)
Thus, the demarcation line ao is a hyperbola. Segment ob. Analysis of the dependence γ(δ) is complex and requires, in addition to relations (6) and (7), a consideration of the behavior of the phase trajectory (4), which leads to very cumbersome approximate expressions. Figure 6 presents the results of a numerical analysis of (4) (curve I). Since the calculation is conducted in a limited region between the points o(0.62; 0.38) and b(e; 0) and the dependence is single-valued (in the absence of additional parameters) assuming an exponential dependence γ(δ), we can use the approximating formula γ(δ) = γ0
1 − exp[1.5(e − δ)] , 1 − exp[1.5(e − δ0 )]
(13)
where δ0 = 0.62 and γ0 = 0.38 are the coordinates of the point o. Given the numerical values, we write γ(δ) = 0.017(exp[1.5(e − δ)] − 1)
(14)
[in Fig. 6, curve (14) is shown by curve II). The general form of the parameters γ=
RT02 αS E cρRT02 , δ= exp , QE QEk0 V RT0
(15)
suggests that the motion of the image point along the horizontal line in Fig. 6 corresponds to a change in the heat removal conditions at a given initial temperature of the exothermically reacting system. Let us analyze particular systems. We consider the first-order exothermic reaction of decomposition of iodomethane into iodine and methyl: CH3 I → CH3 + I.
(16)
Fig. 6. Results of calculations of the demarcation dependence ob between regions 5 and 6 (see Fig. 5): curve I refers to the results of numerical analysis; curve II is the approximating dependence (14).
The heat of the reaction is Q = 234 kJ/mole [14], and the activation energy of the decomposition reaction is E = 231 kJ/mole. Then, for T0 = 675 K, we obtain γ ≈ 3.1 · 10−3 . From Fig. 6 it follows that in this case, the critical ignition conditions are in good agreement with Semenov’s theory δ ≈ e. However, if we consider a less exothermic decomposition reaction, e.g., the decomposition of ethyl chloride (Q ≈ 33 kJ/mole): C2 H5 Cl → C2 H4 + HCl
(17)
(E = 228 kJ/mole [15]), then at the same temperature, we obtain a different estimate (γ ≈ 0.05). Then, as follows from Fig. 6, the ignition condition of the system corresponds to δ ≈ 1.85. Thus, burnout has a significant effect on the critical conditions. The latter case is illustrated in Fig. 7, which shows a family of thermograms obtained by numerical analysis (1) for the parameter values shown by points in Fig. 6. In this case, the transition to progressive heating with the two inflections in the thermogram (δ1 → δ2 ) is observed quite clearly. With increasing parameter γ, this transition becomes less pronounced, and for γ → γ0 all features of the explosive nature of the reaction disappear. We determine from (15), the critical heat removal conditions: δ αS E = ck0 exp − . (18) V cr γ cr RT0 Then, for example, for reaction (17) T0 = 675 K, we obtain (αS/V )cr = 4.4 W/(m3 · K). In contrast, for given heat removal conditions, the critical temperature of the system can be determined using (14) and (15) or the calculated curve I in Fig. 6. It follows from the above that the inequality γ > 0.38 is the condition of complete degeneration of
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Fig. 7. Thermograms of heating for γ = 0.05 and various values of δ (see the points in Fig. 6).
thermal explosion. In order to establish the range of the corresponding parameters, we exclude the initial temperature from equalities (15) and obtain the relationship between the parameters: √ (19) δ = γz exp(β/ γ), where z = αS/cρk0 V, β =
cρE/QR.
(20)
Then, the condition of degeneracy takes the form δ0 β z exp − √ = 1.63 exp(−1.62β). (21) γ0 γ0 If condition (21) is satisfied, a thermal explosion is impossible at any initial temperatures. Thus, the parameters z and β allow one to unambiguously determine the possibility of a first-order reaction occurring in the regime of thermal explosion, i.e., they are necessary conditions for thermal explosion. Sufficient conditions are uniquely determined by curve I (see Fig. 6) or the approximate relation (14).
CONCLUSIONS The self-heating regimes for first-order reactions, which are determined by the relation between the Semenov and Todes criteria, were analyzed with the kinetic inhibition effect rigorously taken into account. Accounting for kinetic inhibition made it possible to move from the one-dimensional (Semenov’s theory) to a two-dimensional parameter problem and thereby enhance the understanding of the reaction regimes. There are four main regions of parameters which differ from each other in the temperature dependence of the heating
rate, with two of the regions being limiting ones. The first is characterized by a low reaction rate and a significant effect of burnout and heat removal on the heating rate. The second, in contrast, is characterized by a high self-heating rate and weak kinetic inhibition and is identified as the region of adiabatic or substantially unsteady heating. In the plane, transitional regions between the regimes were determined. Critical conditions are defined as the intersection of the boundaries of the regions in the parameter plane by the trajectory of the image point with a change in the reaction conditions. Necessary and sufficient conditions for thermal explosion were determined. It is shown that in the diagram there is a characteristic point of convergence of the regimes; passage through this point determines the direct transition between the limiting regimes. Using the diagram, one can purposefully change the Semenov and Todes criteria and control ignition regimes.
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