Mathematical modeling of critical conditions in the thermal explosion problem

Transcription

1 Mathematical modeling of critical conditions in the thermal explosion problem G. N. Gorelov and V. A. Sobolev Samara State University, Russia Abstract The paper is devoted to the thermal explosion problem in the case of autocatalytic reaction given both heat transfer and diffusion. By means of the method of integral manifolds there are investigated critical regimes and are found critical values of parameter for plane-parallel and for cylindrical reactors. This work was written while V.A. Sobolev was a visitor at both the Boole Centre for Research in Informatics and Institute of Nonlinear Science, University College Cork. Financial support is gratefully acknowledged.

2 Contents Introduction The Main Problem 3 3 Critical conditions in the lumped model 3 4 Critical conditions of the plane-parallel reactor 6 5 Critical conditions of cylindrical reactor Introduction The problem of evaluation of critical regimes thought of as regimes separating the regions of explosive and nonexplosive ways of chemical reactions is the main mathematical problem of the thermal explosion theory. Investigation of critical phenomena of the thermal explosion theory was hold by N. N. Semenov [], Ya. B. Zeldovich [], D. A. Frank- Kamenetsky [3], O. M. Todes and P. V. Melent ev [4], A. G. Merzhanov and F. I. Dubovitsky [5], B. Grey [6] et al. Because of considerable difference between velocities of thermal and concentrational changes, singularly perturbed systems of differential equations serve as mathematical models of such problems. But in the above works the authors restrict their consideration to the study of zero order approximation. It does not let them explain the strong parametric sensitivity of this problem as well as to examine the transformation of solutions in the vicinity of the limit of selfignition. In the works [7, 8, 9] it was proposed to use the stable-unstable integral manifold as a mathematical model of the critical regime of the autocatalytic reaction within investigation of the lumped model. This approach permits to work out the algorithms of asymptotic representations of the critical values of the parameter of initial conditions and to describe the transfer regimes. Such approach appeared to be fruitful in the case of distributed model in the problem of thermal explosion.

3 The Main Problem Consider nonlinear singularly perturbed parabolic system where ε θ τ = ϕ(ηe θ +βθ + δ D ξθ, ( ε η τ = εϕ(ηe +βθ θ + ρ D ξη, ( D ξ ( = ( ξ + n ξ boundary conditions θ ξ =, θ =, ξ= ξ= (, n =,,, ξ η ξ =, ξ= η ξ = (3 ξ= and initial conditions θ =, η =. (4 τ= τ= This is a mathematical model of the problem of thermal explosion given heat transfer and diffusion in the case of autocatalytic combustion process. Here θ is a dimensionless temperature, η is a dimensionless rate of combustion, τ is a dimensionless time, ε and β are small positive parameters, δ is a Frank-Kamenetsky criterion, that is the scalar parameter, characterizing initial state of the system. Depending on its value, reaction either is explosive or proceeds slowly. The value of parameter δ separating slow and explosive regimes is called critical. Function ϕ(η determines the law according to which the reaction proceeds: for ϕ(η = η we have the first order reaction, for ϕ(η = η n it is the n th order reaction, and for ϕ(η = η( η it is an autocatalytic one. Using the method of integral manifolds, the critical value of δ is calculated as an asymptotic series with respect to degrees of the small parameter ε δ = δ ( + δ ε + O(ε, (5 where critical regimes are modelled by the duck-trajectories. For n = (plane-parallel reactor n = (cylindrical reactor corresponding values of δ and δ are estimated. 3 Critical conditions in the lumped model Before we get down to the common case consider the model where heat transfer and diffusion are not taken into account. Now, we introduce 3

4 some necessary notions and consider that manner of reasoning which will be held in the common case. The system of equations ε dθ dτ = η( ηeθ αθ, (6 dη dτ = η( ηeθ. (7 is a classical mathematical model of the thermal explosion problem in the case of autocatalytic reaction. Here θ, η, τ are just the same as in system ( (, ε is a small positive parameter. Since a small parameter β enters system ( ( regularly, we can account it a zero without loss of generality. Since the point (, is a steady state of system (6 (7, initial conditions in the case of autocatalytic reaction are the following: η( = η, θ( = θ (η, θ. (8 Parameter α characterizes initial state of the system. It plays the same role as the criterion of Frank-Kamenetsky δ for system ( (. At different values of this parameter reaction transfers either into dissipative regime or into regime of autoacceleration, the latter leads to an explosion. The aim is to evaluate that value of parameter α, at which the reaction, having started in the vicinity of the point θ =, η =, leads to the highest rate of combustion of the reactant. Calculating the value of α at which unstable integral manifold is an extension of the stable one, we determine, in doing so, the critical value of this parameter, because in this very case trajectory of system (6 (7 is moving at the rate of the slow variable for the mostly long time separating slow and explosive regimes. Thus the task of estimation of the critical value of parameter α becomes the problem of calculation of such a value at which system (6 (7 has a duck-trajectory [9]. The slow curve of system (6 (7 F = {(η, θ : η( η = αθe θ, η, θ } has different form in dependence of α > 4 e or α < 4 e. For θ <, the part of the curve F is always stable, and for θ > it is unstable. Denote the stable part of the curve F by F +, and unstable by F. At a distance O(ε from F there exist integral manifolds M + M, corresponding to the parts F +, F. System (6 (7 has two steady states: O(, and P (,. Now, we give a qualitative description of the behavior of the phase trajectories of system (6 (7 corresponding to parameter α variation. 4

5 For α > 4 e, the trajectories, having started in the vicinity of O, will pass along the stable branch F +. in doing so, the value of y will stay not more than. These trajectories correspond to the slow modes of the reaction. For α < 4 e, the stable part F + of the curve F becomes -connected, and the phase trajectory, having reached the point S at the rate of variation of F +, in the vicinity of S falls down into the explosive regime. Due to continuous dependence of the right-hand side of the equation on α, in the neighborhood of α = 4 e, there should exist some trajectories between those mentioned in the previous paragraph. the critical one among them. For α = 4 e the slow curve F has the point of intersection (, /, and application of the same approach as in [9] in the case of autocatalytic reaction is impossible. Still, the use of stable and unstable local integral manifolds for modeling of critical trajectories is extremely helpful in this case, too. In this case, the duck-trajectory of system (6 (7 is proposed as a mathematical object, modeling the critical trajectory. Having estimated the value α, for which an unstable integral manifold is a continuation of the stable one M +, we determine by this the critical value of the parameter a, since in this very case the trajectory of system (6 (7 follows the slow curve for the mostly long time. Denote by α the critical value of parameter α for the autocatalytic reaction. In the vicinity of α α < e k ε there exist such duck-trajectories, which, having passed along integral manifold M for some time, fall down either into explosive regime or into the slow one. These curves are the transient trajectories. The critical trajectory OSL corresponds to the value α, and the trajectories OE E P, OD D P correspond to the values α and α respectively. These curves, having started in the vicinity of, pass along M +, then move for some time along M, after that the temperature falls abruptly and in the system there proceeds a slow motion along M +. We search for the critical value of α and the corresponding ducktrajectory of system (6 (7 as an asymptotic expansion in ε: α = α + εα + ε α ε n α n +..., (9 η = v(θ = v (θ + εv (θ ε n v n (θ ( To calculate α i, v i (θ, we substitute expansions (9, ( into the equality η = v (θ θ. Equating, in the obtained relation, the coefficients at the same degrees of ε we find sequentially α i and v i (we restrict ourself to the terms of the order not exceeding ε : 5

6 α = v ( v e θ θ = e θ= 4, v (θ = ± 4 α θe θ, α = (v v e θ v v (θ = θ(α v + α v ( v e θ, α = v ( v e v α v = e, θ= = 49 θ= 36 e, v (θ = θ (α v + α v + v v e θ + v ( v ( v v ( v e θ. After all, we have that the critical value of α may be presented as follows: α = e ( ε ε + O(ε 3. ( It should be noticed that there exists the trajectory of system (6 (7, moving from M to M +, corresponding to the value of parameter α = e ( + ε ε + O(ε 3. ( The interval (α, α corresponds to transitional regime. 4 Critical conditions of the plane-parallel reactor Setting n = in ( (, in the case of plane-parallel reactor we obtain the system ε θ τ = η( ηeθ + θ δ ξ, ε η τ = εη( ηeθ + (3 η ϱ ξ, with boundary conditions θ η ξ =, θ = ; η ξ= ξ= ξ =, ξ= ξ =. (4 ξ= 6

7 One-dimensional slow integral manifold corresponds to the critical regime. This manifold can be found in a parametric form θ = θ(v, ξ, ε = θ (v, ξ + εθ (v, ξ + O(ε, η = η(v, ξ, ε = η (v, ξ + εη (v, ξ + O(ε, dv dτ = V (v, ε = V (v + εv (v + O(ε. The coefficient δ will be found also as asymptotical expansion (5 Taking into account (5 we obtain for (3 δ = δ ( + εδ + O(ε. (6 ε θ v V = η( ηeθ + θ δ ξ, ε η v V = εη( ηeθ + η ϱ ξ. (7 The problem for zero order approximation of the integral manifold (5 is derived from (7 under ε = : θ ξ + δ η ( η e θ =, η ϱ ξ =, (8 with boundary conditions θ ξ= ξ =, θ =, (9 ξ= η η ξ =, ξ= ξ =. ( ξ= The second equation in (8 with boundary conditions ( has the solution η = η (v. It is convenient to choose parameter v as η (v, ξ v. Thus, the first equation in (8 takes the form θ ξ + δ v( v e θ =. ( Consider the auxiliary boundary value problem y + ae y =, y ( = y( =. ( 7

8 The solution of ( is y = (ln chσ ln chσξ, (3 where σ is the solution of transcendental equation ch(σ = a σ. (4 Under some a = a the last equation possesses the unique solution σ = σ, under a > a there are no solutions, and under a < a there are two solutions of this equation. The corresponding approximate values for a and σ are a = , σ = (5 It is clear that the boundary value problems (3, (9 and (, (9 are coincident under a = δ v( v. The maximal value of the right-hand side of last equality is δ /4 under v = /. The zero approximation of the critical value of the coefficient δ is thus seen to be 4a, and for δ we obtain the approximate value δ = 4a = (6 By this means under the condition δ < δ the boundary value problem (, (9 possesses two solutions. Upper solution corresponding to high temperatures is unstable, and lower one is stable. The solutions coincide at the point v = / under the condition δ = δ, and under the condition δ > δ there exists such interval (v, v that there are no solutions on it. The following expressions as a zero order approximation for the slow one-dimensional integral manifold are obtained: θ = θ (v, ξ, η = v, where θ is a chosen solution of the boundary value problem (, (9 under the condition δ = δ. Substituting (5, (6 in (3, (4 and equating coefficients at ε we obtain the problem for the first order approximation θ ( ξ +δ v( ve θ θ θ = δ v V ( ve θ η δ v( ve θ, η ξ + v( veθ = V, (7 with the boundary conditions 8

9 θ ξ= ξ =, θ =. (8 ξ= η η ξ =, ξ= ξ =. (9 ξ= Integrating (7 over ξ from to and taking into account (9 we obtain V = v( v To calculate the integral in (3 we use (3 and get e θ (ξ,v dξ. (3 θ (ξ, v = ln ch(σ(v ln ch(σ(vξ, (3 where σ(v is the solution of (4 under a = v( vδ and, therefore, From the last formula we have e θ (ξ,v = ch (σ(v ch (σ(vξ. (3 V (v = v( v ch (σ(v ch (σ(vξ dξ = (σ(v v( vch th(σ(v = σ(v = v( v ( ch(σ(vσ(v σ(vth(σ(v. (33 It should be noted that under ν = / taking into account the identity from (4 we obtain σ th(σ =, (34 ( V = ( ch(σ σ th(σ = 4 σ 4 a =. (35 δ To define the first order approximation of the critical value of the parameter δ we consider the first equation (7 under the condition v = / and obtain θ ξ + δ 4 eθ θ = f(ξ, ( θ (ξ, f(ξ = δ ( V v δ 4 eθ (ξ, The corresponding homogeneous boundary value problem. (36 9

10 θ ξ + δ 4 eθ θ =, θ ξ= ξ = θ =, ξ= (37 has nontrivial solution ϕ (ξ, which is an eigenfunction corresponding to zero eigenvalue: is ϕ (ξ = σ ξth(σ ξ. (38 Thus, the solvability condition for the boundary value problem (36 f(ξϕ (ξdξ = (39 or ( θ (ξ, ( V δ v 4 eθ (ξ, ( σ ξth(σ ξdξ =. (4 Note that (4 exactly coincides with the condition (?? of continuity of integral manifold at the point v = / (... Let us calculate now functions contained in (4. Formula (3 implies θ = (th(σ(v th(σ(vξ σ v v. (4 It is convenient to rewrite (4 in the form ch(σ =, [v( v]. (4 σ δ Differentiating (4 with respect to v we obtain ( sh(σ ch(σ σ σ σ v =, [v( v] 3 ( v. (43 δ Taking into consideration that under ν = / the equalities sh(σ ch(σ =, v =, (44 σ σ hold, and after iterated differentiation, we obtain under v = /

11 ( ch(σ sh(σ + ch(σ ( σ σ σ σ 3 v v= = [v( v] δ The last expression may be simplified to the form 3. (45 v= ch(σ sh(σ + ch(σ = σ σ σ 3 = ch(σ + ch(σ σ σ ( 3 σ th(σ = ch(σ. σ By virtue of (4 equation (45 takes the form (46 or ( σ v Consequently, we obtain v= = [v( v] v= = 4 (47 σ v = ±. (48 v= θ v = ±4(th(σ ξth(σ ξ = ± 4 v= σ ( σ ξth(σ ξ, (49 Note that the sign + corresponds to the transfer from the stable part of slow curve to unstable one and the sign corresponds to transfer from the unstable part to the stable one. Hence, trajectories that passed at first along the stable part and after that along the unstable part (canards correspond to the positive value ( θ / v v=/. It follows from (4 that δ = 4 θ (ξ, ( V ( σ ξth(σ ξdξ v = ch (σ ch (σ ξ ( σ ξth(σ ξdξ = ± 3 δ σ ( σ ξth(σ ξ dξ ch (σ ch (σ ξ ( σ ξth(σ ξdξ ±.. (5

12 Thus δ = δ ( + δ ε + O(ε (5 corresponds to a canard and gives the required critical condition for a thermal explosion (first limit of self-ignition. The value δ = δ ( δ ε + O(ε (5 corresponds to the false canard and gives the second limit of self-ignition. The interval (δ, δ corresponds to transitional regimes of combustion. For the difference of the values δ and δ we have δ δ = δ δ ε + O(ε 5.58ε + O(ε (53 For the first order reaction one may use the generalization of the algorithm, worked out in [7], but in this case it may be applied in the numerical form only. Thus, for ρ = we have at ε =. the value of δ =., and at ε =. the value of δ = Critical conditions of cylindrical reactor Now, we put n = in ( ( and investigate the critical conditions of thermal explosion for the cylindrical reactor. In doing so, we have ε θ τ = η( η eθ + ( θ δ ξ + θ, ξ ξ ε η τ = εη( η eθ + ϱ with boundary conditions θ ξ = θ =, ξ= ξ= ( η ξ + ξ η ξ, (54 η ξ = η ξ= ξ =. (55 ξ= In the same manner as in the previous Section, we try to find the slow one-dimensional stable-unstable integral manifold in a parametric form θ = θ(v, ξ, ε = θ (v, ξ + εθ (v, ξ + O(ε, η = η(v, ξ, ε = η (v, ξ + εη (v, ξ + O(ε, (56 dv dτ = V (v, ε = V (v + εv (v + O(ε. The factor δ will be calculated as asymptotic expansion with respect to degrees of the small parameter ε: δ = δ ( + εδ + O(ε. (57

13 Given (56, system (54 results in: ε θ v V = η( η eθ + δ ε η v V = εη( η eθ + ϱ ( θ ξ + ξ ( η ξ + ξ θ ξ η ξ,, (58 Setting ε = in (7 we obtain the problem for the zero order approximation of the integral manifold (5: θ ξ + ξ θ ξ + δ η ( η e θ =, η ξ + ξ η ξ =, (59 with boundary conditions θ ξ =, θ =. (6 ξ= ξ= η η ξ =, ξ= ξ =. (6 ξ= Function η = η (v is a solution of the second equation in (59 with boundary conditions (6. Since we have some freedom of choice of the parameter v, we put η (v, ξ v. The first equation in (59 takes the form θ ξ + ξ Now, we consider an auxiliary problem θ ξ + δ v( ve θ =. (6 y + ξ y + ae y =. y ( = y( =. (63 It can be easily verified that function ( a a ± a ( ± a y = ln ( = ln + ξ a ± a a + ξ ( ± a (64 is a solution to (63. Obviously, for a = the last equation has a single solution, for a > there are no solutions, and for a < there are two 3

14 solutions. Of these two solutions, the lower, corresponding to smaller temperatures, is stable, and the upper one is unstable. Problems (63 and (59, (6 coincide at a = δ v( v. The greatest value of the right-hand side of this relation equals to δ /4 for v = /. That is why the critical value of coefficient δ equals to 4a in its zero order approximation, i. e., we obtain the approximate value for δ : δ = 4a = 8. (65 Thus, for δ > δ system (6, (6 has no solutions, and for δ = δ there are two solutions θ + (ξ, v = ln θ (ξ, v = ln ( + 4v( v 4v( v + ξ ( + 4v( v, (66 ( 4v( v 4v( v ξ ( + 4v( v, (67 which stick together at the point v = /; here θ is stable, and θ + is unstable. For δ < δ there are two solutions. The following expressions are the zero order approximations for onedimensional slow integral manifold: θ = θ (v, ξ, η = v, where, as θ, should be chosen one of solutions of boundary problem (6, (6, regarded for δ = δ. To obtain the problem for the first order approximation, we substitute (56, (56 into (54, (55 and equate the coefficients at ε: θ ξ + ξ θ ξ + δ v( ve θ θ = = δ ( θ v V ( ve θ η δ v( ve θ ( η ϱ ξ + ξ η + v( ve θ = V, ξ, (68 with boundary conditions θ ξ =, ξ= θ =. ξ= (69 η ξ =, ξ= 4 η ξ =. (7 ξ=

15 Multiplying the second equation in (68 by ξ, integrating by ξ from to, and taking into account (7, we derive whence ξv dξ = v( v ξe θ (ξ,v dξ + ϱ ( ξ η dξ, (7 ξ ξ V = v( v ξe θ (ξ,v dξ. (7 Now, we calculate the integral in the right-hand side of (7 for v = /: Hence, ξe ln +ξ dξ = 4 ξ ( + ξ dξ = + ξ =. (73 ( V = 4 =. (74 For v = / the first equation in (68 is as follows: θ ξ + ξ θ ξ + δ 4 eθ (ξ, θ = f(ξ, ( θ (ξ, f(ξ = δ ( V δ v 4 eθ (ξ,. (75 Let us find θ / v v=/ (δ = 8. θ (ξ, v = ln(α± α ln(+ξ (α± α + ln α, α = δ [v( v] = [v( v] Now, we differentiate the first equation in (76 by v : θ ± α ξ v = (α± ( α ± α α + α α± α α + ξ (α± α α v = { = α + ± α ±ξ (α ± } α α ( + ξ (α± α ( α v. (76 (77 5

16 For the sake of brevity, introduce the following denotations: κ(α = α ± α (κ( =. After this, equality (77 is reformulated as follows: { θ v = α + ± α ±κ ξ { α = α ± κ ξ + κ ξ ( + κ ξ ( α } α α v. } α v = (78 Taking into consideration, that α v = 3 [v( v] ( v = 4 4 (v [v( 3 v], (79 α = = 4v( v 4v( v = [v( v], (8 4v + v v we have for (78: θ v = ± { ξ v= + ξ v( v v = ± v } ξ v= + ξ, θ v = ±4 (8 ξ v= + ξ. Here sign + corresponds to the duck-trajectory. The homogeneous boundary problem (75, (69 has a nontrivial solution ( i. e. an eigenfunction, corresponding to the zero eigenvalue ϕ (ξ = ξ + ξ. (8 Hence, the solvability condition of nonhomogeneous boundary problem (75, (69 is as follows: f(ξϕ (ξdξ = (83 or ( θ (ξ, ( V δ v 4 eθ(ξ, ( ξ + ξ dξ =, (84 6

17 whence δ = 4 θ (ξ, v V ( ξ +ξ dξ e θ(ξ, ξ + ξ dξ ( ξ dξ = ±8 ( ξ dξ + ξ 4 (+ξ ξ + ξ dξ = (85 = ± + ξ ξ ( + ξ 3 dξ = ±6 4 π 4 + π ±.93. After all, we have the following: δ = δ ( + δ ε + O(ε (86 corresponds to the duck-trajectory and gives a desired critical condition of thermal explosion, and the value corresponds to the second limit of selfignition. For the difference δ δ we have δ = δ ( δ ε + O(ε (87 δ δ = δ δ ε + O(ε 3.77ε + O(ε. (88 As for the plane-parallel reactor, in the case of the first order reaction for the cylindrical reactor, one has to apply numerical algorithms. In doing so, for ρ =, we have, for example, the following values of δ: δ =. for ε =., and δ =.33 for ε =.. 7

18 References [] Semenov N. N. On some problems of chemical kinetics of reactional capacity. Moscow: AN SSSR, 959. [] Zeldovich Ya. B., Barenblatt G. I., Librovich V. B., Makhviladze G. M. Mathematical theory of combustion and explosion. Moscow: Nauka, 98. [3] Frank-Kamenetsky D. A. Diffusion and heat transfer in chemical kinetics. Moscow: Nauka, 967. [4] Todes O. M., Melent ev P. V. The theory of thermal explosion. // Journal of physical chemistry 939. V. 3, 7. P [5] Merzhanov A. G., Dubovitsky F. I. The modern state of the theory of thermal explosion. // UChN, 966. V. 35, 4. P [6] Gray B. F. Critical behaviour in chemical reacting systems:. An exactly soluble model //Combust.& Flame P [7] Gorelov G. N., Sobolev V. A. Duck-trajectories in a thermal explosion problem // Appl. Math. Lett., 99, V. 5, 6. P [8] Gorelov G. N., Sobolev V. A. Mathematical modeling of critical phenomena in thermal explosion theory // Combust. & Flame, 99. V. 87. P. 3. [9] G. N. Gorelov, V. A. Sobolev, E. A. Shchepakina Singularly Perturbed Combustion Models. Samara: SamVien,