Combustion, Explosion, and Shock Waves, Vol. 43, No. 6, pp. 688–690, 2007

Incompleteness of Conversion in a Traveling Polymerization Wave in the Presence of Heat Loss Yu. I. Babenko1

UDC 536.46

Translated from Fizika Goreniya i Vzryva, Vol. 43, No. 6, pp. 75–77, November–December, 2007. Original article submitted January 11, 2007.

The steady-state propagation of a polymerization wave in an infinite bulk is studied. It is assumed that the heat release law is specified by a truncated Arrhenius function. It is shown that in the presence of heat loss, complete conversion of the substance is impossible. Key words: polymerization, combustion, conversion, incompleteness of conversion.

INTRODUCTION

FORMULATION OF THE PROBLEM

The study of the wave solutions of systems of parabolic equations was pioneered by Kolmogorov et al. [1]. Problems related to combustion theory have been studied in a number of papers (see, for example, [2–6]). The effect of incomplete combustion in the presence of heat loss was found by Khudyakov [7] using the method of joined asymptotic expansions, which is physically equivalent to the separation of the spatial region into the reaction and heating zones (which is justified for high heats typical of combustion reactions). We note that similar questions were considered in [8, 9]. Lyubchenko [8] determined the critical conditions for the existence of flame under the assumption that the incompleteness of combustion was specified in advance. Ulybin [9] studied a combustion problem in a semi-infinite formulation presuming the existence of a flame front, on which boundary conditions were specified. In this case, the incompleteness parameter should be introduced to satisfy the indicated conditions, so that the incomplete conversion effect is not related to heat loss. Babenko and Moshinskii [10] studied the propagation of a polymerization wave in an infinite bulk under the assumption that the heat release rate is specified by a truncated Frank-Kamenetskii function. The present paper solve a problem similar to [10] but with heat loss taken into account.

Let us consider the following system of equations describing the fields of the temperature T and the relative concentration of the final product C in a coordinate system attached to a moving wave of chemical conversion:   dT E d2 T + Tad k exp − a 2 −U dx dx RT0   E(T − T0 ) − 1 − B(T − T0 ) = 0, × (1 − C) exp RT02

1

  E dC + k exp − dx RT0   E(T − T0 ) − 1 = 0, × (1 − C) exp RT02 −U

T = T (x),

C = C(x),

T (±∞) = T0 ,

x ∈ (−∞, ∞),

C(−∞) = 0.

Here a is the thermal diffusivity, E is the activation energy, R is the universal gas constant, k is the preexponent, U is the wave propagation velocity, B is the heat release coefficient, and Tad = Q/cρ is the adiabatic temperature, which is uniquely determined by the heat effect of the exothermic reaction Q, the heat capacity c, and the density of the medium ρ. The indicated quantities and the parameters T0 and C0 are assumed to be constant.

“Applied Chemistry” Russian Scientific Center, St. Petersburg 197198; [email protected]ffe.ru.

688

(1)

c 2007 Springer Science + Business Media, Inc. 0010-5082/07/4306-0688 

Incompleteness of Conversion in a Traveling Polymerization Wave We use the Frank-Kamenetskii heat-release function obtained by linearization of the Arrhenius law in the exponent [11]. In addition, unity is subtracted from the exponent in system (1) because the problem is wellposed mathematically only in this case. Physically, this means that the heat-release rate is set equal to zero at T = T0 . A similar procedure was first used in [12]. It is assumed that the reaction is of the first order. After introduction of the dimensionless quantities Θ= ω2 =

E (T − T0 ) , RT02

Ux , a

ξ=

U2 , ak exp (−E/RT0 )

β=

Θad =

B , k exp (−E/RT0)

dΘ 1 + 2 (1 − C)[exp Θ − 1] = 0, dξ ω

Θ = Θ(ξ),

C = C(ξ),

Θ(±∞) = 0,

(2) (3)

C(−∞) = 0;

Θad , β, ω = const > 0.

(4)

It is required to obtain the fullest possible information on the behavior of the function C(ξ).

REDUCING THE SYSTEM TO ONE EQUATION

(5)

Integrating (5) in the range (−∞, ξ], we have an analog of (10) from [10]: ξ Θdξ.

(6)

−∞

In view of (3), we perform the transformation C Θdξ = −∞

Θ 0

C = ω2 0

C ∈ [0, C∞ ];

Θ(0) = 0.

(9)

Let us consider the case of low heats Θad  1, which often occurs in practice in polymerization processes. In this case, Θ < Θad  1 and we can set exp Θ − 1 ≈ Θ, after which Eq. (8) becomes (10)

Redenoting the variables u = C and ϑ = Θ/Θad and introducing the parameters λ = Θad /ω 2 and γ = β/Θad , we write (10) in a form most convenient for our analysis: ϑ − u − γ ln(1 − u) dϑ = , du λ(1 − u)ϑ u ∈ [0, C∞ ];

ϑ(0) = 0;

(11)

λ, γ = const > 0.

d2 Θ dΘ βΘ dC + Θad − 2 = 0. − 2 dξ dξ dξ ω

ξ

Θ = Θ(C),

ϑ = ϑ(u),

Multiplying (3) by Θad and subtracting the result from (2), we obtain the equation

dΘ β = Θ − Θad C + 2 dξ ω

(8)

0

Θ − Θad C − β ln(1 − C) 1 dΘ = . ω 2 dC (1 − C)Θ

ξ ∈ (−∞, ∞),

2

(1 − C)[exp Θ − 1] dΘ = Θ − Θad C ω2 dC C ΘdC , +β (1 − C)[exp Θ − 1]

CASE OF LOW HEATING

system (1) becomes

−

Dividing (6) by (3) and using (7), we have

The range of the independent variable specified in (9) is not known in advance. It will be determined below during the solution. It is only clear a priori that C∞  1.

ETad , RT02

d2 Θ dΘ Θad βΘ + 2 (1 − C)[exp Θ − 1] − 2 = 0, − 2 dξ dξ ω ω

689

dξ dC dC

ΘdC . (1 − C)[exp Θ − 1]

In [10], Eq. (11) was considered for γ = 0. It turned out that as u increases from 0 to 1, the value of ϑ also increases in the range of 0 to 1. For γ > 0, the nature of the solution of Eq. (11) changes qualitatively. From physical considerations, it follows that as the concentration u increases, the quantity ϑ increases from zero to a certain maximum value ϑmax < 1 and then decreases, tending to zero at the end of the interval considered. Relation (11) immediately implies that, for ϑ = 0, the conversion u = 1 because, otherwise, we obtain the absurd result dϑ/du = ∞. The indicated derivative should remain finite, and, hence, for ϑ = 0 the numerator should be equal to zero. From this, we have the following expression, which gives the maximum possible value u = u∞ : u∞ + γ ln (1 − u∞ ) = 0.

(7)

(12)

The transcendental equation (12) has a unique solution u∞ < 1 provided that γ ∈ (0, 1). Physically, the absence of a solution in case γ  1 means that in the case of intense heat loss, a traveling wave cannot exist.

690

Babenko

For the limiting cases, the following expansions are valid:     exp(−2/γ) 1 u∞ = 1 − exp − +O , γ → 0, γ γ 

u∞

 1−γ 2 + O(1 − γ) , γ → 1. = 2(1 − γ) 1 − 3

CASE OF HIGH HEATS The above consideration for the case of low heats actually implies that the temperature dependence of the reaction rate is linear [see the transformation from (8) to (10)]. In this case, only the quadratic nonlinearity (1 − C)Θ is retained in the initial equations (2) and (3). In the case of arbitrary values of Θad , it is necessary to use Eq. (8). It is easy to see that as ξ → ∞ and as Θ → 0, the quantity C cannot be equal to unity since dΘ/dC = ∞. Thus, the conversion incompleteness effect also occurs for high heats if a steady-state traveling wave exists.

CONCLUSIONS The known fact of conversion incompleteness in a steady-state traveling wave of chemical reaction in the presence of heat loss was confirmed. For the case of small heat effects (for example, in a polymerization reaction), a transcendental equation defining the degree of the maximum attainable conversion was obtained. For the case of small heat effects, the range of the heat loss parameter in which a traveling wave can exist was found. I thank V. B. Ulybin and A. I. Moshinskii for discussions of the work.

REFERENCES 1. A. N. Kolmogorov, I. G. Petrovskii, and N. S. Piskunov, “Investigation of the equation of the diffusion coupled to an increase in the amount of substance and its application to one biological problem,” Bull. Mosk. Gos. Univ., Ser. A: Mat. Mekh., 1, No. 6, 1–26 (1937). 2. Ya. B. Zel’dovich, Nonlinear Waves. Propagation and Interaction [in Russian], Nauka, Moscow (1981). 3. D. A. Vaganov and S. I. Khudyaev, “A steady-state problem of combustion theory,” Fiz. Goreniya Vzryva, 5, No. 2, 167–172 (1969). 4. S. I. Khudyaev, “On the asymptotic theory of a steadystate combustion wave,” Khim. Fiz., 6, No. 5, 681–691 (1987). 5. V. A. Vol’pert and A. I. Vol’pert, “Existence and stability of traveling waves in chemical kinetics,” in: V. I. Bykov (ed.), Dynamics of Chemical and Biological Systems [in Russian], Nauka, Novosibirsk (1989), pp. 56–131. 6. S. I. Khudyaev, “One class of integrable equations in problems of combustion and hydrodynamics,” Mat. Model., 7, No. 1, 35–60 (1995). 7. S. I. Khudyaev, “Asymptotic behavior of a steady-state nonadiabatic combustion wave,” Khim. Fiz., 10, No. 6, 838–847 (1991). 8. V. I. Lyubchenko, “Steady-state combustion of nongasifying condensed substances in the presence of heat loss to the ambient medium,” Zh. Fiz. Khim., 61, No. 4, 964–971 (1987). 9. V. B. Ulybin, “Influence of the incompleteness of combustion on the stability and parameters of steady-state deflagration,” Dokl. Akad. Nauk SSSR, 312, No. 4, 922– 924 (1990). 10. Yu. I. Babenko and A. I. Moshinskii, “Determination of the propagation velocity of a polymerization wave,” Combust., Expl., Shock Waves, 33, No. 5, 565–575 (1997). 11. D. A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, Plenum, New York (1969). 12. J. W. Enig, “Critical conditions in time-dependent thermal explosion theory,” J. Chem. Phys., 41, No. 12, 4012–4013 (1964).