Some Non-Equilibrium Phenomena in Kinetics of
Chemical Reactions of High Energy Molecules
GJ. Dynnikova*, S.F. Gimelshein**, M.S. Ivanov***, N.K. Makashev*
^Central Aerohydrodynamic Institute (TsAGI),
Moscow Region, Zhukovsky 140160, Russia
**George Washington University, Washington, DC 20052
***Institute of Theoretical and Applied Mechanics, Novosibirsk 630090, Russia
ABSTRACT
The paper studies the effect of the of inelastic molecular collision models on the results of high threshold
reaction calculations
REVIEW OF CONSIDERED PROBLEMS AND OBTAINED RESULTS.
When molecules or atoms with translation or excitation energies large compared to the
corresponding "thermal" or "mean" values participate in gas-phase reactions and relaxation processes,
these reactions are called "high threshold". The non-equilibrium kinetics of such reactions is determined by
molecular distribution function tails. It was partially described in Proceedings of International Symposia on
Rarefied Gas Dynamics (Novosibirsk 1982,Oxford 1994, and Marseille 1998).
The calculation of distribution function tails of reacting molecules and the macroscopic rates of the
corresponding reactions can be efficient only if it is performed using special methods for solving kinetic
equations for the case of high molecular energies [1].
These problems were investigated (analytically and numerically) using the following basic
assumptions:
1) the energy of a single molecular degree of freedom contributes to overcoming the reaction
barrier of a high-threshold reaction under consideration;
2) the relaxation time is small compared to the reaction time;
3) the value of the energy barrier of corresponding reverse reaction is of the order of the bulk
thermal energy; the reaction is therefore considered equilibrium.
In this paper we show the results of calculations of molecular distribution functions obtained for
two cases not considered in literature.
First we assume that the high energy threshold of thermal dissociation of diatomic molecules
(considered under conditions of a non-isothermal boundary layer in gas of inert atoms with a trace species
of diatomic molecules) is overcome by the energy of different molecular degrees of freedom (i.e. not only
vibrational, but also rotational and translational energies)[2]. The corresponding thermal dissociation
models were considered (V, TV and TRV-models hereafter). The latter two models show a weak dependence
of the non-equilibrium reaction rate on flow conditions and molecular transport processes in a nonisothermal boundary layer. The reason for that is a finite energy participating in overcoming the reaction
threshold that comes from the thermal motion and molecular rotations (which are at equilibrium) and strong
diminution in variation of vibration level populations due to the diffucion of exited molecule.
It should be noted that the described modifications of the conventionally used "step-by-step"
mechanism of thermal dissociation from upper vibration levels were motivated by problems in reaching the
CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis
© 2001 American Institute of Physics 0-7354-0025-3/01/$18.00
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agreement between the measured and calculated reaction rate constants using conventional reaction
models. The obtained results raise the question of the role of non equilibrium molecular vibrations in thermal
dissociation process.
The second case assumes that both forward and reverse reactions have high energy thresholds.
For the sake of simplicity it was assumed that the forward and reverse binary reactions have equal
thresholds. As a result, the temperature of the gas mixture remains constant.
The direct simulation Monte Carlo method was used in computations. The molecular collisions
were calculated with the majorant frequency scheme [3]. The variable hard sphere model [4] was used with
oc=0 (hard spheres) and oc=0.5 (pseudomaxwell molecules). The reaction cross-section was proportional to
step function ^(1 - gR I g) .
The DSMC results show a typical behavior of velocity distribution functions for reagents, strongly
disturbed by forward reactions [5]. For the distribution functions of molecules participating in reverse
reactions an extremely large duration of kinetic stage of processes was observed. Significant nonmonotonuous changes of calculated distribution functions of forward reaction products were shown. These
results are important for the formulation of the perturbation methods of approximate kinetic equation
solutions.
MODELS OF INELASTIC COLLISIONS
The kinetics of VT relaxation of excited molecules was described in an adiabatic Morse oscillator
approximation [4]. One-level transitions were considered for all vibration levels according to the probabilities
P((a + 1) -» a) - Pg+l = (a + l)pf exp^a).
The accuracy of similar expressions for two- and more level transitions is significantly worse. In
this work only two-level transitions for upper vibration levels are included. The expression for the
corresponding probabilities is written as
P((a + 2) -» a) = P((a + 1) -» a) ij exp[A(E% - Q)lkT] ,
where A<1 andT| < 1 are free parameters.
Let us introduce several dissociation models utilized in this paper. The two following expressions
for the microscopic reaction rate W^ and the macroscopic constant of thermal dissociation K, describe a
widely used "step-by-step"(or V) reaction model (the dissociation is assumed to occur from the upper
vibration level with the energy E a ~ Q, a < j8. )
Here, Zam = na<Jam*j8kT/7Ql -is the collision frequency, na,n, na = nYa - are the number
densities of inert atoms, molecules, and exited molecules with vibrational energy Ej% , respectively. The
coefficient P£ is determined as the ratio of the dissociation cross-section to the collision cross-section. It
may be of order of 10 or smaller for the model under consideration.
For TV and TRY models it is assumed that the dissociation may occur from an arbitrary vibrational
level [2]. The energy to make up to the reaction threshold comes in this case either from the relative
translational energy or the sum of relative translational and rotational energies. For the reaction crosssection of
the calculations give the following expression for the above calculated micro- and macroscopic values (TV
model)
= (Zam lna)^(-QlkT)Ya
a
exp(
667
IkT)
For the TRV model it is assumed that the molecules in the gas are stable when the molecular
internal energy is bounded by the value of Q. It is also assumed that the energy needed to overcome the
reaction threshold comes the relative motion of colliding particles. As a result we obtain the same expression
as for the TV model where the energy Ej% is replaced by the total internal energy Eaj . The molecule are
stable when the rotational energy Ej (oc)< Q-E^ . This expression defines the value of y(°0max- The
final expressions are
W
a
The probability of dissociation for the TV and TRV models is therefore close for all vibrational
levels. This completes the formulation of the kinetics of dissociation in a vibrationally exited gas.
RESULTS AND DISCUSSION
The problem of vibration-diffusion-dissociation interaction was numerically solved using level-bylevel kinetics with simplifications connected with small values of relaxation times and applicability of strong
non-equilibrium local solutions of considered vibration kinetic equations, i.e.[l]
|VlnF a |«|V]n7 a |=e a |Vlnr|,% =EValkT>\,Fa = Ya/Y$
Figure 1 presents the results of calculation of IQ (cm3/mol s) in a motionless gas and various
temperatures T(K) for V, TV, TRV models (curves 2,3, and 4, respectively) with the assumptions
Pa = P = 1,0. T|a = 0. The curve 1 shows the experimental values of dissociation rate constant. The
results show the large discrepancy between V and TV models and the experiment.
The influence of the model and parameters on calculated Kj is shown in Fig. 2 for T=8000K. It is
seen that only TV and TRV models allows one to obtain reliable results at a sufficiently large P. The
influence of two-level transitions is rather weak (T|a = 0, solid lines, T|a = 1,0 , dotted lines).
r\
/•>
The value of correction F^ = K^(A q )/K^ to the reaction rate constant at T=8000K is
presented in Fig. 3. Here, q = Ql kT » 1 . .The parameter
A2 ^D(Vy\nT)2(Zampfei)-l-TVT
/t?«l
was introduced in [1]. It describes the influence of diffusion of excited molecules on upper vibration level
populations. The effect of gas motion in the boundary layer on the non-equilibrium dissociation kinetic is
small when the TRV-model is used and the value of P is large enough for an accurate calculation of K $ .
Figures 4 and 5 show the distribution functions obtained by the DSMC method for a motionless
gas mixture with high reaction thresholds. The reagent distribution functions are shown in Fig. 4, and the
product distribution functions are shown in Fig. 5. The results presented were obtained for pseudomaxwell
spheres and the thresholds value of the relative velocity g^ = 4, 5v 2^7". The numbers denote the time
duration in the units T0 / 7, T0 — molecular free path time.
REFERENCES
1. Makashev N.K. Pros, of 19-th Int. Symp. on Rarefied Gas Dynamics. Oxford Univ. Press, 1995, 1, 3.
2. Shatalov O.P., Losev S.A. AIAA Paper.97-2579, 1997.
3. Ivanov M.S., Rogasinsky S.V. Sov. J. Numer. Anal. Math. Modeling, 1988, 2(6), 453.
4 Bird G.A. Progr. Astro. Aero., 74: Part 1, 239.
5 Gordiets B.F., Osipov A.I., Shelepin L.A. "Kinetic processes in the gases and the molecular lasers". Moscow:
"Nauka", 1980, 512 p (in Russian)
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Figure 1
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Figure 2
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Figure 4
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Figure 5
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