E Kondratyev, A Gerasymenko, K Merkotan - Behavior of nonlinear reactive system under external harmonic perturbation nearby a critical state - страница 1

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ISSN 1024-588X. Вісник Львівського ун-ту. Серія фізична. 2011. Випуск 46. С. 63-70 Visnyk of the Lviv University. Series Physics. 2011. Issue 46. P. 63-70

УДК 544.3.032.4

PACS 82.33.Vx, 64.60.Ht, 05.45.-a



E. Kondratyev1, A. Gerasymenko1, K. Merkotan2, V. Rusov2

1 Odessa National University named after 1.1. Mechnikov Dvorjanskaya str., 2, 65026 Odessa, Ukraine e-mail: kenphys@ukr.net

2 Odessa National Polytechnic University Shevchenko av., 1, 65044 Odessa, Ukraine e-mail: merkotankir@ukr.net

We consider behaviour of critically strained reactive system under external harmonic perturbation. While a reactor works nearby such a state, the role of external fluctuations increases. Because of non-linearity of the system response on external perturbations, the reactor crosses over from a steady operative state into an unsteady one, which is explosive. Then a question arises of how the value of the perturbation provoking the conversion from the steady state to the unsteady one, depends on its frequency. It is shown that in the immediate vicinity of the critical state, the basic equation of the reactor warming-up can be brought to the Riccati equation with a harmonic source. It is found with the parametric calculation method that, under condition of constant amplitude, the transition time increases with increasing frequency. A linear dependence of the external transition energy on the perturbation frequency is revealed.

Key words: reactor, external harmonic perturbation, nonlinear system, crit­ical state, Riccati equation.

The productivity of the system, which can be, for example, a homogeneous reactor or just a particle of combustible, increases on approaching the critical state. However nearby the critical state the risk of a stability loss of the reactor operating condition grows under the perturbation effect that provokes the conversion to the unsteady and difficult controlled state. By this reason, from the scientific point of view the response of the reactor as a nonlinear system on external harmonic perturbation is the most

© Kondratyev E., Gerasymenko A., Merkotan K., Rusov V., 2011

interesting [1]. In the works [2,3] the similar task was examined in a wide temperature range.

The task of a present work is to find out a connection between the perturbation parameters, which are amplitude and frequency, when the system converts from the steady state to the unsteady one.

Taking into account the heat supply due to reactions and heat sink from the system to the environment the equation of the homogeneous reactor warming-up can be expressed as:

dT     QpcV      e      otS /m   m     otS m , .

= ^—ze-ret--(T - T—) +-TA cos wt, (1)

dt       cpm cpm cpm

Here the first and second summands in a right part of the equation (1) define physically the heat supply and the heat sink of the system accordingly, the third


a reaction thermal effect; z is a particle generation rate in a reaction; p is a density of agent, c is its concentration; E is an activation energy, i.e. that energy which the mole of substance must possess to be responded; R is a universal gas constant; T is a reactor temperature; o is a heat-transfer coefficient; T— is an environment temperature; V is a volume where a reaction takes place; S is a surface area limiting the volume; w


of the external perturbation; cp is a specific heat capacity at constant pressure; m is mass of the reactive system.

Supposing that the reactive volume has a spherical form for which, as is generally known, the Nusselt number approximates by 2. Then the equation (1) has the next type

= Qzce- ret--_(T - T00) +-_Ta cos wt. 2

dt      cp Cppr_ Cppr_



We will make the equation (2) dimensionless. For this purpose we will do the known change of variables by Frank-Kamenetskii [4]:

9=m(T -   =REr. (3'

Considering expressions (3) the equation (2) can be brought to the dimensionless temperature form:

d9     1   _e_     9     TA   E , ч

= -x x1+вв - ~r + -г; rtoo cos wt

, Cppr2 , ,      , CpRT_    e    , ,

where Tr = -i—-— is relaxation time; тх = e RT<=° is time of the chemical 3A QzcE


For the wide row of substances a dimensionless parameter (3 is negligible quantity about 0.05.


time t the equation (4) can be presented like that:


— = ке6 - 0 + 0A cos Jt. (5)


Tr E Such denotations are here accepted: к = is a Semenov's parameter, 0A = TA——7r

is a dimensionless amplitude of the external harmonic perturbation, t = t/Tr is di­mensionless perturbation time, J is dimensionless frequency

J = JTr . (6)

The dimensionless frequency by its physical meaning is a ratio of two characteristic process times which are the relaxation time and period of the perturbation effect.

As is generally known [4], critical conditions are implied by not only equality of functions of heat supply and heat sink in tangency point but also their derivatives:

{ к * e6* = 0*,к*вв* = 1. (7) From (7) we find the critical parameters 0*, к* as follows:

{ 0 * = 1,к* = e-6* = e-1. (8)

Then the equation corresponds to the system that is reactive nearby a critical state:


= e6-1 - 0 + 0A cos Jt. (9)


Because of a littleness of 0 1 we will expand the exponent in a Taylor series up to second order. Thus nearby criticism the equation (9) is brought to the Riccati equation with a source of external harmonic perturbation:

d0 1

d~ = -(0 1)2 + 0a cos Jt. (10) dT 2

An analytical solution of the equation (10) does not exist because the right part contains the source written in a harmonic form. Therefore in the work the numeri­cal parametric calculation of the equation solution with a variation of temperature amplitude 0a and frequency J of the perturbation has been carried out (fig. 1).

At each set of parameters induction time ti was found, under which duration of the nonlinear critically strained system transition is implied from the stable initial state to the unsteady one that is explosive.

The results of dependence of the induction time Tj on amplitude 0A and frequency


In the first of them the drop-down dependence of the induction time on the perturbation amplitude is visible at the different fixed frequencies. This result looksбб

£. Kondratyev, A. Gerasymenko, К. Merkotan, V. Rusov ISSN 1024-588X. Вісник Львівського ун-ту. Серія фізична. 2011. Вип.46

obvious because for the nonlinear system of the exothermic reactive environment its response on the phase of warming-up is stronger than the response on the phase of the system cooling down.

On the contrary, in a fig. 3 there is an increase of the induction time with growth of frequency at the fixed external perturbation amplitude. It is consequent that wi­th growth of frequency the nonlinear system response on the external perturbation decreases by virtue of its thermal inertia (6).

Thus the induction time is a function of amplitude and frequency t = ті(6а,ш') of the perturbation. At the fixed transition time the dependence of amplitude on frequency is found out. A form of this dependence is brought in the fig. 4 for different transition times. It is shown that curves 1-3 approximate with high accuracy (~ 1 %) by a linear function.

In the fig. 5 we can see the cross-section of 3D plot by planes at constant induction time. The general property of heavy lines is the linear dependence between perturbati­on energy and frequency.

So, the energy of the external perturbation, which is enough for the system conversion from the initial steady state to the unsteady explosive one, is proportional to the perturbation frequency. This result calls to the well known quantum-mechanical relation in which also, as in our case, energy is in proportion to frequency in the first degree.




0 20 40 60


Fig. 1. Temperature-time curve of the strained system under the external perturbation with в a = 0, 2 J = 1,3

Fig. 2. Dependence of induction time on external perturbation temperature at fixed frequency. Values of frequencies 0,8, 1, 1,3 correspond to the curves 1, 2, 3

Fig. 3. Dependence of induction time on external perturbation frequency at fixed amplitude. Values of amplitudes 0,1, 0,15, 0,2 correspond to the curves 1, 2, 3

Fig. 5. The set of solutions Ті = Ті(6а,ш') of the equation (10) forms 3D plot. Heavy lines correspond to the curves of dependences of amplitude on frequency of external perturbation in cross-sectional planes at constant induction time

1. Жаботинский А. М. Колебания и бегущие волны в химических системах / А. М. Жаботинский, X. Огмер, Р. Филд ; пер. с англ. - М. : Мир, 1988. - 720 с.

2. Kondratyev Е. N. Frequency Response of Ignition for a Mg-particles under Peri­odic Thermal Perturbation / E. N. Kondratyev, A. V. Korobko, A. N. Zolotko // XXV International Symposium on Combustion. - USA, Irvine, 1994. - P. 1.

3. Кондратьев E. H. Частотный отклик критически обостренной системы на внешнее гармоническое воздействие / Е. Н. Кондратьев, А. М. Герасименко, К. К. Меркотан // XXIV научная конференция стран СНГ "Дисперсные си­стемы". - Одесса, 2010.

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