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Microscopic theory of combustion of solid fuel
Volume 148, number 5
PHYSICS LE’ITERS A
20 August 1990
Microscopic theory of combustion of solid fuel S.O. Gladkov Institute of ChemicalPhysics. USSR Academy ofSciences, Kosygin Street 4, Moscow, USSR Received 31 January 1990; revisedmanuscript received 16 March 1990; accepted for publication 3 May 1990 Communicated by D.D. Hoim
The concept of microscopic “preparation of the fuel for combustion” is introduced, which makes it possible to describe the power dependence ofthe combustion time upon the temperatureofthe heatingsurface. The result obtained describes numerous experimentaldata which were earlier considered within the framework of a phenomenological approach.
The numerous experimental data on the determination of the delay time of the process of cornbustion of solid fuel (see refs. [1—10])suggest the
[11—13]aimed at explanation ofthe dependence (1) were based upon this exponential law since an additive of the type Q0exp ( —AfT) had been intro-
decrease of this time (let us call it r.,~)with the ternperature increase of the metal heating plate (see fig. 1). The authors of the above papers propose the following approximation for ; ( T8),
duced into the thermal conductivity equation. This approach enables one to describe the evolution ofthe fuel temperature at short times over the interval
; ( T5) = r0 exp (4/ 7”~)
(1)
,
where 4 is the characteristic energy of the potential barrier, and t~ the delay time at T~~ 4. It is interesting to note that the theoretical studies
TS
(2) If t~tis larger than r we can introduce an activation summand of the type Q~exp( AfT5) in the righthand side of the equation ofheat conductivity. Here —
r is the quantum relaxation time which depends on the phonon mechanism in the case of dielectric materials [14]. of experimental data presented in refs. [2—6]showed that often the temperature dependenceanalysis of the Themost computer and surface scrupulous delay time ;( T8)simulation on the heating is not an exponential function but the power function A
____________
;=~,
\\ \
‘N
~
2
“N
____________
Fig. 1. (1) Heating surface igniting the fuel, (2) fuel, T1 is the temperatureofthe contacting surface, 0375-9601/90/S 03.50 © 1990
—
fi=2—4.
(3)
when the determination of the delay time will be possible within one millisecond precision) the above Thus analysis questioned (at least until the moment experimental data and put forward a new theoretical description in the form of relationship (3). In order to consider eq. (3) theoreticallywe should propose a certain hypothesis which does not contradict the general laws of physics and major postulates of the theory of combustion. Let us begin with the
Elsevier Science Publishers B.V. (North-Holland)
253
Volume 148, number 5
PHYSICS LETFERS A
20 August 1990
fact that according to its physical meaning ; is the time of preparation of the region ô for combustion or in other words ; is the time of heating. Since combustion is a chemical reaction resulting in the release ofa great amount of heat it is evident that the
ginning of the volume combustion. Such variation of the energy of microelements of the fuel is given in fig. 2. The gist of this diagram is the following. If at the initial moment the particle occupies state 1 haying obtained energy *w~where co~=c5k is the fre-
beginning of combustion occurs only when this reaction is developed not only on the surface ofthe fuel but in a certain volume close to the surface. The latter means that the preparation of a microscopic ensemble of chemically active atoms or molecules chaotically distributed over the fuel and capable to react due to the input of heat from an external source would occur only in the case when they (the active elements) have information concerning the beginning ofheating ofthe surface. Let us explain this idea. Let us assume that the contact ofthe heating plate with the fuel is immediate. In this case the two-dimensional phonon excitation immediately attains the temperature T5. In its turn the interaction of bulk phonons with the surface results in the fact that the bulk phonons attain the temperature T~in a certain microscopic time interval Tph. At the next stage the bulk phonons having an increased energy would prepare chemically active atoms for reaction by exciting the latter to the states with high energy. In our consideration this is a necessary condition for the be-
quency ofthe phonon, h is the Planck constant, c~ the velocity of sound, k the wave vector of the phonon, it will shift to a state I’ with probability W11 This is followed by the heat transition to the state 2 above the potential barrier with probability Q1. At the next stage the particle at energy level 2 again absorbs a phonon and “jumps” to the state 2’. The probability of this transition is W22. Further, with probability Q2 it attains state 3. This process will continue till the activation energy of these particles is enough to initiate the chemical reaction with another active element of fuel. This process is relatively slow. The higher probability of preparation for reaction which we shall now discuss involves many-phonon absorption. It has the following mechanism. As is shown in fig. 3, the atom should absorb n phonons in order to attain the state with energy e’. This means that .
N
e’ =
~ ~lco~.
(4)
I
The activation energy is about l0~K and the Debye
3’
,—---‘
/1
\\
I
I
/ / 7/
3
/ 1’
/
2
Fig. 2. Schematicsofthe gradual preparationofthe fuel for combustion. The dashed arrow indicates the thermal activation ofthe active element in the neighbouringpotential well and the solid arrow corresponds to the probability of atom transition to the state with higher energydue to interaction with phonons.
254
Volume 148, numberS
PHYSICS LETTERS A
/
/ ~‘tl
\\
______
regime characterized by a short time of the phonon
/
//
20 August 1990
relaxation compared with the time of transition of atoms to the state e’ we may put in (5) When we are concerned with the high temperature fkJk--~= (7)
/
In this case the collision integral is considerably simplified and we get
_______
Fig. 3. Approximation of the one potential well for a two-level system.
L (ç~)=
2xT~~ ~ {k}
)o(
(4
temperature °D (we recall that
0D
= hc
5f a, where a is a certain average interatomic distance which can be easily measured by X-ray technique) is about 40—200 K for conventional types of fuels. This implies that eq. (4) four is valid only for a relaxation mechanism involving phonons. In order to describe the situation in mathematical terms we shall make use of the method given in ref.
COkE ~(Op~
‘
x A(~n=1 ~ k~
4
C’ — C—Il
~
Consequently, the transition probability of atoms per unit time is 2 k~) kv({k})1 Wft’ = 2,tT {*} ~ (‘)k~ r3~k4 \ 1
A~ fl
—
—
n=I Ilwk 5).
[15]. We shall introduce a certain distribution function ofactive atoms with respect to energy which will be denoted as q. Then for the collision integral describing the transition of atoms from state a to the state a’ involving four phonons we find L (~) =
2x -~-
~
{k}
~ ~v({k}) ~ [(1 ~
x (1 +J~)(1 +Jj4 )q. (1 —
+f~1)(1 +f~)
\n=1
—
~
(5)
*C0k5), i
wheref* is the phonon distribution function. Thus in case of thermodynamic equilibrium ~ = [exp(hwk/T5)— l]’. We can write for the scattering amplitude —
In order to calculate expression (9) we have to analyse the energy and momentum conservation laws simultaneously. This analysis is too sophisticated and unfortunately cannot be made in the fuel because of
sums of k (as is easily seen from the conservation law analysis and from the expression under the integral (see below)) results in the wave vectors k1 = k2 = k3 = ~ 2a0/fzc5, where a0is the initial en-
—
5=1
(9)
X
the complexity of the appearing algebraic expressions. Nevertheless, as the largest contribution in the
~( 1 —q’~)f*tfi2fa3fai
xA( ~ kn)5(a_a’
(8)
fl=I (Ok5).
Il2 2 (ek,)(ekkIWInWk3W~ 2)(ek3)(ek4) 4 p~T/
‘
(6) where p~is the density of the fuel, V the volume of the sample, e the polarization vector ofthe phonons, ~ a dimensionless constant accounting. for the interaction of atoms with the crystal lattice.
ergy of the active atom levels, expression (9) can be calculated approximately. In fact, after the transition from the summation of k,, k2, k3, k4 and a’ to integration according to the rules V 3k, ~ (.•.)= (...)d f~° (...)exp(—a’/a 0)da’ Coo ~ ,~ exp(—a’/a0)da’
(2x)31
we easily obtain the following:
255
Volume 148, numberS
PHYSICS LETTERS A
hp~c~a
0
$ _
X
x
J _
kf dk1
0
01
k~dk2
0
J
We see that 1/; behaves as follows:
N T~(~0D)2exp(—a/aO)
x
20 August 1990
f
1 ~ N(Eo\ ;hp~c~V x T~ at T5 >> fo,
V
k~dk3
0
XT~/a0, at O2m.,x
sin 02 dO2
j” d ~
(14)
T5’<
2~
sin 03 dO3 0
Thus we see that the ignition delay time defined above depends on the time of preparation of active atoms for chemical reactions and is described by expression (13), i.e.
0
Xexp{ (hc5/ao) [k1 +k2 +k3 +Ø(02,03,i~qi)]}, —
(10)
Cs,
Z(pf,
~r~4
where
(a0+T5),
(15)
1
Ø(O2,03,)=(k~+k~+k~+2k,k2cos02 2, +2k,k3cos03+2k2k3cos~)”
where
cos w=cos 02 cos 03—sin 02 sin 03 cos ~
3. Let, for example, pf=2 and v0=cVfN~lO_22 cm g/cm3, 5=6x l0~cm/s, O~=40K, T5=600 K, a~=600 K, ~ =10 18 then the ignition delay time ; ~ 0.01 s, which is in accordance with experiment. As was already mentioned the obtained function describes numerous experimental data on the vanation of the time ; with the temperature ofthe heating surface. Moreover, the proposed expression predicts an additional dependence of ; on several microscopic and macroscopic parameters ofthe fuel. The latter and the intensive dependence of; on the density (;~p~),in particular, must be proven
z(pf, c 5, ü)=
9’3.
~
To calculate expression (10) we note that the maxima of the integrands lie approximately at k1 = k2= k3 = k~,, 20/hc5. In this case the integrals of k,,2,3 are calculated with the saddle point method and as a result we obtain 2 N ~ (11) exp( /~) T~ (COD) 8lE9”2 *p~c~ao
(
—
i~~)
—
where
J
S2(~ç)
2s
J=
d(b.~,)
0
$
SI(02,h.~)
J
sin 02 dO
2
0
sin 03 dO3
N is the number of atoms in the fuel sample. Since the integral J depends only on the angular variables we may assume that J—~1. Then, after the introduction of the renormalization constant = J~f81 ~9/2 and averaging over energies ‘in accordance with the formula ,fJ’V~exp(—a/T5)da
Tc
1
256
— —
j
experimentally. (12)
—
we have
(16) ~
0
x exp [ ~ (Ox, 03~A9,)] ,
<~> = f exp( —afT5) da
h,4c~vo~ ~ \
‘
2~ ‘~, ~9N h( ~“T~0 0+T5)p~c~t.~Il~5) -~.
(13)
References [1] C.R. Rogers and N.P. Suh, ALAA J. (1970) 1501. [2] F.Yu. Karabanov, G.T. Afanas’ev and V.!. Bobolev, in: Chemical physics of processes ofcombustion and explosion (Chernogolovka, 1977) p. 6 [in Russian]. [3] LG. Strakovskii, P.!. Ulyakov and E.I. Frolov, in: Chemical physics of processes of combustion and explosion (Cheniogolovka, 1977) p. 8 [in Russian]. [4) V.S. Berman and V.M. Shevtsova, Fiz. Goreniya Vzryva 16 (1980) 20. [5)I.G. Assovskii and 0.!. Lcipunskii, Fiz. Goreniya Vzryva 16(1980)3. [6] I.S. Lyubchenko, V.V. Matveev and G.N. Marchenko, Dokl. Akad.NaukSSSR26O(1981) 664. [7]I.G.DikiiandA.B.Zurer,Fiz.GoreniyaVzryval8(1982) 16.
Volume 148, number S
PHYSICS LETTERS A
[8] V.E. Aleksandrov, A.V. Dolgolaptev, V.B. lotTe, V.M. Koval’chuk, B.V. Levin and A.P. Obraztsov, Fiz. Goreniya Vzryva 19 (1983) 17. [9] A.M. Baranovskii, Fiz. Goreniya Vzryva 19 (1983) 95. [10) 1.0. Assovskii, Z.G. Zakirov and 0.!. Liepunskii, Fiz. GoreniyaVzryva 19(1983)41. [II] L.G. Starkovskii, Fiz. Goreniya Vzryva 21(1985) 41.
20 August 1990
[12] V.E. Aleksandrov, A.V. Dolgolaptev, V.B. lotTe and B.V. Levin, Fiz. Goreniya Vzryva 21(1985) 58. [131 V.!. Lyubchenko, Z.A. Yudina and G.N. Marchenko, Zh. Fiz. Khim. 23 (1987) 953. [141 S.O. Gladkov, Physica A 159 (1989) 420. [15) 5.0. Gladkov, Phys. Rep. 182 (1989) 211.
257
PHYSICS LE’ITERS A
20 August 1990
Microscopic theory of combustion of solid fuel S.O. Gladkov Institute of ChemicalPhysics. USSR Academy ofSciences, Kosygin Street 4, Moscow, USSR Received 31 January 1990; revisedmanuscript received 16 March 1990; accepted for publication 3 May 1990 Communicated by D.D. Hoim
The concept of microscopic “preparation of the fuel for combustion” is introduced, which makes it possible to describe the power dependence ofthe combustion time upon the temperatureofthe heatingsurface. The result obtained describes numerous experimentaldata which were earlier considered within the framework of a phenomenological approach.
The numerous experimental data on the determination of the delay time of the process of cornbustion of solid fuel (see refs. [1—10])suggest the
[11—13]aimed at explanation ofthe dependence (1) were based upon this exponential law since an additive of the type Q0exp ( —AfT) had been intro-
decrease of this time (let us call it r.,~)with the ternperature increase of the metal heating plate (see fig. 1). The authors of the above papers propose the following approximation for ; ( T8),
duced into the thermal conductivity equation. This approach enables one to describe the evolution ofthe fuel temperature at short times over the interval
; ( T5) = r0 exp (4/ 7”~)
(1)
,
where 4 is the characteristic energy of the potential barrier, and t~ the delay time at T~~ 4. It is interesting to note that the theoretical studies
TS
(2) If t~tis larger than r we can introduce an activation summand of the type Q~exp( AfT5) in the righthand side of the equation ofheat conductivity. Here —
r is the quantum relaxation time which depends on the phonon mechanism in the case of dielectric materials [14]. of experimental data presented in refs. [2—6]showed that often the temperature dependenceanalysis of the Themost computer and surface scrupulous delay time ;( T8)simulation on the heating is not an exponential function but the power function A
____________
;=~,
\\ \
‘N
~
2
“N
____________
Fig. 1. (1) Heating surface igniting the fuel, (2) fuel, T1 is the temperatureofthe contacting surface, 0375-9601/90/S 03.50 © 1990
—
fi=2—4.
(3)
when the determination of the delay time will be possible within one millisecond precision) the above Thus analysis questioned (at least until the moment experimental data and put forward a new theoretical description in the form of relationship (3). In order to consider eq. (3) theoreticallywe should propose a certain hypothesis which does not contradict the general laws of physics and major postulates of the theory of combustion. Let us begin with the
Elsevier Science Publishers B.V. (North-Holland)
253
Volume 148, number 5
PHYSICS LETFERS A
20 August 1990
fact that according to its physical meaning ; is the time of preparation of the region ô for combustion or in other words ; is the time of heating. Since combustion is a chemical reaction resulting in the release ofa great amount of heat it is evident that the
ginning of the volume combustion. Such variation of the energy of microelements of the fuel is given in fig. 2. The gist of this diagram is the following. If at the initial moment the particle occupies state 1 haying obtained energy *w~where co~=c5k is the fre-
beginning of combustion occurs only when this reaction is developed not only on the surface ofthe fuel but in a certain volume close to the surface. The latter means that the preparation of a microscopic ensemble of chemically active atoms or molecules chaotically distributed over the fuel and capable to react due to the input of heat from an external source would occur only in the case when they (the active elements) have information concerning the beginning ofheating ofthe surface. Let us explain this idea. Let us assume that the contact ofthe heating plate with the fuel is immediate. In this case the two-dimensional phonon excitation immediately attains the temperature T5. In its turn the interaction of bulk phonons with the surface results in the fact that the bulk phonons attain the temperature T~in a certain microscopic time interval Tph. At the next stage the bulk phonons having an increased energy would prepare chemically active atoms for reaction by exciting the latter to the states with high energy. In our consideration this is a necessary condition for the be-
quency ofthe phonon, h is the Planck constant, c~ the velocity of sound, k the wave vector of the phonon, it will shift to a state I’ with probability W11 This is followed by the heat transition to the state 2 above the potential barrier with probability Q1. At the next stage the particle at energy level 2 again absorbs a phonon and “jumps” to the state 2’. The probability of this transition is W22. Further, with probability Q2 it attains state 3. This process will continue till the activation energy of these particles is enough to initiate the chemical reaction with another active element of fuel. This process is relatively slow. The higher probability of preparation for reaction which we shall now discuss involves many-phonon absorption. It has the following mechanism. As is shown in fig. 3, the atom should absorb n phonons in order to attain the state with energy e’. This means that .
N
e’ =
~ ~lco~.
(4)
I
The activation energy is about l0~K and the Debye
3’
,—---‘
/1
\\
I
I
/ / 7/
3
/ 1’
/
2
Fig. 2. Schematicsofthe gradual preparationofthe fuel for combustion. The dashed arrow indicates the thermal activation ofthe active element in the neighbouringpotential well and the solid arrow corresponds to the probability of atom transition to the state with higher energydue to interaction with phonons.
254
Volume 148, numberS
PHYSICS LETTERS A
/
/ ~‘tl
\\
______
regime characterized by a short time of the phonon
/
//
20 August 1990
relaxation compared with the time of transition of atoms to the state e’ we may put in (5) When we are concerned with the high temperature fkJk--~= (7)
/
In this case the collision integral is considerably simplified and we get
_______
Fig. 3. Approximation of the one potential well for a two-level system.
L (ç~)=
2xT~~ ~ {k}
)o(
(4
temperature °D (we recall that
0D
= hc
5f a, where a is a certain average interatomic distance which can be easily measured by X-ray technique) is about 40—200 K for conventional types of fuels. This implies that eq. (4) four is valid only for a relaxation mechanism involving phonons. In order to describe the situation in mathematical terms we shall make use of the method given in ref.
COkE ~(Op~
‘
x A(~n=1 ~ k~
4
C’ — C—Il
~
Consequently, the transition probability of atoms per unit time is 2 k~) kv({k})1 Wft’ = 2,tT {*} ~ (‘)k~ r3~k4 \ 1
A~ fl
—
—
n=I Ilwk 5).
[15]. We shall introduce a certain distribution function ofactive atoms with respect to energy which will be denoted as q. Then for the collision integral describing the transition of atoms from state a to the state a’ involving four phonons we find L (~) =
2x -~-
~
{k}
~ ~v({k}) ~ [(1 ~
x (1 +J~)(1 +Jj4 )q. (1 —
+f~1)(1 +f~)
\n=1
—
~
(5)
*C0k5), i
wheref* is the phonon distribution function. Thus in case of thermodynamic equilibrium ~ = [exp(hwk/T5)— l]’. We can write for the scattering amplitude —
In order to calculate expression (9) we have to analyse the energy and momentum conservation laws simultaneously. This analysis is too sophisticated and unfortunately cannot be made in the fuel because of
sums of k (as is easily seen from the conservation law analysis and from the expression under the integral (see below)) results in the wave vectors k1 = k2 = k3 = ~ 2a0/fzc5, where a0is the initial en-
—
5=1
(9)
X
the complexity of the appearing algebraic expressions. Nevertheless, as the largest contribution in the
~( 1 —q’~)f*tfi2fa3fai
xA( ~ kn)5(a_a’
(8)
fl=I (Ok5).
Il2 2 (ek,)(ekkIWInWk3W~ 2)(ek3)(ek4) 4 p~T/
‘
(6) where p~is the density of the fuel, V the volume of the sample, e the polarization vector ofthe phonons, ~ a dimensionless constant accounting. for the interaction of atoms with the crystal lattice.
ergy of the active atom levels, expression (9) can be calculated approximately. In fact, after the transition from the summation of k,, k2, k3, k4 and a’ to integration according to the rules V 3k, ~ (.•.)= (...)d f~° (...)exp(—a’/a 0)da’ Coo ~ ,~ exp(—a’/a0)da’
(2x)31
we easily obtain the following:
255
Volume 148, numberS
PHYSICS LETTERS A
hp~c~a
0
$ _
X
x
J _
kf dk1
0
01
k~dk2
0
J
We see that 1/; behaves as follows:
N T~(~0D)2exp(—a/aO)
x
20 August 1990
f
1 ~ N(Eo\ ;hp~c~V x T~ at T5 >> fo,
V
k~dk3
0
XT~/a0, at O2m.,x
sin 02 dO2
j” d ~
(14)
T5’<
2~
sin 03 dO3 0
Thus we see that the ignition delay time defined above depends on the time of preparation of active atoms for chemical reactions and is described by expression (13), i.e.
0
Xexp{ (hc5/ao) [k1 +k2 +k3 +Ø(02,03,i~qi)]}, —
(10)
Cs,
Z(pf,
~r~4
where
(a0+T5),
(15)
1
Ø(O2,03,)=(k~+k~+k~+2k,k2cos02 2, +2k,k3cos03+2k2k3cos~)”
where
cos w=cos 02 cos 03—sin 02 sin 03 cos ~
3. Let, for example, pf=2 and v0=cVfN~lO_22 cm g/cm3, 5=6x l0~cm/s, O~=40K, T5=600 K, a~=600 K, ~ =10 18 then the ignition delay time ; ~ 0.01 s, which is in accordance with experiment. As was already mentioned the obtained function describes numerous experimental data on the vanation of the time ; with the temperature ofthe heating surface. Moreover, the proposed expression predicts an additional dependence of ; on several microscopic and macroscopic parameters ofthe fuel. The latter and the intensive dependence of; on the density (;~p~),in particular, must be proven
z(pf, c 5, ü)=
9’3.
~
To calculate expression (10) we note that the maxima of the integrands lie approximately at k1 = k2= k3 = k~,, 20/hc5. In this case the integrals of k,,2,3 are calculated with the saddle point method and as a result we obtain 2 N ~ (11) exp( /~) T~ (COD) 8lE9”2 *p~c~ao
(
—
i~~)
—
where
J
S2(~ç)
2s
J=
d(b.~,)
0
$
SI(02,h.~)
J
sin 02 dO
2
0
sin 03 dO3
N is the number of atoms in the fuel sample. Since the integral J depends only on the angular variables we may assume that J—~1. Then, after the introduction of the renormalization constant = J~f81 ~9/2 and averaging over energies ‘in accordance with the formula ,fJ’V~exp(—a/T5)da
Tc
1
256
— —
j
experimentally. (12)
—
we have
(16) ~
0
x exp [ ~ (Ox, 03~A9,)] ,
<~> = f exp( —afT5) da
h,4c~vo~ ~ \
‘
2~ ‘~, ~9N h( ~“T~0 0+T5)p~c~t.~Il~5) -~.
(13)
References [1] C.R. Rogers and N.P. Suh, ALAA J. (1970) 1501. [2] F.Yu. Karabanov, G.T. Afanas’ev and V.!. Bobolev, in: Chemical physics of processes ofcombustion and explosion (Chernogolovka, 1977) p. 6 [in Russian]. [3] LG. Strakovskii, P.!. Ulyakov and E.I. Frolov, in: Chemical physics of processes of combustion and explosion (Cheniogolovka, 1977) p. 8 [in Russian]. [4) V.S. Berman and V.M. Shevtsova, Fiz. Goreniya Vzryva 16 (1980) 20. [5)I.G. Assovskii and 0.!. Lcipunskii, Fiz. Goreniya Vzryva 16(1980)3. [6] I.S. Lyubchenko, V.V. Matveev and G.N. Marchenko, Dokl. Akad.NaukSSSR26O(1981) 664. [7]I.G.DikiiandA.B.Zurer,Fiz.GoreniyaVzryval8(1982) 16.
Volume 148, number S
PHYSICS LETTERS A
[8] V.E. Aleksandrov, A.V. Dolgolaptev, V.B. lotTe, V.M. Koval’chuk, B.V. Levin and A.P. Obraztsov, Fiz. Goreniya Vzryva 19 (1983) 17. [9] A.M. Baranovskii, Fiz. Goreniya Vzryva 19 (1983) 95. [10) 1.0. Assovskii, Z.G. Zakirov and 0.!. Liepunskii, Fiz. GoreniyaVzryva 19(1983)41. [II] L.G. Starkovskii, Fiz. Goreniya Vzryva 21(1985) 41.
20 August 1990
[12] V.E. Aleksandrov, A.V. Dolgolaptev, V.B. lotTe and B.V. Levin, Fiz. Goreniya Vzryva 21(1985) 58. [131 V.!. Lyubchenko, Z.A. Yudina and G.N. Marchenko, Zh. Fiz. Khim. 23 (1987) 953. [141 S.O. Gladkov, Physica A 159 (1989) 420. [15) 5.0. Gladkov, Phys. Rep. 182 (1989) 211.
257