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On the extinction of diffusion flames
ON
THE EXTINCTION OF ~IFFUSTON FLAMES bl. .I. MANTON
I~epartmrnt
of Mathematic\.
Monash
University.
Melbourne.
.Au\trali:l
I. rN’rR~DU~TI(~N
THE ~~LI~.~KE
of ;I Iaminar counterflow diffusion flame is studied experimentally by Twji and Yamaoka [ I. 21. By varying the mass fluxes of the reactants, they find that the extinction of the flame in the upstream stagnation region of a porous cylinder is caused by two different mechanisms. First, thermal quenching of the flame occurs as the mass flux of fuel from the cylinder decreases and the flame approaches the cylinder. The second extinction mechanism is due to “chemical limitations on the combustion rate in the flume zone” [?I. As the advection rate CL of air past the cylinder increases, the fuel rn;i>s flus must be increased to maintain a stable diffusion flame. For U-, greater than ;I certain value however a diffusion flame cannot be obtained. Clearly. this second mechanism is a Damkohler number effect; that is. extinction occurs because the air and fuel reactants are swept away by the flow before chemical reaction can take place. The purpose of the present note is to demonstrate that these extinction mechanisms can he explained by a simple analysis if it is assumed that extinction occurs when the flame temperature falls below some critical value, T,,. Hence, the analysis is extended to consider the problem of flame extinction from the upstream face of a burning droplet of fuel in :I free stream of air. In this case. the only adjustable parameters are the Damkohter number E ’ and the ratio T, of the free-stream air temperature to the boiling point of the fuel. It is found that, for large activation energy reactions, the nonuniqueness of the flame temperature implies that extinction should occur for flame tempertrtures greater than T,.,. The analysis of diffusion fames has developed considerably and the relevant literature is reviewed by Williamsr~]. The effect of n finite Damkohler number is usually investigated either by numerically integrating the non-linear equations or by means of a singular perturbation expansion in the region of the diffusion flame [4, -51.However, the inner representation of the perturbation expansion also is given by a non-linear ordinary differential equation which must be solved numerically. In the present work. the dependence of the flame temperature on Damkohler number is found by approximating the reaction rate term in the thern~[~dyn~~mic equations by a Dirac delta function centred in the diffusion flame region. In this manner. the phenomenon is described by a linear system.
.? II(~li.lI’lOhS
AIOI‘IO'L \UI)
0:
‘I l1t:KMOD~‘K~M1(‘S
We consider stagnation
the heha~~iour of ;I chemical reaction in the neighbourhood of the region of the How p:r\t ;I bluff body. The Gtuation i\ described most WC-
cinctly by the steady state Sh\A-Zeldovich the upstream side of the st;lgn;lticw point.
system the no-slip
[S]. Because
the flame appears
on
h:,und;rrlz condition at the \urf;lce of the body does not dominate the tiou ne;Ir the flame. Hence. the fluid is assumed to hc inviscid, and the equ;ltion\ ft,r ;he conservation of momentum and mass are
where U* is the velocity :;nd p” is the pressure at the point x* in space: p is the fluid density. The combustion proces\ i> :~pproximatrcJ by the simple one-step reaction I’, w,
-t v-w
-
7’: w:.
(9.2)
where V, is the stoichiometric coefficient and W, is the molecular weight of the ith species. We let i = I correspond ?o the oxygen and i = 2 correspond to the fuel. The conservation of mass for the i th species I$ described by T
(pv * Y,
pD0 Y, ) =
(2.3)
N’,.
Y, is the mass fraction of species i; II is the common mass diffusion coefficient the species present; ~9, i\ the rate of production of the i th species. Following Fendell 14). we assume that the density p and the diffusion coefficient D are constants. Further. the J>ewis number A /pDC,,. where A is the thermal conductivity and C,, is the overall specific heat of the fluid. is taken to be equal to unity. Thus. the conservation of energy is erpre\scti by the equation where
of
all
where 7’* is the temperature and 11, i\ the heat of formation of the i th species. The rate of production of the i th species by the chemica! reaction is M', =
S,l',W,W".
(2.5)
where si =
1 for i = I. 1; 1 for i -3;
and w* = BT*np+f+>
’ (Y,/W,)“’
( Yz/Wz)”
Here, B and (Y are constants. E* is the activation energy universal gas constant. For a strictly one-step reaction,
e E*‘Yl’r*.
of the reaction, the exponents
and R,, is the w! equal
the
stoichiometric coefficients I/i. However, any physical process is invariably a multi-step reaction, and so we must assume that the exponents I_L,are determined empirically. Equations (2.1)~(2.5) form a closed system which describes the fluid dynamics and thermodynamics of the problem. To simplify the analysis, we now normalize the equations. Dimensionless variables are introduced such that
where L is a representative length scale of the bluff body. I;, is the temperature of the surface of the body, and U, characterizes the velocity of the flow far upstream of the body. Putting (2.6) into (2.1)-(2.3, the equations of motion become
where
(2.7)
b = - c s, W;hi/C,,Tu W. I I
The important
dimensionless
parameters
are the Peclet number
G = LU,lD.
which represents the ratio of the advection rate of material and heat. and
(2.8)
rate by the flow to the molecular diffusion
which is the ratio of the rate that material undergoes chemical reaction to the rate of molecular diffusion of material. 3. FLAME
UPSTREAM
OF POROUS
(‘YI.INDER
We consider first a model of the flame configuration studied by Tsuji and Yamaoka[l, 21. They find that a diffusion flame is established on the upstream face of a porous cylinder which is situated in a free-stream of air and from which fuel is ejected under pressure. From the model, the flame temperature is calculated as a function of the mass flux of fuel and the Damkohler number, and compared with the flame extinction criteria found by Tsuji and Yamaoka. Because the stand-off distance of the flame is much less than the radius of curvature
M. .I. MANTON
510
of the bluff body, a porous cylinder, dimensional flow approaching the plane rectangular co-ordinates. The appropriate tions of the system (2.7) is
the geometry is approximated by a twoboundary x = constant. where (x, y, z) are solution of the momentum and mass equa-
v = (u, v, M’) = (
x. y. 0)
and
(3.1) p = - j(x’+
11’)+ constant.
Equation (3.1) corresponds to a flow approaching the boundary x = - x, from the positive x-direction. Thus, u = x, at the surface of the body. This implies that the net mass flux of fuel is equal to PU~X,~, which is fixed for a given experiment. To solve the thermodynamic equations of (2.7) the boundary conditions on the dependent variables must be specified. At the surface of the body there is no flux of oxygen and the fuel flux is given: that is,
GUY,-iiY,;Qatx=-x 8X
$9
(3.2)
and GUY? Far upstream Hence,
of the body the concentration
Y,+ The temperature
$$ =Gu,
at x = - x,.
of oxygen
is known
Y,andY,+Qasx-+x.
of the body and the free-stream
temperature
and there
is no fuel.
(3.3) also are fixed such that
T = 1 at x = - x,, and
(3.4)
T -+ T, as x -+ SC. It is seen from (3.1) that the boundary conditions (3.2)-(3.4) are independent of _V and z. Hence, it is found that the thermodynamic equations of (2.7) reduce to the ordinary differential equations
(3.5)
$-
Gus=
~ Fwb.
Now the Peclet number G is invariably very much greater than unity when a diffusion flame is present. Therefore, the solution domain (-x5, m) may be divided into an inner region where x is of order G--“*, and an outer region where x is away from the stagnation point. In the outer region, equations (3.5) reduce to first order equations: that
On the extinction
of diffusion
flames
511
is, the effects of diffusion are neglected. However, our interest lies essentially in the inner region where the flame occurs. Thus, we introduce a stretched independent variable ^rl= (G/2)“‘x,
(3.6)
so that, from (3.1), the system (3.2)-(3.5) becomes
(3.7)
with
(3.8) T = 1 at q = - q,,
Y, -+ Y*, Yz -+ 0, T -+ T- as 7f + =; where E -’ = F/G is the Damkohler number, and r), = (G/2)“% is the normalized mass flux of fuel. The parameter 7, is formally of order unity. We seek the solution of the system (3.7)-(3.8) corresponding to a diffusion flame; that is, the case of large Damkohler number only is considered. It is seen from (3.7)-(3.8) that the functions p(T)) =
Y,- (aJa,)Yt (3.9)
and /37(q) = T + (bla,)Y, satisfy linear systems [6f. In particular.
we find that
P(~)=il-~~f(77~)-(l+~l)erf(~))/{1+f(~,)}
(3.10)
B~(~)~~~n+(T~+az)erf(~,)i(T.,+a~-~~r)erf(~)}f(i+erf(~~)},
(3.11)
an d
where
The constant & depends upon the value of Y, at the boundary q = - 8,. Thus the complete solution for Y,. Y? and T is given from (3.9)-(3.11) if the non-linear system (3.7)~(3.8) can be solved for Y,.
M. .I. Mr\N’TO~
512
For a diffusion flame, the chemical reaction is confined essentially to ;L thin flame region, and hence the non-linear term w is negligible outside this region. Thus we approximate this term such that
where S(E) is to be determined centred at the position ?f where P(T,) = 0. Equation (3.10) shows
Putting
and C?(V) is the Dirac delta function. The flame iy the reactants are in stoichiometric proportions. i.e. that Tf is given by
(3.12) into (3.7)-(3.8), Y,(v)
={Y,-
we find that the mass
S[I -erf(77f)l}(f(r),)4erf(~)}/{l
+ S{ erf(77) -erf(qf)l where H(s) is the Heaviside (3.9) and (3.14). The constant T(- qs) = 1; hence PO= I +a$1
fraction
of oxygen
is
+f(77,)1 (3.14)
H(rl - qfI/),
unit step function. Therefore, Y, and T are given from p,, is determined from the boundary condition (3.8) that
-II
-erf(rlr)l(S/Y,)}g(77,)/(1
(3. IF1
+f(77,)1.
where
The values of the dependent equations (3.9)-(3.14) to be yl(q)={(l
Y2(q,)=
variables
+,,)Y,
at the centre
-a,[1
a,{(1 -ta,)Y.
of the flame are calculated
-t-f(~~)lSll(1 +
- a,[1 + f(7j\)lS)/Y~(l
from
(YIY. +QI).
(3.16)
~(~~)={B,,+(T,+~~)erf(~,)+(T.+cu,-P11)erf(?7f)} /{l+erf(rl,)}-cu~{(l+ai)Y,
~LY1[l+f(rl\)lS}IY,(l+cuI)~.
In the limit E -+ 0, equation (3..7) implies that w is identically zero. Hence. the mass fraction of both reactants goes to zero at the flame centre. Now (3.16) shows that
To account for a finite Damkohler pansion of the form
number,
s = s,,-
we assume
that S has an asymptotic
(3.18)
E”S, t O(E” ).
where the index p and the constant S, are to be determined. Y,, YZ and T at the flame centre have the representations
ex-
Thus,
from (3.15)-(3.18).
31.4
Y,(,,) = Y7h(7J> )E” + O(E’)). Y,(Tf) = a,h(q,)EP
+ O(EP),
(3.19)
T,-T(~i)~=T,~(~~)--~Zh~~,)~‘J{1--cr,~~~,)/[1+erf(77,)1~+~(~P). where
It is usual to define the Dirac functions
where
171. We therefore
the flame thickness
delta function
I(E) h:ts the asymptotic
and the index y is to be determined. the production
Because
Equations
term w at the flame centre
these two representations
A second relation
as the limit of a sequence
of regular
take
between
representation
(1.7). (3.17) and (3.17)-(3.20)
can be written
must he identical.
show that
as
we ha\,e
the indices p and q is found from the conservation
of
species equation (3.7). Within the fl:tme region. the rate of diffusion of a reactant into the region. which is cjf order E” ?“.just balances the rate .)f consumption of the reactant hy the chemical reactic,n. which is of order E”‘~~“.“’ ‘. Therefore.
p -2q Equations
(3.3 1) and (3.23)
-(p,+p,)p
- I.
(3.23)
may be solved to y,ieltl
p = q = l/(1 + cc, + cl-).
(3.24)
Thus. the flame thickness is of order c “‘I “‘l’u~’ . and this reduces to the result obtained by a formal singular perturbation expansion when p, = p2 = 1 [4].
M. .1. MANTON
51‘4
For a given fuel mass flux v< and inverse Damkohler number E, the flame temperature T, can now be calculated from equations (3.13), (3.19). (3.22) and (3.24). The free-stream temperature T, and oxygen mass fraction Y, are assumed to be fixed. Two limiting cases are of interest: (i) E + 0 with vy fixed and (ii) q, + x with E tixed. Case (i) corresponds to the diffusion flame limit of infinite Damkohler number. Thus, (3.19) shows that the flame temperature T, equals 7’,,(~,) which decreases as T\ decreases. If we assume that flame extinction occurs when Tffalls below some critical value T,,, then it is clear that extinction takes place when ~~ is less than r),, where T,,,(T~,) = T,.,. As indicated by Tsuji and Yamaoka[l. 21, this case corresponds to thermal quenching of the flame as it approaches the surface of the bluff body. Thus, when the reaction rate completely dominates the rate of advection of the reactants by the flow (i.e. E -+ 0), a flame can be maintained only if the normalized mass flux of fuel is greater than Q,. It is seen from (3.13) that as q, approaches infinity, the distance of the flame from the stagnation point ^~lrapproaches a finite value q, such that erf(_rl.)=(I-cw,)/(l+cu,). Hence
the flame temperature
in case (ii) is given
by
T,= T,,,(x) - azh(xk", where
T,,,(x) = (T.+ aI+ az)l(l + a,> and h(x) ={[Cl
+ a,)Y:
‘I/efi”.,,]/[2~u,a/+“~T’,‘,e’lj]}“(’,’”~’.
Now Tf decreases as E increases from zero; that is, as the advection rate of the reactants increases relative to the reaction rate. Extinction occurs when E is greater than that value which corresponds to T,= T,,. Because the chemical reaction occurs at a finite rate, the reactants must remain in proximity for a finite time in order to maintain a diffusion flame. As the advection rate increases however, extinction takes place because the reactants are swept away by the flow before a significant reaction can be established. Thus the two mechanisms for flame extinction observed by Tsuji and Yamaoka [ I, 21 would seem to be explained, qualitatively at least, by assuming that extinction occurs when the flame temperature falls below some critical value. Thermodynamic data are not readily available to make possible a qualitative comparison with experiments. However Fendell[4] gives data relevant to the burning of acetone in air, and Fig. 1 shows the family of curves of constant T,as a function of 77%and E corresponding to this data. In particular, we take CYI= 0,1056; cyz = 5.712; Y, = 0.233 1;
E = 41.164; (Y =o:
T,= 0.8751: PI=
a, = 0.6882:
/-LZ= 1,
where the surface temperature Tois set equal to 594”R, the boiling point of acetone. The particular curve of Fig. 1 which corresponds to the extinction curve (i.e. to T,= T,) could be predicted from a stability analysis, or it could be found experimentally, The term w in (2.7) which accounts for the chemical reaction is particularly sensi-
On the extinction
Fig. I. Curves
XKI normalized
of diffusion
51.5
flames
of constant flame temperature T, as functions of inverse Damkohler number t fuel mass flux q\ for an acetone-air diffusion flame up-stream of a porous cylinder. Numbers refer to the values of T,.
tive to the exponential function involving the normalized activation energy reaction E. Hence, a simple extinction criterion might exist of the form
of the
T,, lE = constant. 4. FLAME
UPSTREAM
OF BURNING
DROPLET
for the extinction of the diffusion flame which can be established on the upstream face of a burning droplet of fuel moving relative to a stream of air. The mass flux of fuel is now controlled by the rate of evaporation at the surface of the droplet, and so it depends upon the free-stream temperature and the Damkohler number of the system. The body in the air flow (i.e. the droplet) is essentially spherical. Thus, provided that the stand-off distance of the flame is much less than the radius of the droplet, the geometry may be approximated by an axi-symmetric flow approaching the plane boundary x = constant, where (x, r, 0) are cylindrical polar coordinates. The appropriate stagnation flow solution of the momentum and mass conservation equations of (2.7) is It is of interest
to obtain
a criterion
v = (u,v. w) = (- x. ir. 0) and p = -4(x’+:
r’) + constant.
(4.1)
As in section 3, we take the surface of the body as the boundary x = - x,, which implies that the mass flux of fuel is pU,x,. The boundary conditions on the temperature T and the reactants Y, are given by (3.2), (3.3) and (3.4), together with the additional condition that the mass flux of fuel is equal to the rate of evaporation of fuel from the surface of the droplet. Hence [4],
c= dx where
GHu. at x = - .y0 -’
H = H*/C,,T,, and H* is the latent
heat of vaporization
(4.2) of the fuel. It is clear that
the equations (2.7) now reduce to the ordinary differential equations (.:.5). 1 heref~~r-r, the thermodynamics of the flame upstream of a burning droplet are governed hy the x) \tenl (3.7)-(3.8), together with the boundary condition. from (4.2).
The section fuel Q E. Thus, to
solution of the system (3.7)-(3.8) and (4.3) is identical to that obtained in 3, with an additional constraint imposed by (4.3) which fixes the mass flux of in terms of the free-stream temperature T, and the inverse Damkohler number from equations (3.9), (3.1 l), (3.14) and (3.17)-(3.19), it is seen that (4.3) reduce< T,=l+H{1+erf(~,)}/g(~~)-~~{l--(If~~,)h(~,)~’~f0(~~‘)}.
Substituting
(4.4) into (3.19), we find the flame temperature T,=l+H{[l+erf(~,)]lg(~,)-cu,}/(l+n,)~(~(~”!.
14.4) to be (4.5)
Equation (4.5) shows that the fuel mass flux 77, and the flame temperature Tf are related uniquely, independently of the Damkohler number. Hence, the free-stream temperature T, corresponding to vS and the inverse Damkohler number E may be found from (4.41, where the function h(~,~) is calculated from (3.22) with T,,, equal to Tr correct to O(E” 1. The flame position Q is given by equation (3.13). The function (1 + erf(q,)}/g(q,) increases monotonically with vV. while h(q,) decreases with increasing 7,. Thus, for T, and E fixed, equation (4.4) may be satisfied by two distinct values of T.~.This non-uniqueness is expected particularly when the activation energy E is large. Then the function exp(E/T,), and hence h (7, ), varies greatly for small changes in 77,. Therefore, the flame temperature is not necessarily a unique function of T, and E when the reaction involves a large activation energy. This effect is discussed by Fendell[4] and Chung et u/.l.C] on the basis of non-linear numerical analyses. However, the present analysis uses essentially linear methods to obtain the same result. Also, for T, fixed, there is a critical value E,,,, SLY. of the inverse Damkohler number such that (4.4) has no solution for q\ when E is greater than E.,. As the advection rate of the free stream increases, the residence time of the reactants in the flame region decreases. and so the flame temperature decreasea. Hence. the temperature gradient at the droplet surface is reduced and this, in turn, causes the mas’\ flus of fuel to the flame region to decrease. Thus, the flame temperature is lowered even further. Therefore. because of this coupling between the Damkohler number and the fuel mass flux. ;I diffusion flame may not be maintained as E increases. Clearly, the critical value E,, increases as the activation energy of the reaction E is decreased. But equation (4.4) is valid only for small values of E”. Thus, it would seem that E.~ can be estimated from (4.4) only when E is large. In fact. analysis of the full system [5] shows that there is no critical inverse Damkohler number for low activation energy reactions; that is, q.? is a unique function of E and T Figure 2 shows curves of constant T, as functions of T, and c for the reaction of acteone burning in air. The data are as for Fig. I. together with the normalized latent heat of vaporization H = 0.9139. Because E is very large for this reaction. T, is not LI unique function of T, and E. Thus, assuming that the diffusion flame corresponds to the
6 Tf 5
L -11
-12
-13
-10
-e
-9
-7
lO9lOE
temperature I ;I\ functions of inverse Damkohler Fig. 2 Curwe\ of conxtant free-\tream number t and fame temperature r, for a diffusion flame upstream of a droplet of acetone burning in air. Numherj refer to the values of T..
iarger value of T,, it would seem that flame extinction the
vaitte
c,,(
T
) at
which
(~T,/c&),,
is infinite.
must occur when E is greater than From
the discussion
of section
3
however. it appears that extinction should occur also when T, is less than a certain value. 71,. Combining these two criteria. we find that flame extinction should occur when
where ( iI’I_, /i)c ), -+ Y ;I\ E - til with T, -7 T I. and where T, = T, when t = E,,. ‘That i4. for large c (small Da~~~kohlcr number), extinction takes place at ;I flame temperature greater
th;lrl
'1:,.
A reaction
involving
a
relatively
turc M hich i\ ;I unique function ion U~QIIC! \t‘em
to he \imply
of T
low activation
energy produces
a Hame tempera-
and e [.S]. In this case. the flame extinction
criter-
T, s T,,. REFERENCES
[i] P. M (~‘HIIYG I-. t.. F’ENDELL and .I. F. HOLT, AlAAJ 1, IOZO (1966). IhI F. A. ‘WII.I.IAMS. ~‘rwnh~r,sfion 7‘hrory. Addison-We\leg(1965). 171 h4 I I .IGH’I‘HIi .I.. Fourir,r Andvsis untf C~eneralisrd Functions. Cambridge
UniverGty
Pre\\
(1962).
518
M. II, MAN’I‘ON
flamme soit simplement que la temperature de la Ramme joit inferieure a une \ aleur critique Jonnee. I .or\que I’analyse est &endue a la caracterisation de la Ramme sur la face amont d’une gouttelette de marout cn combustion. on trouve que le critere d’extinction dnit &tre modifie si la reaction implique une haute knerpie
d’activation.
Zusammenfassung-Genaue
Analyse
von
Diffusionsfiammen
fordert
die
Losung
eines
nicht-linearen
Di-
fferentialsystemsr Durch Annlherung des Reaktionsgeschwindigkeitsgliedes in den thermodynamischen Gleichuneen durch eine Dirac’sche Deltafunktion wird aber das Differentialsystem linear. Folglich wird die AbhPngig~eit der Flammentemperatur von der Damkohler-Zahl untersucht. Vergleich der Resultate der Anafyse mit einigen Beobachtungen von Flammen in der vorderen Stauzone eines nor&en Zylinders durch Tsuii und Yamaoka deutet darauf hin, dass ein Kriterium fur Flammenerl~schen einfach darin besteht, dass die ~iammentemp~ratur niedriger als ein kritischer Wert ist. Wenn die Aoalyse ausgedehnt wird, die Flamme an oberstromigen Seite eines brennenden Brennstofftropfens zu modellieren. wird gefunden. dass das Erloschenskriterium modifiziert werden muss. wenn die Reaktion eine hohe Aktivierungsenergie einberieht.
Sommariu--Un’analisi dettagliata delle fiamme di diffusione richiede la soluzione di un sistema differenziale non lineare. Per&, approssimando il termine della velocita di reazione nelle equazioni termodinamiche a mezzo di una funzione a delta di Dirac, il sistema differenziale diviene lineare. Viene studiata percih la dipendenza della temperatura della fiamma da1 numero di Damkohler. II confronto dei risultati di questa analisi con alcune osservazioni sulle fiamme nella regione di ristagno anteriore di un cilindro poroso effettuate da Tsuji e Yamaoka suggerisce the un criteria per I’estinzione della fiamma e semplicemente the la temperatura della fiamma sia inferiore ad un certo valore critico. Quando I’analisi viene estesa per ottenere il modello della hamma sulla faccia a monte di una goccioletta access di combaustibile, si trova the il criteria di estinzione deve essere modificato se la reazione produce una forte energie di attivazione.
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THE EXTINCTION OF ~IFFUSTON FLAMES bl. .I. MANTON
I~epartmrnt
of Mathematic\.
Monash
University.
Melbourne.
.Au\trali:l
I. rN’rR~DU~TI(~N
THE ~~LI~.~KE
of ;I Iaminar counterflow diffusion flame is studied experimentally by Twji and Yamaoka [ I. 21. By varying the mass fluxes of the reactants, they find that the extinction of the flame in the upstream stagnation region of a porous cylinder is caused by two different mechanisms. First, thermal quenching of the flame occurs as the mass flux of fuel from the cylinder decreases and the flame approaches the cylinder. The second extinction mechanism is due to “chemical limitations on the combustion rate in the flume zone” [?I. As the advection rate CL of air past the cylinder increases, the fuel rn;i>s flus must be increased to maintain a stable diffusion flame. For U-, greater than ;I certain value however a diffusion flame cannot be obtained. Clearly. this second mechanism is a Damkohler number effect; that is. extinction occurs because the air and fuel reactants are swept away by the flow before chemical reaction can take place. The purpose of the present note is to demonstrate that these extinction mechanisms can he explained by a simple analysis if it is assumed that extinction occurs when the flame temperature falls below some critical value, T,,. Hence, the analysis is extended to consider the problem of flame extinction from the upstream face of a burning droplet of fuel in :I free stream of air. In this case. the only adjustable parameters are the Damkohter number E ’ and the ratio T, of the free-stream air temperature to the boiling point of the fuel. It is found that, for large activation energy reactions, the nonuniqueness of the flame temperature implies that extinction should occur for flame tempertrtures greater than T,.,. The analysis of diffusion fames has developed considerably and the relevant literature is reviewed by Williamsr~]. The effect of n finite Damkohler number is usually investigated either by numerically integrating the non-linear equations or by means of a singular perturbation expansion in the region of the diffusion flame [4, -51.However, the inner representation of the perturbation expansion also is given by a non-linear ordinary differential equation which must be solved numerically. In the present work. the dependence of the flame temperature on Damkohler number is found by approximating the reaction rate term in the thern~[~dyn~~mic equations by a Dirac delta function centred in the diffusion flame region. In this manner. the phenomenon is described by a linear system.
.? II(~li.lI’lOhS
AIOI‘IO'L \UI)
0:
‘I l1t:KMOD~‘K~M1(‘S
We consider stagnation
the heha~~iour of ;I chemical reaction in the neighbourhood of the region of the How p:r\t ;I bluff body. The Gtuation i\ described most WC-
cinctly by the steady state Sh\A-Zeldovich the upstream side of the st;lgn;lticw point.
system the no-slip
[S]. Because
the flame appears
on
h:,und;rrlz condition at the \urf;lce of the body does not dominate the tiou ne;Ir the flame. Hence. the fluid is assumed to hc inviscid, and the equ;ltion\ ft,r ;he conservation of momentum and mass are
where U* is the velocity :;nd p” is the pressure at the point x* in space: p is the fluid density. The combustion proces\ i> :~pproximatrcJ by the simple one-step reaction I’, w,
-t v-w
-
7’: w:.
(9.2)
where V, is the stoichiometric coefficient and W, is the molecular weight of the ith species. We let i = I correspond ?o the oxygen and i = 2 correspond to the fuel. The conservation of mass for the i th species I$ described by T
(pv * Y,
pD0 Y, ) =
(2.3)
N’,.
Y, is the mass fraction of species i; II is the common mass diffusion coefficient the species present; ~9, i\ the rate of production of the i th species. Following Fendell 14). we assume that the density p and the diffusion coefficient D are constants. Further. the J>ewis number A /pDC,,. where A is the thermal conductivity and C,, is the overall specific heat of the fluid. is taken to be equal to unity. Thus. the conservation of energy is erpre\scti by the equation where
of
all
where 7’* is the temperature and 11, i\ the heat of formation of the i th species. The rate of production of the i th species by the chemica! reaction is M', =
S,l',W,W".
(2.5)
where si =
1 for i = I. 1; 1 for i -3;
and w* = BT*np+f+>
’ (Y,/W,)“’
( Yz/Wz)”
Here, B and (Y are constants. E* is the activation energy universal gas constant. For a strictly one-step reaction,
e E*‘Yl’r*.
of the reaction, the exponents
and R,, is the w! equal
the
stoichiometric coefficients I/i. However, any physical process is invariably a multi-step reaction, and so we must assume that the exponents I_L,are determined empirically. Equations (2.1)~(2.5) form a closed system which describes the fluid dynamics and thermodynamics of the problem. To simplify the analysis, we now normalize the equations. Dimensionless variables are introduced such that
where L is a representative length scale of the bluff body. I;, is the temperature of the surface of the body, and U, characterizes the velocity of the flow far upstream of the body. Putting (2.6) into (2.1)-(2.3, the equations of motion become
where
(2.7)
b = - c s, W;hi/C,,Tu W. I I
The important
dimensionless
parameters
are the Peclet number
G = LU,lD.
which represents the ratio of the advection rate of material and heat. and
(2.8)
rate by the flow to the molecular diffusion
which is the ratio of the rate that material undergoes chemical reaction to the rate of molecular diffusion of material. 3. FLAME
UPSTREAM
OF POROUS
(‘YI.INDER
We consider first a model of the flame configuration studied by Tsuji and Yamaoka[l, 21. They find that a diffusion flame is established on the upstream face of a porous cylinder which is situated in a free-stream of air and from which fuel is ejected under pressure. From the model, the flame temperature is calculated as a function of the mass flux of fuel and the Damkohler number, and compared with the flame extinction criteria found by Tsuji and Yamaoka. Because the stand-off distance of the flame is much less than the radius of curvature
M. .I. MANTON
510
of the bluff body, a porous cylinder, dimensional flow approaching the plane rectangular co-ordinates. The appropriate tions of the system (2.7) is
the geometry is approximated by a twoboundary x = constant. where (x, y, z) are solution of the momentum and mass equa-
v = (u, v, M’) = (
x. y. 0)
and
(3.1) p = - j(x’+
11’)+ constant.
Equation (3.1) corresponds to a flow approaching the boundary x = - x, from the positive x-direction. Thus, u = x, at the surface of the body. This implies that the net mass flux of fuel is equal to PU~X,~, which is fixed for a given experiment. To solve the thermodynamic equations of (2.7) the boundary conditions on the dependent variables must be specified. At the surface of the body there is no flux of oxygen and the fuel flux is given: that is,
GUY,-iiY,;Qatx=-x 8X
$9
(3.2)
and GUY? Far upstream Hence,
of the body the concentration
Y,+ The temperature
$$ =Gu,
at x = - x,.
of oxygen
is known
Y,andY,+Qasx-+x.
of the body and the free-stream
temperature
and there
is no fuel.
(3.3) also are fixed such that
T = 1 at x = - x,, and
(3.4)
T -+ T, as x -+ SC. It is seen from (3.1) that the boundary conditions (3.2)-(3.4) are independent of _V and z. Hence, it is found that the thermodynamic equations of (2.7) reduce to the ordinary differential equations
(3.5)
$-
Gus=
~ Fwb.
Now the Peclet number G is invariably very much greater than unity when a diffusion flame is present. Therefore, the solution domain (-x5, m) may be divided into an inner region where x is of order G--“*, and an outer region where x is away from the stagnation point. In the outer region, equations (3.5) reduce to first order equations: that
On the extinction
of diffusion
flames
511
is, the effects of diffusion are neglected. However, our interest lies essentially in the inner region where the flame occurs. Thus, we introduce a stretched independent variable ^rl= (G/2)“‘x,
(3.6)
so that, from (3.1), the system (3.2)-(3.5) becomes
(3.7)
with
(3.8) T = 1 at q = - q,,
Y, -+ Y*, Yz -+ 0, T -+ T- as 7f + =; where E -’ = F/G is the Damkohler number, and r), = (G/2)“% is the normalized mass flux of fuel. The parameter 7, is formally of order unity. We seek the solution of the system (3.7)-(3.8) corresponding to a diffusion flame; that is, the case of large Damkohler number only is considered. It is seen from (3.7)-(3.8) that the functions p(T)) =
Y,- (aJa,)Yt (3.9)
and /37(q) = T + (bla,)Y, satisfy linear systems [6f. In particular.
we find that
P(~)=il-~~f(77~)-(l+~l)erf(~))/{1+f(~,)}
(3.10)
B~(~)~~~n+(T~+az)erf(~,)i(T.,+a~-~~r)erf(~)}f(i+erf(~~)},
(3.11)
an d
where
The constant & depends upon the value of Y, at the boundary q = - 8,. Thus the complete solution for Y,. Y? and T is given from (3.9)-(3.11) if the non-linear system (3.7)~(3.8) can be solved for Y,.
M. .I. Mr\N’TO~
512
For a diffusion flame, the chemical reaction is confined essentially to ;L thin flame region, and hence the non-linear term w is negligible outside this region. Thus we approximate this term such that
where S(E) is to be determined centred at the position ?f where P(T,) = 0. Equation (3.10) shows
Putting
and C?(V) is the Dirac delta function. The flame iy the reactants are in stoichiometric proportions. i.e. that Tf is given by
(3.12) into (3.7)-(3.8), Y,(v)
={Y,-
we find that the mass
S[I -erf(77f)l}(f(r),)4erf(~)}/{l
+ S{ erf(77) -erf(qf)l where H(s) is the Heaviside (3.9) and (3.14). The constant T(- qs) = 1; hence PO= I +a$1
fraction
of oxygen
is
+f(77,)1 (3.14)
H(rl - qfI/),
unit step function. Therefore, Y, and T are given from p,, is determined from the boundary condition (3.8) that
-II
-erf(rlr)l(S/Y,)}g(77,)/(1
(3. IF1
+f(77,)1.
where
The values of the dependent equations (3.9)-(3.14) to be yl(q)={(l
Y2(q,)=
variables
+,,)Y,
at the centre
-a,[1
a,{(1 -ta,)Y.
of the flame are calculated
-t-f(~~)lSll(1 +
- a,[1 + f(7j\)lS)/Y~(l
from
(YIY. +QI).
(3.16)
~(~~)={B,,+(T,+~~)erf(~,)+(T.+cu,-P11)erf(?7f)} /{l+erf(rl,)}-cu~{(l+ai)Y,
~LY1[l+f(rl\)lS}IY,(l+cuI)~.
In the limit E -+ 0, equation (3..7) implies that w is identically zero. Hence. the mass fraction of both reactants goes to zero at the flame centre. Now (3.16) shows that
To account for a finite Damkohler pansion of the form
number,
s = s,,-
we assume
that S has an asymptotic
(3.18)
E”S, t O(E” ).
where the index p and the constant S, are to be determined. Y,, YZ and T at the flame centre have the representations
ex-
Thus,
from (3.15)-(3.18).
31.4
Y,(,,) = Y7h(7J> )E” + O(E’)). Y,(Tf) = a,h(q,)EP
+ O(EP),
(3.19)
T,-T(~i)~=T,~(~~)--~Zh~~,)~‘J{1--cr,~~~,)/[1+erf(77,)1~+~(~P). where
It is usual to define the Dirac functions
where
171. We therefore
the flame thickness
delta function
I(E) h:ts the asymptotic
and the index y is to be determined. the production
Because
Equations
term w at the flame centre
these two representations
A second relation
as the limit of a sequence
of regular
take
between
representation
(1.7). (3.17) and (3.17)-(3.20)
can be written
must he identical.
show that
as
we ha\,e
the indices p and q is found from the conservation
of
species equation (3.7). Within the fl:tme region. the rate of diffusion of a reactant into the region. which is cjf order E” ?“.just balances the rate .)f consumption of the reactant hy the chemical reactic,n. which is of order E”‘~~“.“’ ‘. Therefore.
p -2q Equations
(3.3 1) and (3.23)
-(p,+p,)p
- I.
(3.23)
may be solved to y,ieltl
p = q = l/(1 + cc, + cl-).
(3.24)
Thus. the flame thickness is of order c “‘I “‘l’u~’ . and this reduces to the result obtained by a formal singular perturbation expansion when p, = p2 = 1 [4].
M. .1. MANTON
51‘4
For a given fuel mass flux v< and inverse Damkohler number E, the flame temperature T, can now be calculated from equations (3.13), (3.19). (3.22) and (3.24). The free-stream temperature T, and oxygen mass fraction Y, are assumed to be fixed. Two limiting cases are of interest: (i) E + 0 with vy fixed and (ii) q, + x with E tixed. Case (i) corresponds to the diffusion flame limit of infinite Damkohler number. Thus, (3.19) shows that the flame temperature T, equals 7’,,(~,) which decreases as T\ decreases. If we assume that flame extinction occurs when Tffalls below some critical value T,,, then it is clear that extinction takes place when ~~ is less than r),, where T,,,(T~,) = T,.,. As indicated by Tsuji and Yamaoka[l. 21, this case corresponds to thermal quenching of the flame as it approaches the surface of the bluff body. Thus, when the reaction rate completely dominates the rate of advection of the reactants by the flow (i.e. E -+ 0), a flame can be maintained only if the normalized mass flux of fuel is greater than Q,. It is seen from (3.13) that as q, approaches infinity, the distance of the flame from the stagnation point ^~lrapproaches a finite value q, such that erf(_rl.)=(I-cw,)/(l+cu,). Hence
the flame temperature
in case (ii) is given
by
T,= T,,,(x) - azh(xk", where
T,,,(x) = (T.+ aI+ az)l(l + a,> and h(x) ={[Cl
+ a,)Y:
‘I/efi”.,,]/[2~u,a/+“~T’,‘,e’lj]}“(’,’”~’.
Now Tf decreases as E increases from zero; that is, as the advection rate of the reactants increases relative to the reaction rate. Extinction occurs when E is greater than that value which corresponds to T,= T,,. Because the chemical reaction occurs at a finite rate, the reactants must remain in proximity for a finite time in order to maintain a diffusion flame. As the advection rate increases however, extinction takes place because the reactants are swept away by the flow before a significant reaction can be established. Thus the two mechanisms for flame extinction observed by Tsuji and Yamaoka [ I, 21 would seem to be explained, qualitatively at least, by assuming that extinction occurs when the flame temperature falls below some critical value. Thermodynamic data are not readily available to make possible a qualitative comparison with experiments. However Fendell[4] gives data relevant to the burning of acetone in air, and Fig. 1 shows the family of curves of constant T,as a function of 77%and E corresponding to this data. In particular, we take CYI= 0,1056; cyz = 5.712; Y, = 0.233 1;
E = 41.164; (Y =o:
T,= 0.8751: PI=
a, = 0.6882:
/-LZ= 1,
where the surface temperature Tois set equal to 594”R, the boiling point of acetone. The particular curve of Fig. 1 which corresponds to the extinction curve (i.e. to T,= T,) could be predicted from a stability analysis, or it could be found experimentally, The term w in (2.7) which accounts for the chemical reaction is particularly sensi-
On the extinction
Fig. I. Curves
XKI normalized
of diffusion
51.5
flames
of constant flame temperature T, as functions of inverse Damkohler number t fuel mass flux q\ for an acetone-air diffusion flame up-stream of a porous cylinder. Numbers refer to the values of T,.
tive to the exponential function involving the normalized activation energy reaction E. Hence, a simple extinction criterion might exist of the form
of the
T,, lE = constant. 4. FLAME
UPSTREAM
OF BURNING
DROPLET
for the extinction of the diffusion flame which can be established on the upstream face of a burning droplet of fuel moving relative to a stream of air. The mass flux of fuel is now controlled by the rate of evaporation at the surface of the droplet, and so it depends upon the free-stream temperature and the Damkohler number of the system. The body in the air flow (i.e. the droplet) is essentially spherical. Thus, provided that the stand-off distance of the flame is much less than the radius of the droplet, the geometry may be approximated by an axi-symmetric flow approaching the plane boundary x = constant, where (x, r, 0) are cylindrical polar coordinates. The appropriate stagnation flow solution of the momentum and mass conservation equations of (2.7) is It is of interest
to obtain
a criterion
v = (u,v. w) = (- x. ir. 0) and p = -4(x’+:
r’) + constant.
(4.1)
As in section 3, we take the surface of the body as the boundary x = - x,, which implies that the mass flux of fuel is pU,x,. The boundary conditions on the temperature T and the reactants Y, are given by (3.2), (3.3) and (3.4), together with the additional condition that the mass flux of fuel is equal to the rate of evaporation of fuel from the surface of the droplet. Hence [4],
c= dx where
GHu. at x = - .y0 -’
H = H*/C,,T,, and H* is the latent
heat of vaporization
(4.2) of the fuel. It is clear that
the equations (2.7) now reduce to the ordinary differential equations (.:.5). 1 heref~~r-r, the thermodynamics of the flame upstream of a burning droplet are governed hy the x) \tenl (3.7)-(3.8), together with the boundary condition. from (4.2).
The section fuel Q E. Thus, to
solution of the system (3.7)-(3.8) and (4.3) is identical to that obtained in 3, with an additional constraint imposed by (4.3) which fixes the mass flux of in terms of the free-stream temperature T, and the inverse Damkohler number from equations (3.9), (3.1 l), (3.14) and (3.17)-(3.19), it is seen that (4.3) reduce< T,=l+H{1+erf(~,)}/g(~~)-~~{l--(If~~,)h(~,)~’~f0(~~‘)}.
Substituting
(4.4) into (3.19), we find the flame temperature T,=l+H{[l+erf(~,)]lg(~,)-cu,}/(l+n,)~(~(~”!.
14.4) to be (4.5)
Equation (4.5) shows that the fuel mass flux 77, and the flame temperature Tf are related uniquely, independently of the Damkohler number. Hence, the free-stream temperature T, corresponding to vS and the inverse Damkohler number E may be found from (4.41, where the function h(~,~) is calculated from (3.22) with T,,, equal to Tr correct to O(E” 1. The flame position Q is given by equation (3.13). The function (1 + erf(q,)}/g(q,) increases monotonically with vV. while h(q,) decreases with increasing 7,. Thus, for T, and E fixed, equation (4.4) may be satisfied by two distinct values of T.~.This non-uniqueness is expected particularly when the activation energy E is large. Then the function exp(E/T,), and hence h (7, ), varies greatly for small changes in 77,. Therefore, the flame temperature is not necessarily a unique function of T, and E when the reaction involves a large activation energy. This effect is discussed by Fendell[4] and Chung et u/.l.C] on the basis of non-linear numerical analyses. However, the present analysis uses essentially linear methods to obtain the same result. Also, for T, fixed, there is a critical value E,,,, SLY. of the inverse Damkohler number such that (4.4) has no solution for q\ when E is greater than E.,. As the advection rate of the free stream increases, the residence time of the reactants in the flame region decreases. and so the flame temperature decreasea. Hence. the temperature gradient at the droplet surface is reduced and this, in turn, causes the mas’\ flus of fuel to the flame region to decrease. Thus, the flame temperature is lowered even further. Therefore. because of this coupling between the Damkohler number and the fuel mass flux. ;I diffusion flame may not be maintained as E increases. Clearly, the critical value E,, increases as the activation energy of the reaction E is decreased. But equation (4.4) is valid only for small values of E”. Thus, it would seem that E.~ can be estimated from (4.4) only when E is large. In fact. analysis of the full system [5] shows that there is no critical inverse Damkohler number for low activation energy reactions; that is, q.? is a unique function of E and T Figure 2 shows curves of constant T, as functions of T, and c for the reaction of acteone burning in air. The data are as for Fig. I. together with the normalized latent heat of vaporization H = 0.9139. Because E is very large for this reaction. T, is not LI unique function of T, and E. Thus, assuming that the diffusion flame corresponds to the
6 Tf 5
L -11
-12
-13
-10
-e
-9
-7
lO9lOE
temperature I ;I\ functions of inverse Damkohler Fig. 2 Curwe\ of conxtant free-\tream number t and fame temperature r, for a diffusion flame upstream of a droplet of acetone burning in air. Numherj refer to the values of T..
iarger value of T,, it would seem that flame extinction the
vaitte
c,,(
T
) at
which
(~T,/c&),,
is infinite.
must occur when E is greater than From
the discussion
of section
3
however. it appears that extinction should occur also when T, is less than a certain value. 71,. Combining these two criteria. we find that flame extinction should occur when
where ( iI’I_, /i)c ), -+ Y ;I\ E - til with T, -7 T I. and where T, = T, when t = E,,. ‘That i4. for large c (small Da~~~kohlcr number), extinction takes place at ;I flame temperature greater
th;lrl
'1:,.
A reaction
involving
a
relatively
turc M hich i\ ;I unique function ion U~QIIC! \t‘em
to he \imply
of T
low activation
energy produces
a Hame tempera-
and e [.S]. In this case. the flame extinction
criter-
T, s T,,. REFERENCES
[i] P. M (~‘HIIYG I-. t.. F’ENDELL and .I. F. HOLT, AlAAJ 1, IOZO (1966). IhI F. A. ‘WII.I.IAMS. ~‘rwnh~r,sfion 7‘hrory. Addison-We\leg(1965). 171 h4 I I .IGH’I‘HIi .I.. Fourir,r Andvsis untf C~eneralisrd Functions. Cambridge
UniverGty
Pre\\
(1962).
518
M. II, MAN’I‘ON
flamme soit simplement que la temperature de la Ramme joit inferieure a une \ aleur critique Jonnee. I .or\que I’analyse est &endue a la caracterisation de la Ramme sur la face amont d’une gouttelette de marout cn combustion. on trouve que le critere d’extinction dnit &tre modifie si la reaction implique une haute knerpie
d’activation.
Zusammenfassung-Genaue
Analyse
von
Diffusionsfiammen
fordert
die
Losung
eines
nicht-linearen
Di-
fferentialsystemsr Durch Annlherung des Reaktionsgeschwindigkeitsgliedes in den thermodynamischen Gleichuneen durch eine Dirac’sche Deltafunktion wird aber das Differentialsystem linear. Folglich wird die AbhPngig~eit der Flammentemperatur von der Damkohler-Zahl untersucht. Vergleich der Resultate der Anafyse mit einigen Beobachtungen von Flammen in der vorderen Stauzone eines nor&en Zylinders durch Tsuii und Yamaoka deutet darauf hin, dass ein Kriterium fur Flammenerl~schen einfach darin besteht, dass die ~iammentemp~ratur niedriger als ein kritischer Wert ist. Wenn die Aoalyse ausgedehnt wird, die Flamme an oberstromigen Seite eines brennenden Brennstofftropfens zu modellieren. wird gefunden. dass das Erloschenskriterium modifiziert werden muss. wenn die Reaktion eine hohe Aktivierungsenergie einberieht.
Sommariu--Un’analisi dettagliata delle fiamme di diffusione richiede la soluzione di un sistema differenziale non lineare. Per&, approssimando il termine della velocita di reazione nelle equazioni termodinamiche a mezzo di una funzione a delta di Dirac, il sistema differenziale diviene lineare. Viene studiata percih la dipendenza della temperatura della fiamma da1 numero di Damkohler. II confronto dei risultati di questa analisi con alcune osservazioni sulle fiamme nella regione di ristagno anteriore di un cilindro poroso effettuate da Tsuji e Yamaoka suggerisce the un criteria per I’estinzione della fiamma e semplicemente the la temperatura della fiamma sia inferiore ad un certo valore critico. Quando I’analisi viene estesa per ottenere il modello della hamma sulla faccia a monte di una goccioletta access di combaustibile, si trova the il criteria di estinzione deve essere modificato se la reazione produce una forte energie di attivazione.
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