C O M B U S T I O N A N D F L A M E 39: 211-214 (1980)
21 !
Regime Classification of an Exothermic Reaction with Nonuniform
Initial Conditions
YA. B. ZELDOVICH
Institute o f Chemical Physics, Vorobyevskoye Chosse 2b, 117977 Moscow V-334, USSR
The initial (t = 0) temperature distribution in a reacting mixture enables one to calculate the induction period t i to the
adiabatic explosion in each particle of the mixture. Isosurfaces ti(x, y, z) = t give the location of the explosion front at the
moment t i if no physical interaction of adjacent layers occurs; the inverse gradient of the induction period [ grad t i [ - 1
determines the propagation velocity of the intensive reaction zone. The comparison of this rate with the rate of normal
flame propagation and the Chapman-Jouguet detonation speed enables conclusions to be drawn about the effects of heat
conductivity or substance movement on chemical reaction under given initial conditions.
INTRODUCTION
particular, extra adiabatic gas heating in the shock
The theory of combustion and detonation has
grown on the basis of studies of particular idealized cases. Of great importance was the development of the theory of uniform flame propagation
and of detonation theory. Taking the initial state
to be independent of space coordinates (or in a
tube-independent of the coordinate along the axis)
we get the simplest situation where a solution of
wave.
In order to have an exact solution of the type
the type F ( x -- ut) (in a tube F ( x -- ut, y , z)) is
possible, which describes propagation with the
constant rate u. In the simplest case (x, t ) t h e
equation with partial derivatives is converted into
an equation with one independent variable ~ = x -ut; in the general case the number of variables is
reduced by one. The rate u here is not given in
advance; it is determined from the solution of the
problem, as a rule from the condition that the solution satisfy all boundary conditions at ~ = -+ oo
that are consistent with those at x = _+oo.
Solutions ofthe"running-wave" type are strictly
divided into two classes: (1) "combustion" ("deflagration") with a subsonic rate of propagation
and the leading role of heat conductivity and diffusion and (2) "detonation" with a supersonic rate
and the dominating role of gas dynamic factors, in
Copyright © 1980 by The Combustion Institute
Published by Elsevier North Holland, Inc.
52 Vanderbilt Avenue, New York, NY 10017
F(~), one must satisfy, for chemical kinetics of the
reaction considered, the condition ¢(x = oo) =
~[T = T(oo), c = c(oo)] = 0, where ¢ is the reaction
rate; T, the temperature; and c the concentrations.
It is evident that otherwise at large distances from
flame or detonation front independent chemical
reaction would take place, leading to heating and
completion of reaction before the arrival of the
front. Afterward, the flame or detonation will
extinguish.
At present solutions of the type F(~) are known
to be intermediate asymptotic. In the general case
when the condition ¢(x = oo) = 0 is not satisfied
the solution F(~) is a good approximation in a certain limited space range and during limited time.
One condition is that the effect of the ignition impulse must be destroyed (this gives t > t l ) and the
other condition is that the independent reaction
ahead of the front must not be completed (t < t~).
The next natural approximation is F(~, t), where
~ = x -- f u d t , which means that the wave regime is
considered taking into account, the variability of
the propagation rate, since the conditions ahead of
the front of the reaction wave are variable. Even
0010-2180/80/080213+04501.75
212
Ya. B. ZELDOVICH
the idealized case t = 0, T = const = TO at x > x o
does not lead to u = const. Therefore it is natural
to consider the general problem in which there is
no uniformity of temperature distribution in space
at the initial time. Assume that some initial distribution T(x, y , z, t = O) = To(x, y , z) or To(x )
in the one-dimension case, 1 is given at t = 0. In
this case at certain time intervals and in certain
space ranges intermediate asymptotics F(~)with
variable rates apply. The rate change is dependent
not only on the presence of chemical reaction
ahead of the combustion or detonation front but
also on the initial distribution of parameters. Behind the front, states with variable parameters
arise and relaxation processes of equalization of
pressure, temperature, and hydrodynamic velocity
also occur,
Most interesting are the processes taking place
in the unburnt gas. Let us consider the extreme
case of large space lengths and small gradients so
that one may neglect interaction between adjacent
volumes of the reacting substance,
In this case in each particle of the substance,
thermal explosion occurs independently. The explosion time t i for a fixed initial distribution of
parameters is a function of position.
RELATIONSHIP BETWEEN SPONTANEOUS
PROPAGATION, NORMAL DETONATION
AND NORMAL FLAME
Let us define in space the surface ti(X, y , Z) =
const and consider the change of the position of
this surface with time in accordance with ti(x, y,
z) = t.
It is possible to find the direction and the rate
of explosion propagation at each point: it is e v i dent that the direction is normal to the surface t i =
const and that the value of u is in inverse proportion to the modulus of the gradient of t i.
These two statements may be combined in a
vector form
~_
grad t i
(grad ti) 2'
Un = [ U'I = [ grad t i [ - 1
•
1 In the general case, concentration, density, and velocity
distributions are added,
Thus along with physically caused propagation of
the reaction front (by heat conductivity, or shock
waves) one more regime is possible, which we call
"spontaneous." For this regime a strong dependence of propagation rate on initial conditions (in
particular, on initial temperature) is characteristic,
as well as its independence of heat conductivity of
the gas and of the sound speed. The concept of
"spontaneous propagation" (SP) is not new.
Qualitatively SP has already been discussed in the
period of development of modern theory of combustion and explosions, in the 1930s.
In a famous work Kolmogorov, Petrovsky, and
Piskunov [1] showed that with reaction kinetics
having ~ = 0, d ~ / d T --/=0 at x = o% the spectrum of
possible values for flame speed uf is continuous
but limited from below by unain ; ur >1 u m ~ .
The authors proved that under strictly uniform
initial conditions the Cauchy problem solution
leads asymptotically to the regime u = uminPropagation with ue > Um~ represents a process
with causally independent successive ignition of
layers of the original mixture. Thus spontaneous
propagation relative to deflagration was essentially
identified.
Having analyzed possible regimes of detonation
in 1941, the author pointed out [2] that weak
detonations (with a speed above the ChapmanJouguet D > Del , but with pressure less than that
at the Jouguet point p < P e l ) might occur if chemical reaction would start in the initial state without
preliminary heating of the substance by the shock
wave.
Recently [3-4], a reaction has been considered
in a medium with a linear initial temperature distribution, and qualitative assumptions have been
discussed about the effect of initial temperature
gradient on the nature of the expected picture of
the phenomenon. But if one considers the general
problem of the evolution of a reacting system with
given initial parameters, then the introduction of
the strict concept of the effective rate of spontaneous propagation is the decisive step.
Different cases may be isolated.
1) Usp > De1 ==-ua, whereDel is a normal ("classical") detonation rate, which satisfies the ChapmanJouguet condition. In this case the propagation of
a weak detonation wave, with the pressure behind
CLASSIFICATION OF EXOTHERMIC REACTION
the wave in the range Po < P < Pj is obtained,
where po is the final pressure of combustion at
constant volume, p j (index J meaning Jouguet) is
the pressure of normal detonation after completion of the reaction. The difference between spontaneous weak propagation and normal detonation
is that in the first case there is no shock wave,
which compresses the gas in the second case. In
the limit grad t i ~ O, Usp ~
one recovers adiabatic explosion at constant volume, Praax = Pv.
2 ) A t Usp < uj the combustion o f initial portions o f the substance results in a shock wave. This
propagates into the unbumt gas, and stationary
normal detonation is formed after the transient
process, with the shock wave ahead of the front.
This regime is studied numerically in the papers
already mentioned [3-4].
3) uf < Usp ~ a < u j, where a is the speed o f
sound in the unburnt medium, which is always less
than the rate of normal detonation uj. Here uf
is the normal rate of flame propagation. In this
case the reaction propagates at a rate usp , which is
small enough so that the pressure has sufficient
time to equalize. The movement of gas that arises
is necessary for pressure equalization, but this
movement is slow subsonic; the pressure differences in sonic and shock waves that accompany
this movement is small, and the inverse effect of
oo
these waves on the induction period and the rate
of flame propagation is not significant,
From the point of view of the theory of slowflame propagation, the first part of the condition
uf < Usp means the notion of that the reaction
front is determined by initial conditions and does
not depend on heat conductivity of the mixture,
The possibility Usp > uf was first mentioned explicitly by Kolmogorov, Petrovsky, and Piskunov
in Ref. [1]. However, it should be noted that this
work was concerned with the degenerated case in
which at infinity the temperature tends to constant To, while the rate of chemical reaction ¢(T)
tends to zero in accordance with the linear law
q5 = K ( T -- To). This case is interesting because
there exists an exact solution of the type T(x -ut), with T ~ To at ~ = x -- ut -~o~
The existence of such a solution is seen from
the following. The induction period may be ex-
213
pressed by the integral (C is heat capacity).
f
ti = C
dT
C
¢b(t) = - - -KI n
( T - - To) + const
and the condition
Usp
i/dti \ - 1
\dx
x
= const,
ti
- -
+ const
U
is satisfied by the function
T = To + c o n s t e - k x / u s p c .
In the general case there are no reasons to believe
that ~b(To) = 0, in particular if the initial temperature itself depends on coordinates. However, here
the induction period is finite everywhere, and the
flame propagation period is limited. In other
words, there is no exact solution of the type
T(x -- ut) if ¢(To) q: 0. The flame propagation (in
this case the regime of spontaneous propagation) is
an intermediate assymptote, valid up to timax ,
which corresponds to the minimum of To.
4) At usp < u r. In this case the regime of normal
flame is obtained, which is caused by heat conductivity and diffusion even without external ignition.
Assume that rapid reaction occurs anywhere at a
given moment tl. In the adjacent layer, if isolated
from the first one, the reaction would occur at the
moment t 2 = t 1 + x2xusp-1 ' where x21 is the distance between the layers. But the condition usp <
uf means that due to heat conduction, the reaction
in the second layer takes place earlier. This means
that it is impossible to neglect heat conductivity,
and therefore the regime of spontaneous propagation does not occur; it is superseded by normal
flame propagation.
There is a formal difficulty: When ¢(To) ~ 0
there is no exact solution of the type T(x - ut)
and, strictly speaking, the normal rate of flame
propagation u~, cannot be calculated.
In 1938 [5, 6] during first calculations of flame
rates, u~, with realistic kinetics, this difficulty was
overcome by a very simple but effective method,
When the rate of chemical reaction at initial ternperature ¢(To) is small, specifically, ¢(To) ,~ ¢(T'),
214
Ya. B. ZELDOVICH
7' ~ Tb - - R T b 2 / A , the reaction rate may be neglected in some finite interval near To, for T <
To + e, where e is small compared with the cornplete temperature change Tb -- To. It has been
shown (especially clearly in a recent work [7]),
that the choice of e within reasonable limits does
not affect the value uf.
However, such a primitive approach to the phenomena considered causes doubts concerning
whether it is logically reasonable to neglect ¢(To)
and even ¢(To + e) in the calculation of u~ and at
the same time to calculate Usp, which depends on
~(To) itself. Is it possible to compare two values
calculated under different assumptions?
Recently [8], a treatment has been developed
that enables one to clarify this question. Without
repeating all the arguments, we note that flame
propagation is considered from the very beginning
as the intermediate asymptote of the phenomenon; the temperature distribution in the flame is
written in the form
T(x, t) = T 1 (t) + I T b - - T 1 (t)]
X
(x:)
--
u dt, t
'
where T l ( t ) is the temperature of the original reacting substance that varies because ¢(To) :~ 0. In
other words, it is supposed in advance that the
substance ahead of the flame front is changed due
to the reaction, and hence the flame speed changes
with time. In Ref. [8] the equation is given for the
dimensionless function 0, indicating the temperature distribution in the flame front: It appears that
solution for 0 exists only for a definite u(t). The
equation for 0 is remarkable in that a modified effective rate of heat release Ce(t) = ~(T) -- (1 -O)¢(T1) replace are true ¢(T). The modification
leads to ~e = 0 at 0 = 0 and 0 = 1, and therefore an
exact solution exists. The conditions that are
necessary for calculation of uf are fulfilled, as
described in more detail in Ref. [8]. Therefore
comparison of uf and Usp is made possible.
CONCLUSION
A classification is given for propagation o f intense
chemical reaction in reacting mixture under given
initial conditions. The oversimplified one-dimensional case is considered. The real situation, however, is much more complicated, depending on
ignition conditions, geometry of the vessel (for
example, on roughness of the walls [9]), initial
hydrodynamic field, and hydrodynamic diffusionheat instability [9-12].
' Nevertheless, the consideration of the induction-period field and of the corresponding spontaneous-rate field of reaction front propagation is
a reasonable and useful first step in the consideration of any particular problem concerning exothermic chemical reaction under arbitrary initial
conditions.
REFERENCES
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G.M., and Sivashinsky, G. I., On Development of
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Received December 1978; revised 29 May 1979