COMBUSTION A N D F L A M E 85:185-194 (1991)
185
Comparison Between Shock Initiations of Detonation Using
Thermally-Sensitive and Chain-Branching Chemical Models
J. W. DOLD
School of Mathematics, University of Bristol, Bristol BS8 ITW, United Kingdom
and
A. K. K A P I L A
Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York 12180-3590, USA
This article shows that, unlike the situation in many other combustion problems, the progress towards detonation
behind an initiating shock wave when the chemistry is modeled using a radical chain-branching mechanism is
fundamentally different from the progress found when a global one-step model is used. In the latter case, a
uniformly supersonic (shockless) wave of chemical activity is found to emerge at the end of an induction process.
Only after this "induction flame" slows down significantly is it possible for a genuine propagation mechanism to
create a Zeldovich-von Neumann-DSring detonation structure. When dominated by chain branching, the reaction
wave produced by the induction process is subsonic, and can therefore be influenced by propagating thermodynamic
disturbances (as originally pointed out by Strehlow). The resulting chemical-hydrodynamic interaction causes an
acceleration of the wave prior to the emergence of the detonation. It is suggested that the existence of these two
distinct forms of initiation may provide a means of testing for the relative importance of possible chain-branching
and state-sensitive mechanisms in the initiation of detonations, especially where little kinetic data are yet available
(as, for example, in condensed explosives).
INTRODUCTION
Recent asymptotic analyses of the initiation of
detonation using a large-activation-energy, onestep model for the chemistry [1-4] have revealed
an interesting sequence of events, but one that is
not always in qualitative agreement with experimental observations [5, 6]. In seeking an explanation, one finds that quite different physical events
can be predicted if a radical chain-branching
model for the chemistry is invoked. This article
seeks to investigate this difference and to point
out its implication concerning both the use of
global, one-step chemical models for detonation
problems and the experimental investigation of
the chemical kinetics of explosives. Some discussion of the underlying physical and chemical reasons for the difference is also presented.
If a strong piston-driven (or reflected) shock
wave raises the temperature of a medium that is
reacting via a single exothermic temperature-sensitive chemical step to a point at which the reaction becomes significant, then during an initial
induction stage, the reaction-rate increases relatively slowly as a thermal runaway begins to set
Copyright © 1991 by The Combustion Institute
Published by Elsevier Science Publishing Co., Inc.
in. After a while, this induction process leads to a
rapid and substantial energy release that starts at
the piston face and always moves as a supersonic
wave of chemical activity into the compressed
medium.
This wave is not, however, a Zeldovich-von
Neumann-DSring detonation. In the manner
identified by Zeldovich and Kompaneets [7],
Clarke [8], and others [4, 9], it represents a "fast
flame" that is dominated throughout its structure
by reactive and supersonically convective processes. As such, the flow through the wave involves no shock and the wave assumes the form
of a weak detonation (at least to begin with).
Although Zeldovich and Kompaneets imagined
such a wave as being initiated by a rapidly moving source of ignition (such as a very fast sequence of sparks), no such "thought experiment"
is required to explain the reaction wave that
emerges in the present context.
Diffusive effects being insignificant, the wave
should also not be confused with any other, laminar or turbulent, flame that is sustained by thermal and mass transport. On the contrary, the
reaction wave that initially appears for a one-step,
0010-2180/91/$3.50
186
temperature-sensitive model of the chemistry simply travels too fast for its motion to be attributable to any propagation mechanism--it is a
wave of chemical activity that moves faster than
any acoustic signal within the system.
Instead, the movement of this fast flame appears by virtue of the fact that chemical and fluid
interactions during an induction process cause
different points within the reacting medium
rapidly to release their chemical energy at different times. This is somewhat analogous to Zeldovich's less precise concept of a "spontaneous
flame" [10], which is notionally defined only in
terms of homogeneous induction times that are
calculated separately for every point based only
on initial data.
Because it therefore emerges from a spatially
varying induction process, and not through any
mechanism for propagation, it may be appropriate to refer to this particular reaction wave as an
induction flame. This term is introduced purely
for convenience at this stage in order to facilitate
discussions below concerning the detailed nature
of reaction waves that emerge within the shocked
medium. In particular, the term is used to emphasize the role of induction, rather than any propagation mechanism, in determining the nature and
movement of a wave of chemical activity. The
term has also been usefully applied elsewhere in
studying the inductive initiation of combustion
with diffusive rather than compressible interactions [11, 12].
For a single-step, temperature-sensitive chemistry, the speed of the induction flame is actually
found to be infinite at its moment of birth near the
piston face (so that neighboring points explode
almost simultaneously). The flame also continues
to move supersonically--after one point explodes, the signal that it produces does not have
time to reach any other point before it too explodes. However, the flame does slow down as it
moves forwards, until its speed reaches the Chapman-Jouguet detonation speed with respect to the
compressed, unburned medium ahead of it. This
would typically happen well before it catches up
with the initiating shock-wave [13].
At this stage, acoustic information begins to
travel faster than the induction flame in the hottest,
burned part of the medium, causing a shock-wave
to form somewhere near the end of the chemical
reaction. As this shock wave overtakes (and de-
J . W . DOLD AND A. K. KAPILA
stroys) the now slower induction flame, it further
increases the sound speed in its wake and so
maintains a mechanism for propagation. A short
but quasi-steady transient leads to the establishment of an underdriven Zeldovich-von Neumann-DSring detonation.
A brief outline of the model and the essentials
of the theoretical description of this sequence of
events are presented below, greater detail being
available in references [1-4]. Recent numerical
analyses of detonations initiated by piston-driven
shock waves in media that react via a one-step
exothermic reaction of moderately large activation energy are in excellent agreement with this
description [14, 15].
The fact that some experimental observations,
both in gases and condensed explosives [5, 6],
reveal a qualitatively different sequence of events,
suggests that a different mathematical model is
required to explain the actual progress to detonation in some materials. In these experiments, the
wave of strong chemical activity that emerges
after an induction stage is not found to be supersonic. Disturbances associated with the observed
flame are thus genuinely able to self-propagate
and may be able to steepen sufficiently to form a
secondary shock wave before catching up with
the leading or initiating shock.
By examining a fairly simple three-step model
for the chemistry, a description that is qualitatively closer to these experimental findings can be
obtained. A numerical study of a similar system
was first carried out by Strehlow and co-workers
in the context of reflected shock initiation, and a
summary of their results appears in Strehlow's
textbook [6]. Our more generalized model consists of two radical-producing temperature-sensitive reactions (that may be endothermic), representing an initiation step and a chain-branching
step. The third reaction is an exothermic radicalconsuming step of negligible activation energy
that completes the chemical change.
This scheme admits some useful asymptotic
analysis and qualitatively mimics some of the
features of hydrogen-oxygen as well as hydrocarbon combustion. Asymptotic solutions are obtained for the early behavior of the reaction-wave
that emerges. This is sufficient to demonstrate the
manner in which propagation mechanisms can
operate and lead to an acceleration of the flame.
The distinguished roles of the temperature sensi-
SHOCK INITIATION OF DETONATION
tivities of each reaction are also easily understood. It is worth noting that while this reaction
model and analysis are open to further generalization, they are sufficient to establish the principal
objectives of this article in a relatively simple
way.
Since the results broadly agree with some experimental observations for detonation initiation
[5], they may provide some clues about the actual
chemical processes that take place in some condensed explosives, for which little kinetic data
are yet available, For instance, it is by no means
clear, a priori, that any intermediate reactants in
such explosives would be produced via a chainbranching (self-catalyzing) process rather than
(say) thermal decomposition. Indeed, bearing in
mind the enormous pressures involved in the
detonation of condensed explosives, one might
anticipate a mechanical element in the decomposition of larger molecules, more sensitive to the
thermodynamic state (including pressure) than to
the presence of active radicals. The model examples of mole-preserving, thermally sensitive, and
chain-branching reactions, which are studied here,
serve primarily to illustrate the behavior at opposite ends of a spectrum of possible chemical
pathways.
After briefly outlining the reasons for the appearance of a supersonic induction flame through
a single temperature-sensitive chemical step, we
discuss the nature of the corresponding flame
arising as a result of the three-step scheme. Only
in one special case, for which the chain-branching reaction is insensitive to temperature, does
this flame represent an induction flame--its path
and chemical structure being determined only by
spatially varying chemical kinetic induction processes and not by any propagation mechanism, in
spite of the fact that it travels subsonically. However, if the reaction is sensitive to temperature,
then thermal changes brought about by the propagation of pressure waves increasingly influence
the movement of this flame from the moment of
its birth at the piston face.
With neither the one-step nor the three-step
model do we take the analysis as far as considering the behavior of the flames as they approach
the initiating shock-wave. Rather, the main intention of this article is to make use of asymptotic
arguments to highlight the two fundamentally
different possible types of reaction waves that can
187
emerge at the end of an induction period for
different kinetic models. Particularly for condensed explosives, this may offer an experimental
means of examining the relative importance of
radical self-catalyzing (chain-branching) reactions
and global thermodynamic sensitivity.
MODEL
Fluid-Dynamic Equations
In order to model the nonlinear pressure, velocity
and thermal interactions that take place in a detonating medium it is convenient to focus attention
on the Euler equations for ideal gas-dynamic
interactions. For values of the adiabatic coefficient 3, as high as three or more, it is believed
that these equations provide a qualitatively adequate model for dynamic interactions in condensed explosives [16]. At the high velocities and
short time-scales found in detonations, it is highly
realistic to ignore diffusive effects, except perhaps within the internal structure of strong shock
waves [17]. For these it is adequate to use Rankine-Hugoniot shock jump conditions.
Also assuming that the chemical process preserves the number of molecules, the set of dimensionless equations becomes
7-1
PTt -
- -
Pt + P q t ,
"7
" Y - ' P t + PPUv, = P q t ,
(1)
ut + "r-lp,~ = 0
P = aT,
in which p is the density, P the pressure, T the
absolute temperature, u the velocity, and q the
amount of energy released by the chemistry. Subscripts are used to denote partial differentiation.
Time is represented by t while ~k represents a
Lagrangean (mass) coordinate defined by ~b =
f~Xtp dx. The second of these equations (representing the effects of compressibility) can be derived from the mass conservation equation Pt +
p2 u~ = 0 by using the first (energy conservation)
equation and the last (gas state) equation to eliminate Pt and Tr The remaining (third) equation
describes momentum conservation.
We suppose that a solid piston of constant
188
J.W. DOLD AND A. K. KAPILA
dimensionless velocity u i drives a shock-wave of
initial Mach number m i > 1 into the cold stationary unreacted explosive medium. Choosing the
initial state of the medium behind the shock-wave
to be the reference state for nondimensionalization (so that T = P = p = 1 at the initial instant
behind the shock wave), the dimensionless state
of the medium ahead of the shock is then given
by the Rankine-Hugoniot conditions
7+1
= 23¢M2 + 1 _ 7 '
P=Pi
One-Step Chemistry
A simple one-step model for the chemistry can be
postulated in the form
R ~ P.
If r represents a normalized mass-fraction of the
reactant R, that is always taken to be unity
immediately ahead of the shock, then with an
Arrhenius model for the reaction rate the chemical evolution is governed by the equation
r
2/M 2 + 3'P=Pi
=
rt +
1
(2)
M/2-1
Ui=2
('7 + 1 ) M i
As the system evolves, driven by changes in
the released chemical energy q, the shock Mach
number M generally changes from its initial
value. As this happens, the state of the medium
immediately behind the shock becomes
2",tM 2 + 1 - "y
P=P°=
2"YMi2 + 1
2/ME
-7,
+ 7-
1
P = Po = 2 / M 2 + 3' -
1,
m 2-
e B(l-1/r) = 0
(7)
O/3
'7+1
with
(6)
(3)
1
u = u o = 2 (7 + 1 ) M V/ffii/Pi"
while at the piston, the relevant boundary condition remains
for an appropriate choice of the dimensionless
time-scale t. (It may be recalled that T = 1,
initially, immediately behind the shock wave.)
Strong temperature sensitivity is represented by
the dimensionless activation temperature /3 being
large. The system of equations is then closed by
noting that the released chemical energy q is
given by
q= Q(1-
r)
(8)
for a dimensionless constant Q, representing the
total chemical energy available in the unreacted
mixture.
With this model, taking r and the other dynamic variables T, P , and so on to be fixed
ahead of the shock wave relies on assuming that
the reaction rate ahead of the shock, i.e., exp[/3(1
- 1/T/)], is negligibly small. Since 1 - 1/T,. is
negative, this assumption relies in /3 being large
and is thus consistent only with an asymptotic
description for 13 ~, 1.
Three-Step Chemistry
It is useful also to identify the dimensionless
mass-flux through the shock-wave, given by
A simple model for the chemistry, involving radical production, chain branching, and exothermic
completion of the reaction can be represented by
the following three steps:
m = M
I: F + M ~ Y + M ,
U --
(4)
U i.
~
with m i = M i v / & P i ,
(5)
having m i < 1 for any M i > 1. In this way, the
shock boundary conditions (Eq. 3) and the piston
boundary condition (Eq. 4), are all parameterized
by the single quantity m i (or, equivalently, M i )
for a given value of 7.
B: F+
Y"
2Y,
kt
= e A(1/rt-
k B = e°°/rB-1/r)
C: Y + M--* P + M,
kc=
l/T)
(9)
1.
Taking f and y to represent normalized mass
fractions of F and Y, respectively, the chemical
SHOCK INITIATION OF DETONATION
189
log k
t
1/T
T
I
I/TI
i
I/TB
llTi
kzf
Fig. 1. Variation of k t, k n, and k c with T. Crossover
temperatures Tt and TB are on opposite sides of the initial
postshock temperature of unity. Preshock temperature is T / <
rB.
evolution can then be modeled by the equations
ft + f(kt
+ Y k s ) = 0,
"Yt = k t f + Y ( f k s
(10)
- kc),
again for an appropriate choice of the time-scale
t. The released chemical energy can be written as
q = Q(1 - f ) - ( Q + D ) y ,
uniformly very slow. For the purposes of this
investigation, we may also consider cases for
which the activation energy of the branching reaction B is moderately large 0 ~, 1, although it is
also generally relatively small when compared
with A.
Provided the temperature in the medium ahead
of the shock T/ is below TB, chain-branching is
not able to build up the concentration of active
radicals and a quasi-steady approximation for y
can be justified, showing that ahead of the shock
one should find
(11)
where Q has the same meaning as previously,
and D represents an amount of endothermic energy absorbed by the initiation and chain-branching reactions 1 and B in breaking down the
reactant F into the energetic radical Y. Representing a recombination of active radicals, the
completion reaction C should meet no significant
activation energy barrier although other possibilities could be considered. It should be emphasized
that this is only a sample model scheme that has
been selected to illustrate the role of chain
branching using an asymptotic approach.
Figure 1 illustrates the ways in which the
Arrhenius reaction "constants" k z ( T ) , k s ( T ) ,
and k c ( T ) are taken to vary with temperature.
The "crossover" temperatures, T, and T s, at
which the initiation and branching reactions become as fast as the completion reaction are taken
to be, respectively, greater than and less than the
dimensionless initial postshock temperature of
unity. The initiation reaction I generally has a
large activation energy, A ~, 1, required to destroy the relatively strong chemical bonds of the
explosive F. As a result, this reaction tends to be
Y = Yi
kc
_ kB f
kt(T/),
(12)
which is typically very small indeed unless T,. is
close to Tn--representing a preshocked temperature only slightly below the level at which chainbranching would begin. The second asymptotic
approximation in Eq. 12 is only valid if T~ is not
close to TB and 0 ~, 1 so that k B can be neglected in comparison to k c = 1. The rate of
change of f is then equally as small, so that it is
reasonable to take f to be fixed at unity in the
medium immediately ahead of the shock, with no
significant change occurring until the shock raises
the temperature of the medium above Ts. Once
this happens, chain branching can dramatically
accelerate the chemical process.
ONE-STEP INITIATION
Since many of the details of the way in which a
detonation would arise from a single one-step
large activation energy reaction appear elsewhere, only a very brief description is offered
here,
Induction
In order to analyze the thermally driven induction
stage that is relevant to this model, it is worth
noting that order /3-1 perturbations cause significant changes in reaction rate. It is therefore
appropriate to use the asymptotic expansions
T-
1 +/3-'~b(~b, t ) ,
u - u i + /3-1v(~b, t ) ,
Pr-
(13)
1 + ('g/3)-Ip(~b, t ) ,
1 -- ( Q f l ) - l w ( f f ,
t)
190
J.W. DOLD AND A. K. KAPILA
for large [3, with which one obtains the reduced
model
4~t = wt + ( 7 Pt
+
v¢, =
1)Pt,
Wt ,
v t + p~ = 0,
(14)
w t = e ¢'.
Boundary conditions for these equations need to
be applied at the piston face tk = 0, at which we
have simply that
v(0, t) -- 0,
(15)
and at the leading shock-wave, which lies at
~k - m i t + [3- ' f ~ m 1 dt. Equations 3 can be used
to deduce that we must have the shock boundary
conditions (at ~k = m i t to leading order):
t) = A p m l,
v(mit,
t) = A u m l ,
w(mit,
t) = O,
(16)
in which the shock mass-flux perturbation m~
appears parametrically. The constants AT(m/),
A p ( m i ) , and A u ( m i ) are easily found by differentiating Eqs. 3 and 5.
It is worth noting that Clarke's equation [18] is
derived by eliminating p, v, and w between Eqs.
14,
(4a t - "ye6)tt = (dp t - e~')~¢~.
(17)
Under the bounded initial and boundary conditions (Eqs. 15 and 16) (or many more general
conditions), Eq. 17 leads to a blowup in finite
time [13, 19, 20]. In particular, for this shock-initiation problem, one always finds that 4~ --' oo as
the time t approaches a supersonic path in space
and time, t = 7(~b), as follows:
lim 4~(~b, t) = oo,
with r~(~k) =
-- t I
+ KII~ (23/-1)/7,
(19)
so that the path of blowup initially has infinite
speed, since 7 > 1, and the mass-flux r~ decreases as the singularity moves forwards (remaining supersonic).
Induction Flame
The correct interpretation of the singular growth
in the perturbation quantities defined in Eqs. 13 is
that changes in T, P , u, and r cease to be small
(of order [3-1) close to the singularity path t =
7(4).
¢k(mit, t) = A r m ,,
P(mit,
through the system at supersonic speeds. This
finding is a direct result of the way in which the
induction proceeds under the influence of Eqs.
14-16. Moreover, with blowup happening first at
the piston-face, one finds [19] that
> 1.
The fact that the mass-flux rh, which is swept
through the singularity path, is necessarily greater
than unity indicates that the singularity moves
Since reaction and heat-release rates become
very fast indeed according to the model (Eq. 7)
when T is not close to unity, a quasi-steady
structure can be analyzed for the resulting induction flames, confirming the Rayleigh line and
Rankine-Hugoniot relationships
P-
1-
- ' y t h 2 ( V - 1)
and
(3' + 1 ) ( P V - 2
1) + ( 7 -
1)(V-tP t
Q(1 - r ) ,
respectively, in which the specific volume is V =
1/0. Most significantly, the induction-flame conforms to "weak detonation" solution paths in the
Rankine-Hugoniot diagram, as illustrated by the
Rayleigh line A - B in Fig. 2.
Shock Formation--Propagating Flame
As the induction flame slows down, its structure
is determined by Rayleigh lines of decreasing
negative slope as th decreases. When the slope of
this line decreases to the point of tangency with
the burned Hugoniot curve (the line A - C in Fig.
2) a sonic point is produced in the burned medium.
At this point, acoustic information travels forwards as quickly as the induction flame itself and
any further slowing down in the induction flame
causes a shock wave to form in the hottest parts
SHOCK INITIATION OF DETONATION
191
line. Initially, most of the chemical change takes
place within an induction flame between these
two lines. The solid line indicates the path of a
shock wave that propagates ahead of the slowing
induction flame, transforming it into a Zeldovich-von Neumann-D/Sring detonation.
P
"~~."""..
. """-.""..
"~,.....
C
THREE-STEP INITIATION
A
//
Induction
V
Fig. 2. Quasi-steady, weak-detonation structure of an induction flame on a Rankine-Hugoniot diagram. Point A represents compressed, almost unreacted state ahead of the induction flame. As reaction proceeds, the state of the medium is
described asymptotically by changes along a Rayleigh line
(such as A - B), that passes through a sequence of intermediate Hugoniot curves (shown dotted), each corresponding to
a partially completed stage of the reaction, towards the final
burnt Hugoniot curve (shown solid). As flame slows down,
point B moves from right to left on burned Hugoniot curve.
Rayleigh line A - B corresponds to an instant when the
flame is supersonic, and A - C to an instant when the flame
is sonic, in relation to the following flow.
of the medium, thus introducing a mechanism for
propagation. For details about this transition, the
reader is referred to Refs. 1, 3, 4, and 21.
An illustration of the transition process is presented in Fig. 3, where the dotted line represents
the singularity path t = 7(~b) in the (~b, t) plane.
Close behind this line, the reaction rate reaches a
locally maximum value, represented by the dashed
t
The induction process with the three-step model
of the chemistry is fundamentally different from
the one-step thermally driven induction. Once the
initial temperature behind the shock is raised
above TB (initially to unity according to the
nondimensionalization), the difference f k B - k c
in Eq. 10 becomes positive. Indeed if 0(1 - Ta)
is large, then k B becomes large, so that this
difference is positive for a wide range of values
of f . This leads to a rapid exponential chainbranching growth in the values of y.
On the other hand, the values of y begin to
grow from the tiny initial value Yi and are forced
to increase initially at the very small rate k/(1).
Thus, the values of y remain small for a prolonged period during which changes in temperature would also be small. As a result, temperature
variations can be neglected during most of the
chain-branching induction process, leading to the
following composite asymptotic solutions for y
and f (in which the subscript 1 denotes evaluation at T = 1):
y-
e(km-kcO(t-trf~/rn,) - k i l / ( kB1 -- k c l ) ,
E BI e (kin -
kct)(t-
t 1- ~// mi)
f-1k m - kcl
+
(21)
kole-(kn,-kct)tt
k m - kc1
+
ktl - kcl
ks1 -- k c l
(t-
~b/mi),
where
Fig. 3. Sketch of transition to a Zeldovich-von NeumannDSring detonation structure in (~b, t) plane. Dotted line represents the singularity path t = t', dashed line represents the
path of maximum reaction rate, and solid line is the locus of a
shock wave that is born when the induction flame has slowed
down just below the Chapman-Jouguet speed. Precursor shock
is not shown.
t z = In ktl "~-~-/(ffn~ --- k c l )
( k a l -- k c l )
A(1 - 1 / T I ) + O ( 1 . / T B - 1)
e O(1/Tn-l) -- 1
+ O(e-Z°(~/rB-').
(22)
192
J.W. DOLD AND A. K. KAPILA
In this, the boundary conditions y = Yi and f = 1
have been applied at the leading-order shock path
~b = m d . It may be noted that these asymptotic
results require only that k m > k c l and that T
be nearly constant. They do not rely on 0 being
large, except insofar as the final estimate for tz in
Eq. 22 becomes more accurate in this limit.
The buildup in the normalized radical massfraction y first reaches order unity values near
the piston face, ~b = 0, during the time
(1)
t-
tI = 0
(23)
kai - kcz
Particularly if 0 is large, this time is relatively
short in comparison to the chain-branching induction time t z.
Because A is large, Eq. 22 identifies a distinguished limit. By the definition of k c in Eq. 9
the dimensionless radical recombination time is
of order unity. The relative time scale of this
exothermic reaction to the initial overall chainbranching induction time depends intimately on
the value of
(24)
~o = a e - ° " / r B - 1 ) .
For order unity values of 0 this quantity is large,
indicating that the time for radical recombination
would be short compared with the induction time•
Only for moderately large values of 0, of the
order of In A, do these times become comparable, or even reversed, in their relative orders of
magnitude. Thus, depending on the values of A
and 0, it can be seen that a wide range of possible
chemical and thermodynamic structures would be
parameterized by the quantity w.
Induction Flame
In the absence of any propagation mechanism,
Eqs. 21 show that values of y would increase
rapidly along the path
t =
tI +
~b/m i + 0
the one-step induction flame described in the previous section.
However, since m i < 1, this path is subsonic.
As a result, pressure-wave disturbances produced
by temperature changes associated with order
unity changes in the value of y can propagate
ahead of the path (Eq. 25). Unless the reaction
constants k I, k B, and k c are (somehow) independent of temperature, these waves will then modify the Eqs. 21 and render the induction flame-path
(Eq. 25) physically meaningless.
It is worth noting that a variant of the model
(Eq. 9), in which one could assume (say) a
constant value for k B > 1 for all T _ TB, would
produce a flame that follows the path of Eq. 25
until significant pressure disturbances catch up
with the leading shock wave--after a time of the
order of m i t t ~ ( 1 - mi). This follows from the
observation that the initiation reaction I is only
significant in Eqs. 10 when y = O ( k i ) . Thus the
solutions (Eqs. 21) only tend to be sensitive to'
changes in k t ( T ) when y is still very small, as it
is close to the initiating shock-wave, and would
thus be changed significantly only when propagating disturbances reach the shock wave.
kBi
1
kcz
)
(25)
Effect o f Pressure-Waves
To illustrate the way in which pressure waves
cause the region of strong chemical activity to
move ahead of the induction flame path (Eq. 25)
we only consider the very early behavior of the
chemical wave in the case D = 0.
If 0 is large, we can also consider the radical
concentration y to increase towards unity in a
vanishingly thin region about some path t = i(~b),
which is initially approximated by ~(~b) = t z +
~b/m o + O(~b2). Following the passage of this
chemical wave, the recombination reaction C
causes heat to be released at the initial rate qt =
Q + O ( t - t).
At sufficiently early times, pressure and velocity variations are small so that linearized characteristic solutions of Eqs. 1-4 give
T-
1
P-
7-1
followed by a region of radical recombination.
Since this result follows entirely from the solutions describing the induction process, it identifies an "induction flame" in an analogous way to
1
7
3-2m
o
( u - u,)
l+mo
3/2 - m o
- (t-
tI -
~b)Qm o
(26)
2
1 -
m o
in the region where ~b <_ t - t I < ~ / m o. At least
SHOCK I N I T I A T I O N OF D E T O N A T I O N
193
for a while, no significant disturbance propagates
ahead of the characteristic ~b = t - t I.
The reaction model (Eqs. 9 and 10) now gives
Yt ~ Y k m e°O'- 1),
II.1"
(27)
after linearizing the Arrhenius exponent and neglecting all but leading order terms. Using the
solutions (Eq. 26), this equation can be integrated
to give
y - exp
( e ~ t - t ' - ' ~ ) - 1) - k e l
mi
j,
(28)
with
3/2 - m 0
K =
-
1)Qmo
2
Fig. 4. Sketch of effect of pressure waves on induction flame
in 3-step model. Precursor shock is identified by S, and
dashed line parallel to it represents induction-flame path when
pressure waves are not taken into account. Pressure waves
accelerate the path of the reaction wave t = ~(~b) in the
manner illustrated by the solid curve. Dotted line represents
the leading characteristic t = t I + ~b.
1 - mo
after using Eqs. 21 to set the values of y at the
leading characteristic ff = t - t z.
This solution is, of course, only valid at early
times, but it is sufficient to show that the path
i(ff), near which values of y increase substantially, has the asymptotic structure
1
~-t1+~+
(
In 1 - r
1-m/~
/
mi
]
structure of the recombination region following
t -- t. Alternatively, a numerical study (comparable to Strehlow's work [6]) could be carried out.
The simple analysis above, however, is sufficient
to illustrate the mechanism by which propagating
disturbances would immediately modify a subsonic induction flame structure that arises when
the three-step reaction scheme (Eqs. 9) describes
the chemical process.
(29)
- t,+C//m
i-
~
mi
]
for small values of ~b. It can thus be seen that
rn o = m r In other words, the path starts out
being close to the induction flame path (Eq. 25),
but it moves ahead by an amount that progressively increases as propagating disturbances increase the rate of chain branching ahead of the
wave.
For comparison with Fig. 3, Fig. 4 illustrates
(schematically) the movement of the path t =
t(~k) in the (~k, t) plane. Temperature disturbances produced by the exothermic chemistry in
the region t > i increase the rate of chain
branching in the region between i and the leading
characteristic ~b = t - t t and so cause the path
t = i(ff) to advance forwards.
A proper analysis of the further evolution of
the reaction wave would require a fully nonlinear
modeling of the compressible flow, coupled with
the general form of the path i(~b) and the full
CONCLUSION
These fairly simple analyses serve to show that a
quite different form of detonation initiation can be
found when an exothermic change in a chemical
state progresses at a rate that depends not only on
thermodynamic quantities but also (through chain
branching) on a radical concentration. Because
the latter is independent of any thermodynamic
quantities, induction times that are determined
primarily by chain-branching processes become
more-or-less uncoupled from any pressure and
velocity interactions during an inductive evolution. Because such a coupling is inherent in systems that involve thermodynamically sensitive,
one-step chemistries, quite different behavior is
predicted.
A significant part of the chemical change that
takes place in many gaseous combustible systems
is known to involve the production and reproduction of chemical radicals by chain-branching processes. It is therefore perhaps surprising that the
194
modeling of such systems using a simple one-step,
temperature-sensitive Arrhenius reaction has led
to numerous qualitatively correct solutions [22]
describing a variety of combustion processes,
such as, for example, wrinkled laminar flames or
ignition and extinction criteria.
The arguments presented here, however, show
that significant qualitative differences arise in the
modeling of detonation initiation. The notion of
an 'induction flame' (presented in the introduction) that can be analyzed quite well using a
one-step large activation energy model does not
persist in the same way when the three-step model
(Eqs. 9) is used. In particular, the solutions of the
relevant equations describing the induction process predict the emergence of reaction waves that
move supersonically in the former case, but subsonically in the latter case. Quite different patterns of observable chemical and pressure-velocity waves are thus predicted.
In the case of the model with chain-branching,
the propagation of these waves is able significantly to modify the corresponding chemical wave
after a very short time. The amount of modification depends both upon the sensitivity of the
branching chemical reaction and the size of the
thermodynamic disturbances at any stage. Only,
initially, before the latter have grown sufficiently
to modify significantly the radical branching rate,
does the notion of a subsonic induction flame
apply moderately well. In the case of a one-step
model, the supersonic induction flame remains
immune to modification by forwards propagating
thermodynamic waves until it slows down below
the speed of these waves in the hottest part of the
medium.
This major difference between the results obtained using the two models of the chemistry
indicates that a one-step model is probably not
adequate for describing detonation initiation in
explosives which are known to burn via a radical
chain-branching process. To some degree, it also
provides a means of experimentally testing the
nature of the dominant chemical kinetic mechanism that may be involved in the detonation of an
explosive, particularly (as in the case of many
condensed explosives) when little kinetic information is yet available.
This work benefited from useful discussions
with Paul Clavin, Forman Williams, and John
J.W. DOLD AND A. K. KAPILA
Bdzii. Research cooperation between the authors was facilitated by a NA TO collaborative
research grant. J.W.D. is supported by a fellowship from the British Science and Engineering Research Council and was a visitor at the
Institute for Mathematics and its Applications,
Minneapolis, during the preparation of this
paper. A.K.K. received partial support from a
research contract with the Los Alamos National Laboratory.
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Received 3 May 1990; revised 8 September 1990