Importance
of NonFourier Heat Conduction
Solid-Phase Reactions
in
This work wmhlishcb that nonFourier heat conduction can be importam for predicting solid-phase Arrheniw
reaction rates. The importance stems from nonFourier transient rempmrunx that are rignifcamly biir
than temperatures haed on the cla+d
model of Fourier’s law. In rum. tbcsc higher temperatures
dram;ltically incxnrc reaction rates. To establish the importance. this work analyzes thermal ignition of a
semi-infinite solid @‘,vernedhy nonFourier “hypcrholic conductiun” while subjected to a constant heat flux at
its surhcc. The slid could rcprcsenl. for example. a granular pmpcllant since recent experiments provide
widcncc of nonFourier conducrion in some granular materials. The workdevelops an a?protimate expression
fur ignition time and soJvcs the governing cquatiom for the rcactmg solid with the method of characteristics.
Results mclude hishcr surfacr‘ trm~eralures than Fourier VBIULIS.
These hi&r tem~xraturn cause a 70%
:C IYCJXhy The Comhusluw lnrtitutc
INTRODUCTION
Evidence
of NonFourier
Conduction
Fourier’s
law is the classical model of heat
conduction
for predicting
temperatures
in solids. In general terms the law states that heat
Rux is directly
proportional
to temperature
gradient.
Although
the law is typically very
accurate, recent experimental
studies provide
evidence of nonFourier
transient conduction
in
some inert nonhomogeneous
solids including
granular
materials such as sand [l], and processed meat [2]. This evidence raises the possibility that nonFourier
conduction
may occur in
some reactive solids, as described shortly.
The studies il. 2] propose nonFourier
conduction occurs in inert nonhomogeneous
solids
by thermal wave propagation
with sharp wave
fronts separating heated and unheated zones.
The nonhomogeneous
structures
of the solids
apparently
induce waves by delaying the response between heat flux and temperature
gradient. For example, this delay may represent
time needed to accumulate
energy for significant heat transfer between structural
elements
[I]. During this delay, heat flux gradually adjusts (relaxes) to the value given by Fourier’s
law. Thus, a wave front is the location where
relaxation begins in response to a thermal disturbance (e.g., heating of a solid).
In contrast,
Fourier’s
law does not lead to
prediction
of these thermal waves because its
CCWBL’S??C?N.4.Y~ N 4&W 117:72Y-341 (1998)
0 1998hy Tbc Combustion Institu:e
Published by Elsevicr Science Inc.
direct proportion
permits heat flux to immediately adjust to changes in temperature
gradicnt. However,
“thermal
wave” is often used
with Fourier conduction.
For example, Fourier’s law underlies “parabolic
thermal waves”
associated
with a nonlinear
dependence
of
thermal conductivity
on temperature
and the
“thermal
wave method” of measuring thermal
diffusivity,
as well as various
“combustion
waves.” These Fourier waves are not the same
as the nonFourier
waves described here.
It is important
to note nonFourier
amduction is usually associated
with “microseale”
apphcatlons
such as submicron
size electronic
devices. However,
the experiments
[l, 21 with
nonhomogeneous
soiids and the work reported
here involve “macroscale”
nonFourier
conduetion where length and time scales are relatively
large.
Importance
of NonFourier
Conduction
The experiments
[l, 21 with inert nonhomogeneous solids suggest deviation from Fourier’s
law can be large. A measure of this deviation is
“thermal
relaxation time” 7, which is approximately the time needed for heat flux relaxation. For instance, 7 = 20 s for one type of
sand [l]. In comparison,
I = 0 for Fourier’s law
since it permits immediate relaxation.
Relaxation times for nonhomogeneous
solids
should depend on their structural
details (e.g.,
OrJlo-2180/98/$19.fm
PII scmlo-218lx97x1o131-4
330
distribution
of sand grain diameters). Unfortunately, there are not yet enough experimental
data to quantify the relation between these
details and relaxation
times for inert solids,
and no data are reported for reactive solids.
Although
there are no data for reactive
solids, the evidence of nonFourier
conduction
in inert solids such as sand [l] raises the possibility that it may occur in some reactive solids
because of similar nonhomogeneous
structures. One possible example is granular propellants used in, for instance,
rocket motors
and guns. This type of propellant is a porous
solid formed by interconnected
grains of reactive solid (e.g., pentaerythritol
tetranitrate,
or PETN [3]) with voids between grains. Thus,
these grains play the role of sand grains. A
second possible example is with the emerging
technology of “self propagating high temperature synthesis” (SHS) for producing
ceramics
[4, 51. SHS involves ignition and self-sustained
reaction of solid reaetants to give solid products.
Typically,
powdered reactants in SHS (e.g.,
Ti and C) are mixed together, then compacted
to form a porous cylinder. One end of the
cylinder is ignited with a heated metal coil or
laser. After ignition, the reaction propagates
down the cylinder
leaving behind a porous
product (Tic). Hence, the powdered reactants
of SHS resemble sand grains. Also, grain sizes
in granular propellants and SHS are often similar to those (approximately
100 pm) in the
experiment with sand [l].
Hence, exploring
the potential
effects of
nonFourier
conduction
on reactive solids is
important,
because this work shows the conduction can play a critical role in predicting
solid-phase Arrhenius
reaction rates. The role
results from nonFourier
temperatures
that are
significantly higher than Fourier values. In turn,
these higher temperatures
dramatically
increase reaction
rates because the rates increase exponentially
with temperature.
One potential benefit of predicting reaction
rates with nonFourier
conduction, when appropriate, is improved accuracy compared to predictions based on Fourier’s law. However, the
greatest benefits may result from exploiting
nonFourier
behavior
to develop novel pro-
PAUL
cesses not anticipated through
described in the conclusion.
J. ANTAKI
Fourier’s
law, as
Objective
Despite the importance of nonFourier
conduction, its potential effects on solid-phase reactions are nearly unexplored. One study [6] analyzes a reacting, deforming solid with heat flux
relaxation where the effects of relaxation and
deformation
are coupled. In contrast, the work
reported here takes a first step toward exploring the isolated effect of relaxation by analyzing thermal ignition of a rigid semi-infinite
solid governed by nonFourier
hyperbolic
conduction. (There is no relation between the
terms “hyperbolic”
conduction
and “hypergolic” propellant.)
Hyperbolic
conduction
is adopted here because it accounts for heat flux relaxation,
as
described shortly. Also, experiments
appear to
support predictions
of hyperbolic
conduction.
Specifically, data with inert granular materials
[I] correlate well with its predictions
and the
study with processed meat [2] shows excellent
agreement
between predicted
temperatures
and measured values. However,
because this
work is only a first step, it does not determine
whether hyperbolic conduction
actually occurs
in reactive solids such as granular propellants.
The objective of this work is to determine
the effect of heat flux relaxation on ignition
time of the semi-infinite
solid governed by
hyperbolic conduction.
To induce ignition, the
solid is subjected to a constant heat flux at its
surface. Ignition occurs when surface temperature exhibits “thermal runaway” caused by heat
release that increases exponentially,
analogous
to the classical Fourier ignition problem for a
semi-infinite
solid [7, 81. Although more complex heat fluxes often occur in actual ignition
scenarios [9], a constant flux is adopted here to
most easily accomplish the objective. However,
several potential consequences
of more complex heat fluxes are discussed in the section on
opposing views.
In addition to the objective just stated, a key
question is: If nonFourier
conduction occurs in
some reactive solids, why has Fourier conduction apparently
been adequate for predicting
NONFOURIER
CONDUCTION
IN SOLID
REACTIONS
solid-phase
reaction rates? A possible answer
is: NonFourier
effects may have been masked
by using Fourier conduction
in predictive models to obtain “best fits” of experimental
data.
Thus, this work is a starting point for evaluating this answer by identifying
some conditions
under which nonFourier
effects would be noticeable. Then, knowledge
of these conditions
could help design experiments
to provide a
definitive answer.
HYPERBOLIC
Governing
CONDUCTION
Equations
Hyperbolic
conduction
[lo- 131 accounts
for
heat flux relaxation with the Cattaneo-Vernotte
model relating heat flux to temperature
gradient,
In Eq. 1, q is the heat flux vector and W’ is the
temperature
gradient, I and A arc time and
thermal conductivity,
respectively,
and r(dq/
at) represents heat flux relaxation. This model
reduces to Fourier’s
law by setting 7 = 0 (immediate relaxation), and for steady-state
conditions (dq/dt
= 0) even with 7 z 0.
The statcmcnt
of energy conservation
for an
incompressible
(rigid) solid is
iIT
pcx=s-v.q,
(2)
where p and C are its density and specific
heat, respectively,
and S is the volumetric
rate
of heat generation
or absorption
within the
solid.
Combining Eqs. I and 2, then eliminating q,
gives the hyperbolic heat equation for the solid.
For constant p, C. h. r, and thermal diffusivity
u = A/( PC), this equation is
(31
which is mathematically
classified as hyperbolic
[l4]. Alternatively,
solving the hyperbolic
system of Eqs. I and 2 is equivalent to solving Eq
3.
331
In Eq. 3, b’T/dt’
represents wave propagation of heat damped by dT/dt
representing
heat diffusion. The term dS/d~ is an apparent
heat generation or absorption that appears because of relaxation,
since changes in S are
thermal disturbances
that induce waves. Also,
thermal wave speed c, is related to relaxation
time by T = a/c,‘.
Equation
3 reduces to the Fourier
heat
equation for steady-state
conditions
or + * 0.
In particular,
I = 0 corresponds
to c, + =,
implying thermal waves propagate at infinite
speed for Fourier conduction.
The transient
Fourier equation obtained for 7 = 0 is classified as parabolic [l4].
Solutions to hyperbolic
conduction
problems
are ddmped, nonFourier
thermal
waves. In
contrast. solutions to Fourier problems permit
conduction
only by diffusion.
Further,
solutions to hyperbolic
problems with S = 0 (e.g.,
inert) converge to corresponding
Fourier solutions for sufficiently
large times after thermal
disturbances.
This convergence
occurs because
of wave damping (decay). Here, “corresponding” means all aspects of nonFourier
and
Fourier problems are identical except for their
different conduction
models.
Key Features
To introduce
key features of hyperbolic
conduction relevant to ignition, qualitative
hyperbolic and Fourier predictions
for heating of an
inert semi-infinite
solid are compared here. In
this inert problem [IS] the rigid solid occupies
the half-space x > 0 and is initially at a temperature
of 0°C. At time t = Of a constant
heat flux is applied uniformly along the surface
at x = 0. All this flux is absorbed at the surface and no heat is lost to the surroundings.
Immediately
upon application
of the heat
Rux, the inert hyperbolic
prediction
gives a
jump in surface temperature.
This jump is a
consequence
of heat flux relaxation
delaying
conduction
into the solid by initially confining
heat to the surface.
In contrast,
the inert
Fourier prediction gives no immediate increase
in surface temperature
because, with no conduction delay, heat is distributed
throughout
the solid.
PAUL
J. ANTAKI
LOCATION: x
Fig. 2. Quatitativu comparison of inert hyperbolic and
Fuururierinternal tempereture profiles.
ooL-
TIME: t
Fig. I. Quatitativc comparison ot inert hyperbolic and
Fourier s!~rfacc tcmpcraturcs YStime.
More specitically,
Fig. 1 shows a qualitative
comparison
of surface tcmpcrature
vs time for
the hyperbolic
and Fourier cases. The hyperholic prediction
(solid line) shows its jump in
surface
temperature
at I = O+, while the
Fourier surface temperature
(dashed line) initially remains at 0°C. However, the difference
hehwen
temperatures
decreases
as time
in-
The
highest
“hyperbolic
temperature”
occurs
at the surface.
Although not shown in Fig. 2, the hyperbolic
temperature
profile converges
to the Fourier
profile as time increases. This convergence
occurs because the propagating
wave decays by
distributing
its energy over an increasing volume.
In summary, one key feature of hyperbolic
conduction
is higher temperatures
compared
to Fourier
values. Second, the highest hyperbolic temperatures
occur at the surface of the
solid. Thus. the rate of heat release for the
reactive
solid with hyperbolic
conduction
will
creases, since the hyperbolic
prediction
converges to the Fourier
prediction.
Thus, the
greatest difference
hehveen surface temperatures occurs immediately
upon application
of
the heat flux.
Next, Fig. 2 shows a qualitative
comparison
of hyperbolic and Fourier internal temperature
profiles at time t, shortly after application
of
the heat flux. According
to the Fourier predic-
be highest at its surface. Consequently,
ignition with hyperbolic
conduction
is determined
by behavior of surface temperature,
similar to
the Fourier ignition prohlem [7, XI.
tion, the temperature
everywhere
in the solid
has been affected since there is no conduction
delay. In contrast,
the hyperbolic
prediction
shows a thermal
wave propagating
into the
solid because of delayed conduction.
The wave
front, located at x,, = CJ,, separates
heated
and unheated zones. Hence, ahead of the wave
(x > x,)
the temperature
has not ye1 been
affected.
Behind the wave (x < x,,), however,
Eqs. I and 2 for the behavior
of surface temperature
leading to thermal
runaway.
and developing
an approximate
expression
for ignition time using Eq. 3.
the temperature
is higher than the Fourier
value because the same amount of energy as
the Fourier case occupies a smaller volume.
IGNITION
ANALYSIS
This ignition analysis determines
the effect of
heat flux relaxation on ignition time by solving
Problem
The
Formulation
reactive
semi-infinite
solid occupies
and is initially at temperature
constant heat flux q,, is applied
the surface at x = 0. All this
at the surface and no heat is
x > 0
T,. At t = 0’ a
uniformly along
flux is absorbed
lost to the sur-
NONFOURIER
CONDUCTION
IN SOLID
REACIIONS
roundings. Heat release is assumed to occur by
a single-step,
irreversible,
exothermic
Arrhcnius reaction,
s = p,,“e-l/“’
(4)
where H, v, and E are the constant heat of
reaction, pre-exponential
factor and activation
energy, respectively,
and R is the universal gas
constant. To isolate the effect of relaxation.
the reaction product is a solid with the same
properties
and relaxation
time as the reactam.
Further, the value of p is taken as constant in
Eq. 4 since the effect of reactant
depletion
should be small compared to the cffcct of the
exponential
term during ignition [7. 81. Hence.
the surface at .Y = 11is fixed in position. Also.
A, C, and r are constant, with T treated as a
parameter.
For this one-dimensional
problem
with
T(x,
I)
and qb, I). the ~dtt~ltV2O-VWl‘ltt+2
model (Eq. I) in dimensionless
form is
where Hln, f I = T/T; and JI(n, 6 1 = o/r/,, arc
dimensionless
tempcraturc
and heat flux. respectively.
Also,
1) = xq,/(hT,)
and
6 =
r~,*/(ApCT,‘)
arc dimensionless
location and
time, respectively.
The dimensionless
relaxation time is c = rrlO’/(hpCT,‘I
The dimensionless
form of the energy equation (Eq. 2) that incorporates
Ey. 4 is
ae
a*
_ + - - & -PI" = ,,,
a.5
37
O(q,O)
= I
(7a)
$(7J,O)
= 0
(7b)
lNq + -, 6) = I
= 1
(7c)
fort>
It is interesting
to note the initial heat flux
of zero given by Eq. 7b implies heat release
occurring
for H = I is zero throughour
the
solid. since a rcleasc of heat would require a
nonzero Hux to maintain 0 = I. Aclually, this
zero initial heat flux is an approximation
because heat release is not zero for H = I, as
shown hy the exponential
term in Eq. 6.
The more accurate initial condition for heat
flux corresponding
to H = I is obtained from
Eq. 6 using Jfl/J.$ = 0 at t: = 0: rj~fn.0) =
Se a?. (Here. Jn/ag = 0 implies temperature
is not changing at 5 = 0.1 This more accurate
condition
shows $(n,Ol + 0 as /3 + x. Thus,
for p + I (the case here) Eq. 7b should he a
good approximation.
In fact, solving Eqs. 5-7d
using this more accurate
initial condition
in
place of Eq. 7h shows no noticeable change in
the results described later.
Solution
Method
Equations
5-7d are solved numerically
with
the method of characteristics
used previously
for several inert hyperbolic problems (e.g., [1611.
As applied here, the method 1171 transforms
Eqs. 5 and 6 into finite difference
forms that
arc “central diffcrcncc”
in space and “forward
difference”
in time. where An and A.$ are
uniform space and time steps, respectively,
and
the stability condition is A(/fe”‘An)
2 1.
The finite difference equations for temperature H and heat Hux @ at the surface n = 0
are
(6)
where S = pHvAT,/q,,’
and p = E/(RT,)
arc
dimensionless
heat of reaction and activation
energy, respectively.
The initial and bounddty conditions for Eqs.
5 and 6 are
t/1(0,&)
333
0.
(7dl
The hyperbolic ignition problem given by Eqs.
5-7d reduces to the corresponding
Fourier ignition problem [7, 81 for E = 0.
(I,‘* ’ = I.
(9)
where n = (s”’
+ A?/21
and b = Cc’/’ hq/2).
In Eqs. 8 and 9, subscripts
I and 2
denote the surface node lo = 1) and fiat interior node (n = 21, respectively,
on the finite
difference
grid [16]. The superscripts
denote
time steps, where i = 1 corresponds
to 5 = 0.
For instance, the initial condition given by Eq.
7a is represented
at n = 2 by 0,’ = 1. Also,
Eq. 9 represents the constant heat flux applied
to the surface (see Eq. 7dl.
33J
PAUL
At interior
und l/f are
8”,I
nodus (II r 2) the equations
’ = $[w;
I + (2: I ) + /J( h::
+Elqy.lg(e
for 0
I - +,L,:+ I )
a/~~:> + &> PiL)]*
(IO)
#,),,;1 ’ = ;[(o.:
I - Y: I I )
+ b( $,:: / + $4::+ I )
When developing these finite difference
equations, the terms jodq that arise arc approximated with the trapezoidal rule (“second order
approximation”
[171), while the rectangle rule
approximates
je -@/‘dq
(“first
order approximation”
[171X Although
the rectangle rule is
less accurate, it avoids solving transcendental
equations
for B that would result from the
trapezoidal cule. The “trade off” for this avoidance is smaller A[ and A,?1 in the calculations
to achieve a desired convergence.
The solution of Eqs. 8-11 approaches
that
of the corresponding
Fourier problem as the
relaxation
time becomes small (e --) 0). However, setting E = 0 would change the problem
from hyperbolic
to parabolic,
requiring
a diffcrcnt solution technique. For S = 0 (zero heat
of reaction),
Eqs. 8-11 reduce to those for
inert hyperbolic problems [16].
The jump in surface temperature
at time
5 = Ot caused by the applied heat flux is conveniently obtained from Eq. 8 by setting &’ =
1 and I&’ = 0 (initial conditions),
then allowing A7 + 0 since the thermal wave has not yet
propagated
into the solid. The temperature
resulting from the jump of magnitude
(1/Z is
e lump = , + ew
(12)
which is the same as for the inert hyperbolic
problem [16]. Finally, the location of the wave
front is 7jw = t/e ‘I*, obtained by transforming
X, = c, 1 into dimensionless
variables. This location is the same as the inert problem [16]
since c, is constant here.
Approximation
for Ignition
J. ANTAKI
Time
To help determine the effect of relaxation time
c on ignition, a rough approximation
for ignition time is developed in the Appendix.
The
physical criterion fdr this rough approximation
is the same as the Fourier ignition problem [X,
181: Ignition occurs when the rate of internal
heating by reaction becomes equal to the rate
of external heating by the applied heat flux.
The rough approximation
for “hyperbolic
ignition time” (,, is the implicit expression
where
B = [I + (SH/EHIJ&,/2E)
and, IN, and I, are modified Bessel functions of
the first kind, of order zero and one, respectively. Reference
[IS] gives the corresponding
rough approximation
for Fourier ignition time
6. Equation 13 reduces to .$ for E + 0.
4. RESULTS
Typical
results for surface temperature
are
shown here because ignition depends on its
behavior. The Appendix
shows corresponding
results for internal temperature.
Values for Calculations
For convenience,
typical values of properties
for solid propellants [7] are used here: T, = 300
K, A = 0.21 W mmi K-‘, C = 1.55 kJ kg-’
Km’, p = 1.6 x 10’ kg m-j, E = 83.19 x 10’
kJ kgmol-’
and Hv=
2.62 x 10’ kW kg-‘,
along with the representative
value [7] of 9, =
41.87 kW m-‘. These values give 8 = 1.5 x 10s
and p = 334, representing
approximate
lower
bounds for heat of reaction and activation energy, respectively.
Finally, selecting 7 = 6.66 s
gives the dimensionless
relaxation time of E =
0.25. This selection seems reasonable as a convenient initial value based on 7 = 20 s measured for sand [l]. Hence, this value of E could
be realistic for propellants with structures simi-
NONFOURIER
CONDUCTION
IN SOLID
REACTIONS
lx to the sand in [I]. As noted previously,
however, measurements
of r have not yet been
performed
for propellants
and other reactive
solids. Consequently,
r is considered a parameter here.
Hyperbolic
Calculations
The “hyperbolic
calculations”
for Eqs. S-11
were performed
with a FORTRAN
program
using A.$ = cli2 An to satisfy the stability condition. The space and time steps were An =
0.001 and At = 0.0005, since convergence
studies showed less than a 1% change in B
with smaller steps. As described shortly, hyperbolic ignition occurred at time 5 = 0.4 with the
wave front located at n,? = 0.8. Thus. n = 1.0
was selected as a convenient maximum for the
calculations
since no tempcraturc
change occurred for TJ > 9,.
These calculations
were tested two ways.
First, results for the inert hyperbolic problem
(8 = 0) agreed with its exact solution [IS]. Second, results obtained from Eqs. 8-I I in the
limit of the Fourier ignition problem (8 # 0.
E + 01 agreed with independent Fourier calculations, described next.
Fourier
Calculations
The Fourier ignition problem wts detined by
Eqs. 3 and 4 in dimensionless
form with l = 0
and Eqs. 7a, c, and d. The problem was solved
numerically
with the heat conduction
module
of the COSMOS/M
finite element package
obtained from the Structural
Research and
Analysis Corporation.
This package uses the
Crank-Nicholson
method for time stepping
and Newton-Raphson
procedure for solving the
nonlinear element equations arising from the
Arrhenius
reaction rate. The location n = 20
was the maximum value used in these calcufations since numerical experiments
showed this
value accurately
simulated n + = (the initial
conditions were undisturbed
at n = 20 for all
calculations).
Convergence
studies led to using 100 one-dimensional elements with a “spacing ratio” (SRI
of 80 and time step of A[ = O.OOI, since more
elements and smaller time steps gave less than
a 1% change in 8. The SR minimized
the
335
number of elements required to achieve this
convergence
by concentrating
the smallest elements near the surface n = 0 where the reac
tion was most rapid and temperature
gradients
were steepest. With SR = 80, spacing between
the first two elements from I) = 0 was 80 times
less than the spacing between the last two
elements at n = 20.
In turn, these finite element calculations
were tested by checking
for agreement
with
the exact solution for the inert Fourier problem (6 = 0) [19]. Also, Fourier ignition times
resulting from the reactive finite element calculations
(S # C) agreed with ignition times
from previous numerical and asymptotic analyscs. as dcscribcd shortly.
Surface
Temperature
Figure 3 compares surface temperature
vs time
for the reactive Fourier and hyperbolic
cases.
In the figure, 0 = 1 is the initial temperature,
and the heat flux is applied at time 5 = 0’. For
convenicncc.
vertical reference lines are drawn
at c = 0.4 and 0.6. For brevity. this figure is
the only comparison
of surface temperatures
given here because the results shown are qualitatively the same as results for other values of
E. 6, and p.
In the figure. ,$; and 5; are ignition times
for the Fourier and hyperbolic
cases, respec-
0
0.1
0.2
0.3
0.4
0.5
0.6
DIMENSIONLESSTIME: E
Fig. 3. Comparison of reactive hyperbolic snd Fourier
surfam temprratures leading to ignition.
PAUL
336
tively, obtained through the calculations
just
summarized.
The rough approximations
for
Fourier and hyperbolic
ignition times, tfi and
c,,, respectively,
are discussed later.
The dashed line in Fig. 3 shows the Fourier
surface temperature.
At sufficiently
small times
(e.g., 5 = 0.2) and low temperatures,
the increasmg
temperatrurc
+hXis
.c
inert heating
since heat release is negligible at these low
temperatures.
Consequently,
the applied heat
flux dominates
heating of the solid at small
times. However, at sufficiently
large times (e.g.,
.$ = 0.5) and high temperatures,
the temperature begins to exhibit the rapid increase associated with thermal
runaway since heating is
now dominated by the exponentially
increasing
rate of heat release.
Selecting a Fourier ignition time is somewhat arbitrary [7], since thermal runaway shows
a continuous
increase in temperature
rather
than a step increase that would be easily identified. Thus, the Fourier
ignition time is selected as 6: = 0.6 because thermal runaway
becomes obvious there. This selection is reasonable, being within approximately
2 and 10%
of ignition times determined
by previous numerical 171 and asymptotic
181 analyses, respectivcly. The Fourier ignition temperature
is 0,*
= 2.16.
Next, the hyperbolic
surface
temperature
(solid line) shows f$,,,,,, = 1.5 at (= 0’. As
time increases the temperature
first retIects
inert heating, then exhibits thermal runaway.
The hyperbolic
ignition time is selected as .$$
= 0.4, since the corresponding
hyperbolic
ignition temperature
H,*, = 2.20 is nearly the same
as the Fourier value tf$ = 2.16). Correspondingly, thermal runaway becomes obvious at .$,$
J. ANTAKI
If this analysis included reactant depletion,
the surface temperatures
of both cases would
“level off’ during thermal runaway instead of
always increasing with the exponential
dependence shown in Eq. 4. Then, the temperatures
would approach inert heating behaviors
since
heating by reaction would vanish when reactants at the surface became depleted. However, the hyperbolic
ignition time would still be
less than the Fourier ignition time because of
higher hyperbolic
temperatures.
Approximate
Ignition
Time
Figure 4 shows that increasing
the relaxation
time E decreases the hyperbolic
ignition time
relative to the Fourier value. Specifically,
the
figure compares
the hyperbolic
and Fourier
rough approximations
Q and &, respectively,
vs E, for fixed heat of reaction 6 and activation
energy p. Equation
13 gives En, while .$ is
obtained
from its expression
in [Ml. The
Fourier
ignition time (P = 0, dashed line) is
constant at .$ = 0.47 since there is no relaxation effect. Also, the figure shows hyperbolic
ignition time .$,, (solid line) approaches
the
Fourier value for E 4 0, as expected.
The hyperbolic
ignition time decreases as P
increases because higher jumps in surface temperature
at 5 = 0’ accompany
larger l (see
Eq. 12). In turn, these higher jumps cause
increased heat release and faster ignition. For
0.5 L
= 0.4.
The key difference
between cases in Fig. 3 is
the 33% reduction
in ignition time for the
hyperbolic
case relative to the Fourier case.
This reduction
in ignition time results from
higher hyperbolic
temperatures
caused by heat
flux relaxation
and. consequently,
larger heat
release at earlier times. In particular,
the initial jump in hyperbolic
surface temperature
is
the principal
contributor
to this reduction.
Hence, heat flux relaxation can have an important effect on ignition time. (Percent reduction
is computed with (4 - ,$$;T,/$ x 100.)
01
0
I
0.1
I
0.2
0.3
I
0.4
I
0.5
DIMENSIONLESS RElAXUlON TIME: E
Fig. 4. Effect of relaxation time on rough approximations
for hyperbolic and Fourier ignition times.
NONFOURIER
CONDUCTION
IN SOLID
example, at E = 0.25 the figure shows approximately a 33% reduction
in ignition time relative to the Fourier value. However, increasing
l to 0.5 shows ignition
time reduced by about
80%. Further, the value E = 0.5 could be realistic since it corresponds
to r = 13 s, similar to
the value measured for sand [I].
Although the approximations
shown in Fig. 4
are only rough indicators of the more accurate
ignition times obtained by solving Eqs. 8-11.
these approximations
can give rather good estimates of re~/ucrioris in the more accumte values. With E = 0.25, for instance, the reduction
in ignition time using the rough approximations is about 33%, almost identical
to the
reduction using the more accurate values given
in Fig. 3 (obtained by solving Eqs. 8-1 I). Further, with e = 0.5 the reduction
using rough
approximations
is about 80%. reasonably close
to the 70% reduction obtained by solving Eqs.
8-11. Hence, using the rough approximations,
rather than solving Eqs. X- Il. is a convenient
way to quickly estimate reductions
in ignition
times caused by heat flux relaxation.
AN OPPOSING
VIEW
The results just illustrated
show the importance of heat flux relaxa,.ion in reducing
ignition
time. However,
several
effects
can
decrease this importance
by moving the hyperbolic ignition time toward the Fourier ignition
time. Several examples of these effects are
cited here to permit an objective assessment of
whether the relaxation effect is sufficiently
important to merit further study.
One effect is an increase in time required to
achieve ignition. By increasing
this time, the
hyperbolic surface temperature
converges more
closely to the Fourier surface temperature
during the period of inert heating (see Fig. II.
Then, subsequent
“hyperbolic
thermal
runaway” and ignition are also closer to Fourier
behavior. For example, increasing
the activation energy p increases the ignition temperature and, consequently,
the time required
to
achieve this temperature.
More specifically, the
previous results pertain to a lower bound of
activation
energy for solid propellants
(p =
33;), showing about a 70% reduction
in ignition time with a relaxation
time of E = 0.5.
REACTIONS
337
However, an activation energy close to an upper bound, say p = 70 171, shows only about’s
5% reduction
in ignition time with l = 0,5.
Similarly,
decreasing
the heat of reaction
6
reduces !hc importance
of relaxation
by increasing the time required to achieve ignition.
Here, ignition time increases because more
time is needed to release enough heat to raise
the solid to its ignition temperature.
A second effect decreasing
the importance
of relaxation
is a reduction
in the hyperbolic
temperature
jump that occurs upon application
of the heat flux. In particular,
the previous
results correspond
to the largest possible jump
since all flux is absorbed at the surface. However. the jump is likely to be smaller with, for
example, radiation heat loss from the surface
[20] or in-depth absorption
of heat flux 1211.
A third effect tending to reduce the importance of relaxation
is an ignition mechanism
that shifts the controlling
events of ignition
away from solid-phdsc
heat conduction
and
reaction.
One example is propelldnt
ignition
controlled
by gas-phase reactions.
With gasphase control.
details of solid-phase
conduction and reaction are less important
[q].
Finally. changes in the structure
of a solid
during
ignition
could
rcducc
or eliminate
structural
characteristics
that
appear to induce
heat flux relaxation.
For instance. melting of
reactant
particles
during SHS or propellant
ignition could fill voids between particles. eliminating granular characteristics
that appear to
induce relaxation.
Despite
these effects, heat llux relaxation
can potentially
remain important.
With SHS
ignition, for example, the jump in hyperbolic
surface temperature
can occur before melting.
thus preserving
the key contributor
to the reduction in ignition time. Hence, further studying the role of relaxation
may provide stratcgies to overcome
effects tending to reduce its
importance.
CONCLUSION
El&t
of Heat Flux Relaxation
In conclusion,
heat flux relaxation can cause a
large reduction
in ignition time of a reactive
solid relative to the classical Fourier ignition
338
PAUL
time. This reduction is a consequence
of higher
surface temperatures
that, in turn, cause higher
Arrhenius
reaction rates and heat rclcasc. The
reduction
is most important
for conditions
of
large relaxation
time, high heat release and
small activation energy.
In practical terms, the reduction
suggests it
may be possible to ignite a solid faster while
using less cncrgy, since the ignition heat flux
would be applied over a shorter period of time.
In addition, heat flux relaxation could have an
important
effect on other processes
involving
solid-phase
Arrhenius
reaction
rates such as
thermal
explosion
(autoignition)
and cxtinction.
Hence, accounting
for heat flux relaxation
with hyperbolic
conduction,
when appropriate,
may improve the accuracy of predictions
for
ignition and othrr processes that have traditionally been based on Fourier’s
law, Perhaps
more importantly,
occurrence
of relaxation offers the potential
of developing
novel processes not anticipated through use. of Fourier’s
law. These processes
could exploit, for example, “thermal
resonance”
(maximum
amplification
of thermal
waves) predicted
for
hyperbolic
conduction
[22]. For instance,
laser-induced
ignition with a heating cycle oscillated to excite resonance
would produce
higher temperatures
compared to nonresonant
heating, and faster ignition. (This hyperbolic
resonance
is not the same as resonance
of
Fourier combustion
waves referred to, for example, in [23].)
Further
Studies
Although several effects can reduce the importance of heat flux relaxation,
as discussed previously, the potentially large reduction
in ignition time provides motivation
to further study
the effects of hyperbolic
conduction
and begin
studies with other nonFourier
mod&.
One route for further study is to determine
the relative importance
of hyperbolic and other
nonFourier
models that may apply to nonhomogeneous
solids, e.g., [24, 251. These other
models can also predict higher temperatures
than Fourier
values. Also, experiments
are
needed with inert and reactive solids to better
understand the mechanism of nonFourier
con-
J. ANTAKI
duction and test nonFourier
models. In addition, these experiments
could evaluate the possibility of deliberately
inducing
nonFourier
conduction
to exploit its behavior.
Finally, the ignition analysis of this work
deals with large values of relaxation
time T.
However,
nonFourier
conduction
is essential
to the emerging
field of microengineering
where r is small [21]. This microscale
nonFourier conduction
pertains to small-scale and
short-time
events not adequately
modeled by
Fourier’s
law. For example, nonFourier
conduction can occur in short pulse (IO ” s) laser
heating of silicon thin films (< 1 pm) where
7 - 0 (10~”
Sk
In microengineering,
many solid-phase
rate
processes are described by Arrhrnius
kinetics.
One process is laser annealing of silicon thin
films during integrated
circuit
fdbrication
to
repair datttdgc to the films [26]. Here, the early
stages of annealing can be analogous to ignition of a reactive solid. Thus, the ignition analysis of this work may provide a convenient
starting point for related studies in microengineering.
Eileen Foy suggested
this article.
many
improremenfs
to
REFERENCES
I. Kaminski, W., J. Heat Tntm/& II2555 (1990)
2. Mitra. K.. Kumar. S., Vedavarz. A., and Mosllcmi.
M. K.. 1. Hem Trmufir 117~568(1995).
3. Kuo, K. K.. and Kookcr, D. E., in Nonsfoxly LJuntbrg
and Co>nbl:rfion Subilify of Solid fmpdllunts (L.
DeLuca. E. W. Price, and M. Summertield, Eds.),
American Institute of Aeronautics and Astmnautics,
Inc., Washington. DC, 1992, p. 494.
4. Mcrzhanov, A. G., In,. J. SHS 4323 (19%).
“srma,
A.. and Lebra,. 1. P., C,~e,n.Eng. Sci. 47x2179
5.
(19y2).
6. Knyrvcva, A. G., and Dyukarev. 8. A., Co~nbrcrr.
Ex/h~. Shock WUIVS3I:3114(1995).
7. Bradley. H. H., Jr.. Combats/. Sci. Tednol. 2: I I (1970).
8. Liibn. A., and Williams. F. A., Comhrrsr. Sci. Tecknol.
3:91 (19771).
9. Hermilnce, C. E., in Fw~damcnrals of Solid /‘ropNonr
Combrrsrion (K. K. Kuo and M. Summerfield, Eds.),
American Institute of Aeronautics and Astronautics,
Inc., Washington. DC, 1984,pp. 239-304.
Ill. Antaki. P. J. Paper No. 95.2044, Thirtieth Thermaphysics Conference, American Institute of Aeronautics and Astronautics. Inc.. Washineton DC. 1995.
II. &isik, M. N., and Tzou, D. Y., J. Hwl Transfer
116526 (1994).
NONFOURIER
CONDUClXON
IN SOLID
I’?.
II.
14.
IS.
Ih.
17.
IX.
I?).
?(I.
?I.
22.
23.
24,
2s.
339
REACTIONS
(IYW.
Frischmuth. K., and Cimnwlli. V. A. Irzr. 1. Et‘,rp?g
sci. 33:2nv (Ivw.
In this inert period, surface temperature
is
approximated
by me inert hyperbolic soluti&t.
For large times, nowever, surface temperature
is high enough for heating by reaction to dormnate and cause thermal runaway. Hence, this
rough approximation
defines “ignition time” as
the moment when rates of internal heating by
reaction and external heating by flux become
equal. This approximation
is rough because,
for example, heating by the flux could still be
important
for times greater than this ignition
time.
This rough approximation,
and part of the
development
of its expression,
are analogous
to the rough approximation
for the Fourier
ignition problem [S, 181. However, developing
the expression for hyperbolic
ignition time requires
an assumption
not needed for the
Fourier problem, as described shortly.
The development
begins by defining T,, as
the hyperbolic
ignition
temperature
in this
rough approximation,
then stating an equality
of heating by internal reaction and external
flux [lSl,
26.
pf,ukT,e-“/‘RTd
l
=
,
(1A)
4,k2
APPENDIX
Equations and the figure number followed by
A refer to those appearing only in this Appendix. Other equations
and figures, and all
references
and dimensionless
variables,
refer
to those from the main body of this article.
Approximation
for Ignition
Time
The rough approximation
for hyperbolic
ignition time of the solid takes advantage of expected behavior
for its surface temperature:
For small values of time after application
of
the external heat flux, internal heating by reaction should be small compared
to heating by
the flux, since surface temperature
and reaction rate arc still relatively
low. Thus, these
small time values correspond
to a period of
neady inert heating dominated by the external
heat flux.
which is satisfied at the rough ignition time.
Further,
placing Eq. 1A into dimensionless
form defines the dimensionless
hyperbolic ignition temperature
@a,
6 = eP/“,,,
(2A)
where S,, > 1 and 6 t 1.
The next step is to use Eq. 3 to deduce the
time at which Eq. 2A is satisfied, since ignition
occurs at this time. Placing Eq. 3 in dimensionless form, then using Eq. 2A to eliminate 6
leads to
where
(4N
340
PAUL
Equation
3A reduces to the inert hyperbolic
heat equation
for r = 0 and ear/@
= 0.
Thus, finding the values of temperature
0, and
corresponding
values of time 6, that reduce
Eq. 3A to the inert equation defines the inert
heating period. At the end of this inert period,
hyperbolic ignition occurs at &. These values
of 0 are partly found using Eq. 4A by noting
for 0 < fI, and B > t3,, r tends to be small
c-s 1) and large (a I), respectively, where r =
1 for 0 = 0,. Hence, neglecting small r in Eq.
3A for 0 < e, partially reduces it to the inert
equation.
For the Fourier
ignition problem (E = O),
times corresponding
to small r completely define the inert heating period [S]. For the hyperbolic problem, however, fully reducing Eq. 3A
to the inert hyperbolic
equation for 8 < 0,
requires e aI’/&$ also be small.
Unfortunately,
~ar/rl.$
may not be small
for 0 < 0,. Referring
to Fig. 3, for example,
let an estimate for hyperbolic ignition time he
& = 0.3 with 0, = 1.8, since inspection of the
figure shows the effect of reaction first becomes obvious there. Then, with 6, and 0,
defined as a smaller time and corresponding
temperature
in the figure,
ar
rH- 6
cag=E( 4r. 1
J. ANTAKI
where ignition occurs when 5 = .& and &,
= e,, marking the end of inert heating. Hence,
substituting
Eq. M into Eq. 2A leads to the
rough approximation
for hyperbolic
ignition
time given by Eq. 13 in the main body of this
article.
Finally, assuming E d r/dg
is small to arrive
at Eq. 13 for &, artificially extends the time
allowed for inert heating. This extension occurs because 6” (as an approximate
end of
inert heating) should be smaller than the value
given by Eq. 13 to ensure l dT/d(
is truly
small, instead of being small by assumption.
Fortunately,
this extension of the inert heating
period gives a better approximation
for & as
an ignition time by moving it closer to the
“true” ignition time characterized
by thermal
runaway (e.g., 82 = 0.4 in Fig. 3).
Internal
Temperature
Figure 1A compares internal temperature
profiles for the hyperbolic reactive and inert cases,
and reactive Fourier case, at the time of hyperbolic ignition e$ = 0.4. At this time, the thermal waves for both hyperbolic
cases have advanced to nN, = 0.8.
(5A)
Choosing
.$, = 0.275 with 0, = 1.77 to compute r, with Eq. 4A, Eq. 5A gives c dr/dt
=
2.7, which is not small. (However,
cdr/dt
does become small for sufficiently small 5, or
E.)
Thus, the assumption
needed to conveniently obtain an expression for hyperbolic ignition time is: Although
B dr/d[
may not be
small for t7 < 9,,, it is assumed small to fully
reduce Eq. 3A to the inert hyperbolic
equation. The consequence
of this assumption
is
evaluated shortly.
After reducing Eq. 3A to the inert equation
its solution gives the exact expression [15] for
surface temperature
of the solid, Bi,,,,, during
the inert heating period,
eincrr = 1 + C”*e-f’**([l
+(5/E)l,(f/2e)l,
+
(5/s)lf,,(y2E)
(6A)
DIMENSDNLESS LOCATION: ‘I
Fig. IA. Comparison of hyperbolic and Fourier intrml
temperature
profiles at time of hyperbolic
ignition.
NONFOURIER
CONDUCTION
IN SOLID
The higher temperatures
of the reactive hyperbolic case, compared to its inert case, result
from heat released by reaction.
Further,
the
figure shows the largest impact of heat release
occurs relatively
close to the surface
1) = 0.
Consequently,
the reactive case converges
to
the inert case as TJ increases.
However,
the
difference
between hyperbolic
cases would increase if properties
and thermal wave speeds
were permitted to vary with temperature.
REACTIONS
Finally, the figure shows temperatures
of the
reactive Fourier case are less than temperatures of the inert hyperbolic case. These loruer
temperatures
reflect inert heating behavior for
the Fourier
case at this value of time. Although not shown, at later times the increasing
rate of heat release for the Fourier case causes
its temperatures
at and near I) = 0 to become
higher than the inert hyperbolic values.