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Importance of nonFourier heat conduction in solid-phase reactions
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
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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.
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.