Theoretical models for the combustion of alloyable materials

Theoretical Models for the Combustion of AIIoyable Materials

ROBERT ARMSTRONG

The purpose of this work is to extend a theoretical model of layered (laminar) media for SHS combustion presented in an earlier article ~] to explore possible mechanisms for after-burning in SHS (i.e., gasless) combustion. As before, our particular interest is how the microscopic geometry of the solid reactants is reflected in the combustion wave and in the reaction product. The model is constructed from alternating lamina of two pure reactants that interdiffuse exo- thermically to form a product. Here, the laminar model is extended to contain layers of differing thicknesses. Using asymptotic theory, it was found that under certain conditions, the combustion wave can become "detached," and an initial thin flame propagates through the media, leaving a slower, thicker flame following behind ( i .e . , afterburning). Thin laminae are consumed in the initial flame and are thick in the secondary. The thin flame has a width determined by the inverse of the activation energy of diffusion, as found previously. The width of the afterburning zone, however, is determined by the absolute time of diffusion for the thicker laminae. Natu- rally, when the laminae are all the same thickness, there is only one thin flame. The condition for the appearance of afterburning is found to be contingent on the square of the ratio of smallest- to-largest thicknesses being considerably less than unity.

I. INTRODUCTION

THIS article is principally concerned with the flame propagation induced by the exothermic interdiffusion of two solid substances (usually metals). Systems of this class are potentially important to the production of high- quality ceramics and belong to the larger topic of combustion synthesis. In a previous article, ~1] a theoretical model showed that lamina of alloyable substances would give rise to a self-propagating combustion wave. It was shown that an activation energy barrier in the diffusion coefficient would behave much as the chemical-reaction activation barrier in well-known models of premixed gas flames, t21 Diffusion in the SHS systems is essential, however, since there is no analogue to a "premixed" flame here. An activation energy barrier to interdiffusion keeps the differentiated particles of alloyable material distinct until the thermal wave raises the temperature to the point that interdiffusion can contribute to its propagation. Though both of these systems can have aspects of mass diffusion and reaction, the gas flame derives its propagation mechanism from exothermic, locally homoge- neous chemical reaction, while the alloying reaction relies on mixing one constituent with another. Afterburning re- fers to the general characteristic exhibited by many of these systems that, after the thin flame zone has passed, a residual quantity of material continues to react long after.

Ceramics have desirable properties for use in combustion machinery, such as turbine blades in jet engines. However, they are notoriously difficult to reform (by grinding, etc.), and thus, it would be of considerable use to form the shape of the desired product first and then undergo a combustion process to the desired ceramic

ROBERT ARMSTRONG, Technical Staff Member, is with the Combustion Research Facility, Sandia National Laboratories, Livermore, CA 94551.

This article is based on a presentation made in the symposium "Reaction Synthesis of Materials" presented during the TMS Annual Meeting, New Orleans, LA, February 17-21, 1991, under the auspices of the TMS Powder Metallurgy Committee.

in situ.[3] The final strength and other properties are dependent on the details of the interdiffusion process during the time that the flame passes. [4]

Typically, the dynamics of the flame involve a bal- ancing between the large length-scale parameters (temperature, etc.) and microscopic parameters (particle size, diffusion, etc.). There are a number of investigations that treat the powder media in a "mesoscopic" fashion, de- riving global rate of conversion of reactants by treating the microscopic geometry of the powder as a phenom- enological continuum, tSl Aldushin and Khaikin, t6] on the other hand, have studied the "flat particle" problem, in which the geometry of the alloying system is idealized. Their model assumes that the powdered media is equiv- alent to alternating laminae of alloyable material. This simplification allows the analytic calculation of a flame speed while keeping all of the microscopic detail of the interdiffusing laminae. Afterburning is usually modeled by inserting an exponential inhibition to the phenome- nological relation for the reaction rate. [5] This is often identified with a growing inhibition to diffusion as the reaction proceeds. 171 Aldushin et al. tS] used a "flat particle" numerical model to look at the role of particle size in SHS combustion. They found that powders containing a wide range of particle sizes will exhibit afterburning.

Oscillatory instabilities are often observed in experiment, [31 because the length of thermal interaction (which defines the size of the instability) is much larger than more common systems involving solids (e.g., solid pro- pellants[9'~~ Margolis et al. t~l] have extended stability analyses to get bifurcation results that give a good geometric description of these instabilities. Filtration combustion t~2j involves percolation of a reactive gas to the flame zone, where it reacts with the solid powder to yield a useful material (e.g., titanium nitride). Models for this system generally consist of an overall reaction rate and a detailed accounting for the gas percolation to the reaction zone.

The intention of this article is to investigate configurational-geometry influences on afterbuming. We

METALLURGICAL TRANSACTIONS A U.S. GOVERNMENT WORK VOLUME 23A, SEPTEMBER 1992--2339 NOT PROTECTED BY U.S. COPYRIGHT

will use mixtures of different lamina thicknesses in a model similar to previous work. t~] It has been shown t~l that models of this type give a good indication of combustion characteristics of three-dimensional random media (e .g . , powders).

I I . THEORY

A modified form of activation-energy asymptotics will be used where the large activation energy is identified with diffusion processes rather than reaction rates. In classical premixed flame theory, a reaction, dependent only on the reactant concentration and not its spatial derivative, is activated by a temperature approaching the flame temperature. In SHS combustion, mass diffusion processes will be activated in the same fashion; however, mass diffusion, by its very nature, will depend on the local concentration gradient. There is no correspond- ing analogue in SHS combustion for a premixed flame. Because the reactants are all condensed-phase constituents, the Lewis number for these systems is extremely large (> 1 0 3) a t the flame temperature, t61 Though the Lewis number is often considered infinite in condensed-phase reactions like propellant systems, It3] here mass transport is the only vehicle for heat release. Thus, we require the characteristic dimension over which mass diffusion will occur to be very small compared to thermal transport dimensions. As shown previously, t~] the Lewis number is large enough at the flame temperature that this mass diffusion distance will be much smaller than the flame thickness and can be used as an additional scaling pa- rameter for the asymptotic analysis. The thickness of the flame is proportional to an inverse Zeldovich number, TM

and the distance over which diffusion occurs is proportional to an inverse Lewis number.

As in the previous work, I~ two constituents will be considered, labeled "A" and "B," which will codiffuse, releasing heat simply by virtue of mixing on the molec- ular scale. We assume that A and B form a solution and that all compositions are traversed in the progress of "reaction" from domains of pure A and pure B to a ho- mogeneous 50-50 mixture. There are some SHS systems that clearly conform to this assumption tl41 and some that do not. ItS1 It is expected, however, that SHS systems that are diffusion-controlled will all be qualitatively similar to the model employed here. One expects the diffusion-controlled assumption to break down as the laminae become thinner. There are, however, examples of exceptionally thin lamina that still fit into this cate- gory. Clevenger et alJ ~61 have observed combustion waves in the Ni /Si system for lamina ~100 .~ that are appar- ently still diffusion-controlled. In addition, it is assumed that the thermal diffusivity, heat capacity, and mass diffusivity are composition-independent and that thermal diffusivity is independent of temperature as well. The mass diffusivity is assumed to have a simple Arrhenius temperature dependence, and it is also assumed that there is no substantial volume change with composition.

Choosing C as the mass concentration of A, the general mass diffusion equation translating in the negative :? direction at a velocity t~ can be written

0 C a C . . . . ----v + i f - - - V . D V C = 0 [1] Ot O~

Since there is no volume dependence on concentration, the presence of A indicates the absence of B, and the concentration of B is Co - (~, where Co is the concentration of pure A. The volumetric heat generation rate can be obtained from the enthalpy as a function of composition, which must have a maximum somewhere between C = 0 and (~ = C0. The enthalpy of the system is assumed to be a function of composition and temperature:

- H0 = fi( 7~ - A((~) [21

where/-7 is the enthalpy of the system, and ~ is a function that relates the enthalpy's dependence on composition. Overall, the enthalpy is conserved, and thus, the material derivative of enthalpy is equal to the negative divergence of the thermal flux.

_

a? ~ ai [31

The thermal transport equation may be written

(p + a - - - X V 2 1 f - = + a - - [41 a s o t o.~

where T is the temperature, )t is the thermal diffusivity, ( is the heat capacity (assumed constant), and fi is the density. A constant flame speed ff can be anticipated, and the concentration, t~, can be normalized by 1/2 Co. In addition, the spatial coordinates are nondimension- alized by the thermal length ( A / f f ) and time by the thermal time (X/ff2), while the temperature is normalized by the flame temperature and mass concentration is normalized by its maximum:

OC OC - - + - - - V . L e - I V C = 0 [5] Ot Ox

- - + - - - - ~72T = + A(C) [61 Ot Ox

where C = 1 is the state at which everything has com- pletely interdiffused or, equivalently, totally reacted. (The terms "interdiffusion" and "reaction" are, for the pur- poses of this article, semantic equivalents.)

It is expected that the Lewis number, Le = A//~, will exhibit an Arrhenius dependence and will be defined here as

L e - l = A e x p - [71

Leo ~ = A e x p - [8]

where Leo ~ is the maximum that Le -~ attains at the flame temperature. In Section III, we will seek a steady state so that the time dependence in Eqs. [5] and [6] can be neglected.

We choose to stack the laminae perpendicular to the flame propagation direction, or equivalently, the mass diffusion direction is perpendicular to the flame direction, which demands a truly two-dimensional model, though

2 3 4 0 - - V O L U M E 23A, SEPTEMBER 1992 METALLURGICAL TRANSACTIONS A

a steady solution can be expected. We seek a solution that consumes the thin laminae in a thin flame and the thick laminae in an afterburning zone. Figure 1 shows the general scheme of the semi-infinite lamina problem and serves as a definition sketch. The mass diffusion equation of A into B is

Ox ~xx + ~ . L e - ' i--Ox + ~ = 0 [91

where R and 5' are the unit vectors in the x and y directions. Here, the boundary conditions are stated in terms of

limits far in front of and in back of the flame:

C"~Co as x--+ - ~ [10]

C----~ 1 asx----~ +oo [11]

For convenience, we choose the reference states for A to be such that

A(0) = A(2) = 0 [121

Note that there is an energy conservation constraint: (2A(1) -- (A(2) + A(0)))/2 = 1 - To (at the flame temperature, T = 1). Anticipating that the heat is released over a length much smaller than the thermal scale, we assumed that the y variation in temperature is negligible and can be replaced by an average over y:

Of [ O ' ~ 2 f = OA(C)

kTJ [131 Ox Ox

where the overbar denotes the y-direction average:

L

A(C) = l i m - L dyA(C) L~oo 2 L

[14]

Most of the mass diffusion for the 8-sized laminae, and hence, heat generation, will occur when Le -1 Le-~(Tr at the temperature in the initial thin flame zone. In the &sized laminae, it is certain that the distance over which diffusion remains active is much thinner (define it to be 0 ( e ) ) than the thermal induction length (1 in nondimensional units). The time that a point in the re- acting media spends in the flame is the flame thickness divided by its velocity ( X / a e ) / a or simply - e . It is nec- ess<a~ in order for the initial thin flame to exist that Leo ~ 6Z/e. In other words, mass diffusion in the smallest &sized laminae must occur appreciably over its thickness in a length of time of order e. From reasonable values t~T1 for diffusion coefficients in SHS systems, this implies 8 '~ e. To obviate this scaling, we redefine the Lewis number:

Le - t = A - e x p - - [15]

where A is an "eigenvalue" to be determined and from which the flame speed will be derived. Since diffusion only exists in a small e-sized region, external to this region there is an outer solution identical to the classical,

chemical-reaction flame results (locating the flame at x = 0 ) :

= = ; -< 0, To + (1 - T0)~b exp (x) (region I); > 0, T(x ) (region III)

[161

where T O is the initial unburned solid temperature and & is the volume fraction reactant contained in the &sized laminae. The thermal behavior for x -> 0 above is yet to be determined for region III. This defines the temperature of the initial flame (region II):

T ~ = To + (1 - T0)~b [171

We anticipate that behind the e-sized initial flame, only the thin &sized laminae are consumed, and before it, nothing is consumed. This assumption will be checked later for consistency. Figure 2 represents pictorially the scaling for which we seek a solution to the overall combustion wave. Anticipating the results of Aldushin et al.,tsl we look for a solution that exhibits a thin primary flame associated with the small-scale slructure and a much wider afterburning zone associated with the large-scale structure. This assumes that the combustion process is dis- tributed between two zones: a thin initial flame at temperature Tr and a broad afterburning zone that com- pletes at the flame temperature T = 1.

I I I . I N I T I A L T H I N F L A M E ( R E G I O N II)

For the inner region, using the reasoning above, x should be scaled on a length e:

x = e ~ [ 1 8 ]

Because the diffusion processes will be much slower for the large laminae ( -Le(Tr 2) than the small ( - L e ( T r ~2), we introduce the stretched coordinates:

y = 677 [19] y = Lh

where t$ ~ L ,~ l , r/ is identified with the lamina of thickness t~, and h is identified with the lamina of thickness/7. It is recognized that the total repeat distance, Z, is arguably a more appropriate distance by which to scale diffusion. However, if L is not a significant fraction of Z, the contribution of the thick laminae to the overall flame temperature is also not significant and, therefore, negligible. Thus, we will assume L - Z and choose L

consistency with the other stretched to maintain coordinate 77.

Choosing

e = T/~ [201

and

f = T ~ - e/? [21]

we can write down the equations for the inner region as

METALLURGICAL TRANSACTIONS A VOLUME 23A, SEPTEMBER 1992--2341

(a)

(b) Fig. 1 - - ( a ) Schematic of the thermal profile of SHS combustion with afterburning. (b) Represents the one repeat unit of the laminar structure in the regions identified by, (a). Black represents component A, white laminae represent component B, and crosshatching represents inert product P. Distinct regions are identified: (I) preheat region, (II) thin combustion zone at T ~ Tg, characterized by rapid combustion of the thin laminae, and (III) afterburning zone characterized by slow conversion of the thick laminae. (b) Serves as a definition sketch for lamina dimensions 3, L, and Z.

2342--VOLUME 23A, SEPTEMBER 1992 METALLURGICAL TRANSACTIONS A

o=

E

,= Direction of Flame Propagation

I . ~ - - 1 =I I= - g ' =1

:11

/

allerbural

~' thin flame

l Distance

x=O

IV

p~t II~me

Fig. 2 - - F i g . l (b) redrawn wi th the sca l ing parameters appropriate to the var ious regions in the f lame identif ied.

l O C

62 { 1 0 1 O 1 ~ } C . A - - g -0) ~ - - - + ~ - - - + ~

e eO~ LOh -6 = 0

[22]

[23] + - 0 - 10 (c)

e e 0s c

Note that 7 ~ and, hence, 0 are independent of y and, hence, h and ~7. The left matching boundary conditions can be derived from Eq. [16] for the above:

lim 0 = + ~ [24]

lim - - = - ( 1 + T0)~b [25]

Thus, the initial condition Co can be written

Co = 1 + Or(h) + Os(r/) [26]

where OL and | are simple functions that take on the values - 1 and 1 and represent the thick and thin laminae, respectively (Figure 3). Since the two types of lamina cannot occupy the same place at the same time, we define O8 as

O8 = Sq(r/) (1 - [OL(h)l) [27]

where Sq is a square wave of period 4 8 to be compatible with earlier work. m The term O8 should be considered principally as a function of r I parameterized by the coordinate h. As will be shown, this " h " functionality will only manifest itself on averaged values, similar to that of Eq. [141.

By construction, OL is nonzero (i.e., has support) only in the domain of the thick laminae and O8 is nonzero (i.e., has support) only in the domain of the thin laminae. Thus, the domains of support for the two O's are disjoint (i.e., a thin laminae cannot occupy the same space as a thick one). Keeping the terms of lowest order in Eqs. [22] and [23],

OC Ae_O C = 0 [28] of

z _ ~ [291

Note that/~ and, hence, 0 are independent of 7/and h. Defining an integration variable,

/ ~ d~'e -0(~') [30]

and its inverse,

= - l o g \ 0 s o / [311

the boundary conditions under this transformation become

lim ~ = 0 [32]

lim - - = 0 [33]

The Green's function solution for Eq. [28] becomes

C(,~, TI) = (*~d~?'G(~, ~? - r / )C0(~/ ' ) [34]

where G is defined as

1 G ( ~ , r/) - _ _ e - ~ 7 2 / 4 A ~ ; [35]

2V- y

From Eq. [26], we have

C(~,Tl , h) = 1 + |

+ f / ~ d r f G ( ~ , ~7 - ~7')08(~7 ') [36]

Note that since 1 + OL(h) only takes on the values of 0 and 2 and that @L has support only where 08 does not, and by definition A(0) = A(2) = 0, domains of h-thickness laminae will contribute nothing to the heat release and act as a diluent. Diffusion will cause "blurring" at the edges of the domains of support for | and | however, but in any y-direction average, this difference will contribute, at most, - ( 1 - qb)6/L. Thus, any significant but finite domain in 7/contributes C(6) to the average:

L if_ z~(.~, ~7,h) = l i m - L d y A ( C ( ~ , y ) ) t--,= 2 L

L if_ = z~(.~) = 4~ lim - L dr I L ----~~176 2 t

+ ~ (8 )

d~' G( ~ , 77 - n')Sq(n'))

[37]


1-

O~, Sq

~ - 26 ---~]

(a)

[, t,] k

-1- U

(b)

Fig. 3 - (a) and (b) The functions used to represent the thin and thick laminae, respectively: Co = 1 + OL(h) + 0~(~). | is magnified, and thus, the scales between (a) and (b) are not comparable. In addition, O~ is a square wave of wavelength 4t5 but vanishes anywhere Ot is nonzero.

Integrating Eq. [29],

O0 - - = A - (1 - T0)(h [381

using the matching condition from Eq. [16] z~ ~ 0 as ~ ----+ --00.

Substituting integrating,

o#

the definition of ~ (Eq. [31]) and

- - - - s d ~ ' ( A ( ~ ' ) - ( 1 - T o ) ~ b ) [39]

The fight-side boundary condition will put an additional constraint on Eq. [39], and from this, the flame speed eigenvalue and, hence, the flame speed will be determined. This is the limit: ~ --+ 0% and it is clear that G(~, ~7) requires that all of the thin laminae are consumed in this limit. We do not know the boundary conditions at this limit because we have not yet determined the thermal profile in the afterburning region (III) where the L-sized laminae are consumed.

IV. A F T E R B U R N I N G ( R E G I O N III)

We now look for a solution in region III that admits consistent boundary conditions between the thin-flame zone and the afterbuming zone and, from this, compute a flame speed. Since, using the above reasoning, all of the &sized laminae are consumed in the limit we seek,

C and T must be independent of 77. This motivates seek- ing the solution in this region in terms of the following coordinates:

x = q [40] y = Lh

From Eq. [9],

{0 10 / 10}c + S' "Le-1 Oq ~qq LO'-h :x + Yzoh = 0

[41]

For the thicker L-sized laminae, it is convenient to recast the Lewis number into a form that reflects its maximum in the afterburning region (III):

01 where

/x -- L 2 / L e o I ~ 1 [43]

defines the length in the flame structure that significant diffusion will occur in the L-sized laminae. Note that, by definition, length scales of order one vary over lengths similar to the thermal induction zone (region I). Other scalings than the one chosen for /* are possible, but in view of the fact that the thermal induction thickness is typically less than 100 pm, it is unlikely that such flame structure could be identified experimentally. In any case, it is unlikely that it would be identified as afterburning. Choosing a new stretched variable that represents this distance,

q = / x ~ [44]

substituting into Eq. [41] and keeping lowest order terms,

OC e(-(r~}/~) ((~/f)-l)) C = 0 [45] o=.

casting into a thermally weighted coordinate similar to Eq. [30],

dp=s [46]

with these and Eq. [45], we have

C(@, rl) = 1 + f ~ dh'G(@, h - h')Om(h') [47]

Equation [13], under the stretched variable ,= to lowest order, becomes

dT 0A - [48]

0=. o_=

or

T = A + To [49]

The overbar still denotes the y-direction average (Eq. [14]); here, the average must be taken over the h


coordinate. Equation [48], given the definition of qb, can be rearranged into

OF. - exp 1 [501 B + ~(+)

The above renders the solution as a single quadrature for which we will not seek a formal solution, since it is ines- sential to obtaining a flame speed. This enables us also to state the matching conditions between afterburning zone and the preceding thin flame and thus determine the flame speed:

o? o~ o? - - = l i m - - = l i m / x - - [ 5 1 ] Ox +-+: 0~ _=--+o aE

Since /z is small and O? /0~ is finite, to lowest order in/~,

l i m - = 0 [52]

Thus, /~ must be a constant in this limit, and since it is also monotonically decreasing over its entire domain, Eq. [21] allows

lim/~ = 0 [53]

or

lim - - = 1; lim ~ = o0 [54]

Using this result in Eq. [39], the "eigenvalue" (A) and, hence, flame speed can be determined.

Now we choose a simple parabolic form for the enthalpy of reaction A for comparison to earlier work: tu

A(C) = (1 - To) (1 - (1 - C) 2) [55]

which conforms to Eq. [ 12]. Using the definition of Co from Figure 2 and Eq. [37], the above becomes

( 8 ~176 1 -2A~:(nlr/2)2 ) 7 7 . 2 z~=(1 -T0)~b 1 Z - - [56] ,~{1.3,5....} n2 e

From the matching condition drawn from Eq. [39], the flame speed can be calculated:

1 1

A = g 6(1 - To) = 6 ( T r To)

or

RT,y gfi~ 6X2___z_ ~ e - / ,r

E a~ = ~ re - ~0 [57]

r~

This result is effectively the same as the flame speed calculated in previous work, [u except that here the thick laminae act as a diluent, lowering the effective flame temperature from 7~/to Tr This produces a lower flame velocity than if all of the laminae had been 6-sized.

An important result is a criterion for the appearance

of afterburning: tz >> 1. That is, the length of the combustion zone for the slower burning material is significantly larger than the thermal induction zone (region I) ahead of the thin flame (region II). If the derived flame speed (Eq. [57]) is used in the definition of/~ (Eq. [43]),

6RT~

(~I 2 if" e_g/f~<<,/~r ~ l

C-?o [581

V. CONCLUSIONS

We have shown, using a simple laminar model for SHS combustion, that afterburning phenomena can be pro- duced from only geometric considerations. The model presented here exhibits an initial thin flame that leads a much thicker combustion zone following behind. This is obtained simply by providing a mixture of two different thicknesses of laminae and assuming that diffusion is controlling the combustion process. No resort is made to kinetic arguments, such as chemical "inhibition," often found in the literature, tT] The initial thin flame is associated with the combustion of the thin laminae. It is assumed that the thin laminae constitute a substantial fraction of the whole ( - 2 0 pct). The thicker afterburning region following the thin flame is associated with the thicker laminae. The thickness of the afterburning zone is not dependent on the width of the thicker laminae per se but is directly proportional to the diffusion time necessary to alloy these laminae.

The properties of the initial flame are most like that models. The of previous work dealing with laminar �9 tl]

initial thin flame is found to have properties similar to a system in which the thick laminae are behaving as a diluent--lowering the flame temperature because of their heat capacity while not participating in any exothermic reaction. The thickness of the primary flame is found to be proportional to the inverse of the diffusion activation energy. Since the width of the initial flame is set by the activation energy, the flame velocity is determined by this width divided by the shortest diffusion time ( i .e . , the time associated with the thin laminae).

The afterburning zone that follows the initial flame has a very different structure. It appears much more as an ignition/explosion problem. The thick laminae are ig- nited by the initial flame but very little is consumed over the time that the flame passes. This is due mainly to the long diffusion time associated with the thick lamina. This diffusion time determines the length that the afterburning region trails the initial flame. This length is the product of the velocity of the initial thin flame and the diffusion time for the thick lamina.

It is clear that the two different laminar thicknesses must be widely different--when their sizes approach one another, the two-flame structure must merge into one. The disparity in thicknesses here is gaged by the disparity in diffusion times between the two types of laminae. Note that the diffusion time is roughly proportional to the square of the laminar thickness. Since the diffusion coefficient is not dependent on the progress of reaction, the ratio of the squares of the thin-to-thick laminae widths has to be much less than unity ((5//5) 2 ~ 1) for


afterburning to be significant. This condition, although necessary, is not sufficient. In order for afterburning to be observed, this ratio must also be significantly less than the expression that appears in Eq. [58]. This expression is effectively a ratio of two Lewis numbers: the first evaluated at the temperature of the initial flame (where the small-scale structure is consumed) and the second evaluated at the final combustion temperature (where the large-scale structure is consumed).

Practical experimental systems are different in significant ways than the model pursued here. Experimental systems, except for those noted in the Introduction, usually employ powders, and thus, the ideal laminar structure assumed here is lost. It has been shown in previous work, m however, that a more complicated analysis involving random (powder) media yields results that are qualitatively identical to the laminar model. In this case, the largest coherent domain size (correlation length) is identified with the laminar thickness. Thus, if we sup- pose that there is a wide range of particle sizes in the random powder media, we would expect afterburning to appear. Afterburning is often observed in experiment, and differing particle sizes have been identified as a possible explanation in Ti /C systems. For small-sized Ti particles, the Ti /C reaction proceeds as a thin flame, but for large-sized Ti particles, afterburning appears, t18]

It is clear that a laminar configuration has been chosen in this work because it is more amenable to analysis. Experimentalists also find laminar systems attractive for SHS combustion, and often, for the very same reason: it renders the geometric contribution to the flame structure more understandable and the resulting product of combustion is more uniform. Anselmi-Tamburini and Munir tl4~ have performed some laminar experiments for sandwiched Ni/AI foils, and Holt tl8J is preparing experiments involving sputtered laminae of various substances. Practical application of laminar SHS is actively being considered for thin, highly crystalline ceramic coatings on metal surfaces. 118]

It is likely that in many practical circumstances, the width of the reaction zone (i.e., thin flame + afterburning region) could be larger than that of the experimental reactor. Indeed, if the reactor is a small fraction of the size of this zone, then the initial flame passes through the entire vessel before virtually any of the thick laminae (large particles) are consumed. The thick laminae (large particles) then burn homogeneously, and its diffusion time, being much larger than any other time associated with the system, becomes approximately the total reaction time. The picture here is an initial flame that passes rapidly through the mixture, igniting slower burning components than burn afterward at a spatially uniform monotonically increasing temperature. This sce- nario is tantamount to replacing the spatial derivative on the left-hand side of Eq. [48] with a derivative of a suit- ably stretched time coordinate.

Future work will focus on producing afterburning in random medium with the idea that this is similar to the powders used in experiment.

C

LIST OF S Y M B O L S

Arrhenius prefactor defined in Eft. [7] nondimensional concentration (C/Co)

C Co ( /5

G /-7 h Le

Leo

R T To

T

To

t

7 ff x

Z

Y Y $, 6 A

q5

A X /z

P =_

concentration [=] g /cm 3 concentration of pure A [=] g /cm 3 heat capacity [=] cal/g mass diffusivity [=] cmZ/s activation energy for mass diffusion [=] cal convenient timelike coordinate defined in Eq. [30] diffusion propagator defined in Eq. [35] enthalpy of reaction [=] cal /cm 3 y-direction diffusion distance (Eq. [19]) Lewis number, ratio of thermal diffusivity to mass diffusivity, X//) Lewis number evaluated at the flame temperature x-direction coordinate defined in Eq. [40] gas constant [=] atm cm3/gmol k nondimensional temperature: 7~/7~ I nondimensional initial unburned temperature:

initial flame temperature (region II, Figure 1) temperature [=] deg K flame temperature initial unburned temperature initial flame temperature (region II, Figure 1) nondimensional time time [ - ] s flame speed [=] c m / s x coordinate x = /7.f/A x coordinate [= ] cm unit vector in the x direction y coordinate y = ffy/A y coordinate [=] cm unit vector in the y direction length scale of diffusion 6 = ffr/A nondimensional heat release, A = A/fig heat release [=] cal /cm 3 afterburning region x coordinate (Eq. [46]) scalin_gfactor for the reaction zone

: R r # e scaled diffusion distance in the y direction defined by Eq. [19] temperature perturbation in the flame zone flame speed eigenvalue thermal diffusivity [=] cm2/s scaling factor for afterburning flame width (Eq. [43]) scaled reaction zone thickness in the x direction x = e~ density of the burning media [=] g /cm 3 scaled x-coordinate introduced by Eq. [44]

A C K N O W L E D G M E N T

This work is supported by the United States Department of Energy, under Contract No. DE-AC04-76DP00789.

R E F E R E N C E S

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18. J.B. Holt: Lawrence Livermore Laboratories, Livermore, CA, personal communication, 1992.


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Theoretical models for the combustion of alloyable materials