Radiant ignition of a combustible solid with gas-phase exothermicity

Acta Astronautica.

VoL 2, pp. 955-979. PelBamon Press 1975. Printed in the U.S.A.

Radiant ignition of a combustible solid with gas-phase exothermicity M. KINDELAN AND F . A . W I L L I A M S Department of Applied Mechanics and Engineering Sciences, University of California, Dan Diego, La Jolla, CA 92037, U.S.A. (Received 9 July 1975)

Abstract--An asymptotic analysis is presented for the ignition of a solid fuel that gasifies endothermically then reacts exothermically in the gas phase through a one-step Arrhenius process. The endothermic gasification was analyzed in an earlier paper; the present paper employs those results in treating the gas-phase reaction. It is shown that depending on values of the chemical parameters, the gas-phase ignition reaction may begin either at the time that gasification begins or during the later stage of transport-controlled gasification. In the former case, results for the gas-phase ignition time are similar to those for condensed-phase ignitions. In the latter case, a mixing layer develops in the gas-phase and moves away from the surface, the exothermic reaction being negligible until thermal runaway occurs somewhere within the mixing layer. Both early-time and late-time ignitions within the mixing layer are analyzed by solving partial differential equations derived from perturbations for large activation energies and a small ratio of gas to solid thermal responsivity. The theory serves to indicate how ignition by gas-phase reactions may be distinguished experimentally from ignition by condensed-phase or heterogenous reactions.

Introduction

IGNITION THEORY is concerned with solving sets of partial differential equations that describe the transient response of a chemically reactive material to an ignition stimulus. There are two broad classes of approach, each having different merits. In one the original differential equations are discretized and solved b y numerical methods. In the other, analytical methods are e m p l o y e d to develop a p p r o x i m a t e solutions of various types. The f o r m e r a p p r o a c h has the advantage of producing accurate results for specific materials, if definite values for all of the properties of the material are known. With sufficient effort, the numerical methods also can be made to provide parametric results, through either graphical presentation or a p p r o x i m a t e fitting of formulas, utilizing results f r o m a large n u m b e r of numerical calculations. The analytical approaches, on the other hand, yield parametric results m u c h more readily, and w h e n they are correct, they also greatly clarify physical understanding of the ignition processes. The main objections that have been raised against analytical methods concern their accuracy, which often has been questionable. H o w e v e r , asymptotic methods recently have developed to a stage at which they can give quite accurate analytical solutions to ignition problems. T h e s e methods are justified principally b y the factual observation that the nondimensional activation energy, the ratio of the overall activation energy for a chemical process to the thermal energy of the medium, is a large number, seldom less than ten during ignition. Analytical solutions to the partial differential equations are obtained by limit process expansions in which the p a r a m e t e r 955

956

M. Kindelan and F. A. Williams

representing the non-dimensional activation energy b e c o m e s large. The analytical procedures are systematic, not ad hoc. The analytical results thereby are elevated in accuracy, often to levels exceeding those of ab initio numerical methods. The present paper contains a further exploitation of these a s y m p t o t i c methods. A considerable amount of study has been given to the problem of ignition of a h o m o g e n e o u s reactive material that occupies a three-dimensional Euclidian half-space, initially has a constant t e m p e r a t u r e and is subjected to a constant rate of energy input per unit area at its planar surface, beginning at time zero, This problem is believed to a p p r o x i m a t e well the ignition of a thick slab of reactant, one of whose faces is exposed to a constant flux of visible or infrared electromagnetic radiation for which the material has a high absorbtivity. Probably no other ignition problem is closer to having been analyzed completely. Results already are available for the case in which an exothermic reaction, distributed within the condensed reactant, is responsible for ignition, and also for the case in which an e x o t h e r m i c heterogeneous reaction at the surface of the material produces ignition. Currently lacking is a parametric study of the case in which the material must first gasify endo-thermically then experience the exothermic ignition reaction in the gas phase. The first parametric results of such a study are reported herein. The primary motivation for analyzing ignitions under identical conditions of geometrical a r r a n g e m e n t and of external stimulus but under differing assumptions concerning the site of the ignition reaction, is the hope of being able to ascertain experimentally from ignition m e a s u r e m e n t s the location of the chemical kinetic steps that are important during ignition. Is it possible from ignition experiments to distinguish between solid-phase, heterogeneous and gas-phase ignition theories? All three types of theories must be available for the same e x p e r i m e n t before this question can be answered. The information provided in the present paper finally allows the question to be addressed. Analyses for gas-phase ignitions are more complicated than those for c o n d e n s e d - p h a s e or heterogeneous ignitions because at least two kinetic p r o c e s s e s must be considered in the gas-phase case, the endothermic gasification and the exothermic gas-phase reaction. This sequence of two kinetic steps m a k e s the theoretical problem challenging and may tend to impart greater physical reality to the model for both polymer flammability and propellant ignition. In a previous p a p e r we have completed a parametric study of the endothermic gasification (Kindelan and Williams, 1975). The present w o r k adds to this earlier study a treatment of the exothermic reaction in the gas. The relevant literature has been cited and discussed in the earlier paper, and therefore repetition of this b a c k g r o u n d information will not be given here. In the previous paper gasification was treated as an Arrhenius process that occurred either in depth or at the surface. The principal finding was that within the context of an asymptotic analysis for large values of the nondimensional activation energy associated with this process, three distinct stages occurred. Initially there was a stage of inert heat conduction within the solid, during which no gasification occurred. This inert stage was followed by a short stage of transition to gasification, which was characterized by a rapid modification of

Radiant ignition of a combustible solid with gas-phase exothermicity

957

temperatures in a thin layer near the surface of the solid and by a sudden diminution in the rate of increase of surface temperature. Following this transition stage was a longer stage of transport-controlled gasification, during which rates of surface regression increased and the gasification process influenced temperature profiles to greater depths within the solid. At very large times, the transportcontrolled gasification approached a steady state. The previous work produced formulas and graphs for the time to onset of gasification. This onset time was compared with ignition times calculated earlier for condensed-phase and heterogeneous ignitions. In these earlier asymptotic analyses that treated the nondimensional activation energy associated with the exothermic condensed-phase or heterogeneous ignition reaction as a large parameter, again inert and transition stages had been identified, but the exothermic transition stage described a transition to ignition, during which the rate of increase of surface temperature accelerated rapidly. In considering the transition to ignition, a very natural and precise definition of ignition time emerges as the time of thermal runaway that is calculated (within the context of the asymptotic analysis) to occur during this transition stage. It seems likely that experimentally measured ignition times, the easiest type of ignition data to find, will correspond with those defined by this thermal runaway, provided that the condensed-phase or heterogeneous model mechanisms are valid. The possibly surprising result of Kindelan and Williams (1975) was that for the same values of thermal and kinetic parameters, the onset time for gasification in the endothermic case agreed closely with the runaway time for ignition in the exothermic cases. This result has definite implications with regard to the ignition time in gas-phase ignition theory. To achieve ignition by a gas-phase mechanism, the fuel must first enter the gas. Therefore the gas-phase ignition cannot occur during the inert stage. If the gases are sufficiently reactive, then in a first approximation the gas-phase ignition occurs as soon as gasification begins. A first approximation to the ignition time then becomes the time to onset of gasification, and it is concluded that under these conditions ignition times are approximately the same for condensed-phase (or heterogeneous) and gas-phase ignition theories. The gas-phase ignition time will differ greatly from the time to onset of gasification only if the gases are less reactive. During the transport-controlled stage that follows the onset of gasification, the vaporized fuel occupies a layer in the gas of increasing thickness and mixes by diffusion with the ambient gas. A transient diffusion layer thereby is established in the gas during the transportcontrolled stage, and a sufficiently nonreactive gaseous fuel eventually undergoes an ignition reaction somewhere in this layer. The present paper is concerned with analysis of the ignition process that occurs in the transient diffusion layer. The results will show in analytical and graphical form how ignition times in gas-phase ignition theories can differ greatly from those in solid-phase or heterogeneous ignition theories. From the preceding description it is seen that the present analysis focuses entirely on processes occurring during the transport-controlled state of the previous paper. It will be necessary first to amplify the analysis of the transport-controlled stage, so as to provide a better description of the nonreacting

958

M. Kindelan and F. A. Williams

transient diffusion layer. Within the context of an asymptotic analysis for large values of the nondimensional activation energy for the gas-phase ignition reaction (which is treated here as an Arrhenius process of first order with respect to both fuel and oxidizer), it is found that this inert mixing continues to occur with the rate of heat release increasing gradually until at some time during the transportcontrolled stage, and at some position in the transient diffusion layer, a thermal runaway resulting in gas-phase ignition develops. An analysis is given herein of this gas-phase ignition process, and corresponding ignition times are derived from the results. The analysis is facilitated greatly by the existence of an earlier analysis (Lififin and Crespo, 1975) of the structure of unsteady diffusion flames. Lififin and Crespo calculated the ignition which occurred during inert mixing of two purely gaseous reactants. Their analysis of this ignition is quite similar to ours, the most important difference being that we account for the possibility of oxidant initially being contained within the fuel. The analysis which parallels that of Lifi~m and Crespo is justified best when thermal runaway occurs late in the transport-controlled stage. This is the case of greatest interest because it corresponds to situations in which the ignition time differs most from the time to onset of gasification. However, cases in which ignition occurs early in the transport-controlled stage also may be of interest. Such cases are considered briefly in the present paper, and corresponding ignition-time formulas are derived. Finally, it is worth emphasizing that even when the gas is sufficiently reactive for ignition to occur during the stage of transition to gasification, there may exist appreciable differences between the ignition time and the time to onset of gasification. These differences cannot be very great if the reactivity is such that ignition occurs late in the stage of transition to gasification, since in that case the analyses given for ignition in the transport-controlled stage soon become applicable, and it becomes improper to view ignition as occurring during transition to gasification. H o w e v e r , a very reactive gas can ignite very early in the stage of transition to gasification and can give rise to ignition times that are appreciably shorter than the previously derived time to onset of gasification. To avoid excessive length of the paper, the analysis of this case is not given here. Its deletion precludes comparison with results of numerical integrations of the partial differential equations (Kashiwagi, 1974) for the radiant ignition problem, since those integrations have been performed almost entirely under conditions such that ignition occurs during transition to gasification. Another important difference is that the results of Kashiwagi account for in-depth absorption of radiation, which is excluded here. In a subsequent paper we shall report our results for both in-depth absorption and ignition occurring during the stage of transition to gasification. Formulation

The assumptions and nomenclature of Kindelan and Williams (1975) are adopted here. The equations for conservation of energy in the solid, energy in the gas, oxidizer mass in the gas and fuel mass in the gas respectively become

Radiant ignition of a combustible solid with gas-phase exothermicity

aT aT O (A OT ~ P C ~ + PVC-~x = O'--x\ OX ] dTs

07"8

OYo+

OYo

0 { A OT,X

(1) weQ

w___o+ 0 / 2 OYo\

pv O~tt = P,

~

959

~p, Os--~-~),

(2)

(3)

and

o YF 4- pv OYF = WF+ 0 Ot O* ~ ~

/

~

OYF\

[P" D,-ff~),

(4)

where the gas-phase reaction has been taken to be of the form F + nO ~ products, and the multicomponent diffusion coefficients for fuel and oxidizer have been set equal. The domains are t > 0, x > 0, qP < 0; symbols are defined in the list of nomenclature. Subject to the approximation of a constant average molecular weight, the Arrhenius kinetics are written as

MFwo/nMo = WF = -- M~Ap~ YFYo e - ~ % ,

(5)

in which the overall order, as measured by the pressure dependence, may differ from two if A is allowed to depend on the pressure p. The initial and boundary conditions are T ( O , x ) = T ( t , oo)=T,(O, gt)= T~(t,-oo) = To, T(t,0) = T~(t,O), YF(0, ~ ) = YF(t,-o0) = 0, Yo(0,~) = Yo(t, -oo)= Yo®,

0I:01

2

- lvp = - vpYo + Ps Dg--~-~ s' --vp(1-1)=--vpYF

+

p,

2DOYF ""~ls'

aT , +Ps A~ aT~ 4 = - A -~x -~, -pvL.

(6) (7)

(8)

Since we have shown (Kindelan and Williams, 1975) that a correspondence can be established such that there is little difference between distributed and surface gasification, in the present work only surface gasification is treated, the condition v = - B e -E/Rr

(9)

being employed. The parameter l accounts for the possibility of oxidizer being contained in the solid fuel; the value ! = 0 may be used to describe ignition of a polymer capable of burning only by a diffusion-flame mechanism o r ignition of a potentially self-sustaining propellant in an atmosphere containing an oxidizer that is much more reactive than the propellant's oxidizer. The good approximation psDs = Ag/c~ (a Lewis number of unity) is introduced,

960

M. Kindelan and F. A. Williams

and the quantities p, c, A, c~, p~A~ and p~A are taken to be constant. The last of these approximations has been generalized by Kindelan (1975) to allow for a power-law dependence of A on temperature with an arbitrary exponent of order unity. In the nondimensional variables of the transport-controlled stage, the equations become

atO F

---

--

OOtO2+ aYe-

--+ Oto

atO_o%

v'--

(10)

v' Otog_ O ~ w' O~ +~'

v

, OY r

O~g

-

c] 2 Y r ,

O~',~

- w'

(11)

'

(12)

and 3 Y o + v' O Y o _ OeYo

with initial and boundary conditions tO(- 1, () = tO(w, oo) = tOg(to,oc) = Y r ( - 1, ~g ) = Yr(to,°c)=O, Y o ( - 1 , ~ ) = Y o ( t o , ~ ) = Y( .... tO.~=tO., ( O Y o / O ~ g ) . = v ' t Y , , ~ - l ) . (OYv/O~)s = v ' ( Y w + l - 1) and ~ \ / ~ a ' ( l , + F!r~] ' l + ( O ' - l ) t o ' ~ =01[1 + ~ ' l n [ 2 ( - l T ~ - ) - F - ~ v ' / ] '

(14)

and (15) and w' = A ' Y F Y o exp {- E"/[1 + (0, - l)tO. 1}.

(16)

Transport-controlled mixing Kindelan and Williams (1975) have analyzed temperature fields during the transport-controlled stage, in the absence of gas-phase reaction, for the case in which F is small. Since this case is encountered most often in practice, the approximation of small F is retained here. Within the context of an expansion for large values of E ' , it was found in the previous paper that time histories of profiles of the nondimensional solid temperature tO depended only on the single parameter a ' . Numerical solutions for these profiles were obtained for various values of a ' , and analytical expressions for the nondimensional temperature gradient at the surface, (OtO/O~L, were derived for all to in the case a ' -- 0 and for both small and large values of to when a ' was nonzero. Here it is necessary to extend the previous results by developing expressions for gas-phase temperature and concentration profile histories, for use in the subsequent ignition analysis.

Radiant ignition of a combustible solid with gas-phase exothermicity

961

If qJ is treated as a known function of to, ¢ and a ' , then for small F eqn (15) shows that 2 F v ' / a ' r X ~ - 1 is a known function of to and a'. This function is plotted in Fig. 7 of Kindelan and Williams (1975). With w' = 0, the problem for determining $8, YP and Yo in lowest order then becomes *~{$s} = 0, -~{Y~} = 0, ~ { y o } = 0, ¢s(to, oo) = y~(to, oo)=0, Yo(tO, OO)= Yo~, qJ,(to, 0) = 1, YF(to, 0) = 1 - l , Yo(to, 0) = l, and YF(0,~8) = 0 , Yo(0, G) = Yo® and g~(O, ~ ) = exp ( - ¢-~) - - ~ ¢ ,

erfc (-~),

(17)

where ~ _ _ ~ + v , _ _ _ _asr a _ a~.~. 8z

(18)

The quoted boundary conditions at ~g = 0 are obtained from eqn (14) and the two equations preceding it by taking the limit E ' = 0% F = 0. The initial conditions at to = 0 are obtained through matching to the inert stage, as described in the previous paper. Initially the fuel concentration vanishes and the oxidizer concentration coincides with that at infinity because no gasification occurs during the inert stage; however, heat conduction in the gas occurs during the inert stage and produces the initial temperature profile given in eqn (17). This difference in initial conditions prevents an immediate correspondence from being established between temperature and composition fields. For the problem as specified there exists an exact linear relationship between YF and Yo. Specifically, if the definition Y = YF/(1 - i) is introduced, then the problem *~{Y} = 0, Y(0, ~g) = Y(to, oo) -_-0, Y(to, 0) = 1

(19)

YF = Y(1 - / ) , Yo = (! - Yo®) Y + Yo®.

(20)

is obtained, and

For the temperature, the problem is .~{~bR} = O, ~b~(to,oo) = O, ~b,(to, O) = 1,

(21)

with the initial condition given by eqn (17). The problems for Os and Y may be solved separately since they are not coupled. Aside from the parameter or, which appears in the functional form of v', there remains only one parameter in each problem, viz. e = 2F/o~'rXF~, which enters explicitly through c' and which is taken to be small. For small e, the convective term in eqn (18) is large whenever ev' is of order unity. Equations (11) and (51) of Kindelan and Williams (1975) show that

[2/ev'

[(I+~')-

:o+; a'

+'..

to small (22) to large

962

M. Kindelan and F. A. Williams

for a ' = 0, eqn (44) shows that ev' = (2/~r) t a n ' (V~o). Cases of small a ' are not likely to h a v e e small and are not treated here because they seldom arise in practice. Equation (22) shows that ev' is of order unity when w is of order unity or large. It follows that convection is a dominant effect in the gas in the cases of greatest practical interest. This convection serves to establish a growing region of nearly constant conditions in the gas adjacent to the surface. Farther f r o m the surface, a moving transient diffusive zone is established, and this zone will be analyzed in the present section. Before proceeding with the analysis, it is helpful in understanding the p h e n o m e n o n to first consider briefly the situation when w is small. The convection term vanishes for ~o = 0 but b e c o m e s significant when w is small. Use of eqns (18), (19) and (22) reveals that in the stretched variables d~ = 2w/~re, ~ = X/2~'./X/-~e, the problem for Y may be written to the lowest order in e as 8Y

OY

= b X/~

32Y

-

, Y(0, 2) = Y(dJ, o~) = 0, Y(&, 0) = 1,

(23~

in which transient, convection and diffusion terms all remain. There exists a series solution of the f o r m Y = Yo + ,SY, + • • -+ where Yo, Y, . . . . are functions of ~/~/~. Specifically+

~e_e2,++(l+~?~)[l_Cf,, i/v~

Y=erfc(__~ ~ + 2¢b 2x/g/ 3~/g t

( l + x 2 ) =e ~ / 2 d x ] } + . " + (24)

where C = [fo(l + x ~) 2e-~~]2dx] ' = 1.5958. For use at larger values of 6), eqn (23) was solved numerically. The results given in Fig. 1 show the beginning of the d e v e l o p m e n t of a fuel layer at the surface of the solid, with any arbitrarily defined b o u n d a r y of this layer having ~, proportional to X/~ initially and increasing more rapidly with w later. The t e m p e r a t u r e problem for small o) is fundamentally different from the ,

i

E

i

i

i

,

i

l

i

+

I

2

5

4

5

6

7

8

9

I0

II

~

i

~

t

i

i

i

i

! 12 13 14 15 16 ]7 18 I

20

i-Fig. 1. Mass fraction profile early in the transport-controlled stage.

Radiant ignition of a combustible solid with gas-phase exothermicity

963

concentration problem because eqn (17) introduces an initial structure over a distance ~g of order unity. To describe the early evolution of this profile, it is permissible to stretch co but not ~. The stretched time variable o5 = co/(31re /4) 2~3 produces the transient convective balance aqJ~/a03 + (3/2)X/-~aqJ~/0~ = 0 to lowest order in e. The characteristics for this first-order equation are lines along which ~g _03m remains constant. The solution subject to the initial and boundary conditions in eqns (17) and (21) is that qJg = 1 for ~ < 033~2and that qJ~ is given by eqn (17) with ~8 replaced by ~ g - 033~z for ~s > 033/2. This solution possesses a discontinuity in slope along the curve ~ = 033~2.The discontinuity means that the diffusion term will be important in the vicinity of this curve. With ~ = (~g - 033/2)/(3~re ]4) 1/3 and q~ = (1 - qJ~)/(3~'5/2e/32) 1/3, the initial-value problem

Oq_ Og2'

{L

g) = O,

g< 0

(25)

in the infinite space - ~ < ~ < oo emerges to lowest order in e, for describing the temperature field in the vicinity of the curve of discontinuous slope. By use of Green's functions for example, the solution is found to be

(26)

The results of the characteristic solution and of the modification thereto produced by eqn (26) are shown in Fig. 2 for a particular value of e. The establishment of a region of nearly uniform elevated temperature in the gas adjacent to the solid is seen clearly in Fig. 2. These small-co results b e c o m e inaccurate when co is of order unity or larger because of neglect of the second and succeeding terms in the small-~o expansion in eqn (22). When to is large compared with unity, as it will be at the onset of a transition to ignition whenever the ignition time is appreciably larger than the time to onset of gasification, the large-co value of ev' in eqn ( 2 2 ) m a y be employed. When this is done, it is inappropriate to retain eqn (17), since modifications to the temperature field such as those shown in Fig. 2 will have occurred before the large-~o value is attained. Nevertheless, it is possible to d e v e l o p an asymptotic solution for large co. Moreover, unlike the Small-co case, the limiting solutions for ~g and Y at large co will be similar, since the times are so long that the layer of nearly u n i f o r m properties will have grown to a thickness large c o m p a r e d with the thickness of the gas-phase region that is heated during the inert stage. For this reason, attention i s focused on eqn (19) with the knowledge that i n t h e first approximation ~0g = Y. Since v' is constant at large to, the convection tends to expand the heated fuel region away from the surface at a constant velocity. T o view a region in which appreciable changes in composition and temperature occur, it is therefore desirable to adopt a co-ordinate system that moves at this constant velocity. In this moving frame, a diffusion process occurs in which profiles spread with.time in

964

M. K i n d e l a n and F. A. W i l l i a m s

- -

similar solution (large 04 v~:5

----

similar solution with corrections

....

characteristic solution (small w)E:I ..... characteristic solution with corrections(=.I

.

.

.

.

.

vL:5

(~:

to:.O8

.06

\ 'x '\\ \\ '\"~ \

.,~

'\ 'L\ \\\

w : .04

\, \ \\

'\\

~:.02 ~o=o

\

• \

"I

\

\

' \

"*\L \\ '\

\

\'L

\9

\

.8

'('~.\\

.7 I

\

t

•6

\

'\\ \ \\\ "\\ \\\\\

~,

~5

.4

\\ ~

\ ', \ "" \ ' ,

"" \

\

.,

\

'\ \ \ '\\\ \ \

'\.

.\ \

%

'N I

2

3

Fig. 2. T e m p e r a t u r e profiles e a r l y in the t r a n s p o r t - c o n t r o l l e d stage.

proportion to X/~. Therefore it is c o n v e n i e n t to normalize the space variable with respect to ~ . Specifically, introduce ~ = (ff~ - v%o)/2X/~ and treat ~o and rl as the i n d e p e n d e n t variables. B y use of eqn (18) it is seen that eqn (19) b e c o m e s

OY

02Y

OY

4 o J - - =----~-+ 2 r / - Or/'

V(w, o o ) = 0 , Y ( o , - v ' ~ / ~ / 2 ) = 1,

(27)

with the appropriate initial condition to be obtained f r o m the solution for ~o of order unity. Since v' is large for small e, it is a very g o o d approximation to apply the b o u n d a r y condition Y = 1 at "0 = -oo. The large-co solution to eqn (27) then b e c o m e s simply 1 Y = ~ erfc ('0).

(28)

There are three things that introduce corrections to eqn (28) if ~o is not large.

Radiant ignition of a combustible solid with gas-phase exothermicity

965

One is the fact that the condition Y = 1 should be applied at , / = - v'V'-~/2 instead of at rl = -oo. Another is the fact that v' becomes to-dependent, departing from the constant value ¢-1(1 + o~')-L The third is the influence of the initial conditions. The first of these effects is very small even when to is of order unity because of the factor ~-'. It may be inferred from Fig. 7 of Kindelan and Williams (1975) that unless a ' is small, the second effect also is not large when to is of order unity or larger; to obtain analytically the next term in the expansion for large to in eqn (22) is a difficult problem that might best be attacked by employing a Laplace transform in ~ to generate an integral equation for v '. The third effect is expected to be the major one in most cases, and it has been analyzed by Kindelan (1975). In that analysis it turns out to be convenient to introduce a time translation of order unity in to, thereby defining a virtual origin whose position is selected to minimize the perturbation to the solution given by eqn (28). An eigenvalue problem emerges, with eigenfunctions involving Hermite polynomials. The results are

0 =~leffc(,o)+ ~ 4 e - " ' [1

27/~- 1 48(to + 2/3)

]

+"

""/d

(29)

and

1

Y = ~ erfc (7) - */e-'2/[3V~(to + 2/3)1 +" •

(30)

where to + 2/3 must replace to in the denominator in the definition of ~ (the virtual-origin effect). The solutions given by eqns (28) and (29) are plotted in Fig. 3 for a particular value of yr. Also shown for comparison is the early-time characteristic solution. The first correction to the similar mixing-layer solution given by eqn (28) is quite i

i

i

---

simUor solution (largeu} vl=5

.... .......

similar solution with corrections similar solution (sr~ll ~) e = .I

~'-

.~,

V',' I

,,,

\

\

\ \ t0:l.i

",,

",..

-,,,

\'\ 2

3

4

Fig. 3. Temperature profilesfor to near unity.

966

M. Kindelan and F. A. Williams

small even for values of to well below ten; it starts to become significant at values of order unity, as is seen from Fig. 3. The first correction produces a small, nonphysical overshoot of 0 above 0, on the fuel side of the mixing layer. This effect is too small to be seen in Fig. 3, but it is evident in Fig. 2, where the large-to solutions are included for comparison purposes even though they are expected to be quite inaccurate. It is seen from Fig. 3 that the large-to and small-to solutions agree reasonably well for to of order unity. Since the solution given by eqn (28) remains reasonably good for values of to as small as unity, the following analyses of ignition employ only this solution. A first correction readily can be derived merely by introducing the virtual origin. In summarizing the structure of the stage of transport-controlled mixing, it may be stated that a diffusion wave starts moving from the solid surface into the gas at the beginning of this stage. For the temperature field, this wave is acceleratory, its standoff distance increasing in proportion to to~¢2. For the composition fields the wave initially is deceleratory, with standoff distance proportional to V ~ . Very quickly both waves achieve a constant velocity, and when to becomes order unity or larger, they travel together, their standoff distance increasing in proportion to to. Throughout their histories, the wave thicknesses increase in proportion to V%, the usual diffusional spreading. Thus, as time increases the standoff distance increases with respect to the thickness in proportion to ~/~, and in a frame moving with the wave the system develops the classical structure of unsteady mixing of two half-spaces of gaseous fuel and oxidizer.

Ignition in the transport-controlled stage At some time during the mixing just analyzed, the gaseous reaction term, w'~ will b e c o m e large, and a transition to an ignited state will occur. As indicated in the introduction, the results of greatest interest are those for which this thermal runaway occurs after the diffusion wave develops a constant speed, i.e. after eqn (28) becomes valid. Earlier ignitions are treated in the appendix. In the present section, ignition at large values of to is analyzed. Full use will be made of the results of Lififin and Crespo (1975), since these authors treated completely ignition in the classical unsteady mixing layer. A general relationship between ~ , YF and Yo can be developed from eqns (11)-(13) and from the boundary conditions applicable during the transportcontrolled stage, provided that the time is sufficiently large for influences of differences in initial conditions on temperature and compositions to be negligible. From eqns (11)-(13), it is seen that in general & e { Y v +/3'0~} = ~{Y~- - Y o / J , } = O, and when E ' is large and E is small, the boundary conditions tk~(to,~) = Y F ( t o , ~ ) = O, Y o ( w , ~ ) = Y o = , O , ( t o , O) = 1, Y ~ ( t o , O) = 1 - I and Y o ( w , O) -- l are obtained. It follows that if the resulting solutions are compatible with initial conditions, then (31)

Y ~ + [3'tO, = ( ~ ' + I -- I) Y

and Y~ - (Yo

-

Yo~)/v

= [1 -

I - (I -- Y o ~ ) / v ]

Y,

(32)

Radiant ignition of a combustible solid with gas-phase exothermicity

967

where Y is the solution to the problem defined in eqn (19) and is given by eqn (28) for the large-oJ case of interest. Equations (31) and (32) enable Y~ and Yo t o be expressed in terms of 0~, constants and the function erfc (71), so that w' can be expressed in terms of these quantities through eqn (16). Equation (11) then becomes the only equation requiring solution, a single and explicit equation for the nondimensional gas-phase temperature 08. It is convenient to work with the departure of the gas temperature from the value that it would have in the absence of the reaction. To produce a correspondence with the notation of Lifi~n and Crespo (1975), define ,p = (01- 1)(~8 - Y)E"/012. To achieve proper ordering of the reaction term w', it is desirable to introduce the new time variable u = to(E"lO,2)(O,- 1)(1 - l)(Yo®/~')A 'e -r'°'.

(33)

With u and ~/ as independent variables, eqn (11) then becomes O~ Ou

~ O~ 1 a2~ [~ - ( 1 - Y)(O,-1)E"IO, 2] 2u a~O 4u ~ = yFyo exp L Y + (1 - Y)IO, + q~O,IE" J'

(34)

where yF = Y - ~13'0121[E"(01- 1)(1 - l)]

(35)

yo = 1 - Y(1 - l l Y o ® ) - q~lS'vO,21[E"(O,- 1)Yo®].

(36)

and

Initial and boundary conditions for eqn (34) are q~(0, ~) = ¢(u, oo) = ~(u, -oo) = 0, the last being obtained by first observing (as in the previous section) that for small the solid surface is located approximately at ~/= -0% then studying eqn (10) and interface conditions to prove that when E ' is large the gas-phase reaction exerts a negligible influence on the surface temperature during ignition. The problem defined by eqn (34), with the stated boundary conditions and with eqns (35), (36) and (28) defining quantities appearing therein, has been analyzed fully by Lififin and Crespo (1975) for the case l = 0, as an asymptotic expansion for large values of E". Two distinguished cases occur, one in which 0~- 1 is of order unity and the other in which (01- 1)E" is of order unity. The former is of primary interest here since the increase in temperature during the inert stage is unlikely to be so small that the latter case applies. When 01- 1 is of order unity, the Arrhenius exponent causes the reaction rate to be negligible unless Y is near unity. This establishes inert conditions in the gas everywhere except near the hot boundary, and a problem with two spatial regions emerges. In the reaction region it is convenient to employ z = (1 - Y)(01 - 1)E"/O, 2 as the independent stretched space variable. In this region, to lowest order in E", it is seen that yF = 1, and eqn (34) becomes

1, r E " ( 0 , - 1)] 202~0_ _ 2 0, az yoe" ",

(37)

968

M. Kindelan and F. A. Williams

w h e r e extensive use of the a s y m p t o t i c expansion of erfc (~/) has b e e n m a d e (see Lififin and Crespo, 1975). The b o u n d a r y condition at r / = - oo implies that q~ = 0 at z = 0, and matching to the transient inert region can be shown to require that Oq~/Oz a p p r o a c h zero as z a p p r o a c h e s infinity. Through the definitions, it is possible to verify a p o s t e r i o r i that the calculated location of the reaction region, z of order unity, lies on the gas side of the solid-gas interface (as it must) w h e n e v e r the order of magnitude of o) exceeds e2(l + a')2 In (E"), which typically is quite small. The f o r m explicitly taken by eqn (37) depends on the magnitude of l / Y,,~. For l / Y o ~ = O , eqn (36) b e c o m e s yo = ( z - / 3 q ~ ) O ~ 2 / [ E " ( 0 1 - 1)], where /3 = / 3 ' v / Y o = , and the p r o b l e m is identical to that solved by Lifihn and Crespo (1975). The value of q~ at z = ~ is denoted by q~ and measures the m a x i m u m t e m p e r a t u r e in the gas. This value depends parametrically on /3 and is a function of the nondimensional time

' ' -,,i} E " ( O , - - 1)

[ 2k/7r0,

I

(38)

called ~', by these authors. The quantity q~= is plotted as a function of ~-, for various values of /3 in their paper. For /3 > 1 this function is single-valued, the heat released in the gas-phase reaction being insufficient to produce a singular event identifiable with a well-defined time of ignition. H o w e v e r for/3 < I, as is usually true for ignition problems, q~ is a double-valued function of ~-,, there being a m a x i m u m value of ~-i b e y o n d which no solution exists. The upper branch of the double-valued function is not meaningful, and neglect of the transient term in eqn (37) b e c o m e s unacceptable as ~-, a p p r o a c h e s its m a x i m u m value. The m a x i m u m value of -rl defines an ignition time in a natural way through a criterion of thermal runaway. If l / Y o ~ is small, of order 1 / E " , then the definition s = E " ( O , - l ) l / O , 2 Y o ~ may be introduced, and eqn (36) b e c o m e s yo = (s + z -/3q~)O,2/[E"(O, - 1)] in lowest order. The p a r a m e t e r s modifies the structure of the reaction zone. The b o u n d a r y condition ¢ (0) = 0 must be replaced by ~ (0) = s//3, since the basic requirement at this b o u n d a r y of highest t e m p e r a t u r e and ample fuel is that the oxidizer concentration vanish (yo = 0). More specifically, the limiting behavior q~ --, z//3 as z a p p r o a c h e s zero (found in Fig. 8 of Lift,in and Crespo, 1975) b e c o m e s q~ --* (s + z)//3. An implication of this condition is that the oxidizer contained in the fuel burns completely before reaching the main gas-phase reaction zone; there is so little oxidizer in the fuel in this case that its complete consumption is insufficient to cause an ignition event or to raise the t e m p e r a t u r e appreciably. Application of this new b o u n d a r y condition can be justified only if there exists a solution in an inner zone of small z, adjacent to the solid surface, in which ~p goes f r o m zero to s 1~3. Kindelan (1975) has shown that such a zone exists and has solved for its structure, which involves a (transient) convective-reactive balance with diffusion negligible. The equation for q~, as well as the boundary conditions, can again be reduced to the problem solved by Lififin and Crespo by redefining q~ --

Radiant ignition of a combustible solid with gas-phase exothermicity

969

(01 - 1)(Og - Y - l//3Yo®)(E"/O, 2) a n d

uO'---~2e'--/a-~ln rE"(O'-1)1~--'

,,#

L 2~0,:

E"(O,-1)(

(39)

JJ "

With these definitions, all of the conclusions obtained for the case s = 0 are still valid. In particular, values of ~ of ignition for a given/3 can be obtained from Fig. 9 of Lifi6n and Crespo (1975) by looking at the maximum value of ~-1for a given/3. From a correlation given by Lifl~m and Crespo (1975) for their numerical results, a good approximation to the ignition time in terms of to is inferred to be

1)]

toA 'e -E'/'' =

2(/3 I v ) ( 2 " / 3 ) e2+('/~>(1 - 1)(1 - / 7 ) 2 In L 2~/~0,2 J"

(40)

This correlation is plotted in Fig. 4, where excellent agreement is found with additional numerical integrations performed by Kindelan (1975). i

i

i

i

i

i

------

correlation formula (40] for defined by equation (38)

- -

correlation formula (40)for ~ defined by equotion (39)

,C

9

x

I

r s =.l

/i

IllXl //'

numericalintegrations for ~ defined by equation (39)

8

i

// .~/ s:.5

6 I~. 5 4

:1

2 d ~

I

.I

.2

~-,=--~ .3

.4

..,...,..,,.---,_..,.,,./ .5

.6

.7

.8

.9

Fig. 4. C o r r e l a t i o n f o r m u l a relating "F o f ignition t o / 3 (cqn 40). C o m p a r i s o n to n u m e r i c a l integrations.

If l/Yo® is of order unity, then the character of the problem changes. To lowest order, eqn (36) becomes yo = l/Yo®, and eqn (37) no longer possesses solutions satisfying both the matching condition that O~o/Oz approach zero as z approaches infinity and an appropriate boundary condition at z = 0. This might be expected on physical grounds because in the absence of reaction, maxima of temperature and fuel concentration now occur at the solid surface with plentiful oxidizer. The mixing zone becomes unimportant, and reaction leading to thermal runaway will start near the surface. With the definitions y = ~g(E"/O,2)(O~- 1)(1 - l)(l//3')A'e-E"/e'/v '

(41)

970

M. Kindelan and F. A. Williams

and U = ul/Yo~, eqn (11) b e c o m e s simply Oq~ + Oq~= e~

OU

(421

Oy

to lowest order in e and in 1/E". The diffusion term has disappeared because it is smaller than the convection term at least by order E 2 if ignition occurs when o~ is of order unity or larger. Also, quantities other than ~ in the exponential term of the reaction rate have been dropped because when co is of order unity or larger they are of a magnitude no larger than e c/~2, with c of order unity. Lines along which y - U remains constant are characteristics of eqn (42). By transforming to U - y and y as independent variables, it is easily seen that q~ = - l n ( 1 y) is the solution subject to the b o u n d a r y condition that q~ = 0 at y = 0. This solution is appropriate only for y < U ; the trajectory of the diffusion front at large o~ is ff~ = v'w, which c o r r e s p o n d s to y = U, and for values of ~', (or y) larger than this, there is a negligible a m o u n t of fuel in the gas (y~ ~ 1), causing = 0 to be the solution in a first approximation. In the vicinity of the line y - U there will be a thin region across which q~ decreases continuously to zero. This region is not analyzed here since it is not needed in deriving the first approximation for the ignition time. The d e v e l o p m e n t of the q~(y) profiles with time is illustrated schematically in Fig. 5. F r o m the solution it is seen that diverges at y = 1, and f r o m Fig. 5 it is clear that the earliest time at which this occurs is U = 1. The solution necessarily b e c o m e invalid for U > 1. The time of ignition, by a thermal r u n a w a y criterion, is simply U = 1. The solution developed here describes the physical situation in which the element of premixed reactants that first enters the gas at oJ = 0 travels along the diffusion front and experiences a h o m o g e n e o u s explosion having r u n a w a y at U = 1. In terms of o) the ignition time

/

f I

-E~ A 2 g

Fig. 5. Temperature profiles late in the transport-controlled stage when l/Yo~ is of order unity.

Radiant ignition of a combustible solid wilh gas-phase exotherraicity

971

is to = [(E"/O,2)(Ol- 1)(1 - l)(l/~8')A 'e-E'/'] -1.

(43)

It is interesting to note that in all three cases analyzed here, viz. l/Yo® zero, small and of order unity, the location of the ignition event is well removed from the solid surface, being located at a value of y or z of order unity. Parametric results for ignition times

In the preceding section formulas for calculating the ignition time were developed (eqns (40) and (43)). It is of interest to generate from these formulas curves of nondimensional ignition time which can be compared with earlier theoretical results. An effort is made here to present those curves in forms that parallel typical experimental data and with coordinates that help to indicate how condensed-phase and gas-phase ignitions may be distinguished. To aid in understanding the theoretical predictions from a physical point of view, it should be observed that the nondimensional ratio of the rate of heat release by the gas-phase reaction to the rate of heat absorption by the surface gasification is approximately A " e <2w-~we', where A " = ¢rMFAQp, c~tTo(1 + F)2(! + Yo®)(1 - l) 4 B 2cspL 2

(44)

The parameter A" is independent of the incident energy flux, independent of both activation energies, and incorporates the complete pressure dependence, being proportional to p raised to a power given by the overall order of the gas-phase reaction minus one, when F is small. From eqns (40) and (43), combined with definitions appearing in the list of nomenclature, it may be shown that in the original nondimensional variable T, the ignition time is given by q'ts. = rl[1 + (@/A")e-(2w-uw°l],

(45)

where (1 + a')22(2-/8) In [E"(O,- 1)/2~,~0, x] = (O, - 1)e'(1 - / 3 ) 2 exp [(I/~SYo®)E"(Ol - Die,']

(46)

when eqn (40) is valid, while = (1 + o~')2(1 + Yo®/l) ( 0 1 - 1)'E"/Ol 2

(47)

when eqn (43) holds. The factor q~ is nearly of order unity; specifically, it is of order In E " when there is little or no oxidizer in the solid (i.e. whenever eqn (46) holds), and it is of order 1/E" when the oxidizer content of the solid is comparable with or greater than that of the gas.

972

M. Kindelan and F. A. Williams

U p o n recalling that r, is the nondimensionat time to onset of gasification, it is seen f r o m eqn (45) (and f r o m the fact that the order of qb does not differ greatly f r o m unity) that the ignition time is a p p r o x i m a t e l y equal to the time to onset o[ gasification if the rate ratio A " e ~2z' F,wo, is of order unity or larger. The ignition time exceeds the time to onset of gasification only if the order of magnitude of this rate ratio is less than unity. Such a prediction is quite understandable. 11 is also physically reasonable that the secondary influence of dp is to shorten the ignition time at high activation energy when the fuel contains its own oxidizer and therefore need not mix to ignite. The usual e x p e r i m e n t s generate graphs of the ignition time as a function of the incident energy flux. With the present nondimensionalizations, this corresponds to plotting B'2r~, as a function of I/x/-B~ for fixed values of A", E ' , E" and other p a r a m e t e r s . Such a plot c o r r e s p o n d s to Fig. 13 of Kindelan and Williams (1975). It is constructed f r o m eqn (45) by making use of the relationship b e t w e e n B' and 0, given in the definition of 0, (keeping a ' / ( 0 , - 1) constant). F r o m eqns (46) and (47) it is seen that in principle the curves depend on the values of a'/(O, - 1), l / Y o . and /3, and also on whether eqn (46) or (47) is valid. H o w e v e r , these effects are not appreciable and therefore need not be considered. Moreover, curves with different values of E ' and E " practically coincide, so long as the ratio E ' / E ' is fixed. A similar independence of E ' in this graph was o b s e r v e d and discussed by Kindelan and Williams (1975). Figure 6 shows the results for the case E " / E ' = 1. The solid lines are those of r~ given by Kindelan and Williams, and the broken lines show the corrections introduced by the gas-phase reaction. The prediction of eqn (45) is seen to depart f r o m the r, lines at a value of 1/',/B-r which depends on the value of A ' . After departure the lines rapidly assume a different constant slope. The error committed by approximating all curves with the same ratio E ' / E ' by a single curve is less than 5% in log (B'2r,,,). Therefore, in plots for other values of I

i

1

100\

-

F

~

I

i

/

c,

!60

[

F

"%'-"'---._ E ::: 33.3

"~'~' " ~ .., }A,,- jo-2O

/

~"_-

-.

i

i

-,o

1

-.

-;

A": 10-8~ I0 10-4 I 1 o

Log ( I t.,/-B' )

Fig. 6. Theoretical prediction of ignition time as a function of heat flux, for several values of the nondimensional parameter A".

Radiant ignitionof a combustiblesolid withgas-phase exothermicity

973

E"/E', only single lines are shown. F r o m Fig. 7 it is seen that the slope after departure is less for E"IE' = 1/3 than it is for E"/E' = 1. Figure 8 shows that for E"/E' = 3/2 the slope is greater. According to Fig. 9, for E"/E' = 2 the slope is the same as that of the ~', curve, there merely being an A " - d e p e n d e n t upward displacement. Since it has been o b s e r v e d that T, closely a p p r o x i m a t e s the results of condensed-phase ignition theory, it m a y be concluded that the d e p e n d e n c e of ignition time on incident flux cannot distinguish gas-phase and condensed-phase ignitions if E " / E ' = 2. Figure 10 illustrates that for E"/E' > 2 the slopes after departure f r o m the ~-, curve are greater in magnitude than that of the r, curve. For E " / E ' < 2 the gas-phase reaction b e c o m e s relatively slow at incident fluxes in I

I

~

,

I

T

r

,

E,,.2

_

60

~" 3

,,

,50

_A.=lO-a4

,

2o

o -,'2 -,; -; Log (l/Vre')

Fill. 7. Theoretical prediction of ignition time as a function of heat flux, for several

values of the nondimensional parameter A".

I

I

I

I

I

I

i

I

I

-" 1.5

60

4o%

"-

"~

20 "I=10

""C~-d=l°-* [0 16 z

-=6

-t4

-t2

-Io

-6

Log

-4

-

(tfB')

Fig. 8. Theoretical prediction of ignition time as a function of heat flux, for several values of the nondimensional parameter A".

974

M. K i n d e l a n a n d F. A. W i l l i a m s

I

I

q-- .......

] ~ - - - ~

,,,~..

--

........ r - - - - I

I

l

-~60

E"_~ ~,~,, "~'-,,.

150

t -~.-..

-~

" , ~ . . ~"~,o~ ~o I

I

I

~

I

I

44

-12

-Io

8

-6

-4

Log

{I/V'~')

I

-2

Fig. 9. Theoretical prediction of ignition time as a function of heat flux, for s e v e r a l values of the nondimensional parameter A ".

A%IO4 •

,:,08 _

,,>

80

x\

?0

", ~\

,,

\\

\ ~,

\

~ \

•

"'-',~

3o N

20

t,° I

44

_112

t

40

I

8

I

-6

i4

-

b

1

-2

Lo~ (I/,/B') Fig. 10. Theoretical prediction of ignition time as a function of heat flux, for several values of the nondimensional parameter A"

e x c e s s of a critical value, while for E"/E'> 2 the gas reaction is slow at low incident fluxes. T h e s e results are understandable from eqn (45) if it is realized from the definitions of B ' and 01 that the Arrhenius factor e -~'/°' for gasification is proportional to the incident flux q during quasisteady gasification. Since the rate ratio is proportional to e ~2~'-~'~/°', increasing the flux increases the gas-phase reaction rate by a proportionally greater amount w h e n E"/E' > 2. It is clear from

Radiant ignition of a combustible solid with gas-phase exothermicity

975

eqn (45) that when ~'~8, is large compared with T,, it becomes proportional to Tl[exp (-E'/01)] 2-E'tE', which varies as q-E"/~'. Thus, the fact that the slopes of the dashed lines in Figs. 6-10 is - E " / E ' can be traced to a mechanism in which the surface temperature 01 achieved during radiant-induced gasification is proportional to - E ' divided by the logarithm of a constant times t~, while the ignition reaction proceeds at a rate proportional to e -E'/°'. When the gas-phase ignition time is long compared with the time to onset of gasification, the slope ( - E " / E ' ) of a log-log plot of ignition time versus heat flux may differ from the slope (-2) that corresponds to condensed-phase ignitions. This provides one potential means for distinguishing experimentally between gas-phase and condensed-phase ignitions. If ignition occurs in the gas-phase regime analyzed herein, then the slope also provides a way to measure experimentally the ratio of gas-phase to interface overall activation energies. Another approach to distinguishing experimentally between gas-phase and condensed-phase ignitions is to measure the dependence of the ignition time on either pressure or ambient oxidizer concentration at a constant value of the incident heat flux. Since A" is proportional to Apg(1 +F)2Yo=, if the present theory is applicable then the curve generated in either of these experiments is related directly to a plot of B'2~',g, as a function of A" for fixed values of B', E', E" and other parameters. The pressure dependence of A" stemming from Apg has been indicated earlier; the factor (1 + F) 2 is independent of p at ordinary pressures but becomes proportional to p at very high pressures. The value of A" is expected to be independent of ambient oxidizer concentration when the oxidizer comes from the fuel but otherwise proportional to the ambient oxidizer concentration raised to a power given by the overall order of the ignition reaction with respect to oxidizer. Representative curves of nondimensional ignition time as a function of A" are given in Fig. 11. At sufficiently large values of A" this parameter has no influence, 36

i

p

]

~

32 ~ =_ " ' - . . , . . ~ 30 ~'~'"..~. 28 ~ ~.~...~. 26 ~ - ~ 24 ;. %'.% I-- 22 20 ; l

".....

]

i

t B'=IO 5

F

i

~

r

I

T

E'= E"= 66 6 E'=E"=333 - - E' - 6 6 6 E"=33 3 ........ E'=333 E"=666 . . . .

.%. ~'"~..

'~ '"- •-...

""~ %'.. -~

6~=loJo - ~- ,%.

...... - . .

-.

,6

1412

". . . . .

~

"~.

B, IO6

"~'~"

I0

8

6 4 2 0 -20

B,=I02

J -18

I -16

I -14

I -12

I -I0

; -

J -6

I -4

J -2

I 0

I 2

I 4

Log A"

Fig. 1 l. Theoretical prediction of ignition time as a function of pressure or oxidizer concentration, for several values of the nondimensional p a r a m e t e r B ' .

976

M. Kindelan and F. A. Williams

~0 g~ 4 -12

20! 24 28

32 36 -40

Log B' Fig. 12. Value of A" at which the ignition time of Fig. 11 starts to deviate from the gasification time. This value is shown as a function of B' for different values of the ratio of activation energies E"/E'.

since the ignition time approaches the time T, for onset of gasification. At smaller values of A", Ti~, becomes inversely proportional to A", as is evident from eqn (45). This behavior translates into a power-law dependence of ignition time on p or Yo... The power for Yo~ will be - 1 if the reaction is of first order with respect to oxidizer. At ordinary pressures the power for p will be one minus the overall order of the reaction. The break points for the curves in Fig. 11 of course correspond to those of Figs. 6-10. At these points the rate ratio is of order unity, and the definition of 0, may be used to show that approximately In B ' = - In A " / ( 2 - E " / E ' ) at the break. The value of A" at the break points of Fig. 1 t is plotted as a function of B ' for various values of E " / E ' in Fig. 12. In Fig. 11, as a first approximation the value of the slope may be taken to be - 1 for values of A" below those plotted in Fig. 12 and zero above.

Acknowledgement--This

work was supported by the U.S. Army Research Office, Durham, under Granl No. ARO-D-31-124-73-G 108.

Nomenclature

s

c specific heat I oxidizer mass fraction in the propellant n stoichiometric coefficient p pressure flux of external heat r specific heat ratio {r = c./c}

t u r v'

measure of oxidizer concentration in the propellant {s ~ E"(O, I)l/O~2Y,.} time nondimensional time in ignition analysis. defined in eqn (33) surface regression rate dimensionless regression rate defined in eqn (15)

Radiant ignition of a combustible solid with gas-phase exothermicity

w, rate of production of species i w' dimensionless gas-phase reaction rate defined in eqn (16) x space coordinate y nondimensional space variable for ignition when oxidizer is contained in the solid, as defined in eqn (41) z nondimensional space variable in the reaction zone {z = (1 - Y)(O~- 1)E"18~z} A frequency factor for gas phase reaction A ' dimensionless frequency factor { A ' = MvAp.pcATo=(1 + F)2%/q2} A" nondimensionai parameter determining ratio of gas-phase reaction rate to surface gasification rate, defined in eqn (44) B frequency factor for gasification reaction B' dimensionless frequency factor {B '=

pLBI~I} D,

molecular diffusivity

E E'

activation energy dimensionless activation energy { E ' = EIR°To} dimensionless activation energy {E"= E, IR°To} heat of vaporization molecular weight heat of combustion per gram of fuel consumed universal gas constant temperature nondimensional time variable for ignition when oxidizer is contained in the solid {U = ullYo=} mass fraction dimensionless inverse heat of gasification {a' = (0, - 1)cTolL} ratio of thermal enthalpy increase per unit mass of the gas during the inert stage to the heat released per unit mass of ambient atmospheric gas consumed in the ignition reaction {/3 = [3'vlYo.}

E" L M Q R° T U

Y a' /3

977

/3' dimensionless inverse heat of gas-phase reaction {~' = (O~- 1)c.To/Q} expansion parameter {e = 2Fla' r ~,~} dimensionless space coordinate for the transport controlled stage {~ = ~/~v~,} diffusion-zone similarity variable {71 = (~. v'o~)/2x/~} 0 dimensionless temperature {0 = T/To} 0, dimensionless long-time surface temperature given by solution of (1+ ¢~')B'e-e'~*, 1 -

=

A coefficient of thermal conductivity dimensionless space coordinate for the solid {f = iJx/kTo(l + I')} dimensionless space coordinate for the gas {t, = -rq'Clp, A, ro(l + r)} v stoichiometric coefficient {v = MonlMF} p density dimensionless time {T = fft2/pcXTo=(1+ F)2} ¢, lowest order value for dimensionless time to onset of gasification {T, = (0~ - 1)2~r/4} see eqn (39) dimensionless temperature deviation {q~ = ( O , - 1)(gJ. - Y)E"IO, 2} g, dimensionless temperature in transport controlled stage {g~ = (0 - 1)/(0, - 1)} ¢o dimensionless time for transportcontrolled stage {co -- ~ / ~ , - 1} F ratio of thermal responsivities { F = see eqns (46) and (47) steam function { ~ = f;p= dx}

Subscripts g ign F 0

gas ignition fuel oxidizer o initial s surface

References Kashiwagi, T. (1974) A Radiative Ignition Model of a Solid Fuel, Combustion Science and Technology 8, 225-236. Kindelan, M. (1975) An Asymptotic Approach to Gas-Phase Ignition Theory, Ph.D. Thesis, University of California, San Diego. Kindelan, M. and Williams, F. A. (1975) Theory for Endothermic Gasification of a Solid by a Constant Energy Flux, Combustion ScOnce and Technology 10, 1-19. Lifl~n, A. and Crespo, A. (1975) An Asymptotic Analysis of Unsteady Diffusion Flames for Large Activation Energies, Combustion Science and Technology to appear,

978

M. Kindelan and F. A. Williams

18 I7 161 15~4 b E3

/"~

/// "~= ,, /

" iI i

/ # . . ~ ~:4

Y0~ 'o

I

,,

e -9

'"

"'\

e

/ ~ " ~

oo7

=5.5 "\,

~ / /

5

"\\

. 4 z _ / / 7 ~ "'3 / / / / )<./~ _ \

I

2

3

4

5

6

L\

", "\X

'\ \

7

'\, ~

8

\ ,..

9

\,\ "X

I0

\",

., ,\

ii

"\.

12

13

:

14

15

Fig. 13. Evolution of the t e m p e r a t u r e profiles when ignition occurs early in the transport-controlled stage.

Appendix

Ignition early in the transport-controlled stage It has been o b s e r v e d in the main text, that for small values of ~o the transport-controlled stage p o s s e s s e s different characteristic variables for temperature and concentrations. In the characteristic variables for changes in concentration, the frozen t e m p e r a t u r e (no gas-phase reaction) is a p p r o x i m a t e l y a constant. Introducing d~ = 2oJ/~-~, ~ = \ / 2 ~ / k / - ~ e and q~ = ( 0 , - l)(t0, - I)E"/O~~ and neglecting reactant consumption, the energy conservation equation (11) becomes

(A1) where E" wE A ' R = ~--~( 0 , - l ) w - ,~-, Yo~(l - l ) exp (-- E"/O,) ol

(A2)

and Y is given by the solution of eqn (23). The boundary and initial conditions are @(0, ¢b) = q~(~c, ¢3) = ¢(~, 0) = 0.

(A3)

Equations (A1) and (23) have been simultaneously integrated numerically, for different values of R and l/Yo=. For convenience the variables dJ and ~ / \ / ~ are used in the integration, which is carried out by the C r a n k - N i c o l s o n procedure. The nonlinear term is quasilinearized by taking e x p ( q ~ [ l + 0 . 5 , J ] ) = exp (q~ [I, J]){1 + 0.5(q~ [I + 1, J] - q~[L J])} where ! stands for time and J for space. In this way a tridiagonal linear s y s t e m of equations is solved in each new time

Radiant ignition of a combustible solid with gas-phase exothermicity

'it/'""'

'"'

7

3

979

'

t_°

I

2

5

4

I

t

I

I

I

i

1--

5

6

7

8

9

I0

II

12

@ign

Fig. 14. N o n d i m e n s i o n a l ignition t i m e as a f u n c t i o n of the n o n d i m e n s i o n a l p a r a m e t e r R for different v a l u e s of llYo.. T h e s e results are valid for ignitions early in the

transport-controlledstage. step. After obtaining the values of the function at a new time step by quasilinearization the accuracy of the solution is improved by iterating in the nonlinear term. That is, the preceding solution is used in the nonlinear term, to make it a known factor, and the resulting linear system of equations is solved. Usually two iterations are enough to achieve convergence at each time step. It is possible to obtain an asymptotic solution for large values of ~/X/-~, with the result that dY

---

d(~/~/o3)

Y---

¢

(A4)

2~/~

and the same for q~. This condition replaces the condition at o0 and allows the integration to be carried out to a large finite value of ~/~'-~. Profiles of Y and q~ are shown in Figs. 1 and 13. The ignition time was calculated by plotting exp ( - q~max)as a function of tb and extrapolating until exp ( - ~,~,) = 0. In Fig. 14 o3,g, is shown as a function of R for various value of l / Yo-. These results are to be used only over the limited range of conditions having to positive and of order ~ at ignition.