Asymptotic analysis of laminar flame propagation: Review and extension

ASYMPTOTIC ANALYSIS OF LAMINAR FLAME PROPAGATION: REVIEW AND EXTENSION WILLIAM Department

of Applied

Mechanics

and Engineering

B. BUSH Sciences,

University

San Diego. La Jolla.

of California.

CA 92093. U.S.A. Abstract-The structure of a steady Idimensional isobaric detlagration is analysed in the physical plane for the case of a direct first-order one-step irreversible exothermic unimolecular decomposition under Arrhenius kinetics. In particular, the eigenvalue giving the speed of propagation of the laminar flame into the unburned gas is sought for Lewis-Semenov number unity. Limit-process expansion techniques are applied for the physically interesting limit of activation temperature large relative to the adiabatic flame temperature. Through consideration of the higher-order contributions to the asymptotic solutions for the downstream reactivediffusive zone and the upstream convectivediffusive zone of the laminar flame, the physical-plane anslysis brings out details of the flame structure, including the two-region character of the downstream zone, not brought out in the conventional phase-plane analysis. Applications of the results of this analysis to more general detlagration problems are discussed.

I. INTRODUCTION THE

PROTOTYPE

of all aerothermochemical

sional burning of a combustible Previous

work on this problem

and approximate Nachbar Wilde,

forms

and Johnson; Klein,

and Burke,

problems

iterative

has involved

procedures

use of variational

closed-form

approximate

solution

Kamenetski

and Semenov.

and/or

by Hirschfelder,

and de Sendagorta;

and von Karman,

is the steady plane

I-dimen-

under laminar conditions.

for the mass flux eigenvalue

Boys and Corner,

Emmons,

deflagration

premixture

Millan

(for Lewis

techniques

Penner and von KQrman, Adams,

and numerical

integrations

and de Sendagorta; number

identically

All these works are reviewed

to yield bounds

flame speed by Rosen and by by Friedman

and, of particular

interest,

unity)

by Zeldovich,

Frank-

and referenced

by Williams[ll

and/or

Glassman [2]. More

recently,

asymptotic temperature

Bush and Fendell

analysis

is large relative

the above-mentioned

deflagration

to the adiabatic temperature,

& the non-dimensional

narrow downstream

reactivediffusive

a relatively

upstream

stream region solutions because the expansion region solutions, variable profiles. A review higher-order

for the flame speed, using an

coordinate,

variable),

rather than in physical

as the independent

in T-space by subdivision

zone near the hot boundary,

convectivediffusive

the activation

to carry out the analysis in phase space

as the independent

spatial

kinetics,

i.e. /3 = T,/T,, + =. In [3], as in

flame temperature,

wave for j3 --)a may be described thicker

expressions

that, in the Arrhenius

papers, it was found “convenient”

(with T, the non-dimensional space (with

derived

based on the assumption

zone, where

variable).

into two regions:

where (1 - T) = 0(8-l); T = O(l).

The a

and

Because the down-

can be obtained independently of the upstream region solutions, and for the flame speed eigenvalue can be found from the downstream

the upstream

region

solutions

yield

further

of ]3] indicates

that, in the transformation

approximations,

from phase space to physical

details

about

of the results, especially

the dependent those for the

space, certain ambiguities

arise. In

order to remove these ambiguities and to obtain uniformly valid solutions for the structure of a steady planar deflagration, it has been determined that it is necessary to carry out the analysis in physical space.t In the present paper, such a physical-space analysis is performed. The physical-space analysis presented here, through its study of further details concerning the structure of the flame, should be considered a complementary one to that presented by Williams[5]. In Section tin a later paper (Bush and FendellI41) on the structure of a steady planar detonation, leading orders of approximation, to carry out the analysis in physical space.

it was found necessary,

even to

598

W. B. BUSH

2, the model deflagration boundary-value problem is formulated, with the approximations employed in the formulation outlined. Particular attention is directed to the justification of the isobaric approximation for the low-Mach-number deflagration. and the demonstration that, for this deflagration, the temperature is a function that is monotonically increasing with distance downstream. In Section 3, the details of the t-space analysis for the downstream and upstream regions are presented. Here, the effects of the higher-order contributions of the I-space analysis, in the determination of the structure of the flame, as well as in the determination of the burning-rate eigenvalue, are given. In Section 4, the present t-space analysis and that of Williams[5] are compared, to leading order of approximation, and applications of the leadingorder deflagration theory to more complex flow geometries are cited. 2. THE

MODEL

DEFLAGRATION

BOUNDARY-VALUE

PROBLEM

Consider the steady planar l-dimensional deflagration wave produced by an N-component mixture of ideal gases. Let x be the spatial variable in the direction of the flow in a coordinate system at rest with respect to the wave; IJ the velocity component of the gas in the x-direction; p, p and T the hydrostatic pressure, density, and temperature; Wi, I’,, El, hi and wi the molecular weight, mass fraction, mass fluux fraction, specific enthalpy, and rate of production by chemical reaction of species i, with i = 1,2, . . . , N: and CL”,A, and d = pfl the longitudinal viscosity, thermal-conductivity, and mass-diffusion coefficients. The equations of conservation of mass, momentum, species, and energy, and the equation of state for the flow of such a gas are (Williams[ll): pu = const. = pooo= pa,, p + pu2 -

de. w. ‘=L dx pu, pU($, &+iu2

dT

-A-&-p%-&

p”: =const. = p. + poui = pm + pa?,

(2.lb)

with 3 Yi, 3 l= 1,

ei= Yi-E$,

i=l

i=l

(2.lc)

do N

N =

(2.la)

COnst. = ~OUO2 Ei.ohi.0 + i U:

i=l

p = pR”T 2 ( YJ Wi), i=l

C

i=I

ci.xhi.m+ i 0: 7 (2. Id)

R” = universal gas const.

(2.le)

In the above, the subscripts 0 and 00refer to the cold and hot boundary states, respectively. A model for the deflagration is now adopted. This model simplifies the system of equations (2.1). and, yet, retains the essential physical features of the problem. The mixture is taken to be a binary mixture, whbse chemistry is described by a direct first-order one-step irreversible exothermic unimolecular reaction R + P, with no inert species present.? For such a mixture, then, N = 2; and WI = W2= W; Y, = I - Y2, l, = 1 - l2; and wI = - w2,where the subscripts I and 2 refer to the reactant R and product P, respectively. It is presumed that Y,, = 1 - Y,.o, c2,0= I - el.o = 1 - Y,,o,and Y,., 7 E,.== 0, Y,,, = c2,== 1. Further, with respect to the heat capacities of the species, it is taken that c,, = c, = c, = const. This, in turn, means that the specific enthalpies may be expressed as h, = hy+ c,(T - To), h2 = h; + cP( T - To), where hy, h! are the specific enthalpies of formation at the cold boundary state. In addition, for the model, the longitudinal viscosity, thermal-conductivity, and mass-diffusion coefficients are taken to be (CL”/&)= (AlAo)= @/Do) = (T/To)“, with w = const. For the Arrhenius kinetics of the model, the reaction rates are given by wI = - w2= - p(BT”)(I - Y2)exp {- T,J( T - To)}, where B = amplitude of the frequency factor = const., II = temperature exponent of the frequency factor = const., and TA = activation temperature = const. In this formulation, the reaction rates go to zero exponentially as the cold boundary tit has beenshown(Clarke[6.7]: Carrier, Fendelland Bush(81) that. in the “fuel-lean” limit. the monopropellant reaction R + P. studied here, effectively models the bipropcllant reaction vFY,= + voYo + vPYP, for the case of vF = I.

Asymptotic

599

analysis of laminar flame propagation

temperature is approached (Friedman and Burke[9); Bush and Fendell[31), and the “cold boundary difficulty*’ is, thus, circumvented. Next, it is convenient to introduce non-dimensional variables. The spatial variable is defined by x = (&/poo,-,)n. The velocity, density, temperature, and pressure, respectively, are given by v = o&, p = pas = pdll4), T = To& p = p on = (poR”T,JW)(sO) = (poRwToXO/4).The mass fractions and mass flux fractions are written as Y, = Y,.&l - Y), Yz= 1 - Y&l - Y), and E, = Y,.o(l - l), lz = I - Y,.dl - l). The specific total enthalpy is expressed as H = (cpT + f d) = Ho+(H,-

Ho)7 = Ho+ Y,.o(h:-

I&

= Ho+ q7.

For the postulated model, the governing equations, thus, are: B”d~=-~of[(4-l)(l-vM:4)-(8-,)l.

4+1

&,e-c=0b), 0 0 dn $&=%$(I-

Y+Oas

1)M:

d4 4 5J

7)+-m,

= (7 - E), 7+0as

as n+=,

(2.2a)

Y+lasn+a, l+l as f7+“,

l+Oas n-)-z,

Y)exp(&],

(I -Pr’$o(Q

4+4=

as7)+-=.

t)+-=,

T+I as 7)+x,

(2.2c) (2.2d)

with

e=i+Q7-~(y-i)fkf$42-i).

e+lasn-+-3C,

e-+e,as

f7+“.

(2.2e)

In (2.2) the parameters introduced are: y = cJ(cp - R,), the ratio of the specific heats; MO= ~ol(~RwTo)‘~,the initial Mach number; Q = q/c,T,, the normalized heat release; Leo = Ao/c,&,, the initial Lewis number; Pr’i = &$A~, the initial (longitudinal) Prandtl number: %, = (poA~cp)(BT~)/(pooo)2, the initial first Damkohler similarity parameter; and eA= T,/T,, the normalized activation temperature. The quantities 4- and 8, are determined from the thermodynamics relating the bounding states. Specifically, for n + =? (2.2a) and (2.2e) yield (e,-

I)-(4,-

I){(, -rM:)-

te,-

l)+ty-

l)M$4,-

gdg4,-

I){1 t:(&-

i))=o,

(2.3a)

I)}= Q;

(2.3b)

and, in turn, (4x - I){(1 - M:) - f (y + l)M$4, - I)} = Q.

(2.3~)

From (2.3) it is determined that

(1-M;)

(‘=-

‘I=

(2.4a)

(y+ l)Mi

(&-1)=(4x-,){(I-yM:)-yM&$-I)} Q - (y - I)(1 t yM@[ I - [I -2’;J,j$

Q}li2]).

(2.4b)

For the deflagration, with Mi+O. the results of (2.4) may be expanded to yield (,#,x-I)-O[I+M:(l+;(y+I)Q)++Q>O:

(2Sa) (2Sb)

600

W. B. BUSH

Thus, for the deflagration, the final values for the (non-dimensional) velocity and temperature are greater than the initial values by a factor of O(1). In the limit of Mi+O, the quantity n,(= (fL/&J) is given by

7rm-[I-M~{~Q}+~~~l: (l-P,)-yM~Q(l+...)~o.

(2.6a)

Thus, the final (nondimensional) pressure is less than the initial pressure by a factor of O(M& and the description of the deflagration as an essentially isobaric process is justified. Finally, for Mi-+O, ML(= (&/&)M$ is determined to be Mi-(1+Q)Mi

1+M ~[~V+I)+lgy+l)Q)t....]: C U+Q)

(~-,)-Q[,+M:{~r+13+~(31+l)Q)+...]-Q>o.

(2.6b)

For Pr”, Leo = 1, with the introduction of & the modified spatial variable, defined by

(2.7) (such that (d/dq) = (l/e”)(d/d[)), and with the introduction of cp and 6, modified temperature variables, respectively, defined by

(v >QQ*

q5=I+(q5s-1)Q=1t

= l+Q

velocity

and

(2.8a)

T-(Y_I)M:(@$+{l+;(~)Qq)],

(2.8b)

the model reducest to

!!f=_LV* d5

rM:

0

V ’

U=(~-i)+;(ytl)M~(~)2QQ(,-Q),

(2.9a) cp+O as t-+---,

‘p+l as t+m;

de ds = AJ( I - T) exp (- PK):

p

=

A=&exp(-p), (e, - 1)’

E-0

as

5+-m,

7-O as t-f-03,

E-1

as

5-m;

(2.9b)

7-l

as

(4m.

(2.9~)

It is noted that, in (2.9b), eM (and/or &) remains to be determined. tFor PI’;,

Leo = I. consistent

with the definitions

of and the boundary

conditions

on Y and 7, it follows

that

Y = 7.

Asymptotic

601

analysis of laminar flame propagation

For the deflagration, it is taken that the temperature is a function that is monotonically increasing with distance (downstream, such that Bu = 8, (and, hence, au = 1). To demonstrate this, consider the first and second (spatial) derivatives of 8, namely:

(2. lob) If &# 0,, specifically, f%,> 8,, then, the derivatives of (2.10), evaluated for f(l) = f(,&) = fM, with f = 0, l, c$,can be written as

de

(Ghf>= t [(eM - ed + yai

- 4

- (y - iw

- 4dIi

- yMi%9= - 1) - (4~~- 4dll=

0; (2.1 la)

(~)M=~[~~~hl(~e~-e~)-~~-l)~:~~~-~~)[l+~~r-l)-~(h-~~)))

+~{l-vM~~~-l)+yM~d~-~M,l

Alternatively, (2.1 la) and (2.11b), respectively, can be rewritten in the following forms :

(e, - es) = KY - 1)(4=- 4di

-

YMS~ 1)- (6 - hdl- YQU- dl

= l(y - IX& - &){1+ CW:, (dX - &))I - YCW- %)I >O; yAe&

+ ow - ld,(b-

~)MY

+y

- wh-

4d{1+

wG.wb

{I + OW@}[(& - &Ml + OM,

- ddl-

(2.12a)

YQU - dl

(& - &f))} - Q(1 - GfM)l < 0. (2.12b)

An examination of (2.12) indicates that the two inequalities cannot be satisfied simultaneously, and, hence, it follows that eM=e,,...: hf9CPM, (2.13) nh 64 = 1. For eM = em, p = e,/(e, - 1) and the functions J and K (of (2.9b)) become, specifically, 1+Q J=

[

r-(y-l)M:(~)“(l+~(~)Q,]]~ (2.14a) [1+((@$+]

’

In the following sections, asymptotic solutions to the boundary-value problem of (2.9), for the deflagration wave, defined by Q

_

4

-

cpTo

Y1.0(~~-~9_ O(1)

c, To

M2’

’

d y&To

-0

’

(2.15a)

are presented for the case of large activation temperature (relative to the maximum temperature

W. B. BUSH

602

achieved in the reaction), i.e. (2.15b) Here, it is noted that, from (2.4b) and/or (2Sb) and from (2.15) (&- 1) = fnc(Q, M$Q[l + owa1O(1), and hence, P = fnc(BA, Q, Mi) - (e,/Q)[l + O(M@]+m, for the deflagration wave. In general, solutions should be sought in terms of the two small parameters, Mg and p-‘, involved. However, from consideration of typical magnitudes of these parameters, the case of (2.16a)

M:P-+O

is found to be the one of most practical interest. Indeed, in what follows, solutions are found for

M;prn+O,

with m +w.

(2.16b)

In this limit (of Mi = 0),it is found that (2.17)

cp-7

and that the governing boundary-value problem reduces to that studied by Williams[l] and by Bush and Fendell [3], namely:

$=(T-E),7+0as de -=A(l--.)exp[-L?v), d.5

5+-m,

7’1

as t-+m;

l+Oas t-+-m,

(2.18a)

E-+1 as 6-m.

(2.18b)

It is noted that (2.18b) follows, for (2.19a)

P - (e&N1 + Wf@l= ub/Q);

l+Qdl+W@lrl ’

-

1+ @[l + O(M:)]

K_

'

[I+ W&)1 - T[I + O(M;)] ~ (1 - 7) d I+ O(M$l

7

*

(2.19b)

Since the quantity A contains ti(= pOuO), the mass flux (mass per cross-sectional area per time) of combustible mixture through the flame, the principal unknown, i.e. A=n,exp(-p): Q

0

=

(Po~olc,w-t3_ =4 - W~w)(~ol~S)~ (POUO)

(POUO)

(2.20)

fil ‘...’

A plays the role of an eigenvalue. Explicitly, A@, Mi)= A(p) is sought. Based upon the results of Bush and Fendell[3], here, it is taken that A(& M:) = A(p) = /?‘A*@) = p2(Ao+ P-IA, + P-‘A2+. . .),

(2.21)

with Ao,A,, A2,. . . = consts. (to be determined). In Bush and Fendell[3], the two-equation boundary-value problem in physical space was reduced to a single-equation problem in phase space, with T as the independent variable, namely: de --_A dT

EeXp(-By],

E+Oas

7’0,

l+l

as

7’1

(2.22)

Asymptotic

analysis

of laminar

603

flame propagation

In turn, with the determination

of c(= E(T)), from the above equation,

deflagration

from

wave is determined

the “structure”

of the

(2.23)

The deflagration

wave

regions:

a narrow

layer

relatively

thicker

layer

expansions

are adopted.

for j3 +z

downstream, upstream,

in T-space by subdivision

into two

near the hot boundary, where (I - T) - O(p-‘); and a where 7 - O(l). For the solution of (2.22). systems of

For the downstream

l(7; while, for the upstream

may be described

region,

+. . .;

B) = C(7; p) = &o(?) + p-‘C,(?)

(2.24a)

region, r* =

E(T;p)=E*(T*;p)=@eXp

-B~~,~~(i*)+Q-‘rl(~*)+...,.

(2.24b)

1

Throughout

the flow field, it is taken

downstream

expansion

that the expansion

for A is that given

in (2.21). The

must satisfy Z0(?)+p-‘C,(7)+...+1

while, the upstream expansion

as 5-O;

(2.25a)

must satisfy

/3exp{--@%$I

7*-O.

[C~(T*)+/_-‘C~(T*)+.*.]+O~~

(2.25b)

Further, the expansion solutions must match in a domain of common validity (where ?-+a and T*-* 1). In Bush and Fendell[3], the details that lead to the determination of Cd?), C,(7),

. . . , &T*),

Ef(T*),

. . .

,and A,j, A,, . . . are presented.

that the leading terms in the flame speed eigenvalue

In principle, determined

For example,

it is determined

are

A, = (3 - I), . . .:

A0 = ;,

I=

expansion

I

0x[l-{l+(l+t)exp(-~)}~~2]dt=l.344...~

the “structure”

of the downstream

(2.26)

and upstream

regions, respectively,

can be

from t(T: @) = tb + p-‘&;

dt

p):

I

1

~=((I-8-‘7)-~)~‘(l-~~)-p-‘(~,+?)+...’ (2.27a)

(2.27b) However,

examination

of (2.27) reveals that, aside from the qualitative

result that the ratio of

604

W. B.BUSH

the thickness of the downstream region to that of the upstream region is of O(p-‘), only “ambiguous” quantitative results can be obtained from this phase-plane approach, especially if the behaviors for the “higher-order corrections” are desired. Such results suggest that, although it has been considered conventional and convenient to analyse the deflagration wave in phase space, it would be better to analyse the deflagration wave in physical space, i.e. solve the (original) boundary-value problem of (2.18). 3.ASYMPTOTICANALYSlSOFTHEMODELDEFLAGRATION BOUNDARY-VALUEPROBLEMINPHYSICALSPACE

In this section, an asymptotic analysis of the model deflagration boundary-value problem in physical space, i.e. (2.18) is presented. In this analysis, the scalings developed in the course of the solution of the phase-plane boundary-value problem are used for guidance. 3.1. The downstream region Based upon the results presented in Bush and Fendellpl, it is taken that the appropriate independent and dependent variables for the downstream region are r = 5(Z: P1 M?) = 4Yr;P) = P(5 - 4); T = ~(5; P, M;) = ~(5; P) = 1 - P-‘G(5; P)

(3.la)

= 1 - @-‘[G,,(S)+ p-‘G,(l) + p-*G&) + * . *I, fz = (5; P. m

= 45; PI = Jx;

P)

(3.lb)

= Eo(() + p-‘E&y

+ p-%*([)

+ **..

As in the phase-plane development, it is taken that the eigenvalue A is of the form A(& M;) = A(p) = @A*(p) 2 P*[A,,+ P-IA, + P-‘A2+ +. .I.

(3.2)

Substitution of (3.1) and (3.2) into the equations of (2.18) yields %=-(1-E),

(3.3a)

$$=(E,+G,),...; .

!$? = AoGGexp (- Go), ~=AOGOexp(-GO)[(~)+~G,-G~],.

(3.3b)

The boundary conditions for these equations are Go,Gi ,... +O,

Eo+l,

El ,... +Oas 5-a.

(3.4)

The leading order functions, Go({) and E,(l), are determined from dGo z=-(l-E&

T=AoGOexp(-G&

Go+O,

E,+l

as l-m.

(3.5)

As an intermediate step, (3.5) may be written in phase-plane form (Bush and Fendell[3]) as

dEo

Go pexp

dGo=-Ao(l-Eo)

(- Go), Eo+ 1 as Go+O.

(3.6a)

This equation may be integrated to yield E. = Eo( Go) = 1 - (2Ao)1’2{1 - (1 + Go) exp (- G0)]“2

= 1 - {l - (1 + Go) exp (- G,,)}1’2 for A0= f.

(3.6b)

Asymptotic

The asymptotic

behaviors

analysis of laminar flame propagation

of this solution

I l-3G,+...

I

1 (

=I-

pG,,

are

for &Go)

6 - {I - (2A0)“2} + (2.M”‘(

-1 as Go+Ofor

for&,=;.

for matching yield E,-*O as Go-‘=,

that the requirements

(3.7a)

i (Go + I) exp (- G,)( I + . . .)}

=f(G~+I)exp(-G$(l+...)~OasG~~w

fn anticipation

A,=;;

(3.7b)

it is taken that

A”=;. Once EdG,-,) is known, [(Go) is determined 1

di

dGo= -

I - Eo(G,J

,J=l(Go)=-

From this solution,

dr

G8 (1 -

5 -, 0

from

= - {1 - (I t Go) exp (- Go)}-‘12;

GO I

(3.8)

‘%I(~))

=

-

I

(3.9a)

G”{l-(1tt)exp(-r)}~“2dr. 08

as Go--, Gg, const. to be determined,

it is found that, at the “downstream

(3.9b)

edge” of this region,

(3.lOa) G,-G0hexp(-~5)(1+...)jOas

[+=,

with (3. lob)

In a similar manner, it is found that, at the “upstream edge” of this region

Go--(-l)+G,,+..

.-_)r:

as J-P -

(3.1la)

m,

(3.llb) behaviour for Gd5) determined from (3.lOb) and (3.1 lb), the asymptotic behaviors for Edl) are found to be

With the asymptotic

1

&-l-~GObexp &-;M+(Gou+l)}

(l+~~.)-tlas(+m:

exp(-GOY)exp{-(-r)}(I+...)-,Oas

(3.12) I--W.

(3.13)

W. B. BUSH

606

Now the functions G’(l) and E,(l) are determined from ‘$

= (E’ + Go), $!

= hoGo exp (- Go){(2) G,, El-,0

+v

G’ - Gi), (3.14)

as l*w.

With arrangement, the first equation of (3.14) and its derivative with respect to Go can be rewritten as (3.15a)

E,=$-G,=-{(1-E@+G,,];

(3.15b) The second equation of (3.14) can be rewritten as (3.16) Equations (3.15b) and (3.16) can, then, be combined to yield the following differential equation for G’(Go) (3.17)

&-(%&($))=-ro[l-z&-G:)],

when the notation To= (1 - Eo) = {I- (1 + Go) exp (- Go)}“’ is employed. The first integral of (3.17) is determined to be ~o~dr-A,~~+3~~{l-Go(1~~Go)$‘]]. 0

(3.18) 0

For Go-O, (3.18) yields the following behavior for G’(G3: G, _ Go _(2”2;‘-

2

‘)

(3.19)

log(~)+o,+O({log(~)+IjGo)],

where a, = const. (to be determined). From (3.15a) and (3.19) it is determined that EdGo), as Go+O, is given by _ (2”2h’ + 1 + 2%‘) + O([log (&) + l}Go)]. 2

(3.20)

Thus, to the orders of approximation considered, it is found that, as Go+0 and/or [+m, lo,(&)+.,+.

. .)+.

. .]

(3.2la)

From (3.21) it is seen that, for (2”*h, - 1) # 0, these solutions for T and E are not uniformly

Asymptotic

analysis

of laminar

607

flame propagation

valid as l-+m and /3+a, such that p-‘[ = (6 - &,) is of order unity. This nonuniformity suggests that the boundary-value problem under consideration is not a two-region problem, but, rather is a three-region problem. As a three-region problem, the downstream region (now under consideration), with 5 fixed, would be considered to be a nearfield downstream region, and would be supplemented by a fartield downstream region, with .$,=/S ‘5 fixed. Based upon (3.21), the asymptotic solutions for 7 and e, in such a farfield region, take the forms

(3.22a) 1 e%l-pG,,bexp = l-IfnG,,,,exp(-$)[(exp(-(2”2~-l)&)}+G(/K,)].

(3.22b)

For this particular problem, the concept of a three-region formulation is introduced for mathematical completeness and does not affect the results of physical interest, such as the determination of A,, etc. However, this concept can be important in the determination of the flame structure, when more general flow fields are considered. From (3.22), it is seen that, for this farheld region, for 6, fixed, (I - T) - 0(/X’ exp (- /3))-0; (I - E) - O(exp (- p))+O. From an asymptotic analysis (see Appendix), it is determined that, insofar as the determination of the leading correction terms for T and E is concerned, this fartield downstream region is characterized by a diffusion-convection-reaction balance. For Go+“, from (3.18), it is determined that G,(G,,) is given by

C

G, - ~G~tc,Go+d,-~(k3G~tk2G~+k,Gotk,,)exp(-Go)t~~~

I

,

with CI, d, =

k3=;,

consts. (to be determined), k,=c,t;,

k,=4c,td,t8-2/i,,

k,=6c,td,tl2-4A,.

(3.23)

Specifically, it is determined that c, = A, - (3 - I),

(3.24a)

where I=

I0

wE,,(r) dr =

I0

~]l-{1-(Itt)exp(-r)}“2]dt+1.344.

(3.24b)

In turn, from (3.lSa) and (3.23) it is determined that E,(Go) as Go+=, is given by E,--;

(m~G~+m2G~tm,Gotmo)exp(-Go)t~~~,

for cl = 0, d, = const. (to be determined),

m3=k3=i 2,

m2 = (k, - 3&jt 1) = 2,

m,=(k,-2k2+1)=d,t4(1-:A,).

mo=(4--k,)=4(1-;A,).

(3.25)

W. B. BUSH

608

It is taken that cl = 0 in order that E, +O (exponentially)

as ~&-)a.

It is seen that the taking of

cl = 0 yields the value of A,, namely A,=(3-I)+

1.656.

Thus, the results A0 = l/2, A, = (3 - I), obtained recovered.

This is not too surprising,

through the consideration

in the original

phase-plane

as these results were obtained

of the behaviors

Thus, to the orders of approximation

(3.26)

of &Go),

considered,

analysis (2.26) are

in the present analysis

E,(G,,), as Go+=. it is found that, as Go+=,

(3.27a)

e - i exp (- Go)[(GO + I +. . .) - ~-‘(m~G~ + mzGt + m,GO+mO+..~)t..~].

To express

these results

intermediate

(spatial) variable,

x -

in terms of a spatial

it is convenient,

to introduce

the

x, defined by

(56- I) _ (- 5)

ev

variable,

(3.27b)

Pm’

with 49 + 0. pa(@)+ 3~ as p

-+ 3,

(3.28a)

such that (- 5) = ~U(B)X + = for x fixed, /3 + 3~. In terms of this intermediate

(3.28b)

variable

(3.29) Introduction

of (3.29) into (3.27) yields

7-

[ I-(u*)+;(q)2-+.. .It/.-‘[l-(q)+.

forAo=f.

A,=(3-I),

GoU=_l,

. .]+3‘2[1 t..

.I+.

. .,

(3.30)

d,+

That A0 = l/2, A, = (3 - I) has been justified; that Go. = -I, d, = 3/2 awaits justification from the matching of these solutions for the downstream region with those of the upstream region. It is noted that, for Go. = -1.

Gi can now be evaluated.

From (3.1 lb), it is seen that Gi must

(3.31)

With the determination

of Gt. the evaluation

3.2. The upstream region For the upstream region,

of GOh follows

the independent

variable

from (3.10b).

is 5, and the appropriate

dependent

Asymptotic

variables

609

analysis of laminar flame propagation

are (Bush and Fendell [3])

(3.32)

The eigenvalue

A is, now, determined

to be

A@; M$ = A(P) = @[/IO t p-‘A,

Substitution

+ * . .]=p’[~+(3-I)p-‘t...].

of (3.32) and (3.33) into the equations

(3.33)

of (2.18) yields

(3.34a) FO

r,=A,

r”=Ao(ltFo)T’ The boundary

conditions

F”

(ItF,)’

for these equations

+A

(I--FoMl+F,)

’

(ItFo)’

as 5 +-al.

n

that the solutions

of (3.34a).

F&)=Aoexp(-O-

=

Ad1

which

A0

+

=

exp

(3.35),

are

EM)

(3.36a)

(tbb),

=

A

I exp

F2 = A*(1 + F&l)) = A2 exp for A0 = exp (&b),

(#b

-

0,

(3.36b)

Al = const. (to be determined),

for AO = exp (&),

For A0 = exp (&),

satisfy

(3.35)

1 =exp(tb-8-1,

for

6(I)

(3.34b)

are

Fo, F,, F2.. . , +=, with +rO(l)forn=O,I,... It is found

““’

(‘$6

-

8,

A2 = const. (to be determined),

(3.36c)

. . ..

it is seen that (3.37a) (3.37b)

Introduction and l yields

of the solutions

obtained

for Fo, F,, F2, . . . and To, r,, . . . into the expansions

=&$l-j3-‘A,-/3-‘(A2-A:)-..+

for r

(3.38a)

610

W. B. BUSH

pFo

To express these results in the limit of Fo-+0 and/or I- &,, it is convenient to re-introduce the intermediate spatial variable, x. defined by (3.28). With (,$ - ,$)= u(@X +O for x fixed, p+m, F,=exp(~~-5)-I-(ox)[l+~(ox)+...]-to,

(3.39)

and, in turn, I-(ox)+;(Vx)2-+...

I

[I +/!-‘-p-2+.

. .],

6 - f _ePbx) exp {- P(o;u)} [I -i @(a~)~- k (ax) + (3 - 21)p+ +. . .I, fOhio=f.

A,=(3-I),

A,=-I,

AZ=2.

(3.40)

That ho= l/2, Ai = (3- I) has been justified previously; that A, = -I, A2 = 2 is justified by the matching of these solutions for the upstream region with those for the downstream region (see (3.30)). Here, it is important to note that consideration of higher-order contributions to the downstream-region solutions must be complemented by consideration of higher-order contributions to the upstream-region solutions-in both the phase-plane and physical-plane analyses. Thus, for example, the upstream representation for the temperature function T, to the orders of approximation considered is 7 a

exp (- (& - t)} . (I + @-’- /F + . . -).

(3.41)

However, for

)

. *

(3.42a)

the upstream representation for the temperature function reduces to 7 = exp (I). 4. FURTHER

RESULTS

(3.42b)

AND DISCUSSION

An examination of (2.18) indicates that this boundary-value problem can be re-expressed as d2r d7 =-A(l-r)exp{-By], @q r+O.

d7 G+Oast+-m,

7-1,

*do df

as

(-*cc;

(4. la)

de -=A(l-7)exp[-/3?). dZ (4. Ib) In this form, the solution for 7([;. . .) can be determined first, with the solution for c(.$; . . .) following in turn. It is essentially the boundary-value problem of (4.Ia) that provides the point of departure for the physical-space analysis of Williams[5]. In terms of the downstream region variables of (3.1) and (3.2) the boundary-value problem

Asymptotic

analysis

of laminar

flame propagation

611

of (4.la) takes the form d2G

(4.2)

v-P

Thus, the boundary-value problem for the leading order approximation in this downstream region is

d2Go -p- _ - bGoexp( -

dGo

Go); Go, z+O

as 5+m.

(4.3)

The first integral of this equation, satisfying the downstream boundary conditions, is determined to be ’ + (2A0)(G,-,+1)exp (- Go) = (const.) = (2&J,

(4.4a)

which, in turn, implies that $$ = - (2A,,)“‘[1 - (G,, + 1) exp (- Go)]“‘.

(4.4b)

An integral relation then, follows for &Go). Here, it is more appropriate to note that (4Sa) (as required); and that dGo

z

- - (2Ao)“‘[ I- k Go exp (- Go) + * * *] + - (2Ao)“2 as Go+ cc and/or 5 + _ 30, (4Sb)

In terms of the upstream region variable of (3.32), (4.la) can be re-expressed as

(4.6) The boundary-value problem for the leading-order approximation is, thus d27, dr,, p-*=0;

dT0 ro,g+O

as t--x,

where TV=

(4.7)

The first integral of this equation, satisfying the upstream boundary conditions, is dlo G

-

7.

=

const. = 0,

(4.8a)

such that

TO =

2

=

CO

exp

(6)

=

CO

exp

(fb)

’ exp

k

(‘fb

-

6))

=

DO

exp

{-

(6,

-

6%

Thus, to leading orders of approximation, the downstream and upstream solutions for KC.9 Vol. 17. No. L-H

(4.8b) 7 are,

W. B. BUSH

612

respectively T&;

P) =

1- P-‘[Go(S)+ . . .I

-1-~-‘[{(2Ao)“*(-~)+~~~}+~~~]as~+-m; Q)(& P) p Q(5) +

* * ’ =

DO

exp

{-

(66

-

(4.9a)

8)

-{DO[l-(5b_5)+“‘]}+..‘as

(4.9b)

[+tb.

Matching of these leading-order downstream and upstream solutions as 5 + - m and ,$+ 56, respectively, yields, just as in the analysis of Section 3, I& = 1, (2&J”* = 1

(4.10)

the burning-rate eigenvalue, as determined originally in the phase-space analyses. From the definitions of the variables for these two regions, it follows that

(4.11) The concept that results of the leading-order solution(s) of the thinner reaction zone can be used to provide boundary conditions for boundary-value problems for the thicker preheat zone for more complex flow geometries is now being employed. In principle, a physical-plane analysis, based on (4.la), can be carried to higher orders of approximation, just as in the present physical-plane analysis, based on (2.18), and will yield the appropriate corrections to the burning-rate eigenvalue, i.e. A, = (3 - I), . . . (Williams [S]), as determined originally in the phase-plane asymptotic analysis of Bush and Fendell [3]. However, it is possible only through consideration of a physical-plane analysis, that is carried to higher orders of approximation, such as the one presented in Section 3, to establish the two-region character of the downstream reaction zone. This two-region character of the reaction zone is lost in the phase-plane analysis of the deflagration problem. This concept of an upstream and a downstream zone of order-unity thickness in t-space, separated by a thin flame zone (of order /.!I-‘),is now being applied to nonadiabatic deflagration problems. Acknowledgements-The work presented in this paper was supported in part by the U.S. Air Force Office of Scientific Research under Grant AFOSR 77-3362, and in part by the National Science Foundation under Grant ENG 77-18730. The author gratefully acknowledges the encouragement and kind advice of Dr. F. A. Williams, University of California, San Diego and Dr. F. E. Fendell, TRW Systems and Energy Group.

REFERENCES

[I] F. A. [2] [3] [4] [S]

[6]

WILLIAMS, Combustion Theory. Addison-Wesley, Reading, Mass. (1%5). 1. GLASSMAN, Combustion. Academic Press, New York (1977). W. B. BUSH and F. E. FENDELL, Cornbust. Sci. Tech. 1,421 (1970). W. B. BUSH and F. E. FENDELL, Cornbust. Sci. Tech. 2, 271 (1971). F. A. WILLIAMS, AGARD Con/. Proc. No. 164. III-I (1975). J. F. CLARKE, Cornbust. Sci. Tech. 10, I89 (1975). J. F. CLARKE, Prog. Aerospace Sci. 16, 3 (1975). G. F. CARRIER, F. E. FENDELL and W. B. BUSH, Cornbust. Sci. Tech. IS, 33 (1978).

[7] [8] [9] R. FRIEDMAN

and E. BURKE, 1. Chem. Phys. 21,710 (1953). (Received 6 September 1978)

APPENDIX:

THE

FARFIELD

DOWNSTREAM

REGION

Based upon (3.22). it is taken that the appropriate independent and dependent variables for the farlield downstream region are 6, = B-‘I = (I-

5lJ:

(Ala)

Asymptotic

analysis of laminar flame propagation

~(~;@)~l-P-‘exp

c(f: fi) =

(

-3

1- exp -3 (

)

613

[gd5,)+B-‘g,(f,)t.‘.l.

>

[f&1)+

B-‘f,(5r)

+.

.l.

(Alb)

Further, it is taken that A is of the form (A2) Substitution of (Al) and (A2) into the equations of (2.18) yields 1

pgo=h 1

A=~7io. f,-2 It is seen that the zeroth-order

I

dgo

dfo

1

(A3a)

pg,-;iTI=f,-IO’...; m

(A3b)

~=~g,t2”*(3_I)g,,....

equations of both (A3a) and (A3b) yield

fo-$o=o; while, the first-order equations of (A3a) and (A3b). respectively,

(A4)

yield

+rzB’=-($&$

(ASa)

f, -j&g,

(A5b)

= P($t

(3 - I)g,).

In turn, (A4) and (AS) can be combined, with the result that dg, + 2”*(3 -I) 2 d5, g&J

- I

= 2”2fo(51)= g,(O) exp

g,=OJ - 2”2(3;

‘)-

I s,),

where g,(O)= Gab, in order that these fartield solutions match with the nearfield solutions of (3.21). It is seen that the determination of the zeroth-order solutions, g&r) and f&,), requires the consideration of the first-order equations. In the sense that these first-order equations contain the convection contribution(s), the determination of these zeroth-order solutions can be said to involve a diffusion-convection-reaction balance.