Flame spread in an opposed flow with a linear velocity gradient

COMBUSTION AND FLAME 50:287-304 (1983)

287

F l a m e S p r e a d in a n O p p o s e d F l o w w i t h a L i n e a r V e l o c i t y G r a d i e n t I. S. WICHMAN Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544

A theoretical model is developed for estimating the rate of flame spread under conditions of heat-transfer control with account taken of the fact that the gas velocity is not uniform. It is shown that the functional dependence of the spread rate on parameters such as the external gas velocity is modified from that obtained in the classical study of deRis. The modified formula is capable of lending interpretation to previously anomalous experimental observations.

1. INTRODUCTION

2. FORMULATION

A model problem focused on energetics and equivalent to the problem of thermally controlled spread of flames over a semiinfinite solid, with an Oseen approximation for the gas velocity (a uniform velocity profile), has previously been investigated [1]. The model problem is best understood on the basis of a heating-element interpretation in which the distribution of the heat-release rate along the downstream half of the gas-solid interface is to be obtained subject to a prescribed temperature distribution there. This model problem is modified here by replacing the Oseen prof'de by a linear velocity profile with a no-slip condition at the interface, as shown in Fig. 1. Results of the modification should be more realistic, for example, in a forced gas flow over a plate with its leading edge sufficiently far upstream from the position of inception of heating. Consequently the physical problem that the present model may be expected to describe is one for which the velocity boundary layer is much thicker than the thermal boundary layer, the flame being embedded inside the velocity boundary layer. Experimental evidence exists to suggest that often this situation actually occurs near the leading edge of the flame.

The equations for conservation of energy in the solid and gas phases in a coordinate system FLxed to the flame front are

Copyright © 1983 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc, 52 Vanderbilt Avenue, New York, NY 10017

02Tit + 02Tg 0Tg ~'gx 03X2 ~'g Y --~)yg2 -- pgCpg(Us + ayg) --Ox = 0

(1) and ]ksx a 2 T s

a2Ts

~JTs

ax 2 + x~ ay---f - p s c p , u , a---~= 0,

(2)

where a is the velocity gradient in the gas that specifies the linear profile. The boundary conditions at y = 0 are

T,(x, 0) = r,(x, 0),

x < 0,

a t , (x ' o )' ~,~ -a -t , (x, o) = -x,~ Ty,

ayg

(3) x
(4)

and

T,(x, 0)-- T,(x, O) = T,,,

x/> o.

(5)

O010-2180/83/04287 + 18503.00

288

I.S. WICHMAN Yg

GAS PHASE

FLAME A.,...,.

sional coordinates. In the subsequent discussion the subscripts g and s will be eliminated from r/ since the phase will be obvious from the context. The nondimensional equations and boundary conditions are r ~ + r,m - (e + n)r~ = 0,

........................

- - ~ < ~ < oo

I~x\\\\\O

a~+onn-p*a~=O,

,7/>0,

(6)

---oo<~
r/~>0 (7)

SOLID PHASE

and

U S

Y$

r = o,

Fig. 1. Flow configuration in a coordinate system fixed to the flame front.

r=o=

% =--Aan, 1,

rl=O,

7=0=0, Far from the interface the conditions Tg(-Oo, y g ) = Too, Tg(x, co) = Too, Tg(% yg) bounded and Ts( --°°, Ys) = Too, Ts(x , co) = Too and Ts(oo, Ys) bounded must apply. Equations (1) and (2) represent a balance between streamwise and transverse diffusion of heat and streamwise convection of heat in the gas and solid phases. The heat release does not appear explicitly in these equations because it is confined to yg = 0, x / > 0 and is determined in the course of solution. The boundary conditions given by Eqs. (3) and (4) for x < 0, y = 0 express the fact that there is no release of heat occurring upstream of the flaming region. As a consequence the interfacial temperatures and heat fluxes must be continuous. However, the boundary conditions given by Eq. (5) for x > 0 require only the attainment of the constant elevated temperature Tv at the surface for both gas and solid. Since the flame is assumed to lie along the surface for x ~> 0, the surface heat fluxes into the gas and solid are not expected to be continuous here. It will be seen in the subsequent analysis that consideration of surface heat fluxes for x ~> 0 leads ultimately to a flame spread-rate formula. To nondimensionalize Eqs. (1)-(5) put r = (Tg - T**)/(T v - T**) and o = (T s - Too)/(T v T**) as the nondimensional temperatures in the gas and solid, respectively, and ~ = x/L, rlg = y g ~ / L , rls = Ys X~sJksy/L, where L = (kgx/pgCpga)1/2(~gx/~kgy) 1/4 as the nondimen-

r/-+ oo,

~ = 0,

~<0,

~>0, ---~
(8) (9) (10)

where subscripts denote partial derivatives and P* = L/(?Vsx/psCpsUs), A = (ksxksy/~.gxXgy) 112, ande=

usx/Xgx/Xg~/aL. The parameter e represents the ratio of the spread rate u s to a characteristic velocity Ug* for the gas, the latter being the square root of the product of a with the thermal diffusivity X/kgx~.gv/Cogcpg). To avoid considering merely a perturbation of the Oseen model, e is not allowed to be large. At certain points in the analysis the approximation e = P* will be made for convenience, but most interest centers on the limit e -+ 0, in which P* plays the role of the effective ratio of gas-phase to solidphase Peclet numbers, with the gas velocity in the gas-phase Peclet number being Ug*. The conductivity ratio A is the same parameter that appeared previously [ 1 ].

3. TRANSFORM SOLUTION Introduce Fourier transforms in the streamwise coordinate ~, defined by

r_(v, n) =

# (

r(~, n)d ~ d~i,

oo

r+(m n) =

r(~, n)e ~ d~,

(11)

THEORETICAL MODEL OF FLAME SPREAD with similar notation for o. The subscripts + and denote analyticity in upper and lower halves of the v plane, respectively. The inverse Fourier transform is

r(~, rl) = 2rr

f

-oo

The transformed equations (6), (7) and boundary conditions, Eqs. (8)-(10) are then

+0 [r+(v,n)+r-(v,n)l =0,

v(v-

<,e(0)=q (19)

[r+(v, n) + r_ (v, 7)]e -iv~ dv. (12)

-

tions of the first and second kind, Ai(O) and Bi(O), by [2]:

= k I ,z[Ai(--O) ~ iBi(--O)],

~

1

289

(13)

he*) [o+(v, 7) + o_(v, 7)1 = 0, (14)

r-(v, O) = o_(v, 0),

%-(v, 0) =--A%-(v, 0),

(15) r+(v, 0) = o+(v, 0) = i/v,

r+(v, ,~) + r_(v, n)

-- [T+(~, 0) + T (v, 0)1 he(O)/he(Oo),

(20)

(16)

and

r+(v, oo) + r_(v, oo) = O,

where kl. 2 = (12) 1/6eZ-in/6. With the cuts selected in the v plane, the functions hl,e(O ) have branch points at v = 0 and singularities at v = i(e + 7), a point that moves from v = ie at 7 = 0 to v = i~ as r~ ~ o~. As [0 I ~ % he(O) vanishes uniformly while hl(0 ) becomes unbounded in the region - 2 n / 3 < arg 0 "< 0, which corresponds to - n / 2 < arg v < 37r/2 (but with most of the upper-half v plane excluded); also, the zeros of he(O ) lie along the ray arg 0 = 2n/3, which corresponds to the positive imaginary axis of the v plane (Figs. 2, 3). Thus, the solution to Eq. (13) satisfying Eq. (17) is a constant times he(O ) . Evaluation of the constant at = 0 gives

o+(v, oo) + o_(v, oo)=O,

(17) where 0 = v 1/a [v - i(e + 7)] is introduced in place of ~ into the transformed gas-phase equation to obtain an Airy equation. Branch cuts in the v plane are drawn from v = /P* to v = i~ and from u = 0 to v = - i oo. Then Re {X/v(v - / P * ) } ~> 0, and the solution to Eq. (14) subject to Eq. (17) is

o+(v, 7) + o_(v, 7) = [a.(v, O) + a_(v, 0)1 e-x/~(v-u'*)n.

where 0 o = v I la(v - ie). The function ha(O ) possesses a power-series expansion about v = 0 in powers of v l/a and is not analytic in the lower half-plane. The singularities are such that the inversion contour must pass through the overlap region, 0 < Im{v} < min(P*, e + 7?), of the upper and lower halves of the v plane (Fig. 3). By differentiating Eqs. (18) and (20) with respect to the appropriate transverse coordinate 7 and then setting r/= 0 and using the boundary conditions given by Eqs. (15) and (16), one finds that

rn+(v, O) + A % _ ( v , O) -- ivlla K(v)[ i/v + a_(v, 0)]

(18)

The general solution to Eq. (13) can be written as a linear combination of hi(0 ) and he(0), the modified Hankel functions of order 1/3, related to the Hankel functions of the first and second kind of order 1/3, /-/(1~2)(0), and to the Airy func-

(21)

and

%÷(v, O) + on_(v, O) = --X/v(v -- iP*)[i/v + a_(v, 0)1,

(22)

where K(v) =- d ( l n [ h e ( e o ) ] }/dOo. Equations (21)

290

I.S. WICHMAN

ZEROS of

h2lOx~ )----~'x

ZER~~ ..lO~... x~

Fig. 2. Behavior of the functions hl(O), h2(O ) in the complex 0 plane.

J

/

Fig. 3. Mapping of the 0 plane onto the v plane. Shown are the upper and lower halves of the v plane, the overlap region, 0 < Im(u) < (e + ~), and the angles # and ¢~.

THEORETICAL MODEL OF FLAME SPREAD and (22) represent two equations in the four unknowns o n _ , %+, %+, and o_. These equations (or any two independent linear combinations of them) must both be "split" in obtaining a solution by the Wiener-Hopf procedure. Present objectives, however, are not so comprehensive, and consistent with the heating-element interpretation of flame spread [1], only the functional form of the interfacial heat flux rn+(v, 0) + Aon+(v, 0) is sought. Multiplication of Eq. (22) by A, addition to Eq. (21), and subsequent division of the result by v1/2 yields ~-u-u)+ [ rn+(v, 0) + Aort+(v, 0)]

291 where the fact that the quantity in the curly brackets is analytic in the upper half-plane may be shown to follow from the definition of C. Without the approximation e = P* a factor involving a square-root term multiplies v 1/6 in Eq. (25), introducing branch points at v = ie and at v =/P* that remove the indicated analyticity; the requisite cut from ie to/P* seems relatively inconsequential for the gross results to be derived. The factoring needed now may be obtained by inspection, viz., (~)_(//v). = [v'=~(i/,,)]

= + + [(i/~)vS=-~*

= _ [ i ~ 1 / ~ K ( ~ ) + A( vx/b--=/e* )_1 X [i/v + o_(v, 0)].

-- X/'~;~(qv)] _. (23)

For solution by the Wiener-Hopf technique [3, 4] the right side of Eq. (23) must be factored into components that are analytic in the upper and lower halves of the v plane. 4. SUBSTITUTE KERNEL

Substitution of Eq. (26) into Eq. (25) yields [rn+(V, 0) + Aon+(V, 0)]/((1 + A)v I/a x [v~/6 + ic/O + A)]}+ + x/z~(i/v)+ = E(v) = --[(i/v)x/v -- iP* -- (i/v)x/'z-'~ l _

The mathematical complexity of the function K(v) requires the introduction of an approximate

kernel. Three criteria for a good substitute kernel [5] are that the value and derivative agree at v = 0 and that the values tend to agree as Iv I -+ o~. The approximation K(v) -~ ( C - iv 1 1 6 ) X / - ~ ie

(24)

with C = x / ~ K ( 0 ) = (e-27r i/a/V~) [31/a r'(.]-)/p({)] satisfies the first and third of these criteria exactly, and the second is nearly satisfied, in that the approximation gives K ' ( v ) ~ v - 5 / 6 as v -+ 0, while the exact behavior is K ' ( v ) ~ v - Z / s . Substitution of Eq. (24) into Eq. (23) produces an equation that is still too difficult to factor by inspection. For simplicity e = P* may be employed in Eq. (24) prior to substitution into Eq. (23), whence upon rearrangement one obtains

- vx/7 = - / P * a _ ( v ,

0),

where E(v) is an entire function (analytic in the extended complex v plane). In view of the known behaviors as I vl -+ oo of the functions on either side of the above equation, E(v) - 0 by Liouville's theorem. Hence, with the use of Eq. (15), one may write rn+(v, O) + % _ ( v , O) + A[%+(v, 0) + %_(v, 0)]

X/v- (1 +A)

iCvr~--~ v21a ,

(27)

the Fourier inversion of which for ~ > 0 is %(~, o) + a % ( ~ , o)

[rn+(V, O) + Aon+(V, 0)] / ((1 + A ) v l l a l v lie + iC/(1 + A)]}+

(I + A ) ~

3~la - - -

= --( -x/77-~-)_[(//v)+ v + a_(v, 0)],

(26)

(25)

r(-})~ 1/a

•

(28)

292

I.S. WlCHMAN

Reintroduction of dimensional quantities leads to

aT,

aT, I

X,s

r_(v, O) = - ( i/v )(x/ff-Z-~ -- VrA~ )/X/v -- ie . (31) The inverse transform of Eq. (30) in dimensional variables is

= -- [N/PsCps)ksyUs + V,O gCp g ~-gyu S

L ~gy 213(pgCpga)l/3 +

and

~kgy ~Tg

q

r(½)(x/3)i/3|_J (Tv -

T..),

(29)

V~s~,psCpsU s

and

and C = v

K(O)

---- V %/pgCpg~kgyus

ayg Yg=O of

where

~

~

=

=

L N/~ ~kgy2/3 (,OgCp ga) 1/3

r(~Xx/3)l/3

(rv - r.)

(32)

have been employed.

forx > 0. The correspondence of Eq. (32) with Eq. (29) for Xsy = 0 supports the acceptability of neglecting the cut from v = ie to v =/P* in the previous development. Moreover, with the solution obtained for r, the solid-phase problem becomes

5. SOLUTIONS FOR P* >> 1 Since there are questionable approximations in the derivation of Eq. (29) that has been given here, and since those approximations are most questionable for P* >> I, it is worthwhile to pursue an alternative approach for this case to check Eq. (29). When P* >> 1 the characteristic length for heat transfer in the gas is much larger than that in the solid, and the nondimensionalizations for x and yg that lead to Eq. (6) are appropriate. However, in the solid the effects of streamwise conduction are expected to be small, and therefore the modified definition ~s = YsVcP~ ~ y / L is helpful. This changes Eq. (7) and the second of Eqs. (8) to o,7,7 = a t + O(1/P*) and rn = - R * o n, where R * = A~/P-~-- {~ksy(PsCpslAs)/[~kgy(JggCpglAg*)]

)1/2,

analogous to the previous R [1], represents the ratio of the ease of heat removal (through downstream convection and transverse conduction) in the solid to that in the gas. In the limit R* -> 0, the gas removes most of the heat generated at the solid-gas interface, and the boundary condition r n = - R * a n (~ < 0) becomes approximately r,~ = 0, so that e is the only parameter that remains in the problem. The gasphase problem now may be solved first by a method paralleling that above, with the substitute kernel, Eq. (24). Results are "rr/+(P , 0) = - - ~

+

Vr'~-~C/p 2/3

(30)

on n = o t '

--oo<~
a(~, O) = ~

_

r~>0,

r(v, O)e - w t dr,

~ < O,

oo

a(~, O) = 1,

~>0,

o(~,oo)=O, (33)

from which it may be shown by Fourier-transform methods that

on(G, 0)=--(V~-/2rr)f~_ [v(v--ie)] - 1 / 2 × e - i v t dv

(34)

for ~/> 0, giving as ~ -->oo the inversion

~Ts ys=0 -

~/)ksYpsCpsUs'

r.),

(35)

which when added to Eq. (32) gives Eq. (29). In the limit R* --> 0% the solid removes most of the heat, and the gas-phase and solid-phase problems are again uncoupled. The latter, now defined by Eq. (33) with its second line replaced by on(G, O) = 0 for ~ < 0, may be solved first, giving o = 0 for

THEORETICAL MODEL OF FLAME SPREAD t < 0 and (r = erfc[r//(2X/~)] for t :> 0, whence on(t, 0) = - 1 / ~ / ~ for t > 0. The boundary conditions at 77 = 0 for the gas-phase problem, Eq. (6), then become r = 0 for t < 0 and r = 1 for >/ 0. Although this problem is soluble by transforms without use of the Wiener-Hopf methods, the approximation, Eq. (24), is needed for evaluating the inverse transform, and the result Tn(t, 0) = --X/Q/X/-~ -- 31/3/[P(½)tl/a] is thereby obtained. From these formulas for 0,7 and T,~, Eq. (29) again follows, thus suggesting its approximate applicability for all R* when P* >> 1. 6. SOLUTIONS FOR P* < 1

When P* <~ 1 most of the energy transferred forward of the point of inception of heating is expected to pass through the solid. Therefore nondimensional coordinates are redefined so that all coefficients in the solid-phase energy equations are unity. Let } = P*x/L and r/s = P * y s ~ y / L so that Eq. (7) is replaced by o ~ + o,~,7 = of. In the gas, to maintain a balance between transverse diffusion and streamwise convection, let r~g = P*l/aygx/~-gx/Xgy/L. Then Eq. (6) becomes rnt / r/7"~ = ep*l/aT"~ p*4IaT~.~ ~ 0 for small P*. The only change in Eqs. (8)-(10) is that % = -R*p*I/6o,~ for 77 = 0, t < 0. A one-parameter problem results, with R' :- R*P .1/6 being the relevant parameter. As R' --> 0% the solid-phase problem may be solved first; it is identical to a problem solved earlier [6]. This limit gives, for the gas-phase problem, r - rTr~ (_oo < t < 0% r7 ~> 0), r(t.~>0) = e r f c ~ : ~ (} < 0), T(t, 0) = 1 (t ~> 0), and r 0 as r~ -+ oo. The solution, through a Duhamel superposition, is -

-

-

r(t, ,7)

fo~~e-(t-~) I~ffi fee+looai~(r?sl/3Ai(O) )_io $ :1 fo°Oe-t[ 1 fc+'~Ai(nsl/a) X--- ds dt,

eS(~÷O 1 dt, $

X--ds

293 which results in

~,~(t, o) = (3~/3/[~-r(})] X

)

t--l/2(t + t) - l / a e - t dt

for } > 0. For large t, the combination of this result with that for on(t, 0) gives again Eq. (29). As R' --+ 0, the gas-phase problem may be solved first. This solution defines a solid-phase problem that has been solved previously [ 1]. Equation (29) is recovered again from these solutions. Since these calculations for P* ~ 1 do not involve introduction of a substitute kernel, it may be concluded that Eq. (29) is not influenced strongly by the approximations in Eq. (24). Thus Eq. (29) may be employed for the entire range of parameters. The formula appears to become exactly correct as R* ~ oo and to represent an expansion for large } otherwise. This is comprehensible in view of the inaccuracy in K'(v) for small v in Eq. (24).

7. AN APPROXIMATE FLAME-SPREAD FORMULA

From Eq. (29) an approximate flame-spread formula may be derived in a manner analogous to a previous study [1]. The approach involves first selecting a representative value of t or x because unlike [1], differing functional dependences on x appear in Eq. (29). The left side of Eq. (29) is equated to -q/x I/z, the heat flux entering the surface by diffusion through the gas-phase velocity gradient in the region x ~> 0. By noting that generally (PsCpsks)/(pgCpgXg)>> 1 the gas-phase term X/pgCpgXgUs/V:~ in Eq. (29) may be neglected in comp~.rison with the corresponding solid-phase tenn. Equating these two representations of the heat flux into the surface for x/> 0 gives, approximately,

}<0

~>~0,

(36)

V~PsCps~,sy//s +

all3E~ 1 ~.r-gCpg ~,Sy 2~1/3 / q

x (Tv-

T~) ~ - -

X1/3

,

294

I.S. WICHMAN

whereby reintroduction o f x = ~L and use of Xgx = ~,gy = ~kg yields

us=k~113%~gPgCpg;kgI~vv--Tvl 2 ' p s C p s ~ s~y T=

(37)

which defines the spread rate u s in terms of the gas-phase velocity gradient and thermodynamic and transport properties of the two media, once is specified. Here k = 32/a/r/P(~) 2, and T~ = T~ + q [F(½)/3113 ]/a 1/3(pgCpgXg2)X/3 is defined as an effective flame temperature. To demonstrate that this is a reasonable definition, the rate of heat release per unit area at the surface is written as the product of the heat of reaction per unit mass of oxygen consumed, Cpg(Tf - T ~ ) / Y o ~ , with the diffusive mass flux of oxygen to the surface pgDg(aYo/ay)o , where Yo is the mass of oxygen and Dg its coefficient of diffusion. In the flow with linear velocity gradient it may be assumed within reason that the x dependence of the oxygen diffusion rate to the surface matches the x dependence of the heat release rate there, whereby (aYo/ay)o ... Yo~/Xl/3L z/a for x > 0, where L = (Xg/pgcpga) 1/2. Consequently the energy release per unit area per second as controlled by the diffusion of oxygen is [cpg(Tf -- T~)/Yo ~ ] [pgDgO Yo/aY)o ] (Tf --

T~.)(pgCpg~g2)ll3all3/x113~ q/x 113,

where Le = 1 has been used to eliminate Dg. Therefore the formula

(Tf- T~ )(~,g2pgCpg)ll3a113 xlla

[p(½)1311a]xlla

for the heat release rate is indeed consistent with the view that heat is supplied by the diffusioncontrolled combustion of oxygen transported through the gas phase velocity gradient to the burning surface. The principal remaining obstacle to a thorough understanding of Eq. (37) is the ~l/a factor contained therein. At the present stage of development, a representative constant value of this factor must be selected to determine us. In the fol-

lowing section an alternative simplified model is discussed which is capable o f lending physical interpretation to this factor. 8. MODEL PROBLEM WITH NEGLIGIBLE STREAMWISE DIFFUSION The successful results obtained by neglecting streamwise diffusion terms in an Oseen-flow model problem in a previous study [1] lead one to expect that valuable insight may likewise be gained by invoking the same approximation in the case of the model problem with linear gas-phase velocity gradient. One notes that streamwise diffusion terms are also discarded (but only in the gas) in the customary boundary-layer formulation of the flamespread problem; the present model imposes an additional constraint through the use of a velocity field with a linear gradient. In the formulation of the relevant equations and boundary conditions for this model problem the gas-phase diffusion flame is allowed to lift off the fuel surface at a finite nondimensional downstream distance Go from the orgin (see Fig. 4). This modification, although physically desirable for flame-spread problems, departs from the surface heating-element interpretation identified in the introduction and was introduced only after attempts to obtain solutions without flame lift-off suggested its possibility. The nonphysical aspects of solutions obtained without flame lift-off indicated its inclusion to be desirable. Concerning the describing equations, the presence of the diffusion flame in the gas requires the use of the equations for conservation of energy, fuel, and oxidizer; for this purpose the gas phase reaction is taken to be F + VlO o P on a mass basis. Concerning the boundary conditions, the presence of the flame along the surface in the region 0 ~< ~ ~< Go indicates that the boundary condition for energy conservation here is a balance between energy release due to diffusive influx of oxidizer from the free stream and conductive efflux of heat to both media, viz. (in dimensional form),

p g D g ( q o - L) OY1 = aTg (I)1 - Yls) ayg --Xgy -ay - -s-

aT s ~sy -ay. -'

0 ~
(38)

THEORETICAL MODEL OF FLAME SPREAD

2~)5

Ii I¸¸ i

~?g

_//

Fig. 4. Physical configuration of model problem without streamwise diffusion.

Downstream from } = }o the appropriate boundary condition for energy conservation is represented by a balance between the energy required for vaporization of the solid fuel, the energy influx by heat conduction through the gas (from the flame), and the energy efflux by heat conduction into the solid. Thus,

pgDgL 3Y2 -

Xgy---X~y--,

3Tg

3T~

1 -- Y2s ~)'g

3yg

3ys

x > x o. (39)

These boundary conditions are nonlinear in the general case, because of the presence of Yls or Y2~ in the denominator arising from convective contributions. To facilitate solution, Eqs. (38) and (39) are linearized by neglecting Y, in the denominator, thereby enabling a straightforward application of the Schvab-Zeldovich procedure. The details of the mathematical solution, including the specification of the flame location in the gas and the derivation of the flame-spread formula, are given in the Appendix. A contour plot for the nondimensional temperature field r and the oxidizer (Y1) and fuel (Y2) concentration fields is shown in Fig. 5 for the case in which the free-stream oxidizer concentration is unity (i.e., YI= = 1). It is seen from Fig. 5 that the nondimensional temperature field is fairly real-

istic, with the flame temperature rising from T = 1 at Go to its adiabatic value rat far downstream. However, tinearization of the boundary condition for energy conservation for ~ > ~o causes Y2 > 1 to occur far downstream, while the condition given. by Eq. (38) generates a singularity in the concentration of oxidizer at the origin, where YI(0, 0) = _oo. This last result is caused by the fact that the gas-phase thermal gradient is infinite at the origin (because of the discontinuous jump in temperature there), which requires similar behavior on the part of the oxygen concentration gradient [through Eq. (38)] and consequently in Ya itself. With r = 1 at the surface the temperature field remains everywhere bounded despite the infinite gradient at the origin; the observed singular behavior for YI is permitted because the value of Y1 is not specified at the surface [and indeed cannot be specified if Eq. (38) and the value of r both are given]. The linearization of Eq. (38) is not responsible for the singularity; in fact it mitigates its strength by producing a larger multiplicative heat-release factor (since Yls < 0 here), thereby allowing smaller values of DY1/Oyg. The negative values of Y1 represent an unphysical aspect of the parabolic problem with the linear velocity profile and would be expected to vanish if streamwise diffusion were included. That the flame does indeed lift into the gas at = Go, despite linearization of the boundary condition for energy conservation, is a direct mani-

296

I.S. WICHMAN

Q=32.9 M=3.9 Y1=1.92 Y~co:1 ~¢0= 4

6

I

4 ~g

.

.~ t . ~ 1 t / 1 r=.l

~

~'~.~

/

/ /

//

j"

I

.I/

:31-

I",,;"

.7

.'/

I

./

. .,~

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..,'" ~

1|i/ I//-

~~.,~1I ~ _2° .~>~- " J~

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,t-: ~/~"-

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~"

0

YI|: - ~3;I"-4.._

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~'~ ~ , ~

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--- --

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-

8 =0 65~ "

.

of

-

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~

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. ...~"'~ .. . . . .

~ L - ~- ....... ~ ~,--, ~..II~ I~ "'~'" 4 5 10

-'r=9 . . . .

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- - I

-~

.

.- ./

/

/ /

.

..

. ~ . .-~

Y1='9

"/

21I/ I'/

_/.-d I// _ <.~'" U .f/

.

//

~

.

. . . . . . / ~

"

. ----.. -

~.----------: . . . . . I 15

~=,-~ ~'~

I 20

I 25

Fig. 5. Contour plot for the model problem without streamwise diffusion when Y1- = 1. Shown are contours for z, Y1, and Y2 in the upstream region (0 < ~ < ~o), the

vicinity of the lift-off point (~ - ~o < 1), and the far downstream region (~/~o >> 1). The dashed lines between the lift-off point vicinity and the far downstream region are interpolated.

festation of the importance of the velocity profile near the surface. In the Oseen-flow model studied previously [1, 7] the flame laid always along the fuel surface downstream of the inception point ~ = 0. The direct cause for this behavior was not clear, because of the simultaneous use of both the Oseen-flow approximation and the linearized form of the boundary condition for energy conservation. Since the Oseen-flow assumption is discarded while the linearized boundary condition is retained in the present model, the presence of the flame in the gas clearly demonstrates the importance in the spreading process of the fluid-dynamic flow configuration near the surface and suggests that the linearization of the boundary condition for energy conservation may be of secondary importance. The linearization becomes significant only well downstream from the lift-off point, where it is responsible for values of Y2 in excess of unity.

9. CRITERION FOR FLAME LIFT-OFF AND FOR SPREAD RATE

The equation for the rate of flame spread obtained in the Appendix [Eq. (A.18)] is repeated here for convenience, viz.,

us=K~oll3vrd~g9gcpg~g~~ T~Tvl2 p~Cp~,sy

LT~-

(40)

Tool

Equation (40) suggests that the ~1/a factor in Eq. (37) may in some sense be interpreted physically as ~o l/a, the cube root of the downstream distance for flame lift-off, even though the analysis leading to Eq. (37) assumed the heat release to be distributed along the interface. Although Eq. (40) and the discussion preceding it indicate that Go represents the nondimensional downstream distance at which the flame lifts off the surface (when there is no streamwise diffusion), appropriate

THEORETICAL MODEL OF FLAME SPREAD means for determining its magnitude have not yet been indicated. It was not clear how to develop an independent criterion for fixing both u s and to simultaneously for the model problem considered in the Appendix. Thus, in contrast to the Oseen-flow problem, an additional physical argument is necessary to fix to and to determine the spread rate; this may be viewed as a consequence of the fact that the possibility of the flame lift-off introduces an additional degree of freedom, the lift-off point, the value of which can change in response to an imposed change in the spread rate with all other quantities held fixed, unless an additional physical condition is appended. A somewhat different view would be obtained by considering the solution given in the Appendix with the heat release always at the solid surface; in this case diffusioncontrolled burning can not occur downstream from a point ~* at which the oxidizer concentration at the surface vanishes, and a change in u s can be accommodated by a change in ~*. The similarity of the velocity profiles in the gas and solid phases for the Oseen-flow problem enables a diffusioncontrolled burning condition to be applied at r7 = 0 for all ~ > 0 even in the parabolic case and thereby allows determination of a unique spread rate through an overall energy balance; but with differing velocity profiles in the parabolic problem without flame lift-off the diffusion-controlled criterion could be applied at best in some sort of an averaged sense. Putting ~ in Eq. (37) equal to a specified constant is equivalent to selecting such an average for use in applying an overall energy balance with a heating-element interpretation. In a strict heatingelement interpretation it is not possible to consider ~ in Eq. (37) as a representation of the nondimensional downstream lift-off distance (since the flame lies along the surface in that interpretation), but it is likely that if the theory with streamwise diffusion were augmented by addition of species conservation equations, such an interpretation of could be developed. In the following paragraph a physical argument is presented for determining the downstream lift-off distance under the assumption that in Eq. (37) ~ actually represents the downstream nondimensional distance for flame lift-off. This development departs from both

297 parabolicity and a simple overall energy balance and takes advantage of the lift-off possibility (unavailable in the Oseen problem) to introduce additional physics which involves upstream heat conduction in an essential way. To obtain an expression for ~o it seems reasonable to introduce the fact that in reality quenching by heat loss to the solid prevents heat release from occurring upstream from the lift-off point. With this modification the effects of streamwise diffusion must be included in the region between ~ = 0 and the lift-off point. Thus a heat balance between forward heat conduction from the flame (anchored at to) required to maintain the elevated surface temperature Tv, and downstream convection, is constructed in the region 0 ~< ~ ~< to, giving Xgx(~Tg/OX ) ~ pgcpgug(T v - T~). Replacing ~kgx(~Tg/<)x ) with Xg~(Tf - Tv)/X/ag/a~o and using Ug ~ aS, where ~5 is the thickness of the thermal diffusion layer, yields

p gcpgacS( Tv

--

T=)

Xgx(T~

--

Tv)

/~

.

(41)

to

The thickness 8 of the thermal diffusion layer is approximately 6 ~ x/agy t, where t is a representative convection time, viz., t = distance/velocity = ~L/(u s + ay) = ~L/aL(r~ + e) ~ ~L/a8 (assuming e "~ 1), whereby 8 ~ (agy/a)t/Z~l/a; the thickness of the thermal diffusion layer grows as ~l/a in the presence of the linear velocity profile. Substituting this expression for 6 (evaluated at ~o), and using ?,gx =kgy in Eq. (41), leads to

to

.+ (rr_- rv )3,, \Tv-T=

(42)

Equation (42) shows that to is a function of stoichiometry through Tf; as Ya~ increases (decreases) to likewise increases (decreases). This is understandable since an increase in YI~ increases the driving temperature difference for forward heat transfer, in comparison with the temperature rise needed to maintain the constant surface temperature Tv in the quenched region 0 ~< ~ ~< to- The exponent in Eq. (42) can vary with the details of the

298

I.S. WICHMAN ,,I////

physical argument; an alternative development replaces 3/4 by 3/2. Equation (42) suggests, for use in Eq. (37), the choice

/ / /

H

/

FLAME

/ / / /

/ / / / / / / / /

(43)

K

(a)

where D is a constant of order unity and m is a constant greater than zero. By use of Eq. (43) the flame spread rate given by Eq. (37) becomes

p,cp,x,

p,c,,,x,,

/ / / /

~L

Tf--Tv)m,

,,

//

DOWNSTREAM BOUNDARY LAYER

A*I°

v, - rv ]

£ (44)

-- ~-----"r'--~-- VISCOUSPROFILE COMPOSITE PROFII g

where A = KD a/3 is a constant. It seems likely that Eq. (44) represents an improvement over Eq. (37) or (40) for the spread rate; Eq. (44) is not obtainable solely from an overall energy balance and moreover involves upstream conduction of heat in the gas. However, it is very similar to Eq. (37), which is based on the overall energy balance. Therefore the balance appears to continue to play an important role in influencing the (conductive and convective) transport-controlled rate of spread along thermally thick solid fuels. 10. COMMENTS ON COMPARISONS WITH EXPERIMENTS The present study has placed predominant emphas•s o n gas-phase fluid-dynamic processes occurring near the surface; the flame-spreading mechanism is assumed to be controlled by local conditions of the flow field. The important implication here is that flame spread rates us measured in two (or more) different experiments may be identical even though the external flow conditions of these experim ents differ significantly (all other factors, such as free-stream oxidizer concentration Yz~ and solid fuel material, remaining the same). The only requirement, from Eq. (44), is that the surface velocity gradients a be identical. For example, a comparison is now possible between rates of flame spread into an opposed (forced) fully developed channel flow [8] and into

k~- INVtSCID PROFILE

•f•

UPSTREAM BOUNDARY LAYER

(b)

Fig. 6. Physical configurations for opposed-flow flame spreading in (a) the channel-flow geometry of Ray [8] and (b) in the free-convective geometry of FernandezPello and Williams [91. a buoyancy-induced vertical free-convection flow [9]. Such comparison would be unjustified on the basis of an Oseen-flow theory of the flame-spread process. In the channel flow the surface velocity gradient is given by a = 4uc/H, where u e is the centerline velocity and H the height of the channel (see Fig. 6a). In the free-convection boundarylayer problem the boundary layer within which the flame lies generates an inviscid outer flow, as indicated in Fig. 6b, which in turn induces a boundary layer ahead of the flame, also indicated in Fig. 6b. From Eqs. (19), (32), and (33) of [9] it can be shown that a = 0.236u~a/2(pg/pg~)gl/6/ u= 5/6 at the point of flame inception, where g is the acceleration of gravity and u~ is the inviscid flow velocity evaluated at the origin (Fig. 6b). Equating these two expressions for a gives the required relation between the velocities u= and u e in the outer flow field for these two different physical configurations. By using H = 2.54 cm as a representative channel height, T~ = 298°K, and

THEORETICAL MODEL OF FLAME SPREAD T --- 1264°K [which for air is the arithmetic mean of ambient and dissociated flame temperatures for spread over polymethylmethacrylate ~ M M A ) ] to evaluate pg for air at p = 1 atm, one finds u= u~ 21a', thus when u~ ~ 100 cm/s, u= ~ 20 cm/s. The surface velocity gradient a for the buoyancyinduced free-convective flow configuration may also be estimated by tile results for the familiar problem of free convection over a heated vertical plate at constant temperature Tv [10]. Thus a 0.016 Umaxl/2[[Jg(Tv T~)]3/4/p,,~ 1/2, where Hma x is tile maximum velocity in the convection profile and ~3e = 1/T is the coefficient of thermal expansion in the gas. Using T = 1264°K (as above) in fig and equating the resulting expression for a with the previous channel flow result yields//max ~ 0.0025ue 2, which gives values OfUma x that are of the same order of magnitude as u~, obtained from [9], over the range 50 cm/s ~< u c £ 150 cm/s. Thus, while the external flow fields for the channel-flow and buoyancy-induced free convection flow configurations are quite different the surface velocity gradients a and spread rates u s [according to Eq. (44)] may coincide. This prediction agrees approximately with experimental results [8, 9]. Further study [11] of experimental results demonstrates the superiority of Eq. (44) to previously available spread rate fornmlas for spread controlled by conductive and convective heat transfer.

20O

aY2 02 Y2 p~(u~ + ayg) - 2 = = pgDg -Oyg - 2 - w, OX --°°
3T s PsCpsUs - OX

02T~ = ~ S y 0ys - -

---oo< x < oo, OY1 _ 0Y z 83'g

FORMULATION

X "( 0,

Y2 = O,

pgCpg(U s + ayg) --0Tg : ?~gy __02Tg+ aX Oyg 2 q ° w " ---°° N x "~ c'°,

0 Yx Pg(u s + ayg) Ox

---~ < x < o°

y g ~ O,

02 Y1 pgDg Oyg2 yg~>0,

Pl W,

(A.1)

0yg

Oy s

Ys = Yg = 0,

pgDg(q o -- 1_) 0 Ya

Ts=Tg=Tv,

Pl -- Yls

or. =--Xgy - -

0yg

Y1 = 0 ,

0)'g

OT~ -- Xsy - - ,

0y s

O<~x<~xo,

Y s = Y g =0,

r~= r~= rv,

[pgDg

a Y2

1 - - Yzs

Oyg

ars x at.,

=--~ksy-_ gyal, s

T s = 7'11 = T ~ ,

Allowing for the presence of a diffusion flame in the gas phase, the equations for conservation of fuel, oxidizer and energy in the gas and solid phases, along with the appropriate boundary conditions, are

ys~>0, 0Tg _ ST, kay - - - - - X s y - -

- O,

AND SOLUTION

OF PARABOLIC PROBLEM

2

~}.Fg

X)Xo, APPENDIX:

yg~>0,

Yg = +~,

~yg Ys =Yg = 0 ,

Y1 = Y1 ~,

Y2 = 0 at x = - ~ ,

Ys = +oo

where x o is the assumed downstream distance for flame detachment. These equations are nondimensionalized by putting ~ = x/L, rlg = yg/L, rls = Ys/L, r = (Tg T=)/(T v - T=), and o = (T s - T ~ ) / ( T v - T~), where L = x/Xgy/pgCpga. The boundary condition for energy conservation in the region x ~> 0, yg = Ys = 0 is linearized by assuming that the surface oxidizer and fuel concentrations are negligibly small. Reaction terms are then eliminated from the equations through the use o f the coupling functions Z 1 = 7"+ ( Q / V l ) ( Y 1 - Y I ~ ) and Z 2 = r + QY2 for

300

I. S. WlCHMAN

the gas-phase diffusion flame, and the problem becomes (r~ + e)Zl~ = Z l n n ,

- - ~ < } < ~,

r~>O,

('0 + e ) Z 2 ~ = Z2.qrT,

---~ < } < ~,

~/~>0,

P ' a t = %n,

---~ < ~ < ~,

r/~>0,

Z1 n = Z 2 n = - A o n ,

}<0,

Z2=l ,

o<~
¢ = r/a/9~J, Co = 31/6p(1/3)2/2nr(5/6), g(~b) = 1 -- P(1/3) - 1

fj

h(¢) = xFl(1/2; 2/3; ¢) --

o=l,

Z 1 = 1 - QYloo/Vl,

o =1,

Z2,~ = (1 - Q/M)ZI,~ --

n=o,

77=0,

(AQ/M)on,

~=0,

~ = --~,

Z 1 = Z 2 : O : O,

r/= oo, (A.2)

where subscripts denote partial derivatives. Here Q = qo/Cpg(Tv - T ~ ) , g = L/cr, g(Tv - T ~ ) , A = = X/-Xgy/PgCr,ga/(3,~y/PsCpsUs), and e = us/aL; Lewis numbers of unity have been as~kSy Dkgy, P*

sumed. Solutions to Eqs. (A.2) are obtained by Laplace transform methods. In the solid phase

I °Ei:l erfc

,

(A.3) /j > O.

In the gas phase the regular perturbation limit e 0 is considered, and by use of tables of inversion formulas for Laplace transforms [12] it is found that Z 1 = Z 2 -- 0 for ~ < 0 (as expected), that

z2(L n) =g(~),

n~>o, o<~<~o, M

+g(¢)l'

t - 2 / a e - t dt,

(A.6)

(--P(--1/3))P(5/6) 1"(1/3)1"(1/2)

× ¢ 1 1 a l F l ( 5 / 6 ; 4/3; ¢),

n=0,

}>Go,

~>0,

o=

where

(A.4)

[./-~-, Q e-Oh(¢) LCoAVP,

n~>0, 0 < ~ < ~ o . (h.5)

in which 1F1 denotes the confluent hypergeometric function, and that the solution in the region } > }o is expressible in tenns of integrals. This last solution is sufficiently complicated to motivate development of expansions for large a n d small values of } - }o- Prior to giving these expansions it is of interest to indicate various properties of Eqs. (A.4) and (A.5). The functions g(¢) and h(¢) are plotted in Fig. 7. Since Y2 = 0 in the region 0 ~< } ~< }o, Eq. (A.4) gives r = g($). From Eq. (A.5) one observes that l i m ~ o ZI(~, 0) = _0% the oxygen mass fraction Y1 becomes singular at the origin, exhibiting discontinuous behavior in r/ [since limt--, o Z I ( ~ , 7"1)= 0 for r / > 0]. A monotonic increase of Y1 with increasing } at the surface occurs thereafter, with Y1 passing through zero (at a value of ~ temporarily denoted by }*) and eventually approaching a value less than unity as } approaches infinity. Since a diffusion-controlled reaction needs Yls = 0, a condition achieved in the parabolic Oseen problem [ l ] , the variation in Yls found here, necessary for satisfaction of the energy balance, has the physically undesirable effect of preventing diffusion-controlled burning at the fuel surface for } > ~* and physically unacceptable negative values of the oxygen mass fraction for 0 < ~ < ~*. This exemplifies a significant shortcoming of the parabolic formulation for problems in which gas-phase and solid-phase velocity profiles differ and implies that streamwise diffusion processes now must be retained to provide physical reality of solutions for concentrations in the gas near the solid surface for 0 < ~ < ~*; the discontinuity of the surface temperature at the origin in the parabolic formulation poses constraints to which the Oseen problem can

THEORETICAL MODEL OF FLAME SPREAD

301

h C~I

.2 .I

2

0

Fig. 7. Behavior of the functionsg(~) and h(da).

respond in a physically acceptable manner but the present problem cannot. If the unphysical behavior of Eq. (A.5) for 0 < t < ~* is deemed acceptable for the purpose of achieving a simplified parabolic model (with the idea that corrections may be introduced later), then Eqs. (A.4) and (A.5) may be viewed as approximate solutions for all t > O, there being no lift-off o f the flame. However, an improvement of this surface-burning solution can be achieved by requiring diffusion-controlled burning to occur whenever the solution is physically reasonable, i.e., by putting ~o = t*, giving flame lift-off at Yls = 0 (see Fig. 5). By putting t = to, r/ = 0, and Y1 = 0 in Eq. (A.5) it is seen that to is detemlined (if u s is known) by C o A % / ~ / ~ ° 116

=

(Q _ M ) y l = / u l

Z l ( t , ~) = 0,

t > 6o,

Z I ( t, O) = 1 -- Q Y I = / U l ,

z~(~o,~)-

-1 +

~ >~ 0,

n>O,

g( d~o ) [ ,

J

where q~o = r/a/9to, and Z 2 may be obtained later from the problem ~Z2~ = Z2"oq, Z2(t ' e~) = 0,

~

r/~>0,

to,

t > 6o,

AQ Z2n(t, 0) = (1 - Q / M ) Z l n(t , O) - - - ~ on(t, 0),

- 1 - g2.

To find Z 1 and Z 2 for ~ > to, Eqs. (A.4) and (A.5) are first evaluated at t = to to provide initial values. From Eq. (A.2) it is then seen that for t > to the function Z 1 may be found first, from the problem ~ >~ to,

(A.8)

e-e~o h( dpo )

O-M

(A.7)

~TZI ~ = Za ~ ,

t > 6,,,

t > to, zz(to, n) =g(Oo),

n>0. (a.9)

Far downstream the effects of the initial values Z l ( t o , ~), Z z ( t o , 77) become negligible by compar-

302

I.S. WICHMAN

ison with local effects in the flaming region, and one may put Zl(}o , r/) = Z2(}o, r?) = 0 in Eqs. (A.8) and (A.9); then define a new streamwise coordinate 3' = } - to, take Laplace transforms in 3' and consider tire formal limit t/to -+ 0% to find

-~

1

\

Ul

)

g(~b),

(A. 1 O)

the solution to which gives

Z z (}, r~) ~--g(dp)+ 6~-~ 1 --M Qs2 fo'rg(r73/9t) dt. (A. 13) The approximate downstream flame location is found from Eqs. (A.10) and (A.11) by eliminating r between Z x and Z z and then setting Y1 = Y2 = 0 simultaneously, whereby

MYx~ua (1 --g(~f)) = a

Ig

e-'~fh(O~)'],

(¢f)

(~o~

J

aQ e-e~h(~) (A.11)

M (t/to)l/6

where Eq. (A.7) has been used in Eq. (A.11). Near the lift-off point one may put Z l ( t , r/) = f(¢; }) + A(t, r/), where f(¢: }) is the solution for Z 1 in 0 ~< } ~< }o, and expand the boundary condition at 77= 0 about 3, = 0 to obtain the problem

(A.14) where Cf = 7/f3/9tf. The downstream flame temperature is obtained by putting Y1 = 0 in Eq. (A.10) [or Y2 =0in Eq. (A.1 1), with the assistance of Eq. (A.14)], viz., r~ = g(¢f) +

vIA.r =Ann,

3'>0,

A(3,. ~) =0,

3'>0,

A(0, 77) = 0,

77>~ 0,

QYI~

[1 --g(¢f)].

(A.15)

Pl

rT>0,

The limit of Eq. (A.14) as (~f/to) ~ "~ gives g(¢f) ~ (MYI=/ul)/[QYa=/u I - I], which substituted into Eq. (A.15) gives the nondirnensional adiabatic flame temperature

3` Qn 6to Q - M

A(3,, 0 ) - - - - -

3,>0, Tf -- T~

which may be solved by Laplace transforms in 7 to yield

raf = - -

- (Q-M)--"

Ylo~

Tv-- T~

(h.16)

ul

Use ofEq. (A.16) in Eq. (A.7) gives Z l ( t , "t'l) "-~ --

M I~QM e-Oh(O)+g(o)l Q- M (t/~o)l/6

=(Q-M)--

Yloo Vl

1 6to

Q Q~2 - M fo "rg(rlz/9t)dt.

(A.12)

Use of this result and earlier results in a similar manner for Zg(}, 7)) = g(0) + B(}, 77) leads to r/B.r = Bnn,

3' > 0,

B(3,, ~) = 0,

3' > 0,

B(0, 77) = 0,

o > 0,

Tf-- Tv - 1= - Tv -- Too

The flame spread rate u s is obtained from Eq. (A.7) by use of the definitions of ~2 [given by Eq. (A. 17)], P*, and A, as

us= kx/-d~g~o,/aP"C'gXg"I rf-- Tvl2,

7?> 0,

Ps Cps~,sy

Bn (3`, 0)

3`2/a

3 l/a

(A.17)

Tv ~ T=_]

(A.18) Qf2

4~o P(1/3) M

,

3'>0,

where k = 1/Co 2. By the solutions for Z 1 and Z 2 given by Eqs. (A.12) and (A.13), respectively, an equation for

THEORETICAL MODEL OF FLAME SPREAD the flame shape near the lift-off point to may be derived. By eliminating r between Eqs. (A.12) and (A.13) and setting Y1 = Yz = 0 simultaneously one finds that

QYI~ v1

303 characteristic gas phase length, L =

(Xgx/PgCpga)1/2(Xgx/Xgy)l/4 L

Q. [ae-~, f h(¢f) + g(dpf)l Q--M ( t f / ~ o ) 1/6

M

[*fg(nfa/9t)dt.

p*

1 Q 6toQ-M

Qf2 M Jo

(A.19) qo Near t = to, 7? = 0it may be shown from Eq. (A.19) that

a

r/__ff~ _

R

3'f

M 6~o

3 l/a

t

T

I

, + (rz/3)(-r(-1/3))r(5/6) r(1/2)

(A.20)

U /2*

which shows a linear variation of the lift-off height with downstream distance, i.e., a constant slope of the flame sheet after lift-off, as indicated in Figs. 4 and 5.

W

Cp

C D

1Fl(a;b;x)

gas phase velocity gradient Airy function of first kind Airy function of second kind specific heat constant in substitute kernel, C = vT~K(0) diffusion coefficient confluent hypergeometric function,

(a).

function defined in (A.6) function defined in (A.6) modified Hankel functions of order 1/3

mass per unit volume per second of fuel produced in the gas-phase reaction streamwise coordinate transverse coordinate mass fraction nondimensional coupling function

Greek Symbols a 3'

thermal diffusivity translated streamwise coordinate, 3' =

t-to r(x) 8

gamma function thickness of thermal diffusion layer in gas, • ~ x/-~y t UsX/Xgx/Xgy/aL (= Us/aL in Appendix)

0 X A

p l / 3 [V -- i(6 -I" 7"/)]

Xn

n=o (b), n!

hl,2(x)

X

Y Y Z

aFx(a;b;x) = ~.~

g(¢) h(¢)

temperature velocity characteristic gradient velocity, Ug* =

aL

NOMENCLATURE

Ai(x) Bi(x)

heat release, defined so that -q/x 1/a is heat flux leaving the surface gas phase heat release per unit mass of fuel consumed nondimensional gas phase heat release, Q = qo/Cpg(Tv - T~) ratio of solid to gas ease of heat removal, R = Ax/ff representative convection time, t

~L/a5

Q--M X

a

solid to gas Peclet number ratio,P* =

L /(Xs/PsCvsUs) q

1 (tol/ap(1/3))

[= (Xgy/pgcpga)l Iz in Appendix)] enthalpy of vaporization of solid fuel per unit mass nondimensional enthalpy of vaporization of solid fuel, M = l_/cpg(Tv -

P Pl

thermal conductivity (= XsyDkgy in Appendix) complex Fourier frequency mass-based stoichiometric coefficient (Xsx~ksyDkgx~kgy) 112

304

p o r ¢

I . S . WICHMAN nondimensional streamwise coordinate density nondimensional solid-phase temperature, o = (Is - T**)/(Tv - T**) nondimensional gas-phase temperature, r = (Tg - T**)/(Tv - T.~) nondimensional similarity variable, ¢ = r/a/9~ (Tf

-

Tv)/(T v - T**)

Subscripts f g

flame gas

o 0 s v 1,2

detachment point or plane oxygen solid, surface vaporization oxygen, fuel ambient analytic in upper and lower halves o f the u plane, respectively

+, -

Superscript *

with velocity gradient in gas

This study was supported by the Center for Fire Research o f the National Bureau o f Standards. I am especially indebted to Professor G. F. Carrier for his suggestion o f the substitute kernel, Eq. [24}, and to the two referees, whose comments and criticisms proved very useful. This

study could not have been completed without the direction and patient guidance o f m y advisor, Professor F. A. Williams

REFERENCES 1. Wichman, I. S.,and Williams, F. A.,Comb. Sci. Tech. to appear. 2. Staff of the HarvardComputation Laboratory, Tables of Modified Hankel Functions of Order One-Third and of their Derivatives, Harvard University Press, Cambridge, MA, 1945. 3. Carrier, G. F., Krook, M., and Pearson, C., Functions of a Complex Variable, McGraw-Hill, New York, 1966. 4. Noble, B., Methods Based on the ;qiener-Hopf Technique for the Solution of Partial Differential Equations. Pergamon Press, New York, 1958. 5. Carrier, G. F., Soc. Indust. Appl. Math. 13: 68 (1965). 6. Fernandez-Pello, A. C., Kindelan, M., and Williams, F. A.,Ing. Aeron. y Astron. 135:41 (1974). 7. deRis, J. N., Twelfth Symposium {International) on Combustion, The Combustion Institute, 1969, pp. 241-252. 8. Ray, S. R., Ph.D. Thesis, Princeton University, Princeton, NJ 1980. 9. Fernandez-Peilo, A. C., and Williams, F. A., Combust. Flame 28:251 (1977). 10. Ostrach, S., in Theory of Laminar Flows (F. K. Moore, Ed.), Princeton University Press, Princeton, NJ., 1964, pp. 546-550. 11. Wichman, I., Williams, F. A., and Glassman, I.: paper presented at the Nineteenth Symposium (Interna. tional) on Combustion, 1982. 12. Roberts, G. E., and Kaufman, H., Tables of Laplace Transforms, W. B. Saunders Co., Philadelphia, 1966.

Received 17May 1982; revised 21 September 1982