# Spread of a laminar diffusion flame

## Spread of a laminar diffusion flame

SPREAD OF A LAMINAR DIFFUSION FLAME J. N. DE RIS National Bureau of Standards, Washington, D. C. A theoretical description is presented for a laminar...

SPREAD OF A LAMINAR DIFFUSION FLAME J. N. DE RIS

National Bureau of Standards, Washington, D. C. A theoretical description is presented for a laminar diffusion flame spreading against an air stream over a solid- or liquid-fuel bed. Both a thin sheet and a semi-infinite fuel bed are considered. The burning process is described as follows: The hot flame heats the unburned fuel bed, which subsequently vaporizes. The resulting fuel vapor reacts with the oxygen supplied by the incoming air, thereby producing the heat that maintains the flame-spread process. The formulated model treats the combustion as a diffusion flame, for which the details of the reaction kinetics can be ignored by assuming infinite reaction rates. The model includes the chemical stoichiometry, heat of combustion, gas-phase conductive heat transfer, radiation, mass transfer, fuel vaporization, and fuel-bed thermal properties. The radiation is mathematically treated as a heat loss at the flame sheet and a heat gain at the fuel-bed surface. The calculated flame-spread formulas are not inconsistent with available experimental data. These results reveal much of the physics involved in a spreading flame. For instance, the flame-spread rate is strongly influenced by (1) the adiabatic stoichiometric flame temperature, and (2) the fuel-bed thermal properties, except for the fuel-bed conductivity parallel to the propagation direction. I. Introduction

Fire Spread Many fire-spread situations involve the following basic burning processes: First, heat from the hot flame is transferred to the unburned fuel bed (see Fig. 1). When an unburned fuel-surface element becomes sufficiently hot, additional heat vaporizes it. The resulting gaseous fuel reacts upon coming into contact with the oxygen supplied by the air flowing into the flame zone. Some of the released chemical energy is transferred to the unburned fuel bed, thus completing the energy cycle necessary to maintain the fire-spread process. Each of the above processes takes place by a variety of physical mechanisms depending upon the particular fire-spread situation being con8idered. This paper considers a particular idealized flame-spread situation, for which a somewhat simplified linear exploratory model is formulated and solved.

The Problem This work considers the problem of a gas-phase laminar diffusion flame spreading against an air stream over a solid- or liquid-fuel bed. A thin241

sheet fuel bed and a semi-infinite fuel bed are considered. In general, the forward heat transfer to the unburned fuel bed takes place by conduction through both the gas-phase and the fuel bed as well as by radiation. There is no convective forward heat transfer, because the air stream opposes the flame spread. For simplicity, the thin fuel-bed problem includes only the gas-phase conductive heat transfer, while the semi-infinite fuel-bed problem includes the gas-phase and fuelbed conduction as well as the effects of radiation. An exponentially decreasing form for the forward radiative heat transfer is postulated. The thin fuel-bed model is solved approximately, while the semi-infinite model is solved exactly. The combustion, which presumably takes place only in the gas-phase, is regarded as a diffusion flame that comes very close to the fuel surface, as shown in Fig. 1. While some finite reaction-rate broadening of the diffusion flame actually occurs near the fuel bed, the present theory assumes that the combustion rate is primarily controlled by the mass transfer of reactants to the flame rather than by the chemical kinetics. Such a diffusion flame can be analyzed with the Schvab-Zeldovich~ diffusion flame theory, thereby avoiding the effects of highly nonlinear reaction kinetics. This treatment should be valid when the quenching

242

INDUCED AIR VELOCITY

'

/

/

/

ix~,.

J---'~~

/

BED

/

/

/

\FUEL

/

/

/

/

/

/

/

/

/VAPORIZING

/ / / / / / / / / / / /

FIG. 1. Physical description of a flame spreading over a stationary fuel bed.

distance (between flame and fuel bed) is small relative to the characteristic gas-phase heattransfer distance.

Related Literature The general subject of fire spread has been discussed by Emmons2 Friedman 3 recently reviewed the currently available knowledge of idealized flame spread over surfaces. The thin fuel-bed experimental results reported therein appear to be consistent with the predictions presented in this paper. Tarifa and Torralbo 4 have considered the flame spread over a senti-infinite fuel bed. By neglecting the fuel-bed conductivity parallel to the fuel surface, it was possible to calculate the flame spread produced by an exponentially decreasing forward heat-transfer profile. By so postulating the heat transfer, analysis of the gas-phase cornbustion could be avoided. The resulting solution is a special case of the semi-infinite fuel-bed solution presented here. They also considered a fully conducting fuel bed, which led to a very complicated solution. The mathematical techniques used here permit an exact solution of the fully conducting fuel-bed problem. The present solution also includes the gas-phase combustion :rod solves for, rather than assumes, the forward heattransfer profile (although the radiative heattransfer profile is still assumed). By postulating the mass flux of fuel leaving the vaporizing surface, and also assuming that the distance of forward heat transfer is a constant, McAlevy and Magee s isolated the gas-phase combustion problem from the fuel bed. Incorporating the effects of chemical kinetics, they predicted that V,-~ ( Y o J P ) ~, where P is the pressure. Although their extensive experimental results are of this same general form, the predicted values of s and/3 differ significantly from

the experimental values. Their experiments correspond to the semi-infinite problem considered here EEq. (29)1. Unfortunately, it is difficult to compare these results, because of the uncertainty in estimating the air velocity V~ in the experiments.

2. Qualitative Description Figure 2 shows some features of the mathematical model. By locating the origin of the coordinate system at the base of the flame, the flame-spread process is made stationary. 2 From the viewpoint of this coordinate system, the fuel bed is fed into the flame with the flame-spread velocity V.

Heat Transfer The heat-transfer process to the fuel bed is of crucial importance, because the fuel bed must receive heat before it can vaporize. The forward gas-phase conduction is influenced by the air velocity. In keeping with the exploratory character of this study, the gas-phase is presumed to have constant properties as well as a uniform velocity profile, as shown in Fig. 2. The downstream convection is included in both the thin and semi-infinite models, while the effects of radiation are included only in the semi-infinite model. The fuel-bed thermal properties are important; they influence the total heat required to raise the fuel-bed surface temperature to its vaporization temperature. The thin-sheet fuel bed, with insulated bottom surface, is regarded as being thin enough to have an essentially uniform temperature distribution across its thickness. For thicker fuel beds, the fully conducting semi-infinite fuelbed solution should provide a better description.

SPREAD OF A LAMINAR DIFFUSION FLAME

_V_o§ INDUCED AIR PROPERTIES

p,Cp A.,D

y

243

l T. ,Yo.

~- - - - b

/ //////// v'/, / / / FUEL BED (0) THIN SHEET "T', ,,Ow, Cpw, L

(b)

SEMI- INFfNrTE

pw,Cpw,/~.wx, ~wy,

L

FIG. 2. Mathematical model--fuel bed being fed into ,~ stationary flame.

Fuel Vaporization In most practical fire-spread situations, the fuel is initially a solid or a liquid. Before it can burn in the gas phase, it must first vaporize. Here, we consider a fuel that is initially entirely in its condensed phase. That is, the fuel's initial vapor pressure is negligible compared to its leanlimit vapor pressure. As the fuel approaches the flame front, its surface temperature increases rapidly. ~The fuel vapor immediately above a liquid or subliming solid fuel surface is essentially in thermodynamic equilibrium with the fuel surface below, since the gas-phase molecular mean free path is several orders of magnitude smaller than the gas-phase mass-transfer characteristic distance.-T Since the equilibrium vapor pressure increases strongly with temperature, one expects a very rapid increase in fuel vapor pressure beneath flame front. [Using the Clapeyron-Clausius relation, the ideal gas law, and the fact that the fuel-bed density is much larger than the fuel-vapor density, one has

(TIPs) dP~,/dT =- L/(~T, where P~ is the vapor pressure, (R is the universal gas constant, and L is the heat of vaporization. Since the ratio L/(~T is usually greater than 10, the vapor pressure is strongly dependent on temperature.~ Downstream, the fuel-vapor pressure is limited by the ambient pressure. Consequently, the surface temperature does not exceed the boiling point associated with the ambient pressure. Actually, the presence of combustion products and other gaseous inerts at the surface reduces the

d o w n s t r e a l n s u r f a c e v a p o r p r e s s u r e a u d tenll)erature.

The idealized mathematical model developed here considers a fuel-bed surface that does not vaporize until it reaches its so-called "vaporization temperature" T~,. After reaching this ten> perature, the fuel surface continues to vaporize at this same temperature with a constant heat of vaporization L.

Mass Transfer The mathematical model permits the diffusion and convection of the pertinent species--namely, fuel vapor, oxygen, products, and inerts. The mathematical model can be readily solved for a uniform convective velocity profile parallel to the fuel surface; thus, for consistency, one nmst approximate the nonlinear perpendicular convection nmving away from the vaporizing downstream fuel bed. The model assumes that all the I)erpendimflar mass transfer takes place by diffusion; however, the mass-transfer boundary condition along the vaporizing fuel bed is suitably linearized so that it provides a good approximation of the perpendicular convection. By expressing the flame-spread solution in terms of the adiabatic stoiehiometric flame temperature, Tft, the result becomes independent of the specific mass-transfer linearizing constant.

Gas-Phase Combustion The combustion is presumed to take place only in the gas-phase. The oxygen and fuel vapor are regarded as reacting instantaneously upon coming into contact. This means that the combustion

244

rate is controlled by the mass transfer of reactants to the flame rather than by the chemical kinetics. The resulting diffusion flame lies between the fuel and oxygen sources. The fuel vapor diffuses from the vaporizing fuel bed beneath the flame, while the oxygen is supplied by the air convected from infinity. The flame touches the fuel bed at the point where it starts to vaporize (i.e., at the origin), and goes out into the gas phase in a somewhat downstream direction due to the air motion. Since the oxygen and fuel-vapor zones do not overlap, one can formulate a linear combustion model using the Schvab-Zeldovich diffusion flame theory. The present flame-spread problem is inherently two dimensional; both the perpendicular and parallel transfer of heat and mass play fundamental role in the flame propagation. To date, the only successful analytical solutions of diffusion flames, which include reaction kinetics, have been for inherently one-dimensional problems (i.e., problems reducible to ordinary differential equations). Since reaction kinetics are probably unimportant for a significant class of flame-spread situations, it appears inappropriate to include them in this exploratory analysis.

3. G o v e r n i n g E q u a t i o n s

The governing equations for the thin-sheet fuel-bed problem are now formulated. The equations, which can be expressed as two WienerHopf integral equations, are solved approximately in Sec. 4. The gas is presumed to move with a uniform and constant velocity V~ parallel to the fuel bed surface. The gas phase is considered to have constant properties: density p, pressure p, conductivity ),, specific heat Cp, and specie diffusivity D. A unit Lewis number ~/(pC~D) is assumed. The mass diffusion of species is presumed to be driven only by specie concentration gradients. Let Y~ be the mass concentration (i.e., mass fraction) of specie "i", and let m / " be the net rate of specie "i" mass generation per unit volume. The specie conservation equation is

pV~ 0 Y~/Ox = pD~O2Y~/Ox2 + 0 2Y~/Oy2] --}-~h/", (1) where x and y are the coordinates parallel and normal to the fuel-bed surface. The thermal energy equation can be written as

where qchem 't! is the net rate of chemical heat release per unit volume, and ~ d ' " is the net rate of radiative heat loss per unit volume. The terms rhi", ~r and q r J " are important only near the thin-flame sheet. The single global reaction, presumed to take place at the front, is ~v' (Fuel) -4- uo' (Oxygen) --~ pPlt' (Product 1 ) -k- ui,2" (Product 2) -4- Heat, where ~/ and u/~ are the usual stoichiometric coefficients. Define Q as the heat released by the combustion of us ~ moles of fuel--i.e., Mrus' grams of fuel, where M s is the molecular weight of the fuel. Equivalently, Q is the heat released by the combustion of Mouo ~ grams of oxygen. I t will be assumed that a constant fraction X of the heat released by combustion will be in the form of radiation--that is, ~ a " = Xqchem""'. Now consider the dimensionless function

a~(x, y) = ~Cp(T-- T~)/L] + ~(Yo -- Yo~o)Q(1 - X)/(Mo~'o'L)~,

where L is the heat of vaporization of the fuel, and the constants T~ and Yo~ are, respectively, the ambient temperature and oxygen concentration at infinity. By combining Eqs. (1)-(3) with unit Lewis number, one has ( p G vo/~,) o~l/Ox - x ~ i / O x ~ - o:,~/Ox ~ =

(1 -- X) k- ~ho"Q (1 -- X) XL 3Iovo'LX

q e h e m H!

= 0;

for

--~
(4)

o~2(x, y) = ['YFQ(1 -- X)/(MFvF'L)-] -- [(Yo

-- Yo~)Q(1 --

X)/(Mo~'o'L)],

(5)

The fuel and oxygen specie Eqs. (1) and (5) provide

(pG VdX ) O~2/Ox- 02,~2/0x2 - o2,~2/Oy~ _ rhF'"Q (1 -- X)

MF,Ft L~ (2)

~,y>0.

The right-hand side of Eq. (4) is zero, because the heat, Q (1 - x ) / M o , o ' , is released into the temperature field when unit mass of oxygen is consumed. Similarly, consider the dimensionless function

p G v . OT/Ox = ~,[-O2T/Ox:+ 02T/Ou~3 + qCh(.m'' -- ~ d " ,

(3)

=0;

for

r h o " Q (1 -- X) Mouo'LX

--~ O;

(6)

SPREAD OF A LAMINAR DIFFUSION FLAME Again, the right-hand side is zero because MFVF' grams of fuel are consumed with Moro' grams of oxygen. Following Zeldovich, 1 and exl)ressing the conservation equations (4) and (6) in terms of al and a2, one eliminates the highly nonlinear reaction kinetic terms rhi'", qchCm'", and Or~d". However, to determine the three distributions of temperature, oxygen, and fuel vapor from the two functions al and a2, one nmst find an additional relationship. Instead of solving either of the nonlinear equations (1) or (2), one notes that, for an infinite chemical kinetic reaction rate there will be no fuel vapor (oxygen) to the left (right) of the flame. This infinite reaction-rate assumption also simplifies the boundary conditions that will now be formulated.

Boundary Conditions

m, a 2 ~ 0

as

y ~

or

x-+--~.

(7)

(In the downstream wake it is mathematically sufficient that the ai's be bounded.) The equations derived up to this point apply to both the thin and semi-infinite problems. The rest of this section formulates the boundary conditions for the thin fuel-bed problem that does not include radiative effects (therefore, set X = 0). The heat transfer to the unburned fuel bed causes a temperature increase as the fuel approaches the flame front. Neglecting the forward radiative heat transfer, one has

p~C~rVOT/Ox = hOT/Oy;

spread over it. This can be expressed as lira c~e= G ( T v , , , - - T~)/L.

f o r x < 0, y = 0+,

where pw, C~, and r are the fuel-bed density, specific heat, and thickness, respectively. Noting that there is no fuel vapor above this surface and no mass transfer across it, one obtains,

(11)

x~0, y~:0+

The vaporizing fuel bed is presumed to vaporize at the constant temperature T,.~p. Since Yo is zero along this surface,

oq = ~Cp(Tva, -- T:,)/L~ -- rYo~Q/ (3lovo'L)] --=---B; f o r x > _ 0 , y = 0+;

(12)

B is the customary mass transfer driving "force". The final boundary conditions relate the heat transfer hOT/Oy to the fuel mass transfer #t"; that is,

#l" = (k/L)OT/Oy;

The c~i's have been defined so as to be zero at infinity outside the downstream wake; that is

245

for x ~ 0, y = 0+.

(13)

This mass flux ~it" is convected and diffused into the gas phase,

ri~" = ~ht'YF -- pD OYF/Oy;

for x ~ 0, y = 0+.

The nonlinear convection term above can be approximated by assuming that the perpendicular (i.e., y axis) mass transfer, ~h" takes place only by diffusion. One can adequately compensate for the convective term by increasing the diffusion term by the factor B/ln (1 + B), where B is defined above. [-This same linearization factor (or "heat-blockage" factor) has been successfully used to correlate the mass transfer in laminar combustion boundary layers3 This factor, which is predicted by the stagnant-film boundary-layer approximation, also predicts the correct mass transfer for the one-dimensional diffusion flame parallel to vaporizing fuel surface. It also provides a good approximation of the one-dilnensional flame temperature and positionY-] Thus we have

~h" = -- [B/ln (1 + B)]pD 0 YF/Oy; bOa2/Ox= Oo~2/Oy; f o r x _ < 0 , y = 0+,

(8)

Ool2/Oy = O,~i/Oy; for x < 0, y = 0+,

(9)

where

b = pwCw,rV/X.

(10)

The dimensionless quantity b contains the flame-spread velocity V; it is the "eigenvalue" of the problem. This eigenvalue is ultimately determined by invoking the flame-spread condition-namely, that the temperature of an unburned fuel element nmst equal T ~ v before the flame can

forx_> 0, y = 0+.

(14)

Combining Eqs. (3), (5), (13), and ( 1 4 ) w i t h X/(pCvD) = 1, and noting that there is no oxygen above this surface, one has

Oa2/Oy= ( 1 - - K ) 0 o q / 0 y ;

forx>0,

y = 0+;

(15) where K = ['B/ln (1 + B)~-~Q,/(MFvv'L). This completes the thin fuel bed problem formulation.

246

FLAME SPREAD AND MASS FIRES 4. Solution of Thin-Sheet Problem

The thin-sheet problem formulated in the last section is well set. There are two elliptic partial differential equations (4) and (6) for the unknowns al and a2 in the gas-phase region (y > 0). There are two boundary conditions along each boundary of this region, namely, Eqs. (7) at infinity, Eqs. (8) and (9) along the unburned fuel bed, and Eqs. (12) and (15) along the vaporizing fuel bed. The eigenvalue relationship (11) determines the flame-spread "eigenvalue" b. One can nondimensionalize the problem by defining

= pC~V~x/(2X), = pC, V,y//(2X). The governing equations (4) and (6) become 2 (0~jo~) = 0 ~ / o ~ ~ + ~ / o ~ ; oQ,~>0.

(16)

Under this coordinate transformation, one just replaces x and y by ~ and Vin the boundary conditions (7), (8), (9), (12), (15), and in the eigenvalue relationship (11). Notice that, under this transformation, the thin-sheet equations are independent of the velocity V,.

Fourier Transformed Problem [The mathematical principles are described in Refs. 8 and 9.~ By taking the Fourier transform of the governing partial differential equations, (16), one obtains ordinary differential equations (in y) which are easily integrated. Using the Fourier transformed boundary conditions, one obtains two simultaneous Wiener-Hopf integral equations. These integral equations can be approximately solved 7 using a substitute kernel. The solution for b >> 1 is

pwCwrV/X =---b~_'~/2(Tf-- T,~,p)/(T,,,p- T,~), (17) where

Vf= T ~ + (BL/C~)(I--1/K) -- ( T ~ p - - T~),

Tft= T~

+

and

fori= 1,2,--~ <~<

rate needed to raise the fuel-bed temperature to its vaporization temperature. The right side is the gas-phase conductive heat-transfer rate fronl the flame forward to the unburned fuel bed. The flame-spread chemistry is principally contained in the temperature Tf. This is also the adiabatic stoiehiometric flame temperature calculated for a diffusion flame with linearized mass transfer. It is possible to calculate the true downstream asymptotic flame temperature Tft, ineluding the nonlinear convective mass transfer, ~

(18)

is the downstream asymptotic flame temperature. Rearranging Eq. (17), one has

p,oC~rV(T.~,-- T~)~V2X(Tf-- T.~,). (19) Equation (19) has a meaningful physical interpretation. The left side is the heat-transfer

{[Q/(MFv/L)J+ Tv~,-- T~-- [L/Cp~} (20) { 1 + [Movo'/(3/FVFtL)]}

Tft is also the adiabatic stoichiometric temperature calculated with nonlinear mass transfer. Since Tf and Tft are usually close numerically, it is quite plausible physically to interpret Tf in Eq. (19) as the true temperature Tft. This interpretation eliminates the mass transfer linearizing factor K from the solution. The approximate flame-shape curve, shown in Fig. 3, was calculated by noting that both the oxygen and fuel concentrations are zero at the flame. The dashed curve is the downstream asymptotic flame shape calculated with the exact (integral equation) kernel (with linearized mass transfer). The calculation using the approximate substitute kernel does not quite approach the dashed curve. Nevertheless, the calculated flame shape corresponds to what one might intuitively expect.

Comparison with Experiment Equation (19) has not yet been established experimentally, however some observations are encourageing. It was reported '~ that the flame-spread rate is inversely proportional to the thin-sheet thickness r. This is consistent with Eq. (19) and it establishes the importance of the forward heat transfer to the unburned fuel bed. Equation (19) is essentially independent of pressure. It was reporte& that the flame spread over thin sheets did not, vary over the limited pressure change, 0.507 to 1.013 X 105 N/m ~ (i.e., 380 to 760 mm Hg), in a 46% 02-54% He atmosphere. (This same pressure independence has been reported over nmeh larger pressure ranges; however the test sample edges were not inhibited.) Equation (19) is independent of air velocity V~. This might help explain the observation1~ (in enriched oxygen atmospheres) that the flamespread rate down a vertical thin sheet and across a horizontal sheet are essentially equal.

SPREAD OF A LAMINAR DIFFUSION FLAME I uJ 0

Z

4-

I

I

I

I

I

I

f

~"~

DOWNSTREAM ASYMPTOTIC FLAME SHAPE ~

247

U~

2-

/

KERNEL

FLAME SHAPE

r

00.

[

I

I

1

I

[

I

2

3

4

5

6

DIMENSIONLESS

I 7

B

DISTANCE "E

FIG. 3. Flame shape.

Since the present theory assumes an infinite reaction rate, it probably provides a poor description of flame spread for low oxygen concentrations (i.e., near the extinction limit). Reaction kinetics become important when the volume (per unit width) required for combustion (X/pCpVa) (pVa/r) is a significant fraction of the volume (per unit width) available for combustion (X/pC~V~)2, where r is the characteristic reaction rate (mass of air)/(time X volume). Thus, the reaction kinetics must be considered when the ratio p2V~Cp/(Xr) approaches unity. The influence of V~ on the magnitude of this term is evident. This corresponds to the common experience of blowing out small flames with a gust of air.

a =

p~c~x~Vpc~xv~

as well as

~ = C p ( T - T~o)/L the fuel-bed equation becomes

2a a ~ / a ~

-

u a~/o~

~-

-~

o ~ / a ~ ~ = o;

<}<

co,~
(21)

with

5. Semi-Infinite Problem This section extends the analysis of the preceding section to include the effects of fuel-bed conduction as well as radiation (i.e., X ~ 0). Although this problem includes more physical processes, one obtains an exact solution with less labor.

Governing Equations

a3---~O

as

co,y~0,

~'--~--r

or s~---~ - - co .

(22)

Let the net radiative heat-transfer flux received by the fuel bed be,

R(x) = Rlexp (x/ll), = R2,

The gas-phase combustion is described, as before, by the partial differential equations (16). Likewise, the boundary conditions (7) still apply. One must now formulate the fuel-bed equations and the fuel-surface boundary conditions. The governing equation for the fuel bed is

p,.C~ V (OT/ax) - - X ~ (a 2T/ax ~) --X~(a2T/Og 2)= 0; --co < x <

where the x and y conductivities Xw~and Xwyare presumed to be constant. By defining the "eigenvalue",

upstream x < 0, downstream x > 0,

where RI, R2, and 11 are constants. Along the unburned fuel bed surface, conservation of energy provides

Oc~ljO~=Oa2/O~; --co < ~ < 0 , ~ = 0 + ;

(23)

--~ <}~0,~=0+,~=0_;

(24)

248

where rl = 2RJ(pV~L) a n d / ~ = 2h/(pC~V~l~). Since the gas and fuel-bed temperatures are identical at the surface, one has a~=a~;

--r

temperature, T/, is

T / = T~, -'F (B'L/Cp) (1 -- 1/K') -- ( T v ~ , -

(25)

T~).

(30)

Since the vaporizing fuel-bed surface temperature equals T~,, one has

(The primes on B t, K t, and T/indicate that the effects of radiation are included, i.e., X ~ 0).

a~ = [Cp(Tv~p- T~)/L]

Discussion of Solution

-- EYo~Q(1 - x)/Mo~,o'L)] ---- --B'; 0_~<

a3= C~(Tw,-- T~)/L;

r

0<~<

(26)

oo,~= 0_. (27)

Relating the net heat trausfer to the linearized mass transfer along the vaporizing fuel bed, one has oc~2/ov = (1 -

K')Oc~/Ov - - K'r2 + K ' O,~3/O~;

0_< ~ < ~ , V = 0 + , ~ =

0_;

(28)

where

re = 2R2/ (pVaL ) and

K' = EB/ln (1 -F B)J-'{Q(1 -- X)}/MFv/L)}. Here, the "eigenvalue" relationship means that the temperature and fuel concentrations are both continuous at the origin, that is, oe and o~3are continuous. One has a well-set problem using Eqs. (16) and (21) with boundary conditions (7), (22) to (28), and the "eigenvalue" relationship. These equations can be converted to three simultaneous Wiener-Hopf integral equations, which are solved exactly/producing,

-- I T ~ - T~, 2R~F(2~/pCpV~l~) [~----~ {-pC~V~(T~.p- Too) 2R2 + ~pCpV~(Tv~, -- T~)

}2

(29)

'

with the definition,

F(z) = ['89

sin-1 ( z - 1 ) / ~ ( 2 z -

Z2)1/2~, 0
= E89 (~ -

~)-'~]

[:--lqXln --1--

(z2 - 2z) '/2] (z2 - 2 z ) I / 2 j '

2_
where the mass transfer linearized adiabatic flame

The left-hand side of the exact solution FEq. (29)-] contains the flame-spread velocity V. The first term on the right describes the effect of forward gas-phase conductive heat transfer, while the middle and last terms describe the effects of forward and downstream radiative heat transfer, respectively. Notice that this equation does not include the parallel fuel conductivity ~ ; the heat that enters the vaporizing fuel bed and is conducted forward to the unburned fuel bed remains within the flame-spread energy cycle. However, V is inversely proportional to k ~ ; as k ~ increases, more heat is conducted into the fuel-bed interior. The form of Eq. (29) without radiation (i.e., R1 -- Re - 0) can be obtained qualitatively directly from the thin-fuel-bed result [-Eq. (19)-]. If ~,i is the effective depth of heat penetration into the semi-imCinite fuel bed, then the total forward heat-transfer rate equals

X (2h/pCpV~),

(31)

where (Trap -- T~)/r~i is the characteristic fuelbed-temperature gradient, and 2h/pCpVa the characteristic gas-phase length over which heat is transferred. Solving for v~i above, and inserting it into Eq. (19), one obtains Eq. (29) with R1 = R2 = 0. One sees that, as V~ increases, the heat transfer distance 2h/pCpV~ decreases, thus r~ decreases in Eq. (31), thereby increasing V in Eq. (19). Thus, one sees why the flame-spread velocity V in Eq. (29) increases with Va. One can eliminate the gas-phase combustion effects (but retain the postulated radiant heat transfer) by letting k-->0 in Eq. (29) [-using lim,~0 F(z)-= (2z)-11~-]. One obtains the result that Tarifa and Torralbo 4 calculated for h ~ = 0,

pwCpwkwyV= R12/1/(Trap -- T ~ )3. This present result, calculated with hw, ~ 0, assumes a constant downstream fuel-bed-surface vaporization temperature. The semi-infinite problem calculations ~ show that the flame lies quite close to the vaporizing

SPREAD OF A LAMINAR DIFFUSION FLAME fuel surface. As the downstream radiation R2 increases, the flame moves away from the surface. For no radiation, the flame lies directly on the surface, thereby possibly extinguishing the flame. This might help explain the observation that sometimes flames do not spread over very thick fuel beds, because the thick fuel bed can drain too much heat to the interior. 6. Conclusion

By formulating a flame-spread model that ignores reaction kinetics, two physically revealing flame-spread formulas were obtained: Eq. (19) for thin-sheet fuel beds, and Eq. (29) for semiinfinite fuel beds. The latter equation includes the effects of radiation and fuel-bed conduction. See the sections following Eqs. (19) and (29) for detailed results. These equations require further experimental verification. In particular, the interrelated roles of reaction kinetics and air velocity V~ need to be examined. Hopefully this exploratory analysis reveals and interrelates some of the important parameters governing idealized flame spread. Nomenclature B, B'

Le K, K r M

Q

R1, R2 Tf, T / Tft Tw, V V~ Y b 11 x, y ~,%~ ~I, 0~2, 013

pi t , pi tt

p

Mass transfer driving "force", Eqs. (12), (26), [ - - ~ Lewis Number = X/pCpD E--~ Mass transfer linearizing coefficients, Eqs. (15), (28), F--~ Molecular weight [-IV[/mol7 Heat liberated with consumption of vf' moles of fuel [-E/mol~ Upstream and downstream radiative heat fluxes [E/L2T~ Mass transfer linearized flame temperatures, Eqs. (18), (30) [-0~ True adiabatic stoichiometric flame temperature, Eq. (20), [-0J Vaporization temperature [-0~ Flame-spread velocity ILl T~ Air velocity w.r.t, stationary flame

[L/T~

Mass concentration [---]

p~c.o,v/x [-3 Characteristic length of forward radiation, [-L~ Parallel and normal coordinates [-L-] = pC, V~x/(2X), n = pC, g.y/(2X), = pC, V.y/(2hwu); ~--3 Distribution functions, Eqs. (3), (5), (21), [ - - ~ Conductivity [ E/L TOJ Stoichiometric coefficients Fmol~ Density [M/ L~

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Thin fuel-bed thickness [-L~ Proportion of chemical energy released in form of radiation [---

T

X

Subscripts F O w co

Fuel Oxygen Fuel bed Ambient ACKN OVVLEDGMENTS

I express my sincere thanks to Professor H. W. Emmons for his guidance and encouragement. This work was supported in part by the National Science Foundation under Grant G.K.-165, by the Division of Engineering and Applied Physics of Harvard, and by Factory Mutual Engineering Corp. Part of this work was done at Harvard University. REFERENCES 1. ZELDOVICR, YA. B.: On the Theory of Combustion of Initially Unmixed Gases, NACA TM 1296, 1951. 2. EMMONS, H. W.: Tenth Symposium (International) on Combustion, p. 951, The Combustion Institute, 1965. 3. FRIEDMAN,R.: Fire Research Abstr. Revs. 10, 1 (1968). 4. TARIFA, C. S. AND TORRALBO, A. M.: Eleventh

Symposium (Internalional) on Combustion, p. 533, The Combustion Institute, 1967. 5. McALEvY, R. F. AND MAGEE, R. S.: Flame Spreading at Elevated Pressures Over the Surface of Igniting Solid Propellants in Oxygen/Inert Environments, Stevens Institute of Technology, Hoboken, N.J.; NASA Grant No. NGR-31-003-014, Oct. 1967. 6. SPALDING, D. B.: Convective Mass Transfer, p. 144, McGraw-Hill, 1963. 7. DE RIS, J. N.: The Spread of a Diffusion Flame Over a Combustible Surface. Ph.D. thesis, Harvard University, 1968. 8. CARRIER, G. F., KROOK, M., AND PEARSON, C. E.: Functions of a Complex Variable, p. 376, McGraw-Hill, 1966. 9. NOBLE, B.: Wiener-Hopf Technique, p. 154, Pergamon, 1958. 10. HUGGETT, C., VON ELBE, G., AND HAGGERTY, W. : The Combustibility of Materials in OxygenHelium and Oxygen-Nitrogen Atmospheres, Brooks Air Force Base, Report SAM-TR-66-85, Dec. 1965. Prepared by Atlantic Research Corp. 11. KINBARA,T.: The Use of Models in Fire Research, p. 270, Publ. No. 786, Nat. Acad. Sci.-Nat. Res. Council, 1961.

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COMMENTS R. F. McAlevy III, Stevens institute of Technology. The author is to be praised for his fine penetration into the problem of flame spreading under the influence of forced convective motion. However, I question the validity of the data he selects for support of his predicted lack of pressure dependence on flamespreading velocity over thin specimens (Ref. 10). They are sparse, scattered, and obtained over a limited range of pressure variation. We have obtained data recently, as yet unpublished, which demonstrates that thin (30-rail) specimens of a cellulose have identical flame-spreading characteristics as thick (0.125-in.) specimens between 4 and 415 p s i a - - t h a t is, a strong influence of pressure and oxygen mole fraction in the surrounding atmosphere (see my paper in This Symposium). If the author claims that 30-rail specimens are still too thick, we would appreciate learning from him the threshold thickness to clarify this point experimentally. J. N. de Ris. The question as to whether a particular fuel bed is thick or thin is indeed important. The question can be approached as follows. Solving Eq. (31) for the effective depth of heat penetration into a semi-infinite solid, Tsi, one has

X E(T,,~p- T~)/(T/-- Tv~p)]. (32) This formula ignores radiation. The air velocity Va, relative to the advancing flame, is unknown. However, if one assumes that the semi-infinite results FEq. (29) with R~ = R2 = 0~ is valid, one obtains Va as function of V, yielding

~ - ~ (V-~x/pwO,~Vs~)

X [ ( T I - T,,~p)/(T,,~p- m~)], where V,i is the flame spread rate over a thick fuel sample. Taking typical values for a flame spreading over cellulose acetate in pure oxygen (at 5 psia, Ref. 10), one obtains a r,i of about 2 X 10 -2 em (or 8 mils). This thickness will decrease with pressure since V,i increases with pressure. Therefore, the flame spread rate over a sheet should follow the semi-infinite result at very high pressures, and follow the thin fuel bed result at very low pressures. The crossover point is given by the above formula. In view of the present uncertainties, more experimental results are needed; however, it is

important to insure that the sample is smooth, has inhibited edges, and is insulated along its back surface.

A. F. Roberts, Safety in Mines Research Establishment. Can you explain in simple physical terms why the rate of spread of flame for a thin fuel bed should be independent of V~, while the rate of spread for a semi-infinite fuel bed is proportional to V~, in your model. I would have thought that the differences between the two cases would have resulted from events occurring in the solid phase. J. N. de Ris. Consider first a thin fuel bed. One can rewrite the thin fuel bed flame-spread formula, Eq. (19), as pwC~rV (T,,~p -- T~)

Noticing, from Eqs. (4) and (6), that the characteristic gas-phase distance 2h/pC~Va applies both to the x and y directions, one sees that I-T1 -- Tv~,]/(2h/pCpVa) is the characteristic gas-phase temperature gradient and that 2k/pCpVa is also the characteristic forward distance over which the heat is transferred. Since these lengths cancel, the total rate of forward heat transfer will be independent of Va. Thus, for thin fuel beds, V will be independent of Va. For a semi-infinite fuel bed, Eq. (32), above, shows that the effective depth of heat penetration into the fuel bed rsi decreases with increasing Va; thus, less heat is absorbed by the fuel bed as the flame spreads a given distance. Since the total rate of forward heat transfer is independent of V~ (see previous paragraph) one expects V to increase with Va for a semi-infinite fuel bed. Of course, radiative effects, if important, would complicate this situation.

C. F. Hermance, University of Waterloo, Canada. I would like to compliment the author on his mathematical manipulations and the resulting solutions. However, I question the formulation of the two models. Have you evaluated the relative magnitudes of your V and the vertical velocity induced in the air above the flame? One would suspect that the

SPREAD OF A LAMINAIr DIFFUSION FLAME induced vertical velocity is not negligible, making its inclusion in the differential equation a necessity. That being the case, it may be that the problem must be split into two parts, one part upstream of the origin, and the second part from the origin downward, which includes the vertical induced velocity in the flame-position solution. This might change some of your results, particularly with respect to the pressure effect on the spreading velocity. Additionally, it is interesting to note that the thin fuel bed solution is basically an energy balance at the bottom of the flame, and looks very much like a Mallard-LeChatelier laminar flame-speed solution. A discussion of this possible coincidence would be interesting.

J. N. de Ris. To answer the first question, gravity will, of course, induce vertical gas motion downstream of the flamefront. However one must determine the effect of gravity on the upstream heat transfer, since this is the controlling mechanism. An increase in the downstream gas motion will induce an increase in the characteristic oncoming upstream air velocity V,. For thin fuel beds, it was shown that V is independent of V~; however for semiinfinite fuel beds, V depends strongly on V~. Thus, one can speculate that gravity will not affect the thin fuel bed result while it strongly influences the semi-infinite result. Therefore, the semi-infinite solution cannot be regarded as complete until the influence of gravity on V~ is understood. Speculating that the characteristic length driving the free convection is 2X/pC~V~, one calculates the conventional vertical plate characteristic laminar free convection velocity as V: ~_ 0.60 {g[ (TI -- T::p)/T+] (2X/pCpV:) }1/2. First, solving this equation for V,, and then substituting into Eq. (29) (with RI = R2 = 0), one obtains

p~C~Xw~V pCp~ (0.60)2[3{ gE (Vf -- Tva p )/Vco-] (2~/pCp) }1/3

= [-(TI-- T ~ , ) / ( T v ~ p - T~o)]2. The measurements of McAlevy and Magee indicate that the flame-spread velocity varies with the 0.69 to 0.78 power of pressure for polymethyhnethacrylate. In the formula above, V varies with pressure (i.e., p) with the 2/3 power. McAlevy and Magee I also measured the distance of forward heat transfer which varied as

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the --0.5 power of pressure. The above formula for Va gives

2X/pCpVa = (0.6) -2/8 (2~/pCp) 2/3 X {gE(Ti-- T.~p)/T+o]} -I/3, which varies as the - - 2 / 3 power pressure. It is interesting that these speculations do have nearly the correct trends; however, further justification is required. As regards the second question; both the present thin fuel bed result and the MallardLeChatelier results can be arrived at with a simple energy transport argument. REFERENCE 1. McALEvY, R. F. AND MAGEE, R. S.: Surface

Temperature Distribution Ahead of Spreading Flames, presented at the 1968 Spring Meeting of the Western States Section of The Combustion Institute, 29-30 April, Los Angeles, Calif.

P. H. Thomas, Fire Research Station, U.K. I t is very encouraging that these solutions do, in a sense, confirm an approximate over-all energy balance. I t may be that such simplified approaches will have to be used for more complicated systems of fire spread, just as, for example, profile methods can be used in certain flame-spread and boundary-layer problems. One such complication is radiation, which is amenable to a simple exponential approximation for its distribution ahead of the flame; it is interesting to consider its role and the author's result for a semi-infinite fuel bed. Consider the burning of a fuel bed with flaming zone length D supported by pyrolysis. The forward radiation will be a function of V, but not necessarily a constant fraction of the heat release, because of the dependence of emissivity on flame thickness. R will increase with V for thin flames, but, for thick flames, the emissivity is constant and the radiation will only then become independent of the length of the flaming zone. Introducing such a dependence into de Ris's result could possibly lead to three values of V, a slow stable spread characterized by thin flames and a fast stable spread characterized by thick flames. There would then be an intermediate unstable velocity. Some consideration should be given to the possibility of jumping from one mode to tlle other and to their practical relevance.

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