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 liquidfuel bed. Both a thin sheet and a semiinfinite 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 flamespread 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, gasphase conductive heat transfer, radiation, mass transfer, fuel vaporization, and fuelbed thermal properties. The radiation is mathematically treated as a heat loss at the flame sheet and a heat gain at the fuelbed surface. The calculated flamespread formulas are not inconsistent with available experimental data. These results reveal much of the physics involved in a spreading flame. For instance, the flamespread rate is strongly influenced by (1) the adiabatic stoichiometric flame temperature, and (2) the fuelbed thermal properties, except for the fuelbed conductivity parallel to the propagation direction. I. Introduction
Fire Spread Many firespread 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 fuelsurface 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 firespread process. Each of the above processes takes place by a variety of physical mechanisms depending upon the particular firespread situation being con8idered. This paper considers a particular idealized flamespread situation, for which a somewhat simplified linear exploratory model is formulated and solved.
The Problem This work considers the problem of a gasphase laminar diffusion flame spreading against an air stream over a solid or liquidfuel bed. A thin241
sheet fuel bed and a semiinfinite fuel bed are considered. In general, the forward heat transfer to the unburned fuel bed takes place by conduction through both the gasphase 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 fuelbed problem includes only the gasphase conductive heat transfer, while the semiinfinite fuelbed problem includes the gasphase and fuelbed conduction as well as the effects of radiation. An exponentially decreasing form for the forward radiative heat transfer is postulated. The thin fuelbed model is solved approximately, while the semiinfinite model is solved exactly. The combustion, which presumably takes place only in the gasphase, is regarded as a diffusion flame that comes very close to the fuel surface, as shown in Fig. 1. While some finite reactionrate 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 SchvabZeldovich~ diffusion flame theory, thereby avoiding the effects of highly nonlinear reaction kinetics. This treatment should be valid when the quenching
242
FLAME SPREAD AND MASS FIRES
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 gasphase 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 fuelbed 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 sentiinfinite fuel bed. By neglecting the fuelbed conductivity parallel to the fuel surface, it was possible to calculate the flame spread produced by an exponentially decreasing forward heattransfer profile. By so postulating the heat transfer, analysis of the gasphase cornbustion could be avoided. The resulting solution is a special case of the semiinfinite fuelbed 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 fuelbed problem. The present solution also includes the gasphase 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 gasphase 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 semiinfinite 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 flamespread process is made stationary. 2 From the viewpoint of this coordinate system, the fuel bed is fed into the flame with the flamespread velocity V.
Heat Transfer The heattransfer process to the fuel bed is of crucial importance, because the fuel bed must receive heat before it can vaporize. The forward gasphase conduction is influenced by the air velocity. In keeping with the exploratory character of this study, the gasphase 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 semiinfinite models, while the effects of radiation are included only in the semiinfinite model. The fuelbed thermal properties are important; they influence the total heat required to raise the fuelbed surface temperature to its vaporization temperature. The thinsheet 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 semiinfinite 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 modelfuel bed being fed into ,~ stationary flame.
Fuel Vaporization In most practical firespread 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 gasphase molecular mean free path is several orders of magnitude smaller than the gasphase masstransfer 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 ClapeyronClausius relation, the ideal gas law, and the fact that the fuelbed density is much larger than the fuelvapor 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 fuelvapor 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 fuelbed surface that does not vaporize until it reaches its socalled "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 speciesnamely, 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 masstransfer boundary condition along the vaporizing fuel bed is suitably linearized so that it provides a good approximation of the perpendicular convection. By expressing the flamespread solution in terms of the adiabatic stoiehiometric flame temperature, Tft, the result becomes independent of the specific masstransfer linearizing constant.
GasPhase Combustion The combustion is presumed to take place only in the gasphase. The oxygen and fuel vapor are regarded as reacting instantaneously upon coming into contact. This means that the combustion
244
FLAME SPREAD AND MASS FIRES
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 fuelvapor zones do not overlap, one can formulate a linear combustion model using the SchvabZeldovich diffusion flame theory. The present flamespread 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 onedimensional problems (i.e., problems reducible to ordinary differential equations). Since reaction kinetics are probably unimportant for a significant class of flamespread 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 thinsheet fuelbed 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 fuelbed 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 thinflame 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 fueli.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 radiationthat 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 righthand 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 righthand 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 reactionrate 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 semiinfinite problems. The rest of this section formulates the boundary conditions for the thin fuelbed 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 fuelbed 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 "heatblockage" factor) has been successfully used to correlate the mass transfer in laminar combustion boundary layers3 This factor, which is predicted by the stagnantfilm boundarylayer approximation, also predicts the correct mass transfer for the onedimensional diffusion flame parallel to vaporizing fuel surface. It also provides a good approximation of the onedilnensional 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 flamespread velocity V; it is the "eigenvalue" of the problem. This eigenvalue is ultimately determined by invoking the flamespread conditionnamely, 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 ThinSheet Problem
The thinsheet 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 gasphase 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 flamespread "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 thinsheet 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 WienerHopf 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~)(I1/K)  ( T ~ p   T~),
Tft= T~
+
and
fori= 1,2,~ <~<
rate needed to raise the fuelbed temperature to its vaporization temperature. The right side is the gasphase conductive heattransfer rate fronl the flame forward to the unburned fuel bed. The flamespread 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 heattransfer
{[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 flameshape 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 flamespread rate is inversely proportional to the thinsheet 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% 0254% 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 fuelbed equation becomes
2a a ~ / a ~

u a~/o~
~
~
o ~ / a ~ ~ = o;
<}<
co,~
(21)
with
5. SemiInfinite Problem This section extends the analysis of the preceding section to include the effects of fuelbed 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 heattransfer flux received by the fuel bed be,
R(x) = Rlexp (x/ll), = R2,
The gasphase combustion is described, as before, by the partial differential equations (16). Likewise, the boundary conditions (7) still apply. One must now formulate the fuelbed equations and the fuelsurface 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
FLAME SPREAD AND MASS FIRES
where rl = 2RJ(pV~L) a n d / ~ = 2h/(pC~V~l~). Since the gas and fuelbed 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 fuelbed 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 wellset 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 WienerHopf 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
sin1 ( 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 lefthand side of the exact solution FEq. (29)] contains the flamespread velocity V. The first term on the right describes the effect of forward gasphase 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 flamespread energy cycle. However, V is inversely proportional to k ~ ; as k ~ increases, more heat is conducted into the fuelbed interior. The form of Eq. (29) without radiation (i.e., R1  Re  0) can be obtained qualitatively directly from the thinfuelbed result [Eq. (19)]. If ~,i is the effective depth of heat penetration into the semiimCinite fuel bed, then the total forward heattransfer rate equals
X (2h/pCpV~),
(31)
where (Trap  T~)/r~i is the characteristic fuelbedtemperature gradient, and 2h/pCpVa the characteristic gasphase 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 flamespread velocity V in Eq. (29) increases with Va. One can eliminate the gasphase 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 fuelbedsurface vaporization temperature. The semiinfinite 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 flamespread model that ignores reaction kinetics, two physically revealing flamespread formulas were obtained: Eq. (19) for thinsheet fuel beds, and Eq. (29) for semiinfinite fuel beds. The latter equation includes the effects of radiation and fuelbed 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~ Flamespread 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~
249
Thin fuelbed 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. NGR31003014, Oct. 1967. 6. SPALDING, D. B.: Convective Mass Transfer, p. 144, McGrawHill, 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, McGrawHill, 1966. 9. NOBLE, B.: WienerHopf Technique, p. 154, Pergamon, 1958. 10. HUGGETT, C., VON ELBE, G., AND HAGGERTY, W. : The Combustibility of Materials in OxygenHelium and OxygenNitrogen Atmospheres, Brooks Air Force Base, Report SAMTR6685, 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 (30rail) specimens of a cellulose have identical flamespreading characteristics as thick (0.125in.) 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 30rail 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 semiinfinite 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 semiinfinite 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 semiinfinite 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 semiinfinite 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 flamespread formula, Eq. (19), as pwC~rV (T,,~p  T~)
Noticing, from Eqs. (4) and (6), that the characteristic gasphase distance 2h/pC~Va applies both to the x and y directions, one sees that IT1  Tv~,]/(2h/pCpVa) is the characteristic gasphase 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 semiinfinite 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 semiinfinite 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 flameposition 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 MallardLeChatelier laminar flamespeed 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 semiinfinite result. Therefore, the semiinfinite 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 flamespread 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, 2930 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 overall 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 flamespread and boundarylayer 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 semiinfinite 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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FLAME SPREAD AND MASS FIRES
J. N. de Ris. This is indeed an intriguing comment. Let us consider the specific situation of a flame spreading horizontally over a thin sheet of fuel as shown below. Roughly speaking, the length D of the burning zone will be proportional to V, assuming a constant vaporization rate. If the flames are turbulent, their height H will vary perhaps as D 2/3 following Thomas. I For optically thin flames, the total forward radiative heat transfer will be proportional to the flame volume DH,., V 5/8. On the other hand, for optically thick flames, the forward radiative heat transfer will be proportional to the height or V 2/3. One must also include the forward conductive heat transfer. REFERENCE 1. THOMAS,P. H. : Ninth Symposium (International) on Combustion, p. 844, Academic Press, 1963.
R. H. Essenhigh, Pennsylvania State University. This comment is addressed to the authors of all three papers dealing with flame spread. The three authors have sufficiently different descriptions or models so that one would expect to find significant differences between their various predictions, or between prediction and experiment, unless the parameters predicted, such as flame~spread rate, do not depend significantly upon the precise mechanism. I t is not always appreciated that this circumstance can arise when the measured variables depend only on the boundary conditions (generally introduced as integration constants or integral limits) and
the nature of the function being integrated is largely irrelevant. This is often the case in steadystate flame problems, where surfaces can be defined across which certain heat and mass balance requirements must be satisfied quite independently of the nature of any reaction mechanism inside the volume enclosed by the defining surfaces. Two of the authors will be aware of an application of this requirement in an analysis of flame spread down paper cylinders by J. P. Stumbar and this commentator (presented at the Pittsburgh, November 1967, meeting of The Combustion Institute's Eastern Section). This successfully predicted the rate of flame spread as a function of diameter by a nonmechanistic or phenomenological analysis, showing that certain relationships must exist independently of any mechanism. J. N. de Ris. I t is not clear that the three flamespread concepts are really so different, despite their difference in emphasis, approach, and approximations employed. If one could clearly define a control volume wherein energy or mass is conserved, one might then develop a simple energy or massconservation argument. However, this is not possible, since energy and mass are continuously lost downstream. The problem of flame spread is fundamentally mechanisticthe forward heat transfer is one controlling mechanism (of many). The mechanism of forward heat transfer (whether by gasphase conduction, fuelbed conduction, or radiation) strongly influences the rate of flame spread, especially for semiinfinite fuel beds which lose energy into their interior.