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Diffusion flame analyses
Fire Safety Journal, 3 ( 1 9 8 0 / 8 1 ) 2 7 3 - 2 8 5
273
© E l s e v i e r S e q u o i a S . A . , L a u s a n n e - - P r i n t e d in t h e N e t h e r l a n d s
Diffusion Flame Analyses PATRICK
J. P A G N I
Mechanical Engineering Department and Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720 (U.S.A.) and Basic Research Department, Factory Mutual Research Corporation, Norwood, MA 02062 (U.S.A.)
SUMMARY
Nine classic diffusion flames, i.e., combustion systems with initially separate fuel and oxidizer, are synthesized in a search for comm o n dimensionless parameters which may serve as indices o f fire hazard. The problems examined are: planar and cylindrical B u r k e Schumann; droplet burning; planar and cylindrical stagnant film; and forced, free, mixed, and stagnation p o i n t combusting boundary layers. Similarity solutions in pyrolysis regions permit identification o f flame locations and quantification o f excess pyrolyzate, i.e., fuel which is not consumed locally. Numerical solutions in downstream regions give explicit expressions for flame extensions in terms o f the n u m b e r and the B number, with the former dominating. This key parameter, the physically available oxygen-to-fuel ratio divided by the chemically required oxygen-to-fuel ratio or inverse equivalence ratio, emerges as indicative o f the fire hazard associated with flame extension. Full scale tests have shown that flame extension is related to fire spread beyond the compartm e n t o f origin. Small r means large flame. Polystyrene with r = 0.12, for example, has an order o f magnitude longer flame than w o o d with r = 0.6. Other polymers fall between these extremes. Much progress remains to be made in the areas o f flame soot, radiation, turbulence, and compartment interactions.
INTRODUCTION
Fire is the primal diffusion flame. By "diffusion" is meant that the characteristic time for transport is much longer than the characteristic time for reaction, so t h a t diffu-
sion is rate controlling. Since transport requires orders of magnitude more molecular collisions than does reaction, all flames which have an oxidizer initially separated from a fuel are diffusion flames. The alternate of initially mixed is called a premixed flame. These two categories are all-inclusive. This paper attempts to review and synthesize some aspects of classical diffusion flame analyses which are pertinent to fire research. Problems of fire safety are complex and require multi-disciplinary solutions. Prevention through material standards plays a small but vital role in overall control strategies. The objective here is to find fundamental bases for fire hazard assessment standards. The motivation for this study came from two full-scale test house fires conducted by Professor Williamson of U.C. Berkeley in conjunction with the Sacramento, Califomia, Fire Department. The first test, as shown in Fig. 1, led to total involvement of the structure in less than 3 min. In the second test, the disaster depicted in Fig. 1 never occurred .... the fire self-extinguished within the room of origin. In each case a polyurethane mattress and bumper in a baby crib next to a gypsum interior wall was ignited with a match. The difference between these tests was that: in the first, the crib was made of molded polystyrene; in the second, the crib was made of wood. What is it that causes polystyrene to be so quickly destructive while wood, in a similar exposure, self-extinguishes? What is the set of material properties which distinguishes hazardous materials from nonhazardous materials? In all the diffusion flame studies considered, the following assumptions are made [1]. The fluid flow is steady, laminar, low-speed, twodimensional, and non-radiative; transport is governed by Fick's Law with the same diffu-
274
Fig. 1. House test fire ~<5 rain after m a t c h ignition of a p o l y u r e t h a n e mattress and b u m p e r in a polystyrene baby crib. Similar test with a w o o d e n baby crib self-extinguished.
sion coefficient for all species and a unit Lewis number. All reactions are fast. These assumptions were first introduced in 1928 by Burke and Schumann [2] who solved the problem of flame locations in concentric duct burners. Similar ideas were used b y Shvab [3] and by Zeldovich [4] in more detailed analyses of the temperature and velocity fields of gas burners. Spalding [5, 6] addressed the problem of fuel pyrolysis due to combustion released energy transport. Emmons [ 7 ] quantified fuel pyrolysis with an elegant, forced flow boundary layer analysis. This reference begins the modern era of diffusion flame studies with fire safety applications. Advanced treatment o f free flow [8 - 10] and mixed flow [11] b o u n d a r y layers followed. The constant thickness b o u n d a r y layer near a stagnation point [12 - 15] shows promise for extending diffusion flame studies to include slow kinetics [16 - 19] and radiation [20, 2 1 ] . This list is n o t meant to be definitive; it only follows a single thread within the broad fabric of flame research. Since synthesis is sought, emphasis is on c o m m o n elements in diffusion flame analyses.
The n e x t section presents a c o m m o n problem formulation which identifies k e y parameters and presents general results. Solutions to the Burke-Schumann problems in planar and cylindrical geometries are followed by the related problems of droplet and stagnantfilm burning. Four boundary-layer problems axe then considered: forced, free, mixed, and stagnation point. The concept of excess pyrolyzate is introduced. In the applications section, explicit flame height expressions are given which distinguish hazardous from nonhazardous materials. The conclusion consists of a summary and suggestions for future work.
COMMON
FORMULATION
Under the assumptions stated, the governing conservation equations are: Continuity:
(1)
V'fio~) = 0, M o mentum: pv.V_v - - V'(pV_v) = E ( P - -
P-) - - Vp,
(2)
275
Species: pJ,l " Il¥; -- 1l"(pDIl¥;) Energy: PJ,l" Ilh - Il"
=
C
Ilh) =
rh;", i
=
1... N,
(3)
J
q'",
'Y - 'Y~
= {3 -
'Y~
~w -
'Y w
-
(3~ .
(10)
~~
All of eqns. (3) and (4) reduce to
p
(5)
Energy-species variable
Simplification is gained by recognizing that eqn. (5) restricts the fuel, oxidizer and energy to . ,,, -m.", q. ", ·--mf o
(6)
Q i.e., fuel and oxygen must be consumed with heat release as specified by eqn. (5). This suggests thinking of oxygen as negative fuel [2] and defining, vfMf
=
(4)
where all symbols are defined in the nomenclature. Secondary effects such as Soret, etc., are discussed elsewhere [1] . The chemistry plays a very important role, even when reactions are fast, through the overall balance,
--=--=-,
ence between its maximum and minimum the same energy-species variable results,
voMo
(7) This variable does not change upon reaction and eqn. (3) for ~ becomes homogeneous and linear. The standard heuristic for nondimensionalizing ~ is to subtract {3's minimum value, {3~, the value in the region of highest oxygen concentration, and to divide by the difference between the maximum ~ and the minimum (3. Let (8)
where ~'s maximum value, ~w, occurs in the region of highest fuel concentration. This nondimensional fuel-oxygen variable goes from 1 at the fuel surface smoothly to 0 at ambiance. At the flame it has the value stipulated by Yf = Yo = 0, since fuel and oxidizer react upon contact. The same results are obtained if (9)
is used to combine eqns. (3) and (4) into a single, homogeneous, linear enthalpy-oxygen equation. An equivalent enthalpy-fuel equation provides no additional information. When 'Y is nondimensionalized by subtracting its minimum, 'Y~, and dividing by the differ-
pJ,l"IlJ - 1l"(pDIJJ)
=0
(11)
with J = 0 or 1 on the boundaries. Surface fuel mass fraction
The oxidizer boundary conditions are constant and known: Yc,~, T~, ~~, 'Y~. For a burner, Yfw is the initial fuel mass fraction and Tw = T~. For a pyrolyzing surface, the activation energy associated with pyrolysis is sufficiently large that the surface temperature, Tw , is constant and known. The remaining constant needed to define the maximum ~ is the surface fuel mass fraction, Yfw, which is specified a priori by the energy and fuel balances at the wall, " "L =X-ah m -I cp an w and
(12)
O"y;"" m ft = m Yfw - pD aYfI _.
an
,
(13)
w
respectively, where Yft, the fuel mass fraction in the transferred material, accounts for the inerts in the pyrolyzate. Solving for m" and equating gives X cpL
ah I -pD aYf I an w = Yft - Yfw an w'
(14)
From the definition of J, eqns. (7) - (10),
ah I ( QYc,~) aJ I an w = hw -- voMo an w
(15a)
and (15b) (Yfw + sYc,~) -aJ I . an w Recall Le = XjcppD = 1; substitute eqns. (15) -aYf
I
an w
=
into eqn. (14) to eliminate aJjan and obtain
(16) The left-hand side is B, Spalding's mass transfer number. Solving eqn. (16) for Yfw yields the fuel mass fraction at the wall
276
Yfw = (BYft
-sYo~)/(1
+ B).
(17)
Since Y ooo • Y ft • sand B are all constant and known, Yfw is constant and known a priori. General results Some additional consequences of choosing J as the dependent variable can be quantified. At the flame, the general expression, J
Yf (Yc, - Yc,~q / [Yfw Yc,~ ] = f vfMf voMo J vfMf + voMo (18)
reduces to
Since the reactions are all fast, there is no fuel outside the flame and no oxygen inside the flame. The enthalpy field is obtained from J = [_
r= - - - (20) YfwvoMo is the key parameter which emerges in this synthesis. It is the physically available oxygen-to-fuel mass ratio, Yc,~/Yfw, divided by the chemically required oxygen-to-fuel mass ratio, voMo /vfMf. It may be called a "mass consumption number" or "flame number" since it indexes the location and size of the flame in the same way that the mass transfer number indexes the surface pyrolysis rate. An analogy can also be made with premixed flames, where the equivalence ratio is defined as the actual fuel to air ratio divided by the stoichiometric fuel to air ratio. Here l/r plays the role of an equivalence ratio for diffusion flames. When r is small, the oxygen present is small compared with that needed, i.e., the system is fuel rich. Since J varies from 1 at the fuel boundary to 0 at the ambiance, as r decreases the flame moves away from the fuel toward the oxidizer. This is true for all diffusion flames. Additional results useful for all diffusion flames are obtained from the definitions of J, eqns. (10) or (18) as shown in Table 1.
(Yo - Yo~) ] / voMo
[_ hw + ~ ] . Q voMo (21)
The flame enthalpy is explicitly given by eqn. (21) with Yc, = 0 and J = (1 + r-1)-1, as shown in Table 1, where the dimensionless heat of combustion, Dc
(19)
~_
Q
= QYo~/voMohw,
(22)
enters only via the enthalpy field. These three dimensionless parameters: r, B and Dc dominate diffusion flame problems in the order listed.
PROBLEMS
Burke-8chumann Figure 2 shows the dimensional and nondimensional geometries of both the planar and cylindrical Burke-Schumann problems. Consider an infinitely long duct of dimension BURKE-SCHUMANN
J
0
~
J~l
J
~
0
y 0 00
,
~
x/b.
c
~
alb.
0 '" , '" 1
a .;;
rl
~
C'O.
,,=
ZD/vb
2
Fig. 2. Burke-8chumann problem geometries and definitions.
TABLE 1 General results for species and enthalpy fields in terms of J, r and Dc, valid for all diffusion flames; indicates fuel side and 1] ;;. 1]fl indicates oxidant side of reaction sheet
1] <;; 1]fl
1]
1]fl
o
o o
(1 + Dclr)J
(Dc + r)/(l + r)
(1 +r)J-r
o 1 - (1 + l/r) J Dc - (Dc -1)J
277
b with a semi-infinite duct of dimension a centered in it. Fuel flows through the inner duct and oxidizer in the outer duct with uniform velocity v. With the z origin at the downstream end of the inner duct, Yr = Yrw, Yo = Yo~ and T = T., at z = O. It is assumed that only axial flow and transverse diffusion exist and that pv and pD are each uniform. Under these restrictions, only eqn. (11) is required in the form
(02 J2 -!:- aJ) = O.
v oJ -D
az
ax
(23)
ax
x
Now let 17 = zDlvb 2, ~ = xlb and e = alb so that aJ a2 J K OJ -=-+--
a17
a~2
~ a~
(24)
,
where K = 0 for planar and K = 1 for cylindrical systems. The range of interest is 17 ~ 0 and ~ ~ 0 by symmetry. The boundary conditions are no flux at ~ = 0 and ~ = 1 with the unburned condition at 17 = 0: JC~,
0) = 1,
for 0
J(~,
0)
for e <
~ ~ ~
e, (25)
=
0,
~ ~
1.
07r------.---------, OVERVENTllATED
iJNDE R VENT I LATED
c
~
.0.4
....
--- 01
I
-
~ 0.3
- - 0,9
w
05
I
w 02
" 0.1
~
10 VENTILATION, dC-i_I)
Fig. 3. Flame heights for planar problem.
Burk~chumann
defines critical ventilation; values < 1 are underventilated, values> 1 overventilated. Figure 3 shows planar flame heights for three geometries versus the ventilation parameter. At the ventilated condition, infinitely high flame sheets are predicted. If the system is overventilated, the flame will reach its maximum extension on the centerline; if underventilated, the flame touches the ~ = 1 surface. To make a connection to fires, note that as b ~ 00, e ~ 0, and the ventilated condition is approached as r ~ O. So, small r means large flames, even in this simplest diffusion flame problem.
The cylindrical solution [1,2] is Droplet burning
J(t n) = e 2 + 2e " ~ JIcrne)JoCrn~) X <;, '/ n~ 1 [Jocrn)] 2
rn
X exp(-r~17),
where I n are Bessel functions of the first kind and r n are the positive roots of J1 Cr n) O. The planar solution [2] is JC~, 17) = e
+2
L
(26)
=
(mr)-l sin (n1Te) X
n=l
(27) The flame location is given by eqn. (19) and the species and enthalpy fields are as shown in Table 1. The ventilated condition occurs when the initial mass flux of fuel and oxidant are in stoichiometric proportion,
Yrw aK + 1
vfMf
-----:------:-- - - Yo~CbK+l-aK+l)
voMo In dimensionless form, r(e-(K+l)
-1)
=1
The mass transfer number does not appear in the Burke-Schumann problem because the fuel mass flux is under external control. In the droplet burning problem [1,5,6], heat transfer to the drop surface dictates the fuel mass flux. It is assumed that the drop is small enough to be uniform inside and spherically symmetric outside and yet large compared with the reaction zone. The drop radius is the only time dependent quantity, i.e., the system is quasi-steady. Here pD is constant a removable approximation [1]. The quasisteady mass flux is also a constant, = 41TX 2pV, where x, the radial coordinate, is the only independent variable. Equation (11) becomes
m
'dJ dx
=
~ l(41TP~X2) dJ] dx
m
.
(30)
dx
By defining (28)
(29)
m =, (31) 41TpDx a dimensionless mass flux which maps x = 00 into ~ = 0, the problem for J(~) reduces to
~
278
J" + J' J(~w)
=
0,
= 1,
(32)
J(O)
= 0;
(33)
with the solution,
= (1 -
J(~)
e-~)/(I- e-~w).
(34)
All species and temperature fields can be obtained from eqn. (34) and Table 1. The surface energy balance, eqn. (12), can now be used to find the eigenvalue, appearing here in the guise of ~w' In terms of J, eqn. (12) is
m,
mL
-"1\ (
--=-
41TX;'
cp
Y..,Q ) dJ -hw + - lJoMo
d.x
I
(35)
w'
U sing the definitions of ~ and B, Le = 1 and eqn. (34), eqn. (35) becomes
~ = dJ d~
B
I = (e~w -
1
r 1.
(36)
= In (1 + B);
(37)
m= 41TpDxwln (1 + B).
(38)
The flame standoff distance, using eqns. (19), (31), (34) and (37) is
~w
In [(1 + r)(1 + B) ] 1 +r +B In (1 +B)
(39)
Note that as r ~ 0, the equivalence ratio becomes large, ~fl ~ 0 and Xfl ~ 00. Although the semi-infinite, cylindrical and planar droplet burning problems do not have steady state solutions, the closely related finite domain "stagnant film" problems do. Consider a one-dimensional planar film between two infinite screens. Beyond the screen at x = 0 is a stagnant oxidizing gaseous ambiance. Beyond the screen at x = X w is a solid or liquid fuel reservoir. Assume steady, laminar combustion occurs within the film and the mass flux, mil = pv, is constant for all x. Let pD be constant and J have its usual definition, eqns. (10), (18) and (21). With ~=
rh"x/pD,
(40)
the governing equation, conditions, and solution are given by eqns. (32), (33) and (34), respectively. The mass pyrolysis rate is given by eqn. (37) so that
(41)
The profiles given by Table 1 and eqn. (34) are linear only in the limit B <{ 1. The flame location is given by eqn. (39), so as r ~ 0 the flame moves to the oxidant screen. In the cylindrical problem, which resembles a candle wick, the origin is at the wick center. The fuel pyrolyses at X w and the oxidant screen is at x~. It is assumed that m' = 21TXPV is constant. Other assumptions are the same as in the planar problem and eqn. (11) reduces to
dJ = ~ [(21TPD~) dJ] . (42) d.x d.x d.x Nondimensionalize the radius on its maximum value x~, and define
m'
~ =
so that the droplet pyrolysis rate is
~fl =
pD
w
From eqn. (36) the eigenvalue is ~w
•
mil = -In (1 + B). Xw
-m'ln(x/x~)
(43) 21TpD The governing eqn. (42), boundary conditions, and solution, again reduce to eqns. (32), (33) and (34), respectively. Equation (37) gives the pyrolysis rate,
• m'
=
21TpD In(l + B) In(x~/Xw)
(44)
,
while eqn. (39) gives the flame location. Again, as r ~ 0, the flame moves to the oxidant screen, Xfl ~ x..,.
Boundary layer combustion Now consider four boundary layer problems -- forced, free, mixed, and stagnation point flow. Although the energy-species eqn. (11) simplifies by the usual boundary layer approximation to pu
OJ + pv aJ = ~ (PD OJ),
ax
ay ay
ay
(45)
these systems are more complex, since the continuity, eqn. (1), and the x-momentum, eqn. (2),
au
au a ( au) ---g(p-p~), ap ay ay. ay ax (46)
pu-+pv-= -- 11-
ax
must be considered in detail.
Forced flow Emmons' study [7] of forced boundary layer combustion could serve as the model research paper. It addresses a complex
279
FLAME ~ SHEET / g
/
U - O@/O~,
I
V-= - ~ @ / ~ ,
f ( ~ ) -- ~/~112.
(51b) (51e)
With T = O, eqns. (51) condense eqns. (48) -
i/
(50) to
J" + erfJ' = 0
(52)
and u®
T
T
Q
y-
f'" + ff" = O.
E m m o n s further showed t h a t if Pr = 1, f' = 2 (1 - - J ) and eqn. (52) becomes identical with eqn. (53), so t h a t the combustion problem reduces to the incompressible Blasius problem. The boundary conditions are
y~
Fig. 4. Boundarydayer combustion
problem
geometries and definitions.
problem, introduces physically valid assumptions, simplifies the mathematics, obtains exact and approximate solutions, explores behavior in limits, and compares with experiments. The treatment given here is abbreviated to encourage the reader to enrich himself from the original. Boundary layer flame schematics defining geometries and n o t a t i o n are shown in Fig. 4. Assume the property product, pg, is constant. Define the dimensionless variables Re1/2 /
f - x / l , ~=
l
o
P~ (47)
Op ) -~x dy ,
which reduce eqns. (1), (45) and (46) to OU --
OV + --
O,
o~-
u/u~ = f'/2,
(55)
f'(0) = 0, f'{oo)= 2.
(56)
The energy balance at the surface, eqn. (12), becomes
f(O) = BJ'(O)/Pr,
(57)
since by definition
rh"(x) - 2p~u~Re~/2
(58)
The solution to eqns. (52) and (53), subject to eqns. (54), (56) and {57), provides the velocity field from eqn. (55) and the energy and species fields from Table 1. Of interest is the pyrolysis rate, eqn. (58), which can be made an explicit function of B alone using a curve-fit obtained at Pr = 0.73, (59)
(48)
1 82J Pr ~ 2 ,
0U OU 02U U -- + V + r of o~ o~ 2
(49)
(50)
where r is a dimensionless force due to gravity or an external pressure gradient. Along the pyrolyzing surface, forced flow similarity is introduced via the definitions, -= ~/(4f) 1/2 ,
(54)
and since
--f(0) = 0.47 B 0"578-0"0971nB =
0J 0J U -- + V of
J(0) = 1, J(oo) = 0,
--f(0)
P~ dy,
U_ u/u®, v = Rel/2 ( / l pV+Uo
(53)
(51a)
The flame location within the boundary layer, ~n, is determined by r from eqn. (19) for Jn (nn). A connection may be made with the Burke-Schumann problem. The pyrolyzing surface provides some fuel which is consumed locally and some which is convected downstream between the flame and the surface. This convected fuel plays the same role at the end of the pyrolyzing slab as does the BurkeSchumann fuel at the end of the inner duct. It determines the flame height. It is convenient to calculate this "excess p y r o l y z a t e " ,
280
"g/e, as a fraction of the total fuel pyrolyzate, Yft /~/p, [10]. The downstream fuel flux is ]~/e(X) =
(pU Yf)~ dy.
(60)
0
The total fuel pyrolyzed over 0 ~
(61)
with ~ given by eqns. (51b). The energy species and m o m e n t u m eqns. (49) and (501} reduce to J " + 3PrfJ' = 0,
(65)
f"' + 3ff"-- 2f '2 + ~ = 0.
(66)
The conditions on J are given by eqns. (54) and the velocity conditions are f'{0) = 0 and f'(~o) = 0.
Nondimensionalizing eqn. (60) and dividing by eqn. (61) gives the excess pyrolyzate as a fraction of the total fuel pyrolyzate,
The surface energy balance gives
----
since by definition,
Yft/~/p 1 + (B ~ l)(r-I)
X
I(l + r-l) ,n
f
o
f0?n) +
11.
f(o---T--
(62)
Note that the Re~ dependence in/9/p and Me cancel, so that the ratio is constant along the pyrolyzing surface. Equation (62) holds for all four combusting boundary layer problems. Since f' ~ u and J ~ Y~, the integral is ~Me and --f(0) is ~/~/p. The non-integral terms in the bracket are negligible. The B and r-~ dependencies in the coefficient, both generally >1, effectively cancel. The primary dependence on r arises from t h e upper limit of the integral. As r decreases, Jfl decreases and ~n increases, allowing more fuel to convect downstream.
Free flow The corresponding problem for free flow is identical except for the b u o y a n c y term in the m o m e n t u m equation. Using the nondimensional variables ~ - x//,
~=-l
u-=
Gr~/2------~ 'ul
v =
-f(O) =
21/2xrh" (x ) 3p®v~Gr~14
(68)
(69)
The solutions to eqns. (65) and (66), subject to these conditions, give the energy and species fields from Table 1 and the velocity field from u = 2~Grxl/2f'(~)/x.
(70)
The local pyrolysis rate, eqn. (69), is made explicit via the curve fit, --f(0) = 0.2 r-°'°45B °:767-0"0591nB ,
(71)
The weak r dependence derives from ~ in the m o m e n t u m eqn. (66). Though eqn. (71) was obtained at Pr = 0.73 and De = 4.0, it is a good approximation for all Pr and Dc of interest. The flame location is again given by eqn. (19). The excess pyrolyzate formulation is identical with the forced case, so eqns. (60) and (62) are valid provided the total fuel pyrolysis rate is given by Yft/l}/p(X) = --f(O)P~U~23/2 Gr~/4 Yft-
(72)
Note that the excess fraction of the total pyrolyzate is independent of Gr~ and only a function of r and B.
-P-dy, 0 P~ Gr~ 41p~v®(p v
f(0) = BJ'(O)/3Pr,
(67)
+ u.
f~' -OxOPdyl,]
0
(63) gives the same eqns. (48) and (49) with now, T = ¢ = h/h~ from Table 1 in eqn. (50). The free flow similarity variables are r/ = ~'/(4~) TM ,
(64a)
f(r/) = ~/v~ G r 1/4 (4~) 3/4
(64b)
Mixed flow The preceding limiting cases are valid for Froude numbers, Fr -= Re2/Gr, >>1 and 41, respectively. Intermediate values of Fr probably more accurately represent flows induced by c o m p a r t m e n t fires. The forced flow dimensionless variables, eqns. (47), are used since t h e y give simpler boundary conditions [11]. The governing eqns. (I), (45) and (46) reduce to eqns. (48), (49) and (50)
281
with ~ = ¢/Fr. A similarity solution does n o t exist, and a local similarity solution proved inaccurate, so that direct numerical solution of eqns. (48) - (50) is required with eqn. (54) and U(~, 0) = 0,
V(~, 0)
B~J .
.
.
.
Pr~
U(0, ~) = U(~, co) = 1
(~, 0),
(73)
i°I r=O.I
o
. . . . . . . . . . - -
~_ 0.5
~
.~
Free: Dc=4 S t a g n a t i o n : Oc = 4 , 0 w = 2 Forced
(74) w
as b o u n d a r y conditions. The results [11] indicate that the free limit holds for Fr < 1, while the forced limit is reached at Fr > 102. Smooth interpolation between limits is possible with linearity increasing as r increases. Stagnation point
A constant thickness b o u n d a r y layer develops in the vicinity of a stagnation point due to the acceleration of u~ away from the origin. If the oncoming flow is an oxidizer and the surface is a pyrolyzing fuel, a useful system for flame studies is obtained. Only the planar geometry, with x along the surface and y normal to it, is discussed here. The axisymmetric geometry has been described in detail [ 1 2 - 15]. Several approximations exist for the free stream [15, 22] ; each has u~ linear in x so that --~p/ax = p=u~u~/~x
(75)
in eqn. (46) is linear in x. Using the forced flow eqns. (47) reduces the governing eqns. (1), (45) and (46) where g = 0 to eqns. (48) (50) with -r = 0 -~ T/T~ = 1 + ¢(0w -- 1)
(76)
given by Table 1, assuming cp approximately constant. A similarity problem arises from eqn. (51b) and (77a)
_-- ~/~1/2,
f07) = ~/P~(Ret~) 1/2 .
(77b)
Then eqn. (49) reduces to eqn. (52) and eqn. (50) becomes f,, + ff,, _
f , 2 + 0 = O.
_ _ ~ __2 ~) 0.1
I
--i I
I
l I I Ill _ i.O Mass
Transfer
l No.,
I
I
I I I I J I0
B
Fig. 5. Excess p y r o l y z a t e as a fraction of total fuel p y r o l y z a t e vs. B n u m b e r parameterized in r n u m b e r for three boundary-layer c o m b u s t i o n p r o b l e m s with P r = 0.73. These results are general, i . e . , indep e n d e n t of Re and Gr and a p p r o x i m a t e l y applicable for all parameter values.
~"(x) --f(0)
p.~u~Re1/2 ,
(79)
which is obtained explicitly from a curve fit to numerical results at Pr = 0.73, De = 4 and 0w = 2 a s --f(0) = 0.543
r-°'°2B
0"626-0"06781nB
.
(80)
The dependence on D e and 0w is negligible for this nonradiating system. The excess pyrolyzate is again given by eqn. (62) with the total fuel pyrolysis rate given by eqn. (61). Figure 5 shows the excess pyrolyzate for all the boundary layer problem systems examined. As this inclusive plot shows, the excess fraction of the total pyrolyzate is dominated by r with only a weak dependence on B. The very weak dependence on Pr, D c and 0w may be neglected. There is no dependence on scale through x, Re or Gr. The practical consequence is that 1 / r serves as an index of the u n b u r n e d pyrolyzate convected downstream to fuel flames remote from their source.
(78)
The b o u n d a r y conditions are identical with eqns. (54), (56) and (57), except t h a t f'(oo) = 1 since u / u ~ - f ' . The solution to eqns. (78) and (52) for f and J gives the velocity field and the energy and species fields from Table 1. The mass pyrolysis rate is
APPLICATIONS
Flame heights
What happens b e y o n d the end of the pyrolyzing surface, x > l, is important to fire safety applications and is determined by
282 TABLE 2 E x p l i c i t curve-fits f o r flame e x t e n s i o n b e y o n d p y r o l y z i n g surface in u n i t s o f t h e fuel slab l e n g t h for forced, free, a n d m i x e d , s t e a d y , l a m i n a r b o u n d a r y - l a y e r s in t w o g e o m e t r i e s F o r c e d flow Xfl ~ 0.14 [(1 + r - l ) ( 1 + B -1) B - ° ' 1 5 l n (1 + B ) ] 2
wall-mounted : free-standing:
Xf*l ~ 0.72 Pr- 0.2 X~l_wall
Free flow X~I ~ 0.24 [(1 + r - 1 ) r --0"16 (1 + B - 1 ) B - ° ' ° 6 1 n (1 + B ) Dc-°'°5] 4/3
wall-mounted : free -standing:
X~l
* 0.9 Xfl.wall
M i x e d flow * ~ --0.2 ~* Xfl ~ (1 -- Fr-O'2)X~l.forced + r r Aft.fre e * Xfl ~ Xfl-free *
F r ~> 1: Fr~
returning to the first diffusion flame problem and treating the excess pyrolyzate as the fuel input to a Burke-Schumann-like flame location problem with more complicated fluid mechanics. The abrupt change in b o u n d a r y conditions at x = l prohibits similarity, so eqns. (48) (50) are solved numerically with t w o alternative sets of b o u n d a r y conditions representing, respectively, a wall-mounted or freestanding fuel slab: wall: or
V(0)
=
Y(0)
=
~}J(0)/d~
=
0
POLYSTYRENE
POLYURETHANE
POLYMETHYLMETHACRYLATE
POLYOXYMETHYLENE
CELLULOSE
2
4
6
8
;0
12
14
16
18
2:0 22
24
DIMENSIONLESS FLAME EXTENSION, ×fit-xfit/~ Fig. 6. P r e d i c t e d flame h e i g h t s for c o m m o n fuels.
(80a)
centerline: a U ( O ) / ~ = V(O) = aJ(O)/a~ = O. (SOb) In each of the four b o u n d a r y layers, the flame location is given b y eqn. (19), J f l ( ~ n , ~fl) = ( 1 + r-l) -1. All of the fuel is assumed consumed when the flame sheet intersects = 0. The value o f ~fl there gives the flame height, X ~ - ~fi-- 1, i.e., the flame extension b e y o n d the fuel slab in units of the slab length. Curve-fits for the first three b o u n d a r y layer problem results are given in Table 2; the fourth is plotted in ref. 15. Fire safety
The expressions in Table 2 suffice to explain w h y the polystyrene crib fire p r o d u c e d the totally involved structure shown in F i g . 1, while the w o o d e n crib self-extinguished. Figure 6 shows a bar graph o f flame heights for several c o m m o n fuels c a l c u l a t e d using Tewarson's properties [23] in the curvefits of Table 2. Polystyrene has a flame height
of 24, while w o o d has a flame height of 2. This means that if the cribs were 1 m high, the w o o d fueled flames would just reach a 3 m high ceiling, while the polystyrene flames would extend 25 m up the wall, across the ceiling, o u t the d o o r and d o w n the corridor igniting any combustible materials along the way via radiation and convection from the combusting gases. This scenario is dramatically visible in films of the Sacramento house fires. Other synthetic polymers fall between these extremes. While turbulence and quenching tend to decrease the flame length, the distinction among fuels remains [ 2 4 ] . What is it a b o u t the chemistry of polystyrene which leads to its extreme fire performance? Table 3 shows the detailed properties of six polymers. Note that s, the mass of fuel c o m b u s t e d per mass of oxygen, for polystyrene is a b o u t one third o f s for cellulose. This means that for a given a m o u n t of fuel, three times more oxygen is required and
283 TABLE 3 Predicted flame extensions and ideal material properties for wall-mounted common polymer slabs with an ambiance of T~ = 293 K and Yo~ = 0.23 The GM designation refers to the Products Research Committee general material sample bank available through the National Bureau of Standards. Material (type)
Formula
Polystyrene (foam, GM-49) Polystyrene (granular) Polyurethane (foam, GM-25) Polymethylmethacrylate (granular) Polyisocyanurate (foam, GM-41) Polyoxymethylene (granular) Cellulose (filter paper)
C8H8.400.03 CsH 8
s
wake
11
24
1.3
1.7
0.6
0.12
4.8
0.33
1.7
1.3
0.5
0.14
4.1
9.3
18
C3.1H5.4ON0.22 0.46
1.2
1.5
0.6
0.19
3.6
6.5
11
C5H802
0.52
1.6
1.5
0.6
0.22
5.1
5.2
8.9
C5.3Hs.2ON0.57
0.43
4.5
0.4
0.2
0.43
4.0
1.8
2.8
CH20
0.94
2.4
1.1
0.4
0.50
5.8
1.8
2.4
C6H1005
0.84
3.5
0.8
0.3
0.60
6.9
1.4
1.8
N i n e classic d i f f u s i o n f l a m e p r o b l e m s h a v e been examined: Burke-Schumann -- planar a n d cylindrical; d r o p l e t b u r n i n g a n d s t a g n a n t film - - p l a n a r a n d c y l i n d r i c a l ; b o u n d a r y l a y e r c o m b u s t i o n - - f o r c e d , free, m i x e d , a n d p l a n a r s t a g n a t i o n p o i n t . I t was a s s u m e d t h a t e a c h system had: steady, low-speed laminar flow w i t h u n i t y Le, s p e c i e s - i n d e p e n d e n t D , F i c k i a n d i f f u s i o n , n o r a d i a t i o n , a n d fast c h e m i c a l
Dc
X~l
0.33
CONCLUSIONS
r
plume
B
larger PS f l a m e surfaces are d e v e l o p e d t o p r o v i d e t h e n e e d e d o x y g e n . In a d d i t i o n , t h e n e x t c o l u m n s h o w s a significant d i f f e r e n c e in h e a t o f p y r o l y s i s , t h e e n e r g y r e q u i r e d t o v a p o r i z e fuel. P o l y s t y r e n e ' s L is a p p r o x i m a t e l y o n e t h i r d t h a t o f w o o d . So, f o r a given h e a t i n p u t t o t h e solid, t h r e e t i m e s as m u c h gaseous PS fuel is p r o d u c e d . T h e s e t w o factors provide the chemical and physical bases f o r t h e o r d e r o f m a g n i t u d e l o n g e r PS flames. T h i s e f f e c t is q u a n t i f i e d b y t h e f a c t t h a t t h e r o f PS is o n e - f i f t h o f t h e r o f cellulose. A l t h o u g h t h e d r a m a t i c crib fire accentuates the hazard of polystyrene furniture units, t h e m o r e useful p r a c t i c a l c o n s e q u e n c e is t h e e m e r g e n c e o f r as an i n d e x o f fire h a z a r d - - small r, %1/3, m e a n s large h a z a r d .
Yfw
X~l
L (k J/g)
r e a c t i o n s . T h e f o l l o w i n g c o n c l u s i o n s are drawn from the solutions obtained: (1) m a n y similarities e x i s t a m o n g d i f f u s i o n flame problems; (2} t h e single m o s t i m p o r t a n t d i m e n s i o n l e s s p a r a m e t e r , w h i c h d o m i n a t e s f l a m e shapes, heights, a n d excess p y r o l y z a t e , is r ~- (Yo~/ Y~w)(vfMf/vo Mo), a m a s s c o n s u m p t i o n n u m b e r or inverse e q u i v a l e n c e r a t i o f o r d i f f u s i o n flames; (3) n e x t in i m p o r t is t h e m a s s t r a n s f e r number, B - (QYo~/voMo -- hw)/L, which d o m i n a t e s p y r o l y s i s rates; (4) f l a m e t e m p e r a t u r e s are c o n t r o l l e d prim a r i l y b y D e -- Q Y o ~ / v o M o h w , a d i m e n s i o n less h e a t o f c o m b u s t i o n ; (5) t h e s e t h r e e p a r a m e t e r s , as t h e pertinent, quantitative, material chemical and p h y s i c a l p r o p e r t i e s , are indices o f fire h a z a r d . All o f t h e i n f o r m a t i o n r e q u i r e d t o e v a l u a t e t h e s e p a r a m e t e r s f o r n e w m a t e r i a l s can be o b t a i n e d f r o m small-scale l a b o r a t o r y a p p a r a t u s . T o b e free f r o m h a z a r d , t h e m a t e r i a l w o u l d h a v e large r a n d small B a n d D e . T h e analyses d e s c r i b e d here, w i t h i n t h e constraints of their assumptions, provide the r e q u i s i t e c o n n e c t i o n b e t w e e n fuel p r o p e r t i e s and dynamic flame behavior to permit a p r i o r i a s s e s s m e n t o f s o m e f o r m s o f fire hazard.
284
Future work Radiation is a crucial c o m p o n e n t of most flames of candle size and above [25]. Some progress is occurring [20, 21] toward incorporating radiative terms in these analyses. But an understanding of soot formation and evolution [26] within diffusion flames is necessary before flame radiation can be predicted with the same generality as pyrolysis rates. Turbulence, which plays a key role in all fires, is n o t understood even in nonreacting systems. Non-boundary layer systems, such as candles or pool fires, have not been solved. Quasi-steady flame spread also awaits more detailed explanation [27]. Improvements may be made by eliminating assumptions, e.g., variable or non-unity Le, non-uniform D and multiple flame sheets. Slow, one-step kinetics have been introduced in extinction studies [ 16 - 19]. Extension to multi-step mechanisms [28] and radiative extinction remain as challenges. Finally, these problems address only the burning of a single fuel source in an open ambiance. The geometric and fluid mechanic complexities of interactions with surrounding compartments must be addressed [29 - 33] before predictive capability can be said to exist for fire behavior as a threat to property and life safety.
Dc
Dimensionless heat of combustion,
f Fr g Gr x
Dimensionless stream function Froude number, Re~/Grl Acceleration of gravity Grashof number, g(Tw -- T~)x:~/v2T~
h
Specific enthalpy, f
QYo~/.oMohw
T
c, dT
T~
J
Normalized energy-species variable, (~ - - ~=)/(~w
-
~)
=
(7
--
~)1(7~
-
~)
L Le l m Mi Pr Q
Effective latent heat of pyrolysis Lewis number, k/cppD Fuel length Mass Molecular weight of species i Prandtl number Energy released by combustion of vf moles of gas phase fuel q Heat Re~ Reynolds number, u~x/u~ r Mass consumption number, Yo.s/Y~w s Stoichiometric ratio, vfMf/uoMo T Temperature U Dimensionless streamwise velocity u Streamwise velocity V Dimensionless transverse velocity v Transverse velocity x Streamwise coordinate y Transverse coordinate Yi Mass fraction of species i
ACKNOWLEDGMENTS
It is a j o y to contribute this compendium to a volume honoring P. H. Thomas. This study is based on research sponsored, in large part, by the Center for Fire Research of the National Bureau of Standards, United States Department of Commerce. Writing was completed while the author held the position of visiting scientist at Factory Mutual Research Corporation. Stimulating conversations there with J. de Ris are much appreciated. The author gratefully acknowledges H. W. Emmons, the founder of fire research in the United States, for the energy, encouragement, and excellence he has brought to the field. NOMENCLATURE
B % D
Mass transfer number, (QYo~/voMo --hw)/L Specific heat Species diffusivity
Symbols Species coupling variable, Y~Iv~M~--- YolvoMo ~/ Energy-species coupling variable, - - h / Q - - Yo/voMo
0 K p v
p ¢
Dimensionless transverse coordinate Similarity variable Dimensionless temperature, T/T® Flow factor (0, planar; 1, cylindrical) Conductivity Dynamic viscosity Kinematic viscosity or stoichiometric coefficient Dimensionless streamwise coordinate Density Dimensionless enthalpy, h/hw Stream function
Subscripts f Fuel fl Flame o Oxidizer
285
t w co
T r a n s f e r r e d gas F u e l surface Ambient
Superscripts * M e a s u r e d f r o m d o w n s t r e a m edge o f fuel surface Per u n i t t i m e ' Per u n i t length or derivative w i t h r e s p e c t t o t h e i n d e p e n d e n t variable REFERENCES 1 F. A. Williams, Combustion Theory, A d d i s o n Wesley, R e a d i n g , MA, 1 9 6 5 . 2 S . P . B u r k e a n d T. E. W. S c h u m a n n , First Symp. (Int.) on Combustion~ T h e C o m b u s t i o n I n s t i t u t e , P i t t s b u r g h , PA, 1 9 2 8 , pp. 2 - 11. 3 V. A. S h v a b , Relation Between the Temperature
and Velocity Fields o f the Flame o f a Gas Burner, Gosenergoizdat, Moscow-Leningrad, 1948. 4 Y. B. Z e l d o v i c h , Zh. Tekh. Fiz., 19 ( 1 0 ) ( 1 9 4 9 ) 1 1 9 9 - 1 2 1 0 (Engl. trans. NACA Tech. Memo. 1296, 1 9 6 0 ) . 5 D. B. Spalding, Fourth Syrup. (Int.) on Combustion, T h e C o m b u s t i o n I n s t i t u t e , P i t t s b u r g h , Pa, 1 9 5 3 , pp. 847 - 8 6 4 . 6 D. B. Spalding, Fuel, 29 ( 1 9 5 0 ) 2 5 ; 32 ( 1 9 5 3 ) 169. 7 H. W. E m m o n s , Z. Angew. Math. Mech., 36 (1) ( 1 9 5 6 ) 60 - 71. 8 F. J. K o s d o n , F. A. Williams a n d C. B u m a n , Twelfth Symp. (Int.) on Combustion, T h e C o m b u s t i o n I n s t i t u t e , P i t t s b u r g h , PA, 1 9 6 8 , pp. 253 - 264. 9 J. S. Kim. J. de Ris a n d F. W. Kroesser, Thirteenth Symp. (Int.) on Combustion, T h e C o m b u s t i o n I n s t i t u t e , P i t t s b u r g h , PA, 1 9 7 0 , pp. 9 4 9 - 961. 10 P . J . Pagni a n d T. M. Shih, Sixteenth Symp. (Int., on Combustion, T h e C o m b u s t i o n I n s t i t u t e , Pittsb u r g h , PA, 1 9 7 6 , pp. 1 3 2 9 - 1 3 4 2 . 11 C. M. K i n o s h i t a a n d P. J. Pagni, J. Heat Transfer, 102 ( 1 9 8 0 ) 104 - 1 0 9 ; 100 ( 1 9 7 8 ) 253 - 259. 12 L. K r i s h n a m u r t h y a n d F. A. Williams, Acta Astronautica, 1 ( 1 9 7 4 ) 711 - 7 3 6 . 13 A. Lififin, Acts Astronautica, 1 ( 1 9 7 4 ) 1 0 0 7 1039. 14 K. Seshadri a n d F. A. Williams, J. Polym. Sci., Polym. Chem. Ed., 16 ( 1 9 7 8 ) 1 7 5 5 - 1 7 7 8 . 15 C. M. K i n o s h i t a , P. J. Pagni a n d R. A. Beier, O p p o s e d flow d i f f u s i o n f l a m e e x t e n s i o n s , Eighteenth Syrup. (Int.) on Combustion, in press, T h e C o m b u s t i o n I n s t i t u t e , P i t t s b u r g h , PA, 1 9 8 0 .
16 F. A. Williams, in M. V a n D y k e et al. (eds.), Annu. Rev. Fluid Mech., Vol. 3, A n n u a l Reviews, Palo A l t o , CA, 1 9 7 1 , p p . 171 - 188. 17 H. Tsuji a n d K. Matsui, Combust. Flame, 26 ( 1 9 7 6 ) 283 - 2 9 8 . 18 L. K r i s h n a m u r t h y , F. A. Williams a n d K. Seshadri, Combust. Flame, 26 ( 1 9 7 6 ) 363 - 3 7 8 . 19 J. S. Tien, S. N. Singhal, D. P. H a r r o l d a n d J. R. Prahl, Combust. Flame, 33 ( 1 9 7 8 ) 55 - 68. 20 D. E. Negrelli, J. R. L l o y d a n d J. L. N o v o t n y , J. Heat Transfer, 99 ( 1 9 7 7 ) 212 - 220. 21 C . M . K i n o s h i t a a n d P. J. Pagni, S t a g n a t i o n - p o i n t c o m b u s t i o n w i t h r a d i a t i o n , Eighteenth Syrup. (Int.) on Combustion, in press, T h e C o m b u s t i o n I n s t i t u t e , P i t t s b u r g h , PA, 1 9 8 0 . 22 K. Seshadri a n d F. A. Williams, Int. J. Heat Mass Transfer, 21 ( 1 9 7 8 ) 251 - 253. 23 A. T e w a r s o n , J. L. Lee a n d R. F. Pion, T h e influe n c e of o x y g e n c o n c e n t r a t i o n o n fuel param e t e r s for fire m o d e l i n g , Eighteenth Symp. (Int.) on Combustion, in press, T h e C o m b u s t i o n I n s t i t u t e , P i t t s b u r g h , PA, 1 9 8 0 . See also P r o d u c t s Res. C o m m i t t e e Rep. Materials Bank Compendium o f Fire Property Data, F e b r u a r y , 1980. 24 T. M. Shih a n d P. J. Pagni, Wake t u r b u l e n t flames, ASME paper No. 77-HT-97, 1977. 25 J. de Ris, Seventeenth Symp. (Int.) on Combustion, T h e C o m b u s t i o n I n s t i t u t e , P i t t s b u r g h , PA, 1 9 7 8 , pp. 1 0 0 3 - 1 0 1 6 . 26 J. K e n t , H. J a n d e r a n d H. Gg. Wagner, S o o t f o r m a t i o n in a l a m i n a r d i f f u s i o n flame, Eighteenth Symp. (Int.) on Combustion, in press, T h e C o m b u s t i o n I n s t i t u t e , P i t t s b u r g h , PA, 1980. 27 A. C. F e r n a n d e z - P e l l o , S.R. Ray a n d I. G l a s s m a n , F l a m e spread in a n o p p o s e d f o r c e d flow: t h e effect of ambient oxygen concentration, Eighteenth Syrup. (Int.) on Combustion, in press, T h e C o m b u s t i o n I n s t i t u t e , P i t t s b u r g h , PA, 1 9 8 0 . 2 8 A. Z e b i b , F. A. Williams a n d D. R. Kassoy, Combust. Sci. Technol., 10 ( 1 9 7 5 ) 37 - 44. 29 P. H. T h o m a s , M. L. Bullen, J. G. Q u i n t i e r e a n d B. J. M c C a f f r e y , Combust. Flame, 38 ( 1 9 8 0 ) 159 - 171. 30 P. H. T h o m a s , Fires a n d flashover in r o o m s -- a simplified t h e o r y , Fire Safety J., 3 ( 1 9 8 0 ) 67 - 76. 31 H. W. E m m o n s , in M. V a n D y k e et al. (eds.) Annu. Rev. Fluid Mech., Vol. 12, A n n u a l Reviews Palo Alto, CA, 1 9 8 0 , pp. 223 - 236. 32 H. W. E m m o n s , Seventeenth Symp. (Int.) on Combustion, T h e C o m b u s t i o n I n s t i t u t e , Pittsb u r g h , PA, 1 9 7 8 , p p . 1101 - 1 1 1 1 . 33 J. G. Q u i n t i e r e , in E. E. S m i t h a n d T. Z. H a r m a t h y (eds.), Design o f Buildings for Fire Safety, A S T M STP 685, A m e r i c a n S o c i e t y for T e s t i n g a n d Materials, P h i l a d e l p h i a , PA, 1 9 7 9 , p p . 139 - 168.
273
© E l s e v i e r S e q u o i a S . A . , L a u s a n n e - - P r i n t e d in t h e N e t h e r l a n d s
Diffusion Flame Analyses PATRICK
J. P A G N I
Mechanical Engineering Department and Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720 (U.S.A.) and Basic Research Department, Factory Mutual Research Corporation, Norwood, MA 02062 (U.S.A.)
SUMMARY
Nine classic diffusion flames, i.e., combustion systems with initially separate fuel and oxidizer, are synthesized in a search for comm o n dimensionless parameters which may serve as indices o f fire hazard. The problems examined are: planar and cylindrical B u r k e Schumann; droplet burning; planar and cylindrical stagnant film; and forced, free, mixed, and stagnation p o i n t combusting boundary layers. Similarity solutions in pyrolysis regions permit identification o f flame locations and quantification o f excess pyrolyzate, i.e., fuel which is not consumed locally. Numerical solutions in downstream regions give explicit expressions for flame extensions in terms o f the n u m b e r and the B number, with the former dominating. This key parameter, the physically available oxygen-to-fuel ratio divided by the chemically required oxygen-to-fuel ratio or inverse equivalence ratio, emerges as indicative o f the fire hazard associated with flame extension. Full scale tests have shown that flame extension is related to fire spread beyond the compartm e n t o f origin. Small r means large flame. Polystyrene with r = 0.12, for example, has an order o f magnitude longer flame than w o o d with r = 0.6. Other polymers fall between these extremes. Much progress remains to be made in the areas o f flame soot, radiation, turbulence, and compartment interactions.
INTRODUCTION
Fire is the primal diffusion flame. By "diffusion" is meant that the characteristic time for transport is much longer than the characteristic time for reaction, so t h a t diffu-
sion is rate controlling. Since transport requires orders of magnitude more molecular collisions than does reaction, all flames which have an oxidizer initially separated from a fuel are diffusion flames. The alternate of initially mixed is called a premixed flame. These two categories are all-inclusive. This paper attempts to review and synthesize some aspects of classical diffusion flame analyses which are pertinent to fire research. Problems of fire safety are complex and require multi-disciplinary solutions. Prevention through material standards plays a small but vital role in overall control strategies. The objective here is to find fundamental bases for fire hazard assessment standards. The motivation for this study came from two full-scale test house fires conducted by Professor Williamson of U.C. Berkeley in conjunction with the Sacramento, Califomia, Fire Department. The first test, as shown in Fig. 1, led to total involvement of the structure in less than 3 min. In the second test, the disaster depicted in Fig. 1 never occurred .... the fire self-extinguished within the room of origin. In each case a polyurethane mattress and bumper in a baby crib next to a gypsum interior wall was ignited with a match. The difference between these tests was that: in the first, the crib was made of molded polystyrene; in the second, the crib was made of wood. What is it that causes polystyrene to be so quickly destructive while wood, in a similar exposure, self-extinguishes? What is the set of material properties which distinguishes hazardous materials from nonhazardous materials? In all the diffusion flame studies considered, the following assumptions are made [1]. The fluid flow is steady, laminar, low-speed, twodimensional, and non-radiative; transport is governed by Fick's Law with the same diffu-
274
Fig. 1. House test fire ~<5 rain after m a t c h ignition of a p o l y u r e t h a n e mattress and b u m p e r in a polystyrene baby crib. Similar test with a w o o d e n baby crib self-extinguished.
sion coefficient for all species and a unit Lewis number. All reactions are fast. These assumptions were first introduced in 1928 by Burke and Schumann [2] who solved the problem of flame locations in concentric duct burners. Similar ideas were used b y Shvab [3] and by Zeldovich [4] in more detailed analyses of the temperature and velocity fields of gas burners. Spalding [5, 6] addressed the problem of fuel pyrolysis due to combustion released energy transport. Emmons [ 7 ] quantified fuel pyrolysis with an elegant, forced flow boundary layer analysis. This reference begins the modern era of diffusion flame studies with fire safety applications. Advanced treatment o f free flow [8 - 10] and mixed flow [11] b o u n d a r y layers followed. The constant thickness b o u n d a r y layer near a stagnation point [12 - 15] shows promise for extending diffusion flame studies to include slow kinetics [16 - 19] and radiation [20, 2 1 ] . This list is n o t meant to be definitive; it only follows a single thread within the broad fabric of flame research. Since synthesis is sought, emphasis is on c o m m o n elements in diffusion flame analyses.
The n e x t section presents a c o m m o n problem formulation which identifies k e y parameters and presents general results. Solutions to the Burke-Schumann problems in planar and cylindrical geometries are followed by the related problems of droplet and stagnantfilm burning. Four boundary-layer problems axe then considered: forced, free, mixed, and stagnation point. The concept of excess pyrolyzate is introduced. In the applications section, explicit flame height expressions are given which distinguish hazardous from nonhazardous materials. The conclusion consists of a summary and suggestions for future work.
COMMON
FORMULATION
Under the assumptions stated, the governing conservation equations are: Continuity:
(1)
V'fio~) = 0, M o mentum: pv.V_v - - V'(pV_v) = E ( P - -
P-) - - Vp,
(2)
275
Species: pJ,l " Il¥; -- 1l"(pDIl¥;) Energy: PJ,l" Ilh - Il"
=
C
Ilh) =
rh;", i
=
1... N,
(3)
J
q'",
'Y - 'Y~
= {3 -
'Y~
~w -
'Y w
-
(3~ .
(10)
~~
All of eqns. (3) and (4) reduce to
p
(5)
Energy-species variable
Simplification is gained by recognizing that eqn. (5) restricts the fuel, oxidizer and energy to . ,,, -m.", q. ", ·--mf o
(6)
Q i.e., fuel and oxygen must be consumed with heat release as specified by eqn. (5). This suggests thinking of oxygen as negative fuel [2] and defining, vfMf
=
(4)
where all symbols are defined in the nomenclature. Secondary effects such as Soret, etc., are discussed elsewhere [1] . The chemistry plays a very important role, even when reactions are fast, through the overall balance,
--=--=-,
ence between its maximum and minimum the same energy-species variable results,
voMo
(7) This variable does not change upon reaction and eqn. (3) for ~ becomes homogeneous and linear. The standard heuristic for nondimensionalizing ~ is to subtract {3's minimum value, {3~, the value in the region of highest oxygen concentration, and to divide by the difference between the maximum ~ and the minimum (3. Let (8)
where ~'s maximum value, ~w, occurs in the region of highest fuel concentration. This nondimensional fuel-oxygen variable goes from 1 at the fuel surface smoothly to 0 at ambiance. At the flame it has the value stipulated by Yf = Yo = 0, since fuel and oxidizer react upon contact. The same results are obtained if (9)
is used to combine eqns. (3) and (4) into a single, homogeneous, linear enthalpy-oxygen equation. An equivalent enthalpy-fuel equation provides no additional information. When 'Y is nondimensionalized by subtracting its minimum, 'Y~, and dividing by the differ-
pJ,l"IlJ - 1l"(pDIJJ)
=0
(11)
with J = 0 or 1 on the boundaries. Surface fuel mass fraction
The oxidizer boundary conditions are constant and known: Yc,~, T~, ~~, 'Y~. For a burner, Yfw is the initial fuel mass fraction and Tw = T~. For a pyrolyzing surface, the activation energy associated with pyrolysis is sufficiently large that the surface temperature, Tw , is constant and known. The remaining constant needed to define the maximum ~ is the surface fuel mass fraction, Yfw, which is specified a priori by the energy and fuel balances at the wall, " "L =X-ah m -I cp an w and
(12)
O"y;"" m ft = m Yfw - pD aYfI _.
an
,
(13)
w
respectively, where Yft, the fuel mass fraction in the transferred material, accounts for the inerts in the pyrolyzate. Solving for m" and equating gives X cpL
ah I -pD aYf I an w = Yft - Yfw an w'
(14)
From the definition of J, eqns. (7) - (10),
ah I ( QYc,~) aJ I an w = hw -- voMo an w
(15a)
and (15b) (Yfw + sYc,~) -aJ I . an w Recall Le = XjcppD = 1; substitute eqns. (15) -aYf
I
an w
=
into eqn. (14) to eliminate aJjan and obtain
(16) The left-hand side is B, Spalding's mass transfer number. Solving eqn. (16) for Yfw yields the fuel mass fraction at the wall
276
Yfw = (BYft
-sYo~)/(1
+ B).
(17)
Since Y ooo • Y ft • sand B are all constant and known, Yfw is constant and known a priori. General results Some additional consequences of choosing J as the dependent variable can be quantified. At the flame, the general expression, J
Yf (Yc, - Yc,~q / [Yfw Yc,~ ] = f vfMf voMo J vfMf + voMo (18)
reduces to
Since the reactions are all fast, there is no fuel outside the flame and no oxygen inside the flame. The enthalpy field is obtained from J = [_
r= - - - (20) YfwvoMo is the key parameter which emerges in this synthesis. It is the physically available oxygen-to-fuel mass ratio, Yc,~/Yfw, divided by the chemically required oxygen-to-fuel mass ratio, voMo /vfMf. It may be called a "mass consumption number" or "flame number" since it indexes the location and size of the flame in the same way that the mass transfer number indexes the surface pyrolysis rate. An analogy can also be made with premixed flames, where the equivalence ratio is defined as the actual fuel to air ratio divided by the stoichiometric fuel to air ratio. Here l/r plays the role of an equivalence ratio for diffusion flames. When r is small, the oxygen present is small compared with that needed, i.e., the system is fuel rich. Since J varies from 1 at the fuel boundary to 0 at the ambiance, as r decreases the flame moves away from the fuel toward the oxidizer. This is true for all diffusion flames. Additional results useful for all diffusion flames are obtained from the definitions of J, eqns. (10) or (18) as shown in Table 1.
(Yo - Yo~) ] / voMo
[_ hw + ~ ] . Q voMo (21)
The flame enthalpy is explicitly given by eqn. (21) with Yc, = 0 and J = (1 + r-1)-1, as shown in Table 1, where the dimensionless heat of combustion, Dc
(19)
~_
Q
= QYo~/voMohw,
(22)
enters only via the enthalpy field. These three dimensionless parameters: r, B and Dc dominate diffusion flame problems in the order listed.
PROBLEMS
Burke-8chumann Figure 2 shows the dimensional and nondimensional geometries of both the planar and cylindrical Burke-Schumann problems. Consider an infinitely long duct of dimension BURKE-SCHUMANN
J
0
~
J~l
J
~
0
y 0 00
,
~
x/b.
c
~
alb.
0 '" , '" 1
a .;;
rl
~
C'O.
,,=
ZD/vb
2
Fig. 2. Burke-8chumann problem geometries and definitions.
TABLE 1 General results for species and enthalpy fields in terms of J, r and Dc, valid for all diffusion flames; indicates fuel side and 1] ;;. 1]fl indicates oxidant side of reaction sheet
1] <;; 1]fl
1]
1]fl
o
o o
(1 + Dclr)J
(Dc + r)/(l + r)
(1 +r)J-r
o 1 - (1 + l/r) J Dc - (Dc -1)J
277
b with a semi-infinite duct of dimension a centered in it. Fuel flows through the inner duct and oxidizer in the outer duct with uniform velocity v. With the z origin at the downstream end of the inner duct, Yr = Yrw, Yo = Yo~ and T = T., at z = O. It is assumed that only axial flow and transverse diffusion exist and that pv and pD are each uniform. Under these restrictions, only eqn. (11) is required in the form
(02 J2 -!:- aJ) = O.
v oJ -D
az
ax
(23)
ax
x
Now let 17 = zDlvb 2, ~ = xlb and e = alb so that aJ a2 J K OJ -=-+--
a17
a~2
~ a~
(24)
,
where K = 0 for planar and K = 1 for cylindrical systems. The range of interest is 17 ~ 0 and ~ ~ 0 by symmetry. The boundary conditions are no flux at ~ = 0 and ~ = 1 with the unburned condition at 17 = 0: JC~,
0) = 1,
for 0
J(~,
0)
for e <
~ ~ ~
e, (25)
=
0,
~ ~
1.
07r------.---------, OVERVENTllATED
iJNDE R VENT I LATED
c
~
.0.4
....
--- 01
I
-
~ 0.3
- - 0,9
w
05
I
w 02
" 0.1
~
10 VENTILATION, dC-i_I)
Fig. 3. Flame heights for planar problem.
Burk~chumann
defines critical ventilation; values < 1 are underventilated, values> 1 overventilated. Figure 3 shows planar flame heights for three geometries versus the ventilation parameter. At the ventilated condition, infinitely high flame sheets are predicted. If the system is overventilated, the flame will reach its maximum extension on the centerline; if underventilated, the flame touches the ~ = 1 surface. To make a connection to fires, note that as b ~ 00, e ~ 0, and the ventilated condition is approached as r ~ O. So, small r means large flames, even in this simplest diffusion flame problem.
The cylindrical solution [1,2] is Droplet burning
J(t n) = e 2 + 2e " ~ JIcrne)JoCrn~) X <;, '/ n~ 1 [Jocrn)] 2
rn
X exp(-r~17),
where I n are Bessel functions of the first kind and r n are the positive roots of J1 Cr n) O. The planar solution [2] is JC~, 17) = e
+2
L
(26)
=
(mr)-l sin (n1Te) X
n=l
(27) The flame location is given by eqn. (19) and the species and enthalpy fields are as shown in Table 1. The ventilated condition occurs when the initial mass flux of fuel and oxidant are in stoichiometric proportion,
Yrw aK + 1
vfMf
-----:------:-- - - Yo~CbK+l-aK+l)
voMo In dimensionless form, r(e-(K+l)
-1)
=1
The mass transfer number does not appear in the Burke-Schumann problem because the fuel mass flux is under external control. In the droplet burning problem [1,5,6], heat transfer to the drop surface dictates the fuel mass flux. It is assumed that the drop is small enough to be uniform inside and spherically symmetric outside and yet large compared with the reaction zone. The drop radius is the only time dependent quantity, i.e., the system is quasi-steady. Here pD is constant a removable approximation [1]. The quasisteady mass flux is also a constant, = 41TX 2pV, where x, the radial coordinate, is the only independent variable. Equation (11) becomes
m
'dJ dx
=
~ l(41TP~X2) dJ] dx
m
.
(30)
dx
By defining (28)
(29)
m =, (31) 41TpDx a dimensionless mass flux which maps x = 00 into ~ = 0, the problem for J(~) reduces to
~
278
J" + J' J(~w)
=
0,
= 1,
(32)
J(O)
= 0;
(33)
with the solution,
= (1 -
J(~)
e-~)/(I- e-~w).
(34)
All species and temperature fields can be obtained from eqn. (34) and Table 1. The surface energy balance, eqn. (12), can now be used to find the eigenvalue, appearing here in the guise of ~w' In terms of J, eqn. (12) is
m,
mL
-"1\ (
--=-
41TX;'
cp
Y..,Q ) dJ -hw + - lJoMo
d.x
I
(35)
w'
U sing the definitions of ~ and B, Le = 1 and eqn. (34), eqn. (35) becomes
~ = dJ d~
B
I = (e~w -
1
r 1.
(36)
= In (1 + B);
(37)
m= 41TpDxwln (1 + B).
(38)
The flame standoff distance, using eqns. (19), (31), (34) and (37) is
~w
In [(1 + r)(1 + B) ] 1 +r +B In (1 +B)
(39)
Note that as r ~ 0, the equivalence ratio becomes large, ~fl ~ 0 and Xfl ~ 00. Although the semi-infinite, cylindrical and planar droplet burning problems do not have steady state solutions, the closely related finite domain "stagnant film" problems do. Consider a one-dimensional planar film between two infinite screens. Beyond the screen at x = 0 is a stagnant oxidizing gaseous ambiance. Beyond the screen at x = X w is a solid or liquid fuel reservoir. Assume steady, laminar combustion occurs within the film and the mass flux, mil = pv, is constant for all x. Let pD be constant and J have its usual definition, eqns. (10), (18) and (21). With ~=
rh"x/pD,
(40)
the governing equation, conditions, and solution are given by eqns. (32), (33) and (34), respectively. The mass pyrolysis rate is given by eqn. (37) so that
(41)
The profiles given by Table 1 and eqn. (34) are linear only in the limit B <{ 1. The flame location is given by eqn. (39), so as r ~ 0 the flame moves to the oxidant screen. In the cylindrical problem, which resembles a candle wick, the origin is at the wick center. The fuel pyrolyses at X w and the oxidant screen is at x~. It is assumed that m' = 21TXPV is constant. Other assumptions are the same as in the planar problem and eqn. (11) reduces to
dJ = ~ [(21TPD~) dJ] . (42) d.x d.x d.x Nondimensionalize the radius on its maximum value x~, and define
m'
~ =
so that the droplet pyrolysis rate is
~fl =
pD
w
From eqn. (36) the eigenvalue is ~w
•
mil = -In (1 + B). Xw
-m'ln(x/x~)
(43) 21TpD The governing eqn. (42), boundary conditions, and solution, again reduce to eqns. (32), (33) and (34), respectively. Equation (37) gives the pyrolysis rate,
• m'
=
21TpD In(l + B) In(x~/Xw)
(44)
,
while eqn. (39) gives the flame location. Again, as r ~ 0, the flame moves to the oxidant screen, Xfl ~ x..,.
Boundary layer combustion Now consider four boundary layer problems -- forced, free, mixed, and stagnation point flow. Although the energy-species eqn. (11) simplifies by the usual boundary layer approximation to pu
OJ + pv aJ = ~ (PD OJ),
ax
ay ay
ay
(45)
these systems are more complex, since the continuity, eqn. (1), and the x-momentum, eqn. (2),
au
au a ( au) ---g(p-p~), ap ay ay. ay ax (46)
pu-+pv-= -- 11-
ax
must be considered in detail.
Forced flow Emmons' study [7] of forced boundary layer combustion could serve as the model research paper. It addresses a complex
279
FLAME ~ SHEET / g
/
U - O@/O~,
I
V-= - ~ @ / ~ ,
f ( ~ ) -- ~/~112.
(51b) (51e)
With T = O, eqns. (51) condense eqns. (48) -
i/
(50) to
J" + erfJ' = 0
(52)
and u®
T
T
Q
y-
f'" + ff" = O.
E m m o n s further showed t h a t if Pr = 1, f' = 2 (1 - - J ) and eqn. (52) becomes identical with eqn. (53), so t h a t the combustion problem reduces to the incompressible Blasius problem. The boundary conditions are
y~
Fig. 4. Boundarydayer combustion
problem
geometries and definitions.
problem, introduces physically valid assumptions, simplifies the mathematics, obtains exact and approximate solutions, explores behavior in limits, and compares with experiments. The treatment given here is abbreviated to encourage the reader to enrich himself from the original. Boundary layer flame schematics defining geometries and n o t a t i o n are shown in Fig. 4. Assume the property product, pg, is constant. Define the dimensionless variables Re1/2 /
f - x / l , ~=
l
o
P~ (47)
Op ) -~x dy ,
which reduce eqns. (1), (45) and (46) to OU --
OV + --
O,
o~-
u/u~ = f'/2,
(55)
f'(0) = 0, f'{oo)= 2.
(56)
The energy balance at the surface, eqn. (12), becomes
f(O) = BJ'(O)/Pr,
(57)
since by definition
rh"(x) - 2p~u~Re~/2
(58)
The solution to eqns. (52) and (53), subject to eqns. (54), (56) and {57), provides the velocity field from eqn. (55) and the energy and species fields from Table 1. Of interest is the pyrolysis rate, eqn. (58), which can be made an explicit function of B alone using a curve-fit obtained at Pr = 0.73, (59)
(48)
1 82J Pr ~ 2 ,
0U OU 02U U -- + V + r of o~ o~ 2
(49)
(50)
where r is a dimensionless force due to gravity or an external pressure gradient. Along the pyrolyzing surface, forced flow similarity is introduced via the definitions, -= ~/(4f) 1/2 ,
(54)
and since
--f(0) = 0.47 B 0"578-0"0971nB =
0J 0J U -- + V of
J(0) = 1, J(oo) = 0,
--f(0)
P~ dy,
U_ u/u®, v = Rel/2 ( / l pV+Uo
(53)
(51a)
The flame location within the boundary layer, ~n, is determined by r from eqn. (19) for Jn (nn). A connection may be made with the Burke-Schumann problem. The pyrolyzing surface provides some fuel which is consumed locally and some which is convected downstream between the flame and the surface. This convected fuel plays the same role at the end of the pyrolyzing slab as does the BurkeSchumann fuel at the end of the inner duct. It determines the flame height. It is convenient to calculate this "excess p y r o l y z a t e " ,
280
"g/e, as a fraction of the total fuel pyrolyzate, Yft /~/p, [10]. The downstream fuel flux is ]~/e(X) =
(pU Yf)~ dy.
(60)
0
The total fuel pyrolyzed over 0 ~
(61)
with ~ given by eqns. (51b). The energy species and m o m e n t u m eqns. (49) and (501} reduce to J " + 3PrfJ' = 0,
(65)
f"' + 3ff"-- 2f '2 + ~ = 0.
(66)
The conditions on J are given by eqns. (54) and the velocity conditions are f'{0) = 0 and f'(~o) = 0.
Nondimensionalizing eqn. (60) and dividing by eqn. (61) gives the excess pyrolyzate as a fraction of the total fuel pyrolyzate,
The surface energy balance gives
----
since by definition,
Yft/~/p 1 + (B ~ l)(r-I)
X
I(l + r-l) ,n
f
o
f0?n) +
11.
f(o---T--
(62)
Note that the Re~ dependence in/9/p and Me cancel, so that the ratio is constant along the pyrolyzing surface. Equation (62) holds for all four combusting boundary layer problems. Since f' ~ u and J ~ Y~, the integral is ~Me and --f(0) is ~/~/p. The non-integral terms in the bracket are negligible. The B and r-~ dependencies in the coefficient, both generally >1, effectively cancel. The primary dependence on r arises from t h e upper limit of the integral. As r decreases, Jfl decreases and ~n increases, allowing more fuel to convect downstream.
Free flow The corresponding problem for free flow is identical except for the b u o y a n c y term in the m o m e n t u m equation. Using the nondimensional variables ~ - x//,
~=-l
u-=
Gr~/2------~ 'ul
v =
-f(O) =
21/2xrh" (x ) 3p®v~Gr~14
(68)
(69)
The solutions to eqns. (65) and (66), subject to these conditions, give the energy and species fields from Table 1 and the velocity field from u = 2~Grxl/2f'(~)/x.
(70)
The local pyrolysis rate, eqn. (69), is made explicit via the curve fit, --f(0) = 0.2 r-°'°45B °:767-0"0591nB ,
(71)
The weak r dependence derives from ~ in the m o m e n t u m eqn. (66). Though eqn. (71) was obtained at Pr = 0.73 and De = 4.0, it is a good approximation for all Pr and Dc of interest. The flame location is again given by eqn. (19). The excess pyrolyzate formulation is identical with the forced case, so eqns. (60) and (62) are valid provided the total fuel pyrolysis rate is given by Yft/l}/p(X) = --f(O)P~U~23/2 Gr~/4 Yft-
(72)
Note that the excess fraction of the total pyrolyzate is independent of Gr~ and only a function of r and B.
-P-dy, 0 P~ Gr~ 41p~v®(p v
f(0) = BJ'(O)/3Pr,
(67)
+ u.
f~' -OxOPdyl,]
0
(63) gives the same eqns. (48) and (49) with now, T = ¢ = h/h~ from Table 1 in eqn. (50). The free flow similarity variables are r/ = ~'/(4~) TM ,
(64a)
f(r/) = ~/v~ G r 1/4 (4~) 3/4
(64b)
Mixed flow The preceding limiting cases are valid for Froude numbers, Fr -= Re2/Gr, >>1 and 41, respectively. Intermediate values of Fr probably more accurately represent flows induced by c o m p a r t m e n t fires. The forced flow dimensionless variables, eqns. (47), are used since t h e y give simpler boundary conditions [11]. The governing eqns. (I), (45) and (46) reduce to eqns. (48), (49) and (50)
281
with ~ = ¢/Fr. A similarity solution does n o t exist, and a local similarity solution proved inaccurate, so that direct numerical solution of eqns. (48) - (50) is required with eqn. (54) and U(~, 0) = 0,
V(~, 0)
B~J .
.
.
.
Pr~
U(0, ~) = U(~, co) = 1
(~, 0),
(73)
i°I r=O.I
o
. . . . . . . . . . - -
~_ 0.5
~
.~
Free: Dc=4 S t a g n a t i o n : Oc = 4 , 0 w = 2 Forced
(74) w
as b o u n d a r y conditions. The results [11] indicate that the free limit holds for Fr < 1, while the forced limit is reached at Fr > 102. Smooth interpolation between limits is possible with linearity increasing as r increases. Stagnation point
A constant thickness b o u n d a r y layer develops in the vicinity of a stagnation point due to the acceleration of u~ away from the origin. If the oncoming flow is an oxidizer and the surface is a pyrolyzing fuel, a useful system for flame studies is obtained. Only the planar geometry, with x along the surface and y normal to it, is discussed here. The axisymmetric geometry has been described in detail [ 1 2 - 15]. Several approximations exist for the free stream [15, 22] ; each has u~ linear in x so that --~p/ax = p=u~u~/~x
(75)
in eqn. (46) is linear in x. Using the forced flow eqns. (47) reduces the governing eqns. (1), (45) and (46) where g = 0 to eqns. (48) (50) with -r = 0 -~ T/T~ = 1 + ¢(0w -- 1)
(76)
given by Table 1, assuming cp approximately constant. A similarity problem arises from eqn. (51b) and (77a)
_-- ~/~1/2,
f07) = ~/P~(Ret~) 1/2 .
(77b)
Then eqn. (49) reduces to eqn. (52) and eqn. (50) becomes f,, + ff,, _
f , 2 + 0 = O.
_ _ ~ __2 ~) 0.1
I
--i I
I
l I I Ill _ i.O Mass
Transfer
l No.,
I
I
I I I I J I0
B
Fig. 5. Excess p y r o l y z a t e as a fraction of total fuel p y r o l y z a t e vs. B n u m b e r parameterized in r n u m b e r for three boundary-layer c o m b u s t i o n p r o b l e m s with P r = 0.73. These results are general, i . e . , indep e n d e n t of Re and Gr and a p p r o x i m a t e l y applicable for all parameter values.
~"(x) --f(0)
p.~u~Re1/2 ,
(79)
which is obtained explicitly from a curve fit to numerical results at Pr = 0.73, De = 4 and 0w = 2 a s --f(0) = 0.543
r-°'°2B
0"626-0"06781nB
.
(80)
The dependence on D e and 0w is negligible for this nonradiating system. The excess pyrolyzate is again given by eqn. (62) with the total fuel pyrolysis rate given by eqn. (61). Figure 5 shows the excess pyrolyzate for all the boundary layer problem systems examined. As this inclusive plot shows, the excess fraction of the total pyrolyzate is dominated by r with only a weak dependence on B. The very weak dependence on Pr, D c and 0w may be neglected. There is no dependence on scale through x, Re or Gr. The practical consequence is that 1 / r serves as an index of the u n b u r n e d pyrolyzate convected downstream to fuel flames remote from their source.
(78)
The b o u n d a r y conditions are identical with eqns. (54), (56) and (57), except t h a t f'(oo) = 1 since u / u ~ - f ' . The solution to eqns. (78) and (52) for f and J gives the velocity field and the energy and species fields from Table 1. The mass pyrolysis rate is
APPLICATIONS
Flame heights
What happens b e y o n d the end of the pyrolyzing surface, x > l, is important to fire safety applications and is determined by
282 TABLE 2 E x p l i c i t curve-fits f o r flame e x t e n s i o n b e y o n d p y r o l y z i n g surface in u n i t s o f t h e fuel slab l e n g t h for forced, free, a n d m i x e d , s t e a d y , l a m i n a r b o u n d a r y - l a y e r s in t w o g e o m e t r i e s F o r c e d flow Xfl ~ 0.14 [(1 + r - l ) ( 1 + B -1) B - ° ' 1 5 l n (1 + B ) ] 2
wall-mounted : free-standing:
Xf*l ~ 0.72 Pr- 0.2 X~l_wall
Free flow X~I ~ 0.24 [(1 + r - 1 ) r --0"16 (1 + B - 1 ) B - ° ' ° 6 1 n (1 + B ) Dc-°'°5] 4/3
wall-mounted : free -standing:
X~l
* 0.9 Xfl.wall
M i x e d flow * ~ --0.2 ~* Xfl ~ (1 -- Fr-O'2)X~l.forced + r r Aft.fre e * Xfl ~ Xfl-free *
F r ~> 1: Fr~
returning to the first diffusion flame problem and treating the excess pyrolyzate as the fuel input to a Burke-Schumann-like flame location problem with more complicated fluid mechanics. The abrupt change in b o u n d a r y conditions at x = l prohibits similarity, so eqns. (48) (50) are solved numerically with t w o alternative sets of b o u n d a r y conditions representing, respectively, a wall-mounted or freestanding fuel slab: wall: or
V(0)
=
Y(0)
=
~}J(0)/d~
=
0
POLYSTYRENE
POLYURETHANE
POLYMETHYLMETHACRYLATE
POLYOXYMETHYLENE
CELLULOSE
2
4
6
8
;0
12
14
16
18
2:0 22
24
DIMENSIONLESS FLAME EXTENSION, ×fit-xfit/~ Fig. 6. P r e d i c t e d flame h e i g h t s for c o m m o n fuels.
(80a)
centerline: a U ( O ) / ~ = V(O) = aJ(O)/a~ = O. (SOb) In each of the four b o u n d a r y layers, the flame location is given b y eqn. (19), J f l ( ~ n , ~fl) = ( 1 + r-l) -1. All of the fuel is assumed consumed when the flame sheet intersects = 0. The value o f ~fl there gives the flame height, X ~ - ~fi-- 1, i.e., the flame extension b e y o n d the fuel slab in units of the slab length. Curve-fits for the first three b o u n d a r y layer problem results are given in Table 2; the fourth is plotted in ref. 15. Fire safety
The expressions in Table 2 suffice to explain w h y the polystyrene crib fire p r o d u c e d the totally involved structure shown in F i g . 1, while the w o o d e n crib self-extinguished. Figure 6 shows a bar graph o f flame heights for several c o m m o n fuels c a l c u l a t e d using Tewarson's properties [23] in the curvefits of Table 2. Polystyrene has a flame height
of 24, while w o o d has a flame height of 2. This means that if the cribs were 1 m high, the w o o d fueled flames would just reach a 3 m high ceiling, while the polystyrene flames would extend 25 m up the wall, across the ceiling, o u t the d o o r and d o w n the corridor igniting any combustible materials along the way via radiation and convection from the combusting gases. This scenario is dramatically visible in films of the Sacramento house fires. Other synthetic polymers fall between these extremes. While turbulence and quenching tend to decrease the flame length, the distinction among fuels remains [ 2 4 ] . What is it a b o u t the chemistry of polystyrene which leads to its extreme fire performance? Table 3 shows the detailed properties of six polymers. Note that s, the mass of fuel c o m b u s t e d per mass of oxygen, for polystyrene is a b o u t one third o f s for cellulose. This means that for a given a m o u n t of fuel, three times more oxygen is required and
283 TABLE 3 Predicted flame extensions and ideal material properties for wall-mounted common polymer slabs with an ambiance of T~ = 293 K and Yo~ = 0.23 The GM designation refers to the Products Research Committee general material sample bank available through the National Bureau of Standards. Material (type)
Formula
Polystyrene (foam, GM-49) Polystyrene (granular) Polyurethane (foam, GM-25) Polymethylmethacrylate (granular) Polyisocyanurate (foam, GM-41) Polyoxymethylene (granular) Cellulose (filter paper)
C8H8.400.03 CsH 8
s
wake
11
24
1.3
1.7
0.6
0.12
4.8
0.33
1.7
1.3
0.5
0.14
4.1
9.3
18
C3.1H5.4ON0.22 0.46
1.2
1.5
0.6
0.19
3.6
6.5
11
C5H802
0.52
1.6
1.5
0.6
0.22
5.1
5.2
8.9
C5.3Hs.2ON0.57
0.43
4.5
0.4
0.2
0.43
4.0
1.8
2.8
CH20
0.94
2.4
1.1
0.4
0.50
5.8
1.8
2.4
C6H1005
0.84
3.5
0.8
0.3
0.60
6.9
1.4
1.8
N i n e classic d i f f u s i o n f l a m e p r o b l e m s h a v e been examined: Burke-Schumann -- planar a n d cylindrical; d r o p l e t b u r n i n g a n d s t a g n a n t film - - p l a n a r a n d c y l i n d r i c a l ; b o u n d a r y l a y e r c o m b u s t i o n - - f o r c e d , free, m i x e d , a n d p l a n a r s t a g n a t i o n p o i n t . I t was a s s u m e d t h a t e a c h system had: steady, low-speed laminar flow w i t h u n i t y Le, s p e c i e s - i n d e p e n d e n t D , F i c k i a n d i f f u s i o n , n o r a d i a t i o n , a n d fast c h e m i c a l
Dc
X~l
0.33
CONCLUSIONS
r
plume
B
larger PS f l a m e surfaces are d e v e l o p e d t o p r o v i d e t h e n e e d e d o x y g e n . In a d d i t i o n , t h e n e x t c o l u m n s h o w s a significant d i f f e r e n c e in h e a t o f p y r o l y s i s , t h e e n e r g y r e q u i r e d t o v a p o r i z e fuel. P o l y s t y r e n e ' s L is a p p r o x i m a t e l y o n e t h i r d t h a t o f w o o d . So, f o r a given h e a t i n p u t t o t h e solid, t h r e e t i m e s as m u c h gaseous PS fuel is p r o d u c e d . T h e s e t w o factors provide the chemical and physical bases f o r t h e o r d e r o f m a g n i t u d e l o n g e r PS flames. T h i s e f f e c t is q u a n t i f i e d b y t h e f a c t t h a t t h e r o f PS is o n e - f i f t h o f t h e r o f cellulose. A l t h o u g h t h e d r a m a t i c crib fire accentuates the hazard of polystyrene furniture units, t h e m o r e useful p r a c t i c a l c o n s e q u e n c e is t h e e m e r g e n c e o f r as an i n d e x o f fire h a z a r d - - small r, %1/3, m e a n s large h a z a r d .
Yfw
X~l
L (k J/g)
r e a c t i o n s . T h e f o l l o w i n g c o n c l u s i o n s are drawn from the solutions obtained: (1) m a n y similarities e x i s t a m o n g d i f f u s i o n flame problems; (2} t h e single m o s t i m p o r t a n t d i m e n s i o n l e s s p a r a m e t e r , w h i c h d o m i n a t e s f l a m e shapes, heights, a n d excess p y r o l y z a t e , is r ~- (Yo~/ Y~w)(vfMf/vo Mo), a m a s s c o n s u m p t i o n n u m b e r or inverse e q u i v a l e n c e r a t i o f o r d i f f u s i o n flames; (3) n e x t in i m p o r t is t h e m a s s t r a n s f e r number, B - (QYo~/voMo -- hw)/L, which d o m i n a t e s p y r o l y s i s rates; (4) f l a m e t e m p e r a t u r e s are c o n t r o l l e d prim a r i l y b y D e -- Q Y o ~ / v o M o h w , a d i m e n s i o n less h e a t o f c o m b u s t i o n ; (5) t h e s e t h r e e p a r a m e t e r s , as t h e pertinent, quantitative, material chemical and p h y s i c a l p r o p e r t i e s , are indices o f fire h a z a r d . All o f t h e i n f o r m a t i o n r e q u i r e d t o e v a l u a t e t h e s e p a r a m e t e r s f o r n e w m a t e r i a l s can be o b t a i n e d f r o m small-scale l a b o r a t o r y a p p a r a t u s . T o b e free f r o m h a z a r d , t h e m a t e r i a l w o u l d h a v e large r a n d small B a n d D e . T h e analyses d e s c r i b e d here, w i t h i n t h e constraints of their assumptions, provide the r e q u i s i t e c o n n e c t i o n b e t w e e n fuel p r o p e r t i e s and dynamic flame behavior to permit a p r i o r i a s s e s s m e n t o f s o m e f o r m s o f fire hazard.
284
Future work Radiation is a crucial c o m p o n e n t of most flames of candle size and above [25]. Some progress is occurring [20, 21] toward incorporating radiative terms in these analyses. But an understanding of soot formation and evolution [26] within diffusion flames is necessary before flame radiation can be predicted with the same generality as pyrolysis rates. Turbulence, which plays a key role in all fires, is n o t understood even in nonreacting systems. Non-boundary layer systems, such as candles or pool fires, have not been solved. Quasi-steady flame spread also awaits more detailed explanation [27]. Improvements may be made by eliminating assumptions, e.g., variable or non-unity Le, non-uniform D and multiple flame sheets. Slow, one-step kinetics have been introduced in extinction studies [ 16 - 19]. Extension to multi-step mechanisms [28] and radiative extinction remain as challenges. Finally, these problems address only the burning of a single fuel source in an open ambiance. The geometric and fluid mechanic complexities of interactions with surrounding compartments must be addressed [29 - 33] before predictive capability can be said to exist for fire behavior as a threat to property and life safety.
Dc
Dimensionless heat of combustion,
f Fr g Gr x
Dimensionless stream function Froude number, Re~/Grl Acceleration of gravity Grashof number, g(Tw -- T~)x:~/v2T~
h
Specific enthalpy, f
QYo~/.oMohw
T
c, dT
T~
J
Normalized energy-species variable, (~ - - ~=)/(~w
-
~)
=
(7
--
~)1(7~
-
~)
L Le l m Mi Pr Q
Effective latent heat of pyrolysis Lewis number, k/cppD Fuel length Mass Molecular weight of species i Prandtl number Energy released by combustion of vf moles of gas phase fuel q Heat Re~ Reynolds number, u~x/u~ r Mass consumption number, Yo.s/Y~w s Stoichiometric ratio, vfMf/uoMo T Temperature U Dimensionless streamwise velocity u Streamwise velocity V Dimensionless transverse velocity v Transverse velocity x Streamwise coordinate y Transverse coordinate Yi Mass fraction of species i
ACKNOWLEDGMENTS
It is a j o y to contribute this compendium to a volume honoring P. H. Thomas. This study is based on research sponsored, in large part, by the Center for Fire Research of the National Bureau of Standards, United States Department of Commerce. Writing was completed while the author held the position of visiting scientist at Factory Mutual Research Corporation. Stimulating conversations there with J. de Ris are much appreciated. The author gratefully acknowledges H. W. Emmons, the founder of fire research in the United States, for the energy, encouragement, and excellence he has brought to the field. NOMENCLATURE
B % D
Mass transfer number, (QYo~/voMo --hw)/L Specific heat Species diffusivity
Symbols Species coupling variable, Y~Iv~M~--- YolvoMo ~/ Energy-species coupling variable, - - h / Q - - Yo/voMo
0 K p v
p ¢
Dimensionless transverse coordinate Similarity variable Dimensionless temperature, T/T® Flow factor (0, planar; 1, cylindrical) Conductivity Dynamic viscosity Kinematic viscosity or stoichiometric coefficient Dimensionless streamwise coordinate Density Dimensionless enthalpy, h/hw Stream function
Subscripts f Fuel fl Flame o Oxidizer
285
t w co
T r a n s f e r r e d gas F u e l surface Ambient
Superscripts * M e a s u r e d f r o m d o w n s t r e a m edge o f fuel surface Per u n i t t i m e ' Per u n i t length or derivative w i t h r e s p e c t t o t h e i n d e p e n d e n t variable REFERENCES 1 F. A. Williams, Combustion Theory, A d d i s o n Wesley, R e a d i n g , MA, 1 9 6 5 . 2 S . P . B u r k e a n d T. E. W. S c h u m a n n , First Symp. (Int.) on Combustion~ T h e C o m b u s t i o n I n s t i t u t e , P i t t s b u r g h , PA, 1 9 2 8 , pp. 2 - 11. 3 V. A. S h v a b , Relation Between the Temperature
and Velocity Fields o f the Flame o f a Gas Burner, Gosenergoizdat, Moscow-Leningrad, 1948. 4 Y. B. Z e l d o v i c h , Zh. Tekh. Fiz., 19 ( 1 0 ) ( 1 9 4 9 ) 1 1 9 9 - 1 2 1 0 (Engl. trans. NACA Tech. Memo. 1296, 1 9 6 0 ) . 5 D. B. Spalding, Fourth Syrup. (Int.) on Combustion, T h e C o m b u s t i o n I n s t i t u t e , P i t t s b u r g h , Pa, 1 9 5 3 , pp. 847 - 8 6 4 . 6 D. B. Spalding, Fuel, 29 ( 1 9 5 0 ) 2 5 ; 32 ( 1 9 5 3 ) 169. 7 H. W. E m m o n s , Z. Angew. Math. Mech., 36 (1) ( 1 9 5 6 ) 60 - 71. 8 F. J. K o s d o n , F. A. Williams a n d C. B u m a n , Twelfth Symp. (Int.) on Combustion, T h e C o m b u s t i o n I n s t i t u t e , P i t t s b u r g h , PA, 1 9 6 8 , pp. 253 - 264. 9 J. S. Kim. J. de Ris a n d F. W. Kroesser, Thirteenth Symp. (Int.) on Combustion, T h e C o m b u s t i o n I n s t i t u t e , P i t t s b u r g h , PA, 1 9 7 0 , pp. 9 4 9 - 961. 10 P . J . Pagni a n d T. M. Shih, Sixteenth Symp. (Int., on Combustion, T h e C o m b u s t i o n I n s t i t u t e , Pittsb u r g h , PA, 1 9 7 6 , pp. 1 3 2 9 - 1 3 4 2 . 11 C. M. K i n o s h i t a a n d P. J. Pagni, J. Heat Transfer, 102 ( 1 9 8 0 ) 104 - 1 0 9 ; 100 ( 1 9 7 8 ) 253 - 259. 12 L. K r i s h n a m u r t h y a n d F. A. Williams, Acta Astronautica, 1 ( 1 9 7 4 ) 711 - 7 3 6 . 13 A. Lififin, Acts Astronautica, 1 ( 1 9 7 4 ) 1 0 0 7 1039. 14 K. Seshadri a n d F. A. Williams, J. Polym. Sci., Polym. Chem. Ed., 16 ( 1 9 7 8 ) 1 7 5 5 - 1 7 7 8 . 15 C. M. K i n o s h i t a , P. J. Pagni a n d R. A. Beier, O p p o s e d flow d i f f u s i o n f l a m e e x t e n s i o n s , Eighteenth Syrup. (Int.) on Combustion, in press, T h e C o m b u s t i o n I n s t i t u t e , P i t t s b u r g h , PA, 1 9 8 0 .
16 F. A. Williams, in M. V a n D y k e et al. (eds.), Annu. Rev. Fluid Mech., Vol. 3, A n n u a l Reviews, Palo A l t o , CA, 1 9 7 1 , p p . 171 - 188. 17 H. Tsuji a n d K. Matsui, Combust. Flame, 26 ( 1 9 7 6 ) 283 - 2 9 8 . 18 L. K r i s h n a m u r t h y , F. A. Williams a n d K. Seshadri, Combust. Flame, 26 ( 1 9 7 6 ) 363 - 3 7 8 . 19 J. S. Tien, S. N. Singhal, D. P. H a r r o l d a n d J. R. Prahl, Combust. Flame, 33 ( 1 9 7 8 ) 55 - 68. 20 D. E. Negrelli, J. R. L l o y d a n d J. L. N o v o t n y , J. Heat Transfer, 99 ( 1 9 7 7 ) 212 - 220. 21 C . M . K i n o s h i t a a n d P. J. Pagni, S t a g n a t i o n - p o i n t c o m b u s t i o n w i t h r a d i a t i o n , Eighteenth Syrup. (Int.) on Combustion, in press, T h e C o m b u s t i o n I n s t i t u t e , P i t t s b u r g h , PA, 1 9 8 0 . 22 K. Seshadri a n d F. A. Williams, Int. J. Heat Mass Transfer, 21 ( 1 9 7 8 ) 251 - 253. 23 A. T e w a r s o n , J. L. Lee a n d R. F. Pion, T h e influe n c e of o x y g e n c o n c e n t r a t i o n o n fuel param e t e r s for fire m o d e l i n g , Eighteenth Symp. (Int.) on Combustion, in press, T h e C o m b u s t i o n I n s t i t u t e , P i t t s b u r g h , PA, 1 9 8 0 . See also P r o d u c t s Res. C o m m i t t e e Rep. Materials Bank Compendium o f Fire Property Data, F e b r u a r y , 1980. 24 T. M. Shih a n d P. J. Pagni, Wake t u r b u l e n t flames, ASME paper No. 77-HT-97, 1977. 25 J. de Ris, Seventeenth Symp. (Int.) on Combustion, T h e C o m b u s t i o n I n s t i t u t e , P i t t s b u r g h , PA, 1 9 7 8 , pp. 1 0 0 3 - 1 0 1 6 . 26 J. K e n t , H. J a n d e r a n d H. Gg. Wagner, S o o t f o r m a t i o n in a l a m i n a r d i f f u s i o n flame, Eighteenth Symp. (Int.) on Combustion, in press, T h e C o m b u s t i o n I n s t i t u t e , P i t t s b u r g h , PA, 1980. 27 A. C. F e r n a n d e z - P e l l o , S.R. Ray a n d I. G l a s s m a n , F l a m e spread in a n o p p o s e d f o r c e d flow: t h e effect of ambient oxygen concentration, Eighteenth Syrup. (Int.) on Combustion, in press, T h e C o m b u s t i o n I n s t i t u t e , P i t t s b u r g h , PA, 1 9 8 0 . 2 8 A. Z e b i b , F. A. Williams a n d D. R. Kassoy, Combust. Sci. Technol., 10 ( 1 9 7 5 ) 37 - 44. 29 P. H. T h o m a s , M. L. Bullen, J. G. Q u i n t i e r e a n d B. J. M c C a f f r e y , Combust. Flame, 38 ( 1 9 8 0 ) 159 - 171. 30 P. H. T h o m a s , Fires a n d flashover in r o o m s -- a simplified t h e o r y , Fire Safety J., 3 ( 1 9 8 0 ) 67 - 76. 31 H. W. E m m o n s , in M. V a n D y k e et al. (eds.) Annu. Rev. Fluid Mech., Vol. 12, A n n u a l Reviews Palo Alto, CA, 1 9 8 0 , pp. 223 - 236. 32 H. W. E m m o n s , Seventeenth Symp. (Int.) on Combustion, T h e C o m b u s t i o n I n s t i t u t e , Pittsb u r g h , PA, 1 9 7 8 , p p . 1101 - 1 1 1 1 . 33 J. G. Q u i n t i e r e , in E. E. S m i t h a n d T. Z. H a r m a t h y (eds.), Design o f Buildings for Fire Safety, A S T M STP 685, A m e r i c a n S o c i e t y for T e s t i n g a n d Materials, P h i l a d e l p h i a , PA, 1 9 7 9 , p p . 139 - 168.