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Cellular and turbulent ceiling fires
COMBUSTIONAND FLAME 18, 389-401 (1972)
389
Cellular and Turbulent Ceiling Fires L. ORLOFF and J. DE Ri$ Factory Mutual Research Corporation. 1151 Bo;ro~.Providence Tzrnpike, Aotwood, Mass;, 02062 Lasge scale ceiling rites are characterized by cellular and turbulent flames. "l'hechemical and physical scaling laws governingsuch flows arc obtained through a theory based on turbulent free convective heat transfer measurements. It is shown that the dimemionless burning rate is essentially neat'oiled by the B number, wlilch characterizes liquid and ~olid fuels. The burning rate per unit a~a appears zo be independent of both the surfa~ area of the ceiling and the depth of the flaming zone. A parallel experimental study is presented b, which solid and liquid fuels are simulated with gaseous fuel inert mixtures flowing through sintemd metal burners. Good agreement between theory and experiment is found for 'all experimentally varied chemical and physical pararantnts, which include ceiling r.~c, fuel nfixture supply rates, fuel-inert ma~s fractions, and fuel molecuiar we',Jghts.The accompanying paper shows that turbulent pool rites can be described by u closely related burning model.
~[ntrodnction The present study is an outgrowth of an earlier study of laminar fires beneath ceilings. The laminar re~,irne was both experimentally and theoretically described in terms of scaling laws which showed that the mass transfer driving force, B, was the dominant fuel parameter controlling ibe burning fate. It was shown that for larger diameter ceilings or for more intensely burning fuels (i.e., fuels with larger 8 numbers) a ceUuhr instability appears. It wa~ apparent thdt this cellular burning r e , m e followed different scaling laws and therefore merited an independent study. It was shown [11 that the instability is due to gravitational forces acting on the hot diffusion flame, burning beneath the cold fuel gases issuing from the ceiling above. '/'his "Rayleigh" instability mecbamsm triggers when the diffusion flame is typically at least one centimeter below the ceiling. Markstein [2, 3] considered the cellular instability formed by premLved rich flames. His observed ceil sizes were considerably smaller than the present diffusion llanle cells. The preferential diffusion instability mechanism studied by Markstei'a does not appear to control the present diffus:on flame cells. The calculated ariticzl Ra2deigh number for the present cells as well as
the obs,.~rved flow patterns, cell shape and size, indicate a Rayleigh instability mechanism for ceiling fires. Figure la shows an end view of some two-dirr~ensional ceils. These relatively stationary cells w~:re formed beneath a !5.2 em x 91.4 cm heptane-soaked celutex sheet which was impregnated ~ith a MnSO4 catalyst to induce smoldering smoke tracks. Two mirror-symmetric rotating flaws are apparent in the center cell. ~igure lb shows the conceptual flow of fuel and air as wel.~, as eoml~ustion products which exist both inside and o~ttside of the diffusion flame ,cell. The spiraling motion of the gaseous fuel (in Fig. la)is due to vortex stretching as the cell fuel vapor moves toward ~he ceiling edge. The [,rimary objecVtve of the present study is to quantitatively examine the fully cellular and turbule, at burning regimes rather than the instabil:ity mechanism which at this time is at least qualital.ively understood. Such a fully cellular flow was induced by placing an edge rim along the circumference of the circular downward facing burner used ha the previous laminar and instability ~tudies Various rim sizes were ttsed to determine the effect, if any, of cell geometry on burning r',~tes, l:igure ld shows an incipient cell beneath a Copyright Q 1972 by The Combustion Institute Publish:ed by American Elsevier Publishing Company, Inc,
390
L. O I ~ O F F and J. DE RI$
ir;i:!!;~,!,
^,r
/rig. 1. Experimental Ceiling Fires: ~'a) Two diraensiomll cells; 15,2 x 9L4 cm heptane soaked cellutex, (b) Cellular flow mechanism. (e) Low Raleigh number instability.,; 25.4 em burner. 1,27 era rim, 50%C, H,t/50%N 2, th" = 0.159 × 10 -3 gin/era2 see. (d) Ceil formation at instability: 25.4 em burner, no rim. 50%C~HJS0%N~. rh~ = 0.497 x l0 "~ gin/era ~ see. re) Intermediate cellular flows; 25.4 era burner. 1.27 cm rim, 50%C~ H.JS0%N~, rh" = 0,7 I0 x I0 "s gm/crn a see. (f) Fully developed cellular flow; 25.4 era burner. 1.27 eta tim. 50%Ca HJSO%N~, ,.h" = 1.14 × 10" a ~nlera 2 see.
burner without a rim. Fuel gas issues from central horizontal portion. The combustion gases move rapidly outward until they "fall o f f " the edge and then move out and up over the 45 ° insulated burner side. Figure lc shows some small "goose pimple" cells produced by a burner (with a 1.27 cm edge tim), shown in Figs. 2a and 2b. These "goose pimple" cells, which occur only at low ma~s flow rates, appear to wrinkle the otherwise fiat diffusion flame without ,~'orming the cusps evidenl in Figs. la, and lb. At higher mass flow rates the ceils become two-dimensional waves as in Fig. ~e, and at still higher flow rates the waves break up into isolated cells as shown in Fig. If. At these very high flow rates the cells become les,~ organized and more turbulent. However, we could nol achieve a fully turbulent (i.e., random) flow with the present experimental apparatus. In a~l cases the cells continually form near the burner center and then move out and over the burner edge.
Experimental Arrangement Two burners o f diameter 15.24 and 25.4 cm were used. Cellular instability was induced with circular rims o f various depths attached to the burners. Figure 2a shows the overall enclosure designed to provide a smooth flow o f air to the ceiling fire. Measured rates of fuel and inert gases are mixed and supplied to the burner, Figure 2h shows the burner details. The fuel gas passes downward through a (2.54 cm thick) sintered bronze disc and burns in the ambient air below. The heat transfer back to the disc is measured by a spiral water cooling tube imbedded i~ the sintered disc. A copper guard ring and copper sheath, maintained at the sintered bronze temperature (600C), prevents extrarteous heat transfer to the disc due to flames lapping at the sides o? the burner. Thermocouple measurements al various positions in the burner and plenum show negligible heat exchange between the disc and its holder. Heat losses to the fuel gas above the
CELLULARAND TURBULENTCEILINGFIRE5
..... ,: TTT/I- ='~='~' I
'°':,,7::"
i"
Fig. 2. ExperimentalApparatus: a) Burnerand housing. b) Burnerconstructionhalf section. disc never exceed 2% of the measured flame heat transfer. All experimant',d runs were made while the surface of the sintered bronze was uniformly blackened with soot to minimize variations in radiative heat transfer, The use of gaseous fuels of varying composition and flow rates permits controlled simulation of a wide range of solid and liquid ceiling fuels. The heat of vaporization L for these practical fuels is simulated in this experiment by the quotient of the total heat transfer to the sintered disc divided by the total mass transfer through the disc. For liquid fuels, assuming no loss of heat to the fuel bed interior, the local ratio oi heat transfer ~" to mass transfer in" would be equal to the fuel's heat of vaporization L. The experiment with its uniform mass transfer rate closely approximates this situation with L equal to the ratio of total heat transfer to total mass transfer, c~//n. The dimensionless mass transfer driving force B = QYo,/(Movo'L) - Cp(Ts - T , ) / L usually
391 defines the burning characteristics of solid and liquid fuels. Wood, for example, has a B number less than 2.0; gasoline has a B of about 5.0. It was shown previonsly [1] that gaseous fuels can be used to effectively simulate liquid and solid fuels by measuring ~"..over a range of gaseous mass transfer rates, th". The resulting variation in the "heat of vaporization" L =~"/hT" makes it possible to simulate a range of B numbers for practical fuels independently of their transport ~reperties and fuel thermodynamics. The range of B covered in this study includes nearly all common flammable materials.
Experimental Results Heat transfer vs. mass transfer measurements were made for (I) two ceiling diameters 15.24 and 25.4 cm to examine the effects of ceiling size; (2)several rim sizes (1.27-10.16 cm) to examine effect of cell depth, (3)various C2Hd/N2 fuelinert ratios to examine the effect of the flame temperature and the fuel mixture's effectwe heat of combustion; and (4)a variety of fuel mixture molecular weights to determine the influence of fuel density on the gravitational cellular flow. In general, the effects relating to the fuel4nert ratios and fuel molecular weights were more important than the effects of ceiling diameter and rim size. Figure 3 indicates that die heat transfer per unit area is not noticeably influenced by the burner dianteter, This insensitivity to ceiling diameter, which is supported by the theory, probably extends to much larger diameters; however th~shas yet to be experimentally demonstrated, Table I shows the mass and heat transfer results for the 25.4 cm burner for rim depths 1.27-10A6 cm, i.e., up to 40% of the burner dianteter. The heat transfer rates ate comparatively insensitive to rim depth (i.e., cell depth) despite the eight fold variation in rim depths used. The decrease of heat transfer with increasing mass transfer is more noticeable. The heat transfer variation due to rim depth variation is 6-20%; however that due to mass transfer variation is 31-64%. As described below, the theoretical development includes the effect due to mass transfer; however it predicts negligible itifluenee from rim depth.
L. ORLOFF and ]. DE KIS
392 0.3
~
i
em
i
i
to incipient extinction of the flame, The "goose pimple" flow (similar to Fig. lc was observed at this point. Finally, Fig. 5 shows the importance of fuel molecular weight on the heat a'ansfer, This molecular weight effect is presumably due to the ~fferences in fuel density on the cellular flow. For high fuel densities, gravitational instability is increased producing a more intense rotational flow and ~ e a t e r heat transfer.
i
m~,m mA
~
0.2 MIxT~g f ~ Fraction)
~
.
~A
B~r~r Oio, r:l t5.24cm, gl 25.4cm. Lip Depth: 2,54 crtl. I
I
r
f
.000~ ,O00T ,0~ ,C~2 MASS TRANSFER, rn"(gm/cm =- sec)
~S
~eory At presen~ there is no mathematical theory, based on the bas}c combustion equations, which explains
Fig. 3. Heat transfer to ceiling ¢I" vs, mass transfer of fact. ,'~". Data R)r two ceiling sizes. Ethane-nitrogen fuel mixtures with diffireat ethane molar concentrations were used to determine the effect of flame temperature on the ~ " v s / n " relationship. These gases were used b~causc their molecular weights are close to that of air, thereby minimizing molecular weigbl effects on the flow. The results are shown in Figs. 4 and 5. As expected Fig. 4 shows litt;e change in he~t transfer between 0.8C2H6/O.2Nz and 0.5C=H6/0.5N2 molar fuel mixtures since their respective adiabatic stoichJometric flame temperatur~.s are quite close. However, for still lower ethane mass fractions the flame temperature drops sharply. This produces a drop in heat tra~sfer as shown in Fig. 4. The marked decreaSe in heat transfer for the smallest mass transfer point on the 0.2C2HJ0.8N: plot in Fig. 4 may be due
I
I
i
I
O o i
.ix.® ~-~ =0"
D
OA O A
Bum~ Oio2t5.24crn L~ O,~n: 1.27on Mialum (tide Fccc#.,n)
0
.ooo~
,eCII%I,ZNn +5GIHe/.5NI
ra
I
~x~
I
I
I
.oo2
ooo8 .oo~
~o~
rn" (gm/cm=- sec)
Fig. 4. Effect of flame temperatu:c on heat transfer. The mixtures used zeptcscnt an approximately IO% change in flame temperature.
TABLE l Heat Transfer to Ceiling (25.4 cm burner, 50%C=H.-50%N= ) Fuel mass transfer, m" (gm/cm~s¢c)
1.27cm
2.54 cm
3.81 cm
5.08 cm
6.35 cm
7.62 cm
10.16 cm
0,426 x 1O" 3 0.710 ×I0 "~ 0.995 xlO "3
0.287 0.262 0,238
0.297 0.265 0.239
0,285 0,268 0,250
0.285 0.253 0.243
0.277 0.262 0.253
0.257 0.265 0.243
0,293 0.278 0.259
1.28 1.42 1.80 2.08
0.212 0.202 0.185 0.175
0.221 0.204 0.192 0.183
0.227 0.212 0,204 0.196
0.224 0.210 0.202 0.190
0.236 0.214 0.201 0.194
0.219 0.211 0.202 0.202
0.240 0.221 0.221 0.204
xl0 "3 xtO"s xlO "3 xl0 "=
~'~ (cal/cm~ sec) when tip depth below ceiling is equal to
393
CELLULAR AND TURBULENTCEILING FIRES
[2
0 + O+ ~' 0
o
+ 0
"i
0 ~+
+
0
+ 0
}
V
v
O ,t
O
1 .0005
1Z7
~e~,/an%
at.
1.27
1~, ~o.i
10¢s¢ ~ m
IL~I t~L24
Z~ Z~
.~i . . . . . . . .
I.
I
~ .
..
Fig. 5. Effec~of fuel raixturc raolccular weight (i.e., density) on heat transfer. these experimental results. However, using physical arguments, one can quantitatively explain the observed results with considerable precision. Although these arguments are not strictly rigorous, they are hazed upon a conceptual understanding of the essential physical processes, This agreement between theory and experiment for ceiling fires has encouraged us to examine Corlett's experimental results [4] for the closely related problem of burning above horizontal surfaces. As described in the accompanying paper a similar theory is also under certain circumstances applicable to this latter problem. Spalding [5] has shown that for tbrced convection situations there is an analogy between a burning fuel surface and the corresponding nonreacting mass transfer situation. Here we extend this analogy to free convective situations (i.e., fires). We find that the analogy is not as exact, since, in principle the burning can alter the characte; of the flow field. In order to employ this analogy, we must first predict, for ceilings, the free convective heat and mass transfer without combustion. We can make such a prediction by considering available measurements of the c!~ely related problem of free convective heat transfi'.r (without combustion)
between two horizontal surfaces. To make a quantitative comparison, we correct for: a) variable property effects, b)heat blockage (or blowing) effects, as well as c)the fact that the combustion takes place beneath a single rigid surface, whereas the related free convective heat transfer results are for convection between two rigid surfaces.
a) Analogous Free Convectlo~ Studies: There have been numerous studies [6-t.4] of free convective heat transfer (w/thout cerebration) between two horizontal surfaces heated from below. The available turbulent heat transfer measurements, for air, can be ,'epreset:ted by t
1There is some dispute as to tile exponent 1/3 (and consequently the ptopor~ienality constant) in the above formula. For cxanlple, the measurements of Sflvorton [61, Rossby [ 7 ~, and Goldstein el ~ | 101 suggesta valae closet to 0.3 while Jakob 114] (based on Mull and Reihers measurements Jill) Deardorff et aL [8], and Globe et al. {91 suggest that the value of 1/3 as do tile approximate theories of Ma]kus [121, and Kraitchc. [ 13l. This dispute is of negligible quantitative iraportan¢,.' to the present discussion.
L. ORLOFF and J. DE RIS
394
hd
Fgd3(Tn -
- - = 0.075 /
ke
L
T¢)'] t/3
vmamT,~
l
.J
,
(l)
where the Nusselt number on the left describes the ratio of the total heat transfer to pure conduction, an.:l the Rayleigh number in square brackets on tire right characterizes the convective Gory field which induces these augmented heat transfer rates. In the above formula d is the distance between the surfaces, Tn the lower (hot) surface temperature, T~ the upper (cold) surface temperature, Tm the mean temperature, z,m and am respectively, the kinematic viscosity and thermal diffusivity evaluated at Tm. All the measurements providing .'he above result were made with small temperature differences (TH - To), so that we must use judgment when evaluating flows with strongly varying properties. However, it is physically plausible (by symmetry) to evaluate v ~ , am, and Tm at the mean fluid temperature, and evaluate the thermal conductivity, 3.0 in the Nusseh number at a t.~'mperature midway between Tm and the surface temperature, Ts. Strictly speaking the above turbulent formula is valid only for Rayleigh numbers greater than 3 × 104; however it is also approximately valid (within *-20%) for cellular convection for Rayleigh numbers greater than 2 x 103. In view of its approximate valid':ty for cellular flows, we shall apply this formula to both cellular and turbulent flows. The time-averaged temperature distribution measurements of Deardorff et at. [8], as well as Goldstein et at. [I0] reveal a central core w~th a uniform temperature midway between the two surface temperatures. This central core is bounded at the upper and lower surfaces by thin ~ayers with steep temperature gradients. Since the entire temperature drop occurs across these two surface layers, one can ascribe the entire heat transfer resistance to these layers. Tbe fluid in the central core region is apparently free to move as it countercurrently transmits the heat between the two surface resistanc,.• layers. Indeed measurements [8] indicate that much of the heat is transferred by widely spaced jets which build up at
one surface and then "faR" to the opposite surface. Since each layer has half the total temperature drop, one can characterize the layer thickness, ~, by 2 h~l(Tn-Tc)
and since 4" = h(Tn - Tc), Eq. 1 provides
~ Aft
l
(
u,na,. T,~ y , 3
(2)
which is independent c,f b~th the surface separation distance d and th~ lateral dimension of the surfaces. As will becom~ apparent, this indeponder, cc suggests that the analogous ceiling fire burning rates (and related resistance layer thicknesses) will also be independent of both the tim depth and ceiling diameter. It is interesting to note that the Rayleigh number associated with the distance 28 (i.e., when the two resistance layers touch each other) is
R,,,6
=
rg(Tn - T~ ) (~)3 l L
(v,na=Tml
= 2400,
J
which is quite close to the Rayleigh instability :value 1708. This is not surprising, since the distance 5 is physically related to the Ray~eigh instability of a layer heated from below between horizontal surfaces. Now consider the free convection problem beneath a single horizontal surface with a rim such as r~ed in this experiment. Suppose the surface (i.e, sintered metal disk with zero mass transfer) were at a temperature. T,, far below the ambient temperature T , . For turbulent or cellular flow, we expect cold eddies to freely fall and hot eddies to al?he measurements of Goldstcin cr ~ [ 10] .show that (ess~;ntin!iy in:lependent of Rayleigh number) the tim¢~-averagetemperature approaches within 20% of the central core temperature at the distance 6 from the surface.
395
CELLULARAND TURBULENTCEILINGFIRES rise close to the plate surface. Since the heat flux must cross only one resistance layer, one has xn(T~, - TA
heavier specie. The value of Y varies from YT in the gas plenum upstream of the ceiling surthce, to Y~ at the ceilin 3 surface, and eventually to Y~ in the ambient fluid far from the ceiling surface. YT" does not equal Ys, since the mass flux of the specie upstream of the ceiling surface is
which with Eq. 2 for ~, becomes
L .... T.,"J
key h" "
Here it seems plausible to use the temperature difference ( T , - T , ) for the fluid mechanics expression in the square brackets; since for a single surface it is (.os - O , ) rather than (Pc - a n ) which generates the turbulent kinetic energy in the flow field. The above result with the constant 0.15 replaced by 0.14, becomes identical to Fishenden and Saunders [15] turbulent correlation for heated square plates facing upward. This agreement apparently supports our conceptual understanding. Finally because gravity acts directly on density differences rather than temperature differences, it is appropriate to express the above equation as --
)-]1/3
I *('<'' -o,.:)1
. (.3)
L l,.y ~v P~'A where we have assumed pT constant '~,4fl'l per =(,o,.,os) 1/2 . Here the subscript ~, implies a condition approximately midway between the ambient and surface fluids. b) MassTransfer Effects Consider a porous ceiling, with bounding rim, issuing a dense nonreacting gas mixture into less dense surroundings having the same temperature. For turbulent motion, we again expect a thin resistance layer, as a result of gravity acting on the density differences. In this isothermal case, however, th~ resistance layer opposes mass transfer rather than heat transfer. Suppose that the supplied ges mixture consists of two gas species of differing molecular weights. Let Y be the time average mass frac:ion of the
The fight hand terms describe respectively the gas phase convective and diffusive flux at'the surface. Using the concept of a laminar resistance layer of thickness 8, the diffusive term can be approximated by +9D(Ys - Y.)/8. The predicted mass transfer rate, ~ " , then becomes,
~ ' = (#D),~
[(v,
-
Y= )l(Vr
-
Y~)]
(4)
where fl indicates a condition near the surface. The term in square brackets is commortly known as the mass transfer dri~dng force, B', for such nonreacting mass transfer problems. Later we will redefine this expression for B' when we consider mass transfer with combustion. In general the surface layer thickness, 6, increases as a result of blowing (i.e., mass tramfer). However for very low mass transfer rates 0.e., for B' = (}s - Y®)/( Y T - Y,) < < l)this blowing effect is small and we have,
<,o>,,,.io.,op<,: l, for B" << 1, which is analogous to Eq. 3 with ix replaced by D. For B" large, one can approximately correct for the blowing or mass transfer efi~ct, by multiplying each transport property (i.e., v, a, and L~) by Itn(l + B')/B'], so that in general
L. ORLOFF and J. DE RIS
396
fa, = tpD)sB , ( I n ( ~ B')) = 0o, L.=O~(p, + p~)J )
,,,
This empirical correction procedure, which is in good agreement with the present experimental results, is based on the concept that blowing reduces the influence of all transport properties near the blowing surface. In the past, this same correction proc~'dure has ~so been successful for interpreting: i l) experiments for laminar ceiling fires [1], and (2)an exact theory for laminav fre~e--convectivc burning of vertical and sloping sm'faces [17].
both the initial reactants and £mal products as gas~:s al the ambient temperature. That is, the combustion releases heat, Q, as it consumes the reactant masses MFvF' and ~/oVa'. Thus the molar reaction rate per unit volume is
~=_ Q
~7
~ (Move)
(MFv~)
(s)
Now define ¢ as the sum of the thermal and chemical enthalpies. So that, in terms of
YoQ Eqs. 6-8 become
c) Turbulent Bualmg-SpaldiagAnalogy
Following Spalding [5], we shall show that a redefinition of B extends the mass transfer result to include combustion. For unit Lewis number pCpD[k= 1, the general time-dependent energy and specie combustion equations are,
p~+pu~--=
+ ¢%
ax t
aY, P T7 + "'j
OY,=
f-~-(pD OY,'~ +
or__
"~\
~/
~',
(6)
(v)
T
where h = f~ CpdT is the thermally sensible enthalpy, Yi is t~he specie " i " mass fraction, 4"'is the volumetric heat release rate, fn/" is the specie " i " volumetric mass generatior rate and the xi are the three spatial coordinates. The effects of (1)radiation, (2)high speed f$ow, and (3)specie diffusion driven by pressure and temperature gradients me all ignored. For diffusion flames the heat and mass source terms t~"' and the"' are related through the global stoichiometry relation
provided the Lewis number pCpDfA = 1. Notice that this equation is similar in form to the nonreactive energy and specie equations. Let us now define the "effective latent heat of vaporization," L =~t"/ia~, as the ratio of the heat transfer, 4", toward the surface divided by the mass transfer of fuel mixture, r~", away from the surface. Using this definition the surface boundary condition for O becomes
~aT
_
x
aS
where it is presumed that the presence of the turbulent diffusion flame eliminates all the oxidant adjacent to the fuel surface. Now the concept of a resistance layer near the surface provides
L
t~" (9)
v~,(Fuel) + v~ (Or;de.at) -
v; (Produci',s) + Q(Heat)
where Q is the heat released by burning vF' moles of fuel or equivalently vo' moles of oxidant, with
which is analogous to Eq. 4 for mass transfer without combustion. Here the mast transfer driving force ~' is the te~'m in square brackets; that is,
CELLULARAND TURBULENTCEILINGFIRES
B
397
(Mou~ L)
since by definition h , = 0 and by presumption Yes =0. Here hs ~Cp(T s - T..) and Q is the heat of combustion per mole of fuel provided vF' is set to unity. We see that B = (¢). - ¢s)/L is the total (thermal and chemical) specific enthalpy of the free stream minus the total surface specific enthalpy divided by the heat required to "vaporize" unit mass of fuel mixture. B is usually the dominant chemical parameter for fire situations. Equation 9 ea~ be used to predict ceiling burning rates provided 5 is known. We can use the value of 8 developed for the analogous mass transfer situation, provided tire differences in the fluid flow, for the two cases, do not s~gnificantly influence 8. If this is the case, we have, analogous to Eq. 5,
t~/~ L--F--J 151£P'-°/] ~,
(il)
where D is replaced by ~, p , is replaced by the flame gas density pf, and 3' signifies a temperature and corresponding composition midway between the flame and fuel surface temperatures. d) CrifiqueofBumi~g lhfedlction I. This burning rate prediction involves several important assumptions. The most important, from a q~zantitative viewpoint, is the procedure for evaluating the gas properties used in Eq. 11. While similar choices have been made successfully by others, ou, experience with this problem indicates that a different method of evaluation could easily change the predicted burning rates by as much as a factor of two. (See Fig. 7). This means that the bnrn~g theory involves a somewhat arbitrary, yet quantitatively significant, choice of properties. Such property choices do not affect trends, such as the influence of changes in ,Jimensions, chemistry, ambient pressure, etc.
2. Equation 11 presumes that 8 can be predicted on the basis of a ~ of density Ps entering an ambient fluid of density p$. However the average temperature of th,• gas between y = 5 and the rim is probably considerably lower than T/, Thus we have overestimated ~ e effective density difference and consequently the le,vel of flow t~arbulence. However this effect is small since if we instead were to replace p! in Eq. 11 by ~(Ps "~,of) we would typically reduce the predicted burning rate by only 10%. 3. At very low mass transfer rates (or low Y~.~,), it is possible that some parts of the tm.bulent diffusion flame would be extinguished by the cold fuel surface. In this case some unreacted oxidant would appear at the fuel surface, thereby increasing the total surface enthalpy, 0s, and thus reducing the heat transfer. Such a reduced heat t~ansfer (with one exception) is not evident among the ceiling fire data; however reduced heat transfer is apparent in Corlett's rimless "pool" burning data [4]. It is also possible that at low B or low YFT the fi~h'ne may become embedded within 8. An em]~.'dded flame could significantlyalter the fluid m echanics, thereby violatinga fundamental hypothasis of the Spalding analogy that the fluid mechauics i s uninfluenced by the flame p~itien. In:any case the fact that the ceiling fire data correlation is unaffected by YF~', means that these possibie complicating effects are unimportant for ~e~ling fires studied. 4. The present theory ignores radiation which typically amounts to 10-40% of the to~tl h¢~t : transfer. However, this neglect of radiant heat transfer is partially self-compensating, since flame rad~tion reduces the flame temperatures and consequently also reduces the convective heat transfer,
Comparison of Theory and Experiment The raw data of Eig. S are replotted in Fig. 7 as suggested by Eq. 11. The correlation appears excellent, and in reasonable agreement with theory. However, as explah~ed Uelow the small vertical sep~at/on between theory and experiment is partly fortuitons, since this separation distance
398
L, ORLOFF and ,L D E RIS
E3
THEORY-
fa
~C'
.s / "
~°~
f" 002
' ,. I ,I a i i O~ O,0 so 20 40 oO tOO DIMENSIONLESS MASS TRANSFEROI~IVIN~ FORCE,B
O2
Fig. 6. Effect of property evaluation procedure on agreement of theory and experimenl
~¢,~
' o,4
o;~
I I oe to
-
~l*
I 40
t oo
J
~
Fig. ?. Dimensionless MassTransfq:r:
pIU
~"Le(p.
is quite sensitive to the choice of transport properties, as indicated in Fig. 6. Specifically Fig, 7 plots
I z.o
[31MF.NS;V~LESS M~SS TRANSFER DRIVING FORC~,E~
~'[vo,,~
Ca(T, L,~1. .i
fa" ~v~'a~'P" l'/3 hi"
;S
pr v;pr:~ll"3
.
,
.
.
.
(i3)
(,2)
verses the mass transfer driving force, B. The subscripts B and y indicate the condRions for evaluating the transport properties. In Fig. 7 we evaluated v~,, and ,or al 7 = ½, that is, for the gas mixture composition and temperature midway between the fuel-surface and flame condition..~a on the other hand was evaluated at/3 = ,¼, that is for a gas mixture a quarter-way from the fuel surface to the flame condition. The Appendix describes the detailed procedure for evaluating these properties. In addition we have assumed that the Prandtl number t'r=laCt,/;~=0.75 is independent of both temperatur~ and composition. See Appendix for details. Comparin& Eqs. 1 1 and 12 we see that the, theory predicts
which is the curve in Fig. 7. The choice of ? = ½ and /~ = % represents our best intuitive ct,oice for evaluating the properties. However as shown in Fig. 6 other possible choices would signilicantty shift the position of the experimental points relative to the theory. For example, if we were to evaluate ~a at # = 0 (i.e., at the fuel surface) and P'r and u.r at y = ~ the theory would lie below the experimental data, This theory agrees with all the experimentally examined trends. In particular it predicts: (1)the correct variation with the mass transfer driving toree B (i.e., the correct relationship between in" and ~"), (2) the very weak dependence of #" on the mass fraction of inerts in the supplied fuel mixture, (3) the correct fuel mixture molecular weight dependence, and Iinafly, (4) the
CELLULARAND TURBULENTCEILINGFIRES independence of burning rate and #" on ceiling diameter and rim size. This agreement lends strong support for the validity of the theory, despite the somewhat arbitrary choice of property values used to compare theory and experiment. Conclusions 1) Cellular burning heat transfer is independent of ceiling diameter and rim size. It is anticipated on the basis of the theory, that this finding might be applicable to much larger scale ceiling fires. 2) The cells appear to be o f a Rayleigh character with an internal rotational flow. The cellular flow can be in the form of "'goose pimples," waves, or independent cells with cusps, for increasing mass flow. 3) A theory based on turbulent free confective heat transfer measurements is presented. This theory predicts the correct trends for all experimentally varied parameters which include ceiling size, rim size, fuel mixture supply rates, fuel-inert mass fractions and molecular weights. The dimensionless burning rate is essentially controlled by the B number, which characterizes liquid and solid fuels. 4) The theory is based on constant property (i.e.., Boussinesq) free convection measurements. Therefore considerable care had to be exercised in d(:ciding on the best procedure for selecting property values. Although ~ e selected procedure is somewhat arbitrary, it is consistent with our conceptual understanding of the physical mechanisms and has provided reasonable quantitative agreement with experiment.
Appendix A: Property Evaluation Procedure This appendix describes the procedure used to evaluate the gas properties at any specified condition between the flame and fuel bed. This procedure assumes ( 1 ) a unit Lewis number pC~,)D/?~, (2)negligible radiation, (3)flame thicknesses much less than diffusion distances, and (4) constant effective latent heat of vaporization. Under these conditions one can easily show [17] that there is a unique relationship between the local gas composition and local gas temperature on each side of the diffusion flame. This relationship is independent of the specific fluid motion,
399 whether it is laminar, cellular, or turbulent [17]. In view of this independence of fluid motion, one c a n conveniently deduce the dependence of composition on temperature by considering the relatively simple one dimensional diffusion flame including the effects of variable properties. Let us define ? as the ratio Of specific gas enthalpy at temperature T,t (between fuel surface and flame) over the specific enthalpy of the flame gas, with the fuel surface gas serving as the enthalpy reference. Spt;cifieally let
y =
(A:I)
T,
~i" YirLr~ Cp, dT This definition can be used to calculate T7 for any specified value of" 7. For example, if 7 = ~, T~. is the gas temperature associated with half the flame specific enthalpy. Assuming unit Lewis number, negligible radiation, and constant effective latent heat of vaporization, one can show that the diffusion flame gas specific enthalpy is3
Yir
Cpj dt
L
•
(A-2)
(r + 1)
where the mass transfer driving force is
[3
QYo~ 1 ~% C p d T , MovoL L r.
and the mass of fuel mixture required to react stoichiometrically with a unit mass of oxidant is Yo~F~k (Y~rMo v~) This flame gas specific enthalpy is identical to the adiabatic stoic~ometfic flame specific entbalpy associated with the sapplied fuel and oxidant mixtures. This tbrmala assumes equal speciespecificheats in the temperature range T~ to Ts. This assumption has negligibleeffect when (Ts-T~)/(TI-T.) .¢ I.
L. ORLOFF attd J. DE RIS
400 For a hydrocarbon-inert-fuel mixture burning in air we assume that the reaction goes to complet'on producing carbon dioxide and water vapor with no incomplete products of combustion, .In this case the species present between the flame and fuel bed are nitrogen, supplied inert, carbon dioxide, water vapor, and supplied fuel. Here we "also ignore dissociation species. Under these conditions, the specie mass fractions are in terms of 7:
from which one evaluates P'r for a perfect gas by
o Xx,..,.
Finally the mixture kinematic and dynamic viscosities are approximated [19] by
Y N = [ y ( B - r)/(r + 1) + 1}
v?. py = /.t~,
[~ X,, M?'2]
( 8 + 1)
YN"t =
¥% = Y l r -
Ytr [T (B - r)/(r + D + 1] (B+l)
where the individual ~l.t are given graphically by Spalding [19] as functions of temperature T.r.
YCO27 =
Discussion
Y°=Mc°2vE°s [y(B - r)/(r + 1)+_!!
(B + 1)
MoV~ ~H207 =
¥OcoMU2OV~I20 [7(B " r ) / ( r + I) + I] (B + 1)
Mov"o Y~7
=
Yt.'r-
= X x,..,o
YFr(r +
1)[TtB- r)/(r + 1) *
1]
(B + 1)
Yo~, = O.
For given B, r, 7, oxidant mass fractions, and • supplied fuel-mixture compositions one can immediatel2¢ evaluate the local specie mass fractions Yt.r. Then, using Prothero's [118] sixth order polynomial aI)proximations for the JANAF specie specific heats Cpi together with Eqs. A-I and A-2, one can calculate T.t. The calculation of T.r is included in the computer program used in this study [20]. We now can calculate the gas properties P-t, v't and V,~. The molar fractions Xrr can be expressed in terms of the mass fractions YPt by
Yi~
Y~
Following this procedure the computer program was used to evaluate Pa, Pl, Pa, P~,, and v~, for each data point. Since the ca!culation ignores dissociation effects the computed values of a f are about 10-20% too low; however this discrepancy induces only about a 1% error in the computed value (as-P.t) tpa applied to each data point. For ]3 or 7 < 1 / 2 dissociation effects are negligible. With the possible exception of including the effects of radiation and products of incomplete combustion, the calculation procedure represents our best present estimate of gas properties. These property estimates have improved our correlation of ceiling fire data for fuel mixtures having significant molecular weight differences.
Nomendature
B Cp d O g h L Mi rh"
Pr
QYo~/(Mov'oL) - Cp(T~ - T=I/L Mass Transfer Driving Force, (.). Specific heat of gas, (E/MO). Parallel plate separation, (/). Specie diffusivity, ff2/t). Gravitational acceleration, (lit 2 ). Specific enthalpy, (ELM). Latent heat of vaporization, df'/dn", (E/M). Molecular weight of specie "i". (M/mole). Mass transfer flux, (M/lat). Prandtl number IZCp/A, (.).
401
CELLULAR A N D TURBULENT CEILING FIRES
Heat transfer flux, (E/I 2 t). Heat of combustion, (E/mole). r YO~klFu'F/(YFrMOP'O) stoichiometric ratio, (-). T Temperature, (O). ui Velocity in directioli ' T ' , (l/t). xt Coordinate in direction " f ' , (1). Xt Molar fraction of specie " f ' , (-). y Distance normal to surface, (l). Yi Mass fraction of specie " i ' , (,). a Thermal diffusivity, (/2 It). Resistancelayer thickness, (0. v Kinematic viscosity, C,2/t). ~/~,v~" Stoichiometdc coefficient of reactants and products respectively, (mole). A Thermal conductivity, (E/lOt). p Density, (M/I 3 ). ~b Total (chemical plus thermal) specific enthalpy, (ELM). /~ Dynamic viscosity, (M/lt).
~" Q
SubscriptS C cold f flame ',F Fuel 'Fs Fuel at surface FT Fuel upstream of surface (i.e., in supply plenum) H Hot m mean 0 oxygen s surface T upstream of surface near surface y gas phase-see Eq. A-I ambient
The authors are indebted to Professor Hubbard l¢. Emmons o f Harvard University for suggesting the use o f gaseous fuels to vary the B par,rmeter. We are thankful to Dr. Raymond Friedman for his continued encouragement during the course o f this wort~
References I. Odoff, L., and de RIS, J., Thirteenth Symposium [International] on Combustion, "file Combustion Institute, Pittsburgh, Pa. (1971). Also FMRC Technical Report Self. No. 19720-2, April, 1970. 2. l~atkstein, G. H., Fourth Symposium [International) on Combustlon. The Williams & Wilklns Co., Bal~mote, Md. (1953), p. 44. 3. Maskstein, G. H, and Somet~ L. M., Fourth Symposium [International) on Combustion, The Williams & Wilkins Co., Baltimore, Md. (1953), p. 527. 4. Curlett, R. C., Combustion attd Flame 14, 351 (1970). 5. Spalding, D. B., hit. J. Hen: Mass Transfer t, 192 (1960). 6. S~veston, P. L., Forseh. lug. Wes. 24, 39 and 59 (1958). 7. Ro~by, H. T., Y. Fluid Mech. 36, 309 (1969). 8. Dcatdorff, .I. W,; and Willis,G. E.,J. Fluid Mech. 28, 675 (1967). 9. Globe, S., and Dropkin, D.,J. Heat Transfer 81, 24 (1959). 10. Goldstein, P~ J., and Cha, T. Y., Progressin Heat and Mass Transfer 2, Pergamon Press, New York (1969), p. 55. I1. Mull, W., and Re,her, H., Beiil. Gesundh. lug. I, 28 (1930). 12. Malku~, W. V. R., Peat. Roy. Sue. (London) A225, 185 and 196 (1954). 13. Ktaiehnan, R.H.,Phys. Fluids 5, 1374 (1962). 14. Jakob, M., Heat Transfer 1, Wiley, New York (1949), p. 522. 1.5. Fishenden, M., and Saundets, O. A.,An Introduction to Heat Transfer, Oxfozd, New York (1950). 16. HlnMey, P. L., o~d Wraight, H., J.F.R.O. Npte No. 743, Joint Fire Rescaleh Organization, Botehamwo,~l, llerts, England, 1969. Abstracted in Fire Res. Abst. and Rev. 11,206 (1969). 17. Rim, J. S., de Ri~.,J., and Kl~e~.er, F.W., Thirteenth Sympo~.ura (luternationaO on Combustion. The Combustion Institute,Pittsburgh,Pa.(1971). 18. la~othezo, A., Combustiou aud Flame 12, 399 (1969). 19. Spalding, D. B., Couveetire blurs Transfer, McGrawHill New York (1963), p. 138. 20. Orloff, L., and de Ris, 3., FMRC Technie',d Report Scr. No. 19720-3, Januat'y, 1971.
[Received June 1971; revised version received November 1971)
389
Cellular and Turbulent Ceiling Fires L. ORLOFF and J. DE Ri$ Factory Mutual Research Corporation. 1151 Bo;ro~.Providence Tzrnpike, Aotwood, Mass;, 02062 Lasge scale ceiling rites are characterized by cellular and turbulent flames. "l'hechemical and physical scaling laws governingsuch flows arc obtained through a theory based on turbulent free convective heat transfer measurements. It is shown that the dimemionless burning rate is essentially neat'oiled by the B number, wlilch characterizes liquid and ~olid fuels. The burning rate per unit a~a appears zo be independent of both the surfa~ area of the ceiling and the depth of the flaming zone. A parallel experimental study is presented b, which solid and liquid fuels are simulated with gaseous fuel inert mixtures flowing through sintemd metal burners. Good agreement between theory and experiment is found for 'all experimentally varied chemical and physical pararantnts, which include ceiling r.~c, fuel nfixture supply rates, fuel-inert ma~s fractions, and fuel molecuiar we',Jghts.The accompanying paper shows that turbulent pool rites can be described by u closely related burning model.
~[ntrodnction The present study is an outgrowth of an earlier study of laminar fires beneath ceilings. The laminar re~,irne was both experimentally and theoretically described in terms of scaling laws which showed that the mass transfer driving force, B, was the dominant fuel parameter controlling ibe burning fate. It was shown that for larger diameter ceilings or for more intensely burning fuels (i.e., fuels with larger 8 numbers) a ceUuhr instability appears. It wa~ apparent thdt this cellular burning r e , m e followed different scaling laws and therefore merited an independent study. It was shown [11 that the instability is due to gravitational forces acting on the hot diffusion flame, burning beneath the cold fuel gases issuing from the ceiling above. '/'his "Rayleigh" instability mecbamsm triggers when the diffusion flame is typically at least one centimeter below the ceiling. Markstein [2, 3] considered the cellular instability formed by premLved rich flames. His observed ceil sizes were considerably smaller than the present diffusion llanle cells. The preferential diffusion instability mechanism studied by Markstei'a does not appear to control the present diffus:on flame cells. The calculated ariticzl Ra2deigh number for the present cells as well as
the obs,.~rved flow patterns, cell shape and size, indicate a Rayleigh instability mechanism for ceiling fires. Figure la shows an end view of some two-dirr~ensional ceils. These relatively stationary cells w~:re formed beneath a !5.2 em x 91.4 cm heptane-soaked celutex sheet which was impregnated ~ith a MnSO4 catalyst to induce smoldering smoke tracks. Two mirror-symmetric rotating flaws are apparent in the center cell. ~igure lb shows the conceptual flow of fuel and air as wel.~, as eoml~ustion products which exist both inside and o~ttside of the diffusion flame ,cell. The spiraling motion of the gaseous fuel (in Fig. la)is due to vortex stretching as the cell fuel vapor moves toward ~he ceiling edge. The [,rimary objecVtve of the present study is to quantitatively examine the fully cellular and turbule, at burning regimes rather than the instabil:ity mechanism which at this time is at least qualital.ively understood. Such a fully cellular flow was induced by placing an edge rim along the circumference of the circular downward facing burner used ha the previous laminar and instability ~tudies Various rim sizes were ttsed to determine the effect, if any, of cell geometry on burning r',~tes, l:igure ld shows an incipient cell beneath a Copyright Q 1972 by The Combustion Institute Publish:ed by American Elsevier Publishing Company, Inc,
390
L. O I ~ O F F and J. DE RI$
ir;i:!!;~,!,
^,r
/rig. 1. Experimental Ceiling Fires: ~'a) Two diraensiomll cells; 15,2 x 9L4 cm heptane soaked cellutex, (b) Cellular flow mechanism. (e) Low Raleigh number instability.,; 25.4 em burner. 1,27 era rim, 50%C, H,t/50%N 2, th" = 0.159 × 10 -3 gin/era2 see. (d) Ceil formation at instability: 25.4 em burner, no rim. 50%C~HJS0%N~. rh~ = 0.497 x l0 "~ gin/era ~ see. re) Intermediate cellular flows; 25.4 era burner. 1.27 cm rim, 50%C~ H.JS0%N~, rh" = 0,7 I0 x I0 "s gm/crn a see. (f) Fully developed cellular flow; 25.4 era burner. 1.27 eta tim. 50%Ca HJSO%N~, ,.h" = 1.14 × 10" a ~nlera 2 see.
burner without a rim. Fuel gas issues from central horizontal portion. The combustion gases move rapidly outward until they "fall o f f " the edge and then move out and up over the 45 ° insulated burner side. Figure lc shows some small "goose pimple" cells produced by a burner (with a 1.27 cm edge tim), shown in Figs. 2a and 2b. These "goose pimple" cells, which occur only at low ma~s flow rates, appear to wrinkle the otherwise fiat diffusion flame without ,~'orming the cusps evidenl in Figs. la, and lb. At higher mass flow rates the ceils become two-dimensional waves as in Fig. ~e, and at still higher flow rates the waves break up into isolated cells as shown in Fig. If. At these very high flow rates the cells become les,~ organized and more turbulent. However, we could nol achieve a fully turbulent (i.e., random) flow with the present experimental apparatus. In a~l cases the cells continually form near the burner center and then move out and over the burner edge.
Experimental Arrangement Two burners o f diameter 15.24 and 25.4 cm were used. Cellular instability was induced with circular rims o f various depths attached to the burners. Figure 2a shows the overall enclosure designed to provide a smooth flow o f air to the ceiling fire. Measured rates of fuel and inert gases are mixed and supplied to the burner, Figure 2h shows the burner details. The fuel gas passes downward through a (2.54 cm thick) sintered bronze disc and burns in the ambient air below. The heat transfer back to the disc is measured by a spiral water cooling tube imbedded i~ the sintered disc. A copper guard ring and copper sheath, maintained at the sintered bronze temperature (600C), prevents extrarteous heat transfer to the disc due to flames lapping at the sides o? the burner. Thermocouple measurements al various positions in the burner and plenum show negligible heat exchange between the disc and its holder. Heat losses to the fuel gas above the
CELLULARAND TURBULENTCEILINGFIRE5
..... ,: TTT/I- ='~='~' I
'°':,,7::"
i"
Fig. 2. ExperimentalApparatus: a) Burnerand housing. b) Burnerconstructionhalf section. disc never exceed 2% of the measured flame heat transfer. All experimant',d runs were made while the surface of the sintered bronze was uniformly blackened with soot to minimize variations in radiative heat transfer, The use of gaseous fuels of varying composition and flow rates permits controlled simulation of a wide range of solid and liquid ceiling fuels. The heat of vaporization L for these practical fuels is simulated in this experiment by the quotient of the total heat transfer to the sintered disc divided by the total mass transfer through the disc. For liquid fuels, assuming no loss of heat to the fuel bed interior, the local ratio oi heat transfer ~" to mass transfer in" would be equal to the fuel's heat of vaporization L. The experiment with its uniform mass transfer rate closely approximates this situation with L equal to the ratio of total heat transfer to total mass transfer, c~//n. The dimensionless mass transfer driving force B = QYo,/(Movo'L) - Cp(Ts - T , ) / L usually
391 defines the burning characteristics of solid and liquid fuels. Wood, for example, has a B number less than 2.0; gasoline has a B of about 5.0. It was shown previonsly [1] that gaseous fuels can be used to effectively simulate liquid and solid fuels by measuring ~"..over a range of gaseous mass transfer rates, th". The resulting variation in the "heat of vaporization" L =~"/hT" makes it possible to simulate a range of B numbers for practical fuels independently of their transport ~reperties and fuel thermodynamics. The range of B covered in this study includes nearly all common flammable materials.
Experimental Results Heat transfer vs. mass transfer measurements were made for (I) two ceiling diameters 15.24 and 25.4 cm to examine the effects of ceiling size; (2)several rim sizes (1.27-10.16 cm) to examine effect of cell depth, (3)various C2Hd/N2 fuelinert ratios to examine the effect of the flame temperature and the fuel mixture's effectwe heat of combustion; and (4)a variety of fuel mixture molecular weights to determine the influence of fuel density on the gravitational cellular flow. In general, the effects relating to the fuel4nert ratios and fuel molecular weights were more important than the effects of ceiling diameter and rim size. Figure 3 indicates that die heat transfer per unit area is not noticeably influenced by the burner dianteter, This insensitivity to ceiling diameter, which is supported by the theory, probably extends to much larger diameters; however th~shas yet to be experimentally demonstrated, Table I shows the mass and heat transfer results for the 25.4 cm burner for rim depths 1.27-10A6 cm, i.e., up to 40% of the burner dianteter. The heat transfer rates ate comparatively insensitive to rim depth (i.e., cell depth) despite the eight fold variation in rim depths used. The decrease of heat transfer with increasing mass transfer is more noticeable. The heat transfer variation due to rim depth variation is 6-20%; however that due to mass transfer variation is 31-64%. As described below, the theoretical development includes the effect due to mass transfer; however it predicts negligible itifluenee from rim depth.
L. ORLOFF and ]. DE KIS
392 0.3
~
i
em
i
i
to incipient extinction of the flame, The "goose pimple" flow (similar to Fig. lc was observed at this point. Finally, Fig. 5 shows the importance of fuel molecular weight on the heat a'ansfer, This molecular weight effect is presumably due to the ~fferences in fuel density on the cellular flow. For high fuel densities, gravitational instability is increased producing a more intense rotational flow and ~ e a t e r heat transfer.
i
m~,m mA
~
0.2 MIxT~g f ~ Fraction)
~
.
~A
B~r~r Oio, r:l t5.24cm, gl 25.4cm. Lip Depth: 2,54 crtl. I
I
r
f
.000~ ,O00T ,0~ ,C~2 MASS TRANSFER, rn"(gm/cm =- sec)
~S
~eory At presen~ there is no mathematical theory, based on the bas}c combustion equations, which explains
Fig. 3. Heat transfer to ceiling ¢I" vs, mass transfer of fact. ,'~". Data R)r two ceiling sizes. Ethane-nitrogen fuel mixtures with diffireat ethane molar concentrations were used to determine the effect of flame temperature on the ~ " v s / n " relationship. These gases were used b~causc their molecular weights are close to that of air, thereby minimizing molecular weigbl effects on the flow. The results are shown in Figs. 4 and 5. As expected Fig. 4 shows litt;e change in he~t transfer between 0.8C2H6/O.2Nz and 0.5C=H6/0.5N2 molar fuel mixtures since their respective adiabatic stoichJometric flame temperatur~.s are quite close. However, for still lower ethane mass fractions the flame temperature drops sharply. This produces a drop in heat tra~sfer as shown in Fig. 4. The marked decreaSe in heat transfer for the smallest mass transfer point on the 0.2C2HJ0.8N: plot in Fig. 4 may be due
I
I
i
I
O o i
.ix.® ~-~ =0"
D
OA O A
Bum~ Oio2t5.24crn L~ O,~n: 1.27on Mialum (tide Fccc#.,n)
0
.ooo~
,eCII%I,ZNn +5GIHe/.5NI
ra
I
~x~
I
I
I
.oo2
ooo8 .oo~
~o~
rn" (gm/cm=- sec)
Fig. 4. Effect of flame temperatu:c on heat transfer. The mixtures used zeptcscnt an approximately IO% change in flame temperature.
TABLE l Heat Transfer to Ceiling (25.4 cm burner, 50%C=H.-50%N= ) Fuel mass transfer, m" (gm/cm~s¢c)
1.27cm
2.54 cm
3.81 cm
5.08 cm
6.35 cm
7.62 cm
10.16 cm
0,426 x 1O" 3 0.710 ×I0 "~ 0.995 xlO "3
0.287 0.262 0,238
0.297 0.265 0.239
0,285 0,268 0,250
0.285 0.253 0.243
0.277 0.262 0.253
0.257 0.265 0.243
0,293 0.278 0.259
1.28 1.42 1.80 2.08
0.212 0.202 0.185 0.175
0.221 0.204 0.192 0.183
0.227 0.212 0,204 0.196
0.224 0.210 0.202 0.190
0.236 0.214 0.201 0.194
0.219 0.211 0.202 0.202
0.240 0.221 0.221 0.204
xl0 "3 xtO"s xlO "3 xl0 "=
~'~ (cal/cm~ sec) when tip depth below ceiling is equal to
393
CELLULAR AND TURBULENTCEILING FIRES
[2
0 + O+ ~' 0
o
+ 0
"i
0 ~+
+
0
+ 0
}
V
v
O ,t
O
1 .0005
1Z7
~e~,/an%
at.
1.27
1~, ~o.i
10¢s¢ ~ m
IL~I t~L24
Z~ Z~
.~i . . . . . . . .
I.
I
~ .
..
Fig. 5. Effec~of fuel raixturc raolccular weight (i.e., density) on heat transfer. these experimental results. However, using physical arguments, one can quantitatively explain the observed results with considerable precision. Although these arguments are not strictly rigorous, they are hazed upon a conceptual understanding of the essential physical processes, This agreement between theory and experiment for ceiling fires has encouraged us to examine Corlett's experimental results [4] for the closely related problem of burning above horizontal surfaces. As described in the accompanying paper a similar theory is also under certain circumstances applicable to this latter problem. Spalding [5] has shown that for tbrced convection situations there is an analogy between a burning fuel surface and the corresponding nonreacting mass transfer situation. Here we extend this analogy to free convective situations (i.e., fires). We find that the analogy is not as exact, since, in principle the burning can alter the characte; of the flow field. In order to employ this analogy, we must first predict, for ceilings, the free convective heat and mass transfer without combustion. We can make such a prediction by considering available measurements of the c!~ely related problem of free convective heat transfi'.r (without combustion)
between two horizontal surfaces. To make a quantitative comparison, we correct for: a) variable property effects, b)heat blockage (or blowing) effects, as well as c)the fact that the combustion takes place beneath a single rigid surface, whereas the related free convective heat transfer results are for convection between two rigid surfaces.
a) Analogous Free Convectlo~ Studies: There have been numerous studies [6-t.4] of free convective heat transfer (w/thout cerebration) between two horizontal surfaces heated from below. The available turbulent heat transfer measurements, for air, can be ,'epreset:ted by t
1There is some dispute as to tile exponent 1/3 (and consequently the ptopor~ienality constant) in the above formula. For cxanlple, the measurements of Sflvorton [61, Rossby [ 7 ~, and Goldstein el ~ | 101 suggesta valae closet to 0.3 while Jakob 114] (based on Mull and Reihers measurements Jill) Deardorff et aL [8], and Globe et al. {91 suggest that the value of 1/3 as do tile approximate theories of Ma]kus [121, and Kraitchc. [ 13l. This dispute is of negligible quantitative iraportan¢,.' to the present discussion.
L. ORLOFF and J. DE RIS
394
hd
Fgd3(Tn -
- - = 0.075 /
ke
L
T¢)'] t/3
vmamT,~
l
.J
,
(l)
where the Nusselt number on the left describes the ratio of the total heat transfer to pure conduction, an.:l the Rayleigh number in square brackets on tire right characterizes the convective Gory field which induces these augmented heat transfer rates. In the above formula d is the distance between the surfaces, Tn the lower (hot) surface temperature, T~ the upper (cold) surface temperature, Tm the mean temperature, z,m and am respectively, the kinematic viscosity and thermal diffusivity evaluated at Tm. All the measurements providing .'he above result were made with small temperature differences (TH - To), so that we must use judgment when evaluating flows with strongly varying properties. However, it is physically plausible (by symmetry) to evaluate v ~ , am, and Tm at the mean fluid temperature, and evaluate the thermal conductivity, 3.0 in the Nusseh number at a t.~'mperature midway between Tm and the surface temperature, Ts. Strictly speaking the above turbulent formula is valid only for Rayleigh numbers greater than 3 × 104; however it is also approximately valid (within *-20%) for cellular convection for Rayleigh numbers greater than 2 x 103. In view of its approximate valid':ty for cellular flows, we shall apply this formula to both cellular and turbulent flows. The time-averaged temperature distribution measurements of Deardorff et at. [8], as well as Goldstein et at. [I0] reveal a central core w~th a uniform temperature midway between the two surface temperatures. This central core is bounded at the upper and lower surfaces by thin ~ayers with steep temperature gradients. Since the entire temperature drop occurs across these two surface layers, one can ascribe the entire heat transfer resistance to these layers. Tbe fluid in the central core region is apparently free to move as it countercurrently transmits the heat between the two surface resistanc,.• layers. Indeed measurements [8] indicate that much of the heat is transferred by widely spaced jets which build up at
one surface and then "faR" to the opposite surface. Since each layer has half the total temperature drop, one can characterize the layer thickness, ~, by 2 h~l(Tn-Tc)
and since 4" = h(Tn - Tc), Eq. 1 provides
~ Aft
l
(
u,na,. T,~ y , 3
(2)
which is independent c,f b~th the surface separation distance d and th~ lateral dimension of the surfaces. As will becom~ apparent, this indeponder, cc suggests that the analogous ceiling fire burning rates (and related resistance layer thicknesses) will also be independent of both the tim depth and ceiling diameter. It is interesting to note that the Rayleigh number associated with the distance 28 (i.e., when the two resistance layers touch each other) is
R,,,6
=
rg(Tn - T~ ) (~)3 l L
(v,na=Tml
= 2400,
J
which is quite close to the Rayleigh instability :value 1708. This is not surprising, since the distance 5 is physically related to the Ray~eigh instability of a layer heated from below between horizontal surfaces. Now consider the free convection problem beneath a single horizontal surface with a rim such as r~ed in this experiment. Suppose the surface (i.e, sintered metal disk with zero mass transfer) were at a temperature. T,, far below the ambient temperature T , . For turbulent or cellular flow, we expect cold eddies to freely fall and hot eddies to al?he measurements of Goldstcin cr ~ [ 10] .show that (ess~;ntin!iy in:lependent of Rayleigh number) the tim¢~-averagetemperature approaches within 20% of the central core temperature at the distance 6 from the surface.
395
CELLULARAND TURBULENTCEILINGFIRES rise close to the plate surface. Since the heat flux must cross only one resistance layer, one has xn(T~, - TA
heavier specie. The value of Y varies from YT in the gas plenum upstream of the ceiling surthce, to Y~ at the ceilin 3 surface, and eventually to Y~ in the ambient fluid far from the ceiling surface. YT" does not equal Ys, since the mass flux of the specie upstream of the ceiling surface is
which with Eq. 2 for ~, becomes
L .... T.,"J
key h" "
Here it seems plausible to use the temperature difference ( T , - T , ) for the fluid mechanics expression in the square brackets; since for a single surface it is (.os - O , ) rather than (Pc - a n ) which generates the turbulent kinetic energy in the flow field. The above result with the constant 0.15 replaced by 0.14, becomes identical to Fishenden and Saunders [15] turbulent correlation for heated square plates facing upward. This agreement apparently supports our conceptual understanding. Finally because gravity acts directly on density differences rather than temperature differences, it is appropriate to express the above equation as --
)-]1/3
. (.3)
L l,.y ~v P~'A where we have assumed pT constant '~,4fl'l per =(,o,.,os) 1/2 . Here the subscript ~, implies a condition approximately midway between the ambient and surface fluids. b) MassTransfer Effects Consider a porous ceiling, with bounding rim, issuing a dense nonreacting gas mixture into less dense surroundings having the same temperature. For turbulent motion, we again expect a thin resistance layer, as a result of gravity acting on the density differences. In this isothermal case, however, th~ resistance layer opposes mass transfer rather than heat transfer. Suppose that the supplied ges mixture consists of two gas species of differing molecular weights. Let Y be the time average mass frac:ion of the
The fight hand terms describe respectively the gas phase convective and diffusive flux at'the surface. Using the concept of a laminar resistance layer of thickness 8, the diffusive term can be approximated by +9D(Ys - Y.)/8. The predicted mass transfer rate, ~ " , then becomes,
~ ' = (#D),~
[(v,
-
Y= )l(Vr
-
Y~)]
(4)
where fl indicates a condition near the surface. The term in square brackets is commortly known as the mass transfer dri~dng force, B', for such nonreacting mass transfer problems. Later we will redefine this expression for B' when we consider mass transfer with combustion. In general the surface layer thickness, 6, increases as a result of blowing (i.e., mass tramfer). However for very low mass transfer rates 0.e., for B' = (}s - Y®)/( Y T - Y,) < < l)this blowing effect is small and we have,
<,o>,,,.io.,op<,: l, for B" << 1, which is analogous to Eq. 3 with ix replaced by D. For B" large, one can approximately correct for the blowing or mass transfer efi~ct, by multiplying each transport property (i.e., v, a, and L~) by Itn(l + B')/B'], so that in general
L. ORLOFF and J. DE RIS
396
fa, = tpD)sB , ( I n ( ~ B')) = 0o, L.=O~(p, + p~)J )
,,,
This empirical correction procedure, which is in good agreement with the present experimental results, is based on the concept that blowing reduces the influence of all transport properties near the blowing surface. In the past, this same correction proc~'dure has ~so been successful for interpreting: i l) experiments for laminar ceiling fires [1], and (2)an exact theory for laminav fre~e--convectivc burning of vertical and sloping sm'faces [17].
both the initial reactants and £mal products as gas~:s al the ambient temperature. That is, the combustion releases heat, Q, as it consumes the reactant masses MFvF' and ~/oVa'. Thus the molar reaction rate per unit volume is
~=_ Q
~7
~ (Move)
(MFv~)
(s)
Now define ¢ as the sum of the thermal and chemical enthalpies. So that, in terms of
YoQ Eqs. 6-8 become
c) Turbulent Bualmg-SpaldiagAnalogy
Following Spalding [5], we shall show that a redefinition of B extends the mass transfer result to include combustion. For unit Lewis number pCpD[k= 1, the general time-dependent energy and specie combustion equations are,
p~+pu~--=
+ ¢%
ax t
aY, P T7 + "'j
OY,=
f-~-(pD OY,'~ +
or__
"~\
~/
~',
(6)
(v)
T
where h = f~ CpdT is the thermally sensible enthalpy, Yi is t~he specie " i " mass fraction, 4"'is the volumetric heat release rate, fn/" is the specie " i " volumetric mass generatior rate and the xi are the three spatial coordinates. The effects of (1)radiation, (2)high speed f$ow, and (3)specie diffusion driven by pressure and temperature gradients me all ignored. For diffusion flames the heat and mass source terms t~"' and the"' are related through the global stoichiometry relation
provided the Lewis number pCpDfA = 1. Notice that this equation is similar in form to the nonreactive energy and specie equations. Let us now define the "effective latent heat of vaporization," L =~t"/ia~, as the ratio of the heat transfer, 4", toward the surface divided by the mass transfer of fuel mixture, r~", away from the surface. Using this definition the surface boundary condition for O becomes
~aT
_
x
aS
where it is presumed that the presence of the turbulent diffusion flame eliminates all the oxidant adjacent to the fuel surface. Now the concept of a resistance layer near the surface provides
L
t~" (9)
v~,(Fuel) + v~ (Or;de.at) -
v; (Produci',s) + Q(Heat)
where Q is the heat released by burning vF' moles of fuel or equivalently vo' moles of oxidant, with
which is analogous to Eq. 4 for mass transfer without combustion. Here the mast transfer driving force ~' is the te~'m in square brackets; that is,
CELLULARAND TURBULENTCEILINGFIRES
B
397
(Mou~ L)
since by definition h , = 0 and by presumption Yes =0. Here hs ~Cp(T s - T..) and Q is the heat of combustion per mole of fuel provided vF' is set to unity. We see that B = (¢). - ¢s)/L is the total (thermal and chemical) specific enthalpy of the free stream minus the total surface specific enthalpy divided by the heat required to "vaporize" unit mass of fuel mixture. B is usually the dominant chemical parameter for fire situations. Equation 9 ea~ be used to predict ceiling burning rates provided 5 is known. We can use the value of 8 developed for the analogous mass transfer situation, provided tire differences in the fluid flow, for the two cases, do not s~gnificantly influence 8. If this is the case, we have, analogous to Eq. 5,
t~/~ L--F--J 151£P'-°/] ~,
(il)
where D is replaced by ~, p , is replaced by the flame gas density pf, and 3' signifies a temperature and corresponding composition midway between the flame and fuel surface temperatures. d) CrifiqueofBumi~g lhfedlction I. This burning rate prediction involves several important assumptions. The most important, from a q~zantitative viewpoint, is the procedure for evaluating the gas properties used in Eq. 11. While similar choices have been made successfully by others, ou, experience with this problem indicates that a different method of evaluation could easily change the predicted burning rates by as much as a factor of two. (See Fig. 7). This means that the bnrn~g theory involves a somewhat arbitrary, yet quantitatively significant, choice of properties. Such property choices do not affect trends, such as the influence of changes in ,Jimensions, chemistry, ambient pressure, etc.
2. Equation 11 presumes that 8 can be predicted on the basis of a ~ of density Ps entering an ambient fluid of density p$. However the average temperature of th,• gas between y = 5 and the rim is probably considerably lower than T/, Thus we have overestimated ~ e effective density difference and consequently the le,vel of flow t~arbulence. However this effect is small since if we instead were to replace p! in Eq. 11 by ~(Ps "~,of) we would typically reduce the predicted burning rate by only 10%. 3. At very low mass transfer rates (or low Y~.~,), it is possible that some parts of the tm.bulent diffusion flame would be extinguished by the cold fuel surface. In this case some unreacted oxidant would appear at the fuel surface, thereby increasing the total surface enthalpy, 0s, and thus reducing the heat transfer. Such a reduced heat t~ansfer (with one exception) is not evident among the ceiling fire data; however reduced heat transfer is apparent in Corlett's rimless "pool" burning data [4]. It is also possible that at low B or low YFT the fi~h'ne may become embedded within 8. An em]~.'dded flame could significantlyalter the fluid m echanics, thereby violatinga fundamental hypothasis of the Spalding analogy that the fluid mechauics i s uninfluenced by the flame p~itien. In:any case the fact that the ceiling fire data correlation is unaffected by YF~', means that these possibie complicating effects are unimportant for ~e~ling fires studied. 4. The present theory ignores radiation which typically amounts to 10-40% of the to~tl h¢~t : transfer. However, this neglect of radiant heat transfer is partially self-compensating, since flame rad~tion reduces the flame temperatures and consequently also reduces the convective heat transfer,
Comparison of Theory and Experiment The raw data of Eig. S are replotted in Fig. 7 as suggested by Eq. 11. The correlation appears excellent, and in reasonable agreement with theory. However, as explah~ed Uelow the small vertical sep~at/on between theory and experiment is partly fortuitons, since this separation distance
398
L, ORLOFF and ,L D E RIS
E3
THEORY-
fa
~C'
.s / "
~°~
f" 002
' ,. I ,I a i i O~ O,0 so 20 40 oO tOO DIMENSIONLESS MASS TRANSFEROI~IVIN~ FORCE,B
O2
Fig. 6. Effect of property evaluation procedure on agreement of theory and experimenl
~¢,~
' o,4
o;~
I I oe to
-
~l*
I 40
t oo
J
~
Fig. ?. Dimensionless MassTransfq:r:
pIU
~"Le(p.
is quite sensitive to the choice of transport properties, as indicated in Fig. 6. Specifically Fig, 7 plots
I z.o
[31MF.NS;V~LESS M~SS TRANSFER DRIVING FORC~,E~
~'[vo,,~
Ca(T, L,~1. .i
fa" ~v~'a~'P" l'/3 hi"
;S
pr v;pr:~ll"3
.
,
.
.
.
(i3)
(,2)
verses the mass transfer driving force, B. The subscripts B and y indicate the condRions for evaluating the transport properties. In Fig. 7 we evaluated v~,, and ,or al 7 = ½, that is, for the gas mixture composition and temperature midway between the fuel-surface and flame condition..~a on the other hand was evaluated at/3 = ,¼, that is for a gas mixture a quarter-way from the fuel surface to the flame condition. The Appendix describes the detailed procedure for evaluating these properties. In addition we have assumed that the Prandtl number t'r=laCt,/;~=0.75 is independent of both temperatur~ and composition. See Appendix for details. Comparin& Eqs. 1 1 and 12 we see that the, theory predicts
which is the curve in Fig. 7. The choice of ? = ½ and /~ = % represents our best intuitive ct,oice for evaluating the properties. However as shown in Fig. 6 other possible choices would signilicantty shift the position of the experimental points relative to the theory. For example, if we were to evaluate ~a at # = 0 (i.e., at the fuel surface) and P'r and u.r at y = ~ the theory would lie below the experimental data, This theory agrees with all the experimentally examined trends. In particular it predicts: (1)the correct variation with the mass transfer driving toree B (i.e., the correct relationship between in" and ~"), (2) the very weak dependence of #" on the mass fraction of inerts in the supplied fuel mixture, (3) the correct fuel mixture molecular weight dependence, and Iinafly, (4) the
CELLULARAND TURBULENTCEILINGFIRES independence of burning rate and #" on ceiling diameter and rim size. This agreement lends strong support for the validity of the theory, despite the somewhat arbitrary choice of property values used to compare theory and experiment. Conclusions 1) Cellular burning heat transfer is independent of ceiling diameter and rim size. It is anticipated on the basis of the theory, that this finding might be applicable to much larger scale ceiling fires. 2) The cells appear to be o f a Rayleigh character with an internal rotational flow. The cellular flow can be in the form of "'goose pimples," waves, or independent cells with cusps, for increasing mass flow. 3) A theory based on turbulent free confective heat transfer measurements is presented. This theory predicts the correct trends for all experimentally varied parameters which include ceiling size, rim size, fuel mixture supply rates, fuel-inert mass fractions and molecular weights. The dimensionless burning rate is essentially controlled by the B number, which characterizes liquid and solid fuels. 4) The theory is based on constant property (i.e.., Boussinesq) free convection measurements. Therefore considerable care had to be exercised in d(:ciding on the best procedure for selecting property values. Although ~ e selected procedure is somewhat arbitrary, it is consistent with our conceptual understanding of the physical mechanisms and has provided reasonable quantitative agreement with experiment.
Appendix A: Property Evaluation Procedure This appendix describes the procedure used to evaluate the gas properties at any specified condition between the flame and fuel bed. This procedure assumes ( 1 ) a unit Lewis number pC~,)D/?~, (2)negligible radiation, (3)flame thicknesses much less than diffusion distances, and (4) constant effective latent heat of vaporization. Under these conditions one can easily show [17] that there is a unique relationship between the local gas composition and local gas temperature on each side of the diffusion flame. This relationship is independent of the specific fluid motion,
399 whether it is laminar, cellular, or turbulent [17]. In view of this independence of fluid motion, one c a n conveniently deduce the dependence of composition on temperature by considering the relatively simple one dimensional diffusion flame including the effects of variable properties. Let us define ? as the ratio Of specific gas enthalpy at temperature T,t (between fuel surface and flame) over the specific enthalpy of the flame gas, with the fuel surface gas serving as the enthalpy reference. Spt;cifieally let
y =
(A:I)
T,
~i" YirLr~ Cp, dT This definition can be used to calculate T7 for any specified value of" 7. For example, if 7 = ~, T~. is the gas temperature associated with half the flame specific enthalpy. Assuming unit Lewis number, negligible radiation, and constant effective latent heat of vaporization, one can show that the diffusion flame gas specific enthalpy is3
Yir
Cpj dt
L
•
(A-2)
(r + 1)
where the mass transfer driving force is
[3
QYo~ 1 ~% C p d T , MovoL L r.
and the mass of fuel mixture required to react stoichiometrically with a unit mass of oxidant is Yo~F~k (Y~rMo v~) This flame gas specific enthalpy is identical to the adiabatic stoic~ometfic flame specific entbalpy associated with the sapplied fuel and oxidant mixtures. This tbrmala assumes equal speciespecificheats in the temperature range T~ to Ts. This assumption has negligibleeffect when (Ts-T~)/(TI-T.) .¢ I.
L. ORLOFF attd J. DE RIS
400 For a hydrocarbon-inert-fuel mixture burning in air we assume that the reaction goes to complet'on producing carbon dioxide and water vapor with no incomplete products of combustion, .In this case the species present between the flame and fuel bed are nitrogen, supplied inert, carbon dioxide, water vapor, and supplied fuel. Here we "also ignore dissociation species. Under these conditions, the specie mass fractions are in terms of 7:
from which one evaluates P'r for a perfect gas by
o Xx,..,.
Finally the mixture kinematic and dynamic viscosities are approximated [19] by
Y N = [ y ( B - r)/(r + 1) + 1}
v?. py = /.t~,
[~ X,, M?'2]
( 8 + 1)
YN"t =
¥% = Y l r -
Ytr [T (B - r)/(r + D + 1] (B+l)
where the individual ~l.t are given graphically by Spalding [19] as functions of temperature T.r.
YCO27 =
Discussion
Y°=Mc°2vE°s [y(B - r)/(r + 1)+_!!
(B + 1)
MoV~ ~H207 =
¥OcoMU2OV~I20 [7(B " r ) / ( r + I) + I] (B + 1)
Mov"o Y~7
=
Yt.'r-
= X x,..,o
YFr(r +
1)[TtB- r)/(r + 1) *
1]
(B + 1)
Yo~, = O.
For given B, r, 7, oxidant mass fractions, and • supplied fuel-mixture compositions one can immediatel2¢ evaluate the local specie mass fractions Yt.r. Then, using Prothero's [118] sixth order polynomial aI)proximations for the JANAF specie specific heats Cpi together with Eqs. A-I and A-2, one can calculate T.t. The calculation of T.r is included in the computer program used in this study [20]. We now can calculate the gas properties P-t, v't and V,~. The molar fractions Xrr can be expressed in terms of the mass fractions YPt by
Yi~
Y~
Following this procedure the computer program was used to evaluate Pa, Pl, Pa, P~,, and v~, for each data point. Since the ca!culation ignores dissociation effects the computed values of a f are about 10-20% too low; however this discrepancy induces only about a 1% error in the computed value (as-P.t) tpa applied to each data point. For ]3 or 7 < 1 / 2 dissociation effects are negligible. With the possible exception of including the effects of radiation and products of incomplete combustion, the calculation procedure represents our best present estimate of gas properties. These property estimates have improved our correlation of ceiling fire data for fuel mixtures having significant molecular weight differences.
Nomendature
B Cp d O g h L Mi rh"
Pr
QYo~/(Mov'oL) - Cp(T~ - T=I/L Mass Transfer Driving Force, (.). Specific heat of gas, (E/MO). Parallel plate separation, (/). Specie diffusivity, ff2/t). Gravitational acceleration, (lit 2 ). Specific enthalpy, (ELM). Latent heat of vaporization, df'/dn", (E/M). Molecular weight of specie "i". (M/mole). Mass transfer flux, (M/lat). Prandtl number IZCp/A, (.).
401
CELLULAR A N D TURBULENT CEILING FIRES
Heat transfer flux, (E/I 2 t). Heat of combustion, (E/mole). r YO~klFu'F/(YFrMOP'O) stoichiometric ratio, (-). T Temperature, (O). ui Velocity in directioli ' T ' , (l/t). xt Coordinate in direction " f ' , (1). Xt Molar fraction of specie " f ' , (-). y Distance normal to surface, (l). Yi Mass fraction of specie " i ' , (,). a Thermal diffusivity, (/2 It). Resistancelayer thickness, (0. v Kinematic viscosity, C,2/t). ~/~,v~" Stoichiometdc coefficient of reactants and products respectively, (mole). A Thermal conductivity, (E/lOt). p Density, (M/I 3 ). ~b Total (chemical plus thermal) specific enthalpy, (ELM). /~ Dynamic viscosity, (M/lt).
~" Q
SubscriptS C cold f flame ',F Fuel 'Fs Fuel at surface FT Fuel upstream of surface (i.e., in supply plenum) H Hot m mean 0 oxygen s surface T upstream of surface near surface y gas phase-see Eq. A-I ambient
The authors are indebted to Professor Hubbard l¢. Emmons o f Harvard University for suggesting the use o f gaseous fuels to vary the B par,rmeter. We are thankful to Dr. Raymond Friedman for his continued encouragement during the course o f this wort~
References I. Odoff, L., and de RIS, J., Thirteenth Symposium [International] on Combustion, "file Combustion Institute, Pittsburgh, Pa. (1971). Also FMRC Technical Report Self. No. 19720-2, April, 1970. 2. l~atkstein, G. H., Fourth Symposium [International) on Combustlon. The Williams & Wilklns Co., Bal~mote, Md. (1953), p. 44. 3. Maskstein, G. H, and Somet~ L. M., Fourth Symposium [International) on Combustion, The Williams & Wilkins Co., Baltimore, Md. (1953), p. 527. 4. Curlett, R. C., Combustion attd Flame 14, 351 (1970). 5. Spalding, D. B., hit. J. Hen: Mass Transfer t, 192 (1960). 6. S~veston, P. L., Forseh. lug. Wes. 24, 39 and 59 (1958). 7. Ro~by, H. T., Y. Fluid Mech. 36, 309 (1969). 8. Dcatdorff, .I. W,; and Willis,G. E.,J. Fluid Mech. 28, 675 (1967). 9. Globe, S., and Dropkin, D.,J. Heat Transfer 81, 24 (1959). 10. Goldstein, P~ J., and Cha, T. Y., Progressin Heat and Mass Transfer 2, Pergamon Press, New York (1969), p. 55. I1. Mull, W., and Re,her, H., Beiil. Gesundh. lug. I, 28 (1930). 12. Malku~, W. V. R., Peat. Roy. Sue. (London) A225, 185 and 196 (1954). 13. Ktaiehnan, R.H.,Phys. Fluids 5, 1374 (1962). 14. Jakob, M., Heat Transfer 1, Wiley, New York (1949), p. 522. 1.5. Fishenden, M., and Saundets, O. A.,An Introduction to Heat Transfer, Oxfozd, New York (1950). 16. HlnMey, P. L., o~d Wraight, H., J.F.R.O. Npte No. 743, Joint Fire Rescaleh Organization, Botehamwo,~l, llerts, England, 1969. Abstracted in Fire Res. Abst. and Rev. 11,206 (1969). 17. Rim, J. S., de Ri~.,J., and Kl~e~.er, F.W., Thirteenth Sympo~.ura (luternationaO on Combustion. The Combustion Institute,Pittsburgh,Pa.(1971). 18. la~othezo, A., Combustiou aud Flame 12, 399 (1969). 19. Spalding, D. B., Couveetire blurs Transfer, McGrawHill New York (1963), p. 138. 20. Orloff, L., and de Ris, 3., FMRC Technie',d Report Scr. No. 19720-3, Januat'y, 1971.
[Received June 1971; revised version received November 1971)