Froude modeling of pool fires

Nineteenth Symposium(International) on Combustion/The Combustion Institute, 1982/pp. 885-895

FROUDE

MODELING

OF POOL

FIRES

LAWRENCE ORLOFF AND JOHN DE RIS Factory Mutual Research Corporation 1151 Boston-Providence Turnpike Norwood, Massachusetts 02062 We present an algorithm for the burning of moderate-scale (0.1-0.7 m dia.) pool fires in terms of the pool scale and fuel properties. Previously one could not solve the energy balance at the fuel surface due to lack of information on the feedback radiation. In general this feedback radiation depends on the flame radiation temperature and absorption-emission coefficient, as well as the flame size and shape. Through the use of scanning radiometer measurements on gaseous burner fires it is shown that the flame size and shape are independent of fuel chemistry for a given overall rate of combustion heat release, as suggested by Froude modeling. The average absorption-emission coefficient at an assumed global flame temperature is related to the radiant fraction of combustion heat release. An analytic flame shape expression is developed which provides the mean beam length required for predicting the radiative feedback. The remaining terms of the energy balance are solved using established procedures.

I. Introduction It has been a long established goal of workers in the fire field to model fire behavior for given characteristics of the fuel array. Those characteristics include its spatial configuration, fuel thermophysical properties, and flame radiation properties. In many realistic situations, such as enclosure fires, (l-z/ a mathematical model is required to describe the complex interactions of the fire with its environment. Physical modeling schemes (e.g. Froude (3-5/ and pressure/6) modeling) have been applied to relatively simple fuel structures, for example wall and pool fires. Those schemes are often insufficiently general because they do not properly scale all of the important parameters (e.g. fuel surface reradiation). Mathematical modeling schemes on the other hand are more readily accepted, because they can maintain the correct physical relationships even though their use is presently restricted to simple geometries. An objective of many current fire studies, such as the present work, is to describe particular fire processes with the simplicity required for subsequent incorporation into comprehensive fire models capable of predicting realistic fire development scenarios. We present here a model, applicable to a broad range of fuels, that predicts burning rates and external radiative heat transfer from moderate-scale (.1-~.7 m dia.) pool fires. The results are expressed in terms of pool size and conventionally measured fuel properties. A Froude modeling approach is used to interpret the size and shape of

actual pool flames, which are non-homogeneous and non-isothermal (7-s/ in terms of equivalent radiatively homogeneous properties. Where viscous effects are small, as in turbulent pool fires, Froude modeling theoretically reproduces the temperature distribution and overall flow pattern independently of fuel stoichiometry. The model thus allows flame shape measurements that were obtained in a gaseous fuel burner experiment to be generalized for other fuels.

II. Experiment and Data Reduction Scanning radiometer (s) measurements were made on 43 gaseous fuel pool fires in the present study. Previous results (s) for a 0.73 m diameter PMMA pool fire are also included. Both a representative low-soot (methane)and a sooty (propene) fuel were supplied to two circular water-cooled sintered bronze burners of diameters equal to 0.38 m and 0.76 m. The nearby laboratory walls and the ceiling hood were painted black and water cooled to prevent any external heat feedback. Combustion products were usually removed from the laboratory through a passive centered chimney in the ceiling hood to minimize convective disturbances. A minimum forced exhaust removed smoke from the largest propene fires. Air entered the laboratory through passive louvered vents, The total convective and radiative heat feedback to the burner surface, q~-, was measured from the temperature rise of the water circulating through flattened copper tubing embedded

885

886

FIRE I--MODELING

C*rem;c

f'apgr L.~p ) .~ I!e ion

F;c. 1. Schematic of gaseous fuel burner with (a) no lip and (b) lip. The sintered bronze is maintained at constant temperature by water circulating through imbedded flattened copper tubing, not shown in the figure. in the sintered bronze. Lip effects (13) were studied in three pool boundary configurations: 1) a 45 ~ bevel (zero lip); 2) a circumferential ceramic paper collar extending 13 mm; and 3) 38 mm above the pool surface. Figures la and lb show burner configurations without and with a lip. There is a marked difference between the zero lip and lip fires; the zero lip fires are tall and neck into a narrow column a short distance above the fuel surface, while the lip fires are thicker and shorter due to turbulence tripping at the leading edge. The scanning radiometer measurements consist of groups of one hundred horizontal sweeps by a 13 mm diameter collimated beam radiometer at each of twenty individual flame heights. The vertical separation between successive heights increased with increasing distance from the pool surface to provide more detailed information in the lower part of the pool fire, which includes most of the radiative flame volume and thus controls the burning. Effects due to flame fluctuations were minimized by averaging each set of 100 scans distributed over a one-minute interval. This procedure provides the average flame radiance, N(x,z), at five hundred horizontally-spaced locations, x, at each of the twenty heights, z, above the pool surface. This comprehensive radiation map was used to define the flame shape and height for evaluation of the effects of fuel, combustion heat release rate, pool scale, and lip size. A second, wide-angle radiometer viewing the entire fire provided measurements of radiative flux at a target location five pool diameters from the fire, This measurement was used to verify spatially integrated scanning radiometer results. Both visual observation and scanning radiometer measurements confirm time-average axially symmetric fires without significant flow disturbance. This is due in part to the very stable pool configuration, to which gas-

eous fuel is supplied at a constant rate, and to the absence of personnel inside the laboratory while measurements under computer control were in progress. A considerable amount of data (106 data points) were generated and recorded for each experiment. However, the present model was developed in a manner that would allow radiation measurements with a wide-angle radiometer viewing the flames at sequential heights through a horizontal slit. This procedure simplifies data reduction and permits use of more widely available radiometers. The radiant power per unit height, #'(z), is obtained from the radiance scans:

q'r(Z)= F* N(X,Z)&.

(1)

The effective flame height, Zf, is defined here as the first moment of q'(z) with respect to z,

L

4'(z) zdz

(2)

Z~ = --~

Jo q' (z) dz This definition of Zf has the advantage of not relying on the somewhat arbitrary visual criteria often used to indicate the average height of the flame tip. Further, the flame tip is less meaningful for determining the radiative feedback. The radiative fraction, • of the theoretical combustion heat release rate, Qc, is defined by:

O_r 47"f x,=

--

Qc

=

I~'(z)dz rhAHc

(3)

where Q~ is the flame radiative power output, rh is the fuel mass transfer rate and AH c is the fuel heat of combustion per unit mass. Complete combustion for gaseous fuels is assumed, XA -- 1, where • is the fractional completeness of combustion. This is supported by Tamanini'st14) findings of approximately 97% completeness for propane pool fires. For most liquid and solid phase fuels • lies between 0.6 and 0.95. t9) In this study we denote QA as the actual heat release rate, QA = XAQc' The measured flame volume, Vy., necessarily relies on the assumption of an arbitrary radiation threshold to distinguish between "flame" and "background." For example, a very low threshold yields a cylindrical flame volume of diameter defined by the pool surface. In previous work (13) flame volumes were derived from photographs, with

FROUDE MODELING OF POOL FIRES the threshold equal to a constant fraction of the maximum flame luminous intensity. It was shown that equivalent flame shapes can be derived from scanning radiometer data. Here we assume the flames are homogeneous (constant average absorption coefficient, /~) and isothermal (constant flame temperature, ~'f) and find the flame radius, S(z), required to produce the measured q'~(z):

887

ture. Figures 2a and 2b show reasonably good correlations for each individual lip size, the methane fires exhibiting somewhat higher flame heights than propene or PMMA. Increasing the lip size produces a relatively small decrease in flame height. Froude modeling also suggests that the average volumetric heat release rate, Q" is weakly dependent on Q~. Assuming flames of similar shape, q" - Q ~ / z R 2 - z~'2t z R 2 - z -x'~ - Q2 "~.

J - S(z)

9[1 - exp(-2/~f~c/S'2(z) - x2)] dx. (4) This definition of flame radius avoids the use of a threshold9 Flame volumes were calculated assuming "if = 1200 K and kf = 0.6 m -1 for methane, and ]'f = 1200 K and /~f = 1.3 m -1 for propene fires, / ~oe

vs =

~ |

S%)dz.

(5)

J0 To employ a simple overall radiation model we must choose a single effective flame temperature and remain consistent with this assumption9 easured (7'21) Schmidt temperatures in pool fires range from 900-1470 K. They vary_with height, radius and fuel type. The choice of Tf = 1200 K is plausible, although somewhat arbitrary9 It is important to note that Eq. (4) defines S(z) in terms of both the chosen Tf and measured q'(z). The choice of a higher 3"f would decrease the observed S(z). The mean-beam-length depends solely on S(z). When all put together, the calculated radiative feedback (Eq. (18)) is only weakly sensitive to our chosen Tf. However, one must be careful to be consistent. Thus Tf = 1200 K for all fuels in this study. It is shown below that/~f T} is obtained from measurements of •

~

HI. Froude Modeling Froude modeling predicts that for geometrically similar optically thin turbulent fires (for which viscous effects can be neglected) Qc must scale with the five halves power of the length scale. (a) Thomas, (4) Steward, (5) and Zukoski et al(ts) correlate flame height with the two fifths power of the burning rate, which is consistent with Froude modeling. Figures 2a and 2b show that the dimensionless flame height, Zf/R is correlated with the dimensionless heat release rate, Qc/p=Upl~g~ ~ 1/'2,'2/5,-) /n, where R is the pool radius, g is the gravitational acceleration, and p=, Cp and T= are respectively the ambient density, specific heat, and tempera-

(6)

Figure 3 shows that Q " = Qa/vf is constant across the range of experimental parameters, with the exception of a group (designated as "low Bnumber") of measurements corresponding to 0.76 m dia. pools with low flame volumes9 Those fires exhibit a radically different flow: the flames are uniformly distributed across the pool surface, with no sign of peaking at the mid-axis. Similiar flows have been observed in pool fires on low B-number fuels, such as sand-wick acetone. (16) Those fires entrain air from above as well as laterally. The resulting flame shapes are not similar to the generally conical fires of typical fuels, and are thus not treated in the present study. We find that

QA = ( ~ Wf, where Q~ = 1200 kW/m 3

(7)

which suggests a zeroth, rather than - 1 / 5 t h power dependence for Q". This discrepancy suggests that the flame height, Zy, should be proportional to Q~/3 in the second proportionality of Eq. (6) rather than the Froude modeling 2/5 power. The fraction of Q~ released as thermal radiation, • is conveniently measured, and has potential usefulness as a modeling parameter. We develop here an expression for • in terms of kfffT~ and Q'". In the limit of optically thick flames, (~rthick = ~

(8)

where Q~ is the flame radiative power output, cr is the Stefan-Boltzmann constant and A b the bounding flame surface area. In the optically thin limit,

Q~. = 4 [r For intermediate optical thicknesses, approximately

(9)

[~fLm,one has

Qr = crt}Ab [1 - exp (-]~fLm)],

(10)

where the mean beam length, Lm, (17) is optimally defined by: Lm = 3.6

Vf/A b.

(11)

888

FIRE I--MODELING 2.5

:

t

u

I

n

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I

POOL RADIUS R - . I g m - OPEN SYMBOLS R=.38 m - SOLID SYMBOLS

LIP (mm) PROPENE METHANE

13

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A

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.5

0

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. ;25

I .5

i .75

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I

I

I

1.25

1.5

1.75

2

(Qc/P~CpTcogl/2)

2/5

R

FIG. 2. (a) Dimensionless flame height as a function of dimensionless combustion heat release rate for l a m m and 38 m m lip pool fires, (b) zero lip pool fires.

FROUDE 25,0

MODELING

'i LIP

(mm)

0

L~

13

[] 0

METHANE

0

Olh.

o

SOLID SYMBOLS - 0 . 7 B m DI^.

o/

150

./

Low .B-Numbe.r..~EE-,.

too

J

PMMA

0 0

OPEN SYMBOLS - 0 . 3 8 m

9~

889

t

PROPENE

Be

200

OF POOL FIRES

9

(~,,,=

Q/Vf

=

1200 kW/m 3

,*

50

0

IW

l

0

..

.05

1

!

. l

.15

Vf

I .2

.25

(m 3)

FIc. 3. Actual pool fire h e a t release rate as a function of f l a m e volume. A linear r e g r e s s i o n fit s u g g e s t s a constant volumetric h e a t release rate. Data labelled "'low B - n u m b e r " r e p r e s e n t fires ( s u c h as t h o s e s e e n on low B - n u m b e r fuels) that do not peak toward t h e mid-axis.

45

I

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I

PROPENE

4

l

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[]

[]

II

II II

l)

0

35 0 3 0 25 0 <>

15

@0 V

9 METHANE

LIP (mm) PROPENE

.!

0

POOL RADIUS

METHANE

9

Y

R-.Ig m - OPEN SYMBOLS

o05 0 0

lO0

I

I

I

I

200

300

400

500

QC/Ap

9

3B

R-.3B m - SOLID SYMBOLS

I

BOO

0 I

700

BOO

kW/m z

FIG. 4. Radiative fraction of c o m p l e t e c o m b u s t i o n h e a t release rate, • release rate p e r unit pool surface area.

as a function of c o m b u s t i o n heat

890

FIRE I - - M O D E L I N G flames. The methane fires (/;f = 0.6) would be more nearly represente_d by Eq.-(13) than the optically thicker propene (kf = 1.3) pool fires. Recalling that flame volumes were obtained assuming the same Tf = 1200 K for both fuels, Fig. 4 and Eq. (13) suggest that • is approximately proportional to /~y for these intermediate-scale fires.

The factor 3.6 is conventionally used instead of 4 to produce a better overall approximation for various optical thicknesses. Noting that Qc = Q" Vs we have in general, Xr -

3.6(rT~XA[1 -

QrXA

9 " QAVf

exp (-/~yLm)]

(12)

QA " Lm

or for small f~fLm, IV. Flame Shapes

• -~ 3 6c[~frF~•

(13)

To predict the radiative heat feedback we need a procedure for estimating the mean beam length, Lm, which is associated with the size and shape of the flame. Fig. 5 shows three typical flame shapes obtained from radiation measurements using Eq. (4). The various empirical flame shapes, S(z)/R

Under geometrically similar conditions Froude modeling suggests that Q . . . . Q~-1/5 so we would expect • - Q--1/5 for ophcally thin fires. Figure 4 shows Xr as a function of the heat release rate per unit pool area, Qc/Ap, for propene and methane

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METERS

FIG. 5. Comparison of measured flame shapes with profiles of equivalent flame volume generated by an analytic expression (Eq. (14)); a) pool radius R = 0.19 m, z o = 0.6258, "q = 4.891; b) R = 0.38 m, Zo = 0.4972, xI = 1.956; c) R = 0.38 m, zo = 0.7840, "q = 3.046.

FROUDE MODELING OF POOL FIRES 1.4

I

LIP (mm) 0

1.2

I

I

I

I

Anolytic Florae Shope

0

/l

OPEN SYMBOLS - O. 38m DIA. SOLID SYI4BOLS - O. 76m DIA.

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PROPENE METHANE PMHA A V

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FLAME ASPECT RATIO, ~/ ; TR---~3"~-

FIG. 6. Dimensionless mean-beam-length, Lm/R, as a function of flame aspect ratio, xI. The curve represents results for an analytic flame shape (Eq. (14)) with 0.4 < zo < 3.23, R = 0.38 m. above the lip can be approximated by the following analytical expression,

\

Zo / I/a (14)

( I + Z z o Z L ) [ I+(z-zL]3]\ Zo / J where z L is the lip height and z o is the characteristic height of the curved flame above the lip. The flame volume is

in" at the base due to lateral entrainment, and the downstream cresting at the visual flame tip, The analytical and measured flame shapes in Fig. 5 have identical flame volumes. Froude modeling implies that the flame shapes are similar for scaled (Qc• independent of the fuel chemistry. This suggests that the dimensionless mean-beam-length, Lm/R = 3.6 Vf/RA b is correlated by the flame aspect ratio, ~q = Vf/ (~rR3/3). The flame aspect ratio is defined here as the ratio of the height to the base radius of a right circular cone of the same volume. The bounding flame area is given by Ab = ~r{R 2 + 2 RzL

S(z)[1+ (dS(z)/dz)2]l/Zdz

+ 2 = 7rR2(ZL +

Zo/2),

(15)

where ~ = (z - ZL)/Zo. The exponents 3 and 1/2 were chosen to compress the flame radius at large z, in conformity with observation. The expression generates a hyperbolic-like curve which replicates the "necking-

(16)

L

where S(z) is a polynomial spline fit to the measured flame radius (Eq. (4)). Similarly for a given ZL/Zo the dimensionless mean beam length Lm/R is a unique function of the assumed flame shape P(z/zo), where zo is proportional to the flame volume or aspect ratio.

892

FIRE I - - M O D E L I N G where AHg is the fuel heat of gasification per unit mass; q'r is the average radiative feedback,

TABLE I Algorithm for pyrolysis rates Energy Balance: (~" +

4~n,' =

qr' = O"F~[1 - exp (-/~fLm)].

7/~'tAHg + 0~r

Re-radiative Heat Loss: q", = ~(~-4_ T 4) Convective Feedback: 4'~'oo~= h / C , [AH~(x, - Xa)r/xa - Cp(T,- T=)] E(y)

where E(y) = y/(e ~ - 1) and y = rig' C J h h/Cp = 8 g/mZs for pools without lips and 6.5 g/ m2s for lip fires.

(18)

Here "if = 1200 K and/~y is expressed in terms of the fuel radiation characteristic, Xr (Eq. 12 or 13). q"onv is the average conyective transport to the fuel surface. Stagnant film theory gives its variation with mass transfer as q~onv = h / C p [AHc(• -

•

-

Cp(T~- T=)] E(y) (19)

Radiative Feedback: q" = (r"i~ [1 - exp (-/~L,,)] I -ln /~r =

1

X~ "~"L ~A ~ -I [ 37~ - ' ~ } XAj Lm

Q~' = 12oo kW/m a

Lm/R = C o + C1'q + C2-qz + Ca'q 3 + C4"q4 ~- C5T[5, where r I = 3rh" XAAHc/Q~'R

and Co = -0.71106 x 10 -~ Ca = 0.18214148 • 10 -l C1 = 0.45453596 C4 = 0.15152567 • 10 -2 C2 = -0.11098310 Cs = 0.48573929 • 10 -4 Table I. Algorithm for calculating pyrolysis rates given fuel properties AHo AHg, T,, XA, • and effective pool radius, R. For convenience, a fifth-order least-squares fit to L,,/R vs. "q is provided for zero lip height. The data for this fit was generated from Eqs. 11, 14 and 15. Fig. 6 compares the analytical (for z L = 0) and experimental correlations of L m / R versus flame aspect ratio where the various symbols denote the respective experiments given in the previous figures. Generally the data for larger diameter fires have lower flame aspect ratios on the left. There is less agreement between the analytic and experimental correlations for the lowest aspect ratios due to the "low B-number" character of the measured fires.

V. Algorithm for Pyrolysis Rates Fig. 6 provides the key for solution of the energy balance at the fuel surface given the pool dimension and fuel properties. Neglecting minor terms such as the radiative feedback due to the lip wall, the fuel pyrolysis rate is given by .... fit"= (q~ + qconv

q"" r r )~/AH, / g

(17)

where E(y) = y / ( e Y - 1) and y = rh" C p / h . Os) The value of h / C p was chosen to satisfy 0~ = q" + c~.... where q~- is the measured total heat feedback: h / C n = 8 g/m2s for pools without lips, and 6.5 g/m2s for the more turbulent lip fires. (19/These values of h / C p are quite realistic. The higher value for the zero lip fires derives from their higher laminar heat transfer near the pool base. q~r is the re-radiative heat loss by the surface at temperature T s, q~r

=

(r(T4 - T4) 9

(20)

Throughout this work we have successfully assumed that the effective flame radiation temperature "if = 1200 K regardless of fuel type. This places the burden for calculating the correct flame radiation on the absorption-emission coefficient kf which can be estimated from measurements of Xr using Eq. (12). Thus we are in a position of calculating burning rates rh" only in terms of R, AH c, r, AHg, T,, XA and Xr using Tf = 1200 K and ]~f, L m and ~q as intermediate variables. This is summarized in Table I. Fig. 7 shows the four primary heat transfer components at the surface of a 0.73 m PMMA pool fire. Csl The net surface flux has two zeros or points of energy balance corresponding to unstable burning on the left and stable burning on the right. The net curve demonstrates how sensitive pool fire pyrolysis rates are to their controlling parameters. This model has been applied to a number of reported pool fire configurations for which the fuel properties are well defined. Details of the calculations are provided in Ref. 19.

VI. Concluding Remarks This study, its precursors, and related work in progress, have a common central objective: development of an algorithm for predicting pool fire pyrolysis rates given their scale, configuration and fuel properties. It has been recognized for a decade

FROUDE MODELING OF POOL FIRES

I

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PYROLYSISR^TE g/rn2s FIG. 7. Energy balance at the surface of a 0.73 m diameter, 13 mm lip PMMA poo] fire. Fuel properties used to calculate the heat transfer components are: AHc = 24,9 kJ/g; AHr = 1.611 kJ/g; T~ = 654 K; • = 0.84; • = 0.40; h/Cp = 6.5 g/m~-s; r = 0.1208. that in turbulent pool fires, radiative feedback dominates the surface energy balance. Yet, this term is the most difficult to resolve because it requires treatment of spatially varying radiative quantities (7's) in the flame; both the spatial distribution and the magnitude of the quantities (Tf,kf) strongly influence pyrolysis rates. (7) Two approaches have been undertaken to deal with this difficulty: 1) a complete description (7's) of the spatially varying local radiation properties, and modeling of the distribution with respect to scale and fuel; 2) a simplified treatment in which the flames are assumed to be homogeneous (/~f) and isothermal (q'f). Both approaches have advantages and drawbacl~s: the nonhomogeneous treatment relies on detailed local radiative properties within the flame; the radiative feedback is rigorously calculated. Important non-homogeneous effects such as radiation blockage (v'12) by cold pyrolysis can be ex-

plicitly treated. However, at present, this approach requires considerable experimental measurements and tedious computations. The second approach requires a characterization of three interdependent flame quantities, /~f, ~f and S(z), the flame shape. This treatment is u~'scribed in the present and in previous studies. (13'2~ Its advantages are: 1) it permits use of very simple analytic expressions for radiative feedback and heat transfer to a remote target; 2) it was shown by recourse to Froude modeling that the flame sootiness characteristic, /~/, is proportional to the radiative fraction of combustion power output, • and 3) also by application of Froude modeling principles, the flame shape is defined b y the pool radius and combustion power output, Q, and is not dependent on fuel chemistry. The pyrolysis rate algorithm shown in Table I thus relies on a limited number of conveniently measurable properties: AH c, AHg,

894

FIRE I MODELING

r, T,, XA, XR and the pool radius. A major drawback of the simplified approach is that by assuming homogeneous and isothermal flames, we have inherently ignored non-homogeneous effects such as radiation blockage, and thus limit the range of scale to which the model can be applied. This radiation blockage becomes important for pool diameters greater than 0.7 m. Beyond 1 m the pyrolysis rate levels off/1~ at magnitudes that suggest optically thick flames with substantial radiation blockage. At scales below 0.1 m the flames tend to be laminar, and radiation is less important relative to convective/conductive heat feedback. Modak's17) measurements of local radiation properties in a 0.73 m diameter PMMA pool fire suggest significant radiation blockage near the center of the pool, but considerably less blockage over the larger annular area adjacent to the pool perimeter. With increasing diameter the fuel-rich volume increases, and thus the radiation-blocking pyrolysis gases presumably cloud an increasingly large portion of the fuel surface. The best test of the reliability and application range of the present model resides in comparison of predicted with actual experimental results. There is limited data for pool fires with well-defined fuels burning under carefully controlled conditions. We have achieved reasonably good predictions of PMMA and methanol pyrolysis rates for pool diameters in the 0.3 m--0.8 m range. (19) The present study illustrates an application of Froude modeling principles to the development of a homogeneous fire radiation model. Such principles might be extended to non-homogeneous situations involving significant radiation blockage. However, this is a topic for a future study.

kf

N P

4' q"

Q r R

S

Surface area (m2) Specific heat (J/gK) Gravitational acceleration (m/s 2) Heat transfer coefficient (kJ/m2K) Average absorption-emission coefficient (m -I) Mean beam length (m) Pyrolysis rate (g/2s) Radiance (kW/m) Radius of analytic flame shape (m) Radiative flux per unit length per unit solid angle (kW/m sr) Convective or radiative flux per unit area (kW/m 2) Power output (kW) Fuel/air stoichiometric mass ratio (-) Radius of pool surface (m) Radius of measured flame shape (m)

m" C~/h (-)

zo AHc, AHg ~1 p cr •

Vertical coordinate (m) Flame height (m) Flame shape factor (-) Heat of combustion and total heat of gasification (kJ/g) Flame aspect ratio (-) Density (g/m ~) Stefan-Boltzmann constant (56.7 • 10-12 kW/m2K 4) Actual or radiative fraction of complete combustion power output (-) Flame shape factor (-)

Subscripts A b cony f P r rr S

T

Actual Bounding Convective Flame Pool Radiative Re-radiant Surface Total Ambient

Acknowledgments The authors are indebted to George H. Markstein, who provided the scanning radiometer for use in this study. The burners were fabricated by Lawney Crudup, Jr. This work was supported in part by the National Bureau of Standards.

Nomenclature A Cp g h

Temperature (K) 3 Flame volume (m) Horizontal coordinate (m)

T Vf x y z Zf

REFERENCES 1. EMMONS, H. W., "The calculation of a Fire in

2.

3. 4. 5. 6.

a Large Building," 20th Joint ASME/AICHE National Heat Transfer Conference, Paper 81HT-2 (August 1981). QUINTIERE, J., STECKLER, K., AND MCCAFFREY, B., "A Model to Predict the Conditions in a Room Subject to Crib Fires," First Specialists Meeting (Int.) of the Combustion Institute (July 1981). DE RIS, J., "'Modeling Techniques for Prediction of Fires," Applied Polymer Symposium No. 22, 185 (1973). STEWARD,F. R., Combustion Science Technology, 2, 203 (1970). THOMAS, P. H., Ninth Symposium (Int.) on Combustion, The Combustion Institute, 278 (1963). ALPE~T, R. L., Sixteenth Symposium (Int.) on

F R O U D E MODELING OF POOL FIRES Combustion, The Combustion Institute, 1489 (1977). 7. MODAK,A. T., Fire Safety Journal, 3, 177 (1980/

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COMMENTS Prof. Toshisuke Hirano, University of Tokyo, Japan. For estimating the radiative heat flux from the pool fire, you have adopted kr in your paper. Because the effect of radiation adsorption by the low temperature annular part surrounding the high temperature core, the determination of the mean temperature t (on which kf is considered to depend) seems difficult. How do you decide on T? Does T depend on the scale of the pool fire?

Author's Reply. Modak's measurements on a 730 mm diameter PMMA pool fire indicate a relatively cold core, presumably due to pyrolysis gases. The present model assumes the fire is homogeneous and isothermal. Radiation blockage by the cold core, which is important at pool diameters > 800 mm, is not accounted for. We find that non-homogeneous effects can be neglected when considering the overall radiative heat feedback in pool fires, q", between 100 and 800 mm diameter. Assuming homogeneous flames, ~" is a function of _Tf, ky, and the flame shape S(z). We have selected Ts = 1200 K as a plausible starting point for all scales and fuels; ~f and S(z) are derived quantities which compensate for the im-

posed value of Tj. Our resulting estimate of q" is not sensitive to the choice of Tf. A fixed single value for I"s yields consistent results for fire diameters less than 800 mm.

Dr. Henry Mitler, Harvard University, Cambridge, Mass. Note that you can mimic the effect of radiation blockage by absorption, by increasing the "effective" heat of vaporization so that it is a function of pool size.

Author's Reply. This is a possible engineering representation of the radiation blockage effect. We feel that before one invokes a simplified treatment of radiation blockage one should have a firmer understanding of the nonhomogeneous soot and gaseous specie distributions. We are currently measuring the effects of scale, fuel chemistry and lip size on these quantities with the goal of extending Froude Modeling concepts to very large-scale pool fires where radiation blockage controls the rate of burning.