Fire growth rates in wood cribs

COMBUSTION A N D F L A M E 27,267-278 (1976)

267

Fire Growth Rates in Wood Cribs MICHAEL A. DELICHATSIOS Factor. Mutual Research Corporation, 1151 Boston-Providence Turnpike, Norwood, Massachusetts 02062

The burning history of a wood crib ignited at the center of its base has been investigated theoretically and experimentally. A simple energy-balance model has proven successful in predicting the radial fire-spread rates and mass burning rates for varying crib geometries with accuracies of -+ I S . Exceptions to the validity of the model were only noted for very densely packed cribs for which significant lateral spread occurred simultaneously with vertical fire spread. Cribs consisting of sticks with thicknesses of 0.635 cm, 1.905 cm and 3.17 cm were burned in the present experiments. Analysis of pressure modeling experiments has also shown that pressure modeling cannot, in general, model the fire growth rates in wood cribs.

1. Introduction Wood cribs, which are simple, symmetrical, unenclosed cross piles of wood, have been extensively used [1-6]1 to study the growth and propagation of fires. A typical wood crib is shown in Fig. 1. In most of the previous studies [1-4], ignition was initiated at the entire base area of the wood cribs. After a short growth period, quasi-steady burning was established. Fons et al. [5] and Thomas et al. [6] investigated the flame propagation in elongated wooden cribs, whose short edge was ignited. Little attention has been given to the analysis of fire growth rates, when the fire starts at a "point" at the center of the base area of a wood crib. The use of such kinds of model fires has been reported previously in a study of sprinkler performance [7]. In the present study a model is developed to predict fire growth rates in a wood crib ignited at the center of its base. The ability to predict the rate of fire growth in an ordered fuel configuration of wood, such as a wood crib, is an important first step in developing the capability of predicting fire growth rates in various similar 'The burning rates of wood cribs are also extensively discussed in Thomas' recent work, e.g.: (1) Thomas. P.H., Fire Research Note 965, March 1973: (2) Thomas, P.H., Fire Research Note 999, February 1974; (3) Thomas, P.H., CP 29/74, August 1973--Fire Research Station, Boreham Wood, WD6 2BL England.

fuel assemblies (e.g., wood pallet stacks, commodity storages). A simple energy-balance model is used to estimate the flame propagation velocity and the fire growth rate. Experimental burns of wood cribs with varying stick size and geometry verify the theoretical predictions. Analysis of pressure modeling experiments also shows that pressure modeling [8] cannot successfully model the fire growth rates in wood cribs. 2. Flame Propagation The fire in a wood crib, ignited at the center of

/

/

/

/

/ f

Fig. 1. Typical wood crib.

Copyright © 1976 by The Combustion Institute Published by American Elsevier Publishing Company, Inc.

268

M.A. DELICHATSIOS

its base area, spreads both vertically and radially from the ignition point. It is assumed in this study that the vertical is much higher than the radial fire spread rate. (Exceptions were noted only for very densely packed cribs for which significant lateral spread occurred simultaneously with vertical fire spread). The flame front will be a circular cylindrical surface with the center of its base at the ignition point and a height equal to the crib height. The rate of increase of the radius of this burning cylinder will be equal to the radial fire spread rate. Soon after the fire reaches the lateral edges of the wood crib, all the wood in the crib will be involved in the burning process. The radial flame spread speed can be calculated from the energy conservation equation (see e.g., Emmons [9]). It is assumed that a fuel volume element in the crib will ignite and then burn, when its temperature reaches an ignition temperature. A fuel element in the wood crib is heated up by the energy flux from the flame front. Assuming quasi-steady radial flame spread and observing that the energy flux drops very rapidly away from the flame front at the position of the adjacent stick layer, we obtain from the energy equation the approximate relation: u s A Q = Io '

A fuel volume element inside the crib consists macroscopically of a mixture of wood and gases. The fuel density in Eq. (2) will thus be equal to the apparent wood crib density I6], namely, the ratio of the total mass of the wood to the wood crib volume nb

Py = Pw ~

,

(3)

where pw is the wood density, n is the number of sticks per layer, b is the thickness of a wood stick, and f is its length. The volume averaged temperature 1"~ in Eq. (2) depends on the homogeneity in the heating of a fuel volume element from ambient conditions to its ignition state. For thin fuel elements [6,10], the temperature inside the element is constant everywhere and the volume averaged temperature is equal to the wood pyrolysis temperature. For thick fuel elements (b e 0. I cm we can estimate the averaged volume temperature in the following way. At the moment of ignition, the bulk of the fuel element is essentially at the ambient temperature, T=, whereas the surface layer temperature is essentially equal to the pyrolysis temperature, T,, down to a depth of a therma length [ 10]

(1)

valid for large radial distances compared to the stick spacing. Here,/kQ is the increase in the heat content per unit volume of the fuel elements from the ambient conditions to the ignition state, and Io is the energy flux near the flame front. The increase in the heat capacity of a fuel volume element for essentially dry conditions may be approximated by

(2) where psis the effective density of the fuel, Csis its heat capacity, /'~ is the v o l u m e - a v e r a g e d ignition temperature and T~ is the ambient temperature.

~T -

kw(rp-rW lo

,

(4)

where kw is the wood thermal conductivity. The volume-averaged temperature at the moment of ignition will be approximately

f ; - r=.

(rp - r=.) - - Vr's -

'

(s)

where S is the surface area of the fuel element exposed to radiation and V is its volume. The value of the volume-averaged temperature at the moment of ignition given by Eq. (5) agrees within a factor 0.85 with the value calculated using the analytical solution for the tempera-

FIRE GROWTH RATES IN WOOD CRIBS

269

ture response of a thermally thick solid after sudden exposure to a constant surface heat flux Io [11]. In a crib configuration the sticks have all orientations relative to the advancing flame front; however, using a model where half of the wood element (sticks) have an orientation normal to the flame front, with respect to their longest dimension (with four sides of the sticks exposed to radiation), and half of them have a parallel orientation (with one side of the stick exposed), Eq. (5) becomes: T i - Too ~' (~,T" - Too) 21 -7" " I~.~__~+ ~4£T 1 ,(6a) b

2.5£ T

Ti- T~(Tp

- T~)

b

(6b)

The energy flux Io from the flame front to the adjacent unburned sticks is the sume of radiative and convective heat flux. Convective heat fluxes have been estimated to be considerably less than measured values [6] of radiative fluxes. In addition, the radiation flux from the flame plume over the crib is negligible [6,12]. Combining Eqs. (l), (2) and (3) we find that the quasi-steady radial fire spread rate is

/./$ =

PwCw( T i - T~)

nb

3.

th" [t - r___ ] dAs(r) for t < t O

'°'

(4).

(7b)

Fire Growth Rates

For a steady fire spread rate, we can readily write down the crib weight loss equation due to burning: _

(8)

/2 s

am(t) _ dt Usto

fo

rn " [t- r_~ ] dA s (:.) for t > t o . (9) Us

Here t is the time measured from the moment of ignition, us is the radial fire spread speed, and As(r) is the wood surface involved in the burning inside the volume of a cylinder of a radius, r, and a height equal to the crib height. Furthermore, rn"(t) is the history of the burning rate per unit surface of a wood element involved in fire. Finally, to is the time required for the flame to spread to the outer edges of the crib, approximately equal to

to ~

~X/~2

/ u s,

(10)

for a square base wood crib made of wood sticks of length 4. The surface area of the wood inside the burning cylinder of radius, r, is approximately

As(r)., ~ 4 rr -nh - r2,

z.Spw Cwkw(r p -

din(t) dt

fo

, (7a)

or using Eqs, (4) and (6b)

--

Ust

(11)

where n is the number of sticks per layer in the wood crib and h is the crib height. The burning rate per unit surface area in wood cribs has been measured by previous investigators [ 1-4]. It was found that quasi-steady burning conditions prevailed in these burns. In accord with these observations, we assume that the burning rate in Eqs. (8) and (9) is constant over a period equal to the burning time of a fuel element. The burning period of a wood element is

270

M.A. DELICHATSIOS (12)

Pwb tB - 4rn "

For the case when the burning time is longer than the spread time (tB > t,,), which was observed to be true for all the experiments reported here, we can get, after algebraic rearrangements, using Eqs. (8) and (9)

sitivity were used for the heavy and light cribs respectively. Table 1 lists the cribs burned in the present e x p e r i m e n t s t o g e t h e r with their g e o m e t r i c characteristics and the crib " p o r o s i t y " , defined according to Heskestad [3] by

p_ 1

din(t)

m "A $

dt

_

;r 2

1

din(t)

m"51s

dt

A

v

s~Ab a/2 .

(14)

As (t/to)2 for t_
- 1

for t > t o ,

(13b)

where A~. is the total wood crib surface area. The discontinuity of the burning rate at t =to can be attributed to the transition of the cylindrical surface flame front from circular to rectangular at the edges of the crib. If the burning time is shorter than the flame spread time (tB < to), a burned out char is formed at the central part of the crib. The growth rate can again be calculated from Eqs. (8) and (9) if we assume that a fuel element burns completely in a time t , (cf. Eq. (12)) with a constant specific burning rate, rn".

4. Experiments and Results The purpose of the burning experiments was to measure the radial flame spread rate and the weight loss history of the crib, when ignited at the center of its base. The cribs used in the present experiments were made from sticks of sugar pine wood without knots. Each crib was dried in a furnace at 200 °F for one day before burning so that the moisture content would be negligible. No independent m e a s u r e m e n t s of moisture content were made during these experiments. A simple recording system, consisting of a universal transducing cell and a Honeywell strip chart recorder, was employed to produce a continuous record of the crib weight versus time during burning. Two load cells (of the resistance strain-gage type) of different sen-

Here A,, is the vent area of the crib (free total area of shafts through which the air is entrained) andAs is the total exposed surface area of the wood in the crib. It has, moreover, been suggested by Heskestad [3], on experimental evidence, that the quasi-steady burning rate for any crib may be approximated by

m" =cb'½f(P),

(15)

where the function, f , takes on the value 1 (openly packed crib) for l a r g e P ( > 0.07 cm) and decreases to zero as P approaches zero. Experimental values o f f ( P ) are plotted in Fig. 7. Fire in cribs Nos. 1 to 13 in Table I was initiated at the center of their base by means of a shallow circular pan, full of acetone, of height approximately 1 cm and diameter equal to the stick spacing. The ignition source was removed as soon as the central shaft was involved in burning. In burning experiments Nos. 14 and 15 (Table 1) wood chips, spread over a square area 20 cm × 20 cm underneath the central part of the crib, were used as the ignition source i13]. The density of the dry wood used in the present experiments varied, as indicated from the weight measurements, from 0.36 gm/cm '~ to 0.41 gm/cm 3. An average value of 0.39 gm/cm ~ is used in Sects. 4.2 and 4.3 whenever the experimental results are compared with theoretical predictions. 4.1

Radial Flame Spread Speed Measurements

In almost all experiments listed in Table 1, fire first spread vertically before it expanded toward the edges of the wood crib. In experimen-

FIRE GROWTH RATES IN WOOD CRIBS

271 TABLE 1 List of Cribs Burned

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

b(cm)

£(cm)

0.635 0.635 0.635 0.635 0.635 0.635 0.635 0.635 1.905 1.905 1.905 1.7 3.17 3.17 3.17

25.4 25.4 25.4 25.4 25.4 25.4 25.4 25.4 38.1 38.1 38.1 18.6 22.7 76.2 76.2

n

N

h (cm)

6 6 9 9 9 9 16 16 4 6 8 6 4 10 14

15 40 3 6 10 20 10 20 36 16 6 11 6 12 12

9.52 25.4 1.90 3.81 6.35 12.70 6.35 12.70 68.58 30.48 11.43 18.7 19.8 38.1 38.1

tal runs Nos. 8 and 12, where very densely (very low porosity) packed cribs were burned (cf. with discussion following Eq. (15)), significant lateral spread occurred while the flame propagated vertically. The radial spread speed in all experiments (with the exception of Nos. 8 and 12) was measured by recording the time required for the flame front to reach positions marked on the top area of the crib. The radial spread speed was found to be independent of the instantaneous position of the flame front. In Fig. 2 we have plotted the measured radial flame spread speed versus g/nb (cf Eq. (7a)). It is seen that, for the same stick size, but varying crib geometry, the experimental points lie on a straight line which is in accord with theoretical predictions (cf. Eq. (7a)), if we assume that the energy flux Io from the flame front to the adjacent unburned sticks is independent of the specific geometry of the crib. In Fig. 3 we have again plotted the radial flame spread speed versus ~e/n(cfEq. (Tb)). The experimental points lie on a straight line in accordance with the theoretical Eq. (10b), if the energy flux is assumed independent of the stick size. The slope of the straight line in Fig. 3 is 0.045 sec -1, which is consistent with the value of the coefficient on the right hand side of Eq. (10b), when the following typical values of the

s(cm)

P(cm)

4.32 4.32 2.46 2.46 2.46 2.46 1.02 1.02 10.16 5.33 3.26 1.68 3.16 4.94 2.44

0.132 0.05 0.28 0.14 0.08 0.04 0.018 0.009 0.098 0.08 0.092 0.014 0.04 0.067 0.017

physical properties of sugar pine wood and of the energy flux [61 are used;

Tp - Too = 350 °C, Pw = 0.39 gm/cm3 , Cw = 2.30 J/gm °C, f

0.7

0.6

,

i

i

,

Stick Geometry b(cm) ,t(cm} D 06:35 25.4 ~7 1.905 38 I o :317 22 7 (No f5),76.2

0.5

"~Q4 b =5.17cm

~2No

I 9 0 5 cm

=~05r

No, / /

2

i

02

0.1

i

i

.!

nb Fig. 2.

Radial flame spread speed as a function of the crib length per number of sticks in each layer and stick width.

272

M.A. DELICHATSIOS possible, because cribs not completely dry were used in those experiments. Nevertheless, it should be noted that flame spread rates measured by Fons et al. [5] for a specific crib geometry and varying wood density agree well with the present results, when the moisture content is 4%.

0.7 1"-I

b • 0 . 6 3 5 cm b • 1 9 0 5 ern

06 0

b • 317crn

//

05

®

/

/

/

0. 4

,~ 0.3

4.2

J

0.7'

0.1

7'

4

6

8

10

12

-~-n ( crn / 8tick )

Fig. 3.

Radial flame spread speed versus the crib length per number of sticks in each layer.

k w = 1.25 • 10"3 W/era °C, 10 = 4.18 W/cm 2 . We can make the following remarks concerning the results in Figs. I and 2: (1) Equations (7a) and (7b) describe satisfactorily the variation of flame spread speed with crib geometry for dry cribs. In the present experiments it has been found that u s = 0.045 £/n.

(16)

(2) The energy flux from the flame front to the adjacent unburned wood sticks is independent of fuel geometry. We can conclude that the flame spread speed will increase linearly with #/n (cf. Eq. (10b)) for the same wood quality under dry conditions. However, it should be expected that energy flux losses from the bottom and top openings of the crib will increase with increasing ratio of stick spacing to height. In the present experiments, the ratio of wood stock spacing to wood crib height (s/h) was varied from 0.064 to 1.29. (3) Comparison of the present results with previous flame spread experiments [5,61 is not

Fire Growth Rates

Typical fire growth rates of cribs burned in the present work are presented in Figs. 4, 5 and 6. In plotting the experimental results, it was decided to use dimensionless variables directly deductible from the measurements. In Figs. 4 and 5 the ordinate in the log-log plot is the dimensionless weight loss, (too - m)/mo, and the abscissa is the ratio (t/to) of the time, t, from the beginning of the burning to the characteristic spread time to given by Eq. (10). In calculating the characteristic spread time, to, the experimentally measured radial spread speed us (see Fig. 2) was used in Eq. (10). In Fig. 6, for which data for the burning rate were taken from Alpert I131, the ordinate is the J r t dimensionless burning rate l '_e% m / , and the abscissa is (t/to). L mo .I The burning history of each crib consists essentially of four distinct periods: from ignition of the central shaft until quasi-steady radial spread is established; quasi-steady spread of the flame front to the outer edges of the crib; total involvement of the crib with approximately constant burning rate; and decay of burning rate until final collapse of the crib. The initial burning period, from ignition until quasi-steady fire spread is established, has a relatively short duration, namely, from 0.2 to to 0.5 to (see Fig. 5). The mass burned during this period is at most 5% of the total burned mass. The third period of total involvement of the crib in burning, with an approximately constant burning rate, is quite distinctly present for cribs consisting of sticks 1.901 cm and 3.17 cm in thickness (Figs. 5, 6) whereas, for sticks 0.635 cm thick (Fig. 3) no such period is observed. This follows from the fact that the burning period for a wood element of size 0.635 cm, as defined in Eq. (12), is of the same order as the characteristic spread time to (see Eq. (10)), whereas for the larger size wood elements the

FIRE GROWTH RATES IN WOOD CRIBS

273

'oI

'

NO. X

O z~ o

"U

S G,088 I0 ii

0,I15 0,210

,,o

,2(÷o)' E

Id'

0.1

/

j,/'/~.~l

NO. 5 to=lSI @ec

0,01

0.1

1.0 t

0 # ', ,,(', I0

I00.1

IO 90 0.08 ,~ II 142 0.092

......... 1,0

I0

to

tO

Fig. 4.

o ~ e~ 0,09. &

Fraction of initial weight loss as a function of the time from the moment of ignition over the characteristic spread time (Crib 5).

corresponding burning period is greater than the characteristic spread time to. The weight loss history (Figs. 4, 5) or the burning rate history (Fig. 6) in the present experiments will now be shown to follow remarkably well the theoretical predictions as expressed by Eq. (13a) for the quasi-steady fire spread period and the constant burning rate period, whenever applicable. Equations (13a) and (13b) assume the following form, for the coordinates used in Figs. 4 to 6:

Fig. 5.

Fraction of initial weight loss as a function of the time from the moment of ignition over the characteristic spread time (Cribs 9, 10, 11).

,o

/

J

E .-*t~ ,o-

/oOO°o<:DOCOo

0 0 0 0 0

-/°,, (+o)'

0

/

t

o

m=3k(t/to)2

f o r t < to,

(17a)

NO. 14 t o = 169 ~ c

m

O

10

t°

ffz = 6

m °

k

for t > t o ,

0.1

(17b)

I0

lr

Fig. 6.

where

X=

1.0 1.! to

2rr 3~

fit" PwUs

• -b

(18)

Dimensionless burning rate as a function of the time from the moment of ignition over the characteristic spread time (Crib 14).

Integrating Eqs. (17a) and (17b) over time, we obtain:

274

M.A. DELICHATSIOS m

o

-m

- X ( t / t o )3

fort<

to ,

mo-m

(19a)

m o

m

0

-m

m o

6

h I(t/to)1+7r/61

fort>to.

_ 6

/T/°

7/"

(20)

x (t/to),

for

?r

t/t o > 1 - 7r/6.

(19b) The integration constant in Eq. (19a) has been taken as equal to zero consistent with the assumption of a small burned mass of wood, before quasi-steady fire spread conditions are established. The integration constant in Eq. (19b) has been defined from the condition that Eqs. (19a) and (19b) must give the same burned mass of wood at t = to. First, note that the functional dependence on (t/to) of the measured weight loss history agrees with theoretical predictions. Next, the consistency of the measured proportionality coefficients with the theoretically predicted values will be examined. From the best fit line, as shown in Figs. 4 to 6, through the experimental points during the quasisteady fire spread period, the experimental coefficient X in Eqs. (17a) or (19a) can be determined. Furthermore, using (1) Eq. (10), (2) the value of characteristic spread speed u, (Eq. 16), (3) the characteristic properties of the crib, and (4) values rn" previously obtained [3], theoretical values of h can be calculated. The values of the specific burning rate rh" are plotted in Fig. 7 for ponderosa pine wood cribs [3] as a function of the crib "porosity" (cf Eq. 15). The specific burning rates rn" for sugar pine wood cribs, used in the present experiments, are approximately the same I4] as the values for ponderosa pine wood cribs. Experimental values ofh are compared with theoretical values in Table 2 for all the cribs burned in the present experiments. It is apparent that good agreement is obtained. For the burning characteristics of cribs depicted in Fig. 5, we have drawn a best fit line for times t > to, when a quasi-steady burning rate exists. The slope of this line in the log-log plot of Fig. 5 is unity, in agreement with the theoretical prediction of Eq. (1 lb), which reduces to

(21)

Further, it is readily shown that the proportionality coefficient of the experimental fit (6/7r X, cf Eq. (20)) corresponds approximately to the value obtained from values ofh determined by the best fit line of the experimental points in the quasi-steady fire spread period. Finally, it may also be verified from Fig. 5 that the burning rate for the constant burning rate period (t > to) agrees well with the theoretical prediction (see Eq. (17b)), where the coefficient h has already been determined for the best fit line of the experimental points in the quasi-steady fire spread period (t < to). In conclusion, it can be stated that Eqs. (17) and (19) describe well the burning history of a wood crib ignited at the center of its base. both during the quasi-steady fire spread and the steady burning period. Accepting Heskestad's I3] equation for the specific burning rate (see Eq. (15)) we may obtain, after some rearrangement, from Eq. (13a):

TABLE 2 Comparison of Experimental and Theoretical Values of No. 1 2 3 4 5 6 7 9 10 11 13 14 15

n

P(cm)

~'exp

Xth

6 6 9 9 9 9 16 4 6 8 4 10 14

0.13 0.05 0.27 0.14 0.08 0.04 0.018 0.098 0.08 0.092 0.04 0.067 0.017

0.85 0.70 1.35 1.3 1.2 1.02 1.5 0.088 0.115 0.210 0.064 0.15 0.14

0.93 0.84 1.38 1.38 1.38 1.19 1.52 0.092 0.138 0.206 0.073 0.15 0.14

FIRE GROWTH RATES IN WOOD CRIBS

1.2

275

I

I

I

~ 0.05

t

J

0.10

0.15

1.0

ml~E o

r'? ~

--

0.8

i

0.6

¢,~

"-E

0.4

0.2

0 0

I

I

(Av/A s) s'~b"2"(cm) Fig. 7, Reduced specific burning rate as a function of the crib porosity for ponderosa pine wood cribs (Taken from Heskestad I3]).

dm dt

9_Nb

- m = (4rr~ 2) •

The value of the inverse propagation time, ~:, was m e a s u r e d 0,045 sec -1 in the present experiments. Finally, for P > 0.07 cm, the porosity f u n c t i o n f ( P ) = I.

• m"?=

n ½

(4rre~2) 9.Nb

;(P) t 2 ,

(22)

/7

wherein Eqs. (7), (10), (15) and (16) have been used. In Eq. (22), c varies with w o o d properties but is a p p r o x i m a t e l y equal to 10-3 g m / c m ::/2 sec for dry conditions; ~: = us • rd~, defined as inv e r s e p r o p a g a t i o n time, d e p e n d s on w o o d properties and the energy flux f r o m the vertical flame front (cf Eq. 7b))

I 2 =

o

2.5

P w e w k w ( Z p - Too) 2

(23)

5. Pressure Modeling of Fire Spread and Growth Rates In pressure modeling [8], fuels geometrically similar are burned in an e n v i r o n m e n t at an elevated pressure p with the product p2L3 maintained constant, where L is the linear scale of the fuel. C o n v e c t i v e heat fluxes increase in this w a y a s p 2/3. Proper modeling would be achieved if radiant fluxes would also v a r y with pressure following the s a m e law as c o n v e c t i v e heat fluxes. There is experimental and theoretical justification [8] for expecting f l a m e radiant f l u x e s to increase as pZfZ and thus model properly. Unfortunately, radiant fluxes emitted by hot solid surfaces would not be expected to

276

M.A. DELICHATSIOS

model properly, since these fluxes are controlled by surface temperatures, which are preserved independent of ambient pressure. Finally, it is expected in pressure modeling that the specific burning rate th" should increase as p2/3 (i.e., when radiation fluxes from hot solid surface may be considered negligible). In a recent study by Alpert [], it was found that pressure modeling [8] cannot successfully model the radial fire spread and hence the fire growth rates in wood cribs ignited at the center of their base. However, it was found experimentally that the quasi-steady burning rate (rn") follows the rules [8] of pressure modeling, i.e., it increases as p2/3. The reason for the above discrepancy is that the energy flux from the vertical flame front to the unburned fuel cannot be pressure-modeled, as Alpert [13] also suggested. Following de Ris e t a / , [81 we assume that the energy flux, Io (cf Eq. (1)), is the sum of two terms: (1) an energy flux, from the hot solid surface independent of pressure, and (2) an energy flux, from the flaming gases, proportional to p2/3, where p is the pressure in the model experiments. It follows that the inverse propagation tine, ~:, defined in Eq. (23), should satisfy the relation ~V2 = at + a2 p2/3,

(24)

where the dimensional coefficients al and a,, are independent of the crib geometry (cf with discussion following Eq. (10)) and must be determined experimentally. The measured values [13] of the square root of the quantity ~ versus p2ta are plotted in Fig. 8. In the tests reported by Alpert [13] the product p~L 3 was maintained constant m accordance with the pressure modeling r e q u i r e m e n t s ; three different geometrieswere used: for tests I, 2, 3 and 6 (see table on Fig, 8) the value of f was calculated from the relation f = u8 n~e (cf Eqs. (7b), (23)), the geometric characteristics of the cribs and the measured value [ 13] of the spread velocity, us. For tests 4 and 7, the value of ~: was estimated using Eq, (22), the properties of the cribs and the measured growth rate by Alpert I13,15]. It is seen that Eq. (24) is verified with

G1

= 0.104 (sec"1/2 ) and a2 = 0.096 (sec°1/2 atm'2/3).

The flame spread rate over the value at ambient conditions increases less rapidly with pressure than the specific burning rate (see also Alpert [13]), as suggested by Eqs. (24) and (6b), in contrast to the equation for the specific burning rate [8,13]: (Us)mod

=

(Us)prot [0.104 +0.096" p2/3 1 2 • p-2/3,(25a) 0.2 (m ")rood = (rn ")prot " p2/3.

(25b)

It follows that the situation may arise especially at high pressures, where the burning time (cf Eq. 12)) is shorter than the spread time (cf Eq. (10)) for the model while the opposite is true for the prototype. For example, calculations are made for the spread time and burning time for tests 1 and 4 (see table on Fig. 8) reported by Alpert [13,15] where test 4 pressure-models test 1 (at atmospheric conditions) in an environment with a pressure of 30 atm. Using Eqs. (12) and (10) and the measured values for burning rate and flame spread rate, we obtain for test I: t B = 475 sec,

(26) t o = 169 sec,

while for test 4: t B = 5 . 5 SeC,

(27) t o = 7.2 see.

It follows that the fire growth rate in the prototype will not in general be modeled by the fire growth rate in the model (cf with discussion in Sect. 3). Therefore, we can conclude that

FIRE GROWTH RATES IN WOOD CRIBS

277

1.4

n l(cm) P(atm) b(cm) I10 76.2 I 3.17 0 2 I0 14.3 I 12.1 0.6 1.2

510 4 I0

11.4i 17.1 7.8 3 0

5 14 76.2 ["1 6 14 II. I 7 I0

1.0

0.48 0.33

I "5.1 7 17.35 0.47

7.7

5.10

For all tests N= 12

D/

--Icu I O

0.98

0.8

--IN 0.6

0.4

0.2

I

1

I

I

3

5

7

9

p'~ (at rn ~ )

Fig. 8. Squareroot of the inversepropagationtime(Eq. (23))as a functionof the ambient pressure. pressure-modeling of the fire growth rate in cribs ignited at the center of their base cannot represent the flame spread, because radiation fluxes from the hot solid surfaces (not properly modeled) are just as significant as radiation fluxes from the flames (properly modeled). 6. Conclusions The main accomplishment of the present work is the derivation of a fire growth rate equation in wood crib configurations, which can be used reliably to predict fire growth rates for dry wood conditions. A simple model based on the energy equation was used to predict fire spread rates and weight loss history in cribs ignited at the center of their

base. The inhomogeneities in the fuel material (wood and air) and in the heating of a fuel element from ambient conditions to the ignition state have been properly incorporated in the model. The radial fire spread speed was found to be independent to angular orientation and proportional to the ratio of the stick length to the number of sticks per layer. The proportionality constant is dependent on the radiative properties of the flame front, and on the physical characteristics of the wood. Experiments, in which the radial fire spread rate and the weight loss were measured, agreed very well with the theoretical predictions. Cribs made of sticks of three different sizes (0.635 cm, 1.905 cm, 3.17 cm) were used in the experiments.

278 Analysis of pressure modeling experiments showed that cylindrical flame spread in wood cribs cannot be successfully modeled, because radiation fluxes from the hot solid surfaces (not properly modeled) are just as significant as radiation fluxes from the flames (properly modeled). Fire growth rates in moist wood cribs are a problem of practical importance, which could be examined as a continuation of the present study. The author wishes to thank Dr. G. Heskestad of Factory Mutual Research Corporation for his helpful advice and suggestions during the course of this work. Part of the present work was presented at the Ninth Fall Meeting Eastern Section: The Combustion Institute, November 6-7, 1975, S U N Y at Stony Brook and Brookhaven National Laboratories at Upton, Long Island. References 1. Gross, D., Experiments on the burning of cross poles of wood, J. o f Research 66C, 99(1962). 2. Block, J.A., A theoretical and experimental study of non-propagating free-burning fires, Thirteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1971, p. 971. 3. Heskestad, G., Modeling of enclosure fires, Fourteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1973, p. 1021. 4. Smith, P.G. and Thomas, P.H.,Fire Technology 6, 29 (1970).

M.A. DELICHATSIOS 5. Fons, W.L., Clements, H. B., and George, H.B., Scale effects on propagation rate of laboratory crib fires, Ninth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1962, p. 860.

6. Thomas, P.H., Simms, D.L., and Wraight, H.G.H., Fire spread in wooden cribs. Part II heat transfer experiments in still air, Ministry of Technology and Fire Office's Committee Joint Fire Research Organization, Fire Res. Note, No. 599 (1965). 7. O'Dogherty, M.J., Nash, P. and Young, R.A., A study of the performance of automatic sprinkler systems, Ministry of Technology and Fire Office's Committee Joint Fire Research Organization, Fire Res. Tech. Paper No. 17 (1967). 8. de Ris, J., Kanury, A.M., and Yuen, M.C., Pressure modeling of fires, Fourteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1973, p. 1033. 9. Emmons, H., Fire in the forest, Fire Research Abstract and Reviews 5, 163 (1963). 10. Kanury, A.M., Combust. Sci. Tech. 5, 135 (1972). I I. (Rohsenow, W.M., and Hartnett, T.B., Eds.) Handbook of Heat Transfer, McGraw-Hill, New York, 1973. p. 3-81. 12. MeCarter, R. J. and Broido, A., Radiative and convective energy from wood crib fires, Pyrodynamics 2, 65 (1965). 13. Alpert, R.L., Pressure modeling of transient crib fires, FMRC Technical Report Serial No. 22360-2 (December 1975). 14. Croce, P.A., Private communication, Factory Mutual Research, Norwood, Mass. (September, 1975). 15. Alpert, R. L., Private Communication, Factory Mutual Research, Norwood, Mass. (October, 1975).

Received 12 December 1975; revised 9 March 1976