Combustion Science and Technology Rate of Spread ...

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Rate of Spread of Free-Burning Fires in Woody Fuels in a Wind Tunnel a

a

b

b

W.R. CATCHPOLE , E.A. CATCHPOLE , B.W. BUTLER , R. C. ROTHERMEL , G. A. MORRIS & D. J. LATHAM

b

b

a

University College, University of New South Wales, Australian Defence Force Academy , Canberra, ACT, 2600, Australia b

USDA Forest Service, Intermountain Fire Sciences Laboratory , Rocky Mountain Research Station, Missoula, MT, 59807 Published online: 06 Apr 2007.

To cite this article: W.R. CATCHPOLE , E.A. CATCHPOLE , B.W. BUTLER , R. C. ROTHERMEL , G. A. MORRIS & D. J. LATHAM (1998) Rate of Spread of Free-Burning Fires in Woody Fuels in a Wind Tunnel, Combustion Science and Technology, 131:1-6, 1-37, DOI: 10.1080/00102209808935753 To link to this article: http://dx.doi.org/10.1080/00102209808935753

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Combust. Sci. and Teen.• 1998. Vol. 131. pp. 1-37 Reprints available directly from the publisher Photocopying permitted b~ license only

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1998 OPA (Overseas Publishers Association) Amsterdam B.V. Published under license under the Gordon and Breach Science Publishers imprint. Printed in India.

Downloaded by [UNSW Library] at 20:34 17 July 2015

Rate of Spread of Free-Burning Fires in Woody Fuels in a Wind Tunnel W. R.CATCHPOLE",E. A. CATCHPOLE"'·,B.W. BUTLERb , R.C. ROTHERMEL b , G. A. MORRIS b and D. J. LATHAMb • University College, University of New South Wales, Australian Defence Force Academy, Canberra, ACT 2600 Australia; b USDA Forest Service, Rocky Mountain Research Station, Intermountain Fire Sciences Laboratory, Missoula, MT 59807 (Received 12 June 1995; In linal form 23 October 1997)

We describe the results of 357 experimental fires conducted in an environmentally controlled large wind tunnel. The fires were burned over a range of particle sizes, fuel bed depths, packing ratios, moisture contents and windspeeds. We find that spread rate decreases with moisture content in a way which depends on the fuel type and diameter. It decreases as the square root of the packing ratio. Fuel bed depth has little elfect on spread rate, and fuel diameter has significant effect only for diameters above I mm. The relationship between rate of spread and windspeed is virtually linear. We develop a predictive model for rate of-spread based on energy transfer considerations and the laboratory results. Other laboratory-based models for spread rate are compared with our model, and tested against the laboratory data. The other models have forms similar to ours, but do not predict our data well. Our model predicts well the spread rates for fires burned in windspeeds below 3 mlsec in other laboratories. The scale of the experiments and the similarity of the dependence on windspeed to that found in the field indicate that a field model may be developed from the laboratory model with relatively few modifications. Keywords: Bed depth; moisture content; packing ratio; Rothermel model; spread rate; wind tunnel

1. INTRODUCTION

A series of 357 experimental fires were burned under a wide range of conditions in the large wind tunnel at the U.S. Forest Service's Intermountain Fire Sciences Laboratory (IFSL). ·Corresponding author. E-mail: [email protected].


FREE-BURNING FIRES IN WOODY FUELS

3

energy, as in Rothermel (1972), in Section 4. We test the model against data from Nelson and Adkins (1988) and Fendell et al. (1990) in Section 5. In Section 6 we compare our model with those of Rothermel (1972), Nelson and Adkins (1988) and Carrier et al. (1991).


2. EXPERIMENTAL SET-UP

The test section of the wind tunnel (Fig. 2) has 3 m x 3 m cross-section and an overall length of 12m. For each experiment, a fuel tray 1 meter wide was centered on the floor of the tunnel. The tray length ranged from 5 to 8 m, depending on the anticipated spread rate and length of the flaming zone (Fig. 3). Strips of metal sheeting matching the fuel depth were placed along each side of the tray to mimic a wider fire front by preventing indrafts into '"turtluJence damping

saeenl

T

6.1 m

'----5.2 m - - - 1 - -

1

f-------8m-----FUl!II bed support

FIGURE 2

Large wind tunnel, intermountain fire sciences laboratory.

PhotoOlIn tubes

et o.5m~"",,.~

I Metal ,trips, let 10ma\t:tl fuel depth

O.25m

F~='='9'rP:::IE:i==:::::t--~- 2.54 emceramic boatO

supported on wire mesh

I.

_ _ _ _ 1.0m

_

and angleironframe

End View

Ignltlcfl\J'Ougn

PhotoOllis

Killlllatlc

1h meterIntervals

pressure probes

Side View

Weighing p1arfofm

_--------------8.0m--------------_

FIGURE 3 Fuel bed arrangement.


4

W. R. CATCHPOLE et al.

the combustion zone and by reflecting some of the radiation. The metal sheeting was fastened to 0.25 m high 12mm wire mesh along each side of the tray. A wire mesh spoiler or trip screen, 0.25 m high and with a grid spacing of 6 mm, was placed 2 m upwind of the fuel tray to initiate a turbulent . boundary layer (see Rothermel and Anderson, 1966). This was supplemented by a wire mesh of 12mm grid spacing arranged in Z-folds and placed on the floor of the wind tunnel from wall to wall upwind of the fuel trays. The height of the folds was adjusted to be the same as the fuel depth, in order to simulate a longer reach of fuel and its resulting boundary layer. The enhanced boundary layer caused the fire to reach a steady rate of spread within 1.5m of the ignition line and resulted in a uniform fire-front. Before the introduction of the modifications, the fire-front had a tendency to develop curvature, either leading in the center (without the metal sheeting) or at the side (with the metal sheeting but without the Z-folds). The air flow in the tunnel is described in Rothermel and Anderson (1966). Fuels burned in non-ambient conditions were pre-conditioned to produce the equilibrium moisture content specified for the experiment. Conditions in the wind tunnel were set to the same levels before the fuel was brought into the tunnel. The fuel beds were prepared in place within the tunnel at the specified environmental conditions. Immediately prior to ignition, fuel samples were extracted for moisture analysis by oven drying. By this process, fuel moisture contents were obtained over the range '2% to 33%. Two thirds of the fires however, were burned at ambient conditions (27°C and 20% relative humidity) which produced fuel moistures of 5% to 9% depending on the fuel type. Wind velocity in the tunnel was controlled at constant values ranging from 0.0 to 3.1 m/sec in the free stream above the boundary layer. Natural fuel beds vary in many different ways. In the experiments described here, primary attention was given to varying the fuel moisture content, packing ratio and depth (and hence fuel load). Four different fuels, each with a unique surface-area-to-volume ratio, were used. These comprised two sizes of poplar excelsior (Populus tranulos), regular (approximately 0.8 x 0.4 mm in cross-section) and coarse (approximately 2.5 x 0.8 mm in cross-section), ponderosa pine (Pinus ponderosa) needles and sticks of ponderosa pine heartwood with a 6 mm square cross section. Excelsior (wood wool) consists of wood strands of fairly uniform rectangular cross-section, but variable length, stripped from poplar wood boles and packaged in bales. It can be readily formed into a random fuel array distributed evenly over a fuel bed. The bulk density can be adjusted by

5

FREE-BURNING FIRES IN WOODY FUELS TABLE I

Fuel bed properties Species

Property

Populus tranulos (regular

Pinus

Populus

ponderosa needles

tranulos

excelsior)

heartwood

sticks

108 7596'

85 5710'

144 3092'

630

0.051 2.426

0.101 3.660

0.101 2.730

0.604 2.324

0.0025 0.040

0.020 0.094

0.0064 0.090

0.009 0.035

8, depth (m)

min max Pp, particle density (kgjm 3)

0.025 0.305 398 3

0.013 0.076 510'

0.025 0.152 3983

0.076 0.152 442 3

Qp. heat of pyrolysis (kJjkg)

7114

6094

7114

659

number of fires 5,

surface area to

volume ratio (m- 1) Wo, fuel load (kg/rrr') min max {3, packing ratio min max


(coarse excelsior)

Pinus ponderosa

I determined from 200 measurements of cross-sectional area; (1982).

2

Brown (1970);

J

20

Wilson (1985);

4

4

Susott

distributing the fuel at a predetermined loading (mass per unit area) and depth (Schuette, 1965). The sticks were arranged as parallel rows of evenly spaced vertical sticks by setting the ends in drilled ceramic boards. To produce a uniform packing ratio the sticks were set in a triangular lattice arrangement when viewed from above (Catchpole et al., 1993). The fuel types were chosen to be reasonable approximations to natural fuels- the pine needles to represent pine forest litter and the excelsior to represent fine shrub fuel and grassland. The sticks were used to extend the range of surface-area-to-volume ratios. The range of fuel and environmental conditions are shown in Table I. They were chosen to reflect those which occur in natural fuels, so that, for example, the excelsior was lightly packed, whereas the pine needles were densely packed. The conditions were also constrained by fuel array flammability and by the facility design. As a result the experiments are not balanced across all conditions, for example the correlation between fuel depth and packing ratio is -0.5\. This lack of balance can make the results difficult to interpret. A group of fires were burned at the same environmental conditions and at a packing ratio and depth at which all fires would burn well, so that the effect of differences between fuel types could be tested unambiguously. Physical and chemical properties of the fuels are also given in Table I. The surface-area-to-volume ratios of the two sizes of excelsior, which are approximately rectangular in cross-section, were determined by measuring


6


the dimensions of 200 samples of each size. For the pine needles we used the surface-area-to-volume ratio determined by Brown (1970). The time of arrival and passage of the flame was recorded at 0.5 m intervals by photocells positioned 25 mm above the fuel surface and pointed horizontally across the fuel bed. This information was used to calculate the rate of spread and the duration of flaming combustion. The photocells comprised photovoltaic diodes, responsive to ultraviolet, visible and near infrared radiation, placed at one end of aluminium tubes 200 mm long and 15 mm interior diameter. The viewing end of the tube was capped and a hole drilled to limit the field of view of the photodiode to a 5 degree cone. Video cameras were used to monitor fire-spread from the side and from directly overhead. An image analysis system (McMahon et al., 1986) was used to determine flame length, height, depth and tilt angle. Three methods of measuring spread rate were tried. The first two methods, involving subjective estimates of the location of the fire front (either live or from the video recording) were found to be subject to observer error. The photocells gave the most consistent method for measuring spread rate. Only fires burned after the introduction of the photocells have been included in this paper. Free stream velocity in the wind tunnel was measured with a TSI model 67 hot film anemometer. Temperature and relative humidity were measured with a Phy. Chemical Corp. PCRC-II HPB resistance thermistor and a Phy. Chemical Corp. Humitemp Series B humidity sensor. The velocity, temperature and humidity sensors were located approximately 2 m above the upwind edge of the fuel array. The facility was operated for a sufficient time before introducing the fuel to verify that the specified conditions were being maintained. Once experimental conditions were verified the preconditioned fuel was distributed in the fuel beds. Visual observations of smoke flow and measurements from pitot-static probes indicated that the hot air and combustion products remained close to the wind tunnel ceiling throughout the whole of each experiment.

3. RESULTS

Rate of spread was determined from the photocells. A typical photocell array voltage versus time trace is shown in Figure 4. By comparing video images and mass-loss traces for several fires, we determined voltages at which the onset and end of flaming combustion occurred. Specifically, the onset of flaming was defined as the time when the smoothed photocell trace

FREE·BURNING FIRES IN WOODY FUELS

7

0.25

~

0.20

..

0.15

s g 0>

~

g

0.10

§ 2

0.05

Co

0


60

40

(a)

80

100

time since Ignition (sec)

90 80 70

.

:§:

,E

60 50

40 30

(b)

3

4

5

6

7

8

distance (m)

FIGURE 4 (a) Smoothed photocell responses from II photocells during a single tire, (b) plot of distance versus time from analysis of photocell traces.

reached 0.25 of its peak value and the flaming was defined to have ceased when the output had decayed to 0.]5 of the peak; both of these being calculated after the raw data had been smoothed. Determining the position of the leading edge of experimental fires has not been difficult in the past, using thermocouples, but a repeatable indication of the trailing edge had proved elusive before the introduction of the photocells. Unlike thermocouples the photocells have a reasonably sharp cut-off at the end of the flaming zone. Plots of distance versus time of onset of flaming indicated that 1.5m was sufficient for the fire to achieve steady rate of spread. The average rate of spread was found after omitting the first 3 observations (the first 1.5m). The within-fire coefficient of variation in spread rate at half-meter intervals was about 30% and thus the average rate of spread over 4.5 m has a coefficient

W. R. CATCHPOLE


8

eI

al.

of variation of about 10%. The appendix shows 10 fires under virtually identical conditions: coarse excelsior, packing ratio 0.02, fuel bed depth 0.076 m, windspeed 1.3 tnlsec, ambient temperature and humidity. The between-fire coefficient of variation in average spread rate between these replicated fires was about 12%. Other replicated conditions gave similar results. The relationship between spread rate and fuel bed depth is shown in Figure 5 for fuels under ambient conditions and at a windspeed of 2.7 mlsec. Results at a windspeed of 1.8m/sec are similar. Because of experimental limitations, we had a large range of fuel depths only for the regular excelsior. For the range of conditions shown, the effect of depth on spread rate is so slight as to .be of no practical concern. There are in fact statistically significant reductions in spread rate with increased depth in the heavily packed coarse excelsior' (p = 0.015) and pine needles (p = 0.001). There is also a significant increase in spread rate with increased depth in the sticks (p = 0.0025) All of these trends could also be seen at lower windspeeds. A simple model with rate of spread independent of depth, but dependent on fuel type and packing ratio, accounts for about 90% of the total variation in log(rate of spread) at each windspeed, under ambient conditions. These results contrast with those of Wilson (1990), who found that, for zero-wind fires, spread rate increased as the square root of fuel bed depth. We conjecture that the effect of depth on spread rate depends on the relative importance of convective and radiative heat transfer, and an understanding of these processes is needed to account for the differing behavior. Figure 6 shows the approximately linear dependence of spread rate on windspeed, for different fuels at various packing ratios (under ambient conditions and at similar fuel depths). This figure gives an idea of the relative effects of windspeed, packing ratio and fuel type on rate of spread, and shows in particular the increase in spread rate with decreasing packing ratio. The damping effect of fuel moisture on rate of spread is shown in Figure 7 and is similar to the damping obtained in zero-wind experiments in Wilson (1990) and Anderson and Rothermel (1965). Figure 8 shows the rate of spread plotted against surface-area-to-volume ratio for the four fuel types at the same packing ratio ({3 = 0.02) and depth (8 = 0.075 m), under ambient moisture conditions and at windspeeds of 1.8 and 2.7 m/sec. It is clear from the figure that fuel type has an effect on rate of spread, but that the dependence is not simply due to surface-area-to-volume 1P

is the p-value associated with the test of the statistical hypothesis of no effect.

FREE-BURNING FIRES IN WOODY FUELS (b)

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fuel depth (m)

FIGURE 5 Rate of spread versus fuel bed depth: 27°C and 20% relative humidity, windspread 2.7 mlsec, (a) regular excelsior, (b) pine needles, (c) coarse excelsior, (d) sticks. Packing ratios are as shown.

ratio. Pine needles have a slightly higher equilibrium moisture content than excelsior (7% rather than 6% at ambient conditions) and a relatively high density (Tab. I). Both of these factors could reduce the spread rate. We now construct an empirical model from the above data, but based on the conservation of energy. This is similar to the approach taken by Rothermel (1972).

4. MODEL DEVELOPMENT

The propagating flux, I p , is defined as the average rate of energy transfer from the flame and combustion zones to the unburned fuel, through a unit vertical cross-section of the fuel bed facing the oncoming fire. From conservation of energy, lp is equal to the rate of spread multiplied by the

10

W. R. CATCHPOLE et al. (8)

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1.0

2.0

3.0

wlndspeed (m/sec)

0 0.0

1.0

e

2.0

3.0

wlndspeed (m/sec)

FIGURE 6 Rate of spread versus windspread: 27'C and 20% relative humidity, (a) regular excelsior, fuel depth 6 = 0.1 m, (b) pine needles, 6 = 0.075 m, (c) coarse excelsior, 6 = 0.075 m, (d) sticks, 6 = 0.075 m. Packing ratios as shown.

energy needed to raise a unit volume of the fuel bed to ignition (Frandsen, 1971):

(1 ) Here R is the rate of spread, Pb = /3pp is the fuel bulk density (mass of fuel per volume of fuel bed), Pp is the fuel particle density, /3 is the packing ratio (volume of fuel per volume of fuel bed) and Qr is the heat of ignition of a unit mass of the fuel (Wilson, 1990). The effective heating fraction, Tip (Frandsen, 1973), is the effective proportion of a fuel particle that is raised to ignition temperature by the time that the particle ignites. Following Wilson (1990) we use Qr in place of the heat of pre-ignition Qig used in Rothermel (1972). It may be approximated as a linear function of moisture content (Dunlap, 1912)

(2)

FREE-BURNING FIRES IN WOODY FUELS (a)

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en (;

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FIGURE 7 Rate of spread, on a log scale, versus moisture content as a proportion of ovendry weight: (a) regular excelsior, packing ratio {3 ~ 0.005, fuel depth 6 ~ 0.1 m (wind-aided); 6 = 0.2m (zero-wind, Wilson, 1990), (b) regular excelsior, {3 = 0.02,6 = 0.025m (zero-wind, Wilson, 1990),6 = O.lOm (wind-aided), (c) pine needles, {3 = 0.065,6 = 0.06-0.08m (zerowind, Anderson and Rothermel, 1965), 6 = 0.025m (wind-aided), (d) sugar pine sticks, s = 2520 m- 1, {3 = 0.02, 6 = 0.025 m (zero-wind, Wilson, 1990);excelsior, wind-aided, {3 ~ 0.03, 6 = 0.075m (open triangles and diamonds), 6 = 0.15m (black triangles and diamonds). Windspeeds are as shown in the legend.

where Qp is the heat of pyrolysis of a unit mass of dry fuel, M is the moisture content of the fuel (as a fraction of fuel dry mass) and Qw is the heat of dessication (per unit mass of water). Qp varies weakly with fuel type; values for typical forest fuels are given in Susott (1982) and in Table I for our fuels. In a physical model the propagating flux would be calculated from radiation and convection energy transfer considerations, so that Eq. (I) could be used to predict the spread rate R. Here we calculate the propagating flux from (I) and investigate how it is influenced by fuel and environmental variables. The main environmental influences on propagating flux are windspeed and fuel moisture content. Windspeed affects both the rate of energy production and the efficiencyof the energy transfer to the unburnt fuel. Fuel moisture content affects the propagating flux via (2), but also dilutes the

12

W. R. CATCHPOLE et al. 0.12 02.7 mlsec

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coarse excelsior

sticks

needles

regular excelsior

6000

8000

o o

2000

4000

surface area to volume ratio (11m) FIGURE 8 Rate of spread versus surface-area-to-volume ratio: 27°C and 20% relative humidity, (3 = 0.02, 6 = 0.075 m. Windspeeds as shown.

reacting components, thereby decreasing the rate of energy production (Albini, 1980). In a homogeneous fuel bed this can happen in two ways. First, the combustion interface in wind-aided fires is not vertical and moisture evaporated from unburned fuel in the lower part of the bed may rise through fuel which is undergoing combustion. Second, the outside of the fuel particle may be burning while moisture still inside the particle is being evaporated and forced out where it combines with reacting matter. Anderson and Rothermel (\965) and Wilson (\990) burned fires in the absence of wind over a range of moisture contents to determine the effect of moisture content on spread rate. We have done the same using a range of fuel types, windspeeds, fuel depths and packing ratios. Figure 9 shows the propagating flux plotted against moisture content. Anderson and Rothermel's ponderosa pine needle data and Wilson's excelsior and 6mm Pinus Lambertiana (sugar pine) stick data are included for comparison.

13

FREE·BURNING FIRES IN WOODY FUELS

(a) 2000

~o

00

00

&&~~

200

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I

• 0.0 mJsec 0.9 m/sec lJ. 1.8 mlsec o 2.7 mlsec

(b)

0.10

0.20

0.30

~

0.0

moisture content

•

• 0.10

~/).Q)~

* 0.20

0.30

moisture content

FIGURE 9 Propagating flux, on a log scale, versus moisture content for a range of windspeeds. Symbols have the same meaning as in Figure 7.

For regular excelsior (Figs. 9a and b), moisture has little effect upon the propagating flux for either the zero-wind fires (as also noted by Wilson, 1990), or for the wind-aided fires. The pine needles and coarse excelsior show reductions in propagating flux with moisture content (Figs. 9c and d). Straight lines on this log-linear scale fit these data well, giving an exponential decay of the same form as Wilson (1990): /p

= /p.exp(-kM).

(3)

Here [po is the propagating flux at zero-moisture content, and k is a moisture-damping constant, not depending on M. Models of the form (3), with /po and k depending on windspeed and fuel bed depth, can be fitted in turn to each of the four data sets shown in Figure 9. The resulting analysis of variance indicate that k does not depend significantly on windspeed or fuel bed depth. The estimates of k are shown in Table II.

14


TABLE II Estimates and standard errors for the moisture damping coefficient k from fitting the logarithmic version of Eq. (3) and p-values for a test of k = 0

st.err, (k)

p

-0.73

0.35

0.05

0.41

0.62

0.53

4.05

0.47

< 10-'

1.46

0.28

10-'

2.65

0.53

10-'

k


Regularexcelsior, {3 = 0.005, 6 = 0.20 m Regular excelsior, {3 = 0.02, 6 = 0.10 m Pine needles, {3 = 0.065, 6 = 0.025 m Coarse excelsior, {3 = 0.03, 6 = 0.075 m Coarse excelsior, {3 = 0.03, 6 = 0.15 m

The results of Table II show that k depends either on packing ratio or fuel particle size, or both. Our data are insufficient to distinguish between these. However, a re-examination of Wilson's (1990) data shows that in zero-wind fires k increases with surface-area-to-volume ratio, but does not depend on packing ratio (or fuel bed depth). Wilson (1990) found that k was greater for pine needles than would be expected for bare wood particles of the same surface-area-to-volume ratio. This may be caused by the lower moisture dilfusivity of the pine needles (Anderson, 1990). Our results are similar to Wilson's. Since the packing ratios we have used for pine needles are similar to those found in forest litter, it is not practically important to determine which of these effects predominates: we simply need to use the value determined here. We investigate the dependence of I p on the fuel parameters and windspeed by first correcting for moisture content to determine Ipo, and then examining the effect of the other parameters on I po' A value of Ipis estimated for each fire and then (3) is solved to get I po using values for k of 0 for regular excelsior, 2.03 for coarse excelsior (combined data) and 4.05 for pine needles (see Tab. II). For the sticks, the value k = 4.40, obtained from Wilson's (1990) data, is used, as our data set does not contain high moisturecontent sticks. This method, of first estimating I po' is preferred over fitting (3) directly to the full data set, as we only have good information on the effect of moisture on I p for 5 sets of conditions. The unbalanced nature of the data otherwise results in the moisture dependence in (3) being contaminated by the dependence of I Po on the other parameters. Figure 10 shows the dependence of I po on fuel type. The figure indicates a possible relationship between I po and surface-area-to-volume ratio for s < 3000 m- I . Curves of the form Ipo ex e -bls, shown in the figure, reflect

15

FREE-BURNING FIRES IN WOODY FUELS 1000 . coarse

sticks

needles

regular

excelsior

excelsior

o ---

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--~--0

o

o

o


,/'/ / ,,

,, ,,, t

t

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2000

4000

6000

8000

surface area to volume ratio (11m)

FIGURE 10 The moisture-adjusted propagating flux Ip, from the comparative experiment plotted against surface-area-to-volume ratio s, for the 1.8 and 2.7 mlsec windspeed fires. These fires burned at 27°C and 20% relative humidity (M "" 0.06) with a fuel depth 6 = 0.075 m and a packing ratio of j3 = 0.02. The fitted lines are of the form Ip, ()( e-bl'.

this relationship reasonably well. This is the same form of relationship as Frandsen (1973) used for the effective heating fraction 'T/pFigure II shows the dependence of [Po on packing ratio for four different conditions of fuel type and fuel depth. There is no clear dependence on fuel bed depth and [po can be represented as a power function in f3. We thus fit the model f>C JPo = A i e -bls fJ,

(4)

with a different constant Ai at each windspeed. Model (4) is fitted to the whole data set, with the exclusion of a total of 4 fires, at windspeeds of 2.0, 2.4 and 2.9 mis, which were not replicated enough to estimate Aj accurately. The estimates for band care b = 800 (std. err. = 48) and c = 0.501 (std. err. = 0.020). Estimates and standard errors of the Ai are shown in Table III.

16


1000

800

600


400

200

I

I

I

I

I

I

0.002

0.005

0.01

0.03

0.06

0.1

packing ratio FIGURE II The moisture-adjusted propagating flux I p , plotted against packing ratio on a log-log scale. Windspeed for all fires is 1.8 tnlsec, and environmental conditions are 27'C and 20% relative humidity: pine needles, 8 = 0.025 m (triangles); pine needles, 8 = 0.075 m (diamonds); regular excelsior, 8 = 0.075 m (crossed circles); regular excelsior, 8 = 0.20 m (solid circles).

Since the standard deviation of the propagating flux increases with the mean, we fit the logarithmic form of (4) and correct the Ai values for bias resulting from the logarithmic transform-see Brownlee (1965) for details. In Figure 12 we show the estimates Ai (together with error bars) plotted against windspeed. A cubic polynomial has been fitted to these values using weighted least-squares, with weights inversely proportional to the estimated variance of Ai at each windspeed, The cubic fit is shown as a dashed line in Figure 12, The fitted cubic describes the relationship between spread rate and windspeed for our laboratory data only, The relationship between spread rate and windspeed in the field has been described by McArthur (1966), Rothermel (1972) and Cheney et al. (1993), and in all cases has been less than a cubic power,

FREE·BURNING FIRES IN WOODY FUELS TABLE III

17

Estimates, standard errors and number of observations for the coefficient A in

Eq. (4)

st.err, (A)

Windspeed imlsec)

o

497 1296 2763 3161 3427 4177 5298 7227

0.44 0.89 1.33 1.78 2.22 2.67 3.11


number of observations

45

43

164

10

311 274 288 791 445 1328

15 53 99 3 124 3

8000

6000

4000

2000

0...L----------------------J o

0.4

0.9

1.3

1.8

2.2

2.7

3.1

windspeed (m/sec) FIGURE 12 The estimated coefficients Ai in (4) versus windspeed, showing standard error bars, the fitted cubic polynomial (dashed line) and the power law (solid line).

It is possible that the increase in spread rate at the highest windspeeds shown in Figure 12 is an artifact of the wind tunnel. We accordingly fit a power function of the fonn

A(U)=e+mU n

(5)

18

W. R. CATCHPOLE ct al.


to the data, again using weighted least-squares. The estimated coefficients are i = 495.5 (standard error 50.4), m= 1934 (s.e. = 161) and II = 0.91 (s.e. = 0.11). The estimate of n is not significantly different from I. The fitted power function is shown as the solid line in Figure 12. Although this does not fit our data as well as the cubic curve, it is more consistent with data from wildland fires. Cheney et al. (1997) have found a dependence on windspeed very similar to (5) for grassland fires. Combining (1)-(4), (5) and Frandsen's (1973) model, 7Jp = exp( -453js), produces

(6)

where t, III and n are as estimated above. The fit of this model to our full data set is shown in Figure 13. The fit is good for each fuel type. The two largest outliers in the Figures 13(c) have a fuel depth of 2.5 em, for which an even di.stribution of fuel is difficult to achieve in the coarse excelsior.

5. TESTING THE MODEL

We now test the empirical model (6) on data from laboratory fires in homogeneous fuel collected in Nelson and Adkins (1988) and from Fendell et al. (1990). The sources and the ranges of fuel and environmental parameters are given in Table IV. Values of k were estimated from the surface-area-to-volume ratios using the values in Wilson (1990), except in the case of pine needles where the value of 4.05, found above, was used. Wolff et al. (1991) show that rate of spread increases approximately linearly with the width of the fuel bed, up to a width of I m. Our fuel bed is 1 m wide. We decrease predictions from (6) proportionally to the width of the fuel bed to account for different bed widths in other laboratory fires. Figure 14 shows the predictions from our model plotted against observed values from experimental fires in other laboratories, for windspeeds of less than 3 mjsec, which was close to the maximum windspeed in our data set. The model fits well, except in one fire in 6 mm sticks at a very high packing ratio. Spread rates for Nelson and Adkins' fires in Pinus elliotii were slightly underpredicted, but this may be the the result of overpredicting k for a

FREE-BURNING FIRES IN WOODY FUELS

19

(b)

(a)

0.15

0.30

" 0.25 "§ "0

'c,~"

.

0.10

0.20 0.15

"0

~

0.10

~ c,

0.05

"0

0.05

0

0 0.10

0

0.20

0.30

0


(e)

.'"

0.10

0.15

0.05

0.10

0.15

(d)

0.15

0.15

0.10

0.10

0.05

0.05

~

"0

0.05

~ c,

"0