# Encyclopedia of Wildfires and Wildland-Urban Interface (WUI) Fires

Living Edition
| Editors: Samuel L. Manzello

# Burning Rate

Living reference work entry
DOI: https://doi.org/10.1007/978-3-319-51727-8_50-1

## Definition

Burning rate (BR) is the amount of fuel mass that is consumed per unit of time in the course of a combustion process.

## Overall Combustion

Mathematically the BR is defined by:
$$BR=\left|\frac{dm}{dt}\right|$$
(1)

In this equation, m is the instantaneous value of the mass of the fuel. The modulus sign is used to define a positive value of the BR as the time derivative of mass is essentially negative. To determine this derivative, it is necessary to evaluate the mass of the fuel during the combustion process.

Very often, the combustion process is not constant in the course of time. Typical plots of mass loss curves of a fuel during the process of combustion are given in Figs. 1 and 2: Fig. 1Mass loss curve with a period of constant mass loss rate Fig. 2Mass loss curve with an inflexion point corresponding to the maximum mass loss rate value
There may be a period of constant mass loss rate that can be used as a characteristic BRa of the combustion process. In the case of Fig. 1, this occurs for t1 < t < t2. Considering that m1 and m2 are the mass values at t1 and t2, respectively, the BRa will be given by:
$${BR}_a=\left|\frac{m_2-{m}_1}{t_2-{t}_1}\right|$$
(2)
In the mass loss curve that is shown in Fig. 2, there is not a period of constant mass loss ratio. Point 3 is an inflexion point of the curve m(t) for which the mass loss ratio is maximum. The maximum BRM calculated at this point can be also used as a characteristic BR of the combustion process. BRM is defined by:
$${BR}_M={\left|\frac{dm}{dt}\right|}_{t={t}_3}$$
(3)
If the evolution of mass during the combustion process is not available, a global or overall BRG can be estimated knowing the initial and final values of mass mo and mf, respectively, and the duration tf of the combustion process. The global BRG is given by:
$${BR}_G=\left|\frac{m_o-{m}_f}{t_f}\right|$$
(4)

## Burning Rate of Cribs

Cribs composed by a series of fuel pieces, like wood sticks, stacked in an organized form, are used commonly to perform heat resistance tests and other sorts of experiments involving combustion that may require a standard and replicable heat source. Burning rates of cribs with a wide variety of layouts and geometries were explored by various researchers (Gross 1962; Smith and Thomas 1970; McAllister and Finney 2016), among other purposes to determine whether results from structural fire experiments hold in the wildland context. Comparisons included the effect of stick dimension (length and width) ratios and the effect of spacing distance between the crib and the support platform (vertical gap).

There are several relations in the literature for the burning rate of wood cribs that have been developed over the years. In the literature, the burning rate is divided into two regimes or crib types: open or loosely packed and closed or densely packed cribs. Gross (1962) conducted the first major exploration of the free burning of wood cribs and described the loosely packed as a regime in which burning rate is independent of the “porosity” of the crib, being governed by heat and mass transfer processes near the surface. In the densely packed regime, the burning rate is dependent of the “porosity” of the crib and increases with the spacing between sticks. In this work, the porosity (φGross) is defined as:
$${\varphi}_{\mathrm{Gross}}={N}^{0.5}{b}^{1.1}\frac{A_v}{A_s}$$
(5)
where N is the number of layers, b is the stick thickness (cm), Av is the initial open area of the vertical shafts in the cribs (cm2), and As is the initial total exposed area of the sticks (cm2). The scaled burning rate R of these cribs was thus a function of porosity
$$F.R.{b}^{1.6}= fn\left[{\varphi}_{\mathrm{Gross}}\right]$$
(6)
where F is the ratio of the thermal diffusivity of Douglas fir to the wood tested, and R is the burning rate (g.s−1).
Later, Block (1971) assumed for loosely packed regime that the burning rate was closely related to the burning rate of the individual sticks and defined it based on a Spaulding’s B number analysis as
$$\frac{R}{A_s}=C{b}^{-1/2}$$
(7)
where C is the fuel property constant (g.s−1.cm−1.5) and varies for different wood species and moisture contents. For densely packed regimes, a theory for turbulent burning was developed resulting in a theoretical model that provides expressions for the maximum burning rate in terms of the basic geometric and physical properties of the fuel structure resulting in the following expression:
$$\frac{R}{A_s}=\frac{1}{2} fp{\left\{\left[\frac{\rho_0-\rho }{\rho}\right] gh\right\}}^{1/2}\left(\frac{\lambda -1}{\lambda}\right)\frac{G}{\Psi}$$
(8)
where ρ is the density of air in the vertical shafts (g.cm−3), ρ0 is the density of ambient air (g.cm−3), g is the acceleration due to gravity (cm.s−2), h is the height of the crib (cm), λ is the ratio of the mass flux of gases leaving to the mass flux entering a volume (dimensionless), and G and Ψ are defined in Block (1971). In addition to the development of the theoretical model, Block also compared his model to burning rate experimental results that he obtained.
Heskestad (1973) combined the experimental results from Gross and Block with the theoretical model of Block and developed a new correlation in which the right member of the equation defines the porosity according to Heskestad:
$$\frac{R}{A_s{b}^{-1/2}}= fn\left[\left(\frac{A_v}{A_s}\right){s}^{1/2}{b}^{1/2}\right]$$
(9)

From these three models, other relationships were developed and adapted for particular cases, such as wildland fuels that are characterized by thin fuel elements (e.g., pine needles), which conducts to the necessity to study cribs with a large range of geometries but which are poorly studied. In addition, another important parameter to study is the vertical gap due to the heterogeneous fuels found in forests, as for example ground fuels (needle litter layers) and crown fuels (trees and shrubs) in which the supply of oxidizer varies greatly. One of the last works performed in this subject was from McAllister and Finney (2016) in which the burning rate of cribs with a wide variety of geometries and aspect ratios as well as the vertical gap in the loosely packed regime is explored to test several correlations from the literature: Heskestad model (1973), Block model (1971), and Thomas model (1973). In general, values obtained using cribs with geometries similar to those tested in literature (e.g., Gross, Block) matched predicted values well. However, the burning rate of cribs built with sticks of large length-to-thickness ratios (such as long, thin sticks) was considerably lower than predicted, indicating that there is insufficient airflow inside the crib which is not predicted by current models. The effect of spacing distance between the crib and the support platform (vertical gap) was strongly dependent on the stick length-to-thickness ratio. Experiments indicated that cribs with large length-to-thickness ratios required a substantial amount of airflow through the bottom of the crib. As the crib-platform spacing increased, however, the burning rate of the large length-to-thickness ratio cribs increased to more closely match the predicted values. The effect of other environmental variables, such as the presence of wind or of a chimney effect, is not yet well studied.

In Table 1, average burning rates values for different configurations of wood cribs are given. These correspond to the work of McAllister and Finney (2016) and refer to cribs conditioned at 35 °C, 3% relative humidity, and approximately 2.5% of moisture content. In the present table, only part of the work is presented given the great variety of cribs configurations.
Table 1

Wood cribs characteristics and average burning rate, from McAllister and Finney (2016)

Crib designa

Stick thickness, b (cm)

Stick length, l (cm)

l/b

Number of sticks per layer, n

Number of layers, N

Surface area, As (cm2)

Vertical gap, d (cm)

Average burning rate (g.s−1)

1

0.64

6.35

10

3

10

442.74

1.27

0.5249

2

5

665.32

0.3569

4

1.27

12.7

10

3

10

1770.96

1.27

1.7407

5

5

10

2661.29

1.27

1.9199

7

0.64

6.35

10

2

12

370.97

1.27

0.4985

15

0.64

15.24

24

4

14

2045.16

0

2.2923

0.64

2.7081

1.27

2.8698

2.54

2.8996

7.62

2.7690

33

0.16

25.4

160

16

8

1980.64

0

0.5040

1.27

2.3002

2.54

3.2240

7.62

3.6089

aThe numeration is in accordance to the work of McAllister and Finney (2016)

## Linear Fire Front

In the case of a linear fire front spreading in a solid porous fuelbed, the mass loss rate is defined per unit of fire line length BR′ (kg.m−1.s−1).
$${BR}^{\prime }=\frac{1}{\ell}\left|\frac{dm}{dt}\right|$$
(10)
Average or maximum values of BR′ can be determined for a spreading fire in a similar form that was described above. Considering that mc (kg.m−2) is the initial fuel load (mass of fuel per unit area), mf is the final fuel load and R is the average value of the rate of spread (m.s−1), then the global BR′G is given by:
$$B{R}_G^{\prime }=\left({m}_c-{m}_f\right).R$$
(11)
Considering the combustion efficiency ε as:
$$\varepsilon =\frac{\left({m}_c-{m}_f\right)}{m_c}$$
(12)
$$B{R}_G^{\prime }=\varepsilon .{m}_c.R$$
(13)

The combustion efficiency ε value is in the range 0 < ε < 1. It depends on the fuelbed properties, namely, the surface-to-volume ratio of the particles, the porosity and moisture content, and on the local flow conditions. For normal fuels and burning conditions, a value of ε = 0.8 can be assumed.

## Pool Fire Burning Rates

The burning rate is one of the most important characteristics of pool fires and is derived from the fuel surface regression rates given in mm/min (i.e., the surface is lowered by a number of mm per minute as the fuel is consumed in the combustion process). Generally the burning rate of pool fires is expressed as mass variation per unit area per unit time, kg.m−2.s−1. Another way of presenting the burning rate, also known as mass loss rate, is in mass of fuel per unit of time, kg.s−1.

There are two types of pool fires that can be distinguished in literature: small pool fires and large pool fires. This difference is due to the possibility of two different basic regimes that exist in these fires as shown by the Hottel’s (1959) analysis of Blinov and Khudiakov’s data in which, for small diameters (D), the burning mode is convective and for large D, it is radiative. For example, Babrauskas (1983, 1986) presented four burning modes of pool fires (Table 2).
Table 2

Pool fire burning modes (Babrauskas 1983)

Pool fire type

D (m)

Burning mode

Small

<0.05

Convective, laminar

0.05–0.2

Convective, turbulent

Large

0.2–1.0

>1.0

Burning rates of pool fires were estimated for different fuels and its value can depend, in the simplest analysis, on the pool diameter and fuel type (Babrauskas 1986). For large pool fires (D > 0.2 m), the mass burning rate can be predicted by:
$${\dot{m}}^{{\prime\prime} }={\dot{m}}_{\infty}^{{\prime\prime} }.\left(1-{e}^{- k\beta D}\right)$$
(14)
where $${\dot{m}}_{\infty}^{{\prime\prime} }$$ is the mass burning rate of an infinite diameter pool (kg.m−2.s−1), k is the absorption-extinction coefficient of the flame, β is the mean beam length corrector, and D is the pool diameter (m). Table 3 presents the data from Babrauskas (1983) for burning rates, density, heat of combustion of large pool fires for several fuels.
Table 3

Large pool fire burning rate data (Babrauskas 1983)

Material

$${\dot{m}}_{\infty}^{{\prime\prime} }$$ (kg.m−2 s−1)

Density (kg.m−3)

∆Hc (MJ.kg−1)

(m−1)

Cryogenics

Liquid H2

0.017

700

120.0

6.1

LNG (mostly CH4)

0.078

415

50.0

1.1

LPG (mostly C3H8)

0.099

585

46.0

1.4

Alcohols

Methanol (CH3OH)

0.017

796

20.0

a

Ethanol (C2H5OH)

0.015

794

26.8

a

Simple organic fuels

Butane (C4H10)

0.078

573

45.7

2.7

Benzene (C6H6)

0.085

874

40.1

2.7

Hexane (C6H14)

0.074

650

44.7

1.9

Heptane (C7H16)

0.101

675

44.6

1.1

Xylene (C8H10)

0.09

870

40.8

1.4

Acetone (C3H6O)

0.041

791

25.8

1.9

Dioxane (C4H8O2)

0.018b

1035

26.2

5.4b

Diethyl ether (C4H10O)

0.085

714

34.2

0.7

Petroleum products

Benzine

0.048

740

44.7

3.6

Gasoline

0.055

740

43.7

2.1

Kerosine

0.039

820

43.2

3.5

JP-4

0.051

760

43.5

3.6

JP-5

0.054

810

43.0

1.6

Transformer oil, hydrocarbon

0.039b

760

46.4

0.7b

Fuel oil, heavy

0.035

940–1000

39.7

1.7

Crude oil

0.022–0.045

830–880

42.5–42.7

2.8

Solids

Polymethylmethacrylate (C5H8O2)n

0.020

1184

24.9

3.3

Polypropylene (C3H6)n

0.018

905

43.2

Polystyrene (C8H8)n

0.034

1050

39.7

aValue independent of diameter in turbulent regime

bEstimate uncertain since only two points available

## Fire Whirl Burning Rates

In recent years, the study of fire whirls has been of great importance for the understanding of the mechanisms of formation and development of this phenomenon. One of the most important parameters analyzed is the burning rate in order to assess the amount of energy that is released by the fire whirl. This parameter has been measured experimentally in laboratory for different gaseous, liquid and solid fuels, and several pool/basket diameters (Lei et al. 2011; Pinto et al. 2017; Zhou et al. 2013). One of the methods normally used to measure burning rates (mass loss rate) is done by weighting the fuel package as it burns, using weighting devices like a load cell, using Eq. 3 (e.g., Pinto et al. 2017).

Fire whirl experimental tests with liquid fuels showed that fire whirls are highly stable burning phenomenon with large quasi-steady periods and that their burning rates depend on D similarly to those of general pool fires. However, the transition to turbulent burning occurs sooner as D increases in relation to pool fire tests (Lei et al. 2011). Data from burning rate measured in experimental tests of pool fires and fire whirls with heptane as liquid fuel are presented in Table~4.
Table 4

Burning rate data for pool fires and fire whirl tests with heptane

Reference

D (m)

m″ (kg.m−2.s−1)

m (kg.s−1)

Pool fires

Tarifaa

0.25

0.0280

0.0014

0.50

0.0620

0.0122

Kung and Stavrianidisa

1.20

0.0670

0.0758

1.70

0.0730

0.1657

Klassen and Gorea

0.05

0.0190

0.0000

0.07

0.0230

0.0001

0.30

0.0360

0.0025

0.60

0.0570

0.0161

1.00

0.0660

0.0518

Koseki and Yumotoa

0.29

0.0160

0.0011

0.59

0.0330

0.0090

0.98

0.0410

0.0309

1.99

0.0520

0.1617

6.01

0.0780

2.2128

Zhou et al. (2013, Table 2)

0.30

0.0155

0.0011

0.60

0.0287

0.0081

1.00

0.0409

0.0321

2.00

0.0516

0.1622

6.00

0.0783

2.2144

Fire whirls

Lei et al. (2011)

0.10

0.0446

0.0003

0.20

0.0684

0.0022

0.30

0.0746

0.0053

0.40

0.0742

0.0093

0.50

0.0766

0.0150

Zhou et al. (2013)

0.20

0.0665

0.0021

0.25

0.0703

0.0035

0.30

0.0697

0.0049

0.35

0.0780

0.0075

0.40

0.0780

0.0098

0.45

0.0824

0.0131

0.50

0.0822

0.0161

0.55

0.0762

0.0181

aValues from Table 2 of Zhou et al. (2013)

In tests with forest fuels, due to natural fuelbed variability and entrainment of environmental/forced air, the bottom of the fuel bed could not be ignited uniformly and mass loss rate values changes over time, i.e., mass loss rate values are dependent of time. The fire starts generally from the center and then spreads to the edge of the container. This is different from liquid pool fires, where ignition of the entire surface is rapid due to high flame-spread rates and the burning rate is constant during a long time. After ignition of the fuel bed, the mass loss rate reaches a maximum value or, in some cases the maximum value is constant during some time. So, the value used in burning rate analysis, in the case of forest fuels, is the maximum value or the average of the several highest points (Pinto et al. 2017). The mass loss rate increases with the diameter of the cylindrical container and for a given diameter the mass loss rate increases with the velocity of the forced flow. In Table 5, mass loss rate data are shown using different forest fuels (shrubs, pine needles, straw, and eucalyptus), different basket diameters, and different velocities of forced flow (vin).
Table 5

Mass loss rate data for fire whirl tests with forest fuels (Pinto et al. 2017)

Test Ref.

FMC (%)

Fuel

D (m)

vin (m.s−1)

M (kg.s−1)

GV16

18.2

Shrub

0.35

0.0225

GV15

18.2

Shrub

0.50

0.0321

GV23

18

Shrub

0.80

0.0480

GV31

14.4

Shrub

1.12

0.0856

GV20

18.5

Straw

0.50

0.0294

GV19

15.1

Pine needles

0.50

0.0286

GVV36

8.4

Eucalyptus

0.50

0.0196

GVV30

10.9

Shrub

0.50

0.31

0.0417

GVV6

11.4

Shrub

0.50

0.78

0.0440

GVV20

11.3

Shrub

0.50

1.73

0.0469

GVV17

13.6

Shrub

0.50

3.86

0.0584

GVV23

11.3

Shrub

0.50

4.83

0.0635

In general, regardless of the fuel type and fire dynamics (pool fire, fire whirl), the burning rate increases with the pool/basket diameter. Moreover, for the same pool/basket diameter, the burning rates of fire whirl tests are higher than for simple pool fires.

## References

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