NBSIR 73-115

Analysis of The Behavior of a Freely

Burning Fire in a Quiescent Atmosphere

J. B. Fang

Center for Building Technology

Institute for Applied Technology

National Bureau of Standards

Washington, D. C. 20234

February 1973

Interim Report

Prepared for Office of Research and Technology

Department of Housing and Urban Development

Washington, D. C. 20410

NBSIR 73-115

ANALYSIS OF THE BEHAVIOR OF A FREELY BURNING FIRE

IN A QUIESCENT ATMOSPHERE

J. B. Fang

Center for Building Technology

Institute for Applied Technology

National Bureau of Standards

Washington, D. C. 20234

February 1973

Interim Report

This report is to be superseded by a future publication which

will receive general distribution and should be cited as a reference. Please consult the NBS Office of Technical

Information and Publications to obtain the proper citation.

U. S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts. Director

ANALYSIS OF THE BEHAVIOR

OF A

FREELY BURNING FIRE

IN A

QUIESCENT ATMOSPHERE

J. B. Fang

A mathematical model which describes the physical and geometrical properties of a turbulent buoyant diffusion flame

over a free burning fire has been set up for both axisymmetric

and two dimensional cases. The mathematical simulation of

the flame consists of a combustion zone near its source and

a buoyant plume above and has been developed based on the

assumptions of infinitely rapid rate of oxygen- limited combustion reaction and "top-hat" profiles representing vertical

distributions of the velocity, temperature, and concentration

of the flame gas. Analytical solutions are presented showing

the effects of fuel mass-flow rate, physical properties of •

the fuel and ambient air, and the size and shape of burning

area on the general characteristics of a buoyant flame. Experimental data on visible flame heights and flame temperature profiles obtained from burning several different fuels

have been satisfactorily correlated by the derived expressions.

1
1. INTRODUCTION

An understanding of heat transfer rate from the flame emerging

above a fuel source is of importance in reduction or elimination of fire hazards in buildings. It is known that the ease

of ignition and the rate of fire propagation of the material

involved in the burning area strongly depend upon the intensity,

size, and duration of the exposing fire. The temperature,

velocity, and composition of the gases within the fire plume

determine rates of heat transfer to contents or structures

in a compartm.ent

The objectives of this analytical study are (1) to obtain

a better understanding of the burning behavior of incidental

fires such as those fires which may start in a wastebasket or

a piece of furniture and (2) to serve as the basis for prediction of heat transfer rates from the flame to nearby surfaces using fundamental quantities such as the fuel burning

rate, heat of combustion of combustible volatiles and the

geometry and size of the fuel source. The theoretical results

can also provide useful information on the fire induced turbulent ceiling- jet which is applicable to the design and

placement of fire detectors and suppressors.

In the experimental phase of this project, the burning

characteristics of the contents of wastebaskets and individual

2
pieces of upholstered furniture ivere measured. [1] For economy

and greater repeatability in subsequent experimental studies,

it is planned to use a wood crib array to represent a typical

piece of furniture. The proposed model of a burning wood crib

array permits (a) direct comparisons to be made with experimental

measurements and (b) subsequent extensions to other room

geometries and fire spread situations.

The present analysis is concerned with a fire plume that

exists over a burning area in the absence of wind. A simplified mathematical model describing the behavior of such turbulent buoyant diffusion flames is presented. This model is an

extension of previous studies of buoyant plumes [13] to include

the analytical treatment of two-dimensional fire plume and the

methods of approach to use a characteristic temperature for defining the visible flame tip, a coefficient to account for radiation effect, two separate entrainm.ent constants for the combustion

zone and the convection column, and variable concentrations of

combustible volatiles supplied at the flame base. Laboratory

experiments of free burning fires have been performed. Data

from these experiments as well as those from other investigators

[2-9] have been correlated with the derived functional relationships suggested by the model.

2. MATHEMATICAL ANALYSIS

Analysis of the burning characteristics of a buoyant diffusion

flame to account for tlie influence of various degrees of complete combustion in the fuel stream can be accomplished by

3
using methods similar to those used in previous flame plume

studies [10-13].

The mathematical description is concerned with a gaseous

mixture of combustible volatiles, combustion products and

nitrogen issuing from either an axisymmetric source or a linear source into a stagnant atmosphere. The combustion

reaction takes place instantaneously as the unburned gaseous

fuel combines with oxygen entrained from the surrounding air.

The heat evolved during this chemical reaction raises the

flame temperature and the buoyancy of the gases produced

in the plume causes them to move upward. At the upper

portion of the flame, the combustion reaction is complete

and the temperature of the resulting convection column decreases upon further mixing with the entrained ambient air.

Thus, a turbulent buoyant diffusion flame may be visualized

as a combination of two zones: a combustion zone near its

source and a buoyant plume zone above.

In order to derive relevant equations governing the flow

within the buoyant flame, the following simplifying assumptions

are made:

1. Turbulent flow is fully developed and molecular

effects can be considered negligible in comparison

with the turbulent transport.

4
2. Turbulent mixing in axial direction is negligible

compared to that in transverse direction.

3. No horizontal pressure variation exists between

the flame and the surrounding air.

4. The distributions of the velocity, temperature,

and concentration in the transverse direction can

be represented by the shape of a "top-hat" profile.

5. The transverse momentum flux due to the entrainmcnt

of ambient air is pro-portional to the local upv/ard

mom.entum flux of the rising flame gas.

6. Ambient air is of uniform density and temperature.

7. The gaseous fuel, combustion products and aml'^ient

air are perfect gases with identical values of

temperature- independent specific heat. The molecular

v/eight of the combustion products in the buoyant

plume zone and the ambient air are equal.

8. In the combustion zone, the combustible volatiles

mix with the entrained oxygen and burn instantaneously.

9. Radiative heat transfer can be neglected.

Assumptions 1 to 3 are common to most of the mathematical

analyses of buoyancy- controlled turbulent plumes [10, 13-16].

The field of flow in most buoyant flames v/as found to be unchanging turbulence except in close proximity to the flame

base [17,18]. However, this initial laminar zone is relatively

small compared with the subsequent turbulent region.
Assuiiiption 2 is not valid for largo intense area fires \\rhere

significant vertical entrainraent occurs [19] .

The pressure within the plume is assumed constant and equal

to its arabient value since the lateral pressure variation

due to turbulent transport of momentuin in the transverse

direction is normally sinall. [20].

The assumption of "top-hat" profiles for velocity, temperature,

and concentration distributions has been widely used to give

a reasonably good prediction of the gross behavior of turbulent buoyant flames [10 , 11 , 13 , 15] . For a low velocity fuel

jet, a temperature profile with a maximum at the flame edge

and a local minimum at the flame axis has been reported [20]

The off-center maximum tends to shift to the flame axis as

the unburned fuel reacts and the turbulent flame develops.

It can be expected that velocity and concentration profiles

within the flame plume will be similar to the temperature

distribution. Therefore a "top-hat" profile for the mean

vertical temperature, velocity, and concentration distributions although over-simplified is a reasonable assumption.

The assumption that the entrainment constant for a turbulent

buoyant jet with a density differing from the ambient air

is proportional to the square root of the ratio of ambient

6
density to the average density inside the jet has been supported

experimentally [21] and derived theoretically [22].

Most fires in the open are not sufficiently large enough to

cause significant variation of the temperature and density of

the ambient atmosphere. However, these non-uniformity effects

can be included in a manner similar to that used for natural

convection column above fires [23-25].

The assumption that the gas mixture inside the flame plume

and the surrounding atmosphere consists of ideal gases with

identical specific heat and molecular weight may be justified

where temperatures are not liigh enough to produce appreciable

dissociation and where the buoyant plume above the combustion

zone consists primarily of nitrogen entrained from, the ambient

air

The instantaneous mixing and combustion to idealize combustion

phenomena is frequently used in mathem.atical analysis of fire

plumes [11,13]. The assumption of perfect m.ixing and a single

step overall reaction with oxygen entrainment as a limiting

factor for combustion, is a simplification which seems reasonable as long as the combustible mixture is above its ignition

temperature.

7
The effect of radiation due to the high temperature gases

inside the flame is very significant and thus v;ill modify

temperature and velocity profiles. Radiative heat transfer

within and from a buoyant turbulent plume has been included

in analytical treatments with the following assumptions that:

the plume is a black body emitter [11] , the plume gas is

considered as an optically transparent medium [24], that the

radiation can be divided into lateral and vertical components

[26 , 27], and the radiating gas is a gray medium and the radiative

interchange within the flame can be neglected [10] . All of

these assumptions which account for the radiation effect

require using numerical techniques to solve the resulting

non-linear differential equations. Comuuter solutions cannot

give explicit expressions and tend to be inconvenient for

studying the effects of various variables on flame behavior.

In order to pursue an analytical solution, radiation is ignored

in the present analysis. However, its effect can be approximated by suitably modifying the heat of combustion used in

the non- radiative model.

V'/itii the preceeding assumptions , the governing equations

x>/hich describe the combustion zone and buoyant plume zone

respectively can be obtained from balances of mass , chem.ical

species , momentum and energy along wit]i tlie ideal gas law as

the equation of state. A vertical flame with a differential

8
volume around xvhich the conservation equations are written

is sliown in Figure 1. The flow in vertical flames can be

considered as either axially symmetric flow over a circular

or a square source, or a two-dimensional flow resulting from

a linear fuel source.

2.1 The Combustion Zone

2.1. a Axisym.metric Vlovj

The continuity equation equates the change in mass flux across

the differential volume element with the mass rate of ambient

air entrained through the outer surface area of the element.

The equation of conservation of chemical species states that

the change in mass flux of species x across the element is

balanced out by the local generation of that species and the

transverse transport of that species due to mass entrainment.

The force balance in the axial direction equates the change

in momentum across the differential element with the buoyancy

force acting on the element.

9
With the assumption of oxygen limited combustion, the change

in sensible energy across the differential element is equal

to the energy liberated by chemical reaction plus the sensible

heat carried in by the entrained surrounding air.

Introducing the perfect gas law as the equation of state, this

may be expressed in the form

for the combustion zone

for the ambient air into the energy equation (4) yields the

following simplified form

dZ. 00 i Pc^^ (6)

where CU =^ -j^^(

{

) -A "T^jHT ^ ^ denotes species

including fuel, N2 and combustion products.

The boundary conditions at the flame base are

H«0 ; >a=-'Uo , P=Po , ^x: = ^^o and y^Vo

2.1.b Two Dimensional Flow

The corresponding equations expressing conservation of mass,

species, momentum, and energy for a two-dimensional diffusion

flame are:

/

a

(8)

10
aCPudpTV) ^ rZ. P.jCc^r-,^^) ^7 ^ 'a'avCpTa+ ^ / ^^^^ where Y

is the half width of the flame.

Using tiie ideal gas law as the equation of state, the energyequation (11) can be rev/ritten as

(12)

The boundary conditions for these differential equations are

a=o;>tc=^o, ^^^o.^xr'^io and y = yo

(13)

Introduction of the dimensionaless variables

p ^ E,

^ -

X>

^ - C14)

into the mass balance, conservation of species, force balance

and modified energy equation yield the following.

Axisymmetric Flow —- - - (15)

11
d.{.9'^r) _ a - e'y ^ ^

^^^^ Fr

Two Dimensional Flow

(18)

j^T 7

£=J p <^ (20)

(21)

(22)

with the boundary conditions:

(23)

^ ^Vo Froude number

The solutions to these governing equations are presented in

Appendix A. The dependent variables of primary interest are

the flame geometry, and the distributions of temperature and

concentrations of the component hot gases along the flame axis

.

5

The ratio of the flame height at which stoichiometric mixing

occurs to the source radius or half width may be expressed in

terms of the m.ass flow rate and physical properties of the

gaseous fuel:

12
Axisymmetric Flow

Two-Dimensional Flow

oa.

b The ratio of flame radius or half-width at any height and

(24)

at stoichiometric mixing to its source radius or half-width

expressed as a function of axial distance are also derived in

Appendix A and are respectively stated as follows:

For Axisymmetric Flow 2. s

For Tv;o -Dimens ional Flov;

(2S)

For Axisymmetric Flow

i—

For Two-Dimensional Flow

'OA

J- 3. /O S

(26)

The axial distributions of the flame density at any height

and for stoichiometric mixing in the comJmstion zone are

found to be separately expressed:

For Axisym.metr ic Flov/

p' =Ui-( CO) [^672 (iJ'(-^)\'+ ,]

13
For Two-Dimensional Flov;

where: = ' Qt = j;;55r-^

The temperature profile in the vertical direction can be ob'

tained readily from the expression for the flame density

along with the equation of state for a perfect gas.

For Axisymmetric Flov; jl <r

For Two-Dimensional Flow * ^ ^ £-KJ-f-Cft oi)

(27)

(28)

(29)

The dimensionless temperature of the hot gases along the

flame axis for stoichiometric mixing is found to be expressed as

/ (CO To )

The axial concentration distributions of combustible volatiles

and of combustion products within the combustion zone are

also derived separately in Appendix A:

For Axisymmetric Flow

^ Po(^^ ^ —7—)

^

For Tv/o -Dimensional Flow _ fe^cx^-^ y J

, ^

14

(31)
For Axisynmetric rioi_

For Two -Dimens ional Flow / z

' — ^PO^ —. — —

Nitrogen does not take part in the combustion reaction and

its concentration along the flame axis is found to be

For Axisymmetr ic Case

For Two-Dimensional1 Case race. *> J

The mass fractions of combustion products and mitrogen for

stoichiometric mixing are respectively expressed by:

(33)

O^-^) (34)

The sum of the mass fractions of all of the constituents except

oxygen within the combustion zone must be unity, on the assum.ption that oxygen and combustible volatiles do not exist together.

x^ + x,-.x,= l ^^^^

16
2.2 The Buoyant Plume

C^.eriical reaction is assumed to be terminated at the u-nper

end o£ the combustion zone and no heat is generated v/ithin

the convection plume. Thus, the convection plume above the

combustion zone behaves as a turbulent buoyant plume which

gradually cools upon mixing with the entrained ambient air.

Referring to the differential volume element of a buoyant

flame slioim in Figure 1, the equations of conservation of

mass and momentum are identical to tliose describing the

combustion zone.

Because there is no combustion reaction occurring and the hot

gases inside the plume and those in the ambient atmosphere

are assumed ideal, the conservation equations of species and

energy are reduced to the following form:

2. 2. a Axisymmetric Flow

C37)

(38)

(39)

(40)

16
Using the ideal gas law to modify the energy equation gives

2.2.b Two Dimensional Flow

The equations o£ conservation of mass, momentum, chemical

species and energy for the two-dimensional case are:

The energy balance can be simplified with the use of the

perfect gas law

d^ ^

The boundary conditions at the source of this zone are:

z-H^ ; -u-M^ , P=i?.^ x,-^.:s and y-y^

Defining the dimensionless quantities:

^ - Pa

^

Ta

17

(^1)

(42)

^2. '^'^ (43)

d^ ^ (44)

(45)

(46)

(47)

(48)
and. introducing these variables into the conservation equations

yields:

Axisyinmetr ic Flow

7?^^—- ^H'iT^ ixy

Two -Dimensional Flow

With the boundary conditions

at 2r=Z^

where: c

(49)

(50)

(51)

(52)

oti^ ' (55)

(56)

(57)

18
The solutions to these governing equations for both axisymmetric

and tv/o- dimensional cases are given in Appendix B. One of the

particular quantities of interest is the flame height which is

equal to the sum of the height at xvhich stoichiometric reaction

takes place plus the height at \>rhich the temperature on the

centerline is reduced to a certain value. This total flame

height is been derived to be:

For Axisymmetric Flov;

-^-=(('^9+ 0,916 kJ) \ '^1^^

For Two-Dimensional Flow

2.

where

(58)

19
These equations indicate that for given fuel volatiles , the

ratio of the hei[^ht of a turbulent, buoyancy- controlled

diffusion flame above an axisymmetric source or a linear fuel

source to the radius or half-width of the source is directly

proportional to the burning rate ("^/^/^^ ) raised to the

two-fifths and tv/o-thirds powers, respectively.

The ratio of flame radius or half-width to that at the flame

base in terms of axial distance for the buoyant plume zone is

expressed annroximately as

For Axisymmetric Flow

For Tv/o -Dimensional Flow

2_

s

v/here ^ = ^ \ d' ^oo- i

(59)

Another quantity required for derivation of vertical temperature

distribution within the convection -nlume is the profile of

plume density which is found to be:

For Axisymmetric Flow

_L 2.

p' = i-(i-/|')[/o'?£'c;,'\ %'-k;)+ i] ^

For Two -Dimen s ional Flow

(60)

20
The vertical teinr)erature distribution of plume Pias within the

convection column can be obtained immediately from the plume

density profile and the ideal gas law.

For Axisymmetric Flov/ j_

For Two -Dimensional FloiA.^ " % (*'- Hr)+ l)

At height above the stoichiometric mixing plane, the fuel

volatiles vanish and the oxygen content ( ) along the

flame axis may be expressed as:

For Axisymmetric Flov/ j_ «^ s;

For Two -Dimensional Flow ^ '^S>1-IJ

The concentration distribution of combustion products within

the hot gas is found to be:

For Axisymmetric Flow —^ --^ -3-

For Two-Di.ensiin^al1l";^"

"^<*'^^'-'^'>+

'] ' &')}

21
The axial distribution of nitrogen content inside the convection

plume is also derived in Appendix B and is expressed as:

For Axisymmetric Flow _X -1-3*

For Two -Dimensional Flow

{ [£'<:;^n; *f )

i] - ( I - &') } C64)

The sum of the mass fractions of oxygen, nitrogen, and combustion products within the hot gas plume must be in unity

since all volatile fuels are burned to stoichiometric completion

in the combustion zone.

These equations (equations 57 to 61) along with the corresponding

results in the combustion zone (equations 24 to 35) provide

the shape and size of the visual flame and the profiles of the

temperature and species concentration of hot gases throughout

the entire flame plume. These quantities are necessary information for estimating the heat transfer rate from the overhead

flame of a free-burning fire.

3 . Comparison with Experimental Data

The mathematical analysis indicates that the behavior of

turbulent buoyancy controlled flames in an unconfined atmosphere

depends upon fuel burning rate, shape of the burning area, fuel

properties and ambient conditions. The flame height for a given

22
fuel type can be predicted as a function of burning rate

using equation (58), which when plotted on log-log coordinates

and in the appropriate dimensionless groups (H/vb versus A/^ ) gives straight line with slopes of 2/5 and 2/3 for the

axisymmetric and two dimensional cases, respectively.

The visible flame height and fuel burning rate data obtained

from burning cribs of wood sticks arranged on a square base

[ 6 ] , spreading of fires along long v/ood cribs [ 7 ] , open

trays of liquid fuels [ 8 ] and gaseous fuels emerging from

various burner shapes and sizes [ 5,9] have been previously

reported. These data along with those obtained recently at

NBS with square-based Avood cribs are correlated and presented

in Figure 2 v\?here H/Vo plotted against the modified

in the figure were measured from the base of the crib or the

top of the burner. The flame rising vertically from a long

strip with a source length to width ratio greater than 1.5

were treated as a two-dimensional condition. The density of

ambient air and the specific heat of flame gas used in all

- 3 3 calculations are assumed to be 1.3 x 10 g/cm and

0.24 cal/g respectively. It can be seen from Figure 2 that for a given fuel type or crib design, all the data in

each group fall adjacent to a straight line calculated by the

method of least squares. Table 1 summarizes the results of

Froude number All of the flame heights shown

23
correlation of flame height to rate of burning and fuel source

characteristics. As one would expect the value of coefficient

K is dependent upon the shape and size of the burning area,

the typo of fuels involved and the fuel bed structure. The

flame heights of the crib fires made recently at NBS are significantly shorter than those by Pons et al. [ 7 ] and Gross [ 6 ] as

illustrated in Table 1. This discrepancy may be due to a large

clearance present between the crib base and the platform support,

which allows more air to enter the combustion zone through

the crib bottom.

Also included in Table 1 are the computed values of dimens ionless

flame tip temperatures obtained from equation 58 using the

values of coefficient K, average entrainment constant for both

combustion and buoyant plume zones and fuel properties. The

numerical values of the fuel properties used for the calculation

of flame tip temperature are listed in Table 2.

The pyrolysis of wood involves various complex processes and

tlie composition of volatile matter evolving from pyrolysis

reaction varies considerably depending upon the conditions

of external heating and the extent of degradation [28],

Roberts [29] conducted ultimate analyses v/ith partially decomposed v/ood and observed that over a wide range of volatilization, the mean composition of the volatiles can be represented

24
by formaldehyde. In the present analysis, gaseous formaldehyde

has been used to represent the average pyrolysis product of

wood as employed in a previous paper [30].

Table 1 illustrates that the computed flame tip temperature

for each data set ranges from 290° to 340°C for crib fires

and 400° to 440°C for fires with liquid and gaseous fuels.

These values are reasonably consistent with the measured

values given in the literature [ 3, 8, 9], it may be noted

that the values of the entrainment constant (E) used for crib

fires to obtain the temperature of flame tip depend upon the

construction of the cribs and ventilation conditions at the

crib edge. A value for the entrainment constant for buoyant

flames from gaseous fuels of E=0.14 gave the best fit to the

observed flame tip temperature.

Calculations based on equations 29 and 61 have been made for the

distribution of temperature along the flame axis for the fires

above a square base crib and a pan of n-heptane. The results

of the computations are presented in Figures 3 and 4 where

the ratio of the flame temperature to the ambient temperature

is plotted against normalized vertical distance from the flame

base. Also shown for comparison are the experimental data reported by Tarifa et al [8 ] and those obtained by the author.

Results shown in Figure 3 are based on the idealized condition

25
that all the gaseous fuel produced from pyrolyzing wood can

be visualized as being supplied at the crib base.

As shown in Figures 3 and 4, reasonably good agreement has

been found between theoretical predictions and experimental

measurements. It may be noted that the best fit value of

entrainment constant for the combustion zone differs significantly from that for the buoyant plume. Considering the difference in nature of the two zones and the close agreement with

experimental results, the use of two separate entrainm.ent

constants for a diffusion flame seem.s relavant. A recent

article [31] has reported that the local entrainment of an

axisymmetric turbulent air jet increases with increasing

vertical distance in the initial region and then becomes constant

in the fully developed flox^/ region which is located an axial

distance of approximately fifteen nozzle diameters. This

experimentally observed non-uniformi ty of air entrainment

at the early stage of a free jet may also occur in a buoyant

flame. Values of the plume zone entrainment constant determined

from crib and n-heptane burnings are comparable ^\rith a value,

E'=0.08 reported in a previous paper [21].

The average molecular weight and mass fractions of constituent

gases within a flame as a function of normalized axial distance

for crib and n-heptane fires are computed by use of equations

26
31 to 36 and 62 to 64 and the results are presented in Figures

5 and 6 respectively. Values for carbon dioxide and water

vapor are obtained from the complete combustion of the gaseous

fuel. As sho^^m in the figures, all of the gaseous fuel supplied

at the base of the flame are completely consumed at the

s toicliiometr ic mixing point and the peak temperature also occurs

at this height. It can be seen that the value of mean molecular

v/eight of the flame gas in the buoyant plume region is nearly

equal to that for ambient air.

27
4. Conclusions

The behavior of buoyant diffusion flames above a naturally

occurring fire can be predicted by the postulated mathematical

model based on a number of simplifying assumptions.

The flame heights of radially symmetric and linear diffusion

flames for several different fuels have been correlated

reasonably well with the derived parameters describing the fuel

source characteristics and fuel properties. From the results

of the correlation, the temperature at the visible flame tip

defined as tlie point at which the flame temperature falls to

a characteristic value, has been found to range from 290^ to

340°C for cribs and 400° to 440°C for gaseous and liquid fuels

and are in good agreement v/ith experimental measurements.

The measured temperature profiles along the flame axis for

radially symmetric fires are found to agree with theoretical

predictions

.

It is believed that the gross feature of fire plume such as

the flame shape and the vertical distributions of temperature

and gas composition can be predicted, using the present model

or one similar in nature, for the stated condition of size and

shape of burning area and rate of production of pyrolysis

products.

28
TABLE 1

Values of Coefficient K and Exponent '^t^ for the Modified Form of Equation

58 (H/Vg= K /^i> ) and Flame Tip Temperature

The

Length to Average

Half-width Entrain—

(or Radius) Coeffi- Expo- Flame Tip ment

Fuel Ratio of the cient nent Temperature Constant

Data Source Type Fuel Source K -ru 1*:/Ta. E

3 Yo

Thomas (2) Wood Crib

(white

niis Study Wood Crib

1

Gross (5) Wood Crib

(Douglas — = ? Fir) y, 41 2/5 2.05 0.11

Fons et al. Wood Crib

(7) (White , , Fir) ^<-< 3 35 2/5 1.89 0.14

3 < - 82 2/3 2.01 0.14

pine) 3 < ^ 63 2/3 1„92 0.22

(Hemlock) ~ = 2 26 2/5 1.94 0.20

Putnam and

Speich ( 5 ) City Gas - = 2 60 2/5 2.32 0.14

2xlO-^<^ < 4x10-^ 105 2/3 2.37 0.42

Putnam (9) 1.184N_ "= 2 41 2/5 2.38 0.14

i- Jo

Grinberg and Ethane + -j^ Putnam

^arifa

etal.(8) 7i-Heptane ~ = 2 36 2/5 2c25 0.21

29
TABLE 2

VALUES OF THE FUEL PROPERTIES USED

FOR FLAME TIP TEMPERATURE CALCULATION

Heat of

Stoichiometric Oxygen- Combustion

Fuel Fuel Mass Ratio (Sc, Volatiles ^ (cal/g)

Fraction of Source-Ambient

Heat Loss Temperature

by Radiation Ratio

Wood

City Gas

Ethane

7U -Heptane

1.1

4.0

3.7

3o5

2,500

11,800

11,300

10,700

0.3

0.3

0„3

0.3

1.9

1.0

1.0

1.6

30
ILLUSTRATIONS

Figure 1 Turbulent Buoyant Diffusion Flame with a Differential Volume Element,

Figure 2 Correlation of Flame Height, Burning Rate, and Source

Characteristics for Various Fuels.

Figure 3 Comparison of Predicted and Measured Gas Temperature along the Flame Axis of a Crib Fire. Width

of the Crib- 0.559 m, fuel burning rate - 593 g/min.

,

ambient temperature - 298°K, heat of combustion -

2500 cal/g. , fraction of radiation loss - 0.35, entrainment coefficients - E=0.18 for the combustion

zone, and E'=0.07 for the buoyant plume.

Figure 4 Comparison of Predicted and Measured Gas Temperature Distribution in the Vertical Direction for

n - Heptane fire. Experimental data by Tarifa et

- 3 2 al. (8), fuel burning rate - 5.32 x 10 g/cm -sec,

ambient temperature - 293°K, heat of combustion -

10700 cal/g, fraction of radiation loss - 0.5, pan

diameter - 0.25 m, entrainment coefficients - E=0.25

for the combustion zone, and E'=0.05 for the buoyant

plume

31
Figure 5 Molecular Weight and Concentration Distributions

Along the Flame Axis for Crib Fire.

Figure 6 Molecular Weight and Concentration Distributions

Along the Flame Axis for n - Heptane Fire.

32
AC KNOWLBDG EMENTS

The author wishes to acknowledge the assistance of Mr. C. F. Viertz with the experimental

work. This work was supported by the United

States Department o£ Housing and Urban Development.

33
Nomenclature

heat capacity

entrainment coefficient of the ambient air to the

combustion zone

entrainment coefficient of the ambient air to the

buoyant plume

Froude nunber = fffO

gravitational constant

production rate of i-th species per unit volume

2^

flame height

height of the flame within the combustion zone

molecular weight

mass flow rate of volatile fuel per unit source area

molecular weight of the flame gas in the combustion

zone

modified Froude number = -

f<?L pressure

net heat of combustion of volatile fuel

universal gas constant

absolute temperature

axial velocity of the flame gas

dimensionless volumetric flow rate in the combustion

zone

24
^ dimensionless volumetric flow rate in the buoyant

plume

'U/'' dimensionless momentum flux in the combustion zone

'UT dimensionless momentum flux in the buoyant plume

mass fraction of species

V flame radius or half-\vridth

S. distance along the flame axis

y stoichiometric oxygen to fuel mass ratio = •

/A-X stoichiometric coefficient of species Z

/t). stoichiometric ratio

P density

Subscripts and Superscripts

A denotes ambient properties

denotes volatile fuel

0 denotes oxygen

N denotes nitrogen

O denotes fluid properties at the flame source

P denotes combustion products

iS denotes fluid properties at the point of stoichiometri

mixing ^ denotes flame properties at the flame tip

denotes quantity have been made dimensionless

absence of any scripts denotes the flame properties

35
REFERENCES

1. Gross, D. , and Fang, J. B., Performance Concept in

Buildings , Joint RILEM-ASTM-CIB Symposium Proceedings,

NBS Special Publication 361 (1972), Vol. 1, p. 677.

2. Thomas, P. H. , Ninth Symposium (International) on Com -

bustion , The Combustion Institute, Pittsburgh, Pa.

,

(1963), p. 844.

3. Thomas, P. H. , Baldwin R. , and Heselden, A. J. M. , Tenth

Symposium (International) on Combustion , The Combustion

Institute, Pittsburgh, Pa. (1963), p. 867.

4. Putnam, A. A., and Speich, C. F., Ninth Symposium (International) on Combustion , The Combustion Institute, Pittsburgh, Pa., (1963), p. 867.

5. Putnam, A. ,A. , and Speich, C, F., A Model Study of the

Interaction Effects of Mass Fires , Summary Report No. 2

for Office of Civil Defense through National Bureau of

Standards, NBS Contract CST-717, Battelle Memorial Institute, Columbus, Ohio, March 27, (1963).

6. Gross, D., J. Res . Nat. Bur. Stand. (U.S.) 66C (2) , (1962)

,

p. 99.

7. Fons , W. L., Clements, H. B., Elliott, E., R. , and George,

P. M. , Project Fire Model , Summary Progress Report II,

U. S. Forest Service, Southeastern Forest Experiment Station,

Macon, Georgia (1962)

.

8. Tarifa, C. S., Natario, P. P., and Valdes, E. C, Open
Fires , Final Report for U. S. Forest Service under Grant

FG-SP-114 and FG-SP-146, (1967).

9. Grinberg, I. M. , and Putnam, A. A., A Model Study of

the Interaction Effects of Mass Fires , Summary Report

No. 3 for Office of Civil Defense through National Bureau of Standards, Battelle Memorial Institute, Columbus,

Ohio, September 28 (1964).

10. Masliyah, J. H. , and Steward, F. R. , Combustion and

Flame , 13, 613 (1969) .

11. Nielson, H. J., and Tao, L. N. , Tenth Symposium (International) on Combustion , The Combustion Institute, Pittsburch, Pa. (1965) p. 965.

12. Steward, F. R. , Combustion and Flame , 8, 171 (1964)

13. Steward, F. R. , Combustion Science and Technology , 2^, 203 (1970).

14. Lee, S. L. , and Emmons, H. W. , J. Fluid Mech . , 11 , 353

(1961)

15. Morton, B. R. , Taylor, G. I., and Turner, J. S., Proc .

Roy. Soc. A 234, 1 (1956).

16. Rouse, H., Yih, C. S., and Humphreys, H. W. , Tellus , 4

201 (1952).

17. Alpert, R. L. , Fire Induced Turbulent Ceiling- Jet , Technical Report FMRC Serial No. 19722-2, Factory Mutual

Research Corporation, Norwood, Massachusetts, (1971).

18. Rao, V. K. , and Brzustowski, T. A., Combustion Science

and Technology , 1, 171 (1969).
19. Faure, H. , The Use of Models in Fire Research , Publication No. 786, Natin ~~

tion No. 786, National Academy of Sciences, National

Research Council, Washington, D.C. (1961), p. 50.

20. Lee, S. L., and Ling, C. H. , Eleventh Symposium (Inter -

national) on Combustion , The Combustion Institute, Pittsburgh, Pa, (1967), p. 501.

21. Ricou, F. P., and Spalding, D. B., J. Fluid Mech . , IJ^,

21 (1961).

22. Townsend, A.^i. , ibid . , 26^, 689 (1966).

23. Murgai, M. P., and Emmons, W. H. , ibid , 8, 611 (1960).

24. Murgai, M. P., and Varma, R.^., Int. J. Heat Mass

Transfer , 8^, 833 (1965).

25. Varma, R. K. , Murgai, M. P., and Ghildyal, C. D. , Proc .

Roy. Soc. (London) A314 , 195 (1970).

26. Morton, B. R. , Tenth Symposium (International) on Combustion , The Combustion Institute, Pittsburgh, Pa. (1965), p. 973.

27. Smith, R.^., Eleventh Symposium (International) on Com -

bustion , The Combustion Institute, Pittsburgh, Pa. (1967)

p. 507.

28. Browne, F. L., Theories of the Combustion of Wood and

Its Control - A Survey of the Literature , U. S. Forest

Service, Research Paper FPL 19, Forest Products Laboratory, Madison, Wisconsin (1958)

.

29. Roberts, A. F. , Combustion and Flame , 8_, 345 (1964).

30. Kosdon, F. J., Williams, F. A., and Buman, C, Twelfth

Symposium (International) on Combustion , The Combustion

38
Institute, Pittsburgh, Pa. (1969) p. 253.

Hill, B. J,, J. Fluid Mech. , 51, 773 (1972).

39
Appendix A

Solution of Equations Governing the Combustion Zone

Defining the following convenient variables of dimens ionless

volumetric flovi rate and momentum flux:

For Axisymmetric Flow

For Two Dimensional Flow

Or'-t^V and i^'-pV'V' (A-1)

and substituting into the appropriate dimensionless continuity

equation and equations of conservation of species, momentum,

and energy (15-22) gives

Axisymmetric Flow

(A-2)

(A- 3)

"Xi^ f,

^'

(A- 4)

Two -Dimens ional Flow

(A- 5)

40
a 2:' Fr i^r' (A- 4)

(A- 5)

subject to the boundary conditions at the flame base

-W' = Po , V'= t , P'=Po', X(=X^ and g'=0

^^^^

Division of equation (A- 2) with equation (A-5)and integrating

the resulting equation within the proper limits yields

The entrainment coefficient in equation (A- 3) may be eliminated

by substitution of equation (A- 2) into equation (A-3) and integration of the resultant equation with respect to axial distance

gives the mass flux of species at height , in terms of the

mass flux through the flame base, the rate of production due to

chemical reaction and the mass flux entraining from the surrounding

atmosphere:

Combustion reactions are generally quite complex, typically

involving a multistep path with a number of short lived species.

However, until detailed mixing and combustion data are available,

it is assumed that chemical reaction can be idealized as a single

step overall reaction.
a)^CFU£L) -OoCOKYCtEN) — (COMBUSTION PRODUCTS") (A- 9)

and that the kinetics of the combustion process are fast enough

that the reaction rate is controlled by the rate of oxygen

entrained into the flame.

With these assumptions, the rates of production or comsumption

of the species are determined by the stoichiometric coefficients

for the overall reaction

/^f /^o /^P (A- 10)

where /^^c'^^jc^Ji * stoichiometric coefficient.

Because there is no excess oxygen within the combustion zone,

equation A- 8 for oxygen is expressed by

f

2:' p'^?-

0

^

(A-11)

For the concentration distribution of combustible volatiles,

equation A- 8 becomes

Introduction of the relations indicated by equations A-10 and

A-11 into equation A-12 and using the definition of stoichiometric

oxygen to combustible volatiles mass ratio, y, gives

42
where —77— , the mass ratio of stoichiometric oxygen

to combustion volatiles.

The mass flow rates at which the combustible volatiles are

entirely consumed through chemical reaction with the oxygen

entrained can be determined by setting the fuel concentration

equal to zero in equation A-13.

r^^S \ \ ^

x.^^ ) (A-14)

The dimensionless volumetric flow at stoichiometric mixing can

be obtained through this relation in conjunction with Equation

(A- 7) evaluated at stoichiometric mixing.

Using the relation given by equation (A- 7) to eliminate in

equation A-4 and dividing the resulting equation A-5 yields a total differential equation, as follows:

For Axisymmetric Flo^/

For Two Dimensional Flow

43
Sei3aration of variables and integration over the proper limits

yields:

For Axisymmetric Flov/ 3 -2.

For Two Dimensional Flo^,

(A-17)

44
In order to obtain a simplified functional relationship for

further analysis, it is assumed that the volumetric flow rate

at the flame base is negligible compared with that at

stoichiometric mixing and that the velocity developed within

the flame plume is much greater than that at the flame base.

With these assumptions, equation A- 17 reduces to

For Axisymmetric Flow

-[ /2L E fr

For Tv;o Dimensional Flow

-]

-^-'=1 f£F. J CA-185

Introduction of this relation into equation A- 5 and integrating

the resulting equation yields the flame height measured from

the flame base up to stoichiometric mixing point. These

equations are:

For Axisymmetric F low _l

£• if. r 3 •^'^(i--^)'^^'

3

For Tv;o Dimensional Flow

(A~19)

Substitution for F^ and -t^' and introducing the total mass flux

entering through the flame base

(A-20)

45
"1"

into equation A-19 while neglecting unity compared with i^'term

produces the desired result (equation 24).

Integration of equation A- 5 after substitution for ttr'from

equation A-18 yields

For Axisymmetric Flow ^

s_

For Two Dimensional Flow

The flame radius divided by its source radius can be obtained

by a combination of the definitions of dimensionless volumetric

flow rate and momentum flux:

For Axisymmetric Flow

For Two Dimensional Case

CA-22)

Substitution for pV^, -uT^ and it' from equations A- 7, A-18,

and A- 21 into the above equation yields the desired equation 25.

Equation 26 is obtained from equation 22 evaluated at stoichiometric mixing and substitution for f|-'lf if' a"<3 "^J f^om

equations A-14, A-15, and A-18.

The dimensionless density of the flame plume (equation 27)

is obtained from equation A- 7 in which ir'is replaced by the

46
relation given by equation A- 21.

nivision of equation A-14 by equation A-15 yields the desired

equation 28.

An overall energy balance on the entire combustion zone from

the flame base up to the stoichiometric mixing height results in

is " CO I J (A- 23)

Substitution for ^'of' from equation A-14 into equation A- 23

and rearranging yields the desired result, equation 30.

Rearrangement of equation A- 13 and substitution for Par' and oT'

from equations A-7 and A- 21 gives the desired equation 31.

Substituting equation A-7 and A- 21 into equation A- 8 for the

combustion products and simplifying the resulting equation

with use of equations A-10 and A-11 yields the desired equation

32.

Neglecting the species generation term in eouation A-8 for

nitrogen and replacement of P ir'andir'with equation A-7 and A-21

yields the desired result, equation 33.

The total mass flow rate of combustion products across a plane

at the stoichiometric mixing height is enual to the

mass rate supplied to the flame base plus the mass rate entrained from the ambient atmosphere plus the mass flux generated by combustion. This relationship is expressed as:

47
Substituting for ^ from equation A-14 into equation A-24

yields tlie desired resultant equation 34.

An overall mass balance for nitrogen on the whole combustion

zone, substitution for f^* lJ^' from equation A-14 and rearrangement of the resulting equation gives the desired

equation 35.

48
Appendix B

Solution of Governing Equations in the Buoyant Plume

Introduction of the simplifying variables of dimensionless

volumetric flow rate and momentum flux

For Axisymmetric Flow

2. 2« ^ V*' = and -ur*^ ? y

For Two Dimensional Flow

2.

into the equations of conservation of mass, species, momentum,

and energy yields:

Axisymmetric Flow

2. (E-3)

(B-4)

(B-5)

Two Dimensional Flow

~~T^ V^J p' (B-2)

49
. 2

B-ur

(B-3)

(3-4)

ir^T^ (B-5)

The boundary conditions at the source o£ the buoyant plume

zone are

(B-6)

Dividing equation B-2 with equation B-5 and integrating between

the pertinent limits results in:

Using equation B-5 to divide equation B-4 and eliminating p'

between the resultant equation and equation B-7, one obtains

For Ax i symmetric Flow

/ I- p.' lA- (ft -I)

For Two Dimensional Flow

d 1/* (B-8)

Separation of variables and integration yields:

For Axisymmetric Flow

50
For Two Dimensional Flow

-[/(fg--l-)+t(p;-oVti^I(-^^^)]}

CB-9)

In order to proceed to further analytical treatment, it may be

possible to neglect ^ compared with -xJ' for points far from

the source and to retain the term containing the highest power

of IT . This approximation yields the follov/ing:

For Axi symmetric Flow 2. 2 - L SE'F, J

For Two Dimensional Flow

3 J_ * 3

J (B-IO)

Substituting this relation for

^

into equation B-5 and

integrating the resulting equation, one obtains the total flame

height

For Axisymmetric Flow / ^

51
For Two Dimensional Flow

which is approximated by \ ^ H'=Hr+[-T?^^]V-i] " s L- e (B-ii)

H H5 where W = —r- and W ==

The dimens ionless volumetric flow rate ( IT ) at the fftme tip

can be obtained through an overall energy and mass balance on

the entire convection plume along with using the assumption of

identical heat for both plume gas and ambient fluid, such that:

(B-12)

where

^

S / Substitution for and 1^ from equations 28 and 30 into

equation B-12 results in:

r r a;0- PoV yx^(i-6))-i , oXfeV-Q-i

, V yg'

^ ATi-'; (B-13)

Introduction of the definition of Froude number into equation

B-11 while neglecting unity compared with 1^ because the

volumetric floxsr rate at the flame tip is much greater than

that at stoichiometric mixing yields:

52
For Ax i symmetric Flow

For Two Dimensional Flow

Yo ^ ^ f'yoO-(s) (B-14)

where = ^ , volumetric flow rate at stoichiometric

mixing divided by that at the flame base.

Substituting for 'V^' from equation A- 15, from equation 28,

1^* and H5 from equations B-13 and 24 into equation B-14 along

with the definition of mass flux of fuel supplied to the flame

source (equation A-20) yields the desired result (equation 58).

Using equation B-10 to replace in equation B-5 and integrating the resulting equation, one obtains

For Axisymmetric Flov/ -L -:L

For Two Dimenional Flow ^

Substitution for Par,Kr and or from equation B-7, B-10, and

B-15 into equation A-22 along with the definition of and

approximation of the resulting equation gives the desired

equation 59.

Solving equation B-7 for r and substituting for from

equation B-15 together with the definition of gives the

desired result, equation 60.

53

(B-15)
Dividing equation B-3 by equation B-2, separation o£ variables and

integration between the proper limits yields:

Elimination of r IT between equation B-7 and B-16 for oxygen

and substitution for IT from equation B-15 gives the desired

equation 62.

Equation 63 is obtained by a combination of equations E-7 and

B-16 for combustion products with equation B-15.

Elimination of p V between equations B-7 and B-16 for nitrogen

and substituting for if from equation B-15 yields the desired

results, equation 64.

(B-16)

54
FIGURE 1

GO

O

o

UJX

S2

UJX

GO

o:

o

CD

lO

ro

CO»

c

o

</) c=E

N >?

CM

•^1

(SS9|U0!SU9UU!P)



U.S. OPPT. OF coMt/. 1 1. PI'iU.K A I K^N OR REPORT NO. 2. Gov't Accession

BIBLIOGRAPHIC DATA j J^gSIR 73-115

3. Recipient's Accession No.

4. TITLF ANP' SU H I 1 1 i i

Ansxysis or tne Dcnavioir ol d. rrecxy ijui.iixi.ig rxLc xii d

Quiescent Atmosphere

5. Publication Date

6. Performing Organization Code

7. Ain'HOK

J. B. Fang

8. Perfornung Organization

NBSIR 73-115

9. PERFOR.MING ORGANIZATION NA.ME AND ADDRESS

NAITONAE BUREAU OF .STANDARDS

DEPAK IMEMT OF COMMERCE

Vi ASHlNG l ON, D.C. 20234

10. Proiect. i ask ^Vtork Unit No. 4618382

11. Contrac t ''Grant .\o. IAA-H-37-72 Amendment 10

; ^ —— ' 12. Sponsoring Organization Name and Address

Department of Housing and Urban Development

Washington, D.C.

13. i vpe of Report & Period

overccj

Tnt^rT"! —

1

14. Sponsoring Agency Code

I 15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) A mathematical model which describes the physical and geometrical properties of a turbulent buoyant diffusion ilame over a free burning fire has been set up for

both axisymmetric and two dimensional cases. The mathematical simulation of the

flame consists of a combustion zone near its source and a buoyant plume above and i has been developed based on the assumptions of infinitely rapid rate of oxygen i limited combustion reaction and "top hat" profiles representing vertical distributions^

of the velocity, temperature, and concentration of the flame gas. Analytical solutions

are presented showing the effects of fuel mass-flow rate, physical properties of the

fuel and ambient air, and the size and shape of burning area on the general characteristics of a buoyant flame. Experimental data on visable flame heights and flame

temperature profiles obtained from burning several different fuels have been satisfactorily correlated by the derived expressions. i

I

17. KEY WORDS (Alphabetical order, separated by semicolons'^ i .1 . j . r- r- . ^, buoyant plume; diffusion flame; fire; flame

height; mathematical modeling; turbulent flow.

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