Reference

NBS

Publications

NAT'L INST. OF STAND & TECH NBSIR 82-2473

mot ESHtMH trainment and Heat Flux of

Buoyant Diffusion Flames

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Washington, DC 20234

February 1 982

Final Report

— QO

100

. U56

J.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

82-2473

1982

NBSIR 82-2473

ENTRAINMENT AND HEAT FLUX OF

BUOYANT DIFFUSION FLAMES

NATIONAL BU*I»AU

09 STANDARDS

L1BBABT

MAY 1 7 1982 .

ftot CLCt -

CXCtoo

no.

B. J. McCaffrey and G. Cox

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Washington, DC 20234

February 1 982

Final Report

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

TABLE OF CONTENTS

Page

List of Figures iv

List of Tables iv

Nomenclature v

Abstract 1

1. Introduction 1

2. Analysis 3

2.1 Data 5 2.2 Radiative Power Output 9

2.3 Non-Dissipative Plume ..... 12

3. Results 13

4. Summary and Conclusion 19

5 . Acknowledgments 19

6. References 20

Appendix A. Turbulent Transport 22

iii
LIST OF FIGURES

Page

Figure 1 . r^(AT/T

Q

) plotted against AT/T

q

for various A .... 25

Figure 2. The radiative fraction, x> as a total heat release rate . . . . function of the

26

Figure 3. The radial parameters (a, 6) as a function of A ... 27

Figure 4. Temperature and velocity profile

distances above the burner . . . data at various

28

Figure 5. Calculated mass and energy fluxes as a function of

height 29

LIST OF TABLES

Page

Table 1. Comparison of fire properties determined from two

techniques 8
NOMENCLATURE

A

B

C

P e

F

o

g

H(z)

L

m(z)

M

n

Q

T

AT

V

x

z

z

centerline velocity correlation parameter (Table I)

centerline temperature rise correlation parameter (Table I)

specific heat of gases

combustion efficiency

source buoyancy parameter defined after eq. (10)

gravitational constant

convective heat flux defined by eq. (2)

-2/5 flame length, z' = 0.2 m kW

mass flow rate defined by eq. (1)

constant defined after eq. (14)

power of centerline V and AT correlations

heat release rate of fire

temperature

temperature rise above ambient, T-T

q

vertical gas velocity

radial or transverse coordinate

vertical or axial coordinate

2/5 scaled axial coordinate = z/Q

a,B parameters used for width of velocity profile eq . (11)

a normal entrainment coefficient

e

T density deficit defined by eqs. (3) and (4)

parameter defined after eq. (12)

A ratio of widths of temperature rise and velocity profiles

p gas density

v
a 1/e of centerline value

X radiative fraction

Subscripts

m mean temperature (appendix)

o ambient conditions

p plume region

RAD radiative component

T temperature

TOT total or nominal

v velocity

Superscripts

’ fluctuating component

- time-averaged value ~ non-dimensional centerline temperature rise and velocity

parameters defined by eqs. (9) and (10)

vi
Entrainment and Heat Flux of Buoyant Diffusion Flames

* B. J. McCaffrey and G. Cox

National Bureau of Standards

Washington, D. C. 20234

Measurements of the vertical component of velocity in

buoyant diffusion flames from extended sources by both cross- correlation [1]^ and pressure probe techniques [2] incorporating time-average signal processing appear to overestimate

the transverse size of these systems based on a heat balance

using measured mean flux. By utilizing measurements of the

radiative fraction of the flames, and forcing a mean flux

heat balance, estimates of the transverse variation of

velocity are obtained and expressions for flame entrainment

and convective heat flux are determined. The use of mean

values is seen to lead to both overestimates as well as underestimates of total flux due to turbulent transport.

Key words: buoyancy; cross-correlation; diffusion flames;

entrainment; heat flux; radiation; turbulence.

1. Introduction

Mathematical modeling of fire is being approached by two main

techniques. Field models attempt to solve the partial differential

equation describing the fluid motion throughout the flow field [3] while

zone models attempt the less ambitious task of solving ordinary differential equations within identifiable flow regimes, e.g., near walls, near ceilings, within the plume, etc. [4, 5].

k

Fire Research Station, Borehamwood, England.

^Numbers in brackets indicate the literature references at the end of

the paper.

1
The zonal approach is closer to immediate application but field

models should, in the long term give more general and detailed predictions. It is now possible with zone models to make predictions of the

outcome of a fire in a single compartment given information about the

geometry of the room, properties of the wall material, the nature and

size of the ignition source, the fuel array and its burning characteristics, etc. References [4] and [5] give examples of the kinds of input

required and an assessment of how well these models reproduce reality.

One of the aims of this effort was to make these models modular in form

such that each portion which describes a distinct physical phenomenon,

e.g., flame radiation or turbulence characteristics, could easily be

replaced as new information becomes known. The models could then

incorporate the best physics and chemistry available at the time.

One of the portions or submodels of these computer codes involves

the entrainment of air into the fire plume. Fresh air enters the lower

portion of the enclosure opening, is sucked or entrained into the fire,

becomes vitiated as the combustion process takes place, and is delivered

to the upper portion of the room above the thermal discontinuity (horizontal layer separating the cool fresh air from the hot combustion

products) . From there the products flow out the top portion of the

door. The fire is thus thought of as a pump. Its characterization is

coupled to the hydrostatic equations describing the flow through the

opening via the height of the thermal discontinuity.

To describe fire correctly an accurate knowledge of the air and

gas flow through the enclosure is critical. The hydrostatic model for

doorway flow has already been demonstrated to be quite adequate [6]

.

The fire entrainment portion however appears to be conceptually weak

since it involves, for the most part, formulations based on point source plume theory which should not be expected to hold in the accelerating

region near the base of the flames.

The present work is an attempt to fill this void. A valid

phenomenological expression was sought for the entrained mass flow rate

2
as a function of the fire size and the height above the burning surface.

This was accomplished using the results of two separate measurement

techniques. Differences found in the two measurements were reconciled

by the use of an additional independent observation, namely, a measure

of the radiative fraction of these flames. The results are applicable

to a fire in the open, which is one that is not impeded or otherwise

influenced by any effects due to the presence of an enclosure. The

effects of the enclosure on these results must still be considered.

2. Analysis

Assuming cyclindrical symmetry with axial coordinate z and radial

coordinate x the stationary value of the mass flow rate, m(z) , and

convected energy flux, H(z) at any height z are

m(z)

H (z)

00

pv 2iTxdx

pvAT C 2irxdx

P

( 1 )

( 2 )

where p, v and AT are the time varying density, vertical velocity and

temperature rise above ambient. It is usual now to assume that

(i) pv = p*v and pvAT = p-vAT

(See the Appendix for a discussion of the effect of neglecting

turbulent transport. Above the flame region (AT->0) neglect of

turbulence will lead to underestimates of the flux while in the

flame the errors appear to be in the opposite direction.)

3
(ii) The gas is ideal with no composition changes

p T = p(T + AT)

o o o

(iii) The transverse velocity and temperature rise profiles may be

represented by Gaussian distributions

v - V(z) AT = AT (z)

where a is the radial distance at which the variable falls to 1/e of its

centerline value. Defining A. = a^/o^, substitution in (1) and (2)

yields

m(z)

2

P TTG V(z)

O V iTCzi 1/A

2

T

3 o

(3)

where y

Then

exp and the subscript o denotes ambient conditions.

m(z) = iTa^V(z) p

r (AT/T ) v o A o and

H (z) = 7Ta 2V(z)C Tp fl- T, (AT/T ) v pool A o

(4)

(5)

where has been assumed constant. Comparing with eq. (4) for the

total mass

1

r""(AT/T ) “

A o H(z) = m(z)C T

P o

1 ( 6 )
where T (AT/T ), a function of the centerline temperature rise, depends

A o only upon the transverse variation of temperature, cr^.

Equation (4) states that the mass flow rate is equal to the ambient

2 density times the centerline velocity times a representative area, , modified by T (< 1) which accounts for density decrease due to the fact

that the gases are hotter than the ambient.

For various values of X eq. (3) can be integrated analytically.

Figure 1 shows the variation of T with centerline temperature rise for

A a variety of assumed c? T or X. The limits are T = 1 for = 0, that is,

the gases are at ambient temperature. For a -* 00

, the temperature

profile is completely horizontal or independent of x and equal to the

centerline value. Neither of these two limits is appropriate for a flame. Experiments indicate that X is of order 1, i.e., the temperature

rise profile is of similar width to the velocity profile. For buoyant

flames the cross-hatched area shows the range of experimentally determined X [1, 2], Note these have X < 1 i.e., the velocity profile is

wider than the temperature rise profile, a result contrary to the

classical plume experimental findings of X = 1.16 [7]. At large z, high above the flame tip where AT -> 0, T converges to 1 since most of

A the gas consists of entrained ambient air. Near the burner AT/T

q can be as high as 3 or 4 leading to T of the order 0.5. This would

A produce large errors if the ambient density were used throughout.

2.1 Data

The quantities in the right hand side of eq. (4) have recently

been measured by two independent techniques: Cross correlation of

naturally occurring thermal and ionization fluctuations [1], and an impact pressure probe/sensitive manometer combined with a thermocouple

for density determination [2]. The same porous refractory gas burner

(0.3 m square) chosen to simulate the unwanted fire (a very buoyant

diffusion flame emanating from an area source) was used by both

5
investigators. Natural gas was employed as the fuel and the flow of

gas corresponded to the mass burning rates of realistic materials (150

2 to 600 kW/m ). For these buoyancy dominated systems it is the large

scale fluctuating (in time and space) nature of these flames that makes

measurement very difficult. The complete details of the two experimental

techniques are contained in references [1] and [2],

Based on time-averaged centerline values of V and AT the flame can

be considered to be divided into three distinct regimes: (1) A

continuous flame region, starting from the surface of the burner with V

close to zero at the surface and rising with the height above the

burner, z, to the 1/2 power. AT is approximately constant over this

regime. Higher up is (2) , an intermittent regime, with pulsating flame

(~3 Hz) exhibiting approximately constant V and AT falling with z to

the first power. Still higher is (3) the plume region which is, most

-1/3 -5/3

of the time, free of flames with V ~ z and AT ~ z as predicted

by conventional plume theory. Throughout the three regimes and indistinguishable among these is the consistency of the buoyancy relation,

V//2gzAT/T

o which has a value of approximately 0.9. This states that

throughout these flame systems velocities are derivable from static

pressure differences due solely to buoyancy created from temperature

differences generated from the heat release of the combustion process.

During these studies the scaling for Q, the nominal heat release

rate or net calorific potential has been determined experimentally as have the three regime demarcations. The scaling reduces all the data

from different size fires to a single universal curve. The length

scale is z/Q^^, the velocity scale is V/Q^^, and the temperature

requires only reference to ambient, AT = T - T

q

. The flame regime is

thus independent of Q. The flame - intermittent intersection is at a -2/5

scaled height of 0.08 imkW and the intermittent - plume intersection

-2/5

at 0.20 m*kW . From reference [2] the visible flame height is seen

to correspond to the top of the intermittent regime, which was determined by a change in the character of both the velocity and temperature

6
rise measurements. At scaled z higher than this value, the centerline

velocity and temperature rise begin to fall as dictated by point source plume theory. The combustion zone is thus over at z/Q corresponding

to the intermittent - plume intersection or visible flame height.

Table I contains the experimental centerline and radial flame

parameters required by eq. (4) as determined by the two techniques.

It is quite evident from table I that problems of comparison or significant differences are going to lie in the transverse or radial

measurement. Axial velocities are within 4% of each other and temperature rise measurements fall within 10%. Note that 10% is not a critical

amount since centerline temperature enters eq. (4) only through T . For

A present purposes the centerline measurements are virtually the same. However, significant differences do exist in the radial measurement and

the mass flow rate is a function of to the second power thus amplifying differences in that measurement. Note that and X in the table

are a least-squares fit of considerably scattered data. It is difficult

and may be purely speculative to try to accurately assess the experimental errors in the two techniques for those radial measurements. Near

the centerline velocities have a somewhat preferred direction. However,

as one moves away from the centerline, in the "wings," vertical components of velocity are approaching zero, and temperatures are approaching

ambient. Horizontal components of velocity are the same order of

magnitude as the vertical components due to mixing resulting from the

large scale, coherent eddy motion characteristic of these systems. Any

inaccuracies due to inherent wavering or meandering of the fire would

also be present in both measurements.

The most likely experimental error common to both techniques

results from the assumption that the velocity measured is only that of

the vertical component. In flows with strong recirculation the yaw

insensitivity of the pressure probe and the possibility of broad fronts

of information highly correlated at right angles to their direction of

7
Table

I.

Comparison

of

Fire

Properties

Determined

from

Two

Techniques

Q:

15-60

kW

0)

CO

!-i 0)>

COgc0H

CM IG

CM

w

hJ

Ph

(Ml O CO A m

II CMO' G N I

CQ

cO

co

CM

00o

CMON

CQH

<1

CMO

HZ V W uo H — o

CM

M

O II

W

N

G

HZ V M 00oo

CQ

Om

00

un

w

00

CM o O o • CQ 00

pt-l

O 00

M CM ^

s CO V

~''~ < 13 i—1 m o

toZ m HZ o o "•«. c_> 4 CO O G 00

<j • N cO o

00 Z ca O rH I H<

hQ CQ M

CQ Qi O M

PdO o u

!A

XIG

CO 4-1

4->

X -H O £ u a

coI—I

CM

co

CO

co

CTc O

00

<r

ooiO

a)

!-i 4-1 E-1 4-1 <—> U W O Oi < « U N

Ph

OO

Q S Pd S «-i H p-i

fed uo 8

UNITS:

z,a

,a

[m];V

[m/s]; Q

[kw]

;AT
flow, i.e., flame fronts, passing the sensors at an oblique angle to the

vertical would produce overestimates of velocity. This effect is likely

to be greatest in the wings. It is clear that although the two techniques differ in their measurements of plume spread they both tend to

give an overestimate of total heat flux based on mean flux balance.

Terai and Nitta also reported similar difficulties [8]

What is

needed is an additional method of obtaining the radial information which

would complement the above results and not be subject to the same inaccuracies of those measurements. One such idea involves the convective

heat flux of the fire and utilizes an additional measurement of the

total radiative power output of the fire.

2.2 Radiative Power Output

Assuming complete combustion, the value of the convective heat

flux at the intermittent-plume intersection will be equal to the energy

supplied to the burner, Q, minus what is radiated away over the combustion region from the base of the burner up to the visible flame tip or intermittent-plume intersection. If this radiative fraction is known

from an additional independent measurement, then eq. (5) can be solved

explicitly for o^.

Let us define x> the radiative fraction, as the total radiative

power output of the fire divided by the total heat release rate, Q.

(Q = H + Q . ) radiative

(7)

Substituting this expression into eq. (5) and evaluating all the

quantities in the plume region yields the radial width at the flame tip:

a v

fC T p A z poop

9
(l - x ) ( 8 ) 1/2

Q

275

2/5 -2/5 all evaluated at z

1 = z/Q =0.2 m-kW from table I and figure 1.

It is interesting to note that for x independent of Q, the proper

2/5 transverse scale is Q which is the same as that determined for the

longitudinal scale [2], Markstein [9] in fact has demonstrated the

consistency of x over an extremely wide range of Q for propane. Unfortunately these results were for very large velocity flames issuing from

small nozzles. For the moderate-sized or developing fire buoyancy is

dominating, the processes are not well understood, and the burning rate

per unit area and x turn out to be increasing functions of source area

for a given fuel [10]. Similarly for a fixed source area different

materials will exhibit different radiative characteristics resulting in

different burning rates per unit area and different x- The effect of

fuel in this case was simulated by varying gas flow rates, Q, to the

burner.

The radiative fraction of the present flame system has been

recently measured in a manner similar to that recommended by Modak [11].

Figure 2 shows the radiative fraction, x > as a function of the heat

release rate [12] for this burner. The asymptotic behavior of constant

X can be seen at the higher flow rates. The curve designated HNG for

"historic" natural gas will be used in subsequent calculations. This

gas most closely resembles the gas used for the measurements in determining the gross features of these flames. Using natural gas for these

systems Zukoski [13] finds x to be about 25% based on a method of

collecting the exhaust gases in a hood.

A crude check of figure 2 can be made by applying those results

with the individual data (before scaling) of McCaffrey [2] where Q

varied from 14 to 58 kW. The radiative components can be subtracted

out of the total Q and the axial parameters in the plume region (the

only regime where other data exist) can be determined and compared with

10
literature values. The weighted averages of the centerline results

become:

F

gAT/T

Q

2/3 -5/3

z o

9.1 (9)

F

1/ 3 r

1/3

o

= 3.9

where F = 8 - x)

o p C T

o p o

( 10 )

This result is identical to Yokoi's [14] results for a point source system with negligible radiation.

Although from figure 2 the dependence of the radiative fraction

upon total heat relearo appears significant, the square root dependence

of x in eq* (8) tenc co diminish this effect on a except for small

2/5

V

fires. The asymptotic scaling, i.e., Q therefore can be used

for calculating purposes provided it is not extrapolated to fires with

ratios of flame height to fire base diameter small, i.e., L/D -> 0.

Note the generality of eq. (8) for other fuels with very different

1/2 radiative characteristics must yet be demonstrated. The (1 - x) in

1/2

eq. (8) will be replaced by (e - x) for systems with combustion

efficiencies, e, signficantly less than 1. Using the asymptotic

scaling,

= a (ID

will simplify the problem a great deal since Q will become a natural

scaling parameter for m(z), H(z), etc. (see reference [1]) together with

11
2/5 z/Q for scaled height. Note that a defined in (11) is 6a /5 where

e a^ is the normal entrainment coefficient from plume theory.

2.3 Non-Dissipative Plume

Substituting eq. (11) into eq. (8) will yield an expression

containing three unknowns: a, 8, and X - the three radial parameters

for which the experimental results are in greatest disagreement. It is

presumed that the centerline results A and B are known well enough from

the measurements as is the radiative fraction, x* Equation (8) states

only that the value of - is 1 - x at the flame tip. It says nothing

about the behavior in the plume region. If radiation from the burnt

gases in the plume is negligible compared with the radiation from the

flame then H/Q will remain at the value determined at the flame tip.

Radiative calculations based on estimated CC^ and 1^0 concentrations

(and no soot) in the plume indeed substantiate this assumption. Since

there are no other significant dissipative mechanisms in the plume, H/Q,

for practical purposes, remains constant there and one can obtain an additional relation among the variables. That relation is = 0 in the

plume. For sooty flames zero will be replaced by a small negative

number as will the combustion efficiency change as noted previously.

Taking the derivative of eq. (5) in the plume region and setting

it equal to zero yields a relation for 6/a in terms of X:

k + X

A

K

x /5 - v

( 12 )

where

K

x = (l - rx )/L(AT/T

o + l)

1 - r

x

]

12
Note for the three T functions (ignoring A = 00 ) in figure 1 it

A can be shown that

Having now two expressions for the three unknowns, a and g can be

solved for in terms of A and compared with the experimental results of

table I. Explicitly, a and g become:

and L is the flame length, z' = 0.2 m kW

3. Results

Figure 3 shows the result of plotting eqs. (13) and (14) as a function of A. The data points are the results of the experiments.

Also shown are results from the literature in non-combusting plumes.

Note the literature results with g = 0 are for experiments with point

heat sources or, the measurements were made so far from the source that

the effect of the source size is lost. From figure 3 it would seem

that the preponderance of evidence points to a A < 1 contrary to the

classical result of A = 1.16.

r

A u

2

(13)

(14)

where

-2/5

13
Note the ct/M portion of figure 3 could have been derived from

point source plume theory results, eqs. (9) and (10). Letting

6 , a = -r- a z and

v 5 e

1 -

1 + 1/A

2

1

1 + 2/A

2

Substituting into eq. (6) yields, in the plume limit:

A B a

2e

1 + 1/A

2

where A and B are the dimensionless, centerline experimental results in

plume theory form (Eqs. (9) and (10)). Now, letting — ->• 1 the requisite

relation between a and A is obtained. However, this analysis will yield

no information about the all important intercept, 3.

For the present measurements the slopes (a) appear to behave like

the calculated result exceedingly well. The intercepts (3), on the

other hand, are too wide by a factor of two! In fact, the calculated 3 hardly varies with A at all indicating that for a heat balance and a non-dissipative plume ^

~. 55 (m*kW 2/5^1/6^ a constant> The zero intercept or transverse extent at the burner surface appears to control

the phenomena. If all the assumptions made in the above analysis are correct then both the cross correlation and pressure probe measurements

are following some particular feature of the flame since the measured

slopes follow the calculation. That feature is obviously not the 1/e

point of the velocity profile, since that would require the measured

and analytical 3 to be the same. Perhaps the large scale eddy motion

in the wings which is time averaged somehow tends to broaden the indicated profile. However, the distortion is conserved throughout the

length of the flame since the slopes appear to be correct. Whatever

the reason, figure 3 can be used as a framework for choosing reasonable

14
radial parameters consistent with the experimental results. Before

doing that it might be instructive to look at a graph of the actual data

from table I.

2/5 Figure 4 shows the thermal profile, a^/Q for both sets of flame

and intermittent data plotted against height along with a least-squares

fit. Within the scatter no significant difference exists between the

two measurements which is not surprising since thermocouples, albeit of

different sizes, were used for both data sets. Also shown on figure 4 are least squares fits to the velocity profile data (reference [2] data are presented in the form of eq. (11)). The differences in the two

measuring techniques (and in A seen in table I) are now clearly evident.

The constraints of figure 3 as a function of A can be sketched onto

figure 4 for comparison. They are shown by crosses for the three A of

figure lat z' =0.2. At z* =0 one cross is shown since 6 is essentially constant, independent of A. Note the constraints at the flame

tip can be derived directly from the energy consideration alone, i.e.,

JJ eq. (8) — 1 - x* Whereas, in order to obtain the constraints at the ^ dH intercept, eq. (12), i.e., -> 0 in the plume, must be incorporated

into the analysis.

To continue the analysis it now becomes a matter of choosing a A consistent with the measurements and eqs. (8) and (12). Since both

measured a's seen on figure 3 appear to be consistent with the energy

constraint (line) taking the mean of the measured A seems reasonable.

That will result in A = 0.77 or 1/1.3 which is very close to the recent

measurements of Nakagome and Hirata [15] for a purely thermal plume.

Note the difference between the mean and either measured A will produce

a change in slope of the curves of figure 4 which is small compared

with the shifts produced by the energy constraints reflected in (3. That new position will be a line connecting the cross marks at z' = 0

and z' =0.2. It can be assumed that the cross-correlation and pressure

probe instrumentation systems are responding consistently but to different features of the large scale eddy motion in the wings of the

15
flame. Note the time-averaged temperature results will be shifted a similar amount reflecting the same distortion seen by the other

instruments.

Choosing X = 1/1.3 will result in the following (from lines on figure 3) with x = 0.26:

v

^ 2/5

= 0.128 + 0.011 £m * kW 2/5

J

(11a)

Using the asymptotic scaling for av

eq. (4) becomes:

m(z)

Q

2

r. (AT/T ) A O

(4a)

where A and n and AT are the centerline average velocity and temperature

rise results from table I and T is determined from figure 1 at

A

X = 1/1.3.

Figure 5 is a plot of eq. (4a) for m/Q versus normalized height,

2/5

z/Q"

- . Also shown on the figure is H/Q calculated from eq. (6) . At

the flame tip m/Q - 0.007 kg/s f

kW which for CH^ (50,000 kW jkg/s)

corresponds to approximately 350 kg/s of air per kg/s of CH^ . Therefore

the flame entrains approximately 20 times the stoichiometric air mass requirement (17.2) up to the end of the combustion region. This number

is about 30% higher than the equivalence ratio of 0.067 + 10% measured

by Zukoski [13] using the hood collection technique (1/0.067 = 14.9).

In order to fit his combustion plume model to a large quantity of flame

height data Steward [16] found that only 400% excess air was entrained

up to the flame tip. At the end of the continuous flame region,

2/5 -2/5 z/Q ' = 0.08 m-kW , the flame has entrained 90 times the methane

flow or about 5 times the stoichiometric requirements. Mixing by the

large-scale eddy motion would therefore appear to be the dominant

controlling mechanism as opposed to any chemical effects. Note that

16
2/5

m/Q, H/Q as well as the scaled height, z/Q are based on the total

or net calorific potential heat release rate, Q, and not on H, the

convective component.

H/Q rises rapidly to a value of approximately 0.4 at the

flame-intermittent intersection and gradually rises to 0.74 at the

intermittent-plume intersection. This number is simply 1 minus the

radiative fraction which has been used to set the constants in the

width function, a . v

Also plotted on figure 5 are the point source plume theory results

of Yokoi [14] shown by the dashed lines. At the flame tip the mass flow rate predicted by plume theory is about 40% less than the present

flame results. Also shown on figure 5 are measurements using a thistledown tracer technique due to Thomas et^ a^L. [17] which agree quite well

with the present results in the continuous flame region after which the

shape deviates from both the present and plume results.

A good representation of eq. (4a) for calculating purposes in S.I.

units is the following:

m/Q = 0.053 (z/Q 2/5

) 1,3

(15)

valid up to the flame tip. It would be inappropriate to extend this

result much beyond the flame tip since the radial data from table I are

for the two lower regimes. In the plume region the point source results

5 /3 of Yokoi [14] (m/Q = 0.063 z' ) should begin to become valid, bearing

in mind the turbulence contribution discussed in the Appendix. More

recent analytical characterization [18] using momentum constraints in

the lower regimes as well as the plume energy considerations discussed

here leads to an expression for mass flow rate not very different from

eq. (15) in the upper portion of the flame. Near the burner the results

are somewhat different in that the mass flow rate and especially the

heat flux rises more rapidly with height than do the present results.

17
A Note on Scaling:

Contrary to normal practice the variables plotted in the figures

have, in general, not been made dimensionless. The reason for this is

to emphasize the fact that no satisfactory scaling of buoyant diffusion

1/5 2/5 flames has yet been realized. Note that Q for velocity and Q for

length are experimental results. Froude model scaling, expected to be

valid in the plume, utilizes the following scales, a function of burner

or fire base diameter, D:

1 ~ D V ~ /gD Q ~ p T C /gD D

2 oop (AT ~ T ) o

5/2 2/5 When the Froude number is "high" enough such that L/D~(Q/D ) and

X = constant [12] one would expect this scaling to be valid (note for

2/5 D~Q Froude scaling would be consistent with the present measurements). However, for large enough fires (L/D -> 0), those below the

critical Froude No., flame heights and the radiative fraction become

functions of both heat release rate and base diameter. Hasemi [19] has

recently proposed a constant eddy diffusivity model utilizing the

Boussinesq approximation in which the following scales, as a function

of heat release rate, Q, were derived:

1 ~ V C pK

3

/gBQ V gBQ/C pK AT ~ Q/C npK

Y P \ P P

where K is the turbulent transfer coefficient, U'V* = - K — and

3 AT 3/5

U’T' = -K —— • He shows that for K = kQ where k is constant, the

3x

model will not only reproduce the experimentally determined scales 1/5

and 2/5, but also, will reproduce the detailed behavior of the intermittent regime extremely well. At the lower end of the intermittent regime

near the continuous flame region where density differences become large

the analysis, as expected, becomes weaker.

18
4. Summary and Conclusion

Expressions for mass and convective energy flux of buoyant

diffusion flames from area sources have been obtained by an analysis

using existing mean centerline velocity and temperature rise data from

two different techniques. The method is based on a heat balance

assuming negligible turbulent transport and a non-dissipative plume and

utilizes an independent estimation of total radiative power output.

Basically the width of the velocity profile is fixed by the balance of

total heat release minus the radiative fraction at the flame tip

together with the assumption of negligible radiative flux from the

plume. On this basis the results indicate that both experimental techniques for measuring velocity overestimate the width of the fire plume

by a considerable amount, possibly due to errors resulting from time

averaging of the large-scale coherent structure in the wings. Horizontal or inflow velocity components cannot be discriminated against by

either experimental technique and hence errors could result when the

signal output is assumed to be simply the vertical component.

Although the present results are considerably higher than point source plume theory, further disagreements between measurements and

plume theory calculations noted in the literature are possibly due to

increased entrainment brought upon by disturbances to the free burn

behavior, for example, by the door jet when the fire is located in an

enclosure. Preliminary estimates of this effect are being pursued by

Zukoski [13].

5. Acknowledgments

One of the authors (BJM) would like to thank the director and staff

of the Fire Research Station for their assistance during his stay there

as a guest worker.

19
6. References

[1] Cox, G. ; Chitty, R. A study of the deterministic properties

of unbounded fire plumes. Combustion and Flame. 39: 191; 1980.

[2] McCaffrey, B. J. Purely buoyant diffusion flames: some experimental results. Nat. Bur. Stand. (U.S.) NBSIR 79-1910;

1979 October.

[3] Ku, A.C.; Doria, M. L. ; Lloyd, J. R. Numerical modeling of

unsteady buoyant flows generated by a fire in a corridor;

Sixteenth Symposium (Int.) on Combustion; The Combustion

Institute; 1977. 1373.

[4] Emmons, H. W. The prediction of fires in buildings; Seventeenth

Symposium (Int.) on Combustion; The Combustion Institute; 1978.

1101 .

[5] Quintiere, J.; McCaffrey, B. J. ; DenBraven, K. Experimental and

theoretical analysis of quasi-steady small-scale enclosure fires;

Seventeenth Symposium (Int.) on Combustion; The Combustion

Institute; 1978. 1125.

[6] McCaffrey, B. J. ; Rockett, J. A. Static pressure measurements

of enclosure fires. J. Res. Nat. Bur. Stand. (U.S.). 81: 107;

1977.

[7] Rouse, H. ; Yih, C. S.; Humphreys, H. W. Gravitational

convection from a boundary source. Tellus. 4: 201; 1952.

[8] Terai, T.; Nitta, K. Experiments on plume rising from burning

heat source with finite sizes. Proc. Symp. Arch. Inst.; Japan;

1976.

[9] Markstein, G. H. Scaling of radiative characteristics of

turbulent diffusion flames; Sixteenth Symposium (Int.) on Combustion; The Combustion Institute; 1976. 1407.

[10] Modak, A. T. ; Croce, P. A. Plastic pool fires. Combustion and

Flame. 30: 251; 1977.

[11] Modak, A. T. Thermal radiation from pool fires. Factory Mutual

Research; Norwood, Massachusetts. TR 22361-5; 1976 August.

[12] McCaffrey, B. J. Measurements of the radiative power output of some buoyant diffusion flames. Western States Section Combustion

Institute Paper No. WSS/CI 81-15.

[13] Zukoski, E. E. ; Kubota, T.; Cetegen, B. Entrainment in fire

plumes. Fire Safety Journal. 3: 107; 1981.

20
[14] Yokol, S. Study on the prevention of fire-spread caused by hot

upward current. Report of the Building Research Institute;

Ministry of Construction; Japan; No. 34; 1960 November.

[15] Nakagome, H. ; Hirata, M. The structure of turbulent diffusion

in an axi-symmetrical thermal plume, in heat transfer and

turbulent buoyant convection. Spalding, D. B.; Afgan, N.; eds.

Washington, D. C.: Hemisphere Publishing Corporation; 1977. 361

[16] Steward, F. R. Prediction of the height of turbulent diffusion

buoyant flames. Combustion Science and Technology. 2: 203; 1970

[17] Thomas, P. H. ; Baldwin, R. ; Heselden, A. J. M. Buoyant diffusion

flames: some measurements of air entrainment, heat transfer, and

flame merging; Tenth Symposium (Int.) on Combustion; The

Combustion Institute; 1965. 983-996.

[18] McCaffrey, B. J. Momentum implications for buoyant diffusion

flames; abstract presented at Eastern States Section Meeting;

The Combustion Institute; Pittsburgh; 1981.

[19] Hasemi, Y. Properties of the intermittent flaming region in the

upward current above a diffusion flame; Proc. Ann. Meeting JAFSE;

1981. 13.

[20] Kotsovinos, N. E. ; List, E. J. Plane turbulent buoyant jets.

Journal of Fluid Mechanics. 81: 25; 1977.

[21] George, W. K. ; Alpert, R. L. ; Tamanini, F. Turbulence

measurements in an axi-symmetric buoyant plume. Int. J. of Heat

and Mass Transfer. 20: 1145; 1977.

[22] Cox, G. ; Chitty, R. Unpublished data; 1980.

21
Appendix A. Turbulent Transport

It is instructive to examine the assumption that the turbulent

transport of heat and mass can be ignored. If pressure fluctuations

are assumed to be small then it is possible to show that

where the usual Reynolds decomposition of V, p and AT have been used.

Y

P oT

o

p = is t jie density based on the mean temperature m

AT + T

o _ p T - p

' T

not the true mean density which is p = —

AT + T

o

Only the first term in eq. (A-2) has been used to assess total heat

flux. Inspection of the "extra terms" in these expressions shows that

22
both over and underestimates are possible if the first term alone is

used.

For small AT (y -> 0) the V'T* term dominates leading to an underestimate of total heat flux. The data of Kotsovinos and List [20]

and of George et^ al. [21] support this conclusion. George estimates

that 85% of the vertical heat transport is carried by the mean flow.

However, for different temperature levels more complex phenomena may be

taking place. For example, for AT = T

q

(say), at the flame tip and

assuming the triple correlation is small:

In a conventional plume George et^ aL. [21] measured peak values of

(T»2)1/2^at a t 0.4 and V'T'/V*AT at 0.07. The transverse distribution

of these parameters was wider than for V and AT so that "top-hat"

profiles could be assumed. Although no measurements of V'T* have been

made in fire plumes, recent measurements of (T' ) /AT [22] indicate

maxima at the flame tip of between 0.4 to 0.7. Overestimates of total

flux might then be expected if V'T'/V*AT is less than between .08 to .25.

So both underestimates and overestimates based on mean flux are possible

in different regions of the flame system making the need for time

resolved measurements critical.

For the present any underestimate or overestimate of Q will be

reflected in M, i.e, 1 - x should be increased or decreased. Hence a and 8 will vary by the square root of the change but m(z) and H(z) are functions of a and B to the second power. The change will therefore be

directly reflected in m and H. If for example the lack of accounting

for turbulence leads to a 15% overestimate of H then in figure 5 below

the flame tip H should be reduced by 15%. If George's [21] contention

is correct then somewhere above the flame tip the curves would have to

23
be increased by 15%. Assuming there is a smooth transition between

this decreasing and increasing of the mean result, the mean result will

overestimate in the flame region, pass through the 'correct value' and

then begin to underestimate in the plume region and therefore on a height-averaged basis yield something close to valid results.

24
(0=1 V)

A£>/l£> =

Figure

1.r

(AT/T )

plotted

against

AT/T

for

various
101 0/OVMq

26

Figure

2.

The

radiative

fraction,

x>

as

a

function

of

the

total

heat

release

rate
9/L ( S/jAM/wlW/tf

CO CM

n r

h

CO

_L

CO CM

r<

9/B-(s/Z

M>I/ U, I W/»

27

Figure

3.

The

radial

parameters

(a,

6)

as

a

function

of
o

t/Q

data

0.07

-

Q(kw)

REF.

a

17.9 )x

0.08

-

a

Ain

(s/ 2- m»I

• UJ) s/3 0/

28

Figure

4.

Temperature

and

velocity

profile

data

at

various

distances

above

the

burner
D/H

oo to w <; <*> n

[ ,_M>| • j_s-8>|] D/UJ

29

Figure

5.

Calculated

mass

and

energy

fluxes

as

a

function

of

height
NBS-114A (rev. 2-80

U.S. DEPT. Or COMM. 1 . BIBLIOGRAPHIC DATA

SH EET (See instructions)

4. TITLE AND SUBTITLE

PUBLICATION OR

REPORT NO.

2. Performing Organ. Report No. 3. Publication Date NBSIR 82-2473 February 1982

Entrainment and Heat Flux of Buoyant Diffusion Flames

5. AUTHOR(S)

B. J. McCaffrey and G. Cox

6. PERFORMING ORGANIZATION (If joint or other than NBS, see in struction s) 7. Contract/Grant No. NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS ( Street„ City. State , ZIP)

10. SUPPLEMENTARY NOTES

Document describes a computer program; SF-185, FIPS Software Summary, is attached.

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

bi bliography or literature survey, mention it here)

Measurements of the vertical component of velocity in buoyant diffusion flames

from extended sources by both cross-correlation and pressure probe techniques incorporating time-average signal processing appear to overestimate the transverse size of

these systems based on a heat balance using measured mean flux. By utilizing

measurements of the radiative fraction of the flames, and forcing a mean flux heat

balance, estimates of the transverse variation of velocity are obtained and expressions

for flame entrainment and convective heat flux are determined. The use of mean values

is seen to lead to both overestimates as well as underestimates of total flux due to

turbulent transport.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Buoyancy; cross-correlation; diffusion flames; entrainment; heat flux; radiation;

turbulence

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