A111D7

NBS

PUBLICATIONS

NBSIR 83-2730

On the Significance of A Wall

Effect in Enclosures With Growing

Fires

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Washington, DC 20234

June 1983

U:S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NBSIR 83-2730

ON THE SIGNIFICANCE OF A WALL

EFFECT IN ENCLOSURES WITH GROWING

FIRES

NATIONAL eitreao of STANDARDS

LIBRARY JUL 2 1 1983

Leonard Y. Cooper

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Washington, DC 20234

June 1983

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

Abstract 1

1 . INTRODUCTION 2

2. A DESCRIPTION OF THE WALL EFFECT AND A MEASURE OF ITS

SIGNIFICANCE 2

3. SOME PROPERTIES OF THE WALL FLOW 7

4. IMPLICATIONS OF THE WALL EFFECT CRITERION 10

4.1 Applying Results When Wall Temperatures Are Close to

Ambient 13

5. ESTIMATING THE WALL TEMPERATURE 17

5.1 Wall Heating Due to Radiation 18

5.2 Wall Heating Due to Convection 22

6. QUASISTEADINESS OF THE WALL FLOWS AND A MEASURE OF THE

LIKELIHOOD OF THEIR PENETRATION 25

6.1 The Quasisteady Flow Assumption 25

6 .2 Wall Layer Penetration 26

7 . SUMMARY AND CONCLUSIONS 27

8. ACKNOWLEDGMENTS 28

9. REFERENCES 29

10. NOMENCLATURE 30

-iii-
LIST OF FIGURES

Page

Figure 1. Sketch of the significant features of the wall effect .... 33

Figure 2. Plots of as a function of per eqs. (23) and

(24) for different values of (m /m )/X„ and for w p 2 m = 0, 1.0 and 2.0 under the assumption 6 << 1 34

Figure 3. Plots of I^xjnO/I^Ojra) and ^(xjm)/ 12 ( 0 ;m)

per eqs. (43)-(44) and (48)-(49), respectively, for m = 0, 1 .0 , 2 .0 and 5.0 35

Figure 4. Plots of eq. (51) and (54) criteria for wall layer

quasisteadiness and penetration, as functions of

interface elevation and for tm enclosure fires 36

LIST OF TABLES

Page

Table 1. Thermal conductivity and diffusivity of selected

materials 37

Table 2. Results of calculations on the significance of the wall

effect in full-scale experiments of references 1 and 2 .... 38

-iv-
ON THE SIGNIFICANCE OF A WALL EFFECT IN

ENCLOSURES WITH GROWING FIRES

Leonard Y. Cooper

Abstract

This paper studies the significance of a wall effect

that has been observed during the growth stage of enclosure fires. Relative to the two-layer phenomenon which

tends to develop during such fires, the effect has to do

with the near-wall downward injection of hot upper layer

gases into the relatively cool uncontaminated lower

layer. It is conjectured that these observed wall flows

are buoyancy driven, and that they develop because of the

relatively cool temperatures of the upper wall whose

surfaces are in contact with the hot upper layer gases.

For a growing fire (growth proportional to tm

; t being

time and m 0) in an enclosed compartment, an analysis

of the conjectured mechanism for the wall flow leads to a

time-dependent solution for the ratio of wall flow mass

ejection rate from the upper layer, m^, to the fire plume

mass injection rate _to^ the upper layer, m^. The solution

indicates that in practical fire scenarios m /m can be w p

of the order of "several tenths" even prior to the time

that the upper layer interface has dropped to an elevation midway between the ceiling and fire. In other

words, the results of the analysis indicate the importance of taking the wall effect into account in two-layer

zonal analyses of enclosure fire phenomena.

- 1 -
1 . INTRODUCTION

The zone approach to mathematical modeling has received much attention in

the analysis of enclosure fire environments. In recent years several such

zone models have been developed to predict environments in rooms of fire

origin as well as in single and even multiple adjacent spaces. Because of the

fact that they have each tended to be developed for specific purposes and

classes of problems, there is quite a variation from model to model in the

amount of attention and detail paid to the simulation of different physical

phenomena that come into play. In this regard, however, even the most sophisticated of the models have tacitly ignored a commonly observed and potentially

important wall effect which can lead to significant features of enclosure fire

environments not heretofore simulated.

After describing the wall effect in question this paper will then

identify the conditions under which the effect is significant. Finally, the

paper will propose a general purpose algorithm for including the wall effect

in an overall zonal model analysis.

2. A DESCRIPTION OF THE WALL EFFECT AND A MEASURE OF ITS SIGNIFICANCE

The wall effect under consideration is first described and analyzed

within the context of a generic fire scenario which develops within an

enclosure lacking any significant ventilation openings (any leakage from the

-2 -
enclosure is assumed to occur near the floor) . Later, the relation of the

wall effect to fire environments developing in enclosures with open doors or

windows will be addressed.

A sketch of the significant features of all the conjectured wall effects

to be described below is presented in figure 1.

A fire starts a distance H below the ceiling of an enclosure of floor

area, A, and releases energy at the rate Q(t), where t is time from the effective instant of ignition. Because of their elevated temperature, the products

of combustion from the fire are driven upwards by buoyancy forces, and a

turbulent plume is generated. All along the axis of the plume, quiescent

ambient air is entrained laterally into and mixed with the plume gases. The

plume eventually impinges on the ceiling and spreads radially outward along

this surface forming a relatively thin, turbulent ceiling jet. The ceiling

jet gases redistribute themselves across the entire ceiling area of the enclosure, start to fill the upper enclosure volume, and eventually submerge all

the continuous ceiling jet flow activity. Below this jet activity the now

reduced momentum plume gases form a relatively quiescent, elevated

temperature, upper gas layer which continues to increase in depth as the plume

gas upward filling process continues in time. The elevation above the fire of

the lower interface of this gas layer is designated by Z^(t).

Except for regions near the bounding ceiling and wall surfaces, it is

reasonable to characterize the upper layer as being uniform in composition

with an absolute temperature, T

u

(t) and with concentrations of products of

combustion i, C^(t) (units of product per unit mass of bulk upper layer

mixture)

.

-3-
The ceiling surface temperatures are generally different from T This

u

.

surface is heated from the initial ambient temperature, T , by convection from

the high temperature ceiling jet gases and by radiation from the fire's

combustion zone. Later into the fire, radiation exchanges from the bulk upper

layer gases at T

u and from other bounding surfaces of the enclosure may also

become significant.

In general the temperatures of the wall surfaces are also different from

T

y

. The radiant heating mechanism of these surfaces is similar to that for

the ceiling. However, except for possible locations of strong ceiling jetwall surface flow interactions (i.e., where wall surfaces are close enough to

the plume-ceiling impingement point) , the vigorous convective heat transfer at

the ceiling is generally absent at the walls. Thus, one can anticipate that

the wall surfaces will be heated from the initial temperature, T , at a significantly lower rate than the ceiling surface.

It is conjectured and seems to have been observed 1

» that the condition

of relatively cool walls, at absolute temperature, T , bounding the elevated

temperature upper layer gases, at T

u , would lead to the development of a

classic, two-dimensional, downward-directed, natural convection or buoyancydriven boundary layer flow along the upper portion of the vertical wall

surfaces. The phenomenon would occur in this basic form away from locations

of vigorous ceiling jet-wall interactions. The boundary layer flow would

originate near the ceiling, increase in mass and momentum flux with decreasing

elevation, and, at the upper layer - lower layer interface, be injected into

the lower layer of the enclosure as a downward-directed wall jet of upper

layer gases. The injected gases would have a bulk temperature lower than the

-4-
upper-layer gases but higher than the relatively cool and uncontaminated

lower-layer gases.

Once in the lower layer, the downward-directed wall jet of the now

upwardly-buoyant, product of combustion laden gas would be buoyed back upward

and away from the wall to either mix with and contaminate the lower layer air

or to entrain additional (i.e., in addition to the fire plume) lower layer air

into the upper layer.

Whatever the eventual disposition of the wall flow contaminents, the

basic wall flow phenomenon could significantly alter the rate of development

of life-threatening conditions in the enclosure. If, for example, the flow is

buoyed upward as a wall plume and re-enters the upper layer, then, having

entrained additional lower layer air, the depth of the upper layer will grow

more rapidly than it would have otherwise, albeit at a reduced temperature and

product concentration. On the other hand, if the wall flow is mixed with the

lower layer gas, then the net effect would be a contamination of the lower

layer, less rapid growth in upper layer depth and at increased temperature and

concentration.

When the mass flux leaving the upper layer in the wall flow is large

enough to significantly retard the velocity of descent, dZ^/dt, of the upper

layer - lower layer interface or to lead to significant increases in temperature and/or product of combustion contamination of the lower layer, then this

flow must be taken account of in a mathematical simulation of the overall

enclosure environment.

- 5 -
Let nip be the mass flux of the plume into the upper layer, i.e., at Z^.

Also, let m^ be the total mass flux of the wall boundary layer flow ejected at

from the upper layer. Then, consistent with the above, an analysis of the

enclosure environment which does not include an accounting of the wall flows

• • • • can lead to reliable results only when m « m . Thus when m /m << 1 the w p w p

wall effect can be neglected, and when this is not the case it must be taken

into account. This will be referred to below as the Wall Effect Criterion.

It is noteworthy that the described wall flow phenomenon would not necessarily occur only in rooms of fire origin. Indeed, it can also play an

important role in the redistribution of the products of combustion in spaces

which are adjacent to, or communicate with rooms-of-f ire-origin. Here, the

fire generated plume in the above room-of-f ire-origin scenario would be

replaced, for example, by an analogous upper doorway plume which was generated

by inflowing combustion products. "Smoke layering” in the adjacent space

would be initiated in the usual way. Also, at least until relatively late

into the fire when radiative heating of adjacent space wall surfaces could

become substantial, these surfaces, being only weakly heated by the conjectured buoyancy driven wall flows, would likely remain close to their original

ambient temperature.

The above discussion is from the perspective of the early growth stage of

the fire when a definitive (T

u

- Tw

) temperature differential exists, when no

significant differential between the lower wall temperature and the lower gas

temperature, T^, (both of which are initially at Tj has developed, and when

the overall wall flow activity, as described earlier, is conjectured to occur.

As time moves on, however, a substantial difference between T

L and the

- 6 -
(probably) relatively greater lower wall temperature (heated primarily by

radiation) may begin to develop. At least near the floor of the enclosure

this would (probably) tend to lead to an upward-directed, lower wall flow. It

is also possible that such a lower wall flow would become relatively strong at

a time when the downward-directed, upper wall flow starts to decrease in

strength by virtue of a reduction in (T - Tw

) . Under such a circumstance,

the nature of the interaction of such combined, counter-current wall flows

could be important in an adequate description of the overall enclosure

environment; the early growth stage description of a dominant upper wall flow

would have to be revised.

An analysis of the combined wall flow effect, which may come into play at

later stages of the fire, is outside the intended scope of the present paper.

The combined flow effect will not be discussed further except to note that a

study has been recently initiated on the significance of the phenomenon in

o ventilated enclosure fire scenarios during steady state conditions.

3. SOME PROPERTIES OF THE WALL FLOW

Refering to the wall flow depicted in figure 1, the total mass flux,

m(x) , at a distance x = H - Z below the ceiling (Z above the fire) can be

estimated, for Prandtl Number (Pr) = 0.72 and according to whether the

boundary layer is laminar or mostly turbulent, from

m(x)

P

1.70 y Gr

1/4

2/5 0.102 y Gv 00 x

Laminar:Gr

x

< 0.5(10^)

Turbulent :Grx

> 1.5(10^)

( 1 )

-7 -
where P is the total length of the perimeter of the enclosure, Gr

x is the

Grashoff number

Gr

x

T v

2

00 00

( 2 )

and where dimensionless values of T

u , Tw , and Z have been defined as

T /T ; u 00 <t> T = T /T ; w w 00 Z/H (3)

The above estimates for m(x) were established from results presented in

references 4-6. The Gr

x bounds for a laminar or mostly turbulent boundary

layer are from reference 7. The Bousinesq approximation was used in developing eq. (1), and will be used throughout the analysis to follow. All gas

properties will be taken as those of air at standard ambient conditions. The

wall temperature is taken to be uniform.

From the origin of the boundary layer to some x above the interface (0 <

x < H - Z^), the average rate of heat transfer per unit area, q" , to the wall

can be estimated from

q” (4)

where

Average Nusselt Number =

1/4 5 0.48 Gr , Laminar

x

’

2/5 0.0184 Gr , Turbulent

x

’ 6

(5)

-8 -
and where k is the thermal conductivity of the gas mixture. Below the interface the heat transfer rate will be less than q"

.

The mixing cup temperature, T , of the wall flow at position x is defined

by

T

u Tm

q"?x

mC

P

where is the specific heat.

( 6 )

Using Pr = 0.72, the results of eqs. (1), (4) and (5) in eq. (6) lead to

the following estimate for Tm

:

T - T {0.39, Laminar

-T —

) (7) u w ( 0.25, Turbulent

This last result indicates that Tm

is independent of x and somewhat closer to

T

u than to Tw

. In particular, eq. (7) is valid at the interface position

x = x

i = H - Z

i<

The kinematic momentum flux, K, of the wall flow

K = /ou2dy (8)

where u is the downward velocity and y is the distance from the wall surface,

can be estimated from results of integral boundary layer analyses in references 6 and 7. Thus, for Pr = 0.72

-9 -
K (9)

2.25v2 Gr

3/4

/x,

00 x Laminar

0.0366V2 Gr 9/10

/x, Turbulent 00 x

Finally, the order of magnitude of the downward velocity will be of

interest in later discussion. From results in references 4 and 6 it is

possible to obtain the result that for Pr = 0.72 the maximum velocity in the

boundary layer, umax , for both laminar and mostly turbulent flow can be estimated from

umax 0.55V00 ( 10 )

4. IMPLICATIONS OF THE WALL EFFECT CRITERION

The mass flux of the plume at the upper layer - lower layer interface can

O Q

be estimated from *

m 0.21p„g

1/2

H

5/2

(l - X

)l/3

Q

*l/3

;5/3 !/3

r

' x

o i

T ( 11 )

where

- VH, * Q(t)/Q . Q.

p»c

p

to/

/ 2 h5/2 ( 12 )

and where Q0 is a characteristic energy release rate, and (1 - X

f

) is the

fraction of Q which effectively acts to heat the plume gases and ultimately

drive the plume's upward momentum. X is approximately the fraction of Q lost

by radiation from the combustion zone and plume. For hazardous flaming

O fires, X

f

is typically in the range 0.3 - 0.4. In real fires, the simple

- 10-
description of Eq. (11) has been shown to yield excellent results above the

combustion zone when the "point" location is taken to be at the lowest elevation of the combustion zone or plume above it where "free unrestricted",

lateral entrainment of ambient air is possible.®

In order to invoke the Wall Effect Criterion, eqs. (1) and (11) are used

to estimate m /m with the result w p

m /m w p

oGr (P/H)

0.2l(l-X

r

) 1/3 Q* 1/3

^

/3 (gH

3, 2 /v„ U/2 .1/3

J

(13)

where

Gr = Gr (x x

x. = H-Z ) = (<j) -l)(l-6)(l-£ ) 3

gH

3

/ i i u i 00 (14)

6 = U - l)/(<j> - 1) w u (15)

a = 1.70. _ 1 / /.

a = 0.102, n = 2/5 if Gr > 1.5(10

10

), Turbulent

(16)

To apply eq . (13) to the dynamic fire scenario discussed in the last section

it is now necessary to estimate the values of t.(t), d> (t) and <t> (t).

i u w

In the absence of any wall effect, and from ignition (^ = 1) up to the

time that the interface elevation starts to approach the elevation of the fire

(i.e., becomes small), the solutions for C^(t) and 4>u(t) have been

previously obtained^ for enclosure fires where Q(t) ~ t m (arbitrary m 0)

,

- 11 -
and where any pressure relieving leakage from the enclosure is primarily near

the lower portions of the enclosure boundary. Under such circumstances

u

- { l-(m + 3)eT

3(,n+1)/(m+3)

/[3( m+ l)(l-C

1 )]r

1 (17)

where

e - (l-X

c )l[(n,+3)/3]m

Q*} 2/ ('t3)

/( l-»

r )(’tl)/(,t3)

( 18 )

t = 3[(l-X

r )0*] 1/3

(tH3/2

g

1/2 /A) <m+3)/3

/(m+3) (19)

and where Qq

is explicitly defined as

Qq - (Q/tm

)(H

3/2

g

1/2 /A)"m

(20)

Also, X c is defined as the fraction of Q which is instantaneously transfered

to the internal bounding surfaces of the enclosure. XCQ would include all

heat transfer to surfaces by both convection and radiation, and Xc is

g typically in the range 0.6 - 0.9. For magnitudes of energy release rates and

1 2 enclosure sizes of practical interest , e << 1 and can be estimated from

lim 5 - (1 + 0.140t)“ 3/2

[1 + 0(e)] (21)

e-K)

for 5 and m which include the ranges 0.5 1 and 0 _< m 2 . (For a

more definite idea of the utility of the eq. (21) approximation refer to

figure 2 of reference 12.) Unless noted otherwise all further estimates in

this paper will only be applicable when the above approximation yields an

acceptable estimate for £^(t).

- 12 -
Using eqs. (17) and (21), and eliminating the explicit dependence of

on time leads to

lim (<f> -1)

e+0

U

e(m+3)

3(m+l)(l-c

i

)

3(m+l)/ (m+3)

+0(e) 2

( 22 )

Using eq. (22) in eq. (14) and rewriting eq . (13) eventually leads to

aGr

n

X, m /m w p 0.21xJ

/2 ^/3

m_ (m +

_

3) (23)

Gr = Gr(^

i

; X^ m, 6)

(l-6)(m+3)

f )2 (

+

' 1 )

3 ( m+ 1 ) VA \ 0.140 /

3(m+l)

(m+3)

(24)

where a and n are given in eq. (16), and where X^ and X

2 are defined according

to

(25)

4.1 Applying Results When Wall Temperatures Are Close to Ambient

During the early growth stage of the fire the increase of T

w above its

initial value, T^, will often develop at a much slower rate than will the

increase of T . Thus, for the purpose of evaluating the significance of the

wall effect, it is reasonable to investigate the implications of eqs. (23) and

(24) under the condition

6 « 1 (26)

- 13-
According to these equations, and for in = 0, 1.0, and 2.0, figure 2 presents

plots of C . as a function of Xi with (m /m )/X_ as a parameter, under the

1 W p L condition 6=0. Thus, for 0 m < 2.0, and for a specified X^ fire scenario,

the plots of figure 2 can be used to determine the value of below which the

wall effect starts to become significant (i.e., m /m becomes larger than, w p

say, several tenths).

The results of figure 2 will be used to evaluate the significance of the

wall effect during test runs of two full-scale enclosure fire test programs

reported in references 1 and 2. As mentioned earlier, the wall effect seems

to have played a role in those experiments.

The first evaluation refers to a constant 16.2 kW enclosure fire reported

in reference 2. The fire was an acetylene diffusion flame generated from a

gas burner source 2.06 m below the ceiling and in the center of a square,

"fully" enclosed space (designed to leak from below) with sides of length

3.64 m. Tests on the burner indicated a 50% radiation loss from the sooty

flame (X^ = 0.5). Total loss to the ceiling and walls was estimated to be

84% (A

c = 0.84). In terms of previous definitions, and using the values

T^ = 29 4°K , = 1.2 kg/m3

, = 1.5 (10~ 5

) m

2 /s, C

p = 240 cal/(kg°K), and

g = 9.8 m/s here and in all later calculations, the test run is found to be

characterized by

m = 0 , e = 0 .0036 , X^ 1.4(10

9

), X

2 = 4.0

- 14 -
• • Now assume the wall effect to be significant when m /m > 0.3 , i.e., w p

(m /m )/X„ > 0.075 . Then figure 2 for m=0 leads to the conclusion that the w p z wall effect must be taken account of once drops to 0.62, i.e., once the

upper layer thickness exceeds 0.78 m.

The remaining evaluations will refer to test runs reported in reference 1

in a 2.36 m high, fully enclosed two to three room space (designed to leak

O from below), where one of the rooms, having an area of 14.0 m and designated

as the burn room, contained a centrally located methane burner 2.12 m below

the ceiling. The one to two rooms adjacent to the burn room were corridorlike in configuration, and the total multi-room test space area ranged from

O 40.6 - 89.6 m . For test runs with all rooms freely connected (i.e., full

open doorways), data indicated that after an initial time interval a single

room, two layer mod had some applicability in analyzing the dynamic conditions

within the overall space. It is therefore reasonable to utilize the present

room-of-f ire-origin results in an evaluation of the significance of the wall

effect under consideration. As mentioned earlier, the existence of the wall

effect was observed during the course of both the reference 1 and reference 2

test programs.

The first evaluation of the above test series will involve the smallest

O constant fire, 25 kW, and the largest test space area, 89.6 m of the test

program. This configuration had a total wall perimeter of approximately

81 m. Previous tests on the burner had indicated X = 0.19, and data from

vertical arrays of thermocouples indicated an average value A

c = 0.87. Using

all of the above data the test run is found to be characterized by

- 15 -
m = 0, e = 0.0032, X = 1.3(10 15

9

), X

£ =

• . • Again, assuming the wall effect to be significant when m^/m^ > 0.3 , i.e.,

(m^/m )/X

2 > 0.020 , figure 2 leads to the conclusion that the wall effect

must be taken account of once drops to 0.85, i.e., once the upper layer

thickness exceeds 0.32 m.

The second evaluation of the test series will involve the largest

O constant fire, 225 kW, and the smallest test space area, 40.6 m , of the test

program. This configuration had a total wall perimeter of approximately

43 m. Tests on the burner indicated Xr = 0.24 for burner fires in the range

50 to several hundred kW, and test run data indicate an average value

X = 0.75. All of this leads to

c

m = 0, e = 0.027, X = 1.1(10

10

), X

2 = 12.

Using the previous criterion and figure 2, the wall effect is predicted to be

significant here once < 0.76.

The final two test run evaluations will refer to the largest and smallest

area configurations where the burner was controlled to generate a fire growing

linearly with time according to

Q(t) = (t/2) kW/s (27)

f

Thus, m = 1, and according to eq. (20)

- 16 -
Q ( 28 ) c

= [ A/C2H3/2

g

1/2

> ] kW/s

First the largest space (A = 89.6 P =* 81 m) : Here Ar = 0.24 , and

test data indicate an average value A

^ = 0.81. This leads to

m = 1, e = 0.0064, X = 2.6(10

9

), X

£ = 39.

Using the previous criterion and the m = 1 .0 plots of figure 2, the wall

effect is predicted to be significant once < 0.93 .

O Finally, the smallest space (A = 40.6 m , P = 43 m) : Here A = 0.24, and

test data indicate an average value A^ = 0.71. This leads to

tn = 1, e = 0.0063, Xj = 2.7(10

9

), X

2 = 27.

From figure 2, the wall effect is predicted to be significant once < 0.90.

5. ESTIMATING THE WALL TEMPERATURE

Figure 2 and the above example calculations were based on the assumption

6 « 1 of eq. (26). In this section estimates of 6 will be developed so that

the validity of this assumption can be evaluated, and so that, when required,

• • estimates of m^/m^ under somewhat more general nonambient wall temperature

conditions can be obtained from eqs. (23) and (24).

In order to predict 6, estimates for $ - 1 under each of two different

wall heating conditions will be established; increases of <j> - 1 due to

- 17 -
possible radiation from the fire's combustion zone, and increases of <f> w - 1 at

the upper wall surfaces due to convective heating from the wall layer flows.

For the purpose of this investigation it is assumed that for the fire's growth

stage of present interest, and far enough away from significant ceiling jet -

wall flow interaction, one of these two mechanisms will dominate the increase

of d>T . w

5.1 Wall Heating Due to Radiation

Increases in <J> Wdue to radiation may be the more significant of the

heating mechanisms for enclosure spaces which are configured in such a manner

that a large fraction of the total wall surface is illuminated by the fire's

combustion zone. Under such circumstances and for the purpose of estimating

<|> w

- 1 and comparing it to 4> u“ 1> assume the fire to be a characteristic

distance ( A/tt

)

1/2 from all wall surfaces. Then the average rate of wall heat

transfer due to radiation from the combustion zone, q", can be estimated by R

* Q(t)

•• __ 1 q R 4A (29)

For the purpose of the present estimate, first consider wall materials

which are thermally thick up to some time, tj, of interest. To be definite,

it is reasonable to consider a wall material as thermally thick if its thickness, and thermal diffusivity, <w

, satisfy

L

THICK

> 2/k t_ w I (30)

- 18 -
For tj- of 3 minutes, and using material properties from table 1, the criterion

would be satisfied, for example, with gypsum board walls thicker than 0.011 m

or concrete walls thicker than 0.055 m.

Using the thick wall assumption and eq. (29) in Carslaw and Jaeger's

solution to the appropriate, specified flux, heat conduction problem for the

wall leads to the result

w

- 1

X k

1/2

q r w o AT Tf l/2Ak CO w

rt t - n)

j 0 1/2

n dn (31)

where kw

is the thermal conductivity of the wall material.

For the t m fire growth studied earlier, eq. (31) can be integrated to

yield

Radiation to Thermally Thick Wall:

<(> -1 w

1/4 0.14X

3

Xj / e(m+3) 1/2 r(m+l)

T(m+l/2) (£\ 0 .

_n/o v 3(2m+l)

^

/j-l\2(m+3)

140 /

(32)

6 =

1/4 0.43X

3

X^ (m+l)

1 / 2

r(m+l)

r(m+l/2) (1-c.

\ / 0.140 \ 2(m+3)

(33)

where

X

r /l-X

r \5/4 (<\U2 /k , “(1-V\1-V V

K / \v

Pr

1/2

H

. 1/2 (34)

- 19-
T(x) i 8 the Gamma function, Xj is defined in eq. (25), and < and k are the

thermal diffusivity and conductivity of air at ambient conditions.

The above result can now be applied to the previously discussed 16.2 kW

enclosure fire (gypsum board wall) of reference 2. The geometry of fire

scenario is one where all walls of the enclosure are illuminated by the fire,

and where radiation can be anticipated to dominate the wall surface heating.

Using the previously identified parameters of that fire, and the k, k values

of table 1 in eq. (33) leads to

5 » (4> - 1) / (

4> - 1) = 0.29 at C. = 0.62 w u l

Note that this ratio is small enough to allow the 6 0 estimate of figure 2

to be useful. Indeed, if the approximation of eq. (26) is supplanted by the

newly revised estimate

4> - <{> ° (<J> ~ 1 ) ( 1 - 6) » 0.71(<() - 1) u w u u

then eqs. (23) and (24) would predict m^/m^ = 0.28 at - 0.62 , instead of

• . • the earlier estimate of m /in = 0.30 . w p

If the wall of the reference 2 enclosure had been concrete instead of

gypsum board (and if X would not have been significantly different from the

measured value of 0.84), then eq. (33) leads to the estimate

6 = (4> - l)/(4>u

- 1) = 0.09 at C 1 = 0.62

-20-
Besides thermally thick walls, wall temperatures for thermally thin walls

are also of interest. Again, for definiteness, it is reasonable to consider a

wall material as thermally thin if its thickness satisfies

lthin < 0.6/k t T w I (35)

Using material properties from reference 11 and a t^ of 3 minutes, this

criterion would be satisfied, for example, with mild steel walls thinner than

0.028 m, glass walls (windows) thinner than 0.006 m, or wood panel walls

thinner than 0.004 m.

Using the radiant heat transfer rate of eq. (29) and the t m fire growth

leads to

Radiation to Thermally Thin Rear-Insulated Wall:

w TT x

4

)

3 ( m+ 1

)

(m+3)

(36)

6 = 0.25 x

4

(i - e ) (37)

where

x

4 -tt^t 7 fc) te) (t)

(38)

If the wall of the reference 2 enclosure had been sheet metal construction of thickness 0.0015 m instead of gypsum board (and if X^ was not significantly different than the measured value of 0.84), then eq . (37) leads to the

estimate

- 21 -
6 » 0.13 at » 0.62

and the original small 6 determination of the significance of the wall effect

continues to be relevant.

5.2 Wall Heating Due to Convection

In some enclosure fire configurations, radiant heating of enclosure wall

surfaces from the fire's combustion zone may play a relatively minor role

compared to convection. This would be the case, for example, if direct, lineof-sight, combustion zone illumination of most wall surfaces was blocked by

furniture or segmented partitions, or on account of an alcove-like fire location. Also, in a multi-room enclosure configuration, which is an extension of

the alcove-like single room configuration, radiant wall heating in rooms other

than the room-of-f ire-origin could play a minor role well into the "smokefilling” process.

An estimate for the convection driven 4>^( t ) increase, to be obtained

here, will be based on an eq. (4) heat flux to the wall at every instant of

A tfc ^ mmm time, t , from t = 0 to t = t. h(t ) in eq. (4) will be computed from

eq. (5) with x = H - Z^(t*) and with Gr

x

= Gr(t*) according to eq. (14) or

(24)

.

In the estimate, 5 will be neglected compared to 1 and a laminar wall

layer will be assumed through most of the heating history up to the time, t,

of interest.

-22-
by

All of the above considerations lead to a convective wall flux, q"

— * q"(t ) 0.48 /m+3\5/4

3

5/4H VV [l-?.] 3/2

15(m+l)

4(m+3)

T

j|f where t is evaluated from eq. (19) with t = t . Using this last result

reference 13 solution to the specified flux heat conduction problem for

thermally thick wall leads to the result

Convection to Thermally Thick Wall:

<f> -1 = w

llnrt-9 0

.

1 7 eXj -l \4(m+3)

(m+3) 7/4 r|3(m+1)/4)

(i-C.)!/

2 V °-l 4 /

* (m+3) l/4 r[3(m+3)/A] V°;m)

_ Q- 31x

3

( 1 -q) /2

( m+3 ) 3/4 T[3(m+l)/4] I I^

1 C i ;m J 6 = -2/3 ,d/4

(m+1) 1/4 r [3C m+3 )/^] I^Ojm)

where

Xr l

l - \\

1/4 (\\ 1/2 /k_\

" V ' V Pr

1/2H \

K / Vj

llm+15

I

1 (x;m) = /*

r, . v- 2/3 .i

(l-xn) -l

4(m+3)

i— i fH 1CN CO

/—

\

1X 1

f(l-xn)

2/3

-l (m+3)

[(1-x)

" 2/3

-l_

vl/2

dn

f , x5/3

(1-XTl) 1

t f n • m ^ _ (nH-3) r(3/2)r[3(m+l)/4]

P

U ’ ; 3 T[3(m+3)/4]

given

(39)

in the

a

(40)

(41)

(42)

372

i

(43)

(44)

-23-
and where (x ;m) / 1^ (0 ;m) has been evaluated and plotted in figure 3 for

m = 0, 1.0, 2.0, and 5.0.

The convective wall flux of eq . (39) has also been used to obtain

$ - 1 and 6 for thermally thin wall constructions, viz

Convection to Thermally Thin Rear-Insulated Wall:

w

llm+15

4(m+3)

/ m + 3\ 9/4

T 2

1 C i ;m ) + 1 J

l 2 (0;m)

6 =

(m-3)

4(m+3) /m+3\5/4 g

l ;m ) ym+1

J

l

2 (0;m)

where

-(R)

1/2

L

THIN

HPr m

I 2 (x;m) 'll

(l-xn)

llm+15

-2/3 , "1 4(m+3)

( 1-x)

)-2/3

-l 1

"2/3

-l J

dn

.5/3 3/2 (1-xq) n

and where I 2 (x ;m) /

I

2

( 0 ;m) has been evaluated and plotted in figure 3 for

m = 0, 1.0, 2.0 and 5.0.

(45)

(46)

(47)

(48)

(49)

- 24-
The above results can now be applied to the previously discussed multiroom enclosure fires of reference 1. In those enclosure configurations,

radiant heating of most wall surfaces (i.e., outside the burn room) was likely

to have played a minor role in smoke movement phenomena during the time frames

of the experiments. Accordingly, values of 5 under conditions of convective

wall layer heating have been computed from eqs. (41) and (46). These are

presented in table 2 along with the results of all previous example calculations. As can be seen, for the cases considered, the values of 6 for the

convection driven wall heating examples are never greater than 0.011, and the

original, small 6 determination of the significance of the wall effect

continues to be relevant.

6. QUASI STEAD I NESS OF THE WALL FLOWS AND A MEASURE OF THE

LIKELIHOOD OF THEIR PENETRATION

The validity of two tacitly assumed aspects of the wall flow will be

studied in this section. The first has to do with the question of the time

required to reach quasisteadiness relative to an assumed steady quiescent

upper layer. Related to this, the second has to do with the expectation that

the velocities of the wall flow are large enough to actually penetrate the

dropping upper layer - lower layer interface.

6.1 The Quasisteady Flow Assumption

The time, t

g

, for a wall flow to establish itself as a steady flow

(subsequent to a step change in wall/ambient temperature, and from the leading

edge of the boundary layer to a station, x) has been studied extensively in

the literature. Reference 7 presents a review of this work, and, in terms of

the present nomenclature, recommends the following estimate for t

25-
t

s - 3(1 -

Cl ) 2 H

2 /(Gr 1/2 vJ (50)

When the time required for the upper layer to grow to a given thickness

and temperature is at least as large as the above t_ estimate, then there is

O good reason to expect that the quasisteady wall flow assumption is valid.

This observation together with the previous results for the t m enclosure fire

environment finally leads to the following criterion:

With regard to the development of the wall flow in the upper layer during

t m enclosure fires, the assumption of quasisteadiness is justified when layer

interface elevations, satisfy

The criterion is illustrated in the upper curve of Figure 4, and has been

successfully used to test the validity of the quasisteady assumption in all

previously described fire scenarios.

During a fire scenario it is reasonable to expect that the wall flow will

penetrate the interface when the characteristic velocity of the former, say

umax ecl* (10), at elevation is much larger than that of the latter,

i.e., when

(51)

6.2 Wall Layer Penetration

u,max (Z = Z

±

) » dZj/dt (52)

-26 -
To be definite, the following criterion is adopted:

At the upper layer - lower layer interface, penetration of the wall flow

into the lower layer is expected if

umax<Z “ z

l> > 5dVdt

Using the present nomenclature and the previous results for the

sure fire environment, finally leads to the following version of eq

.

/

-2/3 Al/2

L_ ( _i ) >1 Q HP_

? 5/3 \ 0.140/ - yi-6J yn+3/

AX

2

The criterion is illustrated in the lower curve of Figure 4, and has

successfully used to test the validity of the penetration assumption

previously described fire scenarios.

7. SUMMARY AND CONCLUSIONS

The results of this work point to the significance of buoyancy driven

wall flows on the development of hazardous environments within enclosures

containing practical growing fires. Such flows were seen to be important, for

example, in fire scenarios involving relatively weak fires and/or plan views

with relatively long peripheral dimensions. It would clearly be appropriate

to formulate and carry out an experimental program for quantitative verification of these results. In any event it appears that the wall flow phenomenon

must be accounted for in mathematical enclosure fire models that hope to

(53)

t m enclo-

(53)

(54)

been

in all

-27 -
estimate dynamic environments generally for the purpose of predicting the

response of fire detectors and fire intervention hardware and the time of

onset of conditions detrimental to life safety. In order to carry out such an

accounting, further analytic and/or experimental research on various aspects

of the basic buoyancy driven wall flow may be required, and should be pursued.

However, until the results of such studies are available, it may be prudent to

add wall flow algorithms to existing models on a more timely basis. Such

algorithms would be based on the kinds of calculation, estimates and results

which have been developed in the present work. For engineering purposes, such

algorithms may prove to be adequate, indeed, optimum, even in the long term.

8 . ACKNOWLEDGMENTS

This work was supported by the U.S. Department of Health and Human

Services, and the Bureau of Mines and National Park Service of the U.S.

Department of Interior.

9 . REFERENCES

[1] Cooper, L. Y. , Harkleroad, M. , Quintiere, J. , and Rinkinen, W. , An

Experimental Study of Upper Hot Layer Stratification in Full-Scale

Multi-room Fire Scenarios, J. of Heat Transfer, Vol. 104, 1982.

[2] Mulholland, G. , Handa, T. , Sugawa, 0., Yamamato, H. , Smoke Filling in an Enclosure, paper 81-HT-8, presented at 20th Joint ASME/AIChE National

Heat Transfer Conf., Milwaukee, 1981.

[3] Jaluria, Y. , Buoyancy-Induced Wall Flow Due to Fire in a Room, to appear

as NBSIR, National Bureau of Standards.

[4] Ostrach, S., Laminar Flows with Body Forces, sect. F of Theory of

Laminar Flows , Moore, F. K. , Ed., Princeton, 1964.

[5] Ostrach, S., An Analysis of Laminar Free-Convection Flow and Heat

Transfer about a Plate Parallel to the Direction of the Generating Body

Force. NACA Rept. 1111, 1953.

- 28 -
[6] Eckert, E. R. G. and Jackson, T. W. , Analysis of Turbulent FreeConvection Boundary Layer on Flat Plate, NACA Tech. Note 2207, 1950.

[7] Ede, A. J., Advances in Free Convection, Advances in Heat Transfer,

Vol. 4 , Hartnett, J. P. and Irving, T. F., Eds., Academic, 1967.

[8] Zukoski, E. E., Kubota, T. , Cetegen, B., Entrainment in Fire Plumes,

Fire Safety Journal, Vol. 3, Nos. 2-4, 1981.

[9] Cooper, L. Y. , A Mathematical Model for Estimating Available Safe Egress

Time in Fires, Fire and Materials, Vol. 6, Nos. 3~4 , 1982.

[10] Heskestad, G. , Engineering Relations for Fire Plumes, to appear in Fire

Safety J.

[11] Cooper, L.Y., Convective Heat Transfer to Ceilings Above Enclosure

Fires, 19th Inter. Symp. on Comb., Haifa, 1982.

[12] Cooper, L. Y. , Development of Hazardous Conditions in Enclosures with

Growing Fires, to appear in Combustion Science and Tech.

[13] Carslaw, H. S. and Jaeger, J. C., Conduction of Heat in Solids , Oxford,

1959.

[14] Peacock, R. D. and Breese, J. N. , Computer Fire Modeling for the

Prediction of Flashover, Nat. Bur. Stand. NBSIR, 1982.

- 29 -
10. NOMENCLATURE

A area of enclosure

C

i concentration of product i S

specific heat of air

Grx Grashoff number, eq . (2)

Gr Gr x at x = H -

g acceleration of gravity

H ceiling-to-f ire distance

h average heat transfer coefficient

*1* l2 integrals, eqs. (43) and (48)

K kinematic momentum flux, eq. (8)

k thermal conductivity of air

kw

thermal conductivity of wall material

lthick thermally thick wall thickness

lthin thermally thin wall thickness

m an exponent

•m mass flux

n an exponent, eqs. (13) and (16)

Nu

X average Nusselt number, eq. (4)

P perimeter

Pr Prantl number

Q energy release rate of fire

o

*

o

O'

O'

a characteristic value of Q

dimensionless Qq , eq. (12)

M q rate of heat transfer per unit area

T absolute temperature

*

t, t time from ignition

- 30 -
L a time of interest

I a Lime ui xutereuL

u downward velocity in wall flow

Xj , ^2 , • • • dimensionless parameters, eqs. (25), (34), (38), (42),

and (47)

X distance below ceiling

y distance from wall

z elevation above fire

a a constant, eq. (16)

r gamma function

6 dimensionless temperature difference, eq . (15)

e dimensionless parameter, eq. (18)

C Z/H

n dummy variable

K thermal diffusivity of air

K

W

thermal diffusivity of wall material

Xr fraction of Q lost by radiation

Xc fraction of Q transferred to enclosure surface

y viscosity

V kinematic viscosity, y/p

T dimensionless time, eq. (19)

<t> dimensionless temperature, T/T^

* Q/Qo

Subscripts

c convective

i interface

L lower layer

max maximum

- 31 -
p

R

u

w

plume

radiant

upper layer

wall flow

ambient

-32-
- 33-

Figure

i.

Sketch

of

the

significant

features

ol

Llie

waLJ.

ellect
- 34-

0

and

2.0

under

the
C£> CD

LO

CD

CO

CNJ C5

X

|ui:o)i/(ui:x]i

- 35 -

Figure

3.

Plots

of

(x;m)/

Ij

(0

;

m)

and

I2

(x;m)/

l

2

(0;m)

per

eqs

.

(43)—

(44)

and

(48)

— (

49)

,

respectively,

for

in

=

0,

1.0,

2.0

and

5.0
=

Zj/H

1.0

0.9

0.8

0.6

Figure 4. Plots of eq. (51) and (54) criteria for wall layer

quasisteadiness and penetration, as functions of

interface elevation and for t m enclosure fires

- 36-
Table 1 . Thermal Conductivity and Diffusivity of

Selected Materials

Material Conductivity Diffusivity

m 4-)

air 13 0.024 1 .9(10~ 5

)

gypsum board^ 0.17 1.6(10“ 7

)

concrete (1:2:4)^ 0.92 4.2(1

0

— 7

)

steel (0.1% C) 11 46.0 1.2(10-5

)

- 37 -
Table

2.

Results

of

Calculations

on

the

Significance

of

the

Wall

Effect

in

Full-Scale

Experiments

of

References 1

and 2

co• o co 00 CM rH 'S’ CO a\ 00 >3* o CM

H

vO cr> CM 00 00 CM >3- CO CM f-H CO 00

O' oo CO o o o o O o o o O rH o O

3|

a

•e 1 *s

CM o o o o o o o o o O

O o o • • • • • • • • • • • • • • • o o o o o o o o o o o o o o o

00

vO CO CO o CM o CO X) CM o 'd- rH CO rH CO rH lO O o >3- o o o o CO o o as o o as • • • • • • • • • • • • • • • be o o H o o IT* o o CO o o CM o o

•H

C IICO

NCO

il>3

Nm

Nm 1sD

IIm

Nto

II

x

nlOum

ax> 1m

IIm

nXI £ X X X X X X X X X X X X X X X • • • • • • • • • • • • • • • • • • • • • • • • x X c X J*! a X X c X c Jx! JC c

1-1 o o cj T-C o CJ •H o CJ •H CJ CJ •H CJ CJ *H

*H •H

X •H •H JC •H •H

x •H •H si •H •H

X

TO

x

X HI

X

X Hi

x

x Hi

x x: HI si x: Hi 3 Hi1 4J1 i T3

Hi1 Hi1 i>

Hi1 H»1 l>

Hi1 Hi1 i>

Hi1

HI1 i>

TO X) CO > > c > > c > > c > > c

fl3 CO c c o c c o c c o c c o p U o o CJ o o CJ o o CJ o o CJ CJ o cj CJ CJ CJ CJ CJ

rH rH rH rH rH

0) CJ a> CD CJ rH ai CJ <u a> (J

TO Hi 4J Hi HI Hi rH *H • CJ W • <u CO • a; CO • a) W • aj W

iH l-i T3 Hi ~o Hi X) Hi X) Hi T3 Hi

CO 4)

X 0) E Xl cj E X a> E X3 CJ E X2 ai E 3 XJ IJ E l-l E M E l-l E i-i E

CO • cj • CJ • CJ • CJ • CJ 2 a c 10 ex c m a c CO a c m ex c CO X o • X o • X o • bO o • X o • bo cj bO cj »H bo (J *H CJ bO CJ rH

CO• o

II CM

m sD co

O

sD oo as as

w

p • • • • • •E I *E O o o o o

<SJ•H

O CM • • • • • X CO CM as

rH

H co CM

/-s

/-“N /^\ o x-s /'“N as as »H CTv as o o o o o H rH *H rH rH X V-/ v> V CO rH X • • • • • «H *H CM CM

X> CM Mf lO

co CO

CM

X vO

Cl) o O o o o o o o • • • • o o o o o o CTi

l-l

m CM CM CM

X • • • • • o o o o o

co rH

CJ 00 00 oo r^. r< • • • • • o o o o o

CM • rH

CO * o rH

n> • •* o a II rH o * r» CO II *H •-H XJ E II c CJ Hi B il II CO c (J E

rH 0) CM r* CN S CM E CM

rH

X rH l-l * E 3 E £ E o X (J 3 X •* r» x CM Hi ex X sD X a x> ex sD

«H • CJ o • co • 6 • E • 3 X> o o m as CM o to o co as

23 CO cj CM 00 CM

M l-i 00

-38
NBS-I14A (rev. 2-80)

3. Publication Date

June 1983

U.8. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SH EET (See Instruction s)

1. PUBLICATION OR

REPORT NO.

NBSIR 83-2730

2. Performing Organ. Report No,

4. TITLE AND SUBTITLE

On the Significance of A Wall Effect in Enclosures

with Growing Fires

5. AUTHOR(S) , Leonard Y. Cooper

6. PERFORMING ORGANIZATION (If joint or other than N BS, see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

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

10. SUPPLEMENTARY NOTES

~1 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) This paper studies the significance of a wall effect that has been observed

during the growth stage of enclosure fire experiments. Relative to the two-layer

phenomenon which tends to develop during such experiments, the effect has to do with

the near-wall downward injection of hot upper layer gases into the relatively cool

uncontaminated lower layer. It is conjectured that these observed wall flows are buoyancy driven, and that they develop because of the relatively cool temperatures

of the upper wall whose surfaces are in contact with the hot upper layer gases. For

a growing fire (growth proportional to t ; t being time and m »_ 0) in an enclosed

compartment, the conjectured mechanism for the wall flow leads to a time-dependent

solution for the ratio of wall layer mass ejection rate from the upper layer, m , to

the fire plume mass injection rate _to the upper layer, A . The solution indicates

that in practical fire scenarios A /A can be of the ordEr of "several tenths" even prior to the time that the upper laye? interface has dropped to an elevation midway

between the ceiling and fire. In other words, the results of the analysis indicate

the importance of taking the wall effect into account in two-layer zonal analyses of

enclosure fire phenomena.

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

compartment fires, enclosure fires, fire growth, growing fires, mathematical modeling,

smoke movement, two-layer phenomenon, wall flows.

13. AVAILABILITY

[XI Uni imited

| | For Official Distribution. Do Not Release to NTIS

| | Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

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PRINTED PAGES

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