Research Information Center

National Bureau o£ Standards

Gaithersburg, Maryland 2089s

NBSIR 88-3707 Ntw

1 0 1983

Prediction of Response Time of

Smoke Detectors in Enclosure

Fires

Yukio Yamauchi

Guest Scientist

Hochiki Corporation

Tokyo 194 Japan

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Gaithersburg, MD 20899

January 1988

Sponsored by

Hochiki Corporation

Tokyo 194 Japan

NBSIR 88-3707

PREDICTION OF RESPONSE TIME OF

SMOKE DETECTORS IN ENCLOSURE

FIRES

Yukio Yamauchi

Guest Scientist

Hochiki Corporation

Tokyo 194 Japan

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Gaithersburg, MD 20899

January 1988

Sponsored by

Hochiki Corporation

Tokyo 194 Japan

U.S. DEPARTMENT OF COMMERCE, C. William Verity, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
.
TABLE OF CONTENTS

Page

Abstract 1

1 . INTRODUCTION 1

2 . PLUME REGION 3

2 . 1 Governing Equations . 3

2 . 2 Parameters . 5

2 . 3 Comparison with Experimental Data 5

3 . CEILING- JET REGION 6

3 . 1 Governing Equations . 6

3 . 2 Parameters 9

3 . 3 Computation . 10

3 . 4 Comparison with Experimental Data ....................... 10

4. FILLING PROCESS AND COAGULATION OF SMOKE 11

4.1 Coagulation in Plume and Ceiling- Jet .................... 12

4 .

2

Filling Process and Coagulation ......................... 13

4.3 Simple Model 14

5. COMPARISON WITH EXPERIMENTS IN ROOM CONFIGURATION ......... 16

CONCLUSION - 20

ACKNOWLEDGEMENT 20

NOMENCLATURE 21

REFERENCES . 23

TABLE

Table 1. Effect of Coagulation in Plume and Ceiling- Jet:

Calculation 15

iii
LIST OF FIGURES

Page

Figure 1. Comparison of experimental data with prediction in plume

centerline temperatures. Solid and dotted lines,

calculation; circles, Evans [ 7 . 25

Figure 2. Comparison of experimental data with prediction in

adiabatic (maximum) ceiling- jet temperatures. Solid line,

calculation; circles, Veldman et al

.

[14],

ATm* - AT„ [Cp 2p

a

2 g H5 / (T« Q2 )]1 ' 3 26

Figure 3. Comparison between prediction and experimental

correlation in maximum excess temperatures of ceiling-jet.

Solid line, calculation; dashed line, Heskestad [17].

AT„* - ATm [C

p

2 p a 2 g H5 / (T. Q2 )]1 ' 3 27

Figure 4. Comparison between prediction and experimental

correlation in radial ceiling-jet velocities. Solid line,

calculation; dashed line, Heskestad [17]. V - v„ [C

p

T, p. H / (g Q)] 1/3 28

Figure 5. Comparison of calculated and measured upper-layer

temperatures in enclosure. Solid line, calculation;

dashed line, experiment. H = 3.0 m 29

Figure 6. Comparison of calculated and measured upper-layer

temperatures in enclosure. Solid line, calculation;

dashed line, experiment. H = 4.0 m..... 30

Figure 7. Comparison of calculated upper-layer smoke concentration

with measured ceiling- jet smoke concentration. Solid line,

calculation; dashed line, experiment. H = 3.0 m 31

Figure 8. Comparison of calculated upper- layer smoke concentration

with measured ceiling- jet smoke concentration. Solid line,

calculation; dashed line, experiment. H = 4.0 m 32

Figure 9. Comparison between calculated and measured ceiling- jet

smoke concentration. Solid lines, calculations;

dashed lines, experiments 33

Figure 10. Comparison between calculated and measured particulate

number concentration in ceiling-jet. Solid line,

calculation; dashed line, experiment 34

APPENDIX A. PROGRAM LISTING. . 35

iv
NBSIR 88-3707

Prediction of Response Time of Smoke Detectors

in Enclosure Fires

Yukio Yamauchi

Guest Scientist

Center for Fire Research

National Bureau of Standards

ABSTRACT

In order to predict the response time of smoke detectors in enclosure fires, a computational model is developed for calculating the local particulate

concentration near the ceiling. The large scale smoke movement is approximated

by integral equations for the plume and ceiling-jet, which originates in the

cold lower layer and penetrates into the accumulated smoke layer in the upper

portion of enclosure. The effect of coagulation, which changes the particle

size distribution, is included to enable predictions of an ionization smoke

detector response. This engineering model is designed to be used in

combination with two- layer zone models for obtaining more detailed information

of smoke concentration in the upper layer. Sample calculations have been made

and comparisons with relevant experimental data showed a reasonable agreement

both in the mass concentration and particle number concentration of smoke in

the ceiling-jet.

1. INTRODUCTION

Comprehensive fire models like those of Mitler and Emmons [1], and Jones [2]

have paved the way for the deterministic assessment of potential fire hazards

in buildings. One of the most important elements in performing a hazard

analysis is the time of detection. Earlier detection of fire can provide a better safety margin to evacuat building occupants. Further, early detection

is necessary for rapid notification of the fire department and for rapid

initiation of automatic fire suppression. A great deal of research has

already been done to understand the response of the heat sensitive devices,

such as automatic sprinklers, but fewer studies have been made in relation to

the response of smoke detectors.

Two types of smoke detectors are in common use today: the photo-electric type

and the ionization type. These two types of detectors operate on fundamentally

different physical principles. The photo-electric smoke detectors use the

scattering of light by smoke particles. The ionization smoke detectors use the

reduction of ion current in an ionization chamber caused by the attachment of

1
ions to smoke particles. Both theoretical and experimental studies [3,4] show

that the response functions of these detectors are strongly dependent on the

number concentration and the size of smoke particles. Thus, in order to

develop a predictive method for the response of smoke detectors, one must

include the effects influencing the particulate component of smoke

characteristics in addition to a model for large scale smoke movement.

Smoke aerosol generated at a fire source is carried upward as part of a buoyant plume. The plume then impinges on the ceiling and forms a radial

outward flow under the ceiling. The nature of the flow is generally turbulent

with strong entrainment; this dilutes the smoke aerosol during its way to the

detector. The heat and particulate release rates are the most important

factors influencing the smoke concentration near the ceiling. Further,

accumulation of the smoke layer in the upper portion of the enclosure may vary

the effect of dilution thus affecting the detector's response time.

Another influential effect on the response of smoke detectors is the

coagulation of aerosol. At high concentration, smoke particles frequently

collide and stick together as a result of Brownian motion. Through this

process the average particle size increases while the particle number

concentration decreases.

The first attempt to model the smoke dynamics in an enclosure fire has been

made by Baum et al. [5]. In their model, the large scale fluid motion is

directly calculated from the field equations of motion. Smoke movement is

simulated by tracking mathematical "blobs" of smoke, and the effect of

coagulation is treated as an internal process within each blob. In a previous

study, the author has developed a model in which local particle number

concentration is directly calculated from the field equation of coagulation

[6]. However, there are difficulties in applying these field equation models

to our engineering problems of detection, due to their requirement for very

large computing facilities.

Recently, Evans [7] developed an engineering model for the prediction of the

response time of automatic sprinklers. Based on the commonly used point source

approximation, known as the Morton-Taylor-Turner model [8], Evans obtained an

analytical solution for the plume flow in the two -layer environment of

enclosure fires. A noticeable achievement in Evans' study is that he showed

the prediction of response time of ceiling-mounted detectors is possible

within the framework of two -layer zone models and with sufficient accuracy.

Following Evans' study, prediction of response time of smoke detectors on the

basis of a zone model is now possible. In this report, a method is presented

for approximating the large scale smoke movement in plume and ceiling-jet,

which originates in the cold lower layer and penetrates into the accumulated

smoke layer in the upper portion of the enclosure. The effect of coagulation

is included to enable predictions of the response of ionization smoke

detectors. Results of comparison with relevant experimental data are also

described

.

2
2. PLUME REGION

2 . 1 Governing Equations

Similar to Evans' study, the Morton-Taylor-Turner model is used as a basis for

the present plume model in a two- layer environment.

The major assumptions of the present model are:

(1) The fire plume is axisymmetric and steady with no swirl;

(2) The environment is stagnant and stratified;

(3) The density (temperature) and the particulate concentration in each layer

are uniform;

(4) The vertical velocity, temperature and particulate concentration have

Gaussian profiles in the radial direction;

(5) The ideal gas law is valid;

(6) The Boussinesq approximation is valid.

The profiles of velocity, temperature and particulate concentration can be

written as follows:

u/Ufl, - exp(-r2 /b 2

), (1)

AT/ATm - AC s/ACsm - exp [ -

r

2 /( Ab) 2

] . (2)

where AT = T - Ta and AC S = C s

- C sa represent respectively the temperature

difference and the concentration difference from ambient, and b(z) is the 1/e

width of the plume with respect to the velocity distribution. The profiles of

the temperature and the particulate concentration are assumed to be identical.

With the above assumptions, the governing equations for mass, momentum, energy

and particulate concentration can be written as follows:

dz

r u dr “ b a i%,

dz

r u

z dr

AT

r g dr,

(3)

(4)

3
dz

r u p aC

p

(T - Tr ) dr b a Ujjj p a Cp (Ta

- T r ) ,

0

d

dz

r u (C s- C sa ) dr b a Um C sa ,

(5)

( 6 )

where a is the entrainment constant determined experimentally. Ta and C sa

represent respectively the ambient temperature and the ambient concentration

at each height, and T r is the arbitrary reference temperature.

Integration of Eqs . (3) -(6) yields the following relations governing plume

flow in a stratified ambient:

d

dz

(b 2 u^ = 2 b a Ujjj, (7)

d

dz

(b 2 Uj/) = 2 A

2 b2

ATux m

g ( 8 )

d

dz

pa

cp

AT®) - - (1 + 1A2

) b

2

dTa ^ Pa. ^p

dz

(9)

dz

(b 2

Ujn AC sm ) = - (1 + 1/A 2) b

2

Ujjj

dC sa

dz

( 10 )

Eqs. (7) -(9) are equivalent to the equations used by Evans [7], and Eq . (10)

for smoke concentration has been added in the same form as the energy

equation. The plume flow in the upper layer is calculated directly from the

governing equations by imposing boundary conditions at the interface between

the upper and lower layer based on plume flow in the lower layer. This is

similar to one of the several methods explored previously by Evans [7]. In

deriving boundary conditions at the interface, it is assumed that the flux of

mass, momentum, excess enthalpy, and smoke are conserved across the -interface.

Since each layer may be considered as having uniform temperature and smoke

4
concentration away from the plume flow region, the right hand side of Eqs . (9)

and (10) are non-zero only at the interface.

The plume flow in the upper layer has to be solved numerically. In the program

listed in APPENDIX, the fourth-order Runge-Kutta method is employed in

obtaining a solution for the plume flow in the upper layer.

2.2 Parameters

The values of the entrainment constant a and Gaussian width ratio A are

necessary for evaluation of the plume model. From extensive experimental

study, McCaffrey [9] provided a set of correlations for the centerline

variations of temperature and velocity in buoyant diffusion flames.

McCaffrey's correlations for the plume region (0.20 < z/Q 2

/-’) are consistent

with the analytical solution of Eqs. (7) -(9) for homogeneous environment.

Also, his data are believed to be the most reliable results available.

Therefore, the values of a and A were selected to match the McCaffrey's

correlations

.

Comparison of McCaffrey's correlations for the plume region with the

analytical solution of Eqs. (7) -(9) yields the following values of a and A:

a = 0.118,

A

2 = 1.157. (11)

These values will be used in all calculations to appear in this paper. They

are slightly different but very close to the values used by Zukoski [10]: a =

0.110, A

2 = 1.12. It should be noted that the vertical coordinate z shall be

taken as measured from the flame base to be consistent with McCaffrey's

correlations

.

2.3 Comparison with Experimental Data

Currently, no detailed smoke concentration data for the fire induced plumes

are available. However, it is meaningful to evaluate this model using

temperature data, since the smoke aerosols are believed to follow the same

mechanism of transportation as temperature under adiabatic conditions.

Evans [7] provided one set of experimental data for plume centerline

temperatures in a two -layer environment. Fig. 1 is a comparison of Evans' data

with prediction. Although there is a sharp decrease in calculated plume

temperature at the interface as pointed out by Evans, the overall agreement is

quite good. This discontinuity of temperature can be smoothed out by simply

assuming that the plume carries its lower-layer features to some distance into

the upper layer (as shown by the dotted line in Fig. 1).

5
3. CEILING-JET REGION

3.1 Governing Equations

Incorporating the Boussinesq treatment for buoyancy and the Reynolds' analogy

for convective heat transfer, Alpert [11] developed an integral model for the

ceiling-jet induced by a fire under an unconfined ceiling. Alpert's

theoretical predictions are in good agreement with the empirical correlations

obtained by Alpert himself [12], and Pickard [13].

Assuming that the fire induced ceiling- jet confined in an enclosure behave the

same way as an unconfined ceiling- jet with the environment of the upper layer,

Alpert's model may be applied for calculating the near-ceiling smoke

concentration as a function of radial distance from the fire axis.

The major assumptions are:

(1) The ceiling- jet is axisymmetric and steady with no swirl;

(2) The environment is stagnant and uniform;

(3) The velocity, temperature and particulate concentration have half -Gaussian

profiles in the vertical direction;

(4) The ideal gas law is valid;

(5) The Boussinesq approximation is valid.

A detailed analysis of this type of flow can be found in Ref. [11].. Thus, only

the final forms of the governing equations and a brief description will be

presented here.

The vertical distribution functions of ceiling- jet velocity, temperature and

smoke concentration can be obtained from Eq . (1) and (2) by replacing the

coordinate r by y, and the length-scale b by 2 ; H is the 1/e thickness of the

ceiling-jet; implications for this is that the profiles are assumed to bear

the same relationship as in the plume. The following are the integrated form

of the governing equations for mass, momentum, energy and particulate

concentration, respectively, for an axisymmetric ceiling-jet:

1 d

r dr

1 d

r dr

1 d

r dr

(r v h) = E v,

1

(r v

2 h) = - —

2

r v h

-w v"

1 d

2 dr

S g h

2

AT

( 12 )

(13)

(14)

6
1 d

(r v h AC S ) = 0

,

(15)

r dr

where

w

Pa v2 / 2

(16)

Q

ft

Pa

(17)

The quantities v, AT and AC S appearing in Eqs . (12) -(15) are the spatiallyaveraged values of the corresponding quantities in the ceiling-jet. The

relations between the averaged and the maximum values are as follows:

v

vm J 2

AT AC,

ATmAC sm \ 1 + A'

h

Z \

7T (18)

The quantity S is the shape parameter which depends on the velocity and the

temperature profiles. In the present case, S takes the value:

2 A

2

S = — - 1.0. (19)

7T Ja 2 /(1 + A

2

)

7
By use of the Richardson number

R-i g h AT

v"

( 20 )

the conservation equations of mass, momentum and energy can be combined into

two separate equations for ceiling- jet thickness h and Richardson number Rj_

,

which results as follows:

dh 1 1 h

(1 - S Rt ) = — fw + (2 - — S R l ) E - —

dr 2 2 r v a

( 21 )

h dR i

(1 - S R± ) — —3R t dr

1

2

f 1 1 Q"

1 1 + — s r 5

17

lit -

l 2 J V* 3 v aJ

h

(1 + 2 S R ± ) ——

.

3 r

where a = AT/Ta

.

( 22 )

The entrainment function E can be expressed as a function of the Richardson

number R^ as

E = E

q exp[£ (Rio - R± )] (23)

where /3 is an empirical constant; E Q and R^ Q , respectively, are the values of

E and R^ at the start of the ceiling-jet.

The convective heat loss term Q /(va) can be correlated to the friction factor

fw by use of the Reynolds' analogy between, the skin friction and the

convective heat transfer:

Pa cp

v

Q

is

v a

s t V s 1

(24)

(25)

8
Kw = (T - TW

)/(T - Ta ) characterizes the thermal conditions of the ceiling. Kw

= 0 can be used for the adiabatic ceiling, and Kw = 1 can be used for the

isothermal ceiling at Tw = Ta. The value of Kw

may be calculated using the

results of other fire models on wall and upper-layer temperatures.

Now, making use of Eqs . (23) -(25), and with proper boundary conditions, Eqs

.

(21) and (22) can be solved numerically. The other quantities of interest can

be calculated by the following relationship

v

v.

' r Ri

'

v. r o Rio J

1/3

exp

Kw S. dr

h

ro

AT

AT,

r 0

2 Ri

r^ Rro

1/3

exp

S t

dr

ro

AC,

AC so

h.

h

V R i

r^ R10 J

1/3

exp

r i

3

KwS. dr

h

ro

(26)

(27)

(28)

where the subscript o denotes that they are the values at the exit of the

stagnation region, i.e. up-stream boundary conditions for the ceiling-jet.

3.2 Parameters

The parameters necessary for calculations in ceiling- jets are the friction

factor fw , and E Q and /3 used in the entrainment function. In order to obtain

the local values of fw , Alpert calculated the thickness of the viscous

sublayer formed between the ceiling and the main portion of ceiling-jet.

However, as pointed out by Alpert himself, the value of fw

has little effect

on the flow properties. In more recent experimental studies on fire induced

ceiling- jets [14,15], values of fw

from 0.02 to 0.04 have been found to match

the ceiling heat- transfer data. Further, You [16] pointed out that if the

stagnation point heat flux is correlated, then fw = 0.025 gives a best fit to

their experimental data on ceiling heat transfer. In this study, fw

is assumed

constant and the value 0.025 will be used.

Alpert assumed that the magnitude of entrainment at the start of the ceilingjet is equal to the plume entrainment constant, and used E Q = 0.12 and /3 =

3.9. If the same assumption is applied to the present model, then E Q becomes

0.167 (0.118 for equivalent Gaussian profile). The value of E Q has a significant effect on the radial variation of smoke concentration and

temperature, especially in the distance range of one or two ceiling-heights.

Further, the initial entrainment need not necessarily be equal to the plume

9
entrainment constant. Therefore, several other values are tried in this study

and their effect will be discussed.

3.3 Computation

Alpert provided the upstream boundary conditions for the ceiling- jet in

relation to the impinging plume characteristics, which can be written with the

present notations as. follows:

ho = bp/J 6 >

vo = ump/'I ^ >

ATo 9 ^mp * 1 + A

2

^so 0 ^smp

1 + A

2

sj 0 = h (i + Je/s ^)- 1

(29)

(30)

The forth- order Runge-Kutta method is also used for numerical computations in

the ceiling-jet.

3.4 Comparison with Experimental Data

In order to evaluate the ceiling- jet model, calculations for homogeneous

environments are carried out and the results are compared with relevant

experimental data on unconfined ceiling- j ets

.

Although minor differences exist due to variations in the plume related

constants, almost identical results to Alpert 's calculations are obtained for

the ceiling- jet thickness, temperature and velocity, on the condition that the

same values are used in E 0 and f) (E 0 = 0.12, /3 = 3.9).

Veldman et al . [14] provided adiabatic ceiling- jet temperature data in their

experimental study on ceiling heat transfer. Fig. 2 shows a comparison of

their data with prediction. With E Q = 0.12, ft = 3.9 and Kw = 0 (adiabatic

condition) , the calculation provides a correlation which fits the data quite

nicely

.

10
The data of Veldman et al . were obtained from laboratory scale experiments;

heat release rates are between 1.17 kW and 1.53 kW and ceiling-heights are

between 0.584 m and 0.813 m. On the other hand, Alpert's experimental

correlations [12] were obtained from large scale tests with heat release rates

ranging from 670 kW to 100 MW and ceiling heights from 4.6 m to 13.7 m. Since

the present study is concerned with the very early stage of an enclosure fire,

it is of great interest that prediction is compared with full scale

experiments of small source strength (in the range of 100 kW)

.

Heskestad [17] provided experimental correlations obtained from fire tests

with heat release rates in the range of 5 kW to 375 kW and the ceiling heights

from 1.2 m to 2.4 m. Shown in Fig. 3 is a comparison between prediction of

this model with Heskestad' s correlation in radial variation of maximum excess > temperatures. The agreement is again quite good. However, in obtaining this

result, the initial magnitude of entrainment EQ was set to 0.05 which is about

half of the value used by Alpert. Fig. 4 shows in the same way a comparison

between the prediction and Heskestad' s correlation in maximum velocities. The

experimental and computational conditions are the same as in Fig. 3. Although

general agreement can be seen, a consistent over-prediction is observed in

this case.

The cause of inaccuracy in the velocity variations is not now understood, but

it may be associated with the assumed Gaussian velocity profile. For the

distance range greater than one ceiling height, the experimental correlation

is very close to the calculated velocity values of equivalent top-hat

profiles, which is 1 /J 2 times less than the maximum velocities, suggesting

decay of the Gaussian profile. This inaccuracy in velocity is not considered

important in calculating the smoke detector's response time, since its

dependence on the gas velocity is generally quite small. Thus, the values E Q = 0.05 and (3 = 3.9 will be used in the following calculations of ceiling-jets in

room fire configurations.

4. FILLING PROCESS AND COAGULATION OF SMOKE

At high concentration, the particle number concentration of smoke changes

quite rapidly by frequent particle collisions and sticking together. This

effect of coagulation is specially important in quantifying the response of

ionization smoke detectors, because their response functions are closely

related to the number concentration of smoke aerosol. In the room

configurations, this effect of coagulation may be estimated separately for the

three regions of concern: plume, ceiling- jet and upper smoke -layer.

11
4.1 Coagulation in Plume and Ceiling- jet

Baum and Mulholland [18] presented a theory for the coagulation in a plume.

According to their analysis, the problem of coagulation in the plume can be

transformed into a problem of homogeneous coagulation by introducing the

characteristic time Tp defined by

dfp A

2 + 1 _ = [b (z) UhjCz) ]

“ 2

, (31)

dz 2

where b(z) and u^z) , respectively are the Gaussian width and the maximum

velocity in the plume as defined in Eqs . (1) and (2) of this paper. The

particle number flux 0 as a function of height can then be given by

<f> / 4> 0 - 1 / [1 + r a0

r] , (32)

where T is the coagulation coefficient and <t> Q is the initial number flux.

By applying the same analysis to the ceiling-jet, the following expression can

be obtained for the characteristic time of ceiling-jet:

dr \

A

2 + 1

[r i(r) vm

2 (r)j1 (33)

Eq . (32) also applies for estimating the effect of coagulation in the ceiling

jet.

12
4.2 Filling Process and Coagulation

The problem of quantifying the effect of coagulation in the upper layer can be

simplified by employing the commonly used assumption of two-layer zone models,

i.e. the properties in the upper layer are uniform. The equations describing

the conservation of mass and number concentration of smoke in the upper layer

can generally be expressed as:

dMs _

. “ ' = msp

” mse

dt

(34)

dN f

dt

nsp

“ nse

“ Kc *

V

Kc = r n„ z dV, (35)

w 0

where Ms and N s , respectively, are the total mass and total particle number of

smoke in the upper layer; mS pand nS prepresent the source terms; mse and nse represent the loss of smoke through the doorway; ns is the local number

concentration of smoke; V is the volume of the upper layer.

With the assumption of homogeneous concentration, the coagulation term can be

simplified as

K,

'V

r ns

2 dV = r N s2

/ V,

0

(36)

and the loss terms can be correlated to the total mass M, and the mass loss

rate me of the gas in the upper layer:

mse

m.

M

M1A s >

nse

m.

M

N £ (37)

13
Substitution of Eqs . (36) -(37) into Eqs . (34) -(35) yields:

dM,

dt

f " nu

= msp

M

M

dN,

dt = nsp

p r

M

N c

m.

M

N t

( 38 )

(39)

Expressing the equations in this form has the advantages that p-P is

relatively insensitive to temperature and the model can be easily combined

with other zone models by which values of M and me as functions of time may be

provided. By further substituting N' = Ns /M in equation 39, N' becomes

insensitive to expansion as the gas is heated.

Given the values of M and me , Eqs. (38) and (39) can be solved numerically.

The mass and number concentration of smoke in the upper layer can then be

obtained by dividing the variables by the upper -layer volume V:

C s = Ms / V,

ns = Ns / V. (40)

4.3 Simple model

We conduct a sample calculation here to make a rough estimate on the effect of

coagulation in practical cases of early stage fire scenarios. As a typical

case of ceiling height and detector location, we choose H = 3.0 m and r = 3.0

m, and the fire size, Q

= 100 kW. Eq

.

(31) can be solved analytically for the

case of uniform environment, which has the form:

f 9

T p = 3 ( + 1) a

2 / 3

f g Q

>. Pa. *3p >

-2/3

(z 0 -V3 . z

‘ I/ 3

) (41)

14
Assuming that the flame height z Q is given by the relation 0.08 = z q /Q 2 / 3

(McCaffrey [9]), and z by Eqs . (30), a value r p = 3.3 s 2 /m3 can be obtained

for z Q = 0.5 m, z = 2 . 84 m and Q = 100 kW. Numerical calculation of Eq . (33)

gives t-; = 2.1 s

2 /m3 at r = 3 . 0 m for the same source strength. The typical

value or coagulation coefficient may be T = 1.0 x 10

m

3 /s (Lee and

Mulholland [19]). Now, substitution of the total characteristic time, r = r p +

Tj into Eq . (32) gives the ratio of initial and local particle number flux

4> / 4> 0 , as illustrated in Table 1.

Table 1 Effect of coagulation in plume and ceiling-jet: calculation.

^o

(particles/s)

<t> / <t> Q number concentration

at r - 3 . 0 m

O (particles/mJ r = 0.0 m(r = r ) p

) r = 3 . 0 m

1.0 x 10 12 0.997 0.995 3.57 x 10 11

1.0 x 10 13 0.968 0.949 3.41 x 10 12

1.0 x 10 14 0.753 0.651 2.34 x 10 13

1.0 x 10 15 0.234 0.157 5.64 x 10 13

1.0 x 10 16 0.030 0.018 6.46 x 10 13

It can be seen from Table 1 that the total effect of coagulation in plume and

ceiling- jet becomes quite important when the particle number flux exceeds the

critical value of 10

1- 4 particles/sec. It also suggests that coagulation in the

plume is more important than in the ceiling-jet, because its initial

concentration is much higher than in the ceiling-jet. The effect of

coagulation becomes even more important when the heat release rate has a smaller value. The critical number flux for a 10 kW source is approximately

10^ 3 particles/sec.

The coagulation in the upper layer also plays a significant role in varying

the smoke detectability, because it changes the environment of plume and

ceiling-jet. If a time scale of 300 sec is assumed for the detector's response

time, then the coagulation effect becomes important even for the number

concentration of 10^- 2 particles/m3

; the sensitivity of the ionization detector

lies in this range.

It is believed that the number flux at the wall, where ceiling- jet ends, is

fed into the upper layer. However, a quantitative model for describing this

effect is not currently available. Therefore, more simplified model will be

used here.

15
In the simple model, the effect of coagulation in plume and ceiling- jet is not

directly calculated, but included in the source term of number concentration

in the upper layer. If the coagulation in the ceiling- jet is neglected, then

the effective particle number flux may be calculated by Eqs . (31) and (32),

assuming the characteristic time r does not change significantly by the

presence of the accumulated upper smoke layer. By numerically integrating Eq

.

(39) using the effective number flux, the average number concentration in the

upper layer can be obtained as a function of time. Now, using the effective

number flux as a source term for the plume, the local number concentration in

the ceiling- jet can be calculated by the same method described in Sec. 2 and 3

of this paper. Note that if the effect of coagulation is neglected, the

equations for the number concentration become identical with the equations for

the mass concentration of smoke. The exact form of equations for the number

concentration can be obtained by replacing the variable C s by ns in Eqs. (6),

(10), (15) and (28).

Based on the models described in Sec. 2, 3 and in this section, a subprogram

has been developed for calculating the temperature, and mass and number

concentration of smoke in ceiling- jet in a two- layer environment. The program

needs input of the upper- layer thickness, average temperature and mass loss

rate of gas through the doorway, which should be calculated by other two -layer

zone models. The current version of the program is designed to be used in

combination with the program FAST. FAST is a comprehensive multi -compartment

fire model whose detailed description can be found in Ref. [2,20]. A complete

list of the subprogram can be found in APPENDIX of this paper.

5. COMPARISON WITH EXPERIMENTS IN ROOM CONFIGURATIONS

Hochiki Corp. of Tokyo, Japan provided temperature and smoke concentration

data collected in their test room for the purpose of evaluating the

performance of smoke detectors. The size of the room is 6.0 m D. x 10.0 m W. x

4.0 m H. The effective ceiling-height can be changed to 3 . 0 m by placing

additional floor-boards 1.0 m above the original floor. The walls are made of

hollow concrete blocks of 25 cm thick, whose effective thickness may be about

5 cm. The ceiling is made of 1.6 cm thick gypsum plaster boards suspended from

the concrete floor slab. The doorways and exhaust vents are sealed during the

tests so that the only effective opening is a 30 cm x 30 cm drain hole located

at the center of the floor. 17 thermocouples and 12 extinction meters are

permanently installed in the test room to measure the radial and vertical

distribution of temperature and smoke concentration. Additionally, one

Measuring Ionization Chamber (MIC) [21] is mounted on the ceiling 3.0 m from

the center. The MIC is a standard measuring device of smoke concentration used

for testing the sensitivity of ionization smoke detectors. The MIC employs the

same physical principle as ionization smoke detectors. The extinction meters

used for the experiments have equivalent sensitivity to UL Standard

specifications [22]. The extinction coefficient K can be obtained by the

following relation from the meter readings:

16
1

K = In (42)

L

where L is the path length, I Q the intensity of incident light, and I the

intensity of transmitted light.

The most generally used combustion materials for testing the performance of

smoke detectors are: cotton, heptane, polyurethane and wood cribs. Among

these, the heptane pool fire has been selected for making comparisons, since

it can be regarded as a constant heat/smoke source with the exception of about

1 min of transient period after ignition.

The size of the pan used for the experiments is 33 cm x 33 cm and 5 cm deep,

which was positioned on the floor at the center of the room. From mass loss

data, the average heat release rate has been found to be approximately 85 kW.

The extinction coefficient and MIC output taken at the point 3.0 m from the

fire axis and 5 cm below the ceiling is used for comparison with prediction of

ceiling-jet calculations.

Using the subprogram listed in APPENDIX in combination with the program FAST,

calculations have been made for the two different ceiling heights, 3.0 and 4.0

m, for which experimental data were available. The heat release rate was

assumed to rise exponentially in the first 1 min after ignition and then

become constant (85 kW) after this transient period. The radiation loss

fraction from the flame zone was assumed to be 0.30, which was subtracted from

the heat release rate to obtain the effective heat flux of the plume. The mass

generation rate of particulate ihp was assumed proportional to the mass flux of

fuel, and the value of 0.0130 (Mulholland [23]) was used for the particle

conversion ratio e = nip/iiif, where ihf is the mass flux of fuel. The particle

number flux was calculated by assuming a volume equivalent diameter Dvof 0.16

^m and the particulate density p s of 2.0 x 10 3 kg/m3

:

nsp = msp / (1/6 * 7r Dv

3

p s ) . (43)

The average particle size Dv

has been determined by the experimental values of

extinction coefficient and MIC output. The exact relation for obtaining the

volume equivalent diameter will be shown in the latter part of this section.

17
There is a well accepted relation between the mass concentration of smoke and

the extinction coefficient during flaming combustion:

K = 8 C s , (44)

where 5 is a dimensional coefficient equal to 7.6 x 10^ m^/kg, which was found

out experimentally by Seader and Einhorn [24], and Lee and Mulholland [19].

This relation is used to interpret the calculated mass concentration into the

extinction coefficient.

The output of MIC is usually expressed by the quantity called Y-value, which

was introduced by Hosemann [3] in his theoretical study on ionization

chambers

Y =

i

(45)

where i Q is the initial ionization current across the positive and negative

electrodes and i is the ionization current in presence of smoke particles.

According to his analysis, the Y-value is proportional to the product of the

particle number concentration and effective particle diameter ns

• d as:

ns d = rj Y, (46)

where rj is the chamber constant defined by

J<*i q

V = 3 — . (47)

C B

In Eq. (47), q is the ion generation rate by the radio-active source, a^_ is

the recombination coefficient of ions and Cg is the Bricard attachment

coefficient equal to 0.3 cm^/sec . The value of I/77 characterizes the

sensitivity of an ionization chamber. In an experimental analysis, Helsper et

al . [25] showed that the value of 1/rj for a Measuring Ionization Chamber is

approximately 3.5 x 10" ^ m^ . This value is used for making comparisons between

the calculated number concentration and the MIC output.

18
Eqs , (44) -(47) have also been used to calculate the average particle size Dv

of heptane smoke from the experimental values of extinction coefficient and

MIC output. Given the instantaneous values of extinction coefficient and Yvalue , the volume equivalent diameter Dv

can be obtained by the following

relation:

Dv "

K

Y <5 *7 ( 1/6 • 7r p s ) "

1/2

(48)

In obtaining the relation of Eq . (48), it has been assumed that .the volume

equivalent diameter Dv

is equal to the effective average diameter d used in

the response function of ionization smoke detectors. The only unknown,

parameter in Eq . (48) is the ratio of extinction coefficient to Y-value, K/Y.

This parameter can be understood as a characteristic quantity influenced by

the size and shape of smoke particles. The exact value of K/Y depends on the

materials and the conditions of combustion.

In order to verify the calculations by FAST, the calculated upper- layer

temperatures are compared with the temperature data taken at the points near

the corners of the room and 5 cm below the ceiling. Fig. 5 and 6 show

comparisons of calculated and measured upper- layer temperatures for two

different ceiling heights. The experimental data shown are the average of

temperatures taken at four different points; the distance of each point from

the fire axis is identical (4.5 m) and its distance from the nearest wall is

approximately 1.0 m. For both cases of ceiling height, the agreement between

the prediction and measurement is quite good. From the fact that the

temperatures have been measured near the ceiling, this may explain why the

upper-layer temperatures are somewhat over-predicted. However, it may not pose

an important effect on the calculations of smoke concentration.

Fig. 7 shows, in the same way, a comparison between the calculated and

measured mass concentration of smoke in terms of extinction coefficient. The

calculated result shown here is the output of FAST, thus representing the

average mass concentration of smoke in the upper layer. The experimental data

are for the point 5 cm below the ceiling and 3.0 m from the fire axis.

Although an qualitative agreement can be seen between the two, there is a significant difference in a quantitative sense. This suggests that the smoke

concentration in the ceiling- jet is much higher than the averaged values in

the upper layer. Similar results have been obtained also in the case of 4.0 m

ceiling height (Fig. 8).

Fig. 9 shows comparisons of ceiling-jet calculations with the experimental

data for the mass concentration of smoke. The experimental data shown here are

of the same location as in Fig. 5, and the calculated results are for the same

radial position in the ceiling-jet. Although there are still small differences

between the predictions and experimental data, the agreement is quite

reasonable taking into account the fact that there is a 10 to 20 % uncertainty

in the validity of Eq . (44) [19].

19
Finally, Fig. 10 shows a comparison of calculated and measured number

concentration of smoke in the ceiling- jet in terms of MIC output. Similar to

the results in Fig. 6, the calculation provides somewhat lower values than the

experimental data. The general agreement is again quite reasonable. The

decreasing rate of change of Y with respect to time is a result of a decreasing number concentration resulting from particle coagulation. It should

be noted however that the assumption to set Dv

and d equal may not be

appropriate, since the exact relation between the two size parameters is not

known for the smoke aerosol from flaming combustion, which usually has quite a complicated agglomerated shape. This leaves an uncertainty in the absolute

value of calculated number concentration.

CONCLUSION

A method is described for calculating the local particulate concentration near

the ceiling in an enclosure fire. The large scale smoke movement is

approximated by integral equations for the plume and ceiling-jet, which

originates in the cold layer and penetrates into the accumulated upper smoke

layer. The effect of coagulation is treated by a simple two- layer zone model.

The key source parameters are the fuel- to-particulate conversion ratio, the

volume equivalent diameter of particulate and the heat release rate. Sample

calculations have been made and comparisons with relevant experimental data

showed a fairly encouraging result.

Direct measurement of number concentration in high density smoke, and a more

detailed analysis on the response of smoke detectors especially for the

particle of agglomerated shape, are desirable for validation and refinement of

the model.

Presently, application of the model is limited to a fire source of flaming

combustion. Refinements must be made before applying this model to smoldering

combustion, which often precedes the flaming combustion in an early stage fire

scenario

.

ACKNOWLEDGEMENT

The author wishes to express his sincere thanks to the staff of the Center for

Fire Research, National Bureau of Standards where this study has been

performed. Special thanks are due to Richard Bukowski , George Mulholland, John

Rockett, Walter Jones and Henri Mitler for their valuable suggestions and

encouragement in the course of this study.

The author also wish to thank Hochiki Corp. of Tokyo, Japan for their support

of this study and permission to use their experimental data.

20
NOMENCLATURE

a AT/Ta b 1/e width of plume [Eq. (1)]

Cg Bricard attachment coefficient

Cp specific heat

C s particulate -mass concentration

AC S particulate-concentration difference from ambient

d effective average diameter of particulate [Eq. (46)]

Dv volume equivalent diameter of particulate [Eq. (48)]

E entrainment function of ceiling- jet

fw ceiling friction factor

g gravitational acceleration

h ceiling- jet thickness for equivalent top-hat profile

hw ceiling heat- transfer coefficient

I light intensity

i ionization current

K extinction coefficient [Eq. (42)]

Kc loss of particles by coagulation [Eq. (36)]

Kw parameter of thermal boundary condition at ceiling [Eq. (25)]

L path length of extinction meter

Z 1/e thickness of ceiling-jet

M total mass of gas in upper layer

m mass flux of gas

Ms total particulate-mass in upper layer

ms particulate -mass flux

Ns total number of particles in upper layer

ns particulate -number concentration

ns particulate -number flux

P r Prandtl number

Q heat release rate

Q parameter of ceiling heat transfer [Eq. (17)]

q ion generation rate

R-l Richardson number

r radial distance from fire axis

S shape parameter of ceiling-jet

S t Stanton number [Eq. (24)]

T absolute temperature

AT temperature difference from ambient

u vertical velocity of plume

V volume of upper layer

v radial velocity of ceiling-jet

Y Y-value of MIC output [Eq. (45)]

y vertical distance measured from ceiling

z vertical distance measured from plume source

21
Greek letters

a plume entrainment constant

ion recombination coefficient

/3 parameter of entrainment [Eq. (23)]

F coagulation coefficient

S constant defined by Eq . (44)

€ fuel- to-particulate conversion ratio

r? chamber constant of ionization chamber

A Gaussian width ratio [Eq. (2)]

p density

r characteristic time of coagulation [Eq. (31), (33)]

r w ceiling shear stress

<j> particle number flux

Superscripts _ spatially averaged quantity

Subscripts

a ambient

e mass flux through doorway

f fuel

i interface

j ceiling- j et

1 lower layer

m maximum value

o initial value

p plume

r reference state

s smoke particulate

u upper layer

w wall

22
REFERENCES

1. Mitler, H. and Emmons, H. , Documentation for CFC V, The Fifth Harvard

Computer Fire Code, Nat. Bur. of Stand. (U.S.), NBS-GCR 81-344, Oct. 1981.

2. Jones, W. W. , A Model for the Transport of Fire, Smoke and Toxic Gases

(FAST), Nat. Bur. of Stand. (U.S.), NBSIR 84-2934, Sept. 1984.

3. Hosemann, J. P., On the Theory of Ionization Chambers with Consideration of

Small-Ion Recombination, Staub- Re inhalt . Luft, Vol. 32, No. 7, pp . 13-17,

1972.

4. Mulholland, G. W. and Liu, B. Y. H., Response of Smoke Detectors to

Monodisperse Aerosols, J. Research Nat. Bur. Stand., Vol. 85, No. 3, pp.223-

238, 1980.

5. Baum, H. R. , Rehm, R. G. and Mulholland, G. W. , Computation of Fire Induced

Flow and Smoke Coagulation, 19th Symposium (International) on Combustion/The

Combustion Inst., pp . 921-931, 1982.

6. Yamauchi, Y. , Numerical Simulations of Smoke Movement and Coagulation,

Proc . 1st International Symposium on Fire Safety Science, pp . 719-728, 1985.

7. Evans, D. D., Calculating Fire Plume Characteristics in a Two Layer

Environment, Nat. Bur. Stand. (U.S.), NBSIR 83-2670, Sept. 1983.

8. Morton, B. R. , Taylor, G. and Turner, J. S., Turbulent Gravitational

Convection from Maintained and Instantaneous Sources, Proc. Roy. Soc., Lond.

A. , Vol. 234, pp. 1-23, 1956.

9. McCaffrey, B. J., Purely Buoyant Diffusion Flames: Some Experimental

Results, Nat. Bur. Stand. (U.S.), NBSIR 79-1910, Oct. 1979.

10. Zukoski, E. E., Development of a Stratified Ceiling Layer in the Early

Stages of a Closed-Room Fire, Fire and Materials, Vol. 2, No. 2, pp . 54-62,

1978.

11. Alpert, R. L. , Turbulent Ceiling-Jet Induced by Large-Scale Fires,

Combustion Science and Technology, Vol. 11, pp. 197-213, 1975.

12. Alpert, R. L. , Calculation of Response Time of Ceiling-Mounted Fire

Detectors, Fire Technology, Aug. 1972.

13. Pickard, R. W. , Hird, D. and Nash, P., The Thermal Testing of HeatSensitive Fire Detectors, Fire Research Note No. 247, Fire Research Station,

Borehamwood, Herts, England, 1957.

14. Veldman, C. C., Kubota, T. and Zukoski, E. E., An Experimental

Investigation of the Heat Transfer from a Buoyant Gas Plume to a Horizontal

Ceiling - Part 1. Unobstructed Ceiling, Nat. Bur. Stand. (U.S.)-, NBS-GCR 77-

97, 1975.

23
15. You, H. Z. and Faeth, G. M. , Ceiling Heat Transfer during Fire Plume and

Fire Impingement, Fire and Materials, Vol.3, No. 3, pp . 140-147, 1979.

16. You, H. Z., An Investigation of Fire-Plume Impingement on a Horizonal

Ceiling 2 -- Impingement and Ceiling-jet Regions, Fire and Materials, Vol. 9,

No. 1, pp . 46-56, 1985.

17. Heskestad, G., Physical Modeling of Fire, J. Fire and Flammability, Vol.

6, pp. 253-273, 1975.

18. Baum, H. R. and Mulholland, G. W. , Coagulation of Smoke Aerosol in a Buoyant Plume, J. Colloid and Interface Science, Vol. 72, No. 1, pp . 1-12,

1979.

19. Lee., T. G. K. and Mulholland, G. W. , Physical Properties of Smokes

Pertinent to Smoke Detector Technology, Nat. Bur. Stand. (U.S.), NBSIR 77-

1312, Nov. 1977.

20. Walton, W. D., Baer, S. R. and Jones, W. W. , User's Guide for FAST, Nat.

Bur. Stand. (U.S.), NBSIR 85-3284, Dec. 1985.

21. Electronikcentralen, Horsholm Denmark, MIC Type EC 23095.

22. UL 268 Standard for Smoke Detectors for Fire Protective Signaling Systems,

Underwrites Laboratories Inc.

23. Mulholland, G. W. , private communication.

24. Seader, J. D. and Einhorn, I. N. , Some Physical, Chemical, Toxicological,

and Physiological Aspects of Fire Smokes, 16th Symposium (International) on

Combustion/The Combustion Inst., pp . 1423-1445, 1976.

25. Helsper, C., Fissan, H., Muggli, J. and Sheidweiler, A., Verification of

Ionization Chamber Theory, Fire Technology, pp . 14-21, Feb. 1983.

24
300

25

Fig. 1

Comparison

of

experimental

data

with

prediction

in

plume

centerline

temperatures.

Solid

and

dotted

lines,

calculation;

circles,

Evans

[7].
1.17-1.53

kW

0.584-0.813 m

H II

CD

CMO 3E

•r*l • x i—

CO cO ew 4J

0) o

•rH e

x> cO cQ E

GO

X!

cO "Of—

e •rH 0) o 'O

CO > c w

•H 0) rH c o O x e H *rH X o o

•H . o. T3 e

0) o

Xf 'H

ex 4J

cO -C i—

u 3

•H On 5 r—I -V CO <rt n3 O <- X> /

cO "d

“<N

0)

C

C

rX

cO X) rH EV.

d

C

CD 'O•X *M E rH

®r->l

O

m

X 00 I

CDex • b X W CM CD (DX <5 4-1 3 03 O X

cO

U

C ^ — O <D w ex i •H E H x o <

CO

U

ex ii E x) o <d * O *r—) I 1 H

!a0 <

CM

C

£>0 *H <f •H CD rX

Cl,

O

—

CO CD CM

^IV

26
5-375

kW

1.2-

2.4 m

OON.CDin^COCNJ'^-O

^IV

27

Fig.

3

Comparison

between

prediction

and

experimental

correlation

in

maxim'

excess

temperatures

of

ceilingjet.

Solid

line,

calculation;

dashed

line,

Heskestad

[17].

AT

m

“

=

AT

m

[C

2p

a 2

g

H

5

/

(T

a Q

2

)

]

1

'

•
5-375

kW

1.2-

2.4 m

28

cd t3 —

cdP

cd

C P

•H CO

0) c ^ O w

*H (1) 4-1 E

cd p—i -

0) CD P C P *H O f—

o

"O 1—I (1) cd E p co C cd

CD E 6

>4

C

0) o

Pj *1-1 X P

a) cd Ec

cd 0) C

•H (0

Co

•rH po

•HE0) P E.

Pj *H / . 1—1 c C O

<D 00

CD P &P CO N

<u <u E *H 3 P C *H O O

CO

O

•rH r—

P <D E- cd >

P-6 P C. o <u - o

txO > G

bO *H * *P a) E o
100

O

COOo

CMooo o

CO o

CO o o

CM o

LLJ

h- ( 0 ) 1

Fig, 5

Comparison

of

calculated

and

measured

upperlayer

temperatures

enclosure.

Solid

line,

calculation^

daslied

line,

experiment. H

3.0m
100

G

•H•

CO 6

<du o GP

cdU II 0)a a e

CD P 4JC P a) 0) £

'•r cd V! rH CD i a p

CD 01 a.

p-G aiC as •H

CD rH pG "C

CO ai cd a

CD 01 S tfli a G£

cd 30 G T

CD i) P cfl cd r— rH 3! G c o H 1—1 cc cd 0 o

Cp

a o P

•r c r- o

CO

a

•rH *r p r-i cd 0, a CO sou 0)33

v£> 0;0 . P

b0

C •H 3 pH CD ( 0 ) 1
33x33

cm*

heptane

pool

to

o o

CO

o

CM

o

(0

UJ

(i-w) y\

31

Fig.

7

Comparison

of

calculated

upperlayer

smoke

concentration

with

measured

ceilingjet

smoke

concentration.

Solid

line,

calculation;

dashed

line,
33x33

cm*

heptane

pool

U-W) >|

32

Fig.

8

Comparison

of

calculated

upperlayer

smoke

concentration

with

measured

ceilingjet

smoke

concentration.

Solid

line,

calculation;

dashed

line,

experiment.

H

=

4.0

m.
33x33

cm

heptane

pool

Q

-

85

kW

T T T

O

o

CO

o

o

CM

CD

tu

o

o

10

m o

(l-W) >1

33

Fig.

9

Comparison

between

calculated

and

measured

ceilingjet

smoke

concentration.

Solid

lines,

calculations;

dashed

lines,

experiments.
IeW/SL+3] S N

o o o o o

00 CD TT c\i

CO

111

H

3mVA-A

34

Fig.

10

Comparison

between

calculated

and

measured

particulate

number

concentration

in

ceiling-jet.

Solid

line,

calculation;

dashed

line,
APPENDIX A. PROGRAM LISTING

C

SUBROUTINE SMKINIT

C

COMMON/SMK/ALFA , BETA , DLTT , EJO , FW , GCP , GRV , GRO , HCRD , KK

,

. LMD2 , MGAS , MGOUT , MSDT , MSTTL , NSDT , NSTTL , PI , PMAS , PR , RCOAG

,

. RDCT , RIO , SP , ST , TO

C

REAL KK , LMD2 , MGAS , MGOUT , MSDT , MSTTL , NSDT , NSTTL

C

ALFA = 0.118

GCP = 1.007

GRV = 9.80665

GRO = 1.2046

LMD2 = 1.157

TO = 293.0

C

BETA =3.9

EJO =0.05

FW =0.025

KK =1.0

PR = 0.7030

RDCT =3.0

SP =1.0

ST = FW * PR**(-2./3.) / 2.

C

COAG = 1.0E-15

PDIA = 0.16E-06

RCOAG = GRO * COAG

RSMK = 2.0E+03

C

DLTT =1.0

PI = 3.141592654

PMAS = 1./6. * PI * (PDIA**3) * RSMK

C

MSTTL =0.0

NSTTL =0.0

C

WRITE (7,700)

C

RETURN

700 FORMAT (" TIME ",5X,"TU ",5X,"DTP ",5X S "DTJ " ,5X,"CSU " ,5X,

. "CSP "

, 5X , "CSJ " , 5X, "NSU ",5X,"NSP ",5X,"NSJ " ,5X,

. "UP " , 5X, "VJ ",/)

END

35
c SUBROUTINE SMKTPORT

C

C COMMON BLOCK AND INITIALIZATION FOR THE "FAST" MODEL

C

C

C NR = MAXIMUM NUMBER OF COMPARTMENTS

C NN = NUMBER OF NODES IN THE SEPARATORS (WALLS AND CEILINGS)

C NT = THE MAXIMUM NUMBER OF EQUATIONS TO BE SOLVED (4 * NR)

C NV = MAXIMUM TIME INTERVALS

C NS = MAXIMUM NUMBER OF SPECIES TO BE TRACKED

C NWALL = NUMBER OF DESCRETE WALL SURFACES (CEILING, UPPER WALL . ..)

C MXSLB = MAXIMUM NUMBER OF DIFFERENT MATERIALS IN A WALL

C

PARAMETER (NR=11, NN = 24 , NV = 21, NS = 10)

PARAMETER (NT = 4*NR+2*NR*NS , MXSLB=3 , NWAL=4)

COMMON/F2C17A/GAMMA , G , SIGM , CP , TA , RA , PREF , RGAS , POFSET , PA , TREF

,

. S S ( NR , NR , 4 ) , SA(NR,NR,4) , AS (NR, NR, 4) , AA ( NR , NR , 4 )

,

. SAU ( NR , NR , 4 ) ,ASL(NR,NR,4) , NEUTRAL (6 ,NR,NR) , MINMAS , NLSPCT

,

. QF(2 , NR) ,QR(2,NR) ,QC(2,NR) , QFR(2 ,NR) ,QFC(2,NR)

,

. EMP(NR) , EMS (NR) , EME(NR) , APS (NR) ,NWV(NR,NR) , RADIUS (NR , NR) , IVERS

,

. DDT , LFMAX , QEXP , LFBO , LFPOS , LFBT , QRADRL , HCOMBA , SWITCH ( 10 )

,

. BFIRED(NV) , AFIRED(NV) , HFIRED(NV) , TFIRED(NV) , TFMAXT , TFIRET , IFIRED

,

. MPRODR(NV , NS ) , MFIRET(NS) , P(NT) , PMIN(NT) , ZMIN(NR)

,

. BW(NR,NR,4) , HH ( NR , NR , 4 )

, HL ( NR , NR , 4 ) ,NW(NR,NR) ,WINDC(NR)

,

. NOPNMX , NRFLOW , HHP ( NR , NR , 4 )

, HLP(NR, NR, 4) , WINDV , WINDRF , WINDPW

,

. DELTAT , LPRINT , NSMAX , LDIAGP , LDIAGO , ITMMAX , MAXINR

,

. 1ST , ITMSTP , ID IAG , DERIV(NT) , STIME , NM1 , N , N2 , N3 , N4

,

. BR(NR) , DR(NR) , HR(NR) , AR(NR) , HRP(NR) ,VR(NR) , HRL(NR)

,

. PW , PM , WC , WH , WO , QP , TE , PPMDV ( 2 , NR , NS )

, TAMB(NR) , RAMB (NR) , PAMB(NR)

,

. FKW (MXSLB , NWAL , NR) , CW (MXSLB , NWAL, NR) , RW(MXSLB , NWAL , NR)

,

. FLW( MXSLB, NWAL, NR) , EPW(NWAL, NR) , NDIV( MXSLB , NWAL, NR) , NSLB(NWAL, NR).

.

, TWJ ( NWAL , NR , NN ), TWE(NWAL,NR) , QSRADW(2 , NR) ,QSCNV(2,NR) , HFLR(NR)

,

. MASS (2 ,NR,NS) , TOXICT(2 ,NR,NS) , PPM(2 ,NR,NS) , ACTIVS(NS),

. NCONFG , NDUMPR , LCOPY , CFILE , DFILE , TITLE , TERMXX

CHARACTER CFILE(5)*17, TITLE(50)*1, DFILE*17

REAL MASS , MPRODR , MFIRET , MINMAS , LC50CF , NEUTRAL

LOGICAL ACTIVS, SWITCH

C

COMMON/SMK/ALFA , BETA , DLTT , EJO , FW , GCP , GRV , GRO , HCRD , KK

,

. LMD2 , MGAS , MGOUT , MSDT , MSTTL , NSDT , NSTTL , PI , PMAS , PR , RCOAG

,

. RDCT , RIO , SP , ST , TO

C

REAL KK , LMD2 , MGAS , MGOUT , MSDT , MSTTL , NSDT , NSTTL

C

DIMENSION WRK(2

)

EXTERNAL FSMK

C

OLDMAS = RAMB(l) * TAMB(l) / P(N+1) * P(N2+1)

C

MGAS = MAX ( OLDMAS , MINMAS)

MGOUT = SS( 1,2,1) + SA( 1,2,1)

MSDT = MFIRET ( 9

)

NSDT = MFIRET ( 9 ) / PMAS

36
c WRK(l) = MSTTL

WRK(2) = NSTTL

C

CALL RKCAL (FSMK, TIME, DLTT , 2, WRK)

C

MSTTL = WRK(l)

NSTTL = WRK(2)

C

RETURN

END

C

37
SUBROUTINE FSMK (F, X, Y)

C

DIMENSION F( 2 ) ,Y(2)

C

COMMON/SMK/ALFA , BETA , DLTT , EJO , FW , GCP , GRV , GRO , HCRD , KK

,

. LMD2 , MGAS , MGOUT , MSDT , MSTTL , NSDT , NSTTL , PI , PMAS , PR , RCOAG

,

. RDCT , RIO , SP , ST , TO

C

REAL KK , LMD2 , MGAS , MGOUT , MSDT , MSTTL , NSDT , NSTTL

C

F(l) = MSDT - Y ( 1

) * (MGOUT/MGAS

)

F( 2

)

= NSDT - RCOAG/MC-AS*(Y(2)**2) - Y(2)*(MGOUT/MGAS)

C

RETURN

END

C

38
SUBROUTINE SMKRESLT (TIME)

C

C COMMON BLOCK AND INITIALIZATION FOR THE "FAST" MODEL

C

C

C NR = MAXIMUM NUMBER OF COMPARTMENTS

C NN = NUMBER OF NODES IN THE SEPARATORS (WALLS AND CEILINGS)

C NT. = THE MAXIMUM NUMBER OF EQUATIONS TO BE SOLVED (4 * NR)

C NV = MAXIMUM TIME INTERVALS

C NS = MAXIMUM NUMBER OF SPECIES TO BE TRACKED

C NWALL = NUMBER OF DESCRETE WALL SURFACES (CEILING, UPPER WALL ...)

C MXSLB = MAXIMUM NUMBER OF DIFFERENT MATERIALS IN A WALL

C

PARAMETER (NR=11, NN = 24, NV = 21, NS = 10)

PARAMETER (NT = 4*NR+2*NR*NS , MXSLB=3 , NWAL=4)

COMMON/F2C17A/GAMMA , G , SIGM , CP , TA , RA , PREF , RGAS , POFSET , PA , TREF

,

. S S ( NR , NR , 4 )

, SA(NR , NR , 4) ,AS(NR,NR,4) , AA ( NR , NR , 4 )

,

. SAU(NR,NR, 4) , ASL(NR , NR , 4) , NEUTRAL (6 , NR , NR) , MINMAS , NLSPCT

,

. QF(2 ,NR) , QR( 2 , NR) ,QC(2,NR) , QFR(2 ,NR) ,QFC(2,NR)

,

. EMP (NR) , EMS (NR) , EME(NR) , APS (NR) ,NWV(NR,NR) , RADIUS (NR, NR) , IVERS

,

. DDT , LFMAX , QEXP , LFBO , LFPOS , LFBT , QRADRL , HCOMBA , SWITCH ( 10 )

,

. BFIRED(NV) , AFIRED(NV) , HFIRED(NV) , TFIRED(NV) , TFMAXT , TFIRET , I FIRED

,

. MPRODR(NV,NS) , MFIRET(NS) , P(NT) , PMIN(NT) , ZMIN(NR)

,

. BW ( NR , NR , 4 )

, HH ( NR , NR , 4 )

, HL ( NR , NR , 4 ), NW(NR , NR) ,WINDC(NR)

,

. NOPNMX , NRFLOW , HHP ( NR , NR , 4 ) ,HLP(NR,NR,4) ,WINDV,WINDRF,WINDPW,

. DELTAT , LPRINT , NSMAX , LDIAGP , LDIAGO , ITMMAX , MAXINR

,

. 1ST , ITMSTP , IDIAG , DERIV(NT) , STIME , NM1 , N , N2 , N3 , N4

,

. BR(NR) , DR(NR) , HR (NR) , AR(NR) , HRP (NR) ,VR(NR) , HRL(NR)

,

. PW , PM , WC , WH , WO , QP , TE , PPMDV ( 2 , NR , NS ), TAMB(NR) , RAMB(NR) , PAMB(NR)

,

. FKW (MXSLB , NWAL , NR) , CW(MXSLB , NWAL, NR) , RW (MXSLB , NWAL , NR)

,

. FLW( MXSLB, NWAL, NR) , EPW(NWAL, NR) , NDIV (MXSLB , NWAL, NR) , NSLB(NWAL, NR)

.

, TWJ (NWAL , NR , NN) , TWE(NWAL,NR) , QSRADW( 2 , NR) ,QSCNV(2,NR) , HFLR(NR)

,

. MASS (2 ,NR,NS) , TOXICT(2 ,NR,NS) , PPM(2 ,NR,NS) , ACTIVS(NS),

. NCONFG , NDUMPR , LCOPY , CFILE , DFILE , TITLE , TERMXX

CHARACTER CFILE(5)*17, TITLE ( 50 )*1, DFILE*17

REAL MASS , MPRODR , MFIRET , MINMAS , LC50CF , NEUTRAL

LOGICAL ACTIVS , SWITCH

C

COMMON/SMK/ALFA , BETA , DLTT , EJO , FW , GCP , GRV , GRO , HCRD , KK

,

. LMD2 , MGAS , MGOUT , MSDT , MSTTL , NSDT , NSTTL , PI , PMAS , PR , RCOAG

,

. RDCT , RIO , SP , ST , TO

COMMON/HFOT/HFOT

C

REAL KK , LMD2 , MGAS , MGOUT , MSDT , MSTTL , NSDT , NSTTL

REAL NSJ , NSP , NSU

C

BP = 0 ,,0

CSP = 0 .,0

DTP = 0 . ,0

NSP = 0 ..0

UP = 0 ,.0

CSJ = 0 , 0

DTJ = 0.,0

39
HJ =0.0

NSJ =0.0

VJ =0.0

C

CSU = MSTTL / P(N2+1)

NSU = NSTTL / P(N2+1)

C

TL = P(N3+1)

TU = P(N+1)

TDF = TU - TL

TW = TWJ (1,1,1)

QPLM = QFC (1,1) + ( -QP+CP*(TE-TL) )*EMP(1)

ZJO = ( HR ( 1 ) - HFOT) / (1. + ( 6 . ** . 5 /5 .

)

* ALFA)

ZU = P(N2+1) / AR( 1

)

ZD = (HCRD - (HRP(l) - ZU) ) * 0.5

ZX = HR( 1) - HFOT - ZU

C

IF (QPLM .LT. 0.01) THEN

CSP = CSU

DTP = TDF

NSP = NSU

GO TO 100

END IF

CALL PLUME2 (CSU , QPLM , MSDT , NSDT , NSU , TDF , TL, ZI , ZJO

,

. BP, CSP, DTP, NSP, UP)

100 CONTINUE

C

IF (UP .LT. 0.01) THEN

CSJ = CSU

DTJ = TDF

NSJ = NSU

GO TO 200

END IF

CALL CLGJET (BP , CSP , CSU , DTP , NSU , NSP , RDCT , TDF , TL, TW , UP

,

. CSJ , DTJ , NSJ , HJ , VJ

)

200 CONTINUE

C

WRITE (7,700) TIME , TU , DTP , DTJ , CSU , CSP , CSJ , NSU , NSP , NSJ , UP , VJ

C

RETURN

700 FORMAT (1P12E10. 3)

END

C

40
SUBROUTINE PLUME2 (CSU , Q , QS ,QN,NSU, TDF , TL, ZI , ZJO

,

. BB , CSM , DTM , NSM , UM)

C

COMMON/SMK/ALFA , BETA , DLTT , EJO , FW , GCP , GRV , GRO , HCRD , KK

,

. LMD2 , MGAS , MGOUT , MSDT , MSTTL , NSDT , NSTTL , PI , PMAS , PR , RCOAG

,

. RDCT , RIO , SP , ST , TO

C

REAL KK , LMD2 , MGAS , MGOUT , MSDT , MSTTL , NSDT , NSTTL

REAL NSM, NSU

C

DIMENSION WRK( 3

)

EXTERNAL FPLM

C

DZ =0.1

TU = TL + TDF

ZF = 0.08 * Q**(2./5.)

C

FI = Q*GRV*(LMD2+1. ) / (PI*GRO*GCP*TO)

FS1 = QS*(LMD2+1 . ) / PI

FN1 = QN*(LMD2+1.) / PI

C

IF (ZI .LT. ZF) GO TO 200

C

Z = ZI

IF (ZI . GT . ZJO) Z = ZJO

C

V = (9./5.*Fl*ALFA)**(l./3.) * Z**(2./3.)

W = 6 . /5 . * ALFA * (9./5.*Fl*ALFA)**(l./3.) * z**(5./3.)

BB = W / V

UM = (V*V) / W

DTM = F1/(LMD2*GRV*W) * TL

CSM = FS1 / (LMD2*W)

NSM = FN1 / (LMD2*W)

IF (Z .GT. ZJO-DZ/2 . ) RETURN

C

F2 = LMD2*GRV*W * (DTM - (LMD2+1 . )/LMD2 * TDF) / TL

FS2 = LMD2*W * (CSM - (LMD2+1 . )/LMD2 * CSU)

FN2 = LMD2*W * (NSM - (LMD2+1 . )/LMD2 * NSU)

C

IF (F2 .LT. 1.0E-04) THEN

BB =0.0

CSM = CSU

DTM = TU

NSM = NSU

UM =0.0

RETURN

END IF

C

WRK(l) = W

WRK(2) = V

WRK(3) = F2

C

100 CALL RKCAL(FPLM , Z , DZ , 3 , WRK)

C

41
Z = Z + DZ

IF (Z .LE. ZJO-DZ/2 . ) GO TO 100

C

W = WRK(l)

V = WRK(2)

GO TO 300

C

200 F2 = FI

FS2 = FS1

FN2 = FN1

DLTZ = ((Fl/F2)**.5 - 1.) * ZI

Z2 = ZJO + DLTZ

V = (9./5.*F2*ALFA)**(l./3.) * Z2**(2./3.)

W = 6 ./ 5 . * ALFA * (9./5.*F2*ALFA)**(l./3.) * Z2**(5./3.)

300 BB = W / V

UM = (V*V) / W

DTM = MAX(F2/(LMD2*GRV*W)*TU+TDF, F1/(LMD2*GRV*W)*TL)

GSM = MAX(FS2/(LMD2*W)+CSU, FS1/(LMD2*W)

)

NSM = MAX(FN2/(LMD2*W)+NSU, FN1/(LMD2*W)

)

C

RETURN

END

C

42
SUBROUTINE FPLM (F, X, Y)

C

COMMON/SMK/ALFA , BETA , DLTT , EJO , FW , GCP , GRV , GRO , HCRD , KK

,

. LMD2 , MGAS , MGOUT , MSDT , MSTTL , NSDT , NSTTL , PI , PMAS , PR , RCOAG

,

. RDCT , RIO , SP , ST , TO

C

DIMENSION F(3) ,Y(3)

C

F(l) = 2. * ALFA * Y(2)

F ( 2 ) = Y( 3 ) * Y(l) / Y(2)**3

F(3) = 0.0

C

RETURN

END

C

43
SUBROUTINE CLGJET (BB , CSP , CSU , DTP , NSU , NSP , RMX, TDF , TL, TW , UM,

. CSJ , DTJ , NSJ , HH , VM)

C

COMMON/SMK/ALFA , BETA , DLTT , EJO , FW , GCP , GRV , GRO , HCRD , KK

,

. LMD2 , MGAS , MGOUT , MSDT , MSTTL , NSDT , NSTTL , PI , PMAS , PR , RCOAG

,

. RDCT , RIO , SP , ST , TO

C

REAL KK , LMD2 , MGAS , MGOUT , MSDT , MSTTL , NSDT , NSTTL

REAL NSA , NSAO , NSJ , NSP , NSU

C

DIMENSION WRK( 3)

EXTERNAL FJET

C

C

C

C

C

C

C

C

DR =0.1

TA = TL + TDF

CSAO = LMD2 / (LMD2+1 .

)

* (CSP - CSU)

DTAO = LMD2 / (LMD2+1 .

)

* (DTP - TDF)

HAO = BB / 6. ** . 5

NSAO = LMD2 / (LMD2+1.

)

* (NSP - NSU)

VAO = UM / 2 . ** .

5

RIO = GRV*(DTAO/TA)*HAO / (VAO*VAO)

RO = 3 . ** .

5

* BB

CSA = CSAO

DTA = DTAO

HA = HAQ

NSA = NSAO

R = RO

RI = RIO

SGM =0.0

VA = VAO

IF (RO .GT. RMX-DR/2 . ) GO TO 200

WRK(l) = HAO

WRK(2) = RIO

WRK( 3 ) = 0.0

100 CALL RKCAL ( FJET, R, DR, 3 ,WRK)

R = R + DR

IF (R .LE. RMX-DR/2.) GO TO 100

HA = WRK(l)

RI = WRK( 2

)

SGM = WRK( 3

200 DMY = EXP(-KK*ST/3.*SGM)

VA = VAO * (RI/RIO * R/R0)**(-l./3.) * DMY

CSA = CSAO * RO*HAO*VAO / (R*HA*VA)

DTA = DTAO * HAO/HA * (RO/R*RO/R*RI/RIO)**( 1 . /3 . ) * DMY*DMY

NSA = NSAO * RO*HAO*VAO / (R*HA*VA)

CSJ = CSA * ( (LMD2+1 . ) /LMD2 ) ** . 5 + CSU

DTJ = DTA * ( (LMD2+1 . ) /LMD2)** . 5 + TDF

44
NSJ = NSA * ((LMD2+1.)/LMD2)**.5 + NSU

HH = HA / (PI/2. )**.5

VM = VA * 2 . ** .

5

C

RETURN

END

C

45
SUBROUTINE FJET (F, X, Y)

COMMON/SMK/ALFA , BETA , DLTT , EJO , FW , GCP , GRV , GRO , HCRD , KK

,

. LMD2 , MGAS , MGOUT , MSDT , MSTTL , NSDT , NSTTL , PI , PMAS , PR , RCOAG

,

. RDCT , RIO , SP , ST , TO

REAL KK , LMD2 , MGAS , MGOUT , MSDT , MSTTL , NSDT , NSTTL

DIMENSION F( 3 ) ,Y(3)

EJ = EJO * EXP (BETA* (RIO -Y( 2) )

)

F(l) = (FW/2. + (2.- SP*Y(2)/2. )*EJ - Y(l)/X - SP*Y(2)*KK*ST/2 .

)

/ (1. - SP*Y(2))

F(2) = (FW/2. + (1.+ SP*Y(2)/2.) * (EJ - KK*ST/3 .

)

- (1.+ 2 . *SP*Y (2 ) ) * Y(1)/(3.*X))

/ ((1.- SP*Y(2) ) * Y(1)/(3.*Y(2)))

F(3) = 1./ Y( 1)

RETURN

END

46
FORmNB«-114A (REV. 11-84)

U.s. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBSIR-88 -3707

2. Performing Organ. Report No, 3. Publication Date

January 1988

4. TITLE AND SUBTITLE

Prediction of Response Time of Smoke Detectors in Enclosure Fires

5. AUTHOR(S) Yukio Yamauchi

6. PERFORMING ORGANIZATION (If joint or other than NBS, see in struction s)

NATIONAL BUREAU OF STANDARDS

U.S. DEPARTMENT OF COMMERCE

GAITHERSBURG, MD 20899

7. Contract/Grant No.

8o Type of Report & Period Covered

Final

9.

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

Hochiki Corporation

246 Tsuruma , Machida- shi

Tokyo 194 JAPAN

10.

SUPPLEMENTARY NOTES

7 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, I f document includes a significant

bi bl iography or literature survey, mention it here)

In order to predict the response time of smoke detectors in enclosure fires, a computational model is developed for calculating the local particulate

concentration near the ceiling. The large scale smoke movement is approximated

by integral equations for plume and ceiling-jet, which originates in the cold

lower layer and penetrates into the accumulated smoke layer in the upper

portion of enclosure. The effect of coagulation, which changes the particle

size distribution, is included to enable predictions of ionization smoke

detector response. This engineering model is designed to be used in

combination with two- layer zone models for obtaining more detailed information

of smoke concentration in the upper layer. Sample calculations have been made

and comparisons with relevant experimental data showed a reasonable agreement

both in the mass concentration and particle number concentration of smoke in

the ceiling-jet.

12.

KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) computer programs; tire models; ionization detectors; particle density

(concentration) ; photoelectric detectors; response time; zone models

13.

AVAILABILITY

|X-

\ UW imi ted

|

I For Official Distribution. Do Not Release to NTIS

Order From Superintendent of Documents, U.S. Government Printing Office, Washington, DC

20402.

jy I O-rtder From National Technical Information Service (NTIS), Springfield, VA 22161

14. NO. OF

PRINTED PAGES

52

15. Price

$13.95

USCOMM-DC 85-6006
‘

.

*

.

'

V