NBSIR 88-3734

Estimating the Environment

and the Response of Sprinkler

Links in Compartment Fires with

Draft Curtains and Fusible

Link-Actuated Ceiling Vents - Part I: Theory

Leonard Y. Cooper

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Gaithersburg, MD 20899

April 1988

Sponsored by:

AAMA Research Foundation

2700 River Road, Suite 118

Des Plaines, iliinois 60018

NBSIR 88-3734

ESTIMATING THE ENVIRONMENT

AND THE RESPONSE OF SPRINKLER

LINKS IN COMPARTMENT FIRES WITH

DRAFT CURTAINS AND FUSIBLE

LINK- ACTUATED CEILING VENTS - PARTI: THEORY

Leonard Y. Cooper

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Gaithersburg, MD 20899

April 1988

Sponsored by:

AAMA Research Foundation

2700 River Road, Suite 118

Des Plaines, Illinois 60018

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

TABLE OF CONTENTS

Page

LIST OF FIGURES iv

ABSTRACT 1

1. Introduction 2

2. The Basic Equations 2

3. Mass Flow and Enthalpy Flow Plus Heat Transfer 5 3.1. Flow to the Upper Layer from the Vents 5 3.2. Flow to the Layer from the Plume and Radiation from the Fire 5

3.3. Flow to the Layer from Below the Curtains 7 3.4. Heat Transfer to the Upper Layer 8

3.4.1. Properties of the Plume in the Upper Layer When

yp ire < y ** 3.4.2. General Properties of the Plume in the Upper Layer . . 11

3.4.3. Computing qHT and the Thermal Response of the

Ceiling 11

3.4. 3.1. Net Heat Transfer Flux to the Ceiling's

Lower Surface 13

3. 4. 3. 2. Net Heat Transfer Flux to Ceiling's Upper

Surface 14

3. 4. 3. 3. Solving for the Thermal Response of the

Ceiling and for qHT 15

4. Actuation of Vents and Sprinklers by Near-Ceiling-Deployed Fusible

Links 16

4.1 Predicting the Thermal Response of the Fusible Links 16

4.2 The Velocity Distribution of the Ceiling Jet 17

4.3 The Temperature Distribution of the Ceiling Jet 18

4.4 Dependence of Open Vent Area on Fusible Link-Actuated Vents . . 20

5. Initiating the Solution of the Model Equations 21

6. Concluding Remarks 22

7. Acknowledgements 22

8. References 23

9. Nomenclature 25

iii
LIST OF FIGURES

Page

Figure 1. Fire in a building space with draft curtains and ceiling vents 29

Figure 2 . The fire and the equivalent source in the lower layer and the

continuation source in the extended upper layer . 30

Figure 3. A plot of dimensionless ceiling jet velocity distribution,

VCJ /VMAX , as a function of z/(0„236) per Eq. (49) 31

Figure 4. Plots of dimensionless ceiling jet temperature distribution,

0, as a function of z/(0.235) per Eq. (52) for cases when 9S is < 0, between 0 and 1, and >0 32

IV
I

ESTIMATING THE ENVIRONMENT AND THE RESPONSE OF SPRINKLER LINKS IN

COMPARTMENT FIRES WITH DRAFT CURTAINS AND FUSIBLE LINK-ACTUATED

CEILING VENTS - PART I: THEORY

Leonard Y. Cooper

ABSTRACT

The physical basis and associated mathematical model for estimating the firegenerated environment and the response of sprinkler links in well -ventilated

compartment fires with draft curtains and fusible link- actuated ceiling vents

is developed. Complete equations and assumptions are presented. Phenomena

taken into account include: the flow dynamics of the upward- driven , buoyant

fire plume; growth of the elevated- temperature smoke layer in the curtained

compartment; the flow of smoke from the layer to the outside though open

ceiling vents; the flow of smoke below curtain partitions to building spaces

adjacent to the curtained space of fire origin; continuation of the fire plume

in the upper layer; heat transfer to the ceiling surface and the thermal

response of the ceiling as a function of radial distance from the point of

plume-ceiling impingement; the velocity and temperature distribution of plume

driven near-ceiling flows and the response of near- ceiling- deployed fusible

links as functions of distance below the ceiling and distance from plume

ceiling impingement.

The theory presented here is the basis of a computer model now under

development which will be used to study parametrically a wide range of

relevant fire scenarios. The results of the parametric study will be

presented in the Part II: Applications portion of this paper.

Keywords: algorithms; building fires; compartment fires; computer models;

fire models; mathematical models; vents; sprinkler response; zone models

.

1
1. Introduction

Consider a space of plan area A defined by ceiling-mounted draft curtains with

a fire of time - dependent energy release rate Q(t)>0 and with open ceiling

vents of total time - dependent area Av (t). The curtained area can be

considered as one of several such spaces in a large building compartment.

Also, by specifying the curtains to be deep enough they can be thought of as simulating the walls of a single uncurtained compartment. This paper presents

the physical basis and associated mathematical model for estimating the firegenerated environment and the response of sprinkler links in curtained

compartment fires with fusible link- actuated ceiling vents. The problem

addressed in this work is similar to the problem addressed in [1]. The theory

presented here is the basis of a computer model which will be developed and

then used to study parametrically a wide range of relevant fire scenarios.

The results of the parametric study will be presented in the Part II:

Applications portion of this paper [2].

The overall building compartment is assumed to have near- floor wall vents

which are large enough to maintain the inside environment, below any near- ceiling smoke layers which may form, at assumed initial outside-ambient

conditions. Figure 1 depicts the generic fire scenario of interest. The

assumption of large near- floor wall vents will require that the modeling be

restricted to conditions where y, the elevation of the smoke layer interface,

is above the floor elevation, i.e., y>0 . The assumption also has important

implications with regard to the cross-ceiling vent pressure differential.

This is the pressure differential which drives elevated- temperature upper

layer smoke through the ceiling vents to the outside. Thus, below the smoke

layer (i.e., from the floor of the facility to the elevation of the smoke

layer interface) the inside- to-outside hydrostatic pressure differential will

be zero, while a positive inside - to- outside pressure differential will exist

at all elevations in the reduced-density smoke layer itself (higher pressure

inside the curtained area, lower pressure in the outside environment), the

maximum differential occurring at the ceiling and across the open ceiling

vents

.

2. The Basic Equations

A two -layer zone -type compartment fire model is used to describe the phenomena

under investigation and, as is typical in such models, the upper smoke layer

is assumed to be uniform in density, , and absolute temperature, Ty

. The

following time - dependent equations describe conservation of energy, mass and

the perfect gas law in the upper smoke layer

Conservation of Energy:

(^CEIL - y)pu Tu ACv ]/dt = qu + pAdy/dt ( 1 )
Conservation of Mass:

d KyCE iL ' y>Pu A ]/ dt = mu ( 2 )

Perfect Gas Law:

Pu/R ~ P/R = constant = Py Ty = Pamb^amb > i • e • > Ty = Iamb ^amb /^u ) (^)

where yCEIL is the elevation of the ceiling above the floor, R = C

p -Cv is the

gas constant, C

p and Cv are the specific heats at constant pressure and

volume, respectively, and p is a constant characteristic pressure, say, pATM

at the floor elevation. In Eq . (1) qy is the net rate of enthalpy flow plus

heat transfer to the upper layer and is made up of flow components: qCURT , from below the curtain, qPLUME > from tde plume, qVENT , from the ceiling vent;

and the component qHT , the total heat transfer rate

% ^CURT + ‘Ip L U M E + ENT + ^HT (4)

In Eq.(2) my is the net rate of mass flow to the upper layer with flow

components: mCURT , from below the curtain, mp

u m

1 ^rom tde plume, and mVENT , from the ceiling vent

mu TMC U R T +mPLUME +mVENT

Using Eq. (3) in Eq . (1) leads to

(5)

dy/dt - qy/(AC

p pAMB TAMB )

if (y = yCEIL and qy

5: 0) or (0 < y < yCEIL and arbitrary qy

)

( 6 )

The first of these conditions will be seen below to be consistent with our

earlier, Q>0 assumption. In particular, we will see that qy>0 initially when

y=yCEIL • Therefore, Eq. (6) will always be applicable.

Using Eq.(l), Eq . (2) can be rewritten as

dPu/dt = (my + pyAdy/dt)/[ (yCEIL - y) A] if 0 < y < yCEIL (7)
Since the denominator of Eq . (7) is zero at the initiation of a fire (i.e.,

when y=yCEIL ) - dpy/dt i s indeterminant at that time. However, as mentioned

above, in our Q>0 scenarios qy>0 generally at t=0 and for this reason a nonzero, initial dy/dt value is predicted by Eq . (6). Then, since dpy/dt of Eq

.

(7) is expected to exist for t>0 it is reasonable to require the numerator of

Eq. (7) to be zero at t=0 , thereby obtaining the initial value of p y

. Doing

this leads to

p 0 (t 0) /3 AMB^AMB (^p mu/Clu ( 8 )

The above value for p u (t=0) can be shown to be identical to the initial

average density of the plume flow at the elevation of ceiling impingement

after all heat transfer from the plume- flow-driven ceiling and wall jets have

been taken into account.

Solving for py requires a definite value for dpy/dt at t=0 , i.e., at y=yCEIL , which is not provided in Eq . (7). To obtain the correct value for this

requires the results of a detailed analysis of the limiting small-t values of

my, dmy/dt, qy , and dqy /dt. Such an analysis can only be carried out within

the context of a specific plume model. To avoid this problem, we will take

arbitrarily the value of dpy/dt to be zero at t=0 . Thus, Eq . (7) becomes

dpy/dt

(my + PyAdy/dt)/[(yCEIL - y)A] if 0 < y < yCEIL

0 if t = 0, i.e.

,

when y = yCEIL

(7')

it is expected that by using a robust numerical integrator and relatively

tight tolerances to solve the governing Eqs . (6) and (7') the solution for p y will recover quickly from an incorrect initial value for dpy/dt and the

correct solution for p y will still be obtained for t>0

.

The basic problem of simulating mathematically the growth and properties of

the upper layer for the generic Figure 1 scenario will require the solution of

the system of Eqs. (6) and (7') for y and p y. Using the computed values of p y at any instant of time, the subsidiary Eq . (3) can then be used to solve for

Ty . To integrate Eqs. (6) and (7') requires algorithms for calculating the

various components of qy and my indicated in Eqs. (4) and (5). These will be

obtained below.

4
3. Mass Flow and Enthalpy Flow Plus Heat Transfer

3.1. Flow to the Upper Layer from the Vents

Conservation of momentum across all open ceiling vents as expressed by

Bernoulli's equation leads to

V - c( 2apCEIL /p0

) 1 ' 2( 9 )

mV ENT = “Pu^V^ = 'Av C( 2 pu APce I L )

1 ^ 2

( 10 )

where V is the average velocity through all open vents, C is the vent flow

coefficient (- 0.68 [3]), and ApCEIL is the cross-vent pressure difference.

From hydrostatics

^Pceil s Pu(y=ycEi L ) Pamb

(

y

=ycEiL ) =

(pamb Pu^s^ycEiL y) (ii)

where g is the acceleration of gravity.

Substituting Eq . (11) in Eq.(10) leads to the desired result for mVENT

mvENT -av c

[

2py (pAMB - Pu^s^ycEiL " y)] 1/2( 12 )

which is equivalent to the equations used to estimate ceiling vent flow rates

in [4] and [5]. Using Eq. (12) we can obtain the desired result for qVENT

PvENT mVENT^p^U (13)

3.2. Flow to the Layer from the Plume and Radiation from the Fire

It is assumed that the mass generation rate of the fire is small compared to mENT ’ t^ie rate °f mass of air entrained into the plume between the fire

elevation, yFIRE , and the layer interface, or compared to other mass flow rate

components of my. It is also assumed that all of the mENT penetrates the

layer interface and enters the upper layer. Thus

5
m‘PLUME mENT (14)

and

‘IPLUME mENT^p^AMB + (-*-'^r^Q (15)

The first term on the right side of Eq. (15) is the enthalpy associated with

mENT and A

r of the second term is the effective fraction of Q assumed to be

radiated isotropically from the fire's combustion zone.

It is assumed that the smoke layer is relatively transparent and that it does

not participate in any significant radiation heat transfer exchanges. In

particular, all of the A

r Q radiation is assumed to be incident on the bounding

surfaces of the compartment. Thus, the last term of Eq . (15) is the net

amount of enthalpy added to the upper layer from the combustion zone and its

buoyantly- driven plume. Flaming fires exhibit A

r

'

s

of 0<A r <0.6, e.g., smaller

values for small methane fires and the higher values for large polystyrene

fires. However, for a hazardous fires involving a wide range of common

groupings of combustibles, it is reasonable to approximate flame radiation by

choosing A r =0.35 [6].

A specific plume entrainment model is required to complete Eqs . (14) -(15) for mPLUME an<5 IpruME • The following estimate for mENT , developed in [7] and

discussed in [8] will be adopted here

TMent/

(

kS/ s ) =

0 if y - yF i re ^ o;

0.0054[(1 - A

r ) (Q/kW) ] (y - yF j r e ) /i-r lame

if 0 < (y - yF x RE ) /i-T LAME <

(16)

0 . 071 [ ( 1 - A

r )(Q/kW)] 1/3

((y - yF !RE -LpLAM E )/m + 0 . 166 [ (1 -

A

r

) (Q/kW )

]

2 / 5

} 5 / 3

1 + e [ (1 - A

r

) (Q/kW )

]

2 ^ 3

* { (y - yF !RE -Lflame)/TM + 0 . 166 [ (1 - A

r

) (Q/kW)

]

2 ^ 5

} ~ 5 ^ 3

if (y - yF iR F )/LF lame — f

6
^lame/dfire = -1.02 + 0 . 249 [ ( 1 -A r )(Q/kW)] 2 ' 5 /(DFIRE /m) (17)

e = 0.0054/0.071 - (0.166) 5/3 = 0.0259168209... - 0.026 (18)

where Lp LAME I s the fire's flame length, DFIRE is the effective diameter of

the fire source (ttD| ire /4 = area of the fire source), and e is chosen so that,

analytically, the value of mENT is exactly continuous at the elevation

y'^F IRE =

^FLAME

3.3. Flow to the Layer from Below the Curtains

If the upper layer interface, y, drops below the elevation of the bottom of

the curtains, yCURT , then there will be mass and enthalpy flows from the upper

layer of the curtained area where the fire is located to adjacent curtained

areas of the overall building compartment. The mass flow rate will be the

result of hydrostatic cross-curtain pressure differentials. Provided adjacent

curtained areas are not yet filled with smoke, this pressure differences

increases linearly from zero at the layer interface to ApCURT at y

=

yC uRT

From hydrostatics

APcURT - Pu(y yCURT ) ' PaMB

(

y ycURT ) (PaMB ' Pu)S(ycURT ’ y) (19)

Using Eq . (19) together with well-known vent flow relations (e.g., Eq. (31) of

[3]) niCURT and the qCURT can be estimated from

"curt

0 if y - yc urt

' (Lcurt/^) [^(ycuRT - y ) 3 P\j ( Pam

b

( 20 ) Pu)

g

] 1/2 if y — yc urt

1cURT mCURT C p'i'u (21)

where is that length of the perimeter of area A which is connected to

other curtained areas of the overall building compartment. For example, if

the curtained area is in one corner of the building compartment, then the

length of its two sides coincident with the walls of the compartment will not

be included in L£ URT . Since the generic vent flow configuration under

consideration here is very long and narrow, a flow coefficient for the vent

flow used in Eq . (20) was taken to be 1.

7
3.4. Heat Transfer to the Upper Layer

As discussed above in section 3.2., when the fire is below the layer interface

the buoyant fire plume rises toward the ceiling and all of its mass and

enthalpy flow, mpLUME and q PLUME , is assumed to be deposited into the upper

layer. Having penetrated the interface, the plume continues to rise toward

the ceiling of the curtained compartment. As it impinges on the ceiling

surface, the plume flow turns and forms a relatively high temperature, high

velocity, turbulent ceiling jet which flows radially outward along the ceiling

and transfers heat to the relatively cool ceiling surface. The ceiling jet is

cooled by convection and the ceiling material is heated indepth by conduction.

The convective heat transfer rate is a strong function of the radial distance

from the point of plume/ceiling impingement, reducing rapidly with increasing

radius. It is dependent also on the characteristics of the plume immediately

upstream of ceiling impingement.

The ceiling jet is blocked eventually by the curtains and/or wall surfaces.

It then turns downward and forms vertical surface flows. In the case of wall

surfaces and very deep curtains, the descent of these flows is stopped

eventually by upward buoyant forces and they mix finally with the upper layer.

Here we assume that the plume/ceiling impingement point is relatively far from

the closest curtain or wall surface, say, greater than a few. fire- to-ceiling

lengths. Under such circumstances the ceiling jet - wall flow interactions

will be relatively weak and, compared to the net rate of heat transfer from

the ceiling and near the plume/ceiling impingement point, the heat transfer to

the upper layer from all vertical surfaces will be relatively small.

Define AC0NV as the fraction of Q which is transferred by convection from the

upper layer gas ceiling jet to the ceiling and wall/curtain surfaces

'-Iht '^convQ ( 22 )

Once the values of AC0NV Q and qHT are determined from a time-dependent

solution to the coupled, ceiling jet/ceiling material, convection/conduction

problem the task of determining an estimate for each component of and qu will be complete.

3.4.1. Properties of the Plume in the Upper Layer When yFIRE < y

Consider times when the elevation of the fire is below the interface, i.e.,

when yFIRE <y.

As the plume flow enters the upper layer the forces of buoyancy which act to

drive the plume toward the ceiling (i.e., as a result of relatively hightemperature, low-density plume gases being submerged in a relatively cool,

high- density ambient environment) are reduced immediately because of the

temperature increase of the upper layer environment over that of the lower

8
ambient. As a result, the continued ascent of the plume gases will be less

vigorous, i.e., at reduced velocity, and of higher temperature than it would

have been in the absence of the layer. Indeed, some of the penetrating plume

flow, will actually be at a lower temperature than T0. The upper layer

buoyant forces on this latter portion of the flow will actually retard and

possibly stop its subsequent rise to the ceiling.

The simple point source plume model of [9] will be used to simulate the plume

flow, first immediately below, or upstream of the interface, and then

throughout the depth of the upper layer itself.

The plume above a Reference -[ 9 ]- type point source of buoyancy, where the source is below the interface, will be equivalent to the plume of our fire (in

the sense of having identical mass and enthalpy flow rates at the interface)

if the point source strength is (l-A r )Q and the elevation of the equivalent

source, yEQ , satisfies

m-PLUME = 0.21pAMB g^ 2 (y - yEQ ) 5/2 QEQ

1 / 3 (23)

In the above, Qeq , a dimensionless measure of the strength of the fire plume

at the interface, is defined as

Qeq (1 " ^Q/ [ ^AMB ^p^AMB

S

1 7

2

(y ' Yeq)

572

] (24)

Note that at an arbitrary instant of time in the simulation of a fire

scenario, mpLUME in Eq . (23) is a known value that would be determined

previously from Eqs . (14) and (16).

Using (23) and (24) we solve for yEQ and Qeq

7eq - y - U1 - )Q/ (Qeq^amb ^p TAMB S

1 ' 2 )}2 1

1

(25)

Qeq " [0.21(1 - \)Q/(Cp TAMB mp LUME )] 3/2 (26)

Now as the plume crosses the interface, the fraction, m* , of mpLUME which is

still buoyant relative to the upper layer environment and presumably continues

to rise to the ceiling, entraining upper layer gases along the way, is

predicted in [10] to be

ft m

0 ; -1 < a < 0

( 1 . 04599a + 0.360391a2 )/(l. + 1.37748a + 0.360391a2

) ; a > 0

(27)

9
where the dimensionless parameter a is defined as

o - (1 - <* + Ct Ql Q

2/3 )/(a - 1) (28)

Q = ^u/^amb ;CT =9.115 (29)

and where Q£ q is the value computed above in Eq . (26). Reference [10]

identifies further the parameters necessary to describe plume flow

continuation in the upper layer (i.e., between y and yCEIL ) according to a Reference- [9] -type point source plume. It has been determined that this plume can be modeled as being driven by a non-radiating buoyant source of strength

Q' located a distance

H ycEiL ^SOURCE > yCEIL yF IRE (30)

below the ceiling in a (downward-) extended upper layer environment of

temperature Ty and density p u . The relevant parameters predicted in [10] are

Q' = (1 - A r )Qom*/(l + a) (31)

ysouRCE = y - (y - yEQ )a3 / 5 m

" 2/5 [(I + a)/o] 1 ' 5 (32)

The fire and the equivalent source in the lower layer and the continuation

source in the upper layer are depicted in Figure 2. Times during a fire

simulation when Eq . (28) predicts a»l are related to states of the fire

environment when the temperature distribution above TAMB of the plume flow, at

the elevation of interface penetration, is predicted to be mostly much larger

than (Tu~Tamb ). Under such circumstances the penetrating plume flow is still

very strongly buoyant as it enters the upper layer. The plume will continue

to rise to the ceiling and to drive ceiling jet convective heat transfer at

rates which differ only slightly (on account of the elevated temperature upper

layer environment) from the heat transfer rates which could occur in the

absence of an upper layer.

Conditions where Eq . (28) predicts o<0 are related to times during a fire

scenario when the temperature of the plume at the elevation of interface

penetration is predicted to be uniformly less than Ty

. Under such

circumstances the penetrating plume flow is nowhere positively (i.e., upward)

buoyant as it enters the upper layer. Thus, while all of this flow is assumed

to enter and mix with the upper layer, it is predicted above that none of it

rises to the ceiling in a coherent plume, i.e., Q'=0. For this reason, when

a<0 the existence of any significant ceiling jet flow is precluded along with

significant convective heat transfer to the ceiling surface or to near- ceiling-deployed fusible links.

10
The above analysis assumes that yFIRE <y- However, at the onset of the fire

scenario yF iRE <y

== )

rCE i l and. since they depend on the unknown initial value of

Tb

. , a, a and m* of Eqs. (27)-(29) are undefined. More critical, however, is

the fact that an internally-consistent value for is required, but not

available, to initiate the solution to the governing equations. The solution

to this problem will be addressed it the last section of this work. For now, we assume that we are dealing with t>0 , when Tb is well-known.

3.4.2. General Properties of the Plume in the Upper Layer

When the fire is below the interface, the results of Eqs. (31') and (32')

allow us to describe the fire-driven plume dynamics in the upper layer

according to the point source plume model of [9]. If the fire is at or above

the interface, i.e., yFIRE ^y> then %lume =0> cipLUME =

^-*-'^r )Q> an<^ we continue

to use the point source model to simulate the upper layer plume flow. All

cases can be treated with the following final versions of Eqs. (31') and (32')

Q' -

(i

(i

A

r )Qam*/(l + <0 if Yfire <

A r)Q lf YfIRE * y

(31')

ysOURCE

y - (y - yF iRE ) a3/5m* 2/5 K 1 + <r)/o] 1,s if yFIRE < y;

yp ire ff yFiRE — y

(32')

where m* , a, and a are calculated from Eqs. (25) -(29).

Q' and ys 0URCE of Eqs. (31') and (32') are the strength and location of a source of buoyancy embedded in the extended upper layer environment. This source is equivalent to the actual fire in the sense that Q' , yE ouRCE > an<^ the

point source plume model of Reference [9] describe the dynamics of the fire

plume once it enters into and passes upward through the upper layer to the

ceiling. Such an upper-layer plume description is required and will be used

below to predict ceiling jet-driven convective heat transfer to the ceiling

surface and to vent- and sprinkler-actuating fusible links.

3.4.3. Computing qHT and the Thermal Response of the Ceiling

When the fire is below the interface and the interface is below the ceiling we will use the method of [11] for calculating the heat transfer from the plumedriven ceiling jet to the ceiling and the thermal response of the ceiling.

11
This method was developed to treat generic, confined-ceiling, room fire

scenarios. As in [11] we will solve the confined ceiling problem by applying

the unconfined ceiling heat transfer solution, developed in [12] and [13], and

applied in [14], to the problem of an Eq . (31') -(32') equivalent upper layer

source in an extended upper layer environment. When the fire is above the

interface, the unconfined ceiling methodology' applies directly.

To use the methods of [11] -[14] we consider an arbitrary instant of time

during the course of fire development. We assume that the temperature

distribution of the ceiling material, T, has been computed up to this time and

is known as a function of distance, Z, measured upward from the bottom surface

of the ceiling, and radial distance, r, measured from the constant point of

plume-ceiling impingement. Then the equivalent, extended-upper- layer

,

unconfined-ceiling flow and heat transfer problem is depicted in Figure 2. It

involves the equivalent Q' heat source of Eq . (31') located a distance H below

the ceiling surface in an extended ambient environment of density p w and

absolute temperature Ty

, where H is found from Eqs . (30) and (32).

The objective here is to estimate the instantaneous convective heat transfer

flux from the upper-layer gas to the lower ceiling surface, q^oNV L (r,t), an<^ the net heat transfer fluxes to the upper and lower surface of the ceiling,

qj](r,t) and qL (r,t), respectively. With this information, the time - dependent

solution for the indepth thermal response of the ceiling material can be

advanced to subsequent times. Also, q^ 0 n v l can integrated over the lower

ceiling surface to obtain the desired instantaneous value for qHT .

In view of assumptions in Section 3.3 on the relatively large distance of the

fire from walls or curtains and on the relatively small contribution of heat

transfer to these vertical surfaces, it is reasonable to carry out a somewhat

simplified calculation for qHT . Thus, qHT will be approximated by the

integral of q,j 0NV l over an effective circular ceiling area AEFF with

diameter DEFF , centered at the point of impingement.

f f qHT = "^coNvQ(t) =

J'Iconv l

( ^ ~

o n v , l ( r ’ rdr (33)

A 0

The value AEFF = 7rD| FF /4 will be taken to be the actual area of the curtained

space, A, plus the portion of the vertical curtain and wall surfaces estimated

to be covered by ceiling jet-driven wall flows. An estimate for this

extended, effective ceiling surface area is obtained in [15] where it is

concluded with some generality that ceiling jet-driven wall flows will

penetrate a distance of approximately 0 . 8H down from the ceiling. Thus

A£ ff = ^^EFf/^ = A + 0.8H(P-Lcurt ) + he UR

T

m^ n [ 0 • , (yc E I L

* UR T )

(34)

where P is the total length of the perimeter of the curtained area.

12
3. 4. 3.1. Net Heat Transfer Flux to the Ceiling's Lower Surface

The net heat transfer flux to the ceiling's lower surface, , is made up of

three components, incident radiation, Trad-fire’ convection, Tconv l> an(3 reradiation, qR ERAD ,L

%AD-FIRE + 0 N V , L + ^RERAD.L (35)

As discussed below in Section 3.2 the radiant energy from the fire, A

r Q, is

assumed to be radiated isotropically from the fire with negligible radiation

absorption and emission from the compartment gases. Therefore

^RAD-FIRE {^rQ/t^^yCEIL '^FIRE ^ ^ ^ + t r/ ( Yc E I L

' TV I R E ^ ] 2

^ 3/2 (36)

The convective heat transfer flux from -the upper- layer gas to the ceiling's

lower surface can be calculated from [12,13]

O N V , L = ( ^A D

" TS , L ) ( 37 )

where Ts L is the absolute temperature of the ceiling's lower surface, TAD , a characteristic gas temperature, is the temperature that would be measured

adjacent to an adiabatic lower ceiling surface, and hL is a heat transfer

coefficient. hL and TAD are given by

8.82ReH

' 1/2 Pr~ 2/3 [1 - (5.0 - 0 . 284ReH

0 • 2

) (r/H)

hL /h = - if 0 < r/H <0.2;

0.283ReH '°

3 Pr' 2/3 (r/H)' 1 • 2 (r/H - 0.0771)/(r/H + 0.279)

if 0.2 < r/H

(38)

(Tad * Tu )/(Tu Q^

2 / 3

) 10.22 - 14 . 9 r/H if 0 < r/H < 0.2;

8.39f(r/H) if 0.2 < r/H

(39)

where

13
f (r/H) = [1 - 1 . 10(r/H)°

•

8 + 0 . 808 (r/H) 1 • 6

]

/

[1 - 1 . 10 ( r/H) 0 • 8 + 2 . 20 (r/H) 1 • 6 + 0 . 690 (r/H) 2 4

] (40)

h - />uCpg

1 ' 2 QS1 ' 3

;

Re„ - g

1 ' 2 H3 '*Q; 1 ' 3 /uu ; (41)

Q5 - Q7[p0 c

p VsH) 1/2 h

2

]

In the above, Pr is the Prandtl number (taken to be 0.7) and uU is the

kinematic viscosity of the upper layer gas which is assumed to have the

properties of air. Also, QB

’ a dimensionless number, is a measure of the

strength of the plume and ReH is a characteristic Reynolds number of the plume

at the elevation of the ceiling.

The following estimate for of air [16] will be used when computing ReH from

Eqs. (41)

uu/ (m2 /s) = [0.04128 (Tu /K) 5/2 10" 7 ]/[ (Ty/K) + 110.4] (42)

As the fire simulation proceeds, the ceiling's lower surface temperature,

T$ L

, initially at TAMB , begins to increase. At all times the lower ceiling

surface is assumed to radiate diffusely to the initially ambient- temperature

floor surface and to exposed surfaces of the building contents . In response

to this radiation, and to the direct radiation from the fire's combustion

zone, the temperature of these surfaces also increase with time. However, for

times of interest here it is assumed that the effective temperature increase

of these floor/contents surfaces are relatively small compared to the

characteristic increases of Ts L

. Accordingly, at a given radial position of

the ceiling's lower surface, the net radiation exchange between the ceiling

and the floor/contents surfaces can be approximated by

iRERAD.L ,

L

+ ^/ e FL00R ' ^ ) (^3)

where a is the Stefan- Boltzmann constant and e L and e FL00R are the effective

emittance/absorptance of the assumed-grey ceiling upper surface and

floor/contents surfaces, respectively, both of which will be taken to be 1.

3. 4. 3. 2. Net Heat Transfer Flux to Ceiling's Upper Surface

It is assumed that the ceiling's upper surface is exposed to a relatively

constant- temperature far-field environment at temperature TAMB . Then the net

heat transfer flux to this surface, qL , is made up of two components,

14
convection, qc 0NV u> and reradiation, qRERAD v

% IcONV.U + ^RERAD ,

U

( 44 )

These can be estimated from

Ic o n v , u

( ta m b

" Ts , u ) (45)

(46)

where Ts B is the absolute temperature of the upper surface of the ceiling, hy

is a heat transfer coefficient, and e FAR and ey are the effective

emittance/absorptance of the assumed-grey far-field and ceiling upper surface,

respectively, both of which will be taken to be 1.

The value for hy to be used here is [17]

3. 4. 3. 3. Solving for the Thermal Response of the Ceiling and for qHT

The temperature of the ceiling material is assumed to be governed by the

Fourier heat conduction equation. By way of the lower ceiling surface

boundary condition, the boundary value problem is coupled to, and must be

solved together with the system of Eqns (6) and (7').

Initially the ceiling is taken to be of uniform temperature, TAMB . The upper

and lower ceiling surfaces are then exposed to the radial- and time -dependent

rates of heat transfer qj] and qB , determined from Eqs . (44) and (35),

respectively. For times of interest here, radial gradients of q[] and q[[ are

assximed to be small enough so that conduction in the ceiling is quasi-onedimensional in space, i.e., T=T(Z,t;r). Thus, the two-dimensional thermal

response for the ceiling can be obtained from the solution to a set of one- dimensional conduction problems for Tn (Z , t)=T(Z , t ; r=rn ) , n=l to NRAD , where

NRAD is the number of discrete radial position required to obtain a sufficiently smooth representation of the overall ceiling temperature

distribution.

The parametric study of [14] for the thermal response of unconfined ceilings

above constant and growing fires indicates generally that changes in T occur smoothly over the radial distances of order one when scaled by H, i.e.,

d ( T/TMAX )/d(r/H)=0(l) , where TMAX ( t)=Ts _ L (r=0 , t)=T(Z=0 , t ; r=0) . Thus, it is

reasonable to expect accurate results for the Eq. (33) integral of the

hy / (W/m2

) = 1.675| (Tamb /K) - (Ts ^/K^ 1 ' 3 (47)

15
qcoNV.L distribution, which depends on the radial distribution of

Ts l

(

r ’ t )=T(Z=0 , t ; r) through Eq . (37), by carrying out the following

procedure: Choose NRAD as several times DEFF / [ 2 (yCE j L

- yFIRE )], where the

latter value is assumed to be > 0.2 (here we choose NRAD as the first integer

equal to or greater than 5DE F

F

/ [ 2 (yc E z

L

- yFIRE )]+2); place one node point at

r = 0 and distribute with uniform separation the remaining NRAD radial node

points at and between r = 0,2(yCEIL -yFIRE ) and r = DEFF /2; solve for the NRAD

temperature distributions Tn and use the lower surface values of these, the

Ts L n ( t ) =

'^s L

( r=rn * c ) =Tn (^=0 , t ; r=rn ) , to compute the corresponding discrete

values of qcoNv . l , n ( t ) =clcoNV , l ( r=rn

> from Ecl- ( 37 >; approximate the q” 0NVL

distribution in r by interpolating between the q,Z 0NV L n (here we will use linear interpolation); and carry out the integration indicated in Eq. (33).

The procedure for solving for the Tn is the same as that used in [14]

.

It

requires the thickness, thermal conductivity and thermal diffusivity of the

ceiling material. The solution to the one -dimensional heat conduction

equation involves an explicit finite difference scheme which uses the

algorithm taken from [18,19]. For a given set of calculations, N<20 equalspaced nodes are positioned at the surfaces and through the thickness of the

ceiling at every radius position rn

. The spacing, 8 Z, of these is selected to

be large enough (based on a maximum time step) to insure stability of the

calculation.

4. Actuation of Vents and Sprinklers by Near-Ceiling-Deployed Fusible Links

It is an objective of this work to simulate conditions in building spaces

where ceiling vents and sprinkler links can be actuated by the responses of

near-ceiling-deployed fusible links. The idea is that during the course of a compartment fire a deployed link will be engulfed by the near-ceiling

convective flow of the elevated- temperature products of combustion and

entrained air of the fire-generated plume. As the fire continues, convective

heating of the link will lead to an increase in its temperature. If and when

its fuse temperature is reached, the device(s) being operated by the link will

be actuated.

The near-ceiling flow engulfing the link is the plume-driven ceiling jet,

refered to earlier, which transfers heat to the lower ceiling surface and is

cooled as it traverses under the ceiling from the point of plume - ceiling

impingement. In the case of relatively smooth-ceiling configurations, assumed

to be representative of the facilities studied in this work, the ceiling jet

flows outward radially from this point of impingement and its gas velocity and

temperature distributions, and TCJ , respectively, are a function of radius

from the impingement point, r, distance below the ceiling, z, and time.

4.1 Predicting the Thermal Response of the Fusible Links

We calculate the thermal response of deployed fusible links, up to their fuse

temperature, TF

, by the convective heating flow model developed in [20] . We

16
assume that a link of interest is positioned at a specified radius from the

impingement point, r = rL

, and distance below the lower ceiling surface, z=z L

. We define TL as the link's assumed-near-uniform temperature. Then

instantaneous changes in TL are determined by

where TCJ L and VCJ L are the values of TCJ and VCJ , respectively, evaluated

local to the link position and where RTI (Response Time Index) , a property of

the link and relative flow orientation, can be measured in the "plunge test"

described in [20, 21]. RTI's for ordinary sprinkler links range from a low

values of 22(m-s) 1/2 for quick operating residential sprinklers, to

375(m-s) 1/2 for slower standard sprinklers [22]. The utility of Eq. (48),

which is shown in [23] to be valid typically through the link fusing process,

is discussed further in [22] and actually used there to predict link response

in a parametric study involving two -layer compartment fire scenarios. Also,

the link response prediction methodology of the latter study was used in the

work of reference [22] which reports favorable comparisons between predicted

and measured link responses in a full-scale, one-room, open-doorway

compartment fire experiment.

To compute TL from Eq. (48) for a different link locations requires estimates

of VCJ L and TCJ L for arbitrary link position, rL and z L

.

4.2 The Velocity Distribution of the Ceiling Jet

Outside of the plume/ceiling impingement stagnation zone, defined

approximately by r/H>0.2, and at a given r, VCJ rises rapidly from zero at the

ceiling's lower surface, z=0 , to a maximum, VMAX , at a distance z=0.235, 5( r)

being the distance below the ceiling where V/VMAX =l/2 [15], In this region

outside the stagnation zone, VCJ can be estimated from [15]

when r/H > 0.2:

dTL /dt - (TCJpL - Tl )Vcjl 1/2 /RTI (48)

(8/7) [z/(0.235)

1/7 (1 - [2/(0.235) ]/8) , 0 < z/(0.235) < 1

MAX (49)

cosh-2

{ (0.23/0. 77)arccosh(2 1/2

) [z/(0. 235) - 1]}, l<z/(0.235)

VM

x

v = 0 • 85 (r/H)

1 • 1

; 5/H = 0 . 10 (r/H) 0 9

; V = g

1

Z H1 ' 2 Q^1 ' 3 (50)

17
where Q* is defined in Eq . (41). VCJ /VMAX per Eq . (49) is plotted in Figure

3.

In the vicinity of near-ceiling-deployed links located inside the stagnation

zone, the fire-driven flow is changing directions from an upward- directed

plume flow to a outward- directed ceiling- j et- type flow. There the flow

velocity local to the link, the velocity which drives the link's convective

heat transfer, involves generally a significant vertical as well as radial

component of velocity. Nevertheless, at such link locations it is reasonable

to continue to approximate the link response by Eq . (48) with VCJ estimated by

Eqs . (49) and (50) and with r/H is set equal to 0.2, i.e.,

when 0 < r/H <0.2

vcj = Vcj( r/H -0.2) (51)

This approximation is adopted here.

4.3 The Temperature Distribution of the Ceiling Jet

Outside of the plume-ceiling impingement stagnation zone, i.e., where r/H>0 , 2

,

and at a given value of r, TCJ rises very rapidly from the temperature of the

ceiling's lower surface Ts L , at z=0 , to a maximum, TMAX , somewhat below the

ceiling surface. We assume that this maximum value of Tc j occurs at the

identical distance below the ceiling as does the maximum of VCJ , i.e., at

z=0.235. Below this elevation, TCJ drops with increasing distance from the

ceiling until it reaches the upper layer temperature, Ty

. In this latter,

outer region of the ceiling jet, the shape of the normalized TCJ distribution,

(Tc j -Ty )/(Tmax -T0 ) , will have the same characteristics as does that of

VCJ /VMAX . Also, since we are dealing with a turbulent boundary flow it is

reasonable to expect that the characteristic thicknesses of the outer region

of both the velocity and temperature distributions are the same, being

dictated there by the distribution of the turbulent eddies.

For the above reasons we approximate the velocity and temperature

distributions as being identical in the outer region of the ceiling jet flow,

0.235<z. In the inner region of the flow, between z=0 and 0.235, we approximate the normalized temperature distribution by a quadratic function of

z/(0.235), requiring TCJ =TS L at z=0 and TC j =^max

>

^C j/dz=0 at z=0.235.

Thus ,

18
when r/H > 0.2:

9S + 2(1 - 0S )[ z/(0 . 235 ) ] - (1 - 0s )[z/(O.235)] 2

,

0 < z/(0 . 235 ) < 1

(52)

MAX > 1 < z/(0.235)

(53)

Note that 0S will be negative when the ceiling surface temperature is less

than the upper layer temperature, for example, relatively early in the fire

when the original ambient- temperature ceiling surface has not yet reached the

average temperature of the growing upper layer. Also, 0S will be greater than

1 when the ceiling surface temperature is greater than TMAX . This is

possible, for example, during times of reduced fire size when the fire's near- ceiling plume temperature is reduced significantly, perhaps temporarily, from

previous values, but the ceiling surface, heated previously to relatively high

temperatures, has not cooled substantially. Plots of 0 per Eq . (52) are presented in Figure 4 for cases when 0S is <0, between 0 and 1, and >0.

In a manner similar to the treatment of VCJ /VMAX , for the purpose of

calculating TL from Eq . (48) we approximate 0S inside the stagnation zone by

the description of Eqs. (52) and (53) with r/H set equal to 0.2, i.e.,

when 0 < r/H < 0.2:

With the radial distribution for Ts L and T0 already calculated up to a particular time of interest, only TMAX is required to complete the Eqs. (52)-

(54) estimate for the ceiling jet temperature distribution. This is obtained

by invoking conservation of energy. Thus, at an arbitrary r outside the

stagnation zone the total rate of radial outflow of enthalpy (relative to the

upper layer temperature) of the ceiling jet is equal to the uniform rate of

enthalpy flow in the upper layer portion of the plume, Q' , less the integral

(from the plume -ceiling impingement point to r) of the flux of convective heat

transfer from the ceiling jet to the ceiling surface, i.e.,

0S = 0S (r/H =0.2) (54)

19
when 0.2 < r/H:

00 •£ 2^Pu Cp( Tcj - Tu) vcj zdz = Q' - 27r/qcoNv L (r,t)rdr - (1 - A’ 0NV )Q' (55)

0 0

Aq o n v

i- s tde fraction of Q' transferred by convection to the ceiling from the

point of ceiling impingement to r, i.e.,

r

AcoNv( r ) = [ 27rJqcoNV L (r, t)rdr]/Q' (56)

0

In Eqs . (55) and (56), Q' has been calculated previously in Eqs . (41). Also,

the integral on the right hand sides of Eqs. (55) and (56) can be calculated

by approximating Iconv , l

(

r >

>

as discussed above in section 3. 4. 3. 3, as a linear function of r between previously calculated values of q^oNV L (r=r , t).

The integral on the left hand side of Eq . (55) is calculated using VCJ of Eqs.

(49) and (50) and TCJ of Eqs. (52) and (53). From this the desired

distribution for TMAX is found finally to be

Umax - T„) - 2.6(1 - A’ OBV )(r/H)-« 0 Q* 2 ' 3 T„ - 0.090(TS

- T„ )

,

0.2 < r/H (57)

The above result together with Eqs. (52) and (53) represent the desired

estimate for TCJ . This and the Eqs. (49) -(51) estimate for VCJ are used to

calculate TL from Eq . (48).

4.4 Dependence of Open Vent Area on Fusible Link-Actuated Vents

As discussed above in section 3.1, the influence of ceiling vent action on the

fire- generated environment is dependent on the active area of the open ceiling

vents, Av

. A variety of basic vent opening design strategies are possible and

it is a major purpose of this work to evaluate these within the context of the

developing fire environment. For example, one of the simplest strategies, the one examined in reference [1], assumes that all vents deployed in the

curtained area of interest are opened by whatever means at the onset of the

fire. In general Av will be time - dependent . To the extent that a strategy of

vent opening is dependent directly on the fusing of any one or several

deployed fusible links, the location of these links and their characteristics

(i.e., likely spacings from plume-ceiling impingement, distance below the

ceiling, and the RTI) and the functional relation between link fusing and Av must be specified. These matters, to be examined in the context of different

solutions to the overall problem, will be addressed in [2].

20
5. Initiating the Solution of the Model Equations

The basic independent variables describing the fire - generated environment are

y and p u . These are determined from the solution to Eqs . (6) and (7'),

respectively. The right hand sides of these latter equations depend on the

time -dependent values of and qy which are computed from Eqs. (4) and (5),

respectively, through use of the variety of algorithms described in this work.

The solution to the governing equations requires the initial value of y and

pu

. The first of these is specified to be yCEIL - The second must be

determined from Eq . (8). But this calls for initial values for my and qy which are not available. Therefore, the determination of p D (t=0) requires the

special considerations described below.

my and qy are determined by evaluating the individual terms on the right hand

sides of Eqs. (4) and (5) as follows.

Since y

_ yC EiL =^’ mvENT =® from Eq. (12) and qv e n

t

from Eq . (13). Since

ycu Rx -y<0, i\. URT =0 from Eq . (20) and qCURT =0 from Eq . (21).

From the initial characteristics of the fire, Q>0 , yFIRE , and DFIRE , compute

Lp lame from Ecl- (17). Using this, calculate (y-yFIRE )/LFL ame • Determine

mPL UME fr°m E<1- (l 6 ). mENT fr°m Etl- (I 4 ). and then ClpLUME =mENT C

p TAMB + ( 1 ' A

r )Q

from Eq . (15) .

At this stage of the calculation, use of Eqs. (8) and (3) together with the

above results leads to the following equation for the unknown T0 (t=0)

Ty(t = 0) - [ mENT <7pl> AMB + (1 ” A r)Q + ClHT]/( mENT^v) = 0 (58)

Except for Tu (t=0), qHT is the only remaining unknown in Eq . (58).

Furthermore, at time t=0 of interest qHT is a function of Tu (t=0), i.e.,

qHT =f(Tu). Since q HT is in the range

-(1 - A

r )Q < qHT < 0 (59)

it follows that the unknown T0 (t=0) is the presumably unique root of Eq . (58)

which lies within the range

Iamb < ^u^t ^) < t mENT

p^amb + (1 - A

r )Q]/(mENT Cv )

The functional relationship between qHT and Ty(t=0) is determined from Eqs.

(25) -(34) and (37) -(42) where Ts L =TAMB in Eq. (37). Thus, for each node

point at r=rn on the lower ceiling surface, o n v l I s computed from

21
Eq . (37) and, as discussed in Che third paragraph of section 3. 4. 3. 3, qHT is

obtained finally from Eq . (33).

6 . Concluding Remarks

The theory presented here is the basis of a computer model now under

development. This will be used to study parametrically a wide range of

relevant fire scenarios. The results of the parametric study will be

presented in Part II of this work [2]: Estimating the Environment and the

Response of Sprinkler Links in Compartment Fires with Draft Curtains and

Fusible Link- Actuated Ceiling Vents - Part II: Application (tentative title)

7 . Acknowledgements

The author acknowledges gratefully the AAMA Research Foundation which

supported this work.

22
8 . References

[1]

. Hinkley, P.L., The Effect of Vents on the Opening of the First

Sprinklers, Fire Safety Journal, 11, pp . 211-225, 1986.

[2]

. Walton, D.W., Cooper, L.Y.

,

and Stroup, D.W., Estimating the Environment

and the Response of Sprinkler Links in Compartment Fires with Draft

Curtains and Fusible Link-Actuated Ceiling Vents - Part II: Application

(tentative title) , to be published.

[3]

. Emmons, H.W., The Flow of Gases Through Vents, Harvard University Home

Fire Project Technical Report No. 75, March 16, 1987.

[4]

. Thomas, P.H., et al, Investigations Into the Flow of Hot Gases in Roof

Venting, Fire Research Technical Paper No. 7, HMSO, London, 1963.

[5]

. Heskestad, G., Smoke Movement and Venting, Fire Safety Journal, 11,

pp77-83, 1986, and Appendix A: Guide for Heat and Smoke Venting, NFPA

204M, National Fire Protection Association, Quincy, MA, 1982.

[6]

, Cooper, L.Y.

,

A Mathematical Model for Estimating Available Safe Egress

Time in Fires, Fire and Materials, 6, 3/4, pp . 135-144, 1982.

[7]

. Heskestad, G., Engineering Relations for Fire Plumes, Fire Safety

Journal, 7, pp . 25-32,1984.

[8]

. Hinkley, P.L., Rates of 'Production' of Hot Gases in Roof Venting

Experiments, Fire Safety Journal, 10, pp . 57-65, 1986.

[9]

. Zukoski , e.e., Kubota, T., and Cetegen, B., Fire Safety Journal, 3, p

107, 1981.

[10]

, Cooper, L.Y.

,

A Buoyant Source in the Lower of Two, Homogeneous, Stably

Stratified Layers, 20th International Symposium on Combustion,

Combustion Institute, pp . 1567-1573, 1984.

[11]

. Cooper, L.Y.

,

Convective Heat Transfer to Ceilings Above Enclosure

Fires, Cooper, L.Y.

,

19th Symposium (International) on Combustion,

Combustion Institute, pp . 933-939 (1982).

[12]

. Cooper, L.Y.

,

Heat Transfer From a Buoyant Plume to an Unconfined

Ceiling, Journal of Heat Transfer, Vol . 104, pp . 446-451, Aug. 1982.

[13]

, Cooper, L.Y. and Woodhouse , A. , The Buoyant Plume-Driven Adiabatic

Ceiling Temperature Revisited, Journal of Heat Transfer, Vol. 108, pp

.

822-826, Nov., 1986.

[14]

. Cooper, L.Y.

,

and Stroup, D.W., Thermal Response of Unconfined Ceilings

Above Growing Fires and the Importance of Convective Heat Transfer,

Journal of Heat Transfer, Vol. 109, pp . 172-178, Feb. 1987.

23
[15]

. Cooper, L.Y. , Ceiling Jet-Driven Wall Flows in Compartment Fires, NBSIR

87-3535, National Bureau of Standards, Gaithersburg, MD , April, 1987. [16]

. Hilsenrath, J., Tables of Thermal Properties of Gases , Circular 564,

[17] National Bureau of Standards, Gaithersburg, MD , Nov. 1955.

. Yousef, W.W. , Tarasuk, J.D., and McKeen, W.J., Free Convection Heat

Transfer From Upward- Facing, Isothermal, Horizontal Surfaces, Journal of

Heat Transfer, Vol. 104, pp . 493-499, Aug. 1982.

[18]

. Emmons, H.W. , The Prediction of Fire in Buildings, 17th Symposium

(International) on Combustion, Combustion Institute, pp . 1101-1111

(1979) .

[19]

, Mitler, H.E., and Emmons, H.W., Documentation for the Fifth Harvard

Computer Fire Code, Home Fire Project Tech. Report 45, Harvard

University, Cambridge, MA. 1981.

[20]

. Heskestad, G. and Smith, H.F., Investigation of a New Sprinkler

Sensitivity Approval Test: The Plunge Test, Technical Report Serial No.

22485, RC 76-T-50, Factory Mutual Research Corporation, Norwood, MA,

1976.

[21]

. Heskestad, G., The Sprinkler Response Time Index (RTI) , Paper RC-81-TP-3

presented at the Technical Conference on Residential Sprinkler Systems,

Factory Mutual Research Corporation, Norwood, MA, April 28-29, 1981.

[22]

. Evans, D.D., Calculating Sprinkler Actuation Times in Compartments, Fire

Safety Journal, 9, pp 147-155, 1985.

[23]

. Evans, D.D., Characterizing the Thermal Response of Fusible Link

Sprinklers, NBSIR 81-2329, National Bureau of Standards, Gaithersburg,

MD, 1981.

[24]

. Cooper, L.Y. and Stroup, D.W., Test Results and Predictions for the

Response of Near-Ceiling Sprinkler Links in Full-Scale Compartment

Fires, NBSIR 87-3663, National Bureau of Standards, Gaithersburg, MD

,

1987, to be presented at the 2nd International Symposium on Fire Safety

Science, Tokyo, June, 1988 and published in the Symposium Proceedings.

24
9 . Nomenclature

A

aeff

Av

C

C

p

cT

Cv

^EFF

DF IRE

g

H

h

^F LAME

* m

^URT

mENT

mP L U M E mU mVENT

N

NRAD

P

plan area of single curtained space

effective area for heat transfer to the extended lower

ceiling surface, 7tD| ff /4

total vent area of curtained space

vent flow coefficient (— 0.68)

specific heat at constant pressure

9.115, dimensionless constant in plume model

specific heat at constant volume, C

p /y

effective diameter of AEFF

effective diameter of fire source (

7

tD^ ire /4 = area of

fire source)

acceleration of gravity

distance below ceiling of equivalent source

characteristic heat transfer coefficient

lower, upper ceiling surface heat transfer coefficient

flame length

fraction of mpLUME which is buoyant relative to Tu

mass flow rate from below curtain to upper layer

rate of plume mass entrainment between fire and layer

interface

mass flow rate of plume at interface

net mass flow rate to upper layer

mass flow rate through ceiling vents to upper layer

number of equal -spaced nodes through the ceiling

number of values of rn length of perimeter of single curtained area

25
Pr Prandtl number, taken to be 0.7

p Pamb at fl° or elevation

Pu > Pamb pressure in upper layer, outside ambient

Q energy release rate of fire

Q

f strength of continuation source in extended upper

layer

Qh dimensionless strength of plume at ceiling

Qeq dimensionless strength of plume at interface

^CONV.L’ ^CONV.U convective heat transfer flux to lower, upper ceiling

surface

^CURT enthalpy flow rate from below curtain to upper layer

Pht heat transfer rate to upper layer

^PLUME enthalpy flow rate of plume at interface

^RAD-FIRE radiation flux incident on lower surface of ceiling

^RERAD.L’ ^RERAD u reradiation flux to lower, upper surface of ceiling

qu net enthalpy flow rate plus heat transfer rate to

upper layer

qL > qii net heat transfer fluxes to upper, lower ceiling

surface

f

qv ENT enthalpy flow rate through ceiling vent to upper layer

qCON V , L , n qcoNV , l ( r=rn >

R gas constant, ( 7 -l)C

p /7 = Cp

- Cv

Re H Reynolds number of plume at ceiling elevation

RTI Response Time Index

r radial distance from point of plume-ceiling

impingement

rL r at link

rn discrete values of r

T absolute temperature of ceiling material

26
Ti AD adiabatic lower ceiling surface temperature

T

J temperature distribution of ceiling jet gas

J , L TCJ at link

tmax

(

Ts L (r=0,t) = T(Z=0,t;r=0)

, L >

, U absolute temperature of lower, upper ceiling surface

, L , n

(

Ts,L<r = rn ,t) = Tn (Z=0,t;r=rn )

TU , ^AMB absolute temperature of upper layer, outside ambient

Tn T(Z,t;r=rn )

t time

V average flow velocity through all open vents

VCJ velocity distribution of ceiling jet gas

VCJ , L VCJ at link

VMAX maximum value of VCJ at a given r

y ’ ycEiL > ycuRT > yp ire elevation of smoke layer interface, ceiling, bottom of

curtain, fire above floor

^SOURCE elevation of plume continuation point source in

extended upper layer above floor

z distance into the ceiling, measured from bottom

surface

z, z

L distance below lower ceiling surface, z at link

a Tu/TAMB

7 ratio of specific heats, C

p /Cv

aPceil cross-vent pressure difference

aPcurt cross-curtain pressure difference

5 value of z where VCJ = VMAX

8Z distance between nodes through the ceiling thickness

e constant, Eq . (18)

27
e L

> e U > € FLOOR ’ e FAP emittance/absorptance of lower, upper, floor, and far

field grey surfaces, all taken to be 1

0 normalized, dimensionless ceiling jet temperature

distribution, (Tc

3

-Tu )/(TMAX -Ty

)

0S 0 at the lower ceiling surface, (Ts L -Ty )/(TMAX -Ty

)

*r fraction of Q radiated from combustion zone

^CONV fraction of Q transferred by convection from upper

layer

^CONV fraction of Q' transferred to the ceiling in a circle

of radius r, and centered at r = 0, Eq. (56).

VU kinematic viscosity of upper layer gas

P\j > Pa m b density of upper layer, outside ambient

a dimensionless variable, Eq . (20)

28
CEILING JET

Figure 1. Fire in a building space with draft curtains and ceiling vents

29
30

Figure

2.

The

fire

and

the

equivalent

source

in

the

lower

layer

and

the

continuation

source

in

the

extended

upper

layer
V /V

CJ MAX

1/2

^Sq": \\\\\\\\\\\\\\^^

Z/C0.23 S)

Figure 3. A plot of dimensionless ceiling jet velocity distribution,

VCJ /VMAX , as a function of z/(0.23S) per Eq. (49)

31
Figure 4. Plots of dimensionless ceiling jet temperature distribution,

0, as a function of z/(0.236) per Eq. (52) for cases when 0S is < 0, between 0 and 1, and > 0

32
NBS-114A (REV. 2-80 ^ejTSr^JEPT. op comm. %

BIBLIOGRAPHIC DATA

SHEET (See in struction s)

1. PUBLICATION OR

RFPORT NO.

NBSIR 88-3734

2. Performing Organ. Report No. 3. Publication Date April 1988

4. TITLE AND SUBTITLE

Estimating the Environment and the Response of Sprinkler Links in Curtained

Compartment Fires with Fusible Link-Actuated Ceiling Vents - Part I: Theory

5. AUTHOR(S)

Leonard Y. Cooper

6. PERFORMING ORGANIZATION (If joint or other than N BS. see instructions) 7. Contract/Grant No. NATIONAL BUREAU OF STANDARDS

U.S. DEPARTMENT OF COMMERCE

GAITHERSBURG, MD 20899

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

AAMA Research Foundation

2700 River Road, Suite 118

Des Plaines, Illinois 60018

8. Type of Report & Period Covered

10.

SUPPLEMENTARY NOTES

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

11.

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

bibliography or literature survey, mention it here)

The physical basis and associated mathematical model for estimating the firegenerated environment and the response of sprinkler links in well-ventilated,

curtained compartment fires with fusible link-actuated ceiling vents is

developed. Complete equations and assumptions are presented. Phenomena taken

into account include: the flow dynamics of the upward-driven, buoyant fire

plume; growth of the elevated-temperature smoke layer in the curtained

compartment; the flow of smoke from the layer to the outside though open

ceiling vents; the flow of smoke below curtain partitions to building spaces

adjacent to the curtained space of fire origin; continuation of the fire plume

in the upper layer; heat transfer to the ceiling surface and the thermal

response of the ceiling as a function of radial distance form the point of

plume-ceiling impingement; the velocity and temperature distribution of plumedriven near-ceiling flows and the response of near-ceiling-deployed fusible

links as functions of distance below the ceiling and istance from plumeceiling impingement.

The theory presented here is the basis of a computer model now under development

which will be used to study parametrically a wide range of relevant fire scenarios.

The results of the parametric study will be presented in the Part II: Application

portion of this paper.

12.

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

algorithms; building fires; compartment fires; computer models; fire models;

mathematical models; vents; sprinkler response: zone models

13.

AVAILABILITY

Unlimited

j | For Official Distribution. Do Not Release to NTIS

H Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402.

[in Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

37

15. Price

$11.95

USCOMM-DC S043-P80
.

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'

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