NBSIR 88-3732

Calculating Flows Through

Vertical Vents in Zone Fire Models

Under Conditions of Arbitrary

Cross-Vent Pressure Difference

Leonard Y. Cooper

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Gaithersburg, MD 20899

May 1988

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NBSIR 88-3732

CALCULATING FLOWS THROUGH

VERTICAL VENTS IN ZONE FIRE MODELS

UNDER CONDITIONS OF ARBITRARY

CROSS-VENT PRESSURE DIFFERENCE

Leonard Y. Cooper

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Gaithersburg, MD 20899

May 1988

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
.
TABUS >OJFj iQONOTTfl J

o.sr.'i Page

LIST OF FIGURES . .. .. .. .. 0+V I. —. y'* io 1. w\.( db\wb«) bna •, w- ,2 Ho- aaol lb • .1 aib^MI

ABSTRACT 1

1. Introduction 2

2. The Two-Layer Description of a Compartment Fire Environment 2

3. Cross-Vent Pressure Differences and the Slabs of the Vent Flow .... 3

4. The Mass Flow Rate Across the Vent at an Arbitrary Elevation ..... 4

5. The Total Slab Flow Rate for Relatively Small Cross -Vent Pressures . . 5

6. Total Slab Flow Rate for Arbitrarily Large Cross -Vent Pressures ... 6

7. Properties of the Penetrating Flow 8

8. References 9

9. Nomenclature 10

iii

v i
LIST OF FIGURES

Pape

Figure 1. Plots of f, w, and (dw/de)/w for 7 — 1.40 11
CALCULATING FLOWS THROUGH VERTICAL VENTS IN ZONE FIRE MODELS UNDER

CONDITIONS OF ARBITRARY CROSS -VENT PRESSURE DIFFERENCE

Leonard Y. Cooper

ABSTRACT

In typical compartment fire scenarios, ratios of cross-vent absolute pressures

are very close to 1. When such is the case, algorithms are available to

predict the resulting cross-vent room-to-room flows. There are, however,

important situations where this pressure condition does not prevail. For

example, in fire scenarios involving relatively small penetrations in

otherwise hermetically-sealed compartments of fire origin, cross -penetration

pressure differences can be of the order of an atmosphere and pressure ratios,

outside-to-inside, can be several tenths less than one. It is important for a versatile compartment fire model to have a capability of predicting vent flows

for the entire range of possible cross-vent pressure conditions. This paper

develops a unified analytic description for flows through uniform-width

vertical vents connecting pairs of two- layer room fire environments under

conditions of arbitrary cross-vent pressure difference. The analysis, which

takes advantage of generally useful modeling approximations, leads to a concise result which is not significantly more complicated than the result for

simple, low-pressure-difference cases.

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

computer programs; fire modeling; high-pressure; pressure differential;

pressure effects; pressure vessels; vent flow; zone modeling.

1
1. Introduction

Algorithms are available to predict cross-vent room-to-room flows in

compartment fire environments (e.g., [1]-[A]). These are restricted generally

to fire scenarios where ratios of cross-vent absolute pressures are close to

1. Although such a pressure condition is typical, there are important

situations where it does not prevail, for example, in fire scenarios involving

relatively small penetrations, holes or cracks, in otherwise hermetically

sealed compartments of fire origin. Reference here is to situations where

penetrations are not so large and vent flows are not so vigorous that they

disrupt fire - generated room environments to the point that otherwise -valid

two -layer zone descriptions of such environments are inadequate.

To fix ideas, consider the following two-space fire scenario which involves

the entire range of cross-vent pressure differences. Let two spaces be formed

by a vertical partition which divides an hermetically-sealed compartment.

Assume initially that the separating partition is also hermetically-sealed and

that the environment of both spaces are at standard temperature and pressure.

Let a fire be initiated in one of the two spaces. As an elevated- temperature

smoke layer grows in thickness below the ceiling, assume that the pressure in

the fire space increases to a level above ambient of the order of one atmosphere (see, for example, the sample problem of [5]), while the

environment in the adjacent space remains at Its initial state. At this time assume further that a vertical crack develops and grows slowly in the

partition, possibly as a result of a fracture in the wall, or of a failure of

the pressure seal of a partition door hatch. When the crack first appears it

is possible to estimate the flow across the crack, from the high-pressure

space-of-fire-origin to the adjacent space, using compressible flow relations

and without regard to relatively insignificant, hydrostatically- generated

components of cross-crack pressure difference. Later, the cross-crack

pressure differences will become small compared to the average absolute

pressure of either of the two spaces. The hydrostatic contribution to these

pressure differences will become significant and dominate eventually the flow.

Indeed, one anticipates the development of one or more neutral planes, i.e.,

elevations of zero cross-crack pressure differential and flow, and

corresponding flow direction reversals as one traverses the crack from top to

bottom

.

When conditions are such that zone- type modeling concepts are useful, it is

important that a versatile compartment fire model have a capability of

predicting vent flows for the entire range of possible cross-vent pressure

conditions (e.g., to simulate the above fire scenario environment). This

paper will develop a unified analytic description of flows through uniformwidth vertical vents connecting pairs of two-layer room fire environments

under conditions of arbitrary cross-vent pressure difference.

2. The Two-Layer Description of a Compartment Fire Environment

In two- layer descriptions of compartment fire phenomena the environment in

each room of a compartment is modeled generally as having an elevatedtemperature upper gas layer and a relatively cool lower gas layer where each

layer is assumed to be uniform in temperature, density, and composition.

2
Define

5 - pg(y-y)/p(y) ( 1 )

where y is an arbitrary elevation within a room layer of interest above a datum elevation, y is an arbitrary fixed value of y within the room, p is the

density in the layer, g is the acceleration of gravity, and p(y) is absolute

pressure. In fire scenarios it is common for

\ 6 \ « 1 ( 2 )

The above modeling assumptions and the approximation of Eq. (2) along with the

momentum equation, i.e., hydrostatics, and the perfect gas law lead to

P(y) “ p(y) [ 1+0(5) ] ~ p(y) (3)

and for this reason, the Eq. (3) approximation is adopted implicitly in twolayer zone models. It can also invoked explicitly to justify simplifications

of pressure -work terms in the energy equation for rooms when such terms are non-negligible . For this latter application, however, the Eq. (3)

approximation has not been used widely in practice.

To compute flow through vents, estimates of cross-vent pressure differences

are required. But such differences are typically of the order of magnitude of

the errors in the approximation of Eq. (3). When such is the case, Eq . (3) is

usually too crude for the vent flow calculation and a detailed hydrostatic

pressure calculation

Pi (y) - Pi (yi )-gJpdy (4)

is usually needed, where in each room i , yl is a fixed elevation, say at the

floor

.

Eq. (4) together with the uniform layer density approximation leads to the

result that in each room, p(y) can be approximated by a linear function of

elevation, where the slope of the pressure - elevation curve , determined by the

density, changes only at the elevation of the interface separating the two

layers

.

3. Cross -Vent Pressure Differences and the Slabs of the Vent Flow

Consider an opening or vent in a common vertical wall segment joining two

rooms, designated as rooms 1 and 2 , of a multi-room compartment. Flow through

the vent will be driven at every elevation of the vent by the value of the

pressure - difference function

Api 2 (y) - Pi (y)-p2 (y) (5)

where and p 2 are the absolute pressure distributions in rooms 1 and 2 , respectively. The sign of Ap 12 will determine the direction of the flow at

any elevation, which will be from higher to lower pressure.

3
In view of the description of the general intra-room pressure distribution,

i.e., two piecewise-continuous linear functions of elevation, it can be shown,

e.g., geometrically, that Ap 12 will be described generally by three piecewisecontinuous linear functions (see, for example, Figures 3 and 5 of [1] and

Figures 5 and 6 of [2]). Also, each of the three pieces can change sign,

i.e., the flow can change direction, once in its elevation interval of

applicability. The result of all this is that at any instant of time during a developing fire scenario the flow across a vertical vent will involve from one

to six, stacked, horizontal slabs of cross-vent flow.

Number the flow slabs through a vent from the lowest (N-l) to the highest (N - NSLAB < 6) , and designate the top and bottom elevation of a particular slab N

relative to the datum elevation as yN and yN+1 , respectively. Attention here

is confined to the problem of estimating the rate of cross-vent flow in an arbitrary slab.

Between the yN and yN+1 bounds of each slab the density and temperature of

the flow's source and the direction of the flow are fixed. Also, throughout

the thickness of a slab, p x

, p 2

, and Ap 12 are each simple linear functions of

elevation. Ap 12 can be zero at the bottom or top of a slab, and it can be

identically zero, i.e., a slab with zero cross-vent flow. But Ap 12 cannot

change signs in the slab.

We define Ap and Ap , and, with complete generality we number the two rooms so that the flow in the slab is from room 1 to room 2.

Ap = p 1 -p 2 Ap(y) s APl 2 (y) - Pi (y) -p2 (y) £ o

( 6 )

where p x and p 2 are the uniform densities in rooms 1 and 2, respectively.

Define ApN , pN and ApN+1 , pN+1 as the values of Ap , p x

, at y - yN and yN+1 , respectively. Then Eq . (4) leads to

Pi (y) - Pn -P ig(y-yN )

Ap (y) - ApN- Apg(y-yN

)

Pi(y) - p n+ i

(

y-yN )/(yN+i -y^-PN (y*yn+i >/<yH+i -y^

Ap (y ) - APn+ 1 (y-yN )/(yN+1 -yN )-ApN (y-yN+ 1 )/(yN+ 1 -yN

)

(7)

(8)

(9)

( 10 )

through the slab.

4. The Mass Flow Rate Across the Vent at an Arbitrary Elevation

Adopting the traditional one-dimensional approximation and assuming isentropic

flow, the mass flow rate of a compressible gas across a vent or nozzle per

unit area, m"(y), is given by [6]

4
ra"(y) = C(2p 1 Ap) 1 / 2 w(x) - C(p x p x )

1 ' 2 f (x) ( 11 )

where

W(x) - f(x)/[ 2 (l-x )

]

1/2

( 12 )

r {

[

27/( 7

- 1) ]x2/i [1-x <7 ” 1)/7

] } 1/2 for x > [ 2 /( 7+l) ]

-r / ( tt - 1 ) f(x) =

1 (13)

l ( 7 [ 2 /( 7+l )]C'r + i )/( 7- 1 ) ) i /2 for x <; [ 2/( 7+l)]'T/(7 ' 1)

X “

P2 /P 1 “ 1’AP/Pi (14)

and where 7 is the ratio of specific heats and C is a flow coefficient which

depends generally on Reynolds number and x. For perfectly isentropic flow, C = 1. For well -designed nozzles, C can be very close to 1, of the order of

0.99, when flows have "pipe" Reynolds numbers of 10 6 or more ([ 6 ], p.99). For

sharp-edged orifices, C - C(x) and is approximately linear with x, ranging

from 0.6 at x = 1 to 0.85 at x - 0 (see Figure 4.17, p . 100 of [ 6 ]). For

orifice-like openings typical of vents in compartment fires C ~ 0.68 for x close to 1 [1]. All these results suggest that it is reasonable to

approximate C(x) for vents, holes or slits in compartment fires by

C(x) - 0 . 68x + 0 . 85 ( 1 -x) - 0.85 - 0.17x ; 0 ^ x <; 1 (15)

unless a more precise description of C is known for some particular vent

configuration of interest.

For air, 7 = 1.40 and the change from the first to the second of Eq. (13) occurs at the choked flow condition

x - [ 2 /( 7+l)

]

t / (

t

1 > « 0.528 (16)

f(x) and w(x) are plotted in Figure 1 for 7 - 1.40.

5. The Total Slab Flow Rate for Relatively Small Cross-Vent Pressures

Define

e(y) - Ap/pj “ 1 - x > 0 (17)

and note from Eqs . (11) and (16) that for air and for a specified p x

, p l condition, reductions of p 2 do not change m"(y) once t exceeds the choked

value 0.472.

In most common compartment fire scenarios

I Ap/Pi | “ | e(y) | « 1 (18)

through the depth of any flow slab. From Eqs. (12) -(14) a small- t expansion

of w leads to

5
lim w(x) = lim w(l-e) = 1- [ 3/(4y) ] £ (y)+0( (IS)

£

2

) x-*T £~*0

From Eq . (11) it therefore follows that

lim m" (y) = C(2p x Ap) 1 ' 2

{ 1- [ 3/(47 )

] £ (y)+0(

£

2

) ) (20)

£~*0

Approximating the above by the leading term in the expansion results in the

commonly useful expression for the total mass flow rate, ra, of the slab

through a vent of uniform width, b,

lim m = bC(2 Pl )

1 /

2

/(Ap) 1 1

2

dy (21) max | £ j -*0

where max |

e

| is the maximum value of e in the slab and the integral is over

the thickness of the slab.

Using Eq . (10), the integral of Eq . (21) can be evaluated to yield [1]

lim m - bC[ (8 Pl ) 1/2 /3] (yN+1 -yN ) [ ( |Ap N |+|ApN ApN+1 | 1/2 +|ApN+1 | )/ max | £ | -»0

<I AP»

I

1,2+

I aPk.i l 1/2

)] (22)

which is equivalent completely to mass flow rate calculations in [2] -[4]. Use

of Eq . (22) requires max |

e

| to be small. Since Ap and p : are both linear and

monotonic in y, then £ is also monotonic in y. Thus, the utility of Eq. (22) can be established by showing that

max | Ap/p x | = max | £ (y) | - max|£ N ,£ N+1 | - max | ApN /pN

, ApN

+

x /pN +

x

| « 1 (23)

where £ N and £ N + 1

, the values of e at yN and yH + 1, respectively, can be

computed with the use of Eqs. (9) and (10).

6. Total Slab Flow Rate for Arbitrarily Large Cross-Vent Pressures

There are important situations where £ is not small, for example, in fire

scenarios involving relatively small penetrations connecting otherwise

hermetically sealed compartment spaces. In such cases the basic two-layer

zone model may still be useful and the approximations of Eqs. (2) and (3) will

usually still be valid. In general, whether or not e is small, Eq . (1) and

the definition of e lead to

x (y ) - 1 - £ - p 2 /Pl - x(y) [ 1+0(5 ) ] - (1-e) [1+0(3) ] - l-'«+0(3) (24)

where

6 “ max(6 x

, S

2 ) ; « 1 ; 6

2 «1

( 25 )
and where 8

1 and 62 are values of the 6 of Eq. (1) for rooms 1 and 2,

respectively, y is confined to the range of elevations in the vent slab of

interest, and y, also confined to the slab, is the same fixed elevation for

both rooms 1 and 2.

From Eq. (24) it follows that

e = e+0(5 ) (26)

uniformly through the thickness of a slab. Noting that w is an analytic

function of e near the real € axis in the range

0 < £ < 1 -

[ 2/(-y-+l) ] + / C -r - 1

) (27)

€ is assumed to be in this range and w is expanded at € — e within the small

region | e -e | = 0(6)

lim w = w| + dw/de | 0(6) (28)

6-*0 € = £ £ = €

Since € in any slab is monotonically increasing or decreasing with y we can

insure that € > 0 by choosing y to be at the mid-elevation of the slab of

interest. Thus, using Eqs . (9), (10), and (17)

€ - « (y=y=[yN +yN+ i ]/2) - 1 - x - (ApN +ApN+1 )/(pN +pN+1 ) (29)

Since w is never zero

lira w = w| {1 + [(dw/de)/w]| 0(6)} (30)

6-+0 e-e €“€

(dw/d£)/w was computed from Eqs. (12) -(14) and is plotted in Figure 1. As can

be seen, the absolute value of this term never exceeds 1.04. It is concluded,

therefore, that for arbitrary e the estimate

w - w| [1+0(6)] “ w| - w ( X“X“1 - £ ) - w (31)

£“£ £-£

is valid uniformly in the entire slab. Also, it is evident from the above

that Eq . (31) can be extended to situations where € exceeds the Eq. (27)

range

.

Just as the above analysis of w lead to Eq . (31), a similar analysis of C

would lead to

C - C | [1+0(6)] « C| - C(x-x- 1 - € ) - C (32)

£“£ £-€

Eqs. (31) and (32) are now used in Eq . (11) which is then integrated over the

slab to obtain

m - bCw(2p x ) 1/2 [J(Ap) 1/2 dy]

7

(33)
As was done earlier below Eq . (21), the above integral is evaluated to yieldthe final unified result for m, valid for arbitrary e

.

m - bCu[ (Sp l ) 1,2 /3] (y„ tl -y„) [ ( |Ap„ I + I *Pn AP» t 1 | 1,2 +|ApBtl | )/

(|apN l 1/2 +|ApH+1 | 1/2

)] (34)

where C and w are defined in Eqs . (31) and (32) and can be computed with the use of Eqs. (12) -(15) and (29). To compute ra numerically from Eq . (34), the

limit of Eq . (19) in small e situations is useful in the evaluation of w from

Eqs. (12) - (14)

.

It is noteworthy that when c - 1 - x - 0, i.e., C - C x NC0MPRESS i ble anc* w - 1,

Eq . (34) is identical to the small-cross-vent-pressure-difference flow

equation of Eq . (22). Thus, a simple interpertation of Eq . (34) is that when

compressibility effects are arbitrarily strong or weak, the compartment firegenerated flow rate of a slab of flow, originating from a uniform stagnation

source and passing through a constant-width vertical vent, can be computed by

ignoring elevation-related differences of compressibility in the slab and by

using the characteristic mass flux for compressible nozzle/orifice flow which

corresponds to the mid-slab cross-vent pressure difference.

7. Properties of the Penetrating Flow

In order to use Eq . (34) in any particular room fire model it is necessary to

know the properties of the penetrating jet as it enters room 2. With such

information in hand it would then be possible to invoke the model's adopted

rule for dividing and depositing this incoming flow into the room's upper and

lower layer. The desired result is obtained by observing that the flow across

the penetration is basically of constant total enthalpy. Thus, assuming

deceleration to negligible velocity prior to mixing in the receiving room, the

temperature T2 will be identical to T x

, the stagnation temperature in the

originating room.

T2 = T x (35)

Using this result, the density p 2 can be obtained from the perfect gas law

P 2 “ P i (P 2 /Pi ) ~ P i x < 36 )

For the most common fire scenarios where cross-vent pressure ratios, x, are close to 1, the above leads to the correct result that the properties of the

penetrating flow are identical to the properties of the jet's source ambient

environment in room 1.

8
8. References

[1] Emmons, H.W., The Flow of Gases Through Vents, Home Fire Project Technical Report 75, Division of Applied Sciences, Harvard University, March

16, 1987.

[2] Zukoski, E.E., and Kubota, T., Two-Layer Modeling of Smoke Movement in

Building Fires, Fire and Materials, Vol. 4, No. 1, 1980.

[3] Tanaka, T., A Model of Multiroom Fire Spread, NBSIR 83-2718, National

Bureau of Standards, Gaithersburg, MD, 1983.

[4] Jones, W.W.

,

and Bodart, X., Buoyancy Driven Flow as the Forcing

Function of Smoke Transport Models, NBSIR 86-3329, National Bureau of

Standards, Gaithersburg, MD, 1986.

[5] Cooper, L.Y.

,

and Forney, G.P., Fire in a Room with a Hole: A Prototype

Application of the Consolidated Compartment Fire Model (CCFM) Computer

Code, Proceedings of the Combined 1987 Fall Technical Meeting of the

Eastern Section of the Combustion Institute and Annual CFR Research

Conference, National Bureau of Standards, Gaithersburg, MD, 1987.

[6] Shapiro, A.H., The Dynamics and Thermodynamics of Compressible Fluid

Flow: Volume 1 . Roland Press, 1953.

9
9. Nomenclature

;

b width of the vent

C a flow coefficient

f Eq. (13)

g acceleration of gravity

m" mass flow rate per unit area

m mass flow rate

P absolute pressure

T temperature

w Eq. (12)

X P2 /Pi> Eq. (14)

y elevation above a datum elevation

y a specified value of y

h a specified value of y in room i

7 ratio of specific heats

Ap 12 , Ap Pi (y)

p2 (y)

Ap Pi - Pz

6 Eq. (2)

6 Eq. (23)

e 1 -P 2 /Pi> Eq. (16)

P density

subscripts

i value in room number i, i - 1 on the high-pressure side of a flow

slab

N, N+l value at bottom, top of flow slab N

superscripts

value at y

10
Figure 1. Plots of f, w, and (dw/de)/w for 7 - 1.40

11
N BS-1 14A (REV. 2-6 U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBSIR-88/37.32

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

4. TITLE AND SUBTITLE

Calculating Flows Through Vertical Vents in Zone Fire Models Under Conditions of

Arbitrary Cross-Vent Pressure Difference

5. AUTHOR(S)

Leonard Y. Cooper

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

NATIONAL BUREAU OF STANDARDS

U.S. DEPARTMENT OF COMMERCE

GAITHERSBURG, MD 20899

7. Contract/Grant No.

I. Type of Report & Period Covered

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

10. SUPPLEMENTARY NOTES

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

11. ABST RACT (A 200-word or less factual summary of most significant In formation. If document Includes a significant

bibliography or literature survey, mention it here)

In typical compartment fire scenarios, ratios of cross-vent absolute pressures are close to 1. When such is the case, algorithms are available to predict the resulting

cross-vent room-to-room flows. There are, however, important situations where this

pressure condition does not prevail, for example, in fire scenarios involving

relatively small penetrations in otherwise hermetically-sealed compartments of fire

origin. It is important for a versatile compartment fire model have a capability of

predicting vent flows for the entire range of possible cross-vent pressure conditions.

This paper develops a unified analytic description for flows through vertical vents

between pairs of two-layer room fire environments under conditions of arbitrary cross- vent pressure difference. The analysis, which takes advantage of generally useful

modeling approximations, leads to a concise result which is not significantly more complicated than the result for simple, low-pressure-difference cases.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) algorithms; building fires; compartment fires; computer models; computer programs;

fire modeling; high-pressure; pressure differential; pressure effects; pressure

vessels; vent tlow; zone modeling _________

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