NBSIR 87-3557

An Analysis of the Pressure Rise in

Sealed Maqazine Fires

Howard R. Baum

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Gaithersburg, MD 20899

November 1987

i part by:

100 apons Center

• U 5 6 <e, California

87-3557

1987

C.2

NBSIR 87-3557

AN ANALYSIS OF THE PRESSURE RISE IN

SEALED MAGAZINE FIRES

Research Information Center

National bureau of Standards

GaiiiiersJburg, Maryland 20899

Dc/ DO

. KH)*

Cl^

Howard R. Baum

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Gaithersburg, MD 20899

November 1987

Supported in part by:

Navy Weapons Center

China Lake, California

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

TABLE OF CONTENTS

List of Figures . .. Abstract ..... ......

1 * INTRODUCTION ........

2. MATHEMATICAL MODEL

2.X Solution ................

3. RESULTS AND DISCUSSION

4. ACKNOWLEDGMENTS ..............

3

REFERENCES ...................

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1

1

2

5

8

10

10

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I

.
LIST OF FIGURES

Page

Figure la. Dimensionless layer height Z/H as function of time in

seconds for configuration 1. x is the fraction of the

chemical heat release lost from the gas by radiation ........ 12

Figure lb. Pressure in atmospheres as function of time in seconds

for Configuration 1 e»©»*c«eoc«eeoo.»©eeee*oe 13

Figure lc. Ratio of upper layer temperature to ambient temperature

as function of time in seconds for configuration 1 14

Figure 2a. Dimensionless layer height Z/H as function of time in

seconds for configuration 2. x is the fraction of the

chemical heat release lost from the gas by radiation ........ 15

Figure 2b. Pressure in atmospheres as function of time in seconds

for configuration 2 16

Figure 2c. Ratio of upper layer temperature to ambient temperature

as function of time in seconds for configuration 2 17

Figure 3a. Dimensionless layer height Z/H as function of time in

seconds for configuration 3. x is the fraction of the

chemical heat release lost from the gas by radiation 18

Figure 3b. Pressure in atmospheres as function of time in seconds

for configuration 3 ......................................... 19

Figure 3c. Ratio of upper layer temperature to ambient temperature

as function of time in seconds for configuration 3 20

Figure 4a. Dimensionless layer height Z/H as function of time in

seconds for configuration 4. x is the fraction of the

chemical heat release lost from the gas by radiation ........ 21

Figure 4b. Pressure in atmospheres as function of time in seconds

for configuration 4 22

Figure 4c. Ratio of upper layer temperature to ambient temperature

as function of time in seconds for configuration 4 23

iv-
: ' V ’ . ' - . '
AN ANALYSIS OF THE PRESSURE RISE IN SEALED MAGAZINE FIRES

Howard R. Baum

Abstract

A mathematical model for the pressure and temperature rise in sealed ship

magazines caused by propellant fires is developed. The model is a simplified

version of several existing zone fire models, modified to consider sealed

compartments. The equations describing the model are solved exactly. Sample

results are shown for a variety of magazine geometries.

1 . INTRODUCTION

A propellant fire in a sealed magazine possesses two features not present

in most building compartment fires. First, the propellant contains its own

oxidizer, permitting the fire to be maintained even in the absence of an adequate supply of room air. Second, the magazine is sealed with a structure

designed to resist large pressure differences between the internal and

external environments. As a result, a large pressure rise can occur in a magazine fire, unlike most building compartment fires. Although adequate

computer models for closed building compartments [1], as well as more general

building environments [2,3] exist, their tacit assumption of a limited

pressure rise prevents their use for the present investigation. However, in

most respects, the mathematical model presented below is really nothing more

than a greatly simplified version of what is already implemented in these

codes. The additional simplifying assumptions permit exact analytical solua

tions to be obtained for the pressure and temperature histories in the maga=

zine. These solutions permit the sensitivity of the results to the various

- 1 -
thermophysical and geometric parameters which describe the problem to be

readily assessed. They should be of some interest in their own right as there

are few, if any, other exact solutions to the compartment fire equations.

2. MATHEMATICAL MODEL

The magazine is assumed to be a closed rectangular volume of height H and

plan area A. The propellant fire is represented by a source of heat of

constant strength Q. At any instant of time t the flow pattern consists of a buoyant plume feeding heat and mass into a uniformly mixed hot upper layer of

temperature T(t). Beneath the hot upper layer is a cold layer of gas assumed

to be at the ambient temperature T

Q

. The plume entrains mass from the lower

layer causing the interface height Z(t) to descend towards the floor. The

pressure P(t) is the same everywhere in the magazine at each instant of time.

The theoretical and experimental justification for this picture is analyzed by

Quintiere [4].

In order to predict the evolution of the mean pressure, temperature, and

interface height with time, it is necessary to consider mass and energy

balances in the enclosure. Let and be the gas densities in the upper

and lower layers, respectively. Then, using the perfect gas equation of

state, these quantities can be rewritten as:

- p (t) . - P(t)

pu“ RT(t )

9 PL

"" RT

( 1 )

Here R is the gas constant. Since the room is sealed, and the propellant mass

is much smaller than the room air mass, the total mass of gas in the room can

-2-
be taken as constant. Thus, if P

Q

is the initial ambient pressure:

£P I is=s> + 2_ i - !£ 1 T T J T

( 2 )

Next consider the mass flow into the upper layer. If m(Z) is the mass flux

entrained into the buoyant plume at a height Z above the source, then a mass balance in the upper layer yields

l §

“ i(z) (3)

Cetegen, et al. [5] proposed a simple empirical formula for m based on a comparison of the point source plume theory with a variety of experimental

data. In the present notation:

m(Z) = 0.21 PL

/gZ z

2

{Q/pl

C T VgZ Z

2

j 1/3

(4)

here, g is the gravitational acceleration and Cp the specific heat at constant

pressure.

Finally, consider an energy balance for the entire magazine. The

internal energy per unit volume is P(t)/(y-l), where y is the specific heat

ratio in the gas. Since the chamber is sealed with immovable walls, the first

law of thermodynamics can be written for the whole enclosure in the form:

d AHP ( t ) i / ! _»

•37 1 f - o-x) q - q

cA ( 5 )

- 3-
The first term on the right hand side of eq. (5) is the rate at which heat is

released into the magazine. A fraction x of the total chemical heat release

rate Q is radiated out of the gas. The second term represents the convective

heat lost to the ceiling and walls in the upper layer. The quantity qc is the

convective heat flux per unit area, and it is assumed to be related to the

temperature rise above the ambient ceiling temperature by a heat transfer

coefficient h as follows:

q - h (T(t) -T) (6)

c

v o'

Equations ( 1 )— ( 6 ) constitute the mathematical system to be solved. It

must be supplemented by initial conditions for the interface height Z(t) and

pressure P(t). The natural choice is

Z(0) = H

(7)

P(0) = P

o

The initial temperature cannot be specified but must be determined as part of

the solution. This situation arises because a steady-state plume model is

used in the analysis which delivers hot gas to the ceiling in zero time.

Mathematically, the system is equivalent to a second order system of differential equations, which can only support two initial conditions.

-4-
2.1 Solution

Using eq. (2) to eliminate the temperature, the mathematical problem can

be reduced to the solution of the following pair of differential equations for

P(t) and Z(t)c

_1_ dp

hI

o

( p - P

0

) Qq (1 - X)

v-1 dt +P H - PZ

= AH

o

0.21

A

gQ izL

Y

1/3

(pz) 2/3z

( 8 )

Now introduce non-dimensional variables P(t) and Z(t) in the form

P - P P(x)

o

Z - HZ (x

)

P 1/3

t = —2j

gQH

Equations (7) and (8) then can be rewritten as?

(9)

A

(P - 1)

A A (l - PZ)

= q

5/3 * - a(ZPr/J7P

a = 0.21

(y-l)hT A

H(gQHV) 1/3

( 10 )

-5
q = (l~x)

2/3

(v-1) Q

H '* P

oH

P(0) = Z(0) = 1

A A Rather than work with P and Z directly, it is more convenient to use P and a

new variable x, defined by:

X 5 (ZP)"1/3

(11)

Then, dividing the first of eqs. (10) by the second:

* a dP = 3Px

dx cc q - —

y- A(P - 1) x - 1

at x a 1; P - 1 ( 12 )

Once eq. (12) has been solved, the dimensionless time t can be recovered by

using the second of eqs* (10).

x 2

a

x

/ dssP(s)

1 (13)

Equation (13) also shows that the domain of interest for x is given by

1 < x < «.

The initial value problem defined by eq. (12) can be readily solved by

noting that the equation, though non-linear, is of the Ricatti type. Thus,

A writing P(x) in the form

P(x) = f ( x)
( 14 )

f(x) = (x

3 - l)/x

4

leads to the following linear differential equation for $

d^$ _ 3xq _fj_ 37d x

4

d$

,2 a f a 3 . dx

dx x - 1 This can be immediately integrated once to yield:

[a/f (x)] exp{3qx2

/2a + / dt/f(t)}

Integrating eq. (16) by parts then yields the general expression for

$ s a |exp[u(x)] - / dt (3qt/a) exp [u(t)j + b}

1

x

u(x) 3qx /2a + / dt/f(t)

1

Using eqs. (16) and (17) in the recipe for P(x) given by eq.

evaluating the constant b from the initial condition given in eq.

A final expression for P(x) becomes:

= P(x) * 1 “-^2= [k(x)] * / dssk(s)

a

1

k(x) x - 1 A/a

/x2 + x + 1 exp 3(A+q)x^/2a +

^ arctan

a

/ 3

For convenience, the remainder of the solution can be summarized as <

(15)

(16)

<j>

(17)

(14), and

(12), the

( 18 )

function

of x as follows:
(19)

[P(x) x

3

]

1

(0) = S-- 1 + X

3

(P - l)/(x3 - 1) o

Note that eqs. (18) and (19) constitute the exact solution to eqs. (10)

for any values of the three parameters a. A, and q. Thus, if a somewhat

different formulation of the model were adopted, with different physical

constants of defining a, A, and q, eqs. (18) and (19) would still apply.

Finally, the initial value for the temperature can be extracted from this

solution. Since small t corresponds to x near 1; from eq. (18) for x-1 « 1:

P(x) = 1 - 3q (x-1) / (A+a)

Substituting this result into the second of eqs. (19); the desired result is

obtained:

The above solution was applied to study the pressure and temperature rise

induced by a 33.3 MW propellant fire in four compartments whose planform

geometry is defined in table 1. Each compartment was 3.05 m in height.

( 20 )

3. RESULTS AND DISCUSSION
Table 1 Summary of Magazine Planform Configurations

Configuration

12

3

4

Planform Dimensions (Meters)

4.57 x 4.57

9.15 x 12.2

9.78 x 14.6

12.2 x 24.4

The results are shown in figs. 1-4. Each figure consists of three parts,

showing the time evolution for the first ten seconds of the interface height,

pressure, and upper layer temperature in each compartment. Three curves are

shewn for each quantity, corresponding to differing estimates of the fraction

X of the chemical heat released lost to the boundaries by radiation. These

curves should be treated as empirical calibration curves, to be used in

conjunction with experimental data as it becomes available to determine the

“best fit" value of x« For this purpose, the remaining physical parameters

were held fixed at values given in table 2.

Table 2

Summary of Reference Values for Physical Parameters

Used in Figures 1-4

Value

Pressure P

Q

Temperature T

Q

Heat Transfer Coef. h

Specific Heat Ratio y

Gravitational Acceleration g

1 Atm = 1.013 x 10^ PA

293 K

28.4 W/m2 K

1.4

2 9.81 m/sec^

-9 -
The results indicate a substantial pressure rise in all cases, compared

with the situation usually encountered in compartment fires. Indeed, only the

heavily reinforced nature of the construction of the magazine permits such

pressures to be achieved without rupturing the walls. In the case of configuration 1, it might not be possible to prevent rupture of the compartment

walls if the upper end of the calculated range of pressures proves correct.

Further work using the more detailed models listed in the references, but

modified to account for hermetically or near hermetically sealed-type

enclosures, is clearly needed. The present model can ultimately be most

useful as a test case against which to check modified versions of the computer

codes that implement the more general models.

4. ACKNOWLEDGMENTS

The author and his colleagues are grateful to Dr. Warren Jaul and

Dr. Kent Farmer of the Navy Weapons Center for several discussions which

provided a great deal of useful background information about the problem

addressed in this report.

5. REFERENCES

[1] Cooper, L.Y. and Stroup, D.W. (1985). "ASET - A Computer Program for

Calculating Available Safe Egresss Time", Fire Safety J. 9, p. 29.

[2] Mitler, H.E. (1985). "The Harvard Fire Model", Fire Safety J. 9, p. 7.

- 10-
[3] Jones, W.W. (1985). "Future Directions for Modeling the Spread of Fire,

Smoke, and Toxic Gases", Fire Safety; Science and Engineering, ASTM

STP 882, T.Z. Harmathy, Ed., American Society for Testing and Materials,

Philadelphia, p. 70.

[4] Quintiere, J. (1984). "A Perspective on Compartment Fire Growth",

Combustion Science and Technology 39, p. 11.

[5] Cetegen, B.M., Zukoski, E.E. and Kubota, T. (1984). "Entrainment in the

Near and Far Field of Fire Plumes", Combustion Science and Technology 39,

p. 305.

- 11 -
10.0

Figure la. Dimensionless layer height Z/H as function of time in seconds for configuration

1. X is the fraction of the chemical heat

release lost from the gas by radiation.

12
t(s)

P/Po Bee® ec e e e OOOOOOOO OOO0

Figure lb. Pressure in atmospheres as function of time in. seconds for

configuration 1.

13
T/T0

ro ro

00 ro p p

4^

o b o b b b o

Figure lc* Ratio of upper layer temperature

to ambient temperature as function of time in

seconds for configuration 1.

14
1 T

Figure 2a. Dimensionless layer height Z/H as function of time in seconds for configuration

2. x is the fraction of the chemical heat

release lost from the gas by radiation.

15
p/p0

p -* -j- ro ro co

o b o bi o bi o

Figure 2b. Pressure in atmospheres as function of time in seconds for

configuration 2.

16
T/T0

o

o6CO

Figure 2c. Ratio of upper layer temperature

to ambient temperature as function of time in

seconds for configuration 2.

17
10.0

Figure 3a. Dimensionless layer height Z/H as function of time in seconds for configuration

3. x is the fraction of the chemical heat

release lost from the gas by radiation.

18
Figure 3b. Pressure in atmospheres as function of time in seconds for

configuration 3.

19
10.0

T/T0 obbbbbbo _-* ro co

^

<ji 05 -vi

Figure 3c, Ratio of upper layer temperature

to ambient temperature as function of time in

seconds for configuration 3.

20
10.0

o

Figure 4a. Dimensionless layer height Z/H as function of time in seconds for configuration

4. x is the fraction of the chemical heat

release lost from the gas by radiation.

21

nran
10.0

P/P0

o . _k ro

o oi o cji o

Figure 4b, Pressure in atmospheres as function of time in seconds for

configuration 4.

22
ro

o

CO 4^ oi

T/To

O

Figure 4c. Ratio of upper layer temperature

to ambient temperature as function of time in

seconds for configuration 4.

23
'
UBS.T14A (REV. 2*40)

r '"’

- -

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET {Se« instructions)

1. PUBLICATION OR

REPORT NO.

NBSIK-87/3557

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

November 1887

' 4. TITLE AND SUBTITLE

An Analysis of the Pressure Rise in Sealed Magazine Fires

5. AUTHOR(S)

Howard R® Baum

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions) 7o Contract/' Grant No. National Bureau of Standards

Department of Commerce f. Type of Report & Period Covered

Gaithersburg, MD 20899

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

Navy Weapons Center

China Lake^ CA

| 10. SUPPLEMENTARY NOTES

| 1 Document describes a computer program; SF-185, FIPS Software Summary, is attached | 11. ABSTRACT (A 20Q~word or less factual summary of most significant information . If document includes a significant

bibliography or literature survey, mention it here)

A mathematical model for the pressure and temperature rise in sealed ship

magazines caused by propellant fires is developed® The model is a simplified

version of several existing zone fire models, modified to consider sealed

compartments. The equations describing the model are solved exactly. Sample

results are shown for a variety of magazine geometries.

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

)

compartment fires; fires; mathematical models; pressure rise; r zone models

13. AVAILABILITY 14. NO. OF

PRINTED PAGES

Cx3 Unlimited

FI For Official Distribution. Do Not Release to NTIS 28

1 1 Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402. 15. Price

G3 Order From National Technical Information Service (NTIS), Springfield, VA. 22161 $ 11.95
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