:,] I "" -"• . ,. . .ESL-TR-87-53

MODELING RESPONSE OF TANKSCONTAINING FLAMMABLES TO

c~l FIRE IMPINGEMENT

-- F.A. ALLAHDADI, C. LLJEHR, T. MOREHOUSE, P. CAMPBELL

NEW MEXICO ENGINEERING RESEARCH INSTITUTEP.O. BOX 25UNIVERSITY OF NEW MEXICOALBUQUERQUE NM 87131

JULY 1988 -

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PRONAL AtirT

1P~Eroo 7 A. Alla aie Luenr, Thomas Morenouse, and Phyllis Campbell

Fia eot;A. Thrlloai, 2/n97 KJutta method 113tin mehd

I I N. Heat, transfer, Themonhvdraulic processes, (continued) -

19. ABSTRACT ICuni-n.e or ,,~s fnecesý pa eIey by block .,o,.ber)

A prediction methodoloov has been veoefo ricngtersne 0faakcontainino licuid flammables to a uniform-'xternal heat flux from an accidental spill fire.

tThe thermofluid ohv-sical crocesses for the "wonrst-case" condition, which results in aRoiling Liouid ExpandinQ Vapor Explosion (BLFVE-,~were assumned. This analysis assumes thatthe tank is totally en(10lfed in the flame of a laroqe, intense, turhuleot fire. Thisdevelopment considers the response of thIe tank unde' thermal loading for two possible vente(and unvented tank conficiurations.

c-7

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EXECUTIVE SUMMARY

The following presents the development of a prediction methodology forassessing the response of a tank containing liquid flammables to a uniformexternal heat flux from an accidental spill fire. This analysis considersthe assumed condition in which a tank is completely engulfed by the flame ofa spill fire, thie "worst-case" condition resulting in a Boiling LiquidExpanding Vapor Explosion (BLEVE). Common examples for the"worst-case"situation include a derailed tank car lying in a pool fire of flammablematerial released from a neighboring punctured tank, or an armed,mission-ready aircraft in a shelter or on the deck of an aircraft carrierexposed to an accidenta, jet fuel spill fire.

This model includes some of the inherent heat transfer phenomena thatwere ignored before. It considers the thermal effects caused by:

- Heat fluxes from total engulfment,- Iritial •ransient heating of the liquid inside the tank and formation

of a convective turbulent boundary layer,- Heat and mass transfer across the liquid/gas interface, and- High intensity heat input.

Inclusion of the above parameters in the heat transfer model hasimproved the quality of prediction both in the liquid phase and the gaseousphase. It is recommended that efforts be continued to investigate othergeometrical and structural aspects nf the thermal tankage proi'lemi nclmidinn:

- Tank initial filling density,- Mechanical and physical integrity of the tank, and- Structural response of the tank shell to fire impingement.

This analysis investigates and models the response of the tank underthermal loading for two possible vented and unvented tank configurations.It considers the thermofluid dynamic processes occuring inside the tank,including the initial turbulent boundary layer flow near the wetted wall o)fthe tank; stratification of the tank bulk at the core; transport processesat the liquid/gas interface; convective heat transport in the ullage volume,and transfer of heat fluxes from the bottom of the tank. Each of theseprocesses is defined, and the governing equations are derived and comTbilnedto provide an integrated physical response model. The systeri of governingintegro/differential equations representing dynamics of the physicalprocesses are solved numerically, and a model predicting the pressure riseinside the tank is developed. The prediction code provides the -esponse ofa tank containing flammables to a large turbulent spill fire and estimate.the elallsed time befure reaching the critical pressure for a given tankconfiguration. Also presented in this report is an evaluation of the codepe•erformiance and comparison of its prediction with experii-untal results.

iii(The reverse of this pane is blank.)

'RIJ[AC[

1 i f irl I report w'as prepalred by the NiM,' Mexito I riq irleer i nr Reseuarc.hI nsft i tut (N!11 H [) , UIli V(1- *1 ty ol New~ Mex i o ., Camifpws Box 21), A Ibuquer-qme , Nu--.Mexi .o 81 31 , I rdir ( oritra t 1 29601 134 C 0OF30 (Smjf~l iak 3.0()) 1 or the Air' I or.t2I mli1 ner'inq~ jfdO~ ev e e ter Vn1 neerin and *mlervices, I dhor il o ry( Al i S/[?[)t'I ) , I yrida 11 Ai r I orte Bs, I ]or Kida V203 60001 Ib1is, work wassptunrisored by th0w 'v Ai r Sy'steriw Covinwind (NAVA It<) a nd t he US0 A ir I o r ceI n'pltreTPP-in r n( 1d ýP Ve vice (fi Cllter (Al I-SC) . Mir Josecph I . I k ir ( Al S )C ), ,ain d Ms

l i' 11 C I Catpc i !1 I)( ( N Vi A ! R) v:ie r e t e CIo vr r r ifne ,r i L tn h.ott r i' . d i pr- o ri m fi il r iaj qtr sI rl ri r t si i mi iar i/e w, xr k d o i f, between (, .,r Se e 1 , i itE.,r- 9ý1 :1i ar id 1 1)br i wry 19 813

ilil reItio rt -habs b e cn r ev i e --od b V t he 0 1)tII i (. AlI I ir 1-, 01 if e (P/\i) md ISreleasable) I to t ic Na t ionialIIr lrrhIit.,i I tn Iormo L on Se r v i e (NI tS) . it t N II IS i Lv: I h e (m] I Ilab Ie to 1. tic qjerer ai pu I iih Ii , I cc -lidi moj I ore i (j i t r i ora I,-

I h I t if- hri i.a i i r cim tI , tit s ir i r ev 1 et rid ir is ilipi ruvcm I o r pub I tm, L I Oill

EDWARD T. MOREHOUSE~, Captain.,U4 ~ IAN C III'SI O1Oi JAProJect Officer Iiirer Ior of I oj ireer icy amid S-r vitts

I Ibo r~i to r y

Cl h t f , q 09 I t f., V i 1,1 1 I (, r IIi oil' JI

V

(The reverse of this page is blank.)

TABLE OF CONTE-NTS

Section Title Page

i INTRODUCTION

. . --, ,I I VE......................... ............. ..

. "•' t ' Ar A -' (;)ACH. .................................... . .

. SAC KGROURI) ........................................L'TERATU E V E ................................ . . 6

I MO;)ELING PRINCIPLES ....................................

,. GOVERN I"INW TRANSPORT EQUAlIONS IN T'1 UJLLAG(EVOLUME . ............... ............................. i

R. ,JNWETTED WALL HEAT TRANSFER PROCESS ............... 13C. ENERGY CONSERVATION IN THE GASEOUS PHASE .......... 140. PRESSURE HISTORY IN THE UILLAGE VOLUME ............. 18E. MASS TRANSFER ACROSS THE INTERFACE ................ 19F. ENERGY TRANSFER ACROSS THE INTERFACE .............. ,2)

III ANALYSIS OF THERMOHYDRAULIC PROCESSES OCCURRINGINSIDE THE I.LiQUIU PHASE ................................ .. .1

A. HEATING OF THE WETTED WALLS ....................... 2_33. CONVECTIVE FLOW FIELO INSII)E THE LIQUID ........... 23C. CONSERVATION OF MASS .............................. 24o). ,IIOYANCY CONSERVATION ...........................

t. CUNSERVAI-ION OF MOMENV OF MOMENTUM ................ 27I.. WALL HEAT FLUX TEMPERATURE RELATION ............... 30

G. VARIATION OF CORE TEMPERATURE ..................... 31

IV NUMERICAL SOLLITION ..................................... 33

A. NUJMERICAIL SCHEME .................................. 33

B. COMPUTATIONAL GRID ................................ i5C. MODELING TEMPERATJRE OF THE WETTE-D WALL ........... 390. MODELING OF LIQUID CORE TEMPERATURE ............... 41E. MODEL ING OF CUNVECTvIVyE BOUNDARY LAYER ............. 43F. MODELING OF VAPORIZATION RATE ................. ..

G. M9OF I. N; OF GAS AN ITHE UNWETTET) WALL TEMPERATURE. 49H. MODELING OF PRESSURE AND BOILING TEMPERATURE ..... 52?

I, INIVIAL VALUE CALCULATION ......................... 52J. DESCRIPTION OF THE COOt ........................... 53

V NUMERICAL RESULTS ...................................... 57

VI CONCLUSIONS ........................................... .I

REFERENCES ............................................. 79

vii

TABLE OF CONTENTS (CONCLUDED)

Section Title Page

APPENDIX

A DERIVATION OF FQIIATION OF MASS TRANSFERSTAGNATION FILM THEORY ................................. 83

B PROGRAM LISTING ........................................ 85

C EXFERIME1NTAL EFFORT .................................... 97

viii

LIST OF FIGURES

Fi gure Ti tle Page

I General Arrangement of a DOT 112A340W Tank Car ......... 3

2 Pictorial Representation of BLEVE ...................... 10

3 Heat Transfer interaction in the Ullage Volume ......... 12

4 Thermohydraulic Processes in the Liquid Phase .......... 22

5 Mass Balance for Boundary Layer Elemental Volume ....... 24

6 Energy b~alance for, Boundary Layer Elemental Volume..... 25

7 Staircase Representation of the Core Temperaturein Terms of a Moving and Stationary Grid; n=4 .......... 34

8 Moving and Stationary Grid Systems ...................... 38

9 Wetted Sidewall lewperature Time Profile for ClosedVent Tank Configuration ................................ 58

10 Wetted Sidewall Temperature Time Profile forVented Tank Configuration .............................. 59

11 bottom Wall Temperature Time Profile for ClosedTank Configuration ..................................... 60

12 Bottom Wall Temperature Time Profile for Vented TankConfiguration .......................................... 61

13 Dry Wall Temperature Time Profile for Closed VentConfiguration .......................................... 62

14 Dry Wall Temperature Time Profile for VentedConfiguration .......................................... 63

15 Boundary Layer Heat Flux Time Profile for Closed VentConfiguration ................. ........................ 64

16 Boundary Layer Heat Flux Time Profile for VentedConfiguration ........................ 65

17 Boundary Layer Mass Flux Time Profile for Closed VentConfiguration .......................................... 66

18 Boundary Layer Mass Flux Time Profile for Vented

Configuration ..................................... ... 67

19 Pressure Time Profile for Closed Vent Configuration .... 69

20 Pressure Time Profile for Vented Configuration ......... 70

ix

A.ST OF FIGURI-S (Concluded)

Figure Ti tie Pa q(

21 Mass of Vapor In Ul lage Volume Time Profile forClosed Vent Configuration .............................. 71

22 Mass of Vapor In Ullage Volume Time Profile forVented Configuration .................................. 72

23 Roi 1i nq Temperature Time Profile for Closed VentConfiguration .......................................... 73

?4 Boiling Temperature Time Profile for VentedConfiguration .......................................... 74

25 Gas Temperature Time Profile for Closed Vent

Configuration .......................................... 76

26 Gas Temperature Time Profile for, Vented Configuration- 76

A-1 Stagnant Fil:m Layer .................................... 83

C-I Water-filled 55-gallon Drum ............................ 93

0-2 Pressure Monitoring Assemhly ........................... 99

C-3 Tank Fires with Different Orientations .................. 100

C-4 Vertical and Horizontal Orientation Tests .............. 101

C-m Fire/Tank Test No. I Temperature Profiles .............. 102

C-6 Fire/Tank Test No. 2 Temperature Profbles .............. ... 103

x

LIST OF SYMBOLS

A total unwetted surface area in the ullage volumew

A liquid surface area

C wall specific heatW

C specific heat of air at constant voluneSCv specific heat of vapor at constant volumne

C VP specific heat of vapor at constant pressure

S1I total incident heat flux for the fire

qr reradiation inward from the wallrr 1

""1~ reradiation outward from the tankrr ,o

q" radiative heat flux incident on the liquid surface

q"1 conductive heat flux incident on the liquidc, £

k convective neat tlux trom the unwetted wallC,.

Clqcls convective heat flux from the side walls

qclb convective heat flux from the bottom wall

T gas temperature

T T substrate liquid temperature• • - •., S'

T. liquid temperature at the interface

T temperature of ambient air

T temperature of the wallw

6w thickness of the tank wall

6 boundary layer thickness

Pw density of the tank wall

o • Stefan-Boltzmann constant

; •emissivity coefficient

1h heat transfer coefficient

xi

LIST OF SYMBOLS (Concluded)

i nternal energy of the gas

• i. rate of mass addition due to vaporization

M riass outflow rite through the vent0

laterit heat of vaporization

u velocity distrilution in the boundary layer

U uniform counter flow velocity

- height of liquid

streau function

3 coeffecient of expansion of liquid

•w gqczs

9% T

F .S--

T - T -- -C--

T-0

T (z) - Tc T 0

kinematic viscosity

Pr Prandtl number

xii

SECTION I

INTRODUCTION

A. OBJECTIVE

The objective of this task encompasses two independent but interrelated

parts: (1) development of a model for a large hydrocarbon pool fire,* and

(2) development of a response prediction model for a fire-exposed tank con-

taining flammables. Specifically, we are to determine the conditions under

which a fire-exposed tank containing flammables fails and Boiling Liquid

Expanding Vapor Explosion (BLEVE) occurs, and to determine the elapsed time

be, ore failure. The response of various thermophysical processes that occur

inside a tank as a result of incident heat are combined and an intpgrated

comprehensive computer prediction methodology is constructed. The

prediction is to provide the pressure rise history profiles for a large

range of flammables and fire/tank configurations. The prediction is

essential to on-the-scene assessment of the fire environment and the safety

of fire-fighting crews.

B. SCOPE/APPROACH

The scope of this effort is to develop an accurate heat transfer model

for the liquid and gaseous phases of a fire-exposed, partially filled con-

tainer. In this investigation, the governing equations describing the ther-

mophysical processes that occur inside a tank are developed and solved

numerically. Considered in this investigation are some of the heat transfer

parameters that were ignored before. These include heat input from the tank

bottom wall and initial transient heating of the liquid. The scope of this

effort also requires conducting specially designed fire tarnk ta-sts to vali-

date and assess the model performance.

* Developmen- of a large hydrocarbon pool fire model has been completed. Itwas rer)rted in a 1986 Task Report (NMERI WA3-12) entitled One-Dimensional Hydrocarbon Fuel Fire Model prepared by New MexicoEngineering Research Institute for AFESC Engineering and ServicesLaboratory.

C. BACKGROUND

Large quantities of petroleuin and volatile hazardous chemicals are

routinely extracted, produced and transported, either by highway or rail.

The possiblity of a major accident caused by human error or otherwise during

transportation or maintenance of these facilities is quite real. The

impetus behind this aralysis is to formulate and model the thermodynamic

phenomena that occur in a fire-engulfed tank and to predict the elapsed time

before failure or the occurrence of BLEVE.

Before engaging in detailed mathematics regarding modeling response of

a tank laden with flammables to a large sooty fire, it is instructive to use

an illustrative exajnple to define the sequence of events occurring in a

fire/tank scenario and to demonstrate the processes that ultimately lead to

3LEVF. Not all fire-engulfed flammable tanks conclude in BLEVE. Consider a

large tank, e.g., a rail tank-car (Figure 1), partially filled with liquid

propane and the remaining volume (ullage) occupied by propane vapor and air.

.h! ullage volume in the tank is necessary to allow for swelling and expan-

sion of the liquid that results from incident external heat fluxes.

When a tank partially filled witn volatile liquid is exposed to a large-

turbulent sooty fire, the heat flux from the fire is partially stored in the

walls of the tank, and is partially transferred to the contents, liquid and

gas phases, of the tank. The total heat transferred to the tank consists of

radiation fro,, the pluiie and convection from the combustion gases. The

contents of the tank are heated by natural convection, lhe tank wall next

to the gas phase always has a higher temperature than the walls wetted by

the liquid phase. The.-efore, the tenperature gradient sets up a conduction

heat transport process along the wall of the tank.

The prevailing heat transfer from the tank wetted wall to the liquim is

by convection. The heat transfer fron the wall establishes a turbulent

convective thermal and velocity boundary layer in the liquin next to the

wall. Tie superheated turbulent boundary layer next to the wall raises the

heated liquid along the wall and discharges into the subcooled main core at

the liquid gas interface, creating a convective circulatory motion. This

process continues until a near-uniform temperature field resulting from a

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high rate of mixing is established. At this time, the heat transfer rate is

quite high, and almost all of the incident heat transported to the wall is

transferred to the liquid; very little goes into incr sing tihe temperature

of the wall, and nucleate boilinq occurs. As a result of this, the inner

wall temperature in the liquid wetted region rarely exceeds the temperature

of thle liquid by more than 10 'C (Reference 1i. Also, there is a transi-

tion region between the liquid wetted wall and the vapor wetted wall which

wil,' create a two-phase flow at the gas/liquid interface. The thickness of

this region has not been deter~niined and will depend on the strength of tihe

boiling action. In this region, the naagniitude of the heat transfer rates

viwill he som-ewhere between those for the liquid wetted and vapor wetted sur-

faces, and as a result the wall temperature in this region will also he

e.tween those for tie liquid and vapor wetted wall regions (Reference C ).

Tne prevailing heat transfers in the gas phase in the ullage volume are

by convection and thermal radiation. The initial convective heat transfer

rate in the vapor phase will be low because of the conbined effect of the

ljw thermal conductivity of the gas/vapor and the low flow velocities.

lqiLial therlu', radiation will also he s;nal I because ot the low wail temper-

atures. As the temperature of the walls in the ullage volume increases, the

energy transport from thermal radiation will also increase. The increase in

the energy transport in the ullage volume is predominantly due to the radi-

ation which is proportional to the fourth power of the wall temperature.

The tank internal pressure rises as a result of an increase in the gas-phase

terperature and surface vaporization of the liquid. When thle temperature of

the wetted walls reaches the saturation temperature, corresponding to the

instantaneous tank pressure, vapor is generated rapidly and the pressure

builds up quickly. At this time, the liquid portion of the tank expands as

a result of thermal expansion. If the tank becomes completely filled with

liquid as result of overexpansion, rupture will occur when the critical

pressure is reached. ,he critical pressure is related to the decreasing

An operational tank is equipped with a pressure relief valve which

would open to allow removal of mass from the tank, thus, keeping the internal

pressure below the tank rupture pressure. However, the released vapor from

the relief valve mixes rapidly with the ambeient air, is ignited, and becomnes

4M

an additional source of heat flux incident on the tank. The additional heat

flux that impinges directly on the unwetted wall, which is now increased as

a result of the drop in the liquid level, deteriorates the mechanical

strength of the tank. Under these conditions, depending upon the integrity

of the physical structure of the tank, the tank will either safely continue

to vent the lading or it will rupture. If the tank ruptures, the remaining

liquid in the tank experiences a sudden depressirization. Since the rernai.n-

ing liquid cannot vaporize quickly enough to establish a stable saturated

condition, the liquid becomes superheated. If the liquid is superheated

sufficiently, a BLEVE results.

Some of the processes leading to the BLEVE have been defined by various

researchers. Reference 2 describes this phenomenon in the following forma.

if a saturated liquid at high pressure suddenly experiences a rapid pressure

decrease, as in the case of a rupturing tank car, then the liquid should

boil until the temperature of the remaining liquid is reduced to the satura-

tion temperature for the next pressure conditions. This requires sufficient

nucleation sites for the boiling to take place, and these sites do not exist. .. - * 1 ... .".1 1 , - ^ 4I . . . . . ...-

-,•, ,ufcient qu.'.... t t the u-" , ..... -i"d. Tni c aucuses c •s•mne-

the liquid to become superheated for a brief period after the depressuri za-

tion occurs. If the degree of superheating is sufficiently large, hornogen-

eous nucleation takes place throughout the bulk of the liqjid and the entire

liquid mass vaporizes almost instantaneously. IThis rapid vaporization

causes a vapor explosion which may cause the tank to be ripped open and

sometimes propels tank fragments for larqe distances. In addition to the

shock loads from the vapor explosion, the vapor rapidly mixes with the sur-

rounding air and ignites, causing a fire ball which loads nearby structures

with intense thermal radiation. Reid (Reference 3) suggests that a IRLEVE is

the result of homogeneous nucleation of a metastable superheated liquid.

Consistent with the definitions, occurence of BLEVE, although not

totally independent of a pressure relief valve, is mostly controlled by the

initial liquid filling density arid by the magnitude of the incident heat

fluxes. Various ,nodes of response for tanks containing flammables have been

studied by various researchers. A brief literature review is presented in

the following.

5

0. LITERATURE REVIEW

The response of a tank engulfed in a pool spill fire has been studied

under special conditions by various researchers. R1oherts et al. (Refer-

ence 4) conducted experimental studies of insulated and uninsulated

500.-liter tanks. The tanks were partially filled with liquid and exposed to

a kerosene pool fire. In each test they measured the heat transfer rate to

the total system, the contents of the tank, the boiling regime, and the tank

wall temperature. They demonstrated that a rpspons? time of 3 minutes (the

time needed for the pressure relief valve to activate) for an uninsulated

propane tank could be extenOed to ranges of 12 to 90 ininutes for insulated

tanks of similar material.

Venart et al. (Reference 5) conducted a study of the physical behavior

of tanks containing liquified fuel under accident conditions. They reached

a number of conclusions that are believed to be true only for low-input heat

flux, i.e., heat fluxes under 7 w/cm2 . Among the conclusions reached were:

(1) prior to pressure relief valve action, the primary source of vapor gen-

eration originates in the liquid boundary layer, and (2) simple thermo-

dynamic and heat transfer models may he used to predict the pressure

response of such vessels.

The problem of evaporation rates of liquified natural gas in containers

was zonsidered by Hashemi and Wesson ("eference 6). They treat the system

as a liquid layer heated from below and develop an empirical relationship

between the boiloff rate and the supersaturation* pressure. This equation,

which shows that the boiloff rate varies as the 4/3 power of the supersatu-

i ation pressure, is solved in conjunction with the energy equation for the

liquid in the tank, resulting in an implicit algebraic equation in terms of

evaporation rate.

* The supersatur tion pressure of the liquid is defined as the product ofthe average pressure at the saturation temperature and the total differencebetween the temperature of the bulk of the liquid and the temperature at thesur-face.

6

Note that the natural convection patterns for a liquid layer will not

exist for elevated pressure and the model will not yield realistic solutions

for pressure rise cases.

The natural laminar convection of fluids in an enclosed cylindrical

geometry has been considered by Barakat and Clark (Reference 7). The

governing conservative equations pertinent to the fluid inside the tank were

simplified by the use of stream functions, and numerical techniQues are used

for the solution of these equations. They demonstrated that steep velocity

and thermal gradients exist in the vicinity of the sidewall and, hence, a

houndary layer flow occurs. Analytical and experimental study of transient

natural convection in a vertical cylinder has heen reported by Evans and

Reid (Reference 8). Their experimental effort consisted of monitoring

transient temperature and flow patterns inside an exposed tank partially 1_

filled with a water-glycerin mixture. The tests were conducted for a range

of Prandtl numbers from 2 t.o 8,000, and for L/IO ratio (L = mixture depth,

1 = cylinder distance) and Grashof numbers from i03 to 1 [j) encompassing

both laminar and turul1ent regiie s ThPy considered the problemI analyti-

cally. The system was divided into three regions: (1) a houndary layer

region near the wail, (2) trne mixing region at the tot) of the gas/liquid

interface, and (3) a mmin core region where the bulk fluid slowly falls (lown

in plug flow. The resultinq equations are solved numerically to obtain the

temperature profile in enclosed fluids subject to wall heating. Tt was

observed that after measurement of the initial transient temperature, the

core temperature increased linearly with time--a characteristic of low heat

tlux input.

Transient thermal stratification in heated, partially filled horizontal

cylindrical tanks was considered by Aydemire et al. (Reference c). They

extended the analytical technique for boundary layer analysis to account for

the presence of a covering of aluminum foil mesh on the inside walls of the

tank. Results with and without ExploSafe were compared with experinmerital

data for Freon-113'". Although fairly good agreement between the experi-

mental and analytical results is obtained, the procedure appears to be valid

only for low-input heat fluxes. Virk and Venkataramana (I:eference I0)

studied the transient hoiloff rates of a liqjified natural gas from large

industrial storage tanks subjected to the perLurbation of ambient

atmospheric pressure.

7

Their theoretical model closely tracks the procedure covered in References

6-9. The model demonstrate; that constant heat input to the tank maintains

a steady b-oiloff rate of LNG from th,, tank. It establishes that a natural

convection boundary layer exists in the vicinity of the wall, and that hot

fluid, upon reaching the liquid surface, diffuses radically inward toward

the center, losing part of its mass and energy as boiloff. They also demon-

strated tnat step changes in ambient pressure (i.e., rise or drop) could

affect the boiloff rate by as much as 1000 percent.

Birk and Oosthuizen (Reference 11) outlined a computer model of a rail

tank car with its exterior surface exposed to a spill fire, Their model is

two-dimensional, capable of predicting tank pressure, wali temperatures,

wal• stresses, vapor and liiuid' mean temperatures, flow rates through the

relief valve, and liquid level as a function of time.

f]el ichatsios (Reference 12) conducted analytical and experimental

studies of fire-exposed containers containing different solvents prior to

rupture. He modeled the thermohydraulic process occurring inside the

cuntainer in terms of a set of physical dimecnsionless prameters. He also

measured the container internal pressure and temperature rise resulting from

incident heat for different solvents using 55 -g. 11on steel drums, In this

investigation he concludes that, for most solvents, the time for tank rup-

ture is equal to the time for the tank's wetted walls to reach the solvent

boiling temperature.

3l l I I

SECTION II

MODELING PRINCIPLES

The description and sequence of events that occur inside a fire-

engulfed tank are illustrated pictorially in Figure 2. Figure 2a shows the

fire engulfment of a rail tank-car. As shown in the figure, the tank is

nearly filled with the flammable liquid and the remaining volume is occupied

by vapor. The radiative and convective heat fluxes from the external fire

are conducted through the wall into the tank lading. The heat input raises

the temperature of the wall next to the liquid, and a convective turbulent

thermal and velocity boundary layer is established. With increasing time,

the temperature of the wall next to the gas increases sufficiently to

radiate. The radiative heat flux, combined with the convective current,

causes vaporization of the liquid at the interface. At this time, the

liquid expands and swells, further increasing the internal pressure of the

tank. If the tank is equipped with a pressure relief valve and if the

tank's internal pressure exceeds the valve operational pressure, the valve

, ill open to allo, the venting of lading- This will reduce the internal

pressure of the tank, but at the expense of increasing the surface of the

unwetted wall in the ullage volume (see Figure 2b).

Simultaneous with the expulsion of lading, two important phenoriena that

significantly contribute to occurrence of the BLEVE will take place:

1. The vented combustible vapor mixes with the surrounding air dnd

ignites; therefore, the total radiative heat incident on the tank

is increased.

2. The radiation heat that impinges directly on the continuously

increasing unwetted wall will result in further reduction of the

tank shell mechanical strength.

The additional radiative heat from the unwetted wall causes extensive

surface evaporation, a phenomenon which further raises the pressure consid-

erably. At this time, the wetted wall temperature has reached the satura-

tion temperature, and copious vapor from nucleate boiling is generated.

With increasing time, the tank's internal pressure rises quickly, and rapid

9

ii •

A~'j

'I ' ,

I )

(a) External fire begins. Heat is (b) Relief valve opens. Venting ofconducted into tank loading, of lading increases the vapor

space.

A)11 1,114-~

(c) Wall adjacent to vapor space (d) Wall temperatures rise to aheats up because vapor is a point where the steel weakenspoor conductor of heat, and the wall ruptures.

Figure 2. Pictorial Representation of Bleve (from Ref. 11)

10

generation of vapor and further expulsion of lading occur (Figure 2c). At

this time, the strength of the unwetted wall has deteriorated to its

critical value and the tank will rupture (Figure 2d). At the moment of

rupture when the tank is depressurized to ambient pressure, if the remaining

liquid in the tank is at saturation temperature, a violent explosion, BLEVE,

Occurs .*

The thermofluid mechanics of the physical processes before BLEVE were

discussed earlier. They are deterined by the following interacting

transport phenomena:

1. Heating of the walls of the tank.

2. Heating of the gaseous phase in the ullage volume.

3. Heating of the liquid phase.

4. Transport of heat and mass across the liquid/gas interface.

The analysis of the mathematical model describing each of these pro-

cesses is presented in the following sections.

A. t(0VERNING TRANSPORT EQUATIONS IN THE ULLAGE VOLUM,,E

Figure 3 depicts the cross section of an arbitrary tank. This figure

describes interactive processes that occur in the ullage volume. The para-

meters indicated are:

reradiation outward from the tank

'rri r reradiation inward from the wall

total heat flux from the fire incident on the tankrhnt

nradiative heat flux incident on the liquid

qc,• : conductive heat flux incident an tile liquid_

convective heat transfer from the wall

Aw • total unwetted surface area in the ullage volume

A• : liquid surface area

*For more explanation of the phenomena refer to the paragraph entitled

"Background" in -Section 1.

II11=

f qr~

q C

Liquid/Gas interface -A

Liquid volumei(!quid)

Figure 3. Heat Transfer Interaction in the Ullage Volume

12

T gas temperatureg

Ti liquid temperature at the interface.

T unwetted tank wall temperature.

In the derivation of the governing equations deIscribing the transport

process in the gas phese, the following assumptions have been made:

1. The thickness of the tank wall is very small compared with its

other dimensions. Therefore, the temperature variation across the

thic:kness is negligible.

2. Vapor and air in the ullage volume behave as perfect gases.

3. Temperature and concerntration distributions in the ullage volume

are homogenous, since the mixing occurs relatively quickly.

4. In spite of the large temperature gradients in the wall near the

liquid meniscus, the total heat flow rate by conduction along the

wall is assumed negligible.

5. The heat flux imposed by the fire is uniformly applied on the tank

wal 1 s.

R. UNWETTED WALL HEAT TRANSFER PROCESS

The total heat stored in the walls of the ullage volume is given by the

following thermal energy balance equation:

dT6 w w 9 /I" ) (i

wwwdt r,o rr,i - (c)

where pw, Cw and 6 ware density, specific heat, and thickness of the wall,

respectively. The reradiation heat loss to the outside is given by

qr,o oc (Tl - Tc ) (2)

In Equation (2), T is the ambient temperature, a = 0.1713 x 10"-

BTU/ft 2 hrR' is the Stefan-Boltzmann constant, and E = 0.99 is the

coefficient of emissivity. Similarly, the radiation heat losses to the

interior of the tank impinging on the liquid surface reads

1qrr = c• (14 - T4i ) (3)rr,i w9

13

The total convective heat transfer rate in the til lage volumie is repre-

sented by q" . It can be shown (Reference 13) that the total convectivec ,g" '-

heat transfer for a given surface coefficient, i.e., h, is given by:

q"' = h (T Wg- T ) (41N•

where h is the tank surface heat transfer coefficient, and, as defined

earlier, T and T are the unwetted tank wall temperature and gas tempera-wg g 9ture, respectively. The surface heat transfer coefficient, h, can be calcu-

lated from the data given in Reference 13, and the convective heat transfer

equation becomes

1'c = Cg (T - T )9 /3 (5)c,g gwg g

In Equation (5) the coefficient, C9, depends on the property values of the

gas inside the ullage volmne. The value of the C is defined in Section

IV.

C. ENLRGY CUNSERVAI ION IN THE GASEOUS PHASL -

The conservation of energy in the gaseous phase is a statement of

energy balance between the internal energy of the gas and the energy trans-

port across the boundary and the gas/liquid interface. The conservation of

energy balance reads

dE , - A + .h. (6)d-T =c w c ,9k I

In deriving this equation, it is assumed that gas inside trle ullage volume

does not absorb any thermal radiation. The parameters in Equation (6) are:

F is the internal energy of the gases in the ullage volwe

F1 :fis the mass addition rate of vapor owing to the evaporation

from the liquid surface.

h.: is the enthalpy of the vapor

14

The energy equation in the uilage voltume can be written in terms of air and

vapor mass fractions. It follows that

M =MYa a

M - M Y (7)V V

where

Ma: mass of air in the ullage volume

Mv: mass of vapor in the ullage volumeYaV mass fraction of air in ullage volume

Y : mass fraction of vapor in ullage volume

Note that

Y +y i ()a v

and the total mass of gases in the ullage volume is given by

M = M + M total mass (9)

a v

it is evident that for a closed tank configuration

dM

a 0 (10)

Therefore,

dMdM _ v - ~ (11)

dt dt

In the open tank configuration, the total mass balance equation for the

ullage volume will take the following form:

dM A.= - (1la)

15

where dM/dt is the rate of change of total mass in the ullage volume and il.is the rate of mass addition at the interface due to the boiling and vapor-

ization. The quantity Ao represents the mass rate of outgoing air and

vapor, collectively. The mass rate of change of each species in the ullage

volune is calculated based on the mass fraction of the outgoing gases. For

exaiiple, the mass rate of change of air ancl vapor in the ullage volune is

given by the following equations, respectively:

d ) = -Y , (lib)-

dt (YaM) = a o

d (Y M) = , - Y M (llc07t V 1 v 0

using Eou3tions (Ila), (l1b), and (11c), the rate of change of air :Qdss

fraction can be calculated as

It can be shown from Bernoulli 's law that the outflo.iing mass is related to

the pressure drop in the tank:

0A = A C I ( Patm) p] f)

where A represents the vent cross-sectional area and Cd is the coefficient

of discharge.

Considering the closed tank configuration, the total internal energy

term in thc gas phase can be written as the sun of the specific energy of

air, ea, and vapor, e :

E = M e + M e (12)a a v v

After taking the time derivative of Equation (12) and substituting for the

mass fractions from Equations (10) and (11), one obtains

16

de de Vd-t M a - + Mv + Miev (13)

Substituting for L- from Equations (12) and (13) into Equation (6) resultsdt

i fl

de a de''a dM v q" A - A" A + Mi (h - ev) (14)Ma ýTt + v dt - ' q c c, zk i ...

Because of the high rate of mixing, the air/vapor mixture in the ullage

vo.iwie remains at a uiifort, and homogeneous temperature. Based upon thsis

conclusion and the definitions of internal energy, e, and enthalpy of ideal

gas, h, one can modify Equation (14). We have

de dT de _Ta___ - Cv ' ___ (15)d-T - = Ca,v dt ' dt v dt

ID

h. =C h C T e + ( 16)hi VP vi VP h v PV

ihc time rate of change of gas temperature, i.e., dT /dt, can be obtained

from the energy equation (Equation (14)) after substituting for de a/dt,

de e/dt and hi from Equations (15) arid (16).

It follows that

dT q" A - '" A + iA. [C (T. - T ) + Pv/ pv]g cg W cz t 1 vIP 1 g v v

dt M C M N Ca a,v v v,v

Similarly, for the vented configuration the rate change of gas

temperature becomes

(IT / - "__9_ =vCv IcA -C" A + C . T-Cdt a ,aaVv cg w c t I v,v g

M 0PV/(M + M) (17a)

17

In arriving at the above equation we have made use of the following

thermodynamic relationship

R Pc VI- TCh5T = ___7

vp v v v 0 T

D. PRESSURE HISTORY IN THE ULLAGE VOLUME

The tot-al pressure in the ullage volume at any time is the sum of the

partial pressure of air and the partial pressure of the vapor. Therefore,

the total pressure becomes

P = Pa + P (18)

and the time derivative of the total pressure becomes

dP dP dPa + v (19)dtd a-

using iceali gas law relations, one can write

M RTa = l a V (20a)

rl RT= v x (2 OI)

v V

For a closed tank conf iguration, the ulla,, volumie, V, remains nearly

constant and the time rate of change of total pressure yields

dP =RT 9 + RM +RM~ dT9 ()

aT + (21) ~ ml) TV~o)v -Vi)o [ Vml aj

18

In the derivation of Equation (21), the identities describing a closed tank

configuration, Equations (10) and (11) have been used. For the vented

configuration, the internal pressure of the tank at any time is determined

by

[m t) M (t)]R t

[ (mol) (M ol) V

Here

(tool) : molecular weight of vaporV

(mo1l)a molecular weight of air

R: universal gas constant

E. MASS TRANSFFR ACROSS THE INTERFACE

The gas temperature in the ullage volmne is much nigher than the liquid

temperature. Evaporation at the interface between the liquid and the gas

occurs because of the convective heat transfer in the gas and radiative heat

flux from the unwetted wall impinging on the liquid at the interface with

the vapor volume. By using stagnant film theory (Reference 14), one can

find that the evaporation rate per unit surface area is

M c Ln I + (22)Cplg Leff " r,/o

Derivation of Equation (22) is shown in Appendix A. In Equation (22), the

parameters are

hc : convective heat transfer coefficient from a horizontal

surface

C : specific heat transfer of the gas mixturep ,g

Leff: effective heat of gasification

19

C : specific heat of liquid

T : surface temperature of liquid

T Z substrate liquid temperature

S :latent heat of vaporization

Leff: i + CpZ(Ti - T 1) (23)

F, ENERGY TRANSFER ACROSS THE INTERFACE

The heaL transfer across the liquid vapor interface is a statement of

energy balance between the conduction and radiative heat transfers incident

on the surface and the energy associated with the vaporized mass. Assuming

no radiation energy is absorbed by the vapor in the ullage volumie, the

interface energy equation reads

oi l " r~i" Lemf (24)c ,Z r,• eff

The radiative heat transfer incident on the liquid suirface, q" , is

directly related to the total inward radiation energy emanatinq from the

side walls. It follows that

q" .Aq11 rr,i w

A

where, as defined previously, 6"rri jc(T4 - Tb).,Wg 1

20

SECTION III

ANALYSIS OF THERMOHYDRAULIC PROCESSESOCCURRING INSIDE THE LIQUID PHASE

Heat from the external fire is partially stored in the ý.al1 next to the

liquid and partially transmitted to the liquid through convective currents.

The heat through the vertical walls establishes a turbulent natural convec-

tive velocity and temperature boundary layer. The analyses of physical

phenomena considered in this iivestigation are:

1. Determination of boundary layer flow produced by natural

convection.

2. Evaluation of the core temperature variation.

3. Analysis of liquid top layer where the boundary layer turns hori-

zontally and the flow descends into the liquid mass.

The kinetic energy and the momentum of the boundary layer near the

surface characterize the flow in the core. They determine whether the flow

at the tank core consists of large miixin , eddies or- a slow strat icd motio

(ReFerence 15). The thermohydraulic processes that occur in the liquid

phase are shown in Figure 4. The parameters indicated in the figure are

6 : boundary layer thicknessT c(z) : core temperature (stratification) distribution

T, u : temperature and velocity distribution in the boundary layer

(defined later)

U uniform downward counter flow velocity

H liquid height in the tank

A , L : area and perimeter of the liquid surface

The mathematical models descriptive of the above physical processes are

developed in the following sections.

21

UlIage volume H(gas)

Boundary layer

/ TC(z)

I" i/H(liquid)

q1I

Figure 4. Thermohydraulic Processes in the Liquid Phase

22

A. HEATING OF THE WETFED WALLS

Analogous to the heating of the wall in the gaseous volume, it is

assumed that the temperature across the thickness of the wall is uniform.

Then, the thermal balance* equation for the walls can be written as

w w Tw - "f - 4I"ros - " (25)

In Equation (25), q"cis is the convective heat loss to the liquid and Tws is

the side wall temperature.

Similar to the derivation of Equation (5), the convective, ý"Zs' andcis

"the re-radiation, heat transfer terms can be written respectively asrros'

= T cCr-T)4/3q cis s T(25a)

" = •o T4 _- u -1-4_.l'rros co ( - T-_

where the coefficient C depends on the physical properties of the liquid(Reference 13), given as

Cs = PeCp Jvc g/[(5.3)4 Pr 2] 1/3 (25b)

B. CONVECTIVE FLOW FIELD INSIDE THE LIQUID

A mathematical description of the heating of the liquid by convective

currents is presented. This development ignores the possibility of laminar

boundary layer which may occur late in time. It is also assumed that the

core of the vessel does not contain large eddies, and that the boundary

layer region can be described continuously by turbulent boundary layer

equations. The governing equations for the boundary layer are based on the

integral representation and are appropriately coupled with the core flow.

In view of initially weak buoyant flow in the boundary layer, the Boussinesq

approximation has been implemented.

*It is assumed that the flames surrounding the tank are radiatively thin.

23

C. CONSERVATION OF MASS

The principle of conservation of mass is applied to an elemental voluie

of boundary layer (see Figure 5).

z

Pu

dz - dm

IPu

Figure 5. Mass Balance For Boundary Layer Elemental Volume

The conservation of mass states that the entrainment rate is equal to

the difference between mass efflux and influx in the boundary layer. There-

fore the mass rate, di., entrained in the boundary layer by convective cur-rents is equal to the amount of mass leaving the boundary at any cross

section. It follows that

f [ pudy +-a (pudy) AZ- pudy = d; (26)

where, after simplification, it becomes

d = pudy (27)

dz dz

24

0. BUOYANCY CONSERVATION

The buoyancy equation is a description of the energy associated with

the upward flow in the boundary layer caused by the temperature gradient.

The integral form of the buoyancy equation is written for an elemental

volune of boundary layer (see Figure 6). It follows that tne rate of change

of energy per unit length in the boundary layer is equal to the algebraic

sum of the heat input (Reference 16).

z ;)uC

dz -pc

Figure 6. Energy Balance for Boundary Layer Elemental Volume

The governing buoyancy equation for a tank having an orthogonal cross

section reads

&"dz = d (fo puCpTdy) - diý C T (p8)\0 p / c

In Equation (28), dz is a differentidl length in the vertical direction, u

is tne velocity of the fluid in the boundary layer, and T is the tempera-cture of the entrained core liquid. Equation (28) can be rewritten as

q"/Cp = (jf' ouTdy) - T

25

Substituting for diý/dz from conservation of mass, Equation (27), one

obtains

""/Cp = d puTdy) _ d f, pudy)T (2)

Since the core temperature Tc varies with the liquid height, the last term

in the rigit side of Equation (29) is nodil~ied to reflect this behavior. IL

follows that

d ( pudy) T SoTc pudy - dT o udy.dzKo puy.c .7 o6 dz

Substitution of the above relation into Equation (29) gives the buoyancy

equation

d f'" udy) dT (30)

p

Equation ( 30) can he modified to a morp us•eful form if it is multiplied

through by g/(pT ). Here, g is the gravitational acceleration and T0 is

some reference temperature. This yields

"d 06 - Tc -- "p - d (Tc(Z)-T°)s (31)

ug f -ug-dy g d - !)f5iy(1-d- o T 0 CpCT 0T

Define:

T i-T(z) Tc (z) - TrA cA c o

•" T0 0

A BToA; Aw = (Tws - T C) g (31a)

[gq cis Vc 1/3

w p p (5.3)'Pr2

26

Also, after replacing the y coordinate with a stream function, p, one

obtains,

= o Y fi udy (31b)

where ,t= o udy is the total flow rate.

Then the governinj buoyancy equation reads

S(fot dw) = -%r (32)

F. CONSERVATION OF MOMENT OF MOMENTUM

Because the wall friction parameter in the boundary layer region is not

well known, especially for slow buoyant flows, the motion of the fluid in

the boundary layer is characterized by the moment of momentum instead of the

conventionally used momentum equation (Reference 17). This method is quite

appropriate for flows attached to vertical surfaces. Rigorous derivation

ano justification of this method is beyond the scope of this work. However,

observe that its acceptability may be based on physical argument, intuition,

and considerable recent research (References 18, 19, and 20).

In deriving the moment of momentum equation, it is assumed that:(1) the

flow is a boundary layer type flow, (2) the average pressure in the plume is

the same as the ambient pressure at sea level, which is the consequence of

the Buossinesq approximation, and (3) the flow is two-dimensional. We start

with the Navier-Stokes turoulent boundary layer equation (References 21 and

22).

u a +27 (33)

27

In Equation (33) z is the vertical direction, y is the distance normal to

the wall, p is the viscosity, and the quantity puiv-i-- is the Reynolds Stress.

The coefficient B is the thernal expansion coefficient for liquid, and T0 is

a reference temperature. Every variable in Equation (33) is represented by

its Favre average (Reference 23):

u=u + u1

U _(34)

where the velocity field u is the algebraic sum of mean, u and fluctuation

velocity u". The bar over the symbols denotes the time or the ensemble

average. Again we make use of stream function, i.e., p, to replace the y

coordinate;

QU - )(35)IP U : Po(35

0 g

P v = - PO az

With this change of variable [(z,y) + (z,w)] and neglecting gradients of

stresses in the direction of flow, the momentum equation, Fquation (33),

takes the form

2u = L +B (36)

3z 3a o

which is a more useful for-i. In this equation

= -P+ u" u + (/) (/Po)

A= Ap / p 9

28

To obtain the moment of momentun equation, first integrat.e Equation (36)

over ip from S to t (total flow rate):

au j'- t d Toj't A/udp (37)

Si nce

U0 @

0 ~t

we have

]p•'p +Pt-ft ud, -- + sT I / 6'd' (38)

-41 0s -- -,_

Next inteqrate over p from 0 to t One obtains

rot k•tu *t St -

o a fud i = - fo TdJ + js dl A /dý (39)--

using the Leibnitz rule (Reference 24)

3f ýt r~t3 tdqp + ýT° fo dp f 0 -A / 'd

which after partial integration reads

St 't 4taz o " r(40)

Equation (4() is the desired moment of momentuin equation. In general, the

second terni at the right side of Equation (40) is small relative to the

first term at the right side of the same equation; its magnitude is slightly

negative for turbulent flows.

Equation (40) demonstrates that the core stratification affects the

momentum of the boundary layer only indirectly through the temperature dif-

ference A In Equation (40), 6 is the coefficient of thermal expdnsion for

the liquid and has the units of inverse temperature.

29

In deriving the moment of momentum equation, it is assumed that the

core velocity, u, is significantiy small indeed, since the width of the tank

is much larger than the boundary layer thickness (see Figure 4). Fi nal ly,

the shear stress term in Equation (40), although small, must be included in

the present application because the velocities and the growth of the

boundary layer are slowed down by stratification. An approximate

development of the shear stress is given in Reference 22. It reads

: V T d ,I = 1 O 0 ( F wv ) 1 1

Thus, Equation (40) may he written as

0 t 1/2f~t td, = .o I u~d@ l0•(Fwu) (41)T-z o (o- IO ( WV t (41

where u is the flow velocity in the boundary layer, n is the kinematic

viscosity of liquid, and F is as defined in Equation (31a).

F. WALL HEAT FLUX TEMPERATURE RELATION

Analogous to the principles defined earlier for the gaseous phase, the

wetted wall heat transfer is based on calculation of turbulent free convec-

tion over a vertical plate (Reference 25 and Reference 13, pp. 197-207)o

The convective heat transfer coefficient, h c, for a large range of Prandtl

nuinbers (Pr= C p/k) is given by

Nu = CT(Gr.Pr)I/ 3 (42)

where the synbol Nu is the Nusselt number, (Nu = h L/K), and Gr is the

Grashof number (Gr = gBATL 3 p2 /p 2 ). Using the experimental data given in

Reference 13, one obtains the following wall heat flux temperature

rel at i on:

w= 5.)• V (Pr) 11 2 (43)

30

G. VARIATION OF CORE TEMPERATURE

The variation of the core temperature has been considered for heating

and nonheating of the tank bottom surface. In the absence of the bottom

heating, the core temperature will increase because the hot layer near the

interface moves aown the center of the tank. In this case the variation of

the core temperature can be readily obtained from the general diffusion heat

transfer equation. Neglecting the second-order diffusion terms ann the

viscous effects, the tank core temperature variation equation reads

A _

aT -U(z) a- - (44)

In Equation (44) Lc = ýg(TC- To) where s is the expansion coefficient and

U(z) is the core main counter flow velocity. Note that U(z) can De obtained

from the following continuity argument. Since the boundary layer thickness,

5, is ouch smaller than the tank diameter, the flow velocity in the boundary

layer must be much larger than core velocity. Therefore, based upon the

conservation of mass argiin, ent. the core velocity can he calculated by set-

ting the upward total mass flow ,L z in the boundary eoual to the downward

total mass flow -pA U(z) in the core. It follows that

U(z)p A --- t(z) L (45)

If the bottom of the tank is also exposed to an external heat flux,

additional heat is transferred to the bulk of the linuid. Ir, this case,

similar to the previous analysis, one must consider the additional heat

transfer processes:

1. Heating of the bottom "horizontal" surface (similar to the

analysis for "vertical" wall).

2. Convective heat loss from the bottom to the liquid (similar to

Equation (5). We have

dTwb " - q" - q(3 6w _ = ýf 3rrob czb

31

where

ýlctb T C w,b C Zb)h /3

T : wall temperature at the bottomw,b

Tzb: core liquid temperature near the bottom

The additional heat from the bottom is distributed by the buoyantthermals near the bottom of the tank. We assullre that these thermals are

"buoyantly" strong enough to transfer the heat uniformly through the bulk of

the fluid while the boundary layers remain unaffected. Under this condition

the governing heat transfer equation, Equation ('4), becomes

+3 AC a A(47)

t - -U(z) -z "-pCpVT c b

In Equation (47), Ab is the bottom surface area and VT is the tank volume.

The following empirical expressions (Reference 25) describe the -profiles of velocity and temperature in the boundary layer:

U= 1.8um(-) (1 -(1

-= 1.23 W(2)/ I .L)Pr-1 2 (48)

The governing equations that have been developed for each zone together

with the boundary layer velocity and temperature profiles, Equation (48),

constitute a set of deterministic integro-differential equations that are

solved numerically. The solutions provide internal pressu-e rise history

and boiling temperature which determines the tank response.

"32

SECTION IV

NUMERICAL SOLUTION

A. NUMERICAL SCHEME

A special numerical scheme is developed to solve the governing thermo-

hydrodynamnic equations of processes occuring inside the tank. This scheme

utilizes a stationary and a moving grid that are essential to temporal and

spatial calculations of the liquid parameters under thermal load. The

numerical scheme also includes a rezoning procedure necessary to model the

variation of the temperature in the descending liquid inside the tank as a

function of height.

The governing partial aifferential Equations (1), (10), (11), (17),

(22), (25), (32), (46), (47) are solved, and the height-dependent variabies

are approximated in either of the two grid systems using a time-dependent

staircase (i.e., piecewise constant) function of height z. The staircase

function is restricted to ji#np only at the grid points. Figure 7 illus-

traLeb an i1nitLantdileuUS plOt Of d staircase function of tne core temnperature

T (tz) in a stationary grid z, z1,. .. z n as well as a moving time-depen-

dent grid, - . The prime sign indicates the movingdent grid - zl(t), .. Zn(t) Z+

grid. The vertical axis measures the two corresponding staircase approxima-

tions for the core temperature T (t,z). The time axis, not shown in thisC

figure, is perpendicular to the plane.

Equation (25), describing the temperature of the wetted wall, can be

easily solved by using a stationary grid. However, a moving grid which

follows the motion of the descending liquid is required to evaluate the core

temperature variation. It is important to note that the temperature

T (tz), which is a function of time and height, is the only variable that

needs to be calculated in terms of both grids. This is because of the

convective heat transfer term, "" in Equation (25) which depends on

T (tz). Therefore, Tc (t,z) is calculated once in terms of thL moving gridc cfor the core temperature equation and a second time in terms of the

stationary grid for the wetted wall equation.

33

A rezoning procedure has been devised to couple the two grid systems.

This procedure converts a staircase function in a moving grid to a staircase

function in a stationary grid. Therefore the rezoning procedure enables one

to convert from one grid system to the other. Details of this procedure arepresented in the following section.

Tc (t,z)

II -J

I I I I I , I I 1-zz 0 z I z2 z 3 z4

Figure 7. Staircase Representation of the Core Temperature inTerms of a Moving and Stationary Grid. n = 4

34

B. COMPUTATIONAL GRID

The variation of the wetted wall temperature as a function of height is

calcuiated nunerically, based on a stationary grid given as

zk = kH An for k =0,1,....,n.

In this equation the height of liquid, H is divided into n cells of equal

size in the interval 0 a z 4 H Once the stationary grid is set up, the

wetted wall temperature T can be approximated using a staircase function.wsws

T t,z) = T ws,k(t) Wfor zk < Z < zk

for k= i,2,...,n. s.

As discussed earlier, the staircase function is restricted to jumps only at

points z&,22 ,..., 21.

The variation of the core temperature, Ic (t,z), cue to the motion of

descending liquid is modeled by using the following moving grid,

0 ' < zQ(t) < z(t) < ... < z<(t) < z;+ 1 = HZ.

In this expression the grid moves with the liquid, and zQ=O and z'= H. are

the fixed end points corresponding to the bottom of tKe tank and top of the

liquid, respectively. Then the core temperature Tc can be approximated

using a staircase function in the moving grid

T c(tz) = T' (t) for z0 1 (t) < z < zQ(t)c c,k k•

ior k=1,2,...,n+l.

A similar approach is taken in a related problem by Germeles (Reference 26).

35

The wetted wall temperature equations, Equations (25) ana (25a),

contain a convective heat transfer term which depends on the core

temperature, Tc. Therefore, analogous to the discretization of the wetted

wall temperature, Tws the core is described in the same manner using a

staircase function in a stationary grid. Hence, T in a stationary gridbecomes

Tc (t,z) Tck(t) for Zk < z < z

for k = .1,2,...,n.

A rezoning procedure is employed to convert the staircase function

in the moving grid into a staircase function approximation in a stationary

grid. The application of the two grid systems for calculating t4: liquid

core temperature T (z,t) as a function of height is presented in the

fol lowing.I

Assume the following time grid

tk = tn + kLt for k = 0,i,2,...

and consider the time step from t. to tj+I,

Initially, at time ti, the liquid core temperature, T c(z,t), is known and is

defined in terms of a staircase function in the stationary grid. In other

words, the values of T ck (ti) are known for all k's.

STEP 1. Oefine the moving grid zý in the same time interval, i.e.,

t. < t < tj 1 so that

(a) at the start of the time step, t = tj, the moving grid coincides

with the stationary g,-id for all values of k, i.e.,

zi (tj) Z Zk for k = 0,1,2,...,n

z =zn+1 n

36

(b) the moving grid points zj(t),z 2 (t),.... zn(t) move with the liquid

core, while the endpoints z' and z' remain fixed. Note that the moving0 ~n+1

grid is redefined at the start of each time step.

STEP 2. Next, express the core temperature T (zt) at time t. in terms

of a staircase function in the moving grid . Since the moving grid coin-

cides originally with the stationary grid, it follows that

T' k(tj) = T ck(tj) for k = 1,2,,,..,n.

STEP 3. Iterate the discretized differential equations for T c(Zt)

over time to calculate the staircase expression for the core temperature in

the moving grid, i.e., T' c,k(tj+l) for k = 1,2,...,n+1.

STEP 4. The discretized side wall temperature, Tws, Equation (25), is

solved numerically. This iterative procedure takes the staircase represen-

tatici of T at time t. in a statinnary grid and calculates the expression

for T at the next time- t.... The values of T_ (t.) are needed for thisf-J. %K J

calculation of the side wall temperature.

STEP 5. Using the rezoning method, the staircase function for T c(z,t)

at ti •e tj+I in the moving grid is converted into a staircase function in

the stationary grid. Therefore, the values of T' (t ) are used forc,k j4l

calculating the values of T c,k(tj I) for k = 1,2,....,n in the stationary

grid.

This procedure is repeated to calculate the variables for the subse-

quent time steps,

A detailed description of the rezoning method is presented in the fol-

lowing. Consider the kth stationary cell zk-1 < z < Zk9 and assumie the

simple case where it overlaps only with kth moving cell z' _K< z <z' and the

(k+l)th moving cell z < z < Zk+I. In other words, one has the situation

where

zk_- I Zk-1 < zk < zk < Zk+i'

37

Then the core temperature, Tck' in the kth stationary cell given by the

rezoning method is

T c,k (zk - zki)-l z k _1) T k + (zk - zk)T' k+l1

In the above relation Tck is a weighted average of the temperatures T'ck

and T' in the kth and (k+l)th movino cells, respectively. The weights

are proportional to the lengths of overlap zý - zk-i and zk - zk of the

stationary cell with the moving cells. FigUre 8 shows the relationship

between the moving and stationary grid systems.

Z H ] = H

3H £

z3 - z(t)

-Cn

Z2 =0 2g

HZ I = z 1(

4 t

=0 =0

Figure 8: Moving and Stationary Grid Systems

38

C. MODELING TEMPERATURE OF THE WETTED WALL

The governing temperature equation for the side wall next to the liquid

T ws(t. z) was aerived earlier in Equation (25). It satisfies the following

partial differential equation

aTPwCw~w w - qf - ýrros - ý"czs (49)

where

rros o (Tws .. T

6" = C (T - T )4/3qc ts s Tws

1/3 = CV3 1/3CS Po Cp (5.3)4Pr2 J ivP°C\ r2

-4/3

C V= (5.3) = 0.11

!he above partial differential equdLiU,, contains a deri;vatit, eex

citly with respect to time t but not z. Therefore, it can be replaced by a

system of ordinary differential equations involving ordinary derivatives

with respect to t. Furthermore, the z dependent variables in Equation (49)

are all replaced by their corresponding staircase functions in a stationary

yrid.

The governing expressions for the z dependent variables as staircase

functions in the stationary grid are

Tws (t, z) =Tws,k (t) for zkI < z < zk

TC (t, z) = Tc,k (t) for zk-I < z < zk

"" (t, z) = " (t) for Zkl < z < zkqrros ' rros~k -k

'"s (t. z) q*' , (t) for zkI K z K zk.cs csk<z<

39

and the partial differential equation for T ws(t, z) in the staircasefunction approximation becomes the following system of ordinary differential

equations,

dTC5 ws,k "" - q" "r"~ ($0)#w w w dt -q f ýIrros,k - q Icts,k ( 0

where

(T 54 T 4N)qrros,k = o(ws,k

" " = C s(T ws - T )4/3 for k 1,2qc~ , • , c ,k ..

The ordinary differential equation for the temperature T wb(t) of the

bottom wall, Equation (46), reads

dTCb - o - (51)Pw w w dC ýf -"rro~b ce.b

where

"" £0 (TW. D T 4)

qrrob t

cb Cb (Twb - Tcb)411

C = C h P o C P - / = 1C/

b Pr2] l IV C

C = 0.06

The constants C and C are coefficients within the heat transfer coeffi-lh iv

cients corresponding to the horizontal and vertical walls of the tank, res--

pectively. The quantity Tcb is the liquid core temperature at the bottom.

The quantity Tcb(t) equals the value of T c(t,z) in the first step of its

staircase approximation,

Tb (t) = T_ (t)

40

A fourth-order Runge Kutta numerical method (Reference 27) is used for

solving the set of ordinary differential equations, Equations (50) and (51).

The quantities Tc,k are kept const -nt during the calculation of a single

time step. However, they are up•( ted to new values at the end of each

iteration.

0. MODEL ING OF LIQU ID CORE T MPERATU[RLE

The partial differentia' equations for the liquid core temperature

T (t,z) was derived earlier i Fquation (47); it reads,C --

aTc C Tc Ab L)

-3-+ u - "" (52)at j3z p pV T qcpb

where V is the tank total liquid volume and the vertical velocity U(t, z) of

the core is given by Equation (45) as,

U(t, z) = -ýt(t, z) Lz / (p 0Aj) (53)

Here L is the perimeter of the horizontal cross section and ,t(t, z) is the

boundary layer mass flux, Detailed description of pt(t,z) is presented

in the next section. The left side of the Equation (52) is the material

derivative DT c/Dt (also called total derivative) of T with respect to time.

Therefore, for a grid which moves with the liquid, the total derivative is

replaced in the discretized equations by an ordinary derivative dT ,k/dt.

Hence, the discretized expression for T c(t, z) as a staircase function in

the moving grid becomes

T (t, z) =' (t) for zk (t) < z < z.'(t)c c,k k-i K

for k = 1, 2,...,n + 1.

and the governing partial differential equation, Equation (52), in the

staircase function approximation will Decome the following system of

ordinary differential equations.

dT' Ab ._cTk A b cb for k = 1,2,..,,n (54)

41

Equation (52) must satisfy the boundary condition at the top, z = H ;

Tc (t,Hk) = Tc top(t)

The quantity Tc top' core temperature at the top, is determined by the rate

that the 'nass and heat -.'-e being added to the core top froi,. tne convective

boundary layer flow. C.,responding to the transfornaLion of partial difft,--

ential Equation (52) into the systemi of ordinary differentia' equations

shown in Eauation (54) by application of the staircase approximation, the

above boundary condition will tae the following for," :

(t) = T It)-c,ni- c top

The variables appearing in the liquid core velocity equation, Fquation (53).

are defired in a stationary grid as:

q@t,k~t) = ý'Pt(t, 7 k) l..=

U, ý t) = U (t, z.) for k= 1 -K K

Note that, unlike the core temperature, •t (t,z) and U(t,z) are not being

represented as staircase functions, and are not constant between the jrid

points. Then Equation (53) at the grid points becomes

Uk(t) -t,k(t)L / (DoA) (55)

Euler's method (Reference 27) is used for numerically updating the

values of Tc,k during each time increment At. Also, the values of the mov-cf

ing grid points z' at the end of a time increment for all values of k are

calculated from:

zý(tj+ At) = zL (ti) , At Uk(t.) Zk + At Uk(t .

and z n'n .

42

The rezoning procedure is used to convert T' in the moving grid intoc,k

Tc,k in a stationary grid at the end of each time step.

E. MODELING OF CONVECTIVE BOUNDARY LAYER

In the convective boundary layer, a continuous upward flow of mass and

heat takes place. The motion of the mass in the boundary layer is defined

by the buoyarcy equation, Equation (29). The buoyancy equation was recast

into the more useful form in terms of total mass flux ýt in Equation (32),

In a similar fashion, the upward heat flux is defined and expressed in terms

of 't" The values of mass and heat flux at the top of the boundary are used

to estimate the temperature Tc top(t) at the top of the core. The buoyancy

equation together with the temperature profile (Reference 28) and the heat

flux temperature relations, in the boundary layer, are given in Equations

(56), (57), and (58), respectively.

- (j df ) ptw -F a c (56)

A 1.23 w Pr 1/ (57)

c •4/3 58)

(.3)4Pr2

where

Integrating Equation (57) from p = 0 to ' = 't gives,

t = 0.937 Zw Pr- 1 / 2 Vc1 /4 4t /4 (59)

Using Equations (58) and (59), the buoyancy equation, Equation (56), reduces

to

-z (w t3/4) = C o 4/3 - Ct-P0 W t

43

where

Co = 1/12/[(.937)(5.3)L/3Prl/61PO Poc I •

C, = Prl/2/ (0.937) v/4

This is a form of the buoyancy equation with t (t, z) as a dependent

variable. Note that *t (t, z) -is the total upward mass flux in the bounJary

layer integrated over the thickness, ý, of the boundary layer.

The new form of the buoyancy equation, Equation (60), can be solved for

tne total mass flux, ýt, once the variables Acand Aw are properly

deternined. Analogous to the previous procedure, a:1 • are expressed

in terms of staircase functions in the stationary grid.

,(t, z) = ýw,k (t) for zk_1 < z < zk

and k = 1,2,...,n.

Xc(tz) = ýC'k (t)

where by definition wk(t) and • (t) are

•w,k = (-1ws,k T c,k

c,k = g 9 (Tc'k -To 0

After substituting for w and Xc in Equation (60), and noting that they

are constant in the interval Zk-1 < z < Zk, the buoyancy equation will take

the following form:

a= CoL3w4k for Zk_ < z < zk (61)

Integration of the modified buoyancy equation, Equation (61), results in

', (tZ) = tp/_ 1 (t) + Coý,3w (t)(z - Zk l) for Zk7 K z K Zk

44

and evaluating it at z = zk for all k = 1,2,...,n yields

3,k(t) = [w•f' 1 (t) + C0~ 13(t)(zk - Z )l4/3 (62)tkt) It, k -1 Co A,k k Zk-1 .1i-

Equation (62) is the general solution of the buoyancy equation,

Equation (61). it calculates the value of the total mass flux, i.e. ýtj'

t,2 .... ,at any grid point along the boundary layer. Obviously, since

there is no flow in the bottom of the tank, , tO = 0.

A more accurate form of the bouyancy equation may be obtained once the

variables Aw and tc are expressed as staircase function approximations. The

resulting form of the bouyancy equation, Equation (61), would be supple-

erented by a jump condition on ýt at each zk' i.e., the assumption of conti-

nuity of t at each zk would be relaxed. This jump condition could be

derived as follows: Integrate each term of Equation (60) from zk - 6 to zk.

+ 6 where 0>0. Then let 6-0. The result would be an expression from which

the jump in t at zk can be calculated.

Another quantity essential to calculation of the core top temperature

is the upward heat flux qbnd(t z), with T (t, z) as its reference tempera-

ture, integrated over the thickness, 6, of the boundary layer. It is

qýnd(t, z) C t (T 6dbdpro T9• _

After substituting from Equation (59), the above equality becomes:

f'bnd(t'z) = C2 ýw (t,z)p'ý"(t,z) (63)

where

0.937 v1 /4CC2 = - C P

13 g(Pr) 42

45 -

At the core top the total mass and the convective heat issuing from the

boundary layer are defined as;

't top (t)= ýt (t, H•) (64)

"'( )" ' t,. (65) --bnd top = bnd '

The two quantities given by Equations (64) and (65) determine the tempera-

ture Tc top (t) of the liquid flowing onto the top core from the boundary

layer according to the following equation

"I k d t p(t) ---Tc top (t) = Tc (t, H. I top-- (66)

opZ o top

The eouations describing the flow variables at the core top, Equations

(64), (65), (66), written in terms of their corresponding staircase repre-

sentation, will take the following form; the total mass flux becosjes

St topt) = 't,n(t) (67)

Similarly, the expression for the heat flux at the top of the boundary

layer, written in staircase form after substituting from Equation (63) into

Equation (65), becomes

t)) 4P 3. /4 N (68bnd top( ,t) t) =(t) (68)

Following the same procedure, the expression for the core top temperature in

the staircase form becomes

"'(t)---

T= T ,top(t) = (t,) + qbnd top (69)c, c c) I -C - t top, t) ' 46

P t'to

46i

F. M~ODELING OF VAPORIZAT ION RATE

The relation describing the vaporization and the mass transfer across

the gas liquid interface is given in Equation (22). This equation can be

put in the form

II,

bm

tea lq a in + m) (70)

or

i" : a ln ( + b + (bc

)

ý1" - c_ -

where

h CC -

a =C - -- 0- 9 Ti) 3

pg P,g

=g k Clh1.ýg 1h Vgg

b C pg( T - T i)teff

C rt.

leff

ieff =L + Cp, (T. -T top)

• ý1" Aq,1 rri wrt A• 9--

The surface temperature Ti(t) is estimated according to the following

rel ationship:

Ti = min [(1/2)(Tg + Tc top)' TO],

The boiling temperature Tb is determined by the Clausius-Clapeyron relation

47

and is discussed in the next section.

Parameters riot previously defined are

k = thermal conductivity of the gasgC1h = dimensionless constant for horizontal orientation of surface

6g = thermal expansion coefficient of the gas

= kinematic viscosity of the gasg

S= thermal diffusivity of the gas9

Equation (70) is a nonlinear equation that needs to be solved for i".

This problem is a particular case from a general class of problems in which

roots are to be found for an equation of the form

f(x) = 0

Numerical methods for solving such problems are discussed extensively in the

literature. To convert Equation (70) to this form, transpose the right side

to the left side of this equation, and replace ifi" by the synbol x. This

result, in the following expression for the function f(x).

f(x) = x - a in (I + bX (71)

Then the value x x0 , where f = 0 is the solution, P", of the

transcendental equation, Equation (70), i.e., '"= x0 .

To develop a nujnerical method of solution, some properties of the

function f(x) need to be determined. Assume that a > 0, b > 0, c > 0, which

will normally be the case. Note the properties,

f(x) undefined for 0 < x < c

f + -- as x + c from the right

f -. ÷ as x +

dfx > I for c < x

48

It follows that there is one solution

xo > C

and

f(x) < o for c < x < x0

f(x) > 0 for xo < x

An iterative scheme* which is a combination of Newton's method and the

bisection method (Reference 29) is used in the numerical solution of this

equation. For any iterate x, if f(x) > o, then the next iterate is taken

half way between the points c and x, unless Newton's method gives a point

closer to x. This procedure is continued until an iterate x is reached

where -(x) < 0. After reaching an iterate that gives f < 0, Newton's method

is used for calculating all the remaining iterates. it can be shown that if

f < 0 for one iterate, then all the remaining iterates from Newton's method

will also give f < 0 . -

The input quantity ftol is the absolute degree of accuracy desired in

the solution x If If, < f then it follows that X-X K< f becausedf 0*i t l o-d > 1 for x > c.

G. MODELING OF GAS AND THE UNWETTED WALL TEMPERATURE

The eauation for the unwetted wall temperature T (t) was derivedwgearlier, Equation (1). It satisfies the following ordinary differential

equation,

dT"", C w (72)w w d f rro rri cg

* The method was developed only to the point where it satisfied the present

needs. Further investigation would be desirable to assess its limitationsand its capability. The method of solving the vaporization equation caneasily be improved. The bisection method, which was used in part of thiscalculation, is slow. The number of iterations can be reduced bymodifying this part of the calculation.

49

where

""Ilo = CO T'+ ( - T4

rri= cc (r - T4

r yr i W9 i

""g = C9 (Twg - Tgý'+/3qc9

The coefficient C was defined earlier, it reads,g

Cg = g Clhk~a

The gas temperature equation, Equation (17), is modified to allow alsofor an open vent configuration. The gas temperature T satisfies thegfollowing ordinary differential equation:

dT N-1 r\ =Ma Ca,v+mv Cv~v) c AW - c" Az + .i (Cv, T - cv

-APv/ (M + Mv)] (73)

where

c • -eff rq

The mass inflow rate Mi into the ullage vulume due to the evaporation

is

Mi = A i" (74)

50

The mass outflow rate M 0 from the ullage volume through the vent is

0 for P < Pvnt

M M max(D - P /AP for P vnt< P < P vnt+ Pt (75)

M o) max for P vnt+ vnt

where

(ri),max =) V]d2~ - 1(M /l/2and Pvnt is the vent activation pressure such that the vent is closed andopen for P < Pvnt and P > Pvnt' respectively. The quantity Av nt when it

is set greater than zero, gives a model of a vent that is partly open in tne

Pressure range Pvnt < P < Pvnt + APvnt"

The mUss I IL) drid MaL) of vapor and air, respectiveiy, in tne ui agevolu,;e satisfies the following ordinary differential equations

dM M for P vnt (76)M.-MM ( + o

dt M oMv My a vnt<

and

dMa 0 for P f Pvnt 77)a- MMa/ (Ma+ M V) for Pvnt< P

The four simultaneous ordinary differential equations, numbered (72), (73),

(76), and (77), are solved by a fourth-order Runge Kutta nunerical method

(Reference 27).

51

H. MODELING OF PRESSURE AND BOILING TEMPERATURE

According to Equations (18) and (20), the total pressure, P(t), of the

gas is

P Ma(t) My(t) 1 RT (t) (78)

[t) rI v +

The boiling temperature Tb(t) of the liquid at the total pressure P(t)

is given by the Clausius-Clapeyron equation,

Tb(t) 1 i R in P(t•) -1 (79)[T1I (mol) vL v PI

where Pi is the vapor pressure at the reference temperature T1 ; i.e., T, is

the boiling temperature when the total pressure is P1.

I. INITIAL VALUE CALCULATION

The initial values of the physical paraneters in the ullage volume,

i.e., vapor pressure Pv (0), vapor mass Mv (0), and the amount of air MaO (0)

are calculated using Clausius-Clapeyron and ideal gas law relations.

Using the Clausius-Clapeyron equation, the initial vapor pressure in

th, tank is obtained from the following relation:F (mol) L / )Pv(0) 0 P1 exp V I I )1 (80)R (T1 Tg9(0)

The corresponding initial mass Mv (0) of the vapor in the tank is calculated

by substituting Pv(0) from Equation (80) in the ideal gas law relation;

(mol) vV Pv (0)Mv(O) V (81)

SRi5 (0)

52

Assumning that the initial total pressure inside the tank is equal to the

ambient atmospheric pressure Patm" the initial mass Ma (0) of the air is

obtained from:

M (0) = I )V[Pt - P (Q)(82)a RT (0)

g

J. DESCRIPTION OF THE CODE

A modular code using the FORTRAN 77 language was written to ntsnerically

solve the governing equations. The code was developed in the VUE (VOS UNIX'"

Environment) operating system on a Harris 800 computer. The code consists

of one main program and seventeen subroutines. A description of each of

these program units is presented below. A copy of the source code is given

in Appendix B.

Program TANK is the main program. To allow flexibility, tne program

has tvo time variables, t and t , for gas and liquid, respectively. Theg

Corr -I-espond I ny t me j ricr enrnnL s can 5e re i aten so Ldt at 2 = m At , wferez gai is a positive integer. However-, in this calculation m = 1 was used.

The main program performs the following:

1. Initially, it calis the INPUTI, iNPUT2, INPUT3, and INJTVAL

subroutines.

2. It carries out the iterative nurTerical procedure by calling GKUITAfor each updating of the gas variables by a time increment, -,.9, and by

calling the BUOY, CORE, LKUTTA, and REZCORE subroutines for each updating of

the liquid variables by a time incremert, At .

3. It controls the output time by the variable tpr" When t becomeslarger than tpr, the program calls WL and DVG to complete the calculations.

It then outputs the desired liquid variables as well as the desired gas

variables on the VOS files called OlJTL and OUTG, respectively. Similarly,it outputs the interface and the remaining variables on the VOS file OUTI.

Finally, it increments tpr by the amount Atpr to determine the ncxt output

time.

53

4. The program stops when tpr exceeds t in I .

Subroutine INPUTI assigns and calculates constants and initial values.

The initial values assigned in this subroutine are the initial time, t =

TIMZ, and the initial temperature, T = TZ. To change the values to be

assigned in INPUTI, the source file which has this subroutine is edited and

recompiled.

Subroutine INPUT2 reads the VOS file iNTK to get the following values:

the maximu-n time t = TIMMX; the time increments At : DTIML, At = DTIMG,nmax 9.garid Atpr = DTIMPR; the paraneters T dfto TDFTOL, x toI XTOL,

fto] = FTOL, and n tol= NTOL which determine the accuracy of the iterativecalculation in Subroutine EVNEWT; the number n = N of computational cells in

the stationary grid zo, z.... ,Z ; and the vent parameters Vopt = VOPT =

VENTOPT = ("Y" for vented option, "N" for unvented option), Pvnt = PVNT, and

9 vnt = DPVNT. It then outputs all these input values to the VOS file

OUTTK.

Suoroutine INPUT3 interactively obtains either a fire diameter, DIAiF,

which it converts to heat flux by a table (when provided), or the heat flux,

ý- zQF, directly. It then outputs these input values to the terminal an.

also to the VOS file OUTTK.

Subroutine INIFVAL initializes the temperature variables Twb, Tws,k_

T c,k , Tg and T ct. It also intializes the two time variables t and tIt calls INITM to calculate initial values for some gas variables. It

calculates the stationary vertical grid points zo, Z1 ,...,zn for the liquid

part of the system.

Subroutine INITM calculates the initial values for the vapor pressure

P for the mass M of vapor in the ullage volume, and for the mass M of

air in the ullage volnrie.

Subroutine GKUTTA updates the gas variables T ,Tg Mv and M over a

time increment At g by using a fourth-order Runge Kutta method for solving

the system of simultaneous ordinary differential equations for the gas

variables. It obtains the first derivative with respect to time of the gas

variables by calling subroutine DVG.

54

Subroutine DVG calculates Ai' M 09 and the first derivative with respect

to time of the gas variables Twg, T9 , Mv and M . It calls subroutine WG to

evaluate other variables appearing in the differential equations of the gas

variables. The other variables are functionally dependent on the gas

variables and on Tct.

Subroutine WG calculates the variables which are functionally dependent

on the gas variables T W T , y, M and also on Tot First, it calculates

cg P P P, and T It calls subroutine SURFTEMP to obtain therro cg a v' I

interface surface temperature Ti. It then calculates r r, ano Leff.

It calls subroutine EVAP to obtain the value of the evaporation rate i"

Then it calculates q""

Subroutine EVAP calculates the value of the evaporation rate •". It

calls subroutine EVNEWT to solve the transcendental equation for in".

Subroutine EVNEWT numerically solves the transcendental equation for

the evaporation rate ,i" using a combination of Newton's method and the

bisection method.

Subroutine SURFTEMP gives an estimate of the value of the surface

temperature T..

Subroutine BUOY calculates the boundary layer mass flux

*p (t ) = t (t z k) for k = 0,1,...,n at the stationary grid heights zo,t.,k z t z

z1 9.. , 7r, by using the analytical solution of the discretized buoyancy

equation in which the side wall temperature Tws(t, z) and the liquid core

temperature T c(t, z) are expressed as staircase functions in the stationary

grid. It then assigns the value for the boundary layer mass flux at the top

t top tn(ta) = pt,n (t ). It calculates the value for the boundary layer

heat flux at the top,

qbnd top (t) X qbnd (t ).

55

Subroutine CORE calculates tne grid heights z' z .. at timeI' n

t + At of the moving grid which initially coincides with the stationary

grid at time t and moves with the liquid core during the time step At It .-

then sets z' z . The temperature Tc top (t + At ) at the liquid coren+1 n to L. 2

top is calculated. The temperature T c(t + At , z) of the liquid core at

the end of the time step At is calculated as a staircase function in themoving grid, i.e.

T (t + At,, z) = Tk (t. + At . ) for z',_< z< z

c z 2.k

for k = 1,2,...,n+I.

Subroutine LKUTTA updates the variables Twb and Tws,k for k = 1,...,n

by a time increment At using a fourth-order Runge Kutta method to solve the

system of ordinary differential equations for the variables. it obtains the

first derivative with respect to time of these variables by calling subrou-

tine DVL.

Subroutine OVL calculates the values of the first derivative with

respect to time of the wetted wall temperatures T and T, for k =wb w~1,2,...,n. It calls subroutine WL to evaluate other variables appearing iIl

the differential equations for the wetted wall temperatures. The other

variables are functionally dependent on the wetted wall temperatures and

also on the core temperature Tc,k for k = 1,2,...,n.

Suoroutine WL calculates the variables l'rrob, qc•' krros,k' and

"" k wlhich are functionally dependent on the wetted wall temperaturesqczs,k,w

Twb, Tws,k and the core temperature Tc,k for k = 1,2,...,n.

Subroutine REZCORE takes the following staircase representation of the

core temperature in a moving grid

T (t + At,, z) = TI (t + At for z'..1 < z < z.ccK 2 k

and converts it into the following staircase function in the stationary

grid

Tc (t, + At., z) = Tck (t + At9 ) for zk < z < z.

56

SECTION V

NUMERICAL RESULTS

The physical processes occurring inside a fire-exposed tank containing

liquid flammables were identi fied and analyzed, and the corresponding

governing thermohydrodynamic equations describing the physical phenomena

were derived in Section III. Numerical solutions of the governing equations

and modeling of the tank response prediction to an external thermal icad

were presented in Section IV. This section describes a special numerical

scheme that was developed to calculate the response of the individual corn-

ponents comprising the fire tank systemn. An integrated response methodology

was developed based on the response of wetted and unwetted wall terpera-

Lures, gas and liquid phases, convective turbulent boundary layer, interface

mass evaporation, and the overall tank pressure rise.

Calculations are made to predict the response of a vented and an

unvented 35-gallon (206-1iter), 18-gaqe steel drum to a uniform external

heating of 7 w/cm2 . For the vented cases, the vent diameter is 7.6 mm and

the vent opening pressure is 41 kPa gage. The vent coefficient of discharge

used in the calculations is 1.0 for the Freon- tanks and 0.25 for the water

tanks. The drumns are filled up to 95 percent filling capacity either with

water or Freon-13T". in the calculations, it is assumed that the druns are

completely engulfed and the total incident heat consists of radiation as

well as convection.

The response predictions of the wetted side wall and bottom wall tem-

peratures for vented and unvented configurations are shown in Figures 9 and

10 and 11 and i2, respectively. The calculation shows that the temperature

of the wetted walls strongly depends on the liquid in contact and is not at

all affected by whether the tank is vented or unvented. It is al so noted

that the bottom wall invariably shows higher te'nperature than the side

wails. Unlike the wetted walls, as ,night he expected, the temperature of

the dry walls in the ullage volume is higher than the wetted wall

temperature and remains the same for both liquids (Figures 13 and 14).

The predicted mass and heat fluxes for water and Freon-113" in the

boundary layer for vented and unvented tank configurations are shown in

5-/

-- --- -- - T - - -r CT:-)

Kc I

(v 4-34-)

4-'

I - 0)(Df

SCL)

2DCD)

-4-K \ 0)

1DCD C) C> Ccli c C-')C~i0c qr CY) -IN _anLjdq ~mpspqa

I 5-

0

1 7 1 I

O0) a) 4).

.. JLL- 2

44-

CD-

4-3

CY)

100 C(\j~~ CC)IW c

Ln ClH

59-(

0) a) 4-3

I CD0 4-

M - c0

S-)

C14 0

4-)

C) U

~ E 0

C D C : C D

60~

C)CD

~ QJ 4--

3-

C.L

KD CD C:)

611

T' - S S'FS '-S ---- i_ T-T - -T-I-- - C)

.i14.

0

I--

E0

Ito

S.-

F -F

CDUC-1

C) CD

62

I I I 1 1 T 1 1 t~tT T 1 II ]a

0c

~ w .CD

E-=

009N IaA~)A~La LRA

I -i

4-

0

aIi

4--0 -V)

I * E

4ý 1

4-e

-ic

s-w/r' 'xfLI4 qaq jaA~L kjvpunog

64

C7aJ 4-ý

4.)

4 J

4-3

*1 a)

4-

CDLEr

4- XI- v4 O'Lkpuo65j

K1 0

VCIO

s-

I-

Chi C.)

CI Ct,-CD C;) C

s-ujbý xnt ssw jaýetýjvuno

66 (

C)

a))

j- U- I-,

-~caI

cs- IWOQJ A

rD C)

C) CDC; 0

S-Ll/bý XtlJ ssui -akeL ka u-o

Figures 15 and 16 and 17 and 18, respectively. Figures 15 and 17 show that

the rate at which heat and mass are transported along the boundary layer is

independent of the tank configuration and is controlled by the physical

properties of the lading. It is noted that the Freon-113 Th high rate of ,1mass

flux in comparison with water is evidence of this phenomenon.

The predicted pressure time profiles in the ullage volume are shown in

Figures 19 and 20. As shown in Figure 19, in an unvented tank, the inside

pressure rises quickly (in less than I minute) to a level that no ordinary

container can withstand, and consequently the tank ruptures. However, the

pressure rise inside a vented tank is more manageable. The pressure rises

sharply until it reaches the vent operating pressure value at which venting

takes place (Fiqure 210). The oscillation observed is caused by the opening

and closing of the vent before it becomes stable. .eyond this point the

pressure continues to rise, hut at a slower rate because of the expulsion of

the tank contents. With increasing time, the internal pressure rises

steadily, causing the tank to rupture in less than 200 seconds. The dry

wal" temperature for hoth Freon'T and water was about 900 K for the closed

vent tanks (Fioure 13). For the vented tdnks, the dry wall temperature was

essentially the same as for the closed vent tanks for Freon-113'" arnd

slightly less for water (Figure 14). The wetted sidewall temperatures were

essentially the same for both the closed vent and vented tanks; that is,

about 400 K for water and 500 K for Freon' (Figures 9 and 10). The model

prediction of other physical parameters sucn as mass of vapor in the ullage

volume, boiling temperature, and gas temperature are shown in Figures 21-

The predictions from the calculations shown in Figures 9-26 are off as

a result of inadvertent omission of a constant. The liquid density factor

(RHOZ) was omitted fre the expression for the constant CZ in both the water,

and Freon'" versions of subroutine INPUTI. This resulted in very small pre-

dicted boundary layer mass and heat fluxes. 1he code was corrected and was

also improved to include the effect of venting on the rate of change of the

gas tenperatue, line corrected code predicted a houndary layer mass flux

that appears to be much too large. Because of project resource limitations,

this was not investigated further.

68

I I I

SU) (3.)

W S..

I-4

V)

S.-

S.-a-

LnL

CD .C C) CD CU-)

69

-T---'-TF--7-7---TcT

T1 -F F1F [FTT~ r 1 o

I '4-0

4-)

4--

cJ

o * S.-0

Q> 0.

4--

ai)

C;

Kn Ci)rLn cU.-_-

zuL/q[ '~aflSsaUd

70

C-)

4-3

CDC

C-)

F ~4-0S.-

Q-

C',C

IIj

5.-

4-

r 4 C,)

U-)

~w/ 5b '.1od',A JO SSQWk

71

m~TTrr~r 7 -rT-r--7 7-T

4-.

Ca-

11r I3

Ln

II

F 22

INI,

C C Lw) 0 a:2

-1 1

o cn

o aiFA

0Ku

ja

F--

CdCFl IC1cV;)n eAd,) bU-4O73L

C~) 4--

cW L.d

0

4-

I -4-I

4-

0l

cu 4

CD~

74J

a) -- r I

-CD

E. 77I -(I

CL

CDC

~V)K -s

75-

CD

__jU-4-R-

C:) 4-.

CI

ICDCD --0 )DC)

Ln Ln"It o Cw

IaAqj da se-

76i

SECTION VI

CONCLUSIONS

The govLrniný equations describing the irterac'ive processes occurring

between the fire and the tank have been solved numerically, and -1 moJel

predicting the ther'no~luid proces.es leading to occurrence of BLEVE ha., been

developed. The calculations were made for vented and unvented tanks

containing either water or Freon-113". The prediction demonstrate4 internal

corsis':--ncy amongjst the para:meters indicative of the mooei perfornian:e.

Based Oa. - ,ysis of the calculate' prediction, the following general

obhervatiors cre made-

The ;pessure risn in the ullage volume and occurrence of BLEVL

s stro rr y depend on the tan,. s in iial fiIi ng density.

2. The physical properties of lading, pi)ec:ally the heat of

vapo-izatior, s gnificantly contrihute to the intornal pressure rise.

3. The liquid/qas interface is the most ther;m1ally active region, and

a dýcreas~i ic temperature radblnt exists in tle donw.ard direction.

.. Only the liquid boiling temperature is affected by configuration

(vented -r unvented) of the tank; all other temperatures remain unaffected,

recardles of the tank configuration.

Specif -ally, to delineate several fundamental issues concerning this

developnr•.nt, -hi following preuiction case is considered. The calculated

response for a vented 55-gallon, 18-gage steel drum, engulfed in an intense

fire (7.0 -w__) and filled to 95 percent of its ca.. city with water showedcho2

that

1i The tank failure pressure is reached at about 200 se-onds. The

calculated pressure for, a closed tank approached values much higher than for

the vented tank for the same time.

77

2. The dry wall arid wetted wall temperatures are changed very little

by tank venting.

3. The internal pressure in a vented tank rises much more slowly than

the pressure in a closed tank.

Similar calculations were made using Freon-113t " in place of water while

maintaining the other variables. As was expected, the calculated response

predictions for Freon-113T were quite different when compared with tne case

for water.

The corrected code predicts a significantly higher boundary layer mass

flux. Causes of this problem might be the simplifying assumptions made and

the empirical fits used ir, the derivation of the model equations, or the

numerical scheme used to solve the equations. Project resource limitations

-id not permit letermuination of the cause and the resulting effects on the

predict -ons

The fa-lure response of the tank, and ultimately, the occurence of

13LFVE also depend on the size, and the mechanical, thermal and physical

integrity of the tank. !hose, together with the variations in heat fluxes

from various fire sizes, as well as differences in the heat input caused b-

the relative location of the tank with respect to the fire, indicate that.

every response predi:tion calculation requires a myriad of different inputs.

This becomes a laborious and time-consLIuling operation, not consistent with

the purpose for which the model was develope~d. The mission of this

developreit is to quickly iprovide to the firefighters the essentialpredictions and the risK" asse3sment data necessary for safe operations and

successful extinguishment of a fire.

To achieve this goal without compromising accuracy of prediction, it is

recommended, cnce the model is refined and verified, that prediction

calculations he made for most probable ranges of fire tank situations and

that a B.EVE prediction data base be created. The results from this data

base could then be plotted into a series of reference charts, or stored in a

computer and selected and displayed on the screen.

78

REFFRENCES

1. Birk, A. M., and Oosthuizen, P. H., "Model for the Prediction ofRadiant Heat Transfer to a Horizontal Cylinder Engulfed in Flames,"ASME Paper No. 82 WA/HT-52, The American Society of MechanicalEngineers. National Heat Transfer Conf.rence, l•8•e7 w York, New York,1983.

2. Collier, 3. G., Convective Boiling and Condensation, McGraw-Hi1,lLondon, 1972.

3. Reid, R. C., "Possible Mechanisms for Pressurized Liquid TankExplosions or BLEVEs," Science, Vol. 203, p. 1263, 1979.

4. Robeits, A. F., Cutler, 0. P., and Billinge, K., "Fire-Engulfed Trailswith LPG Tanks with a Range of Fire Protection Methods," Proceedings ofthe 4th International Symposium of Loss Prevention and Safety Promotion-7-T-eTFocess Industries, Vol. 3,_ Che-m-calProcess Hazards, 1983.

5. Venart, J. F. S., Sousa, A. C. M., Steward, F. R., and Prasad, R. C.,The Physical Behavior of Pressure Liquified Fuel Tanks Under AccidentConditions, Fire Science Center, University oTf N--ewL--urnwlK,redrUcton, N•.w Brunswick, Canada,

6. Hashemi, H. T., and Wesson, W. 0., "Evaporation Rate of LNG,"Lt,'d r-a r. POr! cc jingq Vo/l 50, No- , P. 117, 1q71.

7. ,arackat, H, Z., and Clark, J. A., "Laminar Natural Convection in aCylindrical Tank," Proceedings of the Third International Heat TransferConference, Chicago, Illinois, 1966, p. 150.

8. Evans, L. R., and Reid, R. C., "Transient Natural Convection in aVertical Cylinder," Vinerican Institute of Chemical Engineering (AICHE)Journal, Vol. 14, 1o. 2.

9. Aydemire, N. IJ., Sousa, A. C. M., and Venart, J. E. S., "TransientThermal Stratification in Heated Partially Filled HorizontalCylindrical Tanks," ASME Paper No. 84-HT-60, National Heat TransferConference, 1984, The Anerican Society of Mechanical Engineers, NwYork, New York.

10. Virk, P. S., and Venkataramana, M., "Modeling of LNG Tank Dynamics,"Department of Chemical Engineering, Massachusetts Institute ofTechnology, Cambridg-, Massachusetts.

1I. Birk, A. M., and Oosthuizen, P. H., A Thermodynamic Model of a RailTank-Car Engulfed in Fire, Canadian Thft-tute of Guided GroundTransport, Queens University, Kingston, Ontario, Canada.

79

12. Del ichatsi os, M. A., "Exposure of Steel Drums to an External SpilIFire: Practical Tests of Requirements for the Storage of DrumsContaining Flammable Liquids," Plant/Operation Progress, Vol. 1,pp. 37-45, 1982.

13. Rohsenow, W. M., and Choi, H., Heat, Mass, and Momentum Transfer,International Series in EngineeFng, h r--ntce- TTý-ng-lewood Cliffs,New Jersey, 1961.

14. Kaienetski, F., Diffusion and Heat Transfer in Chemical Kinetics,Plenum Press, New York, 1964, p. 184.

15. Baines, W. 0., and Turner, J. S., "Turhulent Buoyant Convection from aSource in a Confined Region," Journal of Fluid Mechanics, Vol. 37,Part 1, pp. 51-80, 1969.

16. Schlichting, H., Boundary Layer Theory, 6th Edition, McGraw-Hill Seriesin Mechanical Engineoring, New York, 1968.

17. Delichatsios, M. A., "Turbulent Convective Plows and Burning onVertical Walls," Nineteenth International Symposium on Combustion, TheCombustion Institute, 1982.

i•. William, K. G., and Capp, S. P., "A Theory of Natural ConvectionTurbulent Boundary Layers Next to Heated Vertical Surfaces," Int. J.Heat and Mass Transfer, Vol. 22, p. 813, 1979.

19. Kutateladze, S., "The Model of Turbulent Free Convection Near aVertical Heat Transfer Surface," Heat Transfer and Turbulent BuoyancyConvection (Spalding and Afgan, eds.-T-Vo-F--11,-p ,-487 -enispherePlbis'hing, Washington, D.C., 1977.

20 . Morto, B. R., Modeling of Fire Plumes, Tenth International Symposium cnCombustion, Th m-e-Cobus-t t--n Ins u-ute, Pittsburgh, Pennsylvania, 1965,pp. 973-98?.

21. lHatcnelor, G. K., Introduction to Fluid Mechanics, Cambridge UniversityPress, 1970.

22. Hinze, J. 0., The Theory and Mechdnism of Turbulence, McGraw-Hill, NewYork, 1950.

23. Bary, K. N., and Libby, P. A., "Interaction Effects in TurbulentPremnixed Flame," The Physics of Fluids, Vol. 19, 1976.

24. Kaplan, W., Advanced Differential Calculus, Vol. 19, p. 1687, Addison-Wesley Publ iýThTg-Coia-ny, Inc., -R-e-a 1n, Massachusetts, 1959.

80

25. bel ichatsios, A., "A New Jntegral Model For Turbulent NaturalConvection Flows Next To The Heated Walls," Fifth Symposium onTurbulent Shear Flows, Ithaca, N.Y., 1985.

26. Germeles, A. E., "Forced Plumes and Mixing Of Liquid In Tanks," Journalof Fluid Mechanics, pp. 11, 601, 1975.

27. Carnahan, 8., Luther, H. A., and Wilkes, J. 0., Applied NumericalMethods, Wiley, New York, 1969.

28. Del ichatsios , M. A., "A New Integral Method For Turbulent NaturalConvection Flows Next To The Heated Walls," Fifth Synposiutn onTurbulent Shear Flows, Ithaca, New York, 1985.

29. Conte, S. D., and de 8oor, C., Elementary Numerical Analysis: AnAlgorithmic Approach, 2nd ed. , McGraw-Hi 1, New York, 1972.

81(The reverse of this page is blank.)

APPENDIX A

DERIVATION OF EQUATION OF MASS TRANSFERSTAGNATION FILM THEORY

Consider a thin stagnant layer of fluid with thickness 6, Figure A-i(Reference 14, p. 135).

Tg

Figure A-i. Stagnant Film Layer

Assume that due to the high temperature gradient, mass is vaporizing at

the free surface. Define the layer thickness so that

K/6 = h (A 1hc (A-i)

The conduction equation in this layer is

2aT T

PC T K L (A-2)2

ay

If r" is the mass rate of vaporization per unit surface area, the conduction

equation (A-2) can be modified using, v = ih"/e, to read

2

Cm" l = K LT (A-3)P Y ay

The differertial equation, Equation (A-3), is integrated. We have

C K exp (Cp Py/k) ' 1 + C (A-4)

83

Applying the following boundary condition, one obtains

at y = o, T = T ; at y 6, T = T

The solution then becomes

KL

T - Ti C - Exp (C i" 6/k) -1 (A-5)1 Cr~i"p

The energy balance at the surface gives

r~i" Leff con + q" =k + " (A-0)

From equation (A-4)

K T- --- ' -

and hence from Fquation (A-6) we have

CIK m" Leff (A-7

Substituting for C,K from Equation (A-?) into Equation (A-5) one obtains

c (T - T) = (Lef ) ep Cpe ii 61/k -1 (A-8)Cp T9 Ti Left q r,2 p.

Equation (A-8) describes the irass transfer rate which can be solved for m"

to give Equation (22).

84

APPENDIX B

PROGRAM LISTING

85

£L.z t1 r;c:3 czr in 1rc: thc ttic Er~ncry)crt,.Cyn a'i, C ur::t!.Is Cccurri n9

nf24 t.' n;r ti r;i -..h t Trc v1i tr ? ifr pI n:Tjmfr t . Tne n~rccrrvr isI,ýtiols .rcue4h t iI1 %Lo -*:tr~r hc-ct titx er fir?3 cizmeter to ba

1n I nr 7r.-r rttr i nt tho r C>: 7 7?1 bn:L'C,0 . Th? codeQ: r flU i r x1 v. r.5O uNix Cnvironn~ent) ooer~tin,7 system

-'s- n z > c:,*ut?r. it ccnzza~ts cf one Trjn PrOCre'.n and

St a . -. *.**.n 1ta. PL**T 1 .

C. rk.ECI, 'J.?rsicn

L~¶O Hv 0% £CETFLCM/ KCc,..-E, E1i,CZsCCE 2C8 CS/CG'CEV

I.2 vO Tc; AVN IC---.xTPV-NT / SVNT

XLL

L~"C--

L F

AV L

S 7 cC~J C

ZNLC 7. 7-

X?.UL 731.4c2

86

C-ILPH = .711ESI

ALPH1j AKCI/(R!iCALFh~*CALO)H)

C 1 i-T C.G6S1 --1 C.Oc

XKG.*ClhT-~( 3tIAG*G/(Xt\UG*ALPHC) )**(1.0/3.C)C:V= X,..G*ClHl*( (EETAG*G/(XNUG.*ALFHGV))**(1.O/3.O) )ICPG

XhNLm= C.93-'#k(5.3)**(4.C/3.C)CpAC1 (XNUGC**(1 .C/12.C))/(XNUM*Pk**(1.O/6.O))C Fý'Ce 0. 37-( XNUC*(1 ~4 .0) *ýCP/(P.R**(0~.5))C, = CFtC1*QHCZ

C2 /( z*GC lvs C. 11ý1I-3 C.6tC0- C S* C1 ~cl vT I' L C.-T7,Z F 7 L '

T 7\=AVI\T = .:clý-5C 'V'T -I .?VNT =1.4209ý5.APVNT 3RE&T u .I

c-N L

SU2:--UTl?'. IIPU1 1C T~V~rsjcn

CCIV'CN /PHY SOM/ RiCiFA'CC ýM o N /C'ECKM/ HL.,XLLI AL,&.-A~-,A,V.,VT

IMCIT' /"VTPCtAI YOLA , X'C L V, rCo kd'(Zt E TA XLVt XLeTlPlCCP,'P.N /11i:ATCPCt'/ CPzCPLeCV PCVVCAV,,CPC-COP '-'ON /,-EATFLC.M/ kCCoWEPSICZ,C2,C~,CS,CG,,CEV

C M "N /V.-?,TCM/ AVKT,CDONT,,PVTtDPVlIT

riL r

XLL .b;3.45*L

E~ A L

v T AL*-LXI'CLA 29.XIAC LV 1 S. C;rlCZ =1.OE3

;;ETA 1 .2"-4XL t;2 25XLV = XLTl = 3-73.0P1 = 1.0132E3CPL =4.6ECP = CPICVF = 1.90EjCVV= 1.44~E3CAV 7 95 .:

C Pt z CVPRCCW 4.2E

EPSI = PSXNLC = C8-PR 6.E

CS R-CZ*CP*( XNUC*eSTA-o,/((5.3**L)-PR~**) )**C1.CI!.C)

87

XC .1 L P 1L 1P , ,

C L. = XK-C/ýT(CA i4 TAGMLI. C¼1) LPm

C= XKG'*C1Hl c (E TAGG/(lX NL - A LPH "C-*103.O ) /CPC,

CX J CLYK. 1 = CA3X(NU3PwP4kCU36C0

CZ = CFLClI=*;iHJ!C2 = CF.Cc./(SETA-C)Clv 1vS C. 11

Cl- .C.O

7C5*l'5/1V

i4 C t1/

'LC'( /T LQ' T-D F 1V L ,XTuL, FTOL, NTCL

5 :N /.' PvNc4 ArCV K-T P V.%\TfA

1 ,~M~t *1N VlkTOFVcrT~~: *3Nvcr

,.rM.(S), P1LC='2CCýCC -,R4A 0*D&TK')

% TA;ý( 5t , * ) T T:ppx'X

wciýO(5T r

W:2(5 1 *)!1) D- P ý T1:p

1:.(5 T 1 ( Tc, T~, T

Rwq;5>* TFTL XTCL TFIL, XTCLTCLC

'.c~(55,88

T*%L XTCL') )T L, '

v.lT k c~ iC T X,,.,(Xl3')))VGTFNT.PN

~ I T 1

SUiýCUINt_ l'PUT3Th.1SsuScroutife oczn xtoel ti-at fluxes from incut fire

C. diernet.rs ty mi ct F~ tire d1irnexer vs h~zt flux t.~ble (%~hen: r cv i ae c. Alse, It-1! su~rCi.tln* i5 Tlexihle erou;?' to irncut

P * 1 N

i F.L C AY F L j C 15L in fIt. x is oot zin-ae re r Tr or t na tir~ 'i r.& ZetL vs hezit f lux tz-ch- (u~h4r. rrcviaec).

PýINil 'C" :ir? Zie~nexcr (nicters) =~35) A

Fi(INT

GC TO cC.15 CGINTINLe

P'FINT *,"IPUT impoSad Heit Flux (watts/mieter.*2)."

PRINT *

PRI.NT 'C"Imijos d r-ezt Flux (wetts/inetar**i) "13.),CPU~NT

WR~ITý(e5, 'C' " 3X.tEl3.5)') CFW.RlTEWc, *

ZC CONTINLEPRINT *,, "'k5SS P :1 LPFN to Start Prc;rim."

R £ 7004"

NjiROUTINE 1NITV.ALP .R AMIR E C r I f, wz-1 CC )

C ýP MON /INI!TCMI T!1ZtTZP.AT!NF

CCMION INCv/ NC CP PO0 IV LCM/ T-It 6 (1 :(M) 7C (CCM),-TPrZC] i-+ ),T I NFe,-TNFS

L WAM 0 N /V'ýCM/ ¾.'jG,lC, XVV, XFAPT:Ki~TC utPm N /wu..Cm/

T 111L z I1M Z

89

TAN TiIC& T1ITOL CTIh

X:M

FF .F'I

PIF = C2NF

1IFSzINF

LCALL 1 K IM

k T UIR N

C LiN C &EYC M/ 'L X L L, 4LA aVV TLA;*C'%N VI7i AtP¼ZCiý XP$LX"NC L V oH,.IL/I1CZ lý U, XLvXLT1 ,P1

C L-N QO. /vCMý/ T `

rV ?'-:A,-( (Xr$ILVsXLVI.On)(1.C/11 - .C/IT))

11V A , 7

ýC Sk-A?, Ta-NKP J.. :4¶v7rt ( I1zwL1CLCCrMC. N / 1hITCM/ VirZ TT Z NrQCr'ML N / 71! 1CM4/ T In'X:T. ,:IvGUTIMPR,-TIMLTIM3,1ImDRCC+PMCN /NCl"I NC> VM C N I VL CMI T i,1$IZ;k)SC C :C M. TPC(IDOM +1 )TI!N F .TI N rSCDI;'".ON /ALCMI PW:,CE rFr 4/~SC1J)-ZL(1MC CPfM\O0N / VG CMP/ t1 17C , XM V , xr1TI KF TC C11ivCN /w*-CPI/ -,FTe'ý;C,.CGFeA,-vPCRRIAQLXL5PFCXMC*CL

PLFS I z1.45C4E-:.4CA4LL I N FU 7iCALL INPUTZC 0L L INP\uT!CALL INITVtLCFU'A65, IEs.ZC.43Tk, PEN (ce;, F LEz -C C CnAR-.'vlG')

WAITE (65,f) T '-L TnS(1 ) TSN)(.Tc l ) c (h)"

wR ITc cr Tt Thi TGxmv x F;A (PSI.) CXMI YXM3,

6) ')

k ITc (cf t'C PSTTOP ý-ANOTC P TCT

To cr TIL IM"P

1C CCNT1NLý

CALL 6L

90

CALL $Vt4

L T:N *, LlLtTV )

fl~lt, C 2x)( 1.) ImG., T.G T-;, VV xvXML,'oP :,zxI,;'Yt~o

Trn T i~r I *

cL CQNT iNL~

L LjN k T~ 7t

CT1, TT M 1 1TML

c .N T *,*

CALL £LCY

CPiLT ', CB

,-LL LSLTT.

CALL &E..CCC±

I T !, L 1- 1 vPW ;T

S-TM *L T .vu:2C.c*1I.) IL TO: 3.

CFC Af.YEC,, 1 .1Z1C T I.CZC2C

CONMON IIN.wI NCOC NMON / V LO C ImT3A, 1 .s 7c ,u: M T P CoM+1 1 ,IIF TIN SC Crfr0ON IVYOM'1 AQI)PTOIM,(:IC~Z (CIZ'OCr' 0 ITONF1MI FT1, I v ".NJo0101F , CIITIp

i~ C 1L <

~S1I( K) z2 ' 1 :I-l.(. .C) + LZ ZZ* LLIT*1./30) )*4.Ot.OC

P Sl ~j P3T( N)

0 7 IP =C 2 ;L T,4F S TC**3.014.C)R -I ' UXN

PARAMETER ( O~~ )COP!'CN IGe:CVCMI/ HLXLL, eLZAA~t, V,'C CIOMON /jYý7 PQP CgA / X 1 LA', X MC LV,~ o 3ýh0tPh0Z,,ItFX L VXL T 1 'P1CCOrMCN /r:ATCPCMI. 0',OFLOVF ,OVV,OAVPC,'

OO?'"O)N /CI', !/

CCV'MON ICVLC.U/ rT.,:,OTh (Uk) ,CT~C

OC . eON IvY,-.l 1('2 :IC),FST(L :I .M),L(I'-: iC.M),LFP(O: 41)

WVC14MCN /TCFCP' 0SIIOP,..e:NY!CPICTul :,,T*

TOT 1L(r,) * :I>L-.zIPC

91

U~c)C('()

!ý(K) *(K)* 6T;FL*U(IU

~.r(ZP(K) .LT. !P(K-1)1) 'OIC 4-C

4C T 1 :m L.

V~. L T I ) N

KK

ý(: 14) ;C

u.. T..ý4L CONTXINLE

ipAJ .GT. 1!-) GC TC t- S 4 (.,(J)2Z0 (.-1))-T'C(j)

S = S Z(G1~(-)*F~S1 + CZ(K)-"P(IH))*TPC(Iw+l)

i. LONIsiNLElr(& .Ll. \~) GCO TO 1L'.

SLt;UU7IN-. L'(UTTAPAk~tmýTER C 12'zlrCCC v P M UN /1~I I ,IC/Tlrp[T 1;4LtCTIFG -ST I NP;ýoTItIL r:~,G-T I M~CC6 !A N iNCV/ NC:2*fCN /VLCv/ V(IUI.41)

LýfrMON /IývLCMI CV( I Cm1)

H9 z TID'L

vl(K) zVC%,)

92

S (K) 2 VCK)V(K) a VI(K) + CV(K)M/2.i

2C CCNTINLECALL OVLýc 3,: K 21 , p

i(K u(K SW+2* *VS02'41 (K) 'rC h)h

Su. CGTINLECALL D'VLOC 40 u,

S~) (K) 2* *YK

4C CONTINLECALL DVLCU 5C Kt1,?'V(K) = V(K) + (S(K)+.V(K))*tl/C.C

.ýL CCNTINLE

ýUtý."0UTINE ZýVL

C~01VO /Hi~bT=LCf/ ýC joPSjCZsCŽCBPCS.CGoCEVCOMMON INCV/ N

(.4LL 1WLGT%; a .; - -RO

wj16 K=I,N5Tw.S(K) CPQS - ýRRS(K) - -.CLS(K) )/PCtDk

1C CONTINLERel LAN

4LJýCLTINJE ALPiAIcTR l-0%'Z:1CC

LCCf''0N /HETFLCrI t'CCWEPS IoCZiC2PCS,CS,CtCtCVCCIly?'OtN INCI/ NCCOrMMON iVLC"I/ jM,6(I-1),TC(1I4)PTPC(IDM+1 ),TtNFeoTTINFS

QC21=CE*TCIFF**(4.i0j3.G)C.C 10 Krl rK0iZRCS(K) !PSI* (71S(K) **4 - TIhFS**4)

7.CLS(aK)()-C()

kcTURN

SLtR0UTI.N~ ýKUTTACC~MC'N /VT1Cf/ VW4CC.VMON /CVCGCM/ CV(4)(.C1tMON iT IPCP/I rIPPrX, 0 ITMLtCT tFG PDT M~k, TIMLPT IMGt IIMDRDI1VENSICN V1(4), S(4)

.CC 10 K:1,4

IC CCTI1NLE(.ALL DVCG00 NU Kz1,4S WZ EV (K)

~1()+ :V (K).-~

93

CA~LL CVZ4

V(K) BVl(K)

,;u 4u kc1,4S(KC) SM£'Q*VK

ý( ) vl(f) * r c ) ý*C C i\T ' NL c

CALL .)VCLi ELI Ka1,4V01) V1(l()

C Co. Ti~.

SUL;CUTI 1 \E ýVG

C .? ty ON /r,'-.ATCPC!-/ L?' Cý LtCVPPCVVI*CA VCPG

C CY i C / w.VCM'/ .~C,,ýT P-X P-OV, .,CXMICo LXLFCXMC

CCb'MON /PXýYscm/ COP41C C If .G /VihTCM/ AVNT, CCVNJTo0VNTfDPVNTC.4LL Wý

UTS=( 1F T - ;- ; - C C 6)/QCC wi X P T M L.L*:XfM(CVP*Tl - CVV*TG)

C,% (C;*AW- !CL~L - --XiTql)/(XIA*CAV + X,.4V*CVV)0XP 4L-CX!M

ZF CP ýLE. FVI.T) ZE1LH.N

IF(P .LT. CVNT*-JDVNT)) 'XY~C = XC(-VT/)Vjk Va CX %4V - ~lX*XtvI- V/ +XM A)

L%~ C7 - 'XMC*;tV/( (XV*X~lA)*(XMA*CZV+XMV*CVV)

COVISONt d:A:TPZPCl/ )'LAXPCLV,;0'pkOZ,6;TAXLVXL,T1 ,Fl(,Crivt)N /mLATCPC'4/ (PCPvCPPCV~VoCtAVCPC.CCA'MCiN IrME.TFLCMI C./C,~ CCSC,~C CIA MON /VGCM/ TibJxv,XmVXAoTINFT

L CP'A'.ON iTCC;'6/ LTC~~.N7~TT1!. 3k a PSh*(TWý*- - TWT4

TOIFF z ~(~C TPC.GTij)CC.TG-TIFF**(4.C/!.C)

PA (XlVA/XM0LA).l(i/v)*1I.PV = P / Y L ) (/ ) t'

;; PA +* P

T6 l.C/( (l.C/Tl) -(e;/(XPLV*XLVr)).LCý.,C/P1l)

CALL SLgFT!MPi.ARI EPSI*(TwC-*64 71>*d)

Ck C99I*AW/.5LXiFa XL + CPLt(TI.ICT)

CALL ZV;P

94

ýCL z C X.*XL5Fc L

zuc;OUTIN- A:VAPCCP'MC-N /r$&ATCPCW/ C0C,C¾,LCV40,CVVCkVCF(ý C MMON I ItC 7 LCý / ýCh ,. 1 9 S IoC Z -C iA ,CioCGSC .E'vCCWMUiN TC'LC'/ T0F1CLtX TCLFTCLNTCLCC i*'fON IuCTM/ Twtu17, XlVXIA,,TINPT

C 1rMO 7OýC!l/ PST1CP4-ZND7C0,lCiTTIT6121F IF F MAX(US.'J, TC-TI)

CQ*T2)ICF/XL 'FF

-z (TZ;:FF .Gt . TC zT.C L) C TO 10

CALL VlEf(aC,,CAPI11F',, Z'L).FC((1FL*ZFL) .LT. NYCI) EýTULjN

1.AT:;(5,I *! t VI fý , 1; , Tu ;cNTIN E V 1P . NTCL CED.

cNCF

CCPrMuN /TCLCM / TDFTCLXTCL, FTOLNJTCLI F(CA .:7. XTOL) TrbkNAVAR A*SCRT(fl*C/(A*4*Awc*j4))X : C + MAX(AvR,.q,4F. C.L)CLz E

11 FL CIŽ L =C

OF 1.0 a*( ( t*c/((1.O+2)*CX-C)4P_*C) )/(X-C))CX x F / CFI F (F GT . 0.3ý) TrEN11F1 F 1L + 1IL((A0) .44T. F I JL) i.%L?L a MIN(JY, 0.5'O-C,))

IefL zI'2%L~E fC I F

IF ý ( %F) .1T. r-T1C) ;T U kt.AaX-C)'

2CZ. C3(NTINL!"?E I Qk N

,..CpViCN /VGCiAI Tw$ i. ',lýXllVPXrATINFTCC1dS'ON I*GCMIC ,ACCpVPrELXLFXC

kALT Lk.N

.....................

95

%TUTCL XTL TOL NTCj.

N (Numter -T CwIii)6

Vi NTOPI PVNI P V14r14 Z6 9E3 .c

96

APPENDIX C

EXPERIMENTAL EFFORT

Specially designed fire/tank tests were conducted to measure the

response of a tank containing liquid to fire impingement. This experimental

effort consisted of measuring effects of the heat input from a large-scale

turbulent sooty fire on a partially filled licjid solvent container. The

goals of this task were: (1) to evaluate the fire/tank interaction

environment, and (2) to obtain the d;ta necessary for preliminary validation

of the numerical computer prediction model.

In these tests, commercially available 55-gallon, 18-gage steel drums

were used. The drums were instrumented to monitor the internal pressire and

temperature rise of a fire-engulfed tank. The drums were equipped with a

network of chromel alumel thermocouples for measuring the variation of the

liquid temperature (Figure C-I). A pressure gage (0-30 lb/in2 ) was used to

monitor the pressure rise rf.sulting from expansion of the vapor (Figure C-

2) and manually vent the tanks before rupture. The tanks were placed about

0.15 m above the surface of a 4.3-m-diameter JP-4 fuel pool fire in both

vertical and horizontal orientations. Prior to the start of the fire, the

tanks were filled with water up to 12 centimeters (5 inches) fron the top of

the tank. The temperatures were read by a datpogger at 5-second intervals,

and each event was recorded continously, using video and still photographs.

Figures C-3 and C-4 show the fire pit and the orientations of the tank witn

respect to the pool fire. Figures C-5 and C-6 show the temperature profiles

for two tests with the tanks oriented vertically.

General analysis of the temperature data and postfire inspection of the

tank shell revealed the following:

1. The tank initial pressure rise is predominantly dizi to the expan-

sion of vapor In the ullage volume,

2. The radiation heat emitting from the dry wall in the ullage v;,&'.. ne

is the main heat flux mechanism in the gaseous phase,

97

/ Thermocouple tree

•To pressure. indicatorTo data logger and relief valve

5 in9 in •, Vapor98in Vapor Water Level

29.5 in

35 in 4

Figure C-1. Water-filled 55-gallon Drum

98

Pressure gage0-30 lb/in 2

Pressure in linefrom drum

.Manual relief valve

Figure C-2. Pressure Monitoring Assembly

99

*t .*...n-

(a) Vertical orientation tank fire

i~

pp.~

(b) Horizontal orientation tank fire

Figure C-3. Tank Fires with Different Orientations

100

(a) Vertical orientation test

- -; " , " : l. ";.,.-.• -.. ; . : . --

(b' Horizontal orientation test

Figure C-4. Vertical and Horizontal Orientation Tests101

T Thermocouple tree__To pressure. indicator

To data logger and relief valve1.5 Ln

005 5 inVapor Water Level

18 in 00429.5 inI 48 in

003 000

18 in'- 002

_tool001

6 in p-Pool Fire Surface

t Water-filled 55-gallon drum

600 1 1 1 1 1500

Pool Fire-(M/N 000)

500 1250

Tank Vapor/ / (M/N 005)

"o 400 - 1000 U_

(nj

300 - 750

200 - Tank Liquid 500/ CM/N 004) u.

100 - .......- 003 250_ _,-. • -- •"--- - M/N 002

M/N 001

0 4 I I l I 00 50 100 150 200 250 300 350

Time (s)

Figure C-5. Fire/Tank Test No. 1 Temperature Profiles

102

SThermocouple tree

To data logger There t To pressure.indicatorTd l .e in---••'-and relief valve

1.5 in _

005& 5 inin Vapor Water Level

18 in 004295i ;48 in -

35 in

n { 002 000

002 18 in

.1 0016 in Pool Fire Surface

t Water-filled 55-gallon drum

600 1500

500 1250Pool Fire(M/N 000)

400 /- Tank Vapor400 000

S300 750

. I--

200 / Tank Liquid M/N 003 500 L(M/N 004)

100 . . - 2 0

M/N 002 /01

0 1 I 1 . I - 00 50 100 150 200 250 300 350

Time (s)

Figure C-6. Fire/Tank Test No. 2 Temperature Profiles

103

3. Discoloration of the wall next to the vapor compared to the wall

in contact with the liquid is indicdtive of a large temperature

difference across the vapor/liquid interface, and

4. The thermocouples placed at various elevations along the height of

the tank showed a decreasing temperature gradient downward from

the top of the tank.

The discoloration of the tank shell suggests a strong conduction heat

transfer in the tank wall at the liquid meniscus which contributes to the

overall thermal activity of the interface and should be considered in the

model.

The predictions of the model proved to be consistent with physical

intuition and agreed well with the overall trend of the experimental data.

However, the predictions from the calculations consistently showed higher

values than the corresponding experimental data. This variation, sometimes

as high as 200 percent, persisted during all test events.

The factors that might cause or contribute to the discrepancy between

the simulation predictions and the tests were evaluated. The assumption of

constant heat flux over the entire tank in the simulation may not have been

duplicated in the tests as a result of wind effects and the overall test

configuration. Also, because of safety considerations the tank was vented

manually, which could have affected the tank response. Project resource

limitations did not permit further testing and validation of the tank

response model.

104