ML19347D163

Models Used in Loctvs Computer Code to Determine Pressure & Temp Response of Vapor Suppression Containments Following Loca.

ML19347D163
Person / Time
Site:	Shoreham File:Long Island Lighting Company icon.png
Issue date:	02/28/1981
From:	Bradbury R, Goodwin E, Hennessy W STONE & WEBSTER ENGINEERING CORP., STONE & WEBSTER, INC.
To:
Shared Package
ML19347D160	List: ML19347D158 ML19347D163
References
SWECO-8101, NUDOCS 8103110312
Download: ML19347D163 (257)

Text

p, LJ STONE ff WEBSTER ENGINEERING CORPORATION 245 SUMMER STREET, BOSTON. M ASS ACHUSETTS ADDRESS ALL COR R E SPONDE NCE TO P.O. BOX 232. BOSTON. MASS. 02107 W U TELEX 94 C001 vu ceM NST4 sCtion tr"" "C ."" .o N S TS?oN

.'OC.L gry.,Ng o...o W AS he4 NGTO N. D C Mr. James R. Miller March 6, 1981 Chief Standardization anc Special Projects Branch Division of Licensing U.S. Nuclear Regulatory Commission Washington, DC 20555

Dear Sir:

STONE & WEBSER ENGINEERING CORPORATION TOPICAL REPORT SWECO 8101 LOCTVS COMPUTER CODE TO DETERMINE PRESSLTE AND TEMPERATURE RESPONSE OF VAPOR SUPPRESSION CONTAINMENTS FOLLOWING LOSS-0F-COOLANT ACCIDENT m We are sending, under separate cover, 50 copies of Stone & Webster Topical

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Report SWECO 8101. A copy of this transmittal letter is attached to each s report copy.

In accordance with procedutes outlined in NUREG-0390, Stoae & Webster Engineering Corporation (S&W) submits S&W Topical Report SWECO 8101. "LOCTVS Computer Code to Determine Pressure and Temperature Response of Vapor Suppression Containments Following Loss-of-Coolant Accident" for review and approval. This report meets the NRC criteria for acceptance as a topical report as concluded in NRC letter from Mr. K. Kniel, dated October 1, 1975 (at that time designated SWECO 7503).

The analytical models described in SWECO 8101 are used in the LOCTVS computer program. The models described are applicable to over-under and horizontal vent type pressure suppression containments.

SWECO 8101 reflects the current status of the LOCTVS program and is currently being used on River Bend Station - Units 1 and 2 (Docket Nos. 50-458, 459) and Nine Mile Point Nuclear Station - Unit 2 (Docket No. 50-410). S&W Topical Report SWND-2 and Supplements 1 and 2, which describe an earlier version of LOCTVS, remain in effect for Shoreham Nuclear Power Station -

Unit 1 (Docket No.50-322) as invoked in its Final Safety Analysis Report.

4 81031103

JRM 2 March 6, 1981 I

If you or members of your staff have any questions on this material or require clarification, please contact Mr. W. P. Hennessy of our Cherry Hill, New Jersey, o f fic'e at (609) 482-3883.

Very truly yours,

/7 R. B. Bradbury Chief Licensing Engineer

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i SWECO 8101 February, 1981 O

MODELS USED IN LCCTVS - A COMPUTER CODE TO DETERMINE

, PRESSURE AND TEMPERATURE RESPONSE OF VAPOR SUPPRESSION CONTAINMENTS FOLLOWING A LOSS-OF-COOLAN'I ACCIDENT d

A Topical Report Prepared By Stone S Webster Engineering Corporation Boston, Massachusetts v

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U f 000m $/$W Chief Licensing Engineer Copyright 1981 By Stone S i!ebster Engineering Corporation All Rights Reserved i

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SWECO 8101 .

DISCLAIMER OF RESPONSIBILITY Neither Stone & Webster Engineering Corporation nor any of the contributors to this document makes any warranty or representation (expressed or implied) with respect to the accuracy, completeness, or usefulness of the information contain ed in this document. Stone & Webster Engineering Corporation assumes no responsibility for liability or damage which may result from the use of any of the information contained in this document.

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SWECO 8101 1

ABSTRACT The analytical models used to simulate the transient response of a pressure suppression containment following a l os s-of-coolant accident are presented. The equations and their derivations along with major assumptions are given.

Comparisons between models and appli:able experimental data are made. The comparisons indicate that the models are conservative for determining drywc 7/ containment pressure

> and temperature transients. It is conaluded that LOCTVS is an effective tool for designing vapor suppression containments because of the conservatisms inherent in the individual models.

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SWECC 8101 MOCELS USED IN LOCTVS - A COMPUTEB CCCE i TO DETERMINE PEESSURE AND TEMPEBATURE FESPONSE OF VAPOR SUPPRESSION CONTAINMENTS FCLLCWING A LOSS-OF-COCLANT ACCIDENT TABLE OF CONTENTS Section Title Page 1 INTECDUCTICN 1-1 2 REACTOR VESSEL MODEL 2-1

.I 2.1 ASSUMPTIONS 2-1 2.2 BICWCOWN 2-1 2.2.1 Main Steam Line Break 2-1 2.2.1.1 Break Flow Area 2-1 2.2.1.2 Froth Level Rise 2-2 2.2.2 Recirculation Fump Suction Line Break 2-4 l s 2.2.2.1 Break Area 2-4 2.2.2.2 Main Steam Isolation valves (MSIV) 2-5 2.2.3 Other Breaks 2-5 2.2.4 Effluent ? low Bate 2-5 2.3 HEAT SCUECES 2-6 2.3.1 Decay Heat 2-6 2.3.2 Power coastdown 2-6 2.3.3 Core-Stored Heat 2-6 2.3.4 Hot Metal Sources 2-7 2.3.5 Zirconium-Water Reacticn 2-11 2.3.6 ECCS Flows and Pump Heat 2-15 2.4 THEEMODYNAMIC STATE 2-16 3 CRYWELL 3-1 3.1 ASSUMPTIONS 3-1 3.2 ENGINEERED SAFETY FEATURES 3-2 3.3 THERMODYNAMIC STATE 3-2

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SWECC 8101 D

TABLE OF CCNTENTS (Cont)

Secticn . Title Page 4 VENT CLEARING AND VENT CISCHARGE FRESSURE 4-1 4.1 HORIZONTAL VENTS 4-1 4.1.1 Assumptions 4-2 4.1.2 Ciscussion 4-3 4.1.3 Comparison with Test Cata 4-4 4.2 VERTICAL VENTS 4-5 4.2.1 Assumptions 4-5 4.2.2 Equations 4-6 4.2.3 Breakthrough 4-9 4.2.4 Ccmparison with Test Data 4-9 5 VENT FLOW 5-1 5.1 MOCEL CESCBIPTION 5-1 5.2 ASSUMPTIONS 5-2 Os 5.3 EQUATIONS FOR VENT FLOW 5-3 5.4 COMPARISON WITH EXPERIMENTAL DATA 5-4 6 WETWELL (HCRIZCNTAL VENT CONTAINMENT) PPESSURE 6-1 6.1 DISCUSSION 6-1 6.2 ASSUMPTIONS 6-1 6.2.1 General 1 6.2.2 Assumptions Prior to Bubble Ereakthrough 6-2 6.2.3 Assumptions Af ter Eubble Breakthrough 6-2 6.3 ECUATICNS 6-2 6.3.1 Prior to Bubble Ereakthrough 6-2 6.3.2 After Bubble Breakthrough 6-4 Q) it

SWECC 8101 TABLE OF CONTENTS (Cont) l Section Title Page 7 CONTAINMENT 7.1 ASSUMPTIONS 7-1 7.1.1 General Assumptions 7-1 7.1.2 Assumptions Applicable Only to Over-Under Containments 7-1 7.1.3 Assumptions Applicable Only to Horizontal Vent Containments 7-1 7.1.4 Other Assumpticns Normally Utilized in Analysis 7-1 7.2 CONTAINMENT ENGINEERED SAFETY FEATURES 7-2 7.2.1 Containment Sprays 7-2 7.2.2 Containment Unit Coolers 7-2 7.2.3 Residual Heat Removal Heat Exchangers 7-6 7.2.4 Drywell/ Suppression Chamber Vacuum Breakers 7-8 I 7.3 THERMODYNAMIC STATE 7-10

%- 7.3.1 Over-Under Containments 7-10 7.3.2 Horizontal vent Containments 7-12 8 HEAT SINK MODEL 8-1 8.1 ASSUMPTIONS 8-1 8.2 APPRCACH 8-1 i

8.3 BOUNDARY CCNDITIONS 8-2 l 8.4 CONDENSATION - REVAPORIZATION MODEL 8-3 8.4.1 Assumptions 8-3 8.4.2 Model Description 8-3 f 9 STEAM BYFASS 9-1 9.1 D::SCUSSION 9-1 l 9.2 ASSUMPTIONS 9-3 9.2.1 Model Assumptiens 9-3 x/ iii

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1 TABLE OF CONTENTS (Cont)

Section Title Page -

9.2.2 Other Assumption Normally Utilized in Analysis 9-3 APPENCIX A INITIAL SLAB CIVISION AND GRIC SPACING CHANGES A-1 APPENCIX E GOVERNING EQUATIONS FOR A HORIZONTAL NOCE WITH HCRIZCNTAL i ANC VERTICAL PCINTS B-1

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l APPENCIX C DERIVhTION OF ECRIZONTAL VENT CLEARING ECUATIONS C-1 APPENCIX C DERIVATION OF THE BOMOGENEOUS VENT FLOW MOCEL (HVFM) E-1

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4-1 PSTF TEST CATA COMPARISCN i

, 4-2 MARK II ERESSURE SUPERESSION TEST  !

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SWECC 8101 h'

LIST OF FIGURES Figure Number Title 1-1 CVER-UNDER PRESSURE SUFFRESSION CONTAINMENT 1-2 HORIZONTAL VENT PRESSURE SUPPRESSION CCNTAINMENT 1-3 SCHEMATIC OF HEAT (Q) ANC MASS (M) FLOW -

SIMULATEC EY LCCTVS 1-4 CONTAINMENT PRESSURE RESFONSE FOR MAIN STEAM LINE EFEAK 1-S_ CONTAINMENT TEMPERATURE RESPONSE FOR MAIN STEAM LINE EREAK 2-1 MAIN STEAM LINE EREAK LCCATICN 2-2 EFFECTIVE MAIN STEAM LINE EREAK AREA 2-3 FRCTH LEVEL 4

2-4 RECIRCULATION PUMP SUCTICN LINE EREAK LCCATION 2-5 FISSION FRODUCT PLUS HEAVY ELEMENT DECAY HEAT 2-6 FOWER COASTECWN HEAT 4-1 MODEL FOR THE EUEELE 4-2 NOCE CONFIGURATION AND SIGN CONVENTICNS 4-3 CNE-VENT TEST,.2 1/8-IN EFEAK, 12-FT SUEMERGENCE, FSTF TEST SERIES 5701

! 4-4 CNE-VENT TEST, 2 1/2-IN EREAK, 8-FT SUEMERGENCE, l FSTF TEST SERIES 5701 4-5 CNE-VENT TEST, 2 1/2-IN BREAK, 12-ET SUBMERGENCE, FSTF TEST SEHIES 5701 4-6 9NE-VENT TEST, 3 5/8-IN EREAK, 12-FT SUBMERGENCE, PSTF TEST SERIES 5701 4-7 TEC-VENT TEST, 2 1/8-IN EFEAK, 12-FT SUBMERGENCE,

-~ FSTF TEST' SERIES 5702 l V' V1

SWECC 8101

, LIST OF FIGUBES (Cont)

Figure Number Title 4-8 TWC-VENT TEST, 2 1/2-IN BREAK, 8-FT SUBMERGENCE, FSTF TEST SERIES 5702 4-9 TWO-VENT TEST, 2 1/2-IN EREAK, 10-FT SUBMERGENCE, FSTF TEST SEBIES 5702 4-10 TWC-VENT TEST, 3 5/8-IN EREAK, 12-FT SUBMERGENCE, FSTF TEST SERIES 5702 4-11 VENT CLEARING MODEL FOR VERTICAL VENTS 4-12 FOCL SWELL /EUBBLE MOCEL FOR VERTICAL VENTS 4-13 FOCL SWELL VELCCITY FOR 4T TEST BUN 31 4-14 WETWELL FRESSURE RESPONSE FOR 4T TEST RUN 31

/'"' 4-15 FOCL SWELL VELCCITY FOR 4T TEST BUN 31 4-16 WETWELL FRESSURE RESFONSE FOR 4T TEST RUN 31 4-17 FOCL SW'LL VELOCITY FOR 4T TEST BUN 29 4-18 WETWELI 'rRESSURE RESFONSE FOR 4T TEST RUN 29

! 4-19 FOOL SURFACE ELEVATICN FCR 47 TEST BUN 29 4-20 FOCL SWELL VELCCITY FOR 4T TEST FUN 27 i

! 4-21 POOL SWELL VELCCITY FOR 4T TEST FUN 35 5-1 COMPARISCN WITH CHOKED FLOW EXPERIMENTAL DATA AT X = 90%

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SWECO 8101 LIST OF FIGURES (Cont)

Figure Number Title B-1 GECMETRY FCR NCDE WITH HORIZONTAL ANC VERTICAL SURFACE OF SEPAPATICN BETWEEN PHASES j

B-2 HORIZONTAL VELCCITY CISTRIBUTION IN NOCE 2 C-1 SUEFACE OF SEPARA5IGN BETWEEN PHASE FOR CLEAR 1 C-2 CONTROL VOLUME USED IN THE DERIVATION OF EQUATIONS BEFORE TOP VENT CLEARS (ONE VENT)

C-3 CONTROL VOLUMC USED IN THE DERIVATION OF EQUATIONS BEFORE TOP VENT CLEARS (TWO VENTS)

C-4 CONTROL VOLUME USED IN THE DERIVATION OF EQUATIONS BEFORE TOE VENT CLEARS (THREE VENTS)

C-5 SURFACE OF SEPARATION BETWEEN PHASES FOR CLEAR 2 4

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SWECC 3101 LIST OF FIGURES (Cont)

Figure Number Title C-14 SUFFACE OF SEFARATICN EETWEEN PHASES FOR CLEAR 7 EEFORE CLEARING CF THE SECCNC FOW CF VENTS C-15 SUFFACE OF SEPARATION EETWEEN PHASES FOR CLEAR 7 AFTER CLEARING OF THE SECOND ECW CF VENTS C-16 COMBINATION CONTAINMENT VENT CONTRCL VCLUMES USED TO EERIVE SOME EQUATIONS IN CLEAR 7 C-17 SURFACE OF SEPARATION EETEEEN PHASES FCR CLEAR 8 C-18 SURFACE OF SEPARATION EETWEEN PHASES

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SWECO 8101 1

SECTION 1 INTRODUCTION The purpose of a containn.ent for a nuclear power plant is to mitigate the consequences of a break in a high energy fluid line. This is achieved by designing the containment to withstand the pressure transient following a postulated loss-of-coolant accident (LOCA) , thus preventing the release of radioactivity to the environment. The containment for boiling water reactors (Bups) is generally of the vapor suppression type, in which high energy steam is channeled through a large pool of water called the suppression pool,

, where the steam is condensed in the pool. Typical vapor suppression systems are shown in Fig. 1-1 (over-under) and 1-2 (horizontal vent) .

The purpose of this report is to document the analytical models used by Stone S Webster Engineering Corporation (SSW) for predicting the transient effects of a LOCA for over-under and horizontal vent vapor suppression containments. These analytical models are incorporated into the LOCTVS computer code, which has been in continual

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The models were derived with the intention of conservatively predicting the containment response for both short- and long-term phases of a LOCA. A conservative prediction in this case is a higher predic-ed peak drywell/ containment pressure than could reasonably be expected to occur. For example, the greater the rates of heat and mass released to the containment, the more conservative the calculation.

Also, the larger the source of heat and mass, the more conservative the calculation.

The enquence of events for the containment design basis accident (DBA) is as follows. A double-ended rupture is postulated to occur in the largest steam or liquid line (depending on which results in the higher pressure) of the reactor coolant system. Immediately after the break, flow from both ends of the rupture accelerates to the maximum rate rossible and hiah energy fluid is released into the drywell. The fluid depressurizes in the drywell. If the

, fluid is liquid, it flashes to steam; if it is steam, it l superheats the drywell atmosphere. In either case, the l- dryw ell presnare and temperature increase. The rising drvwell presrure depresses the water level within the vent

(~s system, rapidly expelling liquid from the vents. After the

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! vent system is cleared of liquid, a mixture of air, steam, 1-1

SWECO 8101

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V and liquid droplets flows from the drywell through the vents and into the suppression pool. The steam condenses, the liquid remains in the pool, and the air rises to the space above the pool. The drywzi: pressure reaches a peak and starts decreasing.

At about 30 sec after the accident, the emercency core cooling system (ECCS) begins to inject suppression pool water into the reactor which cools and depressurizes it. At about 100 sec, the reactor pressure has decreased to the drywell pre s sure , and the blowdown terminates. Safety injection continues, which eventually floods the reactor up to the break level. At this time, the coolant spills out of the break onto the drywell floor. The water on the drywell floor rises, overflows into the vent system, and returns to the suppression pool, thus establishing a closed path for recirculation of the cooling water.

Long-term containment cooling is provided by the residual heat removal (RHR) system. RHR heat exchangers are actuated following the accident and cool the suppression pool water.

In addition, the containment structure (concrete and steel) serves as a heat sink. The containment may have other

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engineered safety features such as sprays or unit coolers to provide additional means for cooling and depressurizing the containment.

Fig. 1-3 shows the overall heat and mass flow paths utilized in LOCTVS. Fig. 1-4 and 1-5 show typical containment pressure and temperature responses, respectively, following a LOCA as calculated by LOCTVS for a horizontal vent containment.

LOCTVS divides the overall analytical modelina of the containment into the following components and subcomponents:

'. Doactor vessel model

2. Drywell model

3. Vent clearing and vent discharge e ressure model

4. Vent flow model

5. Suppression chamber model

6. Heat sinks.

/"'N Each of these is described in detail in the following

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SWECO 81v v When input data to LOCTVS is prepared, the input values are chosen to predict conservative results for the particular design parameter of interest. For example, the height of the suppression pool is assumed to be at its maximum level when predicting peak drywell pressure since a higher vent submergence delays the time when drywell steam and air begin to flow into the suppression pool. On the other hand, the suppression pool height is taken to be at its minimum level when predicting the long-term containment peak pressure and temperature since a smaller pool is less effective for residual heat removal. Likewise, time steps are chosen from sensitivity studies that show an insignificant change in the design parameter of interest with respect to a change in time 3tep.

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SWECC 8101

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( ) SECTION 2 BEACTOR VESSEL MODEL 2.1 ASSUMPTIONS The following assumptions with regard to initial conditions are made:

1. The reactor is scrammed at the begianing of the accident and goes subcritical.

2. The reactor coolant is saturated steam and liquid.

3. The coolant is instantaneously accelerated tc critical flow out of the break.

4. All metal in contact with the reactor coolant is initially at the coolant temperature.

2.2 BLCHECEN "N The initial state of the reactor coolant before rupture is

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essentially that of a 1cwer phase of saturated liquid and an upper phase of saturated vapor in equilibrium. At the instant of rupture, the reactor System starts tc depressurize by discharging ecolant through the break. This depressurization results in flashing of the saturated liquid, creating bubbles in the lower phase. Since these bubtles cannot rise instantaneously to the surface, the lcwer phase becomes a mixture of steam and liquid, or froth.

! While the surface of the lower phase is below the elevation of the break, only the upper phase (vapor) ficws out through the rupture. When the break is covered by the lower phase, a two-phase mixture is discharged.

Ruptures in both the main steam line (initially above the froth level) and the recirculaticn suction line (initially below the froth level) are evaluated to determine which is the CEA.

2.2.1 Main Steam Line Break 2.2.1.1 Ereak Flcw Area Fig. 2- 1 shows the location of the steam line break.

Immediately after the break occurs, the flow frcm both ends es accelerates to the maximum rate possible, i.e., critical

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SWECC 8101 floN. Flcw through the shcrter path tc the break is critical in the steam line nozzle.

Blowdcwn through the cther end cf the break cccurs because the steam lines are interconnected upstream of the turbine by the rain steam header. This interconnection allcws reactor system fluid to pass from the three unbroken stean lines, through the header, and back into the drywell via the broken line.

After the steam inventcry between the break and the flow restrictor cf the broken loop is depleted, ficw through the icnger path is limited by critical f1cw at the ficw restrictor. Typically it is assumed that the stear inventory discharges at a rate cf 0.7 of the critical ficw rate at the initial reactor pressure, using the full pipc area.(1)

The decrease in steam pressure at the turbine inlet initiates closure of the main steam isolaticn valves (MSIV) within 0.2 sec (typical) . Also, MSIV cicsure signals are generated as differential pressures across the steam line flow restrictors increase above set points. The instruments  !

sensing ficw restrictor differential pressures generate f

N isolation signals within 0.5 sec (typical) after the

) accident.

At about 4 sec, the isolation valves in the broken line have closed sufficiently so that the valve flow area equals the I

flow restrictor area. At that time, the critical ficw location changet frca the flow restrictor tc the isolatica valves. Subsequent closure of the valves in the broken line  ;

terminates flow frcm the turbine side of the break.

l Fig. 2-2 shows the effective blowdown area transit for a l typical plant. .

I 2.2.1.2 Frcth Level Fise l I

Immediately following the break, the steam ficw rate leaving the vessel exceeds the steam generation rate in the core. i This misnatch causes a depressurizaticn cf the reactor vessel, and the resultant fornaticn of bubbles within the reacter liquid causes a rapid rise in water (or f roth) level. When the level reaches the top of the steam dryers, i blowdcwn changes from steam tc two-phase.

O)

\

N- 2-2

SWECC 8101 O

( ,) A two-regicn model is utilized: an upper region of vapor and a lower region of vapor and liquid, or frcth. The froth height, h, at any time is deterrined by accounting for the liquid and steam within the lower region, i.e.,

h= [M gb Y g + (1 - x) M V f ]/A

( 2- 1) where:

Mgb = Mass of vapor bubbles in lower region M = Total reactor coolant nass in vessel x = Reactor coolant quality A = Effective cross-section area Vg = vapor specific volume vf = Liquid srecific volume The accumulation of vapor in the lower region (Mgb) is

Mgb = Tctal steam produced - Steam rise tc the (2-2) surface The total steam production rate is found by rearranging the equation for reactor steam mass (M g) change, i.e.,

Ug= Rate of steam producticn - Steam ficw from ( 2-3) the reactor The rate that steam is lost by bubble rise is steam rise rate = Mgb /T (2-"I where:

T = Average time required for bubbles to rise to the surface x 2-3

SWECC 8101 Taking the time derivative of Equation 2-2 and substituting Equations 2-3 and 2-4, we get for the rate of stear accumulation in the icwer region:

M "v =M - gb + W gb g T (2-5) where:

W = Steam flew from reactor

.This equaticn is integrated assuring kg and W are constant for srall time steps.

Solving M

gb2

= [M g + W] At + M gbt exp (-at/:)

(2-6) where:

/"' at = Timestep = ta - tg Mght = Mgb at beginning or time step Mgb2 = Mgb at end of time step 2.2.2 Becirculation Fump Sucticn Line Break 2.2.2.1 Ereak Area Fig. 2-4 dnows the break location. On the reactor side of the break, the reactor coolant escapes to the drywell at a

, critical' flow rate through the recirculaticn suction line.

On the recirculation pump side, the recirculation line inventory 'tetween the break and the jet pump nozzles discharges at a rate of 0.5 (typical) of the critical flow rate, at the initial reactor pressure, using the full pipe area (1). Fcllowing depletion of the line inventory, the reactor coolant is released at the critical flow rate frcs the cleanup line and the jet purp nczzles cf the brcken loop.

Though the recirculation pump sucticn line is initially -

belcw the froth level and, therefore, a two-p hase mixture would be expected to ficw out the break, a liquid cnly

-~x outflow assumption is made until the level drcps below the t

\s) 2-4

.~ _ ,. __ __

SWECO 8101 p

k,m break elevation. At this time the blcwdown becomes two-phase. This assumption is made to maximize the mass and energy release rate.

To effect this assumption, the flashing reactor coolant steam is assumed to rise instantaneously to the liquid surface leaving an all liquid lower region. When the liquid level descends below the recirculation pump suction line elevation, the blowdcwn changes to a homcgeneous mixture of reactor liquid and steam.

2.2.2.2 Main Steam Isolation Valves As in the case of a main steam line break, the MSIVs start closing shortly after the pipe break. In the case of a recirculaticn pump suction line break, they are assumed to be fully clcsed in the minimum amount of tire. A fast closure time re sult s in a higher reactor pressure and, consequently, a higher sass and energy release rate for this break location.

2.2.3 Other Breaks A spectrum of steam and liquid line break sizes is analyzed to ensure that the double-ended rupture of the main stear

(()T line or recirculation pump Intermadiate-size breaks are suction line is the DBA.

analyzed as discussed in Secticns 2.2.1 and 2.2.2.

Blowdown rates for small breaks are not calculated with LOCTVS since LOCTVS does not have the capability to start and stop the various ECCSs based cn reactor water level, nor i can it simulate safety relief valve (SRV) operation. (For large breaks, ECCSs are actuated in ICCTVS after an appropriate time delay and supply water at rates which are based on reactor pressure.) Fcr small breaks, blowdown rates

are calculated by hand with the Noody flow model(2) based on a reactor pressure transient which is either determined for a prescribed cooldown rate (e.g. , 1000F/hr) or obtained frow the reactor manufacturer.

2.2.4 Effluent Flow Bate The frictionless Moody flow model is used at the vessel pressure and discharge line enthalpy for the LOCTVS determinaticn of reactor vessel blowdown ratetr). The ability of the Moody flow model to conservatively predict reactor blowdown flow rates is discussed in Feference 1.

\N 'j 2-5

SWECC 8101 N

\

/ After blowdown is terminated, the reactor begins to refill as a result of ECCS operation. When the liquid le /el reaches the elevation of the break (jet pump nozzles for a recirculaticn line break) , the coolant spills out the break at a rate equal to the ECCS injection flow rate, after being corrected for a slight change in density resulting from heatup in the reactor vessel.

2.3 HEAT SCURCES 2.3.1 Decay Heat The U.S. Nuclear Regulatory Ccmmission (NBC) Standard Review Plan APCSE 9-2 is used for fission product decay heat addition to the reactor core and coolant (3). A typical curve of decay heat generated as a fraction of full power versus time after shutdown is shcwn on Fig. 2-5.

2.3.2 Fower Coastdown After receipt of a scram signal caused by the DBA, the reactor power level decreases to fission product decay levels over a finite period of time depending on the time it takes for centrol rods to be inserted, the half-life of the

(~'N s ) longest lived delayed neutron precursor, the mcderator, the fuel temperature, and the voids in the core during blcwdown.

The released heat is deposited in both the core and the coolant. A typical power coastdown curve is shown on Fig. 2-6.

2.3.3 Core-stored Heat The core has a sizable heat capacity and centains a large amount of stored heat at an average temperature well abcve the average reactor coclant temperature. The heat transfer from the core to the coolant is a function of the amount of liquid in the core and whether boiling or convective heat transfer is occurring. The heat transfer coefficient fron the core tc the coolant is based upon the assumptions that:

1. The fuel-clad thernal contact resistance remains unchanged.

2. The convective heat transfer coefficient from clad to coolant is given by the maximum of:

a. Two-phase convective boiling, or

b. Pool boiling heat transfer coefficients.

%' 2-6

t I

SWECC 8101

/ The latter two quantities denoted by U (t) are scaled down from the input full pcwer heat transfer coefficient U (0) with the forced convection equation (*):

U(t) = U(0) (1 - a/2) [G(t) / G(0)]O.8 (2-7)

- .ie r e :

G (0) = Input full power coolant mass velocity G (t) = Current mass velocity through the core (equal to the coolant blowdown rate) a = Void fraction in the ccre and the pool boiling relation (*3:

U(t) = U p(1-a) where:

Up = Input pool boiling heat transfer coefficient The larger U(t) is chosen from these twc equations for calculating tne core cooling and thus the core temperature.

The core cocling heat is added to the primary system coolant according tc the following equaticn:

0 = U(t) A (T p -T)C (2-9) where:

A = Feactor core heat transfer surface area Ty = Average core (f uel) temperature TC = Average reactor coolant temperature 2.3.4 Hot Metal Sources During a LOCA the pressure and temperature of the reactor coolant are reduced very rapidly. As a result of the rapid decrease in coolant temperature, a temperature difference

N., exists between the coolant and metal components in contact 2-7

l SWECC 8101

( ,) with the coolant.

driving force for transfer of heat stored in the components T.,is temperature difference is the to the coolant; thus, the corponents become sources of heat for the coolant. Typical sources include the core barrel, centrol red drives, and reactor vessel.

At a given time in the program execution, the rate of heat transfer is strongly dependent on whether the source is in contact with the liquid or vapor phase.

As the primary coolant escapes through the rupture, the elevation of the liquid-vapor interface in the primary system decreases, and more het retal is "unccvered." Thus, the heat sink (liquid or vapor) for a particular source is determined by the 1ccation of the source with reference to the liquid level. For this deterrination the reactor vessel and all other hot metal sources are divided into the following three regions: above core, belcw core, and the core regicn.

Whether the source is in contact with coclant liquid or vapor is of importance to the program because it influences:

1. The assignment of the heat released to either

(' liquid cr vapor (in the latter case the (Sj conservative assumption is made that the heat immediately enters the drywell atmosphere) ,

2. The method of calculating the surface temperature of the source, and

3. The heat transfer coefficient used.

The metal scurces are considered to be slabs with nonunifcrm temperature distribution in the thickness direction. The

, one-dimensicnal transient heat conduction calculations are l based on the general numerical method of Cusinherre, which

, treats each source as a slat with one surface exposed tc a varying temperature (5). The slat is divided into a nurter

, of slices of equal thickness and a heat halance is written l on each slice to determine the slice temperatures.

t Slab Civision i Slab division (into slices) is perfclmed as outlined in Appendix A. See also Reference 5.

l

\

! d 2-8 i

SWECC 8101 Temperature Distribution Initial Temperature Distribution The initial temperature distribution is assumed to be uniform and equal to the primary coolant temperature prior to the accident.

Surface Temperatures The temperature of the source surface exposed to the coolant is computed in one of two ways:

1. If the surface is in ccntact with coolant liquid, the surface temperature Ts at the end of the time step is set equal to the coolant liquid temperature (assumes an infinite heat transfer coefficient):

s" c (2-10)

2. If the surface is in contact with the coolant vapor, a lower gas film heat transfer coefficient for that source determines the surface temperature,

"'}

a as follows:

6o T Tc

• hAit s

(2-11) where:

T 7 = Coolant vapor temperature SQ = Heat transferred during previous time interval l At = tength of previous time interval A = Surface area of source b = Input heat transfer coefficient of source The temperature of the slab surface not in ccntact with the coolant (7j) is set equal to the "old" temperature of the adjacent interior slice (Ty_3) :

T{=T_3 y (2-12)

's / 2-9

SWECC 8101 i Interior Temperatures The interior temperatures are determined by the Dusinberre Method as follows:

= T-3 + (M'2} Ti + T +1 i i T]' M (2- 13) where:

T', = Temperature of slice j at the end of the time step 3

Tj = Temperature of slice j at the beginning of the time step and:

M=

aat where:

Ax = Slice thickness At = Timestep a = Thermal diffusivity Heat Transferred Heat released 60 by a given scurce during the time increment At is determined as follows:

60 = O'-Q where:

0 = Total heat released by source from beginning of the accident to end of the time step, n

= pAAXC p T-Tj 3

j=1 ( 2- 15) and TO= Initial temperature of slice j J

(s_ 2-10 e , e -r-- -

w: --e,---+s , - , - < ---ew --ne , n, , ~ ~ - , em w e-w-ew-me-mpe e-ec.- ,v -m we ,~e,-,=vv-- --<reew r, w s - n ~p w ow w e n- ,,s n,,---o-=y ---~wvm

SWECO 8101

)

\d Stability The LOCTVS code contains several features to prevent abnorralities, accumulation of error, and instability. Two such features are mentioned here.

1. The Dusinberre Method requires that:

a. M 2 2 and

b. the number of slices (n) > 1.

The program ensures that both of these conditions are met before the tusinterre Nethod is applied (Appendix A) .

2. Heat released by a source is determined by computing the sum of internal energy changes of all slices s rather than using the heat flux at the

surface. Thus, the heat released is a function of all node temperatures, and an inaccuracy in one node temperature will not be critical.

A computation based on heat flux is dependent only upon the surface and coolant temperatures and is y ) thus more prcne to error accumulation.

2.3.5 Zirconium-Water Reaction The exposure of zirconium to steam results in the oxidation of zirconium provided that the tenperature of both reactants is high enough. Oxidation of zirconiur is accompanied by the release of energy. The breakdown of steam to hydrogen and oxygen absorbs energy. Since the oxidaticn of zirconium releases more energy than the breakdown of steam absorbs, a net release of energy results.

The chemical equation describing the breakdcwn of water is:

2H2 O + heat e 2H2+O g (2- 16)

The chemical equation describing the oxidaticn of zirconium is:

Zr + O > ZrO + ' eat 2 2 (2-17)

/}

2-11

__ .. - __ .-. =. -- __ .- . - ..

SWECC 8101 The heat of reaction for Equations 2-16 and 2-17 can be terived from values of free energy presented in Reference 6,

=,fc11nws:

57,798 calories / gram-mole of HgG reacted 258,000 calories / gram-mole of Zroa formed A chemical balance of the reaction shows that for every mole of zirconium that is oxidized, two molen of water must be reacted. Thus, the net heat that is released during the reaction is:

142,000 calories / gram-mole of zirconium or 2,811 Btu /lbm of zirconium or 2.558 x 105 Btu /lb mole of zirconium The total reaction can be expressed as:

x i

Zr + 2H 2O '

ZrO + 2H + heat 2 2 Thus, the net result of the reaction is loss of steam, evolution of hydrcgen, and generation of heat.

Calculaticnal Techniques Reaction Heat Released The code has the capability to use a constant or variable rate of the zirconium-water reaction as a function of time.

The rate, R, is expressad as a percent of the total available zirconium in the core reacting per unit time.

Thus, the heat released is expressed as 60 = 28.11 M Zr R At

( 2- 18) 2-12

l SWECC 8101 where:  !

60 = Heat released i MZr = Total mass of available zirconium in core R = Beaction rate At = Time increment Volume of Pydrogen Released The volume of hydrogen at standard temperature and pressure, AVna, can te calculated from heat released (as calculated by Equation 2-18) by use of the following relaticnship:

AV R3 H2 2 (2-19) where:

AVn, = Volume of hydrogen (STF) released, cu ft 6Q = Beat released, Btu R = Cu ft of hydrogen /lb-scle of hydrogen R 2 = Lb-mcle of zirconium /Etu R3 = Lb-mole of hydrogen /lb-mcle of zirconium Thus:

I V

H:

= 0.002806 6C (2-20)

Mass of Hydrogen Released Th. rass of hydrogen released, AMg lb is obtained by  !

multiplying the volume at standard preds,ur(e a) nd temperature by the density of hydrogen at standard pressure and temperature:

Av H

= 0.002806 60 (2-21)

\

2-13

(

4 SWECC 8101 O Sensible Heat of Hydrcqen Released The sensible heat of hydrogen released, SCH, 2 is expressed as 60 H =C p AM (Ty -32.)

H 2 2 2 (2-22) where:

T = Temperature of the hydrcgen released H2 Cp = Specific heat of hydrogen Air Equivalent of Mass of Hydrogen Beleased In crder to facilitate code calculations regarding noncondensibles in the containment atmosphere, the hydrogen released is converted to an equivalent number of moles of air. This is accomplished by multiplying the mass of hydrogen released by the ratio of the molecular weight of

. air to that of hydrogen:

~) 28.97 AMH2 AMg =

2.016 (2-23) i Water Reacved on Peaction Since cne mole of steam is consumed for each mole of hydrogen produced (Equation 2-16) , the rass of water consumed is:

18.016 AMH AM

• l HO2 2.016 (2-24)

The heat removed is found by multiplying the pass removed by the enthalpy of the primary ccolant, h:

( MkI2 0 'bl 2O (2-25) l 2-14

SWECC 8101 t%, ) output The net heat and mass transfers are acccunted for in the follcwing manner:

The heat added to th( core is equal to the heat of reaction (Equation 2-18) plus tie heat associated with the water removed (Fquation 2-2s) minus the sensible heat of the hydrogen (Equation 2-22) , i.e.,

k 2 ON 2 (2-26)

It should be ncted that the heat of the zirconium-water reaction is ojded directly to the core and not to the coolant becaw,2 the zirconium oxide is formed in the fuel rod cladding. It is subsequently released by heat transfer tc the core coolant.

The mass added to the ccre by creation of zirconium oxide is neglected. The mass and heat rencved from the ccolant is that asscciated with the stear removed (Equaticns 2-24 and 2-25) , i.e., AMH 2 and 60H2 0-The rass and heat added to the drywell atmosphere are

(']'T g' associated with hydrogen aM H 2

and 60 H 2 *

(Equaticns 2-21 and 2-28 , i.e.,

2.3.6 ECCS Flows and Pump Heat Energency core cooling is assumed to be effective at an input specified time, as given by the reacter manufacturer, after which water is transferred from the suppression pool ,

as a function of reactor system pressure. Incremental mass, heat flow, and pump heat are calculated and added to the reactor ccolant inventory. Emergency cooling is accomplished by the high pressure core spray (HPCS) , low pressure core spray (LcCS) , cr low pressure coolant injection (LPCI) , (residual baat removal), depending on the single active failure assumed. Appror,riate delay times for receipt cf a safety injec icn signal, valve operation, a n c.

pump startup are inputs to tne LCCTVS program.

b 2-15 l

l l

l

[

SWECC 8101 ,

^

~.}

2.4 THERMOCYNA4IC STATE Consider the reactor system below:

REACTOR SYSTEM M

EWl j - EWJ EWihi =

V

=

ZWjh) l Zoi =

E i

where:

M = Reactor system coolant mass i

s E = Reactor system coolant internal energy i

V = Reactor system volume Ei = ECCS ficws, feedwater hi = Ccrrespcnding ECCS enthalpi es l

Hj = Blowdown flow rate (and steam line flow rate j if a recirculation pump suction line break) hj = Blowdown enthalpy (and raactor steam enthalpy if a recirculation pump suct:f on line break Ci = Decay heat, coastdown heat, core stored heat, hot dietal stored heat, 2r-H 2O reaction heat,,

ECCS pump heat

! /

2-16 l

l i

SWECC 8101 O

Q The conservation of rass and energy equations are solved by finite differencing, i.e.,

4

for mass, dM

= I W.- I W.

dt 1 3 (2-27) becomes M'=M+(IW-Df)At f 3 (2-28) where: ,

M' = New mass M = Cid mass at = Time step and for energy, S

dt

= IQ.1+ I W i1 h. - IW jJ h.

( 2-2 9) becomes E' = E + [\IO. 1 + 31.h.

11 - BY.h 3 J/ . lat (2-30) where:

l E' = bew internal energy E = Cid internal energy After solving Equations 2-28 and 2-30, the reactor coolant temperature and pressure are determined for the new thermcdynamic state that ccrresponds to the new values of specific volume (V/N) and specific internal energy (E/M).

The nethcd entails assuming a trial pressure and, with the known specific internal energy, determining from the stear l

tables the associated specific volume. The search continues until a trial pressure is found that satisfies the kncwn fs properties.

i)

V 2-17

i e

i SWECC 8101

s I

References - Section 2

1. The General Electric Mark III Pressure Suppression ,

Ccntainment System Analytical Model, NEDO-20533,  !

June 1974.

2. Moody, F. J. Maximum Flow Fate of a Single Component 1 Two-Phase Mixture, APED-4378, October 25, 1963.

i

3. U.S. Nuclear Regulatory Commission Standard Review Flan AFCSE 9.2, November 24, 1975.

4. Lcttes, P. A., et al. Lecture Notes on Heat Extraction from Boiling Water Fower Feactors. ANL 6063, October 1959, p 22-23.

5. McAdams, W. H. Heat Transmission, third edition, NcGraw-Hill, New York, NY, 1950, p 44.

6. Perry, J. H. Chemical Engineers Handbook, third edition, McGraw-Hill, New York, NY, 1950.

A r

1 l

l t

c 2-18

,<,r- - - - - -.----=w,- -=~,-<,,-----ww--w-.-,w-- --*,--+-. wet + e -,.,--.-,,,,r ----n-, ,.v,-- vo- + r-er - + ~ + - ev -*- y a ve w ,.we-'r'*-----<-r- -'f-a-d "

. .

e.: -

FLOW RESTRICTORS BREAK 2' INBOARD OUTBOARD BYPASS [

dh MSiv MSiv

= x x x REACTOR

'N t

-- X X ~

X PRESSURE TURBINE VESSEL

• Z X X X

/ = x x -

STOP

+ CO NTAIN M ENT VALVES \

DRYWELL +

O FIGURE 2-1 MAIN STEAM LINE BREAK LOCATION O

O O l

5 -

1 4.38 4 -

• ~

l

)

TOTAL FLOW AREA 3.23 g 1 n s

l k 3 -

g 2,60 4

w 5

3:

\ \- STEAM LINE NOZZLE l 32

- 1.78 h x

glNVENTORY EFFECT MSIV 4

\ FLOW RESTRICTOR O.68 i

% -.-. l j l N

N O ' ' ' ' a *

• 0 1 2 3 4 5 6 7 TIME, SEC J

, FIGURE 2-2 l EFFECTIVE MAIN STEAM LINE BREAK AREA .

4

t h

j

'l' l

g l I I

I STEAM -

(UPPER REGION) _--

I i -W

~

l l' .--

l l l

D ll Mg b/r l

]ll O

o oOO O o

0 0 C

o ao O O O O

OO O O O 7 o

0 0 i O O O g O O

\ O O O

TWO - PH A SE -+ 0 0 C O

. MIXTURE O O O h 0 0 0 i (LOWER REGION) 0 0 O 1

O O n O O O O o O c O O

O O O o g O O O I O INTERFACE STEAM PRODUCTION DUE TO PRESSURE DECREASE S ENERGY INPUT O FROTH FIGURE 2-3 LEVEL m

f , ..,.,.,.-r--w.e.-y-,-m - - , , _ _ - - . - - - - , , - - . . - . . . . - , - , . - . ~ , - . -

\

REACTOR VESSEL MAIN STEAM LINE STEAM

- _ _ __ DRYERS

/  %

STEAM SEPARATORS

/ N RECIR CUL ATION LOOP

=

'I CORE

{

JET PUMP t [

O t

( BREAK _/

/ \ d RECIRCULATION PUMP b

TO REACTOR WATER CLEANUP SYSTEM POINT OF CRITICAL FLOW G -RECIRCULATION LINE b -CLEANUP LINE C - AREA OF JET PUMP NOZZLES, ASSOCIATED WITH BROKEN LOOP l

(7, FIGURE 2-4 I

() RECIRCULATION PUMP SUCTION LINE BREAK LOCATION

i l , ' ,  :

J s r oi 01 T

N E

M E

L E

a o

i 0

1 Y

V A

I C 5ET

_ ' E S

HA 2 S E .

U H

_ T N EL

_ E R U

PA Y

D l _

' I C GTC I

C FC E

/ A UD

. D R

E O T R N '

F A

P

=

E N s OM O o OI I i

' i T S

' S I

' F I

2 o

i

-- - - - - ~ ~ - .

- '0' 1

, 2 5 g

og 01 l -

b E JJ5 tOt zO O4Tu. e l- 4u >4oJO u iIl  ! j

l 1

l 4

t

~

O  !

i.0 ,

i -

l

_

i 1

4 w

B -

O  !

i L r s

a -

n 0.1 w i

L -

O ~

Z 9 -

s _

o ,

4 m -

L z _

R o

o i H

-$ COI l 4 -

0 .

u -

m -

l w -

' 3 i o _

E

[

0.001 I I I IIIII I I I IIIII I I I I I III O.I 1.0 10,0 10 0.0 .

[ TIME AFTER ACCIDENT, SEC i

FIGURE 2-6 POWER COASTDOWN HE AT t

i t

,-r,- - ~ , r,-,v,.e,r. n-,-m,,,- -

, ,,-r-,,,-,,n ,,----,ev,,,,-+,-~,---wem,.~ r-,,.---r.,,..e -,--+--n,=,-------,m,r.,,,wn-~, ..,,w

ShECO 8101 SECTION 3 DRYNELL 3.1 ASSUMPTIONS The following assumptions are made with regard to the drywell:

1 For a recirculation pump suction line break, the reactor effluent mixes completely and homogeneously with the atmosphere. Following the calculation o#

the thermodynamic state of the drywell atmosphere, the liquid condensate drops to the floor.

2. For a steam line break, the reactor effluent is initially all steam resulting in a superheated drywell atmosphere. As the reactor froth level rises above the vessel steam dryers, the blowdown changes to a two-phase mixture. The steam portion of the blowdown is assumed to mix honogeneously with the drywell atmosphere. The liquid portion undergoes a pressure flash down to the drywell f'/

(_,

T total pressure.

pressure flash mixes atmosphere. The The steam resulting from the homogeneously remaining liquid phase with mixes the with the liquid on the drywell floor.

It should be noted that the calculational technique described subsequently is conservative in that all liquid in the atmosphere is immediately put onto the drywell floor.

Removing the liquid from the drywell atmosphere reduces the liquid available for mixing during the next time step, thereby resulting in a higher atmospheric temperature.

Although such an assumption is conservative for determining the thermodynamic state of the drywell atmosphere, it is not conservative for determining the flew through the vents.

Therefore, all liquid on the drywell floor is assumed to be suspended in the drywell atmosphere for purposes of calculating the vent flow rate. For the vent flow calculation, a homogeneous nixture of drywell steam, air, and liquid is generally assumed.

3 U 3-1

SWECO 8101 p 3.2 ENGINEERED SAFETY FEATURES The drywell may have an engineered safety feature system such as sprays. This is typicil of over-under type contait..nents. The spray system is enployed for long-term containment atmospheric cooling and is not normally used during the initial phase of the LOCA transient. The system is usually manually actuated and, when operating, draws water from the suppression pool, passes it through a RHR heat exchanger, and sprays it into the drywell and the suppression chamber. The flow rate is determined by the pressure differential between the drywell and suppression chamber and by the system piping resistance.

3.3 THERMODYNAMIC STATE Consider the system below:

DRYWELL N

EWi =

M a ,M w = E W) v - 20)

ZWhj =

We E Wehe - EWjh) m-m r

Mg Ej where:

Ma = Drywell air mass Mg = Drywell atmosphere water (steam plus liquid) mass b

G 3-2

q SNECO 8101 V = Drywell free volume = total - liquid volume '

i E = Drywell atmosphere internal energy M1 = Drywell floor liquid mass E i = Drywell floor liquid internal energy W1 = Reactor effluent, drywell sprays (if any)

Wj = Vent flow We = Steam condensing out of atmosphere, unflashed

portion of blowdown (see Assumption 2) hi,3 = Enthalpy corresponding to Wi,3 i

j 03 = He0t absorbed by drywell structures i

The drywell is divided into two regions:

1. The atmosphere consisting of steam, air, and break effluent, and 1

> 2. The liquid on the drywell floor resulting from incomplete flashing of the blowdown and condensation.

At every time step, heat and mass inventories are updated and a search is made to determine the new state of the i atmosphere. Any condensate in the atmosphere is removed and j added to the liquid on the floor; then the temperature of the liquid on the floor is calculated. If the temperature I of the liquid on the floor is greater than the saturation l temperature corresponding to the total atmospheric pressure (boiling), the two regions are combined and an equilibrium state is found.

Conservation of mass and energy equations for the atmosphere are:

Mass i

dM M

"~

dt J j Ma + +

s 1 (3-1)

~

c 1

i l

. . , - . . ~ . . . . . . . , . . . . - - , - . , - . . . - - - . - . . - . . - - . - - . _ . - . . . - . . - . - . . , , - . - - - , .

1 SWECO 8101 N.

dM M

" ~

dt i jM + +M y a s (3-2)

Energy h"bW h ig -[Wh 33 -[d 3 (3-3)

These equations are solved by finite differences. It is then determined whether the atmosphere is saturated or superheated in the following manner. The atmosphere total internal energy, E, is compared to the energy of the mass of air and the mass of water evaluated at the saturation temperature corresponding to dry saturated vapor at a specific volume of V/Mw (i.e. , assuming 100 percent quality, or no liquid present) . If E is greater than the energy at the dry saturated condition, the atmosphere is superheated.

If E is less than or equal to the saturation energy, the atmosphere is saturated.

' O g j For the superheated case, a search is made for temperature

's and vapor pressure that will satisfy the following equation, subject to the constraint that the vapor specific volume is v/Mg:

E=M av (T - 32) +MU sy (3-4) where:

Cy = Specific heat of air at constant volume U y = Superheated vapor specific internal energy T = Atmosphere temperature, OF Ms = Mass of steam = Mw O

3-4

SWECO 8101 For the saturated case, a search is made for tenperature with the following equation, subject to the constraint that the specific volume of the wet (saturated liquid and vapor) mixture is V/Mg:

E=M (T - 30 + M s U g +MUcf av (3-5) where:

Mc = Mass of liquid condensate in atmosphere = Mg - V/vg Uf = Saturated liquid specific internal energy

• f g = Specific volume of dry saturated vapor Ug = Specific internal energy of dry saturated vapor Ms = Mass of steam = Mg - Mc The mass and energy of the condensate are then added to the liquid on the floor and subtracted from the atmosphere. The equations for the floor liquid are:

Ox dM y . ( M1 I j

dt c M a +M s +M) 1 (3-6) dt cf

~

j + 1 s

+M 1

~

j (Ma 1 floor (3-7) where:

hl = Liquid enthalpy With the atmosphere temperature known, the drywell total pressure, Pt' 18

• +

t a v (3-8) where:

Pa = Air partial pressure = MaRa (T+460)/V Py = Steam partial pressure = saturation pressure at T J

. 3- 5

1 i

i SWECL 8101 i

for the saturated case or vapor pressure that j satisfied Equation 3-4 for the superheated case.

I i

If the water level on the drywell floor is sufficiently high  !

to overflow into the vent system, the overflow mass and its

associated heat are removed from the inventory of water on l the floor and added to the suppression pool. Overflow

., occurs well after the transient peak pressure and, i therefore, this calculation is of consequence only for long-term studies.

I

}

(

j .

i  ;

M i

I I

I -

I i

i 1

i h

l l

l l

I I

i o

i l

I i

3-6 4

l

...-.. . .. - . . - . . - . - ~ _ . - - . . . . . - . - . - - . . , . - . - - . . - , . ~ _ - . _ , , , - . . . - . . - . - . - - - - - . .

SWEc0 8101 SECTION 4 O VENT CLEARING AND VENT DISCHARGE PRFSSURE 4.1 HORIZONTAL VENTS This model is concerned with the horizontal vent containment (Fig. 1-2). The nuclear reactor vessel is surrounded by a capped cylindrical container of relatively small volume called the drywell. Inside the drywell is a short, concentric cylindrical wall called the weir wall. Below the top of the weir wall, the drywell wall has three rows of vents which open to the containment, which is larger than and concentric with the drywell. In normal operation water f ills the annular space between the weir wall inside the drywell and the outer wall of the containment and covers the three rcws of vents.

A break in a reactor coolant system pipe is postulated to occur inside the drywell. The high energy fluid released to the drywell increases the drywell pressure which, in turn, depresses the water level inside the drywell. The water is thereby forced from the drywell through the vents into the containment. When the water level reaches the top row of vents, the gaseous phase of the drywell starts to flow through the vents. The water component, consisting of vapor and suspended droplets, is trapped in the pool. The air component forma bubbles which begin to grow out of the vent, then separate and rise on the containment side of the pool.

Eventually the bubbles rise to the surface and mix with the containment air. This process occurs for each of the three rows of vents.

The pressure at the opening of the vent on the containment side is referred to as vent back pressure or vent discharge pressure; the latter expression will be used here.

The rapidity of dissipation of the high energy fluid coming f rom the drywell depends strongly upon the fluid velocicy in a cleared vent. This in turn is determined by the pressure difference across the vent, and both the drywell pressure and the vent discharge pressure are important parameters.

The inlet pressure is close to the drywell pressure, but the l

vent discharge pressure depends on the dynamics of the i swelling mixture of water and air.

In order to make the problem tractable, several simplifying assumptions 'are considered. For example, it could be assumed that the water vapor condenses instantly at the vent ,.

SWECO 8101

/ exit, and that the air bubbles travel instantly into the (N) containment air. However, this assumption would yield a low vent discharge pressure and is, therefore, nonconservative.

For conservatism LOCTVS assumes that the fluid issuing from the vent consists of air and liquid at the drywell temperature and flows at the mass flow rate calculated for the mixture of air, steam, and liquid. The air is assumed to form a bubble which remains at the vent elevation. The bubble exerts a pressure on the slug of water above it and accelerates the water upward. The bubble pressure is assumed to be the vent discharge pressure and is higher than the elevation head at this location. This assumption is conservative since it yields a higher drywell pressure.

4.1.1 Assumptions The following assumptions are made concerning the vent clearing and vent discharge pressure model (Fig. 4-1) :

1. The air-steam portion of the mixture flowing through the vents is air only.

2. The liquid which is pushed upward (by lower vent clea7.ing) through the lower surface of the bubble s is assured to be entrained upward and to become

,7-s ) part of the layer of liquid above that bubble. The

\/% mechanism of entrainment is not taken into account in this rodel.

3. The state of the air is uniform throughout the bubble, and the equation of state is the perfect gas law. Therefore, the air within the bubble is assumed motionless, and the pressure variation due to gravity is neclected.

4. The model for the liquid motion in the containment pool is vertical slug flow, with no friction along the vertical walls.

5. The bubble model for vent discharge pressure is not used after bubble breakthrougn occurs (i.e., after the bubble breaks through the pool surf ace) . The vent discharge pressure is then determined with the model described in Section 6.

6. The vent clearing model is not used after the bottom row of vents clears.

N--

4-2

. - - . . - - - , , . , _ - _ - , - - , - , - , _ _ , - - - _ , , , ._. .- -- . - - - - , - - . . . . , _ _ ~ _ - - --

SWECO 8101

/'"s 7. The density of each fluid, including the gaseous

( ) phase, is assumed constant when the fluids are tetween the weir wall and the cuter side of the drywell.

8. The vent clearing flow model is one-dimensional; however, the continuity equations for the horizontal vents include the two-dimensionality of the gaseous phase expanding in the vents.

4.1.2 Discussion A consistent set of equations describing the vent clearing phenomenon is derived by using the mass and momentum conservation equations for all flow regions (t,2), except for the gaseous phase forming the bubbles; the latter is described by the mass conservation equation, the ideal gas equation of state, and the first law of thermodynamics ( 33 The suppression pool has been divided into elementary volumes called nodes, which are represented in Fig. 4-2.

Note the sign convention for distance (x) and for velocity (u).

The fluid velocity through a cleared vent is calculated by the vent ficw model (Section 5) and is not included in the (gx) present discussion of vent clearing.

v The assumption of constant density in one-dimensional slug flow in a pipe yields a uniform velocity along the pipe at all times; in particular, the entrance and exit velocities are the same. By inspection, this result is also valid when the surface of separation between two fluids, both of constant density, is perpendicular to the axis of the pipe.

Furthermore, it is shown in Appendix B that this result is also valid when the surface of separation between the phases consists of intersecting horizontal and vertical planes.

Thus, at all times, the horizontal velocities at the entrance and exit of the vents are related hy:

v2 =vs v = v3

v. = vs and only v3, v3, and v s remain as unknowns.

p

_) 4-3

SWECO 8101 The pressure at the initial drywell water level (for

(()s x=x, a see Fig. 4-2) is denoted by Pa and is treated as a known quantity. P can be determined from the drywell pressure, Po, by accounting fcr the pressure losses at the entrance to the annulus: ,

P E=P 1+ u 2

+c u 2 9 9 where:

G = Pressure loss coefficient, with niaximum value of 0.5 Hence, each time Pa appears, one may substitute P =P ~

E + "

D g i The derivation of the vent clearing equations is presented in Appendix C.

-- 4.1 3 Comparison with Test Data

'N / Test results from General Electric's (GE) Pressure Suppression Test Facility ( PSTF) C * ) have been used to verify the vent clearing and vent discharge pressure portions of i the model. The full-scale one-vent, two-vent, and three-vent steam blowdown tests were used. For these comparisons, the vent-clearing model was isolated from the overall LOCTVS code and programmed with the experimentally measured drywell I

and containment pressures and temperatures to show the response of this model alone. The containment pressure plus

, the pool hydrostatic head were used for the vent discharge

! pressure to neglect the effect of the higher backpressure l

presented by the bubble. Those tests that were available to SSW, along with pertinent parameters, are shown in Table 4-

1. In all cases, the calculated vent-clearing time is l

greater than the measured value (i.e. , conservative) and, on i the average, the calculated times are 15 percent greater.

For the vent discharge pressure comparison, the measured i drywell pressure transients were used from the time of l measured vent clearing to drive the model. Comparisons were l made for four one-vent and four two-vent runs. Fig. 4-3 through 4-10 show the results. For all comparisons, the calculated vent discharge pressures are shown to be higher (O

U 4-4 L

SWECO 8101 p than the measured values and are therefore conservative. In

( some cases, the calculated vent discharge pressure differs

\- from the reasured pressure with respect to time. This is because the measured vent-clearing times used in the model do not always correspond to the exact time when the bubble reaches the pressure-measuring device in the vent system.

These comparisons show that the vent-clearir.g and vent discharge pressure models will conservatively simulate the containment transient following a LOCA.

4.2 VERTICAL VENTS This model is concerned with the over-under type containment (Fig. 1- 1) . The drywell is above the suppression chamber which contains the suppression pool. The vents which connect the drywell atmosphere with the suppression pool pass downward through the drywell floor and are referred to as downcomers.

The analytical treatment of the vent-clearing phenomenon is simpler than that for the horizontal vent configuration, since the vertical vents simultaneously clear, but the two configurations are phenomenologically similar.

A

/ ) 4.2.1 Assumptions v

The following assumptions are made concerning the vent clearing and vent discharge model:

1. The initial pool swell transient ends when bubble breakthrcugh occurs (section 4.2.3) .

2. Suppression chamber air space compresses essentially isentropically during the vent clearing and pool swell stages of the LOCA (see Section 4.2.2 for the pclytropic exponent used) .

Thereafter, the air temperature is equal to the pocl temperature.

3. Bubble temperature is equal to the drywell atmosphere temperature.

4. The model for the liquid motion in the pool is vertical slug flow, with no friction along the vertical walls.

(O,) 4-5

SWECO 8101

(' 5. The upper and lower boundaries of the bubble are considered to be horizontal planes extending across the entire area of the pool.

4.2.2 Equations Neglecting friction and the momentum of the gaseous phase, and assuming one-dimensional flow, the momentum equation for the slug of water in the vent (Fig. 4-11) during vent clearing becomes:

h+1 g (P t

-Pf+

p 2

and the displacement x= u dt where:

u = Vent water slug velocity

\m ') x = Vent water slug displacement P3 = Crywell pressure Pa = Vent discharge pressure b = Height of vent water slug = s - x s = Submergence (initial vertical distance between

the vent exit and pool surface) i P= Pool liquid density i

g = Acceleration of gravity l

1 = Added length of water slug to account for virtual mass effect = downcomer diameter During vent clearing, the vent discharge pressure, P2, is the supprescion chamber atmosphere pressure plus the hydrostatic head of the. pool to the bottom of the downcomer.

Once the vents are cleared and the drywell constituents flow into the pool, a bubble forms at the vent exit.

m t,-] 4-6

- _ __ . . _ ~ , - - _ , . , -. . . . _ . . . , _ , . _ _ _ , _ . , , . . . . _ . . . , , . - , _ . . , . . . . _ . . . _ . . _ . .

i

[

SWECO 8101 ,

1

, The momentum equation for the slug of pool water above the bubble (Fig. 4-12) becomes:

l y =

P2 - Ps

, - g Lo and the pool displacement, i

! y= v - dt  !

i where:

?

(

v = Pool velocity .

1 y = Pool displacement t

' P2 = Eubble pressure P3 = Suppression chamber atmosphere pressure i L = Thickness of pool water slug above bubble The bubble pressure, P, 2 i s determined from the ideal cas

law, and is the vent discharge pressure after vents have cleared up until the time of bubble breakthrough. ,

, i l

t I DbRT a DW l

P* = A py where:

r b = Bubble mass I

Fa = Gas constant for air J  !

TDW = Drywell temperature Ap = Pool crcss section area 1

l s 4-7 i

4 4 f L

vgt --v-r e v %,e- e a e- vy v w T==ww=~-w+1rew e =r ew , v.- .y-wwy~ ve+,-,y-yavy-+ w ry- y ,e-g-y -ye,e,- 3-m-,-,,,w-,wwww,,v-e,--,.gy,-,,pyw,-eg-.-ww.ee--w-,-,-,y+,y,-y,,,9,s,,%.

i

! SWECC 8101  ;

I The bubble mass is obtained f rom the air vent flow: i

(

. m = c u y A y dt  !

3 g i

i where:  !

l I

og = Drywell air and steam density u = Vent flow velocity (determined from vent flow .

rodel) [

Ay= Vent area The suppression chamber pressure, P, s is compressed by the rising pool,  ;

[v sc o hk [H sc o hk 4 P s

=P = P i so v sc ) SO ( Hsc o ~Y) l s .

where:  :

i -

P

• Initial suppression chamber pressure 4

s0 .

v sc = Suppression chamber' air volume v sc = Initial v sc o

. Hsc = Initial height of suppression chamber air space o

k= 1.273 (based on deduction made by Mark II Pressure Suppression Test Program 5) y = Pool displacement

After bubble breakthrough, pool fallback flow is governed by gravity, i.e.,

4 I

v=- g dt 4

i

~

k

( 4-8 1

SWECO 8101 e

S 4.2.3 Breakthrough

Following vent clearing, high pressure drywell air enters the suppr ,sion pool cnd forms a bubble. The bubble pushes the water above the downconer exit upward. The submerged air volume increases and grows vertically. The air bubble eventually overexpands due to the upward momentum of the water slug. At this time, the bubble pressure is less than the hydrostatic pressure because of the bubble overexpansion and because of the rapidly increasing wetwell pressure.

The phenomenon of breakthrough in over-under containments is a function of both the water slug collapsing through the bubble due to the overpressure above it and the submerged air bubble breaking through the water surface due to buoyancy. This type of breakthrough results in the absence of significant froth swell for Mark II containments as described in the Mark II Pressure Suppression Test Program report (5).

When the pool surface reaches its maximum height, an appreciable percentage of the original slug length still exists, even though the pcol surf ace velocity has diminished to zero. This is when the breakthrough process begins. The submerged air bubble starta to disperse into smaller bubbles which then rise through the remaining slug length. As the g

process continues, the water slug begins to fall.

L The pool swell transient ends when the pool surface achieves maximum height and has zero velocity. Fallback velocity and elevation are computed assuming that gravity is the only f orce acting on the pool water.

4.2.4 Comparisen with Test Data The ability of the pool swell model described above to predict swell elevation and velocity and suppression chamber air space pressure has been evaluated. Data from the Mark II Pressure Suppression Test (5) was used for comparison with the model. The model was separated from the overall containment analysis program and driven with the measured drywell pres sure , i.e., the test pressure was input as a boundary condition along with the test facility geometry, initial conditions, etc.

Following vent clearing, drywell air and steam flow through the vents into the suppression pool. The steam condenses and the air forms a bubble at the vent exit. Consequently, the greater the air flow, the greater the bubble pressure and pool swell velocity become. Since the primary purpose

/ )

(_/ 4-9

SWECO 8101

(% f of this comparison is to accurately predict pool swell conditions, the air / steam fraction of the vent flow is an

(

'- / important parameter to be considered and was deduced as described below while making pool swell height comparisons.

When using LOCTVS it is generally assumed that the vent flow is initially all air followed by a homogeneous mixture of air and steam. The process by which this occurs is described as follows. At the time of a LOCA, the drywell atmosphere is thoroughly mixed with reactor effluent steam due to the turbulence created by the high velocity fluid discharging from the break, but due to the small cross-sectional area and the long length of the vent pipes, there is little mixing of the vent system air. The rising drywell pressure compresses the vent system air, which in turn effects clearing of the liquid in the vents. As the air column is expelled into the pool, the bubble is formed.

The ensuing vent flow is a homogeneous mixture of drywell air and steam. The steam condenses and the air contributes to the expanding bubble.

The following comparison with experimental results show9 that the vent flow is initially 100 percent air followed by an air / steam mixture. Comparisons are made with the available Mark II Pressure Suppression Test data (pool swell height, velocity, and wetwell pressure transients fror 7-~s

) Reference 5) . The following four different assumptions with

\/ regard to the composition of the vent flow were evaluated in making the nodel test comparisons for Run 31:

1. 100 percent air

2. Uniform mixture of air and steam

3. Uniform mixture with the air / steam fraction existing in the drywell at the time of vent clearing (mix and purge)

4. 100 percent air until the vent system is cleared of air; afterward a uniform mixture of air and steam.

Fig. 4-13 shows pool swell velocity versus height for both Run 31 of the Mark II Pressure Suppression Test Program and the pool swell model using Assumptions 1, 2, and 3 above.

Assumption 1 initially matches the test velocity but continues to increase as the measured values decrease. The peak velocity for Assumption 1 is 40 percent greater than the test value, and the wetwell pressure rise is 100 percent greater as indicated in Fig. 4-14. Assumptions 2 and 3 are

)

(_,/ 4- 10

SWECO 8101 S closer to the data hut are on the Icw side, especially early

) in the transient.

Assumption 4 is initially all hir followed by a uniform mixture. The duration of the all-air flow is based upon the time it takes to deplete the air mass in the vent line and downcomer sections of the 4T test facility. This mass represents 12 percent of the total drywell air mass.

Fig. 4-15 and 4-16 show the close agreement for Run 31. The calculated peak swell velocity slightly overpredicts the measured value, which indicates that there may be some mixing of drywell steam and vent system air at the interface of the drywell steam and vent line section. The greater calculated pool swell velocity after the swell height of 8 ft may be due to the conservatism of Assumption 3 of Section 4.2.1, which says that there is no heat transfer 4

f rom the high temperature bubble to the relatively cool pool water or to the dcwncomer wall.

Other 4T runs with a sionificant amount of reported data include Runs 27, 29, 33, and 35. Fig. 4-17 through 4-21 show comparisons of the available experimental data with model predictions using Assumptic.i 4 for Runs 29, 27, and

35. Table 4-2 is a summary of the peak values for Runs 31, 29, 27, 35, and 33. Comparisons were not made for other

-~ runs because of insufficient reported data.

D I

t l

l l

i l

v q_11 l

SWECC 8101 References - Section 4

1. Liepman, H. W. and Pohko, A. Elements of Gasdynamics.

John Wiley and Sons, Inc., New York, NY, 1963, p 181-183.

2. Rohsenow, W. and Choi, H. Heat, Mass and Momentum Transfer. Frentice-Hall, Inc., Englewood Cliffs, NJ, 1961, p 67.

3. Hatsopoulos, G. and Keenan, J. Principles of General Thermodynamics. John Wiley and Sons, Inc., New York, NY, 1965, p 111

4. Mark III Confirmatory Test Program Phase 1 - Large-Scale Demonstration Tests. Test series 5701 through 5703, October 1974, NEDM-13377.

5. Mark II Pressure Suppression Test Program, NEDO-13442P, May 1976.

i O

i l

i I

4 1

1 1

\

~/ 4-12

?

EWECO 8101 i

TABLE 4-1  ;

PSTF TEST CATA CCMPARISON i

ONE-VENT TESTS t

Test Cescriction Ereak size, in 2 1/8 2 1/2 2 1/2 3 5/8 [

.; vent submergence, ft 12 8 12 12 i

. t

Vent-Clearing Time, Sec i Test 1.66 1.15 1.45 1.30 Model 1.91 1.40 1.71 1.49

.; - TWO-VENT TESTS j 1

j Test rescription  ;

r Break size, in 1 1/8 2 1/2 2 1/2 3 5/8 Vent submergence, ft 12 8 10 12 l.

f- Vent-Clearing Time, Sec l

Top Vent Test 1.27 0.94 1.20 0.99 <

Model 1.52 1.10 1.38 1.20 l i Bottom Vant t Test 1.76 1.43 1.71 1.36 Model 2.18 1.57 1.88 1.53 k

I 1 of 2

l SWECO 8101 l

TABLE 4-1 PSTF TEST DATA COMPARISON (Cont) i THREE-VENT TESTS Test Description Break size, in 2 1/2 2 1/2 3 5/8

Vent submergence, ft 7 11 11 l

Vent-Clearing Time, Sec i

l Top Vent i Test 0.86 1.14 0.99 l Model 1.03 1.35 1.11 F

Middle Vent i

Test 1.15 1.52 1.19 i

Model 1.40 1.64 1.30 i

Bottom Vent t

Test DNC DNC 1.60

-Model DNC DNC DNC i

l l

l l

l l

l DNC: Did not clear o i 2 of 2 i

t I i l

.- . . .~ .. .- . - . . ~ . . - . . _ . . - . . . - - _ .

d i

1 SWEco 8101 1

i TABLE 4-2 1

J 4

MARK II PRESSURE SUPPRESSION TEST DATA COMPARISON

, Max Pool

, Vent clear- Surface Max Pool Time of Peak Drywell j Submergence Venturi ing Time Velocity Swell lleight Breakthrough Floor Upward 1

Run - (ft) j in) (nec) (ft/sec) (ft) (sec) Ap (psi)

! 31 9 3 l Measured 0.56 29 12.9 1.25 -1.5 calculated 0.59 30.7 -14.7 1.32 -1.1 '

t 29 11 2. 5 i Measured 0.78 20 11.0 1.65 0.6 i calculated 0.79 21.3 12.2 1.67 1.7 i

! 27 11 3

! Measured 0.73 23 12.5 NA 'O.75 r Calculated 0.76 25.0 13.2 1.57 1. 5 35 9 2. 5 Measured 0.77 27 10.2 NA -0.2

{ calculated 0.71 23.4 12.9 1.56 1.9 i

j 33 11 3 j Measured 0.74 20 11 NA 2.2 i calculated 0.72 23.4 13.6 1.60 4.9 i

3 i

i 6

I r

J, j

i i

i .

. t NA = Not available i

l 1 of 1 i . __.. , _ . - ._, ._

O

~

CONTAINMENT WALL WEIR WALL \

DRYWELL WALL LIQUID GAS

_ C.sEOUS VENT FLOW LIQUID l

l e

FIGURE 4-1 MODEL FOR THE SUBBLE l

O

CONTAINMENT p CRYWELL

, WEIR x3

/ WALL 8

/ Pi p t r. 2 [T, w

• f1 9

///// l

/

$ vM 2 @V dE i 2 *2 J

! h -

pv2)

=

E D

<c _______._________________.__.y f 8x" @ / S  :

/ "

[

9 ':

s  ;

f/w //A /[) lf !@

e V4 P 3e X q @V 3 P3

[3 $9 Pv3 P3 o

!"'\ /

77777p u3

/

/

g 5

/ E i $ l /

G // 5 w b l }/ / l@ / // y-i

/m@wPv4d j\

1 l

/

flf 5 P4c Pv4e I

4_

f E

/

/ _-------------- ----------

V 69 V Mo u,

[

/

rf 7fff h h s

h s /

s/

N/##/

ME/Z/ MMH ###/###/A l NOTES:

l L THE u AND v ARE VERTICAL AND HORIZONTAL VELOCITIES, RESPECTIVELY.

l 2.THE: AND y ARE VERTICAL AND HORIZONTAL DISTANCES, RESPECTIVELY.

1 TOTAL AREA CF ONE ROW OF VENT: A, l 4. ANNULUS HORIZONTAL ARE A: A l 5. POOL AREA:A p l --- NODE BOUNCARY SURFACE OF SEPARATION BE7 WEEN PHASES (WHEN APPLICABLE)

FIGURE 4-E NODE CONFIGURATION AND SIGN CONVENTIONS

50 .

40 4

V W 30 e

k.

-2 f

M w -

20 0

0.0 02 0.4 0.6 0A TIME AFTER TOP VENT CLEARS,SEC j LEGEND

1. MEASURED DRYWELL PRESSURE

2. MEASURED BUBBLE PRESSURE

3. CALCULATED VENT PRESSURE FIGURE 4 - 3 ONE VENT TEST 2 '/8 IN. BREAK-12 FT SUBMERGENCE

( ,,

PSTF TEST SERIES 5701

50 4

40 i

b E

d e -

J' 5' N y

tu

[ 3 20 i

O

! l0 i

O0 0. 2 04 0.6 0.8 TIME AFTER TOP VENT CLEARS,SEC.

! LEGEND

1. MEASURED DRYWELL PRESSURE

2. MEASURED BUBBLE PRESSURE

3. CALCULATED VENT PRESSURE l

, FIGURE 4-4 ONE VENT TEST Y

2 2 IN. BREAK-8 FT. SUBMERGENCE PSTF TEST SERIES 5701

t. . - . . - _ _ , . . - - . . . . . - . , - - . - _ . - - - . - - . - - - - . - . . . . - , .

l l

so 40 4_

/1 E N E w a

0 3 a:

20 i

O'~ 10 _

00 0.2 0. 4 0.G TIME AFTER TOP VENT CLEARS, SEC LEGEND L MEASURED DRYWELL PRESSWE

! 2. MEASURED BUBBLE PRESSURE

3. Cl4.CULATED VENT PRESSURE FIGURE 4-5 O ONE VENT TEST

( ,/

2 2 IN. BREAK-12 FT. SUBMERGENCE PSTF TEST SERIES 5701 l

50 l

^

40 s

E J \

3 8:

E x, 3 20 G

U 10 0.0 0.2 0.4 0.6 TIME AFTER TOP VENT CLEARS,SEC l

LEGIND

' MEASURED DRYWELL PRESSURE

2. MEASURED BUBBLE PRESSURE

.3 CALCULATED VENT PRESSURE l

FIGURE 4-6 f

ONE VENT TEST

(,,/ 3b IN. BREAK-12 FT. SUBMERGENCE PSTF TEST SERIES 5701 l

l -- -- - - - - - -

50 i

i l

l 40 E

ui E

3 30 5 X TOP VENT 2

%/&

3 20 BOTTOM VENT i 10 l

CN O.0 0.2 0.4 0.6 08 TIME AFTER TOP VENT CLEARS,SEC l

t LEGEND

1. MEASURED DRYWELL PRESSURE

2. WEASURED BUBBLE PRESSURE

3. CALCULATED TOP VENT PRESSURE

4. CALCULATED BOTTOW VENT PRESSURE l

l l

FIGURE 4-7 l O TWO VENT TEST l \'- 2 b8 IN. BREAK-12 FT. SUBMERGENCE i

l PSTF TEST SERIES 5702

O "

40 5

E

. 30 -

g 3

$ 4 s

20 \ ^ '

TOP VENT \

BOTTOM "ENT 10 0.0 02 0.4 0.6 TIME AFTER TOP VENT CLEARS, SEC.

LEGEND

1. MEASURED DRYWELL PRESSURE

2. MEASURED BUBBLE PRESSURE

3. CALCULATED TOP VENT PRESSURE

4. CALCULATED BOTTOM VENT PRESSURE FIGURE 4-8 TWO VENT TEST 2 2 IN. BREAK-8 FT. SUBMERGENCE

('s- PSTF TEST SERIES 5702

, , . . - - . - - - . - . . . . . , . _ . _ - _ - - -, . , - . . ~ , , _ , _ . _ . . , _,,,__--_, ., . . , - _ . , _ . , , _ _ . - _ , , _ . , - , , _ , .

50 .. 1 40 t

5 E

ui  !

e S 30 1

2 3

TOP VENT BOTTOM VENT 10 -

00 02 04 C6 08 .

~..

TIME AFTER TOP VENT CLEARS, SEC.

LEGEND

1. WE ASURED DRYWELL PRESSURE

2. MEASURED BUBBLE PRESSURE 3 CALCULATED TOP VENT PRESSURE 4, CALCULATED BOTTOW VENT PPESSURE FIGURE 4-9 TWO VENT TEST l 2'2 IN. BREAK-lO FT. SUBMERGENCE

)

l s/ PSTF TEST SERIES 5702 l

_. - - . - - , . . - - . --, _ . ~ . - - . , . . - - . . - - , . _- ,_. .. . ..,. _ ,..---,4 . -- - - - -

i 40 N

5 m

i /\ /\m 4 N I .

TOP VENT g BOTTOM VENT to 0.0 0.2 0.4 0.6 08 TIME AFTER TOP VENT CLEARS, SEC.

LEGEND

1. MEASURED DRYWELL PRESSURE

2. MEASURED BUBBLE PRESSURE

3. CALCULATED TOPVENT PRESSURE

4. CALCULATED BOTTOM VENT PRESSURE FIGURE 4-10 TWO VENT TEST

\ 34 8 IN. BREAK-12 FT. SUBMERGENCE j

v PSTF TEST SERIES 5702

, - . -. , _ . . ,  %-,.+-.---v. - - - . - - - , . - - , ,yr--%_ - - - -,---., e--, , -,,, - - - - . , - - , -y,, * -

O DRYWELL ATMOSPHERE RYWELL FLOOR

DOWNCOMER SUPPRESSION CHAMBER P,

n l O SUPPRESSION POOL a

1 y o l

x u S

h WATER SLUG o u .

Pg FIGURE 4- 11 VENT CLEARING MODEL FOR VERTICAL VENTS

- . _ . . ._ - _ ,...._ ____ _ _ _ . _ . . ~ . . _ . _ _ _ . . _ _ _ . .

AIR,STE AM, LIQUID DROPLETS o \ /

DRYWELL V/

/ ,

V /

/ JA Y P3 SUPPRESSION CHAMBER

1. A DOWNCCMER Ak l

Ji L WATER SLUG V

Ak l

l y *P2 AIR BUBBLE t V 1

SUPPRESSION POOL FIGURE 4-12 POOL SWELL / BUBBLE MODEL FOR VERTiCALVENTS

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O O.5 IO 1.5 2.0 TIME,SEC FIGURE 4-14 O WETWELL PRESSURE RESPONSE. FOR 4T TEST RUN 31

O D 6

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& BREAKTHROUGH W na 10 MEAS. - -CALC.

O o VENT CLEARING TIME 0 05 10 15 20 TIME,SEC O FIGURE 4-16 WETWELL PRESSURE RESPONSE FOR 4T TEST RUN 31

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-l O FIGURE 4-18 WETWELL PRESSURE RESPONSE FOR 4T TEST RUN 29

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SWECO 8101 SECTION 5 O' VENT FLOW 5.1 MODEL DESCRIPTION After the vents have cleared ao described in Section 4, the mixture of gases and water existing in the drywell flows through the vents into the suppression pool. The flow rate through the vents is a function of the difference between the drywell and the vent discharge pressures. It also depends on the thermodynamic state of the mixture entering the vents as determined by the pressure, temperature, and mass fractions of air, liquid, and steam in the drywell.

Under some conditions, the flow through the vents might also be limitnd by critical flow rate considerations.

In two-phase flow the average velocity of the liquid is usually less than that of the gas. When the gas and liquid have different velocities, the flow is referred to as slip, or separated flow. It is difficult to assess, on a theoretical basis, the amount of slip occurring between ~

phases (1-8). However, most empirical data taken with suberitical as well as critical flows reveal the existence es of some relative velocity between the phases. In

\ particular, a gas velocity in excess of the liquid velocity

(

A- l is characteristic of mixture flow at high velocity and high vapor quality, such as the vent flow under consideration.

In the absence of detailed information on the flow, the simplest assumption that can be made is that the gas and the suspended liquid droplets have the same velocity. This type of flow is referred to as homogeneous flow.

! When homogenecus flow is assumed, the resulting homogeneous i velocity is greater than the actual liquid velocity and less than the actual gas velocity. The pressure drop based on the larger homogeneous liquid velocity is higher than would be calculated considering slip, and the smaller homogeneous gas velocity relieves the drywell pressure mere slowly.

Therefore, the homogeneous assumption for vent flow is c onse rvative with regard to the drywell peak pressure calculation.

The flow model used in LOCTVS is based on compressible gas flow theory (*-to) extended to include the effect of the incompressible (liquid) phase. The behavior of the mixture can be conveniently described by considering it as an ideal l

1

(}

v 5-1

.- = -

i SWECO 8101 gas with appropriately calculated pseudo-properties (**).

O The flow through the vents follows the Fanno line.

5.2 ASSUMPTIONS The following assumptions are made:

1. The flow is one-dimensional.

2. The flow is quasi-steady state. During any time interval, the flow in the vent pipes is assumed to be steady, i.e., the response of the flow to changes in driving pressure is instantaneous.

This assumption results in a negligible error since the time scale of the pressure changes is relatively long compared to the vent transit time.

{

3. The flow is homogeneous (no slip between phases is a ss umed) .

4. The flow is adiabatic (no heat transfer occurs between the vent and the fluid) .

5. The quality is constant over the length of the vent, i.e., there is no heat or mass transter between the fluid phases.

6. The gaseous phase (air-steam mixture) is considered a perfect gas. Average properties are calculated by weighting the respective properties i of the two constituents. In these calculations, the thermodynamic properties of the water vapor are obtained from steam tables.

i

7. The liquid phase is incompressible.

8. Based on Assumptions 5, 6, and 7, only the vapor phase undergoes isentropic expansion. The specific heat ratio, Y, is based on the air-steam mixture properties, and ranges from 1.1 to 1.4.

9. The ficw rate becomes critical, and is therefore limited, when the mixture velocity equals the sonic velocity of the mixture at the exit.

l

10. Pressure changes within the vent due to gravitational effects are negligible.

l 5-2 O(N l

l

l. - .-- - - --,w , - , - , , , _ - - - . - - . . , , - - , - . , , ..n.- - ~ , , . , . . - , - , , , - . - , . - , _ .

SWECO 8101

()

(N 11.

12.

The flow is accelerated isentropically from the drywell to the vent inlet.

The vent pressure loss coefficient (K) is  !

constant. The K-f actor used in LOCTVS includes the contraction and skin friction losses which are encountered in flow through the vent. The vent exit loss is implicit in the calculation scheme of LOCTVS. In the calculation of the thermodynamic state, the flow velocity within the drywell and suppression chamber are assumed to be zero. Thus one full velocity head is lost at the exit of the vent. This is equivalent to adding an exit K-factor of 1.

13. The single phase K-factors are used in the momentum equation. This results in an implicit homogeneous two-phase friction multiplier due to the use of the mixture density ( S S 3 I,

5.3 EQUATIONS FOR VENT FLOW The vent flow model de scribed above is called the homogeneous vent flow model (HVFM) . The equations presented

-~s below describing HFVM, are derived in Appendix D.

\_- Solving the equations of state, continuity, energy, and momentum with the assumptions outlined in Section 5.2 leads to the following equations:

3 2 2 Y-1 2 2 M 2 -M 1 /Y+1h l 1+ 2 M2 j lM 3 K=1 + l in Y M,2M,* \ 2Y j 1+ Y-1 M* 3 M,*

(5-1)

~

'( \

l b P

o2 1 2+(Y-1)Mi j Mi p r / 21 2 og f Y-1 2 y-1 2+(Y-1)M2 M2

) M3

\ . .

(5-2)

Og 5-3 V

SWECO 8101

-~ where:

U Pos = Stagnation pressure in the source node Poa = Stagnation pressure in the sink node Y = Air-steam mixture specific heat ratio (C p/C y)

= Mach number at the inlet to the vent Ma M2 = Mach number at the exit of the vent K = Vent pressure loss coefficient.

This system of nonlinear equations is solved for Mi and Ma using a Newton-Raphson interation technique. If Mr 2 1, the flow is assumed to be choked. When Ma > 1, Equation 5-1 is sclved for Mg with M, set equal to 1. Once Mg has been determined, the mass flow rate per unit area is determined by the following equation:

Y+1 G= 12M g g

)h I1+ Y-1 2) 2 (1-Y) x (Yo,P g g, e

)

l l

\

2 .M 3

)

l (5-3) where:

G = Mass flow rate per unit area po = Density of mixture in the source node gc = Ccnversion factor The other units are as defined above.

5.4 COMPARISON WITH EXPERIMENTAL DATA l HVFM has been compared with experimental data for choked flow in Reference 12. Fig. 5-1 through 5-3 show the comparison of flow rate (mass velocity) versus choked exit pressure for qualities of x = 0.9, 0.6, and 0.3, respectively.

These figures also compare the HVFM model'with the Fauske l slip critical flow model( 12 ) . The agreement with

% 5-4

)

1 a

i i

r SWECC 0101  !

i experimental data decreases as the quality decreases and the f pressure incre as es. However, the codel provides a  !

consistently conservative prediction of mass velocity (i.e.,

predicts lower mass velocity than experirentally reasured) f or all cases.

?

i l 1

1 l L

i i

l i

h i

l i

I I L l

[

I f

1

L 1

i l

I I

t l l r i

?

l I ,

.I 5-5 .

SWECO 8101

( References - Section 5

1. Martinelli, R. C. and Nelson, D. B. Prediction of Pressure Drop During Forced-Circulation Boiling of Water. Trans. ASME, 70, 1948, p 695-702.

2. Thom, J. R. S. Prediction of Pressure Drop During Forced Circulation Boiling of Water. Int. J. Heat Mass Transfer, 7 1964, p 709-724,

3. Petrick, M. A. Study of Carry-Under Phenomena in Vapor Liquid separation. A.I.Ch.E. Journal, 9, No. 2, 1963, p 253.

4. Janssen, E. Two-Phase Pressure Drop in Straight Pipes and Channels; Water-Steam Mixtures at 600 to 1,400 psia. GEAP-4616, Mal 1964.

5. Zuber, N. and Findlay, J. A. Average Volumetric Concentration in Two-Phase Flow Systems. J. Heat Transfer, Trane. ASME, 53 ries C, 87, 1965, p 453-468.

6. Levy, S. Steam. Slip - Theoretical Prediction from Momentum Model J. Heat Transfer Trans. ASME, Series C, 1960, p 113-124.

I\ 7. Chenoweth, J. M. and Martin, M. W. Turbulent Two-Phase

\s / Flow. Petroleum Refiner, 34, No. 10, October 1955, p 151-155.

8. Huey, C. T. Adiabatic Homogeneous Bubbly Flow in Horizontal Pipes. Canadian Journal of Chemical Eng.,

tecember 1966.

9. Shapiro, A. The Dynamics and Thermodynamics of Compressible Fluid Flow, The Ronald Press Company, 1953.

10. Thompson, P. A. Compressible-Fluid Dynamics, McGraw Rill Book Co., 1972.

11. Wallis, G. B. One-Dimensional Two-Phase Flow, McGraw Hill Book Co., 1969.

12. Boyle, J. C. Homogeneous Two-Phase Compressible Flow Through a Constant Area Vent. Masters Thesis in Nuclear Engineering, University of Phode Island, 1978.

) 5-6 d

I 55 i

t i'

1000 -

. & FA L E Til t

- X* 85 -95 % 3 MOY 1 4

.

• CPUZ  :

i 500 X'83-97% @ KLINGEBIEL I

[

THEORY ( F AUSKE ) (

FOMOGENEOUS MODEL (FAUS<E)  !

VENT FLON NOCEL 4 I

/ {

/ l 7% ~ 100 -

.a

.=

n

/ .

- a, t

. A t

- Y 4.

SC i

s. 300 V -

o M

- 1 i

• 200 - -  !

E i

70

_/ i go wt , ,,  ! , , ,  ! , l  ;

30 50 100 .

, 500 10 N I f f f f 1 ,

15 0 500 1000 2500 5000 t FLOW R ATE G , k g /s r.t ,.

FIGURE 5-l 5 i

COMPARISON WITH CHOKED FcCW EXPERIMENTAL DATA AT X=90%.(BOYLE AND WALTERS,1977) i

-- ._. ., .. . . _ . - , .. __ _ . . .._. . - . . _ _ . . _ _ , _ . ~ . - _ . - . , . . ~ . . - . _ _ . . . . . . . . , _ . _ . . . . . . . . . . . . . - . _ _ . . . _ . . _

SG s

1000 -

~

X = 60-55 % A FALETTI

- X s 65-55% a MOY X s 60 -65% v CRUZ SOO - Xz 55% o FAUSKE Xs 62 -63% @ KLif 4GEBIEL THEORY (FAUSKE) ,

HOMOGENEOUS MODEL (FAU5KE)  ;

WOO .

, _.- VENT FLOW MODEL A ,

700 jon ,_

~

4 s z

a

!s .

,- /

s

~

.if;f-200 -

[,: *

/, / *

[

70 - 10 i I iil e i i l i,e il 30 50 10 0 2

500 1000 j

t t t , ,

150 500 1000 2500 5000 FLOW RATE G , k g /s m 2 l

l l

[ FIGURE 5-2 COMPARISON WITH CHOKED FLOW EXPERIMENTAL DATA AT X=60%.(80YLE AND WALTERS,1977) l I

i

/

i t

I l

I

57 f

i 1000 -

~

a FALETTI X

• 25-35% o MOY

, e FAUSKE r 3oc _. xs 30-32% @ KLitJGEBIEL l

THEORY (FAUSKE) j - ----- HCMOGEt.EOUS MCOEL(FAU5(E)/

VENT FLOW MODEL A /

1400 -

, [o o

, / /

1 700 10 0 - j,' &[f a / ,

5 ~1

,f g,f"

?' O

~

300

= - '

i 200 - -

,/f

e.

• O D'

70 jotg

/ , , , t , , , ,t , , , t 100 500 1000 5000 4

lb/ - f,2 i f f I 500 2500 SOCO 25 000 FLOW RATE G , k g /s rnt 1

i 1

F FIGURE 5-3 COMPARISON WITH CHOKED FLOW EXPERIMENTAL DATA AT X= 30%.(BOYLE AND WALTERS,1977)

O

SWECO 8101 SECTION 6 N

WETWELL (HORIZONTAL VENT CONTAINMENT) PRESSURE 6.1 DISCUSSION In horizontal vent containments, the space between the hydraulic control unit's (HCU) floor and the suppression pool is referred to as the wetwell.

The model described in this section predicts the pressure response below the HCU floor during the pool owell stage of a LOCA in a horizontal vent type containment. The HCU floor is about 20 ft above the suppression pool under normal operating conditions and occupies from 50 to 75 percent of the containment cross-secticnal area. Cata from GE's Pressure Suppression Test Facility (PSTF) show that the pool may swell in a iroth (air / water mixture) mode to the elevation of the HCU floor following a LOCAC1). The flow restriction impoF2d by the floor on this two-phaso flow may cause pressurization of the wetwell, the effect of which is to increase the vent back pressure and, consequently, increase drywell pressure.

b) 6.2 6.2.1 ASSUMFTIONS General The following general at ptions are made:

1. The fraction o. CU floor area that is open for flew ie 0.25 (in 'actice, this is closer to

0. 4-0. 5) .

2. The HCU floor pressure l oss coefficient is 5.0 (chosen to be conservativel, Sigh) .

3. Bubble breakthrough occurs wie.. the suppression pool swells to 1.6 times the initial vent submergence (2). Although test data show that a factor of 2.0 is roughly the mean of the data, a lower value produces slig;tly higher wetwell pressures.

/O 6-1 V

SWECC 8101 6.2.2 Assumptions Prior to Bubble Breakthrough The following assumptions are made prior to bubble breakthrough:

1. The control volume contains air only.

2. The vent discharge pressure equals the bubble pressure.

6.2.3 Assumptiens After Bubble Breakthrough The following assumptions are made after bubble breakthrough:

1. The control volume is a homogeneous mixture of air, vapor, and suppression pool water that is initially above the centerline of the top row of vents.

2. Flcw through the HCU floor opening is homogeneous.

3. The control volume temperature is equal to the suppression pool temperature.

4. The control volume is saturated.

5. The vent discharge pressure is the control volume (s,)/

m pressure plus the hydrostatic head of the collapsed liquid.

6. Flcw through the HCU floor opening is incompressible.

6.3 EQUATIONS 6.3.1 Prior to Bubble Breakthrough Prior to bubble breakthrough, the pool swells in a bulk mode. The control volume is defined as the free volume between the bottom of the HCU floor and the pool surface (Fig. 6- 1 ) . The growing bubble forces the pool water above the cleared vents to flow upward, compressing the air beneath the HCU floor. The pressure, PHCU is determined

from the ideal gas law.

P l HCU *

• ART /V ,

i

/~'N 6-2 l f I l

l

l i

SWECO 8101 .

l where:

[

! m a = Air mass R = Gas constant for air i

T = Control volume temperature V = Free volume beneath HCU floor (decreases with l

time)

Conservation of mass, energy, and volume are used to J

determine m a, T, and V as follows:

i Mass: The change in air mass is due to,the flow of air ,

l through the HCU floor openings, mout. Thus, b

t ma" out + "o

o (6-2) and l A

m 9c P HCU - P) c out"~ N - -

(6-3) l

, where:

A = HCU floor open area

! K = 5.0 l' gc = Ccnversion factor a

l 0 = Air density ma /V ,

Pc = Containment pressure t = Time

! m o = Initial-air mass  ;

4 L

i 6-3 s

--r---,--,,---, gr--,e e ---,rm-,e -,-.,,--,,,,-,-,av-,,w.,eeng.-se,cw,

. ,m- m,,m ,n w - r-e,n-+vw.,----ww.,ww,-ww-rew--,ew,v.,w-m,re,w,ew,,re,,,r,,

SWECO 8101

• s Energy b[N dU "

HCU dV dt "out + 778. dt (6-4) where:

j U = Total internal energy of air in control volume h = Air specific enthalpy  ;

and T = c m + 32.

(6-5)  !

where:

i U is taken to be zero at 320F cy= Constant volume specific heat for air i

volume

"~

11LA P (6-6) where:

x asl = Pool swell velocity (see vent clearing model, Section 4)

Ap = Pool cross-sectional area 6,3.2 After Bubble Breakthrough ,

t At *he time of breakthrough, a new control volume is defined from the bottom of the HCU floor to the centerline of the top row of vents (Fig. 6-2) and the bubble air inventory is added to the control volume.

The wetwell pressure, P HCU, is the sum of the air and steam partial pressures.

6-4 t

N

SWECO 8101 PHCU = Pa+P y

= maRT/V + P v (6-8) where:

ma = Air mass T= Fool temperature V = Free volume (total minus liquid) from top vent centerline to HCU floor Py= Vapor pressure at pool temperature The flow through the HCU floor, in out, is calculated as before with Equation 6-3 but with the density term, P, equal to the homogeneous mixture density of liquid, steam, and air , i. e. ,

M a +M s +M

m p, L i V+M v f

(6-9) l where:

mg = Pool mass above top vent centerline m s

• Y#V 9 vg = Saturated vapor specific volume at T vf = Saturated liquid specific volume at T This results in an initial density of about 20 lb/cu ft or a void fraction of about 70 percent.

l The mass conservation equations are as follows:

dm, dt

= m,av - m,E H

(6-10) 6-!s

i i

j SWECO 8101  ;

i i i t

! where: l 1  !

j m av = Air flow through vents i 3 ma '

l . f )

ma H = Air flow through HCU floor = m out l m +m +m Lj i -

a and

dm , , ,

{

+ -m s -m L dt3 = m,s j

i mL v v H H i (6-11) where:

l hs y

and h g = Steam and liquid vent flow, respectively ,

j h and h g = vapor and liquid HCU floor flow, sH H respectively l When the wetwell pressure decreases to the containment pressure, the two volumes and their corresponding masses and j energies are combined.

i t

l i

I l

l 1 -

1 I

l l

6-6 l

l l

w we ew +=everw w v w - -w-y-+ - v www-ge*+- m + wyr w,=w vy ,y r+ ~

• w wt y a w - wv-tr

• t ev wet ww ww*w += w ,+ -* v ert

• mv ww vw - e -w e-s-----we- , - io seer w mw- =w =* -www

• m -e m m-wer*M w e rw-

i

! SWECO 8101  !

i 1

l References - Section 6 i .

1. Mark III confirmi. tory Test Program Phase 1 - Large-Scale  :

Demonstration Tests, Test Series 5701 through 5703,  :

October 1974, 14EDM- 13 377. l

! 2. Containment Loads Report (CLR) Mark III Containment,  !

General Electric Co., San Jose, CA, January 1980.  !

.I .

I i t

} I i

i I

i t

li 4

I t

i p

[

1 i

[,

n 6-7 P

, ~ , , . . . , _ _ - _ _ _ _ - - _ _ _ - _ - _ = _ _ _ . _ _ - . _ .

y a-

' Pc HCU FLOOR ] J h l

7_________p_q_________q l  : i I I

• P HCU l

l _ Com ol votum m _ _ . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . . _ _ _ _ .i L

j BUBBLE o N

._ m l

FIGURE 6-1 l

HCU FLOOR MODEL PRIOR TO BUBBLE BREAKTHROUGH l

_u - _a.-Lm__-. ,e__. -d.- _ _ _

.* _, m J s _ .a- . a h -- 2. . h m -- .__ - --*

h CONTROL VOLUME l O '

r=

HCU FLOOR l d a n [

f (0 o 1

s 10 o / \ I

! O I

I I O

! o o l I

lo o o

o I O FROTH i

g o. g o

i e o l l O o l I O I I O I

I o o o i o o I I L----------------------- J O El SUPPRESSION i

A FIGURE 6-2 O HCU FLOOR MODEL AFTER BUBBLE BREAKTHROUGH

SWECO 8101 O SECTION 7 CONTAINMENT 7.1 ASSUMPTIONS 7.1.1 General Assumptions The following assumptions are applicable to both over-under and horizontal vent containments:

1. The steam portion of vent flow condenses in the suppression pool. This has been verified in the Bodega Day, Humboldt Bay, and Pressure Suppression Test Facility (PSTF) tests.

2. The liquid portion of vent flow remains in the suppression pool.

3. The air portion of vent flow rises to the space above the pool (following pool swell) at the temperature of the pool.

O

( j 7.1.2 Assumption Applicable Only to Over-Under Containments After bubble breakthrough, the suppression pool and the suppression chamber atmosphere are in thermodynamic equilibrium, i.e., the atmosphere is saturated and at the temperature of the pool.

7.1.3 Assumption Applicable only to Horizontal Vent Containments Heat and mass transfer between the pool and the suppression enamber atmosphere is by means of evaporation (or condensation) and convection, and radiation heat transfer.

7.1.4 Other Assumptions Normally Utilized in Analysis The following assumptions are generally made when preparing input for use in the program:

1. Suppression chamber pressure and temperature and suppression pool temperature are at their maximum normal operating conditions.

2. In the short-term phase of the analysis, maximum f- ss vent submergence is used. This provides the max-

/ \

' \,j 7-1

SWECO 8101 imum vent discharge pressure and, therefore, results in the highest drywell pressure.

3. In the long-term phase of the analysis, minimus vent submergence is used. This provides minimum heat sink capacity for residual heat removal and, therefore, results in the highest suppression pool temperature and suppression chamber pressure and temperature.

7.2 CONTAINMENT ENGINEERED SAFETY FEATURES 7.2.1 Containment sprays Over-under containments utilize cc 'sinment sprays to provide centainment cooling for post- cident conditions.

Water pumped through the RHR heat exchc.igers can be diverted to spray headers in the drywell and in the suppression chamber. The drywell sprays remove energy from the drywell atmosphere by condensing water vapor. The water collects in the bottom of the drywell until the water level rises above the top of the downcomers. The water then overflows into fx tne suppression pool. Approximately 5 percent of the spray water is directed to the suppression chamber spray ring to (d) cool any noncondensible gases in the free volume above the suppression pool.

7.2.2 Containment Unit Coolers S&W horizontal vent containments utilize containment unit coolers as an engineered safety feature. The containment unit coolers remove heat from the containment atmosphere under normal conditions of plant operation and shutdown.

Under accident conditions, continued circulation of the centainment atmosphere through the coolers removes latent and sensible heat from the steam-air mixture, thus assisting in the post-LOCA containment beat removal. The containment unit coolers consist of banks of thin tubes with finned outer surfaces. An electrically powered fan draws containment atmosphere across the finned tubes.

The heat transfer coefficient of the containment unit coolers as a function of partial pressure fraction of steam in the containment atmosphere is input to the program in the form of a curve which is established from experimental data for a specific unit cooler. At each time interval, the temperature and pressure of the containment atmosphere are

, ,_s known. The inlet temperature of coolant water, flow rate of

(

\s-) ,

SWECO 8101 O coolant water, and the flow rate of steam-air mixture across the cooling coils are also known.

Unknowns are the steam-air outlet temperature, coolant water outlet temperature, and heat transfer rate. Heat balances are performed on the coolant water, the steam-air mixture, and the heat exchanger, thus producing three equations in the three unknowns. Because two of these equations are nonlinear, the system of equations is solved by an iterative technique. l The first equation, representing a heat balance on the steam-air mixture, is q=m a 1 -bg) - (S) - S 2) h (7-1) where:

q = Rate of heat transfer across the unit cooler m a = Mass flow rate of dry air os ) hms = Inlet enthalpy of steam-air mixture per unit mass of dry air (G

hma = outlet enthalpy of steam-air mixture per unit mass of dry air St = Specific humidity of inlet steam-air mixture S, = Specific humidity of outlet steam-air mixture i ha = Specific enthalpy of condensed liquid evalu-ated at the average of the coolant inlet and outlet temperatures and the steam-air mixture j inlet and outlet temperatures i

l The inlet and outlet temperatures, Tm and Tmz of the steam-air mixture can be equal to or above the saturation temperature corresponding to the partial pressure of steam at the inlet or outlet (i.e., either saturated or superheated steam) .

l Specific humidity at the inlet is calculated by assuming air

and steam to be ideal gases:

Ps1 "s al a (7-2) 7-3

SWECO 8101

\'

i whether the atmosphere is saturated or superheated.

where:

Ps = Steam partial pressure at inlet Pat = Air partial pressure at inlet Ms = Molecular weight of steam Ma = Molecular weight of air The calculation of specific humidity at the outlet and the determination of whether the outlet is saturated or superheated are interrelated.

1. If the inlet steam-air mixture is saturated, then the outlet mixture is necessarily saturated and Ps2 M s 2" (P T -Ps2) Ma (7-3)

(p N- /

4 where:

Ps2 = Saturation pressure of steam at Tmr PT = Total pressure (air plus steam)

2. If the inlet steam-air mixture is superheated, then the outlet mixture may be either saturated or

superheated. If it is saturated, then Sa 18 calculated as above. If it is superheated, then LOCTVS interpolates with temperature between the following two end points:

a. The inlet temperature, Tma, and specific humidity, S t.

b. The tube wall temperature at the point farthest downstream in the steam-air flow, T g, and the saturated specific humidity corresponding to this temperature, S. g Thus, (T m2 -T)y S2*b w + IE 1 ~S}w (Tm1 -T)w-

N, {7,G)

)

7-4

1 SWECO 8101 n

v where:

P Ms sw 3 ,

w P M (7-5) aw a and P sw = Saturation pressure of steam at T g P aw = P +p ~E sw st a

3. The outlet specific humidity is calculated both ways, and the lesser value is chosen. If the lesser value is that calculated by Equation 7-3, then the outlet is saturated. If the lesser value is that calculated by Equation 7-4, then the outlet is superheated.

The second equation, representing a heat balance on the coolant water, is:

.v

( q=m e C p (T c2 -Tc3) (7-6) where:

m c = Mass flow rate of coolant Cp = Specific heat of coolant Tc2 = Outlet temperature of coolant i Tct = Inlet temperature of coolant The third equation to be satisfied is a heat balance on the unit cooler. The equation for a counterflow heat exchanger is used:

-T c1 I (Tm1 -Tc2) - (Tm2 q=UAii Tm1 -T c2 T -T ~

m2 c1, D

{G 7-5

SWECO 8101 O

V where:

U.1 = Overall heat transfer coefficient based on inside tube surface area, A i A. = Heat exchanger surfSce area based cn tube inside 1

diameter The method of solution is to solve Equation 7-6 for Tcar substitute this into Equation 7-7, and iterate Equations 7-1, 7-6, and 7-7 until the values of g and Tm2 that satisfy these equations are found.

The heat removed per containment unit cooler (q) is then multiplied by the n' amber of containment unit coolers in the system and corrected for the heat added to the exiting steam-air mixture by the power input to the fan. The mass or steam condensed is then subtracted from the water inventory in the containment atmosphere and added to the water on the floor.

7.2.3 Residual Heat Removal Heat Cychangers

p. The RHR system in the containment coolita mode of operation

) removes heat from the suppression pool and limits the long-(\ / term post-LOCA containment pressure.

The RHR heat exchangers are exposed to suppression pool water and to service water; consequently, the heat exchanger duty varies with the suppression pool and service water temperatures. It is convenient to calculate the ratio of the heat duty to the difference in inlet temperatures. This ratio is a function of the capacity and configuration of the heat exchanger and of both flow rates, and must be recalculated only if either flow rafa changes with time. At each time step, this ratio is multiplied by the current inlet temperature difference to obtain the current heat duty.

The calculation follows the NTU Cesign Method, as presented in Kays and LondonCE).

f% )

i

\- /

7-6

SWECO 8101 i ss The heat exchanger heat transfer effectiveness is defined as:

H (THg - H c cout -

c in out , actual Wg (TH in

-T c in I

" min (TH.

in

-T c.in I , q max (7-8) where:

W = Product of hot side mass flow rate and H specific heat i W = Product of cold side mass flow rate and c

specific heat W y = Lesser of WH and Wc TH.in = Hot side inlet temperature T H ut = Hot side outlet temperature o

T c. = Cold side inlet temperature s in TC E out q = Heat duty y = Maximum possible heat duty, corresponding max to a counterflow heat exchanger with an infinite UA The heat exchanger heat transfer effectiveness depends on heat exchanger capacity, heat exchanger configuration, and both flow rates as follows:

Counterflow 1 -e -NTU ( 1 -R) 3 _ pe -NTU ( 1 -R) 7,9)

("'%

O 7-7

_s SWECO 8101 I

\

where:

NTU = UA/W ndn R=Wmin ! max UA = Product of overall heat transfer coefficient and heat exchanger surface area Wmax = Greater of WH and We When R approaches unity, Equation 7-9 reduces to g , NTU 1 + NTU (7-10)

Parallel-Counterflow, Shell Fluid Mixed c= -

-P 1 + R + /1+R2 3 +g

-I

'N 1-e

) - -

(7- 11) where:

T = NTU / +R2 1 (7-12)

Once the effectiveness of the heat exchanger is known, the rate of heat transfer is determined directly from the equation:

actual = CW min (IHin

- T Cin (7-13) 7.2.4 Drywell/ Suppression Chamber Vacuum Breakers l over-under containments utilize vacuum breakers to limit the l upward pressure differential on the drywell floor. 7he vacuum breakers are mounted on the downcomers within the suppression chamber air space and allow air to flow from the suppression chamber into the drywell.

Durina the LCCA, the reactor effluent pushes all the drywell l

air into 'the suppression chamber. After blowdown has terminated, the drywell is filled with steam which starts I ' (<-~ss_-}

7-8 i

f

SWECO 8101 O condensing due to heat losses to the concrete, steel, and drywell sprays (if used) . This causes the drywell to depressurize. When the drywell pressure decreases to about 0.5 psi less than the suppression chamber pressure, the check valves in the vacuum breakers open and allow air to flow back to the drywell. Shortly thereafter, the drywell and suppression chamber pres sures are essentially equal.

The vacuum breaker system is sized to limit the maximum upward pressure dif ferential on the drywell floor to less than the design value.

The total flow rate through the vacuum breakers, W, is determined with the following equation:

^

W=N

^ f ,

x r (7-14) where:

. A = Cowncomer area O = Suppression chamber atmosphere density AP = Suppression chamber minus drywell pressure differential K = Vacuum breaker system loss coefficient N = Number of downcomers with operating vacuum breakers The vacuum breaker flow and its associated enthalpy are removed from the suppression chamber and added to the drywell.

Since SSE horizcntal vent containments do not have drywell (or containment) sprays, the drywell depressurization is a

relatively slow process. The drywell heat loss is due to heat sinks only. Air returns to the drywell when the suppression chamber to drywell differential pressure exceeds the hydrostatic head needed to clear the vents on the l

containment side. This head is on the order of 5 psi, which can be easily withstood by the drywell wall. Thus, these containments do not require vacuum breakers. .

. '_) 7-9 I

i l

~ __ _ __ _ __ ~ _ _ _ . _ _ - _ - . _ _ . . _ _ -

a b

I'

, SWECO 8101 l

7.3 THERMODYNAMIC STATE I 7.3.1 Over-Under Containments

] Consider the following system shown schematically:

. SUPPRESSION CHAMBER ATMOSPHERE

=WVocuum Wair

?

IWi SUPPRESSION POOL ~ ISI i

- EWJ j ZWhii =

M,E - E Wjh) where:

W1 = Steam, air, and liquid vent flow, suppres-sion chamber sprays

! W. = ECCS flow, containment sprays

3 Wg = Air vent ficw hi,3 = Enthalpy of corresponding W1,3 flows q, = Heat absorbed by containment structures ,

J M = Suppression pool mass E = Suppression pool internal energy Ma = Suppression chamber air mass The mass balance for the suppression pool is dt

• Wi -W air" '

"j (7- 15)

\

s )

7-10

SWECO 8101 and the energy balance is dE " h ii- -

gi - Wh dt W)h3 I air) vacuum breakers (7-16)

The mass balance for the suppression chamber air is dMa dt *Wair-Wyacuum breakers (7-17)

After the mass and energy balances are updated each timestep, the pool temperature, Tp, is calculated:

Tp = E/M + 32 (7-18)

The suppression chamber pressure, P, T is P

T =PA+PS (7-19) l where:

f Pg = Air partial pressure using the ideal gas equation at T p P Steam partial pressure from saturated steam g = tables at T p l'

I I

i l

i

7-11

,m,,,, - - - . _ . - - - . _ - - _ _ - _ , - . . _ ._ _ ,,.,,,,,,. ._--. ,, - -

SWECO 8101 b

U 7.3.2 Horizontal Vent Containments Consider the system shown schematically below.

CONTAINMENT ATMOSPHERE Ma,Ms,Ea E9)

W evap Wair 4 cony, rad mmm-IWi -

SUPPRESSION POOL - 29j

' 2WJ ZWhi i Mp ,Ep EWjhj where:

W1 = Steam, air, and liquid vent flow

\

W 3 = ECCS flow hi ,j = Enthalpy of corresponding Wi,$ flows

q. = Heat absorbed by containment structures J and unit coolers in their respective environments Ma,M3,M p = Containment air and steam and suppression poul mass, respectively Ea,Ep = Containment atmosphere and suppression pool internal energy, respectively i W evap = Suppression pool mass, rate of evaporation (if condensation, this term is negative)

" air = Air vent flow y cony, rad

= Suppression pool convection and radiation heat transfer

^

._/

7-12

SWECC 8101

)

The mass balance for the suppression pool is:

dM Wj ~W dt i~ air ~ evap (7-20) and the energy balance is:

dE p ' .

= W.h.-W C +h dt 1 1 air

, p(Tp-32) o ,

=

E W .h .- (Wh) -q L ] ] evap conv, rad (7-21) where:

, Tp = Suppression pool temperature Cp = Constant pressure specific heat of air i

ho = Reference enthalpy of air (specific internal energy of air is defined as 0 at 320F)

The suppression pool temperature is determined from:

Tp = E p/Mp + 32 (7-22)

The mass balance for the suppression chamber atmosphere air is:

dMair dt air (7-23)

Wair is negative when the vents clear on the containment side and air flows back into drywell.

The vapor balance is:

dM s evap

~

unit cooler

~W heat sink dt condensate condensate

(7-24) s Y

J 7-13

-

SWECC 8101 v

and the energy balance is:

dea C p(Tp -32) +b g +

(Wh)evap dt air

+ cony, rad - [ 9i (7-25)

For reverse vent floa the containment atmosphere temperature, Ta , replaces Tp in the above equation.

The thermodynamic state of the containment atmosphere is determined by the same method as that described for the drywell (section 3.3) .

The suppression pool evaporation rate, Wevap, is calculated from the following equation which was presented in Reference 2.

1.2 Wevap = 0.47A p P -P v v p a N (7-26)

)

where:

Ap = Pool cross-sectional area, sq f t Py = Vapor pressure, psia If the containment atmosphere vapor pressure, Py is greater than the pool vapor pressure, Py , the terms, are reversed and condensatio- occurs. P The convection heat transfer rate, b conv , between the pool and the containment atmosphere is derived f rom Reference 3.

-T)a k cony = 0.4 A p p (7-27) where:

T a = Temperature of containment atmosphere O_

7-14

SWECO 8101 9 and for radiation heat standard equation is used:

transfer, 4 rad, the following j

l rad " 6 AP (p -T a (7-28) l-i where:

c = Stefan-Boltzmann constant

e = Emissivity (assumed to be 1.0) i

T = Absolute temperature i

i

G

@ 7-15

I

,I

)

I SWECC 8101 l

i References - Section 7

1. Kays, W. M. and Lowdon, A. L. Compact Heat Exchangers.

McGraw-Hill, New York, NY, 1958, Chapter II.

2. Boelter, L. M. K. ; Gordon, H. S.; and Griffin, J. R.

Free Evaporation into Air of Water from a Free j Horizontal Quiet Surface. Industrial and Engineering i Chemistry, Vct. 38, No. 6, June 1946, p 596-600.

i

3. Friedman, S. J. Heat Losses from Tanks, Vats, and Kettles. Heating and Ventilation Peference Section, April 1948, p 94-108.

E t

5 F

i l '

i 7-16 l

t

.i

-,,.e,-,.,<*,--ew--e-.es . +.e,.n- c.awi.,.---ee- -,.w ,, --w. w..-ow.-,w.cee., ..-oc-.,r,. w -ar-ew,-+w V-+=v-<- =+w es e we- -*e-,-e-wr-*-i-w~rrw- r ev 4 *--"*-e'se- v s -+ w- vv -- r w ve w w r --

SWECO 8101 s SECTION 8

)

s_/ HEAT SINK MODEL 8.1 ASSUMPTIONS The following assumptions are made:

1. Internal heat generation is neglected.

2. Radiation heat transfer is neglected.

3. Tne physical properties of the materials in the heat sink are constant over the temperature rance involved.

4. The heat transfer model is one-dimensional.

S. All heat sinks can be described in rectangular, cylindrical, or spherical geometry.

6. The boundary conditions on the heat sink surface, e.g. , heat transfer coef ficient, are constant over each time-step.

['q_ 7. The physical properties of constant over each time-step.

the atmosphere are

8. Condensate does not accumulate on any heat sink surface (i.e., removed af ter one time step) .

8.2 APPECACH i

The calculational . technique used in the heat sink model is the same as that described in Beferences 1 and 2. The l

technique is a numerical approximation of the heat conduction equaticn:

! 6 c(x) T(x,t) = V*k(x)?T(x,t) i I

--6t . (8-1) where:

c = Volumetric heat capacity T = Temperature k = Thermal conductivity 8-1

< . _ . . - . _ , , , ~ . , . . - - - - ,_ , . _ . , _ . - _ , , , ,,.,,--,y-7 ,,,--..y,

SWECO 8101 x = Space variable

\~' t = Time variable Using the boundary conditiew' as described in Reference 1, the implicit solution fermed is expressed in a nearly symmetrical matrix form and is solved by applying Gaussian elimination techniques.

8.3 EOUNCAFY CONCITIONS The heat sink model considers condensation at the surface of the sinks. The Uchida correlation is utilized for the condensing heat transfer coefficient (3). The value used is obtained by linear interpolation from a table of heat transfer coefficients as a function of air to steam mass ratio.

When the surface temperature of the heat sink, Ts, is greater than the saturation temperature of the atmosphere, T,g a free convection heat transfer coefficient is used.

When T 3 is less than T,g the boundary conditions are determined as follows:

-~ 1. Trial heat transfer rates are determined from:

\ ~,/

9cond = hu A(Tg -Ts) (8-2) and q cony = h cA(T a-T s)

(8-3) where:

Ta = Atmosphere temperature hu = Conden<iing heat transfer coefficient hc = Free convection heat transfer coefficient

2. If qcond is greater than qcqny, then tue bouncary conditions imposed on the sinx are hu and Tg; otherwise the boundary conditions are he and T-a 8-2 o

, , _. - - ~ . ,. . . .-_ . . . - . . ._ . __ . . _ , _ - . _ - - - _ .. _ _ _ - - . . . _ - _ .

SWECO 8101 CONDENSATION - REVAPORIZ ATION MODEL

"'S 8.4

\- The heat sink model considers condensation of steam onto the heat sinks and the subsequent revaporization of the condensate when the sinks are in a superheated atmosphere.

Although the model described below is a partial revaporization model, the equilibrium model (atmosphere and condensate are in thermal equilibrium) and the isolated condensate model (condensate is thermally isolated from the superheated atmosphere) are simply special cases.

This model does not attempt to describe the actual physical effects involved in condensation heat transfer but rather is a result of matching empirical data (*).

8.4.1 Assumptions The following key assumptions are made in developino this model:

, 1. Revaporization only occurs in the presence of a superheated atmosphere.

2. The condensate layer is transferred from the heat

-~ sink to the floor after each time-step.

\._ / 3. A revaporization process can be separately modeled from the condensation model to form an overall condensation-revaporization model.

4 Changes in steam and condensate enthalpy are small over a time-step.

5. No subcooling of the condensate occurs.

8.4.2 Model Description Condensation occurs on a heat sink when the surface temperature of the sink is below the saturation temperature of the atmosphere. The rate at which heat is added to the sink during condensation is given by:

91 =hu(Tg -TsIA (8-4)

O 8-3 (x

SWECO 8101 The rate at which steam is condensed is:

91

'y =

c hdry-hdew where:

h = Specific enthalpy of the dry steam in the d#Y atmosphere hdew = specific enthalpy of the condensate As the condensate falls through the atmosphere or flows along the sink surface, convection heat transfer causes some of the condensate to be revaporized and returned to the atmosphere. The heat removed from the atmosphere is:

92 =hconv(Ta-T g1A (8-6) and x g

=

9:

I h

dry" dew (8-7) where:

qa = Rate heat flows into the condensate film

$r = Pate of revaporization The net rate mass removed from the atmosphere is:

M=M-Mc r (8-8)

The net amount of heat removed from the atmosphere, a is found from the sum of the heat added to the sink, q net,lus p

the energy removed from the atmosphere via the condensate Thus, (fthdew).

9 net " 92 + iih dew (8-9)

O) i x_/

8 - 14 G

, ,, - , , . - ..y . . - , , . . . . , _ - , , , . _ . g, , . , , , . . .-yy., , . , , , ....., .,.~,---.,,,,, -,,,,.,--,y,_r-.,y-,, , , y.m__ ,,_ - , . , , , , . .

.. .. _. . .- ._ . -. .= - _. . . ..

4 l SWECO 8101 The heat sink model provides two options for applying the revaporization model. The first model allows the specification of a fixed f raction of the condensate to be revaporized. This option specifies that M r is found from:

= F'k c r (8-10) where:

F = Specified fraction of the condensate to be revaporized-q is then found from Equation 8-9.

! The second option for revaporization specifies a fixed value for hconv in Equation 8-6. qnet is then found as i described previously, i

1

8-5 i

! . , _ _ _ , _ . . _ . . . . _ _ _ , _ _ . . . . _ - _ . ~ , . . _ . - . _ , . ~ . _ ~ . . - . . . . _ , ~ . . _ _ , _ _ . . . _ , . . . - _ _ _ . . . _ . - . . _ - - . . . - . - -

SWECO 8101

S References - Section 8

'- 1. Wheat, L. L.; Wagner, R. J.; Niederauer, G. F.; and Obenchain, C. F.. CONTEMPT-LT -- A Computer Program for Predicting Containment Pressure-Temperature Response to a Loss-of-Coolant-Accident. Aerojet Nuclear Company, ANCR-1219, June 1975.

2. Wagner, R. HEAT 1 - A One Dirensional Time Dependent or Steady-State Heat Conductior. Code for the IBM-650, ICO-16867, April 1963.

3. Uchida, H.; Ogama, A.; and Togo, Y. Evaluation of Post-Incident Cooling Systems of Light-Water Power Reactors. Proceedings of the Third International Conference on the Peaceful Uses of Atomic Energy held in Geneva, Switzerland August 31 to September 9, 1964, vol.

13, New York, NY, United Nations 1965, p 93-104,

( A/ Conf . 28/p. 436).

4. Frank, S.; Hynek, S. J.; and Stauder, J. W. Superheated Containment Atmosphere Heat / Mass Transfer to Passive Heat Sinks. Presented at ANS 1978 Annual Meeting, San Diego, CA, June 18-23, 1978.

O l

?

l t

h G

8-6

l l

l SWECO 8101 SECTION 9 STEAM BYPASS

. 9.1 DISCUSSION Steam bypass refers to the steam that bypasses the suppression pool by leaking through the drywell boundary into the suppression chamber atmosphere. The maximum acceptable amount of steam bypass is called the allowable bypass capacity and is defined as that amount of leakage which would result in pressurizing the containment to its design value. Bypass capacity is expressed in terms of

_A W

where:

A = Leakage path area K= Flow loss coefficient Bypass capacity is a function of both the magnitude of the pressure dif ferential across the drywell boundary and its s, duration. This can be seen from the following:

2 AP = K Ov7- (g_3) where:

AP = Drywell to suppression chamber pressure differential P = Density of drywell steam

v = Leakage path velocity

9-1

.o

. _ . , . _ . . . - _ . , _ . , . _ . - - . , _ _ _ _ - - - - - - .. . _ . . . . ~ . _ _ - _ , _ . - - . . _ . - . _ _ _ _ .

SWECO 8101

-~ Substituting the continuity equation (m = pvA) and rearrang-

, \,j m=

^

/2 PAP

/E (9-2) or

^ O Am = /2 PAP (9-3)

Thus, for a given A/ [K and density, the total leakage, Am, is a direct function of the pressure differential, Ap, and its duration, At assuming constant density, p.

For a given reactor break size and depressurization rate, the pressure differential and the duration are known. From Equation 9-3, an A/ /lf can be found that allows sufficient steam bypass to pressurize the ccntainment to its design value; this is the allowable A/Vlf or allowable bypass capacity.

ass analysis utilizing LocTVs is performed by varying The the byp/

A /lf in Equation 9-2 until the containment pressure

[N matches the design pressure. This is done for the spectrum

' of break sizes. A typical result is shown below.

ALLOWABLE BYPASS CAPACITY A/[

SMALL INTERMEDIATE LA RGE DER BREAK SI:ZE Because of the long duration of its blowdown, a small steam line break has been found to be the DBA for the steam bypass a nalysis. Large breaks generate a high pressure differential across the drywell boundary but rapidly l

O)

( 9-2 i

N - -

w ,n -

g +---r-- 9en 9 --s-p -,q. -9 y- . =+-ep-- .y -

. -,--p amy- y- -

1 SWECO 8101 depressurize the reactor. This quickly terminates the blowdown and, consequently, the pressure differential. The

[s,)h s minimum value on the above curve is the maximum allowable bypass capacity or the maximum amount of leakage that can be tolerated.

The analysis is complicated by the fact that containment heat sinks and engineered safety features are continuously removing heat and condensing steam in the containment.

Consequently, the total leakage to pressurize the containment to the design limit is a variable and depends on the individual reactor break size and associated transient conditions.

9.2 ASSUMPTIONS 9.2.1 Model Assumptions The following assumptions are made:

1. The bypass flow is steam only.

2. The bypass flow is incompressible.

3. The vent clearing model is reduced to a static O

pressure balance in crder to facilitate the vent flow calculation.

9.2.2 Other Assumpticn Normally Utilized in Analysis Passive containment heat sinks are utilized.

i

, _ , , .. . _ . . , _ . . . . - _ . , _ . . - - _ _ . ,m-.. _m. . . . . - . . . , ...m .,....,~~...._.,m., . . -

I SWECO 8101 i

I APPENDIX A

(

5 INITIAL SLAB DIVISION AND GRID SPACING CHANGES A.1 INITIAL SLAB DIVISION Connider a slab source. A necessary preliminary calculation is the division of the slab into a number of slices for the finite difference computations (Dusinherre) . These calculations are performed as described in Reference 1, p 44:

Let W= Slab width Ax = Slice width n = Number of slices At = Time step K = Slah thermal conductivity C= Slab specific heat capacity P = Slat density a = Slab thermal dif fusivity, K/p C M = Dimensionless modulus, M x) 3 aat The paramete rs E, At, and a are known. Modulus M is temporarily set equal to two, and the slice width Ax is computed as follows:

Ax = / M a At _

The number of slices is then

( A-2)

The value of n is then truncated to the next lower integer.

In addition, a maximum value, corresponding tc the number of A-1 o

SWECC 8101 x core storage locations assigned, is placed on the number of i

slices.

After n has been selected in this manner, the slice width is recalculated:

Ax = E (A-3)

Also, M is recomputed for later use in the Dusinterre Elgorithm:

g ,(Ax)2 a :t (A-4)

Note that the new value of M exceeds 2.

One further requirement is that the number of slices, n, be greater than unity. If this condition is not met, the Dusinterre finite difference method is, of course, not used, and a uniform temperature model is then imposed.

In summary, the above procedure for setting up the grid ensures that the conditions (M 2 2 and n = integer > 1)

~') necessary for the Dusinberre Method are met before the (s_,/ method is applied. If these conditons cannot be met, the the Dusinterre Method is discarded in f avor of setting the entire slab equal to the reactor coolant temperature (assumes infinite thermal dif fusivity) .

A.2 GRID SFACING CHANGES At several points in time during the execution of a computer run the time increment At is increased to save computer time. This requires an increase in grid spacing and recalculation of M, but the calculation is performed in the same manner using the new time increment. After the new grid is established, the old temperature distribution is assigned to the new grid using a slice-to-slice linear temperature interpolation. Again the conditions M22 and n> 1 are examined, and if they cannot be met with the new time increment, then the uniform temperature rodel is used.

A-2

(l v

t

SWECO 8101 2

Reference - Appendix A

1. McAdams, W. H. Heat Transmission, third edition, McGraw-Hill, New York, NY, 1954.

4 J

l i

9 I

i A-3

--,.w..,--,_,_m_-,___-.......-,~.------.-- _ _ _ . . . . - . - ~ . . . ___ _ _ _ _ _ _ _ - - - . _ _ _ - - - . _ _ _ _ - - _ _ _ . _

SWECO 8101 APPENDIX B I GOVERNING EQUATIONS FOR A HORIZONTAL NCDE WITH HORIZONTAL AND VERTICAL POINTS This appendix is concerned with the derivation of the continuity and momentum equations for a horizcntal node when the surface of separation between the phases consists of horizontal and vertical planes.

B.1 CONTINUITY EQUATIONS General Expression (Fig. B- 1)

A convenient way to obtain the continuity equation is to write, for each of the phases, that the rate of increase of the volume occupied by the phase is equal to the net volume flow rate of this phase out of the node. This is legitimate since each phase is assumed to have constant density.

The volume occupied by the gas is

^Y gF The volumetric rate of incoming gas is Av 91 Hence, d

AY g1 dt g AY gF or dA g Av g 1 = A gp v +Y F dt The volume occupied by the liquid is

^1 YF+Av (Yv-YF)

[ B-1

i SWECO 8101 The net flow of liquid into the node is A v -A v 11 v2 Hence, i

d dt ^1YF+Av IYv~YF} "AY R1 ~ ^vv2 or dAg Ag v -A yv +A vy p = Agpv +yF dt Adding the gas and liquid equations yields the result v2 = v3 Since this result is evidently applicable when the node is filled with one phase only, this result is general and, as was noted in Section 4.1.2, va can be replaced by va and be dropped from the list of unknowns.

As vg and vp are among the unknowns, the relevant continuity equation becomes N

=

yF d^g v -v p y9 d

,t Determination of dag (Fig. E- 1) dt

The area, A, can be expressed in terms of the radius of the vent, R, and the angle, 0, where the latter is defined by x 2 -*F cos O E R

The result is I

A =R2 (0- sin 20) 9 .

B-2 i

v I

--- . r y * , ,-,---,-e. s, u ,,,--.,.--.~.,-,-.4.-,..r.,%,. -r , .,,-.v,.,w+. , , , ~ , . , . . . _ 7--~..-, -.,--,--,,--,-v-.- , . . - - . -

i SWECO 8101 1

4 It is convenient to use the folicwing parameters:

Aq YF 4

a--

Ay 4 = cose 0=4n (1-4 ) 2 -

2R i

Therefore,

! 1 4

a=- [0- sin 20]

n j and

! da 1 2  ;

l

- = - n[1-cos20] = -n (1-4 )

de 2

t l

i ,

i l

Since  :

J -l d4

!' g = - sine = - (1-4 2) j 1 i 4

then 1

da

- = - - n2(1-$ ) h d$

2 i

i d

i From the definiticn of 0,

! 1 1 dx p d_B = _

j dt sin 0 R dt l and ,

t dj , _ - 1 dxp. i dt R dt Hence, 1

dx p da 2 2 1

! dt Y [(l ~$ / li d t i

1 a

I B-3 1:

SWECO 8101

, and dA A A F 1 g_ v d q ,1 da 1 (3_4 )h 1R 2

a dt a dt A

g dt A g dt ( Ay j Furthermore, YF dA g=12 (1-$ )51YF d*F2 1

=-0u Ag dt an R dt a F and the continuity equation becomes 1

v,-v -

-0u p

= 0 Velocity Cistribution in the Node For each of the phases, the general continuity equation is (pA) + (pAv) =0 at ay Hence, o is a constant and A does not vary with y; hence fd + =0 Therefore D = constant 3Y Thus, at any arbitrary time, the velocity distribution is linear in any region (Fig. B-1 and B-2) .

The velocity of the fluid just under the vertical front is vo ,Its value can be determined by applying liquid volume continuity for an observer fixed with the front:

A -V ^v ~

f. fD F) 2 F)

') B-4

.J

SWECC 8101  ;

i

i Therefore, 1

V D"Y+ l-a 1

0u p l i

1 V *V up D F+ a (1-a) i The resulting velocity distribution is shown on Fig. B-2. ,

B.2 MOMENTUM j For v varying linearly with y, v (at y=L) 1 vdy = 7 L (v + V,) ,

v (at y=0)  ;

The momentum of the fluid inside the node is [

Yp i og tv av IYv~Y F + ^ EEE 999 r o

j or

~

1 f PvAy(Y g 3 y -Y p ) +7 pag g (V 3 + V p) + P Ag g(V 3 +V)D YF

t l

l

! The momentur flux coming in is I

2 pv A + pv i

91 9 E 1 f\ 37_39) i i -

i k

B-5 i

i k--._...,.,_____,_.,__,-m...,..._..,,,__.-__ _ . _ _ . . . . . - . . _ _ _ - . . . _ - . _ _ _ . . . - -

l 1

i SWECO 8101 f

The momentur flux leaving is f Agv 2 A v The pressure at the entrance of the node is py, and the pressure at the exist is pyre. Hence, the momentum equation ,

is l

. . i I pvAy(Y g 3 y-Yp)

+ Dgg a (V 1

+Y) F +PALLIV

1. V D YF

(

+pvg 2 3 _p y ag _p gy 2(A y-A )

g g g

= (P y -P V c IAv i

which can be rearranged to give IP h 1 IO \*  !

1

- aylg -

1 l +y v +-aylg -1 lv i 2 F gog j v 1 2 F (pg j F i

1/ Pg h I O gh a

=

2 g ll- p i (v1 +vF) (a v F + 0 uF )-a Il- pgj IV 1 gj Py ,-Pv2c ..

+ '

01  !

l l

t i

I. I

. r

• I f

i  !

i.r i

-N = v sea m mm w w -~~ e,w-r,-w, ve v - wo -e m-- eew~e ew, v wn,em, - .. em me m m .-.-mm-v

O AREA A

/

/g F

/ (OCCUP!ED BY GAS) d MMn GAS X F

-._.-._._._._._._._._._, .. _ . . _ .._L.

i 4 LIQUID i

WMMMA (OCCUPIED Y IQUID) i ALONG THE AXIS PERPENDICULAR TO THE AXIS NOTE:

vi AND vs ARE THE INLET AND EXIT VELOCITIES FOR NODE 2 (SEE FIGURE 4-2 ). THIS FIGURE APPLIES TO CLE AR 2 AND CLEAR 3.

A CHANGE OF SUBSCRIPTS RENDERS THE SAME RESULTS VALID

FOR CLEAR 5, CLEAR 6, CLEAR 8, AND CLE AR 9.

FIGURE B-1 GEOMETRY FOR NODE WITH HORIZONTAL AND VERTICAL SURFACE p OF SEPARATION BETWEEN PHASES Q)

. i i

2 l

L

h i h

\

I l

ilV l 1

LIQUID ~~~--~~#

1 VELOCITY D  !

,'  ?

i k

i V2

____________________ y; s , .

! i s t I t $

j 8 l GAS i

l v,r VELOCITY  ;

' I  ! l i

l  !  !

I '

y Y. Yr l l

i NOTE >

v, AN D vs ARE THE INLET AND ExlT VELCCITIES FOR NODE 2 (SEE a

FIGURE 4-2 ) THIS FIGURE APPLIES TO CLEAR 2 AND CLEAR 3. .

I A CHANGE OF SUBSORIPTS RENOERS THE SAVE RESULTS VALIO FOR CLE AR 5, CLEAR 6, CLE AR 9, AND CLE AR 9.

I I

1 I.

I l

FIGURE B-2 i l HORIZONTAL VELOCITY DISTRIBUTION IN NODE 2  ;

j' '

f h'

v ,

l c P

- _ , _ . , - _ _ _ _ , . - _ _ _ _ _ _ - - . . _ . . . . , _ _ , _ . . ~ . -

4 4

SWECO 8101 APPENDIX C DERIVATION OF HORIZONTAL VENT CLEARING EQUATIONS The vent clearing model is divided into a set of FORTRAN subroutines, clear 1 through clear 9, which are called sequentially as the water level in the drywell is depressed.

Figures C-1 through C-18 indicate which subroutine te called when this level has reached a specific position. See C.10 for nomenclature used and Fig. 4-2 for node definition and sign convention.

C.1 CLEAR 1 (SEE FIG. C-1)

Continuity equations Node 1 The surface of separation between the phases is in this node and is assumed to be horizontal.

The volumetric rate of incoming gas is Au s and the volume occupied by the gas increases at a rate Aup.

Hence:

Aug = Aug or us - up =0 (c.1-1) f No liquid comes into the node, and the volume occupied by the liquid decreases at the rate Aup; it escapes into Node 2 at a rate 1/2 A y va and into Node 3 at a rate Aug.

Hence:

Au p' = Au

+fAy v or A

1 v "a ~ "F

• 7 lI "i =0 (C .1 -2) s ~

C-1

, ... . ~ _ _- _ . , . . _ _ _ . - , . - _ _ - _ _ _ _ . _ _ - - . - _ - - _ . _ _ . - - - _ - .

SWECO 8101 N

This node is full of liquid which comes in from Node 1 at a rate Au, and escapes into Node 2, Node 4, and Node 5 at rates 1/2 Av v,a 1/2 Av v 3, and Au s , respectively.

Hence:

Au

=hA y v

+fAy v, + Au, or u - u, - f f (v + v,) =0 (C .1 -3)

Node 5  ;

i The flow pattern is the same as for Node 3, and the corresponding equation is obtained by increasing the indices by one unit for the u and by two units for the v, hence:

u 3

-u g 2 A (v 3 +v) s

=0 (C .1 -4)

\ ,) Node 7 j The liquid comes from Node 5 at a rate Au. and escapes into l Node 6 at a rate 1/2 A yv s, hence:

" ~ Y

• 5 s (C.1-5)

Momentum equations i

Useful relationships are conveniently obtained by evaluating the various terms of the equation obtained by applying the momentum theorem to the material contained in a control volume, as expressed, for 0xarple, in Reference 1:

h(ou)g F + f { pu u$dA) g =- pdAg+ pa dV i

V A j A V This general relationship can be applied to the special case of inviscid fluids.

, 'h.

\

C-2

_ _ __ . . _ ____ __ _ _ _ _ = - _ _ _ . _ _ _ _ . _ _ _ _ _ _ _ _ _ . . _ _ _ _ _ _ _ _ _ . . _ _ .

4 a i 1

SWECO 8101 '

1  :

s In this equation the term I

) 3 (cug ) dV j

< o.

V  !

l is the total rate of change of romentum inside the control  !

volume in the i direction, the term i i

l  !

ou-j 4 u dA.  !

J  :

, A j j i

is the net flux of momentum leaving the control volume, and, j

finally, the terms  ;

+

pdAg; cagdV A V i take into account the applied pressure and bcdy forces. In the present problem, the latter is that due to gravity.

Node 1 [

;

The vertical downward momentum of the fluid inside the node is >

f_

pu 4 dv = c + og A(x: -x,) u, g A(x F -x 1 ) u g E

l V  :

u  ?

The gas and liquid velocities are assumed to te: l t

L 1 1 u = 7 (u, + up ); ug =

7 (u +uF) g  !

r

! thus:

I j

pu.dV = 5 1

p1 A - e(x,-x ) (u +u,) + (x -x,) (u +u,)

L s o  : 1 1 c. 1 x 1 x V - 1 -

t Y

i J I I'

I

{

E I

i C-3 l 1

,  ?

t- [

J v m *- e- y e =-v - e , y, e e em -,-- e wy e w m W Y g =W 7iTV YY

-e w e -ai-t g wme-w ,- n w w e ew e a+-s,a'T * -vet-TW-r9-**-*~v~+vt**w r= w r r e-* m" p--et-*M N w*h t M-'4 EFT y g WV&th TWWW-4WYw -g V9 wir W

l

l SWECO 8101

! l j and:

i I r D +

+ (x -xF) u h +

3 i

at (pui )dV = 2 p L 1

A< (xF -x t )

l p u

1 2 2 l g V . .

p "p - '

i l

l + -p gE (xF -x ) + (x2 -xF) 1 d

F+uF -p g3- (u +uF ) -(u + u) i 2 F j I i

i  !

i j The momentum flux term is

!

put3 u dA3=og A uz

~A g A" i

• P 1 A I" 2

~

1 3

i and the force terms are:

pdAg = A(p,-p,)

j q A

~

p pag dV = A p g (xp -x ) + (x -xp) g

V -

Collecting the various terms and dividing by 0 gA yields:

P h 14 P a (xp -x ) 0 + (x -xp) d (xy -x ) + (x -xp)

+

i + u 7 F' l

I

~ ~

p p

+7uF (u +uF ) - p3-2 2 1

= u og

-u 2 2 (uF+ut )

i g

pp g

A -

+9 .

(xp -x ) + (x ~*F) 1 2

+

p g (C .1 -6)

C-4

(

l l

l l

l.

--,-,.---,---,.,,-,--,.,,,-,,-,,..,,,n,,,n-n.,--n,--.~,wan- - - ,- wn-,n,.,-.n,-.,,..,,.- ,,,, - .,.n - . n., _.--,---,-.,,,,,a

I 1  !

I i j i

, f t

! SWECO 8101 I 1 i

! l Node 3 i

}'  !

i i

The downward momentum in the node is

i i  ;

l I~

1 j y pug dV = 2 (u +u ) p g 2 3 A(x -x )

3 2

),

j where the mass averaged velocity of the fluid is assumed to

, be t

-2 (u +u )

1 2 3

} ,

i Hence: L i

1 1

' 3 1 i' 3t (oui )dV = 2 p E A (x -x ) (u +u )

2 2 3 V

1 i ~

i The mcmentum flux term is:  !

4  !

pgg3 u u dA3=pA g (u,2 -u *)

! A j ,

i and the force terms are:

t pdA g = A(p ~E 2 s

, A l

l

.l-pa dV = p g A(x -x )g i f ,

V'  !

e Collecting the terms yields:

1 P -P 12 (x -x )2 (E +5 3 2

) - = u 2 2 -u 3 3 2

+ g(x _x2 ) +

3 2

-p s

l E (C .1 -7) l P

G c.

L..-__.--,.-_.-,__ _ . _ _ . , _ _ _ _ _ _ - . .

. __. .. . __ -_ =. - - -- _ - .. ..- .-~ .... _ =_ = _

h J

SWECO 8101 1

j Node 5 The flow pattern is the same as for Node 3; hence the

. relevant equation is obtained by adding one unit to the indices:

P-P 7 (x s-x ) ($ +u ) s 6

= u s

2

-u 2

+ g(x* _x ) + s 3

og 6

4 (C .1 -8)

Node 2 containment corbination (see Fig. C- 2)

It is convenient to consider the combination of horizontal momentum in Node 2 with the upward vertical momentum of the control volume shown in Fig. C-2.

i The horizontal momentum of the liquid in Node 2 is:

pu g dV = p g A y yy V, V

The incoming and outgoing momentum fluxes are identical and gravity has no effect; hence the momentum equation for Node 2 is:

P -P V2 V2C yv v =

p 1

g

The various terms of the upward vertical mcmentum for the j control volume represented in Fig. C-2 are

1 7 Dy )u puf dV = pgAp(x g

V A

v

=

oAg p(X g y -(v+v+v) l

-hD) 7P l

I

\

l C-6 g +-e,+-gm= q yy* r e e w- ""-4'*1"""W"8"9--W'M""^^W*"rW*TT W

• FgP*-rWM 8---"R't"n T"PWa7Tyw- V 7*F-TWP---'-N"v'*7-'yg-wwrv 7 - v

- - - . ~ _ _ - _ . . - - - .

i i  !

4 SWECO 8101 i i  !

i or

! 3 g (pu )dV g = o Agp i

V 1

~

A -2'i ,

A v v i

-A (v +v +v ) + -A (v +v +v )

1 '

--D)  !

{(x ti t 2 v P

t a s

- P i s s 1

i I

i and: ,

2

! [A putu$dA) = -p g A u "~0 h IV +V l p 2c s i

A j t p '( P) pdA =A p (p -Ec) 2c

! 1

)

V pag dV = 1(x tlg -

7 D) y oA gp l

f

! Adding up and dividing by p gp A yields: ,

A A v

1-

-Av(v +v +v )

(x lit --D) 2 v

=' -

A v (34 V +v+v) 1 3 5

+ 1 3 3 i P

kP) ,

1 P 2c -E c ,

~9 I* ti t Y D) y + pg i  !

, I This derivation has used the following continuity equations 1 which can be derived by inspection of, for example, Fig. C-1:

Au =AV(v +v +v )

P 11 1 s 5 t  !

i

? 1 '

Au =Av(2v +v +v s )

p 2c 1 a

. l C-7

I r

SWECO 8101 I \

The application of the steady state Bernoulli equation (see, i for example, Reference 2 along the vent centerline from the vent exit to, say, the middle of the containment pool yields:

P P v2c ,7y 1 2 , 2c + g 1y 2 4 pg 2 og e2 2 where Ke is the expansion loss coefficient and the velocity

, urc is neglected by reference to v a, Ke cart be estimated equal to unity, hence:

P = P V2C 2C (The same applies to the other vents as well, and this fact will be used later in this report.)

Pressure losses at the vent 'ntrance are taken ir.to account by the ergirical coefficient 's :

P

~K #

s p

1

" [P t i 1 The momentum equation for the Node 2 containment combination

, of Fig. C-2 is obtained by adding the last equations to the momentum equations derived in this paragraph; the result is:

+y f+ 1 D \/v +h T j

[x\ ggt 12 D}g v/ v g

g [x\ ggg 2 v/\

3 3

/

' 2 2 g .

gg y

=- - 1 (A1y +L (y y +y y )

v'2 _!

4 (Ap 2 (A_p i s i s E -Ec

-g(x g 1

7 Dy) +

2 E

(C .1 - 9)

Node f4 containment corbination (see Fig. C- 3)

The derivation of the momentum equation for this combination follows the same steps as in the previous paragraph.

L

C-8

_ _ . . . . . . . _ . . . _ , , . _ _ _ _ _ _ _ . . - ._ ___ ~ . . _ . _ _ . _ . , . _ , . . _ ..., _ ..... _ . ~ _. _ . . . . _ ,_

t SWECO 9101 The horizontal momentum equation or Node 4 is: t o -o

, v3 v3c

),v 3 c, The upward vertical mcmentum for the control volume shown in Fig. C-3 is:

Ou dV = 0,A i c p (x tic 1

y D,,)

u

+ (x -x ) u 2: 3  :

tr The net momentum flux is:

cu.u.dA. = o.A (-u sc 2) 1 J J r P A3 By inspection of Fig. 4-2, the continuity equation yields:

A u

11

= o[ (v +v +v ) 1 3 5 p

A v

u 13

= --

og (v +v )

3 5 A

v 1 u = --

(2 v +v )

sc n 3 s Hence the upward vertical mcmentur equation is:

A A v . v . .

(x til 1

yDv) -v, A -

+ (x itt .

1

- y D V +x -x ) --

- 3  : a (v +v )

3 5

=

P P (A.

A (v ' + 3-v-1

• 3

+ 2v v +2v v 1 3 1 5

+vv) 3 5 o -c

-g (x ,

D, s

• x -x ) =

nm - 3  :

1 C-9

- ~ . . -

l SWECO 8101 The pressure losses at the vent entrance are taken into account by the empirical coefficient K,:

P P v'

P g

=d-K p g 22 1v 3 2

and, P P v3C 3C The result is:

A A I*

ti t D) y [OP n

+ (x it i

~

v +

• s ~* 2}

P

+Y v 3

~

1 A

v.y 2 1 3 IA v

+ (X, g7 D + x,-x ) =

-v, 7 K

+7 1

/A y)2 (v 2+2v1 v3 +2v1 vs +v vs ) l g (x 7 Dv + x: -x 2) l-1 A 11 1 g pj P ,- Pc A

t (C .1 - 10)

Node 6 containment combination (see Fig. C-4)

The derivation of the momentum eg'.sation for thf.s combination follows the same steps as in the two previous paragraphs.

The horizontal momentum equation for Node 6 is:

- Ev -P vuc v s Pg C-10

- ._ _ _ - - _ - _ . - _ _ -._ - .---._ .. . . - . . _ = . _ . . . . . . - _

I l I

i i

i

SWEco 8101 l

! l i

The upward vertical momentum for the control volume shown in i Fig. C-4 is:  :

1 i

pu dV = p gp i

A (x g 1

7 D y .t u + (x - lu i

i 3

]

+ (x -x )u 4 3 9 j i The net momentum flux is:

-pu gu$ dA 3 = ogA p (-u.c 4

A 3

By inspection:  !

I Ay A

=-vv l

= A- (v +v +v )

4 u u

• 11 1 3 s s A s F

. P P

! A A l v v1 u

to

= A- (v +v ) a s u

6c =7v A s

, p p i

i Hence the upward vertical momentum equation is: ,

t i Ay . Ay . ,

7 v + x -x 2) v A -v + (x ist 1 1 1 (x ti l 7 D) D 3

-V a

i .

l P P l A v- IA v

+ (x 12 17 Dv + x -x 2) 1 A

-V = l 6

p s (Ap j l

~ ~

l l

! (v +v +v )2 _1y 4 2

_ g(x 1 _Y 1

D v + xs -x )  !

l

. I s s s . 1: 2 ,

l

, i l P,c-Pc

, + ,

f.

I L

l C-11 l

i I

1

i SWECO 8101  :

i  !

f i The pressure losses at the vent entrance are taken into l j account by the empirical coefficient E 3:

1 p o I

..V h. =p h - K s 2lv*

pg g s l

! I as  ;

i J

P E vuc sc t

I The result is: i L

A A t 1 v- v 1

1 (x 11 1 --D) 2 v -v A 2

+ (x 1t 1 --D 1

2 y + x -x2 ) -v 3

P c.P g _

v Y v + x -x 2) -+y 1

+ (x D v ti l

• A v s

- P _

3 3

Av 3 2 1 2

a- v

- 5 4 A p) +2 K s e

i

~

(A, )

• l (v 2+2v tvs+2v sv s+2v ivs+v t2)

(I pf s P 6 -Pc i -g (x ii S D

v + x -x ) +

4 2 p g

(C .1 -11) i Displacement Equations t

The level of the surface separating the phases moves downward in the drywell at a rate u p, hence ,

F " "F (C .1 -12) and the pool height increases at a rate u sa, hence A

+ v x

11 E

= -

A (v +v +vs )

1 3 P ( C.1-13 )

c-12 J

s

.,.-...-,,w.,.....-- ...,-.-r,,.,....,.-y....,,,-w...._.-..,.w.-,._,_,-......,...,.,,..,-,.,,-w~..,.r,m,--.y.,,m- -

.. -, m w y ,, ,..y, .,-..,.,.,,,,,,y-

SWECO 8101 T

,, / Recapitulation for Clear 1 Thirteen equations (C.1-1 through C.1-13) have been derived; the corresponding 13 unknowns are:

u , u , u,, u,, v , v , v,, uF'

• ist' *F' E'E' 2 3 E

• l' C.2 CLEAR 2 (SEE FIG. C- 5)

Continuity equations Node 1 The volumetric rate of incoming gas is Au g , the volume occupied by the gas increases at a rate Au p, and the gas esecpes into Node 2 at a rate A g vg; hence:

Au v 1 = AuF+Ag i The cross-sectional area of the vent which is occupied by the gaseous phase Ag is specified in terms of the parameter, p a, such that:

\s- A a 5 ^E v (See Appendix B.1 for relationships between a, x,a x) p hence:

A u -u F -aA1v =0 (C . 2-1)

No liquid enters into the node; the volume occupied by the liquid decreases at a rate Aup; it escapes into Node 2 at a rate (Al -1/ 2 Ay ) v g ; and into Node 3 at a rate Auz ; hence:

Au g = (A g -fA) y v + Au The cross-sectional area of the vent which is occupied by the liquid phase, A y , is also specified in terms of a; hence:

u -uF+ I ~") V 1

"0 (c . 2 -2) 7-~3 ms C-13

l l

1 i i

SWECO 8101 I Node 2  !

i

! The equation is derived in Appendix B.1; the result is:

1 v -v F

-4u = 0

a F (C . 2-3)

Nodes 3, 5, 7 The flow pattern is the same as for Clean 1, and the results are identical:

l I

u -t 7A (v +v ) =0 2 3 i s (C . 2-4)

Ay 1

u -u 3 " 7^ (v +v ' )

=0 (C . 2 -5)

A u, - f [ v, = 0 (C . 2 -6)

Momentum equations i

Node 1 Although the flow pattern is different from clear 1, the i same derivation is still applicable and the result is

! identical; hence:

{p p

[

d (x p -x, ) u + (x -xp) d, + (xp -x ) + (x,-xp) dp ,

s

~ -

P p l =3u o 2 -

u 2

2

,7uF 1

u +u F 2

3p g (u +uF) g i i

~ ~

p p '-p #

+g 3p g (x F-x ) + (x -x F) 1 2

+

p g

(C . 2-7) l' l

l C-14 l

. . - . _ _ . _ _ _ - - _ . _ _ _ _ _ _ . . . . . _ . . . _ . . . _ _ - , _ . _ _ _ _ _ . _ . _ _ . , _ . _ . _ _ . ~ . - _ _ _ _ _ _ _ _ . . . _ _ _

-_ . = -_- - -. . . _ . . . - . . _ . __ _

SWECO 8101 Nodes 3, 5 The flow pattern is the same as for Clear 1, and the resulta are identical:

o -p l(x -x )(d +0 )

3 2 2 3

=u 2

2

-u 2

+ g(x -x ) +

2 2

P 3

3 3 g (C . 2 - 8 )

=u 2

-u 2

+ g(x _x ) #

P, 7, f(x -x ) (0 +0 )

to 3 3 6 3 4 4 3 Pg (C . 2-9)

Node 2 containment combination (see Fig. C-2)

The horizontal momentum equation fcr Node 2 is derived in Appendix B.2; the result is: ,

"Y F -1 l+y y v +fayF ~ 1 F

p [ o 9 -D O I[ 3o )l (v +vF ) (avF+ 0u) -a 1 A)v 2

+ v2 v2c

= 7 (1 gj F o og i

gj i j 1

The pressure losses at the vent entrance are taken into l account by the same empirical coefficient Kg, but the density is averaged over the cross-sectional area at the -

entrance of the vent:

~"

D av V "#

V L""E g 01 Or av ,g, g, h

' A A l ll Q

l hence:

E '

v2 , P;, _ g

_ 1 3_, g 3_ h) y 2 t og og i 2 og i O C-15 i

, , ,,._ - .-. - .--..- - . ,. - . , . - ., _ ._.,.-_ ,. ,,.,.. ,.-.-..._ , _,..~ ., - ,_ ... _ .. ,

i  !

I j

4 .

i SWECO 8101 r I

I

, The upward vertical mcmentum equation for the control volume shown in Fig. C-2 is the same as in Clear 1:  ;

. 1 A

v - - + IA v 3

-v 2

+v v +v v i

(X1I E7 D) v A-p(v +v +v ) =-l 1 s s A

p 4 1 1 s 1 s

-g (x 1

+ P2c"E c 1

i1 1

--D)2 v j og i [

as before l p =p i

2C v2C Adding up as before yields:

~

/0 ) A' I hayF -l j l+Yv

+ (* ii t D)y gl d t

P- ,

3 A IO c \- i 1 v . . 1

+ (x,,g 7 D) (v +v ) +7ayF l { ~1 j l Y F

I

=

\

1-O L) l (v +vI ) (a vI+0u) 1 p

i:

l [

1 ai 1-P 112 K

) _

( t t, j t

/ Av ^v 1

(vivs +vv) 2

+2 K +3 1 i v l  !

3 4(A p j _ Ap j is P, Pc f

-g(x 1 7 D) # + # (

11 1 (C . 2 -10) r i

r Node 4 and containment combination (see Fig. C-3) I Node 6 and containment combination (see Fig. C-4) i O

i I

C-16 i

i

,,,--,-,---,,.w,--n,~,--...-,-__-m,._.,_ ,,,,-,.,-,,,,-,,-,,,,_,.,c,.,.-,,,.,_.-- _.nn,.n.,.,---,.,,.._w,,,,-,---- _

_ m __

i i i

SWECO 8101 i t

1 The flow pattern is the same as for clear 1 and the results i 1

! are identical, hence: j

- . i A

^v .

I v .

i 1

(x Il l 1

7 D) v n -v 1

+ (x 11 1 1

7 D v

+ x -x )

3 2

-+y V V

3

, P _ ^P _

(

1 Ay .

! + (x 11 1 7 D v + x -x2 )A -v s 3  ;

P ,

i

~

l 1 3 I ^v I^v

=-v 2

7 K* +"l \Ap/ - I

-l

("p/

r--

[

li 4

l I

i (v 2+2v1 v3 +2v1 v5 +v3vs )

1 i

l t .

P, -P c I

! -g(x til 1 5- D V + x -x ) +

3 2 og t

t

(C. 2-11) .

i i  !

I t 1

4

! A A i

v. v.

Y v + x -:: )

1

-- v + (x 11 1 i

1 e

! (x Y D) v M D r- v i 11 1 ^ '

i 1 3 2 3

• ;

l n -

+ x -x )a v~+ yv

' - t' 1

l + (x D v i 11 1- 7 Y 6 2 s ,

i

- P '

t 2 - 2 I A A

= -v 3 -v\  !

+7 1

K v

l >

I

- [P / 3

\ P/  :

(v 3 2 + 2v 1v3 + 2v 3vs + 2v 1vs + v *)1 i

?

~  ;

1 ,

p -p  :

c

-g (x 11 17 D v + x -x ) +

4 2 Og (C . 2-12) [

I i'

f i

j. c-17 l j- i l

._ _ ~ . . . . . _ _ . . _ _ . . -- . _ _ - _ _ _ _

I SWECO 8101 i

Displacement equations In addition to the same equations as for clear 1, the horizontal front roves downward at a velocity v p, hence:

F " "F (C . 2-13) hp " V p (C . 2-14) x 11 1=A P (v +v +v )

1 3 s (C . 2-15)

Recapitulation for Clear 2 Fifteen equations (C. 2-1 through C. 2-15) have been derived; the corresponding 15 unknowns are:

j u,,u ,u,,u,,v ,v,,v,,up ,v p,x g,x'Y' F F P,, P 3 , P, t

P C.3 CLEAR 3 (SEE FIG. C-6)

Continuity equations l

Node 1 The node is full of gas coming into the r.sde at a rate Aus from the drywell; it escapes at a rate Aua into Node 3 and 1/2 A yva into Node 2, hence:

1 Au = Au +7 Av y or (C.3-1) u -u 1 2 7 A1v 1

=0 C-18 i

SWECO 8101 Node 3 The surface of separation between phases is in this node.

The gas ccmes at a rate Au2 from Node 1, escapes into Node 2 at a rate (A g-1/2 A )y ve and its volume increases at a rate Au p, hence:

1 Au = (A g 7 Ay )v + Au F or u -u F

(2a-1)v =0 (C . 3 -2)

The volume of the liquid decreases at a rate Aup, it escapes into Node 2 at a rate A vl , a into 11 ode 5 at a rate Au s and into Node 4 at a rate 1/2A vv3, hence:

4 1

Au g =Av g +7 Ay v, + Au, or u,-up+ f(1-a) v +f fv,=0 (C . 3 -3)

Nodes 2, 5, 7

, There is no change in the flow pattern from clear 2, hence:

v -V p

- h4u p =0 A

v 7A - (v : +v) 1 u -u =0 5 s 3

(C . 3 -5)

V v =0

u. 2 A s (C . 3-6) i k/

s C-19

i j

SWECO 8101 i

Momentum equations Node 1 ,

f The node is full of gas, the flow pattern is the same as ,

for, say, Node 3 in Clear 1 with the cas replacing the

liquid

l 4 l 1

-

• E ~P 1

! 5 (x -x ) (u +u ) =u 2-u a +g(x -x ) +

• 2 1 1 2 1 2 1 1 0 9 (C . 3 -7)

Node 3 i The surface of separation between the phases is in this i node, the same derivation as for Node 1 in Clear 1 is  !

applicable, and the result is obtained by changing the indices:

I (xy -x,) u,+(x,-xF)u, + (xF~* 2)+(X s~*F) "F > .

' s g _

= u 2_3,2 + up u,+uF

~

(U2 +uF) l o p -p

+g

_g d0 (x F-x ) + (:: -x )

2 3 y + 2 Og 3

(C . 3-8)

Node 5 and combination of Nodes 2, 4, 6 with containment;

! there is no change from clear 2, hence:

l P-P 2 2 s =

=u -u , g(x _x ) #

f (x -x ) (u +E )

4 3 3 4 3 6 6 3 Og F

(C . 3-9)

C-20

_ . ~ ~ . _ , _ . _ . _ . - . _ . . , _ . . . _ . . . . . _ . . _ . , , _ . . _ - _ _ _ . . . . _ . _ . . , _ _ . _ _ . . - - _ _ . _ _ , . . . . _ . - - _ , . _ . . _ . _ . . ,

i  !

l  ;

i 1

1 l

l SWECO 8101 l l  !

i t

. t faypi -1 I +yy + (x ttg -

hD) y h I

(

+ (x,,g - h Dy) (0, + 0,) + fayp -1 Iv p i >

l

=

fi 1- l (V,+vy ) (avp + 4 up) 1 ai

( y K t 3 K

3(Ayf 1 t) ((1- y-))l + 7+ g l( ppj 1 L

[ ,

vi

, (

l -

l

[A -y )* 1 P,-Pc

-' 1 (v 1v

+v 1v s) -g(x, , g 7 D) y +

gA p) Pg (C.3-10) i l

A 1 v A  !

(x 7 D) + (x 1

+ x -x )A l+yv 0 ti t y

gp 0 i at t 7ov s 2 P

3 A

v* 3(A y42  ;

7 v + x -x )

1

+ (x D -V A

=-V 2 7

1 K + gi -

I ti l s 2 p s s 2 (Apj

• IA v I (v 2 42v v +2v v +v v )

al - 1 1 1 3 1 5 3 5

\ ^p /

P s -Pc

-g (x, g - f D y + x , -x , ) + (C. 3-ll) i C-21 l

r 1

1

SWECO 8101 (x,,g-fD) y P

0,+(x,,g-fD y

+ x,-x,)

P 0,

3 .

+ (x 1 D + x -x )A l+yy v =-v 2 11 1 2 v u a 5 5 p

3 IA v 1 I"v\ (v * + 2v v + 2v v

-l 4( A pj l

+7 K s -lA

( pj l

s 1 s a 5

+ 2v v, + v,2) -g(x,,g-hD+x,-x,) y P w -PC 01 (C . 3 - 12)

Displacement equations

~

There is no change from clear 2, hence:

F " "F (C . 3 -13) hp = v p (C . 3 - 14)

A,

. s x, g = p (v,+v,+v,)

(C . 3 -15)

Recapitulation for Clear 3 Fifteen equations (C. 3-1 to C. 3-15) have been derived; the corresponding 15 unknowns are:

u,u,u,u,v,v,v,u,v,x p p g,x,y, y p P 2 , P 3 , P, The flow patterns for Clear 2 and clear 3 are the same. The sweeping of the surf ace ' of separation between the phases s past the level of the centerline of the vent does not change C-22

SWECO 8101 the flow pattern, and the division of the calculation between Clear 2 and Clear 3 is arbitrary. The same applies for Clear 5, clear 6, and for clear 8, Clear 9.

C.4 CLEAR 4 Introduction The calculation goes into Clear 4 when the level of the surf ace separating the pha ses in the weir annulus lies between the top and middle row of vents. Under the usual range of initial conditions, the top row of vents clears when the calculation is in clear 4, and the top bubble begins.

The ref ore, two flow patterns occur in clear 4, one before s the vent clears (Fig. C-7) , and one after it clears (Fig. C-8) . The case before the vent clears is referred to by the Roman numeral I and the case after the vent clears by numeral II.

Common equations N Continuity equations Node 1 The node is full of incompressible fluid. The volumetric rate of fluid coming in from the drywell is Aug, the fluid escapes into Node 2 at a rate 1/2 Av vg and into Node 3 at a rate Au z , therefore:

Au = Au +hAy v or

~

1 2 1 (C. 4- 1 (ISII) )

r~N \

(b C-23

/

t i

l ,

I e

SWECO 8101  !

[

Node 3 i The surface of separation between the phases is in this node. The volumetric rate of gas coming into the node is  ;

Au, and it escapes at a rate 1/2Avv i into Node 2; the volume l of the gaseous phase increases at a rate AuF, hence:

1

Au = Aug+7 Av y  ;

or

""F"7 2 1 (C. 4-2 (ISII) )

No liquid comes into Node 3. The volume of liquid decreases at a rate of Aup, it flows out into Node 4 at a rate i 1/2 Avv 3 and into Node 5 at a rate Au s , hence: i 1

Au A"

+7 A v v F  : s  ;

or N

u p-u 1 ^v =0 7pv (c. 4-3 (ISII) )

Ncde 5 The node is filled with liquid. The volume flow rate cominq in frcm Node 3 is Au3, the volume rate out into Node 4 is 1/2Av v 3, into Node 6, 1/2Avv s, and into Node 7, Au., hence:

i

, Au 3

= Au 4

+yA 1

• v v

3

+1 Av v 5

or 2 A (v : +v s )

u -u =0 3 $

(c. 4-4 (ISII) )

a Nm- C-24

I

SWECC 8101 Node 7 i

The node is filled with liquid, it comes in from Node 5 at a rate Au. and goes into Node 6 at a rate 1/2Av vs, hence:

1

Au =w A 4 V v

s s or A

1 v u V =0 l

, s 7A s (C. 4- 5) ISII) )

' Mcmentum equations Node 1 The varicus terms of the momentum equation along the vertical direction are:

pu.dv = 0 u A (x -x )

1 9 9 2 1

V ..

o d

l The symbol ug is the mass average gas velocity; it is assumed to. be 1/2 (ua +u s ) , then

{puudA3=p g3 A(u 2-u 2)

A j i -

pdA. = (p -p )A 1 1 2 A 1 oag dV = og (x,-x,)g A  ;

V .

hence:

o -p

! l (x2 -x ) (d +d2 ) +u 2

-u 2 , 't 2

. g(x _x 2

)

1 1 1 2 1

' s, . ,

(C. 4-6 (ISII) L l.. C ': 3 i

I

'_....__,.,_..._..._.,,.___..---_.._.-_...__--..,._:_.__-..__.._.__.,_.4...._,_.-_...__-...-. _ , , , .

_- . . = . _. . . .- .- . .. - - - . , _ . _ - - -

l SWECO 8101 Node 3 The horizontal surface of separation between the phases is in this node; the vertical momentum terms are:

l pu dV = p u A(x ~* ) +

f F 2 Eu g A(x,-xp)

< V The symbol u is the mass average velocity for the gas, taken as 1/2 (u +uz ) and u1 is the mass average velocity for the liquid, ta en as 1/2 (u F+u3) , subsequently:

1 0 -

3 1 1 pug dV = o g A d + u, 7 (x,-xp) g 7 (xF~* ) 2

{ V

+U p h(x-x) p + h(x,-Xp)

~

l _

+hu p (u +u) p - (u +u p >

J and -

2 2

= o g A(u 2 bu 2)

/A {puudA]. = p Aug - p Au -

ij 3 g 2 3 pg 2 3

l

- pdA g = A(p ~E }

2 s A

i

[padV= f p

g A(x F-* 2 ) + P LI *s

~

• F) _

9 V _

1 i

l C-26

l

l l

4 I i

i

SWECC 8101 l i

J l

i i

! Gathering all the terms and dividing by C1A yields:

} . Oc O

(x -xF) i 1 c - 1 - 1 7 y (x F-x 2 )u 2 + 7 (x -x,)u s - s +7 r (x,-x )

- - 2 3 u,-

~7 ,

i i

+ o e

o e P.4 v, (u 2 +u,,) - (u: +u,)

1 2

+ 7 u, - +u --u=

og og

. og . 3 2 .

4 7 i

-o i

+g 1 (x,-x 2) pg -

+ (x -x FI

,

l (c. 4-7 (ISII) )

i l

Node 5 . t

.I T. I

! This node is full of liquid, hence: ,

i i f cug dv = og Au g (x,-x,)

~

l l' j V

. where uy is the mass average velocity assumed to be

' 1/2 (u3+u.) j

' f hence: 5 e

s 1

Og A (x,-x , ) (u , +u,)

) sugdV = 7  !

( ,V

?

l

! and  :

(  !

3 i

et cu.z dv = a1 c.t A (x -x ) (u +u ')

6 3 3 4 i

j l- V l-l -subsequently:  !

i

{ EpuudA1=c1 13 A tu,2-u,:) l l A j' l I f t

C-27  !

t

!~ i i f I'

i

!. l

! I a - . - . -. . . . _ . . - . - . - . _ . _ - . _ - .

I SWECO 8101 and:

~

pdA. = (p -p )A 1 3 4 A

cag dV = pg A (x -x,) g Gathering all the terms and dividing by P 1A yields:

P3 P.

f (x -x ) (d +E 4) 4

=u 2

-u 4

2

+g(x4 _x ) + 3 Pg 3 3 3 (C. 4-8 (ISII) )

Displacement equations The level in the weir wall lowers at a rate u p, hence:

F " "F (C. 4-9 (ISII) )

Equations applicable before the vent clears; Continuity equations Node 2 The surface of separation between the phases is perpendicular to the axis of the vent, hence:

v -v F

= 0 i

(C. 4-10 (I) )

Momentum equations Node 2 containment combination (see Fig. C-2) i i

s- C-28

. . .,---- . . , . . . . ..-, ,- . . - . . _ . - . . , , . _ . , - , , , , . - . - _ . ~ . , . . - , . . _ . . . . . - . - . . - , - - . . - .

l SWEco 8101

! The momentum of the fluid in the node is:

yp+pg (y -y F) j pu gdv = o A" g v1 V - -

hence:

h ( pu )g dV = A y4 p g yp+pg(yy -yp) Ot +(pg -og )vt vp>

V y _ ,

The flux of momentum is 2 2 g pug3 u dA3= pv g _py g

i A J

The resulting momentum equation for Node 2 is

[ P .

" Pv2-Pv2c

~

Yv'YF o; g 1 p g

I Following the scheme used in, for example, Clear 1, this equation is added to:

~

1. The vertical momentum equation for the control volume shown in Fig. C-2:

1 ^v - - +

(x -

7 D) v -A (v +v +v )

s 11 E 1 a P

f A, ) *

=-l -

A (hv 2+y y ,y y) p} 1 1 3 1 s

-g(x 1 P2 c-Pc 11 E - 7 D) v + pg

's C-29

~ . . . - - _ ,

- - . _ . - -- - - . - = . . - . - _ - - - - _ . .- - _ .

}

l .

i' swEco 8101 1

2. The expression for the losses at the entrance of the vent in terms of the empirical coefficient Rg:

P y*

= g-K 4 v 2

Pg P g i Pg 1 1 I

3. The equation

=

P P l

v2C 2C The result is:

yv-yF (1

!b +A p (x ist l

7 D) v v '

f P gj i l

4 2 Ay 1 . .

1 a 3 IA v 7 Dv ) (v s+v s) -+l 2

+ -

(x -

= -v -K A

p i1 E 1 2 1 pg 4 (A p i

/A yT2 P2 -Ec

-1 gA )

i (vIvs +v1v5 ) -g(x 11 E -

3 7 D) v + Pg l (C. 4-11 (I) ) ,

Node 4 containment combination (see Fig. C-3)

Node 6 containment combination (see Fig. C- 4) i I

I I

l l

, C-30 l

i l

l l

...,.w....,-..,..-,. ..-...n...,,. . - - . . . . .-..-._._n...._--.., .....e, ~,,..c...,-.,-nnnn.,. ,...,-n,-,.--n..-+,.--v,--

I SWECO 8101

There is no change from clear 1, 2, or 3, hence:

A A -

IX ~

+ ~

+ ~

+

' 11 1 v 1 11 1 v 3 2 A v 3 P _ P _

_ 2-7D + x -x ) Al O = -v

+(x 11 E 2 1 K + l v 3 2 p 5 3 2 2 (A p _

l Ay )2 A

(v 2+2v1 v3 +2vI vs +v3 vs )

1 pj P3 Pc

-g(x 7 v +x -x ) +

1 D

pg 11 E 3 2 (C. 4-12 (I) )

and A

1 ^v - + (x 11 E 1 v-(x --D) -v 5 v + x -x ) A-vP D

11 1 2 v A 1 ~ 3 2 3 P

p) -

3 _

h + (x 11 t 1 D + xs -x ) 2+yv v 2 v 2 A 5

_ P _

3 /^v 2 -

(^ 2 v }-l

. 1 -

2

= -v -I + -K '

5

_" N P '_ \ ^p /

(v 2+2v v +2v v + 2v v +v 2)- g (x - 1 D + xs -x 2) 3 1 3 3 5 1 5 1 Il t 2 v i

P,-Pc 9%

(C. 4-13 (I) )

l Displacement equations 1

The front in Node 2 moves at a velocity v p, hence:

F"YF (C. 4-14 (I) )

L

. \s C-31 ,

l l

_ . . _ , ._ , _ - _ _ _ . _ _ _ _ . . . . . _ _ . _ _ . _ _ _ . . . - . _ . . . . _ _ _ . . . . _ _ _ _ _ _ . . . - . _ - . . . . . ~ . . _ _ _ . _ . _ . - . . . , . _ . _ - - ,

SWECO 8101 s

and the location of the surface of the pool is specified by:

A

. v x

ti l=A- P(v +v +v ) 1 3 s (C. 4-15) I) )

Equations applicable after the vent clears:

Continuity equations By inspection (see Fig. C-8) , the continuity equation applied to the pool yields:

Au =Av(v +v )

p 10 4 6 or, since v = v3 and v. = vs (see section 4.1.2) :

A Y

u to A (v +vs )

3 P (C. 4- 10 (II) )

Mcmentum equations It is convenient to combine the horizontal momentum O equations for Node 4 with the vertical momentum equation of the control volume shown in Fig. C-9.

The varicus terms of the momentum equation for the latter contrcl volume yields:

1 pugdV = o Ag p (x, x 7 D) y u V

2

• pu u dA g3 3

=p g A (u g -u,g )

A j pdA g =A g (p 3c ~P2c) 1 pag dV = - p g gA p (x,-x g Dy )

V C-32

SWECC 8101 By inspection, the continuity equation in the pool yields (Fig. C-9) :

(3 .h A"c"A p3 v 2 V +V 3

Gathering the above terms and substituting for the velocities u se and u so, one obtains the vertical momentum equation for the control volume of Fig. C-9:

A v - IAv pj (fv 1

(x -x 2 3 7 D) v - (v +v ) +1 A Ap 3 s s

+v) s

, s 2c - g(x -x -

D) y The horizontal momentum equation for Node ti is:

g h -

pAyv gy y

+oAg y(v s ~v s ) "Av(Evs ~P v3c) or Yv 3 v3 ~Pv3c}

Furthermore, the entrance and turning losses at the entrance to the vent are taken into account by the empirical coefficient K: 2 P = P -K pg vs 2 and the accounting for loss at the exit of the vent:

v3C 3C

.v C-33 i

. . , , - - - - , - - - . - - , , . . , , , , - -. ,,, , .,,...a,,,.-n,., , - , , - - . . ---,.n,,.. . . - - . , . - . , . , . . . . . , . - - , . - - .- -n . . . ,

SWECO 8101 By adding the last twe equations and the two momentum equations, one obtains a useful equation to be included into the model:

3 - A (x3 -x 2 --D)v+y 2 1

vA v v + (x -x 3 3 2 1

2D)v A-V v-P . s P

IAh* y p,pc 1 i +

(Apj 1v s (fv+v) s s

=

pg 2

-K 22

-v 3

2 1

- g(x,-x, 7 D) y (C. 4-11 (II) i Another such combination is presented in Fig. C-10. The application of the momentum theorem to the control volume shown in Fig. C-10 yields:

a Tt puidV = o Ag p (x,-x, 1

7 D) y d

+p g Ap (x,-x, )d z

pug3u dAg=pg A p (u to

-u sc )

. A .

i pag d =Ag (p sc ~E 2c}

I 1 l

pa g dV = - p g gap (x,-x 7 D) l V

I l

m 5

'- C-34

.- - -_. . - - - _ _ . . - ._ . - .. - _. _~ . _ . _ . . - .- . -

1 l

! a

- r i

SWICC 8101  !

2 i

L I

By inspection, the continuity equation in the pool yields I

(see Fia. C-10) :

v u = n- (V +v )

' 13 o 3 s i

e c t

i.

! n,,

u s

=dv a- s i

e ,

, t 1

Av u =7-v n, I

4C a s >

s b

- )

5 Substituting and dividing by D,Ac one ohtains the vertical  !

i momentum equation for the contr$1' volume of Fig. C-10: ,

-

. . , f, g (x -x 1 yD)v --o^vV 1 -

• (x -x 2 1 yD)v -- V

^v .

• f "v ) 2

3 6 g 3 4 og s ( ^, .

\ r  !

!- n -n f 3 r uc r re 4

(v 2 +.--- vs 2+2v2 vs )

a

=

- -c(x -x 2 2

1

-D) v i

e s L i

1 j.

I The equation for the horizontal momentum of Node 6 is the same as for Node 4; and the expression for the pressure less {

at entrance and exit of vent are the same as before, hence: [

d t

+

v v - 1 I

v s og (=v. ev.c}  ;

!.  ;

_ ~ 1 2

• ,L y s

=, i Pvr "r 's T  ;

a D =0 j -vwC SC i

1 i ,

t f

t

[

t

! t i

C-35 ,

i e

k

(

)

SWECO 8101 The resulting combined equation is:

A

-jD) [A v -fD)

(x,-x y

+ yy

+[P (x,-x y v, P _

2 l

+!

[Av \ (v . +3 1

v 2

+ 2v v ) =

P u -P 2c

-K 1

-V 2

A pj i 4 s a s pg 32 s 1

1

- g(x -x --D)

= 2 2 v (C. 4-12 (II) )

Bubble equations The control volume and the characteristic quantities for the

! top bubble are presented in Fig. C-11.

The mass of the liquid above the bubble (Node 11) is:

"1"0A1p I*

31

~* iig)

The liquid which is pushed upward from Node 10 into Node 11 is assumed to become part of the liquid above the bubble, hence, the continuity equation for the liquid in the control volume is:

3 dm g , ,

=pA1 p (x 11 1 -x 119) = p.A u dt A Pto l

or x , -x = u 1 1 A- 119 10 (C. 4-13 (II) )

l j

4 I >

C-36 i

. .v.., _ - . ~ . . . _ __,,_..,.._.._,~.,,~,_,-_,_,._v._ , , _ . . , , - - _ _ _ , , . _ . _ , , , _ . , , , _ , _ , _ _ _ _ . _ , _ _ . . - , , _ . _ , . _ . ,

SWECO 8101 e

The various terms of the momentum equation applied along the vertical direction to the liquid above the bubble, are:

pu dV = p A (x i:1 -x itg) u I i Ip it V

2 pug3 u dAg = - og gAu, A .

J pa.dV1 =-p1g (x 115 , -x 11g) A p

V pdA g =

(p c-Ec } A p 2

A where pc is the pressure in the containment.

The result is obtained by gathering the various terms of the momentum equation and substituting the above continuity

[] equation:

QJ -

p -o 1 2 u + 2c'c _g u = u -u 11 x g -x 11g _

I I" II E ,

(C. 4- 14 (II) )

and naturally x E =- u 11 11 (C. 4- 15 (II) )

l The first law of thermodynamics is acplied for the open

, system formed by the gas inside the bubble, using the forrulaticn expressed in Feference 3:

dE - *

-- = m. h.in -m out hout +0-W dt in Here no mass goes out and there is no heat transfer, hence:

=0 5out = 0 and

\s / C-37

- -, . , - - . _ . , _ , .- ,--,,-mm._, .,--,,-,,m.....,.-r,.r,....~r.~-_---.y,w-_m,,,.,-,_m,,,--.,__ ..,_~_r.,,,,. m-.

SWECO 8101 D

j The hence:

work done by the bubble is assumed to be reversible, p (u 11 W = pv = P A x = P A -u )

r 2c p itg 2c to The specific static enthalpy of the incoming fluid is assumed to be that of air at the drywell temperature, hence the specific stagnation enthalpy of P.he incoming fluid is:

V

, hin = C Tp Drywell + 2 The density of the incoming fluid, p l, is selected to be equal to the density of the gaseous phase in the drywell.

' Therefore, the expression for the first law of thermodynamics becomes

. Av

(" ii -u no )

- 1 E

t

=g .

g (C Tp Drywell + 7 p 2 P

9 2

'P 2c 1

(C. 4-16 (II) )

where the, velocity v2 =Vt is calculated by the vent flow model.

The last bubble equation is obtained by applying the perfect .

gas equation of state for the bubble: l

, e pV = NRT i

,- Since:

l E = NCy T The elimination of the product NT yields:

l l E=[C pV =

Y1 PV

-The application of this result to the bubble yields:

t E  :

11 P2 c, Y-1 A X  ;

P 119 (C. 4-17 (II) )  ;

C-38 j l

.< , ..--,-.ww-, ,, - .-.c-...-,-.,..-,..-,,-,,n--,-,~-,. - - . - , . , _ , , , , - - , , , - - - - - . - , . - ,-n--- - . . . . . , . . ~ . -

SWECO 8101 Recapitulation for Clear 4 For the case before the vent clears, 15 equations (C. 4-1 (I) through C.4-15 (I) ) have been derived; the corresponding 15 unknowns are:

u,u , u,, u,, v , v,, v,, up, v p, x F' Y F' *t11' P,,P,,P, For the case after the vent clears, 17 equations (C. 4-1 (II) through C.4-17 (II) have been derived; the corresponding 17 unknowns are:

u, u,u, u ,

v,, v,, u,x'*

p F tt i' *ttg' p'E 2 3' p,, E ,p c, u , u C.5 CLEAR 5 (SEE FIG. C- 12)

Continuity equations Node 1 There is no change from clear 4, hence:

u -u 1 2

=1Ev ^ 1

( (C. 5- 1)

Node 3 The surface of separation between the phases is in this node, but the geometry is different from that for Clear 4.

The volumetric rate of-gas coming in from Node 1 is Au a ; it escapes at a rate 1/2Avv g into Node 2 and Agv3 into Node 4, where A g is the cross-sectional area occupied by the gas at the entrance of Node 4. The volume occupied by the gas increases at a rate Au F, hence:

Au =fA,,y +Av + Au g l

\ss/ C-39 l

l

,.,-r-,, . . . _ , , , - . . , . . . - , . . - - . _ , , . , - - - . _ - - - , - - . - . . , . . - - - . , . . .

t SWECO 8101 l 4

Instead of using the variable A g, it is more convenient to consider a defired by:

A a5 AA v

hence:

.'! A A v 1 v v 1

u -u F --

av 2 A =7A 1 .

(C. 5- 2)  !

The re is no incoming licuid; the rate of decrease of the liquid volume is Aup; the volumetric out flow of liquid into Node 4 is ( 1/ 2Av- Ag) v 3, and into Node 5 Au s; so:

1

b. = M +(2 Ay -A)v s g  :

hence A

u -u F

12A (1-2a) v =0 i l (C. 5-3)

'l l Node 5, 7, control volume Fig. C-10 i There is no change from clear 4, hence:

4 A

1 v

u 3 -u. 2A (v +vs )

s

=0 (C. 5-4) 1 i

A 1 v i u, 7 g- v , = 0 (C. 5-5) l A [

l u to

- -Av(v +v ) i s

=0 (C. 5-6) [

P 1

r N C-40 t

l.

. . . . , , . - - . . . . . - . - - , _ . . . . _ , _ . _ . . . . _ . . . . _ . . . . . , , . . _ . - . . . . _ _ _ _ , . _ , _ - _ - . . . , _ . . . . . . . . . _ , - _ . . . . . . . ~ , , . . _ . . . . _ . . _ . , _ . , - . _ .

4 SWECO 8101 Node 4 When a surface of separation is assumei to be made of a horizontal plane and a vertical plane, it is necessary to account for some three-dimensional effect. It is convenient here to consider the flow pattern as a hydraulic jump. The derivation of the corresponding equation is rather lengthy and is presented in Appendix B.1. The result is 1

v -v F --0u a F

= 0 3

(C. 5-7)

Momentum equations Node 1, 3, 5

There is no change from Clear 4, hence:

P ~P 2 i

1 I (x -x ) (u +u 2)

+ u22 -u 2 = i p

+ g (x -x )

2 1 2 1 1 1 (C. S- 8) h (xF~*2 ) 2

+

(* ~*F) 3 3

+ ~

I*F *2 )+( 3 ~*F) F

~ ~

p p P -P

+1u 2 F 3 pg (u +u F)-(u +uF) 2 3

+u 3

2

_ p Ju g 2 2 , 2 Og s

+g

[P (xp -x ) + (x,-Xp)

_L .

(C. 5-9)

P "P "

f (x -x ) (d + b) 4 3 3 4

+u 4 2

-u * =3 pg

+g (x4 -x 3)

(C. 5- 10)

As for Clear 4, it is convenient to consider the two combinations of horizontal momentum in a vent with vertical momentum in the containment.

C-41

i SWECO 8101

( The first combination yields a result different than for Clear 4. The vertical momentum for the control volume (Fig. C-9) is unchanged:

A [A 2 1 D) l (h +l l (x -x 23 2 v A p i

+h) s (Ap j v

s (fv s

+v) s s 2c

= -g(x -x -

D) y The de:.ivation of the horizontal momentum equation for' Nede 4 is shown in Appendix B.2: the result is:

/p h [p h i jayy -1; +yy 4, + j ayy -1j 4r p )

=

f 1- (v, + vp) (avp + 4 u p)

P s

, ,2 _ ,Pvs Pvsc P

\ "L ) L The pressure loss at the entrance to the vent is taken into account in the usual manner with the empirical coefficient i

K2:

Py3 =P -K 3 2 3 i

where p' is density averaged over the cross-sectional area at the entrance of the vent:

A A p' E p + og = apg + (1-c) p g

[v 9 v hence:

P._,.= 1-ai1 / d p

P P 2

( t A C-42 I

. . ~ - - . , .. ,. - - - . - , . . - . . . . . - - , - , - - - - . - . . . - . . . - - . . . - . - . . . - - . . . . . . . , . . . - ,-- - ,. ,. ..-

I

.  ;

l SWECO 8101 i

i i

, so the equation fcr the pressure losses become: l I

) l f Py3 "P 3 K

hO g 1-3 1-[O )' V,*

\ 1 r

. .  ;

The expression for the loss at the exit of the vent is unchanged from Clear 4, hence:

, Py =P sc [

l Adding the three equations, as for Clear 4, yields:

t-1 4

A v ID c I a

( x -x 3 2

- :r 1

D v) +2 cy 1

l --1 l+y v

v

- P \pl /

3 i I

A v 1 -

1 IO c I.

+ pp (x,-x, J

7 D) y v,

+ 7 ayF ( T I

I -l /l Y F

l /Ay T* (3 I / Cc h

=-  ! v l4 v +v l +2 1

1 1-p l (v 3

+v) F (A-p/ 3

( 2 5/ ( gj f

/ l- Pa)/ 1 h 1 (avy + ?uF I -V " K 3

lY Eh

• 7 l 4 1 P-9c

-g(x,-x yD) y + 3 L

2

+

(C. 5- 11) i The flow pattern for the second control volume combina'; ion is the same as for Clear 4 (II), hence: ,

A A (x -x - 2 D) 3 2 1

v

-v A

v- + yv +v- a (x -x . 2

--D) 2 1

v v

5 I

s P P A  :

y )l - P=-P2 c 1 --

2

+ -

(v , + 34'v ' + 2vavs)

- = -

K 7v' l (Apf Pg l- 3 s 3 5 4

I --

g(x -x -

jD) (C. 5-12) lt C-43

• -e+ . - + , er- - e- = = - - - . - , , , , , . . . . . + --+vr ,,w ,=gr----,-=w,- >w, .,,<w-,g.4+e--a-,--, wer +. - - , e , ,-e e , .- w w W W y t , y.+wg,*="-r*"^d*-

_. . - . . . - - __ =_

SWECO 8101 O\ Bubble eguations The derivation and results for the bubble equations are the same as f or Clear 4 (II) , hence:

5 -

i =u 11 1 119 1o (C. 5- 13) u 2

-u u p ~Pc

, , ec u =

-9 + p E (x -x 11 x g - x,39 11E 119) (C. 5- 14) i1 1* 11 (C. 5-15)

~

dE -A / 1 11

.A l I CTP Drywell + 7 y 2 2

p y - p (u -u )

dt p A 9 2 2c 11 1o P\ (C. 5- 16)

E p ,Y-1 11 2c A x P 119 (C. 5- 17)

Displacement equations The surface separating the two phases in the drywell side is a horizontal plane moving at a velocity up, hence:

Icp =u F

(C. 5- 18) and a vertical plane moving outward in Node 4 at a velocity vp , hence:

yp = V (C. 5- 19) 7 Recapitulation for Clear 5 Nineteen equations (C.5- 1 through C. 5-19) have been derived; the corresp'nding 19 unknowns are:

u,u,u,u,u ,

v,, v,, uF ' #F ' P'p'p'x119, 2 s s x g, Xp, Yp, u ,E , P J C-44

SWECO 8101 a

. C.6 CLEAR 6 (SEE FIG. C- 13)

Continuity equations Node 1 There is no change from clear 4 or Clear 5, hence:

1 ^v v u -u =7A 1 2 1 (C 6-1)

Node 3 The node is full of gas. The volumetric rate of incoming fluid is Au z, from Node 1; the rate of outgoing fluid into Node 5 is Au s, into Node 2 it is 1/2A yvg and into Nede 4, 1/2Avv 3; hence:

1 1 Au = Au, + 7 Ay v +7 A,V or 2 3 3 1

, (C. 6- 2)

Node 5 The surface of separation between the phases is in thit node. The volumetric rate of gas coming from Node 3 is Au s, the volumetric rate of gas outgoing into Node 4 is (Ag -1/2Ay)v3, and'the rate of volume increase is Au p, hence:

1 Au , = Aug+ (Ag -

7 A) y v A

4 Using the variable a E 3., the latter yields:

l v u,-u-hp (2a-1) v,=0 (C. 6-3) l U C-45  ;

l l

[ E

. . . . - - . - . . . - . - . - . , . . . - ..- -. -..-.-- -, - ,- .. - .~.-. - ,.- -

SWECO 8101 No liquid 9.2mes into the node, the rate of vvlume decrease is Aup , the volumetric rates of outflow are: into Node 4, A l v3, into Node 6, 1/2 Av vs , and into Node 7, Au.; hence:

1 Au g =

Au, + Agv +7 Ay v, or A

1 v u p -u, 2 A 2 (1-a) v +v = 0 (C. 6- 4)

Node 4, 7, hybrid control volume (Fig. C-13)

There is no change from Clear 5, hence:

v,- vF

~

"F =0 (C. 6- 5) 1 "v u v = 0 5 2 a ,

5 (C.6-6)

A

- AE (v +v )

\ u =0 to 3 s P (C. 6-7)

Momentum equations Node 1 There is no change from Clear 5, hence:

I 2 2 o p2 Y (x 2-x ) (u +d2) +u 1 1 2

-u 1

=

0 9

4 g(x _x )

2 1 (C.6-8)

Node 3 The romentum balance is the same as for Node 5 in Clear 5 with gaseous phase instead of liquid phase, hence:

I o 2 _p s Y (x -x 2) (d 2+u 3 ) +u 2 -u2 * = w

+g (x -x )

3 3 3 2 9 (C. 6- 9)

C-46

. _- . - _ . _ - _- . _ . . _. - - . . - - .-._ _- .-. .__ = _ - - . -

i SWECO 8101

/

I Node 5 ,

The Inomentum balance is the same as for Node 3 in Clear 5, hence is obtained by a change of indices:

O 1 c 1 - I 7 (xF~* ) 3 3

+

7 I* ~*F) ".

6

- - i

+f (xp -x,) + (x,-xp) $+

F "F -

(" + F)~("w+"F) 3 _

r u 2

+u 2 ,94 3 s

,g b (x p_x ) , (x _x y)

Dg 3 6 Og , Pg 6 (C. 6- 10)

The flow pattern for the two combinations are the same as for clear 5 and the corresponding momentum equations are unchanged, hence:

I (x,-x-fD)y + f ayp l -1 +y y v,+

1 -

1 IP I. IAv (x,-x, 7 D) y v, + 7 ayF j ~1 Y F"~

vl v,+v +h 1- (v,+vp ) (avp+0u) F

~V 3

l

- / 0 I / h ai 1d -g (x -x 1 1 1

, _ \ 9 2)

I i

\

17 K 2) i+

7 K 2

y '

D )

l ,

p _p (C. 6- 11) l l

• t l

l l

l C-47 l

l r'

l l

i

SWECO 8101 5

and A A v

v A .(x -x 3 2 1

7 D) v : + -

A (x 6-x 2

1 7 D) + Y y

V v

s 3

P . p ,

p" p2c

+i (AV) j (v 3 2, 3_ y 2+2v v ) = -K 2 1

(A p j 4 s a s og 7v s 1 1

-g(x6 -x2 - 2 D) v (C. 6- 12)

Bubble equations There is no change f;om Clear 5, hence:

4 . .

x -x =u 11 E 119 1o (C. 6- 13)

. u 2 -u u 9 2C -P C g , 10 10 11

,, g 4 p (x 11 x -x -x 11 E 119 11 1 119) (C. 6- 14) 11 1 11 (C. 6- 15) dE A / )

d

• A p PT Drywell + v* p g

V 2 -P2 c(u g3 -u ,) 3 (C.6-16)

E p , Y-1 11 ac A x P 119 (C.6-17) ,

Displacement equations There is no change -from Clear 5, hence:

ic y =u p (C. 6-18) y p =v p (C.6-19)

C-48 t

. . _ - .. ~ . . . _ . _ _ . . _ - . , _ . , - - . _ _ . . _ _ _ , . _ _ ~ . _ , _ _ . _ _ _ . _ . _ , _ . . . _ . _ . _ . . . . _ , , _ _ _ - _ . . . _ _ _ _ _ _ _

SWECO 8101 O

s/ Recapitulation for Clear 6 Nineteen equations (C. 6-1 through u. 6-19) have been derived for the same 19 unknowns as for Clear 5, that is:

u ,u ,u,,u ,u ,

v,, v,, uF ' VF' 2E'p'p'xilg' s 4

• M ' *F' Y,utt' F 11' 2c C.7 CLEAR 7 Introduction The calculation goes into Clear 7 when the level of the separation surface between the phases in the weir annulus lies between the middle and lower rows of vents. Under the usual range of initial conditions, the second row of vants clears when the calculation is in Clear 7, and the second bubble begins. Therefore, two configurations occur in Clear 7, one before the middle vent clears (Fig. C-14) and one after it has cleared (Fig. C-15) . The first i configuration is referred to by the numeral I and the

,) second, by II.

Common equations continuity equations Node 1 There is no change from Clear 4, clear 5, or Clear 6, hence:

" ~"

t 2 1 (C. 7- 1 (ISII) )

Node 5 For the liquid, the situation is analogous to Node 3 in clear 4, hence:

l

~

"F "s

~

V s" (C. 7-2 (ISII) )

x- C-49 l

. . ~ . - . _ - _ _ - - - . . _ _ _ _ _ . - _ , _ _ , . . _ . _ . - __. . . . - -

l 1

t SWECO 8101 Node 7 There is no change from Clear 4, clear 5, or Clear 6, hence:

A u

• 7A2v s

=0 l

(C. 7-3 (ISII) ) [

!

Momentum equations Node 1 There is no change f rom clear 4, Clear 5, or Clear 6, hence:

! . . P-P 1

(x2 -x ) (u +u ) 1 1 2

= u 1

2

-u 2

2

+g (x2 _x ) + t P

2 f 1 9

l j

(C. 7-4 (ISII) ) ,

Node 3 i There is no change from clear 6, hence:

! . . p p l

-2 (x -x )2 (u +u )

1 3 2 3

= u 2

2

-u 3

2

,g(x _x ) #

3 2 2

P 3

i 9 i

l j (C. 7-5 (ISII) )

C-50

SWECO 8101 j Node 5 The configuration is the same as for Node 3 in Clear 4, as can be seen by comparing Fig. C-7 and C-8 with Fig. C-14 and C-15, hence the only changes in the equation are for the indices:

f (xp -x,) b, + f (x,-x F) "

5

] +

f .

(xp -x,) +x,-xp 6p =

up L

(u,+up) -

E (u,+up)

P P pu

+ P 3u 2

-u 2

+g __S[. gxp _x ) ,x _x ,P s F pg g 3 4

_og 3 e.

(C. 7-6 (ISII) )

Upper bubble equations There is no change from Clear 4, Clear 5, or Clear 6, hence:

d .

x -x

=u 11 1 119 to (C. 7-7 (ISIT) )

P ~

+

" o~"Io 11 ~ 2 c 'Ec 11 x -x Pg(x -x 11 E 119 11 E 119) (C. 7-8 (ISII) )

11 11 (C. 7-9 (ISII) )

A P Drywell + V Ev'p (u -u 11 p 2 g2 W

~

(C. 7- 10 (ISII) )

E p = Y-1 11 2c Ap x 19 (C. 7-11 (ISII) )

l V C-51 i

a 4

SWECO 8101 i

Displacement equations There is no change from Clear 4, clear 5, and Clear 6 for the surface separating the two phases in the drywell, hence:

F * "F (C. 7-12 (IGII) )

Equations for the one-bubbJ e case only (Fig. C-14)

Continuity equations Node 3 The situation is the same as for Clear 4, Clear 5, or Clear 6, hence:

A A 1 v 1 v

" ~ "

2 3 2 A 3

• 7 lI i (C. 7-13 (I) )

Node 4 The situation is the same as for Node 2 in Clear 4, hence:

v F -v =0 Os 3 (C. 7-14 (I) )

Node 5 For the gaseous phase, the situation is analogous to Node 3 in Clear 4, hence:

A 1 v u -u p 7 gp v =0 (C. 7-15 (I) )

Control volume (Fig. C- 10) l A u

I 1 (v +v ) ' 3

=0 l P (C. 7-16 (I) )

Momentum equations As for Clear 4, Clear 5, or Clear 6, it , convenient to l consider combinations of horizontal momentun. '.n a vent with

( vertical momentum in the containment.

i l Node 4 and control volume (see Fig. C-9)

C-52 l' - . - - . . . . . - - . , - . . _ . , . - . . _ . _ . _ _ . - , . , _ , . _ . _ , , . , . _ _ , , _ _ . . _ _ _ _ _ . . . . , _ _ _ . . . . , _ , , _ ,

SWECO 8101 The ve rtical momentum equation for the control volume of Fig. C-9 has been derived previously; it is:

I 2 (x: -x 7 D) v 2 A (d +v )

=-

A v (

s f v +v )

2 p s s pj a s 1

-g (x,-x -fD) y

+ sc 2c The turning losses from the annulus into Node 4 are taken into account in the usual way with the K, coefficient:

Py, "P 3

- K, f P g v,*

which can be conveniently written:

P P p V2 =l-K 2 7Pdv 2

Pg P g g 3

]

! The horizontal equation for Node 4 has been derived in a previous paragraph; it is:

V' V' y v -y F 1-p I v =

gj s pg and, as was shown before:

E sc " E vac Adding the above equations yields the desired equation:

• ~*

~7 v} A *Y v ~YF

~ ~

s 2 3 A { A \* [ \

l 1

+ A2 (x -x 3 2 7 D) v hs

=-

(Ap) 1 vl4 v +v 3

sj p 3(

-g(x -x s 2 -fD)-K v 2 Pg hv*+ s 3

P E

2c (C. 7-17 (I) )

\

C-53

SWECO 8101 Node 6 and control volume Fig. C-10 The flow pattern is unchanged from Clear 4, Clear 5, or Clear 6, hence:

~

A A yy

+[P (x,-x,-hD)y v , + [P (x,-x -fD) y v, (A)*

(v,2+ hv,2 +2v,v,) -K,hv,*

= -l( [P /

P,.~Pa c

-g(x -x -

1 7 D) y + (C.7-18 (I) }

pg Displacement equation The surface of separation between the phases moves with a velocity vp inside Node 4, hence:

= y YF F (C. 7-19 (I) )

3 Equations for the two-bubbles case only (Fig. C-15)

Continuity equatiens Node 3 u -u =h (v +v,)

(C. 7- 13 (II) )

Node 5 For the gaseous phase:

V 3 F" 3 (C.7-14 (II) )

For these nodes both vg and v3 are calculated by the vent flow model. By inspection of Fig. C-15:

Ay

~

e A] 5 (C. 7- 15 (II) )

C-54

SWECO 8101 Mcmentum equations It is convenient to consider the combination of the vertical momentum for the control volume of Fig. C-16 with the horizontal momentum for Node 6.

The application of the momentum equation to the control volume of Fig. C-16 yields:

AAg p(x,-x,-fD)y u, + pgAp u,2-p Apunc 1 [ \

=-AgAg p x,-x, y Dy +1 P 6c"E c l ^p

- - \ sl Substituting:

v V

", " lP s

".c P s

and dividing by plp u cne obtains:

3A [A32 v 3 y l 1 V 2 1 (x-x 6 s 2 D 1 Vj % s

=j (Apj 4 s

-g(x -x

= s I D) v E uc -E

, sc P

t The horizcntal momentum equation for Node 6 is:

Yv b, " (E v

-Evsc) c-55

i SWEco 8101 The turning losses at the entrance to the vent are taken '

into account by:

1p P

vg

= 1 0 g

-K Yv 2

g og s and, as shown before: ,

P sc

=P VgC then the addition of the above morentum yields:

- A- g yy

+(x,-x,-fD) [P y v, = -v,2(A h2  !( + )

_ \ 9) t P.-P sc

-g(x,-x, 7 D) y +

(C.7 's6 (II) )

Middle bubble equations O The control volume shown in Fig. C-15.

for the middle bubble computation is The mass of the liquid is:

m g =pA gp (x ,-x g) l The liquid comes into the node from underneath at a rate:

1,in"P tp u "s

and goes into the upper node at a rate

1,out 01. ^ P :o

. The application of the continuity equation

i dm g , ,

4 dt " "E,in ~ "1,out

C-56

., _ ._ . .. ~ _ . ,-__.__ ,.._____.-..,_ . _ _ _._____ _ _.

t I

(

l SWECO 8101 }

yields:

x =u -u 139 12 9 (c. 7- 17 (II) )  !

e I

The vertical momentum of the liquid is:  !

1 pAA P(x 1

-x 139)u la i

f The scmentup flux Out is 2

p1Au P le I 1  !

4 r 1

l and the sementum flux in is:

1. 1p A u,'

!

i  !

j The pressures below and above the liquid are ps; and pre L

! respectively, hence: [

} . - 4

. . . i ptAu

{

d- p 1: (x Is -x 139) + pAug . la cAu' g pS  ;

~

l i

(

t

'- + A (p

! =- pLAp(x la

-x 13g) p 3c -p2c)

! r 1

I or  ;

-g(x la -x

)-x 139

. 2 2

! u (x 13 -x u = u -u  ;

10 1Cg la 9 IS 1 9) '

F

+ P 3C~E2C l

• L ,

l l .

i or '

1 - ,

i' + 1 2 E sc- E- c u =

x "x U -u u + '

-c i

i_ 10 9 9 18 D [

2* l'9 I (c.7-18 (II) ) l i j j-  !

n

j 1

d!)

\ c-57 I

i

-,ne,., ww-~~r,r.,-n-ns,~re. . me e nwwww. e nmevem m ,,,-m ew . _ -,ow- e m v -

SWECO 8101 i

The first law of thermodynamics is applied to the gas in the

- control volume shown in Fig. C-15.

dE . . .

j dt " "in h in ~

out h

out + 0 - W IIere the only mass transfer is mass coming in at the drywell temperature; there is no heat transfer and the reversible work done by the bubble comes from its increase in volume; hence:

out

=0 in " O g^v s hin

• C Tp Drywell + Y.

Y=P "

P lc A p tog The substitution of the above into the general expression for the first law of thermodynamics yields:

i O. _ _

"A V,2) io p P

A g V. (C Tp Drywell + _P 3cI "io ~" I ,

(C.7- 19 (II) )

where the continuity equation has been used to compute igaa and the velocity v,=v3 is calculated by the vent flow modsl.

The last bubble equation is obtained by applying the perfect gas law to the bubble E

1o P (C.7-20 (II) )

sc " Y-1A P

x 109 C-58

--.----,,n,1- . . . , . . . . . . + . , - ..,. ,. ,w. n,.-,,.,,- - , .. . . - ,,,,,,,n..-.,.,., ,,-n,.,.,-,.. , , . , , , , , . - _ . ,

SWECO 8101 i Recapitulation for Clear 7

For the one-bubble case, 19 equations (C.7-1 (I) through I C. 7- 19 (I) ) have been derived; the corresponding unknowns (

, are:

u,u,u,u,u , v, v,u,v,p, p,p , x, 1 2 3 4 to 3 s F 2 s . F ,

r F'

1 g' *11 1, u ll , E , p ll 2c i

For the two-bubbles case, 20 equations (C. 7-1 (II) through (

C. 7-20 (II) ) have been derived; the corresponding unknowns are:

u,u,u,u,u,v,u,P, 2

• F P,P,x log, u ,

1 3 s s 2 i s to E

le, P3c, x F, x 11 E, x 119, u 11 ,E 11

,P 2c  !

C.8 CLEAR S (SEE FIG. 4-19) i There are no changes from clear 7 for Nodes 1, 3, 7, 9. The i

p situation for Node 5 is the same as for Node 3 in Clear 5, and Node 6 is the same as for Node 4 in Clear 5.

continuity equaticns  ;

, Node 1 .

A I 1 v u -u v 1 2 =2 A 1

, (C. 8- 1)

Node 3 A

u -u 2 s =7Al (v +v s) 1 (C. 8-2)

Node 5 A A v 1 v u -u p = Y 7av 2 A 3 (C. 8-3)

A 1

2 Av (1-2a)

U -u v = 0

= s (C. 8-4)

O C-59 i

l l'

l

, 3 i  !

i  !

SWECO 8101 i

4 i Node 6 1

i i 1

i i

y -v F

--ou a F

=0 j

r s

i (C. 8-5)  :

f i Node 7 i

Ay l 1

= 0

u. - 2 A -V s

(C. 8-6)

Node 9  :

A i u lvA s

=0 i

! s

! (C. 8-7)

Momentum equations .

I i

Node 1  ;

t j . p'-P2  !

+ g (x2 -x ) +

1 (x -x ) (u +u ) =u 2 1 1 2 1 2

-u 2

2 1 P t

9 (C. 8-8)  !

! I -

Node 3 '

, g(x _x ) , P. 0P

+ - 4

1 y

4 (x -x ) (u +u ) =u 2 3 2 2 3 2

-u 3

2 3 2 3

9 1

1 (C. 8- 9)

L Node'5 l

Y (xF -x ) 3 og

+ x -x S F $F + 21 (xy -x ) 3 pg u

3 ,

~

I

+ h (X,-x ) $,=-fuF p I" +u p) - (u,+uF) 3

- -  ;

O p p3 - p=

+du D 2

-u.2,96 J (x _x ) ,x _j pg F 3  % F pg  !

g 3 i

il i I (C.8-10)  !

i i

C-60 i.

l t

P i

l

!------.~-.-- . . . - . - . . ~ . _ _ - - , . - - . ~ , - - . - . ~ - . .=

SWECO 8101 >

\ Node 6 and control volume (see Fig

C-16)

The surface of separation between the two phases consists of horizontal and vertical planes. It is convenient to combine  ;

, the horizontal momentum of Node 6 with the control volume l 1 indicated in Fig. C-16.

j The horizontal momentum equation for Node 6 is the same as for Node 4 in Clear 5, with just a change of indices:

Y "YF h + 12 ay F I

-1 l+y v -1 y = 1-(Pg ) s pg F 7 p gj (vs +vF) (avF+0u) - av 2

[j_3 p h P

vu

-P vuc F 5 p gj p g

The vertical momentum equation for the control volume in Fig, C-9 is:

(x -x -

fD) y P

V =-1

\P/

I fv 2

-g(x -x -

D)y T

,PcP 3e u 0%

The turning losses are accounted for by:

" ~

i o p p

-v" =l-K 1 1-a 1 3)l v 2 P

t O g 32 p gj_ s The addition of the above equations combined with v4C bc I

l l

V C-61

.. . - . , , - , - - . - - , , , . , , , , , ., _ . , . , . . . . . , _ . , , , - . , . . . . . , , , , , ~ , . ~ _ , . - . - - - , - - - . , - , _ , _ . -

l SWECO 8101 yields: I ay F -l +Y v + I* ~* -

D) v

• : 5 (0 [ O \

+ h ay p -1) vp = 1- (v,+vF )("YF+ u)F

~

/ p 3 Ki /A vi +

l K

-v al l d llr1- 2il+3 41 l 7 3 s ( pgjg j gApj ,

P" D) -P' v +

-g(X -X -

2

= 3 og (C. 8- 11)

Displacement equations The displacement equations are the same as for clear 5 or

% Clear 6, hence:

5 -u p (C.8-12) 7

$'y

• VF (C. 8-13)

Bubble equations The bubble equations are unchanged from the two bubbles case for Clear 7, hence:

x -

i =u 11 E 119 1o (C. 8- 14)

. 1 P 2c-E c 2

i 4

u = u -u u + -g i1 x -

X 1o 1o 11 0 11g 119 L E (C. 8- 15) i-1i E " "21 (C.8-16) l I  ;

\

> C-62 h

l ww yW -v r e-e, e g -twww vv au p ' - my a-q-*-qvre-tqwgh yywymnwwwyWy-T- y g e wy y ve-M-eyy-P----g w --g w wmqye y g -- % W og y g gar 799yWMyM W 7w(wwy y yg- ,f9T-t-gMW W W V *"

i SWECO 8101 I $ = A P

Ay [ CT j 2 .

l pv -P (u -u )

11 P Drywell + 7 v2

! A g2 2c 11 1a

1

- P\ .

(C.8-17)

E p , Y-1 11 2c A X P 119 (c. 8-18) i x = u -u I'9 * '

, (C. 8- 19) ,

+

_

u =

1 u 2

-u u + Pc P2 cs

-g to X -X s s to p to 109 E "

(C.8-20) t i

' "A '

y [C T 1 h sc (u 1o -u s )

2 i E =A -

v 10 v g* p A

1 1o P p ( p Drywell + 2 6 (C. 8-21) p = Y-1 E IJL..

sc A :k.

P 109 (c. 8-22)

Recapitulation for Clear 8 Twenty-two equations - (C. 8- 1 through C. 8- 22) have been derived; the corresponding 22 unknowns are:

E

u,u,u,u,u,v,%,v,P,P,P,xtog, 1 2 3

• 9 s F 2 3 s u to ,

lo i

3C 11g' 11 E' 11' 11' 2c' Y' F 1

C-63 s

e., ,,+,..,.e--,~.,,,.y,,,,,*,,-y--g,e-.,.--,._,m3-v.-9,ge,+,w,-.,,,,p...,_,,+ g--.,-.gy,w.wg+,e,-y.,,%w,,,.-.,,,em. w,_., ..,

SWECO 8101 -

C.9 CLEAR 9 (SEE FIG. C- 18)

There are no chenges from Clear 8 for Nodes 1, 3, 6, 9.

Continuity __ equations Node 1 1 Av v u -u ---

i 2 =2A 1 (C. 9- 1 )

Node 3 A

1 v u -u = 2A -- -- (v +v )

s 1 3 (C. 9-2)

Node 5 There is only gas, which comes in at a rate Aus from Node 3, escapes to Nodes 4, 6, and 7 at rates 1/2 Av v 3, 1/2 Av vs, and Au., respectively, hence:

1 1 Au , = Au, + 7 Ay v, + 7 Ay v,

-^ or u -u -ffv s 3 (C. 9-3)

Node 7 Node 7 contains the two phases.

For the gas, the incoming volume flow rate is Aug, the volume occupied by the phase increases at the rate Aup and the outgoing volume flow rate is (Ag-1/2 A y)v s, hence:

Au , = AuF+ (A g -

'A) v Ys or 4 5 C-64

__ _ _ - . . , _ _ _ . ~ . ~ , _ . . . . _._.. _. _,,_ _ -._,. , , . _ _ _ _ _ - . _ . . _ , - . . . . _ _ _ _

_ _ _ _ . _ . _ . _ _ _ . _ . . . . . . _ . _ _ _ . _ _ _ _ _ . . - . . ~ .__. _ _ _ . _ . _ _ _ _ _ . _ . _ _ _ _ _ . . _ . - . . _ . _ _ _ _ . . _ _ . _ .

l l i

!  ;

4 SWECO 8101 i i l t .

1 or i A

' 1 v u -u

F 2 A (2a-1)v 5

=0  ;

(C. 9-4 )

i j For the liquid, the volume occupied by the phase decreases j at the rate Aup and the outgoing volume flow rate is Avls, i j hence:

2  :

Au F

• Al"s i or l

Au (A -A ) v, G

F" v g i or A

1 v u , 2(1-a) v =0 F 2 a s i

(C. 9-5) i j Node 6 j 1 i v -v F

--0u a F

=0 '

s (C. 9-6)

Node 9 A l-u s

2v A s

= 0 i E

i (C. 9-7)

I?? mentum equations l

Nodes 1, 3, 5 contain gas only, hence:

\ NMe 1

- - D -E 2

- (x -x ) (u +u ) =u 2-u 2 + g(x 2_x ) +

1 4 2 1 2 1 2

+t O

1 1 g

(C. 9-8) '

r Node 3 I

12 (x -x 2) (d 2+u ) =u 2-u * + g (x -x ) + D*-O

l 3 3 2 s s 2 0 i

1 (C. 9-9) -

l I

l l 1

l C-65 j' r i

{

l l

SWECO 8101 i

l Node 5 P 3 -P 6 e

(x -x ) (b +b 4)

=

=u 2

-u 4

2

+ g (x _x ) +

4 3 p 3 3 3 g

(C. 9- 10) t There is no change from Clear 8 for the combination Node 6 control volume of Fig. C-16:

(p ) A h ay p I -1 +y y

+ (x,-x,-fD) y h i

+ fayp I -1 Ih p

=fi1- i (v,+vy) (avF + # "F) i . -

/ p / K\ T2

-v a 1- 3-)I l- l '

l + "3_y, / j A,1 K 5

\ PL/ \ / (Ap/ 8 P u -P sc

-g(x -x 3

12 D) v + og (C. 9- 11) i Displacement and Pubtle equations There are no changes from Clear 8, hence:

F " "F (C. 9- 12)

YF"YF (C. 9- 13) x -x =u 11 1 119 1o (C. 9- 14) 1 2 9 2c ~P c 1 u = u -u u + -g 11 x -x to lo 11 p 11 1 119 E ,

(C. 9- 15) l C-66

i

]

i SWECO 8101 I x =u '

11 E 11

(C. 9- 16) 4

~

E

=A p "Av/ -l CT 1 2) p v p c(u 11 -u to )

11 A P Drywell + 7 v 2 l 92 2

- P\ ) -

t

= y_y E,, (C.9-17)

P 2c A x p 119

, (C. 9-18) x =u -u 109 to 9 (C.9-19)

• 1 2 3c P2c t1 =

X -X u -u u + _g 10 3 g to D 10 109 _

E ,

t (C. 9-20) i

'A y [

. 1

)

E =A CTp Drywell

• 7 V PV P r,c -U }

l, p s g, lo s (C. 9-21)

E

,y-1 p 1o

! 3C A X p 109 (C. 9-22)

Recapitulation Twenty-two equations (C. 9- 1 throuch C.9-22) have been derived; the corresponding unknowns are:

u , u , E ps c' u,, u , u ,

v,, uF ' FV '2 P'P'E'xu9, 3 6 u , ,

i i

119' 11 11 2C' 11 E' F' YF C.10 110MENCLATURE A Horizcntal cross-sectional area between outer surfaces of weir-wall and inner surface of drywell wall

! . - ~ _ _ . . . . . . . , _ . . - , , _ _ . - - - , . . . - . . . , - . . - . . . _ , . ~ , _ . , , - , , _ _ _ . _ ,,,,,___..._,m__,,__,,.m_,,, _ . , _ , _ _ . . _ . . , , , , . ,

i SWECO 8101

) Ag Vertical cross-sectional area for one vent occupied by gaseous phase (Appendix B)

Ay Vertical cross-sectional area for one vent occupied by liquid phase (Appendix B) i Ap Horizor.tal cross-sectional area between drywell and contair. ment walls Ay vertical cross-sectional area for one row of vents (For one vent in Appendix B)

Dy Vent diameter r

E Total energy contained in one bubble (Esa for top row, Ego for middle row)

I hg specific enthalpy of gas entering bubble Ka, F, 2 Es loss coefficients at entrance of t'ents p Pressures (see Fig. 4-2 for meaning of '.ndices)

R Vent radius (Appendix B)

/~N T Temperature (usually drywell temperature) u Vertical velocity (see Fig. 4-2 for meaning of indices) v Horizontal velocity (see Fig. 4-2 for meaning of indices) x Vertical distance (see Fig. 4-2 for meaning of indices) y Horizcetal distance (see Fig. 4-2 for meaning of indices) yy Length of vents or drywell wall thickness (see Fig. 4-2) n Ratio E Ay Y Ratio of specific heats for air (Cp/C y) i

]

Pg Gas density P

l Liquid density

[

'(_,)h C-68

.._____x .

i <

1 1

I i I

i SWECO 8101 1

i 4

l References - Appendix C 1 i l i I

i j 1. Liepman, H. W. and Rohko, A. Elements of Gasdynamics,

John Wiley S Sons, Inc., New York, NY, 1963, p 181-183. ,

. i Rohsenow, W.

j 2. and Choi, H. Heat, Mass, and Momentum ,

i Transfer, P.: entice-Hall, Inc., Englewood Cliffs, NJ, I l 1961, p 67. l

3. Hatsopoulos, G. and Keenan, J. Principles of General  !

Thermodynamics, John Wiley & Sons, Inc. , New York, NY, j 1965, p 111. 18.24 I '

1 l '

I f 4

i

! I i

h l

{

l l

I t

I

. i C-69

- , - , - . . . . . . , _ . . - . - . . . - - . . - - , . . . . - - - . . - - . . . . - - ~ . - , - . . - . . . - . . - - -

GAS GAS p

• I x1 n

D" v y,X 2_ , _

LIQUID y

- - . - - . - . - . - _ . . - . - - . _ - . - . 3.*. . -

X4

- . - - . _ . - . _ . - . - . _ . b . - ..

FIGURE C- 1 b) b/

SURFACE OF SEPARATION BETWEEN PHASES FOR CLEAR I xj < xp<x2 ~

OV

%./

GAS, CONTAINMENT 8R ESSU3E [_p, r----------- , _ _ _ _ _ _ _ _ _

8 ' i u I u l

hl LIQUID I

i A #

\

u2c 2 NODE 2

1. v2c P v2

\

- h\ \

NOTE:

--- BOUNDARY OF THE CONTROL VOLUME l

t FIGURE C-2 t

CONTROL VOLUME USED IN THE DERIVATION OF EQUATIONS BEFORE TOP VENT CLEARS O

U l

GAS I  !

/ t l  :

1. 11 1

'- LIQUID x i 2 l

j ,- -___

L.- _._.___.,-__. _.

"uiO j'/ /

j l

lv I

i U

se 6-_----._________.-____-._;

l e

i

[/NODE 4 y

X 3

/ P 3e P,3 p U

ry

/ <

I NOTE:

a

--- BOUNDARY OF THE CONTROL VOLUME FIGURE C-3 CONTROL VOLUME USED IN THE DERIVATION OF

/' EQUATIONS BEFORE TOP VENT CLEARS V)

GAS r_______.________________,

I l i i l i I I i n l l l 1

LIQUID x i jf i x2 1 jf-I i p______. _______

I l V,/

l I

l

"' c d

I lA / X L_3

_. ._g l _. . __ . _ . __ . _ . __ . _. . __ . _. . _ ]

l i us i r i l

(~'\ '

l V  !

I I

I l

" l /

l U 4c l i i X4

t___________,____________J , ,_, _, , _ , j t l

. A NOTE:

--- BOUNDARY OF THE CONTROL VOLUME FIGURE C-4 rm CONTROL VOLUME USED IN THE DERIVATION OF l!s ) EQUATIONS BEFORE TOP VENT CLEARS 1

l l

a l GAS x7 D'" J' **

_. J L._

o LIQUID [

X 3

_._. _.__.__._. _._.__. J!._._

o E E

4 J f._.._

/

/

FIGURE C-5 O I SURFACE OF SEPARATION BETWEEN PHASES FOR CLEAR 2 (J x2 7;Dy < xF < X2

GAS GAS LICUID xijg

/YyQ

/

l12

. . _ . -._. . . .E . - #f 0'AI '3 Jl.._._.

t o M _ . _ . _ . _ _ _ . . . _ . .

ix 4 L ._

/

FIGURE C-6

,r\

( SURFACE OF SEPARATION BETWEEN PHASES FOR CLEAR 3 x2< *F # *2 + Ov

l b

LIQUID xut du

• 2

.3 _ _

33

- _ _._ _.__.._._._._ J L. ._

_ . _ . _ . _ . . _ . . _ . _ . _ . _ _ . _ JL .-

FIGURE C-7 SURFACE OF SEPARATION 'BETWEEN PHASES (V) FOR CLEAR 4 BEFORE VENT CLEARING

gDRYWELL WALL

(] GAS

/ WEIR WALL LIQUID 3 11 2 n X xii, GAS D, j[,{,

h o n'

8 3

_ . _ . _ . _ . _ . _ . _ . _ . _ ._ J f._ . -

LIQUID X4

. _ . _ . _ . _ .. _ . _ . _ . _ ._ JI._._

i l

FIGURE C-8 SURFACE OF SEPARATION BETWEEN PHASES

C'$ FOR CLE AR 4 AFTER VENT CLEARING l x2+h Dy < xF < *3 -

Dy i

o y~

CONTAINMENT SIDE I' DRYWELL SIDE h

GAS D,

-.-.-..-.-.-._.-.-.--.-.-._..-.-.-..-.2 BUBBLE PRESSURE:p2e p

---

q l

LIQUID l

" l /

u 3e l

I i

/4 NODE 4 i

l

• 3

= - -- . - . . _ . JL . _. .-

.w. _ . _ . g-.

?

f n/ s/

VM M/A NOTESi

--- BOUNDARY OF CONTROL VOLUME

- SURFACE OF SEPARATION BETWEEN PHASES

^

FIGURE C-9 COMBINATION CONTAINMENT VENT

/7 CONTROL VOLUMES USED TO DERIVE (d SOME EQUATIONS IN CLE AR 4

LIQUID GAS 8

2

.NG_._._.._._._._..._ dl.

LIQUID 1

3

_. . _ . _ . . _ . . . . _ . . .__. . _ . . _ . _ . _ . _ . _ J E.._

l ug d i *4 u4e NODE 6 i

._.J._._.- _N. _

P4 P4e pv4c p,,

/ I NOTE:

--- BOUNDARY OF CONTROL VOLUME FIGURE C-10 o COMBINATION CONTAINMENT VENT i 1 CONTROL VOLUMES USED TO DERIVE

\M SOME EQUATIONS IN CLEAR 4

l l

1 l

l O CONTAINMENT SIDE DRYWELL SIDE Pc I I

GAS l 1

I n

l I

/ ,

LIQUID i e

i

• III *11g Pc2 / x2 GAS  !

l 9 v2 "_J LIQUID ujo V/

NOTE:

---CONTROL VOLUME FOR TOP BUBBLE (SAME AS NODE 11)

FIGURE C-11 CONTROL VOLUME AND GEOMETRY FOR THE TOP BUBBLE EQUATIONS (3

L QU D GAS x2 h uio LIQUID /

xy JL x 3

_ _ . _ _ . _ _ . _ . _ . _ . _ . _ . _ _ . _ .JL o E x4

3. _

FIGURE C-12 (3 SURFACE OF SEPARATION BETWEEN PH ASES FOR CLEAR 5 x3 -h Dy <x<x p 3

i GAS -

NJ g Alli LIQUID a

_____._.__._._._ JE LIQUID / y, . /

/ I db._ XF If o

-. ..=._._ _._._._.

E X4 J b. _

P FIGURE C-13 SURFACE OF SEPARATION BETWEEN' PHASES FOR CLEAR 6

( '

x 3<xp<x3+ Dy

() /

GAS LIQUID a

GAS x119 x2

_.__._ .__. _ . _ . _ _ . _ JL.

LIQUID 7/ E m

7#MA X3

_ . _ . _ . . _ . _ . _ _ . _ _ . _ _ . _ . . _ J '. _

) U X4

-._-.--. -._.. _ .--.--.- 3. -

/

ll FIGURE C-14

~ h)

Cl SURFACE OF SEPARATION BETWEEN PHASES FOR CLEAR 7 BEFORE CLEARING OF THE SECOND ROW OF VENTS x3 + f Dy <p x <4 x -fD y

(m .

• ^

N.j LIQUID h

x gij X)j g GAS p 2e / /

1 [2 LIQUID

^S

% *io. 1.'_

j L___.___

n

_ _ __. - _ J l_._._._._._._.

X4

-.-._._.-._._._.- b. _

LIQUID

- #####/ ,

NOTE:

---CONTROL VOLUME FOR MIDDLE BUBBLE EQUATION FIGURE C-15 h, SURFACE OF SEPARATION BETWEEN PHASES FOR CLEAR 7 N/ AFTER CLEARING OF THE SECOND ROW OF VENTS x

3+hD<x<X y F 4 Dy

D 4

'\

A D

-- ic.,_ _ __ __ _ _ _, __

, __, _ _ _, _ _j l /4 NODE 6 P Z.v4 __. .

J L_4_

4c E,4e ,

4 s

/

NOTE:

j ---BOUNDARY OF THE CONTROL VOLUME FIGURE C-16 COMBINATION CONTAINMENT VENT CONTROL VOLUMES USED TO DERIVE x SOME EQUATIONS IN CLEAR 7

O /

U 3 AS LIQUID xyg x ng GAS x2

. . _ _ . _ _ . _ . _ . _ . _ . _ . _ . _ . J t._

LIQUID n

• 109 GAS x3 9

r

{N L GAS

?/// *F J L. y*4 LIQUID /

FIGURE C-17 i

) SURFACE OF SEPARATION BETWEEN PHASES FOR CLEAR 8

-fD x <

4 y xF < x 4

GAS

)

b LIQUID d

xiu x

He GAS 12 J '..

LIQUID c Vff,) '

• 309 _ _ _ . _ . _ . _ . _ _ . _ _ . _ . _ . _ . J L. _

'v' GAS l///

K4

_._._.__.__ . _ . _ . _ . _ . _ . _ J!._ u F

FIGURE C-18

() SURFACE OF SEPARATION BETWEEN PHASES FOR CLEAR 9 x4<xF # *4 + Dy

SWECO 8101 5 APPENDIX D DERIVATION OF THE HOMOGENEOUS VENT FLOW MODEL (HVFM)

This appendix presents the derivation of the equations for the Homogeneous Vent Flow Model (HVFM) presented in Section 5.3. Table D-1 defines the nomenclature used below.

D.1 APPROXIMATION OF THE MIXTURE SPECIFIC VOLUME The mixture quality is defined as W "a + "s R

• • W +u f "W, +W , +w f (D.1-1)

The specific volume of the mixture is defined as v-xv + (1 - x) v f 8 (D.1-2)

Equation D.1-2 may be rearranged as h-x+(1-x) g g (D.1-3)

As shown in Reference 1, the quality of the mixture in the drywell following a LOCA may decrease to less than 0.4 but, at typical drywell pressures (less than 50 psia) , the corresponding average void fraction will be in excess of 0.99. Under these conditions vg >> vf and Equation D.1-3 can be approximated by v = xvg (D.1-4)

The error introduced by this approximation is small(2).

D.2 EQUATION OF STATE The ideal gas law is Pvg = RT (D. 2- 1)

> r~N

( D-1 1

~ . . - , = - . . . . . . ,,

i l 1

SWECO 8101 O '

This equation can be modified for the two phase mixture by substituting Equation D.1-4 to get Equation D.2-1 in terms of the mixture specific volume. This yields Pv = xhT (D. 2-2)

If we define R' = xR (D. ?-3) we get Pv = R'T (D. 2-4)

Substituting 1/p for v, and logarithmically differentiating yields

, dP do dT (D. 2-5)

Y- p "7 D.3 SONIC VELOCITY The sonic velocity in a stationary homogeneous-frozen two-phase mixture, neglecting the compressibility of the liquid, is given by Reference 2 as:

\ ]

2 TP 1 T" p a

g ,2 , , gy _ ,) D

\g/ (D. 3- 1)

Where the void fraction, a, of a homogeneous two-phase mixture is defined as:

Xp g

" " (1 - x) p g+ *P f (D. 3-2) l

[ Equation D.3-1 may be rearranged as l

l 2

a E--a 2I Pf i /#f i 2 11 p l+ a1 l l

a p g g/ ( g/ (D. 3-3)

\

l '

l s D-2 i

i l

l_ . _ _ , ._ . . _ _ . _ . _ - . . _ . _ _ . . . _ .,. _ .. __. _ _ _ . . _ _

SWECO 8101 s Substituting Equation D.3-2 into Equation D.3-3 and rearranging yields 2

a p

fa Z (1 - X) l "TP

+2x(1-x)\ # f' +x

\ # ft (D. 3-4)

As shown in Section D.1, vg >> vf. Therefore p g << o and f

Equation D.3-4 can be approximated by:

2 a 1 2 "x

• TP (D. 3-5)

Rearranging 2 2 a

TP "

• 83 (D. 3-6)

It should be noted that experimentally Equation D.3-6 is not correct, i.e., the mixture sonic velocity is actually O approximated more closely by the gas phase sonic velocity (a a wiI$ =thS) emainder r.

However, Equation D.3-6 is internally consistent of the HVFM model, and provides a better correlation to the experimental data for mass velocity (Section 5. 3) than the analogous homogenecus-frozen flow model assuming the a TP * "g(3). Since:

2 a

s"Y I (D. 3-7)

Substituting Equation D.2-3 yields 2

a = Y R'T (D. 3-8)

TP D.4 MACH NUMBER For a homogeneous mixture the mixture velocity u=ug = uf (D.4-1)

D-3

1 SWECO 8101

- The Mach number is defined as ,

u i M= l l "TP (D. 4-2) l Substituting Equation D.3-8 into Equation D.4-2 yields 2

2 "

u M Y R'T i

1 (D. 4-3)

J Logarithmic differentiation of equation (D. 4-3) yields l 2

dM du dT  !

T " uT T- l

M (D. 4-4) l D.5 CONTINUITY EQUATION The continuity equation for steady flow through a constant area vent is G = Pu = constant (D. 5-1)

Differentiating this equation yields

< du do 7fp=0 (D. 5-2)

Noting that 2

. du 1 du '

7"7 u 2

! (D. 5-3) i Equation D.5-2 can be rewritten as hd"2+ =0 u (D. 5-4)

D-4 1

, ,, - e n .n--r., . , - - - , e.4 . we ,,-..,4,,,.w., ,-.y.,e .--.y . , . , - , ,.-..,w,#,-,,,

.se_ ae , , , ,%.w e ,y swe,e,,- , ...,e ,,-..we,,..,m-s,,,,,,,a,...e, ,,e-.m..,,e,w,,_s

SWECO 8101 n,

( /

(/ D.6 MOMENTUM EQUATION For the assumptions outlined in Section 5.2 the momentum equation can be written as pudu + dP + =0 D

h (D. 6- 1)

Cumbining Equations D.2-4 and D.4-3 yields 2

2 "

M =

Y Fv (D. 6-2)

Rearranging and substituting 1/p for v yields P = #"

YM (D. 6-3)

Now, dividing Equation D.6-1 by P f rom Fquation D.6-3, and using Equation D.5-3 yields YM2 du 2 dP_ YM2 fdz

=0 2 2 P 2 D u h (D. 6-4)

D.7 ENEEGY EQUATION Given the assumptions of no external shaft work, no

gravitational effects, and no heat transfer at the walls, I the steady flow energy equation is

! dh + 2

-0 (o,7_1) l The derivative of the enthalpy, dh, can be written as l

dh = xdhg + (1 - x) dhg l For an ideal gas dhg=CPE dT g (D. 7-3) l I

/

l l

SWECO 8101

/

/3 4 For the liquid phase enthalpy the total derivative can be written as a function of temperature and specific volume, thus f

[6 h g j dh dT dv f f

=(6h\ T f+k T f

(D 7-4)

Since in this model the liquid is assumed to be incompressible and adiabatic and therefore isothermal, the change in enthalpy of the liquid is dhf=0 (D. 7-5)

Since the flow is homogeneous Equation D.4-1 holds, and Equation D.7-1 can be rewritten combining it with Equations D.7-2 and D.7-5 as xC dT Pg g

+d"2 =0 (D. 7-6) n Dropping the subscript g and dividing through by x Cp T

( )I yields k.a f+2xCT du P

=0 (D.7-7)

Noting that for an ideal gas YR C =

p Y-1 (D. 7-8 )

Now, Equations U.2-3, D.4-3, C.7-2, and D.7-8 can be j combined to yield f+Y-1 g 2 u

=0 l (D. 7-9) l l

1 s , D-6

1 SWECO 8101

')

D.8 REARRANGING THE EQUATIONS IN TERMS OF MACH NUMBER i

We now have five simultaneous linear algebraic equations in terms which relate six differential variables: dP/P, d O/0 ,

i dT/T, dM2/M2, dur /u r, and fdz/Dh dP do dT T T " T- (D. 2-5) dM du dT M u (D. 4- 4 )

i

{ d"u

+h=0 4 (D. 5-4) 4 YM2 du 2 dP YM fdz T ~~~I u

+ V + T o** h (D. 6-4) dT 4 Y-1 2 du T

3 =0

%) 2 u (D. 7-9)

.3 These equations will now be rearranged making dMr/M2 the i

independent variable since this yields the most convenient

! form for the solution of the vent flow problem. Combining Equations D.2-5 and D.7-9 to eliminate dT/T yields 1

f-+T~ M

=0 (D. 8- 1)

Combining Equation D.8-1 with Equation D.5-4 yields 2

[=-f1+(Y-1)M -

u (D. 8- 2) or

~

du 2 "1 -

Z" u -

2 1 + (Y - 1) M (D. 8-3)

.g D-7 4

1 1

J s -yu _ . ,c-w ,..m,- ry._-,,,,,,,,._,_,-_p,...

-r--..c.,9,---, .,..,y. ,.ra www=e-- %e---vr*- v+ +- ~ ~ + - -e e- - re= =-+*r-- +- ' v e -- v - - -

SWECO 8101 l

(/ Eliminating dT/T from Equations D.4-4 and D.7-9 yields 2 2 dM , du Y-1 2-(D.8-4)

Combining Equations D.8-3 and D.8-4 yields g,_ ' l + (Y - 1) M dM 2 + (Y - 1) M M (D. 8-5)

Substituting Equation D.8-3 into Equation D.6-4 yields f,_YM [1 + (Y - 1) M ] fdz D

2 [1-M ) h (D. 8-6)

Eliminating dP/P from Equations D.8-5 and D.8-6 fdz (1 - M ) dM 9

~

,Y-1 g2g2 D 2 2

(D. 8-7)

The equations for du/u and dp/p can similarly be rearranged in terms of dM2/M2, D.9 EQUATION 5.3-1 Equation D.8-7 can now be integrated from the duct inlet (1) to the duct exit (2) 2 2 2 j fdz D

, (1 3 )

. dM h YM' _1+Y} M

~ ~

(D.9-1)

The right hand side integral is of the form

/ dy y2 (1 + by)

(D. 9-2)

D-8

SWECO 8101 O Equation D.9-2 can be solved by rewriting it as f 2 1~Y dy - f 2 dY

-f dY

'. y (1 + by) y (1 + by) y (1 + by)

! (D.9-3)

Performing the integration results in l

. . 2 .

2 2 . M2 (1 + Y - 1 g)

-1 Y-1 (1 + Y - 1 3 23 M2 fL , 4 in l + fn l l D 2 2y ( 2 / 2 ( 3 2

/ 2 3

.M t . .Mt l (D. 9- 4)

I Evaluating and rearranging yields K - fL -

y 1

M -M

+ [ y + 1l in

\ 2y /

h+2 M2 M D 2 2 .y-1 2 2 M1 M2. _

2 M1 M2 (5. 3- 1)

D.10 EQUATION 5.3-2 Equation D.8-5 can now be integrated from the duct inlet (1) i to the duct exit (2)

~

2 dP 2 1 + (y _ 1) 3 2 " dM2 if y- - il 2 + (y - 1) M . M 2 2 u (D.10-1) l Integrating the lef t hand side yields if f - in l .~.

(D.10-2) l

\ D-9

SWECO 8101 The right hand side is of the form 1 + ay

!y(1-by) I (D.10-3)

Equati.on D.10-3 can be solved by rewriting it as 1 + ay dy ady

!y(1-by)dy-f y(1-by)+!(1-by)

(D.10-4)

Performing the integration results in 2 2 "M2 M2 2 + (y - 1) M in I=

- f in q

+ in[2+(y-1)M (D.10-5) i m.

j Evaluating and rearranging yields

~

~

2- 2~h P 2 + (y - 1) Mi Mi in(2 1

= in

~ 2 + (y - 1) M2 .

2 2 M2

.. (D.10-6) or 2

2~h P

2 .

2 + (T ~ 1) M1 M1 T' -

2 2

1 2 + (y - 1) g g (D.10-7) ,

This relates the pressures of the inlet and exit of the vent to the inlet and exit Mach numbers, but we wish to express the relationship between the upstream and downstream reservoirs.

(N / D-10

i SWECO 8101 I

y To do this we must account for the isentropic entrance acceleration. From the energy equation (D. 7-6) , direct integration between the reservoir (0) and the inlet (1) yields 1 2 ul - u01 xCp (T01 ~

1) " 2 -

(D.10-8)

For the "statica reservoir (aquasi-steady state" assumption) uas =0 (D.10-9)

Therefore 2

u; P ( 01 - 1) " ~ 2 (D.10-10)

Dividing Equation D.10-10 by xc pT, and substituting Equation D.7-8 yields 2

01 Y-1 "1 v [( E i ~ *)/ "

2 y x RT 1

(D.10-11)

Applying the definition of Mach number from Equation D.4-3 yields T~

-1 - M (D.10-12)

For an isentropic expansion T P Y-1 2 2 y T"T1 1

, (D.10-13) l

( D-11 1

e + = , ' - i.-+,vsa+ - ee- - r-%,ws,- - y-- ,.u,,y,yy,-..,-.ar gr.,p--*w--,-,y-,,,e-..,,.,yc,,,,,,-,-c-,,,,,-,-,,.,--,,-w..-,,,-+,--,--.-,-gmy- . .- ,n- , , . y.--=--, + ,*

SWECO 8101 N

1

,/ Applying Equation D.10-13 from the reservoir (0) to the inlet (1) and substituting into Equation D.10-12 yields p ?_

01 p - (1 + Y j 1 M!)_Y~l 1

(D.10-14)

For conservatism the exit pressure P 2 is taken as the stagnation pressure of the exit reservoir.

Pg = Poa (D.10- 15)

Therefore, combining Equations D.10-7, D.10-14, and D.10-15 yields the desired equation:

2 2" b P

02 1

( + (Y - 1) Mg) M g P Y 2 2 1,Y-1 g Y-1 (2 + (y - 1) M 2) M2 (5. 3-2) 2 - -

D.11 EQUATION 5.3-3 From the continuity equation (D.5-1) and the definition of N> Mach number (D. 4- 3) 2' b c=pu gt =o g ,Y R T M g.

(D.11-1) substituting the ideal gas law ( D . 2- 4 ) and 1/p for v yields "YP 1 2'h c-p Mt 1 #

-1 -

(D.11-2)

For an isenthalpic expansion Y Y V

1 P 01 V01" 1 I (D.11-3) or (P3 \ og i -

\ P01/ 01 (D.11-4) sj D-12

SWECO 8101 q, This can te rewritten as Y-1 b , 01 p y

! og p ol (D.11-5)

Combining Equations D.11-2, D.11-4, and D.11-5 yields

~ ~

1 h

1 Y y-1

/P g y P ol y 2 P # 1 01/ 01 (D.11-6) 4 of Pg) /P g )Y~ 2

~

(P 01 / 01 /

(D.11-7)

O

(Q Now substituting Equation D.10-14 into D.11-7 yields 1 .

G = (1 + Y [ M)I~Y y0 01#01(1+ M) M (D.11-8) l l This can be rearrancad as y+1 1/2 1+Y-2 1 2 2 (Y - 1)

G=M g[yp01P01 1 (D.11-9)

Introducing units where pot is in Ibm /ft3, Pog is in psia,-

and G is in Ibm /ft2-sec, converts Equation D.11-9 into i Equation 5.3-3:

. Y+1 1/2 +Y-2 1 2 2 (Y - 1)

G -12 M g(Y0 MI 01 01 8e ' '

(5. 3-3)

N s,

D-13 l

i i

SWECO 8101 References - Appendix D

1. The General Electric Mark III Pressure Suppression containment System Analytical Model, NEDO-20533, 1974.

2. Henry, R. E.; Grolmes, M. A.; and Fauske, H. E.

Pressure Pulse Propagation in Two-Phase, One- and Two-Component Mixtures. Argonne National Laboratory Report ANL-7792, 1971.

3. Boyle, J. C. Homogeneous, Two-Phase Compressible Flow Through a Constant Area vent. Masters Thesis in Nuclear Engineering, University of Rhode Island, 1978.

/l (v>

I l

/%

x_ D-14 l

l l

l

SWECO 8101 9 TABLE D-1 NOMENCLATURE FOR APPENDIX D Symbol Definition a Sonic velocity (f t/se c)

Cp Specific heat at constant pressure (Btu /lbm OF)

Cy Specific heat at constant volume (Btu /lbn OF) dz Incremental length of vent (ft)

Dh Hydraulic diameter (ft) = 4 times vent area:

divided by the wetted perimeter f Darcy friction factor (dimensionless)

G Mass velocity G = u (lbm/sec-ftz) gc 32.1739 (lbm-f t/lbf -s ec2)

^

( '] h Specific enthalpy (Btu /lbm)

(*~" )

L Length of the vent (ft)

M Mach number, M = u/a P Pressure (psia)

! R Gas ccristant (f t-lbf/lbm OR)

R' Modified two-phase gas constant, R' = xR 1

T Temperature (OR) u Stream velocity (ft/b ec) l v Specific volume of the mixture (unless subscripted) (f t3/1hn)

W Mass (lbm) x Mixture quality (dinensionless)

, ^3

'v' 1 of 2

SWECO 8101  !

s TABLE D-1 (Cont) ,

I Greek Symbol Definition j cx Void fraction pimensionless) ,

L Y Specific heat ratio, =Cp/Cy (dimensionless) (

P Density (lbm/f t3) i f

Subscript [

t 0 Stagnation property ,

t 1 Inlet property  !

2 Exit property a Air t

j f Liquid phase

! g Gaseous phase l s Steam i {% l TP- Two-phase '

t k

,  ;

i l

I I

I i

y 2 of 2 p

. _ . . . , . . . . . _ . _ . - - - . . _ . . . . _ . _ . . _ . . , ~ , . _ . _ _ . . . . . . . . . . _ - . . , . _ . . . . _ _ _ , . . _ _ _ _ . ~ _ - _ . - _ . . . . . . , . . - . _ . _ , . , _ . .