ZOCO V, a computer code for the calculation of time- and space-dependent pressure distributions in reactor containments

NUCLEAR ENGINEERING AND DESIGN 23 (1972) 239-272. NORTH-HOLLAND PUBLISHING COMPANY

Z O C O V, A C O M P U T E R

CODE FOR THE CALCULATION

TIME- AND SPACE-DEPENDENT IN REACTOR

PRESSURE

OF

DISTRIBUTIONS

CONTAINMENTS

D BROSCHE Laboratortum fur Reaktorregelung und A nlagens~cherung Garchmg, Reaktorstation, Lehrstuhl f~r Reaktordynamtk und Reaktorslcherhelt, Techmsche Umversttat M~nchen, 8046 Garchmg, Germany

Received 10 August 1972 A new mathematical model and the resulting computer code ZOCO V are described, which allows to calculate the tame- and space-dependent pressure and temperature dlstXibution m containments of water-cooled nuclear power reactors following a loss-of-coolant acodent, which is caused by the rupture of a mare rectrculatlon or steam pipe (MCA) With this computer code, subdiwded full pressure contmnments as well as pressure suppression systems can be treated primarily for the short term behavlour (within the first seconds from the beginning of the accadent until the end of the blow-down when pressure equthbrmm between primary system and containment is reached) and also for the following long term behavlour This model is a multi-pressure node one whereby the number of the pressure nodes, that of the connections between them and that of the heat conducting sohd structures can be arbitrarily chosen Calculated pressure time bastones have been compared with corresponding expertmental results for full pressure containments and pressure suppression systems showing good agreement between theory and experiment.

1. Introduction

1.1. A bout the zmportance o f transient pressure distributions m power reactor containments The containment which encloses the primary system of a nuclear power plant is the third and last barrier, - after the fuel canning and the prtmary system itself - , safeguarding against the release o f radioactive fission products from the nuclear fuel to the surroundings. The outer wall of the containment therefore should not be destroyed by local pressure maxtma which are caused b y the coolant released after a rupture at the primary system. On the other hand, internal structures of a containment should not be destroyed by local pressure differences to avoid the endangenng o f the function o f Important safety systems such as emergency cooling and energy supply systems. The exact knowledge o f the time and space dependent pressure dlstrlbutlon in a containment which is subdivided into several rooms or compartments is therefore very tmportant with respect to reactor safety To calculate tlus tune and space dependent pressure dlstnbutaon a new mathematical model and the resulting digital computer code ZOCO V [ 1, 39] have been developed. Prlmardy tlus computer code calculates the important short term pressure behavlour which takes place wltlun seconds from the beginning o f the rupture at the primary system until pressure equilibrium between the primary system and the containment In addition the long term pressure behavlour which follows and which takes place within minutes and hours can also be treated with this code. The computer code ZOCO V is formulated In such a manner that it can be used for full pressure containments as well as for pressure suppression systems

1.2. Types o f reactor containments As a consequence of its safety function the reactor containment must be designed for the maximum local pres-

240

D Brosche, 4 computer code jor pressure dtstrtbuttons

sures and pressure differences which are caused by the blow-down oI the prlmar3 coolant -lo minimize the equlhbrlum pressure after the termination of the short term pressure behavlour principally two posslbdltles e\lsl Firstly the free volume of the containment can be chosen large enough and secondly the equthbrmm pressure ~an be partly reduced by steam condensation Containments which use the first alternatwe are called full pressme containments and those based on the second principle are called pressure suppressmn systems l 2 1 Fullpressure e o n t a m t n e n t A typical example of a full pressure contmnment Is shown schentatlcally 11"1 fig I whele Its subdlvlsmn into a large number of different connected rooms can be seen Full pressure containments are often used for pressurized water reactors and rarely lot bolhng water reactor, in different variants 1 2 2 Pressure suppresston w s t e m The pressure suppression reactor containment s) stem is often used lot boding water reactors and is shown schematically in fig 2 A pressure suppression system basically consists of a drywell which contains the pressure

Fig 1 Full pressure containment, scheme 1 Reactor pressure vessel, 2 Steam generator, 3 Primary clrcmt, 4 Steam pipe, secundary ctrcult, 5 Compartment, 6 Flap, 7 Containment outer wall, X Possible pipe rupture locations

D. Brosche, A computer code for pressure dtstrlbunons

241

@

o

Fig, 2 Pressure suppression system, scheme, 1 Reactor pressure vessel, 2 Steam generator, 3 Primary ctrcmt pump, 4 Primary ctrcmt, 5 Secundary ctrcmt, 6 Prtmary steam line, 7 Drywell,8 Vent pipe, 9 Wetwell, 10 Condensationpool, ×posstble pipe rupture locattons vessel and which is normally subdivided into many rooms m a strmlar manner as a full pressure containment (this subdivision is not shown in fig 2) and the wetwell with a condensation water pool. These rooms are m turn connected by a vent pipe system. After a primary system rupture, the steam which is released m the drywell is condensed m the pool water (suppression of the steam partial pressure) which leads to a reduction of the total pressure so that the free volume of a pressure suppression system can be smaller than that of an appropriate full pressure containment A new variant of a pressure suppression system is the Ice containment which uses ice surfaces for steam condensat~on. 1.3. Calculational methods and computer codes Calculatlonal methods and computer codes pubhshed m hterature which deal with the calculation of the pressure budd up m a reactor containment following a prtmary clrcmt rupture have been developed separately for full pressure containments and pressure suppression systems. 1.3.1 Full pressure containment First it can be stated that most of the calculatlonal methods and computer codes pubhshed m hterature to calculate the pressure build up m full pressure containments neglect the contamment's subdivision into different rooms [2-10]. Heat conduction processes m sohd structures are treated m refs. [5-10] so that the calculation of the long term pressure behavlour is possible. All these calculatlonal methods and computer codes therefore allow only to calculate the maxunum equlhbrlum pressure after termination of the short term pressure behavlour or to calculate the pressure tune history of this equlhbrmm pressure following a prtmary c~rcult rupture. The tune- and space-dependent maxima of pressures and pressure differences which are to be found m a subdivided containment and which are mostly much larger than this equthbrmm pressure cannot be calculated. Beyond that the calculated pressure tune history of the equthbrmm pressure IS mostly treated as a quasi-stationary process whereas m reality the tune dependent variations of masses, temperatures and pressures are non-staUonary. To overcome the above mentioned deficiencies the digital computer code ZOCO I (ZOnen COntamment) was developed [11, 12]. This code provides for calculation of the space and tune dependent mass, temperature and pressure behavlour m a subdivided full pressure containment. In tlus code the containment Is subdivided into a number of pressure nodes (which sunulate the different rooms) for which the differential and algebraic equations

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D Bro~che, A computer code for pressure distributions

lor the calculation ot the masses of steain, water and air, the temperature and tlae plessure are tormulated with the aid of the basic assumption that these three media are in thermodynamic equlhbrlum The mass flow rates between the connected pressure nodes are calculated as the flow ol a gas through an Ol lrice with experimentally determined discharge coefficients With this computer code a large number of paralnemc studies were pertormed using dltterent numbers ot pressure nodes and connections between them as can be seen m l e f s ~l 1,12] This computer code has been enlarged and improved by the author (ZOCO 11 [13]) For example in the C~)ll~puter code ZOCO I1 the number of the pressure nodes as well as that of the connections between them ,an be aibitrarlly chosen Also a more sophisticated mathematical treatment of variable flow areas controlled by press/ne differences has been included After the publlcatmn of the computer code ZOCO 1 several models slmllal to ZOCO I and ZOCO II have been developed and published [14, l 5, 1b] which all treat the calculation of pressure time histories nl full pressure ccmtamments during the short term pressure behavlour 1 3 2 Pressure suppressto~t © ' s t e m

A first attempt to calculate the time dependent pressure behavlour in the drywell and wetwell ot a pressure suppression system was published In ref [17] Using the computer code ZOCO 1 as a basis a code for pressure suppression systems was developed by the author in three different variants [18], which allowed the subdivision of the drywell into three pressure nodes and the consideration of one to three vent pipes A general survey concerning the design problems of pressure suppression systems and a shol t description of a computer code can be seen in ref [19] In another computer code [20] the pressures in the drywell and wetwell are calculated using an ltelatlve solution method In summary of this section it can be stated that the importance of transient pressure differences in containments (shown in section 1 1), the complicated structural layout of containments construction (shown m section 1 2), and the relatively sunple computer codes and cal~ulational methods (shown in section I 3)which m many cases are inadequate, particularly for the short-term behavlour, demonstrated need for a more sophisticated computer code which provides for the calculation of time and space dependent pressure distributions in reactor containments For this reason the new mathematical model ZOCO V has been developed

2 Mathematical basis of the computer code ZOCO V The flow and thermodynamic processes which occur in a reactor containment following a primary clrcmt rupture are three dimensional and non-stationary and are in addition colnphcated with geometrical conditions Tlu~ space and time dependent dlstrlbutlou of all variables and therefore also of the pressure ill the containment is simulated m the computer code in a manner strmlar to that m ref [13] where the containment is subdivided into an arbitrarily determined number of pressure nodes for which the time dependent changes of all variables (masses, temperatures and pressures) are calculated (multi-pressure node model) Normally a single room m such a subdivided containment is treated as one pressure node In all cases the number of the rooms m a containment is so large that consideration of all rooms m the mathematical model would consume too much computer time so that it is suitable in such cases to simulate several rooms by one pressure node This is especxaUy the case if such rooms are located far away from the room in which the primary circuit rupture takes place On the other hand rooms in the mmaediate area of the primary system rupture can be subdivided into several pressure nodes for a more detailed analysis In this case the boundaries of the pressure nodes are defined by the flow resistances in this room Such a case is presented m section 3 1 The flow processes which occur between the connected pressure nodes are treated as one dimensional flows

D. Brosche, A computer code for pressure d~strtbutugns

243

If one s~mulates the tune and space dependent pressure d~strlbutlon m a containment w~th the aid of such a multi-pressure node model, the mathematical treatment of the pressure nodes as well as that of the mass flows and heat fluxes between them is of crucml tmportance The most important features of the ZOCO V computer code are the following. a) The computer code ZOCO V, m contrast to the hitherto pubhshed codes, is able to snnulate the ttme and space dependent pressure distributions m full pressure containments as well as pressure suppression systems b) The number of the pressure nodes and the connections between them can (as m ref. [13]) be arbltrardy chosen This feature is ltmlted only by the capacity of the used digital computer Ttus means that also arbltrardy subdwlded drywells of pressure suppression systems can be simulated which was not possible m the earlier pubhshed models. c) The computer code allows the stmulatlon of different geometric situations of the contamment because it is based on several different components. d) Time dependent varmble flow areas controlled by pressure differences of different types can be simulated Features a) to d) show that with the aid of this computer code containments of different geometrical shapes can be simulated (concerning their geometrical properties). The different flow and thermodynamic conditions m the containment are taken rote account by the following properties of the model. e) Three different posslbdltles for the mathematical simulation of a pressure node (section 2.1.1) whereby thermodynamic equdlbnum between the phases m a pressure node can be assumed as well as deviations from ~t f) Calculation of mass flow rates of compressible one- and two-phase flows with different shaped flow paths (sectton 2 1.2 and 2.1.3) g) Calculation of heat fluxes and space and time dependent temperature distributions m sohd structures (section 2 2) to simulate the long term behavlour. h) Slmulatton of the pressure vessel which can be arbltrardy sxtuated wlthan the containment (sect. 2 1.1). Ttus samulatlon of the pressure vessel was necessary for example for the comparisons of theory to experiment for full pressure containments and pressure suppression systems as can be seen m the sections 2.1 4 and 2.15. To minimize the need of computer running ttrne for comphcated problems in the different components of the model, different solution methods for the chfferentlal equations are used (section 2 4). Calculat~onal results of the computer code were compared with approprmte experimental results for the case of a full pressure contamment (section 2 1.4) as well as for that of a pressure suppression system (section 2 1.5) Also the equations for the calculation of mass flow rates of a compressible friction one-phase two-component flow have been tested by comparison with expermaental results [40]. 2 1. Short term pressure behamour

The time dependent pressure budd up m a containment can be subdwlded rote two phases, namely the short term and the long term pressure behavlour. During the short term pressure behavlour the local maxima of pressures and pressure differences are build up m a subdwlded containment. Their magmtudes are influenced by the mass flow behavlour between and the thermodynamic behavlour within the pressure nodes. The blow down of the coolant from the prunary system takes place within seconds, and after ttus period pressure equthbrmm between the different pressure nodes m a full pressure containment or m the drywell of a pressure suppression system takes place, which terminates the short term behavlour. The foUowmg non-stationary pressure behavlour m the containment is influenced prxmardy by heat exchange processes described m section 2.2 Compared with the flow processes dunng the short term behavlour these heat exchange processes are much slower and occur within periods of minutes and hours which is why this period of the transient ~s called long term behawour as shown m section 2 2. In this section the equations for the s~mulat~on of the t~me dependent pressure behav~our m and the mass flow

244

D Brosche A computer code jor pressure dtstrtbuttons

rates between the pressure nodes are formulated st) that It ts possible to calculate the space and tline dependenl pressure behaviour in a subdivided containment 2 1 1 MathemattcalJormulatlon o] the pressure nodes In this section the equations for the simulation of the time dependent variations of the pressure and the temperature or temperatures are described as well as equations for the masses of the three media steam, water and ali m a pressure node In accordance with the thermodynamic state m the pressure node, different assumptions are used so that at last three systems of equations are found to describe the time dependent behavmur of the variables m the pressure node a Ltqutd and gaseous phase o)' the pressure node m thermodwtamlc equthbrmm This pressure node model lg used for the case which occurs most often that a sufficient mass of water is present in the pressure node and it ~s assumed that thermodynamic equlhbrium between liquid and gaseous phase is fulfilled (steam, water and air have the same temperature), which can be assumed if the variations o f temperature and pressure are not too fast or too slow The thermodynamic properties of water and steam for saturation (specific volume and specific enthalpy and their derivatives with respect to temperature) are functions of temperature [21] and are used as such functions in the computer code For this pressure node model the following basic equations are used

Mass balances

M D = GDe

GDa + G v ,

(steam)

~]

Mw=Gwe

Gwa - G v ,

(water)

~2)

M L = GLe

GLa

(air)

(3~

The variable G v in eqs (1) and (2) represents the flashing or condensatlng mass flow rate between the two phases water and steam Energy balance GD e tDe + Gwe tw e + GLetLe+ Q _ GDat D _ G w a l w _ GLaIL=MDtD + M w t w + M L t L

Vp

The energy balance eq (4) represents the first law of thermodynamics for an open system and a non-statlonar) process, whereby no mechanical energy is delivered to the surroundings from the system (constant volume of the pressure node) and potential energies are negligible because the flow processes are essentially flows of s t e a m - a i r mixtures with water droplets. GDe, Gwe and GEe in eqs (1) to (4) are the sums of the entering mass flow rates of steam, water and air, respectively, GDa, Gwa and GLa are the sums of the mass flow rates leaving the pressure node These mass flow rates are calculated with the aid of the equations described m the sections 2 1 2 and 2 1 3 The quantity Q In eq (4) describes heat fluxes caused by heat transfer processes If Q is positive then it represents heat flowing rote the node, for example caused by post shut-down heat, m e t a l - w a t e r reacnons (put as time dependent functions rote the computer code) or by heat emission from hot solid structures or walls (described in section 2 2) A negative Q represents heat flowing out of the node such as to cold surfaces, or to adjacent pressure nodes or to the surroundings For both cases the equations to calculate Q are described in section 2 2 Volume balance

M D vD + M W vw = 0

For this equation it is assumed that the volume of the pressure node IS constant

245

D Brosche, A computer code .for pressure dtstrtbunons

In adthtlon to these basic equations an equation is necessary to descrxbe the correlation between the temperature T and the pressure p m a pressure node The total pressure p is the sum of the partml pressures of steam and air P = PD + PL

(6)

The partual pressure of steam PD m eq (6) is obtained from the tables of the properties of saturated steam as shown m ref. [21] PD = PD (T).

(7)

The partial pressure of air is calculated from the equation of state for Ideal gases which is vahd for the relatively low pressures occurring m this connection (8)

PL = M L R L T / V G .

If the pressure node contains water, the volume of the gaseous phase VG is calculated by eq. (9) VG = V - M w vw ,

~

(9)

whereby V is the total volume of the pressure node. With the aid of these eqs. (1) to (9) and various thermodynamic properties and their derivatives with respect to the temperature (whmh all are functions of the temperature) one Finally gets after some rearrangements one ordinary non-linear differential equation for the time-dependent variation of the temperature T as well as an algebraic equation for the mass flow rate G v

I" =

B - C + E + A F / ( v o - Vw) D - FH

(10)

and Gv -

A vw - - vD

(11)

TH,

with the following auxiliary variables A = (GDe - GDa ) v D + (Gwe

-

(lla)

Gwa ) Vw,

B = GDetDe + GwetWe + GLetLe + Q - GDaI D - GwaxW - GLalL,

(llb)

C = (GDe - GDa)t D + (Gwe - Gwa)t w + (GLe - GLa)tL,

(llc) t

D=MDI'D+Mw'w+MLCpL

VR L T ~GI_¢ e-V-Mwvw

VRLM L V_Mwvw

V R L M L T M w vw _ VPD, (V-Mwvw)2

GLa ) 4 ML. Vw . (6w¢ . . - . 6w

(V- Mw Vw)

)] j'

(11d)

(lle)

246

D Brosche, ~t computer code for pressure dtstrtbuttons V R L M L T vw F=XD-xW~

M o v D + M W l'~ H-

( V - M w l'w)-")

r I)

~]l~,g)

vVv

With the aid o f G v from eq ( 1 1 ) M D a n d M w can be determined by eqs (1) and (2) For each pressure node one obtaans therefore four ordinary non-linear differential equat)ons for the time dependent variations of the temperature T and the masses of steam M D, water M w and air M L After antegratlon o) these differential equations and with the aid of eqs (6) to (9) the time-dependent pressure behavaour m the pressure node as known These equations are applicable only if water is present in the pressure node which means that M w is greater than zero Only in tins case a flashing from the liquid phase and a flashing mass flow rate Gv can be defined so that M w does not become negative I f M W in a pressure node becomes zero the masses o f steam M O and air 3I 1 , the pressure p and the temperature T are calculated by the iollowing equations b Gaseous phase oJ the pressure node m superheated condxtton In tile case of a steam lane ruptule at the pr~ mary system, for example, it can be assumed that several pressure nodes m the containment axe an the superheated condmon Pressure and temperature of the gaseous phase are now an contrast to case a two independent variables with the assumption that steam and air have the same temperature winch meal's that only one temperature exists "File temperature of the gaseous phase in this case is hagher than the saturation temperature corresponding to the steam partial pressure so that the gaseous phase as superheated In this case principally the same basic equations are valid as In case a with the distraction that for the flashing process of water which enters the pressure node (e g caused b~ water carry-over from adjacent pressure nodes) certain assumptions must be made Also the thermodynamic properties of steam are now functions of pressure and temperature The following basic equataons are used Mass balances

M D = GDe

GDa + G v '

M L = GLe - GLa

(steam)

( 1 2)

(air)

(i3)

The mass flow rate G v which flashes from the water phase into the gaseous phase is calculated by the following equation Gv = aGwe

(141

The numerical value of a is that fraction o f the entering water winch flashes into the gaseous phase If the ma~s flow rate of water Gwe is known then the value of a can be between zero and one In tins model it is assumed that a 1S o n e

In tins case the whole mass flow rate o f water Gwe flashes Then It depends on the thermodynamic state o f the gaseous phase and the value of the mass flow rate Gwe whether the gaseous phase stays in superheated condxtlon or is cooled down to saturated condition Energy balance GDetDe + GwetWe + GLetLe Volume balance

GDatDue -- GLat L + Q =MDtDu e + M L / L - Vp ,

M D VDue = 0

(15) t16)

D. Brosche,A computer code for pressuredistributions

247

The specific volume of the superheated steam can be calculated as a function of the temperature and the pressure with the foUowmg equation VDue = VDUe(J~D,T)

(17)

The total dlfferentml of VDue yields /0°Oue~ +(0ODuet dVDue -- ~~--~D-)T dP D \ - ~ - ] p D

dT.

(18)

The specific enthalpy of superheated steam IS also a function of temperature and pressure /Due = IDue(PD ' T)

(19)

The total differential of iDu e yields

(alDue~

(aIDue)

(20)

d/Due = [--~PD] T dpD + \ 3T ] PD dT

These partial differential quotients In eqs. (18) and (20) are described by functions of PD and T as shown m ref. [21] and are used as such m the computer code. In the following equations the left hand side of eq (15) is represented by the auxiliary variable A 1 A 1 = GDetDe + GwelWe + GLelLe - GDaIDu e - GLatL + Q

(21)

After performing several mathematical operations one finally obtains the following equations for T and p

=Z / N ,

(22)

with the auxiliary variables

E(aIDue/aPD)T YOue Z=A1 +(GDe-GDa+Gw*)[" ~ ~ T

V VDue

- lo]ucJ

MD(~VDue/OPD)T

+(GLe - GLa) (RLT - tL),

(22a)

and

F{%ue~

(atDue/apD) T (a VDuJaT)pD~ V(aUDue/aT)pD N =M D L~--g-T--]pD (a VDue/OPD)T _J +ML(CpL - RL) + (aVDue/apD) T

(22b)

as well as

R

p -- "v-~L(J~/LT + M L 7") --MDODu~ +MD (aVDue/~ T)'D 7~

MD ( OVDue/aPD) T

(23)

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D Brosche A computer code Jor prexsure dlstrtbutlons

With the aid o f these four ordinary non-hnear dliferential equation~ (12), (13), (22) and 123), the time depe]~dent variations of the masses of steam and aar M D and M I as well as of the pressure p and the temperature T can be calculated c Ltqutd and gaseous phase of the pressure node not m thermodynamw equthbrmm This case means that gaseous and liquid phase have different temperatures with the assumption that the steam and air temperature are the same The mathematical model of a pressure node not m thermodynamic equilibrium shall be used primarily lot parametric studies to investigate the influence of deviations from thermodynamic equlhbrlum between gaseous and hquld phase with respect to the pressures and temperatures Such deviations from thermodynamic equlhbrlum are possible t\)r example in connection with very rapid temperature and pressure variations if thermodynamic equilibrium between gaseous and hquld phase cannot be leachea fast enough or m connection with the long term behavIour, If the gaseous and hquld phase must be treated a~ separate thermodynamic systems In this case again s~mllar basic equations are valid as m case a with the difference that now to~ the gase,ms and liquid phase separate energy balances have to be used This means that the mass exchange between hquid and gaseous phase must be calculated with the aid of certain assumptions and that the thermodynamic p r o p e m e s ar~ as in b, dependent on the pressure and the temperature The following basic equatmns are used Mass balances

M D = GDe

GDa + G ,

(steam)

(24)

M W = Gwe

Gwa

(water)

( 2~ )

M k = GLe

GLa

(air)

t 26)

G

For G v the assumption shall be made that it IS a lunction of the temperature difference between the gaseous phase (TG) and the hqmd phase (Tw) corresponding to the following equation G = ] ( T G - TW)

(2'~

This function f in eq (27) describes a very complicated combined heat and mass transfer process which IS dependent on geometrical, thermodynamic and fluid-mechanic influences and which can only be determined experimentally For the purpose of a parametric study, G v can be varied G can therefore reach the following extreme nummlcal values (l) G v = 0 , which means that between hquld and gaseous phase there is no mass exchange hence no flashing or condensation takes place The gaseous phase acts like an inert gas (u) G v has the numerical value calculated with the aid o f eq (11) In this case gaseous and hquld phase are m thermodynamic eqmhbrlum and the equations m a are used Due to the fact that gaseous and hquld phase have (the latter case excluded) different temperatures but the same pressure, two separate energy balances are used which is contrary to a and b Energy balances GDetDe + GLetLe + GvlW,D + QG - GDatD - GLatL =MDtD +MLtL-- VGP '

(.gaseous phase) (28)

D. Brosche,A computer code for pressuredlstrtbutmns GwetWe - GvtWD +Qw - GWatW = M w i w - VwP

(hqmd phase)

249 (29)

The mdex of/W,D m e a n s that m the case of flastung W should be chosen and for condensation, D must be used VG and Vw are the volumes of the gaseous and the hqmd phase QG and Qw are the heat fluxes exchanged between these two phases and the surroundmgs The sum of eqs (28) and (29) yields an energy balance for both phases smaflar to eq (4)

Volume balance

MDVD +MwOw = 0

(30)

In addmon to these basic equations functions for the specific volumes and specific enthalples of the steam and thetr derwatlves with respect to the temperature and pressure are used as m b because these properties are now dependent on the temperature and pressure. Similar functions exist for the thermodynamic properties of the liquid phase. However because the pressure dependence of vw and tw Is small m comparison to the properties of steam it Is neglected Therefore for the propertles of water funcuons for saturation condition are used With the aid of auxiliary variables A 2 and A 3 A 2 = GDe/De + GLeILe+ Gv/W,D+ QG - GDa/D - GLa/L,

(31)

A 3 = Gwe tWe - GvZW,D + Qw - Gwatw '

(32)

and

three equations for the Ume-dependent variables TG' TW and/~D are obtained TG = C1 + ( Gwe - Gwa)D1 + (GDe - GDa)E1 + (GLe - GLa)F1 Gv(E1 - D1 ) I1 + i1 ,

(33)

AID(aVD/OPD)T

~/'W= - (GDe - GDa) vD + (Gwe - Gwa)OW

AIw

Mwo

AID(OVD/OT)PD _ V W - VD - I"G MWOw + GV MW Ow ,

(34)

and PD - A2 -- (GDe - GDa) J1 - (GLeB1- GLa) (IL - RLTG) _ GVBS _J1 ~.GH1 ,

(35)

with the following auxthary variables

a, -

MDRLAIL TG [00D 1 /~'D vG XaVDZT + M D [ - ~ D I r - VG ,

VwA2 C1=A3+---~1

AID (aOD/~PD) A2 t(,v.~

VwRLMLrG

(35a)

(35b)

D Brosche A computer codeJor pressure dlstrlbuttons

25(I P

twow D1- v~

l~SLI

tW

t

!

El:

zVevD

PwRLMLTGVD

ow

V~

VGZD +RLMLTGV D +

VGB 1

(~5d~

VwR LML T G G1 -

%~

t

Iw ! vw

(35I}

135g)

H1 = ' VGB1 -- \-~f-/PD +BV \~-~ PD BI (cpL

II=MD

J1 -

( aVD1 \ T T J pD G 1

VwR LML ( ~v D -HI[G1M D VG \~PD )T - FW] '

~35h)

tD VG +RLMLTG° D VG

(35xt

With the aid of TG, J~W and PD one gets the time-dependent vanataon of the total pressure p

RLTG I(GL e - GLa) -M- -L- Vw (Gwe - Gwa)1 P - VG + VG

RLMLTGVw V~,

RLM L ,

MLRLTGMw v'w .

v6 (36)

The volume of the hqmd phase Vw is Vw =Mwu w ,

(37}

and that of the gaseous phase VG VG = V

Vw

(38}

Finally there are six ordmary non-hnear differential equanons for the time-dependent variations of the masses of steam, water and air, the temperatures of the gaseous and hqmd phase and the pressure The actual use of these systems of equaUons for the cases a, b, and c as well as the switch from one system to

D. Brosche, A computer code for pressuredistrzbutlons

251

another for a certain pressure node I$ carned out with the aid of appropriate criteria by the code itself or by mput data. d. Transient pressure and mass flow rate o f the reactor pressure vessel during blowdown For the comparison of theory to experunent for full pressure containments and pressure suppresmon systems whmh are described m the sections 2 1.4 and 2.1 5 experunents were used m whmh the reactor pressure vessel was smaulated by a stmple pressure vessel without internal structures. In these experunents the pressure tune histories m the pressure vessels were measured but not the tune dependent behavlour of the mass flow rates and specific enthalpms whmh left the pressure vessels Because these tune-dependent mass flow rates and specific enthalpms from the pressure vessel are required for the calculation of the pressure build up m the contamment, the behawour of the pressure vessel had to be sunulated with the atd of a special pressure node to calculate these tune dependent variables. Beyond that the blowdown from the pressure vessel and the pressure build up m the containment are sunultaneously running processes whmh can be taken into conmderatlon by the sunulatlon of the pressure vessel m thin computer code Up to now these two processes have been sunulated separately with different computer codes. Comparisons of the measured and calculated pressure tune tustorles for the pressure vessel ymlded reformation about the destred time-dependent mass flow rates and specific enthalpms from the pressure vessel The tune-dependent behavlour of the pressure vessel m described basically by mmilar equations as used for the containment pressure nodes (only the air phase is mmsmg) For these equations the assumption Is also made that steam and water phases are m thermodynamic equilibrium The following bamc equations are used Mass balances

Energy balance

MD = - G D a + Gv ,

(steam)

(39)

M w = - Gwa - G v

(water)

(40)

- G D a t D - G w a ' w +Q =MD~ D +Mwt w - VIi.

(41)

With the aid o f Q m eq. (41) for example heat Input to the pressure vessel from hot sohd structures can be taken into account. Volume balance

MD-OD+ Mw'-OW= 0

(42)

With the auxiliary varmble A 4 A 4 = _ GDa/D- GWa/W +O

(43)

one obtains after some calculations the following equation for the time-dependent variation of the pressure/J

¢ ,o-,w ) +Gwa (tW ,o-,w ) Uw

A4 +GDa D

uD

/~-

Id,D 'o-'w

OD

dv,q

UD

Id,w

UW UW

(44)

--owdp - v

and for G v one obtains

Gv =

ODa vD + Gwa °W - (M D dVD/dP + M W duw/dP)/~ VD _ OW

(45)

D Brosche A computer code ]or pressuredtstrtbuttons

252

With the aid of eqs (39), (40), 144) and 145) the tame-dependent behavlour ot M D, M~,~,p ,rod G can be ~alculated ff Goa, Gwa and Q an eqs (39), (401 and ( 4l ) are known These equations however yield no lnformatmn about the local distribution of the (steam and water phase for example can be separated or more Ol less homogdistribution may differ considerably steam and water phase for example can be separated or more or less homogeneously mixed) and has a great influence on the magnitude and composition ot the mass flow rate which leave~ the pressure vessel and therefore again has an influence on the pressure tlme history in the pressure vessel More detailed information about these problems can be found e g in refs. [1, 22, 23, 33 35, 36.39] The mass flow rates GDa and Gwa in general are calculated wltli the aid of a model fl)r crmcal two-phase flow which can be found In the hterature on the subject

2 1 2 Calculauon of the mass flow rates, ortfice flow The second important problem m connection with the calculation of the time-dependent pressure build-up m a sub-divided containment arises after the calculation of the tmae-dependent behavlour of the pressures in the pressure nodes, namely the calculation of the mass flow rates which occur between these pressure nodes These mass flow rates have a very strong influence on the pressure tune histories during the short term pressure behav~our and are dependent on the geometrical conditions of the flow pathes, either orifice flow or pipe flow In the present section the orifice flow process is treated while the pipe flow process is outhned in section 2 l 3 The flow from one pressure node to another in a subdivided full pressure containment in many cases occurs through one or more parallel openings in the walls between these pressure nodes If the flow passes In a pipe like flow channel the calculation of the mass flow rate is performed with the aid of the equations m section 2 I 3 Due to the fact that the flow through an opening m a wall as comparable with the flow through an orifice this kind of flow shall be called orifice flow This kind of flow cannot be described on the basis of avadable theory alone Therefore a correlation with the md of theoretical considerations and experimental results is set up to calculate the mass flow rate for the case of a two-component one-phase flow (steam-air mixture) as well as for that of a two-component two-phase flow (steam-air-water mixture) But first some auxthary varmbles must be defined which are used in the following equations The mass fractions o f steam, water (water droplets) and air of the whole mass flow rate can be calculated with the assumptmn that the mass flow rate has the same composition as the gaseous phase of the pressure node from which it comes

~D = GD/G =MD/M"

~W = Gw/G =MwD/M'

G = GD + Gw + G L ,

and

~L = GL/G =ML/M

146-48)

with M = M D + MWD + M L

(49,50)

From the eqs (46) to (50) one obtains ~D + ~jW + ~jL= 1

(51)

The variables R, K and cp of the gas fraction ( s t e a m - a u mixture) of the total mass flow rate result from appropriate correlations for gas inLxtures of steam and mr The mass fraction o f the gaseous phase ~¢; of the total mass flow rate follows from eqs (46) and (48) I~G = (GD + GL)/G = I~D + ~L = 1 -- 1~w

(52)

D. Brosche, A computer code for pressure distributions

,f//

///t

/-////

~~L ///

253

//,-/f/

flow

dre ico itn

/ [

0 1 l F~g. 3. Orifice flow, 0 state of the flow m front of the entrance of the opening m the upstream pressure node, 1 state of the flow m the cross-sect~on of the opening, 2 state o f the flow m the vena contracta downstream of the orifice

The subscripts for the next equataons are shown m fig 3. Two further variables to be defined are the area ratio m m = F/F 0 ,

(53)

and the contraction number la = F 2 / F

(54)

Ttus kind of flow is generally a crmcal or subcrmcal (because the pressure ratio of adjacent pressure nodes can be very small) two-component two-phase flow (steam-air-water rmxture) and is not treated m hterature. In hterature concerning two-phase flow ordmardy the crmcal two-phase flow of the systems steam-water (onecomponent two-phase flow) and air-water (two-component two-phase flow) are treated by leaving the solution of this problem to the development of new correlations. To treat this kind of a two-phase flow with a reasonable mathematical display the following assumptions have been made which mostly are supported by appropriate expertmental results a) The static pressure m the flow can be calculated by the equation of state for ideal gases with the specific volume of the gaseous phase which generally Is a steam-mr mixture (the volume of the hqmd phase can be neglected because of the low pressures). b) Gaseous and hqmd phases have the same velocity (homogeneous flow). c) Between steam and water phases no phase exchange takes place that means that the mass fraction ~w of the hqmd phase does not change along the flow path because the flow pathes ordinarily are very short d) The somc velocity m the two-phase mtxture is the same as the somc velocity m the gaseous phase as can be seen m refs [24] and [25]. Thas can be assumed because the flow essentially consists of a gas tn which water droplets are dispersed. e) The flow process is quasi-stationary whereby this assumptmn Is confirmed m section 2 1 3 (0 f) The flow is assumed to be one-dtmenslonal, that means that variations of the flow parameters perpendicular to the flow dtrectlon are neghgable m comparison to those m flow direction With the aid of these assumptions the mass flow rate for the two-phase flow can be calculated m the following manner. Using the continuity equation between 0 and 2 one gets G = WoFo/Ozw,O = w2F2/Ozw,2 ,

(55)

with the spectfic volume of the homogeneous two-phase flow" Vzw = v~G + Ow~w .

(56)

D Brosche, A computer code for pressure dlstrlbuttons

254 The energy equation yields

1 2

1 2

gw 0 + (~wCw + ~GCp)To = ~w 2 +/~wC w + ~GCp)T2

157,

In accordance with assumption c) (w and therefore ~G also do not change and for that reason an appropriate index m eq (57) is missing The relations between specific volume, pressure and temperature for ali lsentroplc process ate 00/io2 =

~2/Po)]/~

l SSa)

and (58b~

T2 / T 0 = (lO2/po)(K - 1)/K The equation of state for 1deal gases at the point 0 yields

PoVo = R T °

(59)

With the aid of these equations one obtains after some rearrangement a relation tbr the mass flow rate as a function of the known variables and of the contraction number/1 and the friction factor ~"which take into account the flow contraction as well as the deviation of the real friction flow from the assumed lsentroplc flow 160~

G = azw Fq; zw x/p 0 /u OA ' with

(60a)

O~zw = ~'#/41 - m 2 / 2 , 1 - m2gt2 ~zw =

1 -- 112m2(p2/Poj2/g[VZW,O/{ OO~0- + (.lO2/t70) -I/K VW,2~JW}] 2 2,K

[(P212fl¢ _ ( P 2 t (K+I)/K]

A = I ~ G +(P2t IlK VW,2~Wl 1

"~0 '

0u

(60b)

(60c)

A

Experimental results m [26] and [27] show, that for orifice flows the contraction number/~ and therefore the discharge coefficient a are functions of the pressure ratio P2/Po whereby the variables/a and ~"can be determined experimentally only together Therefore they have been combined and included in the discharge number a an accordance with eq (60a) Informatlons about a can be found in refs [26] and [27], where It is shown to be a function o f the pressure ratio at the flow area. Further information about these problems can be found in [12] These correlations are valid for all flow area ratios m If one considers the effluent from a vessel then m in the above equations becomes zero which means, that the velomty at the point 0 (In the upstream pressure node) disappears

D. Brosche,A computer code for pressuredtstrtbutugns

255

With the aid of tins assumptmn, the denommator m eq. (60a) as well as the first square root m eq. (60b) reduce to one so that the foregoing equatmns become relatwely smaple. 2.1 3 Calculatwn of the mass flow rates, pipe flow Prlmartly with the aid of the following equations the mass flow rates through the vent pipes of pressure suppresslon systems are calculated as well as those through similar shaped flow pathes m full pressure containments First the two cases of a non-statmnary flow which takes into account local as well as convectwe acceleratmn processes and a quasi stationary flow which takes into account only convective acceleratmn processes are distinguished.

(t) Non-stanonary flow Here the two cases of an incompressible and a compressible flow are treated separately (a) Incompressible flow In pressure suppression systems the dryweli, which ordmardy Is subdivided hke a full pressure containment into &fferent pressure nodes, is connected by a vent pipe system to the wetwell whach contams the condensation pool tn which the partial pressure of the steam is suppressed To condensate the steam completely, the end of the vent pipe submerges into the condensation pool. To allow the air fraction of the steamair or steam-atr-water mxxture to reach the free space of the wetwell above the water level the water column in the submerging part of the vent pipe must be pushed out, whereby a part of the condensation water must be accelerated. This flow process is described by the following equatmns. The momentum equatton for a non-statmnary incompressible frlctmn flow yields 2

1. f a w __ v 1 ~ dx

w2-w~ 20

÷

g(H2-HI) ~

o

~-(P2 _ p ,x) + l20

:

2

2

~__~, +l ~ Zghl, X) w2(x)dx .1 20 1

~.vW2= 0 ,

(61)

whereby some symbols in eq.(61) are shown m fig. 4.

Contmmty equatwn

WlF 1 = w2F2 = w(x)F(x) = Go

(62)

In addition, a correlation for lsothermlc variation of the state of the gas volume m the wetwell is used" He

P2=P20V20 / f

F2(x)dx"

(63)

H2

IIe v

~ l-7--t

l

Fig. 4. Wetwell, scheme

D Brosche, A computer code tor pressure distributions

256

With the aid of eqs (61) to (63) one finally gets after several rearrangements and integrations two ordlnar2y non-linear differential equations to calculate the time dependent variations of the mass flow rate and the wate) level in the vent pipe After integration one finally ge~ the point of tmle when the water is forced down out ot the end of the vent pipe (b) Compressible flow If the water is cleared from the vent pipe the flow o f a steam-air or steam-air-water rmxture through the vent pipe from the drywell to the wetwell starts with the assumption that the flow medium m the vent pipe must be accelerated up to the stanonary mass flow rate which corresponds to the pressure difference between drywell and wetwell This flow process of a compressible one dimensional flow is described by the tollowing basic equations

Continuity equation w

Energy equation

3t

(65)

+ 0x -17):0

Momentum equation ~)x

ffxx

v

2v I w l ~ h

0

(O6)

If this system of equations is solved by a numerical solution method, one obtains the time- and space-dependent distribution of the flow variables whereby pressure waves are considered Due to the complicated geometry of the flow pathes and the need for much computer time, this solution method m this connection is not useful Therefore it was shown with the aid of several simplifications and investigations that local acceleration effects for this kind of flow are neghglble because they have nearly no influence on the pressure time histories in the drywell and wetwell Therefore all compressible flow processes m the computer code ZOCO V are treated as quasi statmnary flows which leads to a considerable reduction of the needed computer time and to a simplification of the computer code The validity of this assumption was also shown by theory to experiment comparison for pressure suppression systems as described in section 2 1 5

(n) Quasi-stationary flow Due to the fact that vent pipes of pressure suppression systems as well as certain flow pathes m full pressure containments often are relatwely comphcated the calculation of the mass flow rates is not possible with the aid of the equations m section 2 1 2 so that other more soptustlcated methods have to be used In the computer code such flow pathes can be represented with the aid o f certain standard elements enabling thmr mass flow rates to be calculated In the following the two cases of pipe elements with a one-phase two-component flow (steam-air m~xture) and a two-phase two-component flow (steam-air-water mixture) are treated separately because they require different solution methods a) Pipe with constant flow area with and without isentropic entrance, one-phase two-component flow In this case one has a one-dimensional, adiabatic, friction compressible one-phase flow For this k m d of flow it is possible to find analytical correlations for the local changes of the flow variables so that it is possible to calculate the mass

D. Brosche, A computer code for pressure dlsmbunons

257

flow rate analytically as can be seen m ref. [40]. A numerical solution method which normally contains several disadvantages therefore IS not necessary. For the ~sentroplc entrance of the pipe the following basic equations are valid, in which the index 0 designates the stagnanon con&non, which e.g. describes the state of fluid in the drywell, 1 IS the state of fluid at the entrance and 2 represents the state of fluid at the exit of the pipe. The flow between 0 and 1 can be considered as a frictionless flow without heat exchange with the surroundings and therefore as an ~sentroplc flow, with 1 20 + C p T 0 = l~ w 2+CpT 1 ~w 1

Energy equatton

(67)

Relation between specific volume and pressure for an lsentroplc process

Vo/V1 = (pl[PO) 1/K

(68)

DefinInon of the Mach number

Ma = w / d Z k - f

(69)

Equanon of state for 1deal gases

P = RT/v

(70)

The basic equations for the adiabatic pipe flow are given below. From eq. (70) one gets after logarithmic dlfferentm laon

dP/P= d T / T - dole

(71)

Eq (69) yxelds in the same manner d(Ma2)_ d(w 2) Ma 2

w2

dT T

(72)

For the dtfferenttal energy equation one gets cpdT + d(lw 2) = 0 ,

(73)

and the continuity equation for constant flow area yields 1 d(w 2) 2 w2

dv _ 0 o

(74)

For the momentum equatton one gets "P + ~_rMa2 ~,dx + d~.v + ½rMa 2 d(w 2) = 0 w2

(75)

The term hdx/D h In eq. (75) describes the pressure loss by wall friction with the hydraulic diameter D h of the flow path and d~"v describes local pressure losses, for example entrance or bend losses By introduction of the Ma number according to eq. (72) It Is possible to perform an analytacal mtegranon of this system of equanons along the flow path and therefore to calculate the mass flow rate for critical and subcntlcal flow condmon m a relatively stmple manner [40]. The mass flow rate which results in this manner has been compared with experunents [28] and good agreement between theory'hnd experiment was found [40]

D Brosche, A computep code ¢or pressure dlstrtbuttons

258

b) Pipe with constant flow area with and without lsentroplc entrance, two-phase two-component flow In this case nearly the same assumptions are vahd as for the orifice flow In secnon 2 1 2 exclusive of assumption c) hi this case one can assume m opposition to section 2 1 2 that between the steam and water phase, mass transfer can occur along the flow path This system of equanons has to be solved numerically contrary to the case of the one-phase two-colnponent flow Therefore the flow path is subdivided into a number of axml nodes to obtain the local distribution of the flow variables For the lsentroplc entrance the following basic equations are valid

Energ3' equation

1 ") ~w~ =A

B T1

476}

Wlflt A = (Cw,0 ~jW,0 + Cp,O~G,o)T0 ,

(7o,1)

B = Cw,1 ~w,1 + Cp, 1 ~G,I

( 76bl

and

~W and ~G are calculated with eqs (47) and (52) The contmutty equation yields G=

WlF 1

(77)

OI~G,1 + t)W,1 ~W,I The equatton o f state for Ideal gases is

PlVl = R I T 1 ,

(78l

and the relation between specific volume and pressure for an lsentroplc process

VO/V1 = (pl/Po)l/K

n

(79)

The basic equations for the vent flow are the following From eq (73) one obtains a relation between the points 1 and n ½w2n-I + (Cw,n 1 ~W,n 1 + Cp,n-1 ~G,n- l)Tn _ 1 =2t-w2n + (Cw,n ~W,n + Cp,n~G,n)Tn

180)

~W and ~G are calculated with the eqs (47) and (52) The contmutty equation is

G = WnFn/Vzw,n = Wn_ lien_ l/Vzw,n_l

(Sl)

With the specific volume of the two-phase mixture Vzw,n

Ozw,n = On ~G,n + °W,n~W,n

(82)

D Brosche,A computer codefor pressured~stnbutagns The equatton of state for 1deal gases

Pn On = R n Tn

259

(83)

For the momentum equation one gets from eq (75) w2

Pn-l-Pn-4

w2

V~,n +vzw,n-1 (XAx~b2/Dh + ~ v ) + ½ Vzw,n wn + Vzw,nWn-~l-I (Wn-Wn-1)

(84)

Gravity effects are neglected Ax in eq (84) 1s the &stance between the points n - 1 and n or the space step, and Oh the average of the hydraulic dmmeters at the points n - 1 and n The factor q)2 takes Into account the additional pressure drop for two-phase flow whereby numerical values ofO 2 can be found e g in refs [29, 30] With the aid of these basic equations one finally gets after some rearrangements appropnate correlations for the calculation of the mass flow rate as well as for the critical and subcntlcal flow condition c) Abrupt flow area changes, one- and two-phase two-component flow Abrupt flow area changes (enlargements and contractions) have a very great influence on the mass flow rates and cause great pressure drops so that they must be taken into account very carefully. This IS only possible with the aid of expertmentaUy determined pressure drop correlations Suitable correlations of this kind have been published m [ 31] where it can be seen that the ratios of the total pressures at the entrance and exit of the abrupt flow area change are nearly the same for incompressible and compressible flows contrary to the corresponding pressure drop coefficients. Therefore these correlatxons can also be used for the general case of a two-phase flow With the aid of the energy and continmty equation and the equations of state for ideal gases as well as the experunentally determined total pressure ratios which contain the momentum equation, one obtmns relations which allow the determination of the flow variables on one side of the abrupt flow area change from those at the other side d) DlstrlbuUon of the total mass flow rate into the mass flow rates of the components. In the sections 2.1 2 and 2.1.3 (u) a to c, the total mass flow rates have been calculated which occur between the different pressure nodes. For the calculation of the states m the pressure nodes, as can be seen m section 2.1.1, the mass flow rates of the angle components steam, water and axr are needed. Due to the fact that the composition of the mass flow rates does not change along the flow path, (case b m thas section excluded) the assumption is made that the total mass flow rate has the same composition as the fired m that pressure node from wluch it comes, so that it Is possible to calculate m a simple manner the mass flow rates of the three components from the total mass flow rates.

2.1.4. Compartson of theory to experiment for full pressure containments For this comparison experunentally determined pressure tame histories in the pressure vessel and in the contamment for the tests No 4 and 11 from [32] have been compared with corresponding calculated pressure tune lustones to test some of the relations m sections 2.1 1 and 2.2. This comparison showed good agreement between expertmental and calculated pressure tune histories as can be seen in more detail m refs. [22, 33, 35]. As mentloned m section 2.1.1 the pressure vessel had to be also sunulated in the calculations, wtuch is shown m more detail m refs. [22, 33, 35]. Fig. 5 shows the pressure tune lustories m the containment for the test No. 4 Beyond that, these comparisons rendered unportant reformation about the magnitude of heat transfer coefficients for the ease of condensatmg steam and the presence of non-condensable air as well as about the extent at wtuch the gaseous and liqmd phase m a pressure node are in thermodynamic equilibrium.

2.1.5. Comparison of theory to experiment for pressure suppresszon systems For tlus comparison expertmentally determined pressure tune tustones m the drywell and wetwell of the pressure suppression system for the tests No 25 and 44 from refs. [17] and [34] have been compared with appropriate calculated pressure tune hlstones This comparison showed also a good agreement between theory and experunent as can be seen m more detail in [23, 35, 36]. Figs. 6 and 7 show the pressure time histories for the drywell and

260

D Brosche A computer code for pressure dtstrtbuttons

pfb~} lp

-

--

/

f

.... - - -

i ~p,, it _l,t ih , i

o

4--6

l

Fxg 5 Pressure rime histories in the lull pressure containment, test No 4

~[~]

Expe

'

rh.or,

~

,,\

\\

1

i

70

t5

Ftg 6 Pressure time histories m the drywell test No 44 the wetwell for the test No 44 For these calculations it was also necessary, similarly to section 2 1 4, to simulate the transient pressure and mass flow rate of the reactor pressure vessel during blowdown in the calculations as can be seen m refs [23, 35, 36] Furthermore these comparisons yielded important information about the magnitude of the water carry-over of the flow from the drywell to the wetwell as well as about the extent m Much the gaseous and the hquld phase in the wetwell are m thermodynamic equdlbrmm 2 2 L o n g - t e r m pressure b e h a v l o u r

The long-term pressure behavxour follows the short-term behavlour described in section 2 1 This means that it starts after the termination of the blow-down from the pramary system and pressure equalization between the different pressure nodes in the containment Contrary to the short-term pressure behavlour which occurs wxthm seconds, the long-term behavlour can last minutes, hours or even days Therefore, the time dependent temperature

261

D. Brosche, A computer code for pressure distrtbuttons

pf *d l

I

. . . . .

~cpert.mont Thoor',/

I0

ff

10

t/'TJ

Fig 7 Pressure time histories m the wetwell, test No 44 and pressure behavmur m the containment must be calculated If energy, e.g. as post shut-down heat is added to the containment and heat losses to the surroundings must be taken rote account. For those cases the quesUon must be answered of whether or not the pressure and temperature m the containment exceed the design values ff one does or does not take rote account safety systems hke spray cooling systems etc. rote the calculation. The energy input rote the containment earl occur m different manners, for example by the above mentioned post shut-down heat from the prunary system or heat from metal-water reactions or by heat input from hot structure materials. Energy losses can mainly occur by heating up of solid structures ~nd heat losses through the outer wall of the containment to the surroundings. These different possab~t~es for heat input are provided for m the code by appropriate energy input vs. tune funcUons. On the other hand, the energy losses must be calculated. Therefore, the space and tune dependent temperature chstributaons m the solid structures and the outer wall must be calculated with the md of appropriate heat transfer correlations. 2.2.1. Calculation o f the space and tzme dependent temperature distribution in solut structures and the outer wall These structures can be approxunated by flats walls. Due to the fact that for most problems m this connection heat transfer Is unportant m only one dtrectlon, only the case of the time dependent one-dtmenslonal heat transfer Is treated which means that the one dunensmnal non-stationary heat transfer equation must be solved. Since a wall can be composed of many layers of different materials (for example the outer wall of the containment) it is not possible to solve the heat transfer equatmn analytically therefore a numerical solution method has to be used. The tune-dependent one-dimensional heat-transfer equation for plane plate geometry is a~/Ot = a 02 O/ax 2 ,

(85)

with the thermal dlffuslvlty

(85a)

a = h/co,

with the aid of the following boundary condition - XOOw/a

= a( x, t)

- Owl.

(86)

262

D Brosthe, A computer code ]or pressure dlstrlbuttons

Eq (85) IS solved numerically so that the wall must be subdivided into a number of temperature nodes, as can be seen in more detad m refs [1,331

2 2 2 Heat-transCer correlattons Appropriate heat transfer correlations for the different heat transfeJ conditions can be found m hterature In this connection especially the case o f condensatmg steam m the presence of the non-condensable air is ~mportant because this case is prevalent during the long term pressure behavlour, ff the hot s t e a m - a i r mixture in the containment comes into contact with the cold outer wall or with cold sohd structures This process IS characterized by the fact that the steam must diffuse through the air m the boundary layer between gas atmosphere and cold structure surface to condense So the heat transfer coefficient c~ depends strongly on the air fraction, the pressure and motion ot the s t e a m - a i r mixture relatwe to the surface and the mchnatmn of the surface A statable correlation t oi this heat-transfer coefficient e g can be found in ref [37] because it has been determined experimentally ~we~ ,, relative wide range of air flactlon The heat flux Q into the wall of the containment or Into or from solid structures which is used in the energy equations of section 2 1 1 can then be calculated w~th the following equation Q =aF(O - Ov~ ),

t87}

where t9 is the temperature of the containment atmosphere and 0 w the temperature at the surface of the wall

2 3 Simulation oJ vartable llow areas In this computer code It is also possible to simulate variable flow areas (e g rupture disks, flaps etc ) which are controlled by pressure differences, acceleration processes of flaps are described by the m o m e n t u m equation of mechanics as can be seen in more detad in ref [13] 2 4 Solutton procedure for the system ofequattons Because the whole system of equations consisting of differential and algebraic equations must be solved simultaneously within a rime step to obtain the non-stationary pressure distribution, the combined integration and iteration procedure described m ref [38] was used This solunon procedure has the following attributes a) It contains an iteration process because all time dependent variables are fed back b) A convergence criterion is used to control the numerical precision of the calculation c) The time step s~ze is variable so that it IS possible to use an optimum time step to minimize computer running time reqmrements d) The solution of the system of equations is independent o f the basic time step The integration of the heat-transfer equations can be performed with a simpler solutxon method to save computer running time The couphng of the heat transfer equations with the other system of equations can be performed by a predlctor-corrector method Beyond that for long term mvestlgatmns it is possible to use an mapllclt solution method for the heat transfer equations The integration of the equations for the simulation of the varxable flow areas is also performed with the aid of a s~mpler integration method to save computer running time 2 5 Structure o f the dtgttal computer code The computer code is built m such a way that containments with very different geomemcal features can be simulated This is possible because the code is composed of several subroutines (e g for the simulation of the pressure nodes and the mass flows and heat fluxes between them) the co-ordination of which is performed with the aid o f matrices The co-ordination of the pressure nodes and the mass flow rates between them e g is performed with m a m c e s such as the one shown m fig 8 With these subroutines the following calculations can be performed

D Brosche, A computer code for pressure dtstributions numbor

1

-K

pressure

node

J

N

3

\

K

\ o o

the

J

2 2

2

o£

263

\ \

\ \

1

\ Fig 8 MatrLxfor pressure nodes and flow pathes

a) b) c) d) e) 0 g)

Stmulation of the pressure node condatlons (section 2.1.1), Simulation of the pressure vessel blowdown behavlour (section 2 1.1), Calculation of the orifice flow (section 2.1.2), Calculation of the non-stationary pipe flow (section 2.1.3 0)), CalcalaUon of the quasi-stationary pipe flow (section 2.1.3 (it)), Calculation of variable flow areas (section 2 3), Calculation of the space- and tune-dependent temperature distribution m solid structures, stataonary and nonstationary (section 2.2.1), h) Calculation of heat-transfer coefficients (section 2 2.2), 0 Calculation of mmal values for the differential equations, j) Solution procedures for the differential equations (section 2.4). All other calculations are performed m the mare programme which consists of a stationary (calculation of mittal conditions) and a non-stationary part. The mare programme serves prtmardy for organlsatlon of the flow procedure of the whole code which is controlled by input data and internal switches. 2 6. Flow diagram

The scheme of the programme Is shown m a very stmphfied manner m the flow dlagrarn of fig. 9 to represent clearly the most essential features of the code. Most of the different subroutines a) through j) are very complicated, especially if Iteration processes and internal switches are used. In addition these subroutines often use auxdtory subroutmes, e.g for the calculation of variables by Interpolation from one or two dimensional data arrays from the mput. To slmphfy the flow diagram, such auxdlary subroutines are not shown m fig. 9. With the aid of such one and two dunenslonal arrays of data, functions which are dependent on one or two independent variables can be used without a preceding approximation by function equations. In ttus manner, for example the time-dependent func. tlons which are described in sectaon 2.2 are put rote the programme. In the same manner the main programme consists of many parts, e.g. the calculation of the ttme step s~ze, switches of different kinds etc., which are also not shown m fig. 9. In the flow dmgram m fig. 9 all possible flow pathes are shown which not necessarily are to be used altogether for a distract case. If for example a full pressure containment Is to be stmulated, the subroutine d) and eventually also e) are not needed. In the same manner for cases in whach heat lossed to walls need not be taken mto account (especmlly for the calculation of the short term behaviour) the subroutines g) and 11) are not needed.

264

D Brosche, A computer code lor pressure dtstrlbuttons

$~br~utll~

L

.

.... prQ9.... ~ i ~ i ~ tQIcu~QTlon 0t o, t i m | dlpItncllnt va¢llbtU

,

.

~um~l~ ~

_ ~ _

1 1

I

Print D~rs~tt

I

I

"i [nd ~t Q

tirol tltp

Fig 9 Flow dmgram of the computer code, strongly simplified All decisions about which parts of the programme are to be used are performed by the mum programme with the aid of input data and internal switches The computer code ZOCO V was written in A L G O L and tested on a Siemens 4004/55 digital computer A version of the code written in FORTRAN-IV is m preparation

3 Sample results With the aid of the computer code ZOCO V the space- and trine-dependent pressure distribution m geometrxcally comphcated containments can be calculated and the influence of a very large number of parameters on this pressure & s t n b u t l o n can be investigated To tUustrate this fact two charactensUc examples for the short term and long term pressure behavlour m a full pressure containment of a modern nuclear power plant shall be shown These examples p n m a r d y shall dehver quahtat]ve results therefore a complete hst of the input data is omitted Furthermore it was assumed that the blow-down from the primary system is fimshed after 0 8 sec, because within this period of time most of the maxtma of the pressure differences according to the figs 11 and 12 between the different rooms occur

265

D Brosche, A computer code for pressure dtstrlbuttons

pCb°,J

L

ozs

os

o~s

t[,j

Fig 10 Pressure time history, containment not subdivided

-"-'-

I

f

J

r2 /

9 12

t

oZS

os

oTs t[="]

Fig. t 1 Pressure time lustorles, containment subdivided mto 14 pressure nodes 3.1. Short-term pressure behawour

Fig. 10 shows the pressure tune history following a prunary ctrcmt rupture for the not subdivided containment wluch would be calculated for example by the single room computer codes described m section 1.3. If, however, the space-dependent pressure tune histories for the subdivided containment wtth the same total volume and the same other conditions are calculated one obtains completely different results as can be seen from fig. 11. In this case the contamment was subdwtded into 14 pressure nodes. The rupture of the prunary ctrcult occurred m pressure node 1. The particular pressure maxima especially for the pressure node 1 considerably exceed the pressure maximum m fig. 10. In practtcal cases usually the number of pressure nodes In the calculaUon is considerably lower than that of the rooms m the containment m order to save computer running tune. However, then It zs unportant to adequately simulate the lmmedmte nelghbourhood surrounding the room where the rupture takes place as well as this room itself. If for a more soplustlcated mvestIgataon the pressure node 1 IS subdivided into 4 pressure nodes and all the other pressure nodes and flow pathes are the same as m the above example the pressure tune lustones shown in fig. 12 are obtained. A comparison of the pressure tune lustones shown m the figs. 11 and 12 very clearly demonstrates the great influence of the kind and precision of the subdivision of the containment into pressure nodes on the pressure ttme tustones. In fig. 12 the rupture o f the prunary ctrcmt has also been assumed to take place m pressure node 1.

266

D Brosche A computercode tor pressure dtstrtbuttons

.........

G-

I i

I'

I I

3 I

I 7

//'/

;i

I Z ~

I ii

IO-

-

?S

o

o25

o~

075

t[sJ

Fig 12 Pressure time histories, contamrnent subdwlded into 17 pressure nodes 3 2. L o n g - t e r m pressure behavtour

In fig. 13 the pressure time histories for the &fferent cases 1 to 4 are represented so that the influence of several parameters on the pressure time h~stones can be seen Fig 14 finally shows for the case 1 in fig 13 the time and space dependent temperature distribution m the outer wall o f the containment This outer wall consists o f an tuner steel liner and an outer layer of concrete and an air filled gap between these layers Thereby it can be seen that the steel hner is heated up relatively quick m contrast to the concrete layer m which nearly no temperature rise can be recogmzed within this period of time

4 Conclusions In this paper the new theoretical model and computer code ZOCO V has been described which allows for the calculation of space- and time-dependent pressure distributions m containments of water cooled nuclear power reactors (full pressure containments and pressure suppression systems) following a loss of coolant accident at the primary system not only for the short term but also for the long term pressure behavlour On the one hand the computer code can be used to calculate the space- and time-dependent pressure and temperature maxima m a containment which is given by its geometrical features Hereby with the aid of parametric studies the influence of different parameters on the pressure t~me h~stones can be mvestxgated in order to obtaan

D. Brosche, A computer code for pressure distnbut~ns

267

pA°']

~ase

hmat Input

heat

losses

spray

coolin

/

8 i

1

post

shut-dob12

heat

outer

wall

2

post

shut-do~n

heat

outeF

wall

post

shut-dou11

beat

no yes

D

I I

3 4

4 1

3 Onset

pray

coollng

0 re*

1o'

Io~

lOJ

t[s]

re"

Fig. 13. Pressure time lustones, long term behawour air

,so

Itlup

'[d I

:t// tO0

$1

lO

0

110 "J

1.re"

II ¢0"

Z.IO" ~[m]

F~g. 14. Space and tune dependent temperature distr~ut]on in the outer waU of the containment, case l of fig 13

reformation about the stresses caused by pressures at all points in the containment and therefore reformation about the cnUcal points as well. On the other hand m cases where at distract points of the containment maxima of pressures and pressure differences should not be exceeded It is possible to calculate the reqmred s~ze and shape of flow areas Also the consequences of other sumlar actions can be investigated.

268

D, Brosche, 4 computer code Jor pressure dtstrlbuttons

F u r t h e r m o r e the c o m p u t e r code can be used with the aid ot future experiments dealing with subdwlded cont a m m e n t s to yield quantitative i n f o r m a t i o n about different parameters such as the discharge coefficients for orifice type openings with c o m p h c a t e d geometric c o n d m o n s or about deviations from the thermodynalnlC e q u l h b r m m or the efficiency of c o n t a i n m e n t spray systems l n f o r m a t m n about these parameters are obtainable by' comparisons of theoretically and experimental b determined pressure time histories a n d , b y appropriate parametric studies Such comparisons can also offer better insight into the t h e r m o d y n a m i c and fluid mechanic processes which occur during the pressure vanaUons and which therefore can lead to better control ot them w)th the aid of vauations in the structural layout of the c o n t a i n m e n t

Nomenclature L a t i n letters

a a c D F

= = = = =

thermal dlffuswlty factor specific heat capacity diameter area

[m 2/s] [J/kgK[ [m] [m 2 ]

G g

= mass flow rate = gravitational constant

[kg/s] [m/s 2 ]

H l K M Ma m N p

= = = = = = = =

hmght specific enthalpy n u m b e r of a c o n n e c t i o n b e t w e e n pressure nodes mass Mach n u l n b e r area ratio q u a n t i t y of pressure nodes pressure

[m] [J/kg]

Q R T t V v,u w x

= = = = = = = =

heat flux gas constant absolute temperature time volume specific volume velocity space coordinate

[kg]

[N/m2 I [ W] [J/kgK] [K] [s] [m 3 ] [m3/kg] [m/s] [m]

G r e e k letters

a a 2x ~ 0 0 K X X ~z

= discharge coefficient = heat transfer coefflment = difference = pressure drop coefficient = angle = temperature = lsentroplc e x p o n e n t = friction factor = heat conductivity = contraction n u m b e r

[W/m 2K]

o

[°C]

[W/mK]

D. Brosche, A computer code for pressure dtstnbutmns mass fraction = densRy = two-phase f l o w m u l t l p h e r = effluent f u n c t i o n

269

=

p

[kedm31

Subscripts a b D e G h L m n p T u ue v W WD zw

= = = = = = = = = = = = = = = = =

exit reference steam inlet gas hydrauhc atr mean n u m b e r , space c o o r d m a t e for fired d y n a m i c e q u a t i o n s c o n s t a n t pressure constant t e m p e r a t u r e surroundings superheated flashing or c o n d e n s a u o n , loss water, wall water m gas phase two-phase

Superscnpts 0 -

'

= = = =

mltlal c o n d R l o n mean value derivative w i t h respect to the temperature T derivative w R h respect to the ttme t

References

[1 ] D. Brosche, ZOCO V, Em Rechenmodell zur Berechnung yon zetthchen und ~rthchen Druckverteflangen In ReaktotsxcherheRsbehaitern, Laboratonum fiir Reaktortegelung und Anlagen slcherung, Techmsche UmversRat Munchen, Intemer Bencht (Oktober 1970). [2] J.C. Heap, Equthbnum P- V - T Relauons for Expanding I.aquld-Vapor systems m a Containment Shell, ANL-5828 (Nov 1958) [3] R.O. Bnttan, A Reactor Containment Program for the Atomic Energy C o m m ~ o n , ANL-5851 (April 1958) [4] A.R. Edwards, EsttmaUon of Accident pressure and Temperature Rises m Containment Vessels, Nuclear Safety 4, Nr 1. (1972) 75 ' [5] O. Volgt, Post-Aecadent Pressure and Temperature Transients m a dry Containment - A parametric study, Nuclear Engmeexmg and Design 3 (1966) 451. [6] L.C. ltachardson, LJ. Finnegan, R.J. Wagner, J.M. Waase, CONTEMPT - A computer Program for Predrctmg the Containment Pressure-Temperatme Response to a Loss-of-Coolant Accident, 1DO-17220 (June 1967). [7 ] B.N. Knsttanson, PTHISTRY-A code for predrctmg Lons-term Pressure-Temperature Htsto~y m Secondaxy Contamment of Water-cooled Reactors Following Accident-Induced Blowdown, ANL-7455, May 1968. [8] G.M. Fuh, Containment Trannent Resulting from Acculental Loss of Coolant m Preuurlzed Water Nuclear Reactor, Joint Power Generatton Conference, San Francumo (15-19 September 1968) AED-Conf. 204-003, 1. {9] G.M. Fuls, ACT-I, A DtgRal Program for the Analysts of the Containment Transient Dutmg a Loss of Coolant Accident m LWB, WAPD-TM-693 (May 1967).

270

D Brosche, A computer code ]or pressure dtstrtbuttons

[10] G. Gaggero, P M Germ1, G Leom, J B van Erp, MACACO (Modello Analoglco (ALcolo COntenltore)--PRFS~I, An ~nalog Model and a Distal Code for Containment Studies, EUR 3927e (1968) [11 ] D Brosche, H Karwat, The Development of Pressure Dtfferentmls Across Contamments ol Large Water-Cooled Power Reactors, Proceedings of a Symposium on the Contaanment and Siting of Nuclear Power Plants, Wlen, Conference Paper N1 SM89/12, April 1967, International Atomic Energy Agency, Wren and MRR 32, Instltut fur Mess- und Regelungstechntk Techmsche Hochsehule Munchen (April 1967) [12] D Brosche, H Karwat, Dynamxscher Druckaut'bau m Slcherheltsbehaltern grosser wassergekuhlter Lemstungsreaktorer~ MRR 32-d, Instltut fur Mess- und Regelungsteehnlk, Fechnlsche Hochschule Munchen (Max 1967) [13] D Brosche, ZOCO It, Em Rechenmodell zur Berechnung des dynamlschen Druckaufbaus In Slcherheltsbehaltern grossei wasser~ekuhlter LeJstungsreaktoren, MRR 66, Instltut fur Mess- und Regelungstechntk, Technische Hochschule Munchen (Dez 1969) [14] H G Sexpel, D Melnhardt, Drtferenzdrucke ZWlSChenden Raumen emes Slcherheltsbehalters nach eInem PnmarKreisbrucn. Atomkemenergle 13, 6 (1968) 401 [15 ] D E Alsch, Analytacal model for the prediction ol dfflerentml pressure transients in realtor buildings during a lo~-(>l-toolant accident, Atomkernenergie, 16. 1 ~1970)6 [16] K D Kuper, J Langhans, Eln Programm zur dynamlschen Berechnung des Druckverlauts m gekoppelten Volumma Atomkernenergle 17, 3 (1971) 163 [17] D B Barton, C P Ashworth. E Janssen, C H Robbms. Predicting Maximum Pressures m Pressure Suppression Reactor (ontalnment, Winter Annual Meeting of the American Society ot Mechanical Engjneers, (26 November 1 December 196 l) ASME Paper Nr 61-WA-222. 1 [18] D Brosche, Dynamlscher Druckaufbau in Druckabbausystemen, MRR 44. Institut fur Mess- und Regelungstechntk lec)~nxsche Hochschule Munchen, (Mat 1968) [19] D R Miller, Pressure Suppression Containment Design- Current State ot the Art, Winter Annual Meeting of the American Society of Mechanical Engineers (1-5 December 1968), Paper Nr 68-WA/NE-1.1 [20] C F Carmxchael, S A Marko, CONTEMPT PS, A DLgltal Computer Code for Predicting the Pressure Temperature Hlstor~ within a Pressure Suppression Containment Vessel in Response to a Loss-of-Coolant Accident. Trans Am Nucl, 11 (1968) 363, and IDO-17252, March 1969 [21 ] D Brosche, Approxlmatlonen thermodynamlscher Zustandsgrossen, lnstltut fur Mess- und Regelungstechnak, Techms~he Hochschule Munchen, Interner Berlcht (September 1969) [22] D Brosche, Instatlonarer Druckaufbau m Volldruckslcherheltbehaltern. Verglelch ZWlSChenTheorle und Experiment, Atomkernenergae 19, 1, (1972)41 [23] D Brosche, Instatlonare Druckverlaule m Druckabbausystemen. Verglelch ZWlSChenTheorie und Experiment, Atomkernenergle 19, 4 (1972) 301 [24] R E Henry, M A Grolmes, H K Fauske. Propagation Velocity of Pressure Waves in Gas-Liquid Mixtures, International Symposmm on Research m Concurrent Gas-Liquid Flow, Symposium Series 1 of the Canadran Society tor Chemical Engineering, Inter Dok 4, 10 (1969) [25 ] W G England, J C Ftrey, O E Trapp, Additional Velocity ol Sound Measurements in Wet Steam, lnd Eng Chemlstr~ Process Design and Development, 5 (1966) 198 [26] W Schiller, Uberkrltlsche Entspannung kompresslbler Flusslgkelten, Forschung, 4, 3 (1933) 128 [27 ] J A Perry, Jr . Critical Flow through Sharp-Edged Orifices, Trans ASME, (October 1949) 757 [28] W Frossel, Stromung in glatten, geraden Rohren mlt Uber- und Unterschallgeschwlndlgkett, Forschung 7 2 (1936) 75 [29] G B Walhs, One-dimensional Two-phase Flow, Mc Graw-Hfll Book Company, 19 [30] W L Owens J r , Two-phase Pressure Gradient, International Developments in Heat Transfer, Am Soc Mech Engrs paper 41, 2 (1961) 363 [31 ] R P Benedict, N A Carluccx, S D Swetz, Flow Losses in Abrupt Enlargements and Contractions, Journal ot Engineering for Power (Jan 1966)73 [32] A Kolflat. Results of 1959 Nuclear Power Plant Containment Tests, Nuclear Engineering and Science (onference (4 " April 1960), New York, PreprInt Paper Nr 10, 1 [33] D Brosche, lnstationare Druck- und Temperaturverlaufe m Volldruckslcherheitsbehaltern, Verglelch zwlschen Theorle und Experiment, Laboratormm fur Reaktorregelung und Anlagenslcherung, Technische Umversltat Munchen, MRR 97 (Dec 1971) [34] C P Ashworth, D B Barton, C H Robbins, Pressure Suppression, Nuclear Engineering (1962) 313 [35] D Brosche, Zur Ermlttlung mstatlonaxer Druck- und Temperaturbelastungen von Reaktor~cherhemtsbehaltern nach Kuhlmlttel-verlustunf'aUen, Verglerch zwaschen theoretlschen und experlmentellen Ergebnlssen. First Intern Conf on Struct Mechan in Reactor Technol, Berlin ( 2 0 - 2 4 September 1971), Paper No J 3/2 [36] D Brosche, Instatlonarer Druckaufbau m Druekabbausystemen. Verglelch zw~schen Theone und Experiment, Laboratorlum far Reaktorregelung und Anlagens~cherung, Techmsche Unlversltat Munchen, MRR 98 (Januar 1972)

D Brosche, A computer code for pressure dlstrlbutlons

271

[37 ] C.L Henderson, J.M MarcheUo, Film Condensation m the Presence of a Non-condensable Gas, Journal of Heat Transfer, (Aug. 1969) 447 [38] R. Buhrsch, J Stoer, Numerical Treatment of Ordinary Dtfferentlal Equations by Extrapolation Methods, Numerlsche Mathematuk 8 (1966) 1 [39] D Brosche, ZOCO V, Em RechenmodeU zur Berectmung yon zelthchen und orthchen Druckvertellungen m Reaktorstcherheltsbehaltern, Laboratorlum fur Reaktorregelung und Anlagenslcherung, Technlsche Umversltat Munchen, MRR 104, (April 1972) [40] D Brosche, Besttrnmung des Massenstromes kompresslbler relbungsbehafteter Stromungen, Verglelch Theone-Expenment, Laboratormm fur Reaktorregelung und Anlagenslcherung, Techmsche Umversltat Mtinchen, MRR 108 (August 1972)

Note added m p r o o f

Most of the input data for the sample results described in section 3 are summarized m ref. [39] Beyond that a complete list o f the Input data can be furnished by the author The FORTRAN-IV version o f the code ZOCO V will be available In the near future at the ENEA library (Ispra, Italy)