mod 1

Nuclear Engineering and Design 119 (1990) 67-95 North-Holland

67

QUANTIFYING REACTOR SAFETY MARGINS P A R T 4: U N C E R T A I N T Y E V A L U A T I O N O F L B L O C A A N A L Y S I S BASED ON TRAC-PF1/MOD 1 * G.S. L E L L O U C H E a n d S. LEVY

s. Levy Incorporated (SLI), 3425 S. Bascom Ave., Campbell, CA 95008, USA and B.E. B O Y A C K ( L A N L ) , I. C A T T O N ( U C L A ) , R.B. D U F F Y (INEL), P. G R I F F I T H (MIT), K.R. K A T S M A (INEL), R. M A Y (SLI), U.S. R O H A T G I (BNL), G. W I L S O N ( I N E L ) , W. W U L F F (BNL), N. Z U B E R ( N R C ) Received 12 January 1989

The Nuclear Regulatory Commission (NRC) has issued a revised EmergencyCore Cooling System (ECCS) rule allowing the use of best estimate computer codes for safety analysis. The rule also requires an estimation of the uncertainty in the calculated system response and a comparison of the resulting bound with the acceptance limits of 10CFR Part 50. To support this revised rule the NRC and its consultants and contractors have developed and demonstrated the Code Scaling, Applicability and Uncertainty Methodology (CSAU). The last of the three elements of the methodology - Uncertainty Evaluation - is described in this paper.

1. Introduction This paper summarizes the results of an uncertainty evaluation of a Pressurized Water Reactor (PWR) Large Break Loss of Coolant Accident (LBLOCA) analysis and completes the presentation on the CSAU methodology started with the preceding papers. This paper is in four parts: L Calculation of TRA C uncertainty (CSA U steps 3-12) This section discusses the establishment of the TRAC calculational matrix and reprises some of the work presented earlier [1,2]. It considers how to reduce the number of TRAC calculations to a feasible number and how we develop the Response Surface (RS) using Regression Analysis (RA). Finally, it discusses the establishment of a probability distribution function (pdf). 11. Calculation of biases (CSA U steps 8-10) This section deals with a number of phenomena

which are considered significant but are not modeled or mismodeled in TRAC. We also consider the effect of various parameters not considered individually significant which, because of large uncertainties in their variation, could become so, or which in combination may become so. I l L Combining the uncertainties and biases (CSA U steps 13-14) In this section we present the conditions under which the biases and the TRAC pdf may be combined and establish the final LBLOCA uncertainty table. IV. Comparison with experimental data (CSA U step 13) In this final section we consider the direct comparison of TRAC with data. This is done at the individual transient level and for blowdown and late reflood at the level of the distribution functions.

2. Conclusions * Work supported by the USNRC office of Nuclear Regulatory Research, under DOE Contract No. DE-AC07761301570.

This uncertainty evaluation for a PWR LBLOCA analysis shows that:

0 0 2 9 - 5 4 9 3 / 9 0 / $ 0 3 . 5 0 © 1990 - Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d )

68

G.S. Lellouche et al. / Quantifying reactor safety margins - Part 4 The Peak Clad Temperature (PCT) of a four-loop Westinghouse Nuclear Power Plant (NPP) undergoing a double-ended L B L O C A modeled by TRA C-PF1 / M O D 1 using the specific nodalization discussed in ref 2 and accounting for all phenomena identified as significant, including those not modeled by TRAC, is bounded at the 95th percent probability level by 1572°F, • that is: Pr(PCT < 1572 ° F) = 0.95.

Another important result of this study is related to the timewise propagation of uncertainties. The uncertainty in the P C T (defined here as the difference between the 95th percentile and the mean), at any point or within any interval in time is a consequence of the preceding time behavior. It is not constant with time and generally tends to increase with time. The most significant consequence of this condition is to require uncertainty evaluations for each major phase of a scenario. This can be seen in table 16 where this uncertainty varies from 2 8 5 ° F in the blowdown period to 5 7 8 °F in the late reflood period. This observation is equally true of uncertainties in the mean or in the 95th percentile. Thus we observe that uncertainties determined early in time do not propagate unchanged as time increases and that uncertainties established for one statistic (the mean for example) cannot be applied indiscriminately to another statistic (the 95th percentile for example). A final conclusion is that: Conservative approximations were made in the study particularly with respect to establishing some of the biases; as such there is a strong likelihood that there is at least a I O 0 ° F reduction that could be made in the 95th percentile limit.

3.

TRAC

uncertainties

This section discusses the preparatory work required to produce a pdf using TRAC. The work includes: establishing the dominant sources of calculational uncertainty; - establishing the pdf for each uncertainty parameter; creating the calculational matrix; and determining the structure of the Response Surface. 3.1. Dominant sources of calculational uncertainty (CS.4 U step 3) The PIRT [1] process had the purpose of identifying which phenomena could be expected to have the greatest effect on the peak clad temperature during a

LBLOCA. This process has already been described [1] and has produced seven phenomena which were felt to be dominant; they are found in table 1. Although PIRT produces seven phenomena, they are generally not directly treatable with any computerized calculational procedure and must be translated into the specific physical aspects that are contained in a systems code. This process can lead to the seven phenomena becoming many more than seven. For example, stored energy in the fuel pin is a function of the thermophysical properties of the fuel, gap, and clad as well as those of the coolant and the geometry of the pin cell and other more global factors. The number of potential parameters here exceeds twenty. However, as has also been shown in earlier papers [1,2], this large list can be reduced to a workable size by straightforward, analytically based sensitivity calculations which can be used to order the importance of each parameter. In this case, the stored energy term was expanded initially into seven fuel related input parameters (table 4 of ref. 2) and nine fuel related physical parameters (table 5 of ref. 2) but several of these were combined into a gap conductance term leaving thirteen independent parameters. A search of the literature provided sufficient information to estimate standard deviations and distributional shapes (mostly uniform). Table 6 of ref. 2 then showed the results of a series of calculations which permit an ordering of the sensitivity of the initial stored energy to changes in the parameters from their nominal to the level one values. Thus we end up with the following ordering: gap conductance followed by peaking factors, fuel thermal conductivity, power level, fuel heat capacity, etc. finally, when translated by transient calculations into changes in peak clad temperature only four have an effect in the blowdown phase of altering the PCT by more than 2 ° K : Gap Conductance, Peaking Factor, Fuel Thermal Conductivity, and Convective Heat Transfer Coefficient. When this process is completed the seven phenomena are replaced by ten parameters and one phenomenon (related to noncondensibles). These are shown in table 2 and are the basic parameters which are assumed to dominate the uncertainty in the PCT. Of these the first four phenomena (comprising seven parameters) will be treated by a direct statistical analysis. The last three will be dealt with as biases and will be discussed in Section 4 of this paper. 3.2. Ranging the uncertainty parameters (CSA U elements 7-10) In this portion of the paper we are following a procedure that is consistent with those used in the

G.S. Lellouche et al. / Quantifying reactor safety margins

69

Part 4

-

Element 1

~o.jt

Requirements and code capabilities

4 Providecompletedocumentation , Code manual User guide I Progammers guide Developmentalassessment Model & correlations QE

I Identify and rank phenomena(P RT} 3

I

I

+

| Oe~,~ne 16

.| appli( ability

I assessment Establish I 7 matrix

Element 2 Assessment and ranging of parameters

+

i

+

+ vs. sets using NPP nodalization Document

I

nodalization ,m for NPP calculations

+ Comparecalculations vs. lETSusing NPP J nodalization Document

4

I

I

+

Determine code and experiment accuracy

I

-

Element 3 Sensitivity and uncertainty analysis

=

9

effect of scale

10

Determineeffect of reactorinput parametersand state

11

Perform NPP sensitivity calculations

L,,

-8

J

- I - Combinebiases and uncertainties ,~ Total uncertainty to calculate specific scenario in a specific NPP

12 r

]13

J

Additional margin if warrantedby "-' ~ limitation data base,in L __code, etc.

14

Fig. 1. Code scaling, applicability and uncertainty (CSAU) evaluation methodology.

theory of experimental design [3]. We need to specify the range and distributions for the first seven parameters on the fight side of table 2 because we plan to combine all such uncertainties through a response

surface for the PCT. Basically a response surface is a fit through a set of data and it replaces the data in subsequent analysis. If this response surface is to have meaning for an uncertainty analysis, the range of the parame-

70

G.S. Lellouche et al. / Quantifying reactor safety margins - Part 4

Table 1 Dominant phenomena from the PIRT process

Table 3 TRAC parameter ranges

Break flow Stored energy Fuel rod to fluid heat transfer Non-condensibles

Parameter

Range

Peaking factor Gap conductance Fuel conductivity Heat transfer coefficient Break C D (1) Pump (2) TMIN

+ 5.6% + 80% + 10% - 25%/+ 50% 0/+2S B 0 / + 2Sp -20° C/+100° C

Pump 2-phase flow Steam binding ECCS bypass

ters entering it must be related to a probability. That is, we wish to be able to state that these seven parameters have underlying probability distribution functions. It is well known [4] that real experimental data do not follow any of the classical distribution functions, but in order to proceed we must specify an underlying function. We deal with this problem by using the Uniform Distribution. This choice should not be presumed to imply that we believe that the data really are uniform, The uniform distribution requires only that the end points (the range) be specified and does not imply that any value of the parameter is more likely than any other value; it is often referred to as the distribution which maximizes lack of knowledge. U n d e r such circumstances the choice of the range is most important. In this study it was made conservatively, and was based on data considerations [2]. Table 3 shows the uncertainty range given as relative to the nominal value contained in the T R A C code [2]. The peaking factor variation is based on calculat i o n / e x p e r i m e n t a l comparisons [5] as is the gap conductance; the Fuel Conductivity was taken from M A T P R O 11, while the Break R m and Pump ranges were based

Table 2 Treatable uncertainty parameters Break flow

RM

Stored energy

Gap conductance Peaking factor Fuel conductivity

Surface heat transfer

Heat transfer coefficient

TMI~ Pump 2-phase flow

Head and torque curves

Steam binding

Interfacial drag in: pool, core, upper plenum and hot legs

ECCS bypass

Interracial drag in the downcomer

Non condensibles

Non condensibles

Notes

(1) S B is based on Marviken data. (2) Sp is based on proprietary Westinghouse data.

on Marviken and proprietary Westinghouse data, respectively. The heat transfer coefficient is one of the most important uncertainty parameters and for our purposes consists of five parts: Single-phase liquid (Dittus,Boelter); - Two-phase subcooled nucleate boiling (Chen); Transition; - Annular with dispersed droplet field; and Single-phase vapor (Dittus-Boelter). In determining what ranges to use for these heat transfer parameters we note that all of the surface heat transfer correlations in T R A C are " i n s i d e " rather than "outside" correlations; that is, they relate to pipe flow rather than flow around a fuel rod (in a bundle of such rods). We must first account for the effect of inside versus outside flows; then we may consider the ranges of uncertainty. The effect of changing from an inside to an outside flow correlation for single-phase liquid or vapor flow has been shown [6] to be simply a multiplication of the single-phase Dittus-Boelter correlation by the pitch to diameter ratio ( P / D = 1.32 for a W PWR). On top of this, experimental data indicate that the uncertainty is +40% around this nominal value. For T R A C as it is this would correspond to an uncertainty range of 0.79 to 1.85 for the single-phase heat transfer coefficients. F o r the subcooled and nucleate boiling regimes where the Chen model is used, there are no studies showing how an inside coefficient must be varied in order to become an outside coefficient. F r o m continuity considerations, the Dittus-Boelter portion of the Chert relation should blend smoothly into the single-phase model but no published evidence for this has b e e n located. The annular with dispersed flow field model (referred to as M o d e 4 in T R A C ) implements the Forslund-Rohsenow (F-R) coefficient. Shumway [7] has -

-

-

G.S. Lellouche et al. / Quantifying reactor safety margins - Part 4

shown that, as implemented, this model is significantly optimistic when compared to separate effects rod bundle data. He has also shown that removing the Forslund-Rohsenow coefficient leaves the predictions of data some 30% low. This 30% may be a characterization of the pitch/diameter correction. Alternatively, if we reduce the F-R coefficient by 74%, the comparison to d a t a shows a mean error near zero and a scatter of about - 2 5 % / + 50% leading to multipliers of 0.75/1.5. In our ranging of the complete heat transfer surface we take the multiplicative range to be 0.75/1.75, but in preparing the probability distribution function and the statistics of tables 8 through 17 we use a sampling range of 0.75/1.50. Subsequently we account for the F-R effect by means of a bias (see Section 4). Further, we shall show in Section 6 that on average TRAC, without any F-R correction, predicts Reflood experiments conservatively. The transition heat transfer regime in TRAC makes use of the surface heat flux at CHF and the Mode 4 heat flux defined at a temperature TMIN and interpolates between them. The TMIN data base has been discussed in [2] and the model used in TRAC lies 36 ° F above the most conservative data bound and 3 6 0 ° F below the other bound. As such, TRAC is considerably conservative in its modeling of TMIN. We take the uncertainty range for this parameter as - 3 6 ° F / + 1 8 0 ° F which is still conservative with respect to the data [2]. A complete list of the uniform distribution ranges can be found in table 3. 3.3. Establishing the calculational matrix ( C S A U steps 9-11)

The purpose of this study - in the end - is to estimate the uncertainty in a best-estimate calculation of a LBLOCA. The word "uncertainty" has, however, not yet been defined. The new Rule and Regulation Guide points out that the 95th percentile of the probability distribution (presumably the cumulative distribution of the PCT) is sufficient, and although the 95th percentile is itself not an uncertainty, we shall accept it as an indication that any important uncertainties have been accounted for. There are a number of ways to establish the underlying probability distribution function (pdf) for this PCT. Some require more calculational effort than others. The one we have chosen requires the establishment of a Response Surface (RS [8]). For our purposes an RS is established by Regression Analysis (RA [12]) and this can be viewed simply as a multinomial least squares

71

fitting process of the calculated PCT in terms of the important parameters selected above. The purpose of the RS is to replace the code by a fit to the output of interest (here the PCT). A number of underlying assumptions have to be made concerning the behavior of the output relative to that of the input (e.g., the output is a continuous function of the input) and as we shall discuss below the order of the polynomials is restricted by considerations of the amount of data available and the structure of the calculational matrix. The amount of data needed to describe an RS is discussed in the standard texts [3,8], specialized publications [13,14] and the general technical literature [9,10]. In the present case with seven parameters, a complete description at a three-level polynomial formulation using a central composite design [8] would require more than 2000 TRAC calculations! Similarly, for a five-level polynomial formulation more than 70000 calculations would be needed! Clearly we are not able to provide such a level of data input. Equally important was the need to be aware of the realistic constraints imposed by finite financial resources. With an expectation that only about 10-15 computer runs were possible any "unnecessary" runs had to be eliminated. As a result, the pump and break calculations were limited to only those runs which would increase the PCT (i.e. be conservative) and the effect of the other side of the nominal point was treated probabilistically. That is, 50% of the probability would be absorbed at the nominal point. Similarly, while intermediate levels were chosen for some parameters on the conservative side, such intermediate levels were excluded on the other side. Table 4 shows the final matrix which will generate close to 200 peak clad temperatures. Fortunately TRAC has a capability for carrying along a set of "supplementary rods", which can be allocated to any sector. These rods carry the thermal hydraulics of that sector (mass flow rate, temperature, void fraction, etc.) and permit different fuel and clad properties (including heat transfer coefficient) to be assigned. Thus in table 4 the vertical column describes the changes in fuel and clad properties while the horizontal row describes the variations which alter the thermal hydraulic behavior. Specifically, there are eight TRAC runs listed here and 23 supplemental rods per run yielding 184 clad temperature traces. These 184 traces consist of one nominal point, 21 linear variations, 98 double crossproducts, 57 triple crossproducts, and seven quadruple erossproducts; they provide the basis for the RA which produces the RS. Having defined the conditions under which TRAC will be run, the actual runs were made and tables 5

72

G.S. Lellouche et al. / Quantifying reactor safety margins - Part 4

Table 4 Calculational test matrix Major

Nominal

run

Supplemental rod status Nominal Peaking factor

Gap conductance (Hg)

Break +2L

Pump +IL

Pump +2L

Pump + L Break+ 2L

Tul N -20°C

TMIN +100°C

As built code - 5.6% -2.8% +2.8% +5.6% - 80% - 46% + 46% + 80%

Fuel conductivity (Kf)

- 10% - 5% + 10%

Heat transfer coefficient (Hc)

- 25% + 25% + 50% +75%

Cross products Hg, Kf Hg, Hc Hg, Hc Hg, Hc Hg, Hc Hg, Hc Hg, Hc

Break +IL

-10% -25% + 50% -25% +25% +75% + 46%, -25%

-

46%, 46%, 80%, 80%, 80%, 80%,

t h r o u g h 7 exhibit the P C T for three m o r e or less distinct p o r t i o n s of the time trace of PCT: Blowdown, Early Reflood, a n d Late Reflood. F r o m a time interval p o i n t of view these comprise (for this W e s t i n g h o u s e model) the intervals: 0 - 2 0 s, 2 0 - 6 0 s, 6 0 - 1 5 0 s. Figs. 2 t h r o u g h 6 show several time traces for a n u m b e r of the p a r a m e ter variations. O f interest is the failure of the p u m p variations to have a n y significant effect o n P C T (max 50 ° F at blowdown); n o t s h o w n b u t similar conclusions can b e f o u n d for the f u e l t h e r m a l conductivity variations. 3.4. The response surface ( C S A U step 12) *

Because we are dealing with a c o m p u t e r code certain f u n d a m e n t a l p r o b l e m s arise c o n c e r n i n g a d e t e r m i n a t i o n * Further details on the response surfaces including the final surfaces themselves can be found in the Appendix.

of the quality of the RS, i.e. c o n c e r n i n g the question of Lack-of-Fit. In n o r m a l experimental design this is resolved b y p r o v i d i n g r e d u n d a n t e x p e r i m e n t s at the nominal point; these provide i n f o r m a t i o n t h a t can b e used to p r e p a r e q u a n t i t a t i v e s t a t e m e n t s a b o u t Lack-of-Fit if the underlying, n o t modeled, b a c k g r o u n d uncertainties meet certain statistical r e q u i r e m e n t s [3,12]. W h e n the inform a t i o n comes from a c o m p u t e r code there are n o b a c k g r o u n d uncertainties a n d n o s t a t e m e n t can b e m a d e c o n c e r n i n g Lack-of-Fit; that is, a c o m p u t e r code always gives the same answer with 100% reliability (voltage failures d o n o t count). Some discussion of a n equivalent p r o b l e m is f o u n d in Plackett a n d B u r m a n [10] b u t generally the question of Lack-of-Fit reduces to a n e x a m i n a t i o n of the R o o t M e a n Square ( R M S ) b e t w e e n calculation a n d fit. It is i m p o r t a n t to note t h a t the R M S does not directly indicate a good or p o o r fit since it can always b e driven to zero b y taking a sufficiently e l a b o r a t e fit! T h e r e does n o t a p p e a r to b e any general

G.S. Lellouche et aL

/

73

Quantifying reactor safety margins - Part 4

Table 5 Blowdown peak clad temperatures (o F) for CSAU rods (0-20 s) Rod # 9 11 12 13 14

Pf

Core

X-product CF, Pump

CSAU variation

Nominal

CFM level one

CFM level two

Pump level one

Pump level two

TMIr~ - 36 o F

TMIr~ + 180 o F

Nominal - 5.6% -2.8% +2.8% +5.6%

1103 1080 1092 1114 1126

1285 1252 1269 1299 1315

1321 1283 1299 1332 1348

1121 1096 1108 1135 1148

1166 1139 1153 1180 1193

1090 1069 1080 1103 1114

1125 1099 1112 1137 1150

1290 1258 1274 1305 1321

entrainment

16 17 18 19

Hg

- 80% -46% +46% +80%

1276 1175 1062 1040

1562 1409 1218 1186

1576 1438 1249 1216

1335 1218 1069 1045

1398 1269 1112 1089

1265 1162 1049 1029

1337 1209 1078 1054

1558 1413 1224 1191

20 21 22

Kf

- 10% - 5% +10%

1114 1108 1094

1308 1296 1263

1337 1326 1296

1144 1134 1101

1189 1179 1146

1101 1096 1083

1137 1130 1114

1310 1301 1270

23 24 25 26

Hc

+ + +

1216 982 853 626

1386 1182 1045 867

1429 1206 1081 871

1258 982 860 626

1294 1029 909 664

1305 975 838 626

1251 1013 799 640

1411 1198 1078 896

27 28 29 30 31 32 33

Hg,Kf Hg,Hc Hg,Hc Hg,Hc Hg, Hc Hg, Hc Hg,Hc

-46%, -46%, -80%, -80%, +46%, -80%, -80%,

1195 1328 923 1481 1162 1116 626

1423 1522 1184 1674 1312 1409 882

1449 1553 1227 1695 1357 1431 921

1234 1377 972 1530 1197 1175 626

1285 1414 1035 1566 1233 1242 702

1166 1416 891 1548 1249 1072 626

1225 1373 905 1531 1188 1135 644

1423 1537 1225 1679 1337 1427 919

25% 25% 50% 75% -10% -25% + 50% -25% -25% +25% +75%

J

2000

I

I

- -

NOMINAL

CASE

---.....

RM + lo' VARIATION RM + 20" VARIATION

1500 I..=J n~

w Q..

,'t

1000

I.=.1 I-£3 .,,(

I tl II tl

500

t'~___

I

0 0

I

50

100 Time

(s)

Fig. 2. Effect of critical flow on PCT.

150

74

G.S. Lellouche et aL / Quantifying reactor safety margins - Part 4

Table 6 First reflood peak clad temperatures (o F) for CSAU rods (20-60 s) Rod

9 11 12 13 14

Pf

CSAU variation

Nominal

CFM level one

CFM level two

Pump level one

Pump level two

TMI N

TMI N

- 36 o F

+ 180 o F

Core entrainment

X-product CF, Pump

Nominal - 5.6% -2.8% + 2.8% + 5.6%

997 968 981 1011 1024

1233 1197 1215 1251 1267

1240 1204 1222 1002 1276

988 961 975 993 1015

981 955 968 1008 1006

995 968 981 736 1022

721 693 707 1000 748

988 959 973 1206 1015

1188 1153 1171 1222

16 17 18 19

Hg

- 80% -46% +46% + 80%

1288 1080 963 946

1562 1341 1182 1159

1567 1346 1123 1173

1274 1069 954 939

1276 1065 945 934

1285 1078 963 950

1121 901 676 658

1269 1071 957 945

1530 1294 1141 1119

20 21 22

Kg

- 10% - 5% + 10%

1015 984 984

1258 1245 1213

1265 1252 1220

1004 995 975

999 988 966

1011 1002 982

739 729 707

1002 995 977

1213 1200 1168

23 24 25 26

Hc

+ + +

1206 1033 592 615

1441 1033 849 788

1470 1029 847 756

1211 700 604 628

1216 680 615 610

1254 835 642 624

1224 585 536 507

1220 669 594 613

1441 981 806 707

27 28 29 30 31 32 33

Hg, Kf Hg, Hc Hg,Hc H~Hc Hg, Hc Hg, Hc Hg, Hc

-46%, -46%, -80%, -80%, +46%, -80%, -80%,

1098 1312 828 1542 1157 1072 709

1364 1562 1045 1792 1384 1299 851

1369 1587 1053 1816 1414 1312 824

1087 1312 835 1549 1166 1063 725

1083 1306 813 1526 1175 1067 705

916 1353 837 1567 1211 1071 711

1089 1326 738 1553 1177 910 590

1317 1314 801 1528 1175 1056 702

1555 1018 1774 1386 1267 819

25% 25% 50% 75% -10% -25% +50% -25% -25% +25% +75%

1500

I

I

- --.....

NOMINAL CASE PUMP + lo" VARIATION PUMP + 2o" VARIATION

v

"r~'

1000

or. n l..ul p-

<

500

I

0 0

I

50

100 Time

(s)

Fig. 3. Effect of pump degradation on PCT.

150

75

G.S. Lellouche et al. / Quantifying reactor safety margins - Part 4 Table 7 Second reflood peak clad temperatures ( ° F ) for CSAU rods (60-150 s) Rod # 9 11 12 13 14

Pf

CSAU variation

Nominal

CFM level one

CFM level two

Pump level one

Pump level two

TMts - 36 o F

Nomin~ - 5.6% -2.8% +2.8% +5.6%

959 928 939 973 990

1098 1062 1080 1116 1134

1141 1103 1123 1161 1182

943 912 928 959 973

950 921 936 966 981

981 948 964 997 1013

Core entrainment

X-product CF, p u m p

464 451 457 471 480

1065 1024 1045 1085 1105

1126 1089 1107 1144 1164

Tit s + 180 o F

16 17 18 19

Hg

-80% -46% +46% + 80%

1126 1004 941 932

1308 1159 1069 1056

1353 1207 1116 1105

1110 990 923 916

1128 999 930 923

1161 1029 961 954

1018 567 442 435

1189 1098 1053 1047

1330 1188 1098 1087

20 21 22

Kf

- 10% - 5% + 10%

968 964 952

1112 1105 1087

1157 1148 1132

952 948 936

961 955 943

990 984 973

471 468 460

1071 1069 1060

1141 1334 1116

23 24 25 26

Hc

-25% +25% +50% + 75%

1251 441 428 426

1438 660 471 504

1504 711 554 576

1249 408 403 426

1242 426 392 392

1303 765 424 520

1251 403 363 334

1431 453 433 435

1481 790 486 486

27 28 29 30 31 32 33

H~Kf H~Hc H~Hc Hg,Hc Hg,Hc Hg,Hc Hg,Hc

- 46%, - 46%, - 80%, -80%, +46%, - 80%, -80%,

1013 1328 491 1512 1216 613 457

1173 1530 563 1704 1395 847 543

1222 1593 642 1771 1458 972 601

997 1324 486 1510 1222 662 462

1009 1317 460 1504 1202 707 433

1036 1378 480 1551 1269 869 525

583 1341 403 1549 1213 577 334

1105 1495 504 1647 1402 757 473

1202 1575 552 1760 1436 959 504

- 10% - 25% + 50% -25% - 25% +25% +75%

1500

I

I

1

0.20 * HGAP (-80X)

v

uJ n-

1000

1.1.1

a..

i..,-, _J C)

"'"

\

(+80,.)

500 ~L

I

0 0

I

5O

tO0 T i m e (s)

Fig. 4. Effect of gap conductance on PCT.

150

76

G.S. Lellouche et aL / Quantifying reactor safety margins 1400

I

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1200

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Part 4

I

~ / ~ - O . T S

• "c (-2sx)_

E"

v

tJ

1000

w

800

',

x.~

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,,"'""'i~,.

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600 ',

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fI ,', .,'

::

I.'~,i

,

I

200 0

I

50

100

150

Time (s) Fig. 5. Effect of convective beat transfer on PCT.

theory available here to guide us a n d we follow Box a n d B e h n k e n [9] w h o state that, " t h e highest degree of p o l y n o m i a l t h a t m a y b e fitted to the observations from a p-level factorial is p - l . " A n e x a m i n a t i o n of table 8 shows that the design we have i m p l e m e n t e d is a seven-factor, mixed (three to five) design a n d the m a x i m u m polynomial order is 1500

different for the various factors: - Peaking factor 4th order H e a t transfer coefficient 4th order - Gap conductance 4th order - Break, p u m p , TMIN 2 n d order - Fuel conductivity 3rd order Quartic fits p r o d u c e a n R M S which is in all cases less

i

I

-----

.....

NOMINAL CASE TMIN - 56°F TMtN + 180°K

"E v w rr

1000

n., w 12.

,,

,,

kJ k-

r-, .<

d

500

i,,,,:.,,,,'"

' ,",'"'"~qev~.. --"

'

r, ..............

I

. _ ~

~_

I

50

100

Ti m e

(s)

Fig. 6. Effect of minimum film boiling temperature on PCT.

150

77

G.S. Lullouche et al. / Quantifying reactor safety margins - Part 4

6 p 1460

X 5 1460 P

X :

X

“0 1450

X

x XXXX x

x

X

xxx

X

!2 F 1450 B

X

X X

X

X

X

X xx I

x X

X

:I I ii I I , I ’ ’ : 1440 1 3 1440 ’ 50000 10000 10000 50000 1000 1000 f NUMBER OF MONTE-CARLO HISTORIES NUMBER OF MONTE-CARLO HISTORIES m f ii

Fig. 7. History study for the “best” fourth order fit to the blowdown data.

than 5% of the mean PCT, hence can be considered small and provides little incentive for further improvement. It is also clear from this table that the data becomes more difficult to fit with a polynomial as we move out in time. This is not unexpected. Within the ranges of the underlying parameters, the RS can be looked at as defining a surface of the form PCT=f (Xi,..., X,). We are interested in knowing what fraction of the surface exceeds any given value for PCT; this produces the cumulative distribution function. If the surface is a linear one, there are analytical tools to provide this information; for a complex surface the usual procedure is to randomly pick values of the Xi,. . ., X, (seven at a time) and calculate the PCT. If we do this many times we can prepare a histogram defining the frequency response (so many calculated values of the PCT between 1000 and 1050, etc.). Normalized, this produces an estimate of the pdf. In order to produce a decent estimate of the probability distribution function (pdf) from an RS we must sample the surface in a statistically acceptable way. Because the surface is only algebraic we use a crude Monte Carlo sampler. The SLI code RAM-CAM (Regression Analysis Monte Carlo Analysis Module) was used in this analysis. Statistical sampling of a surface

f Y

1000 10000 50000 NUMBER OF MONTE-CARLO HISTORIES

contains as a free parameter the number of sampling histories. For each sample set a pdf can be produced and the behavior of the pdf with increasing sample size examined. Figs. 7 through 9 show a study varying between 1000 and 50000 histories for each of the three surfaces. While there is still some variation around 50000 histories it is considered acceptably small and we take the 50000 history pdfs as converged. Although we speak about the 50000 history case being converged it is clear that there is still some variability (but only lo-20°F) in the result. Indeed at lo5 or lo6 histories there would still be some variability. It is at this point that we might ask how sure are we that the mean or 95th percentile value is what we say it is. One could introduce additional studies at this point to determine a tolerance or confidence band about the 50000 history values using the non-parametric methods of [4], but this seems to us to be unnecessary. Using the central limit theorem on the mean value shows convergence within f 2-4O F at the 95th percent (one-sided) level. The question of robustness can be also considered from the viewpoint of the RS polynomial order. Thus tables 9 through 11 show the results of 50000 history studies for each of the surfaces previously considered in

: z

10000 1000 50000 NUMBER OF MONTE-CARLO HISTORIES

Fig. 8. History study for the “best” fourth order fit to the early reflood data.

78

G.S. Lellouche et a L /

Quantifying reactor safety margins - Part 4

ii

o

0 13-

770

~

1360

~

1340

rid "r" Fii

o

760

X

X

tlJ

X

Xxx X X

iii

xX

t

1000

X

X

xX

X

750 Z

X X

I

10000

50000

NUMBER OF M O N T E - C A R L O

m 1320 = 1000

HISTORIES

xX X

X X X X X

I 10000

NUMBER OF M O N T E - C A R L O

) 5000( HISTORIES

Fig. 9. History study for the "best" fourth order fit to the late reflood data.

table 8. These studies lead us to believe that the pdfs we have established quite reasonably describe the response of the T R A C code to variation of the seven parameters of table 3, and the complete uncertainty distribution functions for each phase of the transient can be found in fig. 10. It will be noticed that the width of the pdfs increase with time and that the structure becomes more

Table 8 Optimizing the response surface based on minimizing the RMS Stage of transient

Fit

Blowdown

Linear Linear & crossproducts Full second order Quartic l (52 terms) Best quartic (70 terms)

56.5 51.7 26.8 17.4 9.6

Linear Full second order Quartic I (67 terms) Quartic 2 (best, 67 terms)

91.3 50.6 46.8 24.5

Early reflood

Late reflood

RMS ( o F)

Linear Full second order Quartic 1 Best quartic (60 terms)

133.2 86.0 47.2 42.8

Table 9 Fitting the blowdown data (50000 history study) Fit

Linear Linear + Crossproducts Full second order Quartic 1 (52 terms) Best quartic (70 terms)

Sample mean

95th percentile

( o F)

( o F)

1112 1128 1147 1159 1162

1418 1423 1430 1442 1447

RMS ( ° F) 56.5 51.7 26.8 17.4 9.6

complex. For the Late Reflood pdf it appears as if we have a double peak. There is clear evidence that there is an early rewetting portion to the parameter space characterized most nearly by heat transfer coefficients larger than nominal, and a late rewetting portion characterized by those less than nominal (none of the other parameters seem to be important in this regard). We noted at the beginning of this paper that uncertainties do not generally propagate unchanged in time. Studies show that tolerance bands often tend to widen as time increases. In our situation we have another aspect which is also important. The space-time interval included in each of the three phases increases with time hence may be presumed to tend to broaden the pdf.

Table 10 Early reflood response surface development (50history study) Fit

Linear Fun second order Quarfic 1 Quartic 2 (best quarfic)

Sample mean (°F)

95th percentile

904 961 968 978

1314 1387 1394 1399

RMS ( ° F)

(°F)

91.3 5&6 46.8 24.5

Note: All quartics contain 67 terms.

Table 11 Late reflood response surface development (50000 history study) Fit

Sample mean (°F)

95th percentile (°F)

RMS ( o F)

Linear Full second order Quartic 1 Best quartic (60 terms)

724 761 726 758

1219 1219 1338 1336

133.2 86.0 47.2 42.8

79

G.S. Lellouche et aL / Quantifying reactor safety margins - Part 4

4. Calculation of biases (CSAU elements 8-10)

10-2

~

BLOWDOWN

. LATE REFLO00

~0--3

~

10--4

10--5300

, 500

, 700

, 9(30

~ 1 O0

PEAK CLAD T ~ A T U R E

. . . . 1300

1500

1 O0

OF

Fig. 10. Probability distribution functions for the three phases of the LBLOCA (50 000 history stud2¢).

Although we have spent a great deal of time running TRAC and grinding up a lot of numbers, it appears prudent to ask whether there is any relationship between these numbers and experimental data. We discuss this topic in Section 6 where we conclude from comparisons with experimental data that the overall heat transfer process in TRAC adequately accounts for the blowdown and reflood phenomena during a LBLOCA. It is important to stress the word "overall" because we now need to deal with certain specific aspects of the T R A C - P F 1 / M O D 1 code which are in a separate effects sense, less than adequate. Before proceeding we establish a first view of the final uncertainty table. Table 12 shows the means and 95th percentile values for the three phases of the NPP LBLOCA.

1162 1447

Questions were raised concerning the degree of radial nodalization - should we model the hot assembly directly or only as a hot rod? The decision - discussed in previous papers [1,2] - was to remain with the standard nodalization (two radial powered rings, one downcomer ring). The question of hot channel effects would be treated separately. Among the reasons for this choice was a 30% increase in resource allocation to directly treat the hot channel. It has been pointed out that the use of the supplementary rod as a hot channel may be less accurate than we might expect because the hydraulics are incorrect. To utilize a more elaborate nodal model containing a fourth radial zone in the core was beyond the practical resource capability of the project. As such, it was decided to do a simplified calculation which would only require two TRAC runs with four rings. We would

Phase

PCT ( ° F)

Mean value 95th percentile

4.1. The hot channel

Table 13 3 ring/4 ring: the hot channel bias

Table 12 Preliminary view of the final uncertainties

Blowdown

There are three types of effects which were not dealt with when we constructed the test matrix shown in table 4 and defined the system nodalization: - P h e n o m e n a requiting more extensive nodalization than available resources could support: a fourth radial ring describing a central hot channel; - Phenomena which are incorrectly modeled in the code: the implementation of the Forslund-Rohsenow correlation; - P h e n o m e n a which require multiple simultaneous variations of the relevant CRC or which have insufficient experimental data available to justify even a uniform pdf range; these are labeled as "Other Modeling Uncertainties" but may also imply "incorrect" modeling ab initio. The numbers we present in this section are temperature increments which must be (see Section 5) combined with the results of Section 3.

Blowdown Reflood Early

Late

978 1399

758 1336

Early reflood Late reflood

Additive bias ( o F) mean 95th mean 95th mean 95th

63 45 25 - 54 - 14 - 157

80

G.S. Lellouche et aL / Quantifying reactor safety margins - Part 4

create a mini data base and determine the mean and 95th percentile differences between the two 4-ring and the appropriate 3-ring runs. We would then use these results as a bias to be combined with the results of table 12. This set of calculations was carried out and the results are found in table 13. It can be seen that the variation is nearly linear with time period and leads to a strongly decreasing penalty (becoming a benefit) as time unfolds.

PEAK CLAD TEM'°ERA]URE 500

~ --

]

I ?, CCT~ 1200, O SCTF

~ /" ~/

A/

1oo

:3

:= 49.8 ~K

900

-lO~.

800 b

A

7OD i

4.2. M o d e l i n g limitations

6OO ~ L ~~_ ~ .

There are a number of modeling limitations in the code. We have earlier addressed the fact that the heat transfer surface is only valid for inside flows while the dominant area of need (in the core) is for external flows. This is a potential benefit worth 30-50 ° F at the 95th percentile as the LOCA proceeds through its phases, but because of our inability to validate the p / D correction in the pre Critical Heat Flux (CHF) twophase regime, we do not deal further with it. The implementation of the Forslund-Rohsenow (FR) correlation in the mode 4 heat transfer regime has been criticized for some time. Shumway [7] has shown 10 2 Experlmentol Data: ....

8 ~ LHGR £ 10 KW/FT

....

7.5 ~_ LHGR ~_ 10.5 KW/FT CALCULATED USING TRACPF 1/MOD 1

I I EXPERIIvE:N] RESULTS \ L 10_5

\

\

\ \ \

m

\ \

\

\

S00

600

700

800 900 1000 TRAC CALCULATION(K)

! 100

]\\~\,

that the TRAC-PF1/MOD1 Mode 4 heat transfer modeling is considerably but non-uniformly optimistic, and that when the F-R correlation is removed, the result is quite conservative. However, from the viewpoint of an integrated test comparison a completely different view is found: TRAC-PF1/MOD1 as currently formulated and using the F-R correlation is conservative in both blowdown (LOFT fig. 14) and reflood ( C C T F / S C T F fig. 12) comparisons. Shumway has shown that if a multiplier of 0.26 is applied to the F-R correlation that a comparison with experimental data shows a zero mean error. When a series of eight supplemental rods carrying various F-R multipliers is examined, the effect of a 0.26 multiplier is found to be worth between 47 ° F and 1 6 0 ° F as we proceed from Blowdown to Late Reflood. These results can be quantified as shown in table 14. Because of the difficulty of being sure that the slightly conservative TRAC comparison to experiment exhibited by the Loss of Fluid Test (LOFT, Slab Core Test Facility (SCTF) and Cylindrical Core Test Facility

o

Temperature difference ( F) Reflood Early

10-4

~ 400

i

600

800

~

1DO0

i

i

1200

1400

t_

\\\ i 1600

47

Calculation (FR = 0.26) - (FR = 1.0) 1800

MAXIMUM CLAD TEMPIERATURE OF

Fig. 11. Consideration of the effect of the LHGR range on the distribution function for the maximum clad temperature during blowdown.

j

1300

Fig. 12. Comparison of TRAC-PFI/MODI computed PCT with CCTF and SCTF data.

Blowdown

i

1200

Table 14 The effect of Forslund-Rohsenow

\

L.

500

LOFT mean (EXP- TRAC) SCTF/CCTF mean (EXP- TRAC)

-

Late

84

160

23 -

18.5

G.S. Lellouche et aL / Quantifying reactor safety margins - Part 4 I0000

I

I

I

I

I

,

I , I

I

I

I

I

,

I

i i i , i T~ V o l u m e Scale

l I I

0 Seraiscale )~ FLECFII

m"-.. ~. W ~ x " ~ l - - ' ~ ~[-

*'ql, ......

' ' ¢ ' ~

IL"

~'

..

=

o

X PKL "'i"" '"--

-" l ~ a ~

......:..-~5 , " ~ """ 8

:~'

•~ t~

.

- .-.. ; . . ~

,,,,o

.

*~ ++.

v "-. ....

.....

-

la

",.

.....

95% Tolerance lines Cold RelFood Rate = ECC I n j e c l t o n + Core C r o l l ~e¢lion Flow A r e o I

i

i

v..,.

SKtW~L~

"""" v

Regresslo

i

[]ftLCm

m+

--

I I llJ

0,1

,/14!, ,14 ....

......

~,! ~ "'-

1

VELECHI SLAStI

....

P 10

- I 14

i

I

1/1700 I/40(~ 1/48 1121 i121

A LOFT O CCTF + SCtf

"'-.. I000

81

i

A +

"'-..m#

I

I

"'"" """" "--

I i iiI

10

i

i

i

J _ ~

100

COLD REFLOODRATE (IN/S)

Fig. 13. Measured temperature rise data from scaled experimental facilities (217 points) during reflood. (CCTF) experiments will hold at full scale, it has been decided to accept the use of a F - R multiplier of 0.26 and use the difference between F R = 1 and F R = 0.26 as a bias. This is also shown in table 14. 4.3. Other modefing uncertainties There are three " o t h e r " uncertainties that we believe must be dealt with; they are: the effect of N 2 arising from the Accumulator injection process; steam binding phenomena; and - E C C S bypass phenomena. -

-

4.4. The effect of N 2 addition from the accumulators Two aspects are of potential importance here; the first relates to the N 2 pressure in the accumulator, the 1600

I BLOWDOWNpEAKCUD TEMERATLRE TRAC~I/MK~(31VS. LOFTLP~2-6 27 DATAPOINTS((2¢4ETEST) ~) 1400 ~ - 23,,~

9 , ...E~ []

second to the effect on condensation heat transfer. While the details of the analyses and calculations may be found elsewhere [1,2], the results may be stated as follows: - The allowed operating range for N 2 pressure in an accumulator has been determined to be sufficiently small that it would not affect our calculations; - The time necessary for the depressurizing (fizzing) accumulator liquid to reach the downcomer and the m o m e n t u m change at the downcomer are felt to be sufficiently great that most of the dissolved N 2 would separate out and leave the vessel via the by-pass route. However, the effect on condensation is so large that we calculate a 3.7 s delay in the time to reach peak temperature which corresponds to an 18 ° F bias on the early reflood PCT. While the N 2 condensation effect is complete before the start of the Late Reflood time period, it is not clear that the effect does not carry over into the Late Reflood interval. In order to avoid arguments it was decided to account for this bias in both reflood periods. 4.5. Steam binding phenomena

I

o 1200

J

o

b ~ 1000

D []

800 600

[] '"'

[]

./" . ..."~_ 45° LINE

,., ,' ."" o t-" 800

I 1000 PCT

OF

[]

°o

"" "• o

.. ...... o

~2 a.

"'"'/

."/

I 1200

~ L26Roo~o o I V A

2 3

[]

4 I 1400

1600

TRAC--PFI/M(301

Fig. 14. Illustration of test by test determination of code plus experiment uncertainty.

It is clear from an examination of the modeling of interphase friction existing in T R A C that some disagreement between predicted and experimental entrainment can be expected. This means that upper plenum levels, entrainment into the steam generators, and various other interphase drag related phenomena can be expected to be uncertainly predicted. In order to estimate the importance of this aspect of the code, a series of calculations was undertaken to

82

G.S. Lellouche et al. / Quantifying reactor safety margins - Part 4

d e t e r m i n e w h a t modifications (in the form of multipliers to existing correlations) would be necessary in order to r e a s o n a b l y predict two S C T F ( 6 0 1 , 6 0 2 ) tests. Los A l a m o s N a t i o n a l L a b o r a t o r y ( L A N L ) investigated: the R o z e n e n t r a i n m e n t correlation; core e n t r a i n m e n t ; a n d upper plenum entrainment. A series of calculations led to a set of multipliers which adequately allowed prediction of these two transients. U s i n g these multipliers, a single otherwise n o m i n a l T R A C r u n was restarted at the end of b l o w d o w n to d e t e r m i n e a n additive bias. T h e results indicate a slight ( - 9 ° F) benefit d u r i n g early reflood a n d a substantial (106 ° F) penalty d u r i n g the late reflood phase. The size of this last bias indicates the desirability of a more realistic modeling in T R A C .

Table 16 Estimating the probability associated with the ForslundRohsenow bias Case

-

-

Nominal Break 1-L Break 2-L Pump 1-L Pump 2-L TMIN - 3 5 ° TMIN +180 ° Cross product Ave PCT

Late reflood FR=I PCT (°F)

Pr (T
FR=0.26 PCT(°F)

Pr (T
959 1097 1141 943 950 981 464

0.71 0.81 0.83 0.70 0.71 0.74 0.20

1087 1251 1290 1091 1051 1102 779

0.81 0.91 0.93 0.81 0.78 0.82 0.58

1125

0.83

1289

0.93

958

0.71

1117

0.82

4.6. E C C S bypass p h e n o m e n a A n additional e n t r a i n m e n t p h e n o m e n a concerns bypass a n d d o w n c o m e r modeling. Specifically, the questions relate to - delay in the initiation of refill; a n d lower p l e n u m filling rate. T o estimate the effect of these uncertainties a study of U P T F Test 6 data as well as a b o u n d i n g analysis [11] indicates that the T R A C calculation is uniformly conservative d u r i n g the r e f i l l / r e f l o o d process b y 34 ° F. These four biases are exhibited in our second look at the final u n c e r t a i n t y table (table 15). -

Table 15 Intermediate view of the final uncertainties PCT ( o F) Blowdown

Reflood Early

TRAC pdf results: mean 95th percentile Biases Hot channel: A mean A 95th percentile Forslund-Rohsenow Entrainment ECCS bypass Nitrogen Summed biases

Late

1162 1447

978 1399

758 1336

63 45 47 0 0 0

25 - 54 84 - 9 - 34 18

-- 14 - 157 160 106 - 34 18

84

236

110 Mean

4. 7. The effect o f u n m o d e l e d p a r a m e t e r s U n d e r this h e a d i n g we wish to e x a m i n e several effects of the process which led to the final calculational matrix (table 4). The P I R T process only allowed items labeled " m o s t i m p o r t a n t " to leave C S A U element 3 and, as we p o i n t e d out earlier, n o t all of the c o m p o n e n t p a r a m e t e r s entering a class 9 item were carried t h r o u g h to table 4. W e can then ask questions c o n c e r n i n g w h a t i m p a c t some of the items left out m i g h t have, w h e t h e r a large u n c e r t a i n t y or a low i m p a c t item m i g h t m a k e it into (perhaps) a marginally high i m p a c t item, a n d finally we m i g h t ask w h e t h e r a c o m b i n a t i o n of low impact items (or low i m p a c t with high i m p a c t items) m i g h t create synergisms resulting in significantly greater t h a n expected results. Mathematically, we are asking w h e t h e r the sum of small effects b e c o m e large a n d w h e t h e r significant nonlinearities exist. Even the briefest of e x a m i n a t i o n s of the P I R T [1] shows that a detailed answer to these questions is impossible; there are simply too m a n y parameters. W e c a n n o t provide closure o n such questions with m a t h e matical rigor; all we c a n do is provide evidence which we believe offers a r e a s o n a b l e level of surety on a n " e n g i n e e r i n g j u d g e m e n t " level. A series of extra s u p p l e m e n t a r y rods was carried along which provides us with i n f o r m a t i o n concerning: (1) The effect of a n extreme value (50% reduction) of the clad t h e r m a l conductivity o n PCT. (2) The effect of c o m p o u n d e d 2 n d level (end of range) values of clad conductivity a n d specific h e a t a n d fuel specific heat o n PCT.

G.S. Lellouche et al. / Quantifying reactor safety margins - Part 4

(3) The effect of compounded thermal hydraulic runs were made through blowdown between the pump (medium impact) and the break (high impact) to investigate synergisms at this level (rather than only at the low impact levels). (4) The effect of setting the pump form loss coefficient to zero. Except for item (3), none of these studies show any unexpected results: (1) 50% variations in the clad conductivity (5 times the full range variation) is worth less than 3 6 ° F (whether by itself or compounded with pump, break, or TMIN variations) during Blowdown and progressively less as time unfolds. (2) The compounded end of range (2nd level) study is worth less than 13°F (whether by itself or further compounded with pump, break, or TMIN variations) in Early Reflood and less in the other phases of the transient. (3) In all cases the Break 2nd level and Pump 1st level variation had the largest effect and this was carried as the dominant crossproduct through Reflood. However, there is a negative interaction between the Pump and Break which reduces rather than increases the effect of the compounded variation. Thus the order of the PCT is nominal < Pump < Pump + Break < Break as can be seen in tables 5 through 7. (4) The effect of setting the pump loss coefficient to zero had no effect on the PCT at all (pump effects are dominated by the two phase degradation curves). Except for the Pump, Break crossproduct all these and any other parameter variations have been excluded from further consideration (except for those dealt with earlier as biases) and we have concluded that there is no need for further justification of table 4.

5. Combining the uncertainties and biases (CSAU ele-

ments 13, 14) Table 15 shows two types of numbers. The first types comes from a pdf with a specific statistical pedigree. Even the first bias (for the hot channel) has a statistical pedigree, but the others do not. By pedigree we mean that these are statistically viable means for combining them. It is well known that it is "allowable" to add mean values although the underlying rules which permit such addition are seldom understood. Misconceptions also abound as to what the various central limit theorems

83

mean. In order to combine the biases and the pdf results of table 16 two assumptions have to be made. (1) The biases (stated as ApCT) are independent of the TRAC pdf results and independent of each other. (2) The bias is constant over the time interval of the LBLOCA phase under consideration. These two assumptions permit the last three biases to be algebraically added and combined with both the mean and 95th percentile of the appropriate phases. Clearly assumption 2 is incorrect since the bias would have to "jump" between phases; but in general it appears to be reasonable since most of these effects are based [2] on analytical considerations of physical phenomena, not on statistical procedures; that is, we have used bounding analyses to provide maximum effects. The major problem lies in how to deal with the hot channel bias since we have distributional information. Properly speaking, one can combine the hot channel results with the TRAC pdf. The problem lies in the large benefit biases. There is little doubt that the large benefit biases implied by the A95th values will appear as nearly as large benefits after combination, but it is not felt by the TPG that such large benefits are easily justifiable on physical grounds. The same thing is true of course for the large penalty biases (106, 160) in table 15, but from a regulatory perspective, penalties are easier to justify on the grounds of conservatism. Thus the majority viewpoint of the TPG is to disregard the distributional aspects and utilize the mean value, assume it meets the two assumptions and add it to the other biases. This permits us to use the Summed Bias line in table 15. Before proceeding further, we wish to show just how conservative our estimation really is by reconsidering the F-R correction from a statistical viewpoint. A series of supplementary rods were carried along with the F-R correlation multiplied by various values; interpolation produced the specific results for a multiplier of 0.26. For each base CSAU case we could then compare the PCT with and without F-R fully activated. The eight values were averaged and the difference provides an indication of the effect of setting the F-R multiplier to 0.26. Table 16 shows the PCT calculated with TRAC for the Late Reflood phase of the transient along with the cumulative probability up to that value of PCT. The difference between the two averages ( 1 1 1 7 - 9 5 7 ° K ) = 160 ° F is a difference without a firm statistical location except that all values (except one) are well above the mean hence should not be simply added. The final addition of 160 ° F is a number that is perhaps one half too large and indicates a significant degree of conservatism in our treatment of this bias.

84

G.S. Lellouche et aL / Quantifying reactor safety margins - Part 4

Table 17 Estimate of total LBLOCA uncertainties

6. Comparisons to experimental data (CSAU elements 8-10, 13)

PCT ( o F) Blowdown

TRAC response surface Mean 1162 95th 1447 Summed biases 110 Adjusted mean 1272 Adjusted 95th 1557

The question we wish to pursue here is whether T R A C P F 1 / M O D 1 with the specific nodalization chosen is - as it stands - a conservative estimator of both the blowdown and reflood peaks; that is, should we have added the biases at all. To address this we consider first a series of predictions of experiment of L O F T blowdowns and C C T F / S C T F / L O F T / P K L / e t c . reflood experimental data.

Reflood Early

Late

978 1399 84 1062 1483

758 1336 236 994 1572

6.1. B l o w d o w n

In earlier papers [1,2] two scatter plots of data from various experimental facilities were presented and extensive discussion took place concerning the significance of those plots relative to scaling. At this time we should like to use the blowdown scatter plot (fig. 15) to prepare histograms around the value of 9.5 k W / f t . This value is valid for the full scale N P P calculations we have just discussed. Then we shall superimpose the T R A C blowdown pdf from fig. 10. This composite is found in fig. 11. We note first that the two L H G R ranges shown are essentially similar and that the T R A C pdf nicely rides on top of the experimental data and fits a little to the conservative side. We take this correspondence between experiment and calculation as a reasonable confirmation of the adequacy of the T R A C - P F 1 / M O D 1 prediction for the blowdown portion of an L B L O C A at full scale. Fig. 16 shows the same results with the T R A C pdf shifted by the summed biases of table 17; the singular

5.1. Final table o f uncertainties

Table 17 indicates our estimate of the combined L B L O C A uncertainties at the mean and at the 95th percentile. These are obtained by simple addition of the Summed Biases to the results obtained from the 50 000 history studies of the "Best Quartic" response surfaces (RS). There are several things unlikely about the final numbers if we intercompare them. The RS analyses show that both the mean and 95th percentile values move to smaller numbers as time increases, and after addition of the summed biases while the mean continues to move to smaller numbers the 95th percentile hardly moves at all. This implies a fracturing of the time dependent pdf in a way that seems unlikely - if not unreal, and indicates to us that we should further compare these " f i n a l " uncertainties with whatever experimental data we have available.

2300

i

I

........................................

O PBF 1800 ,-

<

~ 1/30OOO

)4E SEMISCALE A LOBI O THTF

1/1700 1/700 1/500

D CSAU-NPP

(1/1}

+ LOFT

""

~ +,.

.

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300

•

..-'""

A

++ " " - ' " ~r--"4- +o .--""...+ + +

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....... #~. .--'" ""'"

~;~= 800

Facility Scaling Cnter,on Power/Volume

~/48

1300

-

ARPENDtX K LIMIT . . . . . . . . . . . . .

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*+'t + +

+ ~,t,.*'~oo

+4-+

.

~

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+ tel~'~.....-~-~ -~.%_e ÷ + + o

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--"

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........

~7}.)+ . ~............ ..... --

95% Tolerance lines L ea egresslon line

I

I

I

5

10

15

20

LINEAR NEAT GENERATION RATE (KW/FT)

Fig. 15. Maximum measured clad temperature from scaled experimental facilities (301 points) during blowdown.

85

G.S. Lellouche et aL / Quantifying reactor safety margins - Part 4 10--2

10--2

EXPERM I ENTADATA: L --

8 & LC-~ ~ 10 KW/FT

..... 7,5 £ LC44~ <_ 10.5 KW/FT/ -

TRAC

-

...... /~

z

TRAC+BA I SES

T ~ ' - L / / A C

o

TRAC + BIASES

i

8

~I~

--- \ \

7-

"

EXPERIMENTAL DATA

DATA

10 - 3 I--

\\

LOFT + IO0OF

,\

AS MEASURED

~ :

\

C3

10_3 03

k\

Z

[-- ~'\ .------[~ .. \\\\

/ 10--4 400

600

I 800

I 1000

I 1200

I 1400

1600

1800

MAXIMUM CLAD TEMF:~RATURE °F"

10--2

AS ~[:~TED AS I:~'EPORTED+ 10,0

5 o Z

10-3

. . . . .

_ _ . _ . i

. . . . . . .

8 o

10--4 4-Q0

I

600 600

I

I

I

I

1000

1200

1400

1600

MAXIMUM CLAD TEMPERATURE

I

10-4

Fig. 16. Consideration of the effect of the LHGR range on the distribution function for the maximum clad temperature during blowdown.

~

~ ,\\\\\

1800

°F

Fig. 17. Comparing the effect of a 100°F adjustment to the LOFT data and its affect on the data histogram when compared to the TRAC distribution function.

400

6O0

800

1000

MAXIMUM CLAD

I 1200

I 1400

1600

18Q0

TEMPERATURE (F)

Fig. 18. Comparing the effect of a 100°F adjustment to the LOFT data and its affect on the data histogram when compared to the TRAC distribution function during blowdown (7.5 ~
increase in conservatism attributable to the simple addition of the biases is evident. Several questions can be raised here relative to tuning and to the adequacy of the LOFT thermocouple measurements. Fig. 14 shows a direct T R A C / L O F T blowdown comparison for 27 maximum clad data points. TRAC predictions are on average 2 3 ° F higher than the experimental values indicating the general adequacy and slight conservatism of TRAC relative to (at least) this LOFT test. Indeed, were we to add the summed biases, TRAC would be 1 3 3 ° F too high. But the LOFT measurements have been called into question as being low by as much as 100 o F. In this case, the TRAC + summed biases would still be 3 3 ° F too high. Consider, however, what would happen to fig. 16. We took the 301 point blowdown data base and added 100 ° F to each LOFT point and then plotted the 7.5-10.5 k W / f t histograms with and without the 100 ° F addition; this is shown in fig. 17. The result is only a slight shift in the histogram. When we now superimpose the TRAC calculations with and without the bias addition in fig. 18 we see that the added biases still produce a significantly conservative aspect. We conclude then that the addition of the summed biases

86

G.S. Lellouche et al.

/ Quantifying

makes the TRAC 95th percentile estimate of the blowdown PCT conservative by approximately 100 ° F compared to experiment and that the addition of the biases may be unwarranted. 6.2. R e f l o o d

Although TRAC shows the occurrence of two major reflood peaks, the experimental data available does not consistently exhibit such phenomena, as such we have a problem in knowing whether the reflood experiments apply to the first, or second peak. In what follows we show that it applies to the last peak and that it is unlikely to apply to the first (Early) reflood peak because the initial conditions for the early peak do not generally correspond to those of the extent experimental data base. First we note that in a direct comparison of 220 CCTF and SCTF experiments TRAC predictions (fig. 12) yielded statistics as follows: Sample Mean Difference (Calc - Exp) = 10.3 ° C (18.5 ° F), Sample Standard Deviation = 49.8 o C (89.6 ° F), indicating TRAC's capacility to calculate the subscale experiments given the correct initial conditions. This implies that - on average - the Forslund-Rohsenow biases are not required to make the TRAC results match (again on average) experiments. The problem lies in the fact that these and all other reflood subscale facilities contain distorted plena and downcomers relative to what could be expected in full size plants and that although TRAC models these facilities on average questions remain concerning scaleup. It is possible, however, to remove the questions of the downcomer and plena effects and only consider the core heatup. Fig. 13 shows "some 213 reflood C C T F / S C T F / LOFT/etc. data points. These show the core temperature rise as a function of the cold reflood rate (total coolant injected divided by core flow area). The base temperature is the clad temperature at the start of core recovery at the elevation which will ultimately be the hot spot. Because the reflood rate used is independent of bypass or other phenomena, the resulting multi-facility data plot is - within its bounds - a scaleless representation of the reflood temperature rise process. That is we expect about 95% of all experimental data points to fall inside these bounds independently of scale. Analysis indicates that a plot of this sort would be expected

reactor safety margins - Part 4

with a negative UR slope hence we believe that within experimental uncertainty this display can accurately be expected to be a scale free representation of the reflood data. Unfortunately, in the range of most interest (4 < UR < 5 inch/s) there are only 12 experimental data points and this makes direct comparisons of the type done for Blowdown (where we had more than 50 data points in the intervals of interest) less than adequate. Therefore we proceed differently and utilize two different methods of comparison between calculation and experiment. Fig. 13 shows a data display with a relatively stable band width over a range of a factor of 30 in UR. We infer from this that the underlying pdf has the same form independently of the magnitude of UR but that the parameters will be functions of UR. Consequently, we rescale each of the experimental data points multiplying them by (URi/4.5) 114. The 4.5 inch/s value is a compromise on the TRAC Late Reflood cold injection rate which varies between 4.27 and 5.09 inches/s. The plot in fig. 13 will be rotated around the regression line at UR = 4.5 inches/s and the mean regression line will be a constant (61°F = 339/(4.5) 1]4) as will the upper 95% line (at about 225 ° F). Now we have produced 213 pseudo data points at 4.5 inches/s from which we create a frequency histogram in fig. 20. The matrix in table 4 provides a nearly uniform covering of all the important interactions and nearly uniformly covers each underlying parameter. As such the raw TRAC output, by itself, provides a basis for the direct establishment of a histogram. Fig. 19, for example, shows the TRAC data as a histogram and the pdf 10-!

D

...................................

PET D£RIVED FROM REGRESSION ANALYSIS OF TRAC RESULTS

10-2

ul • _

10 - 3 i

c:

Lsl

L

! 10 - 4

':

I

i

HI~JTOC4~AM FROM 161 T R A C R E S U L T S

J

F

! Pr(AT < 350>J) = 0 9 5

f

[

.-

I i i

d

10-5

~--

I

'

i i

_

_[_A 500

..... 600

Fig. 19. Distribution for the temperature rise during late reflood derived from the TRAC calculations.

G.S. Lellouche et aL / Quantifying reactor safety

Part

margins -

87

4

10--1

--

------

I

I

....

SC'W/CCl~/O-OFT/wt¢ E ~ A L

DATA

10--2 i

I

]

TRAC-~-I/IaOOI

~CULAT'JON/4L MATRIXRE~U.TS

._~ I

g~__

II

10_3

r-E

1

10--4

10-- 5

I

0

I I 2~ 300 400 LATE ~ L ( X X ) ~ R A T U R E R E °F

1~

I 500

6~

Fig. 20. Comparing the calculated and experimental frequency histograms for the late reflood period. derived from a fourth order RA through the TRAC data (the smooth line). While the RA derived pdf does not capture the 3% of the data at the right side, it does show that the histogram is a very reasonable approximation to the pdf (and vice versa); indeed within one degree F both representations have the same 95th percentile (PR ( A T < 3 5 0 ° F ) = 0.95). We use the TRAC histogram in place of the derived pdf to compare with

] 0 0 0 0

I

'

'

'

~

'

''I

the 213 experimental data points discussed in the previous paragraph. This is done in fig. 20 where both frequency histograms are overlayed. What is novel here is that these histograms have very similar properties except for an offset between 105 o F and 125 o F. From a probabilistic viewpoint they describe - within a constant offset - the same phenomena. One could use nonparametric analysis [3, Chapter 9] to determine a

'

'

'

'

[

'

I'l

'

I

I

I

I

i

i

i

I

' I

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1000

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10

-.....

Regression 95%

line

Tolerance

ATr[ =

ReflOod Rale ECC Injection / Core

1

01

i

Cioss i

i

"'"~

lines

Cold

""'"-. A

SecliOn i

A

i

Flow i

i i i

Area i

i

i

i

i

i

i i I

I

10

i

AI

i

i

100

COLD REFLOOD RATE I~N/S)

Fig. 21. Temperature rise data from TRAC NPP calculation during reflood.

88

G.S. Lellouche et aL / Quantifying reactor safety margins - Part 4

confidence interval for this offset. It suffices here to say that this comparison between the TRAC calculations and the experimental data shows that TRAC is conservative in predicting the Late Reflood temperature rise by approximately 105 ° F at the mean and 125 ° F at the 95th percentile before the addition of any biases. The above demonstration of TRAC conservatism compared to experiment made use of a scale set of experimental data points in comparing with the T R A C calculated values for Late Reflood. Another comparison would be to simply plot all the TRAC calculated results on the experimental data plot. Fig. 21 reprises fig. 13 without the experimental data display, instead we display all the TRAC calculated points for both parts of reflood. If we examine table 4 in conjunction with tables 5 through 7 we will see that all important maximal crossproducts are included. That is, tables 5 through 7 contain about all the maximum values that can be expected within the underlying parameter ranges. Since the test matrix is a nearly uniform overlay on the parameter space (test points are nearly evenly spaced on a percent of range basis) we can expect (as we saw in fig. 19) that the TRAC results are themselves a reasonable representation of the underlying pdf. The 95% experimental data bound indicates that only 5% of any new data should (on average) fall outside the lines. Based on the raw T R A C data some 19% of the Late Reflood data falls above the upper 95th percentile bound. However, if we subtract the scale factor then only between 4.3 and 6.2% would lie above. We see this as a reasonable match (the expected "tail" would be 2.5% for a symmetric distribution). Although the Late Reflood T R A C calculations can rationally be related to the reflood experimental data, this is not true of the Early Reflood T R A C results. Fig. 21 shows some 61% of the TRAC data exceeds the 95% bound, but more importantly there are a significant number of T R A C data points below the mean value. We do not prepare any histograms here, there is little doubt that the T R A C Early Reflood Peak is not represented by the subscale facility reflood data display. The cause of this is quite simply that the initial conditions in T R A C at the start of the Early Reflood period are not representative of the initial conditions at the start of the reflood experiments; the T R A C conditions are strongly transient, those in the experiment are nearly static. For the Late Reflood T R A C cases the starting point is near the minimum between the two peaks hence appears to correspond more closely to the conditions of the experiments. As a result of these considerations we can conclude

that the Late Reflood TRAC results statistically correspond well (within a scale factor) with the experimental data. We can say that the Early Reflood TRAC results are conservative compared to the experimental data but this correspondence may not be meaningful. As a result, while the addition of the biases due to ForslundRohsenow and the others only make the T R A C results still more conservative, the lack of validation for the Early Reflood situation makes their addition reasonable.

Appendix. Statistical methodology and results Although the three sections on Ranging, Establishing the Calculational Matrix, and Response Surface provide everything needed for an engineer to run his own statistical package, it is felt that a more detailed description and detailing of the numerical results might be helpful. The particular code package (RAM-CAM) used was created for internal SLI use and is not documented. The statistical methodology used, however, is quite standard. The regression analysis uses a matric factorization of the normal equations P matrix known as the singular value decomposition (SVD). This is, we believe, the most reliable method for computing the coefficients for general least squares problems and minimizes the effect of data errors, round off, and linear dependence. The Monte Carlo Analysis is also quite standard.

A.1. Regression analysis

The statistical problem of finding a least squares fit to a set of data is dealt with in most statistics texts; the particular problem here is not significantly different from those discussed in the texts except that we use relatively high order (4th) polynomials. If one redefines higher order terms as auxiliary linear terms such as Z 1=xy, Z2 ~ x2, Z3 = y 2 , etc. then any high order polynomial can be reduced to a linear expansion problem. This is the way many software statistics packages deal with the multinomial re-

G.S. Lellouche et aL / Quantifying reactor safety margins - Part 4

89

Table A1 Blowdown Summary os least squares fit obtained for Pk Cld Tmp DATA - # 1 The specified fit has a total of 70 terms and a redundancy of 2.63, Following axe fit coefficients determined by SVDFIT Fit normaliTation constant (quadratic offse0 = 0.0000 Constant Term Linear Term for Param # Linear Term for Param # Linear Term for Paxam Linear Term for Param # Linear Term for Param # Linear Term for Param # Linear Term for Paxam # Quad Term for Param # Quad Term for Param # Quad Term for Param # Quad Term for Param # Quad Term for Param # Quad Term for Param # Quad Term for Paraxn # Cross Term for Params # Cross Term for Params # Cross Term for Params # Cross Term for Params # Cross Term for Params # Cross Term for Params # Cross Term for Params # Cross Term for Params # Cross Term for Params # Cross Term for Params # Cross Term for Params # Cross Term for Params # Cross Term for Paxams # Cross Term for Params # Cross Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Pararns # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params #

1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 1 1 2 2 2 2 2 3 3 3 4 4 4 5 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 2 3 4 5 5

0.86688E + 03 0.12813E + 02 - 0.53412E + 02 0.70288E + 01 - 0.57052E + 02 0.28762E + 03 0.11527E+ 02 0.23213E+ 02 0.58005E + 00 0.35337E + 02 0.70380E + 00 0.38992E + 01 - 0.17036E + 03 0.23051E + 02 -0.83715E+01 0.49428E + 01 0.18896E + 01 0.11819E+01 - 0.66677E + 01 0.20122E + 02 - 0.91521E + 02 - 0.30770E + 02 - 0.11027E + 02 - 0.20021E + 02 0.14269E + 02 0.14899E + 02 0.28157E + 02 0.18322E + 02 0.13235E + 03 - 0.19232E + 02 - 0.15679E + 02 - 0.14732E + 02 0.28346E + 02 0.97824E + 01 0.60311 E + 02 0.18708E+ 02 0.15098E + 02 0.88323E + 01 0.16008E + 02 0.48188E + 01 - 0.20071E + 01 0.64291E + 02 - 0.27140E + 02 0.15160E+02 0.14929E + 03 0.10196E + 02 0.53766E + 01 0.57277E+ 01 - 0.19232E + 02 0.96162E + 01 -

5 6 7 3 4 5 6 7 5 6 7 5 6 7 6 2 2 2 2 5 6 5 6 7 4 4 4 5 6 7 5 5 5 5 6

-

-

-

2 4 5 6 5 6 5 6 7 4 5 7 5 6 7 6 6 6 6 6

-

-

-

-

90

G.S. Lellouche et aL / Quantifying reactor safety margins - Part 4

Table A1 (continued) Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quarti¢ Quartic

Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params #

2 2 2 2 3 4 4 4 2 2 2 2 3 3 4 4 4 5 3 3

2 2 4 4 3 4 4 4 2 2 5 5 4 5 4 5 5 5 5 4

4 5 4 5 6 4 5 7 2 5 5 6 5 5 5 5 6 6 6 5

5 5 5 5 6 7 5 7 6 6 6 6 6 6 6 6 6 6 6 6

0.18468E + 02 -0.17284E+02 0.10753E + 02 - 0.52818E+01 0.16794E + 01 0.47280E + 01 0.43919E + 01 0.54681E + 02 - 0.47644E + 01 -0.10186E+02 0.10196E + 02 0.50979E + 01 0.00000E + 00 0.53766E + 01 0.36643E + 01 0.57277E + 01 0.28639E + 01 - 0.96162E + 01 0.26883E + 01 O.O0000E + O0

Note: Standard (RMS) error of the fit = 0.540E+ 01 ° K.

gression problem. Tables A1, A2, and A3 tabulate the coefficients for the best quartic regressions in the form I,J,K,L

PCT(° K) =

~

aijmXiXjXkX,,

i,j,k,l=O

with i= 1 = 2 = 3 = 4 = 5 = 6

I=J=K=L=7and X0 = 1 o peaking factor variation, o gap c o n d u c t a n c e variation, o fuel conductivity variation, ~ heat transfer coefficient variation, o break R m variation, ~ p u m p head and torque curve variation,

= 7 ~ TMI N variation. The actual values of ~ are those between the limits listed in table 3 in the b o d y of this d o c u m e n t (using fractions not percent, but using ° C for TMIN). The specific values used for the P u m p and Break variations are normalized; this m e a n s that the values entering the regressed muttinomial are n u m b e r s between zero and two (for 2nd level variation). For these parameters we absorb 50% of the probability between 0 and 10 -3 and then use a linear variation up to 2. The cumulative probability table for these parameters looks like: Variation Cumulative probability

:0 :0

0.001 0.5

1.00 0.75

2.00 1.00

The reason for absorbing the probability at zero is, as

discussed in the b o d y of this document, to reduce the total n u m b e r of c o m p u t e r runs and is allowable because the peak clad temperature (for this N P P and this break scenario) is as tests have shown, a nondecreasing function of the increasing positive variation o f these p a r a m eters. It is i m p o r t a n t to p o i n t out that the specific values of the p u m p variation are not given because they are proprietary data of the Westinghouse 1 / 3 scale p u m p . A user may take the values of the temperatures (tables 4, 5, 6 in the b o d y of the d o c u m e n t ) and the values of the variations implied by table 3 and indicated by the 3rd column and the first tow in tables 4, 5, 6 (nominal = 0, 1st level = 1, etc.) and using any regression analysis p r o g r a m p r o d u c e the equivalent of the results given in tables A 1 - A 3 . A.2. M o n t e Carlo analysis

M o n t e Carlo Analysis covers a very wide range of m e t h o d s and the reader should consult any text on r a n d o m sampling methods. Basically, w h e n the underlying probability distribution function is uniform a rand o m n u m b e r generator is used to generate a n u m b e r between 0, 1 (or - 1 , 1 or a, b). The location of the n u m b e r on the allowed range defines the value of the parameter being chosen. If m o r e t h a n one p a r a m e t e r is used at a time (as in a multinomial) then i n d e p e n d e n t r a n d o m n u m b e r s are chosen for each parameter. These

G.S. Lellouche et al. / Quantifying reactor safety margins - Part 4

91

Table A2 Early reflood Summary of least squares fit obtained for Pk Cld Tmp D A T A - Case ~ 2 The specified fit has a total of 67 terms and a redundancy of 2.75 Following are fit coefficients determined by SVDFIT Fit normalization constant (quadratic offset) = 0.0000 Constant Term Linear Term for Param # Linear Term for Param # Linear Term for Param # Linear Term for Param # Linear Term for Param # Linear Term for Param # Linear Term for Param # Quad Term for Param # Quad Term for Param # Quad Term for Param # Quad Term for Param ~ Quad Term for Param # Quad Term for Param # Quad Term for Param # Cross Term for Params # Cross Term for Params # Cross Term for Params # Cross Term for Params # Cross Term for Params # Cross Term for Params # Cross Term for Params # Cross Term for Params # Cross Term for Params # Cross Term for Params # Cross Term for Params # Cross Term for Params # Cross Term for Params # Cross Term for Params # Cross Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Cubic Term for Params # Quartic Term for Params # Quartic Term for Params # Quartic Term for Params #

1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 1 1 2 2 2 2 2 3 3 3 4 4 4 5 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 5 5 2 2 2

0.79213E + 03 0.15536E+ 02 0.45912E + 02 0.93189E + 01 - 0.13298E+ 03 0.40758E + 03 - 0.30644E + 01 0.78826E + 02 0.94964E + 01 0.66242E + 02 0.11514E+ 02 0.21080E + 02 - 0.26304E + 03 - 0.19296E + 01 0.54986E + 02 0.45994E + 01 - 0.77526E + 00 0.16497E + 00 - 0.15709E + 02 -0.13733E+02 02 0.43679E + 01 - 0.27483E + 02 0.16843E + 02 - 0.16757E + 01 0.63822E + 01 0.95489E + 01 0.94540E + 01 0.1339eE + 02 -0.22115E+02 - 0.51258E+ 02 0.46036E + 02 - 0.56437E + 01 0.39521E + 01 0.11795E + 02 0.33121E + 02 0.10658E+02 - 0.11053E+02 0.13100E+02 0.73541E + 01 - 0.97147E + 01 0.40854E + 02 - 0.12509E + 02 0.83784E + 01 0.54269E + 02 - 0.22115E + 02 -0.11057E+02 0.12215E+ 02 0.66851E+ 02 0.25459E + 02 -

-

-

-

5 6 7 3 4 5 6 7 5 6 7 5 6 7 6 2 2 2 2 2 3 5 3 5 7 4 4 5 6 7 5 6 2 2 2

-

-

-

-

-

0

.

4

1

3

1

6

E

+

G.S. Lellouche et al. / Quantifying reactor safety margins - Part 4

92 Table A2 (continued) Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic

Term Term Term Term Term Term Term Term Term Term Term Term Term Term Term Term Term

for for for for for for for for for for for for for for for for for

Params Params Params Params Params Params Params Params Params Params Params Params Params Params Params Params Params

# # # # # # # # # # # # # # # # #

2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4

2 2 2 3 3 3 3 4 4 4 4 3 4 4 4 5 5

3 4 4 3 5 6 7 4 5 5 7 5 4 5 7 6 5

7 4 5 7 5 6 7 7 5 6 7 6 7 6 7 6 6

- 0.19045E+ 02 0.10824E + 02 -0.25353E+02 - 0.33121E+02 - 0 . 1 7 9 8 5 E + 02 0.30843E + 01 - 0.30550E + 02 0.97923E + 01 - 0.16463E + 02 - 0.97388E + 01 -0.90213E+01 - 0.66220E + 01 0.14306E + 02 0.10058E + 02 0.25945E + 02 - 0.62547E + 01 - 0.12509E + 02 -

-

-

Note: Standard (RMS) error of the fit = 0 . 1 3 6 E + 0 2 ° K.

Table A3 Late reflood Summary of least squares fit obtained for Pk Cld Trap D A T A - Case # 3 The specified fit has a total of 60 terms and a redundancy of 3.07 Following are fit coefficients determined by SVDFIT Fit normalization constant (quadratic offset) = 0.0000 Constant Term Linear Term for Param Linear Term for Param Linear Term for Param Linear Term for Param Linear Term for Param Linear Term for Param Linear Term for Param Quad Term for Param Quad Term for Param Quad Term for Param Quad Term for Param Quad Term for Param Quad Term for Param Qu ad Term for Param Cross Term for Params Cross Term for Params Cross Term for Params Cross Term for Params Cross Term for Params Cross Term for Params Cross Term for Params Cross Term for Params

# # # # # # # # # # # # # # # # # # # # # #

1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 1 1 2 2 2 2 2

0.77295E + 03 0.17072E + 02 - 0.63157E + 01 -0.11441E+02 -0.22556E+03 0.22298E + 03 - 0.27400E + 02 - 0.19702E + 03 0.35255E + 01 0.53233E + 02 0.20924E + 01 - 0.23777E + 01 -0.11037E+03 0.27266E + 02 - 0.48828E + 02 0.44219E + 01 0.34157E + 00 - 0.89018E+ 01 - 0.86315E+ 01 0.22588E + 02 0.22435E + 02 - 0.1t149E + 02 0.10767E + 03 -

93

G.S. Lellouche et al. / Quantifying reactor safety margins - Part 4

Table A3 (continued) Cross Cross Cross Cross Cross Cross Cross Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic Quartic

Term for Params # Term for Params # Term for Params # Term for Pararns # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Pararns # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params # Term for Params #

3 3 3 4 4 4 5 2 2 2 2 2 4 4 4 2 4 3 3 3 4 2 2 4 4 4 4 4 4 2 2 2 3 3 3 4 5

5 6 7 5 6 7 6 2 2 2 4 7 4 4 5 5 5 7 5 5 6 2 2 4 4 4 4 4 4 3 4 5 3 3 3 4 5

0.34379E + 01 0.66447E + 01 - 0.59781E + 02 - 0.47766E + 02 - 0.24133E + 02 -0.18513E+02 0.77327E + 01 0.49421E + 02 0.49630E + 01 0.35897E + 02 0.29150E + 02 03 0.28640E + 02 0.12186E + 03 0.34805E + 02 0.63182E + 01 0.14840E + 02 0.69969E + 02 0.83087E + 01 0.67988E + 01 0.20335E + 02 - 0.67295E + 02 -0.51319E+01 0.43012E + 01 0.78349E- 01 0.49327E + 01 -0.15001E+02 -0.38516E+02 0.19959E + 02 0.68423E + 01 0.10338E + 02 0.63183E + 01 0.49197E + 01 - 0.22670E + 02 0.10564E + 02 - 0.16605E + 02 0.3866eE + 01 -

2 3 7 7 7 4 7 5 6 6 7 5 6 6 2 4 4 4 4 6 4 7 5 5 5 5 5 6 5 6

-

-

0

.

1

0

2

0

0

E

+

-

-

-

-

Note: Standard (RMS) error of the fit = 0.237E+02°K.

values of the p a r a m e t e r s c h o s e n are inserted in the regression m u l t i n o m i a l a n d a value of the P C T is f o u n d . T h i s process is r e p e a t e d m a n y times (50000 trials for e x a m p l e ) a n d a table of the P C T ' s is collected. T h e values are a c c u m u l a t e d in preselected b i n s (500 to 5 2 5 ° K ; 525 to 5 5 0 ° K ; 550 to 5 7 5 ° K ; etc.) a n d norm a l i z e d b y the total n u m b e r of trials; the result is a f r e q u e n c y h i s t o g r a m w h i c h is interpreted as a probability d i s t r i b u t i o n function. F r o m this h i s t o g r a m we m a y d e t e r m i n e the s t a n d a r d statistics desired ( m e a n , m o d e , 95th percentile, etc.). Because of the n o n - d e t e r m i n i s t i c n a t u r e of the process, a different choice of starting

r a n d o m n u m b e r (the i n p u t " s e e d " ) or the u s e of a different r a n d o m n u m b e r g e n e r a t o r or a c h a n g e in the n u m b e r of trials will p r o d u c e a slightly different final result. T h i s was d i s c u s s e d in the b o d y of this d o c u m e n t a n d depicted in figs. 7 - 9 . T h e reader t h e n s h o u l d n o t expect to be able to establish a deterministically exact r e p r o d u c t i o n of the n u m b e r s in tables 9 - 1 2 b u t o n l y a statistically equivalent o n e as implied b y the results in figs. 7 - 9 . T a b l e s A 4 - A 6 c o n t a i n the 5 0 0 0 0 trial e s t i m a t i o n of the distrib u t i o n f u n c t i o n s derived f r o m t h e regression s u r f a c e s listed in tables A 1 - A 3 .

94

G.S. Lellouche et aL / Quantifying reactor safety margins - P a r t 4

Table A4 Blowdown distribution functions S t a t i s t i c s f o r P k C l d T m p ( ° K ) b a s e d o n 5 0 0 0 0 histories: Sample mean value = 901.02594 Sample standard deviation = 89.32980 Prob Pk Cld Trap > 1000.00 = 0.13502 P r o b P k C l d T r a p > 1059.41 = 0.05000 P r o b P k C l d T r a p > 1130.21 = 0.01000

Table A5 Early reflood distribution functions Statistics for P k C l d T m p ( ° K ) b a s e d o n 5 0 0 0 0 histories: Sample mean value = 798.62964 S a m p l e s t a n d a r d d e v i a t i o n = 128.35620 P r o b Pk C l d T m p > 1000.00 = 0.07597 P r o b Pk C l d T m p > 1 0 3 2 . 7 4 = 0.05000 P r o b P k C l d T m p > 1128.96 = 0.01000

P k C l d T m p d i s t r i b u t i o n p r e d i c t e d b y fit a n d M o n t e C a r l o

Pk C I d T m p d i s t r i b u t i o n p r e d i c t e d b y fit a n d M o n t e C a r l o

Index

Pk Cld Tmp ( ° K)

PDF ( o K)

CDF

Index

Pk C l d T m p (°K)

PDF (°K)

CDF

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

688.00 712.00 736.00 760.00 784.00 808.00 832.00 856.00 880.00 904.00 928.00 952.00 976.00 1000.00 1024.00 1048.00 1072.00 1096.00 1120.00 1144.00 1168.00 1192.00 1216.00 1240.00 1264.00

0.75000E0.95000E0.56583E 0.12942E0.19642E 0.23700E 0.29400E 0.38216E 0.45891 E 0.46366E 0.44408E 0.38241E0.31408E0.23433E0.18008E0.12733E 0.89834E 0.66167E0.43083E 0.31250E 0.17833E0.64167E 0.13333E0.00000E + 0.00000E +

- 03 0 . 2 4 6 0 0 E - 02 0 . 1 6 0 4 0 E - 01 0 . 4 7 1 0 0 E - 01 0 . 9 4 2 4 0 E - 01 0 . 1 5 1 1 2 E + 00 0.22168E + 00 0 . 3 1 3 4 0 E + 00 0 . 4 2 3 5 4 E 4. 0 0 0.53482E + 00 0 . 6 4 1 4 0 E + 00 0 . 7 3 3 1 7 E + 00 0 . 8 0 8 5 5 E 4. 0 0 0 . 8 6 4 7 9 E 4. 00 0 . 9 0 8 0 1 E 4. 0 0 0 . 8 3 8 5 7 E + 00 0.96013E + 00 0 . 9 7 6 0 1 E + 00 0.98635E + 00 0 . 9 9 3 8 5 E + 00 0.99813 E + 0 0 0 . 9 9 9 6 7 E 4. 0 0 0.99999E + 00 0 . 9 9 9 9 9 E 4- 00 0.99999E + 00

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

524.00 560.00 596.00 632.00 668.00 704.00 740.00 776.00 812.00 848.00 884.00 920.00 956.00 992.00 1028.00 1064.00 1100.00 1136.00 1172.00 1208.00 1244.00 1280.00 1316.00 1352.00 1388.00

0.61111E-05 0 . 1 9 3 8 9 E - 03 0 . 6 5 4 4 5 E - 03 0.14611E-02 0 . 2 1 4 4 4 E - 02 0 . 2 6 4 3 9 E - 02 0 . 3 0 4 0 5 E - 02 0 . 3 0 0 8 9 E - 02 0 . 2 8 4 4 4 E - 02 0 . 2 5 7 3 3 E - 02 0 . 2 2 7 e e E - 02 0 . 1 9 6 7 8 E - 02 0 . 1 5 4 5 6 E - 02 0.10983E- 02 0 . 8 3 9 4 5 E - 03 0 . 6 3 8 3 4 E - 03 0 . 3 8 6 6 7 E - 03 0 . 2 1 2 7 8 E - 03 0 . 1 4 0 5 6 E - 03 0.65000E - 04 0.28889E - 04 0 . 7 7 7 7 8 E - 05 0.111lIE-05 0.00000E + 00 0.00000E + 00

0.22000E 0.72000E 0.30760E 0.83360E0.16056E + 0.255"ME + 0.36520E + 0.47352E + 0.57592E + 0.66855E + 0.75043E + 0.82127E + 0.87691E + 0.91645E + 0.94667E + 0.96965E + 0.98357E + 0.99123E+ 0.99629E + 0.99863E + 0.99967E + 0.99995E + 0.99999E + 0.99999E + 0.99999E +

05 04 03 02 02 02 02 02 02 02 02 02 02 02 02 02 03 03 03 03 03 04 04 O0 00

0.l 8000E

03 02 01 01 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

G.S. Lellouche et aL / Quantifying reactor safety margins - Part 4

Table A6 Late reflood distribution functions Statistics for Pk Cld Tmp ( ° K ) based on 50000 histories: Sample mean value = 676.86517 Sample standard deviation = 176.71304 Prob Pk Cld Tmp > 1000.000 = 0.04828 Prob Pk Cld Tmp > 997.47 = 0.05000 Prob Pk CId Trap > 1087.12 = 0.01000 Pk Cld Tmp distribution predicted by fit and Monte Carlo Index

Pk Cld Tmp (°K)

PDF (°K)

CDF

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

388.00 436.00 484.00 532.00 580.00 628.00 676.00 724.00 772.00 820.00 868.00 916.00 964.00 1012.00 1060.00 1108.00 1156.00 1204.00 1252.00 1300.00

0.83333E - 06 0.53750E - 03 0.19117E- 02 0.30008E- 02 0.26934E - 02 0.20050E - 02 0.16346E - 02 0.14171E - 02 0.13166E-02 0.12583E-02 0.12050E- 02 0.12162E- 02 0.10725E- 02 0.72792E- 03 0.48834E - 03 0.21500E- 03 0.84583E- 04 0.37500E- 04 0.95833E- 05 0.83333E- 06

0.400(~E - 04 0.25840E - 01 0.11760E+00 0.26164E + 00 0.39092E + 00 0.48716E + 00 0.56562E + 00 0.63364E + 00 0.69684E+ 00 0.75724E + 00 0.81508E + 00 0.87346E + 00 0.92494E + 00 0.95988E + 00 0.98332E + 00 0.99364E + 00 0.99770E + 00 0.99950E + 00 0.99996E + 00 0.10000E + 01

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