ML19211A749
| ML19211A749 | |
| Person / Time | |
|---|---|
| Site: | Clinch River |
| Issue date: | 12/14/1979 |
| From: | Friedland A, Markley R WESTINGHOUSE ELECTRIC COMPANY, DIV OF CBS CORP. |
| To: | |
| Shared Package | |
| ML19211A746 | List: |
| References | |
| WARD-D-0050, WARD-D-0050-R3, WARD-D-50, WARD-D-50-R3, NUDOCS 7912200584 | |
| Download: ML19211A749 (162) | |
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{{#Wiki_filter:.. , WARD-D-OO50, REV. 3 WARD-D Category 11 Clinch River Breeder Reactor Plant CRBRP CORE ASSEMBLIES HOT CHANNEL FACTORS PRELIMINARY ANALYSIS SEPTEMBER 1979 P0 3 El8lNAL Prepared for the United States Department of Energy t.nder contracts EY-76-C-15-2395 and EY-76-C-15-0003. Any Further Distribution by any Holder of this Document or of the Data Therein to Third Parties Representing Foreign Interest, Foreign Govern-ments, Foreign Companies and Foreign Subsidi-aries or Foreign Divisions of U.S. Companies Should be Coordinated with the Director, Division of Reactor Research and Technology, United States Department of Energy. W Westinghouse Electric Corporation ADVANCED RE ACTORS DIVISION B0X 158 f C7!L2 M ADISO N, PENNSY LV ANI A 15663 7oscuog
TITLE DOCUMENT NO. CRBRP Core Assemulies CHANGE CONTROL RECORD " " tor WARD-D-0050 Westinghouse Advanced Reactors Division el nal CHANGE REV N0/DATE RELEASE PAGES AFFECTED REMARKS DGCUMENT Rev. 2 - 11/78 N/A Entire Document Document completely rewritten Rev. 3 - iv, viii,1,15,19, Delete mention of "AFMS". 9/79 N/A 27, 31, 49, 58, 59, Clarify applicability to 67, 71, 84, 88, 97, heteroganeous core. 99, 104, 108, 114, Correct typographical 120, 142, 143, 144, errors. 145, Attachment p.17 1630 003
WARD-D-0050, Rev. 3 WARD-D Category II CRBRP CORE ASSEMBLIES HOT CHANNEL FACTORS PRELIMINARY ANALYSIS September 1979 PREPARED: wu A 4'I' 79 A. J.' Friedland, Date Core T&H Analysis APPROVED: eV 9 / !79 R. A. Markley, Mcp'ager Uate Core T&H Analysis 1630 004
i 1 I ARD-0 TX)50 ~1 REV 2 g INFORMATI0f1 CONCERflING USE OF THIS DOCUMENT l l PRELIMIflARY DOCUMENT l This document contains information of a preliminary nature prepared in the course of work for the U. S. Department of Energy. This infonnation is subject to correction or modification upon the collection and evaluation of additional data, it0TICE ' This document was prepared as an account of work sponsored by the United States Government. fleither the U. S. Department of Energy, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness or usefulness of any information, apparatus, product or process disclosed, or represents that its use would not infringe privately owned rights. .. . C WESTINGHOUSE ELECTRIC CORPORATION ADVANCED REACTORS DIVISION B0X 158 MADISON, PENNSYLVANIA 15663 1630 005 n
IARB-94 0050 . : .4 Rt1 2 ,' CONTRIBUTORS: A. Biancheria A. A. Bishop (Univ. of Pgh.) i M. D. Carelli F. C. Engel W. L. Howarth J. A. Lake R. R. Lowrie J. Marshall (WRL) D. Y. Nee H. P. Planchon . T. S. Roth E. C. Schwegler -( D. R. Spencer B. E. Sundquist G. H. Ursin 1630 006 i
- WARD-9905d .y TABLE OF CONTENTS SEV 2 C Page
1.0 INTRODUCTION
AND
SUMMARY
l 2.0 ANALYTICAL PROCEDURES 3 2.1 Treatment of Statistical Factors 5 2.2 Tolerance Interval 6 2.3 Overall Uncertainty 8 2.4 Sample Calculation 10 3.0 FUEL ASSEliBLIES HOT SPOT FACTORS 13 3.1 Fuel Rod Cladding and Coolant Temperature Hot Spot 13 Factors for Fuel Assemblies 3.1.1 Power Level Measurement and Control System 13 Dead Band 3.1.2 Cladding Circumferential Temperature 13 Variation 3.1.3 Reactor AT and Inlet Temperature Variation 14 3.1.4 Loop Temperature Imbalance 20 3.1.5 Inlet Flow Maldistribution 25 p 3.1.6 Flow Distribution Calculational Uncertainty 33 3.1.7 Wire Wrap Orientation 44 3.1.8 Subchannel Flow Area 44 3.1.9 Film Heat Transfer Coefficient 45 3.1.10 Pellet-Cladding Eccentricity 49 3.1.11 Cladding Thickness and Conductivity 49 3.1.12 Coolant Properties 51 3.1.13 Nuclear Uncertainties 53 3.2 Power-to-Melt Uncertainty Factors for Fuel Assemblies 59 3.2.1 Introduction 59 3.2.2 EBR-II Factors 62 3.2.3 Power-to-Melt in CRBRP 69-3.2.4 Typical Power-to-Melt Analysis 71 3.3 Coolant Exit Mixed Mean Hot Spot Factors for Fuel 81 Assemblies c 1630 007 i
V
$gy g TA8LE OF CONTENTS (Con't)
Page 3.4 Fission Gas Plenum Pressure Uncertainty Factors 81 for Fuel Assemblies 4.0 BLANKET ASSEMBLIES HOT SPOT FACTORS 83 4.1 Fuel Rod Cladding and Coolant Temperatures Hot Spot 83 Factors for Blanket Assemblies 4.1.1 Cladding Circumferential Temperature Variation 83 4.1.2 Inlet Flow Maldistribution 83 4.1.3 Flow Distribution Calculational Uncertainty 87 4.1.4 Subchannel Flow Area 88 5 4.1.5 Film Heat Transfer Coefficient 89 4.1.6 Pellet-Cladding Eccentricity 89 4.1.7 Nuclear Uncertainties 89 4.2 Power-to-Melt Uncertainty Factors for Blanket 97 Assemblies 4.2.1 Extrapolition Uncertainty 97 4.2.2 CRBRP Blanket Rod design Uncertainties 108 4.2.3 Total Blanket Rod Uncertainty and Design 108 Criteria 4.3 Coolant Exit Mixed Mean Hot Spot Factors for Radial 113 Blanket Assemblies 4.4 Fission Gas Plenum Pressure Uncertainty Factors for 113 Radial Blanket Assemblies 5.0 PRIMARY CONTROL ASSEMBLIES HOT SPOT FACTORS j14 5.1 Absorber Pin Temperatures Hot Spot Factors for 114 Primary Control Assemblies 5.1.1 Power Level Measurement and Control System j14 Dead Band 5.1.2 Inlet Flow Maldistribution 114 5.1.3 Assembly Flow Maldistributior Calculational 116 Uncertainties 5.1.4 Bundle / Bypass Flow Split 116 5.1.5 Cladding Circumferential Temperature 116 Variation 1630 008 ii
Rtv 2 TABLE OF CONTENTS (Con't)_ Pagg 116 5.1.6 Reactor AT and Inlet Temperature Variation 117 5.1.7 Absorber Ma1 distribution and Conductivity 118 5.1.8 Wire Wrap Orientation 118 5.1.9 Subchannel Flow Area 118 5.1.10 Film Heat Transfer Coefficient 118 5.1.11 Pellet-Cladding Eccentricity 119 5.1.12 Cladding %ickness and Conductivity 119 5.1.13 Gap Thickness and Conductivity 119 5.1.14 Coolant Properties 119 5.1.15 Nuclear Uncertainties 119 5.1.16 Overall Uncertainty 1 21 5.2 Coolant Exit Mixed Mean Hot Spot Factors for Primary Control Assemblies 1 21 5.3 Fission Gas Plenum Pressure Hot Spot Factors for Prinary Control Assemblies 1 21 5.4 Primary Control Assemblies Lifetime Structuraj Calculation Uncertainty Factor 123 6.0
SUMMARY
OF PRELIMINARY RESULTS AND FUTURE TESTS 142
7.0 REFERENCES
ATTACHMENT: STATISTICAL TOLERANCE LIMITS AND PROPAGATI OF ERROR 1630 009 iii
use-e+'5d 1 REV 3 - TABLES e-Page 2.1 Sample Calculation of Fuel Assembly Rod Cladding Miawall 12 Temperature 3.1 Plant Expected Operating Conditions and Uncertainties 18 3.2 Plant Expected Conditions Considered in Heterogeneous Core 19 Thermofluids Analyses 3.3 Results of Four IPFM Flow Distribution Tests at s100% Flow, 29 3 Loops 3.4 Results of Four IPFM Assembly-to-Assembly Flow Distribution 30 Test Results at N100% Flow, 3 Loops 3.5 Flow Assembly Inlet Flow Maldistribution Subfactors (3a) 32 3.6 Ratio of Side Subchannel Axial Average Velocity to Bundle 43 Axial Average Velocity 3.7 Suninary of Density Measurements for Liquid Sodium 52 3.8 Center Temperatures and Powers-to-Melt of Incipient Melt 61 Sections as Computed by LIFE-III Calibration T18/6000 3.9 Spatial Dependence of Power Uncertainty Calculations from 65 Comparison of Measured and Calculated Control Rod Worths 3.10 Values of Uncertainties in EBR-II 68 3.11 CRBRP Fuel Rod Tolerances and Uncertainties 70 3.12 Sensitivity of Power-to-Melt to Design Uncertainties 77 3.13 Statistical Combination of Power-to-Melt Uncertainties 78 Due to Design Uncertainties 3.14 Effect of Melting Point Uncertainty of TM
= 5.8 F on Un- 79 certainty of Power-to-Melt 3.15 Total Uncertainty on CRBRP Reactor Power-to-Melt in Re- 80 actor Power Units Where Nominal Full Power is 1.0 4.1. Results of Four IPFM Flow Distribution Tests at s100% 85 Flow, 3 Loops 4.2 Blanket Assembly Inlet Flow Maldistribution Subfactors (3o) 86 4.3 Blanket Irradiation Tests 98 4.4 Extrapolation Uncertainties for Blanket Rods 109 1630 010 iv
IARD-DT050 StV 2 TABLES (Continued) Page 4.5 CRBRP Blanket Rod Tolerances and Uncertainties 110 4.6 Blanket Rod Sensitivities of Power-to-Melt to Design Un- 111 certainties; Power-to-Melt Uncertainties Due to Individual Design Uncertainties; and the Statistical Combination of These Power-to-Melt Uncertainties 4.7 Total Uncertainty on CRBRP Reactor Power-to-Melt for 112 Blanket Rod in Reactor Power Units Where Nominal Full Power is 1.0 5.1 Control Assembly Inlet Flow Maldistribution Subfactors (3a) 115 5.2 Comparison of Hot Spot Coolant Temperatures Calculated 120 Using Current Uncertainties and Updated Uncertainties 6.lA CRBRP Fuel Assemblies Rod Temperature Engineering Uncer- 124 tainty Factors 6.18 CRBRP Fuel Assemblies Rod Temperature Nuclear Uncertainty 125 Factors 6.2A CRBRP Fuel Assemblies Mixed Mean Exit Temperature Engineering 126 Hot Channel Factors 6.2B CRBRP Fuel Assemblies Mixed Mean Exit Temperature Nuclear 127 Uncertainty Factors 6.3A CRBRP Fuel Assemblies Plenum Pressure Engineering Uncer- 128 tainty Factors 6.3B CRBRP Fuel Assemblies Plenum Pressure Nuclear Uncertainty 129 Factors 6.4A CRBRP Inner / Radial Blanket Assemblies Rod Temperatures 130 Engineering Uncertainty Factors 6.4B CRBRP Inner / Radial Blanket Assemblies Rod Temperatures 131 Nuclear Uncertainty Factors 6.5A CRBRP Inner / Radial Blanket Assemblies Mixed Mean Exit 132 Temperature Engineering Uncertainty Factors 6.53 CRBRP Inner / Radial Blanket Assemblies Mixed Mean Exit 133 Temperature Nuclear Uncertainty Factors 6.6A CRBRP Inner / Radial Blanket Assemblies Plenum Pressure 134 Engineering Uncertainty Factors 6.6B CRBRP Inner / Radial Blanket Assemblies Plenum Pressure 135 Nuclear Uncertainty Factors y 1630 011
. . WARD-D-0050 MV 2 TABLES (Continued) Page CRBRP Primary Control Assemblies Pin Temperatures Hot 136 6.7 Channel / Spot Factors CRBRP Primary Control Assemblies Mixed Mean Exit Temper- 137 6.8 ature Hot Channel Factors CRBRP Primary Control Assemblies Plenum Pressure Hot 138 6.9 Channel Factors CRBRP Primary Control Assemblies Lifetime Structural Cal- 139 6.10 culations Uncertainty Factors Future Core Assemblies T&H Tests 140 6.11 1630 012 vi
WARO-5dOO50 SEV 2 FIGURES Page 2-1 Graphical Illustration of Semistatistical Method 9 3-1 Deviations in Inlet Module Steady State Mixing Ratio from 21 Perfect Mixing for 3 Loop 100% Flow and Injection in Loop
#1 3-2 Deviations in Inlet Module Steady State Mixing Ratio from 22 Perfect Mixing for 3 Loop 100% Flow and Injection in Loop #2 3-3 Deviations in Inlet Module Steady State Mixing Ratio from 23 Perfect Mixing for 3 Loop 100% Flow and Injection in Loop #3 3-4 2a Inlet Module Temperature Uncertainty Based cn a Loop 24 Temperature Imbalance of up to 34 F, for 3 Loop Operation 3-5 Fuel Assembly Inlet Nozzle Flow Resistance Versus Reynolds 26 Number; Orifice Group Number 5 3-6 Core Flow Distribution Map, Run A5.2, i .8%, 3 Loop Flow, 28 140 F 3-7 12 Inch HEDL Experimental Data 34 3-8 Comparison Between COTEC Results and HEDL Data at 24 Inches 35 3-9 Comparison Between COTEC Results and HEDL Data at 36 Inches 35 3-10 ORNL Test Series #2, Test #02, Run #105 36 3-11 ORNL Test Series #2, Test #02, Run #109 37 3-12 ORNL Test Series #2, Test #03, Run #201 38 3-13 ORNL Test Series #2, Test #04, Run #205 39 3-14 Comparison of Predicted and Experimental Results for 46 P/D = 1.3 3-15 Comparison of Predicted and Experimental Results for 47 P/D = 1.25 3-16 Comparison of Predicted and Experimental Results for 48 P/D = 1.2 3-17 LIFE-III Thermal Calibration T18/6000 60 3-18 Control Rod Worths for EBR-II Run 27A 64 1630 013 vii
WARD-D T 50 1 Rit s FIGURES (Continued) Page 3-19 Power History Used in the Evaluation of Power-to-Melt 72 Uncertainties 3-20 Variation of Power-to-Melt with Pellet Diameter 74 4-1 Comparison of Predicted and Experimental Results for 90 P/D = 1.15 4-2 Comparison of Predicted and Experimental Results for 91 P/D = 1.1 4-3 Comparison of Predicted and Experimental Results for 92 P/D < 1.05 4-4 Power-to-Melt Study, IB/A 128, Pin 27, X/L = 0.466 99 EOC4 3c Operating Conditions 4-5 Comparison of Computed and Observed Central Hole Radii and 105 Melt Radii 4-6 Comparison of Calculated and Predicted Central Hole Radii 106 for Mechanical Calibration Pins 1630 014 viii
WARD-D-00$0 REV S
1.0 INTRODUCTION
AND
SUMMARY
The impact of uncertainties associated with theoretical and experimental analyses, instrumentation accuracy, manufacturing tolerances, physical pro-perties and physical phenomena correlations must be considered in predicting the reactor thermal-hydraulic performance to ensure the safe and reliable operation of the CRBRP core and to guarantee that proper margins are pro-vided to meet both lifetime and safety design limits and requirements. The effects of the various uncertainties are accounted for in the " hot channel factors" or " hot spot factors" analyses. The basis of such analyses is to characterize a hot channel as the one affected by the simultaneous occurrence of all uncertainties and thereby predict the temperature of the hot channel or hot spot, which will be compared with the required limits. The use of hot channel factors has evolved over a long period of time, initially in light water reactors and more recently in liquid metal cooled fast breeder reactors. This report documents the methodology, supporting bases including data base and computed values for the hot channel factors used in the heterogeneous core thermal-hydraulic analyses. The analytical procedures are discussed in Section 2.0. The hot channel factors for the fuel and blanket assemblies for the heterogeneous core are discussed in Sections ').O and 4.0, respectively, and are summarized in Sectic 6.0. The not channel ft . ors for the primary control assem-blies are preliminary values which were developed for the homogeneous core and are being updated. They are discussed in Section 5.0 and are also summarized in Section 6.0. The results presented here are preliminary in nature. Changes in plant design and operating conditions. growth of the supporting data base and ongoing development of analytical methods will result in continued updating as the design proceeds. Final results will be presented in the FSAR. The hot channel factors used ir, calculations of the coolant temperature, cladding temperature, and gas plenum pressure for the CRBRP fuel, blanket, and primary control assemblies are presented in Tables 6.lA through 6.10. Results of uncertainty analyses associated with fuel power-to-melt c.alcu-lations are addressed in the appropriate sections of this report. 1630 015
- 21 g0 paid 056 For convenience, the un _ d.ci:o e .ssociated with the fuel and blanket as-semblies are grouped acc. engineering (table numbers with the "A" suffix) and nuclear consiw ons (table numbers with the "B" suffix). Note that 3a statistical values are applied to the plant thermal-hydraulic design conditions when pcri a. ety analyses. The 2a level is applied to plant expected operating c; tions when calculating parameters such as cladding temperature and pressure, which are input to replaceable core as-sembly lifetime analyses. The hot channel factors in this report are ap-plicable at full power steady state conditions. The 3a uncertainties as-sociated with the tt.rmel-hydrau'ic values are conservatively used as the initial conditions fot urgency and faulted transient and safety analyses. The hot channel factors reported in Tables 6.lA through 6.10 are considered to be conservative. This will be further established as additional data become available from tests which are planned or in progress. These include: additional irradiation experiments on fuel, blanket, and primary control rods; engineering mockup critical experiments in the ZPPR; pressure drop' tests of fuel, blanket, and control assemblies and orifice configurations; heat transfer tests on a 61 fuel rod bundle in sodium; cross flow measurements in a 5:1 scale blanket rod bundle sector in air; and heat transfer tests in sodium on a prototypic blanket rod bundle. These data will be included in the FSAR analyses. A more detailed tabulation of the forthcoming T&H tests is given in Table 6.11. 1630 O!6 2
8400F0 2.0 ANALYTICAL Ph2EDURE The impact of theoretical and experimental analytic uncertainties, instru-mentation accuracy, manufacturing tolerances, physical properties and correlations uncertainties must be considered in predicting the reactor thermal-hydraulic performance to ensure the safe and reliable operation of the CRBRP core and to guarantee that proper margins are provided so as not to exceed the design limits and requirements. The effects of the various uncertainties are accounted for in the " hot channel factors" or " hot spot factors" analysis. The basis of such an analysis is to characterize a hot channel as the one offected by the simultaneous occurrence of all uncertainties and thereby predict the temperature of the hot channel or hot spot, which will be compared with the required limits. . Consistent with PWR, FFTF and Westinghouse proposal studies, the semi-statistical hot spot analysis is used for the CRBRP. Use of the semi-statistical method requires the separation of the variables which cause the hot spot temperatures into two principal categories; one of statistical origin and the other non-statistical . The two categories are defined below. A non-statistical (or direct) uncertainty is defined as a variation in a design variable which is known or conservatively assumed to occur, but is not included in the nominal analysis. Computationally, the non-statistical factors represent multipliers applied to the nominal magnitudes of the variables to provide the worst values which will occur. A variable which has a random frequency distri-bution of occurrence is treated statistically. For exartple, experimental data (such as materials properties) are treated statistically since there is random error in their evaluation. Direct and statistical factors employed in deriving CRBRP fuel rod hot channel / spot temperatures, for example, are shown in Tables 6.lA and 6.1B. The di,act factors in these tables are related to effects that are assumed to occur with certainty. That is, power level measurement and control system deadband, inlet flow maldistribution direct bias, subchannel flow distribution 1630 017 3
- 1 IARD-010050 REV 2 I simulation bias, cladding temperature increase beneath the wire wrap, pellet-cladding eccentricity, physics analysis calculational methods and control rod effects are direct consequences of the design and result in known temperature non-uniformities at some time during core life. The statistical factors, on the other hand, are based on random occurrences (i.e., timewise, spatial-wise or data / tolerance distribution) such that the probability of departure from the nominal value is given by a probability distribution. Typical of such random effects are manufacturing variability of fissile fuel content, dimensional tolerances, instrumentation uncertainties, and experimental uncertainties in measured values. In the CRBRP statistical factors, manu-facturing variability and dimensional tolerances affect fissile fuel maldis-tribution, rod diameter and pitch, and cladding thickness; instrumentation uncertainties affect power level measurement; and uncertainties in experi-mental data affect nuclear power distribution, film coefficient, cladding conductivity and power-to-melt.
The hot channel factors may be grouped into categories which indicate the basis for each factor: Statistical - The data base is sufficient to provide a statistical basis for the hot channel factor. Empirical or Bounding - The data base is small and/or only a bounding value can be calculated. Engineering Judgement - Little or no data are available. Analytical - Represents analytical adjustments to the nominal analysis rather than uncertainties. This report provides the basis for each factor and subfactor. 1630 018 4
0050
~
2.1 Treatment of Statistical Factors g Before describing the method of combining direct and statistical factors, some discussion of the treatment of statistical factors is warranted. If a temperature, T, can be expressed as a linear function of independent randosa variables, xj , T=a g+ a) x) + a2 *2 + ... + an *n (2.1) then the variance of T is equal to the sum of the variances of the a jjx 's, 2 o = a a (2.2) i=1 If the incependent variables have normal distributions then T will have a normal dis;ribution. If the indapendent variables do not have normal distri-butions, T will still tend to approach a normal distribution as the number of independent variables becomes large. If T is an arbitrary function of the independent variables, T = T(x), x2 ' *** *n) (2.3) then T takes the value T when each variable is at its mean value, xj . Since the concern is with what happens to the temperatur a when a variable departs from the mean by some arbitrary amount axj , the temperature can be approxi-mated by a Taylor series expansion about T = T in which only linear terms are retained: T-T 2 ax j (2.4) i=1 i where is evaluated at x) = xj for j = 1, 2, . . . n. Dropping the higher order terms results in negligible error provided departures from the mean are relatively small, as is the case with most hot channel factor applications. Substituting Equation (2.4) in (2.2) results in c 2y T
} (Rax)2 o 2
i (2.5) j,) j 1630 Ol9 5
IARD-OtK)50 i REV 2 If T has a normal distribution, then the probability that T + 2aT will not be exceeded is 97.73% and the probability that T + 3a will not be exceeded T is 99.87%. One type of uncertainty comonly encountered is that of a variable, such as a dimension, with specified counds. In such a case, it is reasonab1; to assume that the mean value has a unifom probability of lying anywhere within the bounds. For such a rectangular distribution, the bounds correspond to
+fe . The actual distributions of dimensions of fabricated components will be used where available for the FSAR uncertainty analyses. As previously noted, as the number of independent variables becomes large, the distribution of their sum approaches a normal distribution. If some of the independent variables have rectangular distributions and some have normal distributions, then the 2a or 3a coverage calculated using Equation (2.5) will be greater (more conservative) than if the distributions were all nomal with the same standard deviations. ,
2.2 Tolerance Interval The use of Equation (2.5) implies that the variances of the independent variables are known. This is often not the case, and it is necessary to infer the variance of the population from a sample set of data. If the number of observations is large, the variance of the sample, I I (xjj-x)j s , p1 2 (2.6) n, - 1 approaches the variance of the population. If the number of observations is small, it is still possible to give tolerance limits on the possible values M T. A detailed discussion is given in the Attachment to this report, and is summarized in the remainder of this section. If the distribution of T is normal, and a set of n observations on T is available, statistical theory provides two classical methods for determining the coverage in hypothetical repeat samples. The first type of tolerance interval is called the 8-expectation type and determines the average coverage. The second type is called the 8-content type and determines the probability that the coverage is at least a given amount. The first type will be used 1630 020 6
IARO-B W 59 Rtt ~2 for the hot channel factor analyses. This is felt to be more aporopriate, as the coverage, 8, is more directly related to the coverage for the known variance case. The upper tolera~nce limit is given by T+tg (n - 1) /1 + 1/n S (2.7) w!;&.~e tg (n - 1) is found from a table of the t-distribution. This is a value which an n-1 degree of freedom t-variate will exceed with probability 1-8 For the case where T is a linear function of xj 's (Equation 2.1), with a random sample of size jn on each xj , Satterthwaite's approximation is used to determine the degrees of freedom to be associated with a linear combination of variance estimates. The upper tolerance limit is given by T+tg (v,) S, (2.8) where S, = [ I Sj ) (nj +1)/nj ] (2.9) i=1 v,= (S,2)2f g " (,2 S,2(n,+1)/n,)2/vj ] (2.10) i=1 n i x, = ( r x j3)/n; (2.11)
]=1 S j =( (x 43 - xj )2)/(n$ - 1) (2.12) j=1 Using available t-tables, as a close approximation of the 2a and the 3e limits which are used for known o's, s = 0.975 is taken to correspond to "2c" and 8 = 0.999 is taken to correspond to "3a".
The interval computed by Equation (2.8) will have coverage 8 on the average. Another interpretation is that Equation (2.8) yields a 100s% confidence limit for the next observation on T. 1630 021 7
1ARD-09050 REV 2 8 2.3 Overall Uncertainty Calculation of fuel rod temperatures for the hot channel is shown in Figure 2.1. The nominal temperature difference AT nom is multiplied by the product of the direct hot channel factors to give the temperature rise including non-statistical factors AT Using this value, the nominal temperature T nom dir. represents the mean value about which is increased to Tdir (or T0o). T dir the statistical uncertainties are applied. The nominal channel can be any channel in the assembly, with the nominal channel coolant AT calculated using subchannel analysis codes. In practice, the channel with the highest coolant temperature is singled out for application of hot channel factors, thus calcu-lation of hot channel temperatures. The hot channel factors used in the CRBRP analyses are shown in Tables 6.1A through 6.10. Independent variables appear on separate horizontal lines. Hot channel factors appearing on the same line are not independent, and their overall effect must be calculated directly before combining statistically with the other factors. For example, in Table 6.lA, the statistical effect of pellet-cladding eccentricity affects both film and cladding AT's. Therefore, the overall uncertainty in film and cladding AT's is calculated directly beforo being combined statistically with the remaining statistical temperature uncertainties. This will be illustrated by the sample cal.culation in the next section. It may be noted that a more conservative procedure was followed in the PSAR analyses. In those analyses, the statistical uncertainties for a given com-ponent (coolant, film or cladding) were combined statistically and the resulting temperature uncertainties for each component were added directly to provide the overall temperature uncertainty. This " vertical-horizontal" procedure was conservatively adopted for convenience in calculation and to provide additional margin in the early stages of design. However, the present procedure is more exact, and is being used for future analyses. Several approximations have been mentioned in the analytical procedure: a) the variables are assumed to have normal distributions when calculating the probability of exceeding the 2a or 3a value; and b) the temperature is assumed to be a linear function of the variables. 1630 022 8
IARD-0*0050 REV 2 9 a__ - l A ____ a
^
o
. _------------------- gi ,
a - e , t 2 e
=
s O 2
.e m g 3 g a
e s r 2
=
w y - _U a s
=
, a -
z.-
.2 a
w o O m e ~ o e E E m E o Z Jk 5 o 2 i 1630 023 9926-1 9
gy-pa > Regarding Item (a), the non-normal probability distributions most conmonly encountered are rectangular distributions. As previously noted, this leads to conservative results, i.e., the probability of exceeding the 2o or 3a values is lower if some of the variables have rectangular probability distri-butions than if they all have normal distributions. Regarding Item (b), the most significant non-linearities are encountered in fuel power-to-melt analyses. In Section 3.2, it is shown that even for the most significant non-linearity, the relation between power-to-melt and pellet diameter, the linearization selected is a close approximation of the actual relation over the range of interest. Furthennore, as noted in the previous paragraph, conservatism is provided by the rectangular probability distribution associated with the pellet diameter. For the FSAR analyses, it is planned to include the effects of non-normal probability distributions and non-linearities directly in the analyses. 2.4 Sample Calculation To illustrate the analytical procedure, assume that the 2e cladding midwall (M.W.) temperature for plant expected operating conditions is to be calculated at the 1:,p of the core for a typical beginning-of-life first core fuel assembly not adjacent to a control rod, with a 1.0 ZPPR-7 flux tilt factor and with the cladding circumferential temperature variation factor of 2.1 on the film AT and 0.9 on the cladding (M.W.) AT. Assume that the nominal AT's for the hot channel calculated using COTEC and NICER are: Coolant AT: 420 F Film AT: 12 F Cladding (M.W.)AT: 13 F Applying the direct factors in Tables 6.lA and 6.18, the Oo AT's are: Coolant AT: 459.1 Film AT: 32.4 Cladding (M.W.)AT: 15.0 1630 024 10
- ' ARD 8 * ' Irv a %n55 Applying the statistical hot channel factors from Tables 6.lA and.6.lB to these Oo AT's, the temperature uncertainties shown in Table 2.1 are calculated. These uncertainties are added horizontally, and the resulting independent temperature uncertainties in the right-hand column are combined statistically to give the overall 3a uncertainty of 87.3 F. Since the 2a uncertainty is required, 2/3 of this value is taken ( 58.2*F). Adding the AT's to the first core nominal plant expected inlet temperature of 704 F, results in the follow-ing maximum cladding midwall temperatures in this fuel assembly: Nominal: 1149*F Oo: 1210.5'F 2o: 1268.7'F The above analysis is based on using the variance of a sangle as an estimate of the variance cf the total population. Applying the tolerance interval method previously discussed to those uncertainties which are based directly on the variance of small samples, and referring to the discussions in Section 3.1 on the bases for the hot channel factors, the inlet flow maldistribution factors of 1.059 on coolant aT and ' i on film AT and the coolant properties factor of 1.017 on coolant AT may be broken down into subfactors (where n is the number of data points): HCF (3a) n Coolant AT Film AT 1.050 1.014 - Flow Ma1 distribution 4 1.020 1.005 33 1.024 1.006 7 C001 ant Properties .009 -- - 1.01 5 -- 5 6 Repeating the analysis in Table 2.1 and applying Equations (2.9) through (2.12), v = 13103 e 3Se = 87.6 The effective three standard deviations, 3S,, is nearly identical with that cal ulated in Table 2.1, the effective degrees of freedom, ve, is extremely lankt, and therefore the results are essentially the same as previously ana.yzed. H 1630 025
TABLE 2.1 SAMPLE CALCULATION OF FUEL ASSEMBLY R0D CLADDING MIDWALL TEMPERATURE COMPONENT TEMPERATURES, *F INLET COOLANT FILM CLADDING T AT AT (M.W.)AT TOTAL NOMINAL: 704 420 12 13 1149 Oo: 704 459.1 32.4 15.0 1210.5 3a TEMPERATURE UNCERTAINTIES, *F Reactor AT and Inlet Temperature Variation 62.2 62.2 Inlet Flow Ma1 distribution 27.1 0.5 27.6 Loop Temperature Imbalance 7.4 7.4 % Wire Wrap Orientation 4.6 4.6 Subchannel F1 Area 8.7 8.7 Film Heat Transfer Coefficient 3.9 3.9 Pellet-Cladding Eccentricity 5.6 2.6 8.2 Cladding Thickness and Conductivity 0 0 Coolant Properties 7.8 7.8 Flow Distribution Calculational . Uncertainty (Calibration) 26.6 0.2 26.8 Experimental (Nuclear) 32.1 2.3 1.1 35.5 Criticality 4.6 0.3 0.2 5.1 Fissile Fuel Maldistribution 23.9 1.7 0.8 26.4 Overall 3a = [I(3aj )23 1/2 = 87.3 EE
*=
ms u -o3 O ve o O' N Ch
- ' IARD-O*005n ' RW 2 3.0 FUEL ASSEMBLIES HOT SPOT FACTORS 3.1 Fuel Rod Cladding and Coolant Temperatures Hot Spot Factors for Fuel Assemblies The hot spot factors for evaluation of CRBRP fuel assemblies rod cladding and channel coolant temperatures are presented in Tables 6.l A and 6.18. 3.1.1 Power Level Measureinent and Control System Dead Band This bounding factor is 1.03, which is a reactor system design requirement. It is applied as a direct factor to thermal hydraulic design value (THDV) condi tions . It does not appear directly for plant expected operating value (PE0V) conditions which are used for core assembly lifetime calculations. For PE0V conditions, its subfactors are included statistically in the Inlet Temperature Uncertainty and in the Reactor AT Variation. A total bounding margin cf 15% steady state overpower is provided between the reactor rated power conditions and the first reactor trip. This steady state overpower margin defines the reactor short time design conditions and is used, in conjunction with 3a hot channel factors, to determine if fuel centerline melting is attained. The power level measurement and control system dead band factor is not applied simultaneously with the 15% overpower. 3.1.2 Cladding Circumferential Temperature Variation In a close packed rod bundle array, the coolant velocity and temperature dis-tribution in the subchannel formed by three adjacent rods is not uniform, but varies circumferentially around the rod. Consequently, the cladding tempera-ture has a circumferential distribution, depending on the P/D radio, the rod power rating, the coolant and cladding thermal conductivities and the bulk coolant velocity. The maximum cladding temperature occurs at the minimum gap between the rods, which is consequently the hot spot. The presence of the wire wrap further skews the peripheral temperature distribution, enhancing the hot spot magnitude. Analyses for CRBRP fuel assemblies using the FATHOM-360 computer program showed that the analytical hot spot factor is in the range of 1.6 to 1.9 across the film (depending upon assembly operating con-ditions), approximately 0.8 from cladding OD to midwall and approximately 0.9 from cladding OD to ID. 13 1630 027
~~
- - WARD-D Ongn
- REV 2 The analy as conservatively assumed uniform heat flux at the cladding inner surface and the coolant temperatures were determined by solving the turbulent momentum and energy equations.
A detailed analysis is performed for the lifetime-limiting assembly in each flow zone. In addition, a 20% uncertainty which is a bounding value based on limiting case analyses is applied to the deviations from the nominal case to conservatively account for uncertainties resulting from finite difference modeling, flow distribution uncertainties and flow mixing uncertainties. Hot spot temperatures have been experimentally measured at HEDL[2] and ORNL[3] on wire-wrapped, electrically heated pins of nominal CRBRP dimensions. The HEDL 7-pin tests indicate the ratio of hot spot / nominal film temperature drop is approximately 21*F/18 F. The ORNL 19-pin tests indicate the ratio is approximately 27 F/17 F. Since these data result in film subfactors of about 1.2 and 1.6, the previously mentioned application of 1.6 to 1.9 for the nominal film subfactor is conservative. The effect of the cladding circumferential temperature variation and of the wire wrap is accounted for if calculating the local cladding temperature. For maximum fuel temperature calculations, however, an average cladding inner diameter temperature is the most appropriate boundary condition; thus, the cladding circumferential variation factor in this case, is equal to 1 by definition. 3.1.3 Reactor AT and Inlet Temperature Variation Two sets of plant conditions are used in the thermal-hydraulic design, i.e. , plant thermal-hydraulic design values (THDV) conditions and plant expected operating values (PE0V) conditions. The THDV conditions (730 F inlet /995 F 6 outlet temperature; total reactor flow 41.446 x 10 lb/hr) are the Clinch River rated plant conditions and therefore no plant system uncertainties ap-ply to these values. They are used in: a) analyzing permanent components which have the same 30 years lifetime as the plant's; b) transient and safety analyses, since they are more conservative than the plant expected conditions. This latter set represents the plant conditions at which the CRBRP is expected to operate accounting for the operating conditions of the heat transfer systems such as pump characteristics, primary loop pressure drop uncertainties, fouling y 1630 028
-
- IARD-0-0050 REV 3 and plugging of heat exchangers, etc. Expected operating values for the primary helt iransport system principal parameters (inlet, outlet tempera-ture and AT) are thus evaluated, together with the associated uncertainties.
The results of this study, which comprised a Monte Carlo type analysis, are reported in Table 3.1. The study included: 1) the consideration of the pro-gressive fouling of the heat exchangers during the plant 30 years lifetime, which affects the predicted values of the plant operating conditions, and
- 2) a comprehensive tccounting of .'1 uncertainties affecting plant operation.
Empirical or bounding values were used for the effect of fouling and for the uncertainties affecting plant operation. Plant expected operating conditions are adopted in core thermofluids analyses of replaceable components, such as the core assemblies, chiefly in determining the fuel rod parameters (cladding temperature, fission gas pressure) which are the basis for evaluating the structural behavior and for assessing whether lifetime /burnup objectives are actually met. Plant expected operating conditions adopted in the first and second core thermal-hydraulic analyses are reported in Table 3.2. Following is a brief discussion of the rationale in determining the values reported in Table 3.2 from the ones in Table 3.1. PE0V conditions will be updated for the FSAR. First, the mean values of Table 3.1 are chosen as the nominal values of Table 3.2, thus including the bias factor directly into the nominal values. Since the most critical times for core assemblies is at the end-of-life, when the cladding strain ard damage function are maximum, first core values have been selected as corresponding to two-year fouling and second core values as corresponding to four-year fouling conditions. Due to the fact that four-year fouling conditions were not evaluated, it was conservatively as-sumed that the same difference in plant parameters between year two and year zero rep ts between year four and year two. Again, the selected approach is conservative for two reasons: 1) plant conditions have been considered constant over the two years span and equal to the worst end-of-span con-ditions, thus neglecting the more favorable conditions which exist through-out the core lifetime; and 2) the effect of fouling is not linear with time, but it is rather pronounced at the beginning and then tapers off during the plant lifetime, as can be seen by comparing plant parameters in 1630 029
- 4 IARD-Dhici i REV 2 Table 3.1 for 0, 2, and 30 years. Thus, the assumption that the same dete-rioration of plant conditions which occurs in the first two years (first core) also occurs during the third and fourth year (second core) is very conservative. If primary cold leg temperature (TRi) and primary AT (AT R ) were statistically independent, then the primary hot leg temperature (TRo) variance would be related to the other variances by 2 2 (3.1) cT Ro
" "T Ri 2 + "AT R Since o T is smaller than the value given by Equation (3.1), .it is e,Hent g
that there is a negative correlation between TRi and AT R
, i.e., higher-than-average values of T tend to be associatee with lower-than-average values Ri of ATR . The more general form of Equation (3.1) which covers this case is
+ + ( }
"T T Ri AT TRi, ATR Ro R where o is the covariance between TRi and aTR . From the two-year Tg , ATR fouling data in Table 3.1,
= 169 oT h
oT = 121 Ri c aTp
" I44 from which 2"TRi, ATR The coolant temperature rise in a given fuel asserrbly channel (AT c
) is pro-portional to AT R aT c
= AT R
(3.3) From the definitions of variance and covariance, 2
,AT 2 . c ,AT 2
1630 030 c R m
'M IARD-D"0050 "TR1, ATc TRi, ATR Therefore, the variance in Tcis given by "T T AT TRi, ATc c R1 c
" +C,2 2 + 2C (3.5)
Tg Tg , AT c From the two-year fouling data in Table 3.1, oT 144 C2 - 96 C + 121 (3.6) c Also, ATR = 250 so that Equation (3.6) may be written as 0.002304 AT - 0.384 aTc + 121 (3.7)
"T c c
The o's in Table 3.1 show little variation with fouling and therefore Equation (3.7) wil' be used for the second core as well as the first core. Typical HCF values on AT c are tabulated below: AT "CY (E"} c "Te 250 13 1.156 300 14.6 1.145 400 18.3 1.137 500 22.5 1.135 1630 031 17
e TABLE 3.1 PLANT EXPECTED OPERATING CONDITIONS AND UNCERTAINTIES Clean & Unplugged Heat Fouled & Plugged Heat Exchangers Exchangers Parameter Estimated (2 Year Fouling) (30 Year Fouling) (New Plant) Nominal Mean a T 97.7 Nominal Nan a T 97.7 Nominal &an a T 97.7 Primary Hot Leg 943 946 13 968 950 954 13 976 960 964 13 987 Temperature, TRo ( F) Primary Cold leg 698 697 13 722 705 704 11 725 714 714 12 736 Temperature, TRi ( F) Primary AT, ATR ( F) 245 249 12 273 245 250 12 274 246 250 12 275 Power (MWt) 975 975 1004 975 975 1004 975 975 1004 NOTE: Design and control uncertainties are included The T values are those that are not exceeded with 97.7% probability. If the distribution 97.7 were normal these would equal the mean plus 2a.
~
e se see > U 4 as 19 Y
==
a u N
WARD-07050 '4 REV 3 f '
,e TABLE 3.2 PLANT EXPECTED CONDITIONS CONSIDERED IN HETEROGENEOUS CORE THERM 0 FLUIDS ANALYSES First Core Second Core Nominal Inlet Temperature (*F) 704 711 Nominal Reactor AT (*F) 250 250 1630 033 19
IARD-OiO56 StV 2 3 .1. 4 Loop Temperature Imbalance The maximum cold leg loop-to-loop temperature imbalance is 34*F, which a plant control system design requirement. Loop imbalance mixing tests were performed in the Inlet Plenum Feature Model at HEDL . A test run consisted of injecting an electrolyte in a downcomer and measuring the steady state concentration in the inlet modules. Individual loop in-jection permitted the assessment of how that loop affected mixing in the 61 inlet modules. From these data, inlet module fluid temperatures due to loop temperatures imbalances were calculated. The results are shown in Figures 3.1 through 3.3. A Monte Carlo type of analysis was performed in which the temperature im-balances in two of the loops were chosen at random (all values equally pro-bable) and the third one adjusted so that their sum was zero, subject to the restraint of a maximum difference of 34 F between loops. The resulting inlet module 2o temperature uncertainties are shown in Figure 3.4. The maximum 2a value for ali inlet modules is 4.6 F. The uncertainty in that value because of the experimental error in the measured loop mixing constants is +0.3 F giving an overall maximum 2a bounding uncertainty of 4.9 F on inlet tempera-ture. Even though the analysis shows that this applies to only the worst module (7 core assemblies), it has conservatively been applied to all assemblies. 1630 034 20
WARD-D-0050 REV 2 PERFECT MIXING RATIO = 1/3 00 300 INLET DOWNCOMER 3 2700 IN LET E22 01 33 00WNCOMER1 . 11
.07 A 00 2700 c
Re 1800 PL AN VIEW 15G3 INLET 00WNCOMER 2 Figure 3-1. Desiation in inlet Slodule Steady State Mising Ratio from Perfect Mixing for 3 Loop 100'; Flow and injection in Loop ::1. 9926-2 1630 035 21
.
- WARD-D-0050 REV 2 PERFECT MIXING RATIO = 1/3 00 300 INLET DOWNCOMER 3 El E24 .
E2a E2 2700 INLET . oc 7 DOWNCOMER1 E22 . D1 oo
..t o 03 0 0C11 0 CIO 0C2 B2 0 0ES E6 2100$ M .io .04 . iy 900
@%@,e@ o C6 E8 C7 D7 OE76
.11 011 .04 08 o ODIO e OE10 e CE12 \ l 1500 INLET 1800 00WNCOMER 2 PL AN VIEW Figure 3-2. Desiation in inlet \lodule Stead.s State \lising Ratio from Perfect \lising for 3 Loop 100 Flow and injection in Loop = 2.
9926-3 1630 036 22
- 6 WARD-D-0050 REV 2 PERFECT MIXING RATIO = 1/3 00 300 INLET 00WNCOMER 3 2700 INLET DOWNCOMER1 E22 13 D1 .10
.17 .13 00 2700 s c eo ec
.11 1500 IN LET 00WNCOMER 2 1800 PLAN VIEW Figure 3-3. Deviation in inlet N1odule Steady State .\lising Ratio from Perfect Mising for 3 Loop 100'; Flow and injection in 1;>op =3.
99264 23 1630 037
' IARD-91050 REV 2 2a lNLET MODULE TEMPERATURE UNCERTAINTY IN OF 00 300 INLET 00WNCOMER3 E24 41 NP lb El 4.4 3.3 00WF'_MER1 W"g@ e@
A g69 2700 El.ll:s5c 180a PLAN VIEW 1500 INLET DOWNCOMER 2 Figure 3-4. 2a inlet Module Temperature Uncertainty Based on a Loop Temperature Imbalance of up to 340F,for 3-Loop Operation. 99265 1630 038 24
EARD-D 0050 - 3.1.5 Inlet Flow Ma1 distribution REV 2 This factor accounts for uncertainties in the calculated assembly flow due to flow maldistributjon in the lower and upper plena, internals structure tolerances (both these effects cause variation in the nominal static pressure profile), assembly orificing uncertainties, and loop flow imbalances.
%e individual subfactors are shown in Table 3.5 and can be categorized under four headings:
- 1) Component Hydraulic Characteristics
- 2) Manufacturing Tolerances
- 3) Configuration
- 4) Systematic Error Interaction Component Hydraulic Characterization Variations due to component hydraulic characterization result from pressure drop and flow rate measurement errors du"ing component flow testing and the accuracy with which the test article geametry is measured.
Systematic errors for the fuel assembly inlet and outlet nozzle flow tests [5] are estimated as Flow rate, 200-1200 gpm,11% of reading Differential pressure,10-250 psi,11% of reading Flow area 1 0.5% Fluid density Negligible These uncertainties are judged to represent 2a values based on previous ex-perience with the experimental instrumentation. Therefore for AP=KW2 , the overall 3a uncertainty in flow due to systematic errors is: 1 f[(.01) + (.005)2 + (.005)2 3 2 = 0.018 The orifice configurations tested [5] probably will not be used for the final design because of potential cavitation problems. However, the random errors in tne test data are judged to be typical of what will be obtained for the final configurations. Figure 3.5 shows a typical set of test data. Allowing for lo = 3% on pressure drop based on these previous data, the 3c uncertainty in flow due to random errors is 0.045. 25 1630 039
1ARD-O"0050 1
#fv 2 L
180 CURVE FIT K = mRe 170 HOLE 160 - SIZE m 1 % STO. E RROR 50 -
, ,,,,, .,,,37, , ,,4 3 l 140 - E 199.65 -0.0587 1.160 l F 105.07 +0.0128 0.884 l
130 G 219.15 -0.0186 0.590 l 125 - I i 120 - l l 115 l 3 5-5 5 s-3333 G x I
'!O 2 O i A*A4M '
I
-_ F 80 -
70 -
-e.,_,
60 -
., ,08 9 - - E g
40 - v - D 30 - 25 - l 20 - l i 15 - l DESIGN Re I I 10
' ' ' I I I I ' ' ' ' '
105 106 107 INLET REYNOLDS NO. Figure 3-5. Fuel Assembly inlet Noule Flow Resistance Versus Reynolds Number: Orifice Group No. S. D:M, 1630 040 26
IARD-0 0050 REV 3 f Manufacturing Tolerances A detailed analysis was performed of the major independent geometric parameters contributing to the fuel assembly hydraulic characteristics (inlet hole dia-meter, inlet nozzle diameter, orifice rounding, shield hole diameter). Each parameter was assumed to have a rectangular probability distribution within the tolerance limits specified on the design drawings. The rod bundle region was excluded as it is considered separately in the subchannel flow area analysis. The remaining uncertainties were combined statistically and the resulting bour. ding 3a flow uncertainty due to manufacturing tolerances is 0.013. Configura tion Inlet Plenum Feature Model (IPFM) test E43 data were obtained on flow maldis-tribution from the reactor inlet plenuni to the Lower Inlet Module (LIM) for the homogeneous core configuration. The results of a typical run are shown in Figure 3.6, and the results of four runs are given in Table 3.3. A statistical analysis of the four-run average flow for each LIM showed a 3a variation of 0.0138 based on 33 data points. The effect of loop-to-loop flow variations was determined empirically by interpolati.ig between the IPFM data for 3 loop and for 2 loop opo ation. IPFM 2 loop test b43 data showed a 3a variation of 0.0268 in LIP flow, com-pared to 0.0138 for 3 loop operation. The maximum bounding loop-to-loop flow variation is +3.37% of nominal, resulting in a maximum difference between loops of 6.74% of nominal. Since 2 loop operation is equivalent to the case of 3 loop operation with 100% flow reduction in one loop, the case of 6.74% flow variation may be estimated by interpolating between the 3 loop and 2 loop results: 3a variation in LIM flow including loop-to-loop u- iation = (6.74/100)(0.0260-0.0138)+0.0138 = 0.0146 This conservatively assumes that the loop-to-loop variation is at its maximum value. Allowing for possible differences between homogeneous and heterogeneous core configurations, a value of 0.020 will be used. This is a 27 1630 041
'& OW ER3 A KET M DULE LOW g@@e g@'@g (O @ @
e e p):,e@, ee 5 INDIVIDUAL FUEL ASSEMBLIE8 22.73 2700 INLET OE19 1.08 Q to s s2 [ a2 o 21 os Es i ss / 900 2 INDIVIDUAL BLANKET ASSEMBLIES 5.96 o 20 DOWN- E18 0 83
'D B5
- 0.28 C4 0.12 COMER 1 gg
@g@@@@ ,[@@@@g@@
= e e@@
E13 A-1 DET All 2 1500 INLET CD 1808 DOWNCOMER2 #" g:C PLAN VIEW 7 INDIVIDUAL FUEL ASSEMBLIES 22.61
# --g Figure 3-6. Core Flow Distribution Map, Run AS.2,103.M.3-Loop flow,1400F. o
.a
IARD-0 REV 2 f50 ( TABLE 3.3 RESULTS OF FOUR IPFM FLOW DISTRIBUTION TESTS AT $100% FLOW, 3 LOOPS FLOW DEVIATION FROM AVERAGE, % LIM RUN 1 RUN 2 RUN 3 RUN 4 B1 .00 .20 .21 .19 66 .53 .07 .15 .14 B5 .28 .05 .03 .07 B4 .75 .46 .59 .53 83 .20 .26 .25 .20 82 .21 .15 .09 .12 C1 -1.05 -1.56 .99 .96 C12 .40 .36 .31 .12 C11 .71 .81 .74 .78 C10 .52 .34 .33 .26 C9 .72 .80 .66 .68 O C8 .88 .55 .46 .61 C7 .79 .33 .60 .44 C6 .39 .20 .20 .36 C5 .66 .37 .34 -v37 C4 .12 .40 .28 .34 C3 .62 .09 .16 .01 C2 .18 .10 .07 .04 01 .66 .39 .42 .43 016 .11 .14 .02 .14 D15 .24 .17 .07 .17 D14 .01 .02 .20 .11 D13 -1.50 .97 -1.03 -1.26 D11 .20 .16 .05 .09 09 .60 .59 .20 .50 DB .31 .14 .17 .25 07 .59 .83 .85 .75 D6 .05 .35 .03 .50 05 .20 .06 .05 .01 D4 .08 .11 .07 .16 D3 .29 .01 .01 .00 D2 .12 .27 .24 .29 D10 .26 .02 .23 .04 .r 1630 043
') G
WARD-O-0056 1 arv a r TABLE 3,4 RESULTS OF FOUR IPFM ASSEMBLY-TO-ASSEMBLY FLOW DISTRIBUTION TEST RESULTS AT s100% FLOW, 3 LOOPS Location Flow Deviation From Average, % Assembly Within LIM Run 1 2 3 4 1 Peripheral .40 .49 .45 .36 2 .95 .94 1.07 .91 3 .41 .07 .09 .24 4 .80 -1.18 -1.14 -1.21 5 .57 .60 .61 .73 6 o 1.32 1.20 1.13 1.00 p 7 Central . 08 .12 .09 .08 1630 044 c
W A RD-D* 005'O REV 3 I conservative estimate empirically based upon the observed differences between fuel assembly and radial blanket assembly (Section 4.1.2) LIM flow variations in the IPFM test. Assembly-to-assembly variation was determined by measuring the flow variation for the seven assemblies within a LIM. The data for four runs are given in Table 3.4. The average peripheral assembly flow rate was nearly identical to the central assembly flow rate. The four-run average flow for each assembly showed a 3a variation of 0.0242 based on 7 data points. This value includes random variations among similarly located assemblies as well as any " side effects". Systematic Error Interaction This factor accounts for systematic errors in the calibration of one core component affecting the flow to another core component. For example, if systematic errors cause the blanket assemblies to be overcooled they will result in the fuel assemblies being undercooled, and vice versa. From Table 4.2, blanket assemblies can have 1.8% flow increase (3o) due to systematic errors. An analysis of the reactor flow distribution with this change in the flow impedance of all blanket assemblies resulted in a decrease in fuel assembly flow by a fz.ctor of 1.002. Thus the 3a uncertainty in fuel assembly flow due to systematic error interaction is calculated to be 0.002. Overall The Inlet Flow Maldistribution 3a subfactors are listed in Table 3.5. The overall factor calculated as the root-mean-square of the subfactors (F = 1 + (E(F -1)2)V2) is al:;o .shown. The effect of the small number of 9 data points for several subfactors on the overall tolerance interval is negligible as shown in Section 2.4. An additional direct bias of 1.02 is applied based on engineering judgement to blanket any additional t.ncertainties which may be uncovered in the hetero-geneous core design. Additional data from the fuel assembly flow and vibration test and from the inlet nozzle flow test (see Table 6.11) will be utilized to update these uncertainties for the FSAR. 3, 1630 045
W ARD-09005k III '2 f n TABLE 3.5 FUEL ASSEMBLY INLET FLOW MALDISTRIBUTION SUBFACTORS (3a) Hydraulic Characteristics Systematic 1.018 Random 1.045 Manufacturing Tolerances 1.013 Configuration Among LIM's 1.02 (33)* Among Assemblies 1.024 (7)* Systematic Error Interaction 1.002 Overall (3c) 1.059_
- p.
- Number of data points shown in parentheses 1630 046 c
32
WARD-0-055h - REV 2 i 3.1.6 Flow Distribution Calculational Uncertainty The coolant flow and temperature distribution within the various subchannels of a wire wrapped assembly is calculated by subchannel analysis codes, e.g., COTEC, COBRA and THI-30. Empirical factors in the CSTEC code to model the effects of turbulent mixing, pumping (displacement of coolant by wire wrap), swecping (forced flow following the wire wrap) in the interior channels and swirl in the edge channels are selected by calibrat'on of the code against available experimental data. An extensive comparison of the COTEC code with experimental data from ORNL (19 rod bundle FFM tests)[6] ANL (91 rod bundle mixing tests)[7,8], HEDL (217 rod bundle mixing tests) , ARD (11:1 scale section of a 217 pin wire wrapped rod bundle air flow test)[10] , and J0Y0 (experimental study on coolant mixing effect in J0YO 19 rod blanket assembly) was performed. The 11:1 scale tests 10] were selected as providing the most detailed data for the calibration of the COTEC factors for pumping, sweepin and swirl. These factors were then used in COTEC analyses of the HEDL data 93 where the turbulent mixing factor, which has a secondary effect, was adjusted to provide the best fit. Approximately 3000 data points from the HEDL tests were compared with the COTEC predictions using the final calibration. Typical results are shown in Figures 3.7, 3.8 and 3.9. Figure 3.7 shows experimental data (average of four runs) 12 inches downstream of the injection plane along the row of channels with the highest concentrations. The data for all of the channels at the 12-inch level were used as input for a CBTEC analysis, and the results at 24 inches and 36 inches are compared to the experimental data *n Figures 3.8 and 3.9. CSTEC predictions are also compared with the ORNL dataE in Figures 3.10 through 3.13 for a heated 19-rod bundle in sodium. The present COTEC calibra-tion conservatively overpredicts the peak temperatures in the ORNL tests. Regarding the statistical calibration uncertainty, the calibration analysis treats the differences between the COTEC calculated temperatures and the measured or experimentally determined temperatures as a random 33 1630 047
1AR0-0 105Q r. REV 2 "1 Si E
- E 5
c 3 aE = 6 [e 4
=
=
}
m E S
" =
t 4 c
- E - .
m E ife
- 3 u:
I I I I I 2 s R S S 8 8 2 8 do '3Hil1VH3dW31 y a 1630 048 c 34
~ YARD-85005b 50 REV 2
,r. CDTEC INPUT AT 12 INCHES N 40 - / \ O
/
# /
/
/
\ _/ \
\
\
i 30 -
/
s E /
\
N j COTEC RESULTS s g 20 -
/
- ) HEDL EXPERIMENTAL DATA \
j \ 10
/
/
./ \
; i l l 1 1 I O
83 84 85 86 87 88 89 90 91 CHANNEL NUMBER Figure 3-8. Comparison Between COTEC Results and HEDL Data at 24 inches D 40 COTEC INPUT AT 12 INCHES HEOL E XPE RIMENTAL DATA g# g 3g _
/ \
d COTEC RESULTS -
\g 5m
< / *%\
/
y '%J \
\
10 -
,j i l l l l i I O
g3 84 85 86 87 88 89 90 91 CHANNEL NUMBER Figure 3-9. Comparison Between COTEC Results and HEDL Data at 36 inches 9426 9.10 1630 049 35
IARD-D 0050 - 1 REV 2 1100 e 1000
- e e
y e B 2 e 900 - C$TEC g EXPERIMENT INLEi TEMP. = 830.2'F I' FLOW RATE = 55 89 GPM POWER = 325 KW 800 CHANNEL DUTLET TRAVERSE (O
. \ ,
) .
v-. .
\ l l l Hgure 3-10. ORNL. Test Series #2. Test = 02. Run
- 105 49261I c' 1630 050 36
1ARO-U W 50 i REV 2 '1 r 1000 INLET TEMP. = 628.88F FLOW R ATE = 54.48 GPM POWER = 164 KW 900 W g CCTEC m at . EXPERIMENT IW 800 - C
- e
. . ./. .
CH ANNEL OUTLET TRAVERSE l (
)
l Figure 3-11. ORNL. Test Series = 2 Tesi = 02 Riin e 109 9926 12 g. 1630 051 37
. i IARD-8-003Q t REV 2 '
RJO ,. C FLOW RATE
- 54.58 GPM INLET TEMP. = 5968F POWER = 163 KW CcTEC 800 - e EXPERIMENT
?
af E E 700 - e e e
./e -
ee r.. 600 CH ANNEL OUTLET TRAVERSE
. (. %
e e e
- Figure 3-12. ORNL. Test Series #2. Test #3, Run # 201 99 6.l.1 c 1630 052 38
WARD-O-0050 i REV 2 1 n 1100 INLET TEMP. = 623 4'F FLOW R ATE: = 5.15 GPM POWEH = 59 KW 1000 5 a 5 g .
- 5 N
gg ae e e 6 e CCTEC e EXPEtilME N T 800 CH ANNEL OUTLET TRAVERSE ( . ligur. .1-1.1 ORNt I e-t Series 2 2. Test =04. Rim 205
'H2614 F
1630 053 39
IARD-lp005b i REV 2 I r variable, i.e., analogous to the displacement of a particle from its initial position while undergoing a one-dimensional random walk. The absence of perfect agreement between CDTEC and experimental data is expected and it must be attri-buted to the inability of the COTEC mixing model to completely represent the axial and transverse flows in a wire wrapped bundle. Thus, it can be deduced that the local transverse coolant temperature distribution directly affects the differential increment of temperature calculational uncertainty, while a flat temperature profile would not contribute to a temperature calculation uncertainty. Since the average coolant temperature gradient in the assembly, (Tmax -T is a coarse measure of the coolant temperature profile, the following equationmin)b23 represents the uncertainty associated with the use of COTEC: U2 6Tj ,(x) = C) /* (Tmax(t) - Tmin(C)) dc (3.8) where 6Tj ,(x) = temperature variation (uncertainty at x) Tmax(C) = maximum coolant temperature at elevation x = c n Tmin(() = minimum coolant temperature at elevation x = c x = elevation C = experimentally determined coefficient = 0.4376 x 10~4 in-I . The coefficient C was determined using an earlier COTEC calibration which did not compare as well with the data as the present calibration, and is therefore conservative. The integrand depends upon the transverse gradient of coolant temperature within a one-sixth sector of the wire wrapped 217 pin bundle and will vary from assembly to assembly with flow rate and power shape. Thus, to obtain an uncertainty interval for the coolant temperature and concomitant hot channel factor repre-sentative of all subassemblies of a specified type (e.g., fuel or blanket), it was necessary to envelope the uncertainties by estimating the assembly flow requirement and selecting the most critical assemblies (i.e., those with greatest power skew and with largest total power) for analysis. COTEC calculations were perfomed for these selected assemblies to compute the estimated uncertainty intervals for each critical assembly. 40 1630 054
IARD-D-005Q REV 2 Numerical integration with Euler's method using a 2-inch step size was used to calculate the temperature uncertainties. The temperature uncertainty increases with elevatior; and the uncertainty at the top of the core was used to calculate the hot channel factor. This is the approximate location of the cladding peak temperature elevation. The largest temperature uncertainty and hot channel factor for the fuel assemblies occurs in the peak skew fuel assembly. The analytically derived statistical hot channel factor is 1.054 (3a). Adoption of the maximum calculated value of the hot channel factor as an envelope for all fuel assemblies is obviously conservative. In addition to the aforementioned factors in COTEC, it is also necessary to specify the flow split among the several types of channels (interior, side and corner) because in wire wrapped rod bundles, the flow split is not found to agree with that predicted on the basis of the hydraulic diameter of the channels. The results of a number of experiments on the ratio of side to average channel velocity are shown in Table 3.6. The information in Table 3.6 was obtained from the referenced reports and / rom discussions with the experi-menters. The analysis of these results was limited to the turbulent flow regime, so the laminar flow data of Chen and Todreas[13] (Re = 640) were omitted. The dye tracer data of Pedersen, et. al .bO'I4] were rejected as being inconsistent with the remaining data including Pederson's thermal tracer da ta . The reduced side wire data of Graves and Catton[15] , although confirming the remaining data, were not included in the analysis as not prototypic of CRBRP geometry. For the remaining seven investigations, the mean ratio of side to average channel velocity, Us N , was 1.015 1 0.042 (2a). These results are consistent with Us/U = 1.00 which was selected for the nominal COTEC analyses and with the maximum expected uncertainties reported in Table 3.6. The uncertainty factor on iis will be conservatively taken as 1.02 1 0.05 (2a). Interior channel flow uncertainty is found from the relation (total side channel flow) x (side channel flow uncertainty) = - (total interior channel flow) x (interior channel flow uncertainty). Thus, for a 217 rod bundle, the corresponding uncertainty factor on interior coolant temperature rise, con-servatively neglecting inter-channel mixing is 1.005 1 0.013 (2a) or 1.005 1 0.020 (30), i .e. , direct: 1.005; statistical (30): 1.020, due to flow split uncertainty. 41 1630 055
IARD-0TC56 MV 2 The overall statistical uncertainty combining the COTEC calibration uncertainty and flow split uncertainty is 1.058 (3o). In addition to the statistical uncertainty, a direct factor of 1.03 is also applied. This includes the direct flow split factor cf 1.005 as well as additional margin, which based on previous experience with sodium heat transfer experiments in rod bundles is judged to be sufficient to cover the te'. ting biases and uncertainties which might be uncovered during the heat transfer testing at ORNL of a 61 rod bundle in liquid sodium (See Table 6.11) since the experimental data for the 217 pin fuel assembly were derived from mixing tests which measured salt solution diffusion or from velocity measurement tests in simulation fluids (water, air). The code calibration and uncertainty data from the 61 rod bundle test in sodium will be used in the FSAR evaluation. i630 056 42
TABLE 3.6 RATIO 0F SIDE SUBCHANNEL AXIAL AVERAGE VELOCITY TO BUNDLE AXIAL AVERAGE VELOCITY RODS-IN-BUNDLE EXPERIMENTER (s ) nominal REYNOLDS NUMBER METHOD MAXIMJM EXPECTED ERROR 7 Ginsberg-Lorenz 1.00 t 13,000-17,000 Isokinetic (Reference 16) 19 Graves & Catton 1.00 10,000-90,000 Salt Tracer -+5% (Reference 15) 19 Graves & Catton 1.00 10,000-90,000 Salt Tracer -+5% (Reference 15) 61 Chen-Todreas 1.19 640 Laser Beam +4% 61 Chen-Todreas 1.04 4500 Laser Beam -+4% (Raference 13) O 91 Pedersen, et al. 1.17 5,000-20,000 Dye Tracer +4% 91 Pedersen, et al. 1.03 5,000-20,000 Thermal Tracer -+4% (References 8,14 ) 91 Lorenz, et al. 0.99 9,000-25,000 Isokinetic (Reference 17) u 217 Graves & Catton (Reference 15) 1.00 10,000-90,000 Salt Tracer -+5% O
- 217 Graves & Catton 1.00 10,000-90,000 Salt Tracer -+5%
o ** (Reference 15 ) ( 217 Bartholet, et al. (Reference 18 ) 1.04 73,000 Pitot Tube +5% mm mp
#E Side channel area decreased by reducing wire wrap diameter; ru}a " ection S of 217 rod bundle. U s
converted to full bundle; I All 12 subchannels sampled.
W ' Efre Wrap Orientation Dae to the swirl flow induced by the wire wrap in the peripheral channels, the flow and temperature distribution in the assembly depends slightly on the relative orientation of the wire wrap and the power skew. An analysis conducted with the COTEC code, investigating the power skew range in CRBRP fuel assemblies and the six possible orientations of the wire wrap, indicated a maximum bounding deviation in the hot channel AT of the order of 1%. 3.1.8 Subchannel Flow Area An analysis was made of the effects of rod bundle dimensional tolerances and fuel rod bowing on the hot channel coolant temperature. The rod diameter, wire wrap diameter and duct inside width were each assumed to have rectangular probability distributions within the tolerance limits specified on the design drawings. The actual distributions of dimensions of fabricated fuel assembly components are expected to be available for the FSAR final hot channel factor analyses. All of the rods and all of the wires within a fuel assembly were asstrned to have the same diameters. This assumption is reasonable in that each lot of rod cladding and of wire is sufficient to supply more than one fuel assembly, and leads to more conservative results than allowing the diameters to vary. The average rod spacing, S, for each subchannel was assumed to have a rectangular probability distribution in the range 0 1 S 1 2Snom, and the rod spacings for the various subchannels were assumed to be independent. COBRA I analyses were performed for the cases of a) a single nested subchannel (S = 0), b) six nested subchannels surrounding the hot rod, and c) the hot subchannel and the three adjacent subchannels all nested. From the results of these analyses, the influence of reduced flow in one subchannel on the temperature distribution among all subchannels was detennined. The overall 3a uncertainty in hot channel coolant temperature rise due to uncertainties in rod, wire and duct dimensions and rod spacings, and including the influence of the other subchannels, was calculated by the preceding bounding analyses to be 0.019. This uncertainty applies at beginning of life. The effect of irradiation-induced swelling of rod, wire and duct was analyzed and was found to result in reduced temperatures in the hot channel. Theref ore, it is conservative to apply the above uncertainty throughout life. u 1630 058
IARD-9 0050__ :1 REV 2 There is a tereiency for thennally-induced bowing to cause all of the rods to bow toward the hottest channal. However, the thermal compressive force was A calculeted to be negligible compared with the experimentally determined bundle compressive forces. Therefore, thermal bow will not significantly change the subchannel flow area. In evaluating the corresponding film temperature drop hot channel factor, it Ns found that the effect of the decrease in the unit cell hydraulic diameter was dominant with respect to the effect of the decrease in the Nusselt number. The effect of the change in pitch-to-diameter ratio on the circumferential temperature distribution was minimal. Thus, a higher value of the film heat transfer coefficient and consequently, a film subfactor less than 1.0 will result. Conservatively, a value equal to unity was adopted. 3.1.9 Film Heat Transfer Coefficient 1.iquid metal heat transfer data for parallel flow through rod bundles were reviewed in Reference 19. A conservative design correlation was selected in Reference 19 for the rod pitch-to-diameter ratio range 1.2 1 /D P 1 1.3 cor-responding to that in CRBRP fuel assemblies. The data base is shown in Figures 3.14 to 3.16 together with the design correlation which is designated Curve 1. A 3a uncertainty of 0.12 was recommended based on the deviations of the data points overpredicted by the correlation. The analysis was repeated taking into account the variation *wung investigators, which reduced the number'of data sets to n = 6. However, it was found that applying Equation (2.8), the new analysis resulted in lower cladding tempera-tures at PE0V conditions and lower fuel centerline temperatures than the correlation recomended in Reference 19. Therefore, the latter correlation is conservative and will still be used. As a further indication of the conservatism in the correlation, it may be noted that in the range of interest, the correlation results in lower heat ' transfer coefficients than the theoretical lower limit calculated for the case of turbulent flow through unbaffled rod bundles, under conditions of uniform heat flux at the inside surface of the cladding, with heat transfer by molecular conduction only through the sodium [ 0] , c 1630 059 45
E@4580 e i< N 4 0 1 8 I 6 I 4 O I 2 3 O 1
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c 102 c y g _ SYMBOL INVE STIG ATOR LIQUID METAL h O PASHEK Na 6 - A BO RISH ANS Kil Na 4 _ O so RiSH ANSKii Hg O SU8BOTIN Hg SYMBOL AUTHOR
- 1. ____ FFTF y
- 2. WARD-D 0034 .
- 3. . GHABER AND RIEGER .
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a _ 2 m\ .. 0.0 . b. . 0' 6 - O ' O A "' 1A_.____ OO 4 00 0@ o 2 LN 100 101 l 2 I 4 6 I I 8 102 2 I 4 I 6 I 8 103 I 4 I 6 I 8 104 2 O Pe C r, x x ey > N 1 igure 3-16. Comparison of Predicted and thperimental Results for P/D = 1.2 <@
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.1.10 Pellet-Cladding Eccentricity An ~ eccentric position of the fuel pellet within the cladding will give rise to an increased heat flux in the area of minimum gap with consequent increases in temperature rise through the coolant film and the cladding. Analyses indicated that the effect on the fuel temperature would be to shift the position of the maximum fuel temperature from the centerline; the value of the maximum fuel temperature being actually low for sccentric than for nominal, concentric conditions. Conservatively, a value equal to 1.0 was selected for power-to-melt analyses (Section 3.2).
The effect of pellet-cladding eccentricity on the film and cladding temperature drops was calculated assuming conservatively that pellet-cladding contact will occur, but at some random circumferential orientation. For a typical fresh fuel rod, the circumte.enual variation in heat flux as a result of pellet-cladding eccentricity was calculated using the LIFE-III correlation for gap conductance. The axial variation of local cladding midwall temperature beneath the wire wrap for a centered pellet, and of coolant temperature were obtained from a NICER run. The results were combined to detennine the maxi-mum cladding midwall temperature anywhere along the rod for each direction of pellet displacement. This provided a probability distribution of maximum cladding midwall temperature. The resulting bounding uncertainty in film and cladding temperature drops was the combination of a direct factor of 1.14 and a 3a statistical factor of 1.174. These factors apply only at beginning of life and decrease to unity at about 0.5 to 1.5 atom % burnup. It may be noted that the orientation of the fresh Fuel Assembly during shipping can result in the pellets tending to be displaced in the direction which is downward during shipping. Specifying the fresh fuel assembly orientation during shipping so as to avoid any increase in the pellet-cladding eccentricity factor due to improper orientation is under consideration. 3.1.11 Cladding Thickness and Conductivity The thennal conductivity of 316 SS is given in Reference 21 together with the data base used. The values are based on unirradiated data. The estimated C uncertainty is + 5%. 49 1630 063
}A p050 i The correlation used in the design analysis yields results which are 5% below the data given in Reference 21 1.e., the uncertainty is already conservatively e included as a direct factor.
For maximum fuel temperature (power-to-melt) analyses, the worst case occurs at beginning of life and no additional correction for irradiation effects is required. The effect of cladding ID and OD tolerances results in a 1 - [2( )2] 7 /3 = 0.08 (3a) uncertainty in temperature drop through the cladding. However, the combined statistical uncertainties in thermal con-ductivity and in cladding thickness have a smaller effect en the temperature drop than does the direct factor of 1.05 built into the design correlation. Therefore no additional uncertainty factor is required for maximum fuel temp-erature calculations. Maximum cladding midwall or 10 temperatures are used for fuel rod lifetime analyses. Minimizn fuel rod lifetime occurs when cladding thickness is at its minimum value, which also corresponds to minimum temperature drop through the cladding. Therefore, it is conservative to assume nominal cladding thickness for the temperature calculation. The effect of irradiation is accounted for by noting that for the most highly irradiated fuel assembly at the end-of-life, volumetric swelling at the cladding hot spot is approximately 1.0% 10.5%, based on the correlation of swelling test data given in Reference 21. Since swelling represents dispersed porosity (vacancies) in the cladding, the effect of irradiation swelling at the cladding hot spot is calculated from the Maxwell-Eucken equation to be 1.01510.008. In addition, a decrease in conductivity would be expected due to interstitial lattice defects produced and remaining when vacancies are also produced. Reference 23 gives some experimental data for the effect of irradiation on the electrical conductivity of cold-worked 347 stainless steel, which is expected to be similar to the effect on the thermal conductivity of the cold-worked 316 stainless steel cladding. There was a reduction in conductivity of 1-2% at low neutron exposures (10 18 to 10 l9 fast nyt) and low temperatures. At higher exposure there was a smaller change. At higher ir-radiation temperature (s300 C) the change was reduced by about 50%. Based on this, an additional direct factor of 1.02 was conservatively applied. The irradiation swelling direct factor of 1.015, the interstitial lattice defect direct factor of 1.02, and the combined statistical uncertainties in 50 1630 064
, . WARD-O-0050 REV 2 i 1 irradiation swelling and in thermal conductivity have a smaller effect on the temperature drop than does the direct factor of 1.05 built into the design correlation. Therefore no additional uncertainty factor is required for maximum cladding temperature calculations. 3.1.12 Coolant Properties The enthalpy and the density of sodium are parameters used in the energy and momentum balance for the heated coolant channels to determine the local coolant temperatures. The correlations are obtained from Reference 24. The reconmended correlation for enthalpy was based on the data of Ginnings, et al.[25] They report a probable error of the mean from 0.01 to 0.03% (depending on the temperature). Taking into consideration their estimate of systematic errors, they estimate the probable error of the enthalpy to be between 0.1 and 0.2% and the probable error of the heat capacity to be between 0.3 and 0.4%. For hot spot analyses, coolant temperature changes are large and the uncertainty associated with enthalpy rather than with heat capacity is more appropriate. The upper estimate 0.2% probable error (errors equally likely to be greater as smaller) converts to a 0.9% 3a uncertainty. More recent measurements by Fredrickson and Chasanov[26] showed excellent agreement with Ginnings' data. For a typical 400*F temperature rise in CRBRP, Fredrickson's correlation gives an enthalpy rise within 0.01% of that obtained with the recomrrended correlation. The density correlation was derived by Stone, et al.[27] from the data of References 27 to 33. The deviations of these data and those of Gol'tsova E ') from the reconmended correlation are shown in Table 3.7. For the five investigators covering the present range of interest (References 30, 31(B), 32,33,34), RMS % Avg. Dev. = 0.674
% Avg. Dev. of Mean = 0.674 7 d=0.337
/5 /4 The average deviation of the mean of 0.337% converts to a 1.5% "30" uncertainty.
The effect of the small number of data points on the overall tolerance interval is negligible as shown in Section 2.4. The statistically combined enthalpy and density subfactors yield a sodium properties factor of (0.92 + 1.52)V2 = 1.7%. 1630 065
WARD-O%56 Rt.V 2 TABLE 3.7
SUMMARY
OF DENSITY MEASUREMENTS FOR LIQUID S0DIUM
% Average Deviation:
Temp. Range Obse ved - Correlation Investigator ( F) Correlation NRL[29] mp to 503 10.08 Jackson [30] 937 to 1314 -0.74 Rinck[32] 804 to 1183 +0.15 Hagen[28] mp to 336 10.05 NRL[27] 1577 to 2491 +0.17 Novikov[31](A) 248 to 505 0.14 (B) 275 to 1324 +0.71 Nishibayoshi[33] 486 to 1580 +1.09 Gol'tsova[34] 802 to 2754 10.11 n 1630 066 52
WARD-D*0050 1 Rtv 2 ; 3.1.13 Nuclear Uncertainties r The total power or burnup uncertainty is composed of nuclear design methods uncertainties and/or biases (based on comparisons of calculations and measure-ments of isotopic fission and capture rates and gama heating in ZPPR-7), CRBRP design uncertainties relating primarily to absolute power normalization and fissile content variations, and a general class of ir.odeling uncertainties. In the fuel, the power uncertainty is broken down into a statistical part which is combined in quadrature (root-mean-square) with other statistical uncertainties, and a nonstatistical bias and uncertainty which is applied directly to envelope the upper limits of the peak power density. Due to the limited scope of the available blanket data, only a nonstatistical uncertainty is developed. Uncertainties are provided for the fuel, inner blanket and radial blanket assemblies. Where a basis exists for such, a spatial distri-bution of the uncertainty is provided (e.g., adjacent to, and removed from, the influence of inserted control rods, and by assembly-row in the radial blanket). C Experimental verification of the calculational methods and data used to predict the power distributions in CRBRP is provided by the measurements performed in the ZPPR critical facility. ZPPR is a zero power critical mockup of the CRBRP consisting of a matrix of rectangular drawers containing fuel, steel, and sodium plates loaded to simulate the average compositions of the CRBRP core and blanket regions. CRBRP design methods are verified, and design bias factors and uncertainties are derived, by direct comparison of calculated and measured ZPPR parameters. These parameters include isotopic fission and capture rate and gamma heating distributions in support of power distribution predictions. The design (methods) bias factors and uncertainties are subsequently applied to the calculation of these parameters in CRBRP which are predicted with the ' same calculational tools and data. 1630 067 r 53
- ' WARD-D 0050 4 4 RtV 2 '
The results presented herein are based primarily on the analysis of ZPPR-7 (Phases A-E) measurements. ZPPR-7 modeled the basic characteristics of a heterogeneous CRBRP core with annular rings of inner blanket assemblies. Tables 6.lB and 6.2B list the statistical and nonstatistical uncertainties ap-plicable to the predictions of both local and rod- or assembly-integrated power density in the core fuel assemblies. The uncertainties are divided into those applicable to fuel regions directly adjacent to the partially inserted row 7 corner (R7C) control rods, and those applicable to all other core locations. The uncertainties for fuel assemblies adjacent to inserted R7C control rods are shown for both the side adjacent to the R7C control rod and the far side. Uncertainty values are presented for the heat flux at the peak power position (peak power density near the core midplane), the heat flux at the top of the core (power density at the core / upper axial blanket (UAB) interface) and the channel coolant (rod-channel integrated power) in Table 6.lB and the assembly mixed mean cutlet temperature (assembly integrated power) in Table 6.28. The peak and integrated power der.sities in the fuel are well predicted with the standard.20-synthesis nuclear design techniques. However, the power density at the top of the core is relatively poorly predicted due to diffi-culties in simultaneously modeling the behavior in this region while preserving the integral and peak (core midplane) power in two dimensions. In addition, the accuracy of few group diffusion theory is poorer in the presence of the steep flux gradient and in the region of the core /UAB material discontinuity. These factors are reflected in the larger " heat flux at top of core" unces tainty. The statistical uncertainties consist of experimental uncertainties (fission rates and gama heating) at the 3a level, criticality and control rod insertion uncertainties, and local fuel fissile content variations. 3.1.13.1 Experimental The experimental power uncertainty consists primarily of Pu-239 and U-238 fission rate and gama heating uncertainties. The experimental (statistical) power uncertainty is derived from the +2% (lo) root-mean-square deviation 1630 068 54
' i WARD-D*0050 REV 2 between calculated and measured core fission rate (Pu-239, U-235) distributions in ZPPR-7 covering a variety of reactor critical configurations. There is an observed radial tilt in the calculation-to-experiment ratios for Pu-239 fission in Phases A and B of ZPPR-7 which tends to overestimate (by 3-5%) the Pu-239 fission rate in the area of the fuel " islands" in rows 7, 8 and 9 and speci-fically around the R7C control locations, in the beginning-of-life configuration.
This overprediction, coupled with the application of a +3a power uncertainties, would result in an overly high estimate of the peak power in these locations. Consequently, the beginning-of-life (80Cl, B0C3, 80C5) power in the rows 7, 8, 9 region of the core is biased downward in Table 6.lB (footnote 4) by 1-3% to remove the basic tilted characteristics in the fission rate (power shape). This radial tilt was not observed in the Phase C or D ZPPR-7 experiments with plutonium loaded in the inner blankets, so the aforementioned bias is not applied to the end-of-life power shape. The analysis of the ZPPR-7 experimental U-238 fission rate distribution showed substantial fluctuations in the calculation-to-experiment ratios between fuel and inner blanket assemblies, possibly attributable to the infinite medium C cross-section preparation scheme. Based on these ZPPR-7 results, a one-sided
+20% U-238 fission uncertainty is indicated in the core fuel assemblies (although U-238 fission only accounts for about 6% of the total power in the fuel so that this uncertainty accounts for approximately 1% on the high-side of the total power uncertainty).
Analysis of the preliminary ZPPR-7 gama heating measurements shows a scatter of 110% in the calculation-to-experiment (C/E) ratios in the core regions and a consistent 15-20% underprediction in the radial blanket. There was some evidence in the ZPPR-4 gama heating measurements, however, that this under-prediction behavior at the core / blanket interface is at least in part an anomaly caused by the comparison of homogeneous (cell-average) calculated heating rates with point-measured values in regions of substantial gama fine structure. This uncertainty is therefore considered an upper bound. Gama heating accounts for about 10% of the total fuel assembly power, so the 110% uncertainty accounts for approximately 11% of the total fuel assembly power uncertainty. /~ 1630 069 55
IARD-0-0050 1 RFV 2 'i i The direct combination of 16% (3a) plutonium fission rate uncertainty (=85% of the total core power), 120% U-238 fission rate uncertainty (=6% of the O total core power, and 110% gama heating uncertainty (=10% of the totalcore power) results in a 17% (approximately 3o) total experimental power uncer-tainty in the fuel, in addition to the space-dependent beginning-of-life bias discussed earlier. This uncertainty applies equally to the local pellet, rod, and fuel assembly power predictions. Due to the slight space-dependent trends in the C/E distributions, this uncertainty is not statistically reducible for small groups of fuel assemblies. The experimental fission rate uncertainties are based on well over 100 foil measurements in ZPPR-7. 3.1.13.2 Criticality Uncertainty The uncertainty in the prediction of the hot critical state of the reactor results in an uncertainty in the depth of primary control rod insertion which, in turn, affects the local power distribution (principally the radial and axial distributions in the imediate vicinity of the inserted control rods, and to a lesser extent throughout the remainder of the core). This O uncertainty is derived from a (3o) statistical value of the criticality uncertainty of 10.5%Ak developed from the 4 unrodded critical configurations in ZPPR-7. The resulting power uncertainty is derived parametrically by varying the control rod bank insertion by an amount corresponding to the 10.5%Ak to produce power perturbation maps. The criticality power uncertainty applies approximately equally to the beginning and end-of-cycle power distributions due to a partial cancellation between the lesser influence of the nearly-withdrawn primary control rods and the larger magnitude of the criticality uncertainty which includes burnup reactivity swing uncertainties near the end-of-cycle. This criticality (rod insertion) uricertainty results in a 11% power uncertainty throughout the core with the exception of those assemblies directly adjacent to the inserted control rods. The power distribution in the fuel assemblies directly adjacent to the inserted control rods is strongly influenced by the position (depth of insertion) of those rods so that the criticality uncertainty in these assemblies is both larger and space-dependent across the assemblies as indicated in footnote (3) to Table 6.lB. p 1630 070 56
VARD-D 0050 8 REV 2 3.1.13.3 Pellet Fissile Content Tolerance The specified fuel pellet fissile content (manufacturing) tolerance results in a local fission rate bounding uncertainty of 13%. Because the pellets are fabricated in batches, this uncertainty is not reduced for groups of pellets comprising a single fuel rod or a single fuel assembly. Very large numbers of fuel assemblies are, however, subject to a smaller specified 10.5% fissile content tolerance. For a rectangular probability profile, the 13% uncertainty results in a 3o uncertainty of 5.2%. 3.1.13.4 Modeling Uncertainties Specific model uncertainties in the CRBRP core power distribution predictions include parametric uncertainties arising from 3D effects (bounding), rod-power interpolation (bounding) and radial and axial burnup modeling (engineering judgement). The aforementioned modeling uncertainties are derived parametrically, so no specific o level can be assigned. The values quoted do, however, generally cover the range of variations noted. Modeling uncertainties are combined with other nonstatistical uncertainties directly rather than statistically. CRBRP power, burnup and reaction rate distributions are predicted by a two-dimensional (2D) synthesis technique whereby " radial" shape factors, F , derived from hexagonal calculations, and axial shape factors F , derived from RZ calculations, are superimposed on average power conditions to result in the three-dimensional (3D) power distribution throughout the core and blankets. A bencnmark calculation with consistent 3D and 2D-synthesis models has been used to infer the geometric modeling biases introduced by the 2D-synthesis technique at the beginning-of-life. In general, the agreement between the two models is good (with the exception of the regions strongly influenced by the partially inserted R7C control rods as noted in Table 6.lB) with the peak power density predicted within the range of 13% and rod- or assembly-integrated power pre-dicted within 11% in the core. The polynomial fit to power-by-rod, performed with input from 24 mesh-per-assembly 2DB hexagonal calculations, introduces an additional uncertainty of 11% in the radial power shape within an assembly. Burnup modeling in the core generally introduces little error in the power distribution in high-power locations, although some uncertainty is necessarily 57 1630 071
44 WARD-O*0050 ~ REV 3 i g introduced in the end-of-life axial power shapes due to smoothing. The axial peak-to-average power has been shown to be nominally overpredicted by 2% in ZPPR-4 and in ZPPR-7 The net peak power density modeling uncertainty through-out the core (with the exception of assemblies directly influenced by inserted control rods) is therefore 12% (13% 3D effect plus 11% intra-assembly radial power shape uncertainty less 2% axial peak-te,-ever6ge overprediction). This value increases to +10% on the high-side for the " heat flux at the top of the core" with the inclusion of the 5% fall-off in the prediction of the axial power shape indicated in the IPPR-4 and -7 experiments, and with some additional uncertainty in the tail of the axial power shape. The 12% integrated rod power uncertainty is made up of 41% 3D effect plus the .11% radial shape uncertairtty, whereas the 11% total assembly power modeling uncertainty is entirely 3D effect. The radial and axial power shapes in the vicinity of inserted control rods are particularly sensitive to control rod modeling. Partially inserted control rods in the CRBRP are modeled with an " effective" amount of control volume fraction which results in the same reactivity worth inserted in the 20. hexagonal calculations as would result from an explicit 3D representation. This technique results in a good representation of the average power in the surrounding fuel assemblies, but necessarily introduces some spatial bias in the radial power shape very close to the inserted control rods. In addition, the axial powe-shape is selected to best represent the assembly-average power so that the shape is somewhat more perturbed in the region of the fuel pins directly adjacent to the inserted control rod, and likewise somewhat less perturbed on the side of the assembly farthest from the inserted rod. The combination of these radial and axial shape perturbations, derived primarily from the 3D-2D synthesis comparison, results in a space-dependent modeling bias (un-certainty) which varies across the assembly as indicated in Table 6.18 (foot-note 1). 3.1.13.5 Control Rod Banking The maximum control rod out-of-bank specified tolerance of 11.5 inches intro-duces a potential power asymmetry in the core. The bounding power uncertainty resulting from the control insertion asymmetry is found from full-core para-metric calculations to be 12% except directly adjacent to the inserted rods where the bounding uncertainty is increased to 14%. 58 1630 072
WARD-D7056 REV 3 3.2 isower-To-Melt Uncertainty Factors for Fuel Assemblies 3.2.1 Introduction r This section presents a preliminary analysis of the uncertaf nties in power-to-melt calculations for the heterogeneous core. The results will be refined and updated for the FSAR based on further analysis and additional experimental data. The primary data used for this work were the results. of the short-time. (low burnup) HEDL-P19 and P-20 tests [35,30 which were integral experiments designed to provide thermal performance information. The LIFE-III code was used as the calculational tool. The code is a detailed model for describing the thennal and mechanical behavior of fast reactor cxide fuel pins and has been calibrated and verified with the HEDL P-19 and P-20 tests in addition to a number of intermediate and high burnup pins. Figure 3.17 and Table 3.8 show how well the code represents the P-19 and P-20 data. The code was used to calculate the power-to-melt of the CRBRP AFMS fuel - and blanket rods, and the sensitivities to variations in fuel pin parameters used fo.' uncertainty analysis. In the subsequent sections for fuel applications, the uncertainties in the C code predictions of the power-to-melt arising from data scatter and the overall accuracy of measurements in the EBR-II are first analyzed. The uncertainties which occur when applying the code to the CRBRP conditions are then evaluated. The factors considered here include the tolerances on fabrication parameters and reactor instrumentation and thermal hydraulic and systematic uncertainties. The individual uncertainties were determined by calculating the power-to-melt for the nominal peak rod conditions and design parameters followed by calculating the effect of a variation in each parameter or the nominal code result. All the individual uncertainties are then statistically combined and the probability distribution for the reacMr power-to-melt determined. The design criterion can then be measured by detennining if the 115% of nominal power is three standard deviations below the power-to-melt. A ' 1630 073 59
*ARD-8*0050 't REV 2 1 C
=
2 ie
>=
5
.3 E
a = 0 m = E O c a
=
3 E E 8 k r~ $ ~ 2 3 : O w
.E u.
U b o
- _ e e.
~
5 O e
= =
o oc a-o Z l I l I E
"
- 2 2 2 0: x 031V1031V3
?
1630 074 c 60
WARD-D-0050 REV 2 y I n TABLE 3.8 CENTER TEMPERATURES AND POWERS-TO-MELT OF INCIPIENT MELT SECTIONS AS COMPUTED BY LIFE-III CALIBRATION T18/6000 Calculated Observed Calculated T Power-to-W11t Power-to-Melt Pin N3. 'F _ ('C) max kW/ft (kw/m) kW/ft (kw/m) P19-35 5009 (2765) 16.62 (54.5) 16.69 (54.6) P19-02 4978 (2748) 16.04 (52.6) 15.89 (52.1) P19-24R 5024 (2773) 15.44 (50.6) 15.59 (51.1) P19-27R 4990 (2754) 19.23 (63.1 ) 19.16 (62.8) P19-28 4966 (2741) 19.02 (62.4) 18.82 (61.8) P19-30 5070 (2799) 17.22 (56.5) 17.79 (58.3) O P19-08 5006 (2763) 15.36 (50.4) 15.40 (50.5) P20-07 5004 (2762) 15.72 (51.7) 15.75 (51.7) P20-30 4985 (2752) 16.82 ($5.2) 16.70 (54.8) P2b-33 4982 (2750) 19.07 (62.5) 18.98 (62.2) Average T = 5001 1 28 F r.m.s error (2761 1 16*C) = 0.21 kw/ft (.69 kw/m) std. dev.
= 0.24 kw/ft (.77 kw/m)
Two degrees of freedom lost in calibrating LIFE-III parameters. O 1630 075 61
WARD-D* 005O ) REV 2 1 3.2.2 EBR-II Factors The factors associated with EBR-II experiments which contribute to uncnr-tainties in power to melt measurements are listed and defined below:
- 1. is the standard deviation) is due to variation in fuel pin fab (
fabrication parameters from their nominal values.
- 2. oP IE is due to uncertainty in post-irradiation examination (PIE) measurements.
- 3. o is due to uncertainties in the spatial dependence of neutronics space calculations and local inhomogenieties in the EBR-II core.
- 4. oht is due to uncertainty on overall power level due to variations in EBR-II instrumentation and the uncertainty in the neutronics calculation for a given core loading. This uncertainty causes random fluctuations in quoted power level that vary with time.
- 5. o is due to a difference between actual and quoted overall EBR-II sys power that doesn't change with time. It is known that a systematic shift in EBR-II power level exists and a correction is made by experi-menters. An estimate is required of the uncertainty on this correction for this analysis which is identified here as 0 This uncertainty 9ts .
does not show up as scatter in the data. Such u systematic uncertainty would show up in the scatter of data comparing different reactors. The evaluation of each of these uncertainties will now be discussed.
" time The major contribution to this uncertainty results from fluctuations in the primary and secondary EBR-II coolant loop heat balance. This yields a 2%
standarddeviation[37] Neutronic calculation of y recursors in the P-19 test introduce a further uncertainty of less than 1% 38] and a value of 0.8% was used. Accordingly this factor was evaluated as
" time 2 2%2 + 0.8%2= 2.2% (3.8) 1630 076 62
- ' IARD-DMOO5b 4 REV 2 This uncertainty would not show up as fluctuations in results from a single subassembly since all pins would have been subjected to the same errors in overall power determination during the same times. sys A detailed analysis of the P-19 test indicated that a correction factor of 0.94 must be applied to the calculated EBR-II power leve1 E38) This factor has ban used in all reported analysis of these results and those of the P-20 test and was used for the LIFE-III calibration. Subsequent work suggested a factor of 0.91[37] , while recent analysis by the EBR-II Project indicates 0.96. As an interim position, the 0.94 factor is being retained. An evaluation of the recent analysis by the EBR-II Project will be perfonned when their report is issued. In addition, burnup analysis data will be evaluated with respect to this factor. The EBR-II Project also estimated that further systematic bias should be no more than +2%. For work in this document an uncertainty, sys, of 3% is used which covers both the
.91 and .96 power factors and is more conservative than the 2% estimate by EBR-II. " space The spatial uncertainties in pin powers arise from uncertainties in neutron transport calculations. Calculations have been checked by comparison of measurements of control rod worth [38] Table 3.9 and Figure 3.18 show the result-ing percentage difference between measured and calculated flux, which has a standard deviation, c' space, of 1.7%. In addition local flux peaks produce an estimated uncertainty dspace of about 0.5%. Finally in the highly enriched fuel of P19, differences in the y absorption can produce a +0.7%
to -0.7% variation in pin power going from the center to the outside of the assembly. This introduces a standard deviation of .7//3 2 0.4%. Combining these,
" space ; 1.7 +.52 , ,42 = 1.8% (3.9) 1630 077 63
WARD s* 005'O REV 2 O O contact noo nos. CALCULATED
---- M ASURED 1
1so lk 9 11 _.AI
\
g / 140lh
/
120sh
/
100th h l I p (1
\ k I /
. ~ -
6 2 5 3 4 Figure .1-18. Control-Rod Worths for EBR-il Run 27 A 9926 19 1630 078 64
IARD-9-0050 REY 2 1 rs TABLE 3.9 - SPATIAL DEPENDENCE OF POWER UNCERTAINTY CALCULATED FROM COMPARISON OF MEASURED AND CALCULATED CONTROL R0D WORTHS Control Rod Calculated Minus Number Measured Power (%) 1
.67 2
-1.75 3
-1.8 4
-2.35 5
0. 6 1.85 7 - 2.25 8 1.5 9
.45 10 1.2 11
.7 12 2.25 Average a Power =
.3%
Standard Deviation = 1.7% 1630 079 r 65
BARD-0*005Q , REW 3 1 fab The scatter in burnup measurements as compared to values calculated from EBR-II powers has been analyzed and found to be 1.1%. This analysis BU is from individual subassemblies, and since pins in a subassembly are in the reactor for the same time, they would not reflect the uncertainties otime and osys. The scatter in the burnup data includes a combination of EBR-II power uncertainties and uncertainty in the burnup measurement technique. BU 2 1.1% 1 space 2+o fab 2+(0.5%[ p B) where the 0.5% is the estimated burnup measurement accuracy of the mass spectrometry. The spatial uncertainty space s .5% is that for an indivi-dual subassembly and not the entire core. The burnup uncertainty has also been calculated by HEDL to be 3% (1o). In this work a single power factor for all subassemblies was used. The data scatter due to power fluctuation with time, a time, and core wide spatial uncertainty, ospace, would therefore apply and the 3% standard deviation obtained can be attributed to 3% 2! tim 2+o space 2+ fab +(0.5%)2 (3.11) Using the previously detemined values for time , space and yespace in equations (3.10) and (3.11) the two self-consistent values for ofab am respectively obtained, fab = .85, .82% (3.12) PIE The scatter of the P-19 and P-20 experiments relative to the LIFE calibration has been calculated (see Table 3.8 and Figure 3.17). The standard deviation is about 1.3%. It is interesting to note that only one pin in this group was significantly outside this standard deviation, and that was P-19-30. If P-19-30 had been excluded, the standard deviation would have been 0.8%. 1630 080 66
It might be core reasonable to use 0.8%, but to be conservative, 1.3% will be used. The scatter in P19 and P20 data reflect uncertainties in fabrication, local spatial fluctuations and post irradiation examination measurement uncertainty. Since P20 powers were normalized to P19 results, P20 is not an independent experiment at a separate time and time does not apply. So "P19/P20 ' l.3% ab +
- space PIE (3.13)
Equation (3.13) then gives PIE = 0.9% (3.14) Values of the above uncertainties are sumarized in Table 3.10. Total Uncertainty The total uncertainty is a combination of all the components tot fab 2+# PIE 2+ space 2+o time 2+o sys 2= 4.3% (3.15) Resolution of the EBR-II power factor is expected to reduce the systematic uncertainty to 2%, and will drop to 3.7%. tot tot represents the scatter and uncertainty in the power to melt data. The uncertainty in the average of these data is given by the standard deviation of the mean. The standard deviation of the mean takes into account the nunber of data points N, and is given by o/ 8. In the various components of tot, a separate measurement toward determining o sys w uld require use of a different reactor; a separate measurement for o time would require an experiment done at a different time and a separate measurement for o" space would require another experiment done in a different position in EBR-II. As explained above, P19 and P20 cannot be counted as being done at different times or different positions. So for o " sys' space and ot ime N=1. For the other components N = 10, the nunber of rods used. Thus "mean"/#sys2+o time 2 + o " space 2 + ( fab 2+o PIE 2+ ' space)/10=4.1% (3.16) 1630 081 67
- ' d 1ARD-O OO50 e REV 2 O
TABLE 3.10 VALUES OF UNCERTAINTIES IN EBR-II space 1.8 time 2.2 D afab 0.8 PIE 0.9 "sys 3.0 tot 4.3 "mean 4.1 1630 082 O 68
D3 09050 - o is dominated by mean sys and otime with the other components making a small contribution. The mean of the power to melt data is represented by the LIFE-III calibration and the uncertainty on this mean is given by mean. 3.2.3 Power-to-Melt in CRBRP The probability of melting in CRBRP must be evaluated. To do this the LIFE-III calibration will be extrapolated from P19/P20 to CRBRP conditions. The power-to-melt margin is lowest near beginning of fuel life. Since we are considering only fresh and very low burnup fuel and this is the burnup range covered by P19/P20, the extrapolation is small and it is assumed that no biases are introduced. The additional uncertainty comes from the use of fuel with 33% Pu enrichment instead of 25% enrichment used in the calibration. An estimate of the values and uncertainties of power-to-melt in CRBR with reference fuel can be made. Then the uncertainties in the CRBR power-to-melt are statistically added to the P19/P20 mean uncertainty and to the extrapolation uncertainty to define a probability of melting.
- 1. CRBRP Design Uncertainties The uncertainties in a CRBRP power-to-melt due to the factors listed in Table 3.11 will be considered. Fabrication and irradiation uncertainties arise from design tolerances and uncertainties in neutron physics and thermal hydraulic calculations. These uncertainties can be estimated from the design tolerances by computing their effect on power-to-melt using LIFE-III.
The design tolerances will be for pellet density, cladding I .D., enrichment, instrumentation and pellet diameter. There is also a tolerance on fuel weight per length which prevents certain combinations of density and diameter. To simplify analysis and add the conservatism of not taking credit 'for this weight per length restriction, it will be neglected. Tolerances lead to fabrication of parts whose means may lie with approximately uniform probability anywhere within the fabrication tolerance tounds. The bounds of such a rectangular distribution correspond to + do. The tolerances are listed in Table 3.11. Actual distributions of dimensions and fuel density are expected to be available for use in the FSAR hot channel factor analyses. 69
80~0T050
- R g TABLE 3.11 CRBRP FUEL R0D TOLERANCES AND UNCERTAINTIES f UNCERTAINTIES REPRESENT /To Nominal / Tog Fuel Fellet Diameter .1935 i .0015 in.
Cladding Insile Diameter .200 i .0005 in. Fuel Pellet Density 91.3 11.6% TD Reactor Power
- 15.07%
Cladding 0.D. Temperature il5.6*F Near Midplane ** D
- Statistical combination of nuclear data, criticality, and fissile fuel maldistribution uncertainties.
** Due to thermal hydraulic uncertainties not including uncertainties in power.
1630 084 m 70
vARD-0-0056 ' Rtv 3 f 3.2.4 Typical Power-to-Helt Analysis LIFE-III runs are made to analyze the effect of variations in the parameters of Table 3.11 on the melting of the peak power pin. A progranined reactor startup is specified by giving the steady state reactor power as a function of time, REPOW (t). EPOW (t) is the total reactor power as determined by the reactor control settings, and Q (t) is the corresponding linear power including direct factors of the peak pin near its axial midplane at the location of peak centerline fuel temperature. The reactor power units are normalized to a value of 1.0 at nominal full power, i.e. REPOW (t) is 1.0 when the nominal power of the peak pin at X/L = .45, Q(t), is 12.73 kw/ft. This reactor power history remains the same for all the runs considered below. The power-to-melt is defined in LIFE-III by ramping the reactor power up until melting starts in the peak pin. This is done at various times during the steady state irradiation to obtain the burnup dependence. The progranined startup assumed for this study is illustrated in Figure 3.19. 3.2.4.1 LIFE-III Runs A code run with the nominal conditions of Table 3.11 and the power history of Figure 3.19 defines the mean reactor power-to-melt for the nominal peak pin as follows: REPOWg (nominal t) = C CIM(nminal,t) where Elg is the peak pin linear power at X/L = .45 when the LIFE-III centerline temperature reaches the fuel melting point and R7GI g is the reactor power at that time. CP, = 12.73 kw/ft because of the normalization of units chosen for reactor pcwer. REP 0W is labeled the mean reactor power-to-melt (signified M by the bar) since the mean of P19/P20 has been used to calibrate LIFE-III and the P19/P20 power uncertainty will be subsequently added. Code runs were also made for small perturbations of the design parameters to detennine the sensitivity of power-to-melt to variations in each parameter. The sensitivites are shown in Table 3.12 and the corresponding standard deviations in power-to-melt obtained from the standard deviations in the parameters from Table 3.11 are shown in Table 3.13. 1630 085 71
IARD-09005b '4
. try a s
p. c a
=
.e c
2 5 4 E 4 e, e s= . c W E i i; 5= e 1 a e
- S .$
=
c m e E a g = = = = W320d lifid 1VNIWON 30 %
.m w:<>.2n 1630 086 72
- - WARD-D'0050 1 StV 2 ' It is assumed that reactor power-to-melt can be approximated by a linear function of each parameter about the nominal point. This assumption was checked for pellet diameter variations for which the greatest non-linearities would be expected. Figure 3.20 shows the actual variations of power-to-melt with pellet diameter and the assumed linearizations which were selected to give a close fit over the range where power-to-melt was below nominal . 3.2.4.2 Extrapolation Uncertainty The melting temperatura of (U,25%Pu)0 2 is 2760 C (5000*F) with a bounding 3a uncertainty of 160*C (108'F). The melting temperature is based on an extensive review of all published data and a thorough analysis of the tech-niques and data has been included in the Nuclear Systems Materials Handbook E2 . The uncertainty was conservatively estimated from a review of the original da ta . It is not necessary to use an uncertainty on melting temperature when the LIFE-III code is employed to analyze (U,25%Pu)02 since the code was calibrated with fuel with this plutonium content. The nominal melting point value was used for calibration. Since the current CRBRP fuel contains 33% Pu02 , extrapolation of this code to predict the power-to-melt requires con-sideration of additional uncertainties due to increased Pu content. Based on the calibrated melting point of 2760 C for 25% Pu0 2 fuel, and co.isidering the 13a uncertainty of 160 C for the entire range of (U,Pu)0 2 solid solutions, the uncertainty in extrapolating from 25% to 33% Pu0 2was determined by estimating the uncertainty of the slopes. The 3a uncertainty in extrapolation is 2 x 60/100 = 1.2 C per wt% Pu02 or 9.6 C in extrapolating from 25% to 33% Pu0 fuel (i .e., melting point uncertainty This 2 TM = 9.6 x 1.8/3 = 5.8 F). uncertainty is applied to the nominal melting point for 33% Pu0 2. The sensitivity of power-to-melt changes in melting point. O REPOWg
/aT gp is calculated from the nominal LIFE-III run, and this is used to convert the uncertainty on melting temperature to uncertainty on power-to-melt
/
'extrap = agg7 aTgp o TM i630 087 3.2.4.3 Combining P19/P20, Extrapolation and CRBRP Uncertainties REPOW g is the mean reactor power-to-melt. To get the probability distribution for the reactor power-to-melt, the uncertainty in the P19/ P20 experiments
3 e e IARD-D 0050 RIV 2 c.
\
\ \
\ \
\ 5 i
g 3 0 \
! / /\
\/\ 1I \ c a
5 =s \ t
//\\ \
- i I/ \
- u m
5 x I i
/ \ a
=
I_
' \ sb
-x
'i O ^
\ C
.a -
n a // \ \ :
~
// \ \
JlI
\ \ l
\ \ E lI \ \' -
8 9, 4
/ \ .
II .i. Il l I I I l 5 $ $ 0 0 5 0 N 0d38 1630 088 e 99_6 ;I 74
- - IARD-0-0050 1 RE9 2 i ( mean) and the extrapolation uncertainties (aextrap) are added to the CRBRP design uncertainties (og), resulting in the probability distribution for reactor power-to-melt with standard deviation o REPOW(t)=!cy(t)2+"mean + extrap and a mean of REPOWg (nominal,t), where t is time-in-life. 3.2.4.4 Design Criterion The design criterion is that the reactor control settings be low enough so that 1.15* REPOW(t) < ITP5i g (nominal,t) - 3aREP04(t) for all time, or that a 15% overpower is three standard deviations below melting power. 3.2.4.5 Results of CRBRP Power-to-Melt Uncertainties The results of the analysis outlined above are suninarized in Tables 3.12 through 3.15. In Table 3.12, the derivatives of the power-to-melt with respect to the different parameters are calculated. In Table 3.13, these are used to convert design uncertainties into uncertainties on power-to-melt which are then statistically contined. The uncertainty on melting point due to extrapolation to a Pu enrichment different from P19/P20 is given in Table 3.14. The nominal LIFE _[I run is used to find the effect of a change in melting point on power-to-melt. This is used to convert the uncertainty of melting point into an uncertainty on power-to-melt. In Table 3.15, the uncertainties from the measured power-to-melt (converted to reactor power units), the extrapolation to CRBRP conditions, and the CRBRP desion uncertainties are statistically combined, and the design criterion for melting is examined to sea if it is fulfilled for the programmed reactor startup chosen. 1630 089 75
4 - WARD-0 0050 ret t n P. The positive margin at all times shows that the power-to-melt criterion is met by the selected programed startup. Applying Equation (2.8) to take into account the number of data points used in obtaining oP19/P20 and a~ s pace does not significantly affect the results. 1630 090
\
C' 76
1 WARD-D*0050 ' REV 2 1 r TABLE 3.12 SENSITIVITY OF POWER-TO-MELT TO DESIGN UNCERTAINTIES CHANGE IN POWER-TO-MELT FOR ONE PERCENT CHANGE IN DESIGN UNCERTAINTY [ ART 04/a(%xi n TIME' t X: er hrs. REPOW j pp ,) D pgj D clad T clad 8 0.7 .0040 .112 .098 .0107 .00092 58 0.9 .0017 .043 .062 .0027 .00069 108 1.0 .0051 .051 .060 .0093 .00081 158 1.0 .0051 - 050-
. .060 .010 .00081 O
Derivative taken from Figure 3.20. 1630 091 77
1ARD-0 1050 StV 2 8 C TABLE 3.13 , STATISTICAL COMBINATION OF POWER-TO-MELT UNCERTAINTIES DUE TO DESIGN UNCERTAINTIES ONE SIGMA POWER UNCERTAINTY RESULTING FROM DESIGN UNCERTAINTY STATISTICAL [aj aRU DIT/aX M $
] COMBINATION TIME
- t hrs. P pel Dpgj D Power T REPOW clad clad 'E 8 0.7 .0041 .050 .014 .0337 .0036 .062 85 8 0.9 .0017 .019 .0089 .0085 .0027 .023 108 1.0 .0052 023 .0087 .0294 .0032 .039 C 158 1.0 .0052 .022 .0087 .0315 .0032 .040 1630 092 m
78
1ARD-9*4050 1 RtV 2 9 r TABLE 3.14 EFFECT OF MELTING POINT UNCERTAINTY OF op(5.8'F ON UNCERTAINTY OF POWER-TO-MELT 3%/aTg a extrap t(hrs) kw/ft/'F (kw/ft) (REPOW Units) 8 0.0095 .055 .0043 58 0.0069 .040 .0031 108 0.0061 .035 .0027 158 0.0058 .034 .0026 0 1630 093 o 79
)
TABLE 3.15 TOTAL UNCERTAINTY ON CRBRP REACTOR POWER-TO-MELT IN REACTOR POWER UNITS WHERE NOMINAL FULL POWER IS 1.0 POW M
- 3a REPOW N g TOTAL o M REPOW (nominal,t) t (hrs) REPOW _T 'mean "extrap M -1.15 REPOW
.062 .047 .0043 .078 1.152 .113 8 0.7
.023 .055 .0031 .060 1.335 .120 58 0.9
.039 .058 .0027 .070 1.415 .055 108 1.0
.040 .059 .0026 .071 1.437 .074 158 1.0 8
- NC
==
& G U
O Nf a d o e C
I ARD-D-0050'
'1 RF.V 2 1 3.3 Coolant Exit Mixed Mean Hot Spot Factors for Fuel Assemblies e This Section discusses the hot spot factors which apply to the average coolant temperature (or enthalpy) rise for a single assembly.
Essentially, the same individual factors discussed in Section 3.1 for the enthalpy rise of the hot rod contribute te the coolant exit temperature; how-ever, integrated values over the assembly must be considered for the at.sembly mixed mean exit. The hot spot factors for evaluation of CRBRP fuel assemblies mixed mean exit temperature are presented in Tables 6.2A and 6.2B. 3.4 Fission Gas Plenum Pressure Uncertainty Factors for Fuel Assemblies This Section discusses the uncertainties which apply to calculations of the fuel rod fission gas plenum pressure. The fission gas plenum pressure is calculated through the perfect gas law: p = nRT/V C and consequently, depends on uncertainties affecting the rated value of: e plenum volume e plenum temperature e amount of gas released to the plenum, which depends on burnup (i.e., amount of gas produced) and fractional release. The fission gas release model, its calibration against experimental data from irradiation of mixed oxide fuels in EBR-II and the associated uncertainty analysis will be documented in a separate report for the FSAR. With regard to plenum volume, the minimum volume considering all the geometrical tolerances is conservatively assumed for all plenum pressure calculations. Finally, an uncertainty analysis of the various factors contributing to the plenum temperature and burnup w s conducted. Hot spot factors are reported in Tables 6.3A and 6.38. Since the analysis is used for fuel lifetime calculations, the hot channel factors are evaluated for plant expected operatirg conditions. C' Essentially, the same individual factors discussed in Section 3.1 for tne enthalpy rise of the hot rod are considered, keeping in mind that for plenum 81 1630 095
1ARD-D-005b '
't REV 2 1 pressure calculations integrated values over the rod length and assembly residence time must be considered. The various subfactors are discussed in the following.
3.4.1 Power Level Measurement and Control System Dead Band The power measurement bounding uncertainty including calorimetric power measurement error, setting instrumentation error and drift in instrumentation and electronics is 1.65% (2o). This appears in Table 6.3A as a 1.025 (3o) factor on burnup. The temperature effect appears as part of the reactor P and inlet temperature variation factor. The integrated effect of the 1 2% control system dead band over the assembly lifetime is estimated to be zero. The control rod stop which limits the combined power level and control dead band uncertainty to a maximum of 13% will conservatively be neglected. 3.4.2 Flow Distribution Calculational Uncertainty Two fuel assembly analyses were performed using COTEC with different mixing parameters. These results showed the uncertainty factor on coolant temperature rise in the core region to be greater than the uncertainty on average plenum temperature. It will be conservatively assumed that the factor for the plenum temperature is equal to that for the core region as discussed in Section 3.1.6. 3.4.3 Subchannel Flow Area The uncertainty will be less than that in the core region as discussed in Section 3.1.8 because the gas plenum temperature is the average of the six neighboring subchannels and because temperature uncertainties in the core region are reduced by coolant mixing in the gas plenum region. However,.for the present analysis the uncertainty will be conservatively assumed to be the same as for the core region. 3.4.4 Remaining Factors The remaining uncertainty factors are the same as for the hot rod, as discussed in Section 3.1. 1630 096 82
W A RD-D-00Sc ret t 4.0 BLANKET ASSEMBLIES HOT SPOT FACTORS The inner and radial blanket hot spot factors are evaluated similarly to the corresponding fuel hot spot factors even though the actual numerical value
'r.ay be different, since a different type of assembly is considered. The detailed discussion in Section 3 is still valid and therefore, it not re-peated here; only the major differences are discussed in this Section.
4.1 Fuel Rod Cladding and Coolant Temperatures Hot Spot Factors for Blanket Assemblies The hot spot factors for evaluation of CRBRP inner and radial blanket as-semblies rod cladding and channel coolant temperatures are presented in Tables 6.4A arp 6.4B. 4.1.1 Cladding Circumferential Temperature Variation Analyses of the blanket rod cladding circumferential temperature variation using the FATHOM-360 computer program showed that the hot spot factor is in the range of 3.0 to 3.3 across the film (depending upon a',semtly operating conditions), approximately 0.6 from cladding OD to midwall and approximately 0.7 from cladding OD to ID. A detailed analysis is performed for the life-time-limiting assembly in each flore zone. In addition, a 20% bounding un-certainty is applied to the deviations from the nominal case to conservatively account for uncertainties rasulting from finite-difference modeling, flow distribution uncertainties and flow mixing uncertainties. 4.1. 2 Inlet Flow Maldistribution Inlet flow maldistribution subfactors are the same as for Fuel Assemblies (Section 3.1.5) with the following exceptions. Manufacturing tolerances (excluding the rod bundle region which is considered separately in the Sub-channel Flow Area analysis) resulted in a bounding 3a flow uncertainty of 0.023 for the Radial Bianket. For the Inner Blanket, the same value was conservatively used as for the Fuel Assemblies. IPFM test data for Fuel Assemblies also apply for Inner Blanket Assemblies. Test data for Radial Blanket LIM's are given in Table 4.1. A statistical analysis of the four-run average flow for e 'h LIM showed a variation of 83 1630 097
WARD-D"0050 ' REV 3 l O.0294 based on 26 data points. This was increased to 0.04 on the basis of engineering judgement to allow margin to envelope possible differences between homogeneous and heterogeneous core configurations. This is considered to be conservative since based upon the observed differences between fuel assembly and radial blanket assembly LIM flow variations in the IPFM test, heterogeneous core radial blanket LIM's are expected to show less scatter than homogeneous core radial blanket LIM's. The 3a uncertainty in blanket assembly flow due to fuel assembly systematic error interation is calculated to be 0.011 by a similar analysis to that discussed in Section 3.1.5. An additional uncertainty occurs in the Radial Blanket because the number of orifices in the LIM is varied to achieve the goal assembly pressure drop. This step-wise approximation of the pressure drop requirements results in a bounding 3a flow uncertainty of 0.043. The remaining Inlet Flow Ma1 distribution subfactors are the same as for the Fuel Assemblies. The 3a subfactors for the Inner Blanket and the Radial Blanket are listed in Table 4.2. The overall factor calculated as the root-mean-square of the subfactors is also shown. 1630 098 84
i WARD-D-0050 REV 2 1 ' O TABLE 4.1 RESULTS OF FOUR IPFM FLOW DISTRIBUTION TESTS AT $100% FLOW, 3 LOOPS FLOW DEVIATION FROM AVERAGE, % RUN 1 RUN 2 RUN 3 RUN 4 LIM D17 .62 .53 .19 .48 D12 .00 .41 .07 .52 E1 .95 .91 .84 .52 E24 -1.37 .66 -1.10 .3? E23 1.05 .99 .95 1.70 E22 1.52 1.88 2.48 1.50 E21 .06 .16 .43 .42 E20 .38 .94 .23 .61 E19 1.08 .10 1.02 .22
.80 .56 C E18
.66 83 .88
.49 .42 .14 E17 E16 .67 1.44 1.53 1.94 EIS 1.42 .97 .19 1.03 E14 .10 .02 .05 .29 E13 .29 .17 .03 .01 E12 -2.15 .79 .38 -1.56 Ell -2.19 .89 .15 .84 E10 -1.19 .77 .63 -1.35 E9 .22 .22 .18 .42 E8 .64 98 -1.61 -1.36 E7 -2.65 -1.88 -1.84 -1.74 E6 1.68 1.32 1.35 .93 2.36 1.39 1.12 1.31 E5 E4 .87 -1.49 -1.43 -1.22 E3 1.77 .13 .78 1.08 E2 .02 .21 .15 .16 1630 099 0
85
WARD-0-0050 REV 2 r TABLE 4.2 BLANKET ASSEMBLY INLET FLOW MALDISTRIBUTION SUBFACTORS (3a) Inner Radial Blanket Blanket Hydraulic Characteristics Systematic 1.018 1.018 Random 1.045 1.045 Manufacturing Tolerances 1.013 1.023 Orificing Inflexibility ----- 1.043 Configuration Among LIM's 1.02 (33)* 1.04 (26)* p Among Assemblies 1.024(7)* 1.024(7)* Systematic Error Interation 1.011 1.011 Overall (3o) 1.060 1.084
- Number of data points shown in parentheses.
1630 100 r-86
1ARD-0~0050 ' REV 2 1 4.1.3 Flow Distribution Calculational Uncertainty The flow distribution calculational uncertainty for the blankets was based on extrapolating from the analysis for the fuel asseQ1ies (Section 3.1.6). The value of the coefficient C in Equation (3.8) was increased to 0.6945 x 10-3 in-I to account for a possible bundle size effect. The value was obtained empirically by interpolating between HEDL 217-rod b93 and ORNL 19-rod [6] data. The value of th? COTEC empirical cross flow parameters were determined from data from exra iments in diverse geometries EO-II) . The uncertainty analysis was performed as for the fuel assemblies (Section 3.1.6) and the largest hot channel factors were found to occur in the peak skew assem-blies. The value at the cladding peak temperature elevation was 1.199 (3a). This was combined with the flow split uncertainty of 1.04 (3a) for a 61-rod bundle (Section 3.1.6) to give an overall statistical uncertainty of 1.203 (30). In addition, a direct factor of 1.03 is applied. This includes the direct flow split factor of 1.01 for a 61-rod bundle (by the same method as in Section 3.1.6) as well as additional margin which based on previous experience with sodium heat transfer experiments in rod bundles is judged to be sufficient to cover the testing biases and uncertainties which might be uncovered during the heat transfer testing of a prototypic rod bundle in sodium since the blanket analysis is based on the extrapolation of non-prototypic rod bundle mixing tests. In a recent study [39] , the flow split was measured in a nearly prototypic blanket geometry using the isokinetic sampling technique. The ratio of in-terior-to-average subchannel velocity at the maximum flow rate tested was reported to be 0.88. This study is being evaluated and may require modifi-cation of the flow split used in COTEC blanket analyses and recalculation of AFMS assemblies flow distribution and temperatures. Test data are forthcoming which will provide a more direct measure of the code calibration and uncertainties for the FSAR analyses (See Table 6.11). A 5:1 scale test at ARD of a sector of a blanket rod bundle using air will provide detailed crossflew data. Heat transfer tests in sodium are being performed at ARD on a prototypic blanket rod bundle covering a range of power levels, flow rates, and power skews typical of blanket operating conditions in CRBRP. 87 1630 101
. - IARD-0 0050 , REV 3 I 4.1.4 Subchannel Flow Area COBRA IV[40] analyses, similar to those discussed in Section 3.1.8 for the fuel rod bundle, were performed for the blanket rod bundle. In addition to the nominal case, runs were made for the cases of a) a single nested subchannel, b) the hot subchannel and three adjacent subchannels all nested, c) maximum duct ID and mininum rod and wire 00, and d) minimum duct ID. From these results, the overall bounding 3a uncertainty in hot channel coolant temperature rise due to uncertainties in rod, wire and duct dimensions and rod spacings, and including the influence of the other subchannels, was cal-culated to be 0.053. The uncertainty was increased to 0.07 based on engineering judgement to account for calculational uncertainties. This uncertainty applies at the beginning of life. The effect of irradiation-induced swelling of rod wire and duct was analyzed for a homogeneous core radial blanket assembly assuming maximum distortion rates and resulted in a direct factor of 1.020 at end-of-life. Using homogeneous core fuel assembly swelling calculations, the heterogeneous core inner blanket direct factor was estimated to be 1.029 at end-of-life. Because of the uncertainty in extrapolat-ing from homogeneous to heterogeneous core conditions, a direct factor of 1.05 at end-of-life will be used based on engineering judgement for both inner and radial blankets until the corresponding analyses are performed for heterogeneous core conditions which will be done for the FSAR. 1630 102 88
WARD-0 REV 2 iOch 4.1.5 Film Heat Transfer Coefficient A conservative design correlation was selected in Reference 19 for the rod pitch-to-diameter ratio range 1.05 s P/D s 1.15 corresponding to that in CRBRP blanket and control assemblies. The data base is shown in Figures 4.1 to 4.3, together with the design correlation which is designed Curve 2. A 3a uncertainty of 0.21 was recommended based on the deviations of the data points overpredicted by the correlation. As an indication of the conservatism of the correlation, it may be noted that in the range of interest, the correlation results in lower heat transfer coefficients than the theoretical lower limit calculated for the case of turbulent flow through unbaffled rod bundles, under conditions of uniform heat flux at the inside surface of the cladding, with heat transfer by molecular conduction only through the sodiumb . 4.1.6 Pellet-Cladding Eccentricity The pellet eccentricity factor was assumed the same as for the fuel assemblies. Since it differs from unity only at beginning of life, it does not significantly affect blanket assembly lifetime. 4.1.7 Nuclear Uncertainties Tables 6.48 and 6.5B list the uncertainties applicable to the predictions of local and rod or assembly-integrated power uncertainties in the inner and radial blanket assemblies. Blanket uncertainties are divided into beginning and end-of-life values primarily as a result of the large change in isotopic fission ra tes . Values are presented for the heat flux (power density) uncertainty near the midplane at the location of the peak linear power and near the top of the 36-inch " core" (the approximate location of the peak clad temperature), and the channel coolant uncertainty (the rod-channel integrated power uncertainty for application to the total channel enthalpy rise and to the total rod burn-up) in Table 6.4B, and for the assembly-integrated power uncertainty (for the mixed mean coolant outlet temperature) in Table 6.5B. The blanket uncertainties have been derived to best fit the ecspective high power locations. 1630 103 89
g 102
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vARD-D*0050 M REV 2 4.1.7.1 Experimental Due to the somewhat limited scope of the measured blanket data in Phases A-E of ZPPR-7, it is recommended that the experimental fission and capture rate and gamma heating uncertainty component be combined directly (rather than in quadrature as with the purely statistical uncertainties) with the other uncer-tainties in the inner blanket assemblies, which is a conservative approach. e The experimental uncertainty component is defined to be three times the root mean square combination of the Pu-239 fission rate, U-238 fission rate, U-238 capture rate and ganma heating rate deviations, weighted by their respective fractions of the total inner blanket power for any particular location and for any particular time in life. The beginning-of-life inner blanket power experimental uncertainty is domin-ated by the statistical 110% (3a) uncertainty in U-238 fission rate which con-tributes nearly 60% of the total fresh blanket power (compared to less than 30% of the end-of-life power in the burned blankets). The remaining uncertainty at the beginning-of-life results from the 15-20% uncertainty in the predicted blanket gamma heating which constitutes nearly 40% of the beginning-of-life blanket power and about 20% of the end-of-life blanket power. Pu-239 fission calculation-to-experiment ratios vary +3% to +5% (3a) at beginning and end-of-life, respectively.. In the inner blankets. In addition, the U-238 capture rate (Pu-239 production rate) is substantially (914%) overpredicted, necessitating the application of an end-of-life fission rate bias of less than unity. Pu-239 fission comprises roughly 50% of the end-of-life inner blanket power. The 2% overprediction of axial peak-to-average power, as well as the s5% underprediction of power density at the extremities of the (36") " core", are biased out of the inner blanket experimental uncertainties in Table 6.48. Combining the afore-mentioned reaction rate anc ganma heating uncertainties with their fractional contributions to the total blanket power results in the time-dependent biases and uncertainties shown in Table 6.4B. In particular, the peak power density uncertainty is 110% at beginning-of-life and -515% at end-of-life. 1630 107 93
IARD-090050 REV 2 1 The radial blanket calculation-to-experiment ratios of fission and capture rate data in ZPPR-7 exhibit a characteristic space dependence, falling-off of the order of 10% in the radial direction from the core-blanket interface out to the blanket-reflector interface. Due to the limited number of radial blanket measurements and the space-dependent nature of the data, the radial blanket experimental uncertainties are treated non-statistically for each blanket row. The radial blanket experimental power uncertainty is derived from the limits of the observed variations in blanket fission and capture rates and gama heating. It is assumed that the observed variation limits represent 1.5o limits so that 3a equivalent empirical uncertainties can be obtained by doubling the maximum deviation from the mean of the Pu-239 fission rate, U-238 fission rate, U-238 capture rate and gamma heating rate variation limits weighted by their respective fractions of the total blanket power at any particular radial location and for any particular time in life. The beginning-of-life row 1 radial blanket power experimental uncertainty is again dominated by the empirical +618% uncertainty in U-238 fission rate which constitutes approximately 60% of the fresh blanket power. In the second row,
- the beginning-of-life U-238 fission rate empirical uncertainty is +23120% due to an apparent mis-prediction of the blanket spectral gradient. The remaining uncertainty results from the 120% uncertainty in the blanket gamma heating rate which constitutes about 40% of the fresh blanket power. The 2% over-prediction of axial peak-to-average power, as well as the s5% underprediction of the power density at the extremities of the 36 inch " core", are biased out of the radial blanket experimental uncertainties. The net resultant beginning-of-life radial blanket peak power density experimental uncertainties are therefore +2 1 9% (row 1) and +10 1 14% (row 2).
At the end-of-life, the U-238 fission rate (15-25% of the total radial blanket power) uncertainty increases to +13 i 12% (row 1) and +33 1 20% (row 2). The U-238 capture rate (Pu-239 production rate) is again overpredicted (+1018% in the first row and +4 i 8% in the second row). Additionally, the end-of-life Pu-239 fission rate uncertainty is 14% (row 1) and +714% (row 2). The U-238 capture rate (buildup) and Pu-239 fission rate uncertainties are both p 1630 108 94
WARD-0 0950 . REV 2 1 applied to the 60-70% of the end-of-life radial blanket power attributable to Pu-239 fission. Combining the Pu-239 fission rate, U-238 capture rate, U-238 fission rate and gamma heating rate uncertainties with their fractional con-tributions to the total radial blanket power, and applying the aforementioned 2% axial peak-to-average bias, results in the time and space-dependent experi-mental biases and uncertainties in Table 6.4B. The resulting net end-of-life radial blanket peak power density experimental empirical uncertainty is -417% in the first row and +518% in the second row. 4.1.7.2 Manufacturing Tolerances The 11% blanket heavy metal theoretical density tolerance gives rise to a similar bounding 11% uncertainty in blanket power generation. The U-235 con-tent tolerance in the depleted uranium blanket feed material results in a small (11%) uncertainty in the beginning-of-life blanket power, and a negligible uncertainty in the end-of-life power where U-235 fission contributes only about 2% of the total blanket power. 4.1.7.3 Modeling Uncertainties Blanket modeling bounding uncertainties consist of power-by-rod interpolation uncertainties, intra-assembly burnup uncertainties, axial power shape and buildup uncertainties and 3D geometric effects. The least-squares fit to power-by-rod in the inner blankets is performed with input from 24 mesh-per-assembly 2DB calculations. Pin-by-pin burnup does not substantially increase the peak end-of-life power in the highest power assemblies. The fitting uncer-tainty in the inner blankets is assumed to be 15% at beginning-of-life (where the fresh blanket power is sensitive to both the total flux and neutron spectrum gradients), and 11% of the end-of-life (where the peak inner blanket power is relatively flat and less fast-spectrum sensitive). The axial peak-to-average power shape uncertainty, deduced from RZ burnup calculations, is 11% near the core midplane and 15% at the top of the 36 inch " core" region, primarily due to smoothing at the end-of-life. Additional three-dimensional effects to not contribute a substantial uncertainty in the inner blankets (il to 2%). 1630 109 95
IAR0-0"0050 8 REV 2 f Radial blanket modeling uncertainties consist of power and burnup-by-rod (PUMA) uncertainties, axial power and buildup uncertainties and 30 geometric effects. The rod-fitted flux, cross-section and ganma heating and tht resulting composite power and burnup (buildup) distribution uncertainties are taken to be 15% at the beginning-of-life of 12 to 13% at the end-of-life The axial power shape based on PUMA benchmark verification calculations. uncertainty in the radial blankets is +l% at the core midplane at the beginning-of-life,12% at the core midplane at the end-of-life based on a chopped cosine approximation, and 15% at the top of 36 inch " core" region due to smoothing of the burnup distribution at end-of-life. The 2D power synthesis is arranged to best predict the radial blanket power (11%) near the highest power locations on the sides of the assemblies closest to the core. The net blanket modeling uncertainty is 17%. 4.1.7.s criticality and Control Rod Banking Uncertainties Criticality and control rod banking bounding uncertainties primarily influence the inner blanket power distributions. In addition to the effect of criticality s uncertainties on control rod insertion, and hence, on power distribution as discussed earlier, the uncertainty in fuel enrichment specification results in a flux level uncertainty which ultimately results in a 11% plutonium buildup and end-of-life blanket power uncertainty. 1630 110 96
-
- W A RD-0-0050 *
, REV 3 1 4.2 Power-to-Melt Uncertainty Factors For Blanket Assemblies
- g. 4.2.1 Extrapolation Uncertainty The LIFE-III Code's thermal analysis has been well calibrated against low-burnup (<1.1%) fuel pin PIE measurements. Blanket pin irradiation tests are in progress and additional tests are planned (see Table 4.3), but PIE data to verify the code are not currently available. Therefore, application of the LIFE-III Code to radial blanket pins involves extrapolation of fuel pin data. Three basic types of possible errors are involved.
- 1. Errors due to the larger diameter of radial blanket pins. i.e..
if any model contains an incorrect dependency on length (e.g., the thermal-gradient dependency of pore velocity may not be linear as assumed), there will be an error which will show up in the thermal analysis of a radial blanket pin even though the calibration corrects for this error in fuel pins;
- 2. Errors due to uncertain dependencies of various fuel properties on Pu0 content; 2
- 3. Errors due to extrapolation of thermal fuel pin data to higher burnups C (%1.6%) and/or times (%13,000 hours) that characterize blanket pins when their power is at a maximum.
Each of these three potential error sources is evaluated below. The present analysis uses the specifications and operating conditions of IB/A 128, pin 27. Garmia heating was taken explicitly into account. However, the approximation was made that it all occurs in the fuel and not in the cladding and sodium. This is a conservative assumption. The pin studied has the highest center temperature of any radial blanket pin. The section at X/L = 0.466 was run since a preliminary study showed that this value of X/L has the highest fuel center temperature. A power ramp was imposed on the normal operating history at the end of Cycle 4, at which time the power is highest, to determine the power-to-melt. The results of the LIFE-III runs are given in Figure 4.4 for initial diametral gaps varying from 4 to 9 mils. The powers-to-melt are all about 23.8 kW/ft (39% over-power) with very little dependence on initial gap size. The latest (Revision 5) f' 1630 \\\ 97
- ' WARD-D-0050 REV 2 ! TABLE 4.3 SLANKET IRRADIATION TESTS SCHEDULE FOR DRS NO. FACILITY TEST COMPLETION PURPOSE 31.3.03 EBR-II WBA-20 9/80 Power Jump Test WBA-21 9/83 2a Limit Test 31.3.04 FFTF WBA-40 9/87 RB/A Proof Test WBA-41 9/85 IB/A 2o Limit Test WBA-45/46 9/85 Instrumented Natural Circulation Test; Power-to-Mel t Test 1630 112 98
1ARD-01050 REV 3
** \ = 40
- c. .
POWER TO MELT 23 - 30 22 - 21
>- 5 g 20 g 5 20 -
g IE o 2
- 15% OVERPOWER ABOVE NOMIN AL 19 -
10 ,r-18 __ EOC4 FULL POWE R (3 - 0)
- 0 17 -
t i I I I 16 4 5 6 7 8 9 10 DI AMETRAL G AP, MILS Figure 44. Power-To-Melt Slutly. til A 128. Pin 27 X 'l = 0.466 EOC4
.br Operating Conditions.
*M:6C 5 r
99
i
-
- g3gD1050 cladding swelling equation was used, with a nominal r of 6.3. The NSMH[2d fuel melting temperature (solidus) was used. It takes burnup and Pu0 2
e content explicitly into account. A Pu0 2content at the end of Cycle 4 of 4.5 wt.% was used. This gives a fuel melting temperature of 5057"F. Error Source (1) There are three possibilities for errors of this type: (a) In addition to microscopic roughness, the surface roughness term in the gap conductance equation includes " waviness" and " errors of form" which are likely to be proportional to the fuel pellet diameter even though LIFE-III assumes this roughness to be independent of pellet diameter. To evaluate this error, it was assumed that the worst possible case of zero microscopic fuel surface roughness applies and that the entire fuel surface roughness is composed of waviness and errors of form. Thus, the surface roughness term for radial blanket rods should be increased by the ratio of the blanket pellet diameter to the feel pellet diameter (2.36). The effect of this change on radial blanket power-to-melt was found to be a reduction from 23.58 o to 23.31 kW/ft., i.e., by 1.6% of nominal full power (16.86 kW/ft). The evidence as to the basic character of the surface roughness is in favor of it being composed of microscopic roughr.ess which is determined by the grinding operation (and hence independent of pellet diameter). LITE-III uses roughness values characteristic of grinding operations. (b) The pore velocity equation may not be linear in thermal gradient as assumed. This is highly improbable, however, because all known mechanisms for pore migration (surface diffusion, volume diffusion, vapor-phase transport) are all known to be linear in thermal gradient. (c) The fuel swelling rate may not properly model changes in scale. Thus, there is an uncertainty in the state of the fuel-cladding gap. In the nominal case the gap is open by 0.17 mils (radial) at the start of the power ramp (at the end of Cycle 4). Because of uncertainty, however, the possibility of a significantly larger gap exists. To evaluate the uncertainty in power-to-melt resulting from the uncertainty n 1630 114 100
i WARD-DN0050 REV 2 { in fuel swelling, the swelling rate was reduced by a factor of 10 to represent a bcunding 3a uncertainty. This produced a start-of-jump gap of 0.60 mils, but it increased the power-to-melt to 24.1 kw/ft. The reason for this result is that the fuel ran hotter during its lifetime and consequently increased the central hole from 9.4 mils to 16.2 mils. The fuel swelling rate was then increased by a factor of 10 to cause the gap to be closed at the start of the power jump. This left the central hole essentially unchanged and reduced the power-to-melt to 23.46 kw/ft. This sensitivity yields a bounding uncertainty of 0.71% of nominal full power. Since further fuel swelling rate increases cannot reduce the gap further, this uncertainty represents essentially the maximum that can be obtained. Error Source (2) Consider first the dependence of fuel thermal conductivity, K , on Pu0 f 2 content. To take account of the lower Pu0 2 content (4.5% at end-of-life) of the radial blanket pins relative to the 25% Pu0 content of the calibration 2 pins, the nominal value of K 7was increased by 3.2% in accordance with the accepted dependence E4I3 of K on Pu0 content. To estimate the 3a error f 2 involved in this extrapolation to lower Pu0 content, it was assumed thst 2 the 3a error in the gradient in the Kf vs. Pu02 function is t100%. Thus, the most negative uncertainty in K is -3.2%. This produces a f reduction in power-tc-melt of 3.1% of nominal full power. Next, consider the dependence of gap conductance, Hg, on Pu0 content. 2 There is no obvious reason why gap conductance should be dependent on Pu0 2 content in any but very minor ways. U0 -Pu0 hardness may be somewhat 2 2 dependent on Pu0 content. However, since the stainless steel cladding is 2 the softer material, the hardness of the fuel does not enter the gap conductance equation. Pore migration velocities depend on Pu0 content by virtue of the 2 different diffusivity and vapor pressure of Pu0 . The difference between 2 the vapor pressure of Pu0x and U0x gaseous species over solid U0 -Pu0 at 2 2 a given temperature is very small however. Thus, pore migration velocities should be only slightly dependent on Pu0 content. This conclusion is 2 further supported by the fact that the rate of pore migration is controlled by the migration rate of the least-mobile species. Thus, over a wide range 10 I630 115
IARD-garn50 'T ret 2 s of Pu02 contents, only very little variation in pore-migration velocity is expected. Therefore, uncertainties from this source are taken as negligibly c small in this analysis. y, Error Source (3) The LIFE-III thermal analysis was calibrated and checked-out agdinst fuel pin PIE data that extended out to burnups of 1.1% and times of about 1,000 hours. Radial blanket pins of interest here have peak burnups of about 1.6% and operating times of about 13,000 hours. Thus, extrapolations outside the calibration-checkout data base are implied by LIFE-III thermal analyses of the radial blanket pins. There are four basic extrapolations to be evaluated here: (a) Melting temperature extrapolation An uncertainty in radial blanket power-to-melt calculations arises from the uncertainty in melting temperature. This uncertainty has two components: (a) the uncertainty in the U0 2-Pu02solidus temperature, and (b) the uncertainty in the effect of burnup on melting temperature.
- The lo uncertainty values for these items are 25[42] and 24.5 C[43]
respectively. A statistical combination of these yields a net lo uncertainty of 35 C. This temperature uncertainty translates into a power uncertainty of 1.05 kW/f t. (3o), i .e. , 6.2% of nominal full power. (b) Gap Conductance Extrapolation Gap conductance, Hg, changes with time because the gap surface roughness is reduced by creep (which causes an increase in Hg.) LIFE-III does not model this creep-relaxation and hence predicts no increase in Hg due purely to time. Hg does change with burnup, however, due to (1) fission gas dilution of the plenum gas and (2) the buildup of solid fission products in the gap. Fission gas dilution of the plenum gas is determined by the amount of fission gas released and the initial moles of helium fill-gas. The fission gas release model has been calibrated against fuel pins covering a wide range of burnups and times as part of the calibration of the mechanical r 1630 116 102
1ARD-0 0050 1 MV P analysis of LIFE-III. However, there is some question as to the direct applicability of this model to radial blanket pins. As an estimate of the 3a uncertainty associated with applying the present LIFE-III fission gas release model to radial blanket pins, the nominal fission gas release rate was decreased by a factor of 100. This reduced the overall fission gas release from 85 to 41% which is considered to be about as large an error in fission gas release that differences between fuel and blanket pins could conceivably cause. This was found to reduce the power to melt by 0.36 kW/ft. i.e. by 2.13% of nominal full power (16.86 kW/ft.). The effect of a given amount of fission gas dilution of the plenum gas on gas thermal conductivity has been accurately measured over the full range of gas compositions by Von Ubish, Hall, and Srivastav b44) The small uncertainties in the Von Ubish et al measurements are negligible in comparison with the remaining uncertainties. The buildup of solid fission products in the fuel-cladding gar is modeled only quasi-empirically by using what is referred to as the "CRUDFIL" factor. To simulate the burnup dependence of this factor, the rate of transfer of solid fission products to the gap is taken to be directly proportional to the-rate of release of gaseous fission products from the fuel. The in-reactor behavior of UO 2 r UO2 with lower Pu02 contents (blanket fuel) is very similar to core fuel (s25% Pu0 2-UO2 ). Thus, it is expected that the "CRUDFIL" factor derived from core fuel experiments should apply to blanket fuel. However, there is some uncertainty concerning the exact value. As a bounding estimate of the 3;r uncertainty associated with this model, it is assumed that the rate of transfer of solid fission products to the gap in a radial blanket pin is only 1/4 of the nominal rate used in LIFE-III for fuel pins. The basis for this assumption is that for this and smaller values, it was found that fuel temperatures in mechanical calibration pins become so high, that the fuel becomes too sof t to strain the cladding or develop a center hole as observed in the experiments. Using 1/4 the nominal CRUDFIL factor on radial blanket pin 27 of IB/A 128 is found to reduce the power-to-
- 1 t from 23.58 kw/f t to 22.89 kw/f t, representing a reduction of 4.1% of nominal full end-of-life power (16.86 kw/f t).
103 1630 l17
i
- ' IARD-0-0050 REV 3 f (c) Restructuring Rate Extrapolation Restructuring is a function only of time and temperature. The best r single indicator of tue overall rate of restructuring is the central hole size. The LIFE-III calculation of central hole size has been calibrated not only against low time, low burnup pins in the thennal calibration, but also against a wide range of burnups and times in the calibration of the mechanical analysis. The agreement between calculation and measurement is good over a wide range of burnups (0-16.5%) and times (0-21,000 hours). This is shown in Figures 4.5 and 4.6. The standard deviation for the 37 comparisons was found to be 1.7 mils.
This results in a 3o uncertainty of about 5.1 mils. To relate the uncertainty in central hole radii to a corresponding error in power-to-melt, the power-to-melt of several high temperature fuel pins was determined with various central hole radii obtained by varying the assumed activation energy for as-fabricated pore migration. It was found that the power-to-melt changed by 0.06 to 0.08 kW/ft per 1-mil change in central hole radius. Thus a 5.1 mil uncertainty in central hole radius results in an uncertainty of 0.31 c to 0.41 kW/ft in power-to-melt. For a nominal full power of 13.72 kW/f t this uncertainty is equivalent to 2.3 to 3.0% of nominal full power. This uncertainty of 3% (3a) of nominal full power was applied to the blanket. (d) Thermal Conductivity Extrapolation Fuel thermal conductivity (Kf ) changes with time due to fuel restructuring which changes the porosity. It also changes with burnup due to the buildup of fission products within the fuel. Because of the good correlation of computed and measured central hole radii over a wide range of times and burnups it is unlikely that the extrapolation of the porosity-dependent factor in the LIFE-III equation for Kf to a burnup of about 1.6% involves any significant error. The effect of fission product buildup on Kf has been measured out to only 0.16 atom percent burnup[45] . No change in thermal conductivity 1630 118 104
g
.O ME LT HADil 9
& P20 0 P19 CENIR AL HOLES O P20 e 0.10 -
S e N e e a C
- a S A 0.05 g O
D 0 0 0.05 0.10 ME ASURED R ADil, cm Figure 4-5. Comparison of Computed and Observed Central Hole Radii and Melt Radii 99:6 _6 - r-1630 l19 105
- WARD-0~cosa -
REV 2 mm 0.25 0.50 0.75 1.00 1.25 1.50 1.50 DATA FROM PINS F2-V, F9C 13, F9A 5 P23 A-25, W213. W2-17 40 - - 1.00 e O, a - e 0.75 e g 30 S a, o a
$ O
'A 20 -
e 0.50 0 10 N A O BOTTOM - 0.25 8 e MID-SECTION 2 6 TOP I I I I I Oe O 10 20 30 40 '50 60 ME ASU RE D, mils Figure 4-6. Comparison of Calculated and Predicted Center lloie Radii for \lechanical Calibration Pins. 9926-27 los 1630 120
g0-0 f5b : was observed. For this preliminary analysis, a worst case estimate was made by assuming that one hole percent of fission products is equivalent to one percent of porosit . In the low-porosity limit one percent porosity changes Kf by 1.5%E O (which is about three times the effect of Pu02 .) The porosity created by the 0.246 atoms per fission of Xe and Kr is already considered in the LIFE-III calculation of Kf. This leaves 2.0 .246 = 1.754 fission product atoms per fission to be accounted for. At a burnup of 1.6 a/o burnup this worst-case ap-proximation treatment yields a reduction in Kf of 4.2% (1.6 x 1.75 x 1.5). Such a reduction in Kf is found to reduce power-to-melt by about 4.0% in IB/A 128, pin 27. This is equivalent to a reduction of 5.6% of nominal full power (16.86 kw/ft). (e) Cladding Swelling The uncertainty in the cladding swelling equation used is ex-pressed as an uncertainty in the parameter T. The empirical worst value of 1 is 5.0 (3o), relative to a nominal (best fit) value of 6.3. A LIFE-III run with r = 5.0 and a diametral cold gap of 4 mils resulted in a reduction in power-to-melt of 2% of nominal full power. SUMY As stated previously, extrapolating the LIPE-III code, which has been calibrated against fuel pin data, to radial blanket pins involves three basic types of uncertainties: (I) Geometry effects; (II) Pu0 effects; 2 (III) Burnup extrapolation effects Each of these three categories has been further resolved into its primary , components and these have been analyzed and estimates of the associated uncertainties made. These are now summarized in Table 4.4. 1630 121 107
1 IARD-D-0054 4.2.2 CRBRP Blanket Rod Design Uncertainties As had been done for the fuel rods, the uncertainties introduced by fabrication tolerances and irradiation uncertainties are estimated by LIFE-III runs in which a parameter is varied within its tolerance to see its effect on power-to-melt. Table 4.5 lists the tolerances considered. Table 4.6 shows the sensitivity of power-to-melt to variations in each parameter. These results are then used to connect the fabrication and irradiation tolerances into uncertainties on power-to-melt which are statistically combined to give the CRBR design contribution to power-to-melt uncertainty, o_, which is also shown in Table 4.6. R 4.2.3 Total Blanket Rod Uncertainty and Design Criteria Using the same analysis procedure as for fuel, the EBR-II data base uncertainty mean, the uncertainty due to extrapolation to CRBRP conditions, oextrap' and the design tolerance uncertainties are combined in Table 4.7 to give the total uncertainty in blanket rod power-to-melt. REP 0N. Application of the design criterion of no melting at 115% of nominal f"ull power at a 3a level of confidence shows that the design criterion is satisfied, and an additional margin of 4.9% in power remains. 1630 122 108
. - WARD-0~0050 i TABLE 4.4 9EV 2 EXTRAPOLATION UNCERTAINTIES FOR BLANKET RODS r '
3aj (I) GEOMETRY EFFECTS: - (A) Gap conductance fuel-surface roughness ).6 % NFP (B) Pore velocity % 0 % NFP (C) Fuel swelling (Gap size) O.71 % NFP (II) Pu0 EFFECTS 2 N3.1 % NFP (III) BURNUP EXTRAPOLATION EFFECTS: (A) Melting Temperature extrapolation 6.2 % NFP (B) Gap Conductance extrapolation 4.6 % NFP (C) Restructuring Rate extrapolation 3.0 % NFP (D) . T'hermal Conductivity extrapolation 5.6 % NFP (E) CladifingSwelling 2.0 % NFP c-A statistical combination of the above uncertainties yields 10.81 at a 3o level for the total extrapolation uncertainty associated with applying LIFE-III to radial blanket pins. 1630 123
/
's e
109
1ARD-D 0050 ' SEV 2 ' TABLE 4.5 c CRBRP Blanket Rod Tolerances and Uncertainties Uncertainties represent T3a Nominal dog Fuel Pellet Diameter .470 i .001 in. Cladding Inside Diameter .4760 1 0005 in. Fuel Pellet Density 95.6 i 1.0% Reactor Power
- 0 Cladding 0.D. Temperature Near Midplane"* 25 F e
All power uncertainties included as direct factors in " nominal" rod power. Due to thermal hydraulic uncertainties not including uncertainties in power. 1630 124 c. 110
) 3 1 -
TABLE 4.6 Blanket Rod Sensitivities of Power-to-Melt to Design Uncertainties; Power-to-Melt Uncertainties due to Individual Design Uncertainties; and the Statistical Combination of These Power-to-Melt Uncertainties (a) Change in Power-to-Melt for one percent change in Design Uncertainty, [3REPOWg /a(%X$ )] (b) One Sigma Power Uncertainty Resulting From Design Uncertainty, [oj aREP0WM /aX$ ] Statistical Combination X j o pel D pel D clad Power T clad pt) (a) (b) (a) (b) (a) (b) (a) (b) (a) (b) C m
.0019 .0011 .025 .0031 .0095 .0006 .0073 0.0 0.0 0.0 .0033 U
CD som
~ $:ID a
m W haf
-s 8
o M
I ) ) TABLE 4.7 Total Uncertainty on CRBRP Reactor Power-to-Melt for Blanket Rod in Reactor Power Units where Nominal Full Power is 1.0 REPOWg REP 0Wg -3 REP 0Wg o o extrap. total eREPOW oR mean (nominal,t) M -1.15 REPOW
.0033 .057 .036 .067 1.400 .049 N
- MM D 45t3 w w a us
_a m o
REV 2 4.3 Coolant Exit Mixed Mean Hot Spot Factors for Blanket Assemblies The blanket assemblies mixed mean hot spot factors are evaluated similarly to the corresponding fuel assemblies factors (Section 3.3 and Tables 6.2A and 6.28). The various subfactors are presented in Tables 6.5A and 6.5B. As previously pointed out in Section 4.1.7, different nuclear uncertainties must be considered depending on the assembly radial position and time-in-life. 4.4 Fission Gas Plenum Pressure Hot Spot Factors for Blanket Assemblies The blanket assemblies fission gas plenum pressure hot spot factors are evaluated similarly to the corresponding fuel assemblies factors (Section 3.4 and Tables 6.3A and 6.3B). The remarks in Section 3.4 also applies to this set of hot spot factors. The various subfactors accounted for in evaluating the CRBRP blanket as-semblies rod fission gas pressure are presented in Tables 6.6A and 6.68. 1630 127 113
gjo9050 8 5.0 PRIMARY CONTROL ASSEMBLIES HOT SPOT FACTORS Primary control assemblies (PCA) hot spot factors currently being used for design analyses are shown in Tables 6.7 through 6.10. These factors currently in use were developed for the homogeneous core, and are being updated for the heterogeneous core. Table 5.2 shows a comparison between the current uncer-tainties and a preliminary updating, which shows that the current uncertainties give overall conservative results. The PCA factors are generally more conser-vative than those which have been adopted for fuel and blanket assemblies because of the less advanced status of the primary control assemblies design. The factors will be updated for the FSAR analyses as the design progresses, and more data such as those from the test listed in Table 6.11 are obtained. The primary control assemblies hot spot factors are evaluated in a similar fashion to the fuel assemblies hot spot factors discussed in Section 3, with four exceptions. One difference is the introduction of three separate hot spot factors to account for uncertainties in heat flux through the absorber, the cladding and the coolant. This is due to the fact that, while in the fuel and blanket rods approximately 98% of the heat is generated in the fuel pellet, a sizeable amount (s20%) of heat is generated in the much thicker cladding of the control rod and in the coolant. A second major difference between control and fuel assemblies is the introduction of a new factor to account for un-certainties in prediction of the flow split between absorber bundle and bypass; in fact, only the fraction of coolant flowing through the absorber bundle acts as an effective heat removal medium. A third difference is the inclusion of separate gap and absorber uncertainties, which were combined in the equivalent fuel and blanket pin power-to-melt analyses. A fourth dif-ference is the need for a Lifetime Structural Calculations Uncertainty Factor which will be discussed in Section 5.4. 5.1 Absorber Pin Temperatures Hot Spot Factors for Primary Control Assemblies 5.1.1 Power level Measurement and Control System Dead Band This factor is 1.03 and is identical to that for the fuel assembly as discussed in Section 3.1.1. 5.1.2 Inlet Flow Maldistribution 630 128 The coolant enthalpy rise factor is taken as a direct factor of 1.08. An analysis similar to those for the fuel and blanket assemblies (Section 3.1.5 and 4.1.3) was perfonned, and the results given in Table 5.1. The resulting lla
IARD-D-Obbo REV 2 - n TABLE 5.1 CONTROL ASSEMBLY INLET FLOW MALDISTRIBUTION SUBFACTORS (3o) Hydraulic Characteristics Systematic 1.018 Random 1.045 Manufacturing Tolerances 1.023 Configuration Anong LIM's 1.02 Anong Assemblies 1.00 Systematic Error Interaction 1.011 Overall (3o) 1.058 r-1630 129 115
IARD-090050 i ' REV 2 1 bounding uncertainty factor of 1.058 (3a) is less than the assumed direct facter of 1.08, and therefore the assumed factor is conservative. More data will be forthcoming from the PCA hydraulic test (see Table 6.11) and will be used for the FSAR analyses. 5.1.3 Assembly Flow Ma1 distribution Calculational Uncertainties A direct factor of 1.08 is taken, which based on available wire-wrapped rod bundle data (see Sections 3.1.6 and 4.1.3), is judged to be sufficient to account for uncertainties in the THI-3D E473 calculations of subchannel temper-ature distributions. A film coefficient uncertainty of 1.075 is conservatively adopted, even though the correlation shows no variation with flow in this range (Figure 4.3). This value includes the uncertainties in inlet flow (previous Section) and bundle / bypass flow split (next Section). 5.1.4 Bundle / Bypass Flow Split The control assembly flow splits into absorber assembly and bypass flow com-ponents; only the former contributes directly to absorber element cooling. The uncertainty in predicting the absorber assembly fraction of control as-sembly flow is estimated to be 10% on the following bases: a) the theoretical flow split prediction shows[48] a maximum under-estimate of 6% when compared with HEDL data; b) since the HEDL data were for a 61-rod bundle, based on engineering judgement an additional 4% was included to account for extra-polation to a 37 rod bundle. Future flow testing of the reference control assembly (see Table 6.11) will reduce this uncertainty. 5.1.5 Cladding Circumferential Temperature Variation The analysis of cladding circumferential temperature variation was performed by a similar method to that for the fuel pin. The factor was found to be 1.286 at the cladding midwall . This was increased to 1.40 on the basis of engineering judgement to allow for calculational uncertainties. This analysis will be updated for the FSAR. 5.1.6 Reactor AT and Inlet Temperature Variation Statistical factors (3a) of 1.02 for inlet temperature variation and 1.04 for reactor AT variation are assumed at Plant Expected Operating Conditions. In addition, the assembly inlet temperature is increased by 16*F.
,,, 1630 130
- WARD-D-0050 '
REV 2 l As discussed in Sections 3.1.3 and 4.1.4, the factor should be about 1.14 (30) together with a 7.4 F (30) uncertainty in inlet temperature due to loop tem-perature imbalance. Therefore, the assumed factors are not sufficiently con-serva tive. However, it will be shown in Section 5.1.16 that the updated total uncertainty is less than that assumed in the analysis. Therefore, the overall results are conservative even though the factors for reactor AT and inlet temperature variation are not sufficiently conservative. 5.1.7 Absorber Maldistributton and Conductivity FFTF calculated the statistical hot channel factor for temperature drop through the pellet due to mechanical deviations to be 1.033. The uncertainty on local absorber heat generation is equal to the uncertainty on pellet temperature drop due to mechanical deviations. (Pellet temperature drop due to conductivity and restructuring is considered in a separate subfactor). The above uncertainty value was adopted, pending further analysis for the specific conditions of CRBRP control assemblies (37 vs. 61 pin design, hence larger diameter absorber pellets) which will be done for the FSAR. A 2% uncertainty is assumed for coolant enthalpy rise since approximately 20% of the coolant enthalpy rise is due to cladding gama heating and since the axially integrated effect of local absorber heating variation will be less than the 3.3% maximum local value. The thermal conductivity of the B C 4 absorber in the control assemblies was taken from Reference 14 and is valid for B-10 captures > 5 x 10 20 captures /cc 440 < T < 180 F The maximum uncertainty was reported in Reference 48 to be in the range of 6 to 9%. To conservatively accommodate this possible variation, a bounding un-certainty value of 10% (3o) has been incorporated in the thermal and structural analyses of the absorber pin. Also, the use of an expression based on burnup levels greater than 5 x 10 20 captures /cc provided further conservatism, since for lower burnup, the B 4C thermal conductivity is much higher than that expressed by the equation. Since the absorber centerline temperature is of the order of 2000 F,, a slight extrapolation of the correlation is necessary.
"7 1630 131
WAR 0-0-0050 t REV 2 5.1.8 Wire Wrap Orientation The maximum statistical variation in the hot channel AT is estimated to be 1% based on the discussion in Section 3.1.7. 5.1.9 Subchannel Flow Area A COBRA analysis which reduced to a minimum the area and rod to rod gaps of only the hot channel was compared to a nominal case. The result was a hot channel subfactor of 1.059. In addition to the subfactor for dimensional variation, a value of 1.062 was included statistically to account for pin diametral increase consistent with FFTF analyses. The statistically combined value of dimensional variation and pin diametral increase is 1.086. Since the film coefficient actually increases, a value of 1.00 is con-servatively taken for the film uncertainty. Associated with the minimum channel area is a reduction in gamma heat consequent to the reduced subchannel volume. For conservatism, a value of 1.00 is assumed for coolant heat flux. 5.1.10 Film Heat Transfer Coefficient An uncertainty factor of 1.21 for film heat transfer coefficient was used consistently with the data and recomendations of Reference 19. 5.1.11 Pellet-Cladding Eccentricity The phenomena accompanying pellet-cladding eccentricity in control assembly absorber rods are similar to those discussed in Section 3.1.10 for fuel rods. At the beginning of life with hot absorber pellet and cladding dimensions, the case of maximum absorber pellet to cladding eccentricity in the control assembly rods resulted in film and cladding subfactors of 1.374 and 1.386, respectively. The film and cladding sub-factors are local and, as such, do not apply to fuel temperature calcu-lations. Fuel and gap subfactors of 1.0 were assumed for conservatism (see discussion in Section 3.1.10). 1630 132 118
i 1ARD-D-0050 REV 2 ' 5.1.12 Cladding Thickness and Conductivity A 1.10 uncertainty factor on irradiated cladding thermal conductivity in addition to a 2.2% dimensional variation combines statistically to give a 1.10 hot spot subfactor for cladding AT. This is conservative as discussed in Section 3.1.11. The cladding dimensional variations result in a 1.044 subfactor for local cladding gannia heat generation and a subfactor less than 1.01 for coolant enthalpy rise since gamma heating is only about 20% of the total heating. 5.1.13 Gap Thickness and Conductivity An empirical 10% uncertainty in thermal conductivity of the gap gas combined statistically with a bounding 8.1% statistical uncertainty on hot gap thick-ness results in a 1.128 value for the subfactor. The uncertainty on hot gap thickness includes the thermal expansion uncertainties and cold dimensional tolerances of the cladding I.D. and the absorber pellet 0.D. at beginning of life. The absorber pellets always swell faster than the cladding, so swelling, even with swelling uncertainties included, always results in gap closure and reduced gap AT. Thus, the hot spot gap AT is maximum at the beginning of life when the fluence induced swelling is negligible. 5.1.14 Coolant Properties The statistical coolant properties variation in the control assemblies is identical to that discussed in Section 3.1.12 for fuel assemblies. 5.1.15 Nuclear Uncertainties Uncertainties due to physics analysis calculational methods (15% on coolant enthalpy rise and on absorber, cladding and coolant heat generation) are applied directly on nuclear radial peaking factors. 5.1.16 Overall Uncertainty As an example, assume a nominal plant expected coolant temperature rise of 300 F in the hot channel. Table 5.2 shows a comparison of the coolant temper-ature rise for the cases of current uncertainties from Table 6.7 and updated "9 1630 133
WARD-D-0050 1 REV 3 f TABLE 5.2 COMPARIS0N OF HOT SPOT COOLANT TEMPERATURES CALCULATED USING CURRENT UNCERTAINTIES AND UPDATED UNCERTAINTIES Assume Nominal Plant Expected Coolant AT = 300 F Current Preliminary Uncertainties Updated Direct (Table 6.7) Uncertainties Power Level Measurement and Control System Dead Band 1.03 1.03 Inlet Flow Ma1 distribution 1.08 1.02 Assembly Flow Maldistribution Calculational Uncertainties 1.08 1.08 Bundle / Bypass Flow Split 1.10 1.10 Statistical (3o) Inlet Flow Maldistribution --- 1.058 Inlet Temperature Variation 1.02 1.146 Reactor AT Variation 1.04j Absorber Maldistribution and Conductivity 1.02 1.02 Wire Wrap Orientation 1.01 1.01 Subchannel Flow Area 1.09 1.09 Cladding Thickness and Conductivity 1.01 1.01 Coolant Properties 1.01 1.017 Loop Temperature Imbalance --- 1.025 TOTAL 2o 1.413 1.402 3o 1.459 1.479 Coolant AT, F, above nominal inlet: 2o: (1.413)(300)'= 424 (1.402)(300) = 421 Inlet: 16 --- Total 440 421 3o: (1.459)(300) = 438 (1.479 )(300) = 444 Inlet: 16 --- Total: 454 444 1630 134
WARD-D-0050 t ' REV 2 8 uncertainties from the preceeding Sections. The updated inlet temperature uncertainty of 7.4 F due to loop temperature imbalance has been converted to an equivalent uncertainty of 1.025 on coolant temperature rise. The coolant temperature rise above nominal inlet is found to be 440 F (2a) and 454 F (3a) using current uncertainties, as compared sith 421 F (2a) and 444 F (30) using updated uncertainties. Therefore, the current uncertainties give overall conservative results. 5.2 Coolant Exit Mixed Mean Hot Spot Factors for Primary Control Assemblies The control assemblies mixed mean hot spot factors are evaluated similarly to the corresponding fuel assemblies factors previously discussed in Section 3.3. The various subfactors are presented in Table 6.8. No flow split uncertainty factor is considered in evaluation of the mixed mean outlet temperature, as the flow is judged to be fully mixed at the assembly exit. 5.3 Fission Gas Plenum Pressure Hot Spot Factors for Primary Control Assemblies Table 6.9 summarizes the hot spot factors applied in evaluation of the primary control assembly absorber rod gas plenum pressure. In addition, the minimum plenum volume and the HEDL recommended fission gas re-lease design correlation E493 are conservatively considered in plenum pressure calculations. 5.4 Primary Control Assemblies Lifetime Structural Calculations Uncertainty Factor In perfonning the cladding stress / strain mechanical analyses, proper consideration must be given to swelling of B4C pellets and cladding during the absorber lifetime. Reference 49 reported that the diametral growth of the B4 C pellet is directly proportional to burnup but inversely proportional to the irradiation temperature. To assure conservative evalua-tions: a) hot spot factors on burnup corresponding to the ones developed for the plenum pressure calculations are used; b) due to the inverse relation-ship between pellet growth and temperature, the uncertainties affecting the absorber temperature are such as to yield its minimum value. 121 1630 135
WARD-0 05q t REV 2 Table 6.10 details the uncertainty factors for lifetime structural calculations. The various factors considered in Table 6.10 correspond to the lower bound of the uncertainty range with the following exceptions: e Uncertainties affecting heat generation have been selected for consistency at the higher level, since they are the same un-certainties affecting the burnup. e Uncertainties have been integrated over the lifetime (hence, for example, elimination of the control system dead band effect) and over the pin volume (hence, elimination of the cladding circumferential temperature variation and pellet-cladding eccentricity factors). e In the case of the bundle / bypass flow split, the fact that CRAB and CRSSA hydraulic model slightly overpredicts the bundle flow was taken into account. e In the case of the subchannel flow area, it was judged that the two effects of increase in flow area due to tolerances and decrease due to pin deformation were counterbalancing. Plant Expected Operating conditions in conjunction with hot spot factors outlined in Table 6.10 are used in lifetime structural calculations. 1630 136 122
WARD 'p050 1 REV 6.0
SUMMARY
OF PRELIMINARY RESULTS AND FUTURE TESTS The hot channel factors used in calculations of the coolant temperature, clad-ding temperature, and gas plenum pressure for the CRBRP fuel, blanket, and primary control assemblies which were discussed in the preceding sections are summarized in Tables 6.1 A through 6.10. For convenience, the uncertainties associated with the fuel and blanket as-semblies, are grouped according to engineering (table numbers with the "A" suffix) and nuclear considerations (table numbers with the "B" suffix). Ihte that 3a statistical values are applied to the plant thermal-hydraulic design conditions when performing safety analyses. The 2e level is applied to plant expected operating conditions when calculating parameters such as cladding temperature and pressure, which are input to replaceable core assembly life-time analyses. The hot channel factors in this report are applicable at full power steady state conditions. The 3a uncertainties associated with the thermal-hydraulic values are conservatively used as the initial conditions for emergency and faulted transient and safety analyses. The hot channel factors reported in Tables 6.lA through 6.10 are considered to be conservative. This will be further established as additional data becomes available from tests which are planned or in progress. These include: additional irradiation experiments on fuel, blanket, and primary control rods; engineering mockup critical experiments in the ZPPR; pressure drop tests of fuel, blanket, and control assemblies and orifice configurations; heat transfer tests on a 61 fuel rod bundle in sodium; cross-flow measurements in a 5:1 scale blanket rod bundle sector in air; and heat transfer tests in sodium on a prototypic blanket rod bundle. These data will be included in the FSAR analyses. A more detailed tabulation of the forthcoming T&H tests is given in Table 6.11. 1630 137 123
TABLE 6.lA _ CRBRP FUEL ASSEMBLIES R00 TEMPERATURE ENGINEERING UNCERTAINTY FACTORS . COOLANT FILM CLADDING DIRECT")I Power Level Measurement and Control System Dead Band 1.03 1.0) 1.03(1.0) 1.03(1.0) Inlet Flow Ma1 distribution (Direct Bias) 1.02 Flow Distribution Calculational > 1.006 Uncertainty (Simulation Bias) 1.03j Cladding Circumferential Temperature Varia tion 1.0(+)si.7-2.1(d) 1.0(+)s0.8(**d)@.9(e*d) Pellet-Cladding Eccentricity 1.14(*,4) 1.14(*,4) STATISTICAL (3o)( ) Reactor AT and Inlet Temperature Variation 1.0(sl .14)(a) 1.016 Inlet Flow Ma1 distribution 1.059 Loop Temperature Imbalance (b) Wire Wrap Orientation 1 . 01 Subchannel Flow Area 1 .01 9 1.0 Film Heat Transfer Coefficient Pellet-Cladding Eccentricity 1.1 1.0 *)l.174(,) 1.0(*)l.174(**+) Cladding Thickness and Conductivity 1.00(c) Coolant Properties 1 .017 Flow Distribution Calculational Uncertainty (Calibration) 1.058 1.005
~
(*) For cladding midwall temperature calculations. (+ ) For fuel temperature calculations 4 88
* (4 T Applies to BOL conditions (o) Nuclear Uncertainty Factors are given on Table 6.1B y7C (a) For given coolant AT, HCF (3a) = 1 + (3/AT) (0.002304aT2 - 0.384aT +.121)1/2 ,4 - (b) Maximum 3a uncertainty of 7.4 F in inlet temperature o W Direct factor of 1.05 included in design correlation I Co Calculated for lifetime-limiting assembly in each flow zone with FATHOM. Includes 20% uncertainty in HCF-1.
e For cladding ID temperature calculations.
^
NOTE: Same values of subfactors apply to both Plant T&H and Expected Operating conditions except when two values are given; in this case, the parenthesized values apply to Plant Expected Operating conditions while the non-parenthesized values apply to T&H Operating conditions.
} TABLE 6.lB CRBRP FUEL ASSEMELIES R0D TEMPERATURE NUCLEAR UNCERTAINTY FACTORS WITH AND WITHOUT CONTROL ASSEMBLY INFLUENCE COOLANT HEAT FLUX DIRECT IU)
Physics Modeling 1.02(I) (*)1.02(1.10)II ) Control Rod Banking 1.02(2) 1.02(2) ZPPR-7'F1ux Tilt 1.0(4) 1.0 I4) STATISTICAL (3a)I ) Experimental (Nuclear) 1.07 1.07 Criticali ty 1 .01 (3) 1 .01 (3) _ Fissile Fuel Ma1 distribution 1.052 1.052 If assembly is influenced by adjacent R7C control rod, replace with: COOLANT HEAT FLUX - " PEAK POWER POSITION" " TOP OF CORE" @ BOL E01 ROI FOl ROI FOI Adjacent 1 .04 1.02 1.03 1.02 1.15 1.15
- 1) Physics Modeling Far Side 1 .01 1.02 .95 1.02 1.30 1.15 Adjacent 1.04 1.02 1.04 1.02 1 .01 1 .01
- 2) Control Rod Banking Far Side 1.02 1.02 1.02 1.02 1.01 1 .01 Adjacent 1.04 1.04 1.04 1.04 1.0 1.0
- 3 m I ) Criticality Far Side T
U 1 .01 1 . 01 1.01 1 .01 1.03 1.03 Cm c3 4) ZPPR-7 Flux Til t - Assy's. 9, 10, 13, 14, 15, 16, 17, 23, 25, 37, 38, 41, 42, 43, 44, 45, T Me 51, 53 (0.97 @ BOL,1.0 0 E0L). Assy's. 12,18,19, 40, 46, 47 (0.99 @ BOL,1.0 9 E0L). _f U e (*)Non-parenthesized value applies at the peak power. position (i.e., core midplane). o b Parenthesized value applies at the core lower / upper axial blanket interface except as superseded by note (1). I ) Engineering Uncertainty Factors are given on Table 6.lA
go-07050 t TABLE 6.2A A-CRBRP FUEL ASSEMBLIES MIXED MEAN EXIT TEMPERATURE ENGINEERING HOT CHANNEL FACTORS DIRECT IU) Power Level Measurement and Control System Dead Band 1.03(1.0) Inlet Flow Maldistribution (Direct Bias) 1.02 STATISTICAL (3a)( ) Reactor aT and Inlet Temperature Variation 1.0(sl .14)(a) Inlet Flow Maldistribution 1.059 Loop Temperature Imbalance (b) Coolant Properties 1 .01 7 O (a) For given coolant AT, HCF(3a) = 1+(3/aT)(0.002304aT2 -0.384aT +121)l/ (b) Maximum 3a uncertainty of 7.4*F in inlet temperature (c) Nuclear uncertainty factors are given in Table 6.2B NOTE: Same values of subfactors apply to both Plant T&H and Expected Operating conditions except when two values are given; in this case, the parenthesized values apply to Plant Expected Operating Conditions while the non-parenthesized values apply to T&H Operating conditions. 1630 140 m 126
TABLE 6.2B CRBRP FUEL ASSEMBLIES MIXED MEAN EXIT TEMPERATURE NUCLEAR UNCERTAINTY FACTORS
~
ASSEMBLY EXIT DIRECT IO) Physics Modeling (*}l .01(1.02 9 BOL,1.01 9 EOL) Control Rod Banking (*)l.02(1.039BOL,1.029E0L) ZPPR-7 Flux Tilt 1.0 (4) STATISTICAL (3o)(0) Experimental (Nuclear) 1.07 Criticality (*)l.01(1.02) Fissile Fuel Maldistribution 1.052 (4) ZPPR-7 Flux Tilt - Assy's. 9, 10, 13, 14, 15, 16, 17, 23, 25, 37, 38, 41, 42, 43, 44, 51 5 (0.97 9 BOL,1.0 0 E0L). Assy's. 12, 18, 19, 40, 46, 47 (0.99 9 BOL, Non-parenthesized values are applied for assemblies not adjacent to R7C control assemblies. Parenthesized values are applied for the control _. assembly effect fo. assemblies adjacent to R7C control assemblies. U (o) Engineering uncertainty factors are given on Table 6.2A. , 8g O 4 :=> N*8 A x
TABLE 6.3A . CRBRP FUEL ASSEMBLIES PLENUM PRESSURE ENGINEERING UNCEPTAINTY FACTORS PLENUM TEMPERATURE BURNUP DIRECT IO) Inlet Flow Maldistribution (Direct Bias) 1.02 Flow Distribution Calculational Uncertainty (Simulation Bias) 1.03 STATISTICAL (3a)( ) Power Level Measurement and Control System Dead Band 1.0 1.025 g Inlet Flow Maldistribution 1.059 Loop Temperature Imbalance (b) Reactor AT and Inlet Temperature Variation Sl.14(a) Wire Wrap Orientation 1 .01 Subchannel Flow Area 1 .01 9 Coolant Properties 1 .01 7 Flow Distribution Calculational Uncertainty (Calibration) 1.058 $ I ) Nuclear Uncertainty Factors are given on Table 6.3B (a)For given coolant AT, HCF (3o) = 1 + (3/AT) (0.002304aT2 - 0.384aT + 121)1/2 u cc (b) Maximum 3a uncertainty of 7.4 F in inlet temperature Q$o NOTE: Values of subfactors apply to plant expected operating conditions.
"S
,f N
un c> O
IABLE 6 3B CRBRP FUEL ASSEMBLIES PLENUM PRESSURE NUCLEAR UNCERTAINTY FACTORS WITH AND WITHOUT CONTROL ASSEMBLY INFLUENCE PLENUM TEMPERATURE BURNUP DIRECT I) Physics Modeling 1.02 II) 1.02 II) Control Rod Banking 1.02(2) 1.02(2) ZPPR-7 Flux Tilt 1.0 I4) 1.0 I4) STATISTICAL (3a)(0) Experimental (Nuclear) 1.07 1.07 Criticality 1 .01 (3) 1.01(3) m Fissile Fuel Ma1 distribution 1.052 1.052 O If assembly is influenced by adjacent control rod, replace with: BOL EOL Adjacent 1.04 1.02
- 1) ihysics Modeling Far Side 1 .01 1.02 Adjacent 1.04 1.02
- 2) Control Rod Banking w as
_ Far Side 1.02 1.02 my ch o u Adjacent 1.04 1.04 c3 3) Criticality wf a Far Side 1.01 1.01 -O
- o u (4)ZPPR-7 Flux Tilt - Assy's. 9, 10, 13, 14, 15, 16, 17, 23, 25, 37, 38, 41, 42, 43, 44, 45, 51, 53 (0.97 0 BOL,1.0 0 EOL). Assy's. 12,18,19, 40, 46, 47 (0.99 0 BOL.1.0 0 EOL).
I ) Engineering Umrtainty Factors are given on Table 6.3A. _.,
) TABLE 6.4A CRBRP INNER / RADIAL BLANKET ASSEMBLIES R0D TEMPERATURES ENGINEERING UNCERTAINTY FACTORS COOLANT FILM CLADDING DIRECT (0)
Power Level Measurement and Control System Dead Band 1.03 1.0) 1.03(1,0) 1.03(1.0) Irradiation Swelling 1.00 *)1.05(f) Inlet Flow Maldistribution (Direct Bias) 1.02 Flow Distribution Calculational ) 1.008 Uncertainty (Simulation Bias) 1.03, C1 adding Circumferential Temperature Variation 1.0(+)s3.4-3.8(g) 1.0(+)s0.7 (*'9 )s0.8(h '9 ) Pellet-Cladding Eccentricity 1.14(***) 1.14(***) STATISTICAL (3o)( ) Reactor AT and Inlet Temperature Variation 1.0(sl .14)(3) Inlet Flow Maldistribution 1.060(d )l .084 (*) 1.018(d)1.025I ') m Loop Temperature Imbalance (b) 8 Wire Wrap Orientation 1 . 01 Subchannel Flow Area 1.07(,)1.0(f) 1.0 Film Heat Transfer Coefficient 1 .21 Pellet-Cladding Eccentricity 1.0(+)l .174I***) 1.0(+)1.174(***) Cladding Thickness and Conductivity 1.0(c) Coolant Properties 1 .017 Flow Distribution Calculational Uncertainty (Calibration) 1.203 1.006 (*) For cladding midwall temperature calculations. (+ ) For fuel temperature calculations (4) Applies to BOL conditions a se _ (f) Applies to E0L conditions E$
& (o) Nuclear Uncertainty Factors are given on Table 6.4B u (a ) For given coolant AT, HCF (3a) = 1 + (3/aT) (0.002304aT 2 - 0.384aT + 121)I/2 Md o (b) Maximum 3a uncertainty of 7.4 F in inlet temperature
.ic) Direct factor of 1.05 included in design correlation Qo (d) Inner Blanket *
^~ ' (e) Radial Blanket ~
(g) Calculated for lifelime-limiting assembly in each" flow zone with' FATHOM.InEludes 20% 'uiicErtainty in HCF-1. (h) For cladding ID temperature calculations. . NOTE: Same values oflubfactors apply to both Plant T&H and Expected Operating' conditions except when two values are given; in this case, the parenthesized values apply to Plant Expected Operating conditions while the non-parenthesized values apply to T&H Operating conditions.
) ) ) -
TABLE 6.4B CRBRP INNER / RADIAL BLANKET ASSEMBLIES R0D TEMPERATURE NUCLEAR UNCERTAINTY FACTORS INNER BLANKET RADIAL BLANKET Row 1 Row 2 COOLANT HEAT FLUX COOLANT HEAT FLUX COOLANT HEAT FLUX DIRECT (BOL)( } Physics Modeling 1.06 (* )1.07 (1.11 ) 1.07 (* )1.07 (1.02 ) 1.03 (*)1.07(0.99) Control Rod Banking 1.02 1.02 1.02 1.02 1.02 Experimental (Nuclear) 1.12 I,1 1.10(1.17) 1.13 (,)l .11 ( 1.18) 1.27 (*)1.02 l.24(1.32) Criticality 1.02 1.02 ---- ---- ---- ---- Heavy Metal 1 . 01 1 .01 1 . 01 1 . 01 1 .01 1 .01 U-235 1 .01 1 .01 1.01 1 .01 1.01 1.01 Ta DIRECT (E0L)( ) Physics Modeling 1.02 (* )l .04 (1.12) 1 .01 I*)l.07(1.05) 1 .01 I*)1.07(1.02) Control Rod Banking 1.02 (,)l.02 1.02 l.02 1.02 I,)1.02 Experimental (Nuclear) 1.03 1.00(1.07) 1.05 (,)1.03(1.10) 1.15 l.13(1.21) Criticali ty 1. 01 1.01 ---- ---- ---- ---- Heavy Metal 1 .01 1 .01 1 .01 1 . 01 1 .01 1 .01 U-235 ---- ---- I*)Non-parenthesized values apply at the peak power position (i.e., near core midplane). Parenthesized y values apply at core upper / lower axial blanket interface extension. mg
*8 u *E CD (0) Engineering Uncertainty Factors are given on Table 6.4A fWf f
C b s $
- M
TABLE 6.5A - CRBRP INNER / RADIAL BLANKET ASSEMBLIES MIXED MEAN EXIT TEMPERATURE ENGINEERING UNCERTAINTY FACTORS DIRECT (0) Power Level Measurement and Control System Dead Band 1.03(1.0) Inlet Flow Ma1 distribution (Direct Bias) 1.02 Irradiation Swelling 1.00(*)1.055#) STATISTICAL (3o)(0) Reactor AT and Inlet Temperature Variation 1.0($1.14)(* } g Inlet Flow Ma1 distribution 1.060(c)1.084(d) Loop Temperature Imbalance (b) Coolant Properties 1 .01 7 (o) Nuclear Uncertainty Factors are given in Table 6.58. (a) For given coolant AT, HCF (3o) = 1 + (3/AT) (0.002304aT2 - 0.384aT + 121)1/2 - (b) Maximum 3a uncertainty of 7.4 F in inlet temperature. (c) Inner Blanket (d) Radial Blanket (e) Applies to BOL conditions (f) Applies to E0L conditions
~
w m5". ca O NOTE: Same values of subfactors apply to both plant T&H and expected operating conditions except when two values are given; in this "tb a case, the parenthesized values apply to plant expected operating _f o
& conditions while the non-parenthesized values apply to T&H g operating conditions.
_._. . , d O
)
- s
- TABLE 6.5B CRBR INNER / RADIAL BLANKET ASSEMBLIES MIXED MEAN EXIT TEMPERATURE NUCLEAR UNCERTAINTY FACTORS INNER BLANKET _ RADIAL BLANKET Row 1 Row 2
- ASSEMBLY EXIT ASSEMBLY EXIT ASSD48LY EXIT-DIRECT (BOL)(8)
Physics Modeling 1.01 1.02 l.02 Control Rod Banking 1.02 1.02 1.02 Experimental (Nuclear) 1.12 1.13 1.27
- Criticality 1.02 ---- ----
Heavy Metal 1.01 1.01 1.01 O 1.01 1.01 U-235 1.01 DIRECT (E0L)(0) Physics Modeling 1.01 1.02 1.02 ' Control Rod Banking 1.02 1.02 1.02 1.03 1.05 1.15 Experimental (Nuclear) - Cri ticality 1.01 ---- Heavy Metal 1.01 1.01 1.01 U-235
~
$.5 g
u Md CD I ) Engineering uncertainty factors are given on Table 6.5A. _$O
.n. .cy N
TABLE 6.6A CRBRP INNER / RADIAL BLANKET ASSEMBLIES PLENUM PRESSURE - ENGINEERING UNCERTAINTY FACTORS PLENUM TEMPERATURE BURNUP DIRECT ( ) Inlet Flow Maldistribution (Direct Bias) 1.02 Flow Distribution Calculational Uncertainty (Simulation Bias) 1.03 Irradiation Swelling 1.00(*}l.05II) STATISTICAL (3o)( ) Power Level Measurement and Control System Dead Band 1.0 1.025 m Inlet Flow Maldistribution 1.060(c)l.084(d) N Loop Temperature Imbalance (b) Reactor AT and Inlet Temperature Variation sl.14(a) Wire Wrap Orientation 1.01 Subchannel Flow Area Coolant Properties 1.017 Flow Gistribution Calculational Uncertainty (Calibration) 1.203 m (o) Nuclear Uncertainty Factors are (a) For given coolant AT, HCF (3o) =given on Table 6.6B1 + (3/aT)2 (0.002304aT - 0.384aT + 121)1/2 $$- 4:o u (b) Maximum 3a uncertainty of 7.4*F in inlet temperature g y CD (c) Inner Blanket up (d) Outer Blanket 4e (e) Applies to BOL conditions Q (f) Applies to E0L conditions g NOTE: Values of subfactors apply to plant expected operating conditions.
.. n a
TABLE 6.6B CRBRP INNER / RADIAL BLANKET ASSEMEiCIES PLENUM PRESSURE NUCLEAR UNCERTAINTY FACTORS INNER BLANKET RADIAL BLANKET Row 1 Row 2 PLENUM TEMPERATURE BURNUP PLENUM TEMPERATURE BURNUP PLENUM TEMPERATURE BURNUP_ DIRECT (BOLL (0) 1.06 1.'06 1.07 1.07 1.'03 1.03 Physics Modeling 1.02 1.02 Control Rod Banking 1.02 1.02 1.02 1.02 Experimental (Nuclear) 1.12 1.12 1.13 1.13 1.27 1.27 Criticality 1.02 1.02 ---- ---- ---- 1.01 1.01 1.01 1 .01 1.01 1.01 -.,, Heavy Mbtal 1.01 1.01 1.01 1.01 1.01 1.01 8 U-235 - IO DIRECT (EOLl ) 1.02 1.02 1.01 1.01 1.01 1.01 Physics Modeling 1.02 1.02 Control Rod Banking 1.02 1.02 1.02 1.02 1.03 1.03 1.05 1.05 1.15 1.15 Experimental (Nuclear) ---- ---- Criticality 1.01 1.01 ---- ---- 1.01 1.01 1.01 1.01 1.01 1.01 Heavy Metal ---- ---- ---- ---- U-235
~ $$
4m o (o)Engineeririg uncertainty factors are given on Table 6.6A NC A C
3 TABLE 6.7
} =
CRBRP PRIttARY CONTROL ASSEhBLIES PIN TEMPERATURES HOT CHANNEL / SPOT -FACTORS Maat Gamerate Cetltal @ Cladd%t2 ff2 Absorber Absor6er Clodrig Coelant g(*) Power Level Measurement and Centrol Systen Dead Band 1.03 - 1 03 1.03 1.03 Inlet Flow Maldistribution 1.08 5ebassembly Flow Maldistribution > - - -- * - -- l .07 5 Calculational Uncertainties 1.08 . , Sundle/8ypass Flow Split 1.10, Cladding Circumferential 1.o(0) 1,o(0) Teocerature Variation (Note 1) STATISTICAL (3o)(ol Inlet Temperature Variation .0 Reactor AT Variation 1.02 1.04 M* 1.0 U Absorber Maldistribution , and Conductivity 1.02 1.10 1.03 . Wire Wrap Orientation 1.01
- 2 Subchannel Flow Area 1.09 -
1.0 1,o Film Heat Transfer Coefficient . Pellet-Cladding Eccentricity 1.21(* 1.37 ) 1.39(*) . Cladding Thickness and Conductivity 1.01 1.10 1.04 Cap Thickness and Conductivity 1.13 Coolant Propertjes 1.01 D j r* * - O j, g !j ,*4 , , (+) thcertainties due to, physics analysis calculational methods (15% on coolant enthalpy rise and on absorber cladding and coolant h=st generation) are applied directly on nuclear radial peaking factors. (*)For local cladding temperature calculations. mg ch 8"s a y u (0)For average cladding, absorber temperature calculations. g (o) In addition. the assembly inlet temperature will be increased by 16*F. to account for primary loop temperature coolant g oncertainties. - -- g :3 t (e) Applies to Plant Expected Operating Conditions. (t) Applies to Plant T&M Design Conditions Je O (Note 1) Under the wire wrap local hot spot factors are 3.17, 1.40 and 1.20, for the claddiny 00, o midwall and ID, respectively. Applies to the nominal temperature drop between cladding and bulk- coolant. ._
~
' '1
- WARD REV -k054 TABLE 6.8 - CRBRP PRIMARY CONTROL ASSEHCLIES MIXED MEAN EXIT TEMPERATURE HOT CHANNEL FACTORS DIRECT I +I Power Level Measurement and Control System Dead Band 1.03 Inlet Flow Maldistribution 1.08 STATISTICAL (3o)IO}
' Inlet Terrperature Variation I 1.03*)l.0(*)
~
Reactor t.T Variation 1.04I4)l.0I#I Absorber Maldistribution 1.02 Cladding Thickness 1.01 Coolant Properties 1.01 TOTAL 2o 1.154 ($h.130 (t) D. 3o 1.175 ($ D .140 (t) (+) Uncertainties due to physics analysis calculational methods (15% for both assembly and chimney exit) are applied directly on nuclear radial peaking factors. (+) Applies to Plant Expected Conditions. (t) Applies to Plant T&H Design Conditions. (o) In addition, the assembly inlet temperature will be increased by 16*F (assembly exit) or 10"F (chimney exit), to account for primary loop tem-perature control uncertainties. 1630 151 137 ,
> W A RD-O-005d' REV 2 8 TABLE 6.9 e,
- CRBRP PRIMARY' CONTROL ASSEN8 LIES PLENUM PRESSURE HOT CHANNEL FSCTORS Plenum Temperature Burnup DIRECT I +)
Power Level Measurement 1.02 1.02 Inlet Flow Maldistribution 1.08 Subassembly Flow Ma1 distribution Calculational Uncertainties 1.08 Sundle/8ypass Flow Split 1.10 STATISTICAL (3o)IO) . Inlet Temperature Variation 1.02I *)1.0I *)
. Reactor AT Variation 1.045 *31 .0It)
Absorber Maldistribution 1.02 1.02 Cladding Thickness 1.01 Wire Wrap Orientation 1.01 Coolant Properties 1.01 TOTAL 2a 1.354
.332ff 1.034 3a 1.376 1.356 1.040
(+) Uncertainties due to physics analysis calculational methods (15% on both plenum tenperature and burnup) are applied directly on nuclear peaking factors. (+) Applies to Plant Expected Operating Conditions. (+) Applies to Plant T&H Design Conditions. (o) In addition, the assembly inlet temperature will be increased by 16*F to account for primary loop temperature control uncertainties. 1630 152 r .' . . e B 138
TABLE 6.10 CRBRP PRIMARY CONTROL ASSEMBLIES LIFETIME STRUCTURAL CALCULATIONS UNCERTAINTY
~
Coolant Film Cladding Heat Generation Gap Absorber Absorber Cladding Cooiant - IIRECT }
'ower Level Measurement and Control System Dead Band 1.02 '
l.02 1.02 1.02 niet Flow Ma1 distribution 0.92 tssembly Flow Maldistribution y 0.95 Calculational Uncertainties 0.92 lundle/ Bypass Flow Split 1.0 j , iTATISTICAL (3o)( )
- nlet Temperature Variation 0.96 teactor aT Variation 0.96 bsorber Maldistribution and Conductivity 1.01 0.9 1.02 lire Wrap Orientation 0.99
;ubchannel Flow Area 1.0 1.0 1,0 illm Heat Transfer Coefficient 0.79
- ladding Thickness and Conductivi ty 1.0 0.9 1.03 iap Thickness and Conductivity 0.87
_ oolant
- Properties 0.99
& TOTAL 2a 0.830 u 0.817 0.933 1.034 1.04 1.02 *" ca 3o 0.814 0.'751 0.90 0.87 0.9
.Mb 1.04 1.051 1.02 *E M c's W
U :+) Uncertainties due to physics analysis calculational methods (1.15 on coolant enthalpy rise and on absorber, cladding and coolant heat generation) are applied directly on nuclear radial peaking factors.
,o) o-In addition, the temperature assembly control inlet tegerature will be decreased by 16*F, to accuunt for primary loop uncertai_nties.
TABLE 6.11 . FUTURE CORE ASSEMBLIES T&H TESTS Schedule for Utilization Component DRS No. Test Description Performer Completion of Results Fuel Assembly 31.02.16 F/A Inlet Nozzle Test F/A Orifices HEDL 6/79 F/A Orifice Calibration 31.14.16 in Water; F/A Cavi- AP Calibrations and Inlet Flow Maldis-tation Tests in Water tribution Uncertainty 31.02.03 F/A Flow and Vibration Component and Overall HEDL 12/78 F/A AP Calibration and Test in Water AP Calibration Inlet Flow Maldistribution Uncertainty 02.01 Fuel Failure Potential Heat Transfer Tests ORNL 6/81 F/A Flow and Temperature and Heat Transfer in with a 61 Fuel Rod Distribution Calibration Large Rod Bundles Bundle in Sodium and Uncertainty Radial Blanket 31.03.02 61 Rod Bundle Radial Heat Transfer Tests WARD 9/80 Blanket Assembly Flow Assembly Blanket Heat Trans- with a Prototypic and Temperature Distri-a fer Test in Sodium Blanket Rod Bundle bution Calibration and 5 in Sodium Uncertainty 31.03.08 Flow and Vibration Component and Overall HEDL 12/79 RS/A AP Calibration and Test of Radial Blanket AP Calibration Inlet Flow Maldistribution Assembly in Water Uncertainty 31.03.25 Radial Blanket Flow Orifice AP Calibration WARD 7/79 RB/A Orifice AP Cali-Orificing Test and Flow Maldistribution bration and Inlet Flow Among Assemblies Within Maldistribution Uncer-LIM tainty 31.14.41 Air Flow Test of AP's Axial and Trans- -W R&D 3/79 Blanket Assembly Flow C7s CRBRP Blanket As- verse Velocity Distri- Distribution Cali-l'4 sembly butions in In-Board, bration and Uncertainty sa=a c:) Side, and Corner Chan- rm2" ___ nels on a 5/1 Scale 'C55 m ad. Model Q.s t c Ut C7
~n
~
TABLE 6.11 (Continued) Schedule for Utilization Component DRS_ iio. Test Description Performer Cmipletion of Results Primary Con- 3;.08.07 Primary Control As- Component and Overall -WARD 6/79 PCA AP and Flow Split trol Assembly sembly Hydraulic Test AP Calibration and Calibration and Uncer-Flow Split tainty Za enemme LN CD Ln en m5
<=
ts
-d S
Cs
- EARD-0~0050 1 REV 3
7.0 REFERENCES
- 1. M. C. Chuang, et. al ., "Three-Dimensional Thermal-Hydraulic Analysis of Wire-Wrapped Rods in Liquid-Metal Fast Breeder Reactor Core A ssemblies",
Nucl. Sci. Eng. 64, pp. 244-257 (1977) .
- 2. R. E. Collingham, et. al. , " Development and Results of an Electrically Heated Seven-Pin Bundle Assembly for Thermal Hydraulic Testing in Liquid Metals," Liquid Metal Heat Transfer and Fluid Dynamics, pp. 41-49, American Society of Mechanical Engineers, New York, 19/0.
- 3. M. F. Fontana, et. al . , " Temperature Distribution in a 19-Rod Simulated LMFf>R Fuel Assembly in a Hexagonal Duct (Fuel Failure Mockup Bundle 2A):
Record of Experimental Data," ORNL-TM-4113, September 1973 (Availability: USD0E Technical Information Center).
- 4. P. M. McConnell, F. R. Fisher and R. G. White, " Inlet Plenum Feature Model Flow Tests of the Clinch River Breeder Reactor: Addendum V Resul ts ,"
HEDL-TME 76-33, March 1976 (Availability: USDOE Technical Information Center).
- 5. P. M. McConnell, " Clinch River Breeder Reactor Fuel Assembly Inlet /0utlet Nozzle Flow Tests," HEDL-TME 77-8, February 1977 (Availability: USDOE Technical Information Center).
- 6. M. H. Fontana , et. al . , " Temperature Distribution in a 19-Rod Simulated LMFBR Fuel Assembly in a Hexagonal Duct - (Fuel Failure Mockup Bundle 2 A) - Record of Experimental Data," 0RNL-TM-4113, September 1973 (Availability: USD0E Technical Information Center).
- 7. J. J. Lorenz, T. Ginsberg, and R. A. Morris, " Experimental Mixing Studies and Velocity Measurements with a Simulated 91-Element LMFBR Fuel Assemblj,"
ANL-CT-74-09, March 1974 (Availability: USD0E Technical Information Center).
- 8. D. R. Pedersen, et. al . , " Experimental Investigation of the Hydrodynamic Entrance Length and Subchannel Velocity Profiles in a 91-Element Wire Wrapped Subassembly," ANL/ RAS 74-5, April 1974 (Availability: USD0E Technical Information Center).
- 9. R. E. Collingham, et. al . , "217-Pin Wire-Wrapped Bundle Coolant Mixing Test," HEDL-TME 71-146, November 1971. R. E. Collingham, W. L. Thorne, and J. D. McCormack, " Coolant Mixing in a Fuel Pin Assembly Utilizing Helical Wire Wrap Spacers," Nucl . Eng. Des. 24, pp. 393-409 (1973).
- 10. R. M. Roidt, T. G. Bartholet, and L. J. Harper, " Experimental Deter-mination of Interior Subchannel Crossflow and Axial Flow in a Model of the Clinch River Breeder Reactor Fuel Assembly Rod Bundle with Wire-Wrap Spacers ," ASME Paper 76/WA/HT-83. T. G. Bartholet, R. M. Roidt, and L. J. Harper, " Clinch River Breeder Reactor Plant; 11:1 Scale Wire Wrapped Rod Bundle Air Flow Tests. Interior Subchannels," CRBRP-ARD-0108, January 1977 (Availability: USD0E Technical Information Center).
142 1630 156
- t W A RO-0-T)050 '
RE4 3 I
- 11. K. Takanashi, K. Miyaguchi, and E. Ishibashi, " Experimental Study on Coolant Sodium Mixing Effect in "Joyo" Fuel Assembly with Spiral Wire Spacer II, Result of Test of Blanket Fuel Assembly," JAPFNR-182, October 1974 (Availability: USDOE Technical Information Center).
- 12. E. Parzen, Modern Probability Theory and Its Applicatio_ns, J. Wiley and Sons, New York, pp. 374-376, 1960.
- 13. Y. B. Chen, K. Ip and N. E. Todreas, " Velocity Measurements in Edge Channels of Wire-Wrapped LMFBR Fuel Assemblies," Trans. Amer. Nucl . Soc.
19_, pp. 323-324 (1974).
- 14. D. R. Pedersen, et. al . , " Cross Flow Mixing in a 91-Element Bundle", ANL/ RAS 74-2, February 1974 (Availability: USD0E Technical Information Center).
- 15. A. W. Graves and I. Catton, "An Explicit Method for Predicting the Thermal Performance of FBR ' dire-Wrapped Fuel Rod Assemblies," Trans. Amer.
Nucl . Soc.15_, p. 404 (1972) .
- 16. T. Ginsberg and J. J. Lorenz, " Experimental Mixing Studies in Simulated Wire-Wrap Fuel Assemblies," Reactor Heat Transfer, pp.13-38, Gesellschaft fur Kernforschung, mbH , Karlsruhe, West Germany,1973.
- 17. J. J. Lorenz, et. al ., " Experimental Mixing Studies and Velocity Measure-ments with a Simulated 91-Element LMFBR Fuel Assembly," ANL-CT-74-09, March 1974 (Availability: USD0E Technical Information Center).
- 18. T. G. Bartholet, R. M. Roidt and J. E. Romano, " Clinch River Breeder Reactor Plant,11:1 Scale Wire Wrapped Rod Bundle Air Flow Test, Side Subchannels," CRBRP-ARD-0129, January 1977 (Availability: USD0E Technical Information Center).
- 19. M. S. Kazimi, " Heat Transfer Correlation for Analysis of CRBRP Assemblies,"
CRBRP-ARD-0034, November 1976 (Availability: USDOE Technical Information Center).
- 20. O. E. Dwyer and H. C. Berry, " Turbulent-Flow Heat Transfer for in-Line Flow Through Unbaffled Rod Bundles: Molecular Conduction Only," Nucl .
Sci . Eng. 4_6, 6 pp. 284-303 (1971) .
- 21. " Nuclear Systems Materials Handbook," Hanford Engineering Development Lab. , Richland , Washington, TID-26666.
- 22. M. Jakob, Heat Transfer, Vol .1, p. 85, John Wiley and Sons, New York,1949.
- 23. C. A. Bruch and W. E. McHugh, " Radiation Damage Studies of Seven Non-Fissionable Metals," Preprint 37, Session 26, presented at the Nuclear Engineering and Science Conference, March 17-21, 1958, Chicago, Illinois.
- 24. G. H. Golden and J. V. Tokar, "Thermophysical Properties of Sodium" ANL-7323, August 1967.
- 25. D. C. Ginnings, T. B. Douglas and A. F. Ball, " Heat Capacity of Sodium between 0 and 900 C, the Triple Point and Heat of Fusion," J. Res. Nat.
Bur. Stand. 45_ pp. 23-33 (July 1950). 143 1630 157
1ARD-0-0050 t Rtv 3
- 26. D. R. Fredrickson and M. G. Chasanov, "The Enthalpy of Liquid Sodium to 1505K by Drop Calorimetry," J. Chem. Thermodyn. 6_, pp. 629-633 (1974).
- 27. J. P. Stone, et al . , "High-Temperature Properties of Sodium, NRL-6241, December 1964.
- 28. E. G. Hagen, "Ueber Die Wurmeausdehnung des Natriums, des Kaliums und Deren Leigirung im Festen und im Geschmolzenen Zustande," Ann. Phys.
Chem.19_, pp. 436-474 (1883).
- 29. C. T. Ewing, et. al ., " Measurements of the Physical and Chemical Pro-perties of the Sodium-Potassium Alloys, Quarterly Progress Report, No. 7,"
NRL C-3287, May 1948.
- 30. C. T. Ewing , et al ., " Density of the System K-Na," Appendix C in " Measure-ment of the Physical and Chemical Properties of the Sodium-Potassium Alloy, Quarterly Progress Report, No.1," NRL P-3010, September 1946
- 31. I. I. Novikov, et al., "The Heat Transfer and High Temperature Properties of Liquid Alkali Metals," Atomnaya En. 1, (4), 92 (1957).
- 32. M. E. Rinch and others, "Chimie Physique - Den Sites du Potassium du Sodiume," Compt. Rend. H.189, pp. 39-41 (1929).
- 33. M. Nishibayashi," Density and Viscosity of Molten Materials, Part 1, Density of Sodium and Sodium Hydroxide," WADC 53-308, Part 1, November 1953.
- 34. E. I. Gol'tsova, " Density of Lithium, Sodium and Potassium up to 1500-1600 C ," Teplofiz. Vys. Temper. 4_, No. 3, pp. 360-363 (1966).
- 35. R. B. Baker, R. D. Leggett and D. S. Dutt, " Interim Report: Effect of Burnup on Heat-Rating-to Incipient Fuel Melting - HEDL P-20," HEDL-TME 75-63,1975 (Availability: USD0E Technical Information Center).
- 36. R. D. Leggett, E. O. Ballard , R. B. Baker, G. R. Horn, and D. S. Dutt,
" Linear Heat Rating for Incipient Fuel Melting in UO2-Pu02 Fuel ,"
Trans. Amer. Nucl . Soc.15, pp. 752-753 (1972).
- 37. G. H. Golden and others, " Correlation and Interpretation of Data Relative to EBR-II Power Level," in Irradiation Experimentation in Fast Reactors, pp. 314-34 2, American Nuclear Society, Hinsdale, Ill .,1973.
- 38. L. B. Miller, et al., " Characterization of the Power in an Experimental Irradiation Subassembly of Mixed-Oxide Fuel in EBR-II", ANL/EBR-047, September 1971.
- 39. C. Chiu and N. E. Todreas, " Flow Split Measurements in an LMFBR Radial Blanket Assembly," Trans. Am. Nucl . Soc. 2_6_, pp. 455-456 (1977 ) .
- 40. C. L. Wheeler and others, " COBRA-IV-1: An Interim Version of COBRA for Thermal-Hydraulic Analysis of Rod Bundle Nuclear Fuel Elements and Cores,"
BNWL-1962, March 1976. 144 1630 158
try j T 50 .
- 41. A. B. G. Washington, " Preferred Values for the Thennal Conductivity of Sintered Ceramic Fuel for Fast Reactor Use," TRG - Report - 2236, September 1973.
- 42. J. A. Christensen, R. J. Allio and A. Biancheria, " Melting Point of Ir-radiated Uranium Dioxide," Trans. Amer. Nucl . Soc. _7_, pp. 390-391 (1964).
- 43. J. L. Krankota and C. N. Craig, "The Melting Point of Plutonia-Urania Mixed 0xide Irradiated to High Burnup," GEAP-13515, July 1969.
- 44. H. von Ubish, S. Hall, and R. Srivastav, " Thermal Conductivities of Mixtures of Fission Product Gases with Helium and with Argon," in Pro-ceedings of the Second United Nations International Conference on the Peaceful Uses of Atomic Energy, Geneva,1958, Vol . 7, pp. 697-700, United Nations, Geneva ,1958.
- 45. D. J. Clough and J. B. Sayers, "The Measurement of the Thermal Conductivity of UO2 under Irradiation in the Temperature Range 150 -1600 C, AERE-R-4690, December 1964.
- 46. A. Biancheria, "The Effect of Porosity on Thermal Conductivity of Ceramic Bodies," Trans. Amer. Nucl . Soc. 9_, p.15 (1966).
- 47. W. T. Sha and R. C. Schmitt, "THI-3D: A Computer Program for Steady State Thermal-Hydraulic Mul ti-Channel Analysis ," ANL-8112, December 1975, (Availability: USD0E Technical Information Center).
- 48. M. D. Carelli, C. W. Bach and R. A. Markley, " Hydraulic and Scram Dynamics Analysis on LMFBR Control Rod Assemblies," Trans. Amer. Nucl . Soc.16, -
pp. 218-219 (1973).
- 49. "A Compilation of Boron Carbide Design Support Data for LMFBR Control Elements," HEDL TC-40, December 1973 (Availability: USD0E Technical Information Center).
1630 159 145
W A RD-01)050 i SEV 2 8 ATTACHMENT r STATISTICAL TOLERANCE LIMITS AND PROPAGATION OF ERROR J. A. Marshall e-
SUMMARY
In preparing a safety analysis of the Clinch River Breeder Reactor Plant, the Advanced Reactors Division must take explicit account of various uncertainties. This work leads repeatedly to the need for making a statistical inference in the framework of a propagation of error problem. Unfortunately the problem has no satisfactory general solution. This report deals with a simplified version of the problem for which a reasonable solution can be (and is) produced. Specifically we derive approximate upper (statistical) tolerance limits for a variable given as a linear combination of other variables which have independently and normally distributed uncertainties and for which data are available. These results will serve to produce answers for problems which satisfy the simplified assumptions and as a base from which to consider more difficult problems. The latter will probably have to be approached via computer simulation. 1630 160 i
- IARD-010Sd~
i BEV 2 I
- 1. INTRODUCTION This report deals with the propagation of error problem in a statistical context. In the general problem a (dependent) variable is given as a function of other (independent) variables which have uncer-tain values. This uncertainty propagates through the function to induce a corresponding uncertainty in the dependent variable. In the simplest statistical version of the problem the uncertainties are visualized in terms of probability distributions and one wants to characterize the induced (output) distribution given the (input) distributions of the independent variables. Our problem has the added feature that we want to take explicit account of the effect of incomplete specification of the input distributions on knowledge of the output distribution. This last feature makes the problem one of statistical inference.
The treatment of the above problem that will be given here is meant to provide an approach for certain calculations that arise in the safety analysis of the Clinch River Breeder Reactor Plant (CRBRP). The inference problem is a very difficult one for which no satisfactory general solution exists. We will deal with a simplified version of the problem for which we can produce a reasonable solution. This will serve both to produce answers to those problems for which the simplified assumptions are adequate and as a base from which to consider more difficult problems. The latter will probably have to be approached via computer simulation. Statistical tolerance limits are a natural and of ten advocated concept for use in dealing with inference problems of the type which underlie this report, Consequently our development is in terms of tolerance limits of which two kinds are presented; one is the more common, but the other is offered as the more relevant for present purposes. 1630 161 1
WARD-O.0050 i REV 2 1 The more technical treatment of the basic material is given in Appendix A. Appendix B contains some remarks on the application of the basic material to somewhat more complicated problems. 1630 162 2
1ARD-D 0050 t REV 2 '
- 2. PRELIMINARIES Consider the relationship (1) Y = f(X1 ,X2 , ..., X,) .
If any X is a random variable then Y necessarily becomes one also. The distribution of Y is induced via f by the distribution of the X's, i.e., the " error" in the X's propagates through f to become " error" in Y. If we assign probability distributions to the X's we can ask for the corresponding Y distribution. In general this is not an easy problem. If f is linear and the X's are independently, normally, distributed the problem is easy; Y is then normally distributed with mean and variance given by simple functions of the means and variances (respectively) of the X's. If f is non-linear the distribution of Y will not in general be a "known" distribution. In such cases one can resort to various approx-imate methods to obtain information about the distribution of Y. The problem to be addressed in this report contains an additional difficulty; the distributions of the X's are not known. This of course is typical of applied work. The usual approach in such prob-lems is to base the choice of a family of distributions for each X (e.g., normal, uniform, etc.) on general knowledge and to use data to select the specific members of the family which are reasonable candidate dis-tributions for the X's. The proper use of the data to select a set of reasonable candidates and the implications of a given selection for the problem at hand are the concerns of statistical inference. The specific results in this report will be directed at the problem of making in-ferences about Y when f in (1) is linear, the X's are independently normal, and we have data on each X. The material introduced in connec-
,, tion with this effort will provide background for discussion of more general problems.
1630 163 3
- sARD-D-005D REV 2 9
- 3. LIMITS The type of inference about Y that we are concerned with is to give limits on its possible values. For example if we knew that Y had a normal distribution with mean u and standard deviation a we could say (2a) Pr {Yfu + 1.960} = 0.975 and (2b) Fr {u-l.960 $ Y fu + 1.960) = .95 .
Of course there is no inference problem here since everything is known. Suppose however that Y is normally distributed and its unknown mean and ^ variance are estimated from data. Then the problem of giving limits on Y similar to those in (2) is an inference problem. The resulting limits are called tolerance limits by analogy with the corresponding limits used in manufacturing. The one-sided type of limit as given in (2a) will be emphasized in this report. In this section we will give a basic discussion of tolerance limits when Y is normally distributed and has been observed directly. The necessary modifications when the observations are on the X's will be taken up later. The main purpose here is to introduce the two different types of normal theory tolerance limits and give a practical appreciation of what they are in a simple setting. If the distribution of Y is normal and Y , ..., Y is a random sample of observations on Y then we might be tempted to define the upper limit in (2a) as i + 1.96S where isthesampleaverage([Y)/nand Sisthesamplestandarddeviation{[(Y-5) }/(n-1). The problem however is that while the limit in (2a) covers 97.5% of the distribution of Y, the corresponding limit based on the sample need not do the same 1630 164 4
1ARD-0-0050 t REV 2 f since i and S need not satisfy the relation +kS = p+ko . For given values of p and a the coverage of i+kS could be calculated and for a different sample it would generally be differant. In fact for any multiplier, k, the coverage of +kS is a rendom variable that varies from sample to sample and has a distribution of its own*, Of course in any application we have just one sample and we want a value of k which will do a given job. Statistical theory provides two classical methods for choosing k. Both refer to the distribution of the coverage in hypothetical repeat samples. One controls the average coverage and the other controls the probability that the coverage is at least a given amount. The remainder of this section will consist of a more detailed description of these two approaches. The first type of tolerance interval is called the 8-expectation type. Suppose Y , ..., Y is a random sample f rom a normal distribution 1 of unknown mean and standard deviation. To find the required k one must consult a table of the t-distribution for a value, say, tg(n-1). This is a value which an n-1 degree of freedom t-variate will exceed with probability 1-8. The upper tolerance limit is then given by (3) E+tg(n-1) /l+1/n S . For example if 8 = .99 and n = 10 then (3) becomes i + (2.821)(/11/10) S or Y + 2.959S and the intervals so computed will have coverage .99 on
- To emphasize this fact (which is fundamental to an understanding of the properties of such limits) we follow statistical convention and use upper-case letters (i.e., ,S) to denote random variables. When a f ormula is presented as a recipe f or use with a sample of data then lower case letters are appropriate; they represent values assumed by random variables. Of course these remarks do not apply to those
, symbols that have nothing to do with randomness (e.g., u and k). t 1630 1oS 5
IARD-D D05p f - REV 2 1 the cverage. Aacthet interpretation is possible, namely, (3) yields a 99% confidence limit for the next observation on Y. To sum up, imagine that one repeatedly takes samples of size n from the normal distribution, computes (3), measures its true coverage, and takes an additional observation. Our assertions are twofold: (1) the average of the coverages will be .99 and (2) in 99% of the trials the next observation will be covered by the limit. The other type of tolerance limit is called the S-content type. Here the factor k depends on the non-central t-distribution. This distribution depends not only on a degree of f reedom parameter but also on a non-centrality parameter (6 say). Let us write t (w) for the value of a non-central t-variate with w degrees of freedom which is exceeded with probability 1-y . Also write K for the value of a g standard normal variate (mean 0, variance 1) which is exceeded with probability 1-8. The 8-content tolerance limit is given by (4) i + (t / 6) S where t = t (n-1) and 6 = 6 Kg . Suppose n = 10, 8 = .99, and y = .95. Tables of the non-central t-distribution are difficult to use, but for the present example one can look up k = t / 6 directly in a special tolerance factor table and find that (4) becomes + 3.981 S. The interpretation is that in repeated sampling 95% of the trials will produce a limit whose coverage is at least .99. One says that a limit of the form (4) has coverage .99 with confidence .95. Clearly the two limits are doing something different. By way of comparison if we put 3.981 = tg(9) /ll/10 and solve for tB(9) = 3.7957 we can ask what value of 8 this corresponds to. Interpolation in a good t-table shows that B = .9978. Thus the
.99 content limit with confidence .95 has an average coverage of .9978 or 6
)bb0 \
WARD-O-0050 f ' REV 2 3 alternatively it may be viewed as a 99.78% confidence limit for the next observation. It seems that in many design situations the appropriate inference is to the next " observation", i.e., to the (essentially only) device that will be built using the design. Of course this is not true for, say, production items where the 8-expectation tolerance limit could be criticized for controlling the average and leaving the extremes unattended. In such a case the 8-content approach has merit. But for a single large project it makes sense to use 8-expectation limits. 1630 167 7
1ARD-0-0050 Riv t f
- 4. TOLERANCE LIMITS FOR Y = X +X 2
The formulas of this section extend easily to the case Y=[aX g1with m terms but for convenience they will be presented for the simple sum Y = X y+X2 . Recall that the X's are supposed normal and independent. Now however instead of having observations directly on Y we have ng observations on X and n2 bservations on X . This spoils 1 2 the applicability of formulas (3) and (4) and it is the purpose of this section to supply replacemer.ts based on a good approximation. By way of additional notation let "i 5 1
=
([X))/n j=1 g i and
, n
) - Eg) )/(n g-1)
=
S ( j=1 (X g where X1 ) is the j* observation on Xg (i=1 or 2). Write v = n i-1 for 1 the degrees of freedom for S g. Finally we need a suggestive notation for i +E which 2 will play the role of i in (3) and (4); we will use Y for this. Now the 8-expectation tolerance limit is given by (5) + t B("e) s e where S, plays the role of /1+1/n S in (3) and v, is its effective degrees of freedom. Specifically S, = {Sy(ny +1)/ny + S (n2+1)/"2I and 2 2 v, = (S )2 jggg(n+1)/n]hv1+I g y 2("2+1 "2I "2l* 8 1630 168
' WARD-o-0090 f REV 2 1 The notation tg(v,) was explained in connection with (3).
The 6-content tolerance limit with confidence Y is based on non-central t as described in conaection with (4) in the previous section; we will still be using the notation t (w) and K introduced g there. For the present application modifications to (4) similar to those used in (5) above are required. The limit is (6) i + (t / 5) S where S = {S2y + S }1/2 , N=nn y2 8 /("281 + "18) ' and t =t , Y(v) with 6=K5,g and v = (S ) /{(S ) /v y + (S2 ) /"2}
- We turn now to examples of the use of these formulas in simple situations of the type Y = X y+X 2 . The simgle modifications of (5) and normal) are given (6)thatareneededtocoverthecaseY=[aX1 1 (X1 in Appendix A.
1630 169 m 9
WARD-DyOSO 't REV 2
- 5. EXAMPLES To give a feeling for the implications of formulas (5) and (6) we have applied them to a series of 15 examples. The examples are chosen to show the effect of variations in s y, s2 ' "1, and n2 n the tolerance limits given by formulas (5) and (6). (As per the footnote
$n Secr4.on 3 we have shifted to lower-case letters to represent actual realizations of random variables, e.g., S = s .) Table 1 gives the 1 1 specifications of the 15 examples in terms of the 1s and ng (i = 1,2) and shows the values taken, in each case, by the random variables N, v.
S, v , and S, introduced in the preceding section. The degrees of Table 1. Specifications (s , s.2' "1, and n2 ) and computation of basic quantities (N, v, s) for 15 examples chosen to illustrate tolerance limits. Specifications Computation of basic quantities s n Content Expectation Case 1 2 1 2 N v s v s 1 1 1 = = = = 1.41 = 1.41 2 1 1 40 40 40 78 1.41 78 1.43 3 1 1 20 20 20 38 1.41 38 1.45 4 1 10 10 10 18 1,41 1 18 1.48 5 1 1 10 20 13.33 24.43 1.41 24.02 1.47 6 1 1 10 40 16 29.25 1.41 27.90 1.46 7 1 1 10 = 20 36 1.41 32.8 1.45 6 1 2 10 10 10 13.24 2.24 13.24 2.35 9 1 2 10 20 16.67 26.23 2.24 26.43 2.3 10 1 2 10 40 25 47.95 2.24 47.82 2.28 11 1 2 10 = 50 225 2.24 193.46 2.26 12 1 2 20 10 11.11 13.66 2.24 13.45 2.33 13 1 2 40 10 11.76 13.86 2.24 13.51 2.33 14 1 2 = 10 12.5 14.06 2.24 13.56 2.32 15 1 4 10 10 10 10.12 4.12 10.12 n.32 1630 170 10
1 ARD-D-0050 REV 2 freedom (v ) and standard deviation (S,) for formula (5) are given under the heading " Expectation" in the columns labeled v and s. The columns listed under " Content" apply in a similar fashion to formula (6). Table 2 gives the multiplier (k) and the allowance (ks) for limits of 8-expectation (formula (5)) and 8-content (f omula (6) with y=95%) for different values of B. For each value of 8 the column labeled E(C) is the expected coverage achieved by the 8-content limit. Table 2. Factors and allowances for tolerance limits of S-expectation and S-content for the examples of Table 1 (y=95% for limits of B-content) 8=97.5% C=99.9% Expectation Content Expectation Content Case k ks k hs E(C) k ks k ks E(C) 1 1.96 2.77 1.96 2.77 97.5 3.09 4.37 3,09 4.37 99.9 2 1.99 2.85 2.37 3.35 98.91 3.2 4.58 3.64 5.15 99.97 3 2.02 2.93 2.58 3.65 99.19 3.32 4.81 3.93 5.56 99.98 4 2.1 3.12 2.94 4.15 99.41 3.61 5.35 4.43 6.27 99.97 5 2,06 3.03 2.77 3.91 99.33 3.47 5.08 4.19 5.93 99.98 6 2.05 2.99 2.68 3.79 99.28 3.41 4.97 4.07 5.76 99.98 7 2.03 2.95 2.59 3.67 99.2 3.36 4.87 3.95 5.59 99.98 8 2.16 5.06 3.1 6.93 99.45 3.84 9 4.71 10.53 99.97 9 2.05 4.73 2.71 6.05 99.29 3.43 7.89 4.13 9.23 99.98 10 2.01 4.59 2.5 5.59 99.11 3.27 7.45 3.82 8.54 99.98 11 1.97 4.45 2.25 5.04 98.67 3.13 7.07 3.45 7.71 99.96 12 2.15 5.03 3.06 6.85 99.44 3.82 8.92 4.66 10.43 99.97 13 2.15 5.01 3.05 6.81 99.44 3.82 8.89 4.64 10.38 99.97 14 2.15 5 3.03 6.77 99.43 3.81 8.86 4.62 10.33 99.97 15 2.22 9.62 3.3 13.59 99.48 4.13 17.85 5.03 20.74 99.96 An example will serve to clarify all of this. Consider case (or row) 5 in the panel of Table 2 labeled 8-97.5%. The entry 2,06 is the value of tS("e) f # "e=24.02 (see Table 1), and the entry 3.03 is the product of 2.06 with a *=1.47 (see Table 1). The entry 2.77 is the value of t /d for N=13.33 and v=24.43 ( from Table 1), and the entry 3.91 is the product of 2.77 with s=1.41 (from Table 1) . Finally the entry 99.33 says that the 8-content interval with 8=97.5% and y=95% u 1630 171
. WARD-D-0050 REV 2 Table 2 continued C=95% B=99% Expectation Content Expectation Content Case h ks k ks E(C) k ks k ks E(C) 1 1.64 2.33 1.64 2.33 95 2.33 3.29 2.33 3.29 99 2 1.66 2.38 2.02 2.86 97.53 2.38 3.4 2.78 3.93 99.63 3 1.69 2.44 2.21 3.12 98.13 2.43 3.52 3.01 4.26 99.72 4 1.73 2.57 2.53 3.58 98.66 2.55 3.79 3.42 4.83 99.78 5 1.71 2.51 2.38 3.36 98.46 2.49 3.65 3.22 4.56 99.76 6 1.7 2.48 2.3 3.25 98.32 2.47 3.6 3.13 4.42 99.75 7 1.69 2.45 2.22 3.14 98.15 2.45 3.54 3.03 4.28 99.73 8 1.77 4.15 2.66 5.96 98.78 2.64 6.2 3.62 8.09 99.79 9 1.7 3.92 2.32 5.19 98.36 2.48 5.7 3.16 7.07 99.75 10 1.68 3.82 2.14 4.78 97.94 2.4 5.48 2.93 6.54 99.69 11 1.65 3.73 1.92 4.3 97.09 2.35 5.3 2.64 5.9 99.52 12 1.77 4.12 2.63 5.88 98.75 2.64 6.16 3.58 8 99.79 13 1.77 4.11 2.61 5.84 98.73 2.64 6.14 3.56 7.96 99.79 14 1.77 4.1 2.59 5.8 98.71 2.64 6.12 3.54 7.92 99.79 15 1.81 7.83 2.82 11.64 98.88 2.76 11.93 3.85 15.89 99.79 actually achieves an expected coverage of 99.33%. Note that y=95% applies to the S-content limits and is fixed for all panels of Table 2. Perhaps it would be useful to indicate an interpretation for some of the quantities appearing in Table 1. Since in (5) and (6) is not the average of a set of observations we cannot signify its precision by giving the number of observations on which it is based. The quantity N is the effective number of observations for in the sense that i has the variance that the mean of N observations on Y would have. Notice when ny=n2 that N is in fact the number of observations on Y. Now s (col. 8) is the estimate of the standard deviation of Y=X y+X2 by a very familiar formula. However s is not in general of the form of an estimated standard deviation based on a set of observations (as s and 2 s2are). Therefore s does not have the sampling distribution that such estimates have and on which many techniques of statistical inference are based. The distribution of s is therefore approximated with the closest fitting member of the usual family of distributions for sample variances; 1630 172 12
o IAR0-0-0050
'i REV 2 1 this fitting leads to the degrees of freedom for a represented by v (col. 7).
Notice that when n y=n2 and s y=sy so that the estimate a can be constructed along conventional lines then v=(ny -1) + (n 2-1) as expected. Finally s and v (cols, 9 and 10) bear the same relationship to each other that s and v (cols. 7 and 8) do, but s (col. 10) is the estimate of the variance of (i-Y') where Y' is a new observation (say Y'=Xh+X2v where u=ny +1 and v=n 2 +1) fr m the distribution of Y. The factors k given in Table 2 for S-expectation and 6-content limits cover a broader range of cases than might at first be realized. For any case in the table the values of sy and s2 can be multiplied by the same constant without changing the value of k. Thus, e.g., case 5 applies to all situations where s =s y 2 ' "l=10, and n =20, 2 and case 10 covets all cases where s 2=2s y, n y=10, and n =40. The conventional "20" 2 and "30" limits that are of ten calculated correspond closely to our case 1, Table 2, panels 97.5% and 99.9% respectively. 1630 173 13
-
- I ARD-01305kD f REV 2
- 6. COMPUTATIONAL RECOMMENDATIONS The application of formulas (5) and (6) requires the evaluation of probability points (tg(v) and t ) of the central and noncentral t-distributions respectively. Tables of the probability points of the central t-distribution are widely available, but interpolation is necessary for fractional degrees of freedom. Noncentral t-tables are relatively less available and are much more difficult to use. Thus approximations based on the readily available tables of the standard normal distribution are attractive. There are some rather good normal approximations available (Peizer and Pratt [5], Wallace [6]) for the central t-distribution. Approximations to the noncentral t-distribution have been reviewed by Owen [4]. He finds no overall winner among the
- candidates, but a certain well known approximation (designated J by Owen) is the best one for the parameter values of interest to us (y=.95, 8=.95 to .999, v>10). To emphasize the dependence of the candidates' ranking on the parameters, we mention that Owen found approximation J to be third best (out of four) when Y=.90 with 8 and v as before. This approximation is actually written to produce the k-value for tne S-content interval directly rathcr than the noncentral t-value.
Specific recommendations are given below. Central t - To find a value of tg(v) for use in formula (5) when 8 and v are given, first use a standard normal table to find z such that Pr {Z$z} = 8 (For example Pr {Z$1.96 } = .975.) The desired t-value is given by (7) t g(v) = [v(exp{(z /v)([8v+3]/[8v+1])2)_1))l/2 , This is taken from Wallace [6]. To gauge the quality of this approximation it was used to approximate the entries in Table 2, panels 95% and 99.9%,
~
l630 174 14
T
" R ARD-01050.
REV 2 columns " Expectation-k". These entries in Table 2 were obtained via careful interpolation in a t-table. The comparison is presented in Table 3 where an extra digit is carried to reveal smaller differences. Table 3. Comparison of central t-values obtained from formula (7) with those from Table 2. S=95% 8=99.9% Case Formula Table 2 Formula Table 2 2 1.665 1.665 3.198 3.198 3 1.686 1.686 3.319 3.319 4 1.734 1.734 3.613 3.61 5 1.711 1.711 3.468 3.467 6 1.701 1.701 3.409 3.408 7 1.693 1.693 3.359 3.358 8 1.769 1.769 3.841 3.836 9 1.705 1.705 3.43 3.429 10 1.677 1.677 3.27 3.269 11 1.653 1.652 3.133 3.132 12 1.766 1.766 3.826 3.822 13 1.766 1.766 3.822 3.817 14 1.765 1.765 3.819 3.815 15 1.81 < 1,81 4.138 4.128 Noncentral t - To find a value of k=t //N for use in formula (6) when 8, Y, N, and V are given put (8) k = (Kg + (K -AB} )/A where A = 1-K /2v , Y B = Kg -K /N , and Kp is the value of a standard normal variable which is exceeded with probability 1-P. This is taken from Owen [4]. Table 4 contains a comparison of formula (8) with certain appropriate entries from Table 2 1630 175 15
1 A RD-0105C i REV 2 I (" Content-k"). These entries were obtained by interpolation in a table of the noncentral t-distribution. Clearly this approximation is not as good as the one for central t, but it can still be quite useful. To sum up, fomula (7) is essentially as good as Table 2 for 8=95% while for B=99.9% it loses accuracy in the third decimal place for the smaller degrees of freedom. Formula (8) on the other hand has two decimal place accuracy at best (8=95% and large degrees of freedom) and one decimal place accuracy at worst (8=99.9% and small degrees of freedom). To obtain greater accuracy than formula (8) provides,one evidently must use a noncentral t-table and interpolate for the non-centrality parameter and the degrees of freedom. The resulting value of t can then be used in (8). For y=.95 a suitable table is available in [4]. Table 4. Comparison of k-values (related to noncentral t-values) obtained from formula (8) with those from Table 2. 8=95% B=99.9% Case Formula Table 2 Formula Table 2 2 2.017 2.021 3.635 3.642 3 2.199 2.209 3.915 3.929 4 2.507 2.529 4.405 4.435 5 2.36 2.376 4.17 4.192 6 2.286 2.299 4.054 4.072 7 2.209 2.22 3.937 3.952 8 2.635 2.663 4.672 4.709 9 2.306 2.32 4.106 4.126 10 2.131 2.139 3.81 3.821 11 1.921 1.923 3.444 3.447 12 2.601 2.627 4.628 4.664 13 2.584 2.61 4.607 4.642 14 2.567 2.592 4.587 4.621 15 2.79 2.824 4.986 5.031 1630 176 16
^ MARD-D 0050 i Rtf 9 I
- 7. REFERENCES
[1] G.E.P. Box, "Some Theorems on Quadratic Fonns Applied in the Study of I. Effect of Inequality of Variance Analysis of Variance Problems: in the One-Way Classification", Annals Math. Stat. 25, pp. 290-302, (1954). [2] K. A. Brownlee, Statistical Theory and Methodology in Science and Engineering, 2nd Edition, John Wiley, New York,1965. Classical and Bayesian, [3] Irwin Guttman, Statistical Tolerance Regions: Hafner Press, New York, 1970. [4] D. B. Owen, " Survey of Properties and A[ plications of Noncentral T-Distribution", Technometrics 10, pp. 445-478, (1968). [5] David B. Peizer and John W. Pratt, " Approximating the Binomial, F, and Commonly Used Related Distributions, I", TR-12 (AD-643954), August 1966. [6] David L. Wallace, " Bounds on Normal Approximations to Student's and the Chi-Square Distributions", Annals Math. Stat. 30, pp.1121-1130, (1959). 1630 177 17
W ARD-O- 005p i - REV 2 1
- 8. APPENDIX A DerivationsforthecaseY=[aX gg m
SupposeY=[aXg with the X gindependently N(p ,o ), and 1 X , ..., X is a random sample of size ng on X1 (i=1,... m). We in want an upper tolerance limit on Y. Both the B-content type with confidence y and the S-expectation type will be treated. The basic reference for tolerance concepts will be Guttman . Our basic approach is to modify the usual formulas for the case where Y y, ...,Y is a randem sample from a N(p,o ). The modification comes from using Satterthwaite's formula for the approximate degrees of freedom to be associated with a linear combination of variance estimates. This will be presented first. Notice that with the above setup we have 2 Vary =[aVarX g andforanestimateofVarYonewoulduse[aS . Here Var is the variance operator and S stands for the usual sample estimator of Var X (with n g-1 in the denominator). Now each S has a mean square dis-tribution with n g-1 degrees of f reedom, but the distribution of m 22 (9) S =[a3 1 in general is not a member of the family of mean square distributions. The idea of using a mean square distribution that approximates the dis-tribution of S is developed in, e.g., BrownleeI I. The approximation is based on calculating the degrees of freedom for a mean square 1630 178 18
- WARD-D-0050 '
't -
REV 2 1 m 2 distribution so that its mean and variance match those of S . The result is called Satterthwaite's approximation (original references are given in Brownlee) and in our case it comes to (10) v = (S2 )2j , (3 2)2fv 1 where v = n 1-1 is the degrees of freedom associated with S . This approximation has been studied by Box who gives numerical evidence indicating that it is excellent for positive a f. To develop an upper 8-expectation tolerance limit for Y we use a theorem due to Paulson and given by Guttman which says that a confidence region of confidence 8 for a future observation from the distribution of Y is a S-expectation tolerance region for the dis-tribution of Y. Thus let Y' be a next observation on Y, i.e., let Y' = aXyh+8 2h+*** X where u = n y+1, v = n +1, etc. Writei=[ai and observe that 2 gt Y'-i N N(0, a2,y2 Y Y) where 2 2 o 7,px 1 0 2, Z,22 o 1630 179 y ix and 1
!"i in an obvious notation. Now , -m Y
19
W A RD 0050 t REV 2 I Var (Y'- ) = a2 , ,2 y y
- ao (n +1)/n 1 i and we will estimate this by 22 S2=[ag(,,1)f, 9
where of course S is the usual estimator of Var (X ) with v 1 =n-1 g degrees of freedom. Thus from Satterthwaite's approximation (10) we have v, = (S ) / (a (n g+1)/ng ) (S ) /v g 1 and so approximately S X (v,) Var (Y'-i) v Clearly Y'-i N(0,1)
/ Var (Y '-i) and therefore N
t(v,) where t(v ) stands for a "t" distribution with v degrees of freedom (the required independence between S, and derives from the classical independence between S t and X1and the assumed independence among the Xg ). If we write tg(v) for a t-variate with v degrees of freedom which is exceeded with probability 1-8 then
^N Pr((Y'- )/S <t g(v,)) = 8 1630 180 20
!/R3-Ot050' 1 REV 2 1 m
and by the usual inversion this becomes (11) Pr{Y' < i + tg(v,)S, } = 8 . Therefore i + tg(v )S, is an upper confidence limit for Y' with confidence 8; hence it is also an upper tolerance limit of S-expectation. To recapitulate briefly: Y' is a random variable with the distribution ofY,i.e.,N([au,[ao); gg g isanestimatorof[ap which is independent of Y'; and S2 is an estimator of Var (Y'-?) which is in-dependent of i. Turning now to the 6-content tolerance limit let us write (with some redundancy) Y = [aX
=
[al ii S = [aS Var Y = a y and EY = p y where E is the expected value operator. In the present case the degrees cf freedom for S as taken from Satterthwaite's approximation is given directly by (10). In this development it is convenient to be able to write Var (k) as Var (Y)/N just as though k were the average of N observations on Y. The N so defined is called the effective number of observations. It is given by Var (Y)/ Var (h) or N=([a*o*1 )/([a*o*1/n') . _ 1630 181 21
~
1ARD~Q=0990 i 1 REV 2 1 m If the n had g a common value n then N would equal n and not depend on the unknown variances. Of course if the unknown variances had a common value then N would only depend on the a and n . g In general we must f approximate N by replacing the unknown variances by their estimates S. That is N 2([aS)/([aS/n)g . Let Kg be the value of a N(0,1) variate which is exceeded with probability 1-8. We want a value of k so that Y+kS will be a B-content tolerance limit with confidence y . In other words k must be such that (12) Pr(p +Kogy<k+kS)=y since py +Kogy has coverage exactly 8 and (12) says that i+kS has at
~
least coverage B with probability y. Rewriting (12) we find p -Y NYa + NK B Y (13) Pr < Ek =y
- 2 3
2 y Since E(p-k)/o % N(0,1) , S /c y % X (V)/V , andSisindependentofk,theexpressionontheleftoftheinequality in (13) has a noncentral t-distribution with noncentrality parameter EKgand degrees of freedom v. If t is the value of such a variate which is exceeded with probability 1-y then (13) holds with t = Ek or k=t */ 6. The more complete notation used in the report in place of t 22 1630 182
WARD-D"0050 i ' REV 2 1 m is t (v) where 6= 8K g. The 8-content tolerance limit is (14) i+(t/M)S.
- 9. APPENDIX B More general situations Appendix A treats the problem of tolerance limits on a Y given by Y=f(X , ..
.X,) where the X's are independent and normal, f 1
is linear, and observations are available on the individual X's. The approach taken to this problem is somewhat more general than it might
-~ appear to be. In this appendix the additional generality will be brought out.
Basically what is needed for a normal theory tolerance limit onYisanormallydistributedestimateofthemeanofY,sayi,whose variance is given by S /N (for some N) and for which S is an estimate (independentofE)ofthevarianceofYhavingameansquaredistribution with v degrees of freedom. The classical situation in which these conditions are met is where a random sample 1 Y , ..., Y,is available on Y and we put i= and S ={[(Y-5) }/(m-1). Then N=m and v=m-1. (Strictly speaking N is required only for the content type limits and not for the expectation type.) In the situation treated in Appendix A the tolerance limits had t.o be approximated. Approximation was needed for two reasons: (1) the natural estimate of the variance of Y (or of {Y'-i) in the expectation type) did not have a mean square distribution and so an approximate one was provided and (2) the effective number of observations (N) involved variances that had to be estimated. 1630 183 23
a IARD-01)050 i REV 2 ' The generalities that may be treated with little change in approach include:
- 1) one or more X s determined from a regression on other variables rather than from direct observation, ii) f is nonlinear, iii) X s not independent.
Consider first the case where an X in f(X g , ..., X,) is determined from a previously established regression on some variables Z. If the regression was linear (in the parameters) then for some k matrix of constants (Z) and some vector of estimated coefficients (B) we estimate the X g contribution to k as $ 1=zB. (We are here treating f as linear.) In this notation z is a row vector expressing the values of the z-variables at which the value of Xg is desired and B is a column vector. For example if tne regression were quadratic in a single O z then k =a+b z+c 2z , B ' = {a ,b , c } , and z-{ 1, 2z , z } . The Z-matrix in this 1 case consists of a column of l's, a column of z-values, and a column of z-values squared. Nowtheestimateofthevarianceofk(=zB)isgiven g by Sg = z(Z'Z) 1 z'S2 where S = { (X q k) /(m-p) and X g represents the values of Xg on which the regression is based. These equations are for any linear regression based on m observations and p parnmeters. The degrees of freedom for S is (m-p). This information is sufficient to alter the formulas in Appendix A so that they would apply to a case where one or more X{s are based on a linear regression. If the regression is not linear in the parameters the fit will be 2 carried out by a series of linear approximations. The S and its degrees of freedom from the final iteration can be used as above. 24 1630 184
A d ' 1ARD-D-0050'
~4 REV 2 f O
Consider next the case of a nonlinear f. An approach here would be to linearize f. That is put s . . m . . . Y = f(Xy ..., X,) + [ { f (X1 .y. . , X,) }{ (X f-Xg ) } i=1 where f is the partial of f with respect to coordinate i evaluated at (Xy ,..., X,). This presents Y as a linear combination of variables, which case has been dealt with. The accuracy of this approximation depends on the variances of the distributions of the X and the severity of the nonlinearity of f. In practice one often finds the variances to be sufficiently small that the approximation is a good one. It should be noted that there are really two approximations here; one is the linearization about a point and the other is the estimation of that point by (X ,y . . . , X ) . Finally go back to a linear f and suppose the uncertainties inthenormalX{sarenotstatisticallyindependent. For simplicity O consider Y=X1+X2 . In the absence of independence we have Var Y = Var Xy + Var 2X +2 Cov y(X2 ,X ) where the last term is twice the covariance of X1 with X2. To find Var Y we must obtain Coy (Xy ,X 2
)'
and there are two cases to be considered. In one case the dependence arises from known physical relations and based on these an analysis will show the covariance to be (perhaps approximately) a linear combination of Var X1 and Var X2 . This makes Var Y a linear combination of variances and Satterthwaite's approximation can be used as before. In the other case Cov (Xy ,X 2 ) will be known only through estimation based on data. To be able to make such an estimate we must have pairs of observations (X ,X2j ) . But this is the same as having direct observations on Y (i.e. , Y =X +X ) which avoids any problem about covariances. 1630 185 25}}