ML19320B145
| ML19320B145 | |
| Person / Time | |
|---|---|
| Site: | Calvert Cliffs  |
| Issue date: | 01/31/1980 |
| From: | ABB COMBUSTION ENGINEERING NUCLEAR FUEL (FORMERLY |
| To: | |
| Shared Package | |
| ML19289B046 | List: |
| References | |
| CEN-124(B)-NP, CEN-124(B)-NP-PT02, CEN-124(B)-NP-PT2, NUDOCS 8007090361 | |
| Download: ML19320B145 (75) | |
Text
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CEN-124(B)-NP 1
l STATISTICAL COMBINATION OF UNCERTAINTIES PART 2 JANUARY,1980 J
3 61 E dEsYus COfJ bJSTION ENGINEERING. INC.
~
LEGAL NOTICE L
THIS REPORT WAS PREPARED AS AN ACCOUNT OF WORK SPONSORED BY COMBUSTION ENGINEERING, INC. NEITHER COMBUSTION ENGINEERING NOR ANY PERSON ACTING ON ITS BEHALF:
A.
MAKES ANY WARRANTY OR REPRESENTATION, EXPRESS OR IMPLIED INCLUDING THE WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE OR MERCHANTABILITY, WITH RESPECT TO THE ACCURACY, COMPLETENESS, OR USEFULNESS OF THE INFORMATION CONTAINED IN THIS REPOPT, OR THAT THE USE OF ANY INFORMATION, APPARATUS, METHOD, OR PROCESS DISCLOSED IN THIS REPORT MAY NOT INFRINGE PRIVATELY OWNED RIGMTS;OR B. ASSUMES ANY LIABILITIES WITH RESPECT TO THE USE OF, OR FOR DAMAGES RESULTING FROM THE USE OF, ANY INFORMATION, APPARATUS, a
METHOD OR PROCESS DISCLOSEDlN THIS REPORT.
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o Cell-124(B)-NP STATISTICAL COMBIllATION OF UNCERTAltiTIES l'ETHOD0 LOGY PART 2:
COMBIrtATI0f OF SYSTEf1 PARAI1ETER U:lCERTAlt; TIES Ill THERMAL fMRGIll ANALYSES FOR CALVERT CLIFFS UNITS 1&2 4
C
ABSTRACT _
This report describes the methods used to statistically combine system parameter uncertainties in the thermal margin analyses for Calvert Cl i
Units 1 and 2 cores.
This distributions and response surface techniques used is presented.
report demonstrates that there will be at least 95% probability with at least 95% confidence that the limiting fuel pin will avoid departure frca nucleate boiling (D 8) so long as the minimum DNS ratio found with the best estimate design TORC model remains above 1.23.
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TABLE OF CONTErlTS Page Ti tle i.
Abstract li-Table of Contents IV-List of Figures V-4 List of Tables Vi-Ncmenclature and Abbreviations I-I 1.0 Summary of Results 2-1 2.0 Introduction 2-2 2.1 Deterministic Method 2-2 2.2 Statistical Method 3-1 3.0 Sources of Uncertainty 3-1 3.1 State Parameters Used in the Study 3-2 3.1.1 Method for Selecting State Parameters 3-3 3.1.2 Inlet Flow Perturbation Sensitivity 3-3 3.1.3 Enthalpy Rise Factor Sensiti/ity 3-4 3.1.4 Systematic Pitch Reduction Sensitivity 3-5 3.1.5 Most Adverse State Parameters 3-5 3.2 Radial Power Distribution 3-6 3.3 Inlet Flow Distribution 3-6 3.4 Exit Pressure Distribution o
3-7 3.5 Enthalpy Rise Factor 3-7 3.6 Heat Flux Factor 3-7 3.7 Clad 0. D.
3-8 3.8 Systematic Pitch Reduction 3-8 3.9 Fuel Rod Bow 11.
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TABLE OF CONTENTS (con't.)
h Title 3-8 3.10 CHF Correlation 3-9 3.11 TORC Code Uncertainty 4-1 4.0 MDNBR Response Surface 4-1 4.1 TORC Model Used 4-1 4.2 Variables Used 4-2 4.3 Experiment Design 4-3 4.4 Design Matrix 4-3 4.5 Response Surface 5-1 5.0 Combination of Probability Distribution Functions 5-1 5.1 Method 5-2 5.2 Results 5-2 5.3 Analytical Comparison 6-1 6.0 Application to Design Analysis 6-1 6.1 Statistically Derived MDNBR Limit 6-1 6.2 Adjustments to Statistically Derived MDNBR Limit 6-2 6.3 Application to TORC Design Model 7-1 7.0 Conclusions 7-1 7.1 Conservatisms in the Methodology 8-1 8.0 References Appendix A-1 Appendix A: Detailed TORC Analyses Used to Generate Response Surface 111.
3 LIST OF FIGURES
=
Fig. No.
Title Page 3-1 Inlet Flow Distribution Used to Establish State Parameters for Response Surface 3-10 3-2 Exit Pressure Distribution Used to Establish State Parameters for Response Surface 3-11
- 4 3-3 Core Wide Radial Power Distribution Used to Establish State Parameters for Response Surface 3-12 3-4 Hot Assembly Radial Power Distribution Used to Establish State Parameters for Response Surface 3-13 3-5 Channel Numbering Scheme for Stage 1 TORC Analysis to Establish Response Surface State Parameters 3-14 3-6 Channel Numbering Scheme for Stage 2 TORC Analysis to Establish Response Surface State Par,ameters 3-15 3-7 Third Stage Channel and Fuel Pin Numbering Schemes Used in TORC Analysis to Establish Response Surface State Parameters 3-16 3-8 Inlet Flow Factors for Seized Rotor Analysis of 217 Bundle 14x14 Assembly Cores 3-17 3-9 Exit Pressure Distribution Used in Sensitivity 3-18 Study 4-1 Core Wide Power Distribuiton Used to Genert'.e 4-5 Response Surface 4-2 Hot Assembly Radial Power Distribution Used to 4-6 Generate Response Surface 4-3 Intermediate (2nd Stage) TORC Model Used in 4-7 Generating Response Surface 4-4 Subchannel (3rd Stage) TORC Model Used in Generating 4-8 Response Surface 5-1 Resultant MDNBR Probability Distribution Function 5-5 i
I iv.
LIST OF TABLES Table No.
Title Page.
3-1 Ranges of Operating Conditions for Which 3-19 Response Surface Is Valid 3-2 Nominal and Perturbed Flow for Establishing Sensitivity of Flow Distribution Effects 3-20 on MDNBR to Operating Conditions 4
3-3 Flow Perturbation Effects at Various Operating 3-21 Conditions 3-4 Sensitivity of Enthalpy Rise Factor Effects to Axial Shape Index (Isolated Hot Assembly Model) 3-22 3-5 Sensitivity of Enthalpy P,ise Factor Effects to Operating Conditions (Isolated Hot Assembly 3-23 Model) 3-6 Sensitivity of Enthalpy Rise Factor Effects 3-24 to Axial Shape Index (Core Wide Analysis) 3-7 Sensitivity of Systematic Pitch Reduction Effects 3-25 to Operating Conditions State Parameters Which Maximize Sensitivity of MDNBR to 3-8 3-26 System Parameters Sensitivity of MDNBR to Inlet Flow Distribution 3-27 3-9 3-28 3-10 Sensitivity of MDNBR to Exit Pressure Distribution 3-29 3-11 As-Built Clad 0.D. (inches) Data for 14x14 Fuel 3-30 3-12 As-Built Gap Width Data (inches) 4-1 State Parameters Included as Variables in the 4-9 Response Surface 4-10 4-2 Coefficients for MDNBR Response Surface 4
5-1 Probability Distribution Functions Combined 5-4 by SIGMA A-1 Coded Set of Detailed TORC Cases Used to A-2 Generate Response Surface A-2 Comparison of TORC and Response Surface MDNBR A-13 for Cases Used to Generate Response Surface v.
l a a==
.m
NOMENCLATURE AND ABBREVIATIONS b
coefficient in response surface constant in response surface c
f arbitrary functional relationship number of independent variables in response surface k
number of items in a sample n
p.d.f.
probability distribution function psf pounds per square foot psia pounds per square inch (absolute) x system parameter y
state parameter MDNBR values predicted by response surface z
ASI axial shape index (defined in Table 3-3)
CE Combustion Engineering CHF Critical Heat Flux DNB Departure from Nucleate Boiling DNBR Departure from Nucleate Boiling Ratio F
Fahrenheit Fq engineering heat flux factor a
MDNBR Minimum Departure from Mucleate Boiling T
temperature T-H thermal-hydraulic constant used to code system parameters (Table 4-1) a constant used to code system parameters (Table 4-1) e coded value of system parameters (Table 4-1) n p
'mean standard deviation a
denotes difference between two parameters a
vi.
..= _ -..
s subscripts denotes vector quantity i
index 9
c nditions at reactor core inlet in 4
index
)
superscripts denotes estimate O
degrees e
average value I
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1.0 Summary of Results flethods were developed to combine statistically the uncertainties 4
in reference thermal margin (Detailed TORC) analyses.
These methods were applied to the Calvert Cliffs I, and Calvert Cliffs II This work demonstrated that there will be at least 95 probability cores.
l, with at least 95% confidence that the limiting fuel pin will avoid departure from nucleate boiling (DflB) so long as the flinimum OflB Ratio (MDilBR) found with the best estimate design TORC model remains i
above 1.23.
The 1.23 f1DllBR limit includes allowances for reference analysis input uncertainties but does not take into account uncer-tainties in operating conditions (e.g. monitoring uncertainties).
An improved treatment of operating condition uncertainti'es has been developed in Part 1 of this report (1-l),
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E 1-1.
2.0 Introduction C-E's thermal margin methodology for Calvert Cliffs I and II has been modified by the application of statistical methods.
This part of the report focuses on the statistical combination of reference thermal-hydraulic (T-H) code input uncertainties.
This combination was accomplished by the generation of a Minimum DNBR (MBNBR) response surface and the application of Monte Carlo methods.
A complete description of the methods used in the statistical combination is provided in this report. The remainder of this section outlines the previous deterministic and the new statistical thermal margin methods.
Section 3.0 describes the sources of un-certainty that were considered in this effort.
Section 4.0 describes the MDNBR response surface.
The application of fionte Carlo Methods is discussed in Section 5.0, and results are presented.
Finally Section 6.0 describes the changes in design analyses that result from this work.
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2-1.
r L-___ s.
2.1 Deterministic Method Two types of problem dependent data are required before a detailed T-H code can be applied.
The first type of data, system parameters, describe the physical system under consideration and are not monitored with the detail needed for detailed T-H analysis while the reactor is System parameters describe the reactor geometry, pin by operational.
pin radial power distributions, inlet and exit flow boundary condition, The second type of data, state parameters, describe the operational etc.
State parameters are monitored while the reactor state of the reactor.
is in operation and include the core average inlet temperature, primary
~
loop flow rate, primary loop pressure, etc.
C-E thermal margin methods (2-1) utilize the TORC code (2-2) and the CE-1 CHF correlation (2-3) with two types of models.
The first model, detailed TORC, is tailored to yield best estimate MDNBR predictions in Both a particular fuel assembly for a specific power distribution.
system and state parameter input are used in a detailed TORC model.
only state parameter The second model, design TORC, requires data and may be applied to any fuel assembly for any power distribution that is expected to occur during a particular fuel cycle. System parameters are fixed in the design model so that the model will yield either accurate or conservative MDNBR predictions for all operating conditions within a specified range.
Design model MDNBR results are verified by comparison with results from the detailed model of the limiting assembly in the deterministic method. After the design model is shown to yield acceptable (i.e.
accurate or conservative) results, additional adjustment factors are applied to account for uncertainties in system parameter input to the detailed model.
For example, engineering factors are applied to the hot subchannel of the design model to account for fuel fabrication uncertainties.
These adjustment factors, though arrived at statisti-cally, are applied in a deterministic manner.
That is, although each adjustment factor represents a 95/95 probability / confidence limit that the particular parameter deviation from nominal is no worse than des-cribed by that factor, all factors are applied sinultaneously to the limiting subchannel.
This is equivalent to-assuming that all adverse deviations occur simultaneously in the limiting subchannel.
2.2 Statistical Method The probability of all adverse system parameter deviations from nominal occurring simultaneously in the limiting subchannel is extremely remote.
With a more reasonable, demonstrably conservative method, the probability of system parameter input being more adverse than specified can be taken into account statistically, as described herein.
The improved methodology involves a statistical combination of system parameter uncertainties with the CHF correlation uncertainties to determine i
2-2.
a revised design MDNBR limit. Since uncertainties in system para-meters are taken into account in the derivation of the new f1Df!BR limit, no other allowance need be made for them. A best estimate design TORC model is therefore used with the revised flDNER limit for thermal margin analysis. ~Thi's best estimate design model yields conservative or accurate MDNBR results when compared with a best estimate detailed model.
An increased MDilBR limit is then applied to the design model to account for system parameter uncertainties.
The resultant best estimate design model and increased f1DNBR limit j,
ensure with at least 95% probability and at least a 95% confidence level that the limiting fuel pin will avoid a departure from nucleate boiling if the predicted MDNBR is not below the limit MDNBR.
O G
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O
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2-3.
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'3.0' Sources of Uncertainty Four types of uncertainty are identified in MDNBR predictions from the TORC code:
1) numerical solution parameter uncertainty 11) code uncertainty iii) state parameter uncertainty iv) system parameter uncertainty Numerical solution parameters are required input that would not be necessary if analytic methods could be used (e.g. radial mesh size, axial mesh size, convergence criteria, etc.). The uncertainties associated with these parameters are dealt with in a conservative manner (3-1) in C-E's present methodology. -
The numerical algorithms in the TORC code represent approximations to the conservation equations of mass, momentum, and energy.
Because of the approximations involved, an inherent code uncertainty exists.
This uncertainty is implicitly dealt with in the CE-1 CHF correlation (3-3)(3-4).
State parameters, as explained in section 2.1, define the operational The treatment of uncertainties in these para-state of the reactor.
meters is addressed in reference 3-2.
As explained in section 2.1, system parameters describe the physical environment that the working fluid encounters.
This report establishes the equivalent MDNBR uncertainty that results from a statistical com-bination of uncertainties in system parameters.
3.1 State Paramaters Used in the Study Generation of a response surface whi.ch simultaneously relates MDNBR to both system and state parameters would require an excessive number of detailed TORC analyses.
Consequently a conservative approximation is made and a response surface relating MDNBR to system parameters only is To achieve conservatism, it is necessary to generate the created.
surface for that set of state parameters which maximizes the sensitivity of MDNBR to system parameter variations.
That is, the response surface
.Can be described as.:
l MDNBR = g (x, y_o) where y_o, the vector of state parameters, is., selected such that 3(MDNBR)
--+ maximum 01 X.o 3-1.
The set of state parameters, yo, that satisfies the above relation, is referred to as the most adverse set of state parameters.
The generation of the response surface is discussed in sect 100 4,3, 3.1.1
!!ethod for Selecting State Parameters
-Allowable operating parameter ranges are presented in Table 3-1.
t These ranges are based upon reactor setpoints including measurement uncertainty. The response surface must be valid over these ranges.
As indicated above, a single set of operating conditio'ns is chosen from these ranges to maximize the sensitivity of MDilBR to system parameters.
This set of state conditions is determined from detailed TORC analyses in the following manner.
Two TORC analyses are performed for a single set of oparating conditions.
In the first analysis, nominal system parameters are used and the core average heat flux is chosen to yield a MDf!BR in the neighborhood of 1.19.
The'second TORC analysis uses the same heat flux and operating conditions but has one of the system parameters perturbed.
The MDNBP from the " perturbed" analysis is then subtracted from the " nominal" HDilBR to yield a AMDNBR for the chosen set of operating conditions.
That is AMDilBR "flominal" MDilBR
" Perturbed" MDflBR (3.1)
=
The percent change in MDilBR is then determined according to the relation:
% Change =
(AMDNBR/" Nominal" MDilBR) x 100 (3.2)
This. process is repeated for several sets of operating conditions to establish the sensitivity of the AMDflBR throughout the allowable operating range.
Sets of operating conditions used in this sensitivity study are chosen to envelop the required ranges shown in Table 3-1.
The operating conditions which yield the maximum percent change in f1DNBR are those which maximize the sensitivity of the 11DNBR to the perturbed system parameter. These state parameters are referred to as the "most adverse" state parameters.
Since MD!lBR is a smoothly varying function of these parameters (3-3),
it is likely that the theoretical most adverse state parameters will be similar to the most adverse set found by the method described above.
Similarly, it is also highly unlikely that MDNBR sensitivities observed with the theoretical most adverse set will differ appreciably i
from f1DilBR sensitivities which occur using the most adverse set found by the above method.
The detailed TORC model used in these cases is for one of the limiting f
assembly candidates of the Calvert Cliffs Unit 1. Cycle 3 core.
In this model care geometry is identical, and boundary conditions are similar to the Calvert Cliffs Unit I and Calvert Cliffs Unit II i
i 3-2.
/
cores. Hence, trends in the sensitivity of MDNBR to variations in system parameters at various operating conditions will be the same for all of these cores. The radial power distribution used in the sensitivity study differs from the distribution used to generate the response surface. The sensitivity of MDNBR to state parameters will exhibit the same trends regardless of radial power distribution since the local coolant conditions in the hot assembly will be similar at 1.19 MDNBR.
Hence, the most adverse set of state parameters found in this study may be applied to generate the response surface.
Inlet flow and exit pressure boundary conditions for the model are shown i
in Fig. 3-1 and 3-2.
Core-wide and hot assembly power distributions are shown in Fig. 3-3 and 3-4 respectively.
The detailed TORC analysis (3-1) consists of three stages. A core-wide analysis is done in the first stage, in which each fuel assembly near the limiting assembly is modeled as an individual channel.
Crossflow boundary conditions from the first stage are applied in the second stage to a more detailed model of the neighborhood around the limiting assembly.
Each quadrant of the limiting assembly is represented by a channel in the second stage analysis.
Crossflow boundary conditions from the second stage are applied to the subchannel model of the limiting assembly hot. quadrant.in the third stage, and the MDNBR is calculated.
TORC models for the first, second, and third stages of the model used in the sensitivity study are shown in Fig. 3-5, 3-6, and 3-7 respectively.
'3.1.2 Inlet Flow Perturbation Sensitivity As indicated in Fig. 3-3, the hot assembly occurs in channel 9 of the first stage TORC model. A perturbed model for use in determining the sensitivity of inlet flow distribution effects on MDNBR to operating conditions is created by reducing the inlet flow fraction to the hot assembly and an adjacent assembly.
Inlet flow is also increased accordingly in two assemblies far from the hot assembly to preserve continuity.
Inlet flow fractions for the perturbed and nominal models are presented in Table 3-2.
The sensitivity of flow distribution effects on MDNBR to operating con-ditions found with the above models is presented in Table 3-3.
MDNBR is most sensitive to variations in the inlet ficw distribution for an
' axial shape index,of [
- ].
Greatest sensitivity to flow perturbations is observed with a pressure / temperature / flow combination of C
-].
Hence the greatest sensitivity is expected to occur for the operating conditions:
P e
J 3.1.3 Enthalpy Rise Factor Sensitivity The method described in section 3.1.1 is altered slightly to determine the state parameters which maximize MDNBR sensitivity to the enthalpy 3-3.
a
rise factor. The uncertainties accommodated by the enthalpy rise factor are discussed in Reference (3-4.
Since the enthalpy rise factor affects only the limiting subcha)nnel and adjacent subchannels, an isolated model of the limiting assembly hot quadrant is used to reduce computational time.
The isolated quadrant model is simply the hot quadrant subchannel model shown in Fig. 3-7 with adiabatic, impervious boundary conditions -imposed on the sides of the quadrant.
Observed trends in behavior found with TORC analyses of the isolated quadrant model are confirmed by multistage TORC analyses.
Nominal cases are run with no enthalpy rise factor.
An enthalpy rise factor of 1.03 is applied to the fuel pins which bound the limiting sub-channel in the perturbed cases.
Results found with the isolated TORC model are shown in Tables 3-4 and 3-5.
The data in these tables indicate that MDNBR sensitivity to the enthalpy rise factor is maximized with [
] axial shape indices, corresponding to [
'] power distributions, and the pressure / temperature / flow combination [
]
Data from multistage TORC analyses are presented in Table 3-6.
These data show a similar trend when compared with the isol,ated quadrant model data of Table 3-4, however maximum sensitivity is seen at a [
]
axial shape index. The greatest sensitivity of MDNBR to the enthalpy rise factor is expected to occur for the operating conditions:
3.1.4 Systematic Pitch Reduction Sensitivity Systematic pitch reduction uniformly decreases fuel rod pitch throughout an entire fuel assembly.
Nominal pitch for 14 x 14 C-E fuel is 0.58".
Hot assembly fuel pitch is reduced to[
~ ]in the limiting assembly of the perturbed model used to establish the sensitivity of systematic pitch reduction effects on MDNBR to operating conditions.
Results from nominal and perturbed pitch TORC analyses are shown in Table 3-7.
Based upon these data, maximum MDNBR sensitivity to systematic pitch reduction is expected to occur at:
Data in Table 3-7 also indicate that the sensitivity of MDNBR to systematic 3
pitch [
3-4.
3.'l.5' Most Adverse State Parameters As explained in section 3.1.0, the set of state parameters chosen for use in generating the response surface should maximize MDNBR sensitivity to variations in system parameters; this is the most adverse set of state parameters.
The most adverse set of parameters is chosen for use in generating the response surface so that the resultant MDNBR uncertainty will be maximized.
This introduces conservatism into the overall treatment.
The state parameters which maximize MDNBR sensitivity to various system parameters are listed in Table 3-8.
This comparison indicates that[
] psia and{
] design flow are respectively th'e most adverse system pressure and flow rates.
The most adverse axial shape index (A.S.I.) and inlet temperature (Tin) are not evident from this compari-son. The magnitude and impact of each system parameter uncertainty must also be considered in choosing the most adverse values of A.S.I.
and Tin-Magnitudes of each of the system parameter uncertainties _ce assigned and discussed in Sections 3.2 - 3.8.
The maonitude and impact of the [
] those of the other system parameters.
Therefore, the A.S.I. and Tin which tend to maximize MDNBR sensitivity to the [
] are used to generate the response surface. Although maximum MDNBR sensitivity to
[
] is observed with [
] A.S.I., this sensitivity is only slightly less than the sensitivity observed with a[-
]A.S.I.,
as shown by the data in Table 3-3.
Since sensitivity to enthalpy rise factor increases witn {
]A.S.I.,[
lis selected as most adverse.
The most adverse set of state parameters is thus:
~
where 100% design flow is 370,000 gpm.
3.2 Radial Power Distribution-The PDQ computer code (3-5) is used to predict planar radial power distributions throughout the life of a core for enveloping operating conditions.
Limiting power distributions are selected from the above set and are used as input to TORC DMB analyses.
Comparisons between PDQ predictions and measured data (3-6) show that PDQ overpredicts radial peaking factors in the peripheral regions of the reactor (i.e. the outermost three rows of fuel assemblies).
Inlet flow distributions for four-loop operation and seized rotor accident analysis of CE's 14 x 14 cores are shown in Fig. 3-1 and 3-8 respectively. These distributions manifest the following trend:
the central portion of the core receives higher than average I
3-5.
. inlet flow while the peripheral assemblies receive lower than average inlet flow.
For this reason, the limiting assembly for DilB analysis is found on the core periphery.
Since the PDQ power distributions overpredict power in the peripheral assemblies, and the limiting assembly for DNB analysis is among these assemblies, the use of PDQ data in DNS analyses is conservative.
This inherent conservatism in the thermal margin methodology makes it unnecesscry to account for uncertainties in the radial power distri-butions that are used in TORC DilB analyses.
3.3 Inlet Flow Distribution An inlet flow boundary condition is used in detailed TORC analysis.
Ratios of the local to core average mass velocity are input for every flow channel in the core-wide analysis.
Mean values of the inlet flow splits for three pump operation are presented in Fig. 3-8.
A large part of the uncertainty.in the flow splits results from measurement uncertainty. This measurement uncertainty is considered random and may be characterized by a normal probability distribution function (p.d.f.).
A sensitivity study, conducted to determi-ne the effects of inlet flow variations in assemblies which neighbor the limiting assembly, yields the results presented in Table 3-9. 01annel numbers in this table refer to Fig. 3-5.
Flow in the assemblies diagonally adjacent to the limiting assembly is decreased by 3, 6, and 9 percent for bottom peaked axial power profiles and by 9 percent for a top peaked profile.
These perturbations are in excess of inlet flow uncertainties, yet only minor chances in MDNBR are observed.
The above sensitivity study has shown that MDNBR in the limitinq assembly is unaffected by changes in the inlet flow of assemblies which are diagonally adjacent to the limiting assembly.
Because of this insensitivity, inlet flow in assemblies which are diagonally adjacent to the limiting assembly may be omitted from the response surface. Only inlet flow to the limiting assembly and those assemblies which are immediately adjacent to it are included in the response surface.
3.4 Exit Pressure Distribution Sensitivity studies conducted to establish the impact on IIDNBR of variations 'in the exit pressure distribution are summarized in Table 3-10.
Detailed TORC analyses are performed with nominal and extreme exit pressure distributions, as shown in Fig. 3-9.
The exit pressure in the limiting assembly is increased to the 95% orobability level while the exit pressures in the assemblies adjacent to the limiting assembly are also increased to yield an approximate 95% probability level for the three adjacent assembly exit pressures in the extreme exit pressure distribution.
Channel 4 of Fig. 3-5 is the limiting channel in this study.
Detailed TORC analyses performed with both bottom peaked and top peaked axial power profiles demonstrate that MDNBP, is extremely insensitive to variations in the exit pressure distribution.
Consequently, the exit pressure distribution need not be included in the MDNBR response
's urf ace.
3-6.
3.5 Enthalpy Rise Factor The engineering enthalpy rise factor accounts for the effects of manufact-uring deviations in fuel fabrication from nominal dimensions and specifi-cations on the enthalpy rise -in the subchannel adjacent to the rod with the MDNBR.
Tolerance deviations in fuel pellet density, enrichment, and diameter averaged over the length of thd fuel rods are used to compute this factor.
A survey of as-built data for 14xl4 fuel indicates that the enthalpy rise factor may be characterized by a normal distribution with a mean of 1.0 and i
standard deviation of 0.010.
Since these values are determined using data for every rod in the core, there is >95% confidence that the population mean and standard deviation are no laroer than these values.
Applicability of these data to future reload cycles will be verified.
3.6 Heat Flux Factor The engineering heat flux factor is used to take into account the effect on local heat flux of deviations from nominal design and specifications that occur in fabrication of the fuel.
Random. variation in pellet errichaent in-itial pellet density, pellet diameter, and clad outside diameter contribute to the effects represented by the engineering heat flux factor.
Tolerance limits and fuel specifications ensure that this factor may be characterized conservatively by a normal p.d.f. with a mean of 1.0 and standard deviation of 0.015.
Since these values are based upon tolerance limits, there is
>95% confidence that the population mean and standard deviation are no larger than these values.
3.7 Clad 0.D.
Variations in clad diameter change subchannel flow area and also change the local heat flux.
The impact of both random and systematic variations in fuel clad 0.D. on the local heat flux is accounted for by.the engineering factor on heat flux, discussed in section 3.6. The effect of random variations in clad 0.D. on subchannel flow area is included in the rod bow penalty, discussed in section 3.9.
The effect of systematic variations in clad 0.D.
on the subchannel hydraulic parameters is addressed here.
Manufacturing tolerances on the fuel clad allow for the possibility that the clad diameter will be systematically above nominal throughout an entire fuel assembly.
That is to say, the mean as-built value of the clad 0.D. may differ from the nominal value.
The distribution of the mean clad 0.0. for fuel assemb-lies may be characterized by a normal p.d.f. with a mean equal to the mean clad 0.D.and a standard deviation given by the relation ( 3-7) :
(N-n) o, n(ft-1)
(3.3) where ti is the number of specimens in the parent population and n is the sample size.
3-7.
f
)
, a
As built data for C-E's 14 x 14 fuel are presented in Table 3-11.
The minimum systematic clad 0.D. is [.
] while the maximum systematic clad 0.D. is [
].
Since the adverse effect of clad 0.D. variations is already taken into account by the engineering heat flux factor, and use of a less than nominal clad O. D. weild increase subchannel flow area, benefittina the MDilBR, the maximum value [
] is used in this study.
The standard deviation of the mean at the 95% confidence limit is [
]
in.
The double accounting for both the adverse effect of a decrease in clad 0.D. in the engineering factor on heat flux and the adverse effect of a systematic increase in clad 0.D. on subchannel flow area adds conservatism to the analysis.
5.8 Systematic Pitch Reduction The rod bow penalty, discussed in section 3.9, takes into account the adverse effect on !1DilBR that results from random variations in fuel rod pitch. The rod bow penalty does not take into account the adverse effect of systematic variatims in fuel rod pitch.
This systematic i
pitch reduction effect must be discussed separately.
Manufacturing tolerances on fuel assemblies allow for the possibility that the as-built fuel pitch will be less than nominal throughout an entire fuel assembly.
Thus the systematic pitch refers to the mean value of the pitch in an assembly.
The systematic pitch distribution is assumed to be a normal distribution characterized by the mean value of the pitch and the standard deviation of that n'ean value.
As-built gap width data for C-E's 14 x 14 fuel are presented in Table 3-12.
The minimum systematic qap width is seen to occur in the c.alvert Cliffs 1[.
]
andis[f J. inches. This, combined with the maximum clad 0.D. from sectTon 3.7 indicates that the minimum pitch is [
~
] At the 9Wconfidence level, the standard deviationofthemeanis[s
} inches.
3.9 Fuel Rod Bow The fuel rod bow penalty accounts for the adverse impact on MDflBR of random variations in spacing between fuel rods. The methodology for determining the rod bow penalty is the subject of a C-E topical report (3-8)4 Appendix G of that report (3-9) applies a formula derived by the flRC (3-10) to compute the rod bow penalty for C-E fuel. The penalty at 45,000 mwd /tiTU for CE's 14 x 14 fuel is 0.6% in DilBR.
This penalty is applied directly to the new MDflBR limit derived in Section 5.
3.10 CHF Correlation The C-E 1 Critical Heat Flux (CHF) correlation (3-11) (3-12) is used in the TORC code (3-1) to determine whether a departure from nucleate i
(
boiling (DflB) will occur.
This correlation is based on a set of 731 l
4 l
3-6.
s data points.
The mean of the ratio of observed to predicted CHF using the CE-1 correlation is 0.99983, while the standard deviation of that ratio is 0.06757.
CHF correlation uncertainty may be characterized by a normal distribution with a mean of 0.99983 and standard deviation of 0.06757. This yields a 1.13 MDNBR limit to satisfy the criterion of "95%
probability at the 95% confidence level that the limiting fuel pin does not experience DMS".
However, because the NRC staff has not yet con-cluded its review of the CE-1 correlation, a 5" penalty has been applied; this raises the 95/95 DNBR limit to 1.19.
This penalty may be conserva-tively treated by assuming a displaced Gaussian distribution with a mean of 1.06 and the same standard deviation as above.
~
3.11 TORC Code Uncertainty The TORC computer code (3-1) represents an approximate solution to the conservation equations of mass, momentum, and energy. Simpli fying assumptions were made, and experimental correlations were used to arrive at the algorithms contained in the TORC code.
Hence, the code has associated with it an inherent calculational uncertainty.
Comparisons between TORC predictions and experimental data (3-10)(3-13) have shown that TORC is capable of adequate predictions of coolant conditions.
As explained in Section 5.0 of Reference (3-13),
the TORC code was used to determine local coolant conditions from data obtained during the CE-1 CHF experiments.
These local coolant conditiaSs were then used to develop the CE-l CHF correlation. Thus, any c&lculational uncertainty in the TORC code is implicitly included in the MDNBR limit that is used with the TORC /CE-1 package in thermal margin analyses.
I a
h i
3-9.
l
_ _ _ _ _ _ _ _ _ _ _ _ _. ~. _ _ _ _. _ _ _ _ _ __
l
.1 i
L I
(!
4 t
.}
t i!!'
f 4
i f
i 4
i
]
1 4
i i
i
?
4 4
i i
i l
1 1
i I
s i
i i
i
?
+
r t
FIGURE 3-1 INLET FLOW DISTRIBU TION USED TO ESTABLISH STATE PARAMETERS
}
l
- FOR RESPONSE SURFACE l
I l
i1 3-10 e
n
.a---.a y
_u-..,.-._
a
.a
..w a
__s n
,_w..n
,a s_
u sa_.z_=a
---.-a
_u--
_.n-e
~
g.
- e, e
e t.
b ll g.
i 4
e I
1 1
h 1
i i
i
(
}
i k
I i
c i
i l
1 l
i k
i a
J FIGURE 3-2 a.
l
' EXIT PRESSURE DISTRIBUTION USED TO ESTABLISH STATE PARAMETERS FOR RESPONSE SURFACE r
t-3-11.
j s
1 2
HOT ASSEMBLY, 0.8349 1.041 STAGE 1 TORC ANALYSIS -
3
,4 5
6 7
CHANNEL NUMBER 0.8384 1.067 1.212 1.112 0.9970 ASSEMBLY AVERAGE 8
9
/
10 11 12 13 RADIAL PEAKING FACTOR
-0.9123 1.249 1.038 0.9496 0.9362 1.280 14 15 16 17 18 19 20 0.9087 1.100 0.9971 0.9223 1.213 0.9155 0.9174 21 22 23 24 25 26 27 28 0.8365 1.245 0.9941 1.240 0.8682 0.8736 0.9778 1.030 l
l l
l l
i.035 0.9270 l 0.8683 l 0.8519 1.037 l 1.026 l0.9620 1.065 1
_p__...__+__p____
_ _ 9 _ __, _
!.8320 l.038 0.8364 I.9512 0
0 i
I.8745 I l
l 0
1 1.200 0.9443 1.213 i,. _ _ + _ _ ____,__7______7__
,_,,,, l l
l l
l0.9858 l 1.026
~ - -- i.110 l0.9352 0.9195 0.9523 Md l 0.9538 1
I l
l
^*4-1.038 l- - -- -I- - - -- - -
-t-T- - - - --'--C k- --
0.9932 1.280 0.9259 1.084 0.9356 0.8308 0.9408 0.8234 l
l Note:
Circled channel numbers denote flow channels in which several i
fuel assentlies have been " lumped" into a single channel Q
for T-H analysis.
FIGURE 3-3 CORE WIDE RADIAL POWER DISTRIBUTION USED TO ESTABLISH STATE PARAMETERS FOR RESPONSE SURFACE 3-12.
,e d
7 enotes assembly quadrant average radial peaking fact-
/
/
FR=
FR" 1.185 1.303 LOCATION 9
1.174 1.1S9 1.214 1.243 1.2SG 1.374 1.210 1.231 1.2G1 1.282 1.297 1.335 1.338 1.25G 1.298 1.359 1.370 1.333 1.310 1.313 1.299 1.371 1.384 1.309 1.284 Fn =
p 1.314 1.332 1.319 1.389 1.387 1.304 1.272 i
l I
1.321 1.358 1.409 1.400 1.347 1.293 1.209 1.341 1.357 1.373 1.305 1.335 1.305 1.287 l
FIGUPS 3 h HOT ASSEMBLY RADIAL PO'.VER DISTRIBUTION USED TO ESTABLISli STATE PARAMETERS FOR RESPONSE SURFACE 3-13.
Q i
1 2
CHANNEL NUMBER IN FIRST STAGE MO
-3 4
5 6
7 8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 l
I l
l l
l 1
1 I
1
_. _ _ I _._ _. _ __ i _ _ 4 _ _ _ _ _ __l _ _ _ I _ _ _
I I
l l
l 298 1
30 l
i 31 l
p _ _l_ _ _ _ _ _l _ _ _ i _ _ _ _ _ _l_ _ _l_ _ _
i l
l I
i 1
l
--M l
l l
l l___l___
___l___i___ __ _ _ l _ _ _I _ _ _
i
.l l
i i
I l
i I
I
-k
(.--
i I
I I
I I
f l
l l
1 k
=
FIGURE 3-5 Cl[AIIIIEL ITGGERI !G CCE'E F0h STAGE 1 TORC AITALYSIS TO ESTABLIS!! RESPO:ISE SURFACE STATE PARNETERG 3-14.
~
CHANNEL NUMBER IN 5
6
'e 12 SECOND STAGE MODEL e
/
I, f 13 8
7 2
1 I
CROSSFLOW BOUNDARY 3
4 CONDITIONS APPLIED ALONG THIS BOUNDARY-14 9
10 11
? 15 h
///////////// ////////
16 17 18
-Q FIGURE 3-6 CHA'mEL NU:GERING SCHE S FOR STAGE 2 TORC ANALYSIS TO ESTABLISH RESPONSE SURFACE STATE PARAETERS 3-15.
l..
THIRD STATE TORC ANALYSIS THIRD STAGE TORC ANALYSIS FUEL PIN NUMBER CHANNEL NUMBER
\\
35 n
An I
1 1
2 I I 3
I I
4 l
5 l
l 6
21 11
- 1 7
.I 8
l I
9 l
I 10 1
11 l
12 I
I 13 l
14 15 10 17 18 19 H
27 22 12 2
37 H
21 22 1
23 24 36 32 28 23 13 7
3 2G 27 l
1 28 29 30 14 8
5 4
33 29 24 34 30 25 19 15 9
6 38 39 40 41 42 43 44 l
31 26 20 16 10 38 FIGURE 3-7 l
THIRD STAGE CHANN? i AND FUEL PIN NUMBERING SCHEME USED IN TORC ANALYSES TO ES:.\\BLISH RESPONSE SURFACE STATE PARAMETERS I
3-16.
i l
FIGURE 3-8 INLET FLO',1 FACTORS FOR SEIZED ROTOR A:!ALYSIS OF 217 ButlDLE 14x14 ASSEf'BLY CORES 3-17.
VALUES DENOTE DEVIATION OF ASSEMBLY AVERAGE EXITPRESSURE FROM CORE AVERAGE EXIT PRESSURE (PSF)
G 4
w e
FIGURE 3-9 EXIT PRESSURE DISTRIBUTIONS USED IN SENSITIVITY STUDY 3-18.
Operating Conditions Units Range
-0.55 < A.S.I.< 0.55 Axial Shape Index Inlet Temperature
- F 465 T 5580 5 in System Pressure psia 1750,Psys 12400 5
i System Flow
% design 77< w <120 Notes
- See note (1) on Table 3-3 for definition of axial shape index tThermal Margin design flow = 370,000 gpm h
RA:lGES OF OPERATI!!G C0:lDITIO!!S FOR UHICil RESPO!!SE TABLE 3-1:
SURFACE IS VALID l
l 3-19.
s Inlet Flow Fraction
- Stage 1 Channel Number Nominal Perturbed I
Assembly Inlet Mass Velocity
- Inlet Flow Fraction =
Core Average Mass Velocity TABLE 3-2: NOMINAL AND PERTUR3ED FLOW FOR ESTABLISHING SENSITIVITY OF FLOW DISTRIBUTION EFFECTS Oil MDNBR TO OPERATING CONDITIONS 9
9 f
3-20.
State Parameters H0fl6R Axial Shape inlet System flominal l Perturbed Index Pressure lempera ture Flow Flow Flow
- A
% Change (1) psia "F
% design (2)
(3)
-0.07 2200 550 100
~
-0.02 2200 550 100 0.00 2200 550 100 O.317 2200 550 100 0.337 2200 550 100 0.444 2200 550 100 0.527 2200 550 100
-0.070 2400 580 120
-0.070 1750 580 120 4
-0.070 2400 465 120
-0.070 1750 465 120
-0.070 2400 580 77
-0.070 1750 580 77
-0.070 2400 465 77
-0.070 1750 465 77 0.337 1750 580 77 0.337 1750 465 77 0.337 1750 580 120 o
L/2
' F dz ~ F dz core average
~
z z
F = axial peaking factor s
(1) Axial Shape Index = ~
o = core midplane L/2 L = active core length
[ F dz z
-L/2 (2) At10flBR = "flominal" M0ilBR
" Perturbed" MDflBR (3) O Change in HDilBR = ( AM0flBR /flominal itDilBR ) x 100
- see Table 3-2 TABLE 3-3:
FLOW PERTURBAT10:1 EFFECTS AT VARIOUS OPERATIflG C0!!DIT10!!S 3-21.
l l
MDNBR AXIAL SHAPE ENTHALPY RISE INDEX NOMIflAL FACTOR APPLIED A
% CHANGE (2)
(3)
(1)
-0.527
-0.359
-0.070
-0.020 0.00(C)*
0.00(S)*
0.337 0.444 0.527
-0.317
-0.162 0.317
- Both a cosine (denoted by "C") and saddle (denoted by "S")
Axial Shape were used for 0.00 A.S.I.
OPERATIf;G C0tiDITI0flS:
2200 psia Pressure
=
5500F Inlet Temperature
=
100% design System Flow e
=
See Notes on Table 3-3 SEllSITIVITY OF EtlTHALPY P,ISE FACTOR EFFECTS TO AXI AL SHAPE TABLE 3-4:
INDEX (ISOLATED HOT ASSEMBLY MODEL) 3-22.
a STATE PARAMETERS MDNBR Axial Shape Inlet S/ stem Enthalpy Rise Index Pressure Temperature Flow tiominal Factor Applied a
% Change (2)
(3)
(1) psia F
% design l
i e
(1)
See tiotes on Table 3-3 (2)
(3)
TABLE 3-5: SEf1SITIVITY OF EllTHALPY RISE FACTOR EFFECTS TO OPERATIriG C0tlDITI0'IS (Isolated Hot Assembly Model)
I 3-23.
\\
i MDNBR Axial shape Enthalpy Rise Index Nominal Factor Applied a
% Change (2)
(3)
(1) l' OPERATItlG CONDITI0tlS 2200 psia Pressure
=
0 550 F Inlet Temperature
=
100% design System Flow
=
(1) '
See Notes on Table 3-3 (2)
(3)
TABLE 3-6: SENSITIVITY OF EtlTHALPY RISE FACTOR EFFECTS TO AXIAL SHAPE It!DEX (Core Wide Analysis)
O 3-24.
s STATE PARAMETERS MDNBR Axial Shape Inlet System Reduced Index Pressure Temperature Flow Nominal Pitch 4.
% Change (2)
(3)
(1) psia F
% design 1
5 (1) 3 (2)
(
See flotes on Table 3-3 (3) J TABLE 3-7: SENSITIVITY OF SYSTE!!ATIC PITCH REDUCTION EFFECTS TO OPERATIt;G CONDITIONS 3-25.
j 1
l
s STATE PARAMETERS Axial Shape Inlet System Parameter Index Pressure Temperature System Flow (1) psia F
% Design
~
Inlet Flow Distribution Enthalpy Rise Factor Systematic Pitch Reduction (1) See note on Table 3-3 TABLE 3-8: STATE PARAMETERS WHICH MAXIMIZE MDNBR SENSITIVITY TO SYSTEM PARAMETERS e
l 3-26.
s Axial Shape Index Flow Split Reduction
- MD!lBR Change in MDflBR TABLE 3-9:
MD:lBR SEllSITIVITY TO IrlLET FLOW DISTRIBUTIO:1 e
3-27.
o Axial Shape Exit Pressure Change In Index Distribution MDflBR MDilBR 0.444 nominal 0.444 cxtreme
-0.359 nominal
-0.359 extreme i
TABLE 3-10: SEllSITIVITY OF T1DilBR TO EXIT PRESSURE DISTRIBUTI0tl 1
i et 3.-28.
-1
~
CYCLE 1
2 3
4 5
BATCH A,0,C D
E F
G OMAHA MILLST0tlE CC #2 CC #1 ST. LUCIE 1 MAlf1E YAtlKEE
- Mean
- Standard Deviation
- Standard Deviation of the Mean l-l l
Note: flominal Clad 0.D. = 0.44 inches TABLE 3-11 AS BUILT CLAD 0.D. (inches) DATA FOR 14 X 14 FUEL l
1 3-29.
!l 5
5 0
1 C
3 I
1 I
0 s
C f
l f
i l
C d
2 s n
~
1 t a e
0 n e v
e m l
C m
a e f C
r o us n
)
a o S
e i m t E
I a
I 0
f i C
o v N
1 0
e I
r d
(
C e
A b d T
m r A
u a D
n d n l
a iT t
D
)
s I
xx W
e x :
P ee
(
A nk x x G
in x x aa x x T
MY x x L
I U
B-n S
a A
e 1
t I
t 2
rs 1
ef vf 3
li al E
l iM n
uo th rl oa FC re nb ar 8
7 6
5 4
3 2
1 r
pu SM i'
Y8-
4.0 MUflilR Re,sponse Surf ace A response surface is a functional relationship which involves several independent variables and one dependent variable.
The surface fp created by fitting the constants of an assumed functional relationship to data obtained f rom experiments.
The response surface provides a convenient means by which accurate estimates of a complex or unknown function response may be obtained.
[
Since the response surface is a relatively simple express ion, it may be applied in analytic techniques where more complex functions would make an analytic solution intractable.
j In the present application, a singic detailed TORC analysis is treated as an " experiment".
A carefully selected set of detailed TORC "experi-ments" is conducted, and a functional relationship is fitted to the This response surface is then used in conjunction with 11DilBR results.
Monte Carlo techniques to combine probability distribution functions (p.d.f.'s) for each of the independent variables into a resultant 110tiBR p. d. f..
4.1 TORC ltadel used The inlet flow distribution (shown in Fig. 3-8) is compared with radial power distributions for the Calvert Cliffs Unit 1 and Unit 2 reactors to determine the limiting location for DilB analysis.
For the purpose of generating the response surf ace, the limiting location is defined as the assembly in which the impact of system para-meter uncertainties on f1DilBR is the greatest.
The core wide and limiting assembly radial power distributions used to generate the response sur-face are shown in Fig. 4-1 and 4-2, respectively.
The first stage TORC model used in this analysis is shown in Fig. 3-5.
The limiting assembly occurs in channelf of this model.
Second and
,s third stage models used in this analysis are shown in Fig. 4-3 and 4-4 respectively.
4.2 Variabis used A careful examination of the sources of uncertainty discussed in Section 3 shows that several of these sources of uncertainty can be omitted from the response surface.
The state variables mentioned in section 3.1 are treated in part 1 of this report (4-1).
As explained in Section 3.2, inherent conservatism in the calculation of radial peaking factors makes the need to account for uncertainty in the radial power distribution used in DtlB analyses unnecessary. llence, the radial power distribution was omitted from the response surface.
e x
{
h 4-1.
d
s
)
i The sensitivity study discussed in Section 3.4 indicates that large perturb-ations in the exit pressure distribution have negligible effect on the pre-dicted MDf!BR. Thus, the exit pressure distribution is not included in the response surface.
The heat flux factor (F,.) is applied to the MDf;BR calculated by TORC in q
the following manner:
4
- MDNBR TORC (4.1)
MDNBR
=
F,.
q
~
4 Since the functional relationship between MDNBR and F "
is known, the heat is not used in generating the response s0rface.
Instead, this flux factor is combined with the resultant surface, as explained in section 4.5.
factor A method has already been developed (4-2) to account for rod bow uncertainty.
i included in the response surface.
Instead, the No rod bow effects are rod bow penalty found with existing methods (4-2) is applied to the design limit MDNBR found in the present alalysis.
The calculational uncertainty associated with MDNBR predictions found with the TORC /CE-1 package is implicitly included in the CHF distribution uncert-1 ainty, as explained in Sections 3.10 and 3.11. Hence no explicit allowance for code uncertainty is included in the response surface.
The system parameters included as variables in the response surface are listed in Table 4-1.
4.3 Experiment Desian An orthogonal central composite experimental design (4-3) is used to gen-erate the response surface applied in this study. The total number of exper-iments needed to generate a response surface using this experiment design is 2k + 2k + 1 The desired response surface where k is the number of variables to be considered.
consists of seven variables, hence 143 " experiments" or detailed TORC analyses The results of these were needed for a full orthogonal central composite design.
experiments may then be manipulated by means of the least squares estimator b= {n' n F [n'} z (4.2 )
l 4-2.
l
J 3
where z is the vector of experimental results, to yield the coefficients which define the response surface 7
^b ng + 7 bjj(n -c) + 7 7 2
b b
+ys MD!lBR q i j (4.3) z =
n ti
=
g 7
RS o
i <j In the above equations, the n are coded values of the system parameters (x )
to be treated in the response, surface, as indicated in Table 41.
The b rbp-4 resent the constants found from the TORC results by means of Eq. 4.2, anti c is'a constant determined by the number of experiments conducted.
The number of TORC analyses needed to generate the response surface could gjn n))
be reduced significantly if some of the interaction effects (i.e. b g
were neglected.
ll 4.4 Design Matrix The set of experiments used to generate the response surface is referred to as the design matrix. This matrix, in coded form, comprises the second through eighth columns of the n matrix cited in Eq. (4.2).
Both coded and uncoded versions of the design matrix used in this study are presented in Appendix A along with resultant MDflBR values. The design matrix was con-structed such that each independent variable included in the response surface extends just beyond the 2a range of its associated p.d.f.
4.5 Response Surface _
Equation (4.2) was solved numericelly using the data in Appendix A.
Constants for the response surface as given by Eq.(4.3) are presented in Tabic 4-2.
Comparisons made between TORC predicted f1DilBR and response surface predictions show excellent agreement.
The 95% confidence estimate of the response surface standard deviation is 0.00377.
I' The heat flux factor is included analytically in the response surface by combining Eq. (4.0) with Eq. 4.3).
The final relationship is given by 7
7 2
7 7
]- (4-4)
,1
} b ilDilBR F,,
+t bg ng+ t bgg (ng -c)+ E E
bjj ngn) q g
i=1 i=1 i=1 J=l
.i<j 4-3.
7 7
.a-
.._._u_
The coef ficient of determination, r, provides an indication of how well the response surface explains the total variation in the response variable (4-4). Wnen ral, a true model has been found.
The r value associated with the response surface generated in this work is 0.9995, which indicates that this response surface is a good model.
Another indication of model performance is provided by the standard error The standard error for the response surface is 0.003396.
of estimate (4-5).
The relative error is 0.29%, indicating that this model performs well.
=
O t
e e
i i
4-4.
r
9.
s 1
2 1.1813 1.5170 3
4 5
6 7
1.1444 1.5640 1.d551 1.35G8 1.0570 BOX NUMBER
=8 9
10 11 12 13 ASSEMBLY AVERAGE
= 1.0481 1.5295 1.4035 1.0914 1.2227 0.7662 RADIAL PEAKING FACTOR 14 15 16 17 18 19 20 1.0472 1.0802 1.0914 1.4707 1.0592 0.50GG G.633G 21 22 23 24 25 20 27 28 1.1410 1.5271 1.0931 1.4179 1.1005 0.4147 0.5007 0.5148 1
29 30 31 32 33 34 35 36 1.493G 1.3999 1.4713 1.032 0.84G1 0.7310 0.5170 0.4752 37 38 39 40 41 42 43 44 1.6445 1.08G8 1.05G9 0.4151 0.7312 0.5029 0.5554 0.3023 45 03 46.
47 48 49 50 51 52 53 1.3469 1.21GG 0.504G 0.58GO 0.5132 0.5534 0.3574 0.3953 54 1.5030 55 SG 57 58 59 60 S1 62 Q, - -
1.6443 0.7744 0.6307 0.5026 0.4520 0.3008 0.39G2 0.1861 ---c 9.
Figure 4-1 CORE WlDE POWER DISTRIBUTION USED TO GENERATE RESPONSE SURFAC2 4-5.
4 s
J i
4 l,
i I
l
't 1
I l
Figure 4-2 HOT ASSEMBLY RADIAL POVIER DISTRIBUTION 11 SED TO GENERATE RESPONSE SURFACE 4-6.
.. l
5.
4 a
l 1
1 4
e i
I 1
a t
t 4
I i-i i
f i
i 1
1 i
i 1
i
~
1 o
i 3
i 1
Figure 4-3 I
INTERMEDIATE (2ND STAGE) TORC MODEL USED IN GENERATING RESPONSE SURFACE 4
i.
4-7.
?
., - _. = - _..,. _. _ _ _ _ _ _, _,,
35 l
1 1
2 3
l 4
l l
5 l
6 1
2 3
\\
l l
9 l
1 l
l l
1G 17 18 10 20 8
4 5
7 j
36 21 22 23 24 25 37 LOCATION 9
10 11 12 13 14 W
26 27 28 1
29 30 15 1G 17 18 20 21 22 31 32 33 34 35 3G 37 23 24 25 26 27 28 29 O
3 i
38 j
l 39 40 41 42 43 44 H
30 31 32 33 34 38 f
l Figure 4-4 SUBCHANNEL (3RD STAGE) TORC MODEL USED IN GENERATING RESPONSE SURFACE 4-8.
System Parameter Variable Index(i) Coded Variable **
o<
/G hot 'ssembly) inlet flow factor x1 1
(ch; :el[4*J channel [3*] inlet flow factor x2 2
channel [5*] inlet flow factor x3 3
channel [10*] inlet flow factor xg 4
enthalpy rise factor x3 5
1.0001 0.0119 systematic pitch xs 6
el systematic clad 0.D.
x7 7
L
- channel numbers refer to Figure 3-5
- variables coded according to relation q; = 'h -
where the a5 4
/7; are chosen such that n j = 0 at nominal conditions, and the e are chosen j
such that the range of the response surface will include s2a ranges of each of the system parameters TABLE 4-1: STATE PARAMETERS INCLUDED AS VARIABLES IN THE RESPONSE SURFACE O
e e
e 8
g 6
4-9..
s MDNBR b, +
b; rr, +
b;; ( q' - c) +,,,,,i ;q;q;-
=
RS b;
- .1 i<j TABLE 4-2
- COEFFICIENTS FOR MDNBR RESPONSE SURFACE 4-10.
5.0 Combination of Probability Distribution Functions The MONBR response surface discussed in Section 5 is applied in Monte Carlo methods to combine numerically the system parameter probability distribution functions (p.d.f.'s) discussed in Section 3 with the CHF correlation uncer-tainty.
A new 95/95 MDNBR limit is then selected from the resultant p.d.f.
This new limit includes the effect of system parameter uncertainties and thus may be used in conjunction with a best estimate design TORC model.
5.1 Method The SIGMA code applies Monte Carlo and stratified sampling techniques to combine arbitrary p.d.f.'s numerically (5-1). This code is used with the response surface to combine system parameter p.d.f.'s with the CE-1 CHF correlation p.d.f. into a resultant MDNBR p.d.f.
The methods used to achieve this combination are discussed below.
The effect of system parameter uncertainties on MDNBR is combined with the effect of uncertainty in the CHF correlation by computing a AMDNBR caused by deviation of the system parameters from nominal:
RS - DNBR (5.1)
MDNBR NOM a
=
ggggg is the MDNBR found by substituting the set of system where MDNBR85 is the MDNBR value parameters into the response surface and MDNBRNOM predicted by the response surface with nominal system parameters. A point is then randomly chosen from the CHF correlation p.d.f. and combined with the ohDNBR from Eq. (5.1) to yield a MDNBR value:
(5.2)
MDNBR = MDNBRCHF + OMDNBR This process is repeated by the SIGMA code for 2000 randomly selected sets of system parameters and randomly selected points from the CHF correlation p.d.f.,'and a resultant MDNBR p.d.f. is generated.
The system parameter p.d.f.'s input to SIGMA are listed in Table 5-1.
Both "best estimate" and 95% confidence estimates of the standard deviation are included.
Standard deviations at the 95% confidence level are input to SIGMA to ensure that the standard deviation of the resultant MDNBR p.d.f. is at least at the 95% confidence limit.
1 5-1.
5.2 Resul ts The resultant 11DflBR p.d.f. is shown in Fig. 5-1.
The mean and standard deviation of this p.d.f. are 0.988 and 0.099451, respectively.
As Fig.
5-1 indicates, the resultant MDilBR p.d.f. approximates a normal distri-bution.
5.3 Analytic Comparison An approximate value of the standard deviation of the resultant MDflBR p.d.f.
may be found by analytic methods.
These methods are based upon the assumption that the uncertainties are small deviations from the mean (5-2). Given a functional relationship y = f(x),x '
- n)
(*
2 the effects of small perturbations in x on y may be found from oyedy=
axj +
Ax2+
^*n (5.4)
+
1 Hence, if several normal distributions are combined by the relationship expressed in Eq.(5.3), the variance of the resultant p.d.f. is y2,(M )2 o 2,(M )2 a2,,,,, (M )2 2 y
ax) xj ax 3*n 2
x n
2 where the partial derivatives are evaluated at the mean values of the xj's.
The response surface relates MDilBR to system parameters by the relationship found on Table 4-2:
7 7
7 7
+j,)bnj+j,) bjj(nj -c)+f,) j,)bjj nj nj (5.6)
=b MDNBRRS g
j i<j x -xj j
nj =
(5.7) where Bj t
Applying Eq. 5.5 to the response surface yields the following expression for the variance:
2 (a(MDilBR)30i)
(5.8) 7 2
o o
=r i
RS 1=1 an j ax j
l 5-2.
Differentiating Eq. (5.6) and (5.7) with respect to njand x :j aM0tlBR (5.9)
=bi +2bii ni +J=i+1 b ij n.J 3"i anj j
(5.10) 3Xj j3; Substituting Eq.(5.9) and (5.10) into Eq.(5.8) results in a relation be-tween the resultant MD?tBR variance and the system parameter variances:
=1 (b +2bjj,j+ I_j,) bgj n) )' (
xj
)2 (5.11)
RS j
=
O i This equation is simplified when evaluated at the mean values of the ng: (i.e.ni=0) 7 2
a
,E b
(5.12) x I
RS i=1 g,2 1
The CHF correlation p.d.f. and system parameter p.d.f. 's are related to the resultant MDilBR in Eq.(5.1) and Eq.(5.2), the heat flux factor is related by Eq. ' 4.1). The resultant MDilBR variance is given by 2
2 2
2 o MDilBR, "R S CHF Fq" (5.13) 2 "2
.Substitutinq values from Tables 4-1, 4-2, 5-1, and Section 4.5 into Eq. (5.11) and Eq (5.13) yields
=
MDtlBR 0.09923 l
which is in excellent agreement with the value predicted by the SIGMA code simulation using the response surface.
5-3.
STATIDARD DEVIATION DISTRIBUTIOu MEAN DEST ESTI:1 ATE 95% CONFIDEf!CE hot assembly inlet flow factor (channel [
])
channel [ ] inlet flow factor channel [ ] inlet flow factor channe1[
] inlet flow factor enthalpy rise factor 1.0
.0100+
systematic pitch (inches) systematic clad 0.D. (inches)
.0150**
heat flux factor 1.0 CE-1 CHF Correlation
.998
.0676
.07384
- channel numbers refer to Figure 3-5
+ entire fuel pin population was sampled, hence >95% confidence
- standard deviation based upon tolerance limits, hence >95% confidence TABLE 5-1: PROBABILITY DISTRIBUTION FUtlCTIONS COMBINED BY SIGMA 5-4.
l l
e i
I i
i i
i i
FREQUENCY -
2000 TRUE GAUSSIAN 0.10 n = NUMBER OF POINTS IN INTERVAL O
ACTUAL DISTRIBUTION (DN3R - 1/2 3DNBR, DNBR + 1/2 3DNBR)
Obtained From Itonte Carlo O
O
-> O Code and Response Surface
/
UN
/
g 0.08 j
\\
O6 g
/
\\
\\
E j
(n 3
06
/
0
\\
/
E P
b
/
\\
/
\\
0.04
,h
\\
O\\
/
\\
\\
0 0.02
,0
\\
ON O/
O h
bN\\
OOAV O
i s4 i
o 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 DNBR Figure 5-1 RESULTANT MDNBR PROBABILITY DISTRIBUTION FUNCTION
6.0 @ plication to Design Analyses This section discusses the application of the statistically derived MDilBR p.d.f.
to design analyses.
Deterministic methodolcgy (6-l) involves use of a design model for TORC analysis which includes deterministic allowances for system para-meter uncertainties. These deterministic penalties are replaced with a higher MDilBR limit in the statistical methodology. This higher i1D!lBR limit is used with a "best estimate" design model in thermal margin analyses.
6.1 Statistically _ Derived MDNBR Limit The MDNBR p.d.f. described in Section 5.0 is a normal distribution having a mean of.9875 and a standard deviation of 0.099451. This standard deviation is at least at the 95% confidence level.
A comparison of TORC results and response surface predictions indicates that the lo error asscciated with the response surface is o = 0.003396; at the 95% confidence level, this value is s
"s95 " (.003396 x,/142/115.461 ) =.00377. (See Eqn. 2-3 of Ref. 6 5)
The MDNBR standard deviation was found to be 0.099451 b'y means of Monte Carlo methods. Since a finite number of points (2000) were used in these methods, a correction must be applied to the calculated value. The resultant MDNBR standard deviation, adjusted for the finite sample size used is (0.099451 x
/1999/1896.131)=0.10211.
The root sum square of this adjusted MDNBR standard deviation and the response surface standard deviation at the 95% confidence level
)(0.10211)2 + (0.00377)2
= 0.10218.
The corresponding 95%
o tot confidence estimate of the mean is
(_.9875 + (1.645 x 0'.10211)/ Tl999) = 0.991. (See Eqn. 2-2 of Ref. 6 5)
Since the resultant MDNBR p.d.f. is a normal distribution, as shown in Figure 5-1, the one-sided 95% probability limit is 1.645o.
Hence there is a 95%
probability with at least 95% confidence that the limiting fuel pin will not experience DNB if the bast estimate design model TORC calculation yields a MDNBR value greater than or equal to (0.991 + 1.C45 x 0.lT18)
= 1.16 6.2 '_ Adjustments _ to Statist _ically Derived MDNBR Limit The statistical MDNBR limit derived in Section 6.1 contains no allowance for the adverse impact on DilBR' of fuel rod bowing.
CE has applied an NRC method for taking rod bow into account in DNBR calculations (6-2).
This application shows that the maximum penalty occurs at fuel end-of-life.
For 14x14 fuel, this penalty is 0.6% in MDNBR.
Thus, the new limit, including an allowance for rod bow is (1.006 x 1.160) or 1.167.
The NRC has not yet completed review of the application of the CE-l CHF corr-elation ( 6-3 ) to non-uniform axial heat flux shape data ( 6-4).
Consequently, a 5% penalty was applied tc the 1.13 MDNBR limit by the NRC. The interim MDNBR limit for use with the CE-1 CHF correlation, pending NRC approval of CE's non-uniform axial heat flux shape data, is 1.19.
For the purposes of this study, a conservative application of this penalty is to shift the mean of the MDNBR p.d.f. by 0.06.
This shift results in a MDNBR limit of 1.227.
Thus the new MDNCR limit which contains allowance for uncertainty in the CHF cor.alation and system parameters as well as a rod bow penalty and the interim 5% penalty on the CE-1 correlation imposed by the NRC is 1.23.
i 6-l
I 6.3 App,lication to TORC _ Des _ign fi_o_ del Statistical combination of system' parameter uncertainties into the MDflBR limit precludes the need for deterministic application of penalty factors to the design TORC model. The design TORC model used with the new MDf1BR limit of 1.23 consists of best estimate system parameters with no engineering factors or other adjustments to accomodate system parameter uncertainties. The inlet flo i split will, however, continue to be chosen such that the best estimate design TORC model will yield accurate or conservative MDf1BR predictions when compared with MDilBR values from detailed TORC analyses ( 6-11 i
1 a
5 J
?
6-2
-,c e
,e
,r g
p 7.0 Conclusions Use of a 1.23 f1DNBR limit with a best estimate design TORC model for the Calvert Cliffs Unit 1.and Unit 2 cores will ensure with at least 95% probability and 95% confidence, that the hot pin will not experience a departure fror, nucleate boiling. The 1.23 MDNBR limit includes explicit allowances for system pcrameter uncertainties, CHF correlation uncertainty, red bow, and the 5% interim penalty imposed by the NRC on the CE-l CHF correlation.
7.1 Conservatisms in the tiethodology Several conservatisms are included in the present application. The significant conservatisms include:
1) combination of system parameter p.d.f.'s at the 95%
confidence level to yield a resultant MDNBR at a 95% +
confidence level ii) use of pessimistic (generic) system parameter p.d.f.'s iii) derivation of the new MDNBR limit such that it applies to both 4-pump operation and seized rotor analyses iv) use of the single most adverse set of state parameters to generate the response surface v) application of the 5% interim penalty imposed by the NRC on the CE-1 CHF correlation 7-1
,,. ~.
r.---
--c-m
8.0 References 8.1 Section 1.0 References D-1)
" Report on Statistical Combination of Uncertainties Methodology, Par t 1, C-E Calculated local Power Density and Thermal f4argin/ Low Pressure LSS'. for Calvert Cliffs Units I and II", CEft-124(B)-P, December 13, 1979 8.2 Section 2.0 References (2-1)
" TORC Code:
Verification and Simplified liodeling Models",
CENPD-206-P, January 1977.
(2-2)
" TORC Code: A Computer Code for Determining the Thermal Margin of a Reactor Core", CENPD-161-P, July 1975.
(2-3)
"C-E Critical Heat Flux:
Critical Heat Flux Correlation for C-E Fuel Assemblies with Standard Grids, Part 1:
Uniform Axial Power Distribution", CENPD-162-P, September 1976.
8.3 Section 3.0 References (3-1)
" TORC Code: A Computer Code for Determining the Thermal Margin of a Reactor Core, CENPD-161-P, July 1975, pp. 5-1 to 5-8.
(3-2)
" Report on Statistical Combination of Uncertainties Methodoloay, Part 1, C-E Calculated Local Power Density and Thermal MargirVLow Pressure LSSS for Calvert Cliffs Units I and II", CEN-124(B)-P, December 13, 1979.
(3-3) Combustion Engineering Standard Safety Analysis Report, (System 80),
Docket #STN-50-470F, October 26, 1979, Fig. 4.4-7.
(3-4) ibid, Subsection 4.4.2.2.2.2.C.
(3-5)
W. R. Cadwell, "PDQ-7 Reference Manual", WAPD-TM-678, January, 1968, Westinghouse Electric Corporation.
(3-6)
A. Jonsson, et al., " Core Physics Validation for the Combustion Engineering PUR", Combustion Engineering Technical Paper TIS 6368 Presented at the American fluclear Society Winter Meeting, November 12-16, 1979, San Francisco.
(3-7) Green & Bourne, "Peliability, Technology", Wiley-Interscience, a division of John Wiley & Sons Ltd., p. 326.
(3-E.
" Fuel and Poison Rod Bowing", CENPD-225-P, October 1976.
(3-9) " Fuel and Poison Rod Bowing-Supplement 3", CENPD-225-P, j
Supplement 3, June 1979.
j l
8-1.
f.
t (3-10) Letter from D. B. Vassallo (NRC) to A. E. Scherer (C-E),
June 12,1978.
(3-11)
"C-E Critical Heat Flux:
Critical Heat Flux Correlation fnr C-E Fuel Assemblics with Standard Spacer Grids, Part 1:
Uniform Axial Power Distribution", CEllPD-162 -P, September 1976.
(3-12)
"C-E Critical Heat Flux:
Critical Heat Flux Correlation for C-E Fuel Assemblies with Standard Spacer Grids, Part 2:
lionuniform Axial Power Distribution", CENPD-207-P, June 1976 (3-13)
" TORC Code:
Verification and Simplified Modeling Methods",
CENPD-206-P, January 1977.
8.4 References for Section 4 (4-1)
" Report on Statistical Combination of Uncertainties Methodology Part 1, C-E Calculated Local Power Density and Thernal Margin / Low Pressure LSSS for Calvert Cliffs Units I and II", CEN-124(B)-P, December 13, 1979.
(4-2)
" Fuel and Poison Rod Bowing, Supplement G", CEllPD-225-P, Supplement 3-P, June,1979.
(4-3)
R. H. Myers, Response Surface Methodology, Allyn and Bacon, Inc.,
Boston, 1971.
(4-4)
N. R. Draper, H. Smith, Applied Regression Analysis, John Wiley &
Sons, New York, 1966, p. 62.
(4-5) ibid., p.118.
8.5 References for Section 5 (5-1)
F. J. Berte, "The Application of Monte Carlo and Bayesian Probability Techniques to Flow Prediction and Determination", Combustion Engineering Technical Paper TIS-5122, presented at the Flow Measurement Symposium, sponsored by the flational Bureau of Standards, Gaithersburg, Maryland, February 23-25, 1977.
(5-2)
E. L. Crow, F. A. Davis, M. W. Maxfield, Statistics Manual, Dover Publications, Inc., New York,1960.
8.6 References for Section 6 (6-1)
" TORC Code: Verification.and Simplified Modeling Methods",
l CENPD-206-P, January 1977.
(6-2)
" Fuel and Poison Rod Bowing, Supplement 3", CENPD-225-P, Supplement 3-P, June 1979.
(6-3)
"C-E Critical Heat Flux:
Critical Heat Flux Correlation for C-E Fuel Assemblies with Standard Spacer Grids, Part 1:
Uniform Axial Power Distribution", CEtlPD-162-P, September, 1976.
(6-4)
"C-E Critical Heat Flux:
Critical Heat Flux Correlation for C-E i
Fuel Assemblies with Standard Spacer Grids. Part 2:
flonuniform Axial Power Distribution", CEtlPD-207-P, June,1976.
8-2.
(6-5)
" Statistical Combination of Uncertainties Meti:odology Part 1:
e C-E Calculated Local Power Density and Thermal Margin / Low Pressure LSSS for Calvert Cliffs Units I and II," CEN-124(B)-P, December,1979.
O 4
e o
P e
e O
e 8-3
[
Appendix A:
Detailed TORC Analyses Used To Generate Response Surface 4
An orthogonal central composite experiment design (A-1) was used to generate the response surface (R S) used in this study. All first order interaction effects (i.e. xjxj terms) were retained in the R S.
The R S used in this study included seven variables. The coded set of detailed TORC analyses performed _ to generate the R S is presented in Table A-1; variables were coded as shown in Table 5-1.
The actual values of the input parameters are presented in Table A-2 along with the resultant MDNBR values.
4 References (A-1)
R. H. Myers, Response Surface Methodology, Allyn & Bacon, Inc., Boston, 1971, p.133.
l l
A-1
Ca.de,. _. - _... _ _ _
____[.n Ir_t. I'l c.. _.... i _ _ -. - _. - -....
l
' Ent!:al py l Sysi.euatic l5ystero1.ic
,Ch neip) [ l'[__Cl;'t. :i.i [ ]_4i, oc torr.i liar.::. l[ ]
' 3.; art. i [.
- ] IC e
- c c co -
M t c'i
! C1aJ 0.ii.
i
. jh $ J.
1
-1
-1
-1
-1
-1
-1
-1
,i I.
2
-1
-1 I
-i
-1
-1
-1 i
1 i
e l
l-I 3
-1 1
-1
-1
-1
-1 1
-1 1
I g
i 4
-1
-1
-1
-1
-1 1
1 l
i s
I I
l 5
-1
-1
-1
-1 1
-1
-1 i
6
-1
-1
-1 1
-1 1
-1 1
[
i i
7 l
-1
-1
-1
-1 1
1 6
-1 1
l I
I i
1 8
-1
-1
-1 j
-1 1
1 1
i I
-1
-1
-1 1
-1
-1 9
-1 l
10
-1
-1
-1 1
-1
-1 1
11
-1
-1
-1 1
.1 1
-1 i
12 l
-1
-1
-1 1
-1 1
1 l13 f
-1
-1
-1 1
1
-1
-1 L
i
_.l.-.-.-
J._.__-
..I-.__
. r to l i g. 3-L
- .ce h nte 4-1 for cod (d te'iot.ionships
^ c.lannel nur.. aces r-flote: Coded values determined by methods described in Reference (A-1).
ii.jj e_ A-1,:
Coio set of Detailed '10.4C Cases I! sed to G-ncrc.te 1:esponse Surfac e A-2 1
1 y
Case In t.:t Fhr.-r l'ac tors Enth.2tpy Sys te:N tic L I: umber ; C!' nn ; [ ],f Cf.: nel [ ]; Channel [ J_
u......:: 1 []
itise facwr Pit.rh Clad D.0.
sys ten:a tic
= _
I I
14
-1 l
-1
-1 1
1
-1 1
e l
4 15
-1
-1
-1 1
1 1
-1 i
I l
1 l
1 1
1 16
-1
-1
-1 I
4 l
17
-1
-1 1
-1
-l
-1
-1 1
18
-1
-1 1
-1 j
-1
-1 1
i i
I i
19
-1
-1 1
-1
-1 1
-1 20
)
-1
-1 1
-1
-1 1
l 1
1 21
-1
-1 1
-1 1
-1
-1 t
ll i*
g 22 i
-1
-1 1
-1 1
-1 1
t 23
-1
-1 1
-1 1
1
-1 24 i
-1
-1 1
-1 1
1 1
8' I
l 25
-1
-1 1
1
-1
-1
-1 8
l 26
-1
-1 1
1
-1
-1 1
l...
.. l._...
__.___.--L_.
l i
5.itann. i 'uhlers refer to Fig. 3-5 see lab.a 4 -1 for coded re'lationships Note:
Cade' values determined by methods described in Reference (A-1).
Table %..
voo.:d Set of Detailed TOI:t L ses Used to 'ientrate f.esponse Surface (con't.)
c A-3
[-Case l__
.____I_nl<1t F ov.' f ac tors Enthalpy
- Sy s te..:a ti t - Systetaa t.i c
. ! Mar I. Cl+251E Ti CJ91ael _1.J
. rMell 1 ":'. : [ 7 F.Ln i ai: tor I
Pit (h C16d 0.!..
27 l
-1
-1 1
1
-1 1
-1 e
28
-1
-1 1
1 i
-1 1
l 1
1 29 I
-1
-1 1
1 1
-1
-1 l
30
-1
-1 1
1 1
I
-1 1
i 31 1
-1
-1 1
1 1
1
-1 i,
32
-1
-1 1
1 1
1 1
t l
33 i
-1 1
-1 i
-1
-1
-1
-1 i
l 34
-1 l
1
-1 l
-1
-1
-1 1
i i
i l
I 35
-1 i
1
-1
-1 l
-1 1
-1 l
I 36
-1 i
1
-1
-1
-1 1
1 4
'l l
37
-1 1
-1
-1 1
-1
-1 i
38 l
-1 1
-1
-1 1
-1 1
39
{
-1 1
-1
-1 1
1
-1 I
I_-- - _ _._
L-
- -- _l 8
t..._.--u....._.
- channel nut:.5ers refer to Fig. 3-5 see Table 4-1 for coded rhietionships Note: Coded values determined by methods described in Reference (A-1).
4
,Tp t. :. A;:1 :
Co;!cd Set of Detailed T0!;C Cases tknl to Generate Response Surface (con't)
A-4
Case, -
_ -It1!;t.F
.._...c.
1
! Entha8py Systeattic Systel:+atic,
C.unne s [_1p]u Fac tors _ { C!:aone' [ ].
4.1.....e : [
, Ciad 0.D. ;l Lu.i ber j _Ciag:p 1T.3 i
]l !!ise I act..r Piv..h t
l
-1 1
-1
-1 l
1 1
1 40 41 i
-1 1
-l i
1
-1
-l
-l I
l i
42
-l 1
-1 1
-1
-1 1
l 43
-1 i
1
-1 1
-1 I
1
-1 l
1 44
-1 1
-1 1
-1 1
'i 1
I I
45
-1 l
1
' -l 1
1
-1
-1 i
46 i
-1 1
-1 l
1 1
-1 1
l l
i 47
-1 l
1
-1 1
1 1
-1 l
i 48
-1 I
1
-1 1
1 1
1 49
-1
- 1 1
-1
-1
-1
-1 s
i 50 5
-l 1
1
-l
-1
-1 1
~
I 51
-1 1
1
-1
-1 1
-l l
52
-1 1
1
-1
.- 1 1
1 l
l i
._.. I
.-....- 1. -.--.....___i
- ch innci nu:abers ref er to Fig. 3-5 see Table 4-1.~or coded relotionships Note: Coded values determined by methods described in Reference (A-1).
Tcnie A-1:
Coded Set ct Detailcd TORC La,es Used to Gencrate P.cspo';:.e Surface (con't.)
A-5 I
~
I Case Inist Flo Enthalpy SyW.atic!sysiemaiic
-~
" thr.:h.:r. Cl.ar.nel [ [3 Clia:...ei [ ]y_lia_c_ tors
((~[!
' < e Fac t n-I'i t cl.
C1sd 0. ').
t lianm 1,[ ]l 3.t t ii.
1 I
53 l
-1 1
1 l
-1 1
-1
-1 54
-1 l
1 1
1
-1 1
-1 1
I 1
i i
i 55
-1 1
1
-1 1
1
-1 i
e i
56
-1 1
1 i
-1 1
1 1
i 57 l
-1 1
1 1
-1
-1
-1 1
58 I
-1 1
1 l
1
-1
-1 1
I I
1 i
59 I
-1 1
1 1
-1 1
-1 i
60 i
-1 1
1 1
-1 1
1 g
1 61
-1 1
1 1
1
-1
-1 I
62
-1 '
1 1
1 1
-1 1
l l
}
-1 1
1 1
~
1 1
i
-1 l
63
)
64 l
-1 1
1 1
1 1
1 65 1
-1
-1
-1 l
-1
-1
-1
..L_...
.-.I_.._---..
3.-
't.liannel nw!wr:, refer to 1 ig. 3-5 see Table 4-I for coded relationships Note:
Coded values determined by methods described in Reference (A-1).
] fab _le A-1,:
Coded Set of Detailed '10RL Cases lised to Generale Response Surf ace (con't.)
i A-6
Case I,_.
~~'
i0.i[tfF5 Chei'iel [ _3 Chain.1T 1:.Wtprs
- ~ ~~~~ ~} Er.t.ha1py Sy te..iatic lsysici. otic]
~
cnaina.1El on.:ini. E ); nir.. r. tor P: tei.
i claa o.u.
!!y.iper.1 66 t
1
-1
-1
-1
-1
-1 1
l 67 1
-1
-1
-l
-l 1
-1 1
8 68 1
-1
-1
-1
-1 1
1 1
l 69 1
-1
-1
-1 1
-1
-l 70 1
-1
-1
-1 1
-1 1
71 1
-1
-1
-1 1
1
-l 72 1
-1
-1
-1 1
1 1
73 1
-1
-1 1
-1
-1
-l 74 i
1
-1
-1 1
-1
-l l
1 I
75 l
-1
-1 1
-1 1
-l I'
76 1
-1
-1 1
-1 1
1 77 1
-1
-1 1
1
-1
-1 78 f
1
-1 1
-1 1
1
-1 1
L- -.. ali t
....- --....... ~.
- channel i: umbers refer to rig. 3-5 see Ta:21e 4-1 for coded rel,.itionships Note:
Coded values determined by methods described in Reference (A-1).
Iable A-1:
Coutd 5et of Detailed TORC Cases trsed to Generate Response Surf ace (con't.)
A-7
~
Case Inlet F_loy Factors Entfalpy Systei.ct i c S.v stema tic '
t'v i.or.y, C anncl[_, j '
Liv.nnt 1[, ) l Ch.:nne i[_ _ J,l Ch.:.ne.[..., J, l'.i w Fa c t or Pitch i Clad u.I'.
79 1
-1
-1 1
1 1
1
-1 l
n 80 1
-1 1 1
1 1
81 1
-1 1
-1
-1
-1
-1 82 1
-1 1
-1
-1
-1 1
83 1
-1 1
-1
-1 1
-1 84 1
-1 1
-1
-1 1
1 i
i 85 l
1
-1 1
-1 1
-1
-1 j
86 l
1
-1 1
-1 1
-1 1
i 87 1
-1 3
1
-1 1
1 i
-1 88 1
-1 1
-1 1
1 1
e i
89 1
-1 1
1
'1
-1
-1 i,
i' 90 l
1
-1 1
1
-1
-l 1
91 I
1
-1 1
1
-1 1
-1
.}
1
___.____.L
.u
- thencel nut.Sers refer to Fig. 3 -b see T. ole 4 -l for coded rdictionships Note:
Coded values determined by method described in Reference (A-1).
Table A-1:
Coded Set of Detailed TORC Cases Used te Genereite Response Surface (cont.)
A-8
7-
_ Inlet f Factors
~]'rnthalp~v ~ ' 5ystbEit Systematic Case I Civ nnel [ _]__I:ha.u:elT. lot:J.l _;i o n I I ]
( o.... i.[-
] i:i..e ractor ritcn
. C i c
st.. Table 4-1 for coded relationiships Note:
Coded values determined by method described in Reference (A-1).
l e 5 l e_ A - 1 :
Cie :i Set of Iletailed TORC Cr.:,es Used to Cenci ctc l'esponi,e Surtote (con't.)
A-9
- u::6e.- [ct.mnr. : [
8
[__'
_ ~ tit 17t_7 li C: anni. [.h{fitjEC]~.
~~
i r.n tiral,y l */.t..a;ie 4j:,, a:a tic' Co 5.e i
J I _ g 9 93 i.[
. ct. ee : [
] !!i:- Faiiar,;
l'i tch Ciad fi.D.
I 105 1
1
-1 1
-1 l
-1
-1 1
I 106 l
1 1
-1 1
1
-1
-1 1
i t
l t
e 107 1
1
-1 1
-1 j
1 1
-1 j
i 108 1
1
-1 1
-1 1
1 I
i I
i 1
1
-1
-1 109 1
1
-1 l
I I
i 110 1
l 1
-1 1
1
-1 1
I i
111 1
l 1
-1 1
1 1
i
-1 i
1 112 l
1 1
-1 1
1 1
l 1
l 113 1
1 1
-1
-1
-1
-1 i
174 1
1 1
-1
-1
-1 1
?
I e
e 1
115 1
1 1
-1
-1
- 1 I
116 i
1 l
1 1
-1
-1 1
1 i
j 117 1
1 i
1
-1 1
l
-1
-1 i
-L-.__.._..]
L __..[
.{.._. _
.._I.__-..._.._.._....
l ___ ~
i
% lunnel n':cerr. refer te Fig. 3 5 see Table 4-1 for coded relat.ionships Note:
Coded values determined by method described in Reference (A-1).
Tchie A-1:
Cv:;ed se:. of vetailed TURC Cas.es U::cd to Cri.2: rate I:esponse Surface (con't.)
A-10
Chr.nnei[
J _,rjiar..id{[l ejyiy v1[-],..,,,.,[_], y:
_ _.. _... _ _ _sy r. te...L t i c______l S ysi.rua t u !
Ccce t-In1r t lir., Factors i g n :.1.,,1 ;.,,
!!urt.er.
y_ 3, s g.. _, i:. tc];. _. ~ [C_1p] 0.y. !
118 1
1 1
-1 1
-1 1
i s
119 1
i 1
1
-1 1
1
-1 1
l I
120 1
1 1
-1 1
1 1
i 121 1
1 1
1
-1
-1
-1 i
t 122 1
1 1
1
-1
-1 i
1 i
i l
123 l
1 1
1 1
-1 1
1
-1 t
I t
i 124 j
1 1
1 1
-1 l
1 1
i 125 i
1 l
1 1
1 1
-1
-1
,i 126 1
1 1
1 1
-1 i
1 l
l l
i 127 1
1 1
i 1
1 1
-1 t
I t
e 128 1
1 1
1 1
'l 1
1 f
I i
i 0
0 0
0 0
^
129 i
0 0
l t.
130 1.91 0
0 0
0 0
0 i
__l
__. _ _ l. _. _..._ _ _ __ _.l...
i
_.. __ j
.a _
- c h a nnr-) num!urs re:cr to Fig. 3 - 5 see lo!,le 4 -1 for coded reiotient. hips Note:
Coded values determined by method described in Reference (A-1).
.lt@le.. A : :
Caut d Set of 1;etailed T04. Car.o. Used to tien.. a.e kusuonte Surface (con't.)
A-11
e b
S t~
- ~
~* ~ = ~ *-.-.
.'J,- O. I
+
i' I
t e'
c.
9 9
y 5 *O o
o o
o o
o o
o o
o o
- ol.
q a
an I
2%.
a O
e
~ ~. ~. -
~--.
. ~.
g.
M e.
,6
.'.3 9"*
.]
(
i i
8 j3o o
o o
o o
o o
o a
o u
l L
s o
o
.,I ve i
. i.
s e
,. 0;
"#I att
'M n
, M P=
P=.
I.U ' a. g Q
Q e' "2 s.
.. ' o o
o o
o o
o o
o o
o
=
t=3 a
i t.: !
a s
e a s
e.
l a
v O
~J t
m m
O ;t
.i oc.>
G t
w L
PA N
4
- i..*
lJ. :
r=
e.=
y c.
.c O.
m o
o o
o o
o o
o o
o o
h ~-
1 i
e g
l *.5[
I
.C.
-r.
r
- g. --- - -
=
e W.
l
.Q Co l
t I
't.,J
- r-L
~.*
nr.-
U
$ o.'
~
-C C*
o O
S =!
o o
o o
o o
o o
o o
o g *v, S
.. h
'O
.J O
. ~g'~'
, ~;
c :?
M ---
s -. m.
I
! f l
W h..
e
- g _s C
.O L
4.-
c O
Q
- p_e lh,
e."
O o
o o
o o
o o
o o
I l-E.o L:
lg c,
4.8 C
j b
h L
T F w
J l
g
".' t a
V M A e
t y
I L
W m
l 1;
.' *1
' E;.
f o
o o
o o
o o
o o
o o
o 3
> 3
- i C
U "O
g
,p.q iO V
.-i 3 =
0 h j-y -
I C
U N
m o
m m
N m
m o
g g
c: n m m
m m
m m
m m
m 2
O _.},.
~~
e_. '
'~
r~
r=
r=
e y
A O
U A,
...0 o.;
l=
E l-= }
4
" e
t e
case
_ t e.3n..( grg. gig y n.inneT ~ }lRisoractor Pitcha*
c141 0.n.**
- *'v e.it t:0 Oi:
pa s i r..*
- inlet rinu tactor I Entfulpy Systetsa tic Sys tenu t ic l'et iled Itn C Hesponse TO;C r:.. *i..e r n in...(
j i-I
.9882
.5763
.C00 2
g
.9882
.5763
.004 l
3
.9892
.5788
.003 I
4
.9882
.5788
.002 5 l 1.012
.5763
.003 3
6 1.012
.5763
.C04 7
1.012
.5788
.002 8
1.012
.5788
.002 l
9
.9882
.5763
.C00 10
.9882
.5763
.004 1
11 i
.9882
.5788
.003 t
i 12 #
.9882
.5788
.032 13 1.012
.5763
. C ')
14 1.012
.5763
.004 l
l 15 I
i 1.012
.5788
.002 jL i
_...t..-...
i l
=
- ct.u.. te. i.L.. r u t. re fer to f ig. 35 a at : 1.ysto.. ;. i. :.:e. r. d.s., en.i t.r.i ess c.n. p t % s te...ti c, i t. h end cl.:s; o.u. (in:fus)
TABLE A-2: Cmparison of TURC ant! Response Surface fu!CR for C4% 4 U* ist tu 'Cenerate Response Lv i.ict:
A-13
.~
"""t J ' Riw_ rgene l Systematic l Systematic IEnthalpy lsetailed 10ac Hespor.se ici<c inlet _ri_ow Factor e
case
_ ii ru.nn..q
[,1 ounne:L1 r.
i ir:im kriv.re p.4 4
- 1., -
ejech -
l g,i o.o. -
r.w t.,. c i ei,rn.. il 16 l.0120
.5788
.001 17
.9982
.5763
.003 13
.9882
.5763
.000 19
.9882
.5788
.002 20
.9882
.5783
.C32 21 1.0120
.5763
.033 22 1.0120
.5763
.001 23 1.0120
.5788
.003 24 1.0120
.5788
.001 25
.9882
.5763
.001 26 e
.9882
.5763
.002 i
i 27
.9882
.5783
.033 28
.9382
.5788
.001 29 1.0120
.5763
.0:2 33
~
1.0120
.5763
.032 l
i l
t t
nur. ers refer to fiel. 3-5
- ill >yst m pe,.,% ter s dii.e.icionless cuent sys tenutic p e u.h
- t t.a r.i;..
a and clari 0.il. (inclies; TABLE A-2: Co:ar.arison of It:RC ar.d Responte Surface lib;iiR for Cases U;ed to Gencrete Respo<ise Surf ace (con't.)
A-14 i
1
~
Lnthalpy Systers tic Systematic beteilec IOxC Response func e
i Lasc I Inlet flqw Factor
__g tn,inne.] l i Rise Factor Pitch ** _.,riaet 0.D.**
r.ttr:!;R flit:3it Pr u e.n l i r:.e re i tn...nnij pg.>.... e [.
j, o..,ic t [
a j
F i
=
i 31 1.0120
.5788
.003 I
32 1.0120
.5788
.001 Ii 33
.9882
.5763
.000 34
.9882
.5763
.004 35
.9882
.5788
.033 36
.9382
.5788
.001 37 1.0120
.5763
.C03 38 1.0120
.5763
.004 39 1.0120
.5788
..032 40 1.0120
.5788
.002 1 41
.9832
.5763
.001 1
42
.9882
.5763
.C03 43
.9882
.5788
.002 44 l
.9382
.5788
.032 45 1.0120
.5763
.001
~
_.--...4
- ._._ i I
- een :<. nueters refer to liJ.1-5
- all
.yst... parmee.. tie.ensionicss eu e:.t syst _*a t ic pi tu.
and clad 0.D. (tru b:s)
TABLE A-2:
Ccnparison of 10l:C and Respunte Surface !3d.R fur ( ses th.ed to Cent rate Response Surface (con't.)
A-15
o case I Inlet _F1pt factor _ _
knthalpy Sys tema tic by5temat1C Detailed lORL Ecsponse IOdC l f:or 5 r i o. ann.it J1 char. reg q,s'.hant.. g Jl triane,icil I Rise Factor Pi tch"
- r. t a,I 0.D. "
MT:3R I't**2.R I
P. 4eemt
~
~
46 1.0120
.5763
~
.003
- 47 1.0120
.5788
.002 48 1.0120
.5788
.C02 49 i
.9882
.5763
.002 50
.9882
.5763
.002 51
.9882
.5788
.003 52
.9882
.5788
.002 53 1.0120
.5763
.003 54 1.0120
.5763
.002 55 1.0120
.5788
.003 g
i 55 i
1.0120
.5788
.001 i
57 l
.9882
.5763
.002 58
.9882
.5763
.002 l
59
.9882
.5788
.004 60
.9882
.5783
.000 I
I
.J t___
._ l
- c.e a nna. eu Nr. refer to fig. 3.S
- *all.ysta s.: paranvrters di.censionless except syste.natic piti h and clad 0.D. (im he<.)
I TABLE A-2:
Lomparisc,n of T0!'C and Respon.e Surf ace IFiult for Cases t!'.e d to Generate l'esponse Surface (con't)
A-16
y e
case inact Fim tactor Enthalpy Systeinatic l Systematic Detailed 10,(L Hesponse Icec Rise Factor Pitch **
I Cla.1 0.D.**
MotDR fin mu 9.s i ren41 Ji rn. inn,.6 F li cnann..I r } tl usannei4. -
t:u,6 r rnment I; i
61 1.0120
.5763
.003 62 1.0120
.5763
.002 63 1.0120
.5788
.003 G4 1.0120
.5788
.001 65
.9882
.5763
.002 66
.9882
.5763
.006 67
.9882
.5788
. Col 68
.9882
.5788
.005 i
1.0120
.5763
.002 69 70 1.0120
.5763
.006 e
71 1.0120
.5788
.001 l
f 72 1.0210
.5788
.c02 1
73
.9832
.5763
.003 ll 74
.?882
.5763
.005
.9822
.5788
.000 75 g
~
.- a e_
I l
m oi.i..:s r uai?:ce s rever to f ig.3 5
- a ll.yst<.t pa.aw tu, tii.cn. ion vess empt
.y. ten itic pitt.h and cl.ed 0.11 (inch..)
TABLE A-2: Louparison of TORC at:d Response Surface HN::!t for Cas.s U.ed to (,enerate. IRsponse surf ace (con't.)
A-17
=
y 5
e case inlet Flou Factor Enthalpy Systenatic Systematic Detailed TuxC Resporise 10RC nise Factor Pitch **
C ad 0.0.**
MfYMR Mr'*.t'k I.a. s i du a l n.e-t.a r f.n..nne ll I i rnanneir q_cnannair ] l (.hannnig j 76
.9882
.5788
.005 77 1.0120
.5763
.003 78 1.0120
.5763
.005 79 1.0120
.5788
.000 80 1.0120
.5788
.004 81
.9332
.5763
.031 82
.9882
.5763
.003 83
.9882
.5788
.002 84
.9832
.5788
.CO2 85 1.0120
.5763
.000 86 1.0120
.5763
.003 1
87 1/1120
.5788
.001 88 1.0120
.5788
.001 89
.9882
.5763
.000 g
90
.9882
.5763
.003 l
l I
I I
- c.hanne s nun.bers refer to Fig. 3-5
- all ystem 1, rawter. dis.d:er:loniess except systceatic pitcle a ni' clad 0.D. (ir hes)
TABLE A-2: Co.:.parison of TORC and Response Surface F*ell3R for Cases Used to Cencrate Response in
- (con't.)
A-18
3
~
7 i
e cose inlet flow Factor i Ent alpy Systema tic Sy,te:ra t ic Detailed 10dC r.csponse TORC Pitch **
Clad 0.D.**
TETA t:D*:ER t'esi v1 f:. '.' r Ch.'real F. ll Oi.or.-l f 1.i Ch.in.u Ir 41 tnannoir 3RiseFactne s.
i 6
i 91
.9882
.5788
.002 92
.9882
.5789
.002 93 1.0120
.5763
.001 94 1.0120
.5763
.004 95 1.0120
.5788
.002 S6 1.0120
.5788
.C02 97
.9882
.5763
.003 93
.9882
.5763
.005 99
.9082
.5788
.C:0 109
.9882 H
.5788
.005
' 101 1.0120
'763
.003 1.0120
.5763
.C04 102 103 1.0120
.5788
.001 104 1.0120
.57S8
.004
.9882
.5763
.004 105
~
_J
.. L.. _ _
_t
'<.: antic. rebers re:fer to fig.
3-5
- all.yste. ; ra:... crs Ji.....-i..iess except systenatic pitch and e: lad i.0. (inches)
TA8LE A-2:
Cu:parison of TO::C and Response Surface M:1!!BR f..e Co.s 0..d to Cencrat Response Surface (con't.)
i A-19
. _. =
e
'1 Inle: 11cw Factor __
t.n tha l py Systematic Sys te:'u tic Octatlec 10kt Response iuRC e
s.e se I
_.[i cnai.no t ( j i O. enc I [ g o.annesl
't i ca Factor Pitch **
C14d 0.D.'*
fW:0P.
- n*
- 't a 0-3 sir't 31
- h-* rj i. v..aa. l.
i 106
.9882
.5763
.004 107
.9882
.5788
.002 103
.93S2
.5783
.004 103 1.0120
.5763
.005 g 110 1.0120
.5763
.004 111 1.0120
.5788
.001 112 1.0120
.5788 l
.C03 113
.9882
.5763
.003 114
.9882
.5763
.004 115
.9382
.5788
.003 115
.9882
.5788'
.002 117 1.0120
.5763
.031 118 1.0120
.5763
.005 119 1.0120
.5788
.003 1
120 1.0120
.5738 j
.002
_ i.
u
' L t.. rnee rius.ifiers refer to l'ig. 3-5
.ill yster para:ic Lers dir.4.'n.lotiicss c:. cept systematic pitch or.d clad 0.D. (inclies)
TAE!.E A-2: Cc ;)arison of TORC nd Response Surface MDNBR for Cases used to Co.erate Responte 'urface (con't.)
A-20
4
_ j Enthalpy Systeca tic syste.atic i uctatico iesc l aesponse iusc e
s.a e InletJIcw factor _
_l l t succoil. f Rise Fector Pitch **
Clad 0.D. " _l.
2:"R I
f"nT.R Pa* i.* n1 t:,r i r
( +wim i t 4g_u..etL J l cnar..u I ta a
.9882
.5763
.001 121
.005
.9882
.5763 122
.002
.9882
.5788 123
.032
.9882
.5788 124
.001 1.0120
.5763 125
.005 1.0120
.5763 126
.C32 1.0120
.5788 127
.002 1.0120
.5788 123
.002 1.0000
.5776 129
.010 1.0000
.5776 130
.010 1.0000
.5776 131 i
.001 1.0000
.5776 132
.001
'.f000
.577f 133
.007
'. 0000
.5776 134
.006 1.0000
.5776 135
~
I i.
..1 I
... ~
- ill systen. paranciers dli..ensionless exccpt systematic pitch
- e r :.r.<. : nuie.>cr5 refer to Fig. 3-5 end clad 0.D. (inc.hes)
TABLE A-2: Comparison af T6kC and Responic Surface I:..:;flR for Cases Used to Cencrate Response Surface (con't.)
A-21
q inlet Flots factor Enthalpy Systematic dystematic Detailed 10kt Response 10i.C e
I t.o c I t
Pitcb**
Cl.id 0. D. "
ItD':9R I D.;';i pat u. -)
_1.; L'i. inn 4 F M, elannoir pr aannai[, g Rise Factor I ;,.., r i uirna. L
.001 1.0000
.5776
~
135
.001 1.0000
.5776 137
.001 1.0228
.5776 133
.CC0
.9774
.5776 139
.012 1.0000
.5799 140
.011 1.0000
.5752 14;
.0J5 1.0000
.5776 142
.005 1.0000
.5776 t
143 1
l i
l
- ct:an ici nu. bers refer to Fig. 3-5
- all ystcia psiosu ters dit:ension'iess except 'spie:. tic pitch and clad 0.D. (incties)
TABLE A-2: Cr.:.perir..,n of TORC aiJ Responsr Surface H!'::0R for Car.es used tai Generate Resinnse Surface (con't.)
A-22
,