ML20099A090
| ML20099A090 | |
| Person / Time | |
|---|---|
| Issue date: | 03/31/2020 |
| From: | Geelhood K, Lucas Kyriazidis, Luscher W, Ian Porter, Edgardo Torres Office of Nuclear Regulatory Research, Pacific Northwest National Laboratory |
| To: | |
| M. Bales | |
| Shared Package | |
| ML20099A087 | List: |
| References | |
| DE-AC05-76RL01830 PNNL-29728 | |
| Download: ML20099A090 (136) | |
Text
PNNL-29728 MatLib-1.0: Nuclear Material Properties Library Developed under NQA-1-2017 March 2020 KJ Geelhood, PNNL WG Luscher, PNNL IE Porter, NRC L Kyriazidis, NRC CE Goodson, PNNL EE Torres, PNNL Prepared for the U.S. Department of Energy Under contract DE-AC05-76RL01830
PNNL-29728 MatLib-1.0: Nuclear Material Properties Library Developed under NQA-1-2017 March 2020 KJ Geelhood, PNNL WG Luscher, PNNL IE Porter, NRC L Kyriazidis, NRC CE Goodson, PNNL EE Torres, PNNL Prepared for the U.S. Department of Energy Under Contract DE-AC05-76RL01830 Pacific Northwest National Laboratory Richland, Washington 99352
PNNL-29728 Abstract The U.S. Nuclear Regulatory Commission (NRC) uses the computer code Fuel Analysis under Steady-state and Transients (FAST) to model steady-state and transient fuel behavior to support regulatory decisions. FAST relies on a material properties library (MatLib) that contains the thermal and mechanical properties of the nuclear materials and coolants of interest to support the US commercial nuclear industry. MatLib contains properties for a variety of nuclear fuels, cladding and other structural materials, gases, and coolants.
In this document, material property correlations for the materials contained within MatLib are pre-sented and discussed. When available, comparisons are made between the material property cor-relations and available data. Additionally, uncertainties are quantified on the material properties, which is then used by the NRC to support uncertainty quantification for best-estimate plus uncer-tainty safety evaluation reviews.
This document describes MatLib-1.0, which is the first official version of the MatLib library. This document is one of a series of documents on FAST; the other documents detail the models used by FAST as well as its integral assessment to experiments and commercial data.
Abstract iv
PNNL-29728 Foreword The U.S. Nuclear Regulatory Commission uses the computer code FAST to model steady-state and transient fuel behavior to support regulatory analyses. To effectively model fuel behavior, material property correlations applicable to a wide range of operating conditions (e.g., temperature and burnup) must be available. In this sense, a material property is a physical characteristic of the material whose quantitative value is necessary in the analysis process.
The consolidated resource for material properties cited most often in the literature is MATPRO
[Siefken et al., 2001]. MATPRO is a compilation of fuel and cladding material property correlations with an extensive history of use with fuel performance and severe accident codes. Since 2001, MATPRO has not been updated despite recent advances in understanding of high burnup material properties and recent evolutions in cladding alloys and fuel types. These updates were documented as part of the FRAPCON [Geelhood et al., 2015b] and FRAPTRAN [Geelhood et al., 2015a] codes in a material property handbook [Luscher et al., 2015]. These codes were the predecessor to FAST
[Porter et al., 2020a].
The primary purpose of this report is to document the current material property correlations used by FAST. Documentation includes the mathematical formulas, comparisons to available data, range of applicability, and model uncertainty.
Historically, FRAPCON and FRAPTRAN were applicable solely to commercial BWRs and PWRs with oxide fuel (UO2 and (U,Pu)O2) and zirconium-alloy cladding (Zircaloy-2, Zircaloy-4, M5TM, ZIRLO and Optimized ZIRLOTM). In order to be applicable to future reactors and fuels, currently available material properties for new fuels (uranium metal alloys), claddings (FeCrAl and HT-9),
and coolants (liquid sodium) are included in the MatLib library.
Unlike the UO2-Zr-alloy system, which has a long irradiation history, the development of advanced fuels and materials is ongoing and irradiation data are sparse. Consequently, the applicable ranges of these advanced fuel systems is smaller and the uncertainty is greater. Nevertheless, these cor-relations, supporting data, range of applicability, and uncertainties are documented here.
Foreword v
PNNL-29728 Acronyms and Abbreviations BWR Boiling water reactor CRUD Chalk River Unidentified Deposit FAST Fuel Analysis under Steady-state and Transients LWR Light water reactor MatLib Material Properties Library MOX Mixed oxide MTU Metric ton of uranium NFI Nuclear Fuel Industries NRC U.S. Nuclear Regulatory Commission O/M Oxygen-to-metal PNNL Pacific Northwest National Laboratory PWR Pressurized water reactor TD Theoretical density PuO2 Plutonium oxide Acronyms and Abbreviations vi
PNNL-29728 Contents Abstract................................................
iv Foreword...............................................
v Acronyms and Abbreviations....................................
vi Contents...............................................
vii Figures................................................
xi Tables................................................. xiii 1.0 Introduction...........................................
1 1.1 Objective of MatLib 2
1.2 Relation to Other Reports...............................
2 2.0 Fuel Material Properties....................................
5 2.1 Oxide Fuel Properties (UO2, (U,Pu)O2).......................
5 2.1.1 Thermal Conductivity............................
5 2.1.2 Specific Heat Capacity and Enthalpy...................
11 2.1.3 Melting Temperature 15 2.1.4 Thermal Expansion.............................
16 2.1.5 Emissivity 20 2.1.6 Density....................................
22 2.1.7 Densification.................................
23 2.1.8 Swelling...................................
25 2.2 Metallic Fuel U-Pu-Zr Material Properties......................
30 2.2.1 Thermal Conductivity............................
30 2.2.2 Specific Heat Capacity...........................
31 2.2.3 Density....................................
33 2.2.4 Melting Temperature 33 2.2.5 Eutectic Temperature............................
34 2.2.6 Thermal Expansion.............................
34 Contents vii
PNNL-29728 2.2.7 Emissivity 35 2.2.8 Swelling...................................
35 3.0 Cladding Material Properties.................................
37 3.1 Zirconium-based Alloys................................
37 3.1.1 Thermal Conductivity............................
37 3.1.2 Specific Heat 39 3.1.3 Melting Temperature 42 3.1.4 Thermal Expansion.............................
42 3.1.5 Emissivity 46 3.1.6 Density....................................
47 3.1.7 Youngs Modulus and Shear Modulus...................
48 3.1.8 Meyers Hardness..............................
51 3.1.9 Axial Growth.................................
52 3.1.10 Strain (Creep) Rate.............................
58 3.2 Iron-Chrome-Aluminum (FeCrAl) Alloys.......................
63 3.2.1 Thermal Conductivity............................
63 3.2.2 Specific Heat 66 3.2.3 Melting Temperature 69 3.2.4 Thermal Expansion.............................
69 3.2.5 Emissivity 72 3.2.6 Density....................................
73 3.2.7 Youngs Modulus and Shear Modulus...................
73 3.2.8 Meyers Hardness..............................
75 3.2.9 Axial Growth.................................
75 3.2.10 Strain (Creep) Rate.............................
76 3.3 HT-9 Alloy 78 3.3.1 Thermal Conductivity............................
78 3.3.2 Specific Heat Capacity...........................
80 Contents viii
PNNL-29728 3.3.3 Melting Temperature 81 3.3.4 Thermal Expansion.............................
81 3.3.5 Emissivity 83 3.3.6 Density....................................
84 3.3.7 Youngs Modulus 84 3.3.8 Shear Modulus 85 3.3.9 Meyers Hardness..............................
86 3.3.10 Strain (Creep) Rate.............................
86 3.3.11 Yield Stress.................................
88 4.0 Gas Material Properties....................................
90 4.1 Thermal Conductivity.................................
90 4.1.1 Model Description..............................
90 4.1.2 Comparisons to Data............................
92 4.1.3 Applicability and Uncertainty........................ 100 5.0 Oxide/CRUD Material Properties............................... 101 5.1 Zirconium Dioxide (ZrO2)............................... 101 5.1.1 Thermal Conductivity............................ 101 5.1.2 Specific Heat Capacity........................... 102 5.1.3 Melting Temperature
............................ 103 5.1.4 Density.................................... 103 5.2 CRUD.......................................... 103 5.2.1 Thermal Conductivity............................ 104 5.2.2 Specific Heat Capacity........................... 104 5.2.3 Density.................................... 104 6.0 Fluid Material Properties
................................... 105 6.1 Water.......................................... 105 6.2 Sodium......................................... 106 Contents ix
PNNL-29728 6.2.1 Thermal Conductivity............................ 106 6.2.2 Viscosity................................... 106 6.2.3 Density.................................... 107 6.2.4 Specific Heat Capacity........................... 108 6.2.5 Enthalpy................................... 109 6.2.6 Melting Temperature
............................ 110 6.2.7 Vapor Pressure............................... 110 7.0 References........................................... 112 Contents x
PNNL-29728 Figures 2-1 Model-to-data Comparison for Unirradiated UO2 Thermal Conductivity Correlation..
8 2-2 Model-to-data Comparison for Irradiated UO2 Thermal Conductivity Correlation 8
2-3 Model-to-Data Comparison for Unirradiated UO2-Gd2O3 Thermal Conductivity Cor-relation 9
2-4 Model-to-Data Comparison for Irradiated UO2-Gd2O3 Thermal Conductivity Correlation 9
2-5 Model-to-Data Comparison for MOX Thermal Conductivity Correlation.........
10 2-6 Model-to-Data Comparison for UO2 Specific Heat Capacity Correlation........
13 2-7 Model-to-Data Comparison for MOX Specific Heat Capacity Correlation........
14 2-8 Model-to-Data Comparison for UO2, PuO2, MOX, and UO2-Gd2O3 Melting Temper-ature Correlation.....................................
16 2-9 Model-to-Data Comparison for UO2 Correlation 18 2-10 Model-to-Data Comparison for PuO2 Correlation.....................
19 2-11 Model-to-Data Comparison for Emissivity of Oxide Fuel.................
21 2-12 Model-to-Data Comparison for Densification of Oxide Fuel 24 2-13 Model-to-Data Comparison for Solid Swelling Correlation................
27 2-14 Model-to-Data Comparison for Solid Swelling Rate Correlation.............
28 3-1 Model-to-Data Comparison for Zirconium-based Alloy Cladding Thermal Conductivity Correlation........................................
38 3-2 Model-to-Data Comparison for Zirconium-based Alloy Cladding Specific Heat Corre-lation 41 3-3 Model-to-Data Comparison for for Zirconium-based Alloy Cladding Circumferential Thermal Expansion Correlation.............................
44 3-4 Model-to-Data Comparison for for Zirconium-based Alloy Cladding Axial Thermal Ex-pansion Correlation 45 3-5 Model-to-Data Comparison for Zirconium-based Alloy Emissivity Correlation 47 3-6 Model-to-Data Comparison for Zirconium Alloy Cladding Youngs Modulus 50 3-7 Model-to-Data Comparison for Zirconium-based Alloy Cladding Meyers Hardness Correlation........................................
52 3-8 Model-to-Data Comparison for Zircaloy-2 Axial Irradiation Growth Correlation.....
54 3-9 Model-to-Data Comparison for Zircaloy-4 Axial Irradiation Growth Correlation.....
55 Figures xi
PNNL-29728 3-10 Model-to-Data Comparison for ZIRLO Axial Irradiation Growth Correlation......
56 3-11 Model-to-Data Comparison for M5TM Axial Irradiation Growth Correlation.......
57 3-12 Model-to-data Comparison for RXA Ziracloy Strain Correlation.............
61 3-13 Model-to-data Comparison for SRA Ziracloy Strain Correlation.............
62 3-14 Model-to-Data Comparison for KanthalAPMT FeCrAlAlloy Thermal Conductivity Cor-relation 64 3-15 Model-to-Data Comparison for C35M FeCrAl Alloy Thermal Conductivity Correlation.
65 3-16 Model-to-Data Comparison for C36M FeCrAl Alloy Thermal Conductivity Correlation.
65 3-17 Model-to-Data Comparison for Kanthal APMT FeCrAl Alloy Specific Heat Correlation 67 3-18 Model-to-Data Comparison for C35M FeCrAl Alloy Specific Heat Correlation.....
68 3-19 Model-to-Data Comparison for C36M FeCrAl Alloy Specific Heat Correlation.....
68 3-20 Model-to-Data Comparison for Kanthal APMT FeCrAl Alloy Thermal Expansion Co-efficient..........................................
71 3-21 Model-to-Data Comparison for C35M FeCrAl Alloy Thermal Expansion Coefficient Correlation........................................
71 3-22 Model-to-Data Comparison for C36M FeCrAl Alloy Thermal Expansion Coefficient Correlation........................................
72 3-23 Model-to-Data Comparison for FeCrAl Alloys Elastic Modulus Correlation.......
75 3-24 Model-to-Model Comparison for HT-9 Alloy Thermal Conductivity Correlations 79 3-25 HT-9 Alloy Specific Heat Capacity Correlation 81 3-26 Model-to-Model Comparison for HT-9 Alloy Thermal Expansion Correlations 83 4-1 Model-to-Data Comparison for Helium Thermal Conductivity Correlation 93 4-2 Model-to-Data Comparison for Argon Thermal Conductivity Correlation........
94 4-3 Model-to-Data Comparison for Krypton Thermal Conductivity Correlation.......
95 4-4 Model-to-Data Comparison for Xenon Thermal Conductivity Correlation........
96 4-5 Model-to-Data Comparison for Hydrogen Thermal Conductivity Correlation......
97 4-6 Model-to-Data Comparison for Nitrogen Thermal Conductivity Correlation.......
98 4-7 Model-to-Data Comparison for Steam Thermal Conductivity Correlation........
99 4-8 Model-to-Data Comparison for Gas Mixture Thermal Conductivity Correlation..... 100 5-1 Model-to-Data Comparison for ZrO2 Thermal Conductivity Correlation
........ 102 Figures xii
PNNL-29728 Tables 1-1 Roadmap to documentation of models and properties used in NRCs fuel performance code FAST........................................
3 2-1 Constants Used in UO2, Gd2O3, and PuO2 Heat Capacity and Enthalpy Correlations 12 2-2 Constants Used in UO2, UO2-Gd2O3, and PuO2 Solid-Phase Thermal Expansion Correlations........................................
17 2-3 Phase Transition Temperatures Used in the Specific Heat Capacity Correlations for U-Pu-Zr Metallic Fuel 32 2-4 Constants Used in the Thermal Expansion Correlations for U-Pu-Zr Metallic Fuel...
34 3-1 Example Heat Treatments and Cold Worked Conditions for Different Zirconium-Based Alloys...........................................
37 3-2 Interpolated Values for the Zirconium-Based Alloys Specific Heat Capacity Correlation 40 3-3 Interpolated Values for the Zirconium-Based Alloys Thermal Expansion Correlation 43 3-4 Cladding Cold Work Dependent Parameters for the Thermal and Irradiation Strain Rate Correlations 59 3-5 Nominal Composition of Various FeCrAl Alloys in Matlib.................
63 3-6 Constants Used in the FeCrAl Thermal Conductivity Correlation............
64 3-7 Constants Used in the FeCrAl Specific Heat Correlation.................
67 3-8 Constants Used in the FeCrAl Thermal Expansion Correlation.............
70 3-9 Densities of Various FeCrAl Alloys.............................
73 3-10 Constants Used in the FeCrAl Thermal Strain Rate Correlation.............
77 3-11 Constants Used in the HT-9 Specific Heat Capacity Correlation 80 3-12 Constants Used in the HT-9 Thermal Expansion Correlation 82 3-13 Constants Used in the HT-9 Thermal Strain Rate Correlation..............
87 3-14 Constants Used in the HT-9 Yield Stress Correlation...................
89 4-1 Constants Used in the Gas Thermal Conductivity Correlation..............
91 Tables xiii
PNNL-29728 1.0 Introduction The U.S. Nuclear Regulatory Commission (NRC) uses the computer code FAST to model steady-state and transient fuel behavior to support regulatory analyses. To effectively model fuel behavior, material property correlations must be used for a wide range of operating conditions (e.g., temper-ature and burnup). In this sense, a material property is a physical characteristic of the material whose quantitataive value is necessary in the analysis process. Further, the property may be used to compare the benefits of one material with those of another. Generally speaking, the material properties of interest in thermal-mechanical regulatory analysis of nuclear fuel behavior as per-formed by FAST are mechanical properties such as elastic modulus, yield stress, and creep rate and thermal properties such as thermal conductivity and specific heat.
In this report, the thermal and mechanical properties are included. Other characteristics of the ma-terial (e.g., fission gas release) are considered models rather than properties and are discussed elsewhere [Porter et al., 2020a]. The primary purpose of this report is to document the current material property correlations used in FAST. Material property correlations for oxide fuels, includ-ing uranium dioxide (UO2) and mixed oxide (MOX) fuels are described in Section 2.1. Throughout this document, the term MOX is used to describe fuels that are blends of uranium and plutonium oxides, (U,Pu)O2. The properties for UO2 with other additives (e.g., gadolinia) are also discussed.
Material properties for metallic fuel U-Pu-Zr are discussed in Section 2.2. Material property corre-lations for cladding materials of zirconium-based alloys, iron-based alloys, and HT-9 are described in Section 3.0. Material property correlations for gases used as fill gas are described in Section 4.0.
Properties for oxides and CRUD are described in Section 5.0. Coolant properties for sodium are described in Section 6.0.
In addition to describing the material property correlations used in the subroutines of FAST, this report also shows comparison to experimental data for each material property correlation. Because these correlations are semi-empirical or empirical, the applicability of the correlations is limited to the range of available data. Therefore, based on the data comparison, a range of applicability will be identified and model uncertainty will be given. Model uncertainty is given in terms of either an absolute standard error or a relative standard error. The standard errors are calculated according to the following equations.
abs =
sPn i=1 (xi xmodel)2 n 1 (1-1) rel =
sPn i=1 [(xi xmodel) /xmodel]2 n 1 (1-2)
- Where, abs = absolute standard error (same units as x) rel = relative standard error (fraction) n = number of data measurement xi = value of data point i (various units)
Introduction 1
PNNL-29728 xmodel = model prediction at conditions of data point i (various units)
A determination of which is used is made based on examining the trend of measured and pre-dicted values as a function of the independent variable of interest such as temperature or burnup. In some cases where data are sparse or it is not possible to calculate this standard error, engineering judgement is used to estimate a standard error.
1.1 Objective of MatLib The ability to accurately calculate the performance of light water reactor (LWR) fuel rods under long-term burnup conditions is a major objective of the reactor safety research program being conducted by the NRC. To achieve this objective, the NRC has sponsored an extensive program of analytical computer code development, as well as both in-pile and out-of-pile experiments to benchmark and assess the analytical code capabilities. Historically, the computer code developed to calculate the long-term burnup response of a single fuel rod was FRAPCON. Recently the transient temperature solution and various other transient models from FRAPTRAN have been added to FRAPCON and the resulting code, which is the next evolution of FRAPCON, is FAST. This report describes the material properties used in FAST-1.0.
1.2 Relation to Other Reports The full documentation of the steady-state and transient fuel performance codes is described in three documents. The basic fuel, cladding, and gas material properties used in FAST-1.0 are de-scribed in the material properties handbook (this report). The FAST-1.0 code structure and behav-ioral models are described in the FAST-1.0 code description document [Porter et al., 2020a]. The integral assessment of FAST-1.0 against steady-state and transient test data is given in the FAST-1.0 integral assessment document [Porter et al., 2020b]. Table 1-1 shows where each specific material property and model used in the NRC fuel performance code is documented.
Introduction 2
PNNL-29728 Table 1-1. Roadmap to documentation of models and properties used in NRCs fuel performance code FAST Model/Property FAST-1.0(a)
Fuel thermal conductivity MatLib Document Fuel thermal expansion MatLib Document Fuel melting temperature MatLib Document Fuel specific heat MatLib Document Fuel enthalpy MatLib Document Fuel emissivity MatLib Document Fuel densification MatLib Document Fuel swelling - solid MatLib Document Fuel swelling - gaseous MatLib Document Fission gas release FAST-1.0 Code Description Fuel relocation FAST-1.0 Code Description Fuel grain growth FAST-1.0 Code Description High burnup rim model FAST-1.0 Code Description Nitrogen release FAST-1.0 Code Description Helium release FAST-1.0 Code Description Radial power profile FAST-1.0 Code Description Stored energy FAST-1.0 Code Description Decay heat model FAST-1.0 Code Description Fuel and cladding temperature solution FAST-1.0 Code Description Cladding thermal conductivity MatLib Document Cladding thermal expansion MatLib Document Cladding Youngs modulus MatLib Document Cladding creep model MatLib Document Cladding specific heat MatLib Document Cladding emissivity MatLib Document Cladding axial growth MatLib Document Cladding Meyer hardness MatLib Document Cladding annealing FAST-1.0 Code Description Cladding yield stress, ultimate stress, and plastic defor-mation FAST-1.0 Code Description Cladding failure criteria FAST-1.0 Code Description Cladding waterside corrosion FAST-1.0 Code Description Cladding hydrogen pickup FAST-1.0 Code Description Cladding high temperature oxidation FAST-1.0 Code Description Cladding ballooning model FAST-1.0 Code Description Introduction 3
PNNL-29728 Table 1-1. Roadmap to documentation of models and properties used in NRCs fuel performance code FAST (continued)
Model/Property FAST-1.0(a)
Cladding mechanical deformation FAST-1.0 Code Description Oxide thermal conductivity MatLib Document CRUD thermal conductivity MatLib Document Gas conductivity MatLib Document Gap conductance FAST-1.0 Code Description Plenum gas temperature FAST-1.0 Code Description Rod internal pressure FAST-1.0 Code Description Coolant temperature and heat transfer coefficients FAST-1.0 Code Description Not Developed at PNNL Water-cooled, water-moderated energy reactor fuel and cladding models NUREG/IA-0164 Cladding finite element analysis model VTT-R-11337-06 (a) MatLib Document [Geelhood et al., 2020]
FAST-1.0 Code Description (this document) [Porter et al., 2020a]
NUREG/IA-0164 [Shestopalov et al., 1999]
VTT-R-11337-06 [Knuutila, 2006]
Introduction 4
PNNL-29728 2.0 Fuel Material Properties 2.1 Oxide Fuel Properties (UO2, (U,Pu)O2)
Material property correlations for UO2 and (U,Pu)O2 are described in the following sections. When indicated, some of the correlations also account for the addition of Gadolinia (Gd2O3) in the UO2 fuel pellet.
2.1.1 Thermal Conductivity The thermal conductivity of oxide nuclear fuel is modeled in MatLib as a function of five parameters:
1.
Temperature 2.
Composition 3.
Burnup 4.
Density 5.
Oxygen-to-metal (O/M) ratio 2.1.1.1 Model Description UO2 and UO2-Gd2O3 The thermal conductivity of 95% theoretical density (TD) UO2 and UO2-Gd2O3 is based on the model proposed by Nuclear Fuel Industries (NFI) [Ohira and Itagaki, 1997] and was modified to alter the temperature-dependent portion of the burnup and include a dependency on gadolinia content [Lanning et al., 2005]:
k95 =
1 A + gad + BT + f (Bu) + (1 0.9e0.04Bu) g (Bu) h(T)
+ C T 2 exp
D T
(2-1) h(T) =
1 1 + 396 exp (Q/T)
(2-2)
- Where, k95 = Thermal conductivity of 95% TD fuel [W/m K]
T = Temperature [K]
Bu = Burnup [GWd/MTU]
f(Bu) = Effect of fission products in crystal matrix (solution) = 0.00187Bu Fuel Material Properties 5
PNNL-29728 g(Bu) = Effect of irradiation defects = 0.038Bu0.28 h(T) = Temperature dependence of annealing on irradiation defects (Equation 2-2)
Q = Temperature-dependent parameter (Q/R) = 6380 [K]
A = 0.0452 [m K/W]
B = 2.46 x 104 [m K/W/K]
C = 3.5 x 109 [W K/m]
D = 16361 [K]
= Constant = 1.1599 gad = Weight fraction of gadolinia [unitless]
MOX The thermal conductivity of 95% theoretical density MOX is baesd on the model proposed by Nuclear Fuel Industries (NFI) [Ohira and Itagaki, 1997] and was modified to alter the temperature-dependent portion of the burnup and include a dependency on gadolinia content [Lanning et al.,
2005] and plutoniua content [Duriez et al., 2000]:
k95 =
1 A (x) + gad + B (x) T + f (Bu) + (1 0.9e0.04Bu) g (Bu) h (T)
+ Cmod T 2 exp
D T
(2-3)
- Where, k95 = Thermal conductivity of 95% TD fuel [W/m K]
T = Temperature [K]
Bu = Burnup [GWd/tHM]
f(Bu) = Effect of fission products in crystal matrix (solution) = 0.00187Bu g(Bu) = Effect of irradiation defects = 0.038Bu0.28 h(T) = Temperature dependence of annealing on irradiation defects (Equation 2-2)
Q = Temperature dependent parameter (Q/R) = 6380 [K]
x = 2.00O/M ratio A (x) = 2.85x + 0.035 [m K/W]
Fuel Material Properties 6
PNNL-29728 B (x) = (2.86 7.15x) x 104 [m/W]
Cmod = 1.5 x 109 [W K/m]
D = 13520 [K]
= Constant = 1.1599 gad = Weight fraction of gadolinia [unitless]
Density Adjustment All of the above models are adjusted for the fuel density (in fraction of TD) using the Lucuta rec-ommendation for spherical-shaped pores [Lucuta et al., 1996], as shown in Equation 2-4.
kd = 1.0789k95 d
1 + 0.5(1 d)
(2-4)
- Where, kd = Thermal conductivity adjusted for fuel density [W/m K]
k95 = Thermal conductivity of 95% TD fuel [W/m K]
d = Fraction of fuel TD, including as-fabricated and densification changes [unitless]
2.1.1.2 Comparison to Data Thermal conductivity data have been collected for UO2 from unirradiated samples [Ronchi et al.,
1999] [Lucuta et al., 1996] [Christensen et al., 1964] [Godfrey et al., 1964] [Bates et al., 1967]
[Gibby, 1971] [Weilbacher, 1972] [Goldsmith and Douglas, 1973] [Hobson et al., 1974] and irra-diated [Ronchi et al., 2004] [Carrol et al., 1994]. A comparison between these data for UO2 is presented in Figure 2-1 for unirradiated data and in Figure 2-2 for irradiated data. This comparison demonstrates good agreement between the correlation and the database within range 300 [K] to 2800 [K] and 0 to 90 [GWd/MTU].
Fuel Material Properties 7
PNNL-29728 0
1 2
3 4
5 6
7 8
9 Measured thermal conductivity [W/m-K]
0 1
2 3
4 5
6 7
8 9
Predicted thermal conductivity [W/m-K]
Measured=Predicted Ronchi et al. [1999]
Lucuta et al. [1996]
Christensen et al. [1964]
Godfrey et al. [1964]
Bates et al. [1967]
Gibby [1971]
Weilbacher [1972]
Goldsmith and Douglas [1973]
Hobson [1974]
Figure 2-1. Model-to-data Comparison for Unirradiated UO2 Thermal Conductivity Correlation 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Measured thermal conductivity [W/m-K]
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Predicted thermal conductivity [W/m-K]
Measured=Predicted Ronchi et al. [2004]
Carrol et al. [1994]
Figure 2-2. Model-to-data Comparison for Irradiated UO2 Thermal Conductivity Correlation Thermal conductivity data have been collected for UO2-Gd2O3 from unirradiated [Minato et al.,
2001] [Newman, 1982] [Amaya and Hirai, 1997] [Hirai and Ishimoto, 1991] and irradiated [Minato et al., 2001] [Amaya and Hirai, 1997] samples. A comparison between these data for UO2-Gd2O3 is presented in Figure 2-3 for unirradiated data and in Figure 2-4 for irradiated data. This comparison Fuel Material Properties 8
PNNL-29728 demonstrates good agreement between the correlation and the database within range 300 [K] to 2800 [K] and 0 to 50 [GWd/MTU].
0 1
2 3
4 5
6 7
Measured thermal conductivity [W/m-K]
0 1
2 3
4 5
6 7
Predicted thermal conductivity [W/m-K]
Measured=Predicted Minato et al. [2001]
Newman [1982]
Amaya and Hirai [1997]
Hirai and Ishimoto [1991]
Figure 2-3. Model-to-Data Comparison for Unirradiated UO2-Gd2O3 Thermal Conductivity Corre-lation 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Measured thermal conductivity [W/m-K]
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Predicted thermal conductivity [W/m-K]
Measured=Predicted Minato et al. [2001]
Amaya and Hirai [1997]
Figure 2-4. Model-to-Data Comparison for Irradiated UO2-Gd2O3 Thermal Conductivity Correla-tion Fuel Material Properties 9
PNNL-29728 Thermal conductivity data have been collected for MOX from unirradiated samples [Duriez et al.,
2000] [Philipponneau, 1992]. A comparison between these data for MOX is presented in Figure 2-5. This comparison demonstrates good agreement between the correlation and the database within range 660 [K] to 2800 [K] and O/M ratio of 1.95 to 2.0.
0 1
2 3
4 5
6 Measured thermal conductivity [W/m-K]
0 1
2 3
4 5
6 Predicted thermal conductivity [W/m-K]
Measured=Predicted Duriez et al. [2000]
Philipponneau [1992]
Figure 2-5. Model-to-Data Comparison for MOX Thermal Conductivity Correlation 2.1.1.3 Applicability and Uncertainty UO2 and UO2-Gd2O3 Applicability The thermal conductivity model (Equation 2-1) is applicable to the range of available data:
Fuel types: UO2 and UO2-Gd2O3 Gadolinia content: 0 to 10 [wt%]
Temperature: 300 to 2800 [K]
Rod-average burnup: 0 to 90 [GWd/MTU] for UO2; 0 to 50 [GWd/MTU] for UO2-Gd2O3 As-fabricated density: 90 to 98.6 [%TD]
Engineering judgment should be used if analysis outside of these ranges is needed.
MOX Applicability The thermal conductivity model (Equation 2-3) is applicable to the range of available data:
Fuel Material Properties 10
PNNL-29728 Fuel type: MOX Temperature: 660 to 2800 [K]
Rod-average burnup: 0 to 90 [GWd/MTU] (assumed to be the same as for UO2)
As-fabricated density: 90 to 98.6 [%TD] (assumed to be the same as for UO2)
O/M ratio: 1.95 to 2.00 Engineering judgment should be used if analysis outside of these ranges is needed.
Uncertainty The uncertainty of the correlation is given below for each fuel type as a relative standard error.
UO2: = 8.3%
UO2-Gd2O3: = 8.8%
MOX: = 7.8%
2.1.2 Specific Heat Capacity and Enthalpy The specific heat capacity and enthalpy of oxide fuel are modeled as functions of four parameters:
1.
Temperature 2.
Composition 3.
Molten fraction 4.
O/M ratio 2.1.2.1 Model Description The specific heat capacity and enthalpy of UO2, Gd2O3, and PuO2 are given by:
Cp =
K12 exp
T
T 2 exp
T
1
2 + K2T + Y K3ED 2RT 2 exp
ED RT
(2-5)
H =
K1 exp
T
1 + K2T 2 2
+ Y 2 K3 exp
ED RT
(2-6)
- Where, Cp = Specific heat capacity [J/kg K]
Fuel Material Properties 11
PNNL-29728 H = Enthalpy [J/kg]
T = Temperature [K]
Y = O/M ratio R = Universal gas constant = 8.3143 [J/mol K]
K1, K2, K3 = Constants (Table 2-1)
= Einstein temperature [K] (Table 2-1)
ED = Activation energy for Frenkel defects [J/mol] (Table 2-1)
Table 2-1. Constants Used in UO2, Gd2O3, and PuO2 Heat Capacity and Enthalpy Correlations Constant UO2(a)
PuO2(b)
Gd2O3 Units K1 2.967 x 102 3.474 x 102 3.1586 x 102
[J/kg K]
K2 2.43 x 102 3.95 x 104 4.044 x 102
J/kg K2
K3 8.745 x 107 3.860 x 107 0.0
[J/kg]
5.35285 x 102 5.710 x 102 3.480 x 102
[K]
ED 1.577 x 105 1.967 x 105 0.0
[J/mol]
(a) [Kerrisk and Clifton, 1972]
(b) [Kruger and Savage, 1968]
For a mixture of UO2, Gd2O3, and PuO2, the specific heat capacity of the solid is determined by combining the contribution from each constituent in proportion to its weight fraction.
The specific heat capacity of UO2 in the liquid state (Equation 2-7) was determined by [Leibowitz et al., 1971] and assumed to be valid for PuO2 in the liquid state.
Cp (liquid) = 503 [J/kg K]
(2-7)
When the material is partially molten, the heat capacity is determined similarly with a weighted sum of the solid and molten fractions.
2.1.2.2 Comparison Data Specific heat data have been collected for UO2 from unirradiated samples [Grønvold et al., 1970]
[Hein et al., 1968] [Leibowitz et al., 1969]. A comparison between these data for UO2 is presented in Figure 2-6. This comparison demonstrates good agreement between the correlation and the database up to about 2800 [K]. Beyond this temperature, the data begins to fall lower than the model. This is attributed to partial melting due to a non-uniform temperature distribution within the sample.
Fuel Material Properties 12
PNNL-29728 500 1000 1500 2000 2500 3000 Temperature [K]
0 100 200 300 400 500 600 700 Heat Capacity [J/kg-K]
MatLib [UO2]
Gronvold et al. [1970]
Hein et al. [1968]
Leibowitz et al. [1969]
Figure 2-6. Model-to-Data Comparison for UO2 Specific Heat Capacity Correlation Specific heat capacity data have been collected for (U0.8Pu0.2)O2 from unirradiated samples [Gibby et al., 1974] [Leibowitz et al., 1972] [Affortit and Marcon, 1970]. A comparison between these data for UO2 is presented in Figure 2-7. This comparison demonstrates good agreement with two of the data sets between the correlation and the database up to about the melting point of about 3000 [K]. The third data set is overpredicted above 2300 [K]. Since the Affortit results are known to be generally low in comparison to results from other investigators, the correlation is considered to be in good agreement with the experimental data.
Fuel Material Properties 13
PNNL-29728 500 1000 1500 2000 2500 3000 Temperature [K]
0 100 200 300 400 500 600 Heat Capacity [J/kg-K]
MatLib [MOX]
Gibby et al. [1974]
Leibowitz et al. [1972]
Affortit and Marcon [1970]
Figure 2-7. Model-to-Data Comparison for MOX Specific Heat Capacity Correlation 2.1.2.3 Applicability and Uncertainty The fuel specific heat capacity (Equation 2-5) and enthalpy (Equation 2-6) models are applicable to the range of available data:
Fuel types: UO2, UO2-Gd2O3, MOXrowc Gadolinia content: 0 to 10 [wt%]
Temperature: 300 [K] to the applicable melting temperature (see Section 2.1.3)
Rod-average burnup: No burnup dependence observed As-fabricated density: No density dependence observed Engineering judgment should be used if analysis outside of these ranges is needed.
The uncertainty of the correlation is given below for each fuel type as an absolute standard error.
The uncertainty of the pooled data appears to be relatively constant with temperature. Therefore, an absolute error is given.
UO2 and UO2-Gd2O3: = 26 [J/kg K]
MOX: = 28 [J/kg K]
Fuel Material Properties 14
PNNL-29728 The standard error of the UO2-Gd2O3 is assumed to be the same as that of UO2 based on the small fraction of Gd2O3 in UO2. When excluding the [Affortit and Marcon, 1970] data from the MOX comparison, the standard error is 9.6 [J/kg K].
2.1.3 Melting Temperature The melting temperature of oxide nuclear fuel is modeled in MatLib as a function of two parameters:
1.
Composition 2.
Burnup 2.1.3.1 Model Description The melting temperature of UO2, Gd2O3, and PuO2 is given by:
Tmelt = 3113.15 0.5Bu 4.8XGd2O3 5.41395XPuO2 + 7.468390 x 103X2 PuO2 (2-8)
- Where, Tmelt = Melting temperature [K]
XPuO2 = PuO2 content [wt%]
XGd2O3 = Gd2O3 content [wt%]
Bu = Burnup [GWd/MTU]
2.1.3.2 Comparison to Data Melting temperature data have been collected for UO2, PuO2, MOX, and UO2-Gd2O3 from unirra-diated and irradiated samples [Popov et al., 2000] [Yamada et al., 1999]. A comparison between these data for UO2, PuO2, MOX and UO2-Gd2O3 is presented in Figure 2-8. This comparison demonstrates good agreement between the correlation and the database within range of 0 to 100 [GWd/MTU] for UO2, PuO2, MOX and UO2-Gd2O3 up to 30% Gd2O3.
Fuel Material Properties 15
PNNL-29728 2600 2700 2800 2900 3000 3100 3200 Measured Melting Temperature [K]
0 500 1000 1500 2000 2500 3000 Predicted Melting Temperature [K]
Predicted = Measured UO2 PuO2 MOX UO2-Gd2O3 Figure 2-8. Model-to-Data Comparison for UO2, PuO2, MOX, and UO2-Gd2O3 Melting Tempera-ture Correlation 2.1.3.3 Applicability and Uncertainty The fuel melting temperature model is applicable to the range of available data:
Fuel types: UO2, PuO2, MOX, and UO2-Gd2O3 Gadolinia content: 0 to 30 [wt%]
Rod-average burnup: 0 to 100 [GWd/MTU]
As-fabricated density: No density dependence observed Engineering judgment should be used if analysis outside of these ranges is needed.
The uncertainty of the correlation is given below for all four fuel types as an absolute standard error.
UO2, PuO2, MOX, and UO2-Gd2O3: = 25 [K]
2.1.4 Thermal Expansion The thermal expansion of oxide nuclear fuel is modeled in MatLib as a function of three parameters:
Fuel Material Properties 16
PNNL-29728 1.
Temperature 2.
Composition 3.
Molten fraction 2.1.4.1 Model Description The thermal expansion of UO2, UO2-Gd2O3, and PuO2 is given by:
L/L = K1T K2 + K3 exp
ED kT
(2-9)
- Where, L/L = Linear strain caused by thermal expansion (equal to zero at 300 [K]) [unitless]
T = Temperature [K]
K1, K2, K3 = Constants (Table 2-2)
ED = Energy of formation of a defect [J] (Table 2-2) k = Boltzmanns constant = 1.38 x 1023 [J/K]
Table 2-2. Constants Used in UO2, UO2-Gd2O3, and PuO2 Solid-Phase Thermal Expansion Cor-relations UO2 and Constant UO2-Gd2O3 PuO2 Units K1 9.80 x 106 9.0 x 106
[1/K]
K2 2.61 x 103 2.7 x 103
[unitless]
K3 3.16 x 101 7.0 x 102
[unitless]
ED 1.32 x 1019 7.0 x 1020
[J]
For mixed UO2 and PuO2, the thermal expansion of the solid is found by combining the contribution from each constituent in proportion to its weight fraction.
The fuel thermal expansion model includes terms for partially molten and completely molten fuel.
However, these correlations are not well validated and their use is subject to greater uncertainty.
During melting, an expansion equal to a linear strain of 0.043 occurs. If the fuel is partially molten, the strain due to thermal expansion is given by Equation 2-10:
L/L0 = L/L0 (Tm) + 0.043fmolten (2-10)
Fuel Material Properties 17
PNNL-29728
- Where, L/L0 (Tm) = Thermal expansion strain of solid fuel from equations with T = Tm Tm = Melting temperature [K]
fmolten = Fraction of the fuel which is molten [unitless]
The correlation used to describe the expansion of entirely molten fuel is given by Equation 2-11:
L/L0 = L/L0 (Tm) + 0.043 + 3.6 x 105 (T (Tm + Tm))
(2-11)
The solid-to-liquid phase transition is isothermal only for pure UO2 or pure PuO2. For MOX, the transition occurs over a finite temperature range, denoted in Equation 2-11 by Tm.
2.1.4.2 Comparisons to Data Thermal expansion data have been collected for UO2 from unirradiated samples [Baldock et al.,
1966] [Grønvold, 1955] [Burdick and Parker, 1956] [Hagrman et al., 1981] [Martin, 1988]. A com-parison between these data for UO2 is presented in Figure 2-9. This comparison demonstrates good agreement between the correlation and the database from room temperature to the melting temperature (3000 [K]).
500 1000 1500 2000 2500 3000 Temperature [K]
0.00 0.01 0.02 0.03 0.04 0.05 0.06 Thermal Expansion [m/m]
MatLib Baldock et al. [1966]
Gronvold [1955]
Burdick and Parker [1956]
Other Hagrman References Martin References Figure 2-9. Model-to-Data Comparison for UO2 Correlation Thermal expansion data have been collected for PuO2 from unirradiated samples [Brett and Rus-sel, 1960] [Tokar and Nutt, 1972]. A comparison between these data for PuO2 is presented in Fuel Material Properties 18
PNNL-29728 Figure 2-10. This comparison demonstrates good agreement between the correlation and the database from room temperature to the melting temperature (3000 [K]).
500 1000 1500 2000 2500 3000 Temperature [K]
0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 Thermal Expansion [m/m]
MatLib
[Brett and Russel 1960] and [Tokar and Nutt 1972] References Figure 2-10. Model-to-Data Comparison for PuO2 Correlation 2.1.4.3 Applicability and Uncertainty The fuel thermal expansion model is applicable to the range of available data:
Fuel types: UO2, UO2-Gd2O3, MOX Gadolinia content: 0 to 10 [wt%]
Temperature: 300 [K] to the applicable melting temperature (see Section 2.1.3)
Rod-average burnup: No burnup dependence observed As-fabricated density: No density dependence observed Engineering judgment should be used if analysis outside of these ranges is needed.
The uncertainty of the correlation is given below for each fuel type as a relative standard error. The uncertainty of the pooled data was found to be temperature dependent, increasing approximately linearly with temperature. Therefore, a relative error is given rather than an absolute error.
UO2 and UO2-Gd2O3: = 10.3%
PuO2: = 3.5%
Fuel Material Properties 19
PNNL-29728 The relative standard error for UO2 was calculated by excluding data with very small measured thermal expansion to avoid artificially increasing the relative standard error. In addition, two data with very large deviation were identified as outliers and removed in this calculation.
2.1.5 Emissivity The emissivity of oxide nuclear fuel is modeled in MatLib as a function of one parameter:
1.
Temperature 2.1.5.1 Model Description The emissivity of UO2, MOX and UO2-Gd2O3 is given by:
= 0.78557 + 1.5263 x 105T (2-12)
- Where,
= Total hemispherical emissivity [unitless]
T = Temperature [K]
2.1.5.2 Comparison to Data Emissivity data have been collected for UO2 from unirradiated samples [Held and Wilder, 1969]
[Cabannes et al., 1967]. A comparison between these data for UO2 is presented in Figure 2-11.
This comparison demonstrates reasonable agreement between the correlation and the database within the range of 300 to 2500 [K].
Fuel Material Properties 20
PNNL-29728 500 1000 1500 2000 2500 3000 Temperature [K]
0.0 0.2 0.4 0.6 0.8 1.0 Emissivity [unitless]
MatLib Held and Wilder [1969]
Cabannes et al. [1967]
Figure 2-11. Model-to-Data Comparison for Emissivity of Oxide Fuel 2.1.5.3 Applicability and Uncertainty The emissivity model is applicable to the range of available data:
Fuel types: UO2, MOX and UO2-Gd2O3 Gadolinia content: 0 to 10 [wt%]
Temperature: 300 to 2500 [K]
Rod-average burnup: No burnup dependence observed As-fabricated density: No density dependence observed Engineering judgment should be used if analysis outside of these ranges is needed.
The uncertainty of the correlation is given below for each fuel type as an absolute standard error.
UO2, MOX, UO2-Gd2O3: = 0.072 [unitless]
The surfaces of UO2, MOX and UO2-Gd2O3 are optically very similar. Therefore, it is assumed the uncertainty of the correlation will be applicable to all the variants.
Fuel Material Properties 21
PNNL-29728 2.1.6 Density The theoretical density of oxide nuclear fuel is modeled in MatLib as a function of one parameter:
1.
Composition 2.1.6.1 Model Description The theoretical density of pure UO2 is taken as 10960
kg/m3
The theoretical density of pure PuO2 is taken as 11460
kg/m3
The addition of gadolinia reduces the theoretical density of UO2 by Equation 2-13.
TD = UO2 3860fGd2O3 (2-13)
- Where, TD = Theoretical density of UO2/Gd2O3 mixture,
kg/m3
UO2 = Theoretical density of UO2,
kg/m3
fGd2O3 = Weight fraction of Gd2O3, [unitless]
The theoretical density of MOX is determined based on the weight fraction of UO2 and PuO2 by Equation 2-14.
TD = UO2
1 fPuO2
+ PuO2
fPuO2
(2-14)
- Where, TD = Theoretical density of UO2/PuO2 mixture
kg/m3
UO2 = Theoretical density of UO2
kg/m3
PuO2 = Theoretical density of PuO2
kg/m3
fPuO2 = Weight fraction of PuO2 [unitless]
2.1.6.2 Applicability and Uncertainty The theoretical density model is applicable to the range of available data:
Fuel types: UO2, PuO2, MOX and UO2-Gd2O3 Fuel Material Properties 22
PNNL-29728 Gadolinia content: 0 to 100 [wt%]
Temperature: Room temperature Rod-average burnup: No burnup dependence observed As-fabricated density: Not applicable Engineering judgment should be used if analysis outside of these ranges is needed.
No uncertainty is given on the theoretical density. Uncertainty in the density of pellets is addressed through the input of fraction of theoretical density.
2.1.7 Densification The densification of oxide nuclear fuel is modeled in MatLib as a function of two parameters:
1.
Maximum expected in-reactor densification 2.
Burnup The maximum expected in-reactor densification is calculated using one of two methods:
The re-sintering method uses the density change observed during re-sintering tests (1973 [K]
for 24 [hours] based on Regulatory Guide 1.126 [NRC, 1978]) in a laboratory furnace and is the preferred input for the calculation.
If a re-sintering density change is not input, the sintering temperature based method uses the initial unirradiated density of the fuel and the fuel fabrication sintering temperature and burnup for density calculations.
2.1.7.1 Model Description The densification of UO2, MOX and UO2-Gd2O3 is given by [Rolstad et al., 1974]:
L L
=
L L
m
+ exp [3 (Bu + B)] + 2 exp [35 (Bu + B)]
(2-15)
L L
m
=
100sint (3start) for sint > 0
kg/m3
22.2(100 TD)
(Tsint 1453.15) for T < 1000 [K]
66.6(100 TD)
(Tsint 1453.15) for T 1000 [K]
for sint = 0
kg/m3
(2-16)
- Where, Fuel Material Properties 23
PNNL-29728 L
L = Dimension change [%]
L L
m = Maximum dimension change due to irradiation [%] (Equation 2-16)
Bu = Burnup [MWd/kgU]
B = A constant determined by the code to fit the boundary condition; L L
= 0 when Bu = 0
[unitless]
sint = Resintered fuel density change
kg/m3
T = Fuel temperature [K]
start = Starting (as-fabricated) density
kg/m3
TD = Initial density [percent theoretical]
Tsint = Sintering temperature [K] (default is 1873.15 [K])
2.1.7.2 Comparison to Data Densifiction data have been collected for UO2 and MOX pellets from irradiated samples [Banks, 1974] [Freshley et al., 1979] [Freshley et al., 1976]. A comparison between these data is presented in Figure 2-12. This comparison demonstrates that basing densification on the sintering tempera-ture provides a large degree of uncertainty.
2 0
2 4
6 8
Measured Densification [\\% TD]
1 0
1 2
3 4
5 6
7 8
Predicted Densification [\\% TD]
Measured=Predicted
[Banks 1974]
[Freshley et al. 1978]
[Freshley et al. 1976]
Figure 2-12. Model-to-Data Comparison for Densification of Oxide Fuel Fuel Material Properties 24
PNNL-29728 2.1.7.3 Applicability and Uncertainty The densification correlation used in MatLib is applicable to the range of available data (i.e., fuels with pore size distributions similar to those included in the [Freshley et al., 1976] study). Engineer-ing judgment should be used if analysis outside of these ranges is needed. Due to the scatter in the experimental data, it is difficult to establish a meaningful measure of uncertainty.
2.1.8 Swelling The swelling in the oxide fuels is modeled in MatLib as two different phenomena; solid swelling and gaseous swelling. Solid swelling proceeds at a constant rate with increasing burnup and with no temperature dependence. Gaseous swelling only occurs at high burnup (>40 [GWd/MTU]) and occurs over a specific temperature range (1233 to 2105 [K])
2.1.8.1 Model Description Solid Swelling The solid swelling of UO2 and MOX is given by:
V V
=
0 for Bu 6 [GWd/MTU]
0.00062 (Bu 6) for 6 < Bu 80 [GWd/MTU]
0.00062 (80 6) + 0.00086 (Bu 80) for Bu > 80 [GWd/MTU]
(2-17)
- Where, V /V = Fractional volume change due to solid fission products
m3/m3
Bu = Pellet-average burnup [GWd/MTU]
The solid swelling of UO2-Gd2O3 is given by:
V V
= 0.0005Bu (2-18)
- Where, V /V = Fractional volume change due to solid fission products
m3/m3
Bu = Pellet-average burnup [GWd/MTU]
Gaseous Swelling The gaseous swelling of UO2, UO2-Gd2O3, and MOX is given by:
Fuel Material Properties 25
PNNL-29728 Bu < 40 [GWd/MTU]
L L
= 0 (2-19) 40 < Bu < 50 [GWd/MTU]
L L
=
0 for T < 1233 [K]
Bu 40 10
4.37 x 102 + 4.55 x 105T
for 1233 T < 1643 [K]
Bu 40 10
7.40 x 102 4.05 x 105T
for 1643 T < 2105 [K]
0 for T > 2105 [K]
(2-20)
Bu 50 [GWd/MTU]
L L
=
0 for T < 1233 [K]
4.37 x 102 + 4.55 x 105T for 1233 T < 1643 [K]
7.40 x 102 4.05 x 105T for 1643 T < 2105 [K]
0 for T > 2105 [K]
(2-21)
- Where, L/L = Fractional volume change due to solid fission products
m3/m3
Bu = Pellet-average burnup [GWd/MTU]
T = Pellet ring temperature [K]
2.1.8.2 Comparison to Data Solid swelling increase data have been collected for UO2 from irradiated samples [Garde, 1986]
[Newman, 1986] [Smith et al., 1994] [Dideon and Bain, 1983] [Turnbull, 2001] [Colombier et al.,
2010]. A comparison between these data for UO2 is presented in Figure 2-13. This comparison demonstrates reasonable comparison between the correlation and the database.
Fuel Material Properties 26
PNNL-29728 0
20 40 60 80 100 Burnup[GWd/MTU]
2 0
2 4
6 8
V/V [\\%]
MatLib Garde [1986]
Newman [1986]
Smith et al. [1994]
Dideon and Bain [1983]
Colombier et al. [2010]
Figure 2-13. Model-to-Data Comparison for Solid Swelling Correlation Solid swelling rate data have been collected for UO2 from irradiated Halden tests [Colombier et al.,
2010] [Petiprez, 2002] [Matsson and Turnbull, 1998] [Turnbull, 2001]. A comparison between these data for UO2 is presented in Figure 2-14. This comparison demonstrates reasonable comparison between the correlation and the database.
Fuel Material Properties 27
PNNL-29728 IFA-633 IFA-610 IFA-534.14 IFA-629.4 IFA-629.3 IFA-597.3 IFA-515.10 IFA-515.10 IFA-519.9 IFA-504 IFA-655 Rods 1 and 3 IFA-655 Rods 2 and 4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Swelling Rate [\\% DV/V per 10 GWd/MTHM]
17 GWd/MTU 62 GWd/MTU 62 GWd/MTU 65 GWd/MTU 68 GWd/MTU 70 GWd/MTU 75 GWd/MTU 78 GWd/MTU 90 GWd/MTU 90 GWd/MTU 99 GWd/MTU 99 GWd/MTU Figure 2-14. Model-to-Data Comparison for Solid Swelling Rate Correlation 2.1.8.3 Applicability and Uncertainty The swelling model is applicable to the range of available data:
Fuel types: UO2, PuO2, MOX and UO2-Gd2O3 Gadolinia content: 0 to 10 [wt%]
Temperature: Entire temperature range Rod-average burnup: 0 to 100 [GWd/MTU]
As-fabricated density: 90 to 98 [%TD]
Engineering judgment should be used if analysis outside of these ranges is needed.
The uncertainty of the correlation is given below for each fuel type as an absolute standard error.
UO2, MOX: = 0.00008 V /V per 1 [GWd/MTU] Bu < 80 [GWd/MTU]
UO2, MOX: = 0.00016 V /V per 1 [GWd/MTU] Bu < 80 [GWd/MTU]
Fuel Material Properties 28
PNNL-29728 UO2-Gd2O3: = 0.00008 V /V per 1 [GWd/MTU]
Fuel Material Properties 29
PNNL-29728 2.2 Metallic Fuel U-Pu-Zr Material Properties Material property correlations for metallic fuels are described in the following sections. Metallic fuel is limited to both U-Zr and U-Pu-Zr. U-Zr and U-Pu-Zr are relatively new fuel types for use in new fast reactor designs.
2.2.1 Thermal Conductivity The thermal conductivity of metallic fuel is modeled in MatLib as a function of four parameters:
1.
Temperature 2.
Pu content 3.
Zr content 4.
Porosity 2.2.1.1 Model Description The thermal conductivity of metallic fuel containing U-Pu-Zr is given by Equation 2-22 [Baker and Wilson, 1992]:
k = D1 100
AT + BT 2 2
+ CT 3 3
(2-22)
- Where, D1 = 1 P 1 + 2P (2-23a)
A = 17.5
1 2.23Zr 1 + 1.61Zr 2.62Pu
(2-23b)
B = 1.54 x 102
1 + 0.061Zr 1 + 1.61Zr
+ 0.90Pu
(2-23c)
C = 9.38 x 106 (1 2.70Pu)
(2-23d)
- and, k = Thermal conductivity [W/m K]
P = Fraction of porosity in the fuel [unitless]
T = Temperature [K]
x = Weight fraction of species x in the fuel mixture [unitless]
Fuel Material Properties 30
PNNL-29728 2.2.1.2 Applicability and Uncertainty The thermal conductivity model is applicable to the range of available data:
Fuel types: U-Zr and U-Pu-Zr Temperature: 273 to 1000 [K]
Rod-average burnup: 0 [GWd/MTU]
Engineering judgment should be used if analysis outside of these ranges is needed.
2.2.2 Specific Heat Capacity The specific heat capacity of U-Pu-Zr metallic fuel is modeled as a function of two parameters:
1.
Temperature 2.
Composition 2.2.2.1 Model Description The model for specific heat in MatLib is based on published experimental data produced from measuring calculated specific heats from incremental enthalpies in a drop calorimeter to about 1200 [C] [Savage, 1968]. Equations 2-24, 2-25, and 2-26 present the specific heat correlations for U-Pu-Zr fuel, dependent on the phase.
C+
p
= A0 +
A1 MW T (2-24)
- Where, C+
p
= Heat capacity of U-Pu-Zr fuel [J/kg K]
A0 = Constant = 26.58 A1 = Constant = 0.027 MW = Molecular weight of the metallic fuel mixture T = Temperature [C]
C p = A0 +
A1 MW T (2-25)
- Where, Fuel Material Properties 31
PNNL-29728 C
p = Specific heat capacity of metallic fuel (U-Pu-Zr / U-Zr) [J/kg K]
A0 = Constant = 15.84 A1 = Constant = 0.026 MW = Molecular weight of the metallic fuel mixture T = Temperature [C]
C+
p
= C p C+
p T2 T1 (T T1) + C+
p (2-26)
- Where, C+
p
= Specific heat capacity of metallic fuel in the + phase [J/kg K]
C p = Specific heat capacity of metallic fuel in the phase [J/kg K]
C+
p
= Specific heat capacity of metallic fuel in the + phase [J/kg K]
T = Temperature [C]
T1 = Transition temperature between + and + phases [K] (Table 2-3)
T2 = Transition temperature between + and phases [L] (Table 2-3)
The transition temperature between phases assumes there is no dependence on Zr content and that the behavior is linear between 0 and 19 [wt%] Pu.
Table 2-3. Phase Transition Temperatures Used in the Specific Heat Capacity Correlations for U-Pu-Zr Metallic Fuel Pu Content T1 T1 T1 T2 T2 T2
[wt%]
[K]
[K]
0 935.15 965.15 19 868.15 923.15 2.2.2.2 Applicability and Uncertainty The specific correlations derived from the published data [Savage, 1968] are applicable for:
Fuel types: U-Pu-Zr Phases: +, +, and Metallic fuel outside the bounds of the correlations will be executed and the user will be prompted with an error message.
Fuel Material Properties 32
PNNL-29728 2.2.3 Density The theoretical density of metallic fuel is modeled in MatLib as a function of one parameter:
1.
Composition 2.2.3.1 Model Description The density of metallic fuel is a function of the weight fractions and densities of uranium and zirco-nium:
TD =
1 (1 WZr)
U
+ WZr Zr (2-27)
- Where, TD = Theoretical density of U-Pu-Zr metallic fuel
kg/m3
Wx = Weight fraction of species x [unitless]
Zr = Theoretical density of Zr = 6500
kg/m3
U = Theoretical density of U = 19000
kg/m3
2.2.3.2 Comparison to Data No comparisons to data are provided as these are theoretical quantities.
2.2.3.3 Applicability and Uncertainty No uncertainty is given on the theoretical density. Uncertainty in the density of the pellets is ad-dressed through the input of fraction of theoretical density.
2.2.4 Melting Temperature The melting temperature of metallic fuel in MatLib is a function of one parameter:
1.
Composition 2.2.4.1 Model Description The melting temperature is a function of the weight fractions of Pu and Zr [Baker and Wilson, 1992]:
Tmelt = 1132(1 0.77WPu)(1 0.94WZr) + 273.15 (2-28)
- Where, Fuel Material Properties 33
PNNL-29728 Tmelt = Melting temperature of metallic fuel [K]
Wx = Weight fraction of species x [unitless]
2.2.5 Eutectic Temperature The eutectic temperature is the temperature at the onset of liquid-phase attack between the metallic fuel and cladding. It is assumed constant [Baker and Wilson, 1992].
Teutectic = 973 [K]
(2-29) 2.2.6 Thermal Expansion The thermal expansion of metallic fuel is modeled in MatLib as a function of one parameter:
1.
Temperature 2.2.6.1 Model Description The thermal expansion of U-Pu-Zr and U-Zr is given by:
L L
= A + BT (2-30)
- Where,
L L
= Linear strain caused by thermal expansion [unitless]
T = Temperature [K]
A, B = Constants (see Table 2-4)
Table 2-4. Constants Used in the Thermal Expansion Correlations for U-Pu-Zr Metallic Fuel Temperature AAA BBB
[K]
[unitless]
K1
T < 868 [K]
5.2448 x 103 1.76 x 105 868 [K] T < 938 [K]
5.4462 x 102 7.43 x 105 T 938 [K]
3.6538 x 103 2.01 x 105 Fuel Material Properties 34
PNNL-29728 2.2.6.2 Applicability and Uncertainty The thermal expansion model is applicable to the range of available data:
Temperature: 293 to 1073 [K]
Engineering judgment should be used if analysis outside of these ranges is needed.
2.2.7 Emissivity The emissivity [unitless] of metallic fuel in MatLib is treated as a constant value [Baker and Wilson, 1992]:
= 0.80 [unitless]
(2-31)
- Where,
= Emissivity [unitless]
2.2.7.1 Applicability and Uncertainty The thermal expansion model is applicable to the range of available data:
Temperature: 293 to 1073 [K]
Engineering judgment should be used if analysis outside of these ranges is needed.
2.2.8 Swelling The swelling in metallic fuels is modeled in MatLib as two different phenomena: pre-contact and post-contact. Post-contact swelling is much slower than pre-contact swelling due to the formation and accumulation of solid fission products. Swelling is a function of one parameter:
1.
Burnup 2.2.8.1 Model Description The swelling rate is assumed constant for each region; a no contact region and a post contact region:
V V
=
(
0.05Bu for Pre-Contact 0.009Bu for Post-Contact (2-32)
- Where, Fuel Material Properties 35
PNNL-29728
V V
= Fuel volumetric swelling [unitless]
Bu = Burnup [at%] Note: 1 [GWd/MTM] = 0.1066 [at%]
Fuel Material Properties 36
PNNL-29728 3.0 Cladding Material Properties 3.1 Zirconium-based Alloys Material property correlations for Zirconium-based claddings are described in the following subsec-tions. Unless otherwise specified, the correlations below are applicable to Zircaloy-2, Zircaloy-4, ZIRLO, Optimized ZIRLOTM, and M5TM. Various heat treatments can be accommodated by spec-ifying the cold worked condition of the alloy. Examples of cold worked conditions for the different alloys are provided in Table 3-1.
Table 3-1. Example Heat Treatments and Cold Worked Conditions for Different Zirconium-Based Alloys Alloy Heat Treatment Cold Worked Conditions Zircaloy-2 RXA(a) 0%
Zircaloy-4 CWRSA(b) 50%
ZIRLO CWRSA 50%
Opt. ZIRLOTM pRXA(c)
<50%
M5TM RXA 0%
Recrystallized Annealed Cold Worked, Stress Relief Annealed Partially Recrystallized Annealed 3.1.1 Thermal Conductivity The thermal conductivity of zirconium-based alloy cladding is modeled in MatLib as a function of one parameter:
1.
Temperature 3.1.1.1 Model Description The thermal conductivity of Zircaloy-4, Zircaloy-2, ZIRLO, Optimized ZIRLOTM, and M5TM is given by:
k = 7.511 + 2.088 x 102T 1.45 x 105T 2 + 7.668 x 109T 3 (3-1)
- Where, k = Cladding thermal conductivity [W/m K]
Cladding Material Properties 37
PNNL-29728 T = Temperature [K]
For temperatures greater than or equal to 2098 [K], the thermal conductivity is given by:
k = 36 [W/m K]
(3-2) 3.1.1.2 Comparison to Data Thermal conductivity data have been collected for Zircaloy-2 and Zircaloy-4 from unirradiated and irradiated samples [Anderson et al., 1962] [Chirigos et al., 1961] [Feith, 1966] [Lucks and Deem, 1958] [Powers, 1961] [Scott, 1965] [Krett and Cleveland, 1997] [Gilchrist, 1976] [Bunnell et al.,
1983] [Murabayashi et al., 1975] [Peggs et al., 1976] [Magli et al., 1994]. A comparison between these data is presented in Figure 3-1. This comparison demonstrates a good agreement between the correlation and the database within a range of 285 to 1770 [K].
500 750 1000 1250 1500 1750 2000 Temperature [K]
0 10 20 30 40 50 Thermal Conductivity [W/m-K]
MatLib Anderson et al. [1962]
Chirigos et al. [1961]
Feith [1966]
Lucks and Deem [1958]
Powers [1961]
Scott [1965]
Krett and Cleveland [1997]
Gilchrist [1976]
Murabayashi et al. [1975]
Peggs et al. [1976]
Maglic et al. [1994]
Figure 3-1. Model-to-Data Comparison for Zirconium-based Alloy Cladding Thermal Conductivity Correlation 3.1.1.3 Applicability and Uncertainty The thermal conductivity model is applicable to the range of available data:
Cladding types: Zircaloy-4, Zircaloy-2, M5TM, ZIRLO and Optimized ZIRLOTM Temperature: 285 to 1770 [K]
Rod-average burnup: No burnup dependence observed Cladding Material Properties 38
PNNL-29728 Engineering judgment should be used if analysis outside of these ranges is needed.
The uncertainty of the correlation is given below and is applicable for each cladding type. No vari-ation in thermal conductivity uncertainty is observed with increasing temperature, so an absolute uncertainty is used.
Zircaloy-4, Zircaloy-2, M5TM, ZIRLO and Optimized ZIRLOTM: = 1.9 [W/m K]
3.1.2 Specific Heat The specific heat of zirconium-based alloy cladding is modeled in MatLib as a function of one parameter:
1.
Temperature 3.1.2.1 Model Description The specific heat of Zircaloy-4, Zircaloy-2, M5TM, ZIRLO, and Optimized ZIRLOTMis given by a lookup table. Specific values at a given temperature can found by linear interpolation between these temperatures:
Cladding Material Properties 39
PNNL-29728 Table 3-2. Interpolated Values for the Zirconium-Based Alloys Specific Heat Capacity Correlation Temperature
[K]
Specific Heat Capacity
[J/kg K]
<290 279 290 279 300 281 400 302 640 331 1090 375 1093 502 1113 590 1133 615 1153 719 1173 816 1193 770 1213 619 1233 469 1248 356
>1248 356 3.1.2.2 Comparison to Data Specific heat data have been collected for Zircaloy-2 and Zircaloy-4 from unirradiated samples
[Deem and Eldridge, 1967] [Brooks and Stansbury, 1966]. A comparison between these data is presented in Figure 3-2. This comparison demonstrates good agreement between the correlation and the database within the range 348 to 1300 [K].
Cladding Material Properties 40
PNNL-29728 200 400 600 800 1000 1200 1400 Temperature [K]
0 100 200 300 400 500 600 700 800 Specific Heat[J/kg-K]
MatLib Deem and Eldridge [1967]
Brooks and Stansbury [1966]
Figure 3-2. Model-to-Data Comparison for Zirconium-based Alloy Cladding Specific Heat Corre-lation 3.1.2.3 Applicability and Uncertainty The specific heat model is applicable to the range of available data:
Cladding types: Zircaloy-4, Zircaloy-2, M5TM, ZIRLO, and Optimized ZIRLOTM Temperature: 285 to 1300 [K]
Rod-average burnup: No burnup dependence observed Engineering judgment should be used if analysis outside of these ranges is needed.
The uncertainty of the correlation is given below and is applicable for each cladding type. No vari-ation in thermal conductivity uncertainty is observed with increasing temperature, so an absolute uncertainty is used.
Zircaloy-4, Zircaloy-2, M5TM, ZIRLO, and Optimized ZIRLOTM:
[J/kg k] =
10 for temperatures less than 1090 [K]
25 for temperatures between 1090 [K] and 1248 [K]
100 for temperatures greater than 1248 [K]
Cladding Material Properties 41
PNNL-29728 3.1.3 Melting Temperature The melting temperature of zirconium-based alloy cladding is modeled in MatLib as a constant value.
3.1.3.1 Model Description The melting temperature of Zircaloy-4, Zircaloy-2, M5TM, ZIRLO, and Optimized ZIRLOTM is given by a constant value:
Tmelt = 2123.15 [K]
(3-3)
- Where, Tmelt = Melting temperature [K]
3.1.3.2 Comparison to Data No Comparison to Data are provided as this is a theoretical quantity.
3.1.3.3 Applicability and Uncertainty The melting temperature model is applicable to the range of available data:
Cladding types: Zircaloy-4, Zircaloy-2, M5TM, ZIRLO and Optimized ZIRLOTM Rod-average burnup: No burnup dependence observed Engineering judgment should be used if analysis outside of these ranges is needed.
No uncertainty is given on the melting temperature. Greater uncertainty exists on the prediction of cladding temperature.
3.1.4 Thermal Expansion The thermal expansion of zirconium-based alloy cladding is modeled in MatLib as a function of one parameter:
1.
Temperature 3.1.4.1 Model Description Rolled and drawn zirconium-based alloy products are known to have anisotropy in the thermal ex-pansion. Correlations for thermal expansion in the axial and circumferential directions are provided in MatLib. The thermal expansion of Zircaloy-4, Zircaloy-2, M5TM, ZIRLO and Optimized ZIRLOTM is given by:
Cladding Material Properties 42
PNNL-29728 axial =
(
2.5060 x 105 + 4.4410 x 106 (T 273.15) for 280 < T 1073.15 [K]
8.300 x 103 + 9.70 x 106 (T 273.15) for T 1273.15 [K]
(3-4) diametral =
(
2.3730 x 104 + 6.7210 x 106 (T 273.15) for 280 < T 1073.15 [K]
6.800 x 103 + 9.70 x 106 (T 273.15) for T 1273.15 [K]
(3-5)
- Where, axial = Axial thermal expansion [m/m]
diametral = Circumferential thermal expansion [m/m]
T = Temperature [K]
For 1073.15 T 1273.15 [K] the thermal expansion is given by a lookup table. Specific values at a given temperature can found by linear interpolation between these temperatures:
Table 3-3. Interpolated Values for the Zirconium-Based Alloys Thermal Expansion Correlation Temperature
[K]
axial [m/m]
diametral [m/m]
1073.15 0.00352774 0.00513950 1083.15 0.00353000 0.00522000 1093.15 0.00350000 0.00525000 1103.15 0.00346000 0.00528000 1113.15 0.00341000 0.00528000 1123.15 0.00333000 0.00524000 1133.15 0.00321000 0.00522000 1143.15 0.00307000 0.00515000 1153.15 0.00280000 0.00508000 1163.15 0.00250000 0.00490000 1173.15 0.00200000 0.00470000 1183.15 0.00150000 0.00445000 1193.15 0.00130000 0.00410000 1203.15 0.00116000 0.00350000 1213.15 0.00113000 0.00313000 1223.15 0.00110000 0.00297000 1233.15 0.00111000 0.00292000 Cladding Material Properties 43
PNNL-29728 Table 3-3. Interpolated Values for the Zirconium-Based Alloys Thermal Expansion Correlation (continued)
Temperature
[K]
axial [m/m]
diametral [m/m]
1243.15 0.00113000 0.00287000 1253.15 0.00120000 0.00286000 1263.15 0.00130000 0.00288000 1273.15 0.00140000 0.00290000 3.1.4.2 Comparison to Data Circumferential thermal expansion data have been collected for Zircaloy-2 and Zircaloy-4 from unirradiated samples [Bunnell et al., 1977] [Kearns, 1965] [Scott, 1965] [Mehan and Wiesinger, 1961]. A comparison between these data for circumferential thermal expansion is presented in Fig-ure 3-3. This comparison demonstrates good agreement between the correlation and the database between 300 and 1080 [K].
400 600 800 1000 1200 1400 1600 1800 2000 Temperature [K]
0.000 0.002 0.004 0.006 0.008 0.010 Circumferential Thermal Expansion [m/m]
MatLib Bunnell et al. [1977]
Kearns [1965] Scott [1965] Mehan and Wiesinger [1961]
Figure 3-3. Model-to-Data Comparison for for Zirconium-based Alloy Cladding Circumferential Thermal Expansion Correlation Axial thermal expansion data have been collected for Zircaloy-2 and Zircaloy-4 from unirradiated samples [Bunnell et al., 1977] [Kearns, 1965] [Scott, 1965] [Mehan and Wiesinger, 1961]. A com-parison between these data for circumferential thermal expansion is presented in Figure 3-4. This Cladding Material Properties 44
PNNL-29728 comparison demonstrates good agreement between the correlation and the database between 300 and 1273 [K].
500 750 1000 1250 1500 1750 2000 Temperature [K]
0.000 0.002 0.004 0.006 0.008 Axial Thermal Expansion [m/m]
MatLib Bunnell et al. [1977]
Kearns [1965] Scott [1965] Mehan and Wiesinger [1961]
Figure 3-4. Model-to-Data Comparison for for Zirconium-based Alloy Cladding Axial Thermal Ex-pansion Correlation 3.1.4.3 Applicability and Uncertainty The thermal expansion model is applicable to the range of available data:
Cladding types: Zircaloy-4, Zircaloy-2, M5TM, ZIRLO and Optimized ZIRLOTM Temperature: 300 to 1080 [K] for circumferential expansion; 300 to 1273 [K] for axial expansion Rod-average burnup: No burnup dependence observed Engineering judgment should be used if analysis outside of these ranges is needed.
The uncertainty of the correlation is given below and is applicable for each cladding type. No vari-ation in thermal conductivity uncertainty is observed with increasing temperature, so an absolute uncertainty is used.
Zircaloy-4, Zircaloy-2, M5TM, ZIRLO and Optimized ZIRLOTM:
Circumferential thermal expansion: = 4.6 x 104 [m/m]
Axial thermal expansion: = 4.8 x 105 [m/m]
Cladding Material Properties 45
PNNL-29728 3.1.5 Emissivity The emissivity of zirconium-based alloy cladding is modeled in MatLib as a function of two param-eters:
1.
Temperature 2.
Cladding inner surface oxide thickness 3.1.5.1 Model Description The emissivity of Zircaloy-4, Zircaloy-2, M5TM, ZIRLO and Optimized ZIRLOTM is given by:
1 =
(
0.325 + 0.1246 x 106toxide for toxide < 3.88 x 106 0.808642 50.0toxide for toxide 3.88 x 106 (3-6)
When the cladding temperature is greater than 1500 [K], emissivity is given by:
2 = MAX
0.325, exp
1500 T 300
1
(3-7)
- Where, 1, 2 = Cladding emissivity [unitless]
T = Temperature [K]
toxide = Inner surface oxide thickness [m]
3.1.5.2 Comparison to Data Emissivity data have been collected for Zircaloy-2 and Zircaloy-4 from irradiated samples [Murphy and Havelock, 1976] [Juenke and Sjodahl, 1968] [Burgoyne and Garlick, 1976]. A comparison between these data for is presented in Figure 3-5. This comparison demonstrates good agreement between the correlation and the database up to 1575 [K] and 120 [µm] oxide thickness.
Cladding Material Properties 46
PNNL-29728 0.00000 0.00002 0.00004 0.00006 0.00008 0.00010 0.00012 0.00014 Oxide Thickness [m]
0.0 0.2 0.4 0.6 0.8 1.0 Emissivity [unitless]
MatLib [T < 1500K]
MatLib [T=1575K]
Murphy and Havelock [1976]
Juenke and Sjodahl [1968]
Burgoyne and Garlick [1976]
Juenke and Sjodahl [1968] at 1575K Figure 3-5. Model-to-Data Comparison for Zirconium-based Alloy Emissivity Correlation 3.1.5.3 Applicability and Uncertainty The emissivity model is applicable to the range of available data:
Cladding types: Zircaloy-4, Zircaloy-2, M5TM, ZIRLO and Optimized ZIRLOTM Temperature: 285 to 1575 [K]
Oxide Thickness: 0 to 120 [µm]
Rod-average burnup: No burnup dependence observed Engineering judgment should be used if analysis outside of these ranges is needed.
The uncertainty of the correlation is given below and is applicable for each cladding type. No varia-tion in emissivity uncertainty is observed with increasing oxide thickness, so an absolute uncertainty is used.
Zircaloy-4, Zircaloy-2, M5TM, ZIRLO and Optimized ZIRLOTM: = 0.054 [unitless]
3.1.6 Density The density of Zircaloy-4, Zircaloy-2, M5TM, ZIRLO and Optimized ZIRLOTM is modeled in MatLib as a constant value:
Cladding Material Properties 47
PNNL-29728
= 6520 h
kg/m3i (3-8)
- Where,
= Density of zirconium-based alloy cladding
kg/m3
3.1.6.1 Comparison to Data No comparisons to data are provided as this is a theoretical quantity.
3.1.6.2 Applicability and Uncertainty The density model is applicable to the following:
Cladding types: Zircaloy-4, Zircaloy-2, M5TM, ZIRLO and Optimized ZIRLOTM Rod-average burnup: No burnup dependence observed No uncertainty is given on the density.
3.1.7 Youngs Modulus and Shear Modulus Youngs modulus and the shear modulus of zirconium-based alloy cladding are modeled in MatLib as a function of three parameters:
1.
Temperature 2.
Cladding cold work 3.
Fast neutron fluence 3.1.7.1 Model Description Youngs Modulus The Youngs modulus of Zircaloy-4, Zircaloy-2, M5TM, ZIRLO and Optimized ZIRLOTM is given by:
E =
1.088 x 1011 5.475 x 107T + c1Oxygen + c3CW c2 for T < 1090 [K]
9.21 x 1010 4.05 x 107T for T > 1255 [K]
(3-9)
For temperatures between 1090 and 1255 [K] a linear interpolation between the predictions at 1090 [K] and 1255 [K] is used.
Cladding Material Properties 48
PNNL-29728
- Where, E = Youngs modulus [Pa]
T = Temperature [K]
Oxygen = Input average oxygen concentration excluding oxide layer (hardwired to 0.0012 in MatLib) [kg oxygen/kg Zircaloy]
CW = Input effective cold work (ratio of areas) [unitless]
c1 = 6.61 x 1011 + 5.912 x 108T c2 = 0.88 + 0.12 exp
1 x 1025
c3 = 2.6 x 1010
= Fast neutron (>1.0 MeV) fluence
n/m2
Shear Modulus The shear modulus of Zircaloy-4, Zircaloy-2, M5TM, ZIRLO and Optimized ZIRLOTM is given by:
G =
4.04 x 1010 2.168 x 107T + c1Oxygen + c3 c2 for T < 1090 [K]
3.49 x 1010 1.66 x 107T for T > 1255 [K]
(3-10)
For temperatures between 1090 and 1255 [K] a linear interpolation between the predictions at 1090 [K] and 1255 [K] is used.
- Where, G = Shear modulus [Pa]
T = Temperature [K]
Oxygen = Input average oxygen concentration excluding oxide layer (hardwired to 0.0012 in MatLib) [kg oxygen/kg Zircaloy]
CW = Input effective cold work (ratio of areas) [unitless]
c1 = 7.07 x 1011 2.315 x 108T c2 = 0.88 + 0.12 exp
1 x 1025
c3 = 0.867 x 1010
= Fast neutron (>1.0 MeV) fluence
n/m2
Cladding Material Properties 49
PNNL-29728 3.1.7.2 Comparison to Data Youngs modulus data have been collected for zirconium, Zircaloy-2, and Zircaloy-4 from unir-radiated samples [Armstrong and Brown, 1964] [Padel and Groff, 1976] [Busby, 1966] [Spasic et al., 1968] [Mehan, 1958] [Northwood et al., 1975] [Bolmaro and Povolo, 1988]. This comparison demonstrates a good agreement between the correlation and the database within a range of 293 and 1474 [K].
400 600 800 1000 1200 1400 1600 1800 Temperature [K]
0.0 0.2 0.4 0.6 0.8 1.0 Young's Modulus [Pa]
1e11 MatLib Armstrong and Brown [1964]
Padel and Groff [1976]
Busby [1966]
Spasic et al. [1968]
Mehan [1958]
Northwood et al. [1975]
Bolmaro and Povolo [1988]
Figure 3-6. Model-to-Data Comparison for Zirconium Alloy Cladding Youngs Modulus Since there is limited data available from shear modulus measurements no model-to-data compar-ison is shown here.
3.1.7.3 Applicability and Uncertainty The Youngs modulus and shear modulus models are applicable to the range of available data:
Cladding types: Zircaloy-2, Zircaloy-4, M5TM, ZIRLO and Optimized ZIRLOTM Temperature: 293 to 1474 [K]
Fast neutron flux: 1.5 x 1026
n/m2
Engineering judgment should be used if analysis outside of these ranges is needed.
The uncertainty of the correlation is given below as an absolute uncertainty and is applicable for each cladding type.
Cladding Material Properties 50
PNNL-29728 Zircaloy-2, Zircaloy-4, M5TM, ZIRLO and Optimized ZIRLOTM:
Youngs modulus: = 3.1 x 109 [Pa]
Shear modulus: = 6.2 x 109 [Pa] (assumed to be twice that of the calculated Youngs modu-lus) 3.1.8 Meyers Hardness The Meyers hardness of zirconium-based alloy cladding is modeled in MatLib as a function of one parameter:
1.
Temperature 3.1.8.1 Model Description The Meyers hardness of Zircaloy-2, Zircaloy-4, M5TM, ZIRLO and Optimized ZIRLOTM is given by:
MH =
exp(26.034 2.6394 x 102T
+ 4.3502 x 105T 2 2.5621 x 108T 3) for T 1235 [K]
1.0 x 105 for T > 1235 [K]
(3-11)
- Where, MH = Cladding Meyer hardness [Pa]
T = Temperature [K]
3.1.8.2 Comparison to Data Meyers hardness data have been collected for Zircaloy-2 and Zircaloy-4 from unirradiated sam-ples [Peggs and Godin, 1975]. A comparison between these data is presented in Figure 3-7. This comparison demonstrates a good agreement between the correlation and the database within a range of 350 and 875 [K].
Cladding Material Properties 51
PNNL-29728 300 400 500 600 700 800 900 1000 Temperature [K]
0 250 500 750 1000 1250 1500 1750 Meyer Hardness [MPa]
MatLib Peggs and Godin [1975]
Figure 3-7. Model-to-Data Comparison for Zirconium-based Alloy Cladding Meyers Hardness Correlation 3.1.8.3 Applicability and Uncertainty The Meyers hardness model is applicable to the range of available data:
Cladding types: Zircaloy-2, Zircaloy-4, M5TM, ZIRLO, and Optimized ZIRLOTM; KanthalAPMT, C35M, and C36M (see Section 3.2.8); and HT9 (see Section 3.3.9)
Temperature: 350 to 875 [K]
Rod-average burnup: No burnup dependence observed Engineering judgment should be used if analysis outside of these ranges is needed.
An estimate of the uncertainty in this correlation has not been established due to the limited data.
In FAST this material property is used to determine the fuel-cladding contact conductance and any uncertainty in this value will be reflected in uncertainty in the prediction of the gap conductance.
3.1.9 Axial Growth The axial irradiation growth of zirconium-based alloy cladding is modeled in MatLib as a function of one parameter:
1.
Fast neutron fluence Cladding Material Properties 52
PNNL-29728 Different correlations are given for each specific cladding alloy. It should be noted that these cor-relations are only valid for fuel rod cladding axial irradiation growth and may not represent guide tube growth as these components are under significantly different stress states.
3.1.9.1 Model Description The axial irradiation growth of Zircaloy-2 is given by:
L L
= 1.09 x 10210.845 (3-12)
The axial irradiation growth of Zircaloy-4 is given by:
L L
= 2.18 x 10210.845 (3-13)
The axial irradiation growth of ZIRLO and Optimized ZIRLOTM is given by:
L L
= 9.7893 x 10250.98239 (3-14)
The axial irradiation growth of M5TM is given by:
L L
= 7.013 x 10210.81787 (3-15)
- Where, L
L
= Axial growth increment [m/m]
= Fast neutron (>1.0 MeV) fluence
n/cm2
3.1.9.2 Comparison to Data Axial irradiation growth data have been collected for Zircaloy-2 irradiated samples [Harbottle, 1970]
[Gilbon et al., 2000]. A comparison between these data is presented in Figure 3-8. This comparison demonstrates a good agreement between the correlation and the database up to a fast neutron fluence of 1.0 x 1022
n/cm2
. The [Gilbon et al., 2000] data is from a fast reactor and may not be representative to the behavior in a LWR.
Cladding Material Properties 53
PNNL-29728 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Fast Fluence [n/cm2]
1e22 0.000 0.001 0.002 0.003 0.004 0.005 Axial Growth [m/m]
MatLib Harbottle [1970]
Gilbon et. al [2000]
Figure 3-8. Model-to-Data Comparison for Zircaloy-2 Axial Irradiation Growth Correlation Axial irradiation growth data have been collected for Zircaloy-4 irradiated samples [Newman, 1986]
[Franklin et al., 1983] [Gilbon et al., 2000]. A comparison between these data is presented in Figure 3-9. This comparison demonstrates a good agreement between the correlation and the database up to a fast neutron fluence of 8.5 x 1021
n/cm2
. The [Gilbon et al., 2000] data is from a fast reactor and may not be representative to the behavior in a LWR.
Cladding Material Properties 54
PNNL-29728 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Fast Fluence [n/cm2]
1e22 0.000 0.002 0.004 0.006 0.008 Axial Growth [m/m]
MatLib Newman [1986]
Franklin et. al [1983]
Gilbon et. al [2000]
Figure 3-9. Model-to-Data Comparison for Zircaloy-4 Axial Irradiation Growth Correlation Axial irradiation growth data have been collected for ZIRLO irradiated samples [Irisa and Alonso, 2000] [Sabol et al., 1994]. A comparison between these data is presented in Figure 3-10. This comparison demonstrates a good agreement between the correlation and the database up to a fast neutron fluence of 8.5 x 1021
n/cm2
. Proprietary data indicates that the axial growth of Optimized ZIRLOTM is similar or slightly lower than for ZIRLO. For this reason, the ZIRLO correlation is applied for Optimized ZIRLOTM.
Cladding Material Properties 55
PNNL-29728 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Fast Fluence [n/cm2]
1e22 0.000 0.001 0.002 0.003 0.004 Axial Growth [m/m]
MatLib Irisa [2000]
BR-3; Sabol et al. [1994]
North Anna; Sabol et al. [1994]
Figure 3-10. Model-to-Data Comparison for ZIRLO Axial Irradiation Growth Correlation Axial irradiation growth have been collected for M5TM irradiated samples [Gilbon et al., 2000].
A comparison between these data is presented in Figure 3-11. This comparison demonstrates a good agreement between the correlation and the database up to a fast neutron fluence of 1 x 1022
n/cm2
Cladding Material Properties 56
PNNL-29728 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Fast Fluence [n/cm2]
1e22 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 Axial Growth [m/m]
MatLib Gilbon et. al [2000]
Figure 3-11. Model-to-Data Comparison for M5TM Axial Irradiation Growth Correlation 3.1.9.3 Applicability and Uncertainty The axial irradiation growth correlation is applicable to the range of available data:
Cladding types: Zircaloy-4, Zircaloy-2, M5TM, ZIRLO and Optimized ZIRLOTM Temperature: 530 to 620 [K]
Fast Neutron Fluence: 0 to 1 x 1022
n/cm2
for Zircaloy-2 and M5TM; 0 to 8.5 x 1021
n/cm2
for Zircaloy-4, ZIRLO and Optimized ZIRLOTM Engineering judgment should be used if analysis outside of these ranges is needed.
The uncertainty of the correlation is given below and is applicable for each cladding type. A rela-tive uncertainty was used for all the cladding types except ZIRLO and Optimized ZIRLOTM. For ZIRLO the scatter in the data did not change with fast neutron fluence.
Zircaloy-2: = 20.9%
Zircaloy-4: = 22.3%
ZIRLO and Optimized ZIRLOTM: = 0.0005 [m/m]
M5TM: = 18.6%
Cladding Material Properties 57
PNNL-29728 3.1.10 Strain (Creep) Rate The strain of zirconium-based alloy cladding is modeled in MatLib as a function of six parameters:
1.
Temperature 2.
Effective stress 3.
Fast neutron flux 4.
Fast neutron fluence 5.
Cladding cold work 6.
Time Different correlations are given for each specific cladding alloy. The RXA correlation is used for Zircaloy-2 and M5TM. The SRA correlation is used for Zircaloy-4. An adjustment to the SRA corre-lation is used for ZIRLO. An adjustment to the RXA correlation is used for Optimized ZIRLOTM.
3.1.10.1 Model Description The thermal strain rate of zirconium-based alloy cladding is given by:
th = AE T
sinhaieff E
n exp
Q RT
(3-16)
- Where, E = 1.148 x 105 59.9T (3-17a) ai = 650 h
1 0.56
1 exp
1.4 x 10271.3i (3-17b) th = Thermal strain rate [in/in hr]
A = Constant(see Table 3-4)
E = Youngs Modulus [MPa]
T = Temperature [K]
ai = Fluence term (parameters changed from original Limbck equation [Limbck and Ander-sson, 1996])
eff = Effective stress [MPa] (see Equation 3-26) n = Stress exponent (see Table 3-4)
Q = Activation energy = 201000 [J/mol]
Cladding Material Properties 58
PNNL-29728 R = Universal gas constant 8.314 [J/mol K]
The irradiation strain rate of zirconium-based alloy cladding is given by:
irr = c0c1c2 efff(T)
(3-18)
- Where, irr = Irradiation strain rate [in/in hr]
c0 = Constant (see Table 3-4)
= Fast neutron (>1.0 MeV) flux
n/m2 s
c1 = Flux exponent = 0.85 [unitless]
eff = Effective stress [MPa] (see Equation 3-26) c2 = Stress exponent = 1.0 [unitless]
f(T) = Temperature term T = Temperature [K]
A number of variables for the thermal and irradiation strain rates are dependent on the cladding cold work (refer to Table 3-1):
Table 3-4. Cladding Cold Work Dependent Parameters for the Thermal and Irradiation Strain Rate Correlations Parameter SRA Cladding RXA Cladding Units A
1.08 x 109 5.47 x 108
[K/MPa hr]
n 2.0 3.5
[unitless]
c0 4.0985 x 1024 1.87473 x 1024
(n/m2 s)c1MPac2
f(T) for T 570 [K]
0.7283 0.7994
[unitless]
f(T) for 570 [K] < T <
625 [K]
7.0237+0.0136*T 3.18562+0.006699132*T
[unitless]
f(T) for T 625 [K]
1.4763 1.1840
[unitless]
The thermal and irradiation creep rates may be added together as shown below and used to cal-culate the saturated primary hoop strain, s p.
s p = 0.0216 0.109 th+irr
2 tanh
3.55 x 104
- th+irr
2.05 (3-19)
Cladding Material Properties 59
PNNL-29728 th+irr = th + irr (3-20)
The total strain, H, can then be calculated as a function of time, t [hours].
H = s p
1 exp
52 p
t th+irr
+ th+irrt (3-21)
However, in FAST the strain rate is used, which is obtained by taking the derivative of the equation above. This derivative is presented in the equation below which relates the total creep strain rate to the saturated primary hoop strain, the combined thermal and irradiation strain rates, and time, t
[hours].
H = 26s p
th+irr
t exp
52 p
t th+irr
+ th+irr (3-22)
The effective stress in the cladding is found using the principle stresses at the mid-wall radius using the thick wall formula. The principle stresses can be determined with:
r =
Pir2 i Por2 o + r2 i r2 o (Po Pi) r2 r2o r2 i
(3-23) t =
Pir2 i Por2 o r2 i r2 o (Po Pi) r2 r2o r2 i
(3-24) l = Pir2 i Por2 o
r2o r2 i
(3-25)
- Where, r = Radial stress [MPa]
t = Tangential stress [MPa]
l = Longitudinal stress [MPa]
Pi = Inner pressure [MPa]
Po = Outer pressure [MPa]
ri = Inner radius [cm]
ro = Outer radius [cm]
r = Radius within tube [cm]
Cladding Material Properties 60
PNNL-29728 The effective stress (eff [MPa]) can then be calculated by:
eff =
r 0.5
(l t)2 + (t r)2 + (r l)2
(3-26)
It has been found that the Zircaloy RXA model adequately describes the creep behavior of M5TM
[Gilbon et al., 2000]. The Zircaloy SRA model is used for ZIRLO and Optimized ZIRLOTM with a reduction factor of 0.8 on H. The reduction factor is the result of studies that have shown that ZIRLO exhibits about 80% of SRA Zircaloy-4 creepdown [Sabol et al., 1994].
3.1.10.2 Comparison to Data Irradiation strain data have been collected for RXA Zircaloy samples [Franklin et al., 1983] [Soniak et al., 2002] [Gilbon et al., 2000] [Sontheimer and Nissen, 1994]. A comparison between these data is presented in Figure 3-12. This comparison demonstrates a good agreement between the correlation and the database.
0.002 0.000 0.002 0.004 0.006 0.008 Measured hoop strain [m/m]
0.002 0.000 0.002 0.004 0.006 0.008 Predicted hoop strain [m/m]
Measured = Predicted Franklin et. al [1983]
Soniak et. al [2002]
Gilbon et. al [2000]
ROPE-1; Sontheimer and Nissen [1994]
IFA-585; Sontheimer and Nissen [1994]
Figure 3-12. Model-to-data Comparison for RXA Ziracloy Strain Correlation Irradiation strain data have been collected for SRA Zircaloy samples [Franklin et al., 1983] [Soniak et al., 2002] [Gilbon et al., 2000]. A comparison between these data is presented in Figure 3-13.
This comparison demonstrates a good agreement between the correlation and the database.
Cladding Material Properties 61
PNNL-29728 0.000 0.005 0.010 0.015 0.020 0.025 Measured hoop strain [m/m]
0.000 0.005 0.010 0.015 0.020 0.025 Predicted hoop strain [m/m]
Measured = Predicted Franklin et. al [1983]
Soniak et al. [2002]
Gilbon et al. [2000]
Figure 3-13. Model-to-data Comparison for SRA Ziracloy Strain Correlation 3.1.10.3 Applicability and Uncertainty The strain correlations for zirconium-based alloy claddings are applicable to the range of available data:
Cladding types: Zircaloy-4, Zircaloy-2, M5TM, ZIRLO and Optimized ZIRLOTM Temperature: 570 to 625 [K]
Effective stress: 40 to 130 [MPa]
Fast Neutron Flux: 1 x 1017 to 2 x 1018
n/cm2 s
Engineering judgment should be used if analysis outside of these ranges is needed.
The uncertainty of the correlation is given below and is applicable for each cladding type:
Zircaloy-2 and M5TM: = 21.6%
Zircaloy-4, ZIRLO, and Optimized ZIRLOTM: = 14.5%
Cladding Material Properties 62
PNNL-29728 3.2 Iron-Chrome-Aluminum (FeCrAl) Alloys Material property correlations for FeCrAl alloy based claddings are described in the following subsections. Unless otherwise specified, the correlations below are applicable to Kanthal APMT, C35M, and C36M. The various alloys of FeCrAl have various compositions. Kanthal APMT has 21 [wt%] Cr and 5 [wt%] Al; C35M has 13 [wt%] Cr and 5 [wt%] Al; and C36M has 13 [wt%] Cr and 6 [wt%] Al. Table 3-5 summarizes the nominal composition of the various FeCrAl alloys included in MatLib [Field et al., 2015] [Field, 2018].
Table 3-5. Nominal Composition of Various FeCrAl Alloys in Matlib Alloy Nominal Composition [wt%]
Kanthal APMT Fe-21Cr-5Al-3Mo C35M Fe-13Cr-5Al-2Mo-0.2Si-0.05Y C36M Fe-13Cr-6Al-2Mo-0.2Si-0.05Y 3.2.1 Thermal Conductivity The thermal conductivity of FeCrAl-alloy cladding is modeled in MatLib as a function of one pa-rameter:
1.
Temperature 3.2.1.1 Model Description The thermal conductivity of Kanthal APMT, C35M, and C36M is given by:
k = A0 + A1T + A2T 2 (3-27)
- Where, k = Thermal conductivity [W/m K]
T = Temperature [K]
Ax = Fitting constants (see Table 3-6)
Values for the fitting constants for each alloy used to calculate the thermal conductivity are provided in Table 3-6 [Field, 2018].
Cladding Material Properties 63
PNNL-29728 Table 3-6. Constants Used in the FeCrAl Thermal Conductivity Correlation Alloy A0 A1
1 x 102
A2
1 x 107
Kanthal APMT 6.569 1.5628
-7.223 C35M 8.502 1.537
-19.86 C36M 8.187 1.368
-9.184 3.2.1.2 Comparison to Data Thermal conductivity data have been collected for FeCrAl samples [Field, 2018]. A comparison between these data is presented in Figures 3-14, 3-15, and 3-16 for Kanthal APMT, C35M, and C36M, respectively.
400 600 800 1000 1200 1400 Temperature [K]
0 5
10 15 20 25 Thermal Conductivity [W/m-K]
MatLib Field [2018]
Figure 3-14. Model-to-Data Comparison for Kanthal APMT FeCrAl Alloy Thermal Conductivity Correlation Cladding Material Properties 64
PNNL-29728 400 600 800 1000 1200 1400 Temperature [K]
0 5
10 15 20 25 Thermal Conductivity [W/m-K]
MatLib Field [2018]
Figure 3-15. Model-to-Data Comparison for C35M FeCrAl Alloy Thermal Conductivity Correlation 400 600 800 1000 1200 1400 Temperature [K]
0 5
10 15 20 25 Thermal Conductivity [W/m-K]
MatLib Field [2018]
Figure 3-16. Model-to-Data Comparison for C36M FeCrAl Alloy Thermal Conductivity Correlation 3.2.1.3 Applicability and Uncertainty The thermal conductivity correlation for FeCrAl is applicable to the range of available data:
Cladding Material Properties 65
PNNL-29728 Cladding types: Kanthal APMT, C35M, C36M Temperature: 300 to 1400 [K]
Burnup: unirradiated Engineering judgment should be used if analysis outside of these ranges is needed.
The data used to generate the 2nd order polynomial reports a 7% uncertainty due to the assumed experimental variability in the heat capacity, thermal diffusivity, and density of the FeCrAl alloys
[Field, 2018].
3.2.2 Specific Heat The specific heat of FeCrAl alloys is modeled in MatLib as a function of one parameter:
1.
Temperature In addition, the specific heat takes into account the alloy-dependent Curie temperature.
3.2.2.1 Model Description The specific heat model in MatLib is a two-expression correlation based on the cladding temepra-ture. In Equation 3-28, the two expressions used to calculate the specific heat of non-irradiated FeCrAl alloys are presented. The correlations were developed at ORNL [Field, 2018] and are summarized below.
cp =
aT + bT 2 + cT 3 for T Tc aT + bT 2 + cT 3 + DT 1 + E ln
lT Tcl Tc
for T > Tc (3-28)
- Where, cp = Specific heat capacity [J/kg K]
T = Temperature [K]
a, b, c, D, and E = Fitting constants (see Table 3-7)
Tc = Curie temperature [K] (see Table 3-7)
The Curie temperature describes the materials magnetic properties at a specific temperature.
Above the Curie temperature, materials lose their permanent magnetic property, which is replaced by induced magnetism.
Table 3-7 provides the coefficients used to determine the specific heat for the various FeCrAl alloys used in MatLib [Field, 2018].
Cladding Material Properties 66
PNNL-29728 Table 3-7. Constants Used in the FeCrAl Specific Heat Correlation Alloy Valid Temperature Range [K]
a b
1 x 103
c
1 x 106
D
1 x 103
E Tc
[K]
Kanthal APMT 300 < T Tc 2.54
-4.311 2.982 852 Kanthal APMT Tc < T < Tm 1.840
-1.843 0.643
-5.712
-50.38 852 C35M 300 < T Tc 2.450
-4.002 2.720 870 C35M Tc < T < Tm 1.946
-2.002 0.698
-1.652
-53.93 870 C36M 300 < T Tc 2.995
-5.953 4.516 771 C36M Tc < T < Tm 1.456
-1.296 0.438 26.45
-46.89 771 3.2.2.2 Comparison to Data Figures 3-17, 3-18, and 3-19 show the model-to-data comparisons for the specific heat capac-ity correlation at constant pressure used in MatLib for Kanthal APMT, C35M, and C36M, respec-tively, using experimentally measured data from non-rradiated FeCrAl alloys [Field, 2018]. The large peaks seen in the figures represent the second order phase transition from the materials ferromagnetic to paramagnetic state.
400 600 800 1000 1200 1400 Temperature [K]
0 100 200 300 400 500 600 700 800 Specific Heat Capacity [J/kg-K]
MatLib Field [2018]
Figure 3-17. Model-to-Data Comparison for Kanthal APMT FeCrAl Alloy Specific Heat Correla-tion Cladding Material Properties 67
PNNL-29728 400 600 800 1000 1200 1400 Temperature [K]
0 100 200 300 400 500 600 700 800 Specific Heat Capacity [J/kg-K]
MatLib Field [2018]
Figure 3-18. Model-to-Data Comparison for C35M FeCrAl Alloy Specific Heat Correlation 400 600 800 1000 1200 1400 Temperature [K]
0 100 200 300 400 500 600 700 800 Specific Heat Capacity [J/kg-K]
MatLib Field [2018]
Figure 3-19. Model-to-Data Comparison for C36M FeCrAl Alloy Specific Heat Correlation 3.2.2.3 Applicability and Uncertainty The specific heat capacity correlation is applicable to the range of available data:
Cladding Material Properties 68
PNNL-29728 Cladding types: Kanthal APMT, C35M, C36M Temperature: 300 to 1400 [K]
Burnup: unirradiated Engineering judgment should be used if analysis outside of these ranges is needed.
No uncertainty for the specific heat capacity is reported.
3.2.3 Melting Temperature The melting temperature of the various alloys of FeCrAl is modeled in MatLib as a constant value
[Kanthal, 2018]:
Tm = 1773.15 [K]
(3-29)
- Where, Tm = Melting temperature [K]
3.2.3.1 Applicability and Uncertainty The melting temperature model is applicable over the following ranges:
Cladding types: Kanthal APMT, C35M, C36M Rod-average Burnup: No burnup dependence observed No uncertainty is given.
3.2.4 Thermal Expansion The thermal expansion coefficient of FeCrAl alloys is modeled in MatLib as a function of one pa-rameter:
1.
Temperature The thermal expansion coefficient is assumed to be isotropic.
Cladding Material Properties 69
PNNL-29728 3.2.4.1 Model Description The thermal expansion coefficient correlation in MatLib is based on experimentally measured data at ORNL [Field, 2018]. The measured expansion data was fitted against a third order polynomial (Equation 3-30), where alloy-dependent fitting constants are used to represent the various types of FeCrAl alloys.
= A0 + A1T + A2T 2 + A3T 3 (3-30)
- Where,
= Thermal expansion coefficient,
K1
Ax = Fitting constants T = Temperature, [K]
Table 3-8 provides the values of the fitting coefficients used to determine the thermal expansion coefficient for the various types of FeCrAl alloys [Field, 2018].
Table 3-8. Constants Used in the FeCrAl Thermal Expansion Correlation Alloy A0 A1
1 x 103
A2
1 x 107
A3
1 x 1010
Kanthal APMT 10.27 1.937 9.558 1.771 C35M 9.810 4.530
-17.46 9.095 C36M 10.56 2.535 2.719 3.079 3.2.4.2 Comparison to Data Thermal expansion data have been collected for FeCrAl samples [Field, 2018]. A model-to-data comparison is presented in Figures 3-20, 3-21, and 3-22 for Kanthal APMT, C35M, and C36M, respectively.
Cladding Material Properties 70
PNNL-29728 400 600 800 1000 1200 1400 Temperature [K]
0 2
4 6
8 10 12 14 16 Thermal Expansion Coefficient [1/K]
MatLib Field [2018]
Figure 3-20. Model-to-Data Comparison for Kanthal APMT FeCrAl Alloy Thermal Expansion Co-efficient 400 600 800 1000 1200 1400 Temperature [K]
0 2
4 6
8 10 12 14 16 Thermal Expansion Coefficient [1/K]
MatLib Field [2018]
Figure 3-21. Model-to-Data Comparison for C35M FeCrAl Alloy Thermal Expansion Coefficient Correlation Cladding Material Properties 71
PNNL-29728 400 600 800 1000 1200 1400 Temperature [K]
0 2
4 6
8 10 12 14 16 Thermal Expansion Coefficient [1/K]
MatLib Field [2018]
Figure 3-22. Model-to-Data Comparison for C36M FeCrAl Alloy Thermal Expansion Coefficient Correlation 3.2.4.3 Applicability and Uncertainty The correlation is applicable to the range of available data:
Cladding types: Kanthal APMT, C35M, C36M Temperature: 300 to 1500 [K]
Burnup: unirradiated Engineering judgment should be used if analysis outside of these ranges is needed.
No uncertainty for the thermal expansion coefficient is reported.
3.2.5 Emissivity The emissivity of the various alloys of FeCrAl is modeled in MatLib as a constant value [Kanthal, 2018]:
= 0.7 [unitless]
(3-31)
- Where,
= Emissivity [unitless]
Cladding Material Properties 72
PNNL-29728 3.2.5.1 Applicability and Uncertainty The emissivity model is applicable over the following ranges:
Cladding types: Kanthal APMT, C35M, C36M Temperature: No temperature dependence observed Rod-average Burnup: No burnup dependence observed No uncertainty is given on the emissivity.
3.2.6 Density The densities for the various FeCrAl alloys in MatLib are modeled as constant values per Table 3-9.
Table 3-9. Densities of Various FeCrAl Alloys Alloy Density
kg/m3
Kanthal APMT 7250 C35M 7180 C36M 7180 3.2.6.1 Applicability and Uncertainty The density model is applicable over the following ranges:
Cladding types: Kanthal APMT, C35M, C36M Temperature: No temperature dependence observed Rod-average Burnup: No burnup dependence observed No uncertainty is given.
3.2.7 Youngs Modulus and Shear Modulus Youngs modulus (or elastic modulus) and shear modulus for FeCrAl-based cladding are modeled within MatLib as a function of one parameter:
1.
Temperature Cladding Material Properties 73
PNNL-29728 3.2.7.1 Model Description The Youngs modulus and shear modulus are related by Poissons ratio according to:
G =
E 2 (1 + )
(3-32)
- Where, G = Shear modulus [Pa]
E = Youngs modulus [Pa] (Equation 3-33)
= Poissons ratio [unitless] (Equation 3-34)
Youngs Modulus The Youngs modulus of FeCrAl alloys is given by [Field, 2018]:
E = 199 3.85 x 102T 5.46 x 105T 2 (3-33)
- Where, E = Youngs modulus [Pa]
T = Temperature [C]
Poissons Ratio Poissons ratio for FeCrAl alloys is given by [Field, 2018]:
= 4.46 x 105T + 0.27 (3-34)
- Where,
= Poissons ratio [unitless]
T = Temperature [C]
3.2.7.2 Comparison to Data Youngs modulus data have been collected for C35M and C36M [Field, 2018] and Kanthal APMT
[Kanthal, 2018]. A model-to-data comparison is presented in Figure 3-23.
Cladding Material Properties 74
PNNL-29728 200 400 600 800 1000 Temperature [C]
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Elastic Modulus [Pa]
1e11 MatLib Kathal APMT [Kanthal 2018]
C35M [Field 2018]
C36M [Field 2018]
Figure 3-23. Model-to-Data Comparison for FeCrAl Alloys Elastic Modulus Correlation 3.2.7.3 Applicability and Uncertainty The Youngs modulus and shear modulus models are applicable to the range of available data:
Cladding types: Kanthal APMT, C35M, C36M Temperature: 25 to 800 [C]
Rod-average Burnup: No burnup dependence observed No uncertainty is given.
3.2.8 Meyers Hardness The Meyers hardness model for FeCrAl alloys utilizes the same Meyers hardness model for zirconium-based alloy (see Section 3.1.8).
3.2.9 Axial Growth The axial irradiation growth of FeCrAl alloy cladding is modeled in MatLib as a function of one parameter:
1.
Fast neutron fluence Cladding Material Properties 75
PNNL-29728 3.2.9.1 Model Description The axial irradiation growth of FeCrAl alloy is given by:
L L
= 0.5dpa 3
(3-35)
- Where, L
L
= Axial growth increment [m/m]
dpa = Fast neutron fluence per dispacement per atom =
0.9 1 x 1025 [dpa]
= Fast neutron (>1.0 MeV) fluence
n/m2
3.2.9.2 Comparison to Data No comparisons to measured data is provided in this document because of the limited availability of experimentally measured axial growth data.
3.2.9.3 Applicability and Uncertainty No uncertainty is given.
3.2.10 Strain (Creep) Rate The strain of FeCrAl alloy cladding is modeled in MatLib as a function of four parameters:
1.
Temperature 2.
Effective stress 3.
Fast neutron fluence 4.
Fast neutron flux The strain rate is assumed isotropic.
3.2.10.1 Model Description The thermal strain rate of FeCrAl alloy cladding is given by [Field, 2018]:
th = A0n exp
Q RT
(3-36)
- Where, Cladding Material Properties 76
PNNL-29728 th = Thermal strain rate,
s1
A0 = Constant
MPans1
(Table 3-10)
= Effective stress [Pa]
n = Creep exponent (Table 3-10)
Q = Activation energy [J/mol] (Table 3-10)
R = Universal gas constant = 8.314 [J/K mol]
T = Temperature [K]
Table 3-10. Constants Used in the FeCrAl Thermal Strain Rate Correlation Alloy A0
MPans1
n Q [J/mol]
Kanthal APMT 2.9 x 106 4.5 1.43 x 105 C35M and C35M for T < 873.15 [K]
2.9 x 103 5.5 2.47 x 105 C35M and C35M for T 873.15 [K]
5.96 x 106 5.5 3.92 x 105 The irradiation strain rate of FeCrAl alloy cladding is given by:
irr = Cirr dpa (3-37)
- Where, irr = Irradiation strain rate
s1
Cirr = Coefficient of irradiation strain = 5 x 1012 [dpa/Pa]
dpa = Fast neutron fluence per displacement per atom =
0.9 1 x 1025 [dpa]
= Fast neutron (>1.0 MeV) fluence
n/m2
= Effective stress [Pa]
= Fast neutron flux
n/m2 s
The total strain rate is the sum of the thermal and irradiation strain rates:
s = th + irr (3-38)
Cladding Material Properties 77
PNNL-29728 3.2.10.2 Comparison to Data Compiled thermal strain data for FeCrAl alloys is tabulated in Reference [Field, 2018]. This com-piled list of data presents the strain rate versus applied stresses for various FeCrAl alloys.
It has been noted that Kanthal APMT exhibit excellent strain strength properties when compared to wrought FeCrAl alloys; the MatLib model may over predict strain rates for Kanthal APMT.
3.2.10.3 Applicability and Uncertainty The strain rate model is applicable over the following ranges:
Cladding types: Kanthal APMT, C35M, C36M Temperature: No range specified Effective stress: No range specified No uncertainty on the strain rate is given.
3.3 HT-9 Alloy The following section describes the material property correlations used to model HT-9 alloy cladding properties in MatLib. HT-9 cladding is a ferritic stainless steel alloy cladding that may be used to contain metallic fuels in future nuclear reactors.
3.3.1 Thermal Conductivity The thermal conductivity model in MatLib for HT-9 cladding is a function of one parameter:
1.
Temperature The thermal conductivity of the cladding can also be a function of residual stress levels, crystal orientation, and minor composition differences. These effects are typically secondary and not ad-dressed in the current MatLib model of thermal conductivity. An accurate prediction of the cladding thermal conductivity is required to accurately predict the temperature profile of the fuel, including the centerline fuel temperature.
3.3.1.1 Model Description The thermal conductivity model is given by [Akiyama, 1991]:
k = A0 + A1T (3-39)
- Where, k = Cladding thermal conductivity [W/m K]
Cladding Material Properties 78
PNNL-29728 T = Temperature [K]
A0 = 22.47 [W/m K]
A1 = 4.397 x 103
W/m K2
3.3.1.2 Comparison to Data Due to limited experimental thermal conductivity data for HT-9, no comparison to experimental data is made but a model-to-model comparison is. Two thermal conductivity models are compared in Figure 3-24: [Akiyama, 1991] (used in MatLib) and [Leibowitz and Blomquist, 1988]. Both models are empirical correlations based on experimental measurements.As more data becomes available, data will plotted against the correlations.
300 400 500 600 700 800 Temperature [K]
0 5
10 15 20 25 Thermal Conductivity [W/m-K]
MatLib; Akiyama [1991]
Leibowitz and Blomquist [1988]
Figure 3-24. Model-to-Model Comparison for HT-9 Alloy Thermal Conductivity Correlations Several differences between the two correlations are seen. MatLib utilizes a linear correlation
[Akiyama, 1991] for the prediction of the thermal conductivity, whereas the Leibowitz model [Lei-bowitz and Blomquist, 1988] is developed from a second order polynomial. Comparison between the applicable temperature range is only made as the Leibowitz model provides an expression for above 1050 [K].
3.3.1.3 Applicability and Uncertainty The HT-9 thermal conductivity model is applicable for the following conditions:
Cladding types: HT-9 Cladding Material Properties 79
PNNL-29728 Temperature: 293 to 873 [K]
No uncertainty is given.
3.3.2 Specific Heat Capacity The specific heat capacity at constant pressure for HT-9 cladding is modeled in MatLib as a function of one parameter:
1.
Temperature 3.3.2.1 Model Description The specific heat model is based on experimental data [Yamanouchi et al., 1992]:
Cp = A0 + A1T (3-40)
- Where, Cp = Specific heat capacity at constant pressure [J/kg K]
T = Temperature [K]
Ax = Fitting constants (see Table 3-11)
Table 3-11 provides the values of the fitting constants.
Table 3-11. Constants Used in the HT-9 Specific Heat Capacity Correlation Alloy Valid Temperature Range [K]
A0
[J/kg K]
A1
J/kg K2
HT-9 T < 800.15 [K]
416.642 0.167 HT-9 T 800.15 [K]
69.910 0.600 3.3.2.2 Comparison to Data The MatLib specific heat capacity correlation for HT-9 cladding [Yamanouchi et al., 1992] is pre-sented in Figure 3-25. The two regions are clearly observed. Above 800.15 [K], there is a slight decrease in the rate the specific heat increases as a function of temperature. As more experimen-tal data becomes available, comparisons against the implemented MatLib model will be made and Figure 3-25 will be updated.
Cladding Material Properties 80
PNNL-29728 300 400 500 600 700 800 Temperature [K]
0 100 200 300 400 500 Specific Heat [J/kg-K]
MatLib; Yamanouchi et al. [1992]
Figure 3-25. HT-9 Alloy Specific Heat Capacity Correlation 3.3.2.3 Applicability and Uncertainty The specific heat capacity correlation is applicable over the following range of conditions:
Cladding types: HT-9 Temperature: 298 to 873 [K]
Burnup: unirradiated No uncertainty for the specific heat capacity is reported.
3.3.3 Melting Temperature The melting temperature for HT-9 cladding is modeled in MatLib using the eutectic temperature between HT-9 and metallic fuel as opposed to the melting temperature of pure HT-9 cladding as a eutectic forms between the cladding and fuel at a temperature lower than pure HT-9.
It is assumed constant [Baker and Wilson, 1992].
Tmelt = Teutectic = 973 [K]
(3-41) 3.3.4 Thermal Expansion The MatLib model for the thermal expansion of HT-9 is a function of one parameter:
Cladding Material Properties 81
PNNL-29728 1.
Temperature The thermal expansion is assumed isotropic.
3.3.4.1 Model Description The thermal expansion model is based on a second order polynomial [Yamanouchi et al., 1992]:
= A1 + A2T + A3T 2 (3-42)
- Where,
= Thermal expansion coefficient
K1
T = Temperature [K]
Ax = Fitting constants (see Table 3-12)
Table 3-12. Constants Used in the HT-9 Thermal Expansion Correlation x
Ax 1
2.882 x 103 2
9.226 x 106 3
1.842 x 109 3.3.4.2 Comparison to Data Due to limited experimental data measuring thermal expansion, no comparison to experimental data is made but a model-to-model comparison is. Two thermal expansion models are compared in Figure 3-26: [Yamanouchi et al., 1992] (used in MatLib) and [Leibowitz and Blomquist, 1988]. Both models are empirical correlations based on experimental data. As more data becomes available, data will plotted against the correlations.
Cladding Material Properties 82
PNNL-29728 300 400 500 600 700 800 Temperature [K]
0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 Thermal Expansion [m/m]
MatLib; Yamanouchi et al. [1992]
Leibowitz and Blomquist [1988]
Figure 3-26. Model-to-Model Comparison for HT-9 Alloy Thermal Expansion Correlations Minor differences between the models are seen. Both models follow the same trend but differ slightly in magnitude. Both models similarly predict the thermal expansion of HT-9 at lower temper-ature but as temperatures increase, the models begin to predict different thermal expansion strains.
Comparison between the temperature range of 298 to 873 [K] is made as only the Leibowitz model provides an applicable expression for higher temperature.
3.3.4.3 Applicability and Uncertainty The thermal expansion correlation is applicable over the following range of conditions:
Cladding types: HT-9 Temperature: 298.15 to 1073.15 [K]
No uncertainty is given.
3.3.5 Emissivity The emissivity of HT-9 cladding is modeled in MatLib as a constant value [Dutt and Baker, 1974]:
= 0.9 [unitless]
(3-43)
- Where,
= Emissivity [unitless]
Cladding Material Properties 83
PNNL-29728 3.3.5.1 Applicability and Uncertainty The emissivity model is applicable over the following range of conditions:
Cladding types: HT-9 Temperature: No temperature dependence observed Burnup: No burnup dependence observed No uncertainty is given.
3.3.6 Density 3.3.6.1 Model Description The density of HT-9 cladding is modeled in MatLib as a constant value [Akiyama, 1991]:
= 7750 h
kg/m3i (3-44)
- Where,
= Density
kg/m3
3.3.6.2 Applicability and Uncertainty The density model is applicable over the following range of conditions:
Cladding types: HT-9 Temperature: No temperature dependence observed Burnup: No burnup dependence observed No uncertainty is given.
3.3.7 Youngs Modulus Youngs modulus (or elastic modulus) for HT-9 cladding is modeled as a function of one parameter:
1.
Temperature Cladding Material Properties 84
PNNL-29728 3.3.7.1 Model Description The Youngs modulus model in MatLib is a linear function with fitting constants based on experi-mental measurements [Akiyama, 1991]:
E = A0 + A1T (3-45)
- Where, E = Youngs modulus [Pa]
T = Temperature [C]
A0 = Fitting constant = 2.137 x 1011 [Pa]
A1 = Fitting constant = 1.0274 x 108 [Pa/C]
3.3.7.2 Applicability and Uncertainty The Youngs modulus model is applicable over the following range of conditions:
Cladding types: HT-9 Temperature: 298.15 to 873.15 [K]
Burnup: No burnup dependence observed No uncertainty is given.
3.3.8 Shear Modulus The shear modulus of HT-9 cladding is modeled as a function of one parameter:
1.
Temperature 3.3.8.1 Model Description The shear modulus of HT-9 cladding is given by:
G = A0 + A1T (3-46)
- Where, G = Shear modulus [Pa]
Cladding Material Properties 85
PNNL-29728 T = Temperature [C]
A0 = Fitting constant = 8.964 x 1010 [Pa]
A1 = Fitting constant = 5.378 x 107 [Pa/C]
The shear modulus decreases with increasing temperature.
3.3.8.2 Applicability and Uncertainty The shear modulus model is applicable over the following range of conditions:
Cladding types: HT-9 Temperature: 298.15 to 873.15 [K]
Burnup: No burnup dependence observed No uncertainty is given.
3.3.9 Meyers Hardness The Meyers hardness model for HT-9 cladding utilizes the same Meyers hardness model for zirconium-based cladding (see Section 3.1.8).
3.3.10 Strain (Creep) Rate The strain rate of HT-9 cladding is modeled in MatLib as a function of four parameters:
1.
Time 2.
Effective stress 3.
Temperature 4.
Fast neutron flux 3.3.10.1 Model Description The thermal strain rate model is based on the proposed model by [Akiyama, 1991]. The thermal creep rate is a summation of the primary, secondary, and tertiary thermal creep rates:
th = 1 + 2 + 3 (3-47)
- Where, th = Total thermal strain rate
s1
Cladding Material Properties 86
PNNL-29728 1 = Primary thermal strain rate
s1
2 = Secondary thermal strain rate
s1
3 = Tertiary thermal strain rate
s1
- and, 1 =
C1 exp
Q1 RT
+ C24 exp
Q2 RT
+ C3 exp
Q3 RT
C4 exp (C4t)
(3-48a) 2 = C52 exp
Q4 RT
+ C65 exp
Q5 RT
(3-48b) 3 = 410
C7 exp
Q6 RT
t
3 (3-48c)
- Where, Cx, Qx = Fitting constants (Table 3-13) t = Time [s]
= Effective stress at time t [MPa]
R = Universal gas constant = 1.987 [cal/mol K]
T = Temperature [K]
Table 3-13 shows the fitting constants used to determine the thermal strain rate for HT-9 alloy. The fitting constants are taken from [Akiyama, 1991].
Table 3-13. Constants Used in the HT-9 Thermal Strain Rate Correlation x
Cx Qx 1
13.4 15027.0 2
8.43 x 103 26451.0 3
4.08 x 1018 89167.0 4
1.6 x 106 83142.0 5
1.17 x 109 108276.0 6
8.33 x 109 94233.3 7
2.12 x 107 Cladding Material Properties 87
PNNL-29728 The irradiation strain rate correlation is also an empirical-based model:
irr =
B0 + A1 exp
Qirr RT
1.3
x 1022 (3-49)
- Where, i = Irradiated creep rate
s1
B0 = Fitting constant = 1.83 x 104 A1 = Fitting constant = 2.59 x 1014 Qirr = Fitting constant = 73000.0
= Fast neutron flux (E > 1 MeV)
n/cm2 s
= Effective stress at time t [MPa]
R = Universal gas constant = 1.987 [cal/mol K]
T = Temperature [K]
The total strain rate is the sum of the thermal and irradiation strain rates:
s = th + irr (3-50) 3.3.10.2 Applicability and Uncertainty The strain rate model is applicable for the following conditions:
Cladding types: HT-9 Temperature: 298.15 to 873.15 [K]
Burnup: No burnup dependence found No uncertainty is given.
3.3.11 Yield Stress The MatLib model for the yield stress of HT-9 cladding is a function of one parameter:
1.
Temperature The ultimate tensile stress for HT-9 cladding is assumed to be equal to the yield stress.
Cladding Material Properties 88
PNNL-29728 3.3.11.1 Model Description The yield stress model is given by [Akiyama, 1991]:
y = A1 + A2T + A3T 2 + A4T 3 (3-51)
- Where, y = Yield stress [Pa]
Ax = Fitting constants (see Table 3-14)
T = Temperature [K]
Table 3-14. Constants Used in the HT-9 Yield Stress Correlation x
Ax 1
1.290 x 109 2
3.561 x 106 3
6.371 x 103 4
3.959 3.3.11.2 Applicability and Uncertainty The yield stress model is applicable over the following range of conditions:
Cladding types: HT-9 Temperature: 298.15 to 873.15 [K]
Burnup: No burnup dependence found No uncertainty is given.
Cladding Material Properties 89
PNNL-29728 4.0 Gas Material Properties This section describes material property correlations for gap gases. The modeled gases include:
Helium Argon Krypton Xenon Hydrogen Nitrogen Air Water Vapor 4.1 Thermal Conductivity For gases other than water vapor, the thermal conductivity is modeled in MatLib as a function of one parameter:
1.
Temperature For water vapor, the thermal conductivity is modeled in MatLib as a function of two parameters:
1.
Temperature 2.
Pressure 4.1.1 Model Description For gases other than water vapor, the thermal conductivity is given by:
k = AT B (4-1)
- Where, k = Gas thermal conductivity [W/m K]
T = Temperature [K]
A, B = Constants (see Table 4-1)
Gas Material Properties 90
PNNL-29728 The parameters A and B used for each gas are given in the table below.
Table 4-1. Constants Used in the Gas Thermal Conductivity Correlation Gas A
B He 2.531 x 103 0.7146 Ar 4.092 x 104 0.6748 Kr 1.966 x 104 0.7006 Xe 9.825 x 105 0.7334 H2 1.349 x 103 0.8408 N2 2.984 x 104 0.7799 Air 1.945 x 104 0.8586 Water Vapor The thermal conductivity of water vapor is given by:
For T 973.15 [K]
k =P T
2.8516 x 108 + 9.424 x 1010T6.005 x 1014T 2
+ 1.009 P 2 T 2(T 273.15)4.2 + 1.76 x 103 + 5.87 x 105 (T 273.15)
+ 1.08 x 107(T 273.15)24.51 x 1011(T 273.15)3 (4-2)
For T > 973.15 [K]
k = 4.44 x 106T 1.45 + 9.45 x 105
2.1668 x 109 P T
1.3 (4-3)
- Where, k = Gas thermal conductivity [W/m K]
P = Gas pressure [Pa]
T = Temperature [K]
The thermal conductivity of gas mixtures is calculated by [Hagrman et al., 1981]:
kmix =
n X
i kixi xi + Pn j=1 (1 ij)ijxj (4-4)
Gas Material Properties 91
PNNL-29728
- Where, ij = ij 1 + 2.41(Mi Mj) (Mi 0.142Mj)
(Mi + Mj)2 (4-5) ij =
1 +
ki kj
1/2
Mi Mj
1/42 23/2
1 + Mi Mj
1/2 (4-6)
- and, ij = Kronecker delta = 1 for i = j, 0 otherwise [unitless]
n = Number of components in mixture [unitless]
Mi = Molecular weight of component i [kg]
xi = Mole fraction of component i [unitless]
ki = Thermal conductivity of component i [W/m K]
4.1.2 Comparisons to Data Thermal conductivity data have been collected for helium at various temperatures [Johnston and Grilly, 1946], [Saxena and Saxena, 1968], [Timrott and Totskii, 1965], [Timrot and Umanskii, 1966],
[Zaitseva, 1959], [Cheung et al., 1962], [Kannuluik and Carman, 1952], [Gambhir et al., 1967], [von Ubisch, 1959], [Faubert and Springer, 1973], [Jain and Saxena, 1975], and [Jody et al., 1977]. A comparison between these data for helium is presented in Figure 4-5. This comparison demon-strates good agreement between the correlation and the database between 273 and 2500 [K].
Gas Material Properties 92
PNNL-29728 500 1000 1500 2000 2500 Temperature [K]
0.0 0.2 0.4 0.6 0.8 1.0 Thermal Conductivity [W/m-K]
MatLib [He]
Johnston and Grilly [1946]
Saxena and Saxena [1968]
Timrot and Totskii [1965]
Timrot and Umanskii [1965]
Zaitseva [1959]
Cheung et al. [1962]
Kannuluik and Carman [1952]
Gambhir et al. [1967]
von Ubisch [1959]
Faubert and Springer [1973]
Jain and Saxena [1975]
Jody et al. [1977]
Figure 4-1. Model-to-Data Comparison for Helium Thermal Conductivity Correlation Thermal conductivity data have been collected for argon at various temperatures [Brokaw, 1969],
[Zaitseva, 1959], [Cheung et al., 1962], [Kannuluik and Carman, 1952], [Gambhir et al., 1967], [von Ubisch, 1959], [Timrot and Umanskii, 1966], [Saxena and Saxena, 1968], [Faubert and Springer, 1972], [Springer and Wingeier, 1973], and [Stefanov et al., 1976].A comparison between these data for argon is presented in Figure 4-2. This comparison demonstrates good agreement between the correlation and the database between 273 and 2500 [K].
Gas Material Properties 93
PNNL-29728 500 1000 1500 2000 2500 Temperature [K]
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Thermal Conductivity [W/m-K]
MatLib [Ar]
Brokaw [1969]
Zaitseva [1959]
Cheung et al. [1962]
Kannuluik and Carman [1952]
Gambhir et al. [1967]
von Ubisch [1959]
Timrot and Umanskii [1966]
Saxena and Saxena [1968b]
Faubert and Springer [1972]
Springer and Wingeier [1973]
Stefanov et al. [1976]
Figure 4-2. Model-to-Data Comparison for Argon Thermal Conductivity Correlation Thermal conductivity data have been collected for krypton at various temperatures [Kannuluik and Carman, 1952], [Gambhir et al., 1967], [von Ubisch, 1959], [Saxena and Saxena, 1969], [Stefanov et al., 1976], [Vargaftik and Yakush, 1971], [Zaitseva, 1959]. A comparison between these data for krypton presented in Figure 4-3. This comparison demonstrates good agreement between the correlation and the database between 273 and 2300 [K].
Gas Material Properties 94
PNNL-29728 500 1000 1500 2000 2500 Temperature [K]
0.00 0.01 0.02 0.03 0.04 Thermal Conductivity [W/m-K]
MatLib [Kr]
Kannuluik and Carman [1952]
Gambhir et al. [1967]
von Ubisch [1959]
Saxena and Saxena [1969]
Stefanov et al. [1976]
Vargaftik and Yakush [1971]
Zaitseva [1959]
Figure 4-3. Model-to-Data Comparison for Krypton Thermal Conductivity Correlation Thermal conductivity data have been collected for xenon at various temperatures [Zaitseva, 1959],
[Kannuluik and Carman, 1952], [Gambhir et al., 1967], [von Ubisch, 1959], [Stefanov et al., 1976],
[Springer and Wingeier, 1973], [Saxena and Saxena, 1969], [Vargaftik and Yakush, 1971]. A com-parison between these data for xenon is presented in Figure 4-4. This comparison demonstrates good agreement between the correlation and the database between 273 and 2200 [K].
Gas Material Properties 95
PNNL-29728 500 1000 1500 2000 2500 Temperature [K]
0.000 0.005 0.010 0.015 0.020 0.025 0.030 Thermal Conductivity [W/m-K]
MatLib [Xe]
Zaitseva [1959]
Kannuluik and Carman [1952]
Gambhir et al. [1967]
von Ubisch [1959]
Stefanov [1976]
Springer and Wigneier [1973]
Saxena and Saxena [1969]
Vargaftik and Yakush [1971]
Figure 4-4. Model-to-Data Comparison for Xenon Thermal Conductivity Correlation Thermal conductivity data have been collected for hydrogen at various temperatures [Johnston and Grilly, 1946], [Timrot and Umanskii, 1966], [Saxena and Saxena, 1970]. A comparison between these data for hydrogen is presented in Figure 4-5. This comparison demonstrates good agreement between the correlation and the database between 273 and 2000 [K].
Gas Material Properties 96
PNNL-29728 500 1000 1500 2000 2500 Temperature [K]
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Thermal Conductivity [W/m-K]
MatLib [H2]
Johnston and Grilly [1946]
Timrot and Umanskii [1966]
Saxena and Saxena [1970]
Figure 4-5. Model-to-Data Comparison for Hydrogen Thermal Conductivity Correlation Thermal conductivity data have been collected for nitrogen at various temperatures [Cheung et al.,
1962], [Brokaw, 1969], [Vargaftik and Zimina, 1964], [Faubert and Springer, 1972], [Chen and Sax-ena, 1973]. A comparison between these data for nitrogen is presented in Figure 4-6. This compar-ison demonstrates good agreement between the correlation and the database between 273 and 2500 [K].
Gas Material Properties 97
PNNL-29728 500 1000 1500 2000 2500 Temperature [K]
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Thermal Conductivity [W/m-K]
MatLib [N2]
Cheung et al. [1962]
Brokaw [1969]
Vargaftik and Zimina [1964]
Faubert and Springer [1972]
Chen and Saxena [1973]
Figure 4-6. Model-to-Data Comparison for Nitrogen Thermal Conductivity Correlation Thermal conductivity data have been collected for steam at various temperatures and 1 x 107 [Pa]
[Hagrman et al., 1981]. A comparison between these data for steam is presented in Figure 4-7.
This comparison demonstrates reasonable agreement between the correlation and the database between 600 and 973 [K].
Gas Material Properties 98
PNNL-29728 600 650 700 750 800 850 900 950 Temperature [K]
0.00 0.02 0.04 0.06 0.08 0.10 Thermal Conductivity [W/m-K]
MatLib [Steam]
Steam at P=1e7 Pa; Hagrman et al. [1981]
Figure 4-7. Model-to-Data Comparison for Steam Thermal Conductivity Correlation Thermal conductivity data have been collected for various gas mixtures at various temperatures
[Andrew and Calvert, 1966]. A comparison between these data is presented in Figure 4-8. This comparison demonstrates reasonable agreement between the correlation and the database be-tween 273 and 800 [K].
Gas Material Properties 99
PNNL-29728 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Measured Thermal Conductivity [W/m-K]
0.00 0.02 0.04 0.06 0.08 0.10 0.12 Predicted Thermal Conductivity [W/m-K]
Predicted = Measured Andrew and Calvert [1966]
Figure 4-8. Model-to-Data Comparison for Gas Mixture Thermal Conductivity Correlation 4.1.3 Applicability and Uncertainty The thermal conductivity is applicable for the following range of conditions:
Temperature:
Helium, argon, nitrogen: 273 to 2500 [K]
Krypton: 273 to 2300 [K]
Xenon: 273 to 2200 [K]
Hydrogen: 273 to 2000 [K]
Steam: 600 to 973 [K]
Gas mixtures: 273 to 800 [K]
The uncertainty of the correlation is given below for all gases as an absolute standard error:
Helium: 8.99 x 103 [W/m K]
Argon: 9.66 x 104 [W/m K]
Krypton: 8.86 x 104 [W/m K]
Xenon: 5.34 x 104 [W/m K]
Hydrogen: 1.67 x 102 [W/m K]
Nitrogen: 1.99 x 103 [W/m K]
Steam: 1.75 x 102 [W/m K]
Gas Material Properties 100
PNNL-29728 5.0 Oxide/CRUD Material Properties 5.1 Zirconium Dioxide (ZrO2) 5.1.1 Thermal Conductivity The thermal conductivity of zirconium dioxide (ZrO2) that forms in-reactor on zirconium-based alloy cladding tubes is modeled in MatLib as a function of one parameter:
1.
Temperature 5.1.1.1 Model Description The thermal conductivity of ZrO2 is given by:
k = 1.9599 2.41 x 104T + 6.43 x 107T 2 1.946 x 1010T 3 (5-1)
- Where, k = ZrO2 thermal conductivity [W/m K]
T = Temperature [K]
5.1.1.2 Comparison to Data Thermal conductivity data have been collected for ZrO2 that is prototypic to that found on zirconium alloy cladding [Kingery et al., 1954] [Adams, 1954]. A comparison between these data is presented in Figure 5-1. This comparison demonstrates a good agreement between the correlation and the database within a range of 285 to 1770 [K].
Oxide/CRUD Material Properties 101
PNNL-29728 400 600 800 1000 1200 1400 1600 Temperature [K]
0.0 0.5 1.0 1.5 2.0 2.5 3.0 Thermal Conductivity [W/m-K]
MatLib Kingery et al. [1954]
Adams [1954]
Figure 5-1. Model-to-Data Comparison for ZrO2 Thermal Conductivity Correlation 5.1.1.3 Applicability and Uncertainty The thermal conductivity model is applicable to the range of available data:
Oxide layer on cladding types: Zircaloy-4, Zircaloy-2, M5TM, ZIRLO and Optimized ZIRLOTM Temperature: 285 to 1770 [K]
Rod-average burnup: No burnup dependence observed Engineering judgment should be used if analysis outside of these ranges is needed.
The uncertainty of the correlation is given below. No variation in thermal conductivity uncertainty is observed with increasing temperature, so an absolute uncertainty is used.
ZrO2: = 0.14 [W/m K]
5.1.2 Specific Heat Capacity The specific heat capacity of ZrO2 is modeled in MatLib as a constant value. A range of values were found [AZO Materials, 2015]; however, their applicability to ZrO2 formed in-reactor is unknown. The values found ranged from 420 to 540 [J/kg K]. For conservatism, the upper bound value is used.
Cp = 540 [J/kg K]
(5-2)
Oxide/CRUD Material Properties 102
PNNL-29728
- Where, Cp = Specific heat capacity of ZrO2 [J/kg K]
5.1.2.1 Applicability and Uncertainty An upper and lower bound of 540 [J/kg K] and 420 [J/kg K] have been observed. No uncer-tainty is given.
5.1.3 Melting Temperature 5.1.3.1 Model Description The melting temperature of ZrO2 is modeled in MatLib as a constant value. A range of values were found [AZO Materials, 2015]; however, their applicability to ZrO2 formed in-reactor is unknown. The values found ranged from 2823 to 2973 [K]. For conservatism, the lower bound value is used.
Tmelt = 2823 [K]
(5-3)
- Where, Tmelt = Melting temperature of ZrO2 [K]
5.1.3.2 Applicability and Uncertainty An upper and lower bound of 2973 [K] and 2823 [K] have been observed. No uncertainty is given.
5.1.4 Density The density of ZrO2 is modeled in MatLib as a constant value:
= 5680 h
kg/m3i (5-4)
- Where,
= density of ZrO2
kg/m3
5.2 CRUD Modeling CRUD in a fuel performance code is challenging for a number of reasons. The presence or absence of CRUD is highly dependent on small changes in coolant chemistry and other oper-ational parameters. Additionally, there are several types of CRUD that are observed. Tenacious CRUD is hard and does not easily brush off. This type of CRUD is often included in the measure-ment of oxide thickness but can be modeled separately from oxide. Fluffy CRUD is sometimes observed and can easily be brushed off. The effective thermal conductivity of this layer is large and is typically not modeled in the modeling of nuclear fuel rods. For BWR applications, the effect of Oxide/CRUD Material Properties 103
PNNL-29728 CRUD is not typically modeled as it is assumed that the water can boil through the CRUD layer that is typically observed.
For application in FAST, the following properties are assumed for modeling the thermal effects of the tenacious CRUD layer in PWR applications. Due to the scarcity of data, no attempt has been made to quantify uncertainties or perform data comparisons to any of these quantities.
5.2.1 Thermal Conductivity The thermal conductivity of CRUD is modeled in MatLib as a constant value:
k = 0.8648 [W/m K]
(5-5)
- Where, k = Thermal conductivity of CRUD [W/m K]
5.2.2 Specific Heat Capacity The specific heat of CRUD is modeled in MatLib as a constant value:
Cp = 800 [J/kg K]
(5-6)
- Where, Cp = Specific heat capacity of CRUD [J/kg K]
5.2.3 Density The density of CRUD is modeled in MatLib as a constant value [Wilson and Comstock, 1999]:
= 1200 h
kg/m3i (5-7)
- Where,
= Density of CRUD
kg/m3
Oxide/CRUD Material Properties 104
PNNL-29728 6.0 Fluid Material Properties This section describes material property correlations for the following fluids:
Water Sodium 6.1 Water The thermodynamic water properties contained in MatLib are based off of the 1967 ASME Steam Tables [Meyer et al., 1967].
The water properties package used in MatLib is based on the STH2X Water Properties Subroutines
[Wagner, 1977]. The subroutines derived from the STH2X package include the following:
sth2x0 = Calculates the saturation pressure as a function of temperature sth2x2 = Calculates saturation properties as a function of pressure and quality sth2x3 = Calculates single phase thermodynamic properties as a function of temperature and pressure sth2x5 = Calculates single phase thermodynamic properties as a function of pressure and enthalpy The properties modeled by the package include the following:
1.
Enthalpy 2.
Specific heat 3.
Specific volume 4.
Density 5.
Entropy 6.
Thermal expansion 7.
Isothermal compressibility 8.
Temperature 9.
Saturation pressure and temperature 10.
Quality For more information regarding the water properties, see the references listed above.
Fluid Material Properties 105
PNNL-29728 6.2 Sodium 6.2.1 Thermal Conductivity The thermal conductivity of liquid sodium is modeled in MatLib as a function of one parameter:
1.
Temperature 6.2.1.1 Model Description The thermal conductivity of liquid sodium is given by [Fink and Leibowitz, 1995]:
k = 124.67 0.11381T + 5.5226 x 105T 2 1.1842 x 108T 3 (6-1)
- Where, k = Thermal conductivity of sodium [W/m K]
T = Temperature [K]
6.2.1.2 Applicability and Uncertainty The thermal conductivity model for liquid sodium is applicable over the following ranges of condi-tions:
Material: Sodium Temperature: Tmelt (371.944 [K]) to 1500 [K]
No uncertainty is given.
6.2.2 Viscosity The viscosity of liquid sodium is modeled in MatLib as a function of one parameter:
1.
Temperature 6.2.2.1 Model Description The viscosity of liquid sodium is given by [Fink and Leibowitz, 1995]:
= exp
6.4406 0.3958 ln (T) + 556.835 T
(6-2)
- Where, Fluid Material Properties 106
PNNL-29728
= Viscosity of liquid sodium [Pa s]
T = Temperature [K]
6.2.2.2 Applicability and Uncertainty The viscosity model for liquid sodium is applicable over the following ranges of conditions:
Material: Sodium Temperature: Tmelt (371.944 [K]) to 2500 [K]
The following relative uncertainties should be applied to the viscosity model:
[%] =
(
2.3 + 0.0018T for Tmelt (371.944 [K]) < T 1500 [K]
10 + 0.01T for 1500 [K] < T 2500 [K]
- Where, T = Temperature [K]
6.2.3 Density The density of liquid sodium is modeled in MatLib as a function of one parameter:
1.
Temperature 6.2.3.1 Model Description The density of liquid sodium is given by [Fink and Leibowitz, 1995]:
= c + f
1 T Tc
+ g
1 T Tc
h (6-3)
- Where,
= Density
kg/m3
c = Density at the critical temperature of sodium = 219.0
kg/m3
f = Constant = 275.32
kg/m3
T = Temperature [K]
Fluid Material Properties 107
PNNL-29728 Tc = Critical temperature of sodium = 2503.7 [K]
g = Constant = 511.58
kg/m3
h = Constant = 0.5 [unitless]
6.2.3.2 Applicability and Uncertainty The density model for liquid sodium is applicable over the following ranges of conditions:
Material: Sodium Temperature: Tmelt (371.944 [K]) to Tc (2503.7 [K])
No uncertainty is given.
6.2.4 Specific Heat Capacity The specific heat capacity of liquid sodium is modeled in MatLib as a function of one parameter:
1.
Temperature 6.2.4.1 Model Description The specific heat capacity of liquid sodium is given by [Fink and Leibowitz, 1995]:
Cp = 1658.2 0.8479T + 4.4541 x 104T 2 2.9926 x 106 T 2 (6-4)
- Where, Cp = Specific heat capacity of liquid sodium [J/kg K]
T = Temperature [K]
6.2.4.2 Applicability and Uncertainty The specific heat capacity model for liquid sodium is applicable over the following ranges of con-ditions:
Material: Sodium Temperature: Tmelt (371.944 [K]) to Tc (2503.7 [K])
No uncertainty is given.
Fluid Material Properties 108
PNNL-29728 6.2.5 Enthalpy The enthalpy of liquid sodium is modeled in MatLib as a function of one parameter:
1.
Temperature 6.2.5.1 Model Description The enthalpy of liquid sodium is given by [Fink and Leibowitz, 1995]:
H =
3.6577 x 105 + 1658.2T 0.42395T 2
+1.4847 x 104T 3 + 2.9926 x 106 T
for 371.944 [K] T 2000 [K]
E + FT 0.5Hvap for 2000 [K] < T 2503.7 [K]
(6-5a)
- Where, Hvap = 393.37
1 T Tc
+ 4398.6
1 T Tc
0.29302 (6-5b) and H = Enthalpy of liquid sodium [J/kg]
T = Temperature [K]
Hvap = Enthalpy of vaporization [J/kg]
E = Constant = 2.1284 x 106 [J/kg]
F = Constant = 8.6496 x 102 [J/kg K]
Tc = Critical temperature of sodium = 2503.7 [K]
6.2.5.2 Applicability and Uncertainty The enthalpy model for liquid sodium is applicable over the following ranges of conditions:
Material: Sodium Temperature: Tmelt (371.944 [K]) to Tc (2503.7 [K])
The following relative uncertainties should be applied [Fink and Leibowitz, 1995]:
Fluid Material Properties 109
PNNL-29728
[%] =
1 for 371.944 [K] < T 1000 [K]
0.17 + 8.3 x 104T for 1000 [K] < T 1600 [K]
0.5 + 1.25 x 103T for 1600 [K] < T 2000 [K]
10 for 2000 [K] < T 2400 [K]
38 + 0.02T for 2400 [K] < T 2500 [K]
- Where, T = Temperature [K]
6.2.6 Melting Temperature The melting temperature of sodium is modeled in MatLib as a constant value [Fink and Leibowitz, 1995]:
Tmelt = 371.944 [K]
(6-6)
- Where, Tmelt = Metling temperature of sodium [K]
6.2.6.1 Comparison to Data No comparisons to data are provided as this is a theoretical quantity.
6.2.6.2 Applicability and Uncertainty The melting temperature of sodium is applicable over the following ranges of conditions:
Material: Sodium No uncertainty is given.
6.2.7 Vapor Pressure The vapor pressure of sodium is modeled in MatLib as a function of one parameter:
1.
Temperature Fluid Material Properties 110
PNNL-29728 6.2.7.1 Model Description The specific heat capacity of liquid sodium is given by [Fink and Leibowitz, 1995]:
P = exp
11.9463 12633.73 T
0.4672 ln (T)
(6-7)
- Where, P = Vapor pressure of liquid sodium [MPa]
T = Temperature [K]
6.2.7.2 Applicability and Uncertainty The vapor pressure model for sodium is applicable over the following ranges of conditions:
Material: Sodium Temperature: Tmelt (371.944 [K]) to Tc (2503.7 [K])
No uncertainty is given.
Fluid Material Properties 111
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