JAEA-DataCode
2013-005
July 2013
Japan Atomic Energy Agency 日本原子力研究開発機構
Motoe SUZUKI Hiroaki SAITOU Yutaka UDAGAWA and Fumihisa NAGASE
Light Water Reactor Fuel Analysis Code
FEMAXI-7 Model and Structure
Reactor Safety Research UnitNuclear Safety Research Center
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なお本レポートの全文は日本原子力研究開発機構ホームページ(httpwwwjaeagojp)
より発信されています
独立行政法人日本原子力研究開発機構 研究技術情報部 研究技術情報課
319-1195 茨城県那珂郡東海村白方白根 2 番地 4
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This report is issued irregularly by Japan Atomic Energy Agency
Inquiries about availability andor copyright of this report should be addressed to
Intellectual Resources Section Intellectual Resources Department
Japan Atomic Energy Agency
2-4 Shirakata Shirane Tokai-mura Naka-gun Ibaraki-ken 319-1195 Japan
Tel +81-29-282-6387 Fax +81-29-282-5920 E-mailird-supportjaeagojp
copy Japan Atomic Energy Agency 2013
i
JAEA-DataCode 2013 - 005
Light Water Reactor Fuel Analysis Code
FEMAXI-7 Model and Structure
Motoe SUZUKI Hiroaki SAITOU Yutaka UDAGAWA and Fumihisa NAGASE
Reactor Safety Research Unit
Nuclear Safety Research Center
Japan Atomic Energy Agency
Tokai-mura Naka-gun Ibaraki-ken
(Received March 14 2013)
A light water reactor fuel analysis code FEMAXI-7 has been developed for the purpose
of analyzing the fuel behavior in both normal conditions and anticipated transient conditions
This code is an advanced version which has been produced by incorporating the former
version FEMAXI-6 with numerous functional improvements and extensions In FEMAXI-7
many new models have been added and parameters have been clearly arranged Also to
facilitate effective maintenance and accessibility of the code modularization of subroutines
and functions have been attained and quality comment descriptions of variables or physical
quantities have been incorporated in the source code With these advancements the
FEMAXI-7 code has been upgraded to a versatile analytical tool for high burnup fuel behavior
analyses This report describes in detail the design basic theory and structure models and
numerical method of FEMAXI-7 and its improvements and extensions
Keywords LWR Fuel FEM Analysis Transient Pellet Cladding Fission Gas Release
PCMI Burn-up
ITOCHU Techno-Solutions Corporation Tokyo Japan
ii
JAEA-DataCode 2013 - 005
軽水炉燃料解析コード
FEMAXI-7 のモデルと構造
日本原子力研究開発機構安全研究センター
原子炉安全研究ユニット
鈴木 元衛斎藤 裕明宇田川 豊永瀬 文久
(2013 年 3 月 14 日受理)
FEMAXI-7 は軽水炉燃料の通常運転時及び過渡条件下のふるまい解析を目的と
するコードとして前バージョン FEMAXI-6 に対して多くの機能の追加改良を実施
した高度化バージョンである特にソースコードの整備及び解読の効率化を図るた
めにサブルーチンやファンクションのモジュール化とコメント記述の充実を図りコ
ードのさらなる拡張を容易にしたまた新しいモデルを追加するとともにユーザ
ーの使いやすさにも考慮して多くのモデルのパラメータを整理したこれらにより
FEMAXI-7は高燃焼度燃料の通常時のみならず過渡時ふるまいの解析に対する強力な
ツールとなった
本報告はFEMAXI-7 の設計基本理論と構造モデルと数値解法改良と拡張
採用した物性値等を詳述したものである
原子力科学研究所(駐在)319-1195 茨城県那珂郡東海村白方白根 2-4
伊藤忠テクノソリューションズ株式会社東京
iii
Contents
1Introduction 1
11 Characteristics of Fuel Performance Code 1
111 Features and roles of fuel code 1
112 Prediction of fuel behavior by FEMAXI 2
113 History of FEMAXI code development 3
114 Usage of FEMAXI 4
12 Features of Structure and Models of FEMAXI 5
121 What is model 5
122 What is fuel modeling 6
123 Analytical targets 7
13 Whole Structure 8
14 Features of Numerical Method and Modeling 11
15 Interfacing with Other Codes etc 14
151 Interfacing with burning analysis codes RODBURN and PLUTON 14
152 Re-start function from base-irradiation to test-irradiation periods 14
153 Execution system environments 14
References 1 15
2 Thermal Analysis Models 17
21 Heat Transfer to Coolant and Thermal-hydraulics Model 17
211 Coolant enthalpy increase model 20
212 Determination of cladding surface heat transfer coefficient 23
213 Flow of thermal-hydraulics calculation 33
214 Transition process of coolant condition 35
215 Equivalent diameter and cross-sectional area of flow channel 38
22 Cladding Waterside Corrosion Model 39
23 One-Dimensional Temperature Calculation Model 41
231 Selection of number of elements in the radial direction 41
232 Determination of cladding surface temperature 43
233 Solution of thermal conduction equation 45
JAEA-DataCode 2013-005
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234 Fuel pellet thermal conductivity 50
24 Determination of Heat Generation Density Profile 53
241 Use of RODBURN output 53
242 Use of PLUTON output 54
243 Robertsonrsquos formula 54
244 Accuracy confirmation of pellet temperature calculation 55
25 Gap Thermal Conductance Model 60
251 Modified Ross amp Stoute model 60
252 Bonding model for gap thermal conductance 63
253 Bonding model for mechanical analysis 65
254 Pellet relocation model 66
255 Swelling and densification models 68
26 Model of Dry-out in a Test Reactor Capsule 68
261 Modelling the dry-out experiment 69
262 Thermal and materials properties 73
263 Numerical solution method 74
27 Generation and Release of Fission Gas 78
271 Fission gas atoms generation rate 79
272 Concept of thermally activated fission gas release in FEMAXI 79
273 Thermal diffusion accompanied by trapping 81
274 White+Tucker model of intra-grain gas bubble radius and its number
density (GBFIS=0) 85
275 Radiation re-dissolution and number density model of intra-granular gas
bubbles (GBFIS=1) 88
276 Pekka Loumlsoumlnen model for intra-granular gas bubbles (GBFIS=2) 92
277 Galerkinrsquos solution method for partial differential diffusion equation
(common in IGASP=0 and 2) 94
278 Fuel grains and boundaries (common in IGASP=0 and 2) 99
279 Amount of fission gas atoms migrating to grain boundary
(common in IGASP=0 and 2) 100
2710 Amount of fission gas accumulated in grain boundary 101
2711 Amount of released fission gas 103
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2712 Sweeping gas atoms to grain boundary by grain growth 105
2713 Re-dissolution of fission gas (common in IGASP=0 and 2) 110
2714 Threshold density of fission gas atoms in grain boundary
(Equilibrium model IGASP=0) 111
2715 In-grain diffusion coefficient of fission gas atoms
(common to IGASP=0 and 2) 114
28 Rate- Law Model (IGASP=2) 117
281 Assumptions shared with the equilibrium model 117
282 Growth rate equation of grain boundary bubble 117
29 Swelling by Grain-boundary Gas Bubble Growth in the Equilibrium
Model (IGASP=0) 125
291 Grain boundary gas bubble growth 125
292 Bubble swelling 128
210 Swelling by Grain-boundary Gas Bubble Growth in the Rate-law Model
(IGASP=2 IFSWEL=1) 129
2101 Coalescence and coarsening of bubbles 129
2102 Swelling 132
211 Thermal Stress Restraint Model 132
2111 Option designated by IPEXT 133
2112 Comparative discussion on the gap gas conductance increase during
power down 134
212 High Burnup Rim Structure Model 135
2121 Basic concept 135
2122 Density decrease of fission gas atoms in solid phase of rim structure 136
2123 Effect of rim structure formation on fuel behavior 140
213 Selection of Models by Input Parameters 148
2131 Number of elements in the radial direction of pellet 148
2132 Calculation of thermal conductivity of fuel pellet 149
2133 Selecting swelling model 150
2134 Selecting fission gas release model 151
2135 Bubble growth and swelling 152
2136 Options for the rim structure formation model 153
JAEA-DataCode 2013-005
vi
214 Gap Gas Diffusion and Flow Model 154
2141 Assumptions and methods of diffusion calculation 155
215 Method to Obtain Free-space Volumes in a Fuel Rod 163
2151 Definition of free space 163
216 Internal Gas Pressure 168
2161 Method of assigning plenum gas temperature and cladding temperature 169
2162 Calculation of the variation in internal gas condition during irradiation 169
217 Time Step Control 170
2171 Automatic control 170
2172 Time step increment determination in FGR model 172
2173 Time step increment determination in temperature calculation 173
References 2 174
3 Mechanical Analysis Model 178
31 Solutions of Basic Equations and Non-linear Equations 178
311 Basic equations 178
312 Initial stress method initial strain method and changed stiffness method
for non-linear strain 184
313 Solution of the basic equations in FEMAXI (non-linear strain) 187
32 Mechanical Analysis of Entire Rod Length 194
321 Finite element model for entire rod length analysis (ERL) 195
322 Determination of finite element matrix 199
323 Derivation of stress-strain (stiffness) matrix of pellet and cladding 201
324 Pellet cracking 208
325 Crack expression in matrix 211
326 Definition equations of equivalent stress and strain in FEMAXI 214
327 Hot-pressing of pellet 216
328 Supplementary explanation of incremental method 219
33 Method to Calculate Non-linear Strain 223
331 Creep of pellet and cladding 223
332 Method of creep strain calculation 224
333 Plasticity of pellet and cladding 233
JAEA-DataCode 2013-005
vii
334 Derivation of stiffness equation 242
34 Formulation of Total Matrix and External Force 244
341 Stiffness equation and total matrix 244
342 Upper plenum and lower plenum boundary conditions 247
343 Pellet-cladding contact model 254
344 Axial force generation at the P-C contact surfaces 256
345 Algorithm of axial force generation at PCMI contact surface 257
346 Evaluation of axial force generated at the P-C contact surface Option A 260
347 Evaluation of Axial force generation Option B 265
35 2-D Local PCMI Mechanical Analysis 271
351 Element geometry 272
352 Basic equations 276
353 Stiffness equation 278
354 Boundary conditions 284
355 PCMI and contact state problem 289
356 Axial loading - Interaction between target pellet and upper pellet - 298
36 Mechanical Properties Model of Creep and Stress-strain 301
361 Creep 301
362 Model of cladding stress-strain relationship 305
363 Ohta model 307
364 Method to determine yield stress in FRAPCON and MATPRO
models (ICPLAS=3 6) 313
37 Skyline Method 315
References 3 322
4 Materials Properties and Models 323
41 UO2 Pellet 323
42 MOX Pellet 343
43 Zirconium Alloy Cladding 353
431 Cladding properties 353
432 Cladding oxide properties 368
44 SUS304 Stainless Steel 370
JAEA-DataCode 2013-005
viii
45 Other Materials Properties and Models 372
46 Burnup Calculation of Gd2O3-Containing Fuel 377
References 4 380
Acknowledgement 382
JAEA-DataCode 2013-005
ix
目 次
1序言 1
11 燃料解析コードの性格 1
111 燃料解析コードの特徴と性格 1
112 FEMAXI による燃料挙動の予測 2
113 FEMAXI コード開発の歴史 3
114 FEMAXI コードの利用 4
12 FEMAXI の構造とモデルの特徴 5
121 モデルとは何か 5
122 燃料モデリングとは何か 6
123 解析対象 7
13 全体構造 8
14 数値計算とモデルの特徴 11
15 他コードとの関係その他 14
151 燃焼解析コード RODBURN および PLUTON との接続 14
152 ベース照射から Re-start による試験照射への接続 14
153 実行環境など 14
参考文献 1 15
2 熱的解析モデル 17
21 冷却材への伝熱と熱水力モデル 17
211 冷却材のエンタルピー上昇のモデル 20
212 被覆管表面熱伝達率の決定 23
213 熱水力計算の流れ 33
214 冷却材状態の遷移 34
215 流路相当直径および流路断面積 38
22 被覆管水側腐食モデル 39
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23 一次元温度計算モデル 41
231 半径方向リング要素数の選択 41
232 被覆管表面温度の決定 43
233 熱伝導方程式の解法 45
234 ペレット熱伝導率 50
24 発熱密度プロファイルの決定 53
241 RODBURN 出力の利用 53
242 PLUTON 出力の利用 54
243 Robertson の式 54
244 ペレット温度計算の精度確認 55
25 ギャップ熱伝達モデル 60
251 修正 Ross amp Stoute モデル 60
252 熱伝達に対するギャップボンディングモデル 63
253 力学計算に対するギャップボンディングモデル 65
254 ペレットリロケーションモデル 66
255 スエリングと焼きしまりモデル 68
26 試験炉キャプセル内ドライアウトモデル 68
261 ドライアウト試験のモデル化 69
262 熱的材料的物性値 73
263 数値解法 74
27 FP ガス生成放出モデル 78
271 FP ガス原子生成速度 79
272 熱活性拡散放出モデルの概念 79
273 トラッピングを伴う熱拡散 81
274 粒内バブル半径と数密度の White+Tucker モデル(GBFIS=0) 85
275 粒内バブル半径と数密度の照射溶解モデル(GBFIS=1) 88
276 粒内バブルの Pekka Loumlsoumlnen モデル(GBFIS=2) 92
277 偏微分拡散方程式の Galerkin 解法(IGASP=0 と 2 で共通) 94
278 結晶粒と粒界(IGASP=0 と 2 で共通) 99
JAEA-DataCode 2013-005
xi
279 粒界へ移動する FP ガス原子の量(IGASP=0 と 2 で共通) 100
2710 粒界に溜まる FP ガス量 101
2711 放出されるガス量 103
2712 粒成長に伴う粒界へのガスの掃き出し 105
2713 FP ガスの再溶解(IGASP=0 と 2 で共通) 110
2714 粒界 FP ガス原子濃度の飽和値(平衡論 IGASP=0) 111
2715 FP ガス原子の粒内拡散定数(IGASP=0 と 2 で共通) 114
28 速度論モデル(IGASP=2) 117
281 平衡論モデルとの共通の仮定 117
282 粒界バブル成長速度式 117
29 粒界バブル成長スエリング平衡論(IGASP=0) 125
291 粒界バブル成長 125
292 バブルスエリング 128
210 バブル成長スエリング-速度論(IGASP=2 IFSWEL=1) 129
2101 バブルの合体粗大化 129
2102 スエリング 132
211 圧力抑制モデル 132
2111 IPEXT で指定されるオプション 133
2112 出力降下時の軸方向ガスコンダクタンス増加との比較考察 134
212 高燃焼度リム組織モデル 135
2121 基本概念 135
2122 リム組織固相における FP ガス原子の濃度減少 136
2123 リム組織の燃料ふるまいへの影響 140
213 パラメータによる計算モデルの選択 148
2131 ペレット径方向リング要素数 148
2132 ペレット熱伝導率計算 149
2133 スエリングモデルの選択 150
2134 FGR モデルの選択 151
2135 バブル成長とスエリング 152
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xii
2136 リム組織生成モデルのオプション 153
214 ギャップ内ガスの拡散流動モデル 154
2141 拡散計算の仮定と方法 155
215 燃料棒内空間体積の求め方 163
2151 自由空間の定義 163
216 内部ガス圧力 168
2161 プレナムのガス温度と被覆管温度の設定方法 169
2162 照射途中における内部ガス状態変更の計算 169
217 タイムステップの制御 170
2171 自動制御 170
2172 FP ガス放出モデルにおけるタイムステップ幅の決定 172
2173 温度計算におけるタイムステップ幅の決定 173
参考文献 2 174
3 力学解析モデル 178
31 基本式と非線形力学の解法 178
311 基本式 178
312 非線形歪みに対する初期応力法初期歪み法剛性変化法 184
313 FEMAXI コードでの基本式の解法(非線形歪み) 187
32 燃料棒全長の力学解析 194
321 全長力学解析の有限要素モデル 195
322 有限要素マトリックスの決定 199
323 ペレットと被覆管の応力歪み(剛性)マトリックスの導出 201
324 ペレットのクラック 208
325 マトリックスにおけるクラック表現方法 211
326 FEMAXI における相当応力相当歪みの定義式 214
327 ペレットのホットプレス 216
328 増分法の補足説明 219
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xiii
33 非線形歪みの計算方法 223
331 ペレットと被覆管のクリープ 223
332 クリープ歪みの計算方式 224
333 ペレットと被覆管の塑性 233
334 剛性方程式の導出 242
34 全体マトリックスの定式化と外力 244
341 剛性方程式と全体マトリックス 244
342 上下プレナム境界条件 247
343 ペレット被覆管接触モデル 254
344 ペレット被覆管接触面における軸力発生 256
345 PCMI 接触面における軸力発生のアルゴリズム 257
346 PCMI 接触面における発生軸力の評価オプション A 260
347 PCMI 接触面における発生軸力の評価オプション B 265
35 2 次元局所 PCMI 力学解析 271
351 要素体系 272
352 基本式 276
353 剛性方程式 278
354 境界条件 284
355 PCMI と接触問題 289
356 軸方向荷重―上部ペレットとの軸方向相互作用― 298
36 クリープ及び応力-歪みに関する機械的性質モデル 301
361 クリープ 301
362 被覆管の応力歪み関係式のモデル 305
363 太田モデル 307
364 FRAPCON と MATPRO モデルの降伏応力の決定方法
(ICPLAS=3 6) 313
37 スカイライン法 315
参考文献 3 322
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xiv
4 物性値とモデル 323
41 UO2 ペレット 323
42 MOX ペレット 343
43 ジルコニウム合金被覆管 353
431 被覆管の物性値 353
432 被覆管酸化物の物性値 368
44 SUS304 ステンレス被覆管 370
45 その他の物性値モデル 372
46 燃焼度の単位換算 377
参考文献 4 380
謝辞 382
JAEA-DataCode 2013-005
- 1 -
1 Introduction A light water reactor has occupied the most essential part in nuclear power generation
for years With the advancement in LWR performance fuel improvement has been actively
pursued in nuclear industries worldwide In particular burnup extension has been promoted
recently and high-burnup fuel behavior has been widely known as a key issue in terms of rod
design and reliability
Under these circumstances development of a computer code which can precisely analyze
fuel behavior is required to evaluate the behavior of high-burnup fuels to assess their
reliability and to serve as a safety cross-check tool In the code introduction of new models
is inevitable departing from the conventional fuel analysis methods because it has been
observed that fuel behavior in a high-burnup region of over 40 - 50 GWdt essentially differs
from that in lower burnup region Such difference is generated by fuel thermal conductivity
degradation rim structure formation in fuel periphery and increase in FGR associated with
the structure change etc They can impose a strong influence on the overall behavior of fuel
rod
On the basis of these considerations an integrated fuel performance code FEMAXI-7
has been developed for analyzing the behavior of fuels under normal operation and
anticipated transients (excluding accident conditions) This code has integrated and
extended the capabilities of the predecessor FEMAXI-6 by incorporating new models and
functions
11 Characteristics of Fuel Performance Code 111 Features and roles of fuel code
Fuel rods are the objects in which radioactive substances are highly accumulated and
concentrated in a reactor core Thermal and mechanical behavior of fuel rod is resulted from
complicated interactions among a number of such factors as temperature neutron flux
nuclear fission accumulation of fission products etc which outspread from atomistic level
phenomena to engineering-controllable macroscopic entities
A fuel assembly has to attain concurrently the two roles ie safety and power
generation efficiency Realizing this purpose needs a various RampD efforts based on
irradiation experiences which gives a fundamental cause for the research of fuel behavior in
terms of engineering and safety
On the other hand that fuel behavior is a result of complicated interactions implies that
fuel research is a field where actual irradiation experiences have inevitably a marked
JAEA-DataCode 2013-005
- 2 -
significance In other words fuel research is a field where theoretical and deterministic
models cannot always have a precise predictability some materials properties or behavior
cannot be expressed by such theoretical model as those of reactor physics or cannot be
clearly defined by established physical laws and there are so many unknown factors such as
the effect of small quantity of impurities crystal grain structure dependence on fabrication
process etc Furthermore so many factors are involved in fuel behavior that it is sometimes
difficult to evaluate the contribution of each factor to whole behavior by simply summing up
each factor-wise calculation Accordingly irradiation experiences have an indispensable role
in simulation design and understanding fuel behavior
However irradiation of fuels ie irradiation tests and post-irradiation examinations
needs in general much cost and long period of time and quantity and quality of measured
data are limited for the cost and time The irradiation test cannot give all that are
necessary to understand the inside processes of fuel rod As a result in some cases we are
obliged to depend on a few specific measurable physical quantities such as fuel center
temperature and internal pressure or on the indirectly evaluated quantities such as burnup
distribution in assessing the whole behavior of fuel rod
In these situations making use of a fuel performance code is an important measure to
recognize and reproduce the interlinked structures inside fuel rod to grasp the controlling
parameters for fuel behavior and to predict the fuel behavior on the basis of the ldquodirectly or
indirectly measured physical quantitiesrdquo as if solving an inverse problem in an exploratory
manner FEMAXI is an engineering tool serving for this purpose In other words
development of FEMAXI is a process which constitutes reproduces and predicts the
interaction structure hidden behind observed fuel behavior on the basis of the limited data
or experiences of irradiation
112 Prediction of fuel behavior by FEMAXI A fuel design code of fuel manufacturer often uses empirical correlations which have
been derived from irradiation test data base not depending on the mechanistic models such
as those in FEMAXI and the calculated results are compared with measured data for
validation Therefore its predictions are reliable at least within the scope of the test data
base and its small external region and it is considered that the prediction is effective for fuel
design
On the other hand many mechanistic and complementary empirical models and
parameters to control the models have been implemented in FEMAXI-7 For prediction of
JAEA-DataCode 2013-005
- 3 -
unknown behavior of fuel FEMAXI-7 could bring about so different calculation results
depending on the combination of its models and parameters that the final prediction would
be capable of having a considerable uncertainty
Accordingly it is necessary to find models and data set combinations which are as
generally applicable as possible to the prediction in the process of verification calculations If
a code can reproduce by a set of models and parameters the behavior of fuel A of which
irradiated results are known and if the same set is applied to another fuel rod B and can
result in a successful reproduction of the irradiated state of the rod B predictability of the
set is verified with respect to at least these fuel rods It depends on the number of analyzed
cases on analyzed fuel designs and irradiation conditions how far such verification has been
made ie how many types of models and parameter sets have been tested and how much
generalization of predictability is guaranteed in FEMAXI In this sense verification of
FEMAXI-7 is on-going As some of verification activities test analyses have been
performed in the international benchmark test FUMEX-III(11) and by using the results of
the Halden test irradiations Through these activities refinement of models and
parameters set are being followed up
At the same time FEMAXI-7 can allow by the combination of its various models and
controlling parameters an attempt to reproduce such fuel behavior (irradiation data) that
the conventional models and parameters cannot obtain This is one of the means which
enables us to estimate the unknown linkage structure of factors inside fuel rod
Furthermore the code can be applied to the prediction of fuel behavior in the conditions
which is planned in test irradiation or new design of fuels or even in the conditions beyond
the experimentally feasible scope
113 History of FEMAXI code development History of development up to the latest version FEMAXI-7 is briefly described It is one
of features that up to FEMAXI-7 a number of researchers have been engaged in the
development and their knowledge and experiences have been accumulated in the code The
code can be considered to be based on the combination of wisdom and experiences of
Japanese fuel researchers and engineers
(1) The first version of FEMAXI was developed by Ichikawa (JAERI Japan Atomic
Energy Research Institute) in the Halden Project It featured a 2-D local elasto-plastic
PCMI analysis function (1974)
JAEA-DataCode 2013-005
- 4 -
(2) Creep model was implemented in FEMAXI-II by Kinoshita (CRIEPI The Central
Research Institute of Electric Power Industry) (1977)
(3) FEMAXI-III(12) was developed by an extensive cooperation of JAERI universities
Hitachi Toshiba NFD(Nuclear Fuel Development Co) CRIEPI and CRC(Century
Research Center Co) Development of this version received an award from Japan
Society of Nuclear Science and Technology It exerted a predominant capability in
international benchmarking calculation Released to NEA-DataBank(13) (1984)
(4) FEMAXI-IV(14)-(112) was a version improved by Nakajima et al It was used in
JNES(113) ( former NUPECNuclear Power Engineering Corporation) as one of cross
check codes This version was taken over to MSuzuki in JAERI Ver2 was developed
and released to NEA-DataBank (1996)
(5) FEMAXI-V (114) was developed by Suzuki for the analysis of high burnup fuels
released to NUPEC and NEA-DataBank (2001)
(6) The code was further extended to FEMAXI-6 by Suzuki et al(JAEA) released to
JNES and to NEA DataBank (2003-2010)
114 Usage of FEMAXI
An extended usage of FEMAXI code can be summarized as follows
(1) Cross check code for safety licensing
The former version FEMAXI-6 is a main part and function of the cross-check code used in
JNES for safety licensing of LWR fuel rod
(2) Release to external users
A code package of FEMAXI-7 (source binary program related codes description and
manual) has been released to domestic users without charge through RIST (Research
Organization for Information Science amp Technology httpwwwristorjp ) The package
will be registered in NEA DataBank as a representative fuel performance code of Japan to
be released to foreign users The former versions FEMAXI-VI FEMAXI-V and FEMAXI-6
have been released from NEA DataBank to foreign users without charge and the total
number of the releases is over thirty Also we are positively acquiring external users
responding to questions from users In effect the FEMAXI code has been a
frame-of-reference of Japanese LWR fuel performance code
JAEA-DataCode 2013-005
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(3) One of the means for experimental analysis prediction of experiments and
training of fuel behavior understanding
The FEMAXI code can be used for exploratory analysis of irradiation tests and for the
prediction of state of fuel rods just prior to test-irradiation in accident conditions
Also the code is used for the planning of irradiation test for checking the adequacy of
fuel design Furthermore it is used as a training tool by performing simulations with
various input conditions for a better understanding of fuel behavior
(4) Platform to evaluate irradiation test data and new models
Newly obtained materials properties of irradiated fuels are in itself sometimes not clearly
evaluated in terms of their contribution to fuel behavior change or improvements However
incorporating them in a fuel code putting them in a various interactions of a number of
factors and performing simulations with them will enable us to evaluate their contribution
to or effects on the whole behavior of fuel rods more clearly or quantitatively The FEMAXI
code can function for such purposes
12 Features of Structure and Models of FEMAXI 121 What is model
Before discussing specific models a fundamental question is addressed what is model
as a code component What is the purpose of model development Here if the answer is
that the purpose is to reproduce behavior of a target object as precisely as possible this
answer is insufficient because it does not address the question that ldquowhy is it necessary to
reproduce preciselyrdquo and ldquowhat is the preciseness in this caserdquo To begin with since
model development starts in the situation where knowledge about the object to be modeled is
insufficient the preciseness has to own its limit In other words a model can be a model
because it is an approximation of a real object or an ideational reflection which has
quantitative nature to a considerable extent Therefore model development is upgrading in
the following steps
1) The first stage is a one which gives insight into the underlying mechanism of phenomena
as to what will appear when watching the phenomena through the prototypical model
2) The second stage allows us to estimate the underlying mechanism of phenomena by
incorporating the model in a code and performing calculation
JAEA-DataCode 2013-005
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3) The third stage enables us to predict the fuel behavior by changing model parameters
which will lead to the improvement of fuel performance and be used as a measure to solve
the ldquoinverse problemrdquo
However the most essential role of modeling is considered to make a reflection on
human notion and deepen the insight into the mechanism of phenomena by realizing an
interaction or mutual permeation between the human notion and the real entity which exists
independently from the human notion
122 What is fuel modeling As for the fuel modeling it is quite usual to have discussion in terms of individual
modeling of major phenomena which dominate fuel behavior such as heat generation and
conduction fission gas release Pellet-Clad Mechanical Interaction(PCMI) etc However in
high burnup fuel behavior as interactions become stronger among various phenomena the
modeling is inevitably extended to not only each phenomenon but also a linkage structure to
combine various factors That is a linkage structure of each model itself is required to be
designed This means that to what extent the interaction among the factors is explicitly
modeled
For example coupling of thermal and mechanical analyses or interlinkage among
fission gas bubble growth pellet swelling and thermal conductivity degradation etc are to
be addressed in the code design
However the interlinkage structure cannot be too complicated at the beginning and it is
an usual strategy to start with a simplified assumption or model For example porosity
increase by fission gas bubble growth has an additional decrease effect on the pellet thermal
conductivity though it is set independent from the thermal properties of pellet
On the other hand as nuclear reaction calculation has a weak interaction with thermal
and mechanical behavior the reaction calculation is carried out in a relative isolation from
fuel behavior calculation in fuel performance code That is no direct interaction with fission
reaction is incorporated in the code but it is considered implicitly as the change of pellet
power density profile with burnup or fast neutron flux corresponding to linear heat rate
These quantities are often calculated by a sub-code or an independent burning analysis code
(reactor core physics code) and the results are fed to the main part of fuel code In this
sense the burning analysis codes which give results to FEMAXI are important (Refer to
Refs(115) and (116))
Based on the above discussion features of FEMAXI models can be summarized as
JAEA-DataCode 2013-005
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follows
(1) The analytical object is limited to a single fuel pin and its surrounding coolant being
isolated from the interactions among the fuel pins set in an assembly or thermal and
nuclear behavior of whole reactor core
(2) In designing models and code as engineering tools the highest priority is placed on the
prediction of fuel behavior in a macroscopic scale Accordingly in this sense it is not a
preferential task to make each model based on a microscopic physical principle or theory
The code adopts a combination of mechanistic model and empirical model If a
mechanistic model cannot be designed the code begins with an empirical model
(3) The code pursues fast calculation Nevertheless a low priority is given to the modeling
efforts to carry on a much reduced calculation time for the purpose of a statistical
analysis which handles a number of fuel pins in a reactor core
123 Analytical targets Analytical objects of FEMAXI-7 is quite identical to those of FEMAXI-6 fuel behavior
in not only steady-state normal operation conditions but also such transient conditions as
fast power ramp load-following power-coolant mismatch etc Nevertheless the code does
not cover the accident conditions such as LOCA (Loss-of-coolant accident) and RIA
(Reactivity- initiated accident)
Table 11 shows the phenomena which FEMAXI-7 deals with As for the material
properties including those for MOX fuels and Gd-containing UO2 fuels those available in
open literatures have been adopted as much as possible Major functional differences
among FEMAXI-V FEMAXI-6 and FEMAXI-7are listed in Table 12 Description for each
item will be presented in chapters 2 3 and 4
Table 11 Phenomena analyzed by FEMAXI-7
Thermal process determiningtemperature distribution
Process with mechanical displacement
Pellet
Thermal conduction (heat flux distribution ) Fission gas release Swelling
Thermal expansion elasticity plasticity creep cracking relocation densification swelling hot-press
Cladding Thermal conduction Waterside corrosion
Thermal expansion elasticity plasticity creep irradiation growth
Fuel rod
Gap thermal conduction (mixedgas contact radiation) cladding surface heat transfer gap gas flow
Mechanical interaction between pellet and cladding friction and bonding between pellet and cladding
JAEA-DataCode 2013-005
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Table 12 Upgrades of main functions of versions up to FEMAXI-7
FEMAXI-Vreleased in 1999
FEMAXI-6 released in 2002
FEMAXI-7
Coupling solution of thermal and mechanical analyses
times
Pellet thermal conductivity degradation with burnup
High burnup structure formation in pelletperiphery
Gas bubble swelling model times
Rate-law model of grain boundary gas bubble growth
times times
Transient thermal hydraulics model
Extension of inputoutput and materials properties
Restart function to have a link between base-irradiation and test-irradiation
times times
Modularization of source codes detailed comments embedded selection of the number of elements
times times
13 Whole Structure
FEMAXI-7 consists of two main parts one for calculating the temperature distribution
of fuel rod thermally induced deformation and fission gas release etc (hereafter called
ldquothermal analysis partrdquo) and the other for calculating the mechanical behavior of fuel rod
(hereafter called ldquomechanical analysis partrdquo) Figure 11 outlines the entire code structure
and Fig12 shows the analytical geometry
In the thermal analysis part calculation always covers an entire rod length which is
divided into the maximum of 40 axial segments Namely the temperature distribution is
calculated at each axial segment as one-dimensional axis-symmetrical problem in the radial
direction and with this temperature such temperature-dependent values as fission gas bubble
growth gas release gap gas flow in the axial direction and their feedback effects on gap
thermal conduction are also calculated
Also to take into account thermal feedback among the axial segments which may arise
due to uneven power distributions in the axial direction iteration is performed until
convergence is attained for the entire rod length
JAEA-DataCode 2013-005
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In the mechanical analysis part users can select one of the two modes analysis of the
entire rod length 1-D ERL mechanical analysis only or the ERL mechanical analysis +
2-D local PCMI analysis in one pellet length
In the former the 1-D axis-symmetrical finite element method (FEM) is applied to each
axial segment of the entire rod length in the latter in addition to the 1-D FEM the
axis-symmetrical FEM is applied to half a pellet length for a symmetrical reason and
mechanical interaction between pellet and cladding ie local PCMI is analyzed only in one
axial segment
In the mechanical analysis the magnitude of pellet strain caused by thermal expansion
densification swelling and relocation is calculated first and a stiffness equation is formulated
with consideration given to cracking elasticityplasticity and creep of pellet
Then stress and strain of pellet and cladding are calculated by solving the stiffness
equation with boundary conditions which correspond to the pellet-cladding contact mode
When PCMI occurs and pellet-cladding contact states change calculation is re-started with
the new boundary conditions of contact from the time when the change occurs
In the ERL mechanical analysis the axial force acting on adjacent axial segment is
evaluated and as a result analysis of axial deformation of entire rod length is performed
Also the finite element method is simplified to reduce the total number of degrees of
freedom
The thermal analysis and the ERL mechanical analysis are coupled convergence
between temperature and deformation is obtained in every time step by iteration
JAEA-DataCode 2013-005
- 10 -
Next Time Step
No
Output
Temperature Internal pressure
Input
<Time step control>
Mechanical analysis of entire rod length
Elasto-plastic creep PCMI axial force deformation(stress-strain)
End of Time Step
Yes
Iteration
Local PCMI analysis [half a pellet length]
Elasto-plastic creep ridging stress-strain distribution
Thermal analysis Temperature fission gas
diffusion and release
Fig11 Overview of analytical flow of FEMAXI
JAEA-DataCode 2013-005
- 11 -
Plenum
Segment
LHR
Axial Power Profile
Zircaloy metal
ZrO2 generated bywaterside corrosion
Temperature
Power andBurnupprofiles
FEM mechanical analysisfor pellet and cladding
Nest ofcoaxialrings
Gas pressure
14 Features of Numerical Method and Modeling
(1) FEM element characteristics
Storage region and calculation time are reduced by introduction of rectangular
3-degrees-of-freedom elements into the FEM analysis in order to perform mechanical
calculation for the entire length of the fuel rod
(2) Creep solution An implicit procedure is applied to obtain numerical solution stability for high creep rate
(3) Contact problem Three contact states are dealt with as the contact conditions between pellet and cladding
open gap state clogged gap state and sliding state In the FEM contact conditions at each
node pair of pellet and cladding are determined and the boundary conditions are set in
accordance with the conditions
Fig12 Geometry of FEMAXI for the analysis of one single fuel rod
JAEA-DataCode 2013-005
- 12 -
(4) Matrix solution The coefficient matrix of simultaneous equations of
the FEM (stiffness equations) is a diagonal symmetrical
matrix having a large number of zero elements as
shown in the figure aside Therefore the skyline
method has been adopted to solve the equations and the
memory method of non-zero elements in the matrix as
well as the calculation procedure has been improved
in order to reduce the size of the storage region and
calculation time
In the skyline method only the elements under the
solid line in the figure are stored In the fission gas
release model the procedure has been improved so that the number of computing steps for the
matrix becomes a minimum
(5) Non-equilibrium residue In order to avoid accumulation of non-equilibrium residue generated at each time step
during solution of a nonlinear problem equilibrium condition equations are formulated not in
an incremental manner but in a form which maintains the total balance of loads and stresses
(6) Behavior of cracked pellet Cracking of a pellet is modeled using a decreased stiffness approximation method similar
to the case for FEMAXI-III In order to describe behavior of the cracked pellet which is
relocated under PCMI recovery of the stiffness during compression is expressed as a function
of the amount of relocation
(7) Fission gas release model and bubble swelling model
Fission gas diffusion in fuel grain is calculated by a combination of the least residual
method in equivalent sphere model and gas bubble formation In addition a model of fission
gas release restraint by thermal stress is incorporated Gas bubble growth can be calculated
either by rate-law model or pressure equilibrium model In the both models option is
incorporated to calculate swelling by intra-grain bubble growth and by grain boundary
bubble growth
Nonzero element
Storage region in the skyline method
JAEA-DataCode 2013-005
- 13 -
(8) Gap gas diffusion-flow model Gap gas-diffusion flow is modeled and effects of the released fission gas on gap thermal
conductance are carefully evaluated
(9) Model of oxidation of cladding Cladding waterside oxidation is modeled and change in the thermal conductivity due to
oxidation is calculated
(10) Introduction of thermal-hydraulics model
With respect to heat transfer between a fuel rod and coolant RELAP5-MOD1 and some
other models have been introduced to cover a wide range of conditions of heat transfer to
coolant including a boiling transition
(11) Non-steady phenomena analysis The fuel behavior in a non-steady state can be analyzed by mechanistic treatment of a
non-steady heat transfer model gap gas flow model and fission gas release model Also the
accuracy of prediction is improved and calculation time is reduced through use of
independent time-step controls in each model
(12) Simplification of calculation In the fission gas release model when the same type of calculations are performed both in
a low-temperature region and a high-temperature region calculation results obtained for
elements in one region are used for calculation in another region in order to avoid carrying out
similar calculations twice Also physical properties and other values which are frequently
referred to are stored in a data table and use of special mathematical functions is avoided
By means of the above-mentioned procedures FEMAXI can give highly accurate
solutions within a shorter calculation time
(13) Empirical models of rim structure formation
Empirical models are incorporated for simulating the rim structure formed in the
peripheral region of high burnup pellet
JAEA-DataCode 2013-005
- 14 -
15 Interfacing with Other Codes etc
151 Interfacing with burning analysis codes RODBURN and PLUTON Power profiles in the radial and axial directions of a pellet and the burnup profile both
change with burnup from BOL These profile changes differ depending on the reactor type
(PWR BWR and heavy-water reactor etc) in which fuel rods are irradiated These profiles
naturally affect the temperature distribution of pellets and also affect the material
properties the performance of which depends on the burnup Accordingly FEMAXI-7
similar to FEMAX-6 includes a function which reads the output files of the burning analysis
codes RODBURN(115) and PLUTON(116) and uses them for calculation RODBURN uses
part of ORIGEN(117) and RABBLE(118) and calculates the power profiles burnup and fission
products in pellets and pellet stacks in accordance with a given power history and fuel rod
specifications and then produces files of these results
PLUTON produces isotope distributions and heat generation density profiles accurately
at high speed using an original morphological function (Refer to section 24)
152 Re-start function from base-irradiation to test-irradiation periods
FEMAXI-7 has a function to generate a result file which records the fuel rod states at
EOL ie at the end of base-irradiation This re-start file can be read by FEMAXI-7 to
restart the calculation of fuel rod in test-irradiation which has often a number of power
changes in a relatively short time In the former version FEMAXI-6 in the case where
base-irradiation is conducted with a long fuel rod and test-irradiation is conducted with a
short rod re-fabricated from the long rod users have no other choice but to perform analysis
with the short refabricated geometry of rod from the beginning of base-irradiation to the end
of test-irradiation This may give some inaccuracy in the final results and also
ineffectiveness of doing the whole calculation process every time In FEMAXI-7 these
problems have been eliminated
Furthermore this file can be read by RANNS to be used as one of the initial conditions for
the fuel rod analysis in accident conditions
153 Execution system environments (1) Machine source code and compilers
FEMAXI -V -6 and -7 have been developed to be operated usually in Windows PC while
Linux version of FEMAXI-7 has been developed The source code is written mainly in
JAEA-DataCode 2013-005
- 15 -
Fortran-77 and partly in Fortran-90
Until now Compaq Digital Visual Fortran v6 or upper (Compaq DVF) has been used for a
compiler though this compiler has been discontinued From now Intelreg Visual Fortran
Compiler for Windows version 10xx or upper (Intel VF) (119) are recommended which
includes every function of Compaq DVF
In other words totally equivalent executable programs of the FEMAXI-7 source code can
be built by the three compilers Compaq DVF Intel VF and Linux-GNU Fortran g77
(2) Plotted output
Numerical results of FEMAXI-7 calculation can be obtained in three types of outputs
direct numerical figures plotted figures by using plotting program EXPLOT and Excel
tables corresponding to the plotted figures
(3) Release of package
A code package of FEMAXI-7 consists of the source code executable program compile
option files related programs such as plotting program code model description document
and InputOutput manual The package can be delivered to users in foreign countries
through NEA DataBank
References 1 (11) Improvement of Computer Codes Used for Fuel Behaviour Simulation - FUMEX III
httpwww-nfcisiaeaorgCRPCRPMainaspRightP=CRPDescriptionampCRPID=16
(12) Nakajima T Ichikawa M et al FEMAXI-III A Computer Code for the Analysis of
Thermal and Mechanical Behavior of Fuel Rods JAERI 1298 (1985)
(13) OECD Nuclear Energy Agency DataBank httpwwwneafrdatabank
(14)
(15)
(16)
Nakajima T FEMAXI-IV A Computer Code for the Analysis of Fuel Rod Behavior
under Transient Conditions Nucl Eng Design 88 pp69-84 (1985)
Nakajima T and Saitou H A Comparison between Fission Gas Release Data and
FEMAXI-IV Code Calculations Nucl Eng Design 101 pp267-279 (1987)
Nakajima T Saitou H and Osaka T Analysis of Fission Gas Release from UO2 Fuel
during Power Transients by FEMAXI-IV Code IWGFPT-27259 pp140-162 (1987)
JAEA-DataCode 2013-005
- 16 -
(17)
(18)
(19)
(110)
(111)
Nakajima T and Ki-Seob Sim Analysis of Fuel Behavior in Power-Ramp Tests by
FEMAXI-IV Code Res Mechanica 25 pp101-128 (1988)
Nakajima T Saitou H and Osaka T Fuel Behavior Modelling Code FEMAXI-IV and
Its Application IAEA-T1-TC-659 Paper presented at IAEA Technical Committee on
Water Reactor Fuel Element Computer Modelling in Steady-State Transient and
Accident Conditions Preston England Sept19-22 (1988)
Uchida M Saito H Benchmarking of FEMAXI-IV Code with Fuel Irradiation Data
in Power Reactors JAERI-M 90-002 (in Japanese) (1990)
Nakajima T Saitou H and Osaka T FEMAXI-IVA Computer Code for the Analysis
of Thermal and Mechanical Behavior of Light Water Reactor Fuel Rods
Trans11th Int Conf on SMIRT (Tokyo Japan) 6297 PVC-D pp1-6 (1991)
Nakajima T Saitou H and Osaka T FEMAXI-IVA Computer Code for the Analysis
of Thermal and Mechanical Behavior of Light Water Reactor Fuel Rods
Nucl Eng Design 148 pp41-52 (1994)
(112) Suzuki M and Saitou H Light Water Reactor Fuel Analysis Code FEMAXI-IV
(Ver2) -Detailed Structure and Userrsquos Manual- JAERI-DataCode 97-043 (1997)
(113) Japan Nuclear Energy Safety Organization httpwwwjnesgojp
(114) Suzuki M Light Water Reactor Fuel Analysis Code FEMAXI-V(Ver1)
JAERI-DataCode 2000-030 (2000)
(115) Uchida M and Saitou H RODBURN A Code for Calculating Power Distribution in
Fuel Rods JAERI-M 93-108 (in Japanese) (1993)
(116)
Lemehov SE and Suzuki M PLUTON ndash Three-Group Neutronic Code for Burnup
Analysis of Isotope Generation and Depletion in Highly Irradiated LWR Fuel Rods
JAERI-DataCode 2001-025 (2001)
(117) Bell MJ ORIGEN ndash The ORNL Isotope Generation and Depletion Code
ORNL-4628 (1973)
(118) Kier PH and Robba AA RABBLE A Program for Computation of Resonance
Absorption in Multi-region Reactor Cells ANL-7326 (1967)
(119) httpsoftwareintelcomen-usintel-compilers
JAEA-DataCode 2013-005
- 17 -
2 Thermal Analysis Models In this chapter theory and models in the thermal analysis of FEMAXI-7 are explained
21 Heat Transfer to Coolant and Thermal-hydraulics Model
In the thermal analysis part temperature distribution at each axial segment of a fuel rod
is calculated by one-dimensional cylindrical geometry with the boundary conditions
determined by coolant temperature and pressure The basic assumption and calculation
procedure are as follows
(1) Temperatures and states of coolant for all axial segments depend on the axial distribution
of coolant enthalpy in a previous time step and are determined by temperature pressure and
flow velocity of coolant at the inlet and surface heat flux from cladding at each segment in
the current time step The thermal-hydraulics model of FEMAXI-7 can be applied not only
to steady states but also to transient changes such as increasedecrease in power and
load-following condition under normal operation and in the rapid change of coolant flow
rate
However in the case in which an active (stack) length of rod is modeled by a small
number of axial segments if the flow rate change occurs in a very short period of time some
possibility would be foreseen that gives an unstable or false solution of axial distribution of
coolant temperature (enthalpy) depending on the length of the axial segment flow rate and
time step increment etc
To eliminate this unfavorable possibility the stack length of rod is always divided into
equal length 100 sub-segments in a coolant enthalpy calculation and enthalpy distribution
calculated at these sub-segments is interpolated to obtain the enthalpy at representative
location of each segment to determine axial distribution of coolant temperature This method
enables users to designate the number and length of segments without concern for the
numerical instability or uncertainty in coolant temperatures at each segment elevation
(2) Enthalpy of coolant at each axial segment is calculated on the basis of the enthalpy
distribution at the end of the former time step with the enthalpy of the coolant at the inlet as
the initial value using the increase in the enthalpy obtained from flow velocity and the
amount of heat conducted from each segment of rod Distribution of the coolant temperature
in the axial direction is determined by thus-obtained distribution of the coolant enthalpy
Here as stated in (1) axial distribution of coolant enthalpy is derived from the
JAEA-DataCode 2013-005
- 18 -
interpolation of values calculated at the sub-segments
(3) Next the heat transfer coefficient in each coolant mode is determined Surface temperature
of cladding at each axial segment is then calculated with heat flux and coolant temperature
The temperature profile in the radial direction from cladding surface to pellet center is
calculated in one dimensional geometry with the result of the term (2) above ie coolant
temperature as boundary condition
However heat transfer in the axial direction of a fuel rod is not taken into consideration
assuming that heat transfer in the axial direction due to the power distribution in the axial
direction is suppressed by such thermal resistance as dishes and gap at the end surface of
pellet Actually the temperature gradient in the axial direction is substantially smaller than
that in the radial direction and can be neglected Therefore inclusion of thermal calculation in
the z (axial) direction is not efficient since it increases calculation time markedly
(4) In addition to this normal mode in which cladding surface temperature is calculated by the
heat conduction from inner region of pellet and to the coolant at the cladding surface another
mode is possible in FEMAXI In this latter mode cladding surface temperature history is
specified by input and temperature calculation inside the fuel rod is performed by adopting
this surface temperature as boundary condition This is so-called the Dirichlet problem
solution
The following items will be explained in sections 23 to 27 calculation of
one-dimensional temperature profile in the radial direction gap thermal conductance model
and fission gas release model The flow of thermal analysis is shown in Fig211
JAEA-DataCode 2013-005
- 19 -
Time Step Start
Calculation of burnup power density and fission gas atoms generation
Calc of gas flow in the axial direction
Calc of heat conduction coefficient between cladding surface and coolant
Calc of fission gas release and grain growth
Fuel center temperature has converged
Calc of temperature distribution in the radial direction of rod
Calc of gap thermal conductance
Fission gas release calculation has converged
Calc for one axial segment has been completed
Fuel center temperatures of all the axial segments have converged
Calc of internal pressure and gap gas composition
Next time step
Fig211 Calculation flow of thermal analysis
Iteration with mechanical analysis of the entire length of rod Calc of gap size and contact pressure at all the axial segments
JAEA-DataCode 2013-005
- 20 -
211 Coolant enthalpy increase model
In this model mass flux and pressure during each time step are assumed to be
constant(21) Enthalpy of the coolant (water or steam) H(z t)(Jkg) at an axial position z (m)
and time t (s) can be given by the integral of its history beginning from an inlet of the coolant
as follows
τdzS
zQzVtHtzH
t
t primeprimeprime
+=in )(
)()()0()( in (211)
Here H(0 tin) enthalpy of coolant at the inlet (Jkg)
tm the time (s) when a coolant located at an axial position z at time t passes
through the inlet
( )τzprime axial position (m) of the coolant at time τ which was positioned at z at
time t
V specific volume (m3kg)
Q heat influx at unit length (Wm)
S flow cross section (m2)
τ time
Focusing on the time step between time t0 and t0+Δt Eq (211) can be rewritten as
H z t t H z tV z Q z
S zd
t
t t( ) ( )
( ) ( )
( )0 0 00
0+ = + prime primeprime
+
ΔΔ
τ (212)
where z0 axial coordinate (m) of a coolant at time t0 which is positioned at z at time t0+Δt
Next the active length of fuel rod is divided equally into 100 sub-segments irrespective of
the segment lengths designated by input data Fig212 shows the enthalpy increase model
for this 100-sub-segment in which the axial elevation z is set to be a representative point of
sub-segment j and a sub-segment containing the axial location z0 is set to be k
When a coolant which is positioned at z at time t0+Δt moved from z0 to z if we designate
the time required for the coolant to pass through the segment i to be δti a time step increment
Δt can be expressed as follows
Δt t t t tk k j j= + + + ++ minusδ δ δ δ1 1helliphellip (213)
By rewriting the integral in Eq(212) using the average values of V Q and S of the
segment i Vi Qi and Si and also δti enthalpy of the segment j Hj at t0+Δt can be obtained as
JAEA-DataCode 2013-005
- 21 -
H t t H tV Q
Stj
i i
ii k
j
i( ) ( )0 0 0+ = +=Δ δ (214)
Δti can be obtained as follows using the flow velocity at the segment i as ui (ms)
δ tL
u
L
G V
S L
WVii
i
i
i i
i i
i
= = =Δ Δ Δ
(215)
where
ΔLi i-th sub-segment length (m)
Gi mass flow rate per unit cross sectional area (kgm2s)
W mass flow rate (kgs)
And ΔLi can be obtained as
node j+1
node j
node j-1
node k+1
node k
segm j
segm j-1
segm k
z
zk+1
z0
δ t j
δ t j minus1
δ tk
z
zk
Fig212 Coolant enthalpy increase model
JAEA-DataCode 2013-005
- 22 -
ΔL
z zi j
z z k i j
z z i ki
j j
i i
k
=
minus=
minus ne neminus =
+
+
+
1
1
1 0
2( )
( )
( )
(216)
H0(t0) can be obtained by interpolating the enthalpies at zk and zk+1 as
( ) ( ) ( ) ( )( )H t H t H t H tz z
z zk k kk
k k0 0 0 1 0 0
0
1
= + minusminusminus+
+
(217)
Fig213 shows the calculation sequence Here the enthalpy at a representative point
on a segment is the average value of enthalpies at the upper and lower nodes of the segment
Here FEMAXI does not accommodate single-steam-phase inlet conditions with respect
to the relationship between temperature and pressure When inlet temperature and pressure of
coolant are given by input the steam table incorporated in FEMAXI-7 is capable of
identifying the inlet temperature as a temperature which exceeds the saturation temperature
due to the rounding of figures and numerical errors in the inlet temperature data In such cases
input enthalpy cannot be determined Thus when inlet temperature is determined to be the
saturation temperature or higher at the coolant pressure the inlet temperature is assumed as
the saturation temperature for a single liquid phase in the code and the inlet enthalpy may be
determined (Refer to name-list parameter SUBCL)
Based on the enthalpy distribution obtained at the 100 equal-length sub-segments by the
above method the enthalpy distribution for the input axial segments is derived by
interpolating Eq(217)
A flow of the calculation is shown in Fig213
JAEA-DataCode 2013-005
- 23 -
Inlet enthalpy
Inlet temperature
Pressure
Mass flow rate
Heat flux at a segment
Inlet flow
velocity
A part of a flowing coolant X ascends and when it reaches a position of a node of an axial segment the enthalpy at the part X in the previous time step is calculated using axial distribution of the enthalpy in the previous step and the axial position of the part X When the part X is judged to be positioned below the coolant inlet the enthalpy of the position X in the previous step is set to be equal to the enthalpy of the coolant at its inlet
Calc of enthalpies at the nodes of 100 sub-segments
Enthalpies of the 100 sub-segments at the former time step
Calc of enthalpy at the representative point of
each node of the axial segments by
interpolating the enthalpies of nodes of the 100
sub-segments
Fig213 Process of enthalpy increase calculation
212 Determination of cladding surface heat transfer coefficient
In FEMAX cladding surface heat transfer coefficient is determined mainly by
RELAP5MOD1 model(22)
(1) Classification of boiling state Boiling is classified into saturation boiling and surface boiling (sub-cool boiling)
depending on the liquid temperature into in-tube boiling and pool boiling depending on the
presence of flow and into nucleate boiling transition boiling and film boiling depending on
the mode of phenomenon Figure 214 qualitatively shows the boiling modes and heat
JAEA-DataCode 2013-005
- 24 -
transfer characteristics in terms of the heat flux and magnitude of super-heating of the heat
transfer surface The magnitude of super-heating of the heat transfer surface ΔTsat shown on
the horizontal axis in Fig 214 can be given by
satwsa TTT t minus=Δ (218)
where Tsat is obtained from coolant pressure and temperature using the steam table
Fig 214 Classification of boiling and heat-transfer characteristic curve
(2) Single-phase heat transfer In the single-phase region (non-boiling region) shown in Fig 214 the Dittus-Boelter
equation is used for the FEMAXI calculation of in-tube forced convection heat transfer
coefficient
Dittus-Boelter equation(23)
( )Bw TThq minus= (219)
Here
8040
e
RePr0230D
Kh =
Hea
t flu
x l
og q
D
A Boiling initiation pointB Critical heat flux pointC Local minimum heat
flux point
Single phase
Nucleate
boiling
Transient
boiling
Film boiling
Magnitude of super-heating of heat transfer surface logΔTsat
A
E
B
C
JAEA-DataCode 2013-005
- 25 -
μeRe DG=
where k coolant thermal conductivity (WmK)
De equivalent thermal hydraulic diameter (m)
V coolant velocity (ms)
ρ coolant density (kgm3)
μ coolant viscosity (kgms)
Pr Prandtlrsquos number
TB coolant temperature (K)
K thermal conductivity (WmK)
Re Reynoldrsquos number(-)
G mass flow rate(kgm2s)
(3) Nucleate boiling heat transfer In nucleate boiling temperature of a heat-transfer surface decreases periodically near the
saturation temperature due to rapid vaporization of thin liquid films remaining at the bottom
of bubbles which leads to the realization of extremely high thermal conductivity
In FEMAXI-7 nucleate heat transfer is given by the sum of heat transfer from surface to
water in sub-cooled state and heat transfer from surface to water in nucleate-boiling state
The Dittus-Boelter equation given by Eq(219) is applied to calculate the heat transfer
from heated surface to sub-cooled water and either Chenrsquos equation or Jens-Lottes equation is
used to calculate the heat transfer from surface to nucleate boiling water
The equation for the nucleate heat transfer is
( ) ( ) ( )satWBBWWBW TThTThTThq minus+minus=minus= (2110)
where q heat flux (Wm2)
h cladding surface heat transfer coefficient (Wm2K)
hw heat transfer coefficient to single phase (non-boiling sub-cooled) water (Wm2K) determined by using the Dittus-Boelter equation
hB heat transfer coefficient to boiling water (Wm2K) determined by using either Chenrsquos equation or Jens-Lottes equation
Tw cladding surface temperature (K)
TB coolant temperature (K) and
Tsat saturation temperature (K)
JAEA-DataCode 2013-005
- 26 -
In Chenrsquos equation heat flux of the nucleate boiling is expressed using the overheating of
surface ΔTsat and mainly applied to the coolant with a large void ratio (cross-sectional area
of a flow channel occupied by gas phasetotal cross-sectional area of the flow channel) in
power transient or in the early phase of LOCA
The Jens-Lottes equation can be applied to regions with a low void ratio and is applied
to mainly sub-cool boiling of water or in BWR condition
Chenrsquos equation (24)
SPTH
CKh 750240
sat240g
240fg
290f
50
490f
450
fP790
fw 00120 ΔΔ=
ρμσρ
(2111)
hw cladding surface heat transfer coefficient (Wm2K)
K thermal conductivity (WmK)
CP Specific heat at constant pressure (Jkg)
ρ density (kgm3)
σ surface tension(Nm)
μ dynamic viscosity(kgms)
Hfg latent heat (Jkg)
satTΔ magnitude of super-hearing of surface (K)
coolsat PPP minus=Δ
Psat Pressure at which cladding surface temperature is the saturation
temperature(Nm2)
Pcool coolant pressure (Nm2)
S Chenrsquos suppression factor (see Table21)
Here suffices f and g represent liquid phase and gas phase respectively
Jens-Lottes empirical equation(25)
hP
WW
N= sdottimes
012636 201 106
0 75 exp
φ (2112)
hW surface heat transfer coefficient (Wcm2K)
PW coolant water pressure (Nm2)
φN cladding outer surface heat flux (Wcm2)
JAEA-DataCode 2013-005
- 27 -
Table21(1) CHENrsquoS REYNOLDS NUMBER FACTOR F
10
f
g
50
g
f
90
e
e
1
minus
=μμ
ρρ
x
xr
ex quality (-)
p density (kgm3)
μ dynamic viscosity(kgms)
R F
00 100
01 107
02 121
03 142
04 163
06 202
10 275
20 430
30 560
40 675
60 910
100 1210
200 2200
500 4470
1000 7600
4000 20000
Fitting parameter AKFAC The cladding surface heat transfer coefficient is adjusted using the following equation
(4) Burn-out and critical heat flux A phenomenon in which the heat transfer characteristic deteriorates due to rapid change
in the boiling mode is generally called boiling crisis and the heat flux with which this boiling
crisis occurs is called critical heat flux There are two types of boiling crisis one generated
Table21(2) CHENrsquoS SUPPRESSION
FACTOR S 251
fRe F S
103 1000
104 0893
2x104 0793
3x104 0703
4x104 0629
6x104 0513
105 0375
2x105 0213
3x105 0142
4x105 0115
6x105 0093
106 0083
108 0000
Selection of model by JL The equation to calculate the nucleate boiling heat transfer coefficient is selected using the input parameter JL JL=0 Chenrsquos equation JL=1 Jens-Lottes equation Default=1
JAEA-DataCode 2013-005
- 28 -
from the transition from nucleate boiling to film boiling and another due to the breakage of
ring-type liquid films etc The former boiling crisis is generated in the case of pool boiling
or forced convection boiling in a low-quality or sub-cool region which is defined as departure
from nucleate boiling (DNB) and the critical heat flux (CHF) in this case is generally called
burn-out heat flux In contrast the latter boiling crisis is generated in the case of high-quality
forced convection boiling which is generally called dry-out
Since so-called burn-out is accompanied by the rapid damage of heat-generating body by
burning it is also called high-speed burn-out In contrast increase in the temperature of
heat-generation body after the boiling crisis is generally mild in the dry-out and therefore
dry-out is called low-speed burn-out In FEMAX Bjornard and Griffithrsquos model(26) is used
to distinguish high-speed burn-out generated by the transition from nucleate boiling to film
boiling from low-speed burn-out ie breakage of ring-like liquid film In the model to
distinguish whether the boiling crisis is high-speed burn-out generated in low-quality regions
or low-speed burn-out generated in high-quality regions the flow velocity ratio of saturated
steam Jg and the flow velocity ratio of saturated water Jf are used as a flow velocity to
normalize the generation rate of bubbles
The procedure is as follows
( ) 21gfgee
g
minusminus= ρρρDGxJ g (2113)
( ) ( ) 21gffee
f 1 minusminusminus= ρρρDxGJ g (2114)
gJ velocity ratio of saturated steam(-)
fJ velocity ratio of saturated water(-)
ex quality(-)
G mass flow rate (kgm2s)
g gravity acceleration (ms2) gρ density of saturated steam (kgm3)
fρ density of saturated water (kgm3)
The following conditions are used to identify the high-speed and low-speed conditions
(2115)
12 12g f crit
12 12g f crit
12 12crit g f crit
(low speed)
12 (highspeed)
12 (interpolation between low speedand highspeed)
J J J
J J J
J J J J
+ le + ge times lt + lt times
JAEA-DataCode 2013-005
- 29 -
Here 136 is used for jcrit under normal flow condition
The quality (gas-phase mass flowtotal mass flow) is given by
( ) fgse HHHx minus= quality (2116)
H enthalpy (Jkg)
Hs saturation enthalpy (Jkg)
Hfg latent heat (Jkg)
Next calculation to obtain critical heat flux (CHF) is explained Tong derived the W-3
equation to obtain an optimally fitted curve of experimental values of DNB heat flux qDNB in a
pressurized water channel of uniform heat generation This is used to evaluate CHF in
high-speed burn-out(27)
( ) ( )( ) ( )[ ]( )
( )
( ) inf7
e
e
eee
e78
863DNB
1041292382580
05487124exp8357026640
86901571
03713563117290596114840
1098705517718exp104268117220
10237960222101543
HH
D
x
Gxxx
xPP
PqqDNBW
minustimes+timesminus+times
minustimes++minustimes
timesminustimesminus+
timesminustimestimes==
minus
minusminus
minus
(2117)
Here
qDNB DNB heat flux(Wm2)
P pressure(Nm2)
Hf saturated enthalpy (liquid phase) (Jkg)
Hin inlet enthalpy (Jkg)
The applicable range of Eq(2117) is
69times106≦P≦158times106 (Pa) 14≦G≦68(kgm2s)
ex ≦01 Hin≧93times105(Jkg)
When xegt01 a correction equation of W-3 equation was presented by Hsu and Beckner(28) as
( )inf
e 03DNB 960761HH
xW DNBqq
==minustimes= α (2118)
α void fraction(-)
Meanwhile for evaluation of CHF in low-speed burn-out a modified version of Zuberrsquos
equation(29) is used
JAEA-DataCode 2013-005
- 30 -
( ) ( )[ ]21
gf
gf250gffgDNB 960231640
+minusminus=
ρρρρ
ρρσα gHq (2119)
Hfg latent heat (Jkg)
ρ density(kgm3)
σ surface tension (Nm)
g gravity acceleration (ms2)
Here suffices f and g represent liquid phase and gas phase respectively
(5) Heat transfer in transition-boiling and film-boiling When heat flux q is increased in a nucleate boiling region the boiling mode rapidly
changes and heat transfer deteriorates When heat flux exceeds the critical heat flux (point B
in Fig 214) the heat transfer surface is covered with a steam film and the heat flux value
moves rightward from point B in Fig 214 along the dotted line and jumps onto point D
Point D is in a film-boiling region nucleate boiling suddenly transits to film boiling
Conversely when the heat flux gradually decreases starting from point D it reaches point C
along the curve However when it passes point C the phenomenon suddenly changes into
that of the nucleate boiling where heat flux moves to point E (on A-B) along the dotted line in
Fig 214 Point C is the local minimum heat flux In the region between B-C heat flux
decreases as ΔTsat increases and extremely unstable phenomena are observed This region is
called the transition boiling region
Among these transitions of boiling rapid changes in the boiling mode from nucleate
boiling to film boiling are incorporated in the boiling transition analysis in FEMAXI As the
evaluation equation of heat transfer coefficients in the transition boiling and film boiling after
high-speed burn-out a modified evaluation equation by Condie-Bengston(22) is used that is
( ) satFBTBtotal Thhq Δ+= (2120)
qtotal heat flux(Wm2)
( )DNB
satsatDNBFBDNBTB 22
expDNB T
TTTqqh
T ΔΔ
ΔminusΔminus=Δ
(2121)
hTB transition boiling heat transfer coefficient (Wm2K)
qDNB critical heat flux(Wm2) refer to eqs(2117) and (2118)
DNB
FB Tq
Δ heat flux by film boiling at DNB(Wm2)
DNBFBFBDNB
ThqT
Δ=Δ
DNBTΔ satTΔ at DNB (K)
JAEA-DataCode 2013-005
- 31 -
( )[ ]
( ) [ ]w7
590282e
78420e
1ln2456060040g
30702w
43760g
FB Pr1045041exp89471131
RePr0810330
e
PxD
Kh
xminus
++
timesminus+
=
(2122) hFB film boiling heat transfer coefficient (Wm2K)
K thermal conductivity(WmK)
Pr Prandtlersquos number(-)
Re Reynoldrsquos number(-)
P pressure(Nm2)
Here suffix w represents wall temperature and suffix g represents the value obtained
using saturation temperature
On the other hand as an evaluation equation for the heat transfer coefficient at transition
boiling or film boiling after low-heat burn-out the following equation(22) is used
vaporliquidtotal qqq += (2123)
( ) ( )fwc TThTThq minus=minus= αfwvvapor (2124)
( )
( )
=330
filmfilm
250film
film
c
PrGr738170
PrGr151443max
e
e
D
KD
K
h (2125)
( ) ( )( )2filmfwfilm
3e 2Gr μρβ TTD minus= g (2126)
qvapor heat flux of steam(Wm2)
hv thermal conductance of steam(Wm2K)
Tw cladding surface temperature(K)
Tf coolant temperature(K)
K thermal conductivity (WmK)
Gr Grasshof number(-)
β thermal expansion coefficient(K-1)
ρ density(kgm3)
μ dynamic viscosity(kgms)
Kfilm is obtained using a value at 2
fwfilm
TTT
+=
JAEA-DataCode 2013-005
- 32 -
( ) ( )
( )
ge
leΔ
gtΔminusΔ+
=
9600
1
1960
satDNB
satsatFBTB
liquid
α
α
BTq
BTThh
q (2127)
( ) ( ) satTB
BTTB ehqqBh Δminus
=Δ
minusminusminus= 1FBvaporDNB
sat960712 α (2128)
( ) 250
satge
fggfg3
g
1720
c
eFB 526775
Δminus
=
TD
HKDh
μρρρ
λg
(2129)
( )50
gfc
minus=
ρρσλ
g (2130)
( ) ( )( ) ( )
1 2
1 2
001368 K at 4137000 N m
001476 K at 6205500 N m
B PB
B P
minus
minus
= == = =
B is obtained by interpolation on a log-log graph with respect to P
qliquid heat flux of the liquid phase (Wm2)
hTB heat transfer coefficient at transition boiling (Wm2K)
hFB heat transfer coefficient at film boiling (Wm2K)
Hfg latent heat(Jkg)
σ surface tension(Nm)
Here suffixes f and g represent liquid phase and gas phase respectively
(6) Heat transfer at steam single-phase When the quality xe becomes 1 the coolant is in a steam single-phase (superheated
steam) accordingly Dittus-Boelterrsquos equation (Eq(219)) is applied again
(7) Condensation heat transfer For the heat transfer at condensation two-phase flow Coillierrsquos evaluation equation(210)
is used Here suffixes f and g represent liquid phase and gas phase respectively
JAEA-DataCode 2013-005
- 33 -
( )q h T Tw sat= minus (2131)
h = max(hlaminar hturbulent) (2132)
( )
( )
2503f
min 2960
minusminus
=wsatfe
fggffarla TTD
KHh
μρρρ g
(2133)
ef
ggpfff
e
gg
f
pff
f
ff
turbulent
D
VCK
D
V
K
CKh
μμρ
μμμ
ρ
310857749
02300650
minustimes=
=
(2134)
q heat flux(Wm2)
Tw cladding surface temperature(K)
Tsat saturation temperature(K)
Cp specific heat(Jkg)
V velocity(ms)
213 Flow of thermal-hydraulics calculation The logical flow of calculations of coolant temperature and cladding surface heat transfer
coefficient in FEMAXI is shown in Fig215 As the figure indicates coolant temperature at
each axial segment is calculated from the coolant pressure and enthalpy at each segment by
using the steam table Also thermal conductivity viscosity density and Prandtle number etc
are calculated by referring to the steam table For this purpose FEMAXI has a built-in steam
table
JAEA-DataCode 2013-005
- 34 -
Fig215 Flow of thermal-hydraulics calculation
Thermal conductivity viscosity density Prandtle number etc
Calculation of enthalpy at a point positioned at each
node of the sub-segments in the previous step
Inlet enthalpy
Inlet temperature
Pressure
Heat flux at each segment
Inlet flow
velocity
Enthalpy at the segment
Mass flow rate
Enthalpy distribution of sub-segments at previous time step
Pressure
Temperature at the segment
Low-flow
high-flow
NO
NO
YES
YES
NO
Water single-phase
Is heat flux equal to or less than critical heat flux
Is the quality 099 or higher
Coolant flow rate
Either Chenrsquos or Jens-Lottesrsquos eq+
Dittus-Boelterrsquos eq
Dittus-Boelterrsquos equation RETURN
Dittus-Boelterrsquos eq (steam single phase)
Equation for transition boiling + film boiling (modified
Condie-Bengstonrsquos equation)
RETURN
RETURN
RETURN
Equation for transition boiling + film boiling (RELAP-5)
(
RETURN
YES
Is the magnitude of super-heating of heat transfer surface negative
and the quality positive
Heat transfer equation for the
condensation two-phase flow
(RELAP-5)
RETURN
JAEA-DataCode 2013-005
- 35 -
214 Transition process of coolant condition
Transient boiling phenomenon modeled in FEMAXI is not the transient boiling
accompanying the heat flux change but the transient boiling accompanying the flow rate
change Therefore heat transfer characteristics are obtained from the relationship between
the quality and magnitude of super-heating of the heat transfer surface rather than from the
relationship between the heat flux and magnitude of super-heating of heat transfer
Figure 216 shows the classification of the heat transfer characteristics in FEMAXI
modeling Table 22 lists the conditions set for each heat transfer mode Here using Fig
216 outline of the calculation of the boiling transition performed in FEMAXI is explained
In the figure ΔTsat on the vertical axis shows the magnitude of super-heating of the heat
transfer surface namely the amount of increase from the saturation temperature The quality
xe on the horizontal axis is defined by Eq(2116)
Fig216 Classification of heat transfer characteristics based on quality magnitude
of super-heating magnitude of sub-cooling and magnitude of super-heating
of the heat transfer surface
[ ]E prime Forced
convection to wall
[A] Forced convection to water
(TwgtTB)
(TwltTB)
(TwltTB)
[E] Forced convection to steam
[ ]Aprime Forced convection to wall
satTΔ
1
[C] transition boiling
[Β] Nucealte boiling
[D] film boiling
0
[F] Condensation
two-phase flow
xe quality
(TwgtTB)
Tw cladding surface temperature
TB coolant bulk temperature
x
Superheated steam region
Sub-cool region
JAEA-DataCode 2013-005
- 36 -
Here since xe is quality (= gas phase mass flow ratetotal mass flow rate) 0lexele1
should hold However we obtain the following results of the calculation using Eq (2116)
when HltHs (in a case of sub-cooling water) xelt0
when HgtHs+Hfg (in a case of super-heated steam) xegt1
Accordingly output of the calculation is expressed as QUALITY when 0ltxe lt1
magnitude of super-heating of the steam (MSUPHTG) when xe ge1 magnitude of sub-cooling
(MSBUCOOL) when xe le 0
(1) Normal operation In normal operation the coolant states belong to either region [A] or [B] In other
words when the cladding surface temperature is equal to or less than the saturation
temperature (ΔTsat le 0) the coolant state is in a region of forced convection heat transfer from
the cladding surface to water phase (region [A]) When the cladding surface temperature
exceeds the saturation temperature (ΔTsat gt 0) it is in a region of heat transfer of nucleate
boiling (including sub-cool boiling) (region [B])
(2) Decrease in the coolant flow rate When the heat flux remains constant and the coolant flow rate is decreased the quality
increases and gradually reaches the critical heat flux Consequently the coolant state is
transited to region [C] or [D] In the high-speed burn-out in a low-quality region the
cladding wall temperature rapidly increases whereas in the low-speed burn-out in a
high-quality region the cladding wall temperature gradually increases These behaviors are
determined by the heat flux at the time of change in the flow rate and the value of flow rate
after the decrease in the flow rate
In addition when the coolant flow rate is lowered significantly the coolant is transited to
a single-phase steam region (super-heating) of [E] In region [E] ΔTsat increases with
increasing enthalpy (temperature) of the coolant Namely the cladding surface temperature is
determined by the axial position of coolant and coolant pressure at the time when the coolant
enters region [E]
(3) Recovery of coolant flow rate When the coolant flow rate recovers enthalpy of the coolant rapidly decreases with
decreasing specific volume V (see Eq(211)) In the film-boiling region [D] ΔTsat becomes
large as the quality decreases Accordingly when ΔTsat is constant if the enthalpy decreases
JAEA-DataCode 2013-005
- 37 -
the quality decreases (see Eq(2116)) and the coolant state transits from the film-boiling
region [D] to the nucleate boiling region [B] In region [B] since the heat transfer rate is
very high the cladding wall temperature rapidly decreases and the normal operation condition
is recovered
When the coolant flow rate is recovered in the steam single-phase region [E] the coolant
temperature decreases due to the decrease in enthalpy With this ΔTsat decreases
Consequently the cladding surface temperature decreases However since it is the
temperature decrease under the condition of the quality being 1 the coolant state transits to
the nucleate boiling region [B] under very low ΔTsat value which rapidly decreases the quality
in region [B] That is the water which first enters is consumed to decrease ΔTsat and the
water which enters thereafter decreases the quality When the quality decreases sufficiently
normal operation is recovered
In region [F] (0ltxelt1) there is a flow of droplets and stream which is not related to the
boiling transition
Table 22 Set conditions for each heat transfer mode
Tw cladding surface temperature TB coolant temperature Heat transfer mode Model or eq Set conditions Convection heat transfer (single phase)
Dittus-Boelter
TW
ltTB
When cladding surface temperature le saturation temperature (water single-phase) when quality ge 099 (steam single-phase)
TW
gtTB
When quality = 0 (water single-phase) when cladding surface temperature ge saturation temperature (steam single phase)
Nucleate boiling heat transfer (two phase)
Either Jens-Lottes or Chen1 +Dittus-Boelter2
TW
gtTB
When cladding surface temperature ge saturation temperature (two phase) and heat flux le critical heat flux
Transition-boiling + film-boiling heat transfer (two phase high-flow)
Modified Condie- Bengston
TW
gtTB
When cladding surface temperature ge saturation temperature and heat flux ge critical heat flux and the quality le 099 (two phase) and coolant flow rate is high
Transition-boiling + film-boiling heat transfer (two phase low-flow)
RELAP-5 MOD-1model
TW gtTB
When cladding surface temperature ge saturation temperature and heat flux ge critical heat flux and quality le 099 (two phase) and coolant flow rate is low
Condensation (two phase liquefaction Collier
TW
ltTB
quality is positive and cladding surface temperature le saturation temperature (two phase)
1 0w satT Tminus gt Even if the coolant enthalpy does not reach the saturation enthalpy ie
2 0sat BT Tminus gt if the coolant is unsaturated the flow in the vicinity of cladding surface is
assumed to be a two-phase flow
JAEA-DataCode 2013-005
- 38 -
215 Equivalent diameter and cross-sectional area of flow channel
Equivalent thermal-hydraulics diameter and cross-sectional area of the coolant flow
channel are the parameters required for obtaining cladding surface temperature The
equivalent diameter De is the diameter of a virtual column when the shape of a coolant
channel around the fuel rod is converted to a column De is used in Eq(219) to determine
the cladding surface heat transfer coefficient and the cross-sectional area of the channel is
used in Eq(211) to calculate the enthalpy of each axial node
Usually these two parameters are directly set by input in the case that one of them is not
set or none of them is set a method of converting one value to the other or of calculating both
values using the rod pitch is incorporated in FEMAXI These methods are explained below
(1) Equivalent diameter of the channel De A) When De is specified by input data the input value is used
B) When De is not input but the cross-sectional area of the channel S is input De is obtained
using
DS
re = 4
2π (2135)
where S cross-sectional area of the channel
Name-list input parameter FAREA
r fuel rod outer radius
eD Name-list input parameter DE
C) When neither De nor S is input these values are
calculated using the pitch between fuel rods Assuming
channel III in Fig217 S is given by
S l r= minus4 2 2π (2136)
where l half of the pitch between fuel rods
Rod pitch input name-list parameter PITCH
De is calculated using Eq(2135) after obtaining S
S
Symmetrical line
l1
Fig217 Channel model
JAEA-DataCode 2013-005
- 39 -
(2) Flow cross section area
A) When flow cross sectional area S is specified by input the input value is used
B) When flow cross sectional area S is not specified but the equivalent diameter De is
specified by input the flow cross sectional area S is determined by
4
2 eDrS
π= (2137)
22 Cladding Waterside Corrosion Model
In the high burnup region waterside corrosion (oxidation) of a cladding advances and its
effect on heat transfer and mechanical behavior of the cladding cannot be neglected Growth
rate of oxide layers depends on temperature and it is calculated in FEMAXI as a function of
temperature at the boundary between the metal substrate and oxide layer This boundary
temperature is obtained by the thermal calculation explained in the following section
Thermal conductivity of oxide layers is specified by name-list input IZOX Specific heat
of the oxide layer is given by MATPRO-A(211)(Refer to section 42)
A typical one is the MATPRO-A model as follows(211)
CN = 0835 + 181times10-4T (221)
Specific heat of oxide layers is given by MATPRO-A(211) as
272 1014110116565 minusminus timesminustimes+= TTCP (222)
(300 1478 )T Kle le
The oxide growth (thickness increase) model is an empirical one and designated by input
In this initial thickness of oxide can be specified segment-wise by input In any models of the
oxide growth the corrosion rate is given as a function of temperature at the metal-oxide
interface This temperature is calculated in the thermal analysis
In Eq(221) since the value of the temperature-dependent term is approximately one
order of magnitude lower than the constant term the temperature gradient across the oxide
layer can be approximated by a straight line (Fig222) Therefore in FEMAXI thermal
conductivity is assumed to be uniform inside the oxide layer and to determine the thermal
conductivity in Eq(221) the average value of the oxide surface temperature and the
boundary temperature is designated as the temperature of the oxide layer
Decrease in the thickness of cladding metal phase with progress of corrosion is
JAEA-DataCode 2013-005
- 40 -
quantitatively related to the volume of oxide layer In Zircaloy volume of the oxide expands
approximately 156 times from that of the volume of the metal phase consumed for oxide
generation This ratio is called the Piling-Bedworth ratio (PBR) (Fig 221)
Using PBR decrease in the thickness of cladding metal part and increase in the cladding
outer diameter can be given as SPBR and S(PBR-1)PBR respectively where S is the
oxide layer thickness Both the thermal calculation and the mechanical calculation are
performed with this changing thickness of radial ring element by re-setting the element
thickness at each time step
Name-list input 1) Option of oxidation modelICORRO =0 No oxidation =1 EPRI model(212)
=2MATPRO-09 (PWR) model(213) =3 MATPRO-09 (BWR) model (Default=1)
2) Oxidation rate tuning factorRCORRO (Default=10) Oxidation rate is multiplied by
RCORRO times
3) Oxide thermal conductivity factor OXFAC (Default=10)
thermal conductivity of oxide NC is adjusted by OXFACCC NN times=
4) PX radial fraction of volumetric increase in X=PBR-10 by oxide growth ()(default=99)
5) PBR volumetric increase ratio by oxide growth ie Piling-Bedworth ratio (default=156)
[Input value initial oxide layer thickness OXTH(40)] Initial oxide layer thickness is designated for each segment using OXTH (microm) Standard value is 01 microm
Temperature
Cladding meta l part
Meta l par t a f ter oxidat ion
Oxide layer Cladding
Meta l part
Oxide
Radius
( )Stimes PBR -1S
PBR PBR
Co
ola
nt
Fig 221 Volume increase due to cladding corrosion
Fig222 Temperature distribution inside cladding
JAEA-DataCode 2013-005
- 41 -
23 One-Dimensional Temperature Calculation Model 231 Selection of number of elements in the radial direction
In performing the thermal and mechanical analyses in FEMAXI it is possible to select a
mesh system ie number of ring elements in the pellet stack and cladding of one axial
segment in one-dimensional geometry as follows (this is explained in the mechanical analysis
chapter again)
(1) Number of ring element divisions in pellet stack
As shown in Table 231 a name-list parameter MESH can designate the number of ring
elements of pellet stack The default is ldquoMESH=3rdquo In this iso-volume ring elements of pellet
stack thickness of the ring decreases as a function of radial location of the ring as shown in
Fig231 In the peripheral region of pellet there is a peaking of heat generation density and
burnup and the rim structure is formed so that a fine digitizing of this steep change region
will markedly contribute to the accuracy and preciseness of calculation
However when the HBS model (rim structure model) which will be later described is
activated by input it is necessary to always input MESHgt0 If the HBS model is activated
MESH=3 is automatically set even if MESH=0 is designated by input
Table 231 Number of radial ring elements of pellet stack in thermal and mechanical analyses Parameter value
1-D thermal analysis 1-D mechanical analysis
2-D local PCMI analysis
MESH=0 Iso-thickness 10 ring elements
Iso-thickness 10 ring elements
Iso-thickness 5 ring elements
MESH=1 Iso-volumetric 36 ring elements Iso-volumetric 18
ring elements Iso-volumetric 9 ring elements
MESH=2 Iso-volumetric 72 ring elements
MESH=3 Iso-volumetric 36 ring elements Iso-volumetric 36
ring elements Iso-volumetric 18 ring elements
MESH=4 Iso-volumetric 72 ring elements
[Adjustment parameter FCOPRO]
Corrosion rate can be adjusted by the following parameters ( )FCORROdt
dS
dt
dS += 1
JAEA-DataCode 2013-005
- 42 -
When name-list parameter ISHAPE=1 is selected by input the number of ring elements
in MESHgt0 is doubled which allows a finer calculation
(2) Number of ring element divisions in cladding wall Ring element division in cladding is common to both the 1-D thermal analysis and 1-D
ERL mechanical analysis irrespective of the MESH value designation shown above That is
A) Cladding with no Zr-liner
8 iso-thickness rings for metallic wall + 2 iso-thickness rings for outer surface oxide layer
B) Cladding with Zr-liner
2 iso-thickness rings for Zr-liner + 8 iso-thickness rings for metallic wall + 2 iso-thickness
rings for outer surface oxide layer
In the 2-D local mechanical analysis
C) Cladding with no Zr-liner
4 iso-thickness rings for metallic wall + 1 ring for outer surface oxide layer
D) Cladding with Zr-liner
One ring for Zr-liner + 4 iso-thickness rings for metallic wall + 1 ring for outer surface
oxide layer
However when ISHAPE=1 the number of rings is doubled from the above in the 2-D
local mechanical analysis
Plot output of cladding deformation In the ERL mechanical analysis in outputting the plotted
figures of cladding diameter or radial displacement IDNO=174 refers to the position of metal-oxide
interface and IDNO=7 and 11refer to the oxide outer surface
In the local 2-D mechanical analysis mechanical properties of oxide layer is not considered
JAEA-DataCode 2013-005
- 43 -
The geometry with these 36 or 72 iso-volume ring elements in pellet stack is effective in
performing a precise calculation of thermal conduction by using a detailed power density
profile in the radial direction This holds for example in loading the result file of the
RODBURN code (36 ring elements) or PLUTON code (36 72 or more ring elements) or
even more detailed profile into FEMAXI Furthermore the 36 or 72 ring elements
geometry is indispensable in conducting the HBS (rim structure) model of pellet which is
described in section 2121 However the 36- or 72-ring elements geometry itself can be used
when the rim structure model is not applied The 10-ring elements model can be still used
as it was in FEMAXI-7 In this case the PLUTON file and the HBS model cannot be used
In chapter 3 the ring element number is explained again
232 Determination of cladding surface temperature
In FEMAXI Eq(231) is obtained from the finite difference equation of thermal
conduction inside a fuel rod (see section 233)
φN = ATN + B (231)
where φNcladding surface heat flux (Wm2)
TNcladding surface temperature (K)
N cladding surface mesh number
Fig231 Ring elements and analytical geometry of pellet stack and cladding
of one axial segment
00 02 04 06 08 10
ZrO2
1 2 3 4 5 6 7 8
Cladding
ring element number
Pellet stack 36 equal-volume ring elements
Relative radius of pellet
Gap or Bonding layerCenterline
Relative power density
JAEA-DataCode 2013-005
- 44 -
Here A and B are the coefficients determined by the finite difference equation of thermal
conduction Surface temperature TN can be obtained from the heat transfer coefficient of a
cladding surface determined using Eq(231) That is when coolant temperature is
designated as TB (K) heat flux at cladding surface can be given as
φN = hW(TN ndashTB) (232)
where hW surface heat transfer coefficient (Wm2K)
φN cladding surface heat flux (Wm2)
TN cladding surface temperature (oxide surface ) (K)
TB coolant temperature (K)
TN can be obtained by eliminating φN from Eqs(231) and (232)
However as stated in the previous section FEMAXI has an optional mode in which
cladding surface temperature is not calculated but given by input This mode has two options
option-1 is to specify the cladding surface (oxide surface) temperature and option-2 is to
specify the temperature at metal-oxide interface In the option-1 name-list parameter ITCP=0
is set and rod temperature calculation is performed with the oxide surface temperature as
boundary condition In the option-2 ITCP=1 is set and the temperature at the oxide surface is
obtained by
ln(2 )N
N OXOX OX
rqT T K rπ
= minus sdot
(233)
where
TN cladding surface(oxide surface) temperature (K)
TOX metal-oxide interface temperature given by input (K)
q linear heat rate (Wm)
KOX oxide thermal conductivity (WmK)
rN cladding (oxide) surface radius (m)
rox radius at metal-oxide interface (m)
JAEA-DataCode 2013-005
- 45 -
233 Solution of thermal conduction equation
In the analysis of temperature gradient in the radial direction at each axial segment
one-dimensional thermal conduction equations are used while the heat conduction in the axial
direction is neglected It is assumed that the thermal properties of the fuel at each ring
element are dependent on temperature Based on these assumptions thermal conduction at
one axial segment is described as follows
( ) ( )[ ] ( ) ( ) ( )partpart t
c T r T r t k T r T r t q r tv = nabla sdot nabla + (234)
Here
T temperature (K)
r coordinate in the radial direction (m)
Cν volume specific heat (Jm3)
k thermal conductivity (WmsdotK)
q heat generation per unit volume (Jm3sdots)
n-1 n n+1
kpn
Cvpn
qpn
ksn
Cvsn
qsn
hpn hsn
hpn 2 hsn 2
Fig 232 Ring element model
Name-list Parameter IS ICTP Setting either IS=3 or =4 enters the mode in which cladding surface temperature history is given by input In this mode ICTP=1 specifies the temperature at metal-oxide interface (default) ICTP=0 specifies the temperature at oxide surface When either IS=3 or =4 thermal-hydraulic calculation is essentially unnecessary Therefore if additionally IS3P=0 is set thermal-hydraulic calculation is omitted and calculation time is reduced IS3P=1 thermal-hydraulic calculation is On (default) IS3P=0 Off To plot out the cladding surface heat flux (IDNO=62) and surface heat transfer coefficient (IDNO=63) IS3P=1 should be set
JAEA-DataCode 2013-005
- 46 -
Volume integration is performed with Eq(234) Here the volume to be integrated is
that enclosed by the dotted lines in Fig 232
( ) ( ) ( ) ( ) ( ) V v V VC T r T r t dV k T r T r t dV q r t dVt
part = nabla sdotnabla +
part (235)
Since this is a one-dimensional problem dimensions of the volume are set as unity except
for the radial direction Eliminating the common factor 2π and using the forward finite
difference method for time differentiation the first term of Eq(235) is expressed as
( ) ( ) ( ) ( )
( )1
2 4 2 4
V v V v
m mn n pn pn sn sn
v pn n v sn npn
TC T r T r t dV C T r r t dV
t t
T T h h h hC r C r
h
+
part part asymp part part
minus asymp minus + +
(236)
where Tnm is the temperature at coordinate rn and time tm and 1+m
nT is the temperature at
coordinate rn and time tm+1 The second term of Eq(235) is expressed as
( ) ( ) ( ) ( )( ) ( )1 1 1 1
1 1
[ ]
2 2
V s
m m m mpn n n sn n npn sn
n npn sn
k T r T r t dV k T r T r t d S
k T T k T Th hr r
h h
+ + + +minus +
nabla sdotnabla = nabla sdot
minus minus asymp minus minus + +
(237)
Here continuity conditions of the heat flux at the inner boundary are applied for
evaluation of the plane integration along the boundary surface
The following terms are defined for convenience sake
hh
rh
h rh
hh
rh
hh
rh
pnV pn
npn
snV
nsn
pns
pnn
pnsns
snn
sn
= minus
= +
= minus
= +
2 4 4
1
2
1
2
(238)
D C h C hn v pn pnV
v sn snV= +
Then the heat generation term q(r t) of the third term in Eq(235) is separated with
variables as
( ) ( ) ( )q r t Q r P t= (239)
Here Q(r) is the relative power generation distribution in the radial direction and P(t) is a
standard value of the heat generation density as a function of time Then the third term of
JAEA-DataCode 2013-005
- 47 -
Eq(234) is expressed as
( ) ( ) ( ) V VV pn pn sn sn mq r t dV Q h Q h P t θ+ asymp + (2310)
By adding the right sides of Eqs(236) (237) and (2310) which are approximate
expressions of the terms in Eq(235) Eq(235) is converted into the difference
approximation equation for the n-th mesh point as follows
( ) ( )( )( ) ( )
1
1
1
nm n n m m s
n n pn pn
m m sn n sn sn
V Vpn pn sn sn n
T T DT T k h
t
T T k h
Q h Q h P t
θ θ
θ θ
θ
+ + +minus
+ ++
+
minus= minus minus
Δ+ minus
+ +
(2311)
Upon application of the Crank-Nicholson method a fully implicit method ( 10θ = ) to
Eq(2310) the following equation holds
( ) ( ) ( )
( ) ( ) ( )
T T D
t
T T T Tk h
T T T Tk h
Q h Q h PP t P t
nm
nm
n nm
nm
nm
nm
pn pns
nm
nm
nm
nm
sn sns
pn pnV
sn snV
fm m
+ +minus
+minus
++
++
+
minus=
+minus
+
+ + minus +
+ ++
1 11
11
11
11
1
2 2
2 2
2
Δ
(2312)
Here thermal conductivity kpn and ksn and Tm+1 and Tm are derived from the average of
temperatures at n-1-th and n-th elements and temperatures at element nodes n and n+1
respectively
Through modification of Eq(2312) the differential equation at ring point n in the
medium region is given as
a T b T c T dn nm
n nm
n nm
nm
minus+ +
+++ + =1
1 11
1 (2313)
where
( )
( ) ( ) ( )
ak h t
ck h t
b D a c
d a T D a c T c T
t Q h Q h PP t P t
npn pn
s
nsn sn
s
n n n n
nm
n nm
n n n nm
n nm
pn pnV
sn snV
fm m
= minus = minus
= minus minus
= minus + + + minus
+ ++
minus +
+
Δ Δ
Δ
2 2
2
1 1
1
(2314)
JAEA-DataCode 2013-005
- 48 -
For the pellet-clad gap region the following equations hold as in the case of Eq (2312)
Next the P-C gap region is shown in Fig233
Pellet CladdingGap
n-1 n n+1 n+2
Tn-1 Tn Tn+1 Tn+2
hpn hsn+1
Fig 233 Gap thermal conductance model
For the gap region similarly to Eq(2312)
( ) ( ) ( )( )1
111
11
11
+
+++
+minus
++
+
minus+minusminus=Δ
primeminus
mfVpnpn
gm
nm
nspnpn
mn
mn
nm
nm
n
tPPhQ
hTThkTTt
DTT
(2315)
( ) ( ) ( )( )11
111
11
211
111
11
++
+++
++
+++
+++
++
+
minus+minusminus=Δ
primeprimeminus
mfns
snsns
mn
mng
mn
mn
nm
nm
n
tPPQ
hkTThTTt
DTT (2316)
Holds where 1 1 1V Vn v pn pn n v s n s nD C h D C h+ + +prime primeprime= =
and
gh is the gap thermal conductance
( ) ( )( )
( )
11 1 1 1 1
1 1 1
1 12 1 1 1
1 1 1
m mn n n m m s
n n g p n p n
m m sn n s n s n
Vs n s n m
T T DT T h h h
t
T T k h
Q h P t
++ + + + +
+ + +
+ ++ + + +
+ + +
primeprimeminus= minus minus
Δ+ minus
+
(2317)
Here prime = primeprime =+ + +D C h D C hn v pn pnV
n v s n s nV
1 1 1 1sn p nh h += = gap width and hg is gap thermal
conductance The gap region is shown in Fig 233
The center region of the fuel rod is shown in Fig 234(a) and the gap and cladding
surface region is shown in Fig 234(b)
JAEA-DataCode 2013-005
- 49 -
Fig234(a) Fuel rod center model Fig 234(b) Cladding surface model
Here the oxide layer on the outer surface of cladding has a lower thermal conductivity
than that of the metal phase
For the center of the fuel rod Eq (2312) is transformed as follows
( ) ( ) ( )10000
10
110
01
0+
+++
+minus=Δminus
mfVss
sss
mmsv
mm
tPPhQhkTTCt
TT (2312a)
The boundary equation for the cladding surface region ( Nr r= ) is
( ) ( ) ( )1
11
1
1
++
minus+
+
+minusminusminus=Δminus
mfVpnpnnn
spnpn
mn
mn
Vpnpnv
mn
mn tPPhQrhkTThC
t
TT φ (2318)
The difference approximation equation (2318) is expressed as
a T b T dn nm
n nm
nminus+ ++ =1
1 1 (2319)
Then the coefficients nn ba and nd are given as
a
k h tn
pn pns
= minusΔ
2 (2320)
b C h an v pn pnV
n= minus (2321)
( )( ) ( )
d a T C h a T
t r tQ h PP t P t
d d
n n nm
V pn pnV
n nm
nnm
nm
pn pnV
fm m
n n nm
= minus + minus
minus +
+
+
= prime + primeprime
minus
++
+
1
11
1
2 2
Δ Δφ φ
φ
(2322)
However in the normal operation period the heat flux mnφ at time tm in Eq(2322) is
Pellet center
0 1
T0 T1
hs0
Pellet Surface Cladding outer surface
n-1 n=N
Tn-1 Tn=TN
hpn
φ φn N=
Gap
JAEA-DataCode 2013-005
- 50 -
given by
φ πnm m
nq r= 2 (2323)
qlinear heat rate (Wm)
This is to avoid the accumulation of numerical difference errors which may occur
through repeated re-setting of ring element on the calculated heat flux
Here φN is the cladding surface heat flux Now unknown variables in Eqs(2318) to
(2321) are 1 11 m m
n nT T+ +minus and 1mφ +
Next using coefficients expressed as E and F which are obtained using the forward
elimination of the Gaussian elimination method starting from the first point to the (N-1)th
point the following equation is derived
T E T Fnm
n nm
nminus+
minus+
minus= minus +11
11
1 (2324)
By substituting Eqs(2319) to (2322) into Eq(2318) we obtain
A T Bnm m
11
11+ ++ = φ (2325)
where
( )A b a E dn n n n1 1= minus primeprimeminus (2326)
( )B a F d dn n n n1 1= minus prime primeprimeminus (2327)
However Eq(2325) is equivalent to Eq(231) and in Eq(231) A1 and B1 are known
values Therefore 1m
nT + can be determined as stated in section 232 Hereafter using the
backward substitution of the Gaussian elimination method Tm+1 is sequentially obtained
234 Fuel pellet thermal conductivity (1) Thermal conductivity models UO2 and MOX fuel thermal conductivity decreases with burnup Typical models ie the
NFI model(214) and Halden model(215) are shown in Fig235 The thermal conductivity model
can be chosen by name-list input parameter IPTHCN (See section 41)
Details of equations and literatures are described in section 41
JAEA-DataCode 2013-005
- 51 -
500 1000 1500 20001
2
3
4
5
6
80 GWdtHM
40 GWdtHM
Ther
mal
Con
duct
ivity
(Wm
K)
Temperature (K)
NFI UO2
NFI MOX Pu=10 Halden UO2
Halden MOX0 GWdtHM
(2) Effect of porosity The thermal conductivity is also affected by porosity and density as shown in Chapter 4
Here between porosity p and theoretical density ratio D the following equation holds
Option for fuel pellet thermal conductivity IPTHCN Detailed in Chap4 =1 MATPRO-09 =2 Washington =3 Hirai =4 Halden (Wiesenack) =5 Modified Hirai =6 Forsberg =7 Kjaer-Pedersen =8 BaronampCouty =10 Lucta Matzke ampHastings
=11 Tverberg Amano Wiesenack (UO2 Gadlinia-containing fuel) =13Fukushima (UO2 Gadlinia-containing fuel) =15Daniel Baron (UO2 Gadlinia-containing fuel) =16 CRIEPI (Kitajima amp Kinosita)=17Halden new (Wiesenack) (UO2) =18 PNNL-modified Halden model(UO2) =90 Ohira amp Itagaki (UO2) =91 Ohira amp Itagaki (NFI) latest Model for 95TD UO2 and MOX pellets =92 Ohira amp Itagaki Modified in FRAPCON 33 for 95TD UO2 pellet =30Martin (MOX) =31MATPRO-11(MOX) =32Martin+Philipponneau (MOX) =33Duriez et al(MOX) =34Philipponneau (MOX) =35 Halden (new MOX) =36Daniel Baron (MOX UO2 Gadlinia-containing fuel) =38 FRAPCON-3 PNNL Modified model =39 FRAPCON-3 PNNL-modified Halden model
Fig235 Comparison of fuel thermal conductivity degradation with burnup
JAEA-DataCode 2013-005
- 52 -
p(I)=10 - D(I) (2328)
where I pellet ring element number
D(i) is defined as
D(I)=Di-(3ΔLr) (2329)
where Di initial theoretical density ratio
ΔL radial displacement due to volume change generated
by irradiation
r pellet radius
In FEMAXI it is possible to specify the definition of porosity p (or D) in Eq(2328) by
designating the parameter IPRO (default is 0) as follows
IPRO =0 Initial porosity 0p is used when D(I)=Di initial theoretical density ratio
=1 00
swgVp p
V
Δ= + is used
=2 0
0 0
dens hotV Vp p
V V
Δ Δ= + + is used
=3 00 0 0
swg dens hotV V V
p pV V V
Δ Δ Δ= + + + is used
where 0
swgV
V
Δ volumetric swelling of pellet induced by fission gas bubble growth
0
densV
V
Δ volumetric densification of pellet
0
hotV
V
Δ volumetric shrinkage of pellet due to hot-pressing
Name-list input parameter IPRO (Default=0)
However this calculation is not performed in every ring element but is performed at
each axial segment to derive the average porosity over the segment Plotting output is done by
IDNO=57 pellet density =1 pminus
Another related name-list parameter MPORO is an option to designate the assumed
effect of porosity on pellet thermal conductivity
That is
MPORO=0 pellet thermal conductivity is calculated using the segment-wise average
porosity which is calculated on the basis of IPRO (default setting)
MPORO=1 pellet thermal conductivity is calculated using the ring-element-wise
calculated porosity 00 0
swg densV V
p pV V
Δ Δ= + +
JAEA-DataCode 2013-005
- 53 -
Here definitions of swelling and densification are equivalent to that of IPRO However
the element-wise change of pellet thermal conductivity induced by the activation of HBS
model is independent from designation of MPORO so that effect of element-wise porosity
change by the HBS model can be included in calculation irrespective of the MPORO value
The corresponding plotting outputs are done by IDNO=257 total porosity and
IDNO=260 fission gas bubble porosity
(3) Relationship between porosity and thermal conductivity models The thermal conductivity models explained in section (1) implicitly adopt p=005 as a
default value When the porosity changes from 005 with extension of burnup pellet thermal
conductivity would change To take into account of this effect IPRO is used as above It is to
be noted that some models of pellet thermal conductivity have already included this effect
For example the Halden model (IPTHCN=4 17) has been derived from the measured
data of linear heat rate and cladding temperature of test rod under irradiation Accordingly it
can be considered that this model implicitly includes the effect of porosity change If IPRO=1
2 or 3 is designated for this model calculation takes into account of the effect doubly by the
model and by the FEMAXI algorithm Thus the calculated results will be either over- or
under estimation Therefore to reflect the porosity change in thermal conductivity change it is
necessary to confirm the condition in which thermal conductivity model has been derived
24 Determination of Heat Generation Density Profile
Heat generation density profile in the radial direction of a pellet can be obtained by any
one of the following four methods
1 Designation by input as a function of burnup
2 Using the output of RODBURN(1155)
3 Using the output of PLUTON(116) and
4 Using a distribution function Q(r) of the Robertsonrsquos formula (Halden experimental
equation) (216)
241 Use of RODBURN output
The burning analysis code RODBURN calculates the power density profile burnup
profile and fast neutron flux in the axial and radial directions of a fuel rod from data on fuel
JAEA-DataCode 2013-005
- 54 -
size shape materials temperature power history and thermal neutron flux history and then to
produce an output file FEMAXI can read this file as one of the calculation conditions
242 Use of PLUTON output
The burning analysis code PLUTON(116) calculates accurately and at high speed the
power density profile burnup profile fast neutron flux distribution of Pu isotope production
and other factors such as fission product element production distribution in the axial and
radial directions of a fuel rod from data on the fuel size shape materials temperature and
power history and produces output files FEMAXI can read these files as one of the
calculation conditions
243 Robertsonrsquos formula
Robertson proposed the following equation regarding heat generation density profile of a
pellet in the Halden reactor(216)
( ) ( ) ( )( ) ( )RkK
RkK
RkIRkIRQ sdotsdot
sdotsdot+sdot= 0
11
1101 (241)
Here
I first transformed Bessel function
R1 pellet center hole radius
K second transformed Bessel function
R radial coordinate in pellet
k Inverse of the neutron diffusion distance (cm-1)
Here k is calculated using
( ) ( )k E DR
E Dp
= sdot +
sdot sdot0328 054
050 8
0 82
0 19
(242)
where
E 235U enrichment ()
Name-list input parameter IFLX =-2 input from RODBURN results
=-1 input from PLUTON results =0 using the Robertsonrsquos formula
IFLXgt1 Designated by input as a function of burnup
The heat generation density profile in between the input burnup points is
determined by interpolation in terms of burnup
JAEA-DataCode 2013-005
- 55 -
D pellet theoretical density ratio (minus)
Rp pellet diameter (cm)
The distribution function (241) can be approximated by the following function
( ) ( )( ) ( ) ( )[ ]( )( ) ( )( )22
2
2 1121
112
ahhhahb
harahbrQ
bb
bb
minusminus+minusminusminus+minusminus+= (243)
where
r normalized pellet radius )10( lele r
a heat generation density ratio of inner surface to outer surface of the pellet (minus)
b shape factor of the heat generation density function Q(r) (minus) and
h normalized inner radius of the pellet (minus)
In the case of Halden reactor the shape factor in Eq (243) can be set as b=2 since Eq
(243) can be well approximated with a quadratic function The heat generation density
ratio a is expressed as follows using Eq (241)
( )( )pRQ
RQa
1
11= (244)
244 Accuracy confirmation of pellet temperature calculation
To evaluate the accuracy of the numerical calculation of the temperature distribution in
the radial direction in a pellet in FEMAXI a comparison between analytical solutions and
numerical results by FEMAXI is shown below
(1) Power profile in the radial direction in a pellet The temperature distribution in a pellet can be given as
1po
po pi
T r r
pT r rdT r q dr dr
rλ = sdot sdot (245)
where pλ pellet thermal conductivity (W(cmK))
r location in the radial direction in the pellet (cm)
T temperature (K) at location r
rpi radius of the center hole in the pellet (cm) and
q heat generation density at radial location r(Wcm3)
By normalizing Eq(245) with respect to location r the heat generation density q is
JAEA-DataCode 2013-005
- 56 -
written as a function of location r as q(r)
1 1( )
po
T r
pT r hdT r q r dr dr
rλ = sdot sdot (246)
Here r normalized radial location in the pellet
h normalized radius of the central hole in the pellet
The function of generated heat density distribution Eq (246) q(r) is given by
( ) )(1
)(2
rh
Prq LHR ϕ
πsdot
minus= (247)
where )(rϕ is the heat generation density distribution function and PLHR is the radiation
power Using Eqs(246) and (247) when the pellet thermal conductivity pλ is constant and
the heat generation density distribution function )(rϕ is uniform the temperature
distribution in a pellet can be easily obtained For example for the normalized radius h=0 of
the center hole in the pellet the linear power PLHR=400 (Wcm) the heat generation density distribution function )(rϕ =1 and pellet thermal conductivity pλ =003(W(cmk)) by
substituting Eq(247) into Eq(246) the following equation is obtained
( )
21 1
00
1212
1 1
2
12 2 2 4
po
rT r
LHR LHRpT r r
LHR LHR LHR
rr
P P rdT r dr dr dr
r r
P P r Prdr r
λπ π
π π π
= sdot sdot = sdot
= = = minus
(248)
Accordingly [ ] ( )214
rP
T LHRTTp po
minus=π
λ and analytical solutions for the temperature in the
pellet can be obtained by providing the values of pλ PLHR and the outer-surface temperature
of pellet TPO When TPO=450 (oC) the temperature T at relative radius r is given as
( ) )1(0304
4004501
422 rr
PTT
p
LHRPO minus
sdot+=minus+=
ππλ (249)
Then using Robertsonrsquos formula assuming that the heat generation density ratio of
internal to external surfaces of pellet is a and the form exponent of the heat generation density
function is b the heat generation density distribution function )(rϕ is
( )( ) ( ) ( )[ ]( )( ) ( )( )22
2
1121
112)(
ahhhahb
harahbr
bb
bb
minusminus+minusminusminus+minusminus+=ϕ (2410)
By substituting Eq(2410) into Eq(247) and substituting Eq(247) into Eq(246) the
JAEA-DataCode 2013-005
- 57 -
following equation is obtained
( )4po
TLHR
pT
PdT F rλ
π= sdot (2411)
Here
( ) [ ][ ])1)(1(2))(1(
ln)2)(()1(22)1)()(2(2
)1)(1(4
22
2222
ahhhahb
rhbhaharhabb
rarF
bb
bbbb
minusminus+minusminus
+minus+minus+minusminus+++
minusminus= ++
(2412)
When h=0 Eqs(249) and (2410) are simplified to
( ) ( )[ ]
2
12)(
++minus+=
ba
arabr
b
ϕ (2413)
( ) )2(
)1()2(2
)1)(1(4 22
+
minus+++
minusminus
=
+
ba
rabb
ra
rF
b
(2414)
For example when a=07 and b=2
( )24 03 07( )
34
rrϕ
+= ( ) 4 203(1 ) 28(1 )
34r r
F rminus + minus
=
Accordingly by obtaining F(0) F(0)=3134=0912 The value obtained using
Eq(2412) is 0912-fold that under uniform power distribution
Moreover with respect to )(rϕ according to [ 7030 2 +r ] in which the central value is
07 and the surface value is 1 to set the integral value at 1 it is necessary to multiply )(rϕ
by 434=1176 Here when the pellet radius is divided into 10 equal-thickness rings the
relative power density distributions applied to the rings are 08244 08315 08456 08668
08950 09303 09727 10221 10785 and 11421 from the inside toward the outside of the
pellet These are relative power density distributions obtained by volume averaging the
solution for each ring using Eq(2413)
Because F(r) is obtained using Eq(2412) when λp is constant at pλ =003 (Wcm-K)
the analytical solution for the pellet temperature distribution is obtained from Eq(2410)
Table 241 shows a comparison between the values of analytical and numerical solutions
The latter is obtained using FEMAXI by discretizing the radial distance into 10 and 100
segments The numerical solutions in Table 241 in the case when the power distribution in
JAEA-DataCode 2013-005
- 58 -
the radial direction is given are the results obtained using a table for the relative power
density distribution obtained by volume averaging the solution for each ring as mentioned
above In the 100 mesh system results were obtained by calculating values for 100 points by
interpolating the above 10 data points using the following equations
f(x)=arb+C (2415)
( ) (1 ) PFACf x CSFR r CSFR= minus sdot + (2416)
(Here strictly ( ) ( )2 1( )
2
bb a x af x
ba
+ minus + =+
where a=CSFR and b=PFAC)
Furthermore the numerical solution in the case in which CSFR=07 and PFAC=20 are
given for both the10- and 100-mesh systems is the value obtained by calculating the central
coordinate of r (r is normalized by setting the radius of a pellet to 10) and calculating the
value of )(rϕ corresponding to r using Eq(2411)
As a result as is apparent from Table 241 it has been confirmed that the difference in
pellet temperatures between the analytical solution and numerical solution for the same power
density distribution is 01 or smaller In addition compatibility with the analytical solution
is better when simply the relative power density distribution at the coordinate center (the
numerical solution obtained by providing CSFR=07 and PFAC=20) is used than when the
relative power density distribution obtained by volume averaging (the numerical solution
when the power distribution in the radial direction is given) is used
Table 241 Comparison between values from the analytical and numerical
solutions obtained by FEMAXI
Relative radius
of pellet
Analytical solution of
integral equation
Numerical solutions
obtained by providing power
profile in the radial direction
Same as left (number of rings in the
radial direction of pellet = 100)
Numerical solution when CSFR=07 and PFAC=20 are
given
Same as left (number of rings
in the radial direction of
pellet = 100)
00 01 02 03 04 05 06 07 08 09 10
14174114086713823113380112752111931110907196678819846482245000
141873 140997 138358 133922 127633 119412 109158 96747 82033 64848 45000
141758 140883 138246 133814 127530 119317 109075
96679 81984 64822 45000
141787 140912 138276 133844 127560 119346 109101
96701 82001 64831 45000
141758 140883 138246 133814 127530 119317 109075
96679 81984 64822 45000
JAEA-DataCode 2013-005
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This is because the calculation of temperature distribution in a pellet is one dimensional
in the radial direction the effect of relative power density obtained by volume averaging on
the temperature distribution is greater on the peripheral side (the side with higher relative
power density) where the volume is larger than on the inner side
(2) Thermal conductivity of pellet Since the thermal conductivity of pellet depends on temperature burnup and density the
heat conduction equation for pellet is nonlinear and it is impossible to obtain an analytical
solution Factors such as burnup and density dependences are effectively equivalent to a
difference in sensitivity to temperature dependence temperature dependence alone should be
taken into consideration in the examination of programs and the accuracy of numerical
solutions The remaining issue is a proper coding of physical models in the program The
thermal conductivity equation of Wiesenack(43) was used in the comparison of simulated
solutions with the numerical solutions based on analytical solutions
Ignoring the burnup dependence of the Wiesenack (Halden) equation(43) gives the
following equation
( )41 01148 2475 10 00132exp(000188 ) (WmK)K T Tminus= + times sdot + sdot (2417)
T Temperature (degC)
This equation is applied to FEMAXI calculations In the calculation based on analytical
solutions the error in pellet center temperature on the left-hand side of Eq(2410) was set to
01degC Because the thermal conductivity equation is temperature-dependent the conductivity
cannot be determined accurately unless the temperature distribution is determined
Furthermore unless thermal conductivity is determined temperature distribution cannot be
obtained thus iterative calculations are required In iterative calculations a pellet is divided
into 10 ring elements and the pellet surface temperature is adopted as a boundary condition
In determining the temperature distribution for the first iteration the temperature of the inner
side of the pellet ring element is obtained by calculating thermal conductivity using the
temperature of the outer side of each element In the determination of temperature distribution
for the second iteration and thereafter the average temperatures of the inner side and outer
side of each element are calculated using the results of temperature calculations in the
preceding stage Using these results thermal conductivity is obtained and the temperature of
the inner side is determined As a result of these iterations when the differences in
temperature at the center of the pellet reach 01degC or smaller it is assumed that convergence is
JAEA-DataCode 2013-005
- 60 -
attained
According to Table 242 which was obtained from these results it was confirmed that
the difference between the analytical solution and numerical solution was within 02
Table 242 Comparison of analytical solutions with numerical solutions by FEMAXI
Relative radius of
pellet
Analytical solution of
integral equation
Numerical solution
obtained from power profile in the radial
direction
Same as left (number of rings in the
radial direction= 100)
Numerical solution when CSFR=07 and PFAC=20 are
given
Same as left (number of rings
in the radial direction = 100)
00 137512 137794 137535 137686 137507 01 136408 136688 136430 136581 136403 02 133109 133381 133129 133278 133106 03 127661 127919 127677 127823 127658 04 120151 120388 120163 120302 120149 05 110726 110937 110734 110863 110725 06 99607 99784 99611 99725 99606 07 87088 87227 87090 87182 87087 08 73533 73628 73534 73599 73533
09 59357 59405 59357 59391 59357 10 45000 45000 45000 45000 45000
25 Gap Thermal Conductance Model
The gap model can be selected from 7 options In standard cases Eq(251) of the
modified Ross amp Stoute model (IGAPCN=0) is used while in pellet-clad bonding case
IGAPCN=5 or 6 (bonding gap model) is used
251 Modified Ross Stoute model
Gap thermal conductance is expressed by the following equation which is modified from
the Ross and Stoute model(217)
( ) ( ) 1 22 1 2 05
gas m cr
eff
Ph h
R HC R R g g GAP
λ λ sdot= + +sdot sdot+ + + +
(251)
The first term on the right side of the equation is the gas conduction component the
second term is solid contact conduction and the third term radiation conduction
[Selection of option IGAPCN] IGAPCN=0 modified Ross amp Stoute model =1 MATPRO-09 model =2 Ross amp Stoute model =3 Modified Dean model =5bonding model 1 =6 bonding model 2
JAEA-DataCode 2013-005
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Here
C Pc= minus times sdotminus2 77 2 55 10 8
( ) ( )
( ) ( )λ
λ λλ λm
p po c ci
p po c ci
T T
T T=
sdot+
2 (WcmK)
hT T
T Trp c
po ci
po ci
= + minus
sdot sdotminusminus
minus1 1
1
1 4 4
ε εσ
Tpo pellet outer surface temperature (K)
Tci cladding inner surface temperature (K) λ p pellet thermal conductivity (WcmK)
λc cladding thermal conductivity (WcmK)
λ gas thermal conductivity of the mixed gas (WcmK)
Pc pellet-cladding contact pressure (Pa) [described in section 26(4)]
Reff apparent surface roughness of pellet (cm)
R2 cladding surface roughness (cm)
g g1 2+ temperature jump distance between solid phase and liquid phase (cm)
GAP radial gap width (cm)
R R Reff
22
2
2
+
H cladding Mayerrsquos hardness (Pa) ( H Y= 2 8 σ (213))
σ Y yield stress (Pa) (See section 342)
σ Stefan-Boltzmann constant (WcmK4) ( )σ = times minus5 67 10 12
ε p pellet emissivity (minus)(See chapter 4)
ε c cladding black oxide emissivity (minus)(See chapter 4)
Equations for describing λgas and g1 + g2 are as follows
(1) Thermal conductivity of mixed gas This value can be represented as follows using the MATPRO-09 model (213)
1
1 1ji j
ni
gas nji
iji
x
x
λλφ
=ne
=
= minus Σ
(252)
The roughness Reff and R2 can be designated by the input parameters R1 and R2 (μm) respectively The default values are both 10 μm
JAEA-DataCode 2013-005
- 62 -
Here the equation
( )( )
( )φ
λλ
ij
i
j
i
j
i
j
i j i j
i j
M
M
M
M
M M M M
M M=
+
+
+minus minus
+
1
2 1
12 41 0142
1
2
1
4
2
3
2
1
2
2
(253)
holds where xi molar ratio of gas i
λ i thermal conductivity of gas i (WcmK)
M i molar mass of gas i
The gases of this model are Helium Nitrogen Krypton and Xenon Naturally gas
composition depends on fission gas release which will be explained in section 274
Thermal conductivities of gases (213) are shown
3 0668
4 0701
5 0872
5 0923
4 08462
3366 10
3421 10
40288 10 (WmK) (K)
4726 10
2091 10
He
Ar
Xe
Kr
N
k T
k T
k T k T
k T
k T
minus
minus
minus
minus
minus
= times
= times = times = times = times
(254)
Here it is noted that the thermal conductivity of He increases by only a few even if
pressure of He gas increases from 1MPa to 10MPa(218) (219)
(2) Temperature jump distance at solid-gas inetrface The temperature jump distance is given by
( )5
1 2 1 21
10n
iigas
g g g gP=
+ = Σ + sdot (255)
Here
(g1 + g2) jump distance of gas i between solid phase and liquid phase
(cm)
Pgas gas pressure (Pa)
Name-list input Parameter IAR Gas can be specified among He N Kr and Xe
When IAR=rsquoARrsquo is designated the gas can be specified into Argon instead of Nitrogen
JAEA-DataCode 2013-005
- 63 -
According to the Ross amp Stoute model the values of (g1+g2) for Helium Nitrogen
Krypton and Xenon are 10times10-4 5times10-4 1times10-4 and 1times10-4(cm) respectively
Calculation of Pgas is explained later in section 2167 using Eq(2171)
(3) Pellet- Cladding contact pressure To calculate the gap thermal conductance in Eq(251) it is necessary to calculate the
contact pressure cP between the pellet and cladding In FEMAXI-7 the contact pressure
which is calculated by the entire length mechanical analysis is directly used to calculate the
gap thermal conductance
252 Bonding model for gap thermal conductance
(1) Phenomena In LWR fuel rods the gap between pellet and cladding is closed after average burnup
reaches approximately 40 GWdtU and bonding layer is formed(217)as a result of a chemical
reaction between pellet and cladding After the formation of bonding layer the heat
conductance significantly increases from the values predicted by Eq(251) In addition
regardless of the magnitude of the contact pressure biaxial stress-state analysis is required in
which effect of bonding (anchored) gap states on PCMI is taken into consideration
Therefore modeling of bonding is inevitable in the situation where the importance of
PCMI has revived in high-burnup-fuel behavior By assuming that a bonding layer formation
advances as a function of burnup contact pressure and time and that the resulting changes in
the gap thermal conductance occur the following model is considered
(2) Calculation and assumption of the development of bonding The following is considered based on the mechanical analysis of the entire length of fuel
rod When pellet and cladding are in contact in PCMI the bonding reaction begins and
advances During this period the thermal conductance of the gap changes (increases)
depending on the development of bonding layer
Here bonding development (BD) is defined as
start
t
CtBD P dt X= (0 10)BDle le (256)
where Pc contact pressure between pellet and cladding
tstart time at which contact is initiated
JAEA-DataCode 2013-005
- 64 -
X empirically determined parameter (hourMPa)
Equation (256) is integrated for the period of contact in the entire length mechanical
calculation If contact is interrupted during the course of the analysis the integration is
terminated and when the contact redevelops the integration is restarted the total integration
is monitored from the time of initial contact If the value of BD reaches 10 the bonding layer
is considered to be completed and BD is assumed not to increase from this point
Value of X Name-list input parameter BDX=1000000 hr-MPa (Default)
() Note Regardless of the usage of the above bonding model Eq(256) is always calculated
(3) Gap thermal conductance If IGAPCN=5 or 6 is selected ldquogap conductance (GC) rdquo is defined as shown below
A) When bonding does not develop the GC1=Ross amp Stoute model is used as is usual
B) When bonding has already begun and the gaps are determined to be closed assuming
that the GC given by Eq (251) is OpenGC the gap conductance GC2 under such
conditions is given
GC2=(1-BD)OpenGC + BDBondGC (Wcm2K) (257)
BondGC= ( ) (1 ) ( )p p ox cF T Tθ λ θ λsdot + minus sdot (0 1 0 )Fθle le lt (258)
The terms )( pp Tλ and )( cox Tλ denote the thermal conductivities of pellet and oxide film
of cladding respectively at the outer surface of pellet and inner surface of cladding Here the
thermal conductivity of the cladding oxide layer can be selected and adjusted by using
name-list parameters IZOX and CNOX (See section 43)
Furthermore the thermal conductivity of pellet is set to a value corresponding to burnup
at the outer surface of pellet The terms F and θ are input parameters determined empirically
and have the following default values
Name-list input parameter F FBONDG=100 (Default) θ SBONDG=001 (Default)
C) In a segment at which bonding has already begun and when the mechanical calculations
determine that the gap is open due to power decrease or high internal pressure if
IGAPCN=5 the following GC3 is used
GC3=OpenGC(10+αBD) α adjustment parameter (gt0) (259)
The term α is an input parameter which can be determined empirically and has the
JAEA-DataCode 2013-005
- 65 -
following default values
Name-list input parameter α ALBD=07 (default)
D) In a segment in which bonding has already begun when the mechanical calculations
determine that the gap becomes open due to power decrease or high internal pressure and if
IGAPCN=6 BondGC in Eq(258) is used as it is
That is the mechanical model calculates the gap size on the assumption that the pellet
stack is a solid continuum However in an actual rod pellet fragments are likely to remain
bonded to the inner surface of cladding In this situation the ldquonumerically resulted
re-openingrdquo of gap due to the cladding outward creep deformation or pellet shrinkage can
be considered to correspond to the increase of crack space among the pellet fragments In
this case in FEMAXI gap thermal conductance is given the value which is determined in
the bonded gap state This is the physical implication of IGAPCN=6
253 Bonding model for mechanical analysis
In the entire rod length (ERL) mechanical analysis independent from IGAPCN which
specifies the gap thermal conductance model a name-list input parameter IBOND can specify
the bonding state between pellet and cladding as follows (This is explained in chapter 3
again)
IBOND=0 bonding state is not applied (default)
IBONDge1 bonding state is applied
(1) In case of IBOND=1 A) After the bonding reaction begins it is assumed that contact between pellet and cladding is
kept depending on its progress irrespective of the magnitudes of power or internal pressure
Accordingly in a segment in which bonding has occurred (or is progressing) the clogged
(bonded) state is assumed after a certain value of bonding development (above a certain value
of BD) irrespective of the calculated existence of contact or magnitude of contact pressure
A certain value of BD name-list input parameter BDTR=05 (default)
B) In a segment in which clogged gap has occurred due to bonding the mechanical calculation
is performed with the displacements of pellet outer surface and cladding inner surface always
being kept equal in the axial direction even if the gap re-opens
JAEA-DataCode 2013-005
- 66 -
(2) In case of IBOND=2 After the bonding reaction begins it is assumed that contact between pellet and cladding is
kept depending on its progress irrespective of the magnitudes of power or internal pressure
Accordingly in a segment in which bonding has occurred (or is progressing) the clogged
(bonded) state is assumed after a certain value of bonding reaction progress (above a certain
value of BD) while if the contact pressure drops under PCPRES the bonding (clogged) state
is cancelled and the gap recovers to a normal state before the bonding reaction
PCPRESName-list input parameter PCPRES =001(MPa) (default)
(3) In case of IBOND=3 After the bonding reaction begins it is assumed that contact between pellet and cladding is
kept depending on its progress irrespective of the magnitudes of power or internal pressure
Accordingly in a segment in which bonding has occurred (or is progressing) the clogged
(bonded) state is assumed after a certain value of bonding reaction progress(above a certain
value of BD) while if the gap re-opens more than GWD (mm) in the radial direction the
bonding (clogged) state is cancelled and the gap recovers to a normal state before the bonding
reaction
GWDName-list input parameter GWD =01(mm) (default)
254 Pellet relocation model As one of the factors dominating the pellet temperature FEMAXI model of fuel pellet
relocation having an effect on the gap size is explained
(1) Basic concept of relocation model In the start-up period of a fresh fuel temperature of fuel pellet increases with power and
thermal stress is generated to give rise to radial cracks and pellet is broken into some pieces of
fragments These fragments pushes each other due to thermal expansion and they bring
themselves outward to partially fill the gap space This is referred to as relocation This
relocation magnitude cannot be measured in situ It can be estimated only indirectly from the
temperature and linear power of rod
The relocation model of FEMAXI has been designed to input a constant value in both the
radial and axial directions in the beginning of irradiation Name-list parameter FRELOC
(=035 as default) specifies the decrease ratio of initial hot gap size which is determined by
JAEA-DataCode 2013-005
- 67 -
pellet diameter thermal expansions of pellet and cladding and elastic deformation of cladding
immediately after initial start-up
The gap size decreases with extension of burnup while when gap is closed and PCMI
contact pressure is generated pellet fragments are pushed back by cladding and the initial
relocation is decreasing This process is explained in the mechanical analysis part sections
33 and 34
Irradiation in a test reactor such as the Halden reactor a test rod is subjected to frequent
power changes and cycles This induces a thermal cycle in pellet fragments and at every cycle
the relocation amount is likely to change finely To reproduce this situation FEMAXI has
the following controlling function
(2) Function to decouple thermal relocation from mechanical relocation
In this function the mechanical analysis uses FRELOC as per usual while in the thermal
analysis it is possible to add a modification term DELTAR into the radial gap size GAP in
Eq(251) which is resulted from the mechanical analysis
A) Users can specify by input DELTAR ie the pellet thermal relocation modification amount
(μm) of axial segments at any history point Namely suppose that GAPi is the gap size
resulted in the mechanical analysis at i-step the thermal gap size GAP1i is expressed as
GAP1i=GAPi - Σi(DELTAR) (2510)
where Σi(DELTAR) is the accumulated sum of DELTAR until i-step
B) Input parameters are set as DELTAR(IZ IN) (Default DELTAR(IZ IN)=0
(1 le IZ leNAX 1 le IN leNHIST)) DELTAR is the increment for Σi-1(DELTAR) of the
previous time step For the history points with no designations DELTAR=0 is assumed
C) When GAP1i= GAPi - Σi(DELTAR) = 0 GAP1=0 is applied to calculation even if
DELTARgt0 is specified in the following history points
D) At the instant when PCMI occurs in the mechanical analysis Σi(DELTAR) or the
accumulated sum of DELTAR until then is reset to be null If the gap re-opens after this
Σi(DELTAR)=0 is applied to calculation even if DELTARgt0 is specified in the following
history points
JAEA-DataCode 2013-005
- 68 -
E) Value of Σi(DELTAR) is given as a modification term to the radial gap size in the gap
thermal conductance model Namely the first term of Eq(251) ie GAP value is modified
Here it is noted that DELTAR is effective only when no PCMI occurs In this situation the
second term (solid contact term) is 0 and is not affected by DELTAR at all
255 Swelling and densification models With burnup extension fission products accumulate in a fuel pellet and pellet volumetric
increase occurs This is referred to as swelling This swelling consists of solid fission product
swelling which is proportional to burnup and fission gas bubble swelling which is dependent
on both burnup and fuel temperature The swelling model of FEMAXI is broadly classified
into an empirical model and gas bubble swelling model which are described in detail in
section 27 - 213 in connection with fission gas bubble growth model
26 Model of Dry-out in a Test Reactor Capsule
FEMAXI-7 has implemented a function to perform a predictive analysis of dry-out
experiment in JMTR (Japan Materials Testing Reactor) The analytical geometry is shown in
Fig261 A typical set of analytical conditions is as follows
Fuel rod BWR-9x9A type LHR=about 100Wcm
Heated steam phase inlet conditions= 72MPa quality 10 flow velocity parameter
Flow channel tube double-layer thermal insulated tube made of SUS316
Circulating water 72MPa 50oC 001ms returning condensed water from heating steam
Capsule outer mantle material SUS316
Coolant water 04MPa inlet temp40 oC flow velocity 19mh
JAEA-DataCode 2013-005
- 69 -
Fig261 Geometry of cladding temperature regulation system
261 Modelling the dry-out experiment
(1) Basic assumption A) The boundary condition temperature is fixed either at the inner surface of internal tube or
outer surface of the capsule outer tube Basically it is assumed that the outer tune surface
temperature is fixed at 40oC
B) With respect to the heat exchanging flows only the overheated steam and circulating water
are taken into account and the thermal removal by Helium flow in the Helium insulation layer
is neglected
C) Therefore heat flow in the axial direction is taken into account only by the overheated
steam and circulating water
Fuel pellet
96
Cladding
978
113 Internal tube
Outer tube
Capsule outer m
antle
Heated steam
layer
Helium
layer
Circulating w
ater
Coolant w
ater
180
200
210
230
270
320
Diameter (mm)Center line
JAEA-DataCode 2013-005
- 70 -
(2) Basic equations Fig262 shows a cladding temperature controlling system In this figure j denotes one
segment j of the axial segments
jl
Fixed at 40
Fuel pellet
T2j T3j T4j T5j T6j T7j T1j
R1 Heated steam layer
R2 R3 R4 R5 R6 R7
Cladding
Internal tube
He layerCapsule
outer mantle
Circulating w
ater
Outer tube
Fig262 Cladding temperature controlling system
In the heated steam layer shown in Fig262 assuming that the steam enthalpy is H2j-1 at
the boundary of segment j-1and its upper segment j
( ) )(2
112220
22
1222
2222 jjjj
j
jjjj qRqR
RRHH
V
W
t
TC minus
minus=minus+ minuspart
partρ (261)
holds where
ρ density (kgm3)
C specific heat (JkgK)
T temperature (K)
W mass flow rate (kgs)
V volume (m3)
H steam enthalpy (Jkg)
R radius (m)
q heat flux by heat conduction (Wm2)
By transforming Eq(261) neglecting the mass flow rate change in the axial direction
JAEA-DataCode 2013-005
- 71 -
)(2
)(
112221
22
112
122
1111 jCjC
jj
jjj qRqR
RRz
TC
RR
W
t
TC minus
minus=
minus+
partpart
πpartpart
ρ (262)
is obtained
For the internal tube neglecting the heat flow in the axial direction
)(2
223322
23
222 jCjC
jjj qRqR
RRt
TC minus
minus=
partpart
ρ (263)
is obtained Similarly for the helium thermal insulating layer neglecting the heat flow in the
axial direction
)(2
334423
24
333 jCjC
jjj qRqR
RRt
TC minus
minus=
partpart
ρ (264)
is obtained Also similarly for the outer tube neglecting the heat flow in the axial direction
)(
244552
425
444 jCjC
jjj qRqR
RRt
TC minus
minus=
partpart
ρ (265)
is obtained For the circulating water taking into account of the heat flow in the axial
direction
)(2
)(55662
526
552
526
5555 jCjC
jj
jjj qRqR
RRz
TC
RR
W
t
TC minus
minus=
minus+
partpart
πpartpart
ρ (266)
is obtained For the capsule outer mantle neglecting the heat flow in the axial direction
)(2
66
7726
27
666 jCjC
jjj qRqR
RRt
TC minus
minus=
partpart
ρ (267)
is obtained
7
1 jCjC qq in Eqs(262) to (267) are
( )jNjjjjC BTAq 0
1 )( φminus=minussdotminus= (268)
Here Aj and Bj are the coefficients determined by the difference equations of thermal
conduction as follows
jN φ heat flux at the outer surface of cladding (Wm2K)
jT 0 temperature at the outer surface of cladding (K)
)( 1222 jjjC TThq minus= (269)
)( 2333 jjjC TThq minus= (2610)
)( 3444 jjjC TThq minus= (2611)
)( 4555 jjjC TThq minus= (2612)
JAEA-DataCode 2013-005
- 72 -
)( 5666 jjjC TThq minus= (2613)
)( 677
7 jjjC TThq minus= (2614)
where h thermal conductance (Wm2K)
The thermal conductance h in Eqs(268) to (2614) are given respectively as follows
2
32
2
2
1
2
2ln
11
R
RRR
h
h+
+prime
=
λ
(2615)
1hprime forced convection thermal conductance including radiation in
the superheated steam layer (Wm2K)
2λ thermal conductivity of the inner tube (WmK)
3
43
3
3
32
3
2
33
2ln
2ln
1
R
RRR
RR
RRh
++
+
=
λλ
(2616)
3λ equivalent thermal conductivity of the Helium thermal insulating
layer (WmK)
4
54
4
4
43
4
3
44
2ln
2ln
1
R
RRR
RR
RRh
++
+
=
λλ
(2617)
4λ thermal conductivity of the outer tube (WmK)
554
5
4
55 12
ln
1
hRR
RRh
prime+
+
=
λ
(2618)
5hprime forced convection thermal conductance including radiation in
the circulating water superheated steam layer (Wm2K)
6
76
6
6
5
6
2ln
11
R
RRR
h
h+
+prime
=
λ
(2619)
λ6 thermal conductivity of the capsule outer mantle (WmK)
76
7
6
77 2
ln
1
RR
RRh
+
=
λ
(2620)
JAEA-DataCode 2013-005
- 73 -
262 Thermal and materials properties
(1) Materials properties
The necessary properties in the present model are listed below
A) SUS316
thermal conductivity (non-disclosure data) density ρ=7830 (kgm3)
Isobaric specific heat (non-disclosure data)
B) Helium
thermal conductivity(211) 3 069293 10 (WmK) (K)T Tλ ηminus= times sdot
where η adjustment factor
Density(221) 2 30 (kgm )a b cρ ρ+ + =
( ) ( )( )
4
3
54245 10 1
1890
(K) bar 00207723 bar kg K
Pa b c
T RT
T P R
minus= times + = = minus+
= sdot
Isobaric specific heat (222) ( )( )3519 10 at 773K J kg KPC = times sdot
C) Heated steam and circulating water
Calculated by using the steam table at 50oC for the circulating water
Thermal conductivity 0653 (WmK)λ = Densityρ=73645 (kgm3)
Isobaric specific heat for heated steam ( )( )5456 J kgtimesKPC =
Thermal conductivity
(290 C) (400 C) (600 C) (800 C)0064 0065 0088 0100 (WmK)λ deg deg deg deg=
Density ρ=3766 (kgm3)
Isobaric specific heat for heated steam ( )( )5153 J kgtimesKPC =
(2) Thermal conductance
Thermal conductance hw is given by the sum of convection term and radiation term For
the convection term the heated steam layer is a single phase steam region (quality=1 [mass
flow rate of gas ][total mass flow rate]) and the circulating water is a single phase liquid
region so that the Dittus-Boelter equation(23) is applied to both the terms
JAEA-DataCode 2013-005
- 74 -
Dittus-Boelter equation 8040
e
RePr0230D
KhW = (2621)
where μeRe DG=
K thermal conductivity (WmK)
De equivalent diameter (m)
Pr Prandtle number (-)
Re Reynoldrsquos number (-)
G Mass flow flux (kgm2s)
μ dynamic viscosity coefficient (kgms)
Also the radiative thermal conductance is
21
42
41
1
21
111
TT
TThr minus
minussdotsdot
minus+=
minus
σεε
(2622)
where σ Stephan-Boltzmann constant(Wm2K4) ( )810675 minustimes=σ
1ε emissivity of cladding outer surface or inner tube outer surface (-)
2ε emissivity of cladding inner surface or outer tube inner surface
263 Numerical solution method Based on the modeling method described in section 22 equations are converted into
difference formulae as follows though in this conversion an outlet-representation format is
adopted This is to make a difference in equations for the heat flow in the axial direction from
upper stream with respect to the flow direction and for the heat flow in the radial direction
with respect to the outlet temperature at the corresponding axial segment
(1) Converting the constitutive equation to difference equation
For heated steam layer
( )
))()((2
)(
011
11
22221
22
111
111112
122
111
111
jn
jjnj
nj
njj
njj
j
nj
nj
jj
BTARTThRRR
TCTClRR
W
t
TTC
minussdot+minusminus
=
sdotminussdotminus
+Δminus
++
++minusminus
+
πρ
(2623)
JAEA-DataCode 2013-005
- 75 -
))(2
2
)()(
011111
221
22
22
112
122
222
122
11111112
122
111
jn
jjnj
jjnj
nj
j
jjjnj
j
j
BTARTt
CT
RR
hR
TRR
hR
lRR
CW
t
CT
lRR
CW
minussdot+Δ
=minus
minus
minus+
minusminus
Δ+
minus
+
++minus
minus
ρ
πρ
π (2624)
For inner tube
)()(2 1
11
2221
21
33322
23
21
222
+++++
minusminusminusminus
=Δminus
nj
nj
nj
nj
nj
nj
jj TThRTThRRRt
TTCρ (2625)
( ) nj
jjnj
nj
jjnj T
t
CT
RR
hRT
RR
hRhR
t
CT
RR
hR2
22132
223
33122
223
223322112
223
22 222
Δ=
minusminus
minus+
+Δ
+minus
minus +++ ρρ(2626)
For Heium gas layer
)()(2 1
21
3331
31
44423
24
31
333
+++++
minusminusminusminus
=Δminus
nj
nj
nj
nj
nj
nj
jj TThRTThRRRt
TTCρ (2627)
( ) nj
jjnj
nj
jjnj T
t
CT
RR
hRT
RR
hRhR
t
CT
RR
hR3
33142
324
44132
324
334433122
324
33 222
Δ=
minusminus
minus+
+Δ
+minus
minus +++ ρρ (2628)
For outer tube
)()(2 1
31
4441
41
55524
25
41
444
+++++
minusminusminusminus
=Δminus
nj
nj
nj
nj
nj
nj
jj TThRTThRRRt
TTCρ (2629)
( ) nj
jjnj
nj
jjnj T
t
CT
RR
hRT
RR
hRhR
t
CT
RR
hR4
44152
425
55142
425
445544132
425
44 222
Δ=
minusminus
minus+
+Δ
+minus
minus +++ ρρ(2630)
For circulating water
( )
))()((2
)(
14
1555
15
16662
526
155
115152
526
551
555
++++
++++
+
minusminusminusminus
=
sdotminussdotminus
+Δminus
nj
nj
nj
nj
njj
njj
j
nj
nj
jj
TThRTThRRR
TCTClRR
W
t
TTC
πρ
(2631)
( )
nj
jjnj
nj
j
j
nj
j
jjjnj
Tt
CT
RR
hRT
lRR
CW
TRR
hRhR
lRR
CW
t
CT
RR
hR
5551
625
26
661152
526
155
152
526
556625
26
5555142
526
55
2
)(
2
)(
2
Δ=
minusminus
minus+
minus+
+minus
minusΔ
+minus
minus
+++
+
++
ρπ
πρ
(2632)
JAEA-DataCode 2013-005
- 76 -
For capsule outer mantle
)()(2 1
51
6661
67772
627
61
666
++++
minusminusminusminus
=Δminus
nj
nj
njj
nj
nj
jj TThRTThRRRt
TTCρ (2633)
( ) 72
627
776
66162
627
667766152
627
66 222j
nj
jjnj
jjnj T
RR
hRT
t
CT
RR
hRhR
t
CT
RR
hR
minus+
Δ=
minus+
+Δ
+minus
minus ++ ρρ (2634)
where 7 jT =40 oC is the temperature at the outer boundary of capsule outer mantle
At the inlet boundary of heated steam layer
( )
))()((2
)(
110111
111
122221
22
11111
111
12
122
1111
111111
BTARTThRRR
TCTClRR
W
t
TTC
nnn
nninin
nn
minussdot+minusminus
=
sdotminussdotminus
+Δminus
++
+++
πρ
(2635)
where 11
+ninT is the heated steam temperature at inlet
At the inlet boundary of circulating water
( )
))()((2
)(
14
1555
15
16662
526
155
1552
526
551
555
++++
+++
minusminusminusminus
=
sdotminussdotminus
+Δminus
nN
nN
nN
nN
nNN
ninin
N
nN
nN
NN
TThRTThRRR
TCTClRR
W
t
TTC
πρ
(2636)
where 15+ninT is the water temperature at inlet and N is the number of segments
in the axial direction
(2) Method of solution
The algebraic equations ordered into n+1 steps of temperatures are solved
The unknown quantities are 6timesN ie 1 111 61 n nT T+ + 1
1nNT + and 1 1
1 6 n nN NT T+ +
By set out these unknowns from Eqs(2623) (2626) (2628) (2630) (2632) (2634)
and (2636) the coefficient matrix is obtained as Eq(2637) where Eq(2637) is a case with
four axial segments
JAEA-DataCode 2013-005
- 77 -
(2637)
By solving the above matrix by the Gaussian elimination method n+1 steps temperatures
are obtained By this the axial distribution of heated steam and cladding surface heat transfer
coefficients are determined Thus temperatures of cladding outer surface and fuel can be
obtained
JAEA-DataCode 2013-005
- 78 -
27 Generation and Release of Fission Gas FEMAXI calculates the diffusion of fission gas atoms bubble formation and growth
and release of fission gas on the basis of the temperature obtained in the one-dimensional
thermal analysis at each ring element of axial segment Fig271 shows the flow of analysis
Time Step Start
Calc of grain growth
Updating element nodal coordinate with grain growth
Calc of amount of grain-growth-induced migration and diffusion of fission gas atoms to grain boundary grain boundary gas bubble growth and grain boundary gas density
Solution of the total simultaneous equations to determine the distribution of in-grain fission gas atoms
Re-distribution of density of intra-grain fission gas atoms at updated grain size
Calc of gas release threshold (grain boundary bubble radius grain boundary saturation density of gas etc)
Intra-grain gas atom density calculation converges
Calc of the total amount of released gas by the current time step
Calc for all the ring elements completed Calc for all the axial segments completed
Calc of in-grain gas bubble (size and gas amount) and bubble density
Calc of amount of fission gas release during time step
Calc of element matrix and element vector Formation of entire matrix and entire vector (Eq277-22)
Fig271 Process of fission gas generation and release in the equilibrium and rate-law
models in FEMAXI
JAEA-DataCode 2013-005
- 79 -
271 Fission gas atoms generation rate
Composition of fission gas consists of Xe and Kr while generation of He can be
designated by input The fission gas generation rate in the axial segment j and radial element i
is represented by
PY f q
E Nij
ij j
f A
= sdot sdotsdot
(271)
Here
Pij fission gas generation rate per unit length in element ij (molcm3s)
( )+
minus=
50
502
i
i
ij rdrrf φπ
φ(r) heat generation density profile function in the radial direction Q(r) in Eq (239)
q j average heat generation density of axial segment j (Wcm3) = ( )2 21
jLHR
pout
P
r hπ minus
PLHRj linear heat rate of axial segment j (Wcm)
h normalized radius of the center hole of pellet (minus)
Ef energy generated per one fission 3204times10minus11 J (=200 MeV)
Y fission yield of fission gas (Xe+Kr) = 03
NA Avogadros number 602times1023
【Note】 Iodine release calculation in FEMAXI To evaluate the Iodine-SCC of cladding under PCMI condition FEMAXI has a function
to estimate the release of Iodine In this function Iodine yield ratio is assumed as 005 (0025
for I2) with respect to the yield of Xe and using the atomic weight of Iodine = 1269045gmol
released amount (g) of Iodine is calculated as per the unit axial length (cm) of rod It is
considered that inside a fuel pellet Iodine is not present in the form of single element but
substantially in the form of CsI However as a reference value Iodine concentration (gcm2)
on the inner surface of cladding is determined by dividing the single element Iodine quantity
by the inner surface area (cm2) of unit axial length of cladding
272 Concept of thermally activated fission gas release in FEMAXI 272-1 Equilibrium model and Rate-law model With respect to the calculation method of grain boundary gas bubble growth in the fission
gas diffusion and release model one of the following two types are selected an equilibrium
JAEA-DataCode 2013-005
- 80 -
model assuming that the grain boundary bubble is growing with the gas pressure being kept in
equilibrium with the external compressive pressure of surrounding solid matrix and a
rate-law model assuming that the grain boundary bubble radius is changing in accordance with
the differential pressure across the bubble boundary and temperature
272-2 Assumptions of fission gas release mechanism Assumption of FEMAXI model for fission gas release is as follows
A) The fission gas atoms (Xe and Kr) produced in fuel (UO2 or MOX) grains transfer to
grain boundaries through the two mechanisms described below and that there it
accumulates and forms grain boundary gas bubbles
i) Gas atom diffusion within grains
ii) Sweeping of gas atoms to grain boundaries by grain growth
B) When the amount of fission gas atoms in the gas bubbles reaches a certain saturation
value a number of bubbles connect at grain boundary and a tunnel which leads to a free
surface of pellet is formed
C) After the tunnel is formed additional gas atoms which have diffused from inside of
grains to the grain boundaries enter the tunnel and are released to the free surface The
gas accumulated at the grain boundaries also re-dissolves into the grains at a certain rate
D) Evaluation of formation of the intra-granular gas bubbles includes trapping of gas atoms
by the intra-granular bubbles and re-dissolution of gas from the bubbles into grain
matrix
(1) Gas release in the equilibrium model The equilibrium model in FEMAXI assumes that the fission gas atoms continuously
accumulate in the grain boundary bubbles while the pressure of gas in the bubble is always in
equilibrium with external compressive pressure + surface tension and that when the density
of gas atoms per unit grain boundary area exceeds a threshold value
N fmax
which is
determined by the combination of temperature bubble radius and bubble coverage fraction on
grain boundary etc the excess fission gas over the threshold value is released to free space
outside the fuel pellet That is in a situation in which tunneling has occurred ie grain
boundary accumulation of fission gas atoms has exceeded the value N fmax
the fission gas
atoms enter the tunnel and are immediately released However if the accumulated amount
of gas atoms becomes lower than the value N fmax
due to temperature decrease etc the release
is assumed to cease
JAEA-DataCode 2013-005
- 81 -
(2) Rate-law model of FGR FEMAXI-7 has introduced a new model of FGR rate-law model This model is based
on the same understanding of that of the equilibrium model for fission gas atoms diffusion
accumulation in grain boundary and release However this model calculates the growth or
shrinkage rate of radius of the grain boundary bubble and assumes that if either the bubble
size or grain boundary coverage reaches a certain value gas release begins
(3) Contents shared by the two FGR models A) Calculation of fission gas atoms diffusion from inside of grain to grain boundary is
common in both the models The diffusion calculation addresses the gas atoms trapping by the
intra-grain bubbles and re-dissolution to solid matrix Also the method to calculate the
intra-grain bubble size is different between the two models
B) Direct release of gas by recoil and knock-out from pellet surface assumes 05 of the total
fission gas atoms generation That is even if no FGR occurs from fission gas bubbles 05
release is assumed to take place However this 05 is a default value This can be specified
by name-list parameter FRMIN in input file For example when FRMIN=0 FGR is 0 if
no release occurs from the bubbles
C) FGR from fission gas pores in the high burnup structure (HBS) ie rim structure and a
direct FGR from the HBS are evaluated by an empirical model as an additional amount of gas
release apart from the above two models
273 Thermal diffusion accompanied by trapping (1) Diffusion accompanied by trapping Figure 272 shows states of intra-granular gas bubbles and grain boundary gas bubbles in
a pellet and Fig 273 shows an ideal model of crystal grains of pellet
JAEA-DataCode 2013-005
- 82 -
INTRAGRANULAR
GAS BUBBLE INTERGRANULAR
GAS BUBBLE
Fig 272 Schematics of intra-granular and grain boundary gas bubbles in pellet
Spherical shell containing half the amount of FP gas retained on a grain boundary
Fig273 Model of ideal spherical grains of pellet
The grain shown in Fig273 is illustrated as if it were covered with a spherical blanket
An actual grain boundary is a thin amorphous layer separating adjacent crystal grains All
the descriptions hereafter address one single crystal grain and the parameters accompanying
titles are name-list parameters and its values to control models
For example ldquo (IGASP=0 2)rdquo indicates that calculation is performed in a mode which is
in accordance with either IGASP=0 or 2 A specific explanation of parameters is given in
section 214 and chapter 4
273-1 Partition and diffusion equation of fission gas atoms in intra-grain bubbles (common in IGASP=0 and 2)
It is assumed that the relative amounts of gas atoms dissolved in the matrix and present in
intra-granular bubbles depend on the equilibrium between trapping and re-dissolution The
JAEA-DataCode 2013-005
- 83 -
gas atoms dissolved in the matrix moves toward the grain boundary by concentration-gradient
driven diffusion The gas atoms diffusion rate in a spherical solid matrix with a radius of a is
given by Speightrsquos equation in which trapping and re-dissolution are taken into
consideration(223)
β+prime+minus
partpart+
partpart=
partpart
mbgcr
c
rr
cD
t
c 22
2
(273-1)
Here c number of gas atoms (Xe and Kr) dissolved per unit volume of solid matrix
(atomscm3) D diffusion coefficient of gas atoms (cm2s)
g rate of trapping of gas atoms by intra-granular bubbles (sminus1)
bprime rate of re-dissolution of gas atoms into solid phase (sminus1)
m number of gas atoms per unit bubble volume (atomscm3)
β generation rate of the gas atoms per unit volume of pellet (atomscm3s)
The gas atoms generation rate ijβ (atomscm3s) in a unit volume of one pellet region i j
which is shown in Eq(273-1) is using ijP in Eq(271)
1 12 2
2 2 ( )ij ij A i j i jP N r rβ π + minus= minus (273-2)
where AN the Avogadro constant
12 i jr+ outer radius of pellet region i j(cm)
12 i jrminus inner radius of region i j(cm)
The difference between trapping and re-dissolution is
0m
mt gc b mt
part primeΔ = = minus nepart
(273-3)
It is assumed that calculation at the end of every time step gives the following equilibrium
relation
0gc b mprimeminus = (273-4)
Thus addition of Eq(273-1) to (273-4) gives
( ) 2
2
2c m c cD
t r r r
part part part βpart part part+
= + +
(273-5)
JAEA-DataCode 2013-005
- 84 -
Transforming Eq(273-3) gives
1 0mt
gc b mmb
Δ primeminus minus = prime (273-6)
Here 1
1
1 ii
i
mtb
m bα minus
minus
Δprime= minus prime (273-7)
is defined to determine the initial value of iα at current time step by using the value at the
previous time step i-1 Then
( )i
i
c c mg
αα
= ++
(273-8)
holds
Substituting the above equation into Eq(273-5)
( ) ( ) ( )2
2
2i
i
c m c m c mD
t g r r r
part part partα βpart α part part
+ + += + + +
(273-9)
is obtained Setting the sum of the number of gas atoms in the solid matrix and the number of
those in the intra-granular bubbles as
Ψ = +c m (273-10)
the diffusion equation in the intra-granular spherical coordinate system including trapping is
2
2
2D
t r r r
part part part βpart part part
Ψ Ψ Ψprime= + +
(273-11)
Here
i
i
DD
g
αα
prime =+
(273-12)
and Ψ represents the apparent average number of gas atoms per unit intra-granular volume
(atomscm3) regardless of the location of gas atoms (ie whether it is present in the matrix or
in the intra-granular bubbles) Dprime is the apparent diffusion coefficient which includes the
effects of trapping re-solution and partial destruction of bubbles by fission fragments
Thus the FEM numerical solution of Eq(273-11) and intra-granular bubble radius R
and its number density N are interlinked
Next to solve Eq(273-11) g brsquo R and D are determined assuming that there are N
intra-granular bubbles with R radius in an unit volume of pellet It is likely that the actual
radius R has a size distribution though its weighted average value (representative value) is
JAEA-DataCode 2013-005
- 85 -
considered This R and N are determined by one of the methods described in sections 274 -
276 below
274 White+Tucker model of intra-grain gas bubble radius and its
number density (GBFIS=0)
Assuming White amp Tuckerrsquos model that N intra-granular gas bubbles with an average
radius of R are present per unit volume R and N are related as follows in an equilibrium
state during irradiation(224)
( )2152 oN R Zα πλ= + (274-1)
Here
N intra-granular bubble density (bubblescm3)
R average radius of intra-granular bubbles (cm)
Z0 affected zone (=10-7cm)
α bubble generation rate (= 24 bubblesfission fragment)
λ range (6 times 10minus4 cm)
The initial value of R is determined by a name-list parameter APORE (default=1 nm)
The rate of trapping by intra-granular bubbles g is given by (225)
4g DRNπ= (274-2)
The number density of bubbles N in Eq(274-2) is obtained by Eq(274-1)
The rate of re-dissolution of the gas into the matrix is given by (224)
( )2
0303b F R Zπλprime = + (274-3)
Here
Ffission rate (fissionscm3s)
λ fission distance(=6times10-4cm)
Zoaffected range (cm)
In addition the lowest limit of 1013
(fissionscm3s) is set for fission rate F in Eq(2714) The value is set to suppress the
occurrence of unnatural results when b prime changes suddenly due to a decrease in power
Namelist input BFCT
Re-solution ratio to solid phase can be adjusted by
BFCT prime = prime timesb b BFCT (Default=10)
The rate of re-dissolution brsquo is controlled using
BFCT bprime = bprime times BFCT (Default value 10)
Fitting parameter BFCT
JAEA-DataCode 2013-005
- 86 -
Diffusion coefficient D of fission gas atoms inside a pellet grain is described in section
2715 In the meantime if average radius R of the intra-grain bubbles is given by Eqs(274-1)
(274-2) and (274-3) g brsquo and D are determined and D is derived from Eq(273-12)
Therefore by solving Eq(273-11) Ψ can be obtained Solution method of the partial
differential equation Eq(273-11) is presented in section 277
Here the method used to obtain R in the current time step is explained R is obtained
using van der Waalsrsquo state equation which is
( )PR
V m B m kT0
2+
minus =γ (274-4)
( )
gasP V m B m kTsdot minus =
(274-5) where
R average intra-granular bubble radius (cm)
P0 external force applied on intra-granular bubbles (dyncm2)
γ surface tension (626 ergcm2)
V intra-granular bubble volume ( = 43
3πR cm3)
B van der Waals constant ( = times minus8 5 10 23 cm3atom)
m number of gas atoms per unit bubble volume (atomsbubble)
k Boltzmanns constant ( = times minus1 38 10 16 ergK)
T temperature (K)
As the model assumes that the gas pressure inside the grain boundary bubble is in
balance with the sum of compressive static pressure extP exerted by surrounding solid and
surface tension
2
gas extP PR
γ= + (274-6)
holds Here Pext usually takes a negative value in the mechanical analysis because Pext is
usually a compressive stress Nevertheless in the calculation of Eq(274-6) and related
equations absolute value of Pext is taken
By solving Eq (274-4) with respect to m using 343V Rπ=
( )
3 24
3 2ext
ext
P RRm
B P R kTR
γπγ
+= sdot+ +
(274-7)
JAEA-DataCode 2013-005
- 87 -
is obtained
On the other hand by Eq(273-3) ie the equilibrium relationship between trapping of
gas atoms by intra-gain bubbles and re-dissolution from the bubbles
0gc b m mtprimeminus minus Δ = (274-8)
holds Here assuming 0mtΔ = gives
gc b m= prime (274-9)
By transforming this equation into an equation which appears m by using m=mN and
c m= minusΨ
( )prime + =b g m N g Ψ (274-10)
is obtained
Substituting Eqs(274-2) and (274-3) into Eq(274-10) gives
( )( )2 0303 4 4F R Z DRN m DRπλ π π+ + = Ψ (274-11)
By substituting Eq(274-7) into the above equation
( )( ) ( )3
2
0
( 2 )303 4 3
2ext
ext
R ZR P R
F DRN DB P R kTR
γπλ πγ
+++ sdot = Ψ
+ + (274-12)
is obtained Further by substituting Eq(274-1) to the above equation and arranging with
respect to R
( )( ) ( )
32
0 2
608 ( 2 )303 3
2ext
exto
DR R P RF R Z D
B P R kTRR Z
α γπλγλ
+ + + sdot = Ψ + ++ (274-13)
is obtained Solution of this equation determines the bubble radius R
Eq(274-13) indicates that R is dependent on Ψ The number density N is determined
by Eq(274-1) once R is determined Trapping rate and re-dissolution rate are determined
by substituting R and N into Eqs(274-2) and (274-3)
In actual numerical calculations by designating the m obtained from
Eq(274-7) as 1m and by designating the m obtained from Eq(274-9) as
2m
R which satisfies 2
1 mm = is obtained The range of R is assumed to be 10-15 (cm)
to 10-5 (cm) R is calculated by applying the dichotomy to R10log
JAEA-DataCode 2013-005
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275 Radiation re-dissolution and number density model of Intra-granular gas bubbles (GBFIS=1)
Similarly to the method in the previous section 274 it is assumed that per unit volume
of fuel pellet there are N intra-granular bubbles with average radius of R The following
assumption is set to obtain the changing rate of the number density N (bubblescm3) of
intra-granular bubbles
1) α bubble nucleus is generated from one fission
2) Fission fragment generates the bubble nucleus in the solid volume except the bubble
volume in the solid matrix That is it generates bubble nucleus in the solid volume which
has a fraction of 341
3N Rπ minus sdot
of the total solid volume
3) The generated bubble nucleus is immediately destroyed by punching of other fragments
This destruction rate is assumed as
22 ( ) ( )F R Z f R Nπ λsdot + (275-1)
where ( )f R implies the effect that the bubble is not destroyed by fission fragments when
the bubble has grown to reach a certain size That is if the bubble radius R is under R0
destruction rate is 10 if over R0 it is a partial destruction rate which depends on R as
follows
0
00
10 ( )
( )( )
k
R R
f R RR R
R
le= ge
(275-2)
Namelist Parameters k= KFIS (default=20) R0= RFIS (default= 20 nm)
Based on the above assumption changing rate of the number density N(bubblescm3)
during irradiation is obtained by incorporating ( )f R into the White and Tuckerrsquos equation as
3 20
42 1 2 ( ) ( )
3
dNF N R F R Z f R N
dtα π π λ = minus sdot minus sdot +
(275-3)
where F fission rate (fissionscm3s)
α bubble nucleus generation rate(=24 bubblesfission fragment)
R average radius of intra-granular bubble(cm)
λ fission track distance(= 46 10 cmminustimes )
0Z affected range of fission (=10-7cm)
Then
JAEA-DataCode 2013-005
- 89 -
3 20
3 20
42 2 ( ) ( )
3
2 2
4( ) ( )
3
dNF F R R Z f R N
dt
F FKN
K R R Z f R
α πα πλ
α
πα πλ
= minus + + sdot
= minus
= + +
(275-4)
holds Thus the numerical solution of Eq(275-4) is
1( 2 )exp( 2 ) 2i iN N FK F t FKαminus= minus minus sdotΔ + (275-5)
Assuming that the total number of gas atoms contained in an intra-granular bubble is m
by the number of gas atoms m (atomscm3) in bubbles in unit volume and by the number
density of bubbles N (bubblescm3)
m m N= (275-6)
is obtained Changes of m and m are determined by the difference between the number of
gas atoms entering bubbles by trapping and the number of gas atoms sputtered out by fission
fragment (re-dissolution)
The trapping rate g (s-1) of fission gas atoms by the intra-granular bubbles is given by
4g DRNπ= (275-7)
where D diffusion coefficient of fission gas atoms (cm2s)
Further setting that c is an average number of fission gas per unit volume of fuel pellet
containing the gas bubbles irrespective of whether the number of gas atoms Ψ (atomscm3)
which are dissolved in unit volume of matrix is in a solid phase of the matrix or in
intra-granular bubbles
Ψ = +c m (275-8)
holds Assuming that the re-dissolution rate to solid phase is
20303 ( )b F R Zπλprime = + (275-9)
2
04 303 ( )
( )
dmDRNc F R Z m
dtgc b m g g b m
π πλ= minus + sdot
prime prime= minus = Ψ minus + (275-10)
is obtained Accordingly the numerical solution of Eq(275-10) is
1( )exp( ( ) )i im m g g b t gminus prime= minus Ψ minus + Δ + Ψ (275-11)
At the same time assuming that the state equation can hold for the bubble gas
32 4
3
RP m B m kT
R
γ π + sdot minus sdot =
(275-12)
JAEA-DataCode 2013-005
- 90 -
is obtained However in the present model the intra-granular bubble gas pressure is in balance
with the static pressure extP from external matrix and surface tension then
2gas extP P
R
γ= + (275-13)
holds
Here to determine the values of andm m R N at the i-step a numerical solution is
obtained which simultaneously fulfills Eqs(275-5) (275-6) (275-11) and (275-12)
At the same time since the bubbles increase their volume by influx of vacancies setting
that the vacancy concentration is nv and vacancy diffusion coefficient is Dv
4 2v v P
H gas
dn D RP
dt kT R
π γσ = + minus
(275-14)
gas
v v
m kTP
n=
Ω (275-15)
3
4
3
v vV n m B
Rπ
= Ω +
= (275-16)
are obtained(226 ) where
vΩ vacancy volume ( 29 3409 10 mminussdot )
Pγ UO2 pore surface energy (Jm2)
4 2
041
085 140 10 (Jm )
P SV SV
SV T
γ γ γγ minus
= sdot =
= minus times
GSV (0ltTlt2850C)
Default GSV=041 (227)
Also Turnbull(228) presents
2v V vD k S j C= sdot (275-17)
where S atom jump distance=1
3Ω =34454E-10 m
Ωatomic volume = ( )29 3409 10 mminustimes
( )13 410 exp 552 10 j RTν = minus times
Rgas constant =1987 calmoleK
Ttemperature (K)
Vk adjusting factor (KV=10 default)
C lattice defects concentration to which vacancy jumps
is approximated as 10(228)
JAEA-DataCode 2013-005
- 91 -
------------------------------- 【Explanation of numerical solution】 -----------------------------------
To determine the values of andm m R N at the i-time step a numerical solution is
obtained which simultaneously fulfills Eqs(275-5) (275-6) (275-11) (275-12)
(275-13) and (275-14) For this purpose to determine a new value of ivn in Eq(275-12)
at the time step -i- the followings are set
1i i vv v
dnn n t
dtminus= + sdot Δ (275-18)
1
11
4 2i ii iv vH gasi i
dn D RP
dt kT R
π γσminus
minusminus
= + minus
(275-19)
The determined value of ivn by using these formulations is put into Eq(275-14) and
by using the value of m a new value of iR is determined
Here it is noted that ivn cannot exceed the value of vn which is obtained by 0vdn
dt=
because vn is a value at an infinite time elapse When calculating ivn by using Eqs(275-18)
and (275-19) it is not guaranteed that ivn will not exceed vn Therefore when calculating
ivn by using Eqs(275-18) and (275-19) it is necessary to set a sufficiently small time step
increment tΔ though setting of tΔ size is much difficult because the size depends on state
variables
Accordingly after determining the limiting value of vn paying a special attention not to
exceed this limiting value of vn Eq(275-19) is transformed by the Crank-Nicholson method
That is Eq(275-20) is the transformed equation of Eq(275-19) by the Crank-Nicholson
method
1
1
1
422
22
i ii
i iv
gas gasi ivi iH gasi
R RD P Pdn
PR Rdt kT
π γσ
minus
+
minus
+ +
= + minus +
(275-20)
Naturally when using Eq(275-20) iteration is required to obtain convergence
First calculation of vn derived from 0vdn
dt= is explained
Setting that R obtained from vn is R and gasP is gasP
4 20
iiv vH gasi
dn D RP
dt kT R
π γσ = + minus =
(275-21)
JAEA-DataCode 2013-005
- 92 -
is obtained From this
( ) 2 0iH gasP Rσ γ+ minus = (275-22)
holds Also gasP and R in Eqs(275-21) and (275-22) are defined as follows
gasv
AP
n= (275-23)
3
vR nα β= + (275-24)
To substitute Eqs(275-20) and (275-21) into Eq(275-19) both sides of Eq(275-22)
are cubed
33 3( ) 8H gasP Rσ γ+ = (275-25)
By substituting Eqs(275-23) and (275-24) into Eq(275-25) and rearranging
( ) ( ) ( )3 4 2 3 3 3 2 2 2 3 2 33 8 3 3 3 0H v H H v H H v H vn A n A A n A A n Aασ ασ βσ γ ασ βσ α βσ β+ + minus + + + + + =
(275-26)
is obtained By using Eq(275-23) the solution in 0vn gt is obtained by the N-R method
-------------------------------------------------------------------------------------------------
276 Pekka Loumlsoumlnen model for intra-granular gas bubbles (GBFIS=2)
This model is an option in the rate-law model In this model the intra-granular bubble
radius R and its number density ( )oN N= are predicted by using the Loumlsoumlnenrsquos empirical
correlation(229) by assuming the relationship expressed by Eqs(274-2) and (274-3)
(1) Initial bubble radius is assumed as 0 001R = nm
Then the number density is determined
Number density 0 0
00762 Bu 104( ) ( )
98N Bu T N T
minus sdot += (cm-3) (276-1)
170 ( ) (157 000578 ) 10N T T= minus sdot times (cm-3) (276-2)
Bu burn up (MWdkgU) T Kelvin
However values of Eqs(276-1) and (276-2) are fixed at those at T=2173K above 2173K
(Cut-off)
(2) Average radius of intra-granular bubbles Next an average radius of growing bubbles is determined
JAEA-DataCode 2013-005
- 93 -
1 3
0
3
4
m MR
Nπρ prime
=
(276-3)
where M is molecular weight of gas atoms and gas density 34 (gcm )ρ = is assumed
GROU=40 (default)
Next ldquogas concentration mrsquo (molecm3)rdquo is calculated which is defined as the total gas
(mole) in all the bubbles in one fuel grain divided by the grain volume (cm3)
04 ( ) ( ) ( )g g g
dmRDc r t N r t Fbm r t
dtπprime prime= minus (276-4)
173 10b = times =BBC (cm3) BBC= 173 10minustimes (default)
3 fission rate (1cm s)F sdot
However the initial gas concentration 0mprime is calculated by 0 001R nm= and Eq(276-3)
(3) Coalescence and growth of intra-granular bubbles Coalescence of bubbles is predicted by the following empirical equation when the
temperature exceeds the coalescence threshold mgT
8000
273159
ln0005
mgTBu
= +minus
(276-5)
( )4 4( ) 1 exp( 83 10 ) ( )exp( 83 10 )g F ini gR r t R t R r t tminus minus= minus minus times + minus times (276-6)
100 nmFR = coarsened radius RADMG=100nm (default)
iniR radius before coarsening
Bu burn up (MWdkgU) T Kelvin
t elapsed time from the start of bubble coalescence (s)
In Eq(276-5) if 91Bu le Bu is fixed at 91 By Eq(276-6) the coalescence
terminates after about 15 hour and the bubbles become a single bubble with radius of
RAMDG (nm) Then in the process of this change the number density 0N and bubble gas
concentration mrsquo change though as an approximation mrsquo is assumed to hold the
pre-coalescence value as a kind of equilibrium state That is after coalescence Eq(276-4)
is no longer used and gas atoms influx to and re-dissolution from the bubbles are assumed to
maintain the equilibrium
Here the number density of bubbles at the start and end of coalescence and after that is
JAEA-DataCode 2013-005
- 94 -
expressed as
0 3
3 ( )( )
4g
g
m r t MN r t
Rπ ρprime
= (276-7)
1) R Rρlt 0ρ ρ= Rρ RROU (Default=10 nm)
2) R Rρge 0
N
iniR
R
ρ
ρ ρ =
N ρ NROU (default=10)
When the temperature once exceeds mgT and then falls below mgT the bubble radius is
assumed to remain at 100 nm Shrinkage of the intra-granular bubbles is associated with so
many factors and there are only scarce observation data Consideration of the bubble
shrinkage remains a future task
277 Galerkinrsquos solution method for partial differential diffusion equation (common in IGASP=0 and 2)
Solving the partial differential diffusion equation (273-11) for the diffusion of fission
gas atoms uses the FEM in spherical geometry assuming that the grain is spherical and inside
the elements formulation is made by using the Galerkinrsquos weighted residual method
The Galerkinrsquos method as one of the FEM allows a precise solution with a small number
of elements and accordingly in a short calculation time Also it is easy to treat the
re-dissolution and this method is theoretically adaptable to transient situation
Eq (273-11) can be rewritten as
02 =minusΨnablaprimeminuspartΨpart βDt
(277-1)
where an approximation function of the function Ψ(t r) is set as follows
( ) ( ) ( ) ( )Ψ Ψ Ψt r t r t rj jj
asymp = φ
Here φj(r) is an arbitrary known function called the basic function and Ψ j is an
unknown coefficient where the residuals are represented as
( ) βminusΨnablaprimeminuspartΨpart=Ψ
ˆ 2Dt
rtR (277-2)
The coefficient Ψ j which provides the minimum residuals in Eq(277-2) can be
determined by requiring a condition in which weighted integration becomes zero When the
basic function is selected as a weight function this requirement can be satisfied by the
JAEA-DataCode 2013-005
- 95 -
equation
( )R t r dviV Ψ φ = 0 (277-3)
To satisfy Eq(277-3) for an arbitrary weight function of φ j RΨ(t r)=0 must be
satisfied Therefore Eq(277-3) is equivalent to Eq(277-1)
Integrating Eq(277-3)
2ˆ
ˆ 0iVD dv
tβ φ
partΨ primeminus nabla Ψ minus = part
2ˆ
ˆ 0i iV Vdv D dv
tβ φ φ
partΨ primeminus minus nabla Ψ = part (277-4)
are obtained The second term of Eq(277-4) can also be expressed as
( ) ( ) ( )
( ) ( ) ˆˆ
ˆˆˆ2
nablasdotΨnablaprime+sdotΨnablaprimeminus=
nablasdotΨnablaminusΨnablasdotnablaprimeminus=Ψnablaprimeminus
nabla V ii
V iV iV i
dvDsdD
dvdvDdvD
φφ
φφφ
part
(277-5)
Here when a weight function which satisfies iφ =0 at the grain boundary Vpart is selected the
surface integration term is eliminated and Eq(277-4) is expressed by
( ) ( )ˆ
ˆ 0i iV Vdv D dv
tβ φ φ
partΨ primeminus + nablaΨ sdot nabla = part (277-6)
Then in Eq(277-6) ˆj jφΨ = Ψ is set in a certain finite elementVe Here
jΨ is the
fission gas atoms concentration at three nodes of the element eV and jφ is an interpolating
function inside the element Originally the previously defined iφ is an arbitrary function
group though derivation from Eq(277-3) can hold as-is if this iφ is defined as the
interpolating function as above Accordingly this function is treated as the same function
ie iφ =jφ
Then in Eq(277-6) fission gas generation term βis a constant inside the element so
that
( ) ( ) 0jV j i V j j idV D dV
t
partφ β φ φ φ
partΨ primeminus + nabla Ψ sdot nabla =
(277-7)
can be written In Eq(277-7) concentration at node jΨ and generation term β can be
JAEA-DataCode 2013-005
- 96 -
put out from the volume integral and
0jV j i V i V j i jdV dV D dV
t
partφ φ φ β φ φ
partΨ
primesdot minus sdot + nabla sdot nabla sdot Ψ = (277-8)
is obtained Consequently for a certain element eV Eq(277-9) and the following three
equations are derived or FEM divides Eq(277-7) into the following finite elements
( contents inside sum of Eq(277-9))
( )Ψ Ψj ji j ji ij
E D A H+ prime minus = β 0 (277-9)
where
Veji j i
j iji Ve
i Ve i
E dV
A dVr r
H dV
φ φ
partφ partφpart part
φ
=
=
=
(277-10)
Then the elements in the present geometry is set as quadratic and the following
quadratic functions are adopted as basic functions φ j
( ) ( )
( )
( ) ( )ξξξφ
ξξφ
ξξξφ
+=
minus=
minusminus=
12
1
1
12
1
3
22
1
(277-11)
In introducing FEM a direct use of real coordinates is often inconvenient in
calculation To facilitate the following development of equations an actual
element region is converted into -1 to +1 region in a relative coordinate system
ξ A quadratic element has three nodes In this relative coordinate system the
nodes are atξ= -1 0 and 1 In this case an arbitrary extensive variable ie
fission gas atoms concentration is given by an approximate function and this
function values can be determined by the values at the nodes This means that
the function must be an approximate function that guarantees the values at
nodes Eq(277-11) which is the basic function (interpolating function) in the
quadratic elements takes the following values to guarantee the continuity at
each node
JAEA-DataCode 2013-005
- 97 -
( ) ( )1
11
2φ ξ ξ ξ= minus minus
gives 1 atξ= -1 and 0 at ξ=0
( ) 22 1φ ξ ξ= minus gives 1 atξ=0 and 0 at ξ= -1 and 1
( ) ( )3
11
2φ ξ ξ ξ= + gives 1 atξ=1 and 0 at ξ= -1 and 0
Therefore a basic function within the element ie the interpolating function to express
the gas atoms concentration profile in the radial direction is expressed as
( ) ( ) ( ) ( )21 2 3
1 1( 1 ) 1 ( 1 )
2 2ξ ξ ξ ξ ξ ξΨ = minus minus Ψ + minus Ψ + + Ψ
(277-12)
and
( ) 11Ψ minus = Ψ ( ) 20Ψ = Ψ ( ) 31Ψ = Ψ (277-13)
holds
Figure 274 schematically shows Eq(277-11) by the red broken curve inside the grain
and Eq(277-12) by the red solid line
Using these basic functions Eji Aji and Hi in Eq(277-11) are represented using Rk ie
coordinates of the midpoint of element Ve and using half a element width ΔR as follows
gas atom concentration
boundary layer
Fig274 Gas concentration profiles in FEM element of a grain which is
approximated by sphere
JAEA-DataCode 2013-005
- 98 -
( )
minus
Δ+
Δminus
Δ+Δ=
==
1
1
2
2
4
4
ξξφφπ
φφπφφ
dRRR
drrdVE
kij
RR
RR ijVe ijji
K
K (277-14)
( )
2
1 2
1
4
4
k
Ve k
R Rj ji iji R R
j ik
A dV r drr r r r
R R dR
partφ partφpartφ partφπpart part part partpartφπ partφ ξ ξpartξ partξ
+Δ
minusΔ
minus
= =
= + ΔΔ
(277-15)
( )1 22
14 4
k
Ve k
R R
i i i i kR RH dV r dr R R R dφ π φ π φ ξ ξ
+Δ
minusΔ minus= = = Δ + Δ (277-16)
Next Eq(277-10) is represented by an implicit formula
( )( )1
1 1 0n nj j n n
ji j j ji ij
E D A Ht
θ θ β+
+ Ψ minus Ψ + minus Ψ + Ψ minus = Δ (277-17)
Here
( )( )
D D Dn n
n n
= minus prime + prime
prime = minus +
+
+
1
1
1
1
θ θβ θ β θβ
(277-18)
The reason to use an implicit method here is as follows The diffusion equation is
included in the FGR model and the FGR model is set in a temperature convergence
calculation process Accordingly there are two temperatures Tn at the n-th (previous)
step and Tn+1 at the n+1th (current) step so that the current value is brought into the
convergence process by using the implicit method Also there are njΨ and 1n
j+Ψ for
fission gas atoms concentration The implicit method eg θ =1 allows fission gas
atoms to diffuse by concentration gradient in 1nj+Ψ The diffusion equation in
Eq(277-17) enables us to obtain 1nj+Ψ on the basis of the concentration gradient
derived from the unknown quantity 1nj+Ψ by setting θ =1 in matrix calculation
(implicit method with respect to jΨ ) Introduction of the implicit method allows a
larger time step increment which beneficially reduces computing time
Through rearrangement of the unknown and known values of Eq(277-17)
JAEA-DataCode 2013-005
- 99 -
( ) 1 1 11n n
ji ji j ji ji j ij j
E D A E D A Ht t
θ θ β+ + Ψ = minus minus Ψ + Δ Δ (277-19)
is obtained Therefore the element matrix Wij and the element vector Qi are
1ij ji jiW E D A
kθ= +
Δ (277-20)
( ) 11 n
i ji ji j ij
Q E D A Ht
θ β = minus minus Ψ + Δ (277-21)
Through superimposition of the equations for all elements and setting of the condition of
Ψn+1 = 0 on the surface according to the assumption that the gas atoms concentration is zero at
grain boundary Ψjn+1 can be obtained by solving the simultaneous equations below
[ ] W Qij jn
iΨ + =1 (277-22)
278 Fuel grains and boundaries (common in IGASP=0 and 2)
It should be noted that the diffusion explanation described until now and the logic to
determine ldquoboundary accumulation amount nfn rdquowhich will be discussed hereafter assumes a
situation that there is one single grain exists in space and grain boundary surrounds its surface
In an actual case fuel pellet is poly-crystal
and the ring element of pellet stack consists of a
number of crystal grains Since two neighboring
crystal grains share a boundary it is necessary to
take into account of such actual conditions when
considering the concentration and amount of
fission gas atoms at grain boundary
Here attention is focused on one grain
boundary Since concentration of gas atoms at
the boundary increases by the influxes diffused
from the two adjacent grains the actual
concentration of gas atoms at boundary is 2
times that of the result of diffusion calculation
Now in Fig275A which illustrates the grains by two-dimensional hexagonal shape for
Fig275A Schematic of grain
boundary (1)
A B
JAEA-DataCode 2013-005
- 100 -
simplicity the area of a partial boundary PB of the central grain A with radius ldquoardquo is
one-sixth of the total boundary area The diffusion influx from the adjacent grain B comes into
this PB similarly to the influx from the grain A The rest five-sixth boundary has an identical
situation After all the actual concentration of fission gas atoms accumulated at the boundary
including grain boundary gas bubbles is equivalent to 2-times
that of the diffusion calculation result as shown in Fig275B
However in the boundary gas inventory Eq(279-1)
which is explained in the section 279 below this 2-times
multiplication is not necessary because the amount of gas
atoms which diffused from each grain to boundary is
multiplied by the number of grains contained in one ring
element of pellet Also in the calculation of released gas
amount described in section 2711 the number of grains per
one ring element is taken into account
Furthermore because the gas release threshold N fmax
which is described in section 2714 is simply derived by applying the state equation to the
gas in a representative bubble in which gas pressure is in balance with the compressive stress
from surrounding matrix and surface tension this N fmax
itself does not have to be changed
In other words FEMAXI models that grains and boundaries in each ring element are
represented by one single grain and its surrounding boundary which eliminates the
consideration of the ldquodouble amountrdquo in actual grain boundaries 279 Amount of fission gas atoms migrating to grain boundary (common in IGASP=0 and 2)
Amount of gas atoms 1nfn +Δ (atoms) diffusing to grain boundary from inside of a grain
during time step tΔ is calculated on the basis of the change of fission gas atoms inside grain
as a baseline Assuming that
total amount of fission gas atoms inside grain at time step n ntotalΨ (atoms)
amount of fission gas atoms generation during tΔ V
tdVβΔ (atoms)
total amount of fission gas atoms in grain after tΔ
(at time step n+1)1n
total+Ψ (atoms)
Fig275B Schematic of grain
boundary (2)
JAEA-DataCode 2013-005
- 101 -
1 1n n nf total totalV
n tdVβ+ +Δ = Ψ + Δ minus Ψ (279-1)
is obtained where ntotalΨ and 1n
total+Ψ are determined as follows by using the results of section
277 in the case in which the grain has three elements
( )
( ) ( )
32
1
3
1
ˆ4
ˆ
1
e
k ktotal V
e
k kj j
j
r r dr
r r
k n n
π
φ
=
=
Ψ = Ψ Ψ = Ψ
= +
(279-2)
Here if grain growth occurs the amount of gas atoms 1nfn +Δ which move to grain
boundary during tΔ is determined in the following way setting that the grain radius before
growth is na and the radius after the growth is 1na + which is a baseline radius (refer to the
next section 2710) and using the effective fission gas atoms generation rate β which is
defined by Eq(2712-2) later
Amount of fission gas atoms inside grain at time step n is
3
11 nntotal
n
a
a++
sdot Ψ
(279-3)
where radius 1na + is a baseline and amount of fission gas atoms generated during tΔ is
( )31
4
3 n Vt a tdVβ π β+ sdot Δ = Δ
(279-4)
consequently 3
1 11n n nnf total totalV
n
an tdV
aβ+ ++
Δ = Ψ + Δ minus Ψ
(279-5)
is obtained
2710 Amount of fission gas accumulated in grain boundary
In the fission gas atoms diffusion calculation thus far it has been assumed that the gas is
in an atomic molecule except in the intra-granular gas bubbles However in the discussion
of grain boundary bubbles a gas state is assumed
A value is considered which is obtained by dividing the total amount of fission gas atoms
ie grain boundary inventory invG accumulated at grain boundary of unit volume of pellet
ring element at time step -n- with the total number of grains in the unit volume of ring
JAEA-DataCode 2013-005
- 102 -
element Here all the grain boundaries in the ring element -i- having a volume of iringV is
considered Assuming that the unit volume of pellet consists of igrainN grains with radius
ofga
3
143
igrain
g
Naπ
= (2710-1)
holds
Determination of grain boundary accumulation amount nfn (or 1n
fn + )(atoms) assumes a
situation in which two kinds of movements of gas atoms are present atomic diffusion from
inside of grains and gaseous release to external free space Therefore a relationship n i
inv f grainG n N= (2710-2)
holds Thus invG times the volume of each ring element is the inventory in the element
(1) Equilibrium model (IGASP=0)
If we assume that the amount of fission gas retained in the grain boundary at time step n
is nfn (atoms) amount of gas in the boundary after tΔ will be the smaller one of
1n nf fn n ++ Δ or max
fn ie
1 1 max 1min( )n n n nf f f fn n n n+ + += + Δ (2710-3)
Here maxfn is the maximum amount of gas that can be retained in the grain boundary
Using maxfN (atomscm2) as the saturation value of number of gas atoms in the unit area of
grain boundary maxfn is expressed by
max 1 2 max 114n n
f n fn a Nπ+ ++= sdot (atoms) (2710-4)
The present model assumes that when amount of fission gas at the boundary exceeds maxfn ( max
fN ) excess amount of gas is immediately released to free space of rod and after this
gas transported by diffusion from inner region of grain is released as long as the amount of gas
in the bubble is above maxfn ( max
fN )
In the case of grain growth [see the next section 2712] amount of fission gas retained at
the boundary at time step n is
3
1 nnf
n
an
a+
sdot
(2710-5)
JAEA-DataCode 2013-005
- 103 -
and amount of gas transported to the boundary during tΔ is 1nfn +Δ
Then amount of gas retained at the boundary after tΔ is given by
3
1 1 max 11min n n n nnf f f f
n
an n n n
a+ + ++
= sdot + Δ
(2710-6)
(2) Rate-law model (IGASP=2)
Amount of fission gas retained in the grain boundary after tΔ is a value which is
obtained by subtracting the released amount rlsfn from
1n nf fn n ++ Δ This value is a grain
boundary inventory of one grain when gas release occurs It is
1 1 1n n n rls nf f f fn n n n+ + += + Δ minus
(2710-7)
where 1rls n
f releasen n t+ = timesΔ (2710-8)
and releasen is the gas release rate which is explained in section 28 Total quantity of gas
release from a ring element is given by
1i rls ngrain fN n +times
(2710-9)
In the case of grain growth [see the next section 2712] amount of fission gas retained
in the grain boundary at time step n is
3
1 nnf
n
an
a+
sdot
(2710-10)
and amount of gas transported to the boundary during tΔ is 1nfn +Δ
Then amount of gas retained at the boundary after tΔ is given as
3
1 1 11n n n rls nnf f f f
n
an n n n
a+ + ++
= sdot + Δ minus
(2710-11)
2711 Amount of released fission gas
(1) Equilibrium model (IGASP=0)
Assuming that the amount of fission gas per one grain released to outer free space from
pellet during tΔ is 1nrn +Δ (atoms) in the case of no grain growth we have
JAEA-DataCode 2013-005
- 104 -
( )1 1 max 10n n n nr f f fn n n n+ + +Δ = + Δ le (2711-1)
( )1 1 max 1 1 max 1n n n n n n nr f f f f f fn n n n n n n+ + + + +Δ = + Δ minus + Δ gt (2711-2)
In the case of grain growth [see next section] we have
3
1 1 max 110n n n nnr f f f
n
an n n n
a+ + ++
Δ = sdot + Δ le
(2711-3)
3 3
1 1 max 1 1 max 11 1n n n n n n nn nr f f f f f f
n n
a an n n n n n n
a a+ + + + ++ +
Δ = sdot + Δ minus sdot + Δ gt
(2711-4)
Also decrease in gas concentration 1nrN +Δ (atomscm3) of grain boundary by the release
during tΔ is given by
1
12
1
nn rr
n
nN
aπ
++
+
ΔΔ = (2711-5)
Total quantity of gas release from a ring element is given by
1i ngrain rN n +timesΔ (2711-6)
(2) Rate-law model (IGASP=2)
Refer to the previous section 2710(2)
(3) Fission gas release rate FGR (common to IGASP=0 and 2)
Total quantity of gas released from one grain during tΔ from the 1-th to n-th time steps
is given by
1
nkr
k
n=
Δ (2711-7)
and the gas release rate is assuming that the total amount of fission gas atoms generation in
one grain corresponding to the burnup at n-th time step is ntotalFPG given by
1
nkr
kntotal
nFGR
FPG=
Δ=
(2711-8)
JAEA-DataCode 2013-005
- 105 -
2712 Sweeping gas atoms to grain boundary by grain growth
(1) Equilibrium model (IGASP=0)
During grain growth the gas in grains is swept into the grain boundary The rate of gas
sweeping into the grain boundary is given by
3
1 1ng
n
af
a+
= minus
(2712-1)
Here
1na + grain radius after grain growth (cm)
na grain radius before grain growth (cm)
Figure 276 shows a schematic of the grain growth model The gas present in the
region with hatched lines is released at the grain boundary during grain growth
an+1
an
Fig276 Conceptual diagram of grain growth
The grain growth model uses the following Itohrsquos model as a representative one(230) (231)
da
dtK
a
N N
af f
m
= minus+
1 1 max
(μmh) (2712-2)
( )K RT= times minus times5 24 10 2 67 107 5 exp (μm2h)
a grain radius at current time step (μm)
am maximum grain radius ( )( )= times minus2 23 10 76203 exp T (μm)
R gas constant (=8314 JmolK)
Nf gas atom density at grain boundary (atomscm2)
Nfmax saturation value of the gas atom density at grain boundary (atomscm2)
[Fitting Parameter FGG] The rate of gas sweepings is controlled with FGG as fg = fg times FGG
JAEA-DataCode 2013-005
- 106 -
Fitting parameters AG and GRWFThe grain growth rate can be controlled using AG and GRWF as follows 1) Control of the gas atom density Nf = Nf times AG (Default value AG=15)
2) Grain growth rate GRWFda da
dt dt= times
(Default value GRWF=10)
The option of the grain growth model can be selected using IGRAIN as follows IGRAIN=0 Itohrsquos model(default) =1 Ainscough model =2 MacEwan model =3 Lyons model =4 MATPRO-09 model
Selection of option IGRAIN
In the diffusion calculation using FEM each grain consists of 3 elements as shown in
Fig277 but can have 5 elements at the maximum
Boundary
layer
Node
Fig277 Element division of a pellet grain
The outermost element is the boundary layer which is a special element for
consideration of re-dissolution When grain growth occurs during time step from n to n+1
elements (mesh) in the grains are re-meshed
Setting the boundary layer width as 2ΔR coordinates r at the node of the first and second
elements are given as
11
2
2n
n nn
a Rr r
a R+
+minus Δ=
minus Δ (2712-3)
where
rn+1 coordinates at the node point after grain growth and
rn coordinates at the node point before grain growth
Coordinates at the three node points in the outermost boundary layer are given as
1 2na R+ minus Δ 1na R+ minus Δ and 1na +
The generation term β is changed by taking account of the dissolution only at the
interface element These elements are of ldquothree-node quadratic elementrdquo with the assumed
boundary condition of zero concentration of gas atoms at the grain boundary
JAEA-DataCode 2013-005
- 107 -
The fission gas atoms concentration Ψ at a given set of coordinates after re-meshing is
given by
Ψn+10(r) = Ψn(r) (0 le r le an) (2712-4)
Ψn+10(r) = 0 (an le r le an+1)
where
Ψn+10(r) initial concentration distribution of fission gas after grain growth
Ψn(r) fission gas concentration distribution before grain growth
Namely it is assumed that sweeping of the gas into the grain boundary due to grain
growth occurs instantaneously
Next fission gas atoms concentration at unit area of grain boundary fN is obtained
Expressing Eq (2710-3) again
1 1 max 1min( )n n n nf f f fn n n n+ + += + Δ (2710-3)
Here in the case where no grain growth occurs using 1n na a +=
( )
11 max 1
21
1 max 1
min 4
min
n nf fn n
f fn
n n nf f f
n nN N
a
N N N
π
++ +
+
+ +
+ Δ=
= + Δ
(atomscm2) (27-12-5)
holds However since grain growth is assumed in this section
Δ+sdot=
Δ+sdot
=
+++
+
+
++
+
1max11
1max2
1
1
3
1
1
min
4
min
nf
nf
nf
n
n
nf
n
nf
nf
n
n
nf
NNNa
a
Na
nna
a
Nπ
(2712-6)
holds where24
nfn
fn
nN
aπ= and
11
214
nfn
fn
nN
aπ
++
+
ΔΔ =
Model parameters NODEG and RREL The number of elements in grains is designated by
NODEG The default value is 3 and the maximum value is 5 The width of layers other than
the outermost layer namely the elements of NODEG-1 is designated by the ratio RREL The
default value is 5 1 which means that the ratio of the width of the first to second layers is 51
JAEA-DataCode 2013-005
- 108 -
(2) Rate-law model (IGASP=2)
In the case where the grain growth occurs
3
11
12
1
3
2 11
21
11
4
4
4
n nnf f
nnf
n
n nnn f f
n
n
n nnf f
n
an n
aN
a
aa N n
a
a
aN N
a
π
π
π
++
+
+
++
+
++
sdot + Δ
=
sdot + Δ
=
= sdot + Δ
(2712-7)
holds where 24
nfn
fn
nN
aπ= and
11
214
nfn
fn
nN
aπ
++
+
ΔΔ =
If no grain growth occurs 1n na a += is set in the above equations
(3) How to calculate 1n
fN + (common in IGASP=0 and 2)
Here how to calculate the change nfNrarr
1nfN +
during tΔ ie time step nrarrn+1 when
grain growth occurs is explained To determine 1nfN + it is necessary to take into account of
the transport of gas atoms from inside of grain to the boundary and their re-dissolution during
grain growth period tΔ so that iteration calculation is required Assuming that the initial
value of this iteration is N fn+1 0
( )3
2 2 10114 4n n n nn
total n f total n fn
aa N a N
aπ π ++
+
Ψ + sdot = Ψ + sdot
(2712-8)
holds where Ψtotaln total number of fission gas atoms inside the grain
before grain growth (atoms)
Using Eq(2712-1) and solving Eq(2712-8) with respect to N f
n+1 0 give
10 12
14
ng totaln nn
f fn n
f aN N
a aπ+ +
+
Ψ= + (2712-9)
where 2
14
ng total
n
f
aπ +
Ψ is the amount by grain-growth sweeping and n
fN is the grain boundary
concentration at the beginning of time step
JAEA-DataCode 2013-005
- 109 -
Here if i is supposed to be a count index of iteration
1
1 12
1
4
n
ng totaln i n in
f fr an n
f aN N D t tdV
a a rβ
π+
+ +
=+
Ψ partΨ= + minus sdot Δ minus Δpart (2712-10)
holds The third term of the right hand side of Eq(2712-10) is the amount of fission gas
atoms migrated from inside of the grain to the boundary the fourth term is the amount of fission gas atoms that return to the inside of grain by re-dissolution and iβ is the
re-dissolution rate defined by
( ) ( )1 1 11
2 2i n n i
f f
bN N ADDF
Rβ θ θ + minusprime = minus +
Δ (2712-11)
Eq(2712-11) will be explained in the next section
Now the first and second terms of the right hand side of Eq(2712-10) are known by
Eq(2712-9) Consequently the following process holds
A) Substituting iβ given by Eq(2712-11) into Eq(274-11) and solving the
equation 1 11 1( ) ( )n n
n na a r+ ++ +Ψ Ψ minus Δ and 1
1( 2 )nna r+
+Ψ minus Δ are obtained
B) Using these values the third term of the right hand side of eq(2712-10)
1nr a
D tr
+=
partΨminus sdot Δpart
(2712-12)
is determined
C) Substituting Eqs(27112-11) and (2712-12) into Eq(2712-10) proceeds the
iteration count and 1n ifN + is obtained By this value β in Eq(2712-11) is
updated
Then the steps A) to C) are repeated until convergence is attained ie
1 1 1
1
n i n if f
n if
N N
Nε
+ + minus
+
minuslt is satisfied
However in the actual numerical calculation of N f
n+1 the following equation which is
equivalent to eq (2711-10) is used
( ) 1
1 12
1
1
4
n ng total total Vnn n
f fn n
f tdvaN N
a a
βπ
++ +
+
+ Ψ minus Ψ + Δ= + (2712-13)
Here Ψtotaln +1number of fission gas atoms in grains at the end of time step (atoms)
JAEA-DataCode 2013-005
- 110 -
β Δtdv amount of fission gas atoms (atoms) generated excluding the
amount of re-dissolved quantity during time step tΔ where 3
1
4
3V ndv aπ + = holds
2713 Re-dissolution of fission gas (common in IGASP=0 and 2)
In the model of fission gas atom re-dissolution from grain boundary into grain it is
considered that the boundary separates adjacent grains In sections 288 to 2810 amount of fission gas atoms ( fn fN ) which are accumulated at the boundary of one grain is
considered
As explained in section 288 in the actual grains the amount of gas atoms accumulated at boundary is two times larger than ( fn fN ) due to influxes of gas atoms from the adjacent
two grains However as the re-dissolution occurs for the two adjacent grains concurrently for
each one grain the re-dissolution rate can be evaluated with respect to the amount of fission gas atoms fN per unit area at boundary Namely re-dissolution amount is given by
ADDFfbN sdot (atomscm2s) (2713-1)
where b re-dissolution rate (10-6s)
The re-dissolution means physical transfer of fission gas atoms from the grain boundary
into grains which is virtually equivalent to an additional generation of fission gas atoms in
grains Accordingly when the thickness of regions where fission gas re-dissolves is assumed
to be equal to the boundary layer thickness (200 Aring) the apparent rate of fission gas atoms
generation primeβ in the boundary layer is determined
Therefore primeβ is given by the following equation
( ) ( )1 11 ADDF
2 2n nf f
bN N
Rβ θ θ +prime = minus +
Δ (atomscm3s) (2713-2)
Here Nf
n concentration of fission gas atoms in the grain boundary at time step n (atomscm2)
N fn+1 concentration of fission gas atoms in the grain boundary at time step(n+1)
(atomscm2)
θ interpolation parameter between time steps in the implicit solution
where values of Nfn and N f
n+1 do not depend on the occurrence of grain growth
Accordingly β in Eq (2713-2) for the boundary layer is expressed as
JAEA-DataCode 2013-005
- 111 -
Model parameter RB This parameter controls the thickness of boundary
layer in which fission gas atoms are re-dissolved RB specifies the
thickness Default value is 2times10-6cm
~β β β= + prime (2713-3)
and ~β is defined as the effective fission gas generation rate
2714 Threshold density of fission gas atoms in grain boundary (Equilibrium model IGASP=0)
The gas release model described in the previous section 281 ie White+Tucker model
assumes that when number of fission gas atoms at grain boundary (= area density of fission
gas atoms) exceeds a threshold value the isolated babbles are connected and tunneling path to
free space is formed and the gas atoms diffused from inside of grain are instantaneously
released to free space In other words in the model described in section 271 the diffusion
equation is solved but the growth of grain boundary bubbles is only implicitly considered and
not directly calculated Furthermore the model does not contain the idea of number density of
bubbles in the boundary Nevertheless the model calculates the threshold ie upper limit of
gas atoms concentration in the boundary for the onset of gas release by definition equation
including the grain boundary bubble radius
The method to determine this threshold (saturation) concentration maxfN (atomscm2) is
explained in the followings(224)
(1) Radius of grain boundary gas bubble
(common to equilibrium model and rate-law model)
As shown in Fig278 fr is defined as a radius of curvature of lenticular bubble shape
and br is the radius of the orthogonal projection of the bubble on the grain boundary surface
We have
sinb fr r θ= (2714-1)
The amount of gas re-dissolved is controlled
using ADDF The default value of ADDF is 90
Fitting parameter ADDF
JAEA-DataCode 2013-005
- 112 -
(2) Average concentration and release threshold of grain boundary gas atoms
By applying the state equation of gas to the bubble gas
1gas bubbleF P V m kT (2714-2)
holds where 1m is the number of gas atoms considered in one grain geometry It is the total
amount of gas atoms diffused from inside of one grain to its boundary and the bubble size
corresponds to the condition in which this amount of gas is concentrated in a hypothetical
single gas bubble Here by using the van der Waals constant B of fission gas (Xe+Kr) a
modification factor F is used to take into account of the slight deviation from an ideal gas
1
1 gas
FP B
kT
=
+
(2714-3)
Volume of a bubble is
34( )
3bubble f fV r f (2714-4 )
By the equilibrium assumption
2
gas extf
P Pr
(2714-5)
holds Obtaining the average gas atom density fN by dividing the number of gas atoms m1
by the area which a bubble occupies in the boundary we have
Fig278 Curvature radius of lenticular bubble and its orthogonal projection radius
on the boundary
Grain boundary
JAEA-DataCode 2013-005
- 113 -
12
2
( sin )
4 ( ) 2
3 sin
ff
fext
f
mN
r
r fP F
kT r
π θ
θ γθ
=
= +
(2714-6)
The threshold or upper limit maxfN considered as a quantity for total grain boundary area is
assuming that fb is the covering fraction of lenticular bubbles over grain boundary surface
maxf f bN N f= sdot (2714-7)
From Eqs(2714-6) and (2714-7) we obtain (224)
( )max
2
4 2
3 sinf f
f b extf
r fN f P F
kT r
θ γθ
= +
( )2atomscm (2714-8)
Here N fmax saturation value of the number of gas atoms per unit area at the
grain boundary (atomscm2)
rf intergranular gas bubble radius _=RF (default=05μm)
ff(θ) volume ratio of lenticular bubble to sphere bubble
θ lenticular bubble angle (=50deg)
γ surface tension (626 ergcm2) (220)
k Boltzmanns constant (=138times10minus16 ergK)
fb covering fraction of lenticular bubbles over grain boundary surface
_=FBCOV (default=025)
Pext external force (dynecm2)
The external pressure Pext is obtained using a procedure described below taking into
account the plenum pressure contact pressure between pellet and cladding and pellet internal
thermal stress
Here instead of considering directly the number of grain boundary bubbles or their
actual size the amount of gas moved to grain boundary is considered first and the average gas
atoms concentration in the boundary at the onset of gas release is defined by Eq(2714-8)
In Ref(224) as the specific parameter values to determine N fmax 05μmfr = o50θ =
and 025bf = are presented However PIE observation data suggests that
05 10μmfr = minus and 05bf cong are considered to give more realistic prediction of FGR
JAEA-DataCode 2013-005
- 114 -
2715 In-grain diffusion coefficient of fission gas atoms (common to IGASP=0 and 2) (1) Diffusion coefficient models Details of diffusion coefficients models which are incorporated in FEMAXI-7 are
presented in Chap4 The temperature dependent curves calculated by using some
representative models are shown in Fig279
(2) Fitting function of temperature dependency of coefficient
FEMAXI-7has a function to adjust the temperature dependency of diffusion coefficient
by multiplying a factor FGDIF However when FGDIFX=0 (default) this function is
ineffective and FGDIFX=1 effective
Input parameters OPORO FBSAT ALHOT Fraction of inter-granular bubbles which have always conduit path to free surface is specified by OPORO (default=00) FBSAT is a multiplication factor for the saturation value of inter-granular gas bubbles (Default=10)
500 1000 1500 200010-22
10-21
10-20
10-19
10-18
10-17
10-16
Diff
usio
n C
onst
ant
(m2 s
)
Temperature (K)
Turnbull original Modified Turnbull White 100Wcm White 200Wcm 300Wcm 400Wcm Kogai original Kogai case D White+Tucker Kitajima Kinoshita
Fig279 Comparison of temperature dependency of representative diffusion coefficient models for fission gas atoms in grain
JAEA-DataCode 2013-005
- 115 -
4 5 6 7 8 9 10 11 12 13 14 15 161E-22
1E-21
1E-20
1E-19
1E-18
1E-17
1E-16
1E-15
Diff
usio
n co
nsta
nt (
m2 s
)
10000T (1K)
Turnbull Kitajima K White+Tucker modified Turnbull
Enhanced Turnbulby magnifying factor
(FGDIF0-1)
FGDIF FGDIF010000
9 exp 7
a
a a aT
sdot= minus + sdot sdot minus minus (2715-1)
6a = (default) T (Kelvin)
where name-list parameter a = EFA=6 (default) FGDIF0=10 (default) and FGDIFX =0
(default)
The temperature dependency of this factor is shown in Fig2710 In this case the
diffusion coefficient increases in high temperature region Accordingly as indicated in
Fig2711 in the default condition the original Turnbullrsquos curve is 10-times enhanced above
about 1250K
However this is only an example An optimization on the basis of experimental data is
indispensable
Fig2710 Examples of the values of temperature dependency adjusting factors
JAEA-DataCode 2013-005
- 116 -
5 6 7 8 9 100
2
4
6
8
10
FGDIF0=5 a=6
FGDIF0=10 a=4
FGDIF0=10 a=8
FGD
IF M
agni
fyin
g fa
ctor
10000T (1K)
FGDIF0=10 a=6 default
Fig2711 An example of enhanced diffusion coefficient in high temperature region
of the Turnbull model by applying the temperature dependency adjusting
factor FGDIF
JAEA-DataCode 2013-005
- 117 -
28 Rate-Law Model (IGASP=2)
The rate-law model has been explained only partially in sections 272-2 (2)
2710(2) and 2712(2) Here in this section an essential explanation is described
281 Assumptions shared with the equilibrium model
(1) Calculations of diffusion and intra-grain gas bubble growth
The rate-law model has common model and algorithm to those of the equilibrium
model for the diffusion of gas atoms from inside of grain to grain boundary trapping by
intra-grain bubbles and influence on the effective diffusion rate by growth of intra-grain
bubble growth etc In other words the rate-law model is applied to only the grain boundary
gas bubble growth gas release and coalescencecoarsening of grain boundary bubbles
Also either the irradiation re-dissolution model or Pekka Loumlsoumlnen model which is
explained in section 276 is applied to the intra-grain bubble growth model
(2) Criteria for gas release Gas release from grain boundary gas bubbles is theoretically dependent on the
conductance of gas pass connecting the bubbles and pellet free surface and on pressure
difference between the free space and gas bubbles and on temperature However in the
present model considering that the quantitative evaluation of pass conductance is difficult
and that the known assumption of ldquogas release by tunneling of bubblesrdquo has worked
successfully in predicting FGR behavior as described in section 2714 and below it is
assumed that when either the radius of grain boundary bubble fr or grain boundary
coverage fraction bf reaches a certain threshold level gas in the bubbles begins to be
released to free space
282 Growth rate equation of grain boundary bubble
(1) Grain boundary bubble radius its number density and grain boundary inventory Based on the discussion in section 278 a hypothetical situation is first supposed in
which one isolated crystal grain exists in a space and ldquograin boundaryrdquo wraps its surface
and grain boundary gas bubbles of the grain are generated by the gas atoms diffusion from
inside of the grain
Then bbN the number of grain boundary bubbles of a single grain is defined as
JAEA-DataCode 2013-005
- 118 -
24bb g densN a Nπ= (282-1)
where ga grain radius densN number density of grain boundary bubbles (bubblescm2)
Here it is assumed that
1 densN d= (μm2) d=183x183 (μm2) (282-2)
Namely default value of a name-list parameter NDENS corresponding to densN = is
311x10-2 Also when bubbles coalesce bbN ( densN ) decreases in inverse proportion to
the times of coalescence MGB (See section 282 (3))
Assuming that the bubble radius is fr (cm) internal pressure bP (MPa) of one bubble is
determined by the state equation as follows
34( )
3b fP r f mB mkTπ θ sdot minus =
(282-3)
where
m number of gas atoms per one grain boundary bubble
(atomsbubble) refer to Eq(282-24)
B van der Waals constant for (Xe+Kr) (85x10-23 cm3atoms)
Also grain boundary coverage fraction bf is defined as
2
2 2 2 22
sin 4 sin4
fb f bb g bb
g
rf r N a N
aπ θ π θ= sdot = (282-4)
Here by setting
2 2
2
sinfc
rF
s
π θ= (282-5)
(where s is defined in the next section) ( )c bF f= = FBCOV (282-6)
( )2 2
2 2 2 2 22 2
sin 4 sin sin4 4
f fb bb g dens f dens
g g
r rf N a N r N
a aθ π θ π θ= = = sdot sdot and (282-7)
2 2sin
bdens
f
fN
rπ θ=
sdot (282-8)
hold
Since the value of bf is calculated by Eq(282-4) it varies with fr and ga Here the
present model assumes that gas release from bubbles begins when either one of the
JAEA-DataCode 2013-005
- 119 -
following conditions is satisfied bf becomes equal to FBCOV (input name-list parameter
default=025) by the changes of fr and ga or the bubble radius fr reaches RF (input
name-list parameter default=5E-5 cm) Accordingly FBCOV can be used as a threshold
value for gas release ( Refer to the later section (4))
(2) Rate-law equation to determine grain boundary bubble radius
Radius of grain boundary gas bubble is determined by integrating the growth rate fdr
dt
with time based on the model of Kogai et al(232)
1
nf
f kk
drr t
dt=
= Δ (282-9)
2
22
2FRAD
4 ( )kT
Vf bd gb
b ext ff f
dr s DP P k
dt r f r
δ γθ
Ω= sdot minus minus sdot
(282-10)
where extP gap gas pressure of the axial segment calculated It is set equal to
plenum gas pressure
fr radius of bubble defined in section 2714 (1)
FRADadjustment factor
When 0fdr
dtgt or in a bubble-growing period vacancy moves by grain
boundary diffusion so that FRAD= HP (=20 default) is set
Reference Calculation 1) Assuming 2 11 21 d μm 3 10 (bubblesm )densN = = times (d=183x183μm2) and setting
605μm 05 10 mfr minus= = times and o50θ = we have 2 2(sin50 ) (0766044) 058682= =
and 6 2 11 2(05 10 m) 3 10 m 058682 01383bf π minus minus= times sdot times sdot =
2) Inversely setting 025bf = 605μm 05 10 mfr minus= = times and o50θ = Eq(282-8)
gives 11 25424 10 (bubblesm )densN = times
3) Obtaining the number of gas bubbles K per one grain boundary by referring to
section 291 (1) we have 2
2 2
4
sing b
f
a fK
r θsdot
=
Substituting 025bf = and grain radius 5 μmga = gives K =1704 (bubblesgrain
boundary)
JAEA-DataCode 2013-005
- 120 -
When 0fdr
dtle or in a bubble-shrinking period vacancy moves by lattice-
diffusion so that FRAD= HN (=10 default) is set
Vacancy sink strength of grain boundary bubble is(233)
Vacancy sink strength
22
8(1 )
( 1)(3 ) 2 lnfks
φφ φ φ
minus=minus minus minus
(282-11)
2 2 2
2
sin sinf f br r f
s s
θ θφ
π = = =
(282-12)
3
z rextP θσ σ σ+ += (282-13)
2s (cm2)bubble spacing area or territory area of
one bubble including bubble itself on grain
boundary
2 1densN ssdot = (282-14)
2 2
2 sinf
b
rs
f
π θ= (282-15)
bδ grain boundary thickness = name-list parameter GBTHIK (default=5x10-10 m) V
gbD grain boundary diffusion coefficient of vacancy (=assumed
to be equal to that of Xe atom)
6 4138 10 exp( 2876 10 )VgbD Tminus= times minus times (m2s)
Ω atomic volume of Xe = 409x10-29 m3
Be careful in unit conversion Here since
2 2 2 2 22
sin sin sindens f ff
HN r r H
rφ θ θ θ= = =
φ = 007465x058682=00438 is obtained That is in Eq(282-11) 2 2fs ksdot has a
constant value
JAEA-DataCode 2013-005
- 121 -
(3) Changes of the number density and size of bubbles
The assumption that densN is inversely proportional to the square of bubble curvature
radius fr =RF (default=05E-6m) as shown in Eq(282-8) is supported by observations(226)
That is 2densf
HN
r= holds where H is a proportional constant Also when 025bf = and
05 μmfr = H 01356= is obtained for 11 22
5424 10 (bubblesm )densf
HN
r= = times (Refer
to the reference calculation in the previous section (1) As the bubble is tiny in its embryo
stage it is necessary to circumvent ldquoabnormal numerical resultsrdquo during the embryo stage
For this purpose value of densN is determined in the following method
First the initial value of fr is set as RINIT (default= 001μm=1E-6 cm) Namely
RINITfr ge holds
A) In case of RF (default=5E-5 cm)fr lt
A fixed vale is set 115424 10densN = times (Refer to the above reference calculation 2) In
accordance with the equations of 24bb g densN a Nπ= and inv
bb
Gm
N= not only fr but also bbN
and m change
B) In case of RFfr ge gas release occurs though if the diffusion influx of gas atoms from
inside of grain exceeds the release amount the bubble grows In this situation in a case in
which no coalescence of bubbles is assumed (NGB=1) bf changes in accordance with
Eq(282-8) but the number density is fixed at 115424 10densN = times
In other case in which the coalescence is assumed (NGB≧2) the above two variables
change in inverse proportion to the coalescence number NGB That is after coalescence
the number density becomes
1new
dens densN N = times
MGB
NGB (282-16)
where MGB is the times of coalescence (Refer to section 2101)
JAEA-DataCode 2013-005
- 122 -
(4) Gas release rate from grain boundary bubbles Release of fission gas in the grain boundary bubbles begins in accordance with the van Uffelen model(234) if either 05μmfr = (RF sinb fr r θ= ) or 025bf = (FBCOV) is
【Note】As fr has a very strong dependence on fr numerical instability happens to
occur To circumvent this the following approximate method is adopted in the
calculation
1) For the calculation of Eq(282-9)1
nf
f kk
drr t
dt=
= Δ it is assumed that when the radius
growth rate becomes 0 at some time step ie the bubble grows to reach an equilibrium
0fdr
dt= This is equivalent to obtain the equilibrium size fr when
2b ext
f
P Pr
γ= + holds
In other words
f fr r= is obtained by solving
342
( ) 3
f
f
rP f m b m kT
r
πγ θ
+ sdot minus sdot = (a)
However in the actual situation which the rate-law model addresses f fr rlt
2) Next time required for the growth of bubble from present fr to fr is calculated by
using Eq(282-10) As a first step of calculation the process of change fr rarr fr is
divided equally into 100 steps That is
2 (100 1)100 100 100 100f f f f f f f f
f f f f f f
r r r r r r r rr r r r r r
minus minus minus minusrarr + + rarr + + minus rarr (b)
Then at each divided step time to change ie time to realize the growth amount in
each step is calculated For example if the change at i-th step is expressed as
1f i f ir r +rarr and if change rate of fr is small time itΔ required to the growth from
f ir to 1f ir + ( 1f i f ir r +rarr ) can be obtained by using Eq(282-10) as follows
1
f if i f i i
drr r t
dt+minus = Δ (c)
3) By comparing the time =
Δ=i
jji tt
1
obtained from itΔ with the actual time step
increment tΔ fr is determined as follows
( )( ) ( )iiiifififf ttttrrrr minusminusΔminus+= ++ 11 (d)
Namely tΔ lies between the i-th time it and i+1-th time 1it + of the 100-division
steps so that setting that fr is f ir at it and 1f ir + at 1it + fr を is determined by
Eq(d) If 100tt gtΔ ff rr = holds
JAEA-DataCode 2013-005
- 123 -
satisfied with the following rate
2
FGCND c bb b gbrelease
V N P Sn
kTη= sdot (282-17)
22669
MTης
= gas viscosity (282-18)
4 2 2 2( ) 16b plenumS a P P Lπ η= minus (282-19)
(principle equation)(232)
24gb gS aπ= grain boundary surface area
a capillary inner radius
FGCND adjustment factor of gas conductance in capillary (default=1000)
24bb g densN a Nπ=
Here it is assumed that when fr =RF the number density of grain boundary bubbles is
115424 10densN = times (bubblesm2) and this is proportional to the square of bubble radius
(refer to the previous section (3))
Furthermore the followings are set
Atomic mass of Xe M=131
Its diameter as a solid sphere approximation ς = 4047x10-10 (m)
Capillary volume 0 ( ) ( )c c c eV V f F g σ= (282-20)
4
0 12 32 210 (m )
16 3c
aV
L
π minus= = sdot (282-21)
0( ) 1 exp[1 ( ) ]NFCc c cf F F F= minus minus NFC= 10 default (282-22)
0 0784cFπ= = a baseline value
0( ) 1 exp[1 ( ) ]NSCe e eg σ σ σ= minus minus NSC= 10default (282-23)
0 10 MPaeσ = a baseline value SIG0=10 default
Here it is predicted that after the bubble coalescence the baseline value of 0cF increases
In other words bubble spacing increases and probability of tunneling between bubbles
becomes less Accordingly
After one coalescence of bubbles 0 085cF = is set FC1=085 default
JAEA-DataCode 2013-005
- 124 -
After twice coalescences 0 09cF = is set FC2=09 default
After three times coalescences 0 095cF = is set FC3=095 default
After four times coalescences 0 099cF = is set FC4=099 default
These name-list parameters are given provisional values They should be optimized by
measured data taking into consideration of various factors
(5) Gas amount accumulated in grain boundary (grain boundary inventory)
Here invG is set as fission gas inventory (accumulated amount) at boundary of grains
which are contained in a unit volume of fuel pellet gas amount per a bubble is set as m
(atoms) and the number of grain boundary bubbles is similarly to the previous section (1) set as bbN
As explained in section 278 an actual grain boundary is shared by the two adjacent
grains and the gas in the boundary is also shared However we suppose a situation that one
single grain is isolated in a space and its surface is wrapped by ldquograin boundaryrdquo Assuming that the pellet unit volume contains i
grainN grains with radius ga
iinv bb grainG mN N= sdot (282-24)
holds Therefore gas amount per one bubble m (atoms) is
invi
bb grain
Gm
N N=
sdot (282-25)
Accordingly invG times the volume of each ring element is an inventory value in each
element
(6) Effective stress acting the grain boundary
2 2 2
2 2 2
sin
sin 1h f b c b
ef c
s r P F P
s r F
σ π θ σσπ θ
+ sdot += =minus minus
(282-26)
h ext gapP Pσ = + (282-27)
In these Eqs(282-26) and (282-27) solely the sign is reversed with respect to that in
usual formulation ie compressive stress has a positive value And
2 2 2 2
2 2 2
sin sin
sinf b f
c bf
r f rF f
s r
π θ π θπ θ
= = = (282-28)
is obtained
JAEA-DataCode 2013-005
- 125 -
29 Swelling by Grain-boundary Gas Bubble Growth in the Equilibrium Model (IGASP=0)
291 Grain boundary gas bubble growth In this section a model of gas bubble swelling is explained in the calculation process of
the grain boundary gas bubble growth on the basis of temperature equilibrium pressure and
number of atoms
(1) Calculation of number density of grain boundary bubbles Instead of calculating the growth rate of gas bubbles at grain boundaries the grain radius
which balances with the [external pressure + surface-tension] ie equilibrium model is
calculated by assuming a constant number density at each time step
Here the number density of bubbles K per grain boundary is defined as follows
The orthogonal projection radius of a gas bubble radius fr on the grain boundary
surface is sinfr θ and assuming that the grain radius is gra the total grain boundary
surface area is 24 graπ (Refer to section 2710)
Furthermore assuming that the area number density of bubbles at the moment of gas
release (in the saturated (threshold) condition) is K (though this is a non-dimensional
quantity its physical implication is bubblescm2) the bubble area covering the grain
boundary surface in the saturated condition is given by 2 2sinfK rπ θsdot The fraction of this
area to the total grain boundary surface area is bf thus the following equation holds
2 2 2sin 4f gr bK r a fπ θ πsdot = sdot (291-1)
In other words 2
2 2
4
singr b
f
a fK
r θsdot
= (291-2)
Assuming that the number of bubbles on the grain boundary is inversely proportional to the square of the gas bubble radius fr on the grain boundary and calculating Eq(291-2) by
setting fr=05 μm and θ =50 (223) the following is obtained
2
2
6816 b gr
f
f aK
r
sdot= (291-3)
Namely the number density of bubbles in the saturated condition is given by
Eq(291-3) at the same time the number density of bubbles in the condition prior to the
JAEA-DataCode 2013-005
- 126 -
saturated condition can be obtained by substituting gra bf and fr into Eq(291-3)
Hence the number density of grain boundary bubbles is given by Eq(291-3) K in
Eq(291-3) is adjusted simultaneously with N fmax using the name-list parameter
bf =FBCOV (Refer to 2714 (2) and 282(1))
(2) Growth of bubbles at grain boundaries Assuming that the volume of fission gas remaining on the grain boundary of a grain at
time step n is nfn (atoms) the following holds
nfn K n= sdot (291-4)
Here the term n is the number of fission gas atoms in a bubble and nfn is obtained as
a numerical solution of the diffusion equation More specifically nfn is obtained from
Eqs(279-5) and (2710-3) as
1 maxmin( )n n n nf f f fn n n nminus= + Δ (291-5)
3
1
1
n n nnf total totalV
n
an tdV
aβminus
minus
Δ = Ψ + Δ minus Ψ
(291-6)
Fission gas atoms flow into the micro-bubbles generated at the grain boundary by
diffusion bubble gas pressure increases and bubble growth starts
It is assumed that the gas pressure gasP within the bubble is always equal to the [external
pressure + surface tension pressure]
That is 2
gas extf
P Pr
γ= + (291-7)
The gas pressure within the bubble Pgas during growth is determined by the following equation using fr n B and T
34( )
3f
gas f
rP f n B n kT
πθ
sdot sdot minus sdot = sdot
(291-8)
By solving Eqs(291-3) (291-4) (291-7) and (291-8) simultaneously gasP and fr
are obtained From Eqs(291-3) and (291-4)
2
26816
nf f
b gr
r nn
f a
sdot=
sdot (291-9)
JAEA-DataCode 2013-005
- 127 -
By substituting this into Eq(291-8) and using Eq(291-7)
3 2 2
2 2
42( )
3 6816 6816
n nf f f f f
ext ff b gr b gr
r r n r nP f B kT
r f a f a
πγ θ sdot sdot
+ sdot sdot minus sdot = sdot sdot sdot (291-10)
By dividing both sides by 2fr and letting 26816
nf
b gr
nQ
f a=
sdot
42( )
3f
ext ff
rP f QB QkT
r
πγ θ
+ sdot sdot minus = (291-11)
From Eq(291-11)
( ) ( ) 0223
4
3
4 2 =minus
minusminussdotsdot+sdot QBrQKTQBPfrPf fextffextf γθγπθπ (291-12)
By solving this quadratic equation rf is obtained
In the above equation let
( ) ( ) QBcQKTQBPfbPfa extfextf γθγπθπ 223
4
3
4 minus=minusminussdotsdot== (291-13)
where agt0 and clt0 two real roots are obtained
The larger of the two solutions is the solution sought ie rf
a
acbbrf 2
42 minus+minus= (291-14)
Then using Eq(291-7) Pgas is obtained
As explained the calculation of bubble size on the basis of the equilibrium model is
performed until fr grows to 05μm when it reaches 05μm gas is released as in the
conventional model which is discussed later Bubbles which released gas do not grow beyond
fr =05 μm
(3) In the case grain growth is accompanied
abbreviated Identical to section 2712(1)
(4) Limit of bubble growth due to fission gas release
abbreviated Identical to section 2714
JAEA-DataCode 2013-005
- 128 -
292 Bubble swelling
(1) Modeling of grain boundary bubble swelling
A) Number of grain boundary bubbles of one crystal grain
Writing again Eq(291-3) 2
2
6816 b gr
f
f aK
r
sdot= (291-3)
Here the number of bubbles in the condition prior to the saturated condition can be obtained by substituting gra bf and fr into Eq(291-3)
B) Bubble volume and swelling
The gas bubble volume Vbb and the volumetric bubble swelling are defined for bubble size fr obtained by the above procedure First the volume of one bubble is
34( )
3f
bb f
rV f
πθ= sdot (292-1)
In terms of the grain radius agr the grain volume Vgr is given as
34
3gr
gr
aV
π= (292-2)
Then volumetric gas swelling is
0 2
Bubble
bb
gr
V V K
V V
Δ sdot=
(292-3)
A coefficient of 2 in the denominator on the right-hand side results because one bubble
is shared by two grains (In reality a bubble at a corner of a grain is shared by two or more
grains and in that case the coefficient should be slightly larger than 2 on an average)
In this context 0
001436Bubble
V
V
Δ =
for 5μm 05μmgr fa r= = and 025bf = the
volumetric swelling is 144
(2) Intra-grain bubble swelling Assuming that the number density of intra-grain bubbles is 3(bubbles cm )N and
radius of the intra-grain bubble is intraR (cm) the volumetric swelling is similarly to the
derivation of Eq(292-3)
JAEA-DataCode 2013-005
- 129 -
3intra
0
4
3
BubbleV
R NV
π Δ = sdot
(292-4)
(3) Changes of porosity and density Pellet porosity change pΔ and related density change ρΔ are associated with the total
bubble swelling in the following formula
Total bubble swelling is 3intra
0
4
3
Bubble
bb
grTotal
V V KR N
V Vπ
Δ sdot= sdot +
(292-5)
and 0
Bubble
Total
Vp
V
ΔΔ =
(292-6)
and pρΔ = minusΔ (292-7)
hold
210 Swelling by Grain-boundary Gas Bubble Growth in the Rate-law Model (IGASP=2 IFSWEL=1)
Concerning the bubble growth associated number density of grain boundary bubbles
and gas release are described in sections 281 and 282 Also the bubble growth and gas
release in the case in which grain growth occurs are described in section 2712
Hereafter as part of the model of grain boundary bubble swelling a model of bubble
coalescence and coarsening are described This model is only tentative and has no
verification Here only some basic assumption is presented
[Note] In the equilibrium model (IGASP=0) coalescence and coarsening of grain
boundary bubbles are not dealt with because the equilibrium model assumes that [bubble gas
pressure + surface tension] is always in balance with the compressive force from surrounding
solid matrix However the coalescence and coarsening of grain boundary bubbles are driven
by the excess pressure from the equilibrium pressure Therefore they contradict with the
assumption of the model
2101 Coalescence and coarsening of bubbles
(1) Definition and condition of bubble coalescence Coalescence of grain boundary bubbles cannot occur only by the contact of two isolated
but growing bubbles By contact they are connected and give rise to tunneling leading to gas
JAEA-DataCode 2013-005
- 130 -
release Then by gas release gas pressure decreases inside the connected bubbles and
when the surrounding pressure ie compressive force from solid matrix exceeds the gas
pressure the tunneling path will close by creep down That is tunneling open path of the
connected bubbles is closed After this the connected bubbles will become an isolated and
coalesced bubble by surface tension and the Ostwald growth This process is expressed as
ldquocoalescencerdquo
The above process can be realized when the following two conditions are fulfilled one
is the pellet temperature should rise above the threshold level MGT TMG=1200K (default)
and the other is that the external compressive pressure exceeds the bubble gas pressure This
pressure condition can be expressed by
2b ext gapb
P P Pr
γle + + (2101-1)
03c bF f= ge = FCON (2101-2)
FCON=03 (default)
Here gapP is the gap gas pressure in the axial segment This is approximated
as gap plenumP P= In a future extension of model gapP can be derived by the calculation of gap
gas flow using gap gas flow cross section gapS In this case gapS will vary depending on the
gap state ie open gap closed gap and bonded gap
(2) Number of coalesced bubbles It is assumed that the coalescence is such that M-bubbles merges at one coalescence
and after that the merged bubbles become an isolated single bubble and that this new bubble
grows or shrinks without gas release The bubble number density at grain boundary becomes
1M Growth of bubble would be affected by the Ostwald growth so that the growing rate
should be to some extent deviated from the previous model of ldquodiffusion control driven by
flux from the inner region of grainrdquo
ldquoIGASP=2rdquo condition is controlled by the following parameter
Number of coalescent bubbles by name-list input for ldquoMrdquo NGB default=4
Number of coalescence times by name-list input for ldquoMGBrdquo MGB default=10
In this case when NGB=0 is designated no coalescence occur Also MGB 1ge is
assumed
JAEA-DataCode 2013-005
- 131 -
(3) Geometry of coalescence The above described coalescence is considered
Suppose that M bubbles 1 with curvature radius 1fr r= merge to form a larger single
bubble 2 with radius r2 Here the grain boundary area S which is occupied by the M
bubbles 1 is
2 21 sinS M rπ θ= sdot (2101-3)
Assuming that M does not change and the M bubbles 1 are replaced hypothetically by a
single bubble an equivalent radius r1 of this hypothetical bubble is
1 1r M r= sdot (2101-4)
(4) Change of bubble radius
Next when M bubbles 1 merge into a single bubble 2 with radius r2 it is assumed that
gas pressure surface energy γ and the external pressure extP will remain unchanged before
and after the coalescence and that the coalescence occurs instantaneously so that no influx of
fission gas atoms are diffused into the bubble during coalescence
At this process the lenticular shape is held (exactly speaking the bubbles become a tunnel shape once but this is ignored) increase of radius lessens the surface tension from
1
2γ
r
to2
2γ
r The bubble gas equilibrium pressure would be in a decreasing trend but this change is
so small that it is neglected
Here volume of the bubble 2 is M-times that of the bubble 1 while the radius r2 falls to 3 M times as follows
3 3
1 2
32 1
4 4
3 3r M r
r M r
π πtimes =
= sdot (2101-5)
Therefore the area on the grain boundary occupied by this single bubble is 23 M times
That is the radius of bubble 2 formed by coalescence decreases from the effective radius r1
after coalescence by
31 2 1 1r r r M r M rΔ = minus = minus (2101-6)
Then the number of bubbles on the grain boundary has decreased to 1M from that which is
expressed by K
JAEA-DataCode 2013-005
- 132 -
2
2 2
4
singr b
f
a fK
r θsdot
= (2101-7)
And the fraction of grain boundary coverage has decreased to
2
2 3
1
bb
r ff
r M
sdot =
(2101-8)
(5) Growth of coalesced single bubble
Let us consider the condition of gas re-release by growth of coalesced bubble For the
bubble 2 with radius r2 to grow to the hypothetical bubble with the original effective radius
r1 or to reach the former value of bf the bubble 2 should grow by the rate-law model to
have a radius
2 1 1 sinr r M r θ= = sdot (2101-9)
That is a radius increment is
32 1 1( ) sinr r r M M r θΔ = minus = minus sdot (2101-10)
In this condition the bubble volume becomes M -times larger as the following equation
indicates
32
1
1 rM
M r
=
(2101-11)
2102 Swelling Bubble swelling by the rate-law model is identical to that which is described in the
equilibrium model presented in section 292 Namely if the volumes (radius) of bubbles at
grain boundary and inside grain and their number density are once calculated the swelling is
derived in a similar way to the equilibrium model Relationship between porosity and density
is also identical
211 Thermal Stress Restraint Model
In the equilibrium model of FGR (IGASP=0) a restraint effect on gas release imposed
by the external pressure Pext acting on the grain boundary bubbles can be evaluated by
selection of name-list parameter IPEXT which takes into account of the plenum pressure
PCMI contact pressure and pellet internal thermal stress
JAEA-DataCode 2013-005
- 133 -
For example in Eq(2714-6) increase in Pext will increase the threshold value maxfN
which decreases gas release and vice versa However in the mode ldquoFBSAT=1rdquo if
ldquoFBSATS= maxfN (atomscm3) (default=5E15)rdquo is designated by input max
fN remains
unchanged irrespective of the IPEXT value
In the rate-law model of FGR (IGASP= 2) the gas release threshold is determined by
either the bubble radius fr or grain boundary coverage fraction rf which are indirectly
dependent on extP
However since the gas release rate is a function of bubble pressure plenum pressure and
pellet internal stress as expressed by Eqs(282-16) to (282-22) the restraint effect by Pext
on gas release is implicitly included in the model Also pellet thermal stress is affected by
the swelling model which is selected by IFSWEL When IFSWEL=1 the thermal stress is
affected by FGR which in turn depends on the FGR model selected by IGASP
Thus evaluation of Pext has an indispensable role in the prediction of the amount of
released fission gas Some options in this evaluation are described in the next two sections
It is noted here that Pext has normally a negative value because it is usually a
compressive stress nevertheless
2111 Option designated by IPEXT
Value of name-list parameter IPEXT designates the following mode of external
pressure Pext as the restraint on gas release
=0 Pext =0 =1 Pext =Plenum gas pressure only
=2 Pext =P-C contact pressure only =3 Pext = Max(plenum pressure contact pressure)
=13 Pext = plenum gas pressure + contact pressure
Here the above contact pressure is the PCMI contact pressure which is calculated by the
ERL mechanical analysis
=14(default) Pext = Sav+ plenum gas pressure
Here Sav is the thermal stress (average hydrostatic pressure) calculated by the ERL
mechanical analysis It is defined by Eq(3254) 3
r zST
θσ σ σσ + += The PCMI contact
pressure is always taken into account in the calculation of Sav
JAEA-DataCode 2013-005
- 134 -
Only when IPEXT=14(default) Pext is set as follows
(1) When IGASP=0 Pext = |Sav|+ Plenum gas pressure Here if Sav is positive (tensile
stress) Sav=0 is set
(2) When IGASP=2 Pext =|Sav|+Plenum gas pressure When Savgt0 ie Sav is tensile
stress and 2
0extPr
γ+ = Pext is fixed at Pext =2
08r
γminus times
2112 Comparative discussion on the gap gas conductance increase during power down
In some cases of a test fuel exceeding a certain level of burnup during irradiation tests in
a test reactor when linear power decreases fast concurrently with this power dip or in the
power increase after this power dip a temporary increase of fission gas release has been
observed More exactly an increase of reading of the internal pressure sensor installed in the
plenum region is observed This temporary increase of gas pressure ie substantial increase
in fission gas release can be interpreted by the following two ways
(1) Gas conductance increase in P-C gap During high power period of irradiation pellet temperature is elevated diffusion of
fission gas atoms is enhanced tunneling of grain boundary gas bubbles to outer free space
easily takes place resulting in FGR from grain boundary gas bubbles In this situation due to
thermal expansion of pellets the P-C gasp is either closed or has narrowed markedly or
cracks in the pellet are almost healed As a result the released fission gas from pellet is
retained in a very small space remaining among pellet fragments and cannot flow to reach the
plenum space In other words gas conductance in the axial direction of rod is much limited
Contrarily when linear power decreases pellet thermal expansion mitigates and P-C gap or
crack space are re-opened recovering the gas conductance in the axial direction which
allows the locally retained gas to flow into the plenum space and is observed as an increase in
internal pressure
(2) Change from containment to release of grain boundary bubble gas Similarly to the explanation in the previous section (1) the grain boundary gas
bubbles are growing due to diffusion influx from inside of the grain while the bubbles are
under compressive thermal stress from surrounding solid fuel matrix so that it is difficult to
form a tunnel to release gas to outer free space On the contrary when linear power falls
pellet thermal expansion decreases which facilitates tunneling formation and as a result gas
JAEA-DataCode 2013-005
- 135 -
is easily released to free space and flows to the plenum space through the opened P-C gap and
crack spaces This process is observed as increase in internal pressure This situation
corresponds to the relationship in Eq(2714-8) in which decrease in Pext will decrease the threshold value max
fN leading to gas release This implies a physical meaning of
Eq(2714-8)
【Discussion】 Comparison of the above two mechanisms (1) and (2) suggests that both
have the possibility or that both takes place concurrently in actual fuel rod It cannot be
justified to confine the mechanism interpretation to only one of the two The FEMAXI model
has assumed the second interpretation while modeling on the basis of the first mechanism is
also possible However in fact the FEMAXI model based on the second interpretation has
well reproduced the internal pressure of plenum part successfully at the power dip and
recovery
212 High Burnup Rim Structure Model Formation of the rim structure (the rim phase) affects fuel behavior(226)-(228) In
FEMAXI change in the volume of the rim structure produced (high-burnup structure=HBS)
and its effects on the thermal properties of the pellet are modeled
2121 Basic concept
There has been a number of researches and reports which address such phenomena
associated with the rim structure formation as the porosity FGR density of fission gas atoms
in solid phase swelling and thermal conductivity deterioration and recovery(236)-(243)
Nevertheless up to now no models have been successfully developed which describe
comprehensively the effect of rim structure formation on the macroscopic ie in terms of
engineering level fuel behavior change In addition for the formation of rim structure
expressing a notion that fuel is a ldquocomplex systemrdquo can by itself give no further
understanding of fuel behavior and as such totally meaningless pedantry These situations
are induced by the facts that the rim formation is intrinsically associated with a process in
which many factors are intertwined such as pellet fabrication process burnup temperature
grain size enrichment power history fast neutron flux bonding with cladding etc and also
difficulties in treating high burnup fuels in experiments
Therefore in modeling the rim structure formation in a fuel performance code it costs
too much effort to compose known observations and interactions in a mechanistic manner and
to adjust a number of parameters consistently to obtain at least a reasonable agreement to
several observed data Furthermore it is likely that the resulted predictability could not be
balanced with such tough effort in terms of engineering level
JAEA-DataCode 2013-005
- 136 -
Here noting that the rim structure formation itself gives a relatively independent and
additional effects on the whole aspects of fuel behavior such as a certain degree of fuel
temperature rise or swelling amount the modeling which is simple and based on empirical
correlations and covers the complicated phenomena with some degree of uncertainty can be a
rational strategy in terms of the prediction of whole aspects of fuel behavior
Of course such empirical correlations and model parameters can be easily modified in
combination with the accumulation of observed data
Consequently the present model assumes the followings
A) Rim structure formation ratio
The volumetric ratio Fv of rim structure formation in each ring element of pellet is a
function of burnup Bu and temperature T
B) Porosity and fission gas inventory in rim structure
The porosity is a function of Fv and Bu and takes a fixed value for Fv=100
Gas pore size distribution and gas pressure are not calculated On the other hand fission
gas inventories inside the gas pores and in the solid matrix of fuel and additional FGR by
athermal diffusion etc are determined by empirical correlations
C) Effects of porosity and swelling on thermal conductivity of fuel pellet
Swelling amount is calculated directly from porosity The thermal conductivity
degradation is calculated by Ikatsursquos equation
D) Conditions for diffusion calculation of fission gas atoms
Fission gas atoms diffusion rate is very low at the low temperature peripheral region of
pellet Thus to avoid a redundant calculation the code has an option to duplicate the
calculated results of diffusion in a region at temperature TDIFF (default =600C) to adopt
them to the region with temperature lower than TDIFF However in the case in which the
local temperature in rim region is considerably high adoption of this option would not be
appropriate and need some attention
2122 Density decrease of fission gas atoms in solid phase of rim structure
Although not directly related to the fuel behavior calculation a method is presented first
as the simplest empirical model to evaluate the depletion of fission gas atoms concentration
due to the formation of rim structure This depletion amount is evaluated by the following
three methods while these method are not activated when IRIM=0
JAEA-DataCode 2013-005
- 137 -
(1) Battelle model(236) (HBS=0 IRIM=1) In the model proposed in the Battelle Experimental Program for high-burnup effects
fission gas is represented by Xenon and the following calculation is performed
( )F B B MWd kgUr u u= minus gt0 00625 62 2 62 2 ( ) (2121)
Here Bu is a local burnup at the peripheral (rim region) of pellet Development of the
rim region with burnup is given as a function of the burnup as
( )W Bu= minus2 19 48 8 (2122)
where W is the thickness of the rim region (μm) Equation (2122) shows that even at
local burnup of 150 MWdkg-U the rim region has 220 μm thickness at most and can be
comparable with the thickness of the outermost ring element of pellet Therefore the
athermal fission gas release is calculated only in the outermost ring region Vr
Using Eqs(2121) and (2122) the ratio Rr of xenon which is athermally released in
addition to the release by the thermally activated diffusion from the rim region to that
generated in the outermost region is given by
R V Fr r r= sdot (2123)
( ) ( )V r r W r rr o o o i= minus minus minusπ π2 2 2 2 (2124)
where ri is the inner radius of the outermost region and ro is the pellet outer radius
Assuming that the fission gas detached from the solid matrix of rim structure is released
to outer free space the additional gas release at the outermost ring element induced by rim
structure formation is approximated to occur from grain boundaries to have a compatibility
with the grain boundary gas bubble release model Then the concentration of fission gas
atoms at the grain boundary in the n+1 time step of the grain growth is given by the following
equation rather than Eq(272-13)
( ) ( )1 1 1
1 12
1
1
4
n n n n n ng total total V r rn nn
f fn n
f tdv R M R MaN N
a a
βπ
+ + ++ +
+
+ Ψ minus Ψ + Δ + minus= + (2125)
Here Mn+1 number of fission gas atoms generated in the grains with a radius of an+1 up
to the time tn+1 (atoms)
Mn number of fission gas atoms generated in the crystalline grains with a radius
of an+1 up to the time tn (atoms)
JAEA-DataCode 2013-005
- 138 -
Rrn+1 fission gas release rate due to rim region formation up to the time tn+1
Rrn fission gas release rate due to rim region formation up to the time tn
(2) Cunningham model(225) (HBS=0 IRIM=2) In the Cunningham model fission gas is also represented by xenon and the amount of
additional release is calculated by the following equation
R F Pr r r= (2126)
Here Rr ratio of the amount of fission gas released from the rim region to the
amount of fission gas generated in the entire pellet
Fr (cross-section area ratio of rim to pellet) x (amount of fission gas released
from rim regionamount of fission gas generated in the rim region)
Pr Ratio of the average burnup at the rim region to the average burnup of
pellet ( assumed to be 13)
In addition the following equation holds
F B Br u u= times minus times sdot + times sdotminus minus minus790 10 698 10 834 103 4 6 2 ( Bu gt65MWdkgU) (2127)
where Bu is the local burnup at the pellet peripheral region ie the rim structure region
Fig2121 shows the change of Fr with burnup Thus the amount of additional fission gas
release can be obtained as a ratio to the total amount of fission gas generation which is
calculated by the FGR model
By setting Fr = 0 in Eq (2127) Bu = 1349 and 7020 are obtained as solutions
Accordingly Fr has positive values in the regions Bult1349 and Bugt7020 Hence
because the range of positive Fr values is 7020 MWdKgU or greater in the domain of
Bugt65 MWdKgU the domain of Bu in Eq (2127) is set at Bugt7020 MWdKgU
In addition because fission gas release rate in an entire pellet is obtained by integrating
the release rate of gas in each ring element of pellet formulation of Eq(2126) does not fit
the constraints of FEMAXI Under these circumstances formulation of Eq(2126) is altered
as described below
Eq(2126) gives the ratio of the fission gas released from the rim region to the fission
gas produced from the entire pellet On the basis of this equation calculation of the release
rate Frim of fission gas from the rim structure region is defined as follows
Assuming that the amount of fission gas production is proportional to the burnup
JAEA-DataCode 2013-005
- 139 -
50 60 70 80 90 100 110 120000
001
002
003
004
005
Frac
tion
of ri
m a
rea
Local burnup (GWdt)
Cunningham model
Fig2121 Change of fraction of the rim structure region as a function of pellet local burnup in the Cunningham model
(Pellet cross-section arearim area)
(Pellet average burnupRim average burnup)rim rF R= times
times (2128)
holds
(3) Modified Cunningham model (HBS=0 IRIM=3)
Next in the Cunningham model ratio Pr for the average burnup in the rim region to the
average burnup of the entire pellet is treated as a corrected rate in Eq (2126) thus this
corrected approach is adapted as an option
In other words there are two cases the case in which Pr=13 is adopted and the case in
which Pr calculated in FEMAXI is adopted
A) When Pr=13 is adopted (IRIM=2) Frim is obtained using Eq (2128) In this case the
ratio of the average burnup of pellets to rim phase average burnup calculated in FEMAXI is
used for the value (average pellet burnup)(rim phase average burnup) in Eq (2128)
therefore in general the ratio is different from 113
B) Contrary to this when Pr calculated in the code is used (IRIM=3) the product of Pr in Eq
(2128) and (average pellet burnup)(rim phase average burnup) equaling 1 is assumed
(Pellet cross-section arearim area)rim rF F= times (2129)
In both the cases the area of rim phase is required however the Cunningham model
does not consider the area of rim phase Thus one should determine the area in order to
obtain reasonable solutions Accordingly the area of rim phase is calculated expediently
using Eq(2122)
JAEA-DataCode 2013-005
- 140 -
2123 Effect of rim structure formation on fuel behavior
The FEMAXI has systematically modelled the effects of rim structure formation on
pellet behavior These models are effective when name-list parameter HBS=1 and =2 though
in these cases IRIM=0 is automatically set and as a result the previous depletion model
cannot be applied When HBS=0 designation of IRIM is effective but these models of the
effects are not applied
2123-1 Burnup of rim structure (HBS=1 2)
In considering the rim structure formation one of the important points for modelling is
that the onset local burnup for the rim structure formation and change of the nature of
structure with burnup extension In this two options are set with respect to the burnup of
rim structure formation ie a local burnup and effective burnup to calculate the volume of
rim structure and porosity etc by designating the name-list parameter HBS
(1) Fraction of rim structure transformation Xv(HBS=1 2)
It is assumed that fraction of transformation of a normal high burnup fuel matrix into the
rim structure is Xv This Xv has nothing to do with the models described in the next
sections 2123-2 and 2123-3 though it is a physical quantity to indicate how much the rim
structure has been formed and gives a reference to consider the correlation between model
prediction and observation (plotting output index is IDNO=256)
A) When HBS=1 the transformation fraction Xv is Xv=10 for the ring elements of pellet in
which element local burnup have exceeded the name-list parameter BKONA (default=65
GWdtU) and Xv=00 for the elements in which local burnup is under BKONA
When in the two adjacent ring elements to a certain ring element R local burnup in the
outer element R+1 has exceeded BNONA while local burnup in the inner element R-1 does
[Fitting parameter RFGFAC] The amount of additional fission gas release due to rim region formation can be adjusted using a factor RFGFAC ( default = 10 ) F F R F G F A Cr r= times
[Option Parameter IRIM] A model for additional fission gas release from rim region can be selected from the following three models IRIM=1 Battelle model (default) =2 Cunninghum model =3 Modified Cunninghum model
JAEA-DataCode 2013-005
- 141 -
not Xv in the ring element R is determined by the interpolated area fraction of Xv(gt0) in
R+1 element and Xv(=00) in R-1 element
B) When HBS=2 Xv is determined by ( )1
1tanXv
effB Buα δ
π
minus minus= + (2123-1)
where α=105 (ARIM default=105) δ=052 (DRIM default=052)
effB effective burnup explained later in section (3) Eq (2123-3)
1Bu BURIMS (default=600 GWdtU)
(2) Local burnup (HBS=1) In this option (HBS=1) local burnup localBu in every ring element of pellet is directly
used to the models of rim structure formation and properties change
(3) Concept of effective burnup (JAEA model HBS=2) Formation of rim structure and its volume increase depend not only on local burnup but
also local temperature Because of this a model concept of ldquoeffective burnuprdquo is introduced
on the basis of the following discussion and assumptions
A) The rim structure is generated in the region with a certain level of high burnup eg 50 - 60
GWdt and with time its volumetric fraction increases(236) - (241) This is because the high
burnup region has augmented the density of lattice defects and resulted in enhanced strain
energy which promotes the lattice structure transformation into the rim structure
B) On the other hand the distorted lattice in high burnup region can be annealed by thermal
activation process and recover its lattice to a normal condition With temperature rise this
recovery process is accelerated and as a result the thermal conductivity degradation also
mitigates (242) and the density of lattice defects and strain energy decrease which counteracts
the rim structure formation(243)
C) For these reasons modelling to predict the rim structure formation should be conditioned
by the two terms burnup and temperature history Therefore the radial direction profile of
burnup resulted from a burning analysis code is modified into an ldquoeffective burnup profilerdquo by
assuming the lattice recovery effect due to temperature rise and it is assumed that if in a
region of pellet this effective burnup profile exceeds some threshold level the rim structure is
JAEA-DataCode 2013-005
- 142 -
generated and developed
D) The effective burnup profile is derived from a combination of local burnup extension and
temperature history
First it is assumed that at the n-th time step the effective burnup 1neffB minus up to the n-1 th
time step decreases by the recovery during the current time step increment tΔ That is
recovery process is a thermally activated process so that it decreases as a function of the
local temperature nT at the n-th time step
11 0exp ( )n
eff nB k T T tminus sdot minus minus sdot Δ (2123-2)
On the other hand at the n-th time step the burnup increases by nBΔ so that the
effective burnup at the n-th time step is
11 0exp ( )n n n
eff eff nB B k T T t Bminus= sdot minus minus sdot Δ + Δ (2123-3)
where neffB effective burnup at the n-th time step (GWdtU)
nT local temperature of fuel at the n-th time step (K)
here if 0nT Tle 0nT T= is assumed
0T baseline temperature TSTD=1000K (default)
1k constant 1 KON1= 1E-8 (default)
tΔ time step increment (s)
nBΔ local burnup increment during the n-th time step
Fig2122 shows a comparison of burnup profiles at different average burnups of fuel
rod
00 01 02 03 04 050
10
20
30
40
50
60
70
80
90
Burn
up (G
WdtU
)
Pellet Radius (cm)
Fig2122 Comparison of profiles of burnup and the effective burnup at different
average burnups of fuel rod
JAEA-DataCode 2013-005
- 143 -
The present model assumes that the rim structure is developed in the pellet region in
which the local effective burnup calculated by Eq(2123-3) exceeds a certain threshold
value
2123-2 Fission gas release from gas pores (HBS=1 2)
(1) Model of fission gas atoms migration from rim structure to gas pores In the previous section 2122 depletion of fission gas atoms concentration in rim
structure ie migration into gas pores is expressed by the Battelle and Cunningham models
(HBS=0) The present new model (HBS=1 =2) adopts the Lassmann empirical model(243)
instead of these two models and expresses the fraction of fission gas atoms retained in solid
phase to the total fission gas atoms generated in the rim structure as FSOLID and also
expresses the fraction FPOR of gas which moves from solid phase and gas pores to the free
space outside pellets as a function of burnup Bu (either localBu or effB ) Namely the model
evaluates the partition and released amount to the solid phase and gas phase of fission gas
atoms generated in the solid phase of rim structure
The Lassmann model represents the fission gas atoms in the rim structure by Xe and the
Xe concentration in solid phase 1Xe is
( )1 0 0(wt) FPINF (GEN1 FPINF)exp GEN2 ( )Xe Bu Bu Bu= + sdot minus minus sdot minus (2123-4)
Therefore the fraction FSOLID of gas retained in solid phase is
[ ]1FSOLID Total Xe generation(wt)Xe= (2123-5)
Also the fraction FPOR of gas moved from the solid phase is
[ ]1
FPOR 10 FSOLID
10 Total Xe generation(wt)Xe
= minus= minus
(2123-6)
However when 0Bu Bule all the gas is assumed to be retained in the solid phase
Here
[ ]Total generation(wt) GEN1 (GWdt)Xe Bu= sdot (2123-7)
Default FPINF=025(wt )
GEN1=00145 GEN2=01 0Bu =BURIMXE=600
are defined Fig2123 shows change of Xe concentration in solid phase with burnup
Also assuming that the fraction of gas which migrates directly from the solid phase to
JAEA-DataCode 2013-005
- 144 -
the free space by athermal release is ATHMR (default=00) 0 ATHMR FPORle le holds
Here as an inter-conversion between A wt and B atomscm3 of Xenon
3
3
23
001 [pellet density(gcm )]B (atomscm )=A(wt)
131 (gmol)
6022 10 (atomsmol)
ρtimestimes
times sdot (2123-8)
is used
Here it is better to put it in order though repetition that the gas content GS0 existing in
the rim structure is defined by a result of subtractions of the following three terms from total
amount of gas generated in the rim structure GB1 for the gas atoms moving to the grain
boundaries by thermal diffusion GB2 for the gas atoms moving into the intra-granular
bubbles in grain and GR for the gas atoms migrating from the rim structure at the fraction of
FPOR As a result
GS0 = GG ndash GB1 ndash GB2 ndash GR
holds Therefore when HBS is either =1 or = 2 the following procedure is performed
A) At a certain time step in a certain ring element once the rim structure is formed the
thermal diffusion calculation is ceased at the element At this moment the amounts of gas
atoms in solid phase GS0a and the gas accumulated in the bubbles GB1 and GB2 are
determined by the result of diffusion calculation by that time step
B) After this time step with formation of rim structure neither GB1 nor GB2 changes and
the amount of gas atoms retained in the solid phase is calculated by subtracting GR from
GS0a
0 50 100 150 20000
05
10
15 BURIM=60GWdt at 65 GWdt at 70 GWdt at 75 GWdt
Xe
conc
(w
t)
Local burnup (GWdt)
Total generation rate of Xe
Fig2123 Change of Xe concentration in solid phase by the Lassmann model
JAEA-DataCode 2013-005
- 145 -
(2) FGR from rim structure (HBS=1 2 RMOGR=012) An additional release of fission gas from the rim structure to outer free space is
predicted by using empirical equations while in the case of RMOGR=0 no release is
assumed from the gas pores
Of the total gas generated in the rim structure the Lassmanrsquos model defines the fraction
ldquoFPORrdquo of gas which migrates from solid phase and gas pores of the rim structure into free
space That is FPOR is the fraction of amount of gas detached from the solid phase of rim
structure However the present model described in this section (2) assumes that the gas
which migrates to the pores is all accumulated in the pores and that no release of gas occurs
directly from the solid phase to free space In addition it is assumed that out of all the gas
atoms contained in all the pores only a fraction OPR is released to outer free space Then
fission gas release rate FGRIM from the pores out of the total generation of fission gas atoms
in the rim structure is
FGRIM=OPRFPOR (2123-9)
A fraction of fission gas atoms retained in the matrix of rim structure except pores is
(1-FPOR) of the total generation Therefore FGR of the ring element is
0(Bu) FGRIMFGR FGR= + (2123-10)
Bu local burnup (either localBu or effB ) (GWdt)
where FGR by the thermal diffusion is 0FGR
The open pore fraction OPR is determined by either the A) or B) in the followings
A) Option A to specify the release rate (RMOGR=1 ) By using ATHMR which is explained in section 21232 (1) the open pore fraction is
defines as
OPR=ATHMR (2123-11)
where 0 OPR ATHMR FPORle = le That is after the rim structure is formed a part of gas
out of the generated gas in the rim structure is always released to outer free space at the rate
of (ATHMR x 100) via pores in addition to the FGR induced by the thermal diffusion
B) Option B to specify the open pore fraction (RMOGR=2) The open pore fraction OPR to the porosity of rim structure porosity rimp is defined as a
function of rimp as follows(241)
003OPR (0 023)023 rim rimp p= sdot le le
(2123-12)
OPR 003 (015 003)(100 23) (023 024)rim rimp p= + minus sdot minus le le (2123-13)
JAEA-DataCode 2013-005
- 146 -
20 40 60 80 100 120 1400
5
10
15
20
Min
Por
osity
(
)
Local burnup (GWdt)
Max
Fig2124 Change of porosity with burnup in the NFD correlation equation
OPR 015 (045 015)(100 24) (024 025)rim rimp p= + minus sdot minus le le (2123-14)
where the rim structure porosity rimp is determined by the next section 2123-3
2123-3 Porosity thermal conductivity and swelling (HBS=1 and 2 RMPST=01 and 2)
The porosity change induced by fission gas pore generation in the rim structure its
associated degradation of thermal conductivity and the effect of swelling by the rim structure
are modeled The porosity is specified by a name-list parameter RMPST When RMPST=0
rimp =005 is set not calculating the porosity change
(1) Empirical correlation equation of NFD data (RMPST=1)
The porosity rimp is calculated by the following empirical equation(240) which is shown
in Fig2124
2
2
PROMAX 0005 ( 40) 25
PROMIN 00013 ( 40) 25
PROMAX (10 06) 06 PROMIN
= sdot minus += sdot minus +
= sdot minus + sdotrim
Bu
Bu
p
(2123-15)
Bu bunrup (either localBu or effB ) (GWdt)
Here it is assumed that the upper limit of rimp is set as PMX=0254 which is common to
RMPRS=2 mode
JAEA-DataCode 2013-005
- 147 -
(2) Billaux model ( RMPST=2)
Not calculating the amount of fission gas in pores the rim porosity rimp is obtained by
an empirical correlation with respect to burnup That is by applying partially the Billauxrsquos
correlation equation (246)
as-fabricated porosity=p
AP1 AP2 BP1 BP2 GWdtM= sdot minus le lerimp Bu Bu (2123-16)
PMX BP2 GWdtM= gtrimp Bu (2123-17)
Default AP1=00024 AP2=0106
BP1=65 (GWdt) BP2=150 PMX=0254 Bu local burnup (either localBu or effB ) (GWdt)
are assumed Here it is assumed that after rimp increases and attains a temporary highest
value maxrimp rimp remains at its highest value maxrimp even if temperature rises markedly
(3) Effect on thermal conductivity of fuel pellet (RMPRO=0 1) As one of important effects of the rim structure effect of porosity on the pellet thermal
conductivity is calculated by the equation modifying the thermal conductivity of ring
elements which contain the rim structure as follows here suppose that the thermal
conductivity is rimλ of the elements in which the rim structure has developed one of the
following two equations is selected by input
Ikatsu(247) and Bekker(248) 16(1 )rim Bx rimpλ λ= sdot minus (2123-18)
Billauxrsquos equation 25
1
095rim
rim Bx
pλ λ minus = sdot
(2123-19)
where rimp is the porosity of the ring elements defined by Eqs(2123-15) to (2123-17)
RMPRO=0 Ikatsu model (Default) =1 Billaux model
Here Bxλ is a thermal conductivity of fuel matrix at the effective burnup Bx on the basis
of the consideration that the burnup of matrix would virtually decrease ie effects of thermal
conductivity degradation due to burnup extension is mitigated because development of the
rim structure will induce the strain relief of matrix and depletion of fission gas atoms In
other words the relation can be
( )Bx T Bxλ λ= (2123-20)
where Bx effective burnup BXEQ=40 (GWdtU) (Default)
JAEA-DataCode 2013-005
- 148 -
T local temperature at element (K)
Here RMPRO is independent from the designation of RMPST
(4) Swelling When RIMSWL=1 irrespective of the IFSWEL value the volumetric swelling rate of
ring element is set equal to [porosity + solid swelling rate] (default value of RIMSWL is 0)
213 Selection of Models by Input Parameters
The thermal analysis of FEMAXI determines analytical conditions and model options by
a combination of several input parameters ie name-list input parameters The inter-linkage
among these parameters is explained below in an organized manner Here this explanation
of options designated by name-list parameters is in principle given by the order of analysis methodrarrgeometryrarrmaterial properties (models) Further the following explanation is
on the premise of IFEMOP=2 ie a completely coupled solution of thermal and mechanical
analyses When IFEMOP ne 2 some selections below would not be available
2131 Number of elements in the radial direction of pellet
The number of ring elements of a pellet stack is linked to the selection of burnup
calculation code (heat generation density profile model) and the HBS model Here the Table
231 shown in section 231 is re-written for importance
However when the HBS (rim structure) model is designated to use MESHgt0 is required
Even if MESH=0 is input the calculation automatically sets MESH=3
Here when ISHAPE=1 is designated the number of ring elements doubles for the value
of MESHgt0 which allows a more detailed calculation
Table 231 Number of radial ring elements of pellet stack in thermal and mechanical analyses Parameter value
1-D thermal analysis 1-D mechanical analysis
2-D local PCMI analysis
MESH=0 Iso-thickness 10 ring elements
Iso-thickness 10 ring elements
Iso-thickness 5 ring elements
MESH=1 Iso-volumetric 36 ring elements Iso-volumetric 18
ring elements Iso-volumetric 9 ring elements
MESH=2 Iso-volumetric 72 ring elements
MESH=3 Iso-volumetric 36 ring elements Iso-volumetric 36
ring elements Iso-volumetric 18 ring elements
MESH=4 Iso-volumetric 72 ring elements
JAEA-DataCode 2013-005
- 149 -
2132 Calculation of thermal conductivity of fuel pellet
(1) The case in which rim structure formation is not considered (HBS=0) Regardless of the value of parameter MESH let us assume that the thermal conductivity
of pellet is
100( ) ( )TDF p T Buλ λ= sdot (2131)
where ( )F p is porosity factor and this function is described in Chapter 4
The variable ldquoporosity p rdquo is calculated by following the name-list parameter IPRO and
put into the function ( )F p
IPRO=0 (Default) (refer to section 234)
1) IPRO=0 0p p= (2132)
2) IPRO=1 00
swgVp p
V
Δ= + (2133)
3) IPRO=2 00 0
dens hotV Vp p
V V
Δ Δ= + + (2134)
4) IPRO=3 00 0 0
swg dens hotV V V
p pV V V
Δ Δ Δ= + + + (2135)
where P0 initial porosity
0
swgV
V
Δ fission gas bubble swelling
0
densV
V
Δ densification
0
hotV
V
Δ hot press
Here the content of 0
swgV
V
Δ is determined by the selection of name-list input parameter
IFSWEL
(2) The case in which formation of a rim phase is considered (HBS=1 2)
The thermal conductivity of ring element which have been transformed into the rim
structure is defined by Eqs(2123-18) and (2123-19) previously in section 2123-3 (3)
Ikatsu Bekker equation 16(1 )rim Bx rimpλ λ= sdot minus (2123-18)
JAEA-DataCode 2013-005
- 150 -
Billaux equation
251
095rim
rim Bx
pλ λ minus = sdot
(2123-19)
For the ring element having no rim structure Eq(2131) is applied to the thermal
conductivity
(3) Effects of the decrease in density
Regardless of values of MESH HBS and IPRO when IFSWEL=0 or 1 assuming that
the solid swelling is 0
swsV
V
Δ and the density decrease factor due to solid selling is Df
the corrected thermal conductivity λ is
100 ( ) ( )D D TDf f F p T Buλ λ λ= sdot = sdot sdot (2136)
D HBSfλ λ= sdot (2137)
Also when name-list parameter FDENSF=0 (default)
10Df =
and when FDENSF=1 0
10 swsD
Vf
V
Δ= minus (2138)
For IFSWEL see section 41 Chubb and Zimmermann + FEMAXI-III model or
FEMAXI model
2133 Selecting swelling model Swelling depends on the temperature of pellet At the same time it is a phenomenon
which is directly related to deformation and PCMI it has both thermal and mechanical
characteristics Here option is explained when swelling is taken into consideration in terms
of its relationship to fission gas bubbles
(1) Definitions and Classification a) Swelling due to intra-granular gas bubble formation is defined as intraS
b) Swelling due to gas bubble growth at the grain boundary is defined as interS
c) Total gas bubble swelling is defined as sg intra interS S S= +
d) Solid-state fission product swelling is defined as solid ssS S=
e) Swelling due to increase in porosity caused by the rim structure development is
defined as rimS
These are defined by the swelling model which is selected by the name-list input
JAEA-DataCode 2013-005
- 151 -
parameters combined and calculated as described below
(2) Selection of options of swelling model by IFSWEL
A) IFSWEL=0 sgS = Chubb and Zimmermann model(249) + FEMAXI-III(12)
00025ss
solid
VS SWSLD
V
Δ = = sdot
per 1020 fissionscm3 (2139)
SWSLD=10 (Default)
B) IFSWEL=1 FEAMXI-6 model 1) sg intra interS S S= + is calculated from fission gas bubbles
2) 00025ss
solid
VS SWSLD
V
Δ = = sdot
per 1020 fissionscm3 (21310)
SWSLD=10 (Default)
3) IFSWEL=2 MATPRO-9 model
4) IFSWEL=3 Kosaka model(250)
5) IFSWEL=4 Studzvik model(251)
6) IFSWEL=5 Hollowell model(252) See Chapter 4 for details of each model
7) DENSWEL=1 one of the swelling models and one of the densification models are
merged to calculate the volume change of pellet at every time step
(3) Relation to FGR models
When IFSWEL=0 2 3 4 and 5 swelling is determined independently from the gas
bubbles growth calculation
When IGASP=0 and IFSWEL=1 swelling is determined by the equilibrium bubble
growth model
When IGASP=2 and IFSWEL=1 swelling is determined by the rate-law bubble growth
model
2134 Selecting fission gas release model (1) Selection of model either the equilibrium model or rate-law model is selected
(2) Calculation of intra-granular bubble radius and its number density (IGASP=0 2) A) A strict analytical solution of the White+Tucker equilibrium state-equation model
GBFIS=0
JAEA-DataCode 2013-005
- 152 -
B) Model of re-dissolution by irradiation including bubble destruction and vacancy
migration GBFIS=1
C) Pekka Loumlsoumlnen model GBFIS=2
2135 Bubble growth and swelling
(1) Swelling by grain boundary bubble growth and intragranular bubble growth IFSWEL=1 enables users to calculate the swelling when either IGASP=0 or 2
(2) Porosity (density) Porosity (density) is controlled by IPRO and IDENSWEL They can affect the pellet
thermal conductivity in calculation
(3) Coalescence and coarsening of grain boundary bubbles By IGASP=2 the coalescence and coarsening of grain boundary bubbles are calculated
while if NGB=1 is specified no such calculations are performed
JAEA-DataCode 2013-005
- 153 -
2136 Options for the rim structure formation model
When HBS=0 irrespective of IGASP additional FGR from the rim structure can be
specified by IRIM and RFGFAC
When HBS=1 or 2 Burnup in the rim structure is designated Designation of IRIM
and RFGFAC is ineffective though the rim structure model can be used irrespective of
IGASP values However in the cases of both HBS=0 and 2 designation of MESH=2 or 3 or
4 is required Types of combination of FGR models and swelling models A1 A2 B1 and
B2 are summarized in Table 2131
Table 2131 List of combination of FGR models and swelling models
IGASP IFSWEL Grain boundary bubble Intra-granular bubble Method of calculation GBFIS
A1 0
Equili-
brium
model
0 2
3 4 5
Bubble growth is calculated by the White-Speight- Tucker model with the state equation of gas Not taking into account of the swelling by grain boundary and intra-granular bubbles
Bubble radius is calculated by the state equation
0 default
Bubble radius is calculated by irradiation-re-dissolution model
1
Bubble radius is calculated by Loumlsoumlnen model 2
A2 1
Bubble growth is calculated by the White-Speight- Tucker model with the state equation of gas Taking into account of the swelling by grain boundary and intra-granular bubbles
Bubble radius is calculated by the state equation
0 default
Bubble radius is calculated by irradiation-re-dissolution model
1
Bubble radius is calculated by Loumlsoumlnen model 2
B1 2
Rate-
Law
model
0 2
3 4 5
Bubble growth is calculated by the rate-law model Taking into account of the coalescence and coarsening of bubbles but not considering the swelling by grain boundary and intra-granular bubbles
Bubble radius is calculated by the state equation
0 default
Bubble radius is calculated by irradiation-re-dissolution model
1
Bubble radius is calculated by Loumlsoumlnen model 2
B2 1
Bubble growth is calculated by the rate-law model Taking into account of the coalescence and coarsening of bubbles and swelling by grain boundary and intra-granular bubbles
Bubble radius is calculated by the state equation
0 default
Bubble radius is calculated by irradiation-re-dissolution model
1
Bubble radius is calculated by Loumlsoumlnen model 2
JAEA-DataCode 2013-005
- 154 -
214 Gap Gas Diffusion and Flow Model
Calculation method for the flow of gap gas is described in Fig2141
Time Step Start
Diffused amount of gas between adjacent (lower and upper) segments by concentration gradient
Concentration and composition of gas of each segment after diffusion
Calculated at all the axial segments
Concentration and composition of gap gas at each segment after diffusion
Equilibrium pressure in total space inside fuel rod
Axial transport of gas under pressure gradient (adjustment from lower segment)
Total amount and composition of gas at each axial segment after pressure adjustment
Gas concentration at gap space of each segment
Mutual-diffusion constant of gas
Fission gas release in each axial segment during one time step
Total amount and composition of released gas
Fig 2141 Calculation process of gap gas diffusion and flow
JAEA-DataCode 2013-005
- 155 -
2141 Assumptions and methods of diffusion calculation
In the case of (name-list parameter) IST=0 (default) complete and instantaneous mixing
of gases is assumed while in the case of IST=1 calculation of diffusion and flow of gap gas
is performed on the basis of the following assumptions
A) Helium and Nitrogen (or Argon) as well as released Xenon and Krypton are completely
mixed instantaneously therefore behavior of Nitrogen (or Argon) is represented by that
of Helium and also behavior of Krypton is represented by that of Xenon Thus
calculation is performed for a 2-component mixed gas ie mixture of Helium and
Xenon
B) Counter-diffusion occurs with Helium and Xenon which pass through the gap
C) The counter-diffusion calculation is performed using an implicit solution based on Ficks
first law The diffusion is independent of flow caused by pressure difference in the fuel
rod
D) Pressure in the axial segments of fuel rod is uniform during the diffusion calculation
E) Pressures at different regions inside fuel rod are assumed to be equilibrated
instantaneously Therefore gas flow across the axial segments induced by the
segment-wise pressure difference also occurs instantaneously
In the following sections the method in IST=1 case is described
(1) Calculation method for the gap space The following is a list of symbols used
N gas molar number (mol)
n molar density (molcm3)
D12 counter-diffusion constant (m2s) (D12 = D21)
T temperature (K)
V volume (m3)
C molar fraction ( )( )C n n n1 1 1 2= +
Subscripts are
I kind of gas (1 = Xenon 2 = Helium) j axial segment ( )j P N PL U= 1 2 3 (PL lower plenum PU upper plenum)
In particular subscripts for molar density are
JAEA-DataCode 2013-005
- 156 -
nij density of gas i at axial segment j
nj density of all gases at axial segment j
Figure 2142 shows the model
PlenumSegment j Segment j+1
TpTj+1Tj
2lj 2lj+1
Fig 2142 Gas diffusion and flow model
The total amount of gas i at axial segment j is
N n V
C n VP
RC
V
T
i j i j j
i j j j i jj
j
=
= = (2141)
Here each segment has a channel cross-sectional area of Sj and length of 2l j also the
boundary of segments j and j+1 is indicated by
The amount of gas 1 (Xe) ( )G j j1 1rarr + flowing from segment j to segment j+1 over the
boundary is obtained on the basis of Ficks law
( )G j S Dn n
lj jj
j1
12 1 1rarr =minus
(2142)
Similarly the amount of gas 1 (Xe) flowing into segment j+1 over is represented as
( )G j S Dn n
lj jj
j1 1 1
12 1 1 1
1
1 rarr + =minus
+ ++
+ (2143)
From the continuity condition
( ) ( )G j G j1 1 1rarr = rarr +
holds and
S Dn n
lS D
n n
lj jj
jj j
j
j
12 1 11 1
12 1 1 1
1
minus=
minus+ +
+
+ (2144)
can be obtained By solving this equation for n1 we obtain
nS D l n S D l n
S D l S D lj j j j j j j j
j j j j j j1
121 1 1 1
121 1
1 112 12
1
=+
++ + + +
+ + + (2145)
Using Eq (2142) we obtain the following equation
JAEA-DataCode 2013-005
- 157 -
( ) ( )
( )G j j G j
S D S D
S D l S D ln nj j j j
j j j j j jj j
1 1
121 1
12
1 112 12
11 1 1
1rarr + = rarr
=+
minus+ +
+ + ++
(2146)
Here by setting
FS D S D
S D l S D lj jj j j j
j j j j j j +
+ +
+ + +
=minus1
121 1
12
1 112 12
1
(2147)
Eq(2146) is expressed as ( ) ( )G j j F n nj j j j1 1 1 1 11rarr + = minus+ +
(2148)
Meanwhile regarding gas 2 (He) ( ) ( )G j j F n nj j j j2 1 2 2 11rarr + = minus+ +
(2149)
is obtained
Through addition of Eqs(2148) and (2149) the total amount of inflow mixed gas can
be calculated as ( ) ( )G j j F n nj j j jrarr + = minus+ +1 1 1
(21410)
Here the next equation
n n n n n nj j j j j j= + = ++ + +1 2 1 1 1 2 1
holds
Since in general the gap temperatures at segments j and j+1 differ the following equation
holds in the equilibrium state njnen j+1 (21411)
According to Eq(21410) however even if segments j and j+1 are in equilibrium
imaginary flow from segment j to j+1 is present in the numerical calculation
This is because Fickrsquos first law is applied to a field with a temperature gradient To
eliminate this flow a second term representing the effect of temperature difference must be
added to both Eqs(2149) and (21410)
The second term to be added to Eq (21410) is
( ) ( ) ( ) G j j F n n n nj j j j j jrarr + = minus minus minus+ + +1 1 1 1 (21412)
where n nj jand +1 are given by the ideal gas law as follows
nP
R Tn
P
R Tjj
jj
= =++
1 11
1
(21413)
The amount of flow of each component namely Eqs(2148) and (2149) is given
respectively as follows
JAEA-DataCode 2013-005
- 158 -
( ) ( ) ( ) G j j F n n n nj j j j j j1 1 1 1 1 1 1 11rarr + = minus minus minus+ + + (21414)
( ) ( ) ( ) G j j F n n n nj j j j j j2 1 2 2 1 2 2 11rarr + = minus minus minus+ + + (21415)
Here
nP
R
C
Tn
P
R
C
Tjj
jj
11
1 11
1 = =+
+ (21416)
CN N
V V
n V n V
V Vj j
j j
j j j j
j j1
1 1 1
1
1 1 1 1
1
=++
=++
+
+
+ +
+
(21417)
nP
R
C
Tn
P
R
C
Tjj
jj
22
2 12
1 = =+
+ (21418)
CN N
V V
n V n V
V Vj j
j j
j j j j
j j2
2 2 1
1
2 2 1 1
1
=++
=++
+
+
+ +
+
(21419)
C C1 2 1+ = (21420)
(2) Implicit solution of He-Xe mutual diffusion The flux J R1 of gas 1 (Xe) that diffuses through the boundary surface R (boundary
between segments jminus1 and j) into segment j is given by
( ) ( ) J F n n n nR j j j j j j1 1 1 1 1 1 1 1 = minus minus minusminus minus minus (21421)
The flux J S1 of gas 1 (Xe) that diffuses through the boundary surface S (boundary
between segments j and j+1) into segment j+1 is given by
( ) ( ) J F n n n nS j j j j j j1 1 1 1 1 1 1 1 = minus minus minus+ + + (21422)
Then the change in the amount of Xenon in segment j is given by
Vn
tJ Jj
jR S
partpart
11 1
= minus (21423)
Through differentiation of Eq(21423) using Eqs(21421) and (21422) and the implicit
method the following Equation is obtained Here k denotes time-step number
JAEA-DataCode 2013-005
- 159 -
( ) ( )[ ]
( ) ( )[ ]n n
t VF n n n n
F n n n n
jk
jk
jj j j j j j
j j j j j j
11
1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1
11 2 2 1 1
1 2 2 1 1
+
minus minus minus
minus + +
minus= minus minus minus
minus minus minus minusΔ θ
θ θ θ θ θ
θ θ θ θ θ (21424)
Here
θ 1= interpolation with respect to temperature volume pressure and
equilibrium molar ratio
θ 2= interpolation with respect to molarity
θ 1 and θ 2
are the interpolation parameters in a converging loop in the axial direction and
in the counter-diffusion calculation respectively θ 1 changes in the converging loop in the
axial direction but remains constant in the counter-diffusion calculation
Tuning parameter THG1THG2 θ1 and θ 2 can be specified by THG1 and THG2
respectively Default=10
Therefore an interpolation equation for θ 2
( )n n nj jk
jk
12
2 1 2 111
θ θ θ= minus + + (21425)
is substituted into Eq (21424) and the following equation is obtained
( )[ ( )
( )]
n n
t
F
Vn
F F
Vn
F
Vn
VF n n
n n F
jk
jk
j j
jj
k j j j j
jj
k
j j
jj
k
jj j j
kj
k
j j j j
11
1 2 11 1
1 2 1 1
11
2 11 1
11 2 1 1 1
1 1 1 1
1
1
1 1
1
1
1 1
1
1 1 1
11
1
+minus
minus+ + minus +
minus+
+minus minus
minus +
minus= minus
+
+ + minus minus
minus minus minus minus
Δθ θ
θθ
θ
θ
θ
θ θ
θ
θ
θ θθ
θ θ θ ( )[ ( ) ( )]2 1 1 1 1 1 11 1n n n nj
kj
kj j minus minus minus+ +
θ θ
(21426)
Through rearrangement of Eq (21426)
A n B n C n Dj jk
j jk
j jk
j1 11
11
1 11
minus+ +
+++ + sdot = (21427)
Here
AF
Vtj
j i
j
= minus minusθ θ
θ2 1
1
1
Δ (21428)
B
F F
Vtj
j i j j
j
= + minus+minus +
12 1 1
1 1
1
θ θ θ
θ Δ (21429)
JAEA-DataCode 2013-005
- 160 -
CF
Vtj
j j
j
= minus +θ θ
θ2 1
1
1
Δ (21430)
( ) ( )( )
( ) ( )
DV
F t nV
F F t n
VF t n
VF n n F n n
jj
j j jk
jj j j j j
k
jj j j
k
jj j j j j j j j
= minus + minus + minus
+ minus + minus minus minus
minus minus minus +
+ + + + minus minus
11 1
11
11
1
1
1
1
1 1
1
1
1
1 1 1 1 1
1 2 1 1 1 1 2 1
1 2 1 1 1 1 1 1 1 1 1 1
θθ
θθ θ
θθ
θθ θ θ θ θ
θ θ
θ
Δ Δ
Δ ( )θ1 Δt
(21431)
First the upper and lower axial segments including plenum are considered
Since there is no boundary surface R for the first segment
( ) ( )[ ]V
n
tJ
F n n n n
S11 1
1
1 2 1 1 1 2 1 1 1 2
partpart
= minus
= minus minus minus minus
(21432)
holds therefore the following equation holds
( )
( ) ( )
n n
t
F
Vn
F
Vn
F
Vn
F
Vn
F
Vn n
k kk k k
k
1 11
1 1 2 1 2
11 1
1 2 1 2
11 2
1 2 1 2
11 1
2 1 2
11 2
1 2
11 1 1 2
1
1
1
1
1
1
1
1
1
1
1 1
1
1
++ +minus
= minus + minusminus
+minus
+ minus
Δθ θ θ
θ
θ
θ
θ
θ
θ
θθ
θ
θ
θθ θ
(21433)
This equation is rearranged using
( ) ( ) ( )
BF
Vt C
F
Vt
DF
Vt n
F
Vtn
F
Vn n tk k
12 1 2
11
2 1 2
1
12 1 2
11 1
2 1 2
11 2
1 2
11 1 1 2
1
11 1
1
1
1
1
1
1
1
1
1
1
1 1
= + = minus
= minusminus
+
minus+ minus
θ θ
θ θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
θθ θ
Δ Δ
Δ Δ Δ (21434)
and we obtain
B n C n Dk k1 1 1
11 2
11sdot + sdot =+ +
(21435)
Similarly there is no boundary surface S for the uppermost segment n and thus the
following equation holds
( ) ( )[ ]V
n
tJ
F n n n n
nn
R
n n n n n n
partpart
11
1 1 1 1 1 1 1
= minus
= minus minus minus minusminus minus minus
(21436)
Therefore by setting
A
F
Vt B
F
Vtn
n n
nn
n n
n
= minus = +minus minusθ θθ
θ
θ
θ2 1 2 1
1
1
1
11 Δ Δ
JAEA-DataCode 2013-005
- 161 -
( ) ( )
( ) tnnV
F
ntV
Fnt
V
FD
nnn
nn
knn
n
nnkn
n
nnn
Δminusminus
Δ
minusminus+Δ
minus=
minusminus
minusminus
minus
11
1
1
1
1
1
1
1111
112
1112 1
11
θθθ
θ
θ
θ
θ
θ θθ
(21437)
we obtain
A n B n Dn nk
n nk
nsdot + sdot =minus+ +
1 11
11
(21438)
Combining the above we obtain the following equation for the diffusion of gas 1
B CA B C
A B C
A B CA B C
A B
n
n
n
n
n
n
n n n
n n n
n n
k
k
k
nk
nk
nk
1 1
2 2 2
3 3 3
2 2 2
1 1 1
1 11
1 21
1 31
1 21
1 11
11
sdotsdotsdotsdotsdot
sdotsdotsdotsdotsdot
minus minus minus
minus minus minus
+
+
+
minus+
minus+
+
=
sdotsdotsdotsdotsdot
minus
minus
DDD
DDD
n
n
n
1
2
3
2
1
(21439)
Gas concentration ni jk+1 of each axial segment is obtained by solving Eq (21439)
(3) Mutual diffusion constant
The gas counter-diffusion constant adopts Presents constant(231)
DkT
m n d12
1
2
122
3
8 2
1=
ππ
(21440)
where
mm m
m m =
+1 2
1 2
(m weight of one molecule)
( )d d d1212 1 2= + (d diameter of one molecule)
n n n= +1 2 (molar density)
k Boltzmanns constant T temperature (K)
(4) Pressure adjustment calculation
Setting the total gas molar number as Nj for each axial segment at the end of time step
N Nj iji
= (21441)
JAEA-DataCode 2013-005
- 162 -
holds where i represents kind of gas For the entire volume of the fuel rod inner space the
gas molar number is given by
jj
N N= (21442)
Upperpl enum
Segment
5
Segment
Xe injectionPressure adjustment
4
Counter-diffusion
Segment3
Segment2
Segment1
Lower
pl enum
Previous step
Current step
He Xe He Xe He Xe
After pressure adjustment
Fig 2143 Calculation of pressure adjustment (vertical axis gas molar number)
Since
P N RV
T jj
= sdot sdot
1 (21443)
holds on the basis of the ideal gas law the total molar number of gas primeN j in segment j after
the calculation of pressure adjustment is given by
prime =
NP
R
V
Tjj
(21444)
JAEA-DataCode 2013-005
- 163 -
The discrepancy between Nj and Njrsquo is adjusted by balancing of the input and output flows
between the upper and lower adjacent segments beginning from the lowest part of pellet stack
Figure 2143 shows the pressure adjustment process which consists of three steps
Namely if the following are set
Nirarrj molar number of mixed gas transported from segment i to segment j
fij fraction of segment i in segment j before transport
the gas molar number i for segment j after transport is given by
prime = minus
+rarr rarr N f N N f Nij ij j j kk
il l jl
(21445)
Model parameter IST When using the above model of gas flow and diffusion in gap designate IST=1 Default is IST=0 which gives an instantaneous complete mixture of gas is assumed
215 Method to Obtain Free-space Volumes in a Fuel Rod 2151 Definition of free space
The gap volume in a fuel rod can be classified into the following four components A)
plenum volume B) P-C gap volume C) pellet center hole volume and D) free-gas volume in
the pellet stack This classification is required because the pressure reference temperature for
each volume is different in calculation The free gas volume in the pellet stack consists of dish
volume chamfer volume inter-gap volume between pellet ends (pellet-to-pellet aperture) and
crack gap space inside pellet
When IFEMRD=1 ie in the entire rod length mechanical analysis quantities of the A)
B) and C) components change following the calculated results while changes of ldquoD) free-gas
volume in the pellet stackrdquo ie volume changes of dish space chamfer space and inter-gap
between pellet ends are not taken into account and crack space calculation depends on the
option value of name-list parameter IRELCV (default=0) while when IFEMRD=0 ie in
the local PCMI mechanical analysis it is assumed that of the four components the volumes
of the center hole and free gas in the stack do not change during irradiation
(1) Center hole volume
The center hole volume is given using the initial inner radius of pellet rpi and the length
JAEA-DataCode 2013-005
- 164 -
lZj of axial segment j as
V r lhj
pi Zj= π 2 (2151)
The free gas volume inside the stack includes dish volume chamfer volume
pellet-to-pellet free space volume and void volume in cracks
When the length of one pellet is set as lp dish volume per pellet as Vdish chamfer volume
as Vchem inter-gap volume between pellet ends as Vp and void volume in crack as Vcrack the
equations
V r lp po= π 2 2Δ (2152)
V r u lcrack porel
p= 2π (2153)
hold where
rpo outer radius of pellet
Δl pellet-to-pellet aperture width
urel radial displacement of pellet caused by relocation
The free gas volume V jint in the stack in axial segment j in Fig2151 is given by
( )V V V r l r u ll
lj
dish cham po porel
pZj
pint = + + + sdotπ π2 2 2Δ (2154)
(2) Plenum volume and pellet-cladding gap volume
In contrast to the above plenum volume and gas volume are assumed to change during
irradiation The plenum volume change is obtained from the change in axial segment length
The pellet-to-pellet aperture
lΔ is defined as the distance
between two edges of
adjacent pellets as described
by the right-side figure lΔ
It specifies the tilting of pellets
It is designated by parameter AY
The default value is 20 μm
Parameter AY
pellet
Fig2151 Definition of pellet-to-pellet aperture
JAEA-DataCode 2013-005
- 165 -
inside the stack In segment j setting the thermal expansion strain averaged by pellet
volume as ε thj densification strain as ε den
j and solid swelling strain as ε ssj the segment
length ~
lZ pj including the changes by these strains is
( )~l lZ pj
thj
denj
ssj
Zj= + + +1 ε ε ε (2155)
However when IFEMOPgt0 j
Z pl is obtained by the calculated deformation of
mechanical model In cladding setting the thermal strain as Δε c thj and strain by irradiation
growth as Δε c irrj the segment length
~lZ Cj is
( )~ l lZ Cj
c thj
c irrj
Zj= + +1 ε ε (2156)
Similarly the plenum length Z pll is given by the calculated deformation of mechanical
model as follows (Refer to section 33 ldquoMechanical analysis of entire rod lengthrdquo for details)
( ) 1 pl plZ pl c th c irr Z pll lε ε= + + (2157)
The change in plenum volume ΔVpl is given by
( )2
1
NAXIj j
pl ci Z C Z p Z pl Z plj
V r l l l lπ=
Δ = minus + minus
(2158)
where NAXI represents the number of axial segments and rci the cladding inner radius
Accordingly when the initial plenum volume is set as Vpl the plenum volume is
~V V Vpl pl pl= + Δ (2159)
The gap volume of axial segment j is given as follows using the gap width δ Here δ
= 0 is assumed when numerically δ lt0
~ ~ V r lgap j ci Z p
j= sdot sdot2π δ (21510)
(3) Volumes of dish chamfer and pellet-to-pellet gap space
Space volume of chamfer which is calculated by Fig2152 is
2 3chamfer
Dp aV abπ = minus
(21511)
JAEA-DataCode 2013-005
- 166 -
Dish space volume which is calculated by Fig2153 is
2 2( 3 )6dish
dV d r
π= + (21512)
Fig2153 Volume of dish space
Fig2152 Volume of chamfer
a
b
Dp
Dp
r
d
JAEA-DataCode 2013-005
- 167 -
Space of tilting end faces of pellet which is calculated by Fig2154 is
2 21tan ( )
8tilting inV Dp Dp Dπ θ= sdot minus (21513)
where tanθ=0002 is assumed
(4) Volume of cracks inside pellet Change of crack space volume inside pellet is not taken into account when IRELCV=0
When IRELCV=1 this change is taken into account on the basis of the relocation strain
change inside pellet Here setting as
A= P-C gap space volume change induced by relocation strain and
B=Space volume of cracks inside pellet produced by relocation
a model of ldquocrack space volume inside a pelletrdquo is introduced which assumes that A+B can
remain unchanged irrespective of the change of relocation strain However when
IRELCV=0 is applied A+B will not remain unchanged with the change of relocation strain
Accordingly for the internal pressure calculation IRELCV=1 has to be applied
When IRELCV=1 is applied and change of crack space is taken into account on the basis
of the relocation strain the crack space approaches zero when cladding creeps down inward
P-C gap is closed pellet fragments are pushed back relocation displacement is decreased and
compressive stress is generated inside pellet This is linked with the model of pellet Youngrsquos
modulus The pellet stiffness change model is applied to the pellet volume change by
Fig2154 Space volume by tilting of pellet end face
θ
Din
Center hole
Dp
JAEA-DataCode 2013-005
- 168 -
relocation Details are described in section 323
216 Internal Gas Pressure It is assumed that the gas in a fuel rod behaves as an ideal gas and the pressure in the
fuel rod is uniform in every part of rod Then the gas pressure is calculated as
int
1
Tgas j j jM
pl L pl U gap hj j j
jpl L pl U gap pi av
n RP
V V V V V
T T T T T=
sdot=
+ + + +
(2161)
where
Pgas gas pressure inside fuel rod (Pa)
nT total molar number of all gases (mol)
R gas constant 8314 JK sdot mol
VplL lower plenum volume (m3)
TplL lower plenum gas temperature (K)
VplU upper plenum volume (m3)
TplU upper plenum gas temperature (K)
Tw coolant temperature (K)
Vgapj
gap volume in axial segment j (m3)
Vhj pellet center hole volume in axial segment j (m3)
V jint free gas volume in pellet stack (m3)
Tgapj gap temperature in axial segment j (K) = 05 ( )T Tps
jci
j+
Tpij pellet center temperature in axial segment j (K)
Tpij volumetric average temperature of pellet in axial segment j (K)
Here temperatures used to calculate the gas pressure inside fuel rod are
A) Plenum gas temperature is a sum of temperature of coolant at the plenum elevation and
DTPL (Refer to the next section 2161)
B) Gap gas temperature is an average of pellet outer surface temperature and cladding inner
surface temperature
C) Pellet center hole gas temperature is the temperature of pellet center or pellet center hole
inner wall temperature
D) Free gas temperature inside pellet stack is a volumetric average temperature of pellet
JAEA-DataCode 2013-005
- 169 -
Here the free gas volume V jint includes dish volume chamfer volume inter-pellet free
space volume and void volume in crack V jint is already known by Eq(2154)
2161 Method of assigning plenum gas temperature and cladding temperature
Temperatures at plenums are assigned by two parameters DTPL and ITPLEN
(1) ITPLEN=0 (default) specifies the followings
The lower plenum gas temperature is given as the [coolant inlet temperature + DTPL]
and the upper plenum temperature is given as the [coolant outlet temperature + DTPL] The
value of DTPL can be specified by input parameter DTPL (default value is 25C) Here one
must be cautious with the plenum gas temperature because the plenum gas pressure is
calculated using the [coolant temperature + DTPL] even under a cold state
In addition the upper plenum cladding temperature is set equal to the calculated coolant
outlet temperature The lower plenum cladding temperature is set equal to the coolant inlet
temperature These cladding temperatures are not dependent on the calculation procedures
whether the cladding surface temperature is given by input or by calculation Here the outlet
coolant temperature at the upper plenum section is determined by the heat conduction
calculation at the segment immediately below the upper plenum section of the fuel rod
(2) ITPLEN=1 specifies the followings
The plenum gas temperature is set as the [coolant inlet temperature + DTPL] In this
option the cladding temperatures of the upper and lower plenum sections are given by the
coolant inlet temperature When there is a lower plenum section pl L pl UT T= =[coolant inlet
temperature + DTPL] is adopted for Eq (2171)
2162 Calculation of the variation in internal gas condition during irradiation
In FEMAXI it is possible to change the composition pressure of the internal gas and
plenum volume up to 20 times during irradiation by specifying the name-list input parameters
This option enables users to perform analysis of experiments on fuel rod which is
base-irradiated in a commercial reactor and then test-irradiated in a test reactor after
instrumentation or analysis of creep deformation experiment of cladding by increased internal
pressure during irradiation The name-list input parameters for these purposes are ITIME(20)
JAEA-DataCode 2013-005
- 170 -
GASPRN(20) PLENM(20) and GMIXN(4 20)
Here when a rod is refabricated and instrumented it is assumed that the instrumentation
is carried out at room temperature and with no power Accordingly this input condition has to
designate zero linear power at the irradiation history point which is specified by ITIME
written in input file In this zero power point the rod temperature becomes equal to the
coolant temperature at this history point For example if coolant temperature is 300K and
even if DTPLgt0 temperature of the entire rod including plenum becomes 300K Therefore
the pressure and volume of internal gas should be considered as quantities at 300K
Tuning parameter DTPL Default value of DTPL is 25K As the plenum gas
temperature is always set as [coolant temperature +DTPL] it is to be noted that the
plenum gas pressure is calculated by using the temperature which has additional
DTPL even in the cold state after irradiation
Option parameter ITPLEN Default value is ITPLEN=0
217 Time Step Control The time-step width is automatically controlled in FEMAXI In the entire calculation
the width is determined by the restriction Δt1 while in the thermal analysis it is subdivided
into Δt2 and in the mechanical analysis it is subdivided into Δt3
2171 Automatic control
(1) Restriction for entire calculation Δt1 is the minimum among the values determined by the following four conditions
1) Changes in linear heat rate is within 10 Wcm during one time-step
2) Changes in burnup is within 100 MWdt during one time-step
3) Creep strain rate is limited so that creep strain increment does not exceed elastic
strain
4) Time step width is within 15 times the former time step width
5) Change of coolant flow rate is within 10 of that of the former time step
The condition 1) is to stabilize calculation when power changes more than 10Wcm the
condition 2) is determined by subroutine PHIST to make the burnup increment less than
JAEA-DataCode 2013-005
- 171 -
100MWdtUO2 The condition 3) is a restriction to avoid divergence in creep calculation for
all elements and Δt1 is determined by
EFCOEF1 sdotsdot
leΔcE
tε
σ (2171)
where EFCOEF is a tuning parameter
The condition 4) restricts the expansion of Δt1 within a factor of 15 when rod power
increases at less than 10Wcm during one time step The condition 5) is for the fast change of
coolant flow rate However in the thermal analysis a time step width Δt2 which is smaller
than Δt1 can be used in the following manner
(2) Restriction for sub-division of time step width for thermal analysis Δt2 is used for the gas diffusion time-step width which is the subdivision of Δt1 and
determined as the shortest time among the time required to reach the equilibrium state of gas
flow in each axial segment and the time determined from the maximum amount of transport
of gas between the segments as well as from pressure adjustment conditions with its upper
limit value set as either 100 ms or Δt1
The time required to reach the equilibrium state in each axial segment is derived from
Eq(21423) as
( )
Δtn n V
J Jji j j j
R S
=minus
minus1 1
1 1
(2172)
where the superscript i represents axial segment number
Next the time determined from the maximum amount of transport of gas between
segments is obtained as follows Using the maximum molar number of the gas transported
These factors are related to 1t Δ The term DPXX
is the variation size of linear power within a time step its default value is DPXX=10
(Wcm) The term DPBU is the burnup increase within a time step its default value is
DPBU=1 00 (MWdt O2 ) This default value occasionally causes an extremely long
computation In many cases adopting DPBU=5000 is effective in reducing computation
time Setting DPXX value is effective in reducing the extremely long calculation time
For example it can be set to DPXX=90
Tuning parameter DPXX DPBU
JAEA-DataCode 2013-005
- 172 -
between segments Nmax as standard since ( ) ( ) J JR j S j1 1 1= minus we represent Δt2 as follows
using ( )J R j1
( )ΔtN
Jj
R j
2
1
= max
(2173)
Then the time determined from the pressure adjustment conditions in gas flow is obtained
as follows Due to the assumption of instantaneous equilibrium of pressure inside the fuel
rod numerical instability resulting from gas flow between segments may occur To avoid
this instability we must impose a restriction on the amount of gas transport at each segment
Setting the gas transport rate in each segment as β (mols) the restriction is given by
ΔtN
jj
3 = max
β (2174)
Thus under the conditions
( )Δ Δ Δ Δt t t tn21 11
21 1
2= sdotmin η (2175)
( )Δ Δ Δ Δt t t tn22 12
22 2= min (2176)
( )Δ Δ Δ Δt t t tn23 13
23 3= min (2177)
Δt2 is given by
( )Δ Δ Δ Δt t t t2 21 22 23= min (2178)
Nmax in Eqs(2173) and (2174) and η 2 are the control parameters programmed in the
code
Tuning parameters EFCOEF AMLMX2 AMLX3 and DTPR
Default value of EFCOEF in Eq(2171) is 10 Nmax in Eq(2173) can be specified by AMLMX2 Default value is 10-6(mole)
Nmax in Eq(2174) can be specified by AMLMX3 Default value is 2times10-6(mole)
η2 in Eq(2175) can be specified by DTPR Default value is 001
2172 Time step increment determination in FGR model In the fission gas release model the time step increment
2tΔ given by section 2171 is
further divided by a particular method which uses the equivalent diffusion constant Drsquo and
JAEA-DataCode 2013-005
- 173 -
nodal interval 2RΔ of the second zone of spherical FEM system to give sub-divided time step
as
22
3 005 FMULTR
tD
ΔΔ = times timesprime
(2179)
Adjusting input parameter FMULT Default=10
2173 Time step increment determination in temperature calculation In a non-steady temperature calculation the time step increment 2tΔ given by section
2181 is further divided by a particular method which uses the thermal diffusivity Ki and ring
element thickness rΔ to give sub-divided time step as
( )2
64
min
10i
rt
K
Δ Δ = sdot
(21710)
Here the factor 106 is very large because it is determined by an extremely thin oxide
thickness rΔ and has been empirically determined through sensitivity analysis
In some cases 4tΔ is determined as a very small value with respect to 2tΔ If 4tΔ is less
than 1100 of 2tΔ to proceed a time step of transient thermal calculation the time step period
2tΔ is not divided with equal intervals of 4tΔ but the time step begins to increase with the
initial value of 4tΔ 分 and doubles the former time step tΔ to circumvent the too large
number of time steps
On the other hand as a result of non-steady thermal calculation using 4tΔ if the obtained
temperature values are abnormal or if the iteration does not reach convergence after more
than 20 times iterations which is set as the maximum value in the process of determining a
temperature-dependent materials properties by iteration calculation this result is discarded
and re-calculation is performed with the time step width of 230 times 4tΔ as a new 4tΔ This
230 times is the value determined empirically through sensitivity analysis This procedure
has proved to suppress the numerical instability of temperature calculation
JAEA-DataCode 2013-005
- 174 -
References 2
(21) Uchida M Otsubo N Models of Multi-rod Code FRETA-B for Transient Fuel Behavior Analysis(Final Version) JAERI 1293(1984)
(22) Ransom VH et al RELAP5MOD1 Code Manual Volume 1 System Models and
Numerical Methods NUREGCR-1826(EGG-2070) (1980)
(23) Dittus FW Boelter LMK Univ Calif Pubs Eng2 443(1930)
(24) Chen J A Correlation for Boiling Heat Transfer to Saturated Fluids in Convective Flow Process Design Developments 5(1966)
(25) Jens WH Lottes PA Analysis of Heat Transfer Burnout Pressure Drop and
Density Data for High Pressure Water ANL-4627(1951)
(26) Bjornard TA Griffith P PWR Blowdown Heat Transfer ASME Symposium on the Thermal and Hydraulic Aspects of Nuclear Reactor Safety Vol1(1977)
(27) Tong LS Weisman J Thermal Analysis of Pressurized Water Reactors American
Nuclear Society (1970) 76-HT-9(1976)
(28) Hsu YY Beckner WD A Correlation for the Onset of Transient CHF cited in LS Tong GLBennett NRC Water Reactor Safety Research Program Nuclear Safety 18 1 JanuaryFebruary(1977)
(29) Smith RA Griffith P A Simple Model for Estimating Time to CHF in a PWR
LOCA Transactions of American Society of Mechanical Engineers paper No76-HT-9(1976)
(210) Collier JG Convection Boiling and Condensation London McGraw-Hill Book
Company Inc(1972)
(211) SCDAPRELAP5MOD31 Code Manual Vol4 MATPRO-A A Library of Materials Properties for Light-Water-Reactor Accident Analysis NUREGCR-6150(1995)
(212) Gazarolli F Garde AM et al Waterside Corrosion of Zircaloy Fuel Rods
EPRI-NP 2789 (1982)
(213) MATPRO-09 A Handbook of Materials Properties for Use in the Analysis of Light Water Reactor Fuel Rod Behavior USNRC TREE NUREG-1005 (1976)
(214) Ohira K and Itagaki N Thermal Conductivity Measurements of High Burnup UO2
Pellet and a Benchmark Calculation of Fuel Center Temperaturerdquo Proc ANS Topical Meeting on Light Water Reactor Fuel Performance pp541-549 Portland USA March (1997)
JAEA-DataCode 2013-005
- 175 -
(215) Wiesenack W and Tverberg T Thermal Performance of High Burnup Fuel ndash In-pile Temperature Data and Analysisndash Proc 2000 Int Topical Mtg on LWR Fuel Performance Park City USA (2000)
(216) Robertson JAL kdθ in Fuel Irradiation CRFD-835 (1959)
(217) Ross AM and Stoute RL Heat Transfer Coefficient between UO2 and Zircaloy-2
CRFD-1075 (1962)
(218) Wilson MP Jr GA-1355 (1960)
(219) Hasegawa and Mishima Handbook of Materials for Nuclear Reactor p466 Nikkan Kougyou Shinbunsha (1977) [in Japanese]
(220) Une K Nogita K Kashibe S Toyonaga T and Amaya M Effect of Irradiation-
Induced Microstructural Evolution on High Burnup Fuel Behavior Proceedings to Int Topical Meeting on LWR Fuel Performance Portland USA pp478-489 (1997)
(221) Fujimoto N Nakagawa S Tuyuzaki N et al Verification of Plant Dynamics
Analysis Code for HTTR ldquoASURArdquo JAERI-M 89-195 (1989) [in Japanese]
(222) The Japan Society of Mechanical Engineers -JSME Data Book Heat Transfer 5th Edition- (2009)
(223) Speight MV A Calculation on the Migration of Fission Gas in Material Exhibiting
Precipitation and Re-solution of Gas Atoms under Irradiation Nucl Sci Eng 37p180 (1969)
(224) White RJ and Tucker MO A New Fission Gas Release Model
J Nucl Mater 118 pp1-38 (1983)
(225) Ham FS J Phys Chem Solids 6 p335 (1958)
(226) White RJ Fission Gas Release HWR-632 (2000)
(227) Hall ROA and Mortimer MJ Surface Energy Measurements on UO2 - A Critical Review JNuclMater148 pp237-256(1987)
(228) Turnbull JA Friskney CA Findlay FR Johnson FA and Walter AJ The
Diffusion Coefficients of Gaseous and Volatile Species during the Irradiation of Uranium Dioxide J Nucl Mater 107 pp168-184 (1982)
(229) Loumlsoumlnen P Modelling intragranular fission gas release in irradiation of sintered LWR
UO2 fuel JNuclMater 304 pp29-49 (2002)
(230) Itoh K Iwasaki R and Iwano Y Finite Element Model for Analysis of Fission Gas
JAEA-DataCode 2013-005
- 176 -
Release from UO2 Fuel JNuclSciTechnol 22 pp129-138 (1985) (231) Ainscough JB and Oldfield BW Ware JO Isothermal Grain Growth Kinetics in
Sintered UO2 Pellets J Nucl Mater49 pp117-128 (197374)
(232) Kogai T Modelling of fission gas release and gaseous swelling of light water reactor fuels JNuclMater 244 pp131-140 (1997)
(233) Matthews JR and Wood MH A simple Operational Gas Release and Swelling
Model IIGrain boundary gas JNucleMater91 pp241-256(1980)
(234) Van Uffelen P Parametric Study of a Fission Gas release Model for Light Water Reactor Fuel Proc 2000 Int Topical Mtg on LWR Fuel Performance Park City USA (2000)
(235) White RJ The Development of Grain Face Porosity in Irradiated Oxide Fuel
Seminar on Fission Gas Behavior in Water Reactor Fuels Cadarache France NEANSCDOC-20 (2000)
(236) Central Research Institute of Electric Power Industry Battelle High Burnup Effects
Program Task 3 Report Komae Research Laboratory Report T90802 (1990) [in Japanese]
(237) Walker CT Kameyama T Kitajima S and Kinoshita M Concerning the
microstructure changes that occur at the surface of UO2 pellets on irradiation to high burnup JNuclMater 188 pp73-79 (1992)
(238) Kitajima S and Kinoshita M Development of a Code and Models for High Burnup
Fuel Performance Analysis IAEA TCM on Water Reactor Fuel Element Modelling at
High Burnup and its Experimental Support Windermere UK Sept(1994)
(239) Nogita K and Une K Irradiation-induced recrystallization in high burnup UO2 fuel JNuclMater 226 pp302-310 (1995)
(240) Une K Nogita K Suzawa Y Hayashi K Ito K and Etoh Y Effects of grain size
and PCI restraint on the rim structure formation of UO2 fuels Proc IAEA Technical
Committee Meeting on Nuclear Fuel Behavior Modelling at High Burnup and its
Experimental Support Windermere UK (2000)
(241) Yan-Hyun Koo Je-Yong Oh Byung-Ho Lee Dong-Seong Sohn Three-dimensional
simulation of threshold porosity for fission gas release in the rim region of LWR UO2
fuel JNuclMater 321 pp 249-255 (2003)
(242) Cunningham ME Freshley MD and Lanning DD Development and Characteristics of the Rim Region in High Burnup UO2 Fuel Pellets JNuclMater 188 pp19-27 (1992)
JAEA-DataCode 2013-005
- 177 -
(243) Lassmann K Walker CT van de Laar J and Lindestroem F Modelling the high burnup UO2 structure in LWR fuel J NuclMater 226 pp 1-8(1995)
(244) Yan-Hyun Koo Je-Yong Oh Byung-Ho Lee Dong-Seong Sohn Three-dimensional
simulation of threshold porosity for fission gas release in the rim region of LWR UO2
fuel JNM 321 pp 249-255(2003)
(245) Kinoshita M Ultra High Burnup Characteristics of UO2 Fuel Proceedings to KNSAESJ Joint Seminar YongPyung Korea (2002)
(246) Billaux MR Sontheimer F Arimescu VI and Landskron H Fuel and Cladding
Properties at High Burnup Seminar on Fission Gas Behaviour in Water Reactor Fuels- Executive Summary Cadarache France NEANSCDOC(2000)20 (2000)
(247) Ikatsu N Itagaki N Ohira K and Bekker K Influence of rim effect on fuel center
temperature Proc IAEA Meeting on High Burn-up Fuel Specially Oriented to Fuel Chemistry and Pellet-Clad Interaction Sweden Sept (1998)
(248) Bakker K Kwast H and Cordfunke EHP Determination of a Porosity Correction
Factor for the Thermal Conductivity of Irradiated UO2 Fuel by Means of the Finite Element Method JNuclMater 226 pp 128-143 (1995)
(249) Chubb W Storhok VW and Keller DL Factors Affecting the Swelling of Nuclear
Fuel at High Temperatures Nucl Technol 18 pp231-255 (1973)
(250) Kosaka Y Thermal Conductivity Degradation Analysis of the Ultra High Burnup Experiment (IFA-562) HWR-341 (1993)
(251) Schrire D Kindlund A and Ekberg P Solid Swelling of LWR UO2 Fuel
HPR-34922 Enlarged HPG Meeting Lillehammer (1998)
(252) Hollowell TE The Development of an Improved UO2 Fuel Swelling Model and Comparison between Predicted Pellet Cladding Gaps and PIE Measured Gaps (Gap Meter Measurements) HPR-229 Enlarged Halden Programme Group Meeting on Fuel Performance Experiment and Evaluation Hanko vol1 (1979)
(253) Present RD Kinetic Theory of Gases Mc-Graw Hill NY p55 (1958)
JAEA-DataCode 2013-005
- 178 -
x y z yz zx xyσ σ σ τ τ τ
ε ε ε γ γ γx y z yz zx xy
3 Mechanical Analysis Model First the analysis based on the finite element method used in FEMAXI is explained
31 Solutions of Basic Equations and Non-linear Equations
The governing equations of elastic solid mechanics in analyzing the behavior of objects
are linear This implies the followings
(a) The equation relating strain to displacement is linear
(b) The equation relating stress to strain is linear
However there are many nonlinear phenomena among actual objects Accordingly to
address these phenomena it is necessary to expand the method of numerical calculation to
accommodate nonlinearity In this manner phenomena such as plasticity and creep or other
complex constitutive equations (the equation which expresses the relationship between stress
and strain) which encompass the entire range of solid mechanics can be replaced by simple
and linear elastic relationships This approximation is explained below
311 Basic equations
In FMAXI because constitutive equations are constructed on the basis of infinitesimal
deformation theory (model of small increment) the relational expression of
strain-displacement is linear The problem of nonlinearity occurs in the relational expression
of stress and strain To analyze this problem a general method of replacing the nonlinear
relationship between stress and strain with a linear relationship is discussed
(1) Basic assumption and formulation of mechanical analysis
In general a mechanical analysis of materials comes down to solve the problem to obtain
the following set of 15 components
1) three components of displacement u v w
2) six components of strains
3) six components of stress
Here equations and conditions to obtain these quantities are
1) boundary condition
2) equations of equilibrium in the three principal directions ( x y z or rθ z )
3) six equations of strain-displacement relation and
JAEA-DataCode 2013-005
- 179 -
10r rr rz R
r r z rθ θτ σ σσ τ
θpart minuspart part+ + + + =
part part part
10zrz z rz Z
r r z rθττ σ τθ
partpart part+ + + + =part part part
210r z r
r r z rθ θ θ θτ σ τ τ
θpart part part+ + + + Θ =part part part
0rr
r rθσ σσ minuspart + =
part
rr
u
rε part=
part1r uu
r rθ
θεθ
part= +part
zz
u
zε part=
part
[ ]1( ) pl cr sw
r r z TE θε σ ν σ σ α ε ε ε= minus + + + + +
[ ]1( ) pl cr sw
r z TEθ θε σ ν σ σ α ε ε ε= minus + + + + +
[ ]1( ) pl cr sw
z z r TE θε σ ν σ σ α ε ε ε= minus + + + + +
0r zθε ε ε+ + =
4) six stress-strain constitutive equations
In theory solution can be obtained by numerical calculation which satisfies all these
equations and conditions
These equations are shown below with respect to a cylindrical coordinate system
A group equations of equilibrium in the three directions
Here R Z and Θ are body force such as gravity inertia electromagnetic force etc
B group three equations of strain-displacement relation omitting another three equations
for shear strain
C group four constitutive equations of stress-strain relation omitting another four equations
for shear components
JAEA-DataCode 2013-005
- 180 -
The last equation implies the conservation (invariance) of volume of material Therefore
it holds in elastic plastic and creep deformations It does not hold in phenomenon which
includes irreversible volumetric change or concurrent expansion or shrinkage in the three
directions such as thermal expansion irradiation growth swelling and hot-pressing These
phenomena are dealt with by another method in FEMAXI
Here the approximation and hypothesis in numerical calculation by FEMAXI are as
follows
1) within a small region homogeneity of material and uniformity of temperature are
assumed
2) the body forces such as gravity inertia and electromagnetic force are neglected
3) geometrical symmetry is taken into consideration In a fuel rod axis-symmetry is
assumed
4) Either plane strain condition or plane stress condition is adopted case by case
(2) Incremental method First the idea of a typical [incremental method] is explained below In an incremental
interval linearity is assumed for the relational expression of stress and strain (relational
equation for stress increment-strain increment)
The equation for creep is an example of a nonlinear stress-strain relational equation The
general equation for creep cannot express the stress level explicitly by strain however strain
(or its increment) can be expressed explicitly by stress
In other words the equation
( ) tt
f Δsdotpartpart=Δ σε (311)
holds and in general ( )xf is nonlinear equation
Here Eq(311) is expressed as shown below in an incremental interval
tε αΔ = sdot Δ (312)
Here ( )nt
f σαpartpart=
nσ initial stress in the current increment interval (the stress
obtained in the preceding time step)
In the interval between time steps ( )σt
f
partpart
changes with the stress σ change and if
the incremental time tΔ is sufficiently small resulting change in stress is so small that
JAEA-DataCode 2013-005
- 181 -
approximation ( ) ( )nt
f
t
f σσpartpart=
partpart
can be used and the increase in strain can be determined
independent of change in stress using Eq(312) As shown linearity can be assumed between
the stress increment and strain increment in an incremental interval
Next the analysis of FEM using the incremental method is explained
(3) Outline of a solution using the incremental method First an outline of a FEM solution of the basic equation is presented when a linear
relationship holds approximately over an incremental interval This is a basic method in
elastic solid mechanics when linearity can be assumed
In the incremental method which is explained using the creep strain all strain increments
other than increments of elastic strain are represented as known strain and these are
represented by initial strain incremental vectors 0εΔ
Also denoting external force as F stress as σ and a matrix which connects the
strain ε in an FEM element to the displacement of node u as [ ]B
Then
[ ] B uε = (313)
holds This is equivalent to the ldquoB group three equations of strain-displacement relationrdquo
The nodal external force is set as F which is statically equivalent to the element
boundary stress In FEM element force exerts only at nodal points To formulate the FEM
matrix it is necessary to obtain F For this purpose an arbitrary virtual nodal displacement
is given and for this displacement the equilibrium is set between the work done to the
element by the force and the work done by the stress In other words the work by force and
the work by stress are set as equivalent Such a virtual displacement is set as ν
The virtual work FW done to the element node by the external force F is equal to the
sum of the products of each component of force and corresponding displacement
T
FW Fν= (314)
Similarly a virtual internal work per unit volume done by the stress is
[ ] TT
INW Bν σ= (315)
Assuming that the external work determined by integrating these two equations over entire
element volume V is equal to the internal work (the principle of virtual work) the next
equation is obtained
JAEA-DataCode 2013-005
- 182 -
[ ]
[ ]
TT T
V
TT
V
F B dV
B dV
ν ν σ
ν σ
=
=
(316)
Consequently by using the principle of virtual work the equations of equilibrium which
correspond to the above ldquoA group equations of equilibrium in the three directionsrdquo
[ ] T
VF B dVσ= (317)
is obtained Here since Eq(317) can hold at any stress-strain relationship irrespective of
the type of stress-strain relationship it allows FEM to be applied to not only the elastic
deformation of material but also plastic and creep deformation That is theoretically the
principle of virtual work is equivalent to both the equilibrium equations and mechanical
boundary conditions In other words in FEM work done to elements by the external force is
equal to the strain energy change inside elements In summary FEM assumes the ldquoprinciple of
virtual workrdquo which assumes that external work is equal to internal work
However any type of body forces inside elements are taken into account in the equation
formulation process in the code in terms of strains which are associated with volumetric
expansion or shrinkage induced by temperature or structure changes An initial strain and
initial stress are similarly dealt with in the code
Here considering the equilibrium condition (equilibrium equation) at (n+1) th time step
ie at time tn+1
[ ] ++ = dVBF nT
n 11 σ (318)
holds where
Fn+1 external force vector at the n+1 step and
σ n+1 stress vector at the n+1 step
To clearly distinguish an unknown quantity from a known quantity by separating σ n+1
into known stress σ n and unknown stress increments Δσ n+1 Eq (318) is changed to
[ ] [ ] F B dV B dVn
T
n
T
n+ += + 1 1σ σΔ (319)
[ ] [ ] B dV F B dVT
n n
T
nΔσ σ+ += minus 1 1 (3110)
In Eq(319) the right-hand side is a known value and the left-hand side is an unknown
quantity
Here the relationship between stress and strain (which correspond to the above ldquoC
JAEA-DataCode 2013-005
- 183 -
group four constitutive equations of stress-strain relation omitting another four equations for
shear componentsrdquo) is given by
[ ] 1e
n nDσ ε+ +Δ = Δ (3111)
Even if there are other strain components this relationship always holds Besides as a
result of calculation covering all strain components eg if plastic strain increment plasticεΔ
exists σΔ decreases and elastic strain eεΔ decreases also Plastic strain and creep strain are
quantities determined by σΔ andσ and they have no direct dependences on eεΔ
In Eq(3111) [ ]D is called a strain-stress matrix (stiffness matrix) and shows the
stiffness of an element This consists of Youngrsquos modulus and Poisonrsquos ratio indicating a
stiffness of element ie mechanical resistance against deformation
Here the following equation holds
Δσ n+1 stress increment vector from tn to tn+1
Δε ne
+1 elastic strain increment vector from tn to tn+1
[ ] ( )[ ] [ ]D D Dn n n+ += minus + =
θ θ θ θ1
1
21 and
[ ]Dn Stress-strain (stiffness) matrix at tn
Defining [ ]C matrix as an inverse matrix of [ D ] makes Eq(3111) become
[ ] Δ Δε σθne
n nC+ + +=1 1 (3112)
By substituting Eq(3111) into Eq(3110)
[ ] [ ] [ ] B D dV F B dVT
ne
n
T
nΔε σ+ += minus 1 1 (3113)
Since the following relationship holds among the elastic strain increment Δεne+1 total
strain increment Δεn+1 and initial strain increment as known quantity 01+Δ nε
10
11 +++ ΔminusΔ=Δ nnen εεε (3114)
By substituting Eq(3114) into Eq(3113)
[ ] [ ] [ ] minus=ΔminusΔ +++ dVBFdVDB nT
nnnT σεε 1
011 (3115)
Then transferring the known term [ ] [ ] +Δ dVDB nT 0
1ε to the right-hand side gives
[ ] [ ] [ ] [ ] [ ] minusΔ+=Δ +++ dVBdVDBFdVDB nT
nT
nnT σεε 0
111 (3116)
By substituting the strain-stress incremental relationship [ ] B uεΔ = sdot Δ into Eq(3116)
JAEA-DataCode 2013-005
- 184 -
[ ] [ ][ ] [ ] [ ] [ ]
1
01 1
T
n
T T
n n n
B D B dV u
F B D dV B dVε σ
+
+ +
Δ
= + Δ minus
(3117)
Hence the unknown quantity Δun+1 can be bundled out from the integral Eq(3117)
is the stiffness equation ie an equation presenting the relationship between displacement and
external force in the incremental method Also this equation can be expressed as
[ ] [ ] [ ] [ ] [ ]
T
T TH rel swl den crk c p HOT
B D B dV u F
B D dVε ε ε ε ε ε ε ε
Δ = Δ
+ Δ + Δ + Δ + Δ + Δ + Δ + Δ + Δ
(3118)
Finally by discretizing Eq(3117) into a sum of M elements
[ ] [ ][ ]
[ ] [ ] [ ]
11
01 1
1 1
MT
i i i i ni
M MT T i
n i i n i i n ii i
B D B V u
F B D V B Vε σ
+=
+ += =
Δ Δ
= + Δ Δ minus Δ
(3119)
( ΔVi volume of elementi M total number of elements)
Equation (3112) is a simultaneous linear equation By representing Eq(3112) in the
form of a matrix the unknown quantity of the displacement incremental vector Δun+1 can
be obtained
The above method is applicable when a governing equation can be linearly approximated
in incremental intervals However as explained in the case of creep the assumption (linear
approximation) of ( ) ( )nt
f
t
f σσpartpart=
partpart
is only applicable when stress variations are small
enough ie only when the stress increment σΔ is sufficiently small Therefore in creep
calculation restriction on the size of time increment becomes strong To resolve this problem
use of a more general constitutive equation containing nonlinear strain in FEMAXI is
explained in the next section and thereafter
312 Initial stress method initial strain method and changed stiffness method for non-linear strain
(1) Equation for stiffness As discussed in the preceding section 311-(1) non-linear problems can be converted
into linear elastic problems of small strain which may be formulated using the displacement
method by assuming linearity between stress increments and strain increments in intervals An
equation for stiffness which is expressed by Eq(3119) is obtained
[ ] 0=ΔminusΔ FuK (3120)
In linear elastic problems Eq(3120) is solved and usually the final form of the
JAEA-DataCode 2013-005
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equation is obtained
In formulating the above equation it is assumed that not only the linear equation of
strain-displacement continuity of displacement and balance of force but also the linear elastic
equation is satisfied [ ] ( ) 00 σεεσ Δ+ΔminusΔ=Δ D (3121)
Here 0εΔ initial strain increment
0σΔ initial stress increment
On the other hand in stress-strain equations which pose problems due to nonlinearity in
general assuming that the function ( )xF is nonlinear
( )σε F=Δ (3122)
Therefore by adjusting one or more of the parameters of [ ] 0 εΔD and 0σΔ
contained in Eq(3121) if Eqs(3121) and (3122) provide the same stress and strain as
solutions of Eq(3120) a nonlinear solution can be obtained To realize this iterative
calculation is required
(2) Iterative calculation In iterative calculation which of the three quantities discussed in (1) above should be
adjusted is determined by the following two conditions
A) solution used in equivalent linear elastic problems (method of solving nonlinear
problems by replacing them with equivalent linear elastic problems)
B) characteristics of physical principles defining the stress-strain relationship
When iterative calculation is performed after adjustment of the [ ]D matrix the process
is called the stiffness variation method When either 0εΔ or 0σΔ is adjusted the method
is called either the initial strain method or the initial stress method
In the stiffness variation method with respect to the specific behavior of materials in
terms of the relationship between stress and strain (for example creep behavior and plastic
behavior) the stiffness matrix [ ]D rewritten in the form of Eq(3112) is given as a function
of stress or strain level attained In other words considering the stiffness variation
[ ] [ ] [ ]εσ DDD == (3123)
the process can be applied In this case change in stress or strain alters the stiffness in the
stiffness matrix Therefore in the iteration to obtain convergence of the changes in stress and
JAEA-DataCode 2013-005
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strain solution of stress-strain of deformation is affected by the stiffness matrix thus it is
necessary to iterate until changes in the stiffness matrix converge
In other words the stiffness calculation method is a process which can be explained as
follows It is the method to obtain a deformation solution by adjusting (renewing) the material
constants which satisfy the nonlinear constitutive equation with a new material constant in the
final step after the adjustment in the calculation if it is possible to obtain a solution for a
nonlinear problem iteratively In the stiffness variation method in FEMAXI a method of
adjusting the stress-related material constants by iterative calculation is used
On the other hand in the initial strain method elastic strain increments obtained at each
step in the calculation are compared with strain increments corresponding to the constitutive
equation Eq(3119) the difference is used to calculate residual non-equilibrium force For
example in creep calculation by expressing the increment of creep strain and the increment of
elastic strain in separate forms the correction term for the initial strain increment in each step
can be directly obtained
The difference between the initial stress and initial strain methods is explained using Fig
311 In this figure the relationship between stress and strain obtained by the initial
approximation solution is shown by a Point 1 The solid line in the figure represents the real
behavior of the object described by an empirical equation and material equation In the initial
stress method the initial stress 01σΔ shown in the figure is introduced and the stress is
corrected to an appropriate level This is advantageous when strain sharply increases with
stress because the relationship of stress and strain is corrected using the small stress changes
01σΔ
εΔ
σ
εε
ε
σ
σ
)(a
)(b
1
Fig311 Initail-Strain method and Initial-Stress method (a)Hardening rate decreases with strain (b) Fastening material increases hardening rate with strain
JAEA-DataCode 2013-005
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However in the initial strain method strain is adjusted by the correction represented
by 01εΔ This method is advantageous for fastening materials in which stress sharply increases
with strain For the creep and plasticity of fuel pellets and claddings discussed in FEMAXI
the initial stress method proves better by comparing the initial strain method with the initial
stress method because strain changes sharply with increasing stress
However when the initial strain and initial stress methods are compared with the
stiffness variation method the latter method proves better in reducing the number of iterations
until convergence because the stress-strain relationship can be incorporated in the preparation
of the stiffness matrix In other words in the initial strain and initial stress methods the
convergence is calculated by renewing the load vectors using calculated stress and strain
On the other hand the stiffness variation method predicts changes in stiffness prior to
calculation thus it is equivalent in that the two processes carried out in the initial strain and
initial stress methods are completed in one calculation
Because of this basic equations are formulated using the stiffness variation method in
which the stiffness matrix [ ]D is given as a function of stress level attained (ie
[ ] [ ]σDD = ) in FEMAXI An outline of formulation in FEMAXI is described in the next
section
313 Solution of the basic equations in FEMAXI (non-linear strain) As described in the previous section (2) the stiffness variation method utilizing the stress
for convergence is adopted in the treatment of nonlinear strain in FEMAXI This method is
based on the formulation by the incremental method described in section (1) For example the
equation for creep is given by a relational equation between stress and strain increments as
shown in Eq (3114) which fits for the incremental method
(1) Classification and treatment of strain components Next strain used in FEMAXI code is examined The strain discussed in FEMAXI
includes the strain defined by an stress-strain relationship equation and the strain
independent of stress When the strain is given by the stress-strain equation if strain
changes stress changes and vice versa On the other hand strain independent of stress is the
component in which stress changes by the development of strain however even if stress
changes strain does not
First the strain discussed in FEMAXI is explained
JAEA-DataCode 2013-005
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cn
Pn
HPn
swn
denn
crkn
reln
thn
enn
111
1111111
+++
+++++++
Δ+Δ+Δ+
Δ+Δ+Δ+Δ+Δ+Δ=Δ
εεεεεεεεεε
(3124)
where Δε n+1 total strain incremental vector
Δε ne+1 elastic strain incremental vector
Δε nth+1 thermal strain incremental vector
Δε nrel+1 pellet relocation strain incremental vector
Δεncrk+1 pellet crack strain incremental vector
Δε nden
+1 pellet densification strain incremental vector
Δε nsw
+1 pellet swelling strain incremental vector
Δε nHP+1 pellet hot-press strain incremental vector
Δε nP+1 plastic strain incremental vector and
Δε nc+1 creep strain incremental vector
Of these the strain increments given by the stress-strain equation are en 1+Δε Δεn
crk+1
Δε nHP+1 Δε n
P+1 and Δε n
c+1 the strain-stress equation ( )σε f= is given by an
independent relational equation (material equation empirical equation and models) Of these the elastic strain increment is linear because ( ) σασε Δ==Δ fe The strain
incremental vectors which are independent of stress are denn
reln
thn 111 +++ ΔΔΔ εεε and sw
n 1+Δε
On the basis of this background let us write the relational equation between the
increment of elastic strain and the stress increment at time interval ( )Δt t tn n n+ += minus1 1
considering that [C] depends on temperature (depends on time)
[ ] e Cε σΔ = Δ (3125)
[ ] Δ Δε σθne
n nC+ + +=1 1 (3126)
where [ ] ( )[ ] [ ]
=+minus= ++ 2
11 1 θθθθ nnn CCC
Δε ne+1 elastic strain incremental vector
[ ]Cn+θ compliance (stress-strain) matrix
Δσ n+1 stress incremental vector
θ parameter for implicit solution
In addition to obtain stress increments from Eq(3126) it is necessary to obtain elastic
strain increments The elastic strain increment can be written as
JAEA-DataCode 2013-005
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1 1 1 1 1 1 1
1 1 1
e th rel crk den swn n n n n n n
HP P cn n n
ε ε ε ε ε ε ε
ε ε ε+ + + + + + +
+ + +
Δ = Δ minus Δ minus Δ minus Δ minus Δ minus Δ
minus Δ minus Δ minus Δ (3127)
The strain incremental vector which is independent of stress can be summarized as
follows considering it as the initial strain incremental vector
swn
denn
reln
thnn 1111
01 +++++ Δ+Δ+Δ+Δ=Δ εεεεε (3128)
Here 01+Δ nε initial strain incremental vector
Furthermore as will be discussed in section 317 in detail because the strain
incremental vector from the hot-press of a pellet can be incorporated into the plastic and creep
strain incremental vectors of pellet Eq(3127) can be written as follows
cHn
PHn
crknnn
en 111
0111 ++++++ ΔminusΔminusΔminusΔminusΔ=Δ εεεεεε (3129)
where
PHn 1+Δε plastic strain incremental vector including hot-press strain
cHn 1+Δε creep strain incremental vector including hot-press strain
It should be noted here that all PHn
crkn 11 ++ ΔΔ εε and cH
n 1+Δε are given by ( )σε f=Δ
and the function f is nonlinear
On the other hand the total strain can be related to nodal displacement as follows
[ ] 11 ++ Δ=Δ nn uBε (3130)
where Δun+1 is an incremental vector for nodal displacement
By substituting Eqs(3126) and (3130) into Eq(3129)
[ ] [ ] 01110
111 =Δ+Δ+Δ+Δ+ΔminusΔ +++++++cHn
PHn
crknnnnn uBC εεεεσθ (3131)
In Eq(3131) the treatment of PHn
crkn 11 ++ εΔεΔ and cH
n 1+εΔ which are the nonlinear strain
increments is explained below
Let us assume that the following equations are given to the nonlinear strain increment
1 1 1 2 1 3 crk PH cHn n nf f fε σ ε σ ε σ+ + +Δ = Δ = Δ = (3132)
Here σσ 21 ff and σ3f are known functions obtained from material parameters
empirical equations or tensile tests the relationship between stress and strain is known
JAEA-DataCode 2013-005
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(2) Determining stress by iteration method Here a method of determining σ by solving Eq(3132) is considered In this method
without obtaining an accurate value of stress σ the strain increment of PHn
crkn 11 ++ εΔεΔ and
cHn 1+εΔ cannot be calculated
For this reason the stiffness variation method is applied to obtain numerical solutions
The Newton-Raphson method (NR method) is applied to the iterative calculation to solve this
nonlinear equation
Here the general concept of the NR method is explained It is applied to solve the
following general nonlinear equation having a single variable x
ψ( x)=0 (3133)
In the NR method assuming that ix which is sufficiently close to the exact solution is
obtained while 0)( nexψ still holds the improved test solution 1+ix is obtained in the
following manner
1+ix = ix + Δ 1+ix
Here
i
ii
dx
dxx
minus=Δ + ψψ )(
1
Figure 312 illustrates an iterative calculation using a changing gradient as described above
In the case of applying of the NR method to Eq (3130) x=σ and each )(xψ is
0)()( 111 =Δminus= +crknf εσσψ (3134)
0)()( 122 =Δminus= +PHnf εσσψ (3135)
)(xψ
idx
d
ψ
ix1+ix
)( ixψ
1
Fig 312 Iteration method of the Newton-Raphson method using a changing gradient
JAEA-DataCode 2013-005
- 191 -
0)()( 133 =Δminus= +cHnf εσσψ (3136)
also
)( 11 σσψ fdd = (3137)
)( 22 σσψ fdd = (3138)
)( 33 σσψ fdd = (3139)
As is apparent the three equations from Eqs (3134) - (3136) include the unknown
values 1 1 1 crk PH cHn n nε ε ε+ + +Δ Δ Δ other than stress and the number of unknowns exceeds that
of the equations hence it is impossible to solve these equations simultaneously This implies
that the conditions of 0)( =σψ cannot be obtained The reason is that Eq(3131) is not in a
final form which allows the application of the NR method Therefore it is necessary to
proceed with the formulation
The following discussion is given on the basis of the application of the NR method to
iterative calculations for nonlinear analyses Here i represents the count of iterative
calculations in the NR method When the ith iterative calculation in the NR method is
completed (the variable value in the ith iteration is known ) and the i+1th iteration is being
performed strain displacement and stress other than 01+Δ nε and nσ change in response
to the iterative calculation Therefore Eq(3131) which is the equation of the NR method is
represented using superscripts to indicate the number of iterations i and i+1 as
[ ] ( )
1 11 1
0 1 1 11 1 1 1 1 0
i i in n n
crk i PH i cH i i in n n n n n n
C d B u
C
θ
θ
σ
ε ε ε ε σ σ
+ ++ + +
+ + ++ + + + + +
minus Δ
+ Δ + Δ + Δ + Δ + minus =
(3140)
Here d ni
ni
niσ σ σ+
+++
+= minus11
11
1
In addition the equilibrium condition Eq (3110) is rewritten as
[ ] [ ] V
T
ni
V
T
ni
nB dV B d dV F + ++
++ =σ σ1 11
1 (3141)
In Eqs(3140) and (3141) because the unknown values are the values at the i+1th
iteration they are d niσ ++
11
Δεncrk i+
+1
1 1
1+
+Δ iPHnε 1
1+
+Δ icHnε and Δun
i++
11
Here the
conditional equations to obtain 0)( =σψ in the NR method Eqs(3134) (3135) and
(3136) are given below
0)()( 1111 =Δminus= +
+icrk
nf εσσψ (3142)
0)()( 1122 =Δminus= +
+iPH
nf εσσψ (3143)
0)()( 1133 =Δminus= +
+icH
nf εσσψ (3144)
JAEA-DataCode 2013-005
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By this approach because five equations Eqs(3140)-(3144) are obtained the
unknown values 11
++
indσ Δε n
crk i+
+1
1 1
1+
+Δ iPHnε 1
1+
+Δ icHnε and Δun
i++
11
can be determined To
complete the convergence process by the NR method the addition of an equilibrium
conditional equation (balanced equation between external force and internal stress) described
by Eq(3141) is necessary
Incidentally with respect to the method of correcting the elastic matrix by the stiffness
variation method in FEMAXI suitable formulation algorithms for each characteristic of the
material behavior model are explained after section 32 Fig 313 shows the algorithm for
the treatment of the entire calculation of deformation Alteration of the contact boundary
conditions and conditions required for evaluation of yield and unloading will be explained
again in a later section The evaluation of yield and unloading is performed after completion
of the convergence calculation by the NR method in the algorithm presented in Fig 313
The number of iterative calculations in the
Newton-Raphson method is given by LMAX (The standard value is 20)
However after three or more iterations when the maximum amount of total correction
(the difference in magnitude between the preceding and current results) is 001 or less
iterations are terminated by assuming that the convergence is complete
Model parameter LMAX
JAEA-DataCode 2013-005
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Preparation of nonlinear strain increments
iPHn
1ˆ +Δε and icH
n
1ˆ +Δε
Preparation of elastoplastic creep coefficient matrix [ ]pD
The intercept and contact planes are obtained in the Newton-Raphson method on the basis of the quantity of the state (stress strain) at stage i
For each element Preparation of residualcorrection load inF 1+Δ
from stress and strain
Preparation of stiffness matrix [ ]θ+nK from coefficient matrix
Preparation of entire stiffness matrix and load by summing the contribution from each element The residual load is added to the total load
Solving the stiffness equation
[ ] in
in
in FuK 1
11 +
+++ Δ=Δθ
To confirm whether contradictions developed in the contact boundary conditions (anchoringslipping) based on displacement and load obtained (if there is a contradiction the contradiction should be corrected and recalculations carried out)
The corrections for strain and stress increments 11
++
indε and 1
1++
indσ are
obtained from the solution 11
++Δ i
nu and the estimated values 11
++
inε and
11
++
inσ at the i+1th iteration are obtained
The parameters of the nonlinear process (hardening rule of creeping) are renewed
Evaluation of convergence by the Newton-Raphson method
Evaluation of contact boundary conditions (contact or noncontact) based on calculated displacement and load and evaluation of yield and unloading based on stress and plastic strain increments obtained
Fig313 Algorithm and flow of mechanical analysis in FEMAXI
JAEA-DataCode 2013-005
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32 Mechanical Analysis of Entire Rod Length
First an overview of the total flow is shown in Fig321 before each step is described
Fig321 Flow chart of the Global FEM mechanical analysis
Time Step Start
Contact state judgment at contact node settled
Displacement converged
Break the present time step at the instant of state change Stress and strain are interpolated into the values at the instant
Yielding unloading or contact-non-contact state change occurred
Calc of stress-strain distribution
Proceed to next Time Step
Set pellet and cladding contact state and boundary conditions at nodal points
Calculation of matrix for all the elements completed
Calc of nodal displacement by solving the total stiffness equation
Set initial strains at each element such as densification swelling and thermal expansion
Calc of material matrix of each element
Incorporation of stiffness matrix and nodal vector in each element
JAEA-DataCode 2013-005
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(Note) The 1-D analysis over the entire rod length (ERL) is performed in the
routine FEMROD while the analysis of 2-D local PCMI deformation of rod is
performed in other routines The 2-D analysis can be performed concurrently
with the 1-D analysis by using the input parameter IFEMRD
IFEMRD=0 1-D ERL and 2-D local analyses and IFEMRD=11-D ERL only
321 Finite element model for entire rod length analysis (ERL)
(1) Geometry
As shown in Fig322 the analysis model includes a 1-dimensional axis-symmetrical
system in which the entire length of the fuel rod is divided into axial segments and each
segment is further divided into concentric ring elements in the radial direction In this
system the stress-strain analysis is performed using the finite element method with
quadrangular elements with four degrees of freedom as shown in Fig324
Fig322 Geometry of FEMAXI model
U P PER PL ENUM
S E GMENT( M)
S E GMENT( M- 1 )
S E GMENT( 2 )
S E GMENT( 1 )
L O WER PL ENUM
Pellet
Cladding
Clad Pellet
JAEA-DataCode 2013-005
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(2) Ring elements Fig323 shows the ring element formation of one axial segment in ldquoMESH=3rdquomode
This is the same as Fig231
To help understand this geometry a simplified FEM element structure is shown in
Fig324 in which the number of ring element of pellet stack is 10 and the number of ring
elements in cladding is 4 In this figure dummy space elements for dish and chamfer are
shown which is described in section 322 The inner region three elements 13 14 and 15 of
cladding are metallic part and the outer one element 16 is oxide layer (ZrO2) The outer
surface oxide element is re-meshed to increase in accordance with the growth of oxide layer
(refer to chapter 2 section 22)
Also nodal coordinates in one ring element are shown in Fig325
Uz2
Lz9Lz3
Lz2Lz1
PELLET(10ring-elements) CLADDING
r12
Uz3Uz1
Uz10
r1 r2 r3 r10 r11
Lz10
r13 r14 r15 r16
Lz11Lz12
Lz13Lz14
Uz11Uz12
Uz13 Uz14Uz9
Mz1Mz2
Mz3
Mz9Mz10
Fig324 Ring element of FEM for one axial segment
00 02 04 06 08 10
ZrO2
1 2 3 4 5 6 7 8
Cladding
ring element number
Pellet stack 36 equal-volume ring elements
Relative radius of pellet
Bonding layerCenterline
Fig323 Ring elements geometry for one axial segment
JAEA-DataCode 2013-005
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(3) Degree of freedom of elements
As shown in Fig 324 and Fig325 each node has one degree of freedom in either the
radial or axial direction On the assumption that the axial displacements of all nodes are
identical which is the generalized plane strain condition a degree of freedom of one element
is used as a representative value for degrees of freedom in the axial direction of pellet stack
1 10L Lz z - 1 10
U Uz z- and in the axial direction of cladding 11 14L Lz z- and 11 14
U Uz z - as shown
below
( )( )
( )( )U
CUUUU
LC
LLLL
Up
UUU
Lp
LLL
zzzzz
zzzzz
zzzz
zzzz
====
====
====
====
14131211
14131211
1021
1021
(321)
That is it is assumed that the bottom plane at each segment keeps the same elevation for
both pellet stack and cladding and that the top plane of pellet stack and that of cladding have
a freedom to displace independently from each other in the axial direction with the two top
planes being kept flat
This is required by the characteristics of FEM where ri and ri+1 have a freedom in the
radial direction solely and ZiL and ZiU have a freedom in the axial direction solely It should
be noted that this applies only to the mechanical analysis over the entire length of a fuel rod
Generally in FEM displacement and strain are continuous at nodes and boundaries of each
element while stress is not continuous However derivatives of displacement and strain are
not always continuous
(4) Number of digitized elements of pellet stack in the radial direction
Although this is already explained in Table 231 in chapter 2 it is presented here again
for its importance
ri ri+1
ZiU
ZiL
Fig325 Quadrangle four nodes element with
degree of freedom (ri ri+1 ZiL ZiU)
JAEA-DataCode 2013-005
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As Table 231 has shown the name-list parameter MESH can specify the number of ring
elements in the pellet stack Default value is MESH=3 In the iso-volumetric elements
thickness of ring decreases insomuch as the outer region of pellet stack When ISHAPE=1
the number of elements doubles in the case of MESHgt0 which allows a finer calculation
(5) Number of ring elements of cladding
The 1-D thermal analysis shares the ring elements of cladding with the 1-D ERL
mechanical analysis irrespective of the designated value of ldquoMESHrdquo
That is in the 1-D thermal and mechanical analyses
A) Cladding without Zr liner 8 metallic elements + 2 outer surface oxide elements
B) Cladding with Zr liner 2 Zr-liner elements + 8 metallic elements + 2 outer surface
oxide elements
Also in the 2-D local PCMI mechanical analysis which is later described
C) Cladding without Zr liner 4 metallic elements + 1 outer surface oxide elements
D) Cladding with Zr liner
1 Zr-liner element+ 4 metallic elements + 1 outer surface oxide elements
(6) Cladding oxide layer The volume expansion of the oxide during each time step is added as an axial strain
increment to the oxide layer element The fraction of volume increment X is
X=PBR ndash 10 (322)
Table 231 Number of radial ring elements of pellet stack in thermal and mechanical analyses Parameter value
1-D thermal analysis 1-D mechanical analysis
2-D local PCMI analysis
MESH=0 Iso-thickness 10 ring elements
Iso-thickness 10 ring elements
Iso-thickness 5 ring elements
MESH=1 Iso-volumetric 36 ring elements Iso-volumetric
18 ring elementsIso-volumetric 9 ring elements
MESH=2 Iso-volumetric 72 ring elements
MESH=3 Iso-volumetric 36 ring elements Iso-volumetric
36 ring elementsIso-volumetric 18 ring elements
MESH=4 Iso-volumetric 72 ring elements
JAEA-DataCode 2013-005
- 199 -
and X is assumed to be distributed in the radial direction by PX () and in the axial direction
by 100-PX () respectively PX is given as an input parameter and PBR is the Piling
Bedworth ratio
Accordingly the metal phase thickness decreases and is re-meshed with the increase in
thickness of the oxide layer Mechanical characteristics such as Youngrsquos modulus thermal
expansion coefficient creep rate and irradiation growth rate naturally differ between the metal
phase and oxide layer generating an internal stress in the metallic substrate of cladding
mainly in the axial direction affecting the mechanical behavior of the entire cladding This
stress is included in the mechanical analysis However since no measurement data on the
creep rate and irradiation growth rate of ZrO2 has been obtained creep in the oxide layer is
neglected and the irradiation growth rate is tentatively set as that of Zircaloy multiplied by a
factor RX (input parameter default value is 10) in the calculation
322 Determination of finite element matrix Here various matrices of the finite element method will be determined
(1) Derivation of strain-displacement matrix [B]
Denoting the radial displacement as u and axial displacement as ν and setting ir 1ir+ Miz
Uiz and
Liz as respectively pre-determined values of radial coordinate and axial
coordinate of the nodes of ring element ie fixed values given by input displacement (u v) at
an arbitrary point inside an element shown in Fig324 can be written as
111
1+
++
+
minusminus
+minusminus
= iii
ii
ii
i urr
rru
rr
rru (323)
( )L
M Lii iM L
i i
z z
z zν ν νminus= minus
minus
or ( )M
U Mii iU M
i i
z z
z zν ν νminus= minus
minus (324)
In axis-symmetric geometry the strain component inside the element is
[Input Parameter PBR and PX] Piling-Bedworth ratio is given by PBR Default value is 156 Default value of PX is 800
JAEA-DataCode 2013-005
- 200 -
[ ] r
z
ur
u B ur
z
θ
partpartε
ε εε partν
part
= = equiv
(325)
Using Eq(323) and Eq(324)
( ) ( ) 111
1
111
11
+++
+
+++
minusminus
+minusminus
=
minus+
minusminus=
iii
ii
ii
i
iii
iii
urrr
rru
rrr
rr
r
u
urr
urrr
u
partpart
(326)
Here by putting 1
2i ir r
r ++= as central-point coordinate of the element we obtain
( ) ( )
1 11
11 1
1 1
11 1
2 2
2 21 1
i i i ii i
i ii i i i
i i i i
i ii i i i
r r r rr ru
u ur r r rr r r r r
u ur r r r
+ ++
++ +
+ +
++ +
+ +minus minus= ++ +minus minus
= ++ +
(327)
M Li iM Li iz z z
partν ν νpart
minus=minus
or U Mi iU Mi iz z z
partν ν νpart
minus=minus
(328)
By substituting Eqs(326) (327) and (328) into Eq(325) we obtain
1 1
1
1 1
1 10 0
1 10 0
1 10 0
ii i i i
r i
i
ii i i i L
i
z iM
M L M L ii i i i
ur r r r
u
r r r r
z z z z
θ
ε
εν
εν
+ +
+
+ +
minus minus minus = + +
minus minus minus
Or
1 1
1
1 1
1 10 0
1 10 0
1 10 0
ii i i i
r i
i
ii i i i M
i
z iU
U M U M ii i i i
ur r r r
u
r r r r
z z z z
θ
ε
εν
εν
+ +
+
+ +
minus minus minus = + +
minus minus minus
(329)
Consequently we obtain the following as components of the matrix[B](shape function)
JAEA-DataCode 2013-005
- 201 -
[ ]
minusminusminus
++
minusminusminus
=++
++
Li
Mi
Li
Mi
iiii
iiii
zzzz
rrrr
rrrr
B
1100
0011
0011
11
11
or [ ]
minusminusminus
++
minusminusminus
=++
++
Mi
Ui
Mi
Ui
iiii
iiii
zzzz
rrrr
rrrr
B
1100
0011
0011
11
11
(3210)
Eqs(329) and (3210) are used in all the FEM solutions of the ERL (entire rod length)
mechanical analysis for both the linear and non-linear deformations
The volume of each element is
( )( )Li
Miiii zzrrV minusminus= +
221π
or ( )( )Mi
Uiiii zzrrV minusminus= +
221π (3211)
323 Derivation of stress-strain (stiffness) matrix of pellet and cladding
We obtain strain-stress matrix [C] The stress-strain relationship is
[ ] Δ Δσ ε= D e (3212)
where [D] is the stress-strain matrix that represents the stiffness of an element
When matrix [C] is defined as an inverse matrix of [D] Eq(3212) is expressed as
[ ] Δ Δε σe C= (3213)
Accordingly matrix [C] is given by
[ ]
minusminusminusminusminusminus
=1
1
11
νννννν
EC (3214)
JAEA-DataCode 2013-005
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where E is Youngs modulus and ν is the Poisson ratio
Here solution of the basic constitutive equations for stress-strain analysis is outlined
The external force and initial strain vector are set as F and 0εΔ respectively When
the calculation has proceeded to the n-th time step and calculation for the next (n+1)-th time
step is performed the equilibrium equation
[ ] F B dVn
T
n+ += 1 1σ (3215)
is given using the principle of virtual work
Here
Fn+1 external force vector at (n+1)th time step
σ n+1 stress vector at (n+1)th time step
To distinguish unknown variables from the known we divide σ n+1 into known stress
σ n and unknown stress Δσ n+1 then we obtain transformed equations of Eq(3215) as
[ ] [ ] F B dV B dVn
T
n
T
n+ += + 1 1σ σΔ (3216)
[ ] [ ] B dV F B dVT
n n
T
nΔσ σ+ += minus 1 1 (3217)
The right-hand side of Eq(3217) consists of known values and the left side consists of
unknown values Substituting Eq (3212) into Eq (3217) we obtain
[ ] [ ] [ ] B D dV F B dVT
ne
n
T
nΔε σ+ += minus 1 1 (3218)
Here the following equation holds among elastic strain increment Δε ne+1 total strain
increment Δε n+1 and the known value initial strain increment 01+Δ nε
0111 +++ ΔminusΔ=Δ nn
en εεε (3219)
Accordingly substituting Eq (3219) into Eq (3218) we obtain
[ ] [ ] [ ] minus=ΔminusΔ +++ dVBFdVDB nT
nnnT σεε 1
011 (3220)
Transposing the known value term [ ] [ ] +Δ dVDB nT 0
1ε to the right-hand side
[ ] [ ] [ ] [ ] [ ] minusΔ+=Δ +++ dVBdVDBFdVDB nT
nT
nnT σεε 0
111 (3221)
JAEA-DataCode 2013-005
- 203 -
is obtained Substituting Eq (325) into Eq (3221) we obtain
[ ] [ ][ ]
[ ] [ ] [ ] 1
01 1
T
n
T T
n n n
B D B dV u
F B D dV B dVε σ
+
+ +
Δ
= + Δ minus
(3222)
and the unknown value Δun+1 can be calculated by integration
Namely expressing Eq(3222) in a discrete form with summation of M elements
[ ] [ ] [ ]
[ ] [ ] [ ]
11
01 1
1 1
MT
i i i i ni
M MT T i
n i i n i i n ii i
B D B V u
F B D V B Vε σ
+=
+ += =
Δ Δ
= + Δ Δ minus Δ
(3223)
( ΔVi represents the volume of the element i and M represents total number of elements)
Thus the unknown displacement increment vector Δun+1 is expressed using known values
The above is the basic calculation method In actual calculations a more complex
solution is required due to the use of implicit solutions for creep and plasticity
In the next section modification of the basic equations for the purpose of implicit
solutions will be described
(1) Details of pellet stack elements
In a cylindrical shape pellet solid or hollow thermal stress is induced by temperature
gradient in the radial direction which generates a strong thermal stress inside pellet and
sometimes a ridging deformation of cladding occurs by pellet hour-glass shape expansion
(refer to chapter 35) In actual fuel rods these thermal deformations are markedly mitigated
by the dish and chamfer spaces However a simple solid cylinder stack model which neglects
the dish and chamfer spaces would generate numerically a far stronger thermal compressive
stress in the central region of pellet stack and contrarily a stronger tensile stress in the outer
region This is caused by the generalized plane strain condition adopted in FEMAXI in which
pellet upper plane keeps a flat plane in displacing in the axial direction (Refer to section
321 (2))
To avoid these virtual thermal stresses which would be induced by the model
characteristics and at the same time to keep the generalized plane strain condition dish or
chamfer space can be specified by input in the upper part of the pellet stack by designating a
parameter IDSELM=1 Namely a one pellet stack of segment can consist of three elements
JAEA-DataCode 2013-005
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This is illustrated in
Figs326 and 327 Fig326
shows a pellet stack geometry
of one segment The number of
ring elements is set 10 for
simplicity of explanation
Here while actually a
pellet stack of ldquoone segmentrdquo
length consists of several or a
few tens of pellets in the
present model the yellow
region represents the pellet
solid part and the other region
corresponds to the dish
chamfer and inter-pellet
aperture spaces of all the pellets
that consist of this segment
In other words Fig326 is
a geometry which takes into
account of the effects of dish and chamfer spaces on the
solid cylinder model These ldquovoid space elementsrdquo are given
a very weak stiffness which is shown in the figure to act as
a buffer for the thermal expansion or swelling of pellet
Fig327 is the 7th element which has neither dish nor
chamfer out of the elements in Fig326 Here the upper
non-hatched part is the inter-pellet (IP) space and lower
hatched part is the pellet solid element The division ratio of
this element is that the upper part is BUFSP() of the
segment length and the lower part (100 - BUFSP) of the
segment length and BUFSP is given by input Namely in
this element the segment is divided at BUFSP (100 - BUFSP) ratio
The IP space element is presented in Fig326 as a uniform thickness zl -spaced element
over the dish and chamfer spaces and zl is initially given by input as BUFSP () of the
Fig327 Pellet ring element
without dish and chamfer
MiZ
ir
UiZ
LiZ
1ir +
centerline
Outer surface
Fig326 FEM element division corresponding to the pellet stack of one axial segment
M3Ζ
M7Ζ
Chamfer space
Dish space
U1Ζ U
2Ζ U3Ζ
U4Ζ
U5Ζ U
6Ζ U7Ζ U
8Ζ U9Ζ U
10Ζ
M10Ζ
M1Ζ
M4Ζ
M5Ζ
M2Ζ
L1Ζ L
2Ζ L3Ζ L
4Ζ L5Ζ L
6Ζ L7Ζ L
8Ζ L9Ζ L
10Ζ
M6Ζ
M8Ζ
M9Ζ
Pellet-pellet gap space which is assumed to have a fixed thickness z lz ie BUFSP () of stack length of one axial segment
JAEA-DataCode 2013-005
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segment length That is the pellet shape matrix in the ERL mechanical analysis is composed
with reference to the initial coordinate value of the segment baseline and the IP thickness is
fixed at zl in the axial direction
Here for the pellet stack of one segment space element depth of dish (or chamfer) in the
axial direction is specified in accordance with the ratio of depths of dish and chamfer In
Fig326 depth of dish element is calculated by [ one pellet dish depthtimes2timesnumber of pellets
in one segment] which determines Uiz
Miz and
Liz as initial conditions
The axial length of the chamfer space is similarly determined The concave region and
gap shown in Fig326 indicate the spaces which take into account of this shape feature
They are ldquobuffer spaces with weak spring stiffnessrdquo which allows to calculate a realistic
thermal stress and strain
It should be noted that the axial displacement of the upper coordinate of pellet solid part Miz is calculated in each ring element while the axial shear stress exerting between the
adjacent elements is not taken into account in the present 1-D mechanical model Also the
displacement of the top plane of segment follows the generalized plane strain condition which
is defined by the uniform displacement as U Uiz vΔ = and ( )1 2
U U U Ui pz z z z= = = = =
(Refer to Eq(321))
That is in the calculation of the total matrix the displacement of each solid ring element
of pellet in the axial direction is determined as ( )M Li iv vminus where L
iv is equal to Uv of the
lower adjacent segment Accordingly it is the same in all the ring elements of a segment
(Refer to section (1)) Further
U Uiz vΔ = is determined by the force which is obtained by summing up the
spring forces of each ring element calculated by the concave space depth and axial gap
stiffness (or spring constant) of all ring elements for the values of ( )M Li iv vminus of each ring
element
Also Uv includes implicitly the effect of axial forces which are induced by the PCMI
apart from the above spring forces That is Uv is determined by incorporating all these
interactions among the forces into the matrix
Default value of BUFSP is 10() It is assumed as an uniform thickness of space elementwith BUFSP of the stack length ofone axial segment
Adjustment parameter
Default value of IDSELM is 0 Model parameter IDSELM (used only in mechanical
JAEA-DataCode 2013-005
- 206 -
(2) Assumption of mechanical properties of elements consisting of pellet stack The calculation method by this element geometry is specifically described In Fig326
when the pellet solid element is expanding by thermal expansion or swelling this element can
increase its volume almost freely in the axial direction until the dish space (concave void) is
fully filled and during this expansion the deformation calculation can be performed with the
original mechanical properties of pellet in both the tensile and compressive stress state
Next when the dish space shown in Fig326 is filled the subsequent expansion in the
axial direction fills the inter-pellet gap space elements In this situation stiffness of the gap
space elements is recovering to the original property of pellet so that the solid elements are no
longer able to expand or shrink freely in the axial direction For the chamfer elements an
identical numerical process is conducted
However there is some situation where the spaces of dish and chamfer elements can be
recovered when power is falling and pellet is shrinking During this process stiffness of the
spring is decreasing reversely in accordance with the depth recovery of the inter-pellet gap
space elements
(3) Change of the stiffness (spring constant) in the axial direction The above stiffness change of the dish (chamfer) and inter-pellet gap spaces is modeled
by a single spring in the axial direction for the [dish + gap space] [chamfer + gap space] and
[gap space] However the stiffness recovery of pellet stack element by crack healing is
modeled in another way which will be described in section 324 Here the spring stiffness
in the axial direction is treated by the spring constant change as follows
When the gap space is assumed to be completely replaced with the solid pellet the spring
constant zk of the gap space element is given as
z
zz l
SEk = (3224)
where zE is Youngrsquos modulus of pellet S is initial cross sectional area normal to the center
axis of pellet of the dish or chamfer elements and if the pellet has neither dish nor chamfer
S is initial cross sectional area normal to the center axis of pellet zl is the depth (thickness)
of the gap space element which is uniform in the radial direction as explained in the previous
section Since value of zl remains constant zk is affected only by temperature dependence
of zE and if zE remains unchanged zk keeps a constant value
Next the spring constant zdk which is shared by the dish element and gap space
JAEA-DataCode 2013-005
- 207 -
element (or the chamfer element and gap space element) in calculation is required to take into
account of the stiffness change or recovery which is caused by the decrease or increase of the
dish (chamfer) space elements due to pellet expansion or shrinkage
Here depth (thickness) value of dish (or chamfer) in the axial direction of a certain ring
element i at a certain time step is set as zd il with the upper surface level of inter-pellet gap
space element being a baseline and downward direction is set as positive
A) The spring constant is set as
5 10zd zk k FAC FAC minus= sdot = (3225)
during the period in which dish (chamfer) space is not filled (ie z zd il llt ) This means that the
spring constant zdk through the dish (chamfer) + gap space elements has a very small value
Here it is noted that the dish could become deeper than the initial state depending on
irradiation condition
B) When expansion still continues ie zd i zl lle after the dish or chamfer space is filled
completely the thickness of gap space is decreasing and the spring constant zsk of the dish
(+chamfer) + gap space is approaching zk in accordance with Eq(3226) as the expansion
amount z zd il l lΔ = minus entering the gap space is coming closer to zl
zs zz
lk k
l
Δ= sdot (3226)
C) For the ring elements of pellet which have no dish (chamfer) as described in Figs326 and
327 the elements are entering directly the gap space Accordingly a similar equation to
Eq(3226) expresses the change of spring constant as follows
( )zzs z zd z
z
lk k k k FAC
l
primeΔ= sdot ge = sdot (3227)
where ( )z zl lprimeΔ le is the amount of expansion in the axial direction In the case of shrinkage
zs zd zk k k FAC= = sdot holds
(4) Derivation of stress-strain matrix (stiffness matrix)[D]
Stress-elastic strain relation of each ring element of pellet is expressed as
[ ] enn D 11 ++ = εσ (3228)
[ ]D is as stated in section 31 called a strain-stress matrix ie stiffness matrix which
indicates the mechanical stiffness of elements Defining [ ]C matrix as an inverse matrix of
this [ ]D matrix gives
[ ] 111 +++ = nnen C σε (3229)
JAEA-DataCode 2013-005
- 208 -
Here [ ]C matrix is presented by Eq(3214)
324 Pellet cracking
(1) Model of pellet cracking and stiffness recovery The concept of crack model in the mechanical analysis is described with the schematic
shown in Fig328(14)(110) When a pellet is in a tensile stress in the i direction (axial
circumferential or radial direction) due to thermal dilatation it is assumed that cracks are
generated in the orientation perpendicular to the i direction and the apparent Youngrsquos modulus
in the i direction decreases to approximately 1100 of the original material property value
(Since the Youngrsquos modulus occupies a diagonal position in the FEM matrix of the
detailed mechanical analysis the value cannot be 0)
When pellet is in a compressive stress due to PCMI it is assumed that the elasticity of
the pellet partly recovers in accordance with the decrease in relocation strain That is in the
initial state ie burnup=0 power=0 andε εi iominus = 0 Youngrsquos modulus Ei of pellet is a small
value represented by Ec
StrongInteraction
W eakInteraction
NoInteraction
Ec
E
ε eminusε rel
Fig 328 Stiffness model of pellet with cracks
However FEMAXI does not deal with the pellet cracking in an explicit way It is
assumed that a pellet stays in a cylindrical continuum and when i direction (axial hoop or
radial direction) is in a tensile stress state crack occurs in the plane normal to the stress and
the Youngrsquos modulus in the i direction becomes very small ie about 1100 of the original
JAEA-DataCode 2013-005
- 209 -
value
Also in a compressive stress state it is assumed that the elastic stiffness recovers
depending on the relocation strain That is in an initial state ie no burnup no power and
ε εi iominus = 0 the Youngrsquos modulus Ei of pellet takes a small value Ec because pellet fragments
can move freely in the radial direction
Change of pellet stiffness with increasing power is modeled as follows
A) When pellet is not in contact with cladding thermal stress in pellet is significantly
relieved by the cracks accordingly the pellet can expand almost at its original expansion rate
and the gap is narrowed
B) When pellet is in contact with cladding the compressive strain inside the pellet increases
by the restraint of cladding Then the crack void space generated by relocation is
compressed and stiffness of the pellet increases This process is modeled by linearly changing
the Youngrsquos modulus Namely it is the process in which a mechanical interaction is gradually
enhanced between a cracked and relocated pellet and cladding
C) Next elastic stiffness of pellet completely recovers to the original value ie compressive
stiffness when fragments of cracked pellet are compressed by cladding and fill the void space
which was originally generated by relocation In this situation a strong mechanical interaction
between pellet and cladding is supposed
Youngrsquos modulus of a pellet with cracks is set by ECRAC3 The default value of ECRAC3 is 2times109 (Nm2) which is approximately 1100 of original Youngrsquos modulus of UO2
Fitting parameter ECRAC3
The pellet stiffness model shown in Fig 326 is for the case
IYNG = 1 (default) When IYNG=1 the region between minusεrel and 0 is described by a straight line When IYNG=0 this region is described by a concave quadratic function
Model parameter IYNG
(2) Relationship between relocation and stress-strain Based on the above-described model the relationship between stress and strain of pellet
concerning the relocation is described
The apparent Youngrsquos modulus in each direction Ei(Er Ez Eθ)is defined as follows
JAEA-DataCode 2013-005
- 210 -
( )
minus=lt
ltltminus+minusminus
=lt
=
reli
ei
ei
reliccrel
i
ei
eic
i
E
EEE
E
E
εε
εεεε
ε
0
0
(3230)
where eiε elastic strain in i direction
ε irel initial relocation strain in i direction (input data)
Ec effective Youngrsquos modulus of pellet with cracks in a tensile state
Then sum of the elastic strain eiε in i direction and relocation strain r
iε in i direction is
introduced as
ri
eii εεε += (3231)
In Eq(3231) when 0=eiε rel
iri εε = holds and when rel
iei εε minus= 0=r
iε That is the
relocation strain is ε irel when stress is null ( 0=e
iε ) and when reli
ei εε minus= pellet stiffness
completely recovers and the relocation strain becomes null Accordingly in the elastic strain
range 0ltltminus ei
reli εε the relocation strain changes in the range rel
iireli εεε ltltminus
Fig329 Schematic of relationship between relocation strain and stress in cracked pellet
Elastic region
Elastic strain
Relocation strain
-εrel
εrel
σ
εi
minusσR
JAEA-DataCode 2013-005
- 211 -
In the range relii
reli εεε ltltminus shown in Fig329 since change of the apparent Youngrsquos
modulus is expressed by a linear equation in Eq(3230) and given by εσ dd the
stress-strain relationship derived by integrating the equation of εσ dd is a quadratic
function
Therefore the stress change in relii
reli εεε ltltminus is set as a quadratic function of strain
cba iii ++= εεσ 2
(3232)
Differentiating Eq(3232) gives
bad
dE i
i
ii +== ε
εσ
2 (3233)
Here coefficients a b and c have to be determined Since the boundary condition is that at relii εε = 0=iσ at rel
ii εε = ci EE = and at relii εε minus= EEi = Eq(3232) gives
02 =++ cba rel
ireli εε (3234)
From Eq(3233)
creli Eba =+ε2 (3235)
Eba reli =+minus ε2 (3236)
holds Solving Eqs(3234) (3235) and (3236) gives
reli
ccreli
c EEc
EEb
EEa ε
ε 4
3
2
4
+minus=
+=
minusminus= (3237)
Therefore Eq(3232) becomes
reli
ci
cirel
i
ci
EEEEEE εεεε
σ4
3
242 +
minus+
+minus
minus= (3238)
In Fig329 the Y-axis intercept value of the curve of Eq(3234) is reli
ci
EE εσ4
3+minus=
and it is found that the stress at reli iε ε= minus is rel
ici EE εσ )( +minus= when the Youngrsquos modulus
recovers completely
Characteristics of the present model is to derive the stress-strain relationship including
the pellet stiffness change due to cracking by introducing the equation of stress vs [elastic
strain + relocation strain] not by introducing the equation of stress vs elastic strain
325 Crack expression in matrix As described previously the FEMAXI mechanical model consistently assumes a
cylindrical continuum for pellet and the crack formation is expressed approximately by the
decrease in stiffness of the ring elements of pellet stack That is further direct division of ring
elements of the cracked pellet is not performed
JAEA-DataCode 2013-005
- 212 -
A crack strain increment Δε ncrk+1 which is proportional to the change in Δσ is given
by transforming the elements of [C] matrix which is in Eq(3213) from Eq(3214)
The strain-stress matrix of the pellet [ ]~C is represented as
[ ]~C
E E E
E E E
E E E
r
z
=
minus minus
minus minus
minus minus
1
1
1
ν ν
ν ν
ν νθ
(3239)
Here Er Youngrsquos modulus in the radial direction (Pa)
Ez Youngrsquos modulus in the axial direction (Pa)
Eθ Youngrsquos modulus in the circumferential direction (Pa)
E Youngrsquos modulus of pellet (material property) (Pa)
ν Poissonrsquos ratio (minus)
The apparent Youngrsquos modulus in each direction Ei (Er Ez Eθ) is defined as follows
( )
0
00
0
0
0
c i i
reli ii c c i i irel
i
reli i i
E
E E E E
E
ε ε
ε ε ε ε εε
ε ε ε
lt minus= minus= minus minus + minus lt minus lt minus ltminus =
(3240)
Here ε i strain in i direction
0iε initial strain in i direction
ε irel initial relocation strain in i direction (input data)
Ec effective Youngrsquos modulus of pellet with cracks in a tensile state
From Eq (3213) [ ] Δ Δε σθn
en nC+ + +=1 1 (3241)
and setting the sum of elastic strain increment and crack strain increment as
[ ] Δ Δ Δε ε σθne
ncrk i
n niC+ +
++ +
++ =1 11
11 ~
(3242)
we obtain the crack strain increment vector using Eqs (3241) and (3242) as
JAEA-DataCode 2013-005
- 213 -
[ ] [ ] Δ Δε σ
ν ν
ν ν
ν ν
ν ν
ν ν
ν ν
θ θ
θ
ncrk i
n n ni
r
z
C C
E E E
E E E
E E E
E E E
E E E
E E E
++
+ + ++= minus
=
minus minus
minus minus
minus minus
minus
minus minus
minus minus
minus minus
11
11
1
1
1
1
1
1
~
=
minus
minus
minus
+
minus
++
++
++
+
+
+
Δ
Δ
Δ
Δ
Δ
Δ
σσσ
σ
σ
σ
θ
θθ
r ni
z ni
ni
rr ni
zz ni
ni
rE E
E E
E E
E E
11
11
11
1
1
1
1 1
1 1
1 1
1 1
minus
minus
++
++
++
d
E Ed
E Ed
r ni
zz ni
ni
σ
σ
σθ
θ
11
11
11
1 1
1 1
(3243)
By this process the unknown quantity Δε ncrk i+
+1
1 in Eq(3140) ie crack strain
increment is expressed by the unknown quantities 1 1
1 1i ir n z nσ σ+ +
+ +Δ Δ and 1
1i
nθσ ++Δ and one
unknown quantity is eliminated Here Eq(3243) indicates the process of obtaining the crack
strain increment and is incorporated into the total matrix by Eq(3242) That is it is
formulated by the relationship between the elastic and crack strain increments and stress
increment as shown in Eq(3242)
Substituting Eq(3243) which is resulted as the elimination process above into
Eq(3140) gives
[ ] ( ) ( )
1 1 01 1 1
11 1
1 11 1 1 0
i i in n n n
i i i in n n n n
P i c i i in n n n n
C d B u
C C d
C
θ
θ θ
θ
σ ε
σ σ σ
ε ε σ σ
+ ++ + + +
++ + + +
+ ++ + + +
minus Δ + Δ
+ minus + minus
+ Δ + Δ + minus =
and finally it is converted into
[ ] [ ] [ ] ( )~ ~ C d B u Cni
ni
ni
n nP i
nc i
ni
ni
n+ ++
++
+ ++
++
+ +minus + + + + minus =θ θσ ε ε ε σ σ11
11
10
11
11
1 0Δ Δ Δ Δ (3244)
Fitting parameters FACR and FACZEi = E holds when rel
iii εεε minus=minus 0 however these values are adjusted using
parameters FACR in the circumferential direction and FACZ in the axial direction as ε ε εi i i
relminus = minus sdot0 FACR and ε ε εi i irelminus = minus sdot0 FACZ respectively
The default values of FACR and FACZ are both 1
JAEA-DataCode 2013-005
- 214 -
Fitting parameters FRELOC and EPSRLZ
Relocation strain εrel in the radial direction is given by the value which is calculated from the initial radial gap width multiplied by RELOC and divided by the pellet radius
In the axial direction relocation strain εrel is directly set by EPSRLZ The default values of FRELOC and EPSRLZ are 05 and 0003 respectively
Youngrsquos modulus of cracked pellet is specified by
ECRAC3 Default value is 2 109 2times N m about 1100 of the original Youngrsquos modulus of UO2
Fitting parameter ECRAC3
Default value of IYNG is 10 When
IYNG=10 in the pellet stiffness change model shown in Fig327 minusε rel and 0 is connected by
a straight line When IYNG=0 minusε rel and 0 is connected by a quadratic curve which is convex downward
Model parameter IYNG (used only in mechanical analysis)
It is assumed that a pellet can recover its stiffness up to EtimesEFAC if the pellet has an original Youngrsquos modulus E EFAC is used in only ERL mechanical analysis In the 2-D local PCMI analysis described in chapter 35 EFAC is not a fittting parameter It is fixed at 10
Fitting parameter EFAC
326 Definition equations of equivalent stress and strain in FEMAXI Definitions are shown of equivalent stress and strain which have an important role in
understanding the description of non-linear mechanics after this
(1) Equivalent stress A general form of the equivalent stress or effective stress σ or yield function h of
pellet and anisotropic cladding where shear stress components are assumed to exist in the
axial and radial directions only and three principal stresses are always identical to the hoop
axial and radial stresses
( ) ( ) ( ) ( )
( )
2 22 2
12 2
36
2
3
r z z r zr
r z
h H F GF G H θ θ
θ
σ σ σ σ σ σ σ τ
α σ σ σ
= = minus + minus + minus + + +
+ + +
(3245)
where H F G anisotropy coefficients (in the pellet H=F=G=1 is assumed)
α pellet hot-press parameter of pellet (α =0 is assumed for cladding)
Here α is the hot-press parameter which is explained in section 327 In Eq(3245) if
isotropy is assumed 10F G H= = = so that in ERL mechanical analysis (IFEMRD=1) we
have
JAEA-DataCode 2013-005
- 215 -
( ) ( ) ( ) ( )2222 32
1zrrzzr σσσασσσσσσσ θθθ +++minus+minus+minus= (3246)
and in LCL (local PCMI) mechanical analysis (IFEMRD=0)
( ) ( ) ( ) ( )22222 362
1zrzrrzzr σσσατσσσσσσσ θθθ ++++minus+minus+minus=
(3247)
(2) Equivalent elastic strain in isotropic material In ERL mechanical analysis (IFEMRD=1)
( ) ( ) ( )05
2 2 22 1
3 2e r z z rθ θε ε ε ε ε ε ε = minus + minus + minus (3248)
In local PCMI analysis (IFEMRD=0)
( ) ( ) ( )05
2 2 2 22 1 3
3 2 4e r z z r rzθ θε ε ε ε ε ε ε γ = minus + minus + minus + (3249)
(3) Plastic strain increment (identical formulation for creep strain increment) In ERL mechanical analysis (IFEMRD=1)
2 2 22( ) ( ) ( )
3pl pl pl pl
r zd d d dθε ε ε ε = + + (3250)
In local PCMI analysis (IFEMRD=0)
( )2 2 2 2 22 1( ) ( ) ( ) ( ) ( )
3 2pl pl pl pl pl pl
r z r zd d d d d dθε ε ε ε γ γ = + + + + (3251)
Also total equivalent plastic strain is the sum of the increments from the one at the first time
step to that at the n-th time step as
1
npl pl
ii
dε ε=
= (3252)
Anisotropy parameters H F G for
pellet cladding (SUS) pure Zr and ZrO2 are respectively H0(1) F0(1) G0(1)H0(2)
F0(2) G0(2)H0(3) F0(3) G0(3) and H0(4) F0(4) G0(4) Default values are all 10
Anisotropy parameter H0(4) F0(4) G0(4)
JAEA-DataCode 2013-005
- 216 -
327 Hot-pressing of pellet (1) Phenomenon and equation
Volume compression of pellet by hydrostatic pressure ie hot-pressing is a result of
increasing density due to crushing the void space inside pellet by hydrostatic pressure(31) The
following description is applied to both the ERL analysis and local PCMI analysis
The equivalent stress and yield function of pellet is as follows neglecting the shear stress
components as they are ineffective for hot-pressing in Eq(3235) for the ERL analysis
(IFEMRD=1)
( ) ( ) ( ) ( )
( )
2 22
12 2
3
2
3
r z z r
r z
h H F GF G H θ θ
θ
σ σ σ σ σ σ
α σ σ σ σ
= minus + minus + minus + +
+ + + equiv
(3253)
holds where
hyield function (von Mises function)
σ equivalent stress
H F G anisotropy factors though a pellet is isotropy and H=F=G=1
α hot-pressing parameter (in cladding α=0)
Eq(3243) is added by the term
( )2
23 27
3r z
r zθ
θσ σ σα σ σ σ α + + + + =
In FEMAXI it is assumed that the hot-pressing strain is proportional to plastic and creep
strains Accordingly it is convenient to formulate the hot-pressing strain as being included in
those of plastic and creep strains Thus by giving the equivalent stress with Eq(3253) the
creep strain increment including hot-pressing is obtained In this primeσ is a deviatoric stress
vector including hot-pressing which is given by
( )( )( )
( )
( )
( )
22
322
2 23
2 22
3
r zr z
r ST r z
z rz ST r z r z
ST r zr z
r z
θθ
θ
θθ θ
θ θ θθ
σ σ σ α σ σ σσ σ α σ σ σ
σ σ σσ σ σ α σ σ σ α σ σ σσ σ α σ σ σ σ σ σ α σ σ σ
minus minus + + + minus + + + minus minus prime = minus + + + = + + + minus + + + minus minus + + +
(3254)
Here the term 3r z
STθσ σ σσ + += in Eq(3254) represents a hydrostatic stress
JAEA-DataCode 2013-005
- 217 -
If the material is metal ldquoα=0rdquo in Eq(3254) has and the deviatoric stress is
2
32
03
2
3
r z
r ST
z rz ST
STr z
θ
θ
θθ
σ σ σ
σ σσ σ σσ σ σ
σ σ σ σ σ
minus minus
minus minus minus prime = minus = = minus minus minus
(3255)
In this condition the equivalent stress σ =0 holds and no plastic strain is generated
However in the solid material such as pellet which has gas pores and voids situation is
different For example if there remain voids in pellet after sintering process even in the
condition that the deviation stress is null the voids are pressed to collapse by a compressive
static pressure ( 0)STσ lt which results in pellet shrinkage and volume change ie permanent
strain generation The rate of this shrinkage can be assumed proportional to the creep strain
rate and plastic strain rate Also the shrinkage rate is considered proportional to the static
pressure ( 0)STσ lt so that as shown in Eq(3253)
2
273
r z θσ σ σα + +
is added inside the
square root of equivalent stress formula (31)
Here by considering the plastic flow rule in the iterative calculation of the nth time step
1 1P Pn n
n θ
σε εσ+ +
+
part Δ = Δ part (3256)
holds Assuming the isotropy of pellet material ( )H F G N= = = =1 3 from Eq(3253)
( )3 22
2 3r z
r zr
θθ
σ σ σ σ α σ σ σσ σ
part minus minus = + + + part (3257)
holds Accordingly
1 3 05 3 05 3
11 3 05 3
1 3n
n n Symθ
θ θ
α α ασ α α σσ σ
α+
+ +
+ minus + minus + part = + minus + part
+
(3258)
By substituting Eq (3258) into Eq (3256) and rearranging the result
( ) 1
11 3
1
1
P P
PP P Pnn n
n P P
v v
v v
v vθ θ
θ
α εε σσ
++ +
+
minus minus+ Δ Δ = minus minus
minus minus
(3259)
where v P = minus+
0 5 3
1 3
αα
(3260)
JAEA-DataCode 2013-005
- 218 -
This equation holds in the 2-D local PCMI analysis discussed later
The volume strain increment Δε h nP +1 due to the hot-pressing during the plastic
deformation process is formulated by
( ) ( )( )
( )
Δ Δ Δ Δ
Δ
Δ
ε ε ε ε
α εσ
σ σ σ
α εσ
σ σ σ
θ
θ
θ
h nP
nP r
nP z
nP
nP
Pr z
nP
r z
v
+ + + +
+
+
= + +
=+
minus + +
= + +
1 1 1 1
1
1
1 31 2
9
(3261)
Furthermore Eq (3261) is also applied to the volume strain increment Δε h nc
+1 due to
the hot-pressing during the creep advancement so that similarly
( )Δ Δε α εσ
σ σ σθh nc n
c
r z ++= + +119
(3262)
Accordingly the overall volume change (=hot-pressing strain) Δε h n +1 accompanied by
plastic and creep deformation is given by
( )( )
Δ Δ Δ
Δ Δ
ε ε εα
σε ε σ σ σ θ
h n h nP
h nc
nP
nc
r z
+ + +
+ +
= +
= + + +
1 1 1
1 1
9 (3263)
The method to determine the plastic and creep strains is explained in the next part
(2) Parameters for hot-pressing The name-list parameters to control the hot-pressing calculation are explained
A) BETAX
BETAX is a coefficient correlating the theoretical density ratio and the hot-press
parameter α Value of α depends on the optional parameters IHOT and IHPOP
1) For IHOT=0 α is fixed as α = BETAX (Applied to the ERL mechanical analysis)
2) For IHPOP=0 α is fixed as follows
0=α (non-contact period) α = BETAX (contact period)
(Applied to the LCL mechanical analysis)
3) For IHOT=1 (or IHPOP=1) α at the n+1 time step is defined as a
function of porosity as follows
11
1
BETAX ( )
0 ( )
nn
i
n
D DD D
D D
D D
α
α
++
+
minus= ltminus
= ge (3264)
JAEA-DataCode 2013-005
- 219 -
Here D theoretical density ratio (-) when the hot pressing is completed
1 11n nD p+ += minus
pn+1 porosity inside the pellet
iD initial theoretical density ratio (-)
In this case α advances asymptotically to 0 if hot-pressing and densification advance and
porosity decreases however it does not assume a negative value
B) IHOT An option used to adjust α in the ERL mechanical analysis Default value=0
C) IHPOP An option used to adjust α in the local PCMI mechanical analysis Default
value=0
D) FDENH Theoretical density ratio D(-)when hot-pressing is completed This is applied
to cases when IHOT=1 or IHPOP=1 Default value=10
328 Supplementary explanation of incremental method
When speaking of ldquoelastic strain incrementrdquo does it mean ldquoa strain which includes
non-linear strains such as plastic and creep strains but can be approximated as a linear strain
in an incremental methodrdquo For some users this ldquoelastic strain incrementrdquo might imply that
the matrixes such as Eqs(328) (329) and (3217) are shown first and after that these
matrixes are also applied to plastic and creep calculation
However equations such as Eq(3217) which use the Poissonrsquos ratio cannot hold for
plastic and creep deformation Therefore a method is explained below how to treat plastic and
creep strain increments apart from the elastic deformation
Here an additional explanation is given first with respect to the calculation of elastic
strain increment
(1) Stiffness equation in incremental method The stiffness equation in incremental method is as described in chapter 31
[ ] [ ] [ ]
[ ] [ ]
T
TH rel swl den
T
crk c p HOT
B D B dV u F
B D dVε ε ε ε
ε ε ε ε
Δ = Δ
Δ + Δ + Δ + Δ + + Δ + Δ + Δ + Δ
(3265)
Eq(3264) can be transformed into
JAEA-DataCode 2013-005
- 220 -
[ ] [ ]
[ ] [ ]
T
TH rel swl den
T
crk c p HOT
B D dV F
B D dV
ε
ε ε ε ε
ε ε ε ε
Δ = Δ
Δ + Δ + Δ + Δ + + Δ + Δ + Δ + Δ
(3266)
where the terms in the right-hand side are
THεΔ thermal strain increment
relεΔ relocation strain increment
swlεΔ swelling strain increment
denεΔ densification strain increment
crkεΔ cracking strain increment
cεΔ creep strain increment
pεΔ plastic strain increment
HOTεΔ hot-pressing strain increment
The strain in the left-hand side is total strain increment
e TH rel swl den
crk c p HOT
ε ε ε ε ε ε
ε ε ε ε
Δ = Δ + Δ + Δ + Δ + Δ
+ Δ + Δ + Δ + Δ (3267)
By substituting Eq(3266) into Eq(3265)
[ ] [ ] T eB D dV FεΔ = Δ (3268)
is obtained In Eq(3267) according to the equilibrium equation
[ ] TF B dVσΔ = Δ (3269)
holds so that we obtain
[ ] [ ] [ ] T TeB D dV B dVε σΔ = Δ (3270)
and [ ] eDσ εΔ = Δ finally Eq(3111) is obtained
Here in Eq(3268) if 0FΔ = 0eεΔ = holds
In an actual application when no differences are generated among non-linear strain
increments
0FΔ = (3271)
holds and
0eεΔ = (3272)
holds
However when some difference is generated among non-linear strain increments
JAEA-DataCode 2013-005
- 221 -
loading is produced depending on the difference of strains among elements That is 0FΔ ne
and elastic strain increment takes place Because this FΔ value is not known it is necessary
to solve the stiffness matrix of Eq(3266)
Here in the following relationship
[ ] eDσ εΔ = Δ (3273)
a strain determining the stress is called an elastic strain irrespective of taking account of the
other strain components
(2) Initial strain increment In the initial strain incremental method the following eight components are given at the
start of every time step to solve Eq(3266)
TH rel swl den crk cε ε ε ε ε εΔ Δ Δ Δ Δ Δ p HOTε εΔ Δ
These components are referred to as initial strain increments However the strain
increments which can be defined at the start of time step ie strain change only as a function
of temperature and burnup etc are limited to the following four components
TH rel swl denε ε ε εΔ Δ Δ Δ (3274)
Accordingly since the strain increments which can be defined at the start of time step are the initial strain increments the initial strain increment 0εΔ is set as
0 TH rel swl denε ε ε ε εΔ = Δ + Δ + Δ + Δ (3275)
For the unknown strain increments at the start of time step ie stress-dependent
components crk c p HOTε ε ε εΔ Δ Δ Δ (3276)
determination of these components needs some numerical device so that in the following text
its solution method is explained sequentially
By substituting Eq(3275) into Eq(3267) we obtain an elastic strain increment vector
as Δ Δ Δ Δ Δ Δε ε ε ε ε εn
en n n
crknc
np
+ + + + + += minus minus minus minus1 1 10
1 1 1 (3277)
where HOTεΔ is included in cεΔ and pεΔ
Δε n+1 total strain increment vector
Δε n+10 initial strain increment vector
JAEA-DataCode 2013-005
- 222 -
Δε ncrk+1 pellet cracking strain increment vector
Δε nc
+1 creep strain increment vector
Δε np+1 plastic strain increment vector
The strains by thermal expansion densification swelling and relocation are all collectively
treated as the initial strain In the next section 33 the numerical solution method for these
non-linear strains is explained
JAEA-DataCode 2013-005
- 223 -
33 Method to Calculate Non-linear Strain 331 Creep of pellet and cladding
The creep equation of cladding and pellet is generally represented as
( ) ε σ ε φc Hf T F= (331)
Here
ε c equivalent creep strain rate (1s)
σ equivalent stress (Pa)
ε H creep hardening parameter (minus)
T temperature (K)
φ fast neutron flux (nm2sdots)
F fission rate (fissionm3sdots)
The variation rate of creep hardening parameter ε H is represented as
( ) ε σ ε φH Hf T F= (332)
Eq(331) is usually an equation of creep under uniaxial stress When this equation is
generalized for multi-axial stress state the creep strain rate vector ε c is expressed as a
vector function of stress and creep hardening parameter When this vector function is set as
β ε c is represented as
( ) ε β σ εc H= (333)
where Tφ and F are omitted because they can be dealt with as known parameters
When a calculation at time tn is finished and calculation in the next time increment
Δtn+1 is being performed the creep strain increment vector is represented as
Δ Δε ε β σ εθ θ θnc
n nc
n nHt+ + + + += =1 1
(334)
where
( )
( )σ θ σ θ σ
ε θ ε θεθ
θ
n n n
nH
nH
nH
+ +
+ +
= minus +
= minus +
1
1
1
1
(here 0≦ θ ≦1)
When θ = 0 is adopted the above equation represents the initial strain method This
method adopts the creep rate which is determined by state quantity which has known value at
the starting point of an increment In this case the calculation procedure is simple but the
solution tends to be numerically unstable therefore it is known that the time increment must
JAEA-DataCode 2013-005
- 224 -
be extremely small at high creep rate
In contrast when θ ne 0 is adopted the calculation procedure becomes complex since the
equation contains unknown values however as θ approaches 1 stability of the numerical
solution improves this method of setting θ ne 0 is called an implicit solution
In FEMAXI θ = 1 (complete implicit method) is set to put emphasis on the importance
of numerical stability
Now in a calculation performed from tn to tn+1 when the (i+1)th iteration by the
Newton-Raphson method is being performed after completion of the i-th iteration (see
Eqs(3134) to (3136)) the creep rate vector can be expressed as follows
Since the function form of each principal direction component is identical the creep
Equation (331) is represented as follows using a flow rule and Eq(333)
ε ε σσ σ
σc c d
d
f=
= prime3
2 (335)
Here σ is an equivalent stress which is defined by Eq(3245)
332 Method of creep strain calculation Before developing equations following Eq(335) this section gives an additional
explanation on the relationship between creep strainplastic strain and elastic strain in the
FEMAXI calculation
(1) Principal stress directions and creep strain increment The creep and plastic strain increments in each principal stress direction are determined
in proportion to the magnitude of deviatoric stress following the Prandtle-Reuss flow-rule
An absolute magnitude of the equivalent strain rate is determined by stress temperature creep
strain hardening etc In other words a positive creep strain relative to average stress occurs in
the direction of tensile stress and a negative creep strain relative to average stress in the
direction of compressive stress
At the same time a positive elastic strain is generated in the tensile stress direction and a
negative elastic strain in the compressive stress direction so that the sign of elastic strain
coincides with its principal stress direction The elastic strain occurs in the direction which
【Note】The Prandtle-Reuss flow rule also holds for the creep strain components
(increments) Since creep strain increment is c ct tε εΔ = Δ the flow rule can be
applied to creep strain rate too Refer to the next section 332
JAEA-DataCode 2013-005
- 225 -
enables the strain to accommodate the difference from other strain components and to
decrease the total strain difference which is induced by the displacement difference between
adjacent elements Here ldquodifference from other strain componentsrdquo denotes the difference of
non-elastic strains between adjacent elements in FEM
(2) Difference of strain components Displacement which corresponds to total strain is provided by materials stiffness That is
the total strain differences among all the elements that constitute an entire system are
controlled by the stiffness of each element The component which directly depends on
stiffness is elastic strain whereas the statement ldquothe total strain differences are providedrdquo
holds because the total strain difference includes elastic strain In FEM stress analysis of
FEMAXI displacement increment is pursued and therefore the total displacement increment
is pursued
The elastic strain increment in this total strain increment is provided by the stiffness of
element It follows that in the adjacent elements the element which has a larger non-elastic
strain components than the other element generates a negative elastic strain (compressive
strain) component while the element which has a smaller non-elastic strain components than
the other element generates a positive elastic strain (tensile strain) component Here extent of
elastic strain is determined by the external force imposed on element boundary and by the
magnitude relation of non-elastic strains between adjacent elements The ldquoexternal force
imposed on boundaryrdquo is represented by a loading (pressure difference across the cladding or
contact force in PCMI) in cladding Namely when this loading is exerted on some nodes of
element as external force stress is generated within the element and elastic strain is induced
Also ldquothe magnitude relation of non-elastic strains between adjacent elementsrdquo indicates
a magnitude relation obtained in comparing the element deformations For example when an
inner element has a larger thermal strain (one of the non-elastic strain components) and an
outer element has a smaller thermal strain the inner element expansion is restrained by the
outer element resulting in a negative stress (compressive elastic strain) in the inner element
and vice versa in the outer element
(3) Stress relaxation by creep and strain difference When creep strain occurs as a result that the creep strain is generated in the same
direction of that of the elastic strain ie the two kinds of strains are made to have the same
sign by the flow rule stress is relieved and elastic strain is mitigated Nevertheless the
JAEA-DataCode 2013-005
- 226 -
difference of initial strain components which are other than the elastic and plastic strains
between the adjacent elements remains unchanged
For example four types of strains in cladding elastic creep plastic and initial (=
thermal expansion and radiation growth) strains are considered When creep occurs the elastic
strain is mitigated and stress induced by external force and thermal expansion is relieved This
means that the strain difference of elements consisting of cladding decreases On the other
hand plastic strain will not decrease in general when once generated unless the direction of
stress is reversed and also such stress direction reversal is hardly occurs in fuel rod
Meanwhile during the calculation in one time step whether plastic strain increment is
generated or not is determined by not only the elastic strain increment but also the relation
between creep strain increment and stress The other types of strains are determined by
temperature and burnup etc independently from the stress Therefore during the calculation
in one time step in some cases creep strain mitigates elastic strain relieves stress and
decreases plastic strain increment while the creep strain does not decreases the initial strains
For example the elastic strain increase induced by PCMI stress is mitigated by the creep in
the radial direction of cladding though the plastic strain which has been generated by the
previous time step or thermal strain are not decreased
(4) Method of creep calculation An important point in calculation is how much the creep deformation is generated ie to
evaluate justly the equivalent creep strain rate The equivalent creep strain rate has a high
sensitivity to stress Accordingly to evaluate the rate precisely the code has sufficiently
elaborated its numerical process to predict unknown stress change In the present section
332 and in the next section ldquo333 Plasticityrdquo explanation is given particularly for the
numerical prediction method of unknown stress change Also another explanation is given on
the concept and numerical elaboration method of ldquocreep strain hardeningrdquo for this concept is
one of the restraining factors on the equivalent creep rate change However even if the creep
equation does not include the strain hardening term no problem arises in the FEMAXI
calculation
The essential of creep calculation is to perform a partition evaluation of the strain
induced by stress change between the newly generated creep strain and elastic strain (and
plastic strain if yielding occurs) Needless to say this new creep strain is dependent not only
on strain hardening but also on stress change
JAEA-DataCode 2013-005
- 227 -
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Here we return to the explanation of numerical process By Eq(335) in the (i+1)
iteration during the n-th time step the following equation holds
( ) ( )( )
( ) ( )
ε β σ θ σ ε θ ε
σ σ θ σ ε θ ε
σ σ θ σ
σ σ θ σ
θ θ θ
θ θ
θ
θ
nc i
ni
ni
nH i
nH i
ni
ni
nH i
nH i
ni
ni
ni
ni
d d
f d d
d
d
++
+ ++
+ ++
+ ++
+ ++
+ ++
+ ++
= + +
=+ +
+
times prime +
111
11
11
11
11
11
3
2 (336)
Creep strain rate depends on stress temperature fission density fast neutron flux and
creep strain hardening etc Among these factors the rate equation is formulated assuming that
temperature fission density and fast neutron flux are known
( ) 1 1 11 1c i i i H i H i
n n n n nd dθ θ θε β σ θ σ ε θ ε+ + ++ + + + += + + (337)
is obtained Here the unknown quantities are 11
indσ +
+ and 11
H indε +
+ Since the creep strain rate
obeys the flow rule
( )( ) ( ) ( )
1 11 1 1 1
111
1 11
3
2
3
2
i i H i H in n n nc i i i
n n ni in n
i in ni
n
f d dd
d
f
θ θθ θ
θ
θ θθ
σ σ θ σ ε θ εε σ σ θ σ
σ σ θ σ
σσ
+ ++ + + ++ +
+ + +++ +
+ ++ ++
+
+ +prime= times +
+
prime=
(338)
holds
Eq(338) can be approximated through the first-order Taylor expansion with respect to the
unknown 11
indσ +
+ and 11
H indε +
+ as follows
( ) ( )
( )
( )( )
( ) 11
11
11
11
111
2
3
2
3
2
3
2
3
2
3
2
3
+++
++
++++
+
+++++
+
++++
+++
+
++
+++
+
prime
part
part+
primepartpart+
primepartpartprime+
prime
partpart+prime=prime
iHn
in
i
n
H
H
in
in
in
Hini
n
in
in
in
Hini
n
in
in
Hini
n
in
Hini
n
in
ini
n
df
df
df
dfff
εθσε
εσσ
σθσεσσσ
σθσσ
σεσσ
σθσεσσσ
σεσσ
σσ
θθθ
θθθ
θθθθ
θθθ
θθθ
θθθ
(339)
Then
JAEA-DataCode 2013-005
- 228 -
( )
1 112
1 11 1
11
3 3
2 2
3 3
2 2
3
2
ic i i i i i in n n n n ni i
nn n
i i ii i i i
n n n ni in n nn n
ii H i
n ni Hnn
f f d
ff d d
fd
θ θ θ θ θθθ θ
θ θθ θ θθ θ
θθθ
partσε σ σ θ σσ partσσ
partσ part partσθ σ σ θ σσ partσ σ partσ partσ
partσ θ εσ partε
+ ++ + + + + +
++ +
+ ++ + + +
+ + ++ +
++ +
++
prime prime= minus
prime prime+ +
prime+
is derived and re-arranging this gives
1 11
1 11 1
3
2
3 3
2 2
i iic i c i inn n ni i
n nn n
i iii i H inn n ni i H
n nn n
f fd
f fd d
θθ θ
θ θθ θ
θθ
θ θθ θ
part partσε ε σ θ σσ partσ σ partσ
partσ partθ σ σ θ εσ partσ σ partε
+ +++ + +
+ ++ +
+ +++ + +
+ ++ +
prime= + minus
prime prime+ + prime
(3310)
The above equation is rewritten by using the relation of partσpartσ σ
σ
= prime3
2
( ) [ ]
ε ε
σpartpartσ σ
σ σ θ σ
σpartσpartσ
θ σσ
σ partpartε
θ ε
θ θ
θ θ
θ
θθ
θ
θ θ θθ
θ
nc i
nc i
ni
n
i
ni
ni i j n
i
ni
ni
ni
n
i
ni
ni n
iH
n
i
nH
f fd
fd
fd
++
+
+ +
+
++ +
+
+
+ +++
++
++
=
+
minus
prime prime
+ prime
+ prime
1
2 11
11
1
9
4
3
2
3
2i+1
(3311)
is obtained
Eq(3311) is a result of the first-order Taylor expansion and the process of equation
development up to here is a generally approved process irrespective of the creep equation
type
In the development of equations after this a calculation method is explained concerning
the creep strain hardening which is another factor to evaluate the equivalent creep strain rate
The creep strain hardening is usually observed in a primary creep region (thermally activated
creep) where stress is a function which is strongly affected by creep strain Here the
numerical process is explained by classifying the creep equations in terms of ldquowith or without
strain hardeningrdquo into A) B) and C) as follows
A) Stress is an increasing function of creep strain as in the primary creep equation This case assumes that hardening of material occurs with increasing creep strain in the
primary creep region This primary creep component is expressed as ( )1 1i H in ng θ θσ ε+ +
+ + by
JAEA-DataCode 2013-005
- 229 -
referring to Eq(337) Then the corresponding creep strain hardening increment is expressed
by using Eq(337) as follows
( ) ( )( )
1 1 11 1
1 11 1 1
H i i H in n n n
i i H i H in n n n n
t g
t g d d
θ θ
θ θ
ε σ ε
σ σ θ σ ε θ ε
+ + ++ + + +
+ ++ + + + +
Δ = Δ
= Δ + + (3312)
The above equation is approximated by the first-order Taylor expansion
1 11 1 1
11 1 1
11 1
H i H i H in n n
i ii i
n n n nn n
iH i
n nHn
d
gt g t d
gt d
θθ θ
θ
ε ε ε
part σθ σpartσ σ
partθ εpartε
+ ++ + +
++ + + +
+ +
++ +
+
Δ = Δ +
part = Δ + Δ part
+ Δ
(3313)
From Eq(3313) 11
H indε +
+ is expressed as
( ) 1
1 1 1 1 11
11
i ii H i i
n n n n nH i n n
n i
n Hn
gt g t d
dg
t
θθ θ
θ
part partσε θ σpartσ partσεpartθ
partε
++ + + + +
+ + ++
++
Δ minus Δ + Δ = minus Δ
(3314)
Thus formulation of the unknown quantity 11
H indε +
+ is completed which permits the
elimination of the unknown quantity 11
H indε +
+ of two unknowns 11
++
indσ and 1
1H i
ndε ++ in the
right hand side of Eq(3311)
By substituting Eq(3314) into Eq(3311) and re-arranging
1 1 11 1 2 1 3 1
1
iic i c i i i i
n n i j n n nnn
F d F d Fθ θ θ
partσε ε θ σ σ σ θ σ θ σpartσ
+ + ++ + + + ++
+
prime prime prime prime = + + + (3315)
is obtained where
( )
1
1 2
1
9
4 1
i i
i ni Hn n n
iiin nn
n Hn
f gt
ffF
gt
θ θ θ
θ θθ
θ
part partθpart partε partσpartσ σ partσ θ
partε
++ + +
+ +++
+
Δ = minus + minus Δ
Ffn
i
ni2
3
2= +
+
θ
θσ (3316)
( )1 1
3
1
3
21
ii H i
n n nHn
iin
n Hn
ft g
Fg
t
θθ
θ
θ
part εpartε
σ partθpartε
+ + ++
++
+
Δ minus Δ =
minus Δ
JAEA-DataCode 2013-005
- 230 -
B) Creep rate is gradually decreasing under constant stress as in the secondary creep equation
This case implies that to maintain a positive strain rate of creep stress has to be
monotonously increased In the secondary creep region ie steady creep region no
time-hardening effect exists and strain hardening does not occur with time and strain Here
ldquotime-hardeningrdquo means that stress needed to maintain a certain magnitude of creep strain rate
is increasing with time By replacing this time-hardening effect term with the strain hardening
term creep equation is expressed in FEMAXI
However in the case in which creep equation is made by a combination of primary creep
and secondary creep which has the time-hardening a whole creep region is subjected to strain
hardening eg Zircaloy creep equation of MATPRO-09 Accordingly if the creep equation
is given by the following form
1 2( ) ( ) nc f T t f T tε σ σ= sdot + sdot (nlt1) (3317)
the creep equation is formulated by taking account of the strain hardening In other words
1( )s f Tε σ= in Eq(3317) is a steady rate of creep strain term and this term does not
contribute to strain-hardening so that the hardening term is limited
2 ( ) nH f T tε σ= sdot (3318)
Consequently by differentiating Eq(3318)
12 ( ) n
H n f T tε σ minus= sdot (3319)
is obtained By eliminating time t from Eqs(3318) and (3319)
[ ] ( )1 1
2( ) ( )n
n nH H Hg n f Tε σ ε σ ε
minus
= = (3320)
is derived then Eq(3320) is an equation to define ( )Hg σ ε Accordingly in Eq(3318) if
1( ) 0s f Tε σ= = a whole region is subjected to strain hardening
C) None of power function term of time t is included in creep equation In the case of a steady creep equation (no time-hardening) which has no strain hardening
effect the method of FEMAXI is as follows
The creep strain rate is given by Eq(3315) and 0=partpart
H
f
ε and g=0 in no strain
hardening state Then
( )1 2
9
4
i inii
n nn
f fF θ
θ θθ
partpartσ σσ
+
+ ++
= minus
JAEA-DataCode 2013-005
- 231 -
2
3
2
inin
fF θ
θσ+
+
=
(3321)
3 0F =
are obtained
Since the unknown quantity of the right-hand side of Eq(3315) is only 11
indσ +
+
dividing the known and unknown quantities to express the unknown creep strain increment
gives 1 1 1
1 1 1 1c i c i c i c i in n n n n nt C dθ θε ε ε σ+ + +
+ + + + + + Δ = Δ = Δ + (3322)
where
1 1 3 1 1
3
2c i i i in n n n n ni
n
f t F tθ θθ
ε σ θ σσ+ + + + + +
+
prime primeΔ = Δ + Δ
1 1 2
iic i
n n i j nn
C t F Fθ θ θ
partσθ σ σpartσ+ + + +
prime prime prime = Δ +
(3323)
Here in the formulation of Eq(3322) the hot-pressing is included if the material is fuel
pellet as shown in Eq(3259)
Next to eliminate 11
c inε +
+Δ in Eq(3131) which is transformed from Eq(3244)
[ ] ( )
1 1 0 1 11 1 1 1 1
1 0
i i i P i c in n n n n n
i in n n
C d B u
C
θ
θ
σ ε ε ε
σ σ
+ + + ++ + + + + +
+ +
minus Δ + Δ + Δ + Δ
+ minus =
(3244)
substituting Eq(3322) and re-arranging give the following equation
( ) [ ] ( )
1 1 0 1 1 1 1 1 1
1 0
i c i i i P i c in n n n n n n
i in n n
C C d B u
C
θ θ
θ
σ ε ε ε
σ σ
+ + ++ + + + + + +
+ +
+ minus Δ + Δ + Δ + Δ
+ minus =
(3324)
[ ]( ( )
)
1 11 1 1
0 11 1 1
ˆi i i i in n n n n n
c i P in n n
d D B u Cθ θσ σ σ
ε ε ε
+ ++ + + + +
++ + +
= Δ minus minus
minus Δ minus Δ minus Δ
(3325)
where
[ ] [ ] [ ]( ) 1111
~ˆ minus
+++ += icn
in
in CCD
ldquo^rdquomeans an apparent stiffness 1inC + corresponds to elasticity and [ ]ic
nC 1+ to creep
Eq(3325) is a sum of terms related to the unknown quantity 11
indσ +
+ In this equation
11
c inε +
+Δ has been eliminated and application of the Newton-Raphson iteration to this is due
to the approximation by first-order Taylor expansion That is by performing a first-order
JAEA-DataCode 2013-005
- 232 -
Taylor expansion of the creep strain rate equation with respect to the magnitude of changes of
stress and creep strain hardening a formulation is carried out to predict the change of creep
strain rate which is induced by changes of stress and creep strain hardening and an accurate
prediction of creep strain rate is obtained
Here the creep reversal calculation by the Pughrsquos method(32) is explained When stress
direction is reversed primary creep occurs again Pugh proposed a simple and convenient
method to simulate this reversal as follows
1) Creep hardening induced by creep strain progress is expressed by ε H That is
( ) ε partσpartσ
σ εcHf T=
(3326)
2) When stress is reversed ε εH gt holds and therefore
ˆ 0HHε ε ε= = (3327)
is set
3) When stress is reversed again or reversed after some time elapsed
If ε εH ge ε ε ε= = =H H0 0 is set and
If ε εH lt ε ε εH H= minus is set
Fig331 illustrates these relationships
ε c ε
ε
ε
εH =0
εH =0 εH =0
ε ε εH H= minus
①
②
③
④
Fig331 Creep strain reversal induced by stress direction reversal
Here the stress direction reversal is judged by the following way
Since creep equation is
( ) ε partσpartσ
σ εcHf T f=
ge 0 (3328)
time
JAEA-DataCode 2013-005