Light Water Reactor Fuel Analysis Code FEMAXI-7; Model and Structure

Transcription

1 JAEA-Data/Code -5 Lght Water Reactor Fuel Aalyss Code FEMAXI-7; Model ad Structure Motoe SUZUKI, Hroak SAITOU, Yutaka UDAGAWA ad Fumhsa NAGASE Reactor Safety Research Ut Nuclear Safety Research Ceter July Japa Atomc Eergy Agecy

2 本レポートは独立行政法人日本原子力研究開発機構が不定期に発行する成果報告書です 本レポートの入手並びに著作権利用に関するお問い合わせは 下記あてにお問い合わせ下さい 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 ( 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 9-95 茨城県那珂郡東海村白方白根 番地 4 電話 , Fax , E-mal:rd-support@aea.go.p Ths report s ssued rregularly by Japa Atomc Eergy Agecy Iqures about avalablty ad/or copyrght of ths report should be addressed to Itellectual Resources Secto, Itellectual Resources Departmet, Japa Atomc Eergy Agecy -4 Shrakata Shrae, Toka-mura, Naka-gu, Ibarak-ke 9-95 Japa Tel , Fax , E-mal:rd-support@aea.go.p Japa Atomc Eergy Agecy,

3 JAEA-Data/Code - 5 Lght Water Reactor Fuel Aalyss Code FEMAXI-7; Model ad Structure Motoe SUZUKI, Hroak SAITOU*, Yutaka UDAGAWA ad Fumhsa NAGASE Reactor Safety Research Ut Nuclear Safety Research Ceter, Japa Atomc Eergy Agecy Toka-mura, Naka-gu, Ibarak-ke (Receved March 4, ) A lght water reactor fuel aalyss code FEMAXI-7 has bee developed for the purpose of aalyzg the fuel behavor both ormal codtos ad atcpated traset codtos. Ths code s a advaced verso whch has bee produced by corporatg the former verso FEMAXI-6 wth umerous fuctoal mprovemets ad extesos. I FEMAXI-7, may ew models have bee added ad parameters have bee clearly arraged. Also, to facltate effectve mateace ad accessblty of the code, modularzato of subroutes ad fuctos have bee attaed, ad qualty commet descrptos of varables or physcal quattes have bee corporated the source code. Wth these advacemets, the FEMAXI-7 code has bee upgraded to a versatle aalytcal tool for hgh burup fuel behavor aalyses. Ths report descrbes detal the desg, basc theory ad structure, models ad umercal method of FEMAXI-7, ad ts mprovemets ad extesos. Keywords: LWR Fuel, FEM Aalyss, Traset, Pellet, Claddg, Fsso Gas Release, PCMI, ur-up * ITOCHU Techo-Solutos Corporato, Tokyo, Japa

4 JAEA-Data/Code - 5 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 ( 年 月 4 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 ): 9-95 茨城県那珂郡東海村白方白根 -4 * 伊藤忠テクノソリューションズ株式会社 東京

5 JAEA-Data/Code -5 Cotets.Itroducto.... Characterstcs of Fuel Performace Code..... Features ad roles of fuel code..... Predcto of fuel behavor by FEMAXI..... Hstory of FEMAXI code developmet Usage of FEMAXI Features of Structure ad Models of FEMAXI What s model? What s fuel modelg? Aalytcal targets Whole Structure Features of Numercal Method ad Modelg....5 Iterfacg wth Other Codes, etc Iterfacg wth burg aalyss codes RODURN ad PLUTON Re-start fucto from base-rradato to test-rradato perods Executo system evromets... 4 Refereces Thermal Aalyss Models Heat Trasfer to Coolat ad Thermal-hydraulcs Model Coolat ethalpy crease model..... Determato of claddg surface heat trasfer coeffcet..... Flow of thermal-hydraulcs calculato Trasto process of coolat codto Equvalet dameter ad cross-sectoal area of flow chael Claddg Watersde Corroso Model Oe-Dmesoal Temperature Calculato Model Selecto of umber of elemets the radal drecto Determato of claddg surface temperature Soluto of thermal coducto equato... 45

6 JAEA-Data/Code Fuel pellet thermal coductvty Determato of Heat Geerato Desty Profle Use of RODURN output Use of PLUTON output Robertso s formula Accuracy cofrmato of pellet temperature calculato Gap Thermal Coductace Model Modfed Ross & Stoute model odg model for gap thermal coductace odg model for mechacal aalyss Pellet relocato model Swellg ad desfcato models Model of Dry-out a Test Reactor Capsule Modellg the dry-out expermet Thermal ad materals propertes Numercal soluto method Geerato ad Release of Fsso Gas Fsso gas atoms geerato rate Cocept of thermally actvated fsso gas release FEMAXI Thermal dffuso accompaed by trappg WhteTucker model of tra-gra gas bubble radus ad ts umber desty (GFIS=) Radato re-dssoluto ad umber desty model of tra-graular gas bubbles (GFIS=) Pekka Lösöe model for tra-graular gas bubbles (GFIS=) Galerk s soluto method for partal dfferetal dffuso equato (commo IGASP= ad ) Fuel gras ad boudares (commo IGASP= ad ) Amout of fsso gas atoms mgratg to gra boudary (commo IGASP= ad ) Amout of fsso gas accumulated gra boudary Amout of released fsso gas... v

7 JAEA-Data/Code Sweepg gas atoms to gra boudary by gra growth Re-dssoluto of fsso gas (commo IGASP= ad ) Threshold desty of fsso gas atoms gra boudary (Equlbrum model ; IGASP=) I-gra dffuso coeffcet of fsso gas atoms (commo to IGASP= ad ) Rate- Law Model (IGASP=) Assumptos shared wth the equlbrum model Growth rate equato of gra boudary bubble Swellg by Gra-boudary Gas ubble Growth the Equlbrum Model (IGASP=) Gra boudary gas bubble growth ubble swellg Swellg by Gra-boudary Gas ubble Growth the Rate-law Model (IGASP=, IFSWEL=) Coalescece ad coarseg of bubbles Swellg.... Thermal Stress Restrat Model..... Opto desgated by IPEXT..... Comparatve dscusso o the gap gas coductace crease durg power dow Hgh urup Rm Structure Model asc cocept Desty decrease of fsso gas atoms sold phase of rm structure Effect of rm structure formato o fuel behavor Selecto of Models by Iput Parameters Number of elemets the radal drecto of pellet Calculato of thermal coductvty of fuel pellet Selectg swellg model Selectg fsso gas release model ubble growth ad swellg Optos for the rm structure formato model... 5 v

8 JAEA-Data/Code -5.4 Gap Gas Dffuso ad Flow Model Assumptos ad methods of dffuso calculato Method to Obta Free-space Volumes a Fuel Rod Defto of free space Iteral Gas Pressure Method of assgg pleum gas temperature ad claddg temperature Calculato of the varato teral gas codto durg rradato Tme Step Cotrol Automatc cotrol Tme step cremet determato FGR model Tme step cremet determato temperature calculato... 7 Refereces Mechacal Aalyss Model Solutos of asc Equatos ad No-lear Equatos asc equatos Ital stress method, tal stra method ad chaged stffess method for o-lear stra Soluto of the basc equatos FEMAXI (o-lear stra) Mechacal Aalyss of Etre Rod Legth Fte elemet model for etre rod legth aalyss (ERL) Determato of fte elemet matrx Dervato of stress-stra (stffess) matrx of pellet ad claddg Pellet crackg Crack expresso matrx Defto equatos of equvalet stress ad stra FEMAXI Hot-pressg of pellet Supplemetary explaato of cremetal method Method to Calculate No-lear Stra..... Creep of pellet ad claddg..... Method of creep stra calculato Plastcty of pellet ad claddg... v

9 JAEA-Data/Code Dervato of stffess equato Formulato of Total Matrx ad Exteral Force Stffess equato ad total matrx Upper pleum ad lower pleum boudary codtos Pellet-claddg cotact model Axal force geerato at the P-C cotact surfaces Algorthm of axal force geerato at PCMI cotact surface Evaluato of axal force geerated at the P-C cotact surface: Opto A Evaluato of Axal force geerato: Opto D Local PCMI Mechacal Aalyss Elemet geometry asc equatos Stffess equato oudary codtos PCMI ad cotact state problem Axal loadg - Iteracto betwee target pellet ad upper pellet Mechacal Propertes Model of Creep ad Stress-stra Creep Model of claddg stress-stra relatoshp Ohta model Method to determe yeld stress FRAPCON ad MATPRO models (ICPLAS=, 6)....7 Skyle Method... 5 Refereces Materals Propertes ad Models UO Pellet MOX Pellet Zrcoum Alloy Claddg Claddg propertes Claddg oxde propertes SUS4 Staless Steel... 7 v

10 JAEA-Data/Code Other Materals Propertes ad Models urup Calculato of Gd O -Cotag Fuel Refereces Ackowledgemet... 8 v

11 JAEA-Data/Code -5 目次. 序言.... 燃料解析コードの性格..... 燃料解析コードの特徴と性格..... FEMAXI による燃料挙動の予測..... FEMAXI コード開発の歴史 FEMAXI コードの利用 FEMAXI の構造とモデルの特徴 モデルとは何か 燃料モデリングとは何か 解析対象 全体構造 数値計算とモデルの特徴....5 他コードとの関係 その他 燃焼解析コード RODURN および PLUTON との接続 ベース照射から Re-start による試験照射への接続 実行環境など... 4 参考文献 熱的解析モデル 冷却材への伝熱と熱水力モデル 冷却材のエンタルピー上昇のモデル..... 被覆管表面熱伝達率の決定..... 熱水力計算の流れ 冷却材状態の遷移 流路相当直径および流路断面積 被覆管水側腐食モデル... 9 x

12 JAEA-Data/Code -5. 一次元温度計算モデル 半径方向リング要素数の選択 被覆管表面温度の決定 熱伝導方程式の解法 ペレット熱伝導率 発熱密度プロファイルの決定 RODURN 出力の利用 PLUTON 出力の利用 Robertso の式 ペレット温度計算の精度確認 ギャップ熱伝達モデル 修正 Ross & Stoute モデル 熱伝達に対するギャップボンディングモデル 力学計算に対するギャップボンディングモデル ペレットリロケーションモデル スエリングと焼きしまりモデル 試験炉キャプセル内ドライアウトモデル ドライアウト試験のモデル化 熱的 材料的物性値 数値解法 FP ガス生成 放出モデル FP ガス原子生成速度 熱活性拡散 放出モデルの概念 トラッピングを伴う熱拡散 粒内バブル半径と数密度の WhteTucker モデル (GFIS=) 粒内バブル半径と数密度の照射溶解モデル (GFIS=) 粒内バブルの Pekka Lösöe モデル (GFIS=) 偏微分拡散方程式の Galerk 解法 (IGASP= と で共通 ) 結晶粒と粒界 (IGASP= と で共通 ) x

13 JAEA-Data/Code 粒界へ移動する FP ガス原子の量 (IGASP= と で共通 ) 粒界に溜まる FP ガス量 放出されるガス量 粒成長に伴う粒界へのガスの掃き出し FP ガスの再溶解 (IGASP= と で共通 ) 粒界 FP ガス原子濃度の飽和値 ( 平衡論 ; IGASP=) FP ガス原子の粒内拡散定数 (IGASP= と で共通 ) 速度論モデル (IGASP=) 平衡論モデルとの共通の仮定 粒界バブル成長速度式 粒界バブル成長スエリング : 平衡論 (IGASP=) 粒界バブル成長 バブルスエリング バブル成長スエリング- 速度論 (IGASP=, IFSWEL=) バブルの合体 粗大化 スエリング.... 圧力抑制モデル..... IPEXT で指定されるオプション..... 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 高燃焼度リム組織モデル 基本概念 リム組織固相における FP ガス原子の濃度減少 リム組織の燃料ふるまいへの影響 パラメータによる計算モデルの選択 ペレット径方向リング要素数 ペレット熱伝導率計算 スエリングモデルの選択 FGR モデルの選択 バブル成長とスエリング... 5 x

14 JAEA-Data/Code リム組織生成モデルのオプション ギャップ内ガスの拡散 流動モデル 拡散計算の仮定と方法 燃料棒内空間体積の求め方 自由空間の定義 内部ガス圧力 プレナムのガス温度と被覆管温度の設定方法 照射途中における内部ガス状態変更の計算 タイムステップの制御 自動制御 FP ガス放出モデルにおけるタイムステップ幅の決定 温度計算におけるタイムステップ幅の決定... 7 参考文献 力学解析モデル 基本式と非線形力学の解法 基本式 非線形歪みに対する初期応力法 初期歪み法 剛性変化法 FEMAXI コードでの基本式の解法 ( 非線形歪み ) 燃料棒全長の力学解析 全長力学解析の有限要素モデル 有限要素マトリックスの決定 ペレットと被覆管の応力歪み ( 剛性 ) マトリックスの導出 ペレットのクラック マトリックスにおけるクラック表現方法 FEMAXI における相当応力 相当歪みの定義式 ペレットのホットプレス 増分法の補足説明... 9 x

15 JAEA-Data/Code -5. 非線形歪みの計算方法..... ペレットと被覆管のクリープ..... クリープ歪みの計算方式 ペレットと被覆管の塑性 剛性方程式の導出 全体マトリックスの定式化と外力 剛性方程式と全体マトリックス 上下プレナム境界条件 ペレット 被覆管接触モデル ペレット被覆管接触面における軸力発生 PCMI 接触面における軸力発生のアルゴリズム PCMI 接触面における発生軸力の評価 : オプション A PCMI 接触面における発生軸力の評価 : オプション 次元局所 PCMI 力学解析 要素体系 基本式 剛性方程式 境界条件 PCMI と接触問題 軸方向荷重 上部ペレットとの軸方向相互作用 クリープ及び応力 - 歪みに関する機械的性質モデル クリープ 被覆管の応力歪み関係式のモデル 太田モデル FRAPCON と MATPRO モデルの降伏応力の決定方法 (ICPLAS=, 6)....7 スカイライン法... 5 参考文献... x

16 JAEA-Data/Code 物性値とモデル UO ペレット MOX ペレット ジルコニウム合金被覆管 被覆管の物性値 被覆管酸化物の物性値 SUS4 ステンレス被覆管 その他の物性値 モデル 燃焼度の単位換算 参考文献 謝辞... 8 xv

17 JAEA-Data/Code -5. Itroducto A lght water reactor has occuped the most essetal part uclear power geerato for years. Wth the advacemet LWR performace, fuel mprovemet has bee actvely pursued uclear dustres worldwde. I partcular, burup exteso has bee promoted recetly, ad hgh-burup fuel behavor has bee wdely kow as a key ssue terms of rod desg ad relablty. Uder these crcumstaces, developmet of a computer code whch ca precsely aalyze fuel behavor s requred to evaluate the behavor of hgh-burup fuels, to assess ther relablty, ad to serve as a safety cross-check tool. I the code, troducto of ew models s evtable departg from the covetoal fuel aalyss methods, because t has bee observed that fuel behavor a hgh-burup rego of over 4-5 GWd/t essetally dffers from that lower burup rego. Such dfferece s geerated by fuel thermal coductvty degradato, rm structure formato fuel perphery ad crease FGR assocated wth the structure chage, etc. They ca mpose a strog fluece o the overall behavor of fuel rod. O the bass of these cosderatos, a tegrated fuel performace code, FEMAXI-7, has bee developed for aalyzg the behavor of fuels uder ormal operato ad atcpated trasets (excludg accdet codtos). Ths code has tegrated ad exteded the capabltes of the predecessor FEMAXI-6 by corporatg ew models ad fuctos.. Characterstcs of Fuel Performace Code.. Features ad roles of fuel code Fuel rods are the obects whch radoactve substaces are hghly accumulated ad cocetrated a reactor core. Thermal ad mechacal behavor of fuel rod s resulted from complcated teractos amog a umber of such factors as temperature, eutro flux, uclear fsso, accumulato of fsso products etc. whch outspread from atomstc level pheomea to egeerg-cotrollable macroscopc ettes. A fuel assembly has to atta cocurretly the two roles,.e. safety ad power geerato effcecy. Realzg ths purpose eeds a varous R&D efforts based o rradato expereces, whch gves a fudametal cause for the research of fuel behavor terms of egeerg ad safety. O the other had, that fuel behavor s a result of complcated teractos mples that fuel research s a feld where actual rradato expereces have evtably a marked - -

18 JAEA-Data/Code -5 sgfcace. I other words, fuel research s a feld where theoretcal ad determstc models caot always have a precse predctablty, some materals propertes or behavor caot be expressed by such theoretcal model as those of reactor physcs or caot be clearly defed by establshed physcal laws, ad there are so may ukow factors such as the effect of small quatty of mpurtes, crystal gra structure, depedece o fabrcato process, etc. Furthermore, so may factors are volved fuel behavor that t s sometmes dffcult to evaluate the cotrbuto of each factor to whole behavor by smply summg up each factor-wse calculato. Accordgly, rradato expereces have a dspesable role smulato, desg ad uderstadg fuel behavor. However, rradato of fuels,.e. rradato tests ad post-rradato examatos, eeds geeral much cost ad log perod of tme, ad quatty ad qualty of measured data are lmted for the cost ad tme. The rradato test caot gve all that are ecessary to uderstad the sde processes of fuel rod. As a result, some cases we are oblged to deped o a few specfc measurable physcal quattes such as fuel ceter temperature ad teral pressure, or o the drectly evaluated quattes such as burup dstrbuto assessg the whole behavor of fuel rod. I these stuatos, makg use of a fuel performace code s a mportat measure to recogze ad reproduce the terlked structures sde fuel rod, to grasp the cotrollg parameters for fuel behavor, ad to predct the fuel behavor o the bass of the drectly or drectly measured physcal quattes, as f solvg a verse problem a exploratory maer. FEMAXI s a egeerg tool servg for ths purpose. I other words, developmet of FEMAXI s a process whch costtutes, reproduces ad predcts the teracto structure hdde behd observed fuel behavor o the bass of the lmted data or expereces of rradato... Predcto of fuel behavor by FEMAXI A fuel desg code of fuel maufacturer ofte uses emprcal correlatos whch have bee derved from rradato test data base, ot depedg o the mechastc models such as those FEMAXI, ad the calculated results are compared wth measured data for valdato. Therefore, ts predctos are relable at least wth the scope of the test data base ad ts small exteral rego, ad t s cosdered that the predcto s effectve for fuel desg. O the other had, may mechastc ad complemetary emprcal models, ad parameters to cotrol the models have bee mplemeted FEMAXI-7. For predcto of - -

19 JAEA-Data/Code -5 ukow behavor of fuel, FEMAXI-7 could brg about so dfferet calculato results depedg o the combato of ts models ad parameters, that the fal predcto would be capable of havg a cosderable ucertaty. Accordgly, t s ecessary to fd models ad data set combatos whch are as geerally applcable as possble to the predcto the process of verfcato calculatos. If a code ca reproduce by a set of models ad parameters the behavor of fuel A of whch rradated results are kow, ad f the same set s appled to aother fuel rod ad ca result a successful reproducto of the rradated state of the rod, predctablty of the set s verfed wth respect to at least these fuel rods. It depeds o the umber of aalyzed cases, o aalyzed fuel desgs ad rradato codtos how far such verfcato has bee made,.e. how may types of models ad parameter sets have bee tested ad how much geeralzato of predctablty s guarateed FEMAXI. I ths sese, verfcato of FEMAXI-7 s o-gog. As some of verfcato actvtes, test aalyses have bee performed the teratoal bechmark test FUMEX-III (.), ad by usg the results of the Halde test rradatos. Through these actvtes, refemet of models ad parameters set are beg followed up. At the same tme, FEMAXI-7 ca allow by the combato of ts varous models ad cotrollg parameters a attempt to reproduce such fuel behavor (rradato data) that the covetoal models ad parameters caot obta. Ths s oe of the meas whch eables us to estmate the ukow lkage structure of factors sde fuel rod. Furthermore, the code ca be appled to the predcto of fuel behavor the codtos whch s plaed test rradato or ew desg of fuels, or eve the codtos beyod the expermetally feasble scope... Hstory of FEMAXI code developmet Hstory of developmet up to the latest verso FEMAXI-7 s brefly descrbed. It s oe of features that up to FEMAXI-7 a umber of researchers have bee egaged the developmet ad ther kowledge ad expereces have bee accumulated the code. The code ca be cosdered to be based o the combato of wsdom ad expereces of Japaese fuel researchers ad egeers. () The frst verso of FEMAXI was developed by Ichkawa (JAERI: Japa Atomc Eergy Research Isttute) the Halde Proect. It featured a -D local elasto-plastc PCMI aalyss fucto. (974) - -

20 JAEA-Data/Code -5 () Creep model was mplemeted FEMAXI-II by Koshta (CRIEPI: The Cetral Research Isttute of Electrc Power Idustry). (977) () FEMAXI-III (.) was developed by a extesve cooperato of JAERI, uverstes, Htach, Toshba, NFD(Nuclear Fuel Developmet, Co.), CRIEPI, ad CRC(Cetury Research Ceter Co.,). Developmet of ths verso receved a award from Japa Socety of Nuclear Scece ad Techology. It exerted a predomat capablty teratoal bechmarkg calculato. Released to NEA-Dataak (.) (984) (4) FEMAXI-IV (.4)-(.) was a verso mproved by Nakama et al. It was used JNES (.) ( former NUPEC:Nuclear Power Egeerg Corporato) as oe of cross check codes. Ths verso was take over to M.Suzuk JAERI, Ver. was developed, ad released to NEA-Dataak. (996) (5) FEMAXI-V (.4) was developed by Suzuk for the aalyss of hgh burup fuels, released to NUPEC ad NEA-Dataak (). (6) The code was further exteded to FEMAXI-6 by Suzuk et al.(jaea), released to JNES ad to NEA Dataak. (-)..4 Usage of FEMAXI A exteded usage of FEMAXI code ca be summarzed as follows: () Cross check code for safety lcesg The former verso FEMAXI-6 s a ma part ad fucto of the cross-check code used JNES for safety lcesg of LWR fuel rod. () Release to exteral users A code package of FEMAXI-7 (source, bary program, related codes, descrpto ad maual) has bee released to domestc users wthout charge through RIST (Research Orgazato for Iformato Scece & Techology, ). The package wll be regstered NEA Dataak as a represetatve fuel performace code of Japa to be released to foreg users. The former versos FEMAXI-VI, FEMAXI-V ad FEMAXI-6 have bee released from NEA Dataak to foreg users wthout charge, ad the total umber of the releases s over thrty. Also we are postvely acqurg exteral users, respodg to questos from users. I effect, the FEMAXI code has bee a frame-of-referece of Japaese LWR fuel performace code

21 JAEA-Data/Code -5 () Oe of the meas for expermetal aalyss, predcto of expermets, ad trag of fuel behavor uderstadg The FEMAXI code ca be used for exploratory aalyss of rradato tests, ad for the predcto of state of fuel rods ust pror to test-rradato accdet codtos. Also the code s used for the plag of rradato test, for checkg the adequacy of fuel desg. Furthermore, t s used as a trag tool by performg smulatos wth varous put codtos for a better uderstadg of fuel behavor. (4) Platform to evaluate rradato test data ad ew models Newly obtaed materals propertes of rradated fuels are tself sometmes ot clearly evaluated terms of ther cotrbuto to fuel behavor chage or mprovemets. However, corporatg them a fuel code, puttg them a varous teractos of a umber of factors, ad performg smulatos wth them wll eable us to evaluate ther cotrbuto to or effects o the whole behavor of fuel rods more clearly or quattatvely. The FEMAXI code ca fucto for such purposes.. Features of Structure ad Models of FEMAXI.. What s model? efore dscussg specfc models, a fudametal questo s addressed; what s model as a code compoet? What s the purpose of model developmet? Here, f the aswer s that the purpose s to reproduce behavor of a target obect as precsely as possble, ths aswer s suffcet because, t does ot address the questo that why s t ecessary to reproduce precsely?, ad what s the precseess ths case?. To beg wth, sce model developmet starts the stuato where kowledge about the obect to be modeled s suffcet, the precseess has to ow ts lmt. I other words, a model ca be a model because t s a approxmato of a real obect, or a deatoal reflecto whch has quattatve ature to a cosderable extet. Therefore, model developmet s upgradg the followg steps: ) The frst stage s a oe whch gves sght to the uderlyg mechasm of pheomea as to what wll appear whe watchg the pheomea through the prototypcal model. ) The secod stage allows us to estmate the uderlyg mechasm of pheomea by corporatg the model a code ad performg calculato

22 JAEA-Data/Code -5 ) The thrd stage eables us to predct the fuel behavor by chagg model parameters, whch wll lead to the mprovemet of fuel performace ad be used as a measure to solve the verse problem. However, the most essetal role of modelg s cosdered to make a reflecto o huma oto ad deepe the sght to the mechasm of pheomea by realzg a teracto or mutual permeato betwee the huma oto ad the real etty whch exsts depedetly from the huma oto... What s fuel modelg? As for the fuel modelg, t s qute usual to have dscusso terms of dvdual modelg of maor pheomea whch domate fuel behavor, such as heat geerato ad coducto, fsso gas release, Pellet-Clad Mechacal Iteracto(PCMI), etc. However, hgh burup fuel behavor, as teractos become stroger amog varous pheomea, the modelg s evtably exteded to ot oly each pheomeo but also a lkage structure to combe varous factors. That s, a lkage structure of each model tself s requred to be desged. Ths meas that to what extet the teracto amog the factors s explctly modeled. For example, couplg of thermal ad mechacal aalyses, or terlkage amog fsso gas bubble growth, pellet swellg ad thermal coductvty degradato, etc. are to be addressed the code desg. However, the terlkage structure caot be too complcated at the begg, ad t s a usual strategy to start wth a smplfed assumpto or model. For example, porosty crease by fsso gas bubble growth has a addtoal decrease effect o the pellet thermal coductvty, though t s set depedet from the thermal propertes of pellet. O the other had, as uclear reacto calculato has a weak teracto wth thermal ad mechacal behavor, the reacto calculato s carred out a relatve solato from fuel behavor calculato fuel performace code. That s, o drect teracto wth fsso reacto s corporated the code but t s cosdered mplctly as the chage of pellet power desty profle wth burup or fast eutro flux correspodg to lear heat rate. These quattes are ofte calculated by a sub-code or a depedet burg aalyss code (reactor core physcs code) ad the results are fed to the ma part of fuel code. I ths sese, the burg aalyss codes whch gve results to FEMAXI are mportat. (Refer to Refs.(.5) ad (.6)) ased o the above dscusso, features of FEMAXI models ca be summarzed as - 6 -

23 JAEA-Data/Code -5 follows: () The aalytcal obect s lmted to a sgle fuel p ad ts surroudg coolat, beg solated from the teractos amog the fuel ps set a assembly or thermal ad uclear behavor of whole reactor core. () I desgg models ad code as egeerg tools, the hghest prorty s placed o the predcto of fuel behavor a macroscopc scale. Accordgly, ths sese, t s ot a preferetal task to make each model based o a mcroscopc physcal prcple or theory. The code adopts a combato of mechastc model ad emprcal model. If a mechastc model caot be desged, the code begs wth a emprcal model. () The code pursues fast calculato. Nevertheless a low prorty s gve to the modelg efforts to carry o a much reduced calculato tme for the purpose of a statstcal aalyss whch hadles a umber of fuel ps a reactor core... Aalytcal targets Aalytcal obects of FEMAXI-7 s, qute detcal to those of FEMAXI-6, fuel behavor ot oly steady-state ormal operato codtos but also such traset codtos as fast power ramp, load-followg, power-coolat msmatch, etc. Nevertheless the code does ot cover the accdet codtos such as LOCA (Loss-of-coolat accdet) ad RIA (Reactvty- tated accdet). Table. shows the pheomea whch FEMAXI-7 deals wth. As for the materal propertes cludg those for MOX fuels ad Gd-cotag UO fuels, those avalable ope lteratures have bee adopted as much as possble. Maor fuctoal dffereces amog FEMAXI-V, FEMAXI-6 ad FEMAXI-7are lsted Table.. Descrpto for each tem wll be preseted chapters, ad 4. Pellet Claddg Fuel rod Table. Pheomea aalyzed by FEMAXI-7 Thermal process determg temperature dstrbuto Thermal coducto. (heat flux dstrbuto ) Fsso gas release. Swellg Thermal coducto, Watersde corroso. Gap thermal coducto (mxed gas, cotact, radato), claddg surface heat trasfer, gap gas flow. Process wth mechacal dsplacemet Thermal expaso, elastcty, plastcty, creep, crackg, relocato, desfcato, swellg, hot-press. Thermal expaso, elastcty, plastcty, creep, rradato growth. Mechacal teracto betwee pellet ad claddg, frcto ad bodg betwee pellet ad claddg

24 JAEA-Data/Code -5 Table. Upgrades of ma fuctos of versos up to FEMAXI-7 FEMAXI-V FEMAXI-6 released released FEMAXI Couplg soluto of thermal ad mechacal aalyses Pellet thermal coductvty degradato wth burup Hgh burup structure formato pellet perphery Gas bubble swellg model Rate-law model of gra boudary gas bubble growth Traset thermal hydraulcs model Exteso of put/output ad materals propertes Restart fucto to have a lk betwee base-rradato ad test-rradato. Modularzato of source codes, detaled commets embedded, selecto of the umber of elemets.. Whole Structure FEMAXI-7 cossts of two ma parts: oe for calculatg the temperature dstrbuto of fuel rod, thermally duced deformato, ad fsso gas release etc. (hereafter called thermal aalyss part ), ad the other for calculatg the mechacal behavor of fuel rod (hereafter called mechacal aalyss part ). Fgure. outles the etre code structure ad Fg.. shows the aalytcal geometry. I the thermal aalyss part, calculato always covers a etre rod legth whch s dvded to the maxmum of 4 axal segmets. Namely, the temperature dstrbuto s calculated at each axal segmet as oe-dmesoal axs-symmetrcal problem the radal drecto, ad wth ths temperature, such temperature-depedet values as fsso gas bubble growth, gas release, gap gas flow the axal drecto, ad ther feedback effects o gap thermal coducto are also calculated. Also, to take to accout thermal feedback amog the axal segmets, whch may arse due to ueve power dstrbutos the axal drecto, terato s performed utl covergece s attaed for the etre rod legth

25 JAEA-Data/Code -5 I the mechacal aalyss part, users ca select oe of the two modes: aalyss of the etre rod legth : -D ERL mechacal aalyss oly, or the ERL mechacal aalyss -D local PCMI aalyss oe pellet legth. I the former, the -D axs-symmetrcal fte elemet method (FEM) s appled to each axal segmet of the etre rod legth; the latter, addto to the -D FEM, the axs-symmetrcal FEM s appled to half a pellet legth for a symmetrcal reaso, ad mechacal teracto betwee pellet ad claddg,.e. local PCMI s aalyzed oly oe axal segmet. I the mechacal aalyss, the magtude of pellet stra caused by thermal expaso, desfcato, swellg ad relocato s calculated frst, ad a stffess equato s formulated wth cosderato gve to crackg, elastcty/plastcty ad creep of pellet. The, stress ad stra of pellet ad claddg are calculated by solvg the stffess equato wth boudary codtos whch correspod to the pellet-claddg cotact mode. Whe PCMI occurs ad pellet-claddg cotact states chage, calculato s re-started wth the ew boudary codtos of cotact from the tme whe the chage occurs. I the ERL mechacal aalyss, the axal force actg o adacet axal segmet s evaluated, ad as a result, aalyss of axal deformato of etre rod legth s performed. Also, the fte elemet method s smplfed to reduce the total umber of degrees of freedom. The thermal aalyss ad the ERL mechacal aalyss are coupled; covergece betwee temperature ad deformato s obtaed every tme step by terato

26 JAEA-Data/Code -5 Iput Thermal aalyss Temperature, fsso gas dffuso ad release. <Tme step cotrol> Iterato Mechacal aalyss of etre rod legth Elasto-plastc, creep, PCMI, axal force, deformato(stress-stra). Next Tme Step No Temperature, Iteral pressure Ed of Tme Step Output Yes Local PCMI aalyss [half a pellet legth] Elasto-plastc, creep, rdgg, stress-stra dstrbuto. Fg.. Overvew of aalytcal flow of FEMAXI - -

27 JAEA-Data/Code -5 Pleum Axal Power Profle Nest of coaxal rgs Zrcaloy metal FEM mechacal aalyss for pellet ad claddg ZrO geerated by watersde corroso Segmet Temperature Gas pressure Power ad urup profles LHR Fg.. Geometry of FEMAXI for the aalyss of oe sgle fuel rod.4 Features of Numercal Method ad Modelg () FEM elemet characterstcs Storage rego ad calculato tme are reduced by troducto of rectagular -degrees-of-freedom elemets to the FEM aalyss order to perform mechacal calculato for the etre legth of the fuel rod. () Creep soluto A mplct procedure s appled to obta umercal soluto stablty for hgh creep rate. () Cotact problem Three cotact states are dealt wth as the cotact codtos betwee pellet ad claddg: ope gap state, clogged gap state ad sldg state. I the FEM, cotact codtos at each ode par of pellet ad claddg are determed, ad the boudary codtos are set accordace wth the codtos. - -

28 JAEA-Data/Code -5 (4) Matrx soluto The coeffcet matrx of smultaeous equatos of * * * * the FEM (stffess equatos) s a dagoal symmetrcal * * * * * * matrx havg a large umber of zero elemets, as * * * * * * * * show the fgure asde. Therefore, the skyle * * * * * * method has bee adopted to solve the equatos, ad the * * * * * memory method of o-zero elemets the matrx, as * * * well as the calculato procedure, has bee mproved, * * order to reduce the sze of the storage rego ad *: Nozero elemet calculato tme. : Storage rego the I the skyle method, oly the elemets uder the skyle method sold le the fgure are stored. I the fsso gas release model, the procedure has bee mproved so that the umber of computg steps for the matrx becomes a mmum. (5) No-equlbrum resdue I order to avod accumulato of o-equlbrum resdue geerated at each tme step durg soluto of a olear problem, equlbrum codto equatos are formulated ot a cremetal maer but a form whch matas the total balace of loads ad stresses. (6) ehavor of cracked pellet Crackg of a pellet s modeled usg a decreased stffess approxmato method, smlar to the case for FEMAXI-III. I order to descrbe behavor of the cracked pellet whch s relocated uder PCMI, recovery of the stffess durg compresso s expressed as a fucto of the amout of relocato. (7) Fsso gas release model ad bubble swellg model Fsso gas dffuso fuel gra s calculated by a combato of the least resdual method equvalet sphere model ad gas bubble formato. I addto, a model of fsso gas release restrat by thermal stress s corporated. Gas bubble growth ca be calculated ether by rate-law model or pressure equlbrum model. I the both models, opto s corporated to calculate swellg by tra-gra bubble growth ad by gra boudary bubble growth. - -

29 JAEA-Data/Code -5 (8) Gap gas dffuso-flow model Gap gas-dffuso flow s modeled, ad effects of the released fsso gas o gap thermal coductace are carefully evaluated. (9) Model of oxdato of claddg Claddg watersde oxdato s modeled ad chage the thermal coductvty due to oxdato s calculated. () Itroducto of thermal-hydraulcs model Wth respect to heat trasfer betwee a fuel rod ad coolat, RELAP5-MOD ad some other models have bee troduced to cover a wde rage of codtos of heat trasfer to coolat, cludg a bolg trasto. () No-steady pheomea aalyss The fuel behavor a o-steady state ca be aalyzed by mechastc treatmet of a o-steady heat trasfer model, gap gas flow model ad fsso gas release model. Also, the accuracy of predcto s mproved ad calculato tme s reduced through use of depedet tme-step cotrols each model. () Smplfcato of calculato I the fsso gas release model, whe the same type of calculatos are performed both a low-temperature rego ad a hgh-temperature rego, calculato results obtaed for elemets oe rego are used for calculato aother rego order to avod carryg out smlar calculatos twce. Also, physcal propertes ad other values whch are frequetly referred to are stored a data table, ad use of specal mathematcal fuctos s avoded. y meas of the above-metoed procedures, FEMAXI ca gve hghly accurate solutos wth a shorter calculato tme. () Emprcal models of rm structure formato Emprcal models are corporated for smulatg the rm structure formed the perpheral rego of hgh burup pellet. - -

30 JAEA-Data/Code -5.5 Iterfacg wth Other Codes, etc..5. Iterfacg wth burg aalyss codes RODURN ad PLUTON Power profles the radal ad axal drectos of a pellet ad the burup profle both chage wth burup from OL. These profle chages dffer depedg o the reactor type (PWR, WR ad heavy-water reactor, etc.) whch fuel rods are rradated. These profles aturally affect the temperature dstrbuto of pellets ad also affect the materal propertes, the performace of whch depeds o the burup. Accordgly, FEMAXI-7, smlar to FEMAX-6, cludes a fucto whch reads the output fles of the burg aalyss codes, RODURN (.5) ad PLUTON (.6), ad uses them for calculato. RODURN uses part of ORIGEN (.7) ad RALE (.8) ad calculates the power profles, burup ad fsso products pellets ad pellet stacks accordace wth a gve power hstory ad fuel rod specfcatos, ad the produces fles of these results. PLUTON produces sotope dstrbutos ad heat geerato desty profles accurately at hgh speed usg a orgal morphologcal fucto. (Refer to secto.4.).5. Re-start fucto from base-rradato to test-rradato perods FEMAXI-7 has a fucto to geerate a result fle whch records the fuel rod states at EOL,.e. at the ed of base-rradato. Ths re-start fle ca be read by FEMAXI-7 to restart the calculato of fuel rod test-rradato whch has ofte a umber of power chages a relatvely short tme. I the former verso FEMAXI-6, the case where base-rradato s coducted wth a log fuel rod ad test-rradato s coducted wth a short rod re-fabrcated from the log rod, users have o other choce but to perform aalyss wth the short refabrcated geometry of rod from the begg of base-rradato to the ed of test-rradato. Ths may gve some accuracy the fal results ad also effectveess of dog the whole calculato process every tme. I FEMAXI-7, these problems have bee elmated. Furthermore, ths fle ca be read by RANNS to be used as oe of the tal codtos for the fuel rod aalyss accdet codtos..5. Executo system evromets () Mache, source code, ad complers FEMAXI -V, -6 ad -7 have bee developed to be operated usually Wdows PC, whle Lux verso of FEMAXI-7 has bee developed. The source code s wrtte maly - 4 -

31 JAEA-Data/Code -5 Fortra-77, ad partly Fortra-9. Utl ow, Compaq Dgtal Vsual Fortra v.6 or upper (Compaq DVF) has bee used for a compler, though ths compler has bee dscotued. From ow, Itel Vsual Fortra Compler for Wdows verso.x.x or upper (Itel VF) (.9) are recommeded whch cludes every fucto of Compaq DVF. I other words, totally equvalet executable programs of the FEMAXI-7 source code ca be bult by the three complers, Compaq DVF, Itel VF ad Lux-GNU Fortra: g77. () Plotted output Numercal results of FEMAXI-7 calculato ca be obtaed three types of outputs: drect umercal fgures, plotted fgures by usg plottg program EXPLOT, ad Excel tables correspodg to the plotted fgures. () Release of package A code package of FEMAXI-7 cossts of the source code, executable program, comple opto fles, related programs such as plottg program, code model descrpto documet, ad Iput/Output maual. The package ca be delvered to users foreg coutres through NEA Dataak. Refereces (.) Improvemet of Computer Codes Used for Fuel ehavour Smulato - FUMEX III, (.) Nakama T., Ichkawa M., et al., FEMAXI-III : A Computer Code for the Aalyss of Thermal ad Mechacal ehavor of Fuel Rods, JAERI 98 (985). (.) OECD Nuclear Eergy Agecy, Dataak, (.4) Nakama T., FEMAXI-IV: A Computer Code for the Aalyss of Fuel Rod ehavor uder Traset Codtos, Nucl. Eg. Desg 88, pp (985). (.5) Nakama T. ad Satou H., A Comparso betwee Fsso Gas Release Data ad FEMAXI-IV Code Calculatos, Nucl. Eg. Desg, pp (987). (.6) Nakama T., Satou H. ad Osaka T., Aalyss of Fsso Gas Release from UO Fuel durg Power Trasets by FEMAXI-IV Code, IWGFPT-7.59, pp.4-6 (987)

32 JAEA-Data/Code -5 (.7) Nakama T. ad K-Seob Sm, Aalyss of Fuel ehavor Power-Ramp Tests by FEMAXI-IV Code, Res Mechaca 5, pp.-8 (988). (.8) Nakama T., Satou H. ad Osaka T., Fuel ehavor Modellg Code FEMAXI-IV ad Its Applcato, IAEA-T-TC-659, Paper preseted at IAEA Techcal Commttee o Water Reactor Fuel Elemet Computer Modellg Steady-State, Traset ad Accdet Codtos, Presto, Eglad, Sept.9-, (988). (.9) Uchda M., Sato H., echmarkg of FEMAXI-IV Code wth Fuel Irradato Data Power Reactors, JAERI-M 9- ( Japaese) (99). (.) Nakama T., Satou H., ad Osaka T., FEMAXI-IV:A Computer Code for the Aalyss of Thermal ad Mechacal ehavor of Lght Water Reactor Fuel Rods, Tras. th It. Cof. o SMIRT (Tokyo, Japa), 697 PV.C-D, pp.-6 (99). (.) Nakama T., Satou H., ad Osaka T., FEMAXI-IV:A Computer Code for the Aalyss of Thermal ad Mechacal ehavor of Lght Water Reactor Fuel Rods, Nucl. Eg. Desg 48, pp.4-5 (994). (.) Suzuk M. ad Satou H., Lght Water Reactor Fuel Aalyss Code FEMAXI-IV (Ver.) -Detaled Structure ad User s Maual-, JAERI-Data/Code 97-4 (997). (.) Japa Nuclear Eergy Safety Orgazato, (.4) Suzuk M., Lght Water Reactor Fuel Aalyss Code FEMAXI-V(Ver.), JAERI-Data/Code - (). (.5) Uchda M. ad Satou H., RODURN: A Code for Calculatg Power Dstrbuto Fuel Rods, JAERI-M 9-8 ( Japaese) (99) (.6) Lemehov, S.E. ad Suzuk M., PLUTON Three-Group Neutroc Code for urup Aalyss of Isotope Geerato ad Depleto Hghly Irradated LWR Fuel Rods, JAERI-Data/Code -5 (). (.7) ell M.J., ORIGEN The ORNL Isotope Geerato ad Depleto Code, ORNL-468 (97). (.8) Ker P.H. ad Robba A.A., RALE, A Program for Computato of Resoace Absorpto Mult-rego Reactor Cells, ANL-76 (967). (.9)

33 JAEA-Data/Code -5. Thermal Aalyss Models I ths chapter, theory ad models the thermal aalyss of FEMAXI-7 are explaed.. Heat Trasfer to Coolat ad Thermal-hydraulcs Model I the thermal aalyss part, temperature dstrbuto at each axal segmet of a fuel rod s calculated by oe-dmesoal cyldrcal geometry wth the boudary codtos determed by coolat temperature ad pressure. The basc assumpto ad calculato procedure are as follows: () Temperatures ad states of coolat for all axal segmets deped o the axal dstrbuto of coolat ethalpy a prevous tme step, ad are determed by temperature, pressure ad flow velocty of coolat at the let ad surface heat flux from claddg at each segmet the curret tme step. The thermal-hydraulcs model of FEMAXI-7 ca be appled ot oly to steady states, but also to traset chages such as crease/decrease power ad load-followg codto uder ormal operato ad the rapd chage of coolat flow rate. However, the case whch a actve (stack) legth of rod s modeled by a small umber of axal segmets, f the flow rate chage occurs a very short perod of tme, some possblty would be foresee that gves a ustable or false soluto of axal dstrbuto of coolat temperature (ethalpy), depedg o the legth of the axal segmet, flow rate, ad tme step cremet, etc. To elmate ths ufavorable possblty, the stack legth of rod s always dvded to equal legth sub-segmets a coolat ethalpy calculato, ad ethalpy dstrbuto calculated at these sub-segmets s terpolated to obta the ethalpy at represetatve locato of each segmet to determe axal dstrbuto of coolat temperature. Ths method eables users to desgate the umber ad legth of segmets wthout cocer for the umercal stablty or ucertaty coolat temperatures at each segmet elevato. () Ethalpy of coolat at each axal segmet s calculated, o the bass of the ethalpy dstrbuto at the ed of the former tme step, wth the ethalpy of the coolat at the let as the tal value, usg the crease the ethalpy obtaed from flow velocty ad the amout of heat coducted from each segmet of rod. Dstrbuto of the coolat temperature the axal drecto s determed by thus-obtaed dstrbuto of the coolat ethalpy. Here, as stated (), axal dstrbuto of coolat ethalpy s derved from the - 7 -

34 JAEA-Data/Code -5 terpolato of values calculated at the sub-segmets. () Next, the heat trasfer coeffcet each coolat mode s determed. Surface temperature of claddg at each axal segmet s the calculated wth heat flux ad coolat temperature. The temperature profle the radal drecto from claddg surface to pellet ceter s calculated oe dmesoal geometry wth the result of the term () above,.e. coolat temperature as boudary codto. However, heat trasfer the axal drecto of a fuel rod s ot take to cosderato, assumg that heat trasfer the axal drecto due to the power dstrbuto the axal drecto s suppressed by such thermal resstace as dshes ad gap at the ed surface of pellet. Actually, the temperature gradet the axal drecto s substatally smaller tha that the radal drecto ad ca be eglected. Therefore, cluso of thermal calculato the z (axal) drecto s ot effcet sce t creases calculato tme markedly. (4) I addto to ths ormal mode whch claddg surface temperature s calculated by the heat coducto from er rego of pellet ad to the coolat at the claddg surface, aother mode s possble FEMAXI. I ths latter mode, claddg surface temperature hstory s specfed by put, ad temperature calculato sde the fuel rod s performed by adoptg ths surface temperature as boudary codto. Ths s so-called the Drchlet problem soluto. The followg tems wll be explaed sectos. to.7: calculato of oe-dmesoal temperature profle the radal drecto, gap thermal coductace model, ad fsso gas release model. The flow of thermal aalyss s show Fg

35 JAEA-Data/Code -5 Tme Step Start Calculato of burup, power desty, ad fsso gas atoms geerato Calc. of heat coducto coeffcet betwee claddg surface ad coolat Calc. of gap thermal coductace Calc. of temperature dstrbuto the radal drecto of rod Fuel ceter temperature has coverged? Calc. of fsso gas release ad gra growth Fsso gas release calculato has coverged? Calc. for oe axal segmet has bee completed? Calc. of gas flow the axal drecto Calc. of teral pressure ad gap gas composto Iterato wth mechacal aalyss of the etre legth of rod: Calc. of gap sze ad cotact pressure at all the axal segmets Fuel ceter temperatures of all the axal segmets have coverged? Next tme step Fg... Calculato flow of thermal aalyss - 9 -

36 JAEA-Data/Code -5.. Coolat ethalpy crease model I ths model, mass flux ad pressure durg each tme step are assumed to be costat (.). Ethalpy of the coolat (water or steam) H(z, t)(j/kg) at a axal posto z (m) ad tme t (s) ca be gve by the tegral of ts hstory begg from a let of the coolat, as follows: t V ( z ) Q( z ) H ( z, t) = H (, t ) dτ (..) t S( z ) Here, H(, t ) : ethalpy of coolat at the let (J/kg), t m : the tme (s) whe a coolat located at a axal posto z at tme t passes through the let, z ( τ ): axal posto (m) of the coolat at tme τ, whch was postoed at z at tme t. V : specfc volume (m /kg) Q : heat flux at ut legth (W/m) S : flow cross secto (m ) τ : tme. Focusg o the tme step betwee tme t ad t Δt, Eq. (..) ca be rewrtte as Hzt (, Δt) = Hz (, t) t Δt t V( z ) Q( z ) dτ (..) Sz ( ) where z : axal coordate (m) of a coolat at tme t, whch s postoed at z at tme t Δt. Next, the actve legth of fuel rod s dvded equally to sub-segmets, rrespectve of the segmet legths desgated by put data. Fg... shows the ethalpy crease model for ths -sub-segmet, whch the axal elevato z s set to be a represetatve pot of sub-segmet, ad a sub-segmet cotag the axal locato z s set to be k. Whe a coolat whch s postoed at z at tme t Δt moved from z to z, f we desgate the tme requred for the coolat to pass through the segmet to be δt, a tme step cremet Δt ca be expressed as follows. Δt = δt δt δt δt (..) k k y rewrtg the tegral Eq.(..) usg the average values of V, Q ad S of the segmet, V, Q ad S ad also δt, ethalpy of the segmet, H, at t Δt ca be obtaed as - -

37 JAEA-Data/Code -5 VQ H ( t Δt) = H( t) δ t (..4) S = k z ode segm. z δ t ode segm. - δ t ode - ode k z k segm. k δ t k z ode k z k Fg... Coolat ethalpy crease model Δt ca be obtaed as follows usg the flow velocty at the segmet as u (m/s) δ t ΔL ΔL SΔL = = = u GV WV (..5) where ΔL : -th sub-segmet legth (m) G : mass flow rate per ut cross sectoal area (kg/m s) W : mass flow rate (kg/s) Ad ΔL ca be obtaed as - -

38 JAEA-Data/Code -5 ΔL z z ( = ) = z z ( k ) zk z ( = k) (..6) H (t ) ca be obtaed by terpolatg the ethalpes at z k ad z k as ( ) ( ) = ( ) ( ) ( ) H t H t H t H t k k k z z k zk z k (..7) Fg... shows the calculato sequece. Here, the ethalpy at a represetatve pot o a segmet s the average value of ethalpes at the upper ad lower odes of the segmet. Here, FEMAXI does ot accommodate sgle-steam-phase let codtos wth respect to the relatoshp betwee temperature ad pressure. Whe let temperature ad pressure of coolat are gve by put, the steam table corporated FEMAXI-7 s capable of detfyg the let temperature as a temperature whch exceeds the saturato temperature, due to the roudg of fgures ad umercal errors the let temperature data. I such cases, put ethalpy caot be determed. Thus, whe let temperature s determed to be the saturato temperature or hgher at the coolat pressure, the let temperature s assumed as the saturato temperature for a sgle lqud phase the code, ad the let ethalpy may be determed. (Refer to ame-lst parameter SUCL) ased o the ethalpy dstrbuto obtaed at the equal-legth sub-segmets by the above method, the ethalpy dstrbuto for the put axal segmets s derved by terpolatg Eq.(..7). A flow of the calculato s show Fg

39 JAEA-Data/Code -5 Ilet temperature Pressure Ilet ethalpy Ethalpes of the sub-segmets at the former tme step Ilet flow velocty Mass flow rate A part of a flowg coolat, X, asceds ad whe t reaches a posto of a ode of a axal segmet, the ethalpy at the part X the prevous tme step s calculated usg axal dstrbuto of the ethalpy the prevous step ad the axal posto of the part X. Whe the part X s udged to be postoed below the coolat let, the ethalpy of the posto X the prevous step s set to be equal to the ethalpy of the coolat at ts let. Heat flux at a segmet Calc. of ethalpes at the odes of sub-segmets Calc. of ethalpy at the represetatve pot of each ode of the axal segmets by terpolatg the ethalpes of odes of the sub-segmets. Fg... Process of ethalpy crease calculato.. Determato of claddg surface heat trasfer coeffcet I FEMAX, claddg surface heat trasfer coeffcet s determed maly by RELAP5/MOD model (.). () Classfcato of bolg state olg s classfed to saturato bolg ad surface bolg (sub-cool bolg) depedg o the lqud temperature, to -tube bolg ad pool bolg depedg o the presece of flow, ad to ucleate bolg, trasto bolg ad flm bolg depedg o the mode of pheomeo. Fgure..4 qualtatvely shows the bolg modes ad heat - -

40 JAEA-Data/Code -5 trasfer characterstcs terms of the heat flux ad magtude of super-heatg of the heat trasfer surface. The magtude of super-heatg of the heat trasfer surface ΔT sat show o the horzotal axs Fg...4 ca be gve by Δ = T T (..8) Tsa t w where T sat s obtaed from coolat pressure ad temperature usg the steam table. sat D Heat flux log q E C A Nucleate Traset bolg A: olg tato pot, : Crtcal heat flux pot, C: Local mmum heat flux pot. Sgle phase bolg Flm bolg Magtude of super-heatg of heat trasfer surface logδt sat Fg...4 Classfcato of bolg ad heat-trasfer characterstc curve () Sgle-phase heat trasfer I the sgle-phase rego (o-bolg rego) show Fg...4, the Dttus-oelter equato s used for the FEMAXI calculato of -tube forced covecto heat trasfer coeffcet. Dttus-oelter equato (.) ( ) q = h T w T (..9) Here, h =. K D e Pr.4 Re.8-4 -

41 JAEA-Data/Code -5 Re = G D e where k : coolat thermal coductvty (W/m K) μ D e : equvalet thermal hydraulc dameter (m) V : coolat velocty (m/s) ρ : coolat desty (kg/m ) μ : coolat vscosty (kg/m s) Pr : Pradtl s umber T : coolat temperature (K) K : thermal coductvty (W/m K) Re : Reyold s umber(-) G : mass flow rate(kg/m s) () Nucleate bolg heat trasfer I ucleate bolg, temperature of a heat-trasfer surface decreases perodcally ear the saturato temperature due to rapd vaporzato of th lqud flms remag at the bottom of bubbles, whch leads to the realzato of extremely hgh thermal coductvty. I FEMAXI-7, ucleate heat trasfer s gve by the sum of heat trasfer from surface to water sub-cooled state ad heat trasfer from surface to water ucleate-bolg state. The Dttus-oelter equato gve by Eq.(..9) s appled to calculate the heat trasfer from heated surface to sub-cooled water, ad ether Che s equato or Jes-Lottes equato s used to calculate the heat trasfer from surface to ucleate bolg water. The equato for the ucleate heat trasfer s: ( T T ) = h ( T T ) h ( T T ) q = h (..) where q : heat flux (W/m ) W W W h : claddg surface heat trasfer coeffcet (W/m K), W h w : heat trasfer coeffcet to sgle phase (o-bolg, sub-cooled) water (W/m K), determed by usg the Dttus-oelter equato. h : heat trasfer coeffcet to bolg water (W/m K), determed by usg ether Che s equato or Jes-Lottes equato. T w : claddg surface temperature (K), T : coolat temperature (K), ad T sat : saturato temperature (K). sat - 5 -