Title Improvement o Accuracy and Reliability on BWR Therm Drag Force Models Author(s) 尾崎, 哲浩 Issue Date 2018-09-25 DOI 10.14943/doctoral.k13343 Doc URL http://hdl.handle.net/2115/71808 Type theses (doctoral) File Inormation Tetsuhiro_Ozaki.pd Instructions or use Hokkaido University Collection o Scholarly and Aca
Improvement o Accuracy and Reliability on BWR Thermal-Hydraulic Analysis Code by Newly Modiied Interacial Drag Force Models by Tetsuhiro Ozaki A dissertation submitted in partial ulillment o requirements or the Degree o Doctor o Philosophy in Engineering Division o Energy and Environmental Systems Graduate School o Engineering Hokkaido University Sapporo, JAPAN August 2018
Table o Contents 1. Introduction... 1 1.1 Historical background o thermal-hydraulic saety analysis codes... 1 1.2 Application o best estimate codes and its reliability... 3 1.3 Thesis objectives... 4 1.4 Outline o the thesis... 7 Reerences... 10 2. Constitutive Equations or Vertical Upward Two-Phase Flow in Rod Bundle... 12 2.1 Introduction... 12 2.2 Flow regime map... 13 2.3 Void raction... 17 2.4 Void raction and relative velocity covariance... 20 2.5 Interacial area concentration... 22 2.5.1 Interacial area correlations... 22 2.5.2 Interacial area transport equation... 23 2.6 Wall riction... 24 2.6.1 Single-phase riction actor... 25 2.6.2 Two-phase multiplier... 27 2.7 Conclusions... 29 Reerences... 46 3. Eect o Void Fraction Covariance on Two-Fluid Model Based Code Calculation in Pipe Flow... 52 3.1 Introduction... 52 3.2 Interacial drag term or one-dimensional two-luid model... 54 3.2.1 Derivation o the interacial drag term... 54 3.2.2 Constitutive equations or distribution parameter and drit velocity... 56 3.3 Evaluation o void raction in circular pipe using TRAC-BF1 code... 60 3.4 Results and Discussions... 61 3.4.1 Steady-State Conditions... 61 3.4.2 Eect o covariance on drit velocity term... 62 3.4.3 Momentum equation in one-dimensional two-luid model... 64 3.4.4 Transient Conditions... 67 i
3.5 Conclusions... 68 Reerences... 87 4. Development o One-Dimensional Two-Fluid Model with Consideration o Void Fraction Covariance Eect... 89 4.1 Introduction... 89 4.2 Momentum Equation in One-dimensional Two-luid Model... 90 4.2.1 Closure Relations Considering Void Fraction Distribution... 91 4.2.2 Closure Relations or Uniorm Void Fraction Assumption... 94 4.3 Eect on Code Calculation due to Approximations... 96 4.3.1 Code Model (Constitutive Relations) and Calculation Conditions... 96 4.4 Results and Discussions... 99 4.4.1 Steady-state Condition... 99 4.4.2. Transient Condition... 100 4.5 Conclusions... 102 Reerences... 116 5. Code Perormance with Improved Two-Group Interacial Area Concentration or One-Dimensional Forced Convective... 119 Two-Phase Flow Simulation... 119 5.1 Introduction... 119 5.2 One-dimensional interacial drag model with drag coeicient and interacial area concentration... 121 5.2.1 Constitutive relation or deriving drag orce coeicient... 123 5.2.2 Closure relations to evaluate drag orce term... 127 5.3 Calculation case or validating the interacial drag model with C D approach. 129 5.3.1 Experimental data as separate eect test... 129 5.3.2 Sensitivity analysis accounting or an uncertainty o interacial drag orce term... 130 5.3.3 Analysis o numerical stability with crossing low regime boundary... 131 5.3.4 Transient calculation or nuclear reactor plant... 131 5.4 Results and Discussions... 132 5.4.1 Validation results with separate eect tests data... 132 5.4.2 Sensitivity o interacial drag orce term... 133 5.4.4 Results o transient analysis o nuclear power plant... 135 ii
5.5 Conclusions... 136 Reerences... 160 6. Conclusions... 163 Acknowledgments... 168 iii
Nomenclature a a i b C C i C 0 C D w coeicient interacial area concentration coeicient Chisholm s parameter interacial drag coeicient distribution parameter drag coeicient C wall drag coeicient o liquid phase C wg wall drag coeicient o gas phase C α void raction covariance C α relative velocity covariance C α C d B D base D d D H drit velocity covariance asymptotic distribution parameter bubble departure diameter drag diameter o cap bubble drag diameter hydraulic equivalent diameter D Sm Sauter mean diameter DR F w g g 1 G j j j g La, Lo dumping ratio pressure drop due to wall riction Fanning riction actor gravitational acceleration actor mass lux mixture volumetric lux supericial liquid velocity supericial gas velocity Laplace length M ik interacial drag orce iv
M τ k viscous and turbulent shear stress m d m rel m relab. n N a i mean absolute error mean relative deviation mean absolute relative deviation exponent non-dimensional interacial area concentration N La non-dimensional Laplace length N Re b bubble Reynolds number N vr relative velocity ratio N µ viscosity number N ρ p, P p c Q r R w Re density ratio pressure critical pressure bundle power radial distance radius o a pipe Reynolds number Re2 φ, liquid phase Reynolds number s d t v standard deviation time velocity v gj v r + V gj drit velocity relative velocity between phases non-dimensional drit velocity W X * v gj X + 3 drit velocity ratio mass low rate Martinelli s parameter speciic value o positive third peak v
X + 4 speciic value o positive ourth peak X steady speciic value at an initial steady-state condition x exit area-averaged low quality at the exit o heated region z axial position Greek symbols α α 0 void raction void raction at center o a pipe α t arget target void raction predicted by drit-lux model ρ density dierence ε ε BI ε sh ε w energy dissipation rate per unit mass bubble induced energy dissipation shear induced energy dissipation wall surace roughness φ momentum source deined in chapter 3 2 φ φ S two-phase wall riction multiplier correction actor Φ Phase Change sink/source term o interacial area concentration due to phase change Φ Pressure Change sink/source term o interacial area concentration due to pressure change Φ sink sink term o interacial area concentration due to bubble coalescence Φ source source term o interacial area concentration due to bubble η breakup actor µ viscosity ν kinematic viscosity vi
ρ σ density surace tension τ w wall shear stress ψ ξ ζ shape actor uncertainty actor or drag coeicient actor or relative velocity ratio Subscripts 1 value or group 1 bubble 2 value or group 2 bubble B BB crit ilm g init k L bubbly low condition bulk boiling condition critical value liquid phase liquid ilm gas phase initial value liquid or gas phase laminar region L T transition region between laminar and turbulent regions m max P SB T w w cov w/o cov mixture maximum value pool condition subcooled boiling condition turbulent region wall With covariance Without covariance Superscripts + non-dimensional value vii
Mathematical symbols area-averaged quantity void-raction-weighted mean quantity asymptotic value viii
1. Introduction 1.1 Historical background o thermal-hydraulic saety analysis codes Since the irst nuclear power plant, Dresden-1, began commercial operation in the United States, ensuring a high level o saety has been a primary concern. Sophisticated analytical methodology and system calculation codes have always been explored to accurately predict thermal hydraulic behavior within various components o a nuclear power plant, such as the reactor core, steam separator, steam generator, jet pump, pipe, and so on. Nuclear energy is attractive, compared to other power generation methods, due to high energy density, low uel costs, and suppression o CO 2 emissions. However, limited public acceptance o nuclear power exists due to signiicant saety concerns, especially ater the severe accidents occurring at TMI-2 in 1979, Chernobyl in 1986, and the Fukushima Daiichi nuclear power plant in 2011. Thereore, the implementation o thermal-hydraulic simulations in the analysis o nuclear power plants is crucial to ensure design integrity, develop countermeasures to prevent reactor meltdowns in accident scenarios, and to provide useul inormation or reactor operators. In the 1960 s, low computational perormance restricted the selection o the analytical method to a homogenous approach. Under the homogeneous low assumption, equal temperature and velocity in the gas and liquid phases are assumed. This approximation allows or three undamental equations o mass, momentum, and energy to be solved when determining the thermal-hydraulic behavior o two-phase low. The idea o the drit-lux model, later proposed by Zuber and Findlay [1-1], allowed the gas-phase velocity to be approximated by a drit velocity and distribution parameter, which represent the dierence between gas velocity and mixture volumetric lux, the spatial covariance o mixture volumetric lux and void raction, respectively. Ishii [1-2] extended the drit-lux model to develop the distribution parameter and the drit velocity models, which enhanced the applicability o the drit-lux model in thermal-hydraulic simulations. In the 1970 s, thermal-hydraulic codes implemented the drit-lux model through the use o our undamental equations. In the 1980 s, increases in computational perormance became evident and codes, such as RELAP5, TRAC, etc., began to implement the two-luid model. The two-luid model uses separate mass, momentum, and energy equations or the gas and liquid phase, respectively. The individual treatment o the gas and liquid phase allows or the elimination o the homogenous low assumption. Thereore, this more rigorous evaluation can provide more precise calculation results, but the complexity o the analysis increases signiicantly. Since the two-luid model solves six conservation equations, several constitutive equations representing mass, 1
momentum, and energy transer terms between gas-liquid interace are needed to close the model. Interacial transport o mass, momentum, and energy is dependent on the interacial area concentration and low structure o two-phase low, which depend on low velocity, void raction, geometrical condition, etc. Two-luid analysis codes developed around the 2000 s introduced a low regime map model, which allowed or speciication o appropriate constitutive equations in accordance with speciic two-phase low regimes, such as the dispersed bubbly low, churn low, annular low, and droplet low regimes. Recently, high-perormance computers have allowed or the prediction o two-phase low behavior to be reined by implementing an interacial area concentration equation in two-luid codes. The interacial area concentration equation can estimate the time and spatial dependent value o interacial area concentration by accounting or the change o interacial structure in two-phase low through introducing source terms to model bubble breakup, coalescence, expansion, and phase change. Traditionally, a low regime map model is not necessary i the interacial area concentration equation is introduced in a two-luid code because the value o interacial area concentration is obtained through mechanistically ormulated models representing physical processes o bubble interacial behavior. However, many experimental works must be required to develop reliable constitutive relations or source terms and to obtain suicient databases over the wide range o low conditions needed to evaluate the relations. High computational costs, compared to the existing low regime map based two-luid model, are associated with two-luid codes because additional conservation equations are needed to solve the interacial area transport equation. The historical evolution o thermal-hydraulic simulation codes shows that as computational perormance has increased the implementation o more rigorous methodology has improved prediction o two-phase low behaviors. This improved accuracy has allowed us to increased saety and economic eiciency o a nuclear power plant. However, the number o constitutive relations required to close the conservation equations have increased, resulting in the credibility o results obtained by analysis codes to become largely dependent on the idelity o the implemented constitutive relations. It is important to utilize a computational code that determines the applicability o implemented constitutive relations to the problem o interest and selects proper methodology to achieve the highest level o saety possible. Relevant physical phenomena and computational behavior should be studied and compared to realize improved accuracy in uture saety analysis codes. The developed constitutive relations should then be careully selected based on the recognition o real physical mechanisms and model sensitivity. 2
1.2 Application o best estimate codes and its reliability Two-luid thermal-hydraulic codes like TRAC-BF1, TRACE, RELAP5, and so on are classiied as best estimate codes and are expected to improve code accuracy compared to codes using the homogenous low approach. These codes have been utilized to simulate nuclear power plant behavior during AOOs (Anticipated Operational Occurrences) or anticipated severe accident scenarios. Such best-estimate codes solve the six basic conservation equations with many constitutive equations, which are required to couple the conservation equations. Thereore, the relationship between inputs and results o simulations become quite complicated, and the eect o each constitutive equation on simulation outputs is unclear. The credibility o an analysis code, or the simulation o an anticipated operating scenario, should be assessed through conirmation o proper code design with the intended algorithms (veriication) and the ability to simulate the required physical phenomena through proper mathematical models (validation). The code validation process begins by identiying a speciic power plant and scenario. Then the important phenomena to be considered must be identiied and summarized in a PIRT (Phenomena Identiication and Ranking Table) [1-3]. Highly ranked phenomena, determined by a PIRT, have a signiicant inluence on saety parameters when licensing criteria assessed or a speciic simulation scenario. The eligibility o the physical models to a real power plant and a speciic scenario in highly ranked phenomena determines the credibility o the code. Thereore, it is necessary to validate the physical models o highly ranked phenomena based on appropriate experimental databases. Ideally, these experimental databases should be obtained by reproducing practical conditions. Otherwise, the scalability o a database should be assessed by considering whether the experiment and physical model can be extended to realistic low conditions or the use in the simulation o actual phenomena. Moreover, EMDAP (Evaluation Model Development and Assessment Process), ruled by the US regulatory guide 1.203 and V&V methodology require the identiication o model uncertainties based on the inormation about idelity and pedigree o each physical model, and interaction with the other phenomena. As can be seen in V&V methodology [1-4], the ollowing conirmation process should be perormed to assure the model credibility and identiication o model uncertainty. 1) The physical concept o the model 2) Adequacy o the experimental database 3) Model prediction ability to various experimental data 4) Scalability o experiment and model 3
5) Eect o approximated and neglected phenomena Mechanistic models o physical phenomena implemented in simulation codes are becoming more detailed and complicated, which necessitates increased analysis to determine code credibility. Methodology such as CSAU (Code Scaling, Applicability and Uncertainty), EMDAP, and V&V provide procedures to enhance overall code credibility by selecting the most important physical phenomena that need elaboration, and allow or prevention o time and resource consumption caused by reckless development. 1.3 Thesis objectives Based on the background to enhance the reliability o a nuclear saety analysis code, it becomes necessary to conirm the idelity o the code and to identiy how precisely the code can simulate the anticipated scenario. Traditionally, saety analysis codes have been developed to simulate speciic experimental data. The code models are not rigorous containing approximation and compensating errors may exist i the model parameters have been adjusted to simulate experimental data. Additionally, i the experimental data with prototypic conditions are not available to develop the codes, the scalability o the code models are the problem to assure the codes can simulate prototypic plant behavior as well as the experimental data. Eliminating the approximation o the codes is one o an approach to resolve this problem. Thereore, it is required to improve the reliability o a nuclear saety analysis code by means o the proper usage o basic equation and selection o constitutive equations. Study o a rigorous interacial drag model and improvement o the momentum equation used in one-dimensional two-luid analysis codes are the ocus o this thesis. Figure 1-1 schematically shows the relationship between the research topic o this thesis and existing research about interacial drag models utilized in one-dimensional two-luid analysis codes. A momentum transer term between the liquid and gas phases has a signiicant role in two-luid analysis codes and is used to calculate void raction, a characteristic parameter o two-phase low. It is well known that void raction impacts nuclear thermal power eedback, pressure loss, low distribution within a core, two-phase water level, low induced vibration, etc. in a light water reactor. Depending on the scenarios to be simulated, the phenomenon o interacial momentum transer in a core region may be highly-ranked in a PIRT or many anticipated scenarios. One-dimensional two-luid analysis codes are typically selected to perorm the saety analysis o nuclear power plants, since detailed three-dimensional codes, such as 3D-CFD codes, are 4
inappropriate to simulate such a large and complicated system due to extremely high computational demand. In one-dimensional two-luid analysis codes, the conservation equations are solved, to simulate two-phase low behavior, through use o physical quantities that are area-averaged over a speciic geometrical low area. Thereore, each source term implemented in the area-averaged conservation equations must account or the eect o area-averaging. The interacial drag term, which is an important source term in the momentum equation, can be ormulated based on parameters such as relative velocity between gas and liquid phase, drag coeicient, interacial area concentration, and so on. It is also necessary to consider the eect o void raction and velocity spatial distribution on the area averaged value o the interacial drag term. Andersen and Chu [1-5] proposed an area-averaged interacial drag model based on Ishii and Mishima[1-6] s ormulation o area-averaged relative velocity through the introduction o distribution parameter and drit velocity. Namely, this area-averaged interacial drag model is utilized in one-dimensional two-luid analysis codes by applying the idea o the drit-lux model to consider the void raction and mixture volumetric lux proiles. On the other hand, Brooks et al. [1-7, 1-8] indicates the necessity o introducing void raction covariance (spatial auto-covariance o void raction), which is a result o the rigorous area-averaging o relative velocity. Additionally, Hibiki and Ozaki [1-9], Ozaki and Hibiki [1-10] developed constitutive equations o void raction covariance or piping and rod bundle, respectively. The approximation o uniorm void raction proile, pointed out by Brooks et al. [1-7, 1-8], can be excluded by implementing these developed covariance models into one-dimensional two-luid analysis codes, which contributes to an improvement in the rigorous treatment o saety analysis. However, the other problem still needs to be solved. The wall shear riction term included in the momentum equation o a current existing one-dimensional two-luid analysis code is derived based on the assumption o uniorm void raction proile. No existing knowledge has been ound, regarding the proper expression o the momentum equation with consideration o the void raction covariance model. Additionally, the eect o the uniorm void raction proile approximation should be assessed by comparing the obtained results with a rigorous ormulation treating void raction covariance. These discussions are necessary to judge the validity and credibility o the current code approximation. Since the interacial drag orce term is dependent on the geometrical interacial structure o two-phase low, consideration o low regime allows or the determination o this term. Kelley [1-11], however, pointed out that discontinuity o calculation results and numerical instability was caused by constitutive equation transition as two-phase low changes low regime. In response, to address this problem, the interacial area concentration transport equation (IATE) has been 5
developed. IATE can represent the interacial structure o two-phase low without using a low regime map model, so Talley et al. [1-12, 1-13] implemented IATE in the TRACE code to assess its applicability. The usage o IATE still has the ollowing problems to overcome when accounting or its compatibility to V&V methodology, namely, 1) Results o IATE are strongly dependent on an initial value which cannot readily be determined. 2) Sink/source terms included in IATE require many coeicients, whose validity cannot readily be conirmed. 3) Coeicients o source terms are dependent on the geometrical condition o the low path. 4) Constitutive relations to determine source terms are not scalable to the prototypic operational conditions in a light water reactor (LWR) because these correlations have been developed based on databases obtained under steady-state air-water low at atmospheric pressure. Recently, Ozar et al. [1-14], Schlegel and Hibiki [1-15], and Shen and Hibiki [1-16] developed constitutive equations or predicting interacial area based on the idea o a two-group model. Bubbles in two-phase low are categorized according to a characteristic dierence o interacial drag, and thus spherical bubbles in dispersed bubbly low are considered as group-1 bubbles. Whereas, Taylor bubbles in slug low, cap bubbles in cap bubbly or cap turbulent low, and bubbles in churn turbulent low regimes are considered as group-2 bubbles. The interacial area or group-1 and group-2 bubbles can be identiied by the proposed constitutive equations and possibly by providing an interacial drag orce term in a one-dimensional two-luid analysis code. Although the introduction o these interacial area concentration correlations can resolve some IATE problems, such as the complexity o equations and increased computational cost, the existing work is needed to determine the adequacy o the constitutive equations when implemented in a one-dimensional two-luid code and the eect o uncertainty on the prediction o void raction. Discussion o how the interacial area concentration aects the results o a two-luid analysis code is another goal o this thesis. Sink/source terms implemented in IATE shall be modeled based on physical phenomena like coalescence and breakup o bubbles, bubble expansion due to pressure gradient, phase change, and so on. The existing databases are insuicient to scale the data to the prototypic operating conditions o a real power plant, so the databases should be extended to a range comparable to prototypic low conditions, geometrical condition, etc. In general, the cost to obtain such databases is signiicant. Thereore it should be careully discussed whether the investments or IATE 6
development will contribute to improved accuracy in saety analysis codes. Based on the above mentioned existing works and unresolved issues, the ollowing studies have been perormed to determine useul knowledge about the importance o the interacial drag model in a one-dimensional two-luid analysis code. 1) A rigorous ormulation o the momentum equation or a one-dimensional two-luid analysis code, considering void raction covariance. 2) A comparison between the existing model based on Andersen and Chu [1-5] and the above rigorous approach to clariy the eect o the approximation on uniorm void raction distribution. 3) Development o an interacial drag model or a one-dimensional two-luid analysis code, based on the two-group interacial area concentration correlation models developed by Ozar et al. [1-14] and Schlegel and Hibiki [1-15], and conirmation o the newly developed model s applicability. 4) Clariy the eect o interacial area concentration and drag coeicient on void raction through uncertainty analysis o the interacial drag model, based on the two-group interacial area concentration correlation model. 1.4 Outline o the thesis The ollowing chapters discuss void raction covariance, momentum equation development, and the interacial drag orce term. Chapter 2 reviews existing constitutive equations applied to rod bundle geometry in two-luid analysis codes. The eect o void distribution covariance on a one-dimensional two-luid analysis code has been studied or pipe and rod bundle and is discussed in chapter 3 and chapter 4, respectively. The appropriate ormulation o the momentum equation, with consideration o a void raction covariance model, is also discussed in these chapters. Additionally, the interacial drag orce term may aect transient behavior due to the dierence o characteristics in momentum coupling between phases. Thereore, dierences in calculation results or transient scenarios are investigated to compare cases with and without consideration o covariance. Chapter 5 discusses the ormulation o the interacial drag orce term, based on the two-group interacial area concentration correlations. This methodology, a so-called C D approach, is validated against several representative separate eect tests and determination o numerical instability, which might occur due to low regime transition, is discussed. Chapter 5 also investigates the inluence o uncertainties in interacial area concentration and drag coeicient, since 7
these uncertainties are normally considered to be signiicant. The sensitivity o interacial area concentration and drag coeicient is also quantiied to provide useul inormation to determine urther development required in saety analysis codes. Lastly, the indings obtained in these studies are summarized and concluded in chapter 6. 8
Two-Fluid Analysis Code One-Dimensional model Theme or the Doctoral Study Related Study Fundamental Equations (Mass,Momentum,Energy Conservation Eqs.) Constitutive Equations Knowledge rom PIRT Establish Momentum Equation Accounting or the Covariance Interacial Drag Model Distribution Parameter Model Ishii, 1977 (Pipe) Ozar et al., 2012 (Annulus) Ozaki and Hibiki, 2015 (Rod Bundle) Clark et al., 2014(Low Flow Condition) Andersen Approach CD Approach Solution to the Numerical Instability Problem due to Flow Regime Transition Kelly (1996) Covariance Model Drit Velocity Model Drag Coeicient Interacial Area Concentration Correlation Interacial Area Transport Equation Talley et al., 2011; 2013 Hibiki-Ozaki (2017) Ozaki-Hibiki (2018) Ozar et al.(2012) Schlegel Hibiki (2015) Constitutive Equations or Sink/Source terms Implementation to the Code Evaluation o Eect o Covariance Implementation to the code Model Validation Estimation o Numerical Stability Insuicient Experimental Database Problem o Initial Value Complexity o the Model Chuang-Hibiki (2015) Chapter 3 Chapter 4 Chapter 5 Enhance the Applicability o One-Dimensional Two-Fluid Analysis Code to Nuclear Saety Simulations Figure 1-1 Schematic o the relation between this research and existing research related to one-dimensional interacial drag model. 9
Reerences [1-1] Zuber N, Findlay JA. Average volumetric concentration in two-phase low systems. Journal o Heat Transer 1965; 87: 453-468. [1-2] Ishii M. One-dimensional drit-lux model and constitutive equations or relative motion between phases in various two-phase low regimes. USA: Argonne National Laboratory; 1979. (ANL-77-47). [1-3] Boyack B, Duey R, Griith P,et al. Quantiying reactor saety margins: Application o code scaling, applicability, and uncertainty evaluation methodology to a large-break, loss-o-coolant accident. USA: US NRC; 1989. (NUREG/CR-5249/EGG-2552). [1-4] Atomic Energy Society o Japan. Guidline or credibility assessment o nuclear simulations: 2015. Tokyo: AESJ; 2016. (AESJ-SC-A008:2015). Japanese. [1-5] Andersen JGM, Chu KH. BWR reill-relood program constitutive correlations or shear and heat transer or the BWR version o TRAC. Washington DC: US NRC; 1983. (NURG/CR-2134/GEAP-24940). [1-6] Ishii M, Mishima K. Two-luid model and hydrodynamic constitutive relations. Nuclear Engineering and Design 1984;82:107-126. [1-7] Brooks CS, Ozar B, Hibiki T, et al. Two-group drit-lux model in boiling low. International Journal o Heat and Mass Transer 2012;55:6121-6129. [1-8] Brooks CS, Liu Y, Hibiki T, et al. Eect o void raction covariance on relative velocity in gas-dispersed two-phase low. Progress in Nuclear Energy 2014;70:209-220. [1-9] Hibiki T, Ozaki T. Modeling o distribution parameter, void raction covariance and relative velocity covariance or upward steam-water boiling low in vertical pipe. International Journal o Heat and Mass Transer 2017;112:620-629. [1-10] Ozaki T, Hibiki T. Modeling o distribution parameter, void raction covariance and relative velocity covariance or upward steam-water boiling low in vertical rod bundle. Journal o Nuclear Science and Technology 2018;55:386-399. [1-11] Kelley JM. Thermal-hydraulic modeling needs or passive reactors. Proceedings o the OECD/CSNI Workshop on Transient Thermal-Hydraulic and Neutronic Codes Requirements, Annapolis, Maryland, USA, Nov. 5-8, 1996. [1-12] Talley JD, Kim S, Mahay J, et al. Implementation and evaluation o one-group interacial area transport equation in TRACE. Nuclear Engineering and Design 2011;241:865-873. [1-13] Talley JD, Worosz T, Kim S, Bajorek S, Tien K. Eect o bubble interactions on the prediction o interacial area in TRACE. Nuclear Engineering and Design 2013;264:135-145. 10
[1-14] Ozar, B., Dixit, A., Chen, S.W., Hibiki, T., Ishii, M., Interacial area concentration in gas-liquid bubbly to churn-turbulent low regime. International Journal o Heat and Fluid Flow 38 (2012) 168-179. [1-15] Schlegel, J.P., Hibiki, T., A correlation or interacial area concentration in high void raction lows in large diameter channels. Chemical Engineering Science 31 (2015) 172-186. [1-16] Shen X, Hibiki T. Interacial area concentration in gas-liquid bubbly to churn low regimes in large diameter pipes. International Journal o Heat and Fluid Flow 2015;54:107-118. 11
2. Constitutive Equations or Vertical Upward Two-Phase Flow in Rod Bundle 2.1 Introduction Two-phase lows are encountered in various industrial apparatuses such as chemical reactors, boilers, heat exchangers and nuclear reactors. Detailed three-dimensional two-phase low analyses using two-phase computational luid dynamics (two-phase low CFD) codes have been advanced or design and perormance analyses o industrial apparatuses [2-1]. However, the prediction accuracy o the two-phase low CFD does not reach suicient level or these purposes due to the diiculty o modeling in interacial area concentration, two-phase low turbulence, non-drag orce and wall nucleation source [2-2, 2-3] as well as lack o local two-phase low data to be used or validating the two-phase low CFD [2-4]. In a practical use o two-phase low analyses, one-dimensional analyses are common. For example, a nuclear reactor system is composed o many components such as reactor core, piping and saety components which make the system complicated. In order to simulate some accident scenario, the nuclear system behavior is the ocus. A low channel in each component is area-averaged and one-dimensional ormulation is used in a nuclear thermal-hydraulic system analysis code. In the nuclear thermal-hydraulic system analysis code, the two-luid model is oten utilized as modeled two-phase conservation equations [2-5]. The one-dimensional two-luid model is ormulated by averaging local time-averaged two-luid model over a low channel and is composed o six equations, namely mass, momentum and energy balance equations or gas and liquid phases. The two-luid model is considered one o most accurate two-phase low balance equations because it can treat thermal and kinematic non-equilibrium between two phases. In order to close the mathematical system o the two-luid model, numerous constitutive equations should be given. Figure 2-1 shows a typical code structure and important constitutive equations. Since constitutive equations are oten low-regime-dependent, accurate low regime transition boundaries should be identiied. A drit-lux type correlation is oten used or calculating the area-averaged relative velocity between phases. An interacial area correlation is important or calculating an available area or mass, momentum and energy transers. A correlation to predict a wall riction is indispensable in closing the momentum equation. Continuous eorts have been made to improve the prediction accuracy o these correlations and constitutive correlations have been well-established or a simple geometry such as a pipe. Due to the requirement to use best-estimate codes, CSAU (Code Scalability, Applicability, 12
and Uncertainty) methodology has been established. In the CSAU methodology, the scalability o constitutive correlations in terms o channel geometry (size and shape) and thermal-hydraulic conditions (pressure, temperature and velocity) should be assessed and the uncertainty o the correlations should be evaluated. One cornerstone o the CSAU methodology is to develop the Phenomenon Identiication and Ranking Table (PIRT) [2-7]. Some PIRT evaluation suggests that constitutive equations in a nuclear reactor core analysis may have a signiicant impact on the saety measure. In view o this, important constitutive equations or a rod bundle should be re-assessed and be improved to enhance the prediction accuracy. Recently, several improved constitutive equations have been proposed or low regime transition criteria, void raction, void raction covariance and relative velocity covariance and interacial area concentration in a rod bundle. This chapter discusses the state-o-the-art constitutive equations or low regime transition criteria, void raction, void raction covariance and relative velocity covariance and interacial area concentration in a rod bundle and reviews the constitutive equation or wall rictional pressure drop used in legacy one-dimensional nuclear thermal-hydraulic system analysis codes such as TRACE, RELAP5 and TRAC codes. 2.2 Flow regime map In a dynamic two-phase low, an interacial structure evolves spatially and temporally. Since the interaction between two phases occurs through the interace, the interacial structure signiicantly aects the mass, momentum and energy transers between two phases. The dependence o the interacial structure on low parameters is expressed as a low regime map or a low pattern map. Typical two-phase low regimes observed in a vertical channel are bubbly, slug, churn and annular low. In a large size channel, slug bubbles cannot exist due to its surace instability and the slug low regime is replaced with cap bubbly low and cap turbulent low regimes [2-8]. In a nuclear thermal-hydraulic system analysis code, a two-phase low regime is commonly determined by two parameters such as void raction and mass lux, and low-regime-dependent constitutive equations are used with some interpolation scheme between two dierent low regimes. Table 2-1 lists existing experimental studies in observing two-phase low regime map in vertical rod bundles. From the overall viewpoint, observed two-phase low regimes in the vertical rod bundles are bubbly, inely dispersed bubbly, cap bubbly, cap turbulent, churn and annular low regimes. It should be noted here that Venkateswararao et al. [2-9] adopted slug low regime instead o cap bubbly and cap turbulent low regimes. This may be due to a limited understanding o two-phase low characteristics in a large channel such as large bubble disintegration due to its 13
surace instability as o 1982. The two-phase low regimes observed in a relatively small bundle such as a 3 3 rod bundle may be dierent rom these in a large bundle such as an 8 8 rod bundle, because slug bubbles spanning over the bundle casing can exist in the small bundle. Venkateswararao et al. [2-9] proposed a phenomenological model to predict the transition boundaries between two-phase low regimes including slug low regime. As discussed above, the slug low regime should be replaced with cap bubbly and cap turbulent low regimes. Liu and Hibiki [2-15] perormed extensive literature survey o existing experimental low regime maps and existing two-phase low regime transition criteria model. They developed a phenomenological model to predict the two-phase low regime transition boundaries and demonstrated its validity by comparing their model with existing data taken in vertical rod bundles. The brie summary o the Liu and Hibiki s model is given below. Bubbly-to-cap bubbly low transition The bubbly-to-cap bubbly low transition criterion given by Eq. (2-1) was derived by assuming a signiicant increase in the bubble coalescence rate at the distance between bubbles being smaller than the bubble diameter. j 1 = 1 j g ( 0. 234+ 0. 066 ρ ρ C ) C 0 g 0 v gj (2-1) where j, j, g v, C, gj 0 ρ and g ρ are the supericial liquid velocity, supericial gas velocity, drit velocity, distribution parameter, gas density and liquid density, respectively. and are the area-averaged quantity and void-raction-weighted mean quantity, respectively. The distribution parameter and drit velocity are calculated by Ozaki and Hibiki s correlation [2-16] and Hibiki and Ishii s correlation [2-17], respectively. Cap bubbly-to-cap turbulent low transition The cap bubbly-to-cap turbulent low transition criterion considered "two-group" bubbles, namely, small bubble (group-1) and large-cap bubble (group-2). The transition criterion given by Eq. (2-2) was derived by assuming a signiicant increase in the small bubble coalescence rate at the distance between small bubbles being smaller than the small bubble diameter as well as a signiicant increase in the large bubble coalescence rate at the distance between large bubbles being smaller 14
than the large bubble diameter. j 1 v = 1 j 0. 51C C gj g 0 0 (2-2) Bubbly-to-dispersed bubbly low and dispersed bubbly-to-cap bubbly low transitions Two low transition criteria were proposed based on critical Weber number and a maximum allowable void raction. Criterion based on critical Weber number: σ ρg 5 6 ρ σ 5 9 2 3 ε = µ N 0. 0849 (2-3) where σ, ρ, g and ε are the surace tension, density dierence, gravitational acceleration and energy dissipation rate per unit mass. The viscous number, N µ, is deined by: N µ µ ρσ ρ σ g 1 2 (2-4) where µ is the liquid viscosity. Criterion based on maximum allowable void raction: α = 0. 52 crit (2-5) where α is the void raction. Cap turbulent-to-churn low transition The cap turbulent-to-churn low transition criterion was derived by assuming the void raction averaged over the entire region being larger than that averaged over the cap-bubble section as: 15
α crit 1 0. 813 ( 1) C j + v 0 3 2 { gdh } j + 0. 94 ρgd ρ ρ ρν H gj 1 18 0. 75 (2-6) where j, D and H ν are the mixture volumetric lux, hydraulic equivalent diameter and liquid kinematic viscosity, respectively. Churn-to-annular low transition Two transition criteria were proposed based on low reversal in the liquid ilm section along large bubbles and destruction o large cap-bubbles or large waves by entrainment or deormation. Criterion based on low reversal: 3 ρgdh j = ( α 0. 11 g ) (2-7) 2ρ g Criterion based on destruction o large cap-bubbles or large waves by entrainment or deormation: 1 4 ρσ g j N g 2 µ ρ g 0. 2 (2-8) Figure 2-2 compares Liu and Hibiki s model with a vertical upward air-water low regime map observed in an 8 8 rod bundle under an atmospheric pressure condition [2-15]. Open square, solid diamond, open triangular and solid triangular in Fig. 2-2 indicate bubbly, cap bubbly, cap turbulent and churn low regimes, respectively. Liu and Hibiki s model predicts the observed two-phase low regime transition boundaries well. The validity o Liu and Hibiki s model at the boundaries between bubbly and inely dispersed bubbly low regimes and between churn and annular low regimes are also demonstrated using the data taken by Liu et al. [2-13] and Venkateswararao et al. [2-9], respectively. Liu and Hibiki s model adopts the state-o-the-art two-phase low regime deinitions in a vertical rod bundle and has been validated by several experimental two-phase low regime maps. In view o these, Liu and Hibiki s model is considered 16
the state-o-the-art model to predict two-phase low regime boundaries in a vertical rod bundle. However, since its validation has been done using data taken under atmospheric pressure conditions, the applicability o Liu and Hibiki s model to high pressure conditions should be examined using data to be taken in a uture study. 2.3 Void raction Void raction is one o most important two-phase low parameters in characterizing gas raction o a two-phase low. The accurate prediction o void raction is a key to estimate actual coolant level in a nuclear reactor core under an accident. Void raction is also an important design parameter in various industrial apparatuses. A correlation based on the drit-lux model [2-18], namely a drit-lux correlation, is oten used or predicting area-averaged or one-dimensional void raction. The drit-lux model considers the relative velocity between phases through the drit velocity deined by: v v j (2-9) gj g where v is the gas velocity. g Averaging Eq. (2-9) over a low channel yields one-dimensional drit-lux model as: v g j g = C j + v 0 α gj (2-10) One-dimensional nuclear thermal-hydraulic system analysis codes use a drit-lux correlation to calculate area-averaged relative velocity between phases rom void-raction-weighted mean gas and liquid velocities, v and g v [2-19]. The distribution parameter is modeled by considering a scaling parameter such as a density ratio and a channel geometry. The drit velocity is modeled by a drag law or multi-particles. Table 2-2 lists existing experimental studies in measuring void raction in vertical rod bundles. Based on these existing data, several drit-lux type correlations were developed. Ozaki et al. [2-36] and Ozaki and Hibiki [2-16] perormed an extensive review o the existing drit-lux type correlations including Bestion s correlation [2-37], Chexal and Lellouche s correlation [2-38], Inoue et al. s correlation [2-39], Maier and Coddington s correlation [2-40] and Julia et al. s correlation [2-41]. Ozaki and Hibiki [2-16] developed a drit-lux type correlation based on 17
vertical upward boiling water low data taken in an 8 8 rod bundle under prototypic high pressure and temperature conditions as: ρg C = 1. 1 0. 1 (2-11) 0 ρ ( 1 39 ) { 1 exp ( 1. 39 j )} V = V exp. j + V (2-12) + + + + + gj gj,b g gj,p g where vgj V + gj ρgσ 2 ρ 1 4 and jg j + g ρgσ 2 ρ 1 4 (2-13) The subscripts o B and P denote the bubbly low and pool condition. The drit velocities or bubbly low and pool condition are calculated by Ishii s correlation [2-42] and Kataoka and Ishii s correlation [2-43]. Ozaki et al. [2-36] demonstrated no signiicant eects o the power distribution o a rod bundle on the distribution parameter. They also revealed that the spacer grid eect on the distribution parameter goes away within a short distance rom the spacer grid. Ozaki and Hibiki [2-16] discussed the eect o an unheated rod in a rod bundle on the distribution parameter and recommended Eq. (2-14) or a rod bundle with a large unheated rod. ρg C = 1. 08 0. 08 (2-14) 0 ρ Figure 2-3 compares Ozaki and Hibiki s correlation with void raction measured in a vertical 8 8 rod bundle. Blue broken and red solid lines indicate the calculated values using Eqs. (2-11) and (2-14), respectively. Ozaki and Hibiki s correlation agrees with the data well. The average relative error o Ozaki and Hibiki s correlation is determined to be ±4.36 % based on data taken under a wide range o test conditions such as pressure rom 0.1-12 MPa, mass lux rom 5-2000 kg/m 2 s, rod bundle casing size rom 79-140 mm, hydraulic equivalent diameter rom 9.8-21.7 mm and adiabatic and boiling lows. Ozaki and Hibiki [2-44] used a bubble-layer thickness model [2-45] or deriving the distribution parameter o a subcooled boiling low in a rod bundle as: 18
0. 701 { α } ρ g C = 1. 1 0. 1 1 exp 12. 1 0 ρ (2-15) which is applicable to D P = 0. 7 0. 9. D and P are the rod diameter and pitch 0 0 0 0 between neighboring rods, respectively. Equation (2-15) indicates that enhanced wall peaking in void raction distribution due to subcooled void near the rod lowers the distribution parameter in a subcooled boiling region and asymptotically approaches Eq. (2-11) in a bulk boiling region. Liu et al. [2-46] perormed an experiment using a vertical upward air-water bubbly low in a 5 5 rod bundle under an atmospheric pressure condition and collected void raction data. They developed the ollowing correlation to predict the distribution parameter o an adiabatic bubbly low in a rod bundle as: ρ D g Sm C = 1. 1 0. 1 1 exp 17 0 ρ D H (2-16) where D is the bubble Sauter mean diameter. Sm Equation (2-16) indicates that a lit orce acting on relatively small bubbles pushes bubbles towards the rod resulting in lowered distribution parameter. Chen et al. [2-33] and Clark et al. [2-47] perormed an experiment using a vertical upward air-water low in an 8 8 rod bundle under pool conditions and low liquid low conditions, respectively. As shown in Fig. 2-4, Clark et al. [2-47] ound that the distribution parameter increased due to a secondary low ormed in the rod bundle at low low conditions. Figure 2-4 indicates that the distribution parameter asymptotically approaches Eq. (2-11) with increased mixture volumetric lux. low low conditions. Clark et al. [2-47] developed a drit-lux type correlation applicable to Schlegel and Hibiki [2-48] and Kinoshita et al. [2-49] modiied the Clark et al. s correlation by considering a proper pressure scaling as: C C C ( ) ρ = g 1 0 ρ (2-17) C C L or j j = C H or j > j + + + + C max C max (2-18) 19
C H + 0.1 j ρ g = 1.1+ 1.84e F (2-19) ρ 1 ρ g F = min 1.70 582 max ρ 0 C + ( j C max ) C H 1 = ( j + g ) + 1 + j j C max L + (2-20) (2-21) j = m j + b ; + + C max 1 m = α C ; 1 crit 0 + Vgj α crit b = ; 1 α C crit 0 j j + ρgσ 2 ρ 1 4 ; j j + ρgσ 2 ρ 1 4 (2-22) α + ( ) = min 0.0284 + 0.125, 0.52 (2-23) crit j The notations o the symbols used in Eqs. (2-18) to (2-23) are given in Fig. 2-5. 2.4 Void raction and relative velocity covariance Brooks et al. [2-19, 2-50] have pointed out the importance o void raction covariance, C α, and relative velocity covariance, C α, in one-dimensional two-phase low analyses. In current one-dimensional nuclear thermal-hydraulic system analysis codes, the area-averaged relative velocity between phases is calculated by: v 1 C α 0 = v C v 1 α r g 0 (2-24) Brooks et al. [2-19] pointed out that void raction covariance or relative velocity covariance is missing in Eq. (2-24) and provided the correct orm o Eq. (2-24) as: 20
1 C α 0 v = C v C v r α g 0 1 α (2-25) 1 α C α 1 C α α (2-26) C α 2 α α α (2-27) Since the interacial drag orce term in momentum equations is proportional to the square o the area-averaged relative velocity, the correct orm o the area-averaged relative velocity should be used or an accurate prediction o void raction. In view o this, Ozaki and Hibiki [2-44] developed correlations o void raction covariance and relative velocity covariance or vertical upward two-phase lows in a rod bundle as: ( α) max ( C, C ) or α, SB α, BB 1 C ( α α crit ) ( ) ( ) ( ) 1 αcrit C = α > α α crit C α = α 1+ 1 or α α crit where α (2-28) C α 2 4 ρ g = 1+ { 9. 38( α 0. 5) + 0. 414 } 1 ( 1 α, BB ) ρ (2-29) 0. 190 C α =, SB 0. 855 α (2-30) α = 0. 84 (2-31) crit Figure 2-6 compares Ozaki and Hibiki s correlation with relative velocity covariance measured in a vertical 8 8 rod bundle. The igure shows that the relative velocity covariance reaches 2 at the area-averaged void raction o about 0.8. Ozaki and Hibiki s correlation agrees with the data well. The mean absolute error (bias) and standard deviation (random error) o Ozaki and Hibiki s correlation or the relative velocity covariance are determined to be -0.00241 and 0.0452 based on steam-water data taken under a wide range o test conditions such as pressure rom 21
1.0-8.6 MPa, mass lux rom 280-2000 kg/m 2 s and exit quality rom 0.0-0.25. Ozaki et al. [2-51] derived gas and liquid momentum equations by considering the void raction distribution. 2.5 Interacial area concentration Interacial area concentration, a, is one o most important two-phase low parameters in i characterizing available interacial area or mass, momentum and energy transer between two phases. The inverse o the interacial area concentration is one o the important length scales in a two-phase low characterizing a bubble size. The accurate prediction o the interacial area concentration is a key to estimate mass, momentum and energy transer in a two-phase low analysis. An interacial area correlation is oten used or predicting area-averaged interacial area concentration and the introduction o an interacial area transport equation into a code is also considered or predicting the dynamic behavior o the interacial area concentration [2-52, 2-53]. However, limited work has been conducted on developing an interacial area correlation and an interacial area transport equation in a rod bundle. The existing interacial area correlations and interacial area transport equation developed or simple geometries such as a pipe has been tested or their applicability to a two-phase low in a vertical rod bundle. 2.5.1 Interacial area correlations Hibiki and Ishii [2-54] simpliied an interacial area transport equation and developed a simple interacial area correlation or adiabatic bubbly lows. The developed interacial area correlation predicted 459 adiabatic bubbly low data taken in low channels including pipes and rectangular channels with an average relative deviation o ±22.0 %. Hibiki and Ishii s correlation was also compared with boiling bubbly low data taken in a vertical 3 3 rod bundle and its agreement with the data was airly well. Hibiki et al. [2-55] extended Hibiki and Ishii s correlation [2-54] to boiling bubbly lows. The developed interacial area correlation predicted 569 adiabatic bubbly low data and 343 boiling bubbly low data with averaged relative deviations o ±21.1 % and 31.0 %, respectively. The boiling database included R-12 data taken at 1.46 MPa, which simulated subcooled bubbly low under prototypic PWR condition. The correlations are given by: 0. 335 0. 239 N = 3. 02ηN α N a i La Re b η= 1 or adiabatic bubbly low [2-54] 0 170 0 138 η 1 22 α.. =. N ρ or boiling bubbly low [2-55] (2-32) 22
where 1 3 4 3 ε La ρ σ,,, (2-33) ν ρ ρg N ala N N La a i Re ρ i b g As shown in Fig. 2-7, Hibiki et al. s correlation agreed with the boiling bubbly low data taken in a vertical 3 3 rod bundle airly well. Ozar et al. [2-56] examined the prediction accuracy o interacial area correlations implemented in one-dimensional nuclear thermal-hydraulic saety analysis codes such as RELAP5 and TRAC-P codes using three databases (air-naoh two-phase low in a 25.4 mm pipe, air-water two-phase low in a 48.3 mm pipe, air-water two-phase low in an annulus with the hydraulic equivalent diameter o 19.1 mm). A total number o the data points was 127. A comparison between the correlations and the data demonstrated that the interacial area correlations in RELAP5 and TRAC-P codes ailed to predict the interacial area concentration in bubbly-to-churn low regimes. Ozar et al. [2-56] extended Hibiki and Ishii s correlation [2-54, 2-55] to high void raction region including slug and churn low regimes. The set o the correlation is given in Table 2-3. The notations o the symbols used in Ozar et al. s correlation are given in Fig. 2-8. Ozar et al. s correlation agreed with the above three datasets with an averaged relative deviation o ±30 %. Schlegel and Hibiki [2-48] simpliied an interacial area transport equation and developed a simple interacial area correlation or cap bubbly, cap turbulent-to-churn low in a large diameter pipe. The set o the correlation is given in Table 2-4. Schlegel and Hibiki s correlation agreed with the data taken in large diameter pipes with a bias and root mean square o -4.29 % and 22.6 %, respectively. As shown in Fig. 2-9, Schlegel and Hibiki s correlation was also compared with adiabatic two-phase low data taken in a vertical 8 8 rod bundle and its agreement with the data was airly well. 2.5.2 Interacial area transport equation A general orm o one-dimensional interacial area transport equation is expressed as: a i t a v i i a + = Φ Φ + Φ + Φ Source Sink Phase Change Pressure Change z (2-34) where t, v and z are the time, interacial velocity and axial location, respectively. i Φ Source is the interacial area concentration source term due to bubble breakup accompanied by the increase 23
o the bubble number density, whereas Φ is the interacial area concentration sink term due Sink to bubble coalescence accompanied by the decrease o the bubble number density. Φ PhaseChange is the interacial area concentration source or sink term due to phase change including boiling and condensation. Wall nucleation is accompanied by the increase o the bubble number density but bubble expansion and contraction through interacial heat transer are not accompanied by the change o the bubble number density. Φ is the interacial area concentration source PressureChange or sink term due to pressure change along a low direction, which is not accompanied by the change o the bubble number density. Yang et al. [2-57] assumed similarities between a narrow rectangular channel and a sub-channel in a rod bundle, and applied the models o interacial area concentration sink and source terms developed or a vertical narrow rectangular channel [2-58] with some modiications to a rod bundle. Five bubble interaction mechanisms considered in the model were bubble coalescence due to (1) bubble-bubble random collision induced by liquid turbulence and (2) due to wake entrainment and bubble breakup due to (3) turbulent impact on bubbles, (4) sharing o and (5) surace instability. The bubbles were treated in two-group, namely small bubbles or group-1 bubbles and large bubbles or group-2 bubbles. Figure 2-10 compares the axial development o interacial area concentration calculated by the interacial area transport equation with adiabatic two-phase low data taken in a vertical 8 8 rod bundle and its agreement with the data was airly well. However, the interacial area transport equation may not predict accurate interacial area concentration unless the initial condition is accurately given. 2.6 Wall riction Wall riction is one o important key parameters in momentum equations. In what ollows, constitutive correlations adopted in one-dimensional nuclear thermal-hydraulic saety analysis codes [2-59, 2-60, 2-61] are briely reviewed. wall riction is expressed as: riction In the codes, the pressure gradient due to dp = C v v + C v v w wg g g (2-35) dz where C and w C are, respectively, the liquid and gas riction actors where C = 0 or a wg wg 24
Pre-CHF (Critical Heat Flux) regime and C = 0 or a Post-CHF regime. The pressure gradient w in the Pre-CHF regime is given as: dp = C v v = φ dz riction 2 w 2G G D H ρ (2-36) where 2 φ, and G are the two-phase multiplier, Fanning riction actor and liquid mass lux, respectively. The liquid riction actor is expressed by: C w ( ) 2 2 1 2 α ρ = φ DH (2-37) Constitutive correlations or a single-phase riction actor and a two-phase multiplier are necessary to calculate the pressure gradient. 2.6.1 Single-phase riction actor TRACE code adopts Churchill s correlation [2-62], which is applicable to laminar, transition and turbulent regimes as: 12 8 1 = 2 + Re a b where ( + ) 3/ 2 1/ 12 (2-38) a = 2. 457 log 7 Re 0. 9 1 ε w + 0. 27 D H 16 and b 4 3. 753 10 = Re 16 (2-39) Re and ε are the Reynolds number and wall roughness, respectively. w Churchill s correlation asymptotically approaches the theoretical laminar single-phase riction actor in Re< 2100 and an empirical turbulent single-phase riction coeicient in Re> 3000. RELAP5/MOD3.3 code classiies the single-phase low regime into three regions: (1) Laminar region ( Re 2200 ), (2) Transition region ( 2200 < Re 3000 ) and (3) Turbulent 25
region ( Re > 3000 ). The single-phase riction actor in the laminar region,, is given by: L L where 16 = Re φ S (2-40) φ S is a correction actor considering the geometrical dierence between a pipe and other low channels, which is given by a user input. region,, is calculated by Zigrang and Sylvester s correlation [2-63] as: T The single-phase riction actor in the turbulent 1 εw 2. 51 εw 21. 25 = 4log + 1 14 2 10. log10 0. 9 3. 7D Re D T H H Re (2-41) The single-phase riction actor in the transition region, L T, is given by an interpolation unction as: 8250 = 3. 75 ( ) + Re L T TRe, = 3000 LRe, = 2200 LRe, = 2200 (2-42) TRAC-BF1 code classiies the single-phase low regime into our regions and the single-phase riction actor in each region is given by Pann s correlation [2-64] as: 16 =, or Re < 2300 (2-43) Re 2 1 0. 28 60 =, or 2300< Re < 4 log Re 0. 82 2ε w D H 1. 111 (2-44) = 1 0. 25, 4 2ε w ( 3. 393 0. 805g1 ) g 2. 477 log 1 D H 2ε 0. 87 log 60 D or < Re < 424 1. 111 2ε 2εw w D DH H 2 w H (2-45) 26
2 2εw 0. 87 log 1 0. 25 DH =, or Re > 424 4 2ε 2ε w w 0. 87 log D D H H where g 1 2εw Re D H log 2ε 0. 87 log D w H (2-46) (2-47) 2.6.2 Two-phase multiplier TRACE code gives the two-phase multiplier with respect to each low regime. two-phase multiplier or adiabatic bubbly and slug low regimes is given by: The 2 1 φ = n ( 1 α ) (2-48) where n is an exponent whose value is not speciied in the TRACE manual but is expected to be between 1.72 and 1.8. The two-phase multiplier or boiling low is ormulated by considering increased wall roughness due to bubble nucleation as: ( 1 C ) φ = φ + (2-49) NB where C NB { ( )} 0. α α 62 min 2, 155 d B = D 1 H (2-50) The bubble departure diameter, d, is calculated by Levy s orce balance model [2-65] as: B d D B H σ = 0. 015 (2-51) τd w H where τ is the wall shear stress. w The two-phase multiplier or annular low regime is given by: 27
2 φ = 1 ( 1 α ) 2 (2-52) The single-phase riction actor or a laminar liquid ilm in annular low regime is given by: ilm ( ) 1 / 3 3 3 L T = + (2-53) where L 16 =, or α < 0. 95 Re φ 2, L α 0. 95 16 + 8 0. 99 0. 95 =, or 0.95< α < 0. 99 Re2 φ, (2-54) L 24 =, or α > 0. 95 Re φ 2, The single-phase riction actor or a turbulent liquid ilm in annular low regime is given by Haaland's correlation [2-66] as: T = 6. 9 ε / D w H 3. 6 log10 + Re 3 7 2 φ., 1 1. 11 2 (2-55) RELAP5/MOD3.3 code uses Lockhart and Martinelli s correlation [2-67] and Chisholm s correlation [2-68]. The rictional pressure gradient is expressed by: p z riction 2 p = φg z g and Martinelli s parameter is deined by: p z riction 2 p = φ z (2-56) X p p = φ φ z z g 2 2 g (2-57) Chisholm [2-68] proposed the ollowing simple correlation to calculate the two-phase multiplier as: 2 C 1 φ = 1+ + (2-58) 2 X X 28
The RELAP5 code calculates the parameter, C, using the ollowing correlation developed based on HTFS tests [2-69]. C ( ) T( ) = 2 + G Λ, G (2-59) 1 1 where ( ) G = 28 0. 3 1 G T ( Λ, G) = exp ( log Λ+ 2. 5) 2 1 4 10 2. 4 G 10 (2-60) 0. 2 ρ µ g Λ ρ µ = g G, µ and µ are the mass lux, liquid viscosity and gas viscosity, respectively. g TRAC-BF1 code calculates the two-phase multiplier based on Lockhart-Marinelli s model. The two-phase multiplier is given by Hancox and Nicoll s correlation [2-70] as: { ( ) } 0.. 25 1 = 1 1 + 1+ 1 Λ 2 0 5 φ X RX X where (2-61) p R= 3. 1 1 ( 5 65 10 4 G) p exp. (2-62) p is the critical pressure. c c 2.7 Conclusions In view o CSAU methodology and code V & V (Veriication and Validation), the implementation o most advanced and accurate constitutive equations into a code is indispensable. This chapter reviews the state-o-the-art correlations or predicting key two-phase low parameters in a vertical rod bundle. The reviewed correlations include low regime map, void raction, void raction covariance and relative velocity covariance, interacial area concentration and wall riction. Important conclusions are given as ollows: 29
Flow regime map The identiied low regimes in a rod bundle were bubbly, inely dispersed bubbly, cap bubbly, cap turbulent, churn and annular lows. Existing low regime maps were taken under atmospheric pressure conditions. Liu and Hibiki [2-15] developed low regime transition criteria or a rod bundle. Liu and Hibiki s model agreed with existing low regime maps but its applicability to high pressure and temperature two-phase low should be examined using data to be taken in a uture study. Void raction Five state-o-the-art drit-lux type correlations are identiied. They are the drit-lux type correlations or (1) a rod bundle under prototypic nuclear reactor core conditions [2-16, 2-36], (2) a rod bundle with unheated rod at the bundle center [2-16], (3) a rod bundle under subcooled boiling conditions [2-44], (4) a rod bundle under adiabatic bubbly low [2-46] and (5) a rod bundle at low liquid low under low pressure conditions [2-47, 2-48, 2-49]. All correlations agreed with existing void raction data but the correlation or a rod bundle at low liquid low under low pressure conditions should be urther examined using data to be taken in a uture study. Void raction covariance and relative velocity covariance Modeling o void raction covariance and relative velocity covariance is indispensable or calculating area-averaged relative velocity accurately. Only one model o void raction covariance and relative velocity covariance in a rod bundle [2-44] is identiied. Ozaki and Hibiki s correlation agreed with existing data taken in a vertical 8 8 rod bundle under prototypic nuclear reactor core conditions. Interacial area concentration Accurate modeling o interacial area concentration is important or calculating interacial drag orce and interacial heat transer. The interace structure o large bubbles depends on a low channel size. Slug bubbles spanning over a low channel can exist in a relatively small size channel, whereas cap bubbles created by the disintegration o large bubbles due to their surace instability exist in a relatively large size channel. Two types o interacial area correlations or relatively small and large size channels with simple geometries are identiied [2-56, 2-48]. In bubbly low, Ozar et al. s correlation becomes identical to Hibiki and Ishii s correlation [2-54, 2-55]. The applicability o Hibiki and Ishii s correlation to a rod bundle was partly examined but the 30
applicability o Ozar et al. s correlation to a rod bundle has not been tested. Schlegel and Hibiki s correlation agreed with existing data taken in a vertical 8 8 rod bundle under an atmospheric condition but its applicability to prototypic nuclear reactor core conditions should be tested by data to be taken in a uture study. Wall riction Key constitutive correlations to calculate wall riction are single-phase riction actor and two-phase multiplier. The constitutive correlations used in one-dimensional nuclear thermal-hydraulic system analysis codes such as TRACE [2-59], RELAP5 [2-60] and TRAC-BF1 [2-61] were reviewed. Code improvement by implementing the above state-o-the-art correlations is expected to enhance the code prediction accuracy or two-phase low analyses in a rod bundle. 31
Geometry o Flow Fields, Boundary Conditions Determination o Flow Regime Interacial Area Concentration Wall Friction Wall Heat Transer Interacial Momentum Transer Interacial Heat Transer Heat Flux or Each Phase Wall Conduction and Radiation Heat Transer Relative Velocity Conservation Equations Numerical Solution Figure 2-1 Schematic diagram o typical one-dimensional two-phase low analysis code structure [2-6]. 32