Multiphase Flow Dynamics 5

Transcription

1 Mutiphase Fow Dynamics 5

2 Nikoay Ivanov Koev Mutiphase Fow Dynamics 5 Nucear Therma Hydrauics ABC

3 Author Dr. Nikoay Ivanov Koev Möhrendorferstr Herzogenaurach Germany E-mai: Nikoay.Koev@herzovision.de ISBN e-isbn DOI / Library of Congress Contro Number: c 011 Springer-Verag Berin Heideberg This work is subject to copyright. A rights are reserved, whether the whoe or part of the materia is concerned, specificay the rights of transation, reprinting, reuse of iustrations, recitation, broadcasting, reproduction on microfim or in any other way, and storage in data banks. Dupication of this pubication or parts thereof is permitted ony under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must aways be obtained from Springer. Vioations are iabe to prosecution under the German Copyright Law. The use of genera descriptive names, registered names, trademarks, etc. in this pubication does not impy, even in the absence of a specific statement, that such names are exempt from the reevant protective aws and reguations and therefore free for genera use. Typeset & Cover Design: Scientific Pubishing Services Pvt. Ltd., Chennai, India. Printed on acid-free paper springer.com

4 To my mother! Nordsee, Oct. 005, Nikoay Ivanov Koev, oi on inen

5 Nikoay Ivanov Koev, PhD, DrSc Born , Gabrowo, Bugaria

6 A Few Words about the Second Edition After a break of about 0 years the word started again to modernize the od nucear power pants and to buid new ones. Students, engineers, and scientists need modern books in this fied, refecting the word-wide engineering experience. This expains the considerabe interest in this book, which came very much in time in 008, and now needs a second edition. Severa chapters have been updated and improved. I hope it wi hep young scientists and engineers in their professiona ife of designing better faciities than those created by my generation. Herzogenaurach December 8, 010

7 The Motivation to Write This Book Nucear therma hydrauics is the science that provides knowedge and mathematica toos for adequate description of the process of transferring the fission heat reeased in materias due to nucear reactions into its environment. Aong its way to the environment the therma energy is organized to provide usefu mechanica work or usefu heat. Propery arranged and controed processes achieve this target. Impropery arranged processes or inappropriatey controed processes may ead to damage, osing the investment partiay or totay. If power pants are designed so that in ow-probabiity accidenta processes ony the investment is ost, we speak about safe nucear power pants. Impropery designed power pants that contain the potentia besides osing the investment to destroy the environment and human ives are not acceptabe to human society. Nucear therma hydrauics is a substantia part of the engineering discipine caed nucear reactor safety. Nucear reactor safety is not ony a technica science. It contains the reations between society, with its mature and effective contro mechanisms, and technoogy. Scientists and engineers aone cannot sove the probem of nucear reactor safety. It is a technoogica and simutaneousy a socia probem, as is any probem associated with high-energy technoogies. I wi imit my attention in this work to the scientific part. After about 60 years research and practice we know how to buid technicay safe nucear power pants. The pubic attitude to this subject has had its up and downs. Now the word faces the probem of dramaticay increasing oi and energy prices, making nucear energy inevitabe. At the same time there is a generation change, and a arge army of experienced nucear engineers are retiring. The responsibiity to transfer knowedge to the next generation is what drives me to write this book. I hope it wi hep young scientists and engineers in their professiona ife of designing better faciities than those created by my generation. Herzogenaurach May, 006

8 Summary This monograph contains theory, methods, and practica experience for describing compex transient mutiphase processes in arbitrary geometrica configurations. It is intended to hep appied scientists and practicing engineers to understand better natura and industria processes containing dynamic evoutions of compex mutiphase fows. The book is aso intended to be a usefu source of information for students in the high semesters and in PhD programs. This monograph consists of five voumes: Vo. 1 Fundamentas, 4th ed. (14 chapters and appendices), 78 pages Vo. Mechanica Interactions, 4th ed. (11 chapters), 64 pages Vo. Therma Interactions, 4th ed. (16 chapters), 678 pages Vo. 4 Turbuence, Gas Absorption and Reease by Liquid, Diese Fue Properties, nd ed. (1 chapters), 8 pages Vo. 5 Nucear Therma Hydrauics, nd ed. (17 chapters), 848 pages In Voume 1 the concept of three-fuid modeing is presented in detai from the origin to the appications. This incudes derivation of oca voume- and timeaveraged equations and their working forms, deveopment of methods for their numerica integration, and finay finding a variety of soutions for different probems of practica interest. Specia attention is paid in Voume 1 to the ink between the partia differentia equations and the constitutive reations caed cosure aws without providing any information on the cosure aws. Voumes and are devoted to these important constitutive reations for mathematica description of the mechanica and therma interactions. The structure of the voumes is in fact a state-of-the-art review and seection of the best avaiabe approaches for describing interfacia transfer processes. In many cases the origina contribution of the author is incorporated in the overa presentation. The most important aspects of the presentation are that they stem from the author s ong years of experience deveoping computer codes. The emphasis is on the practica use of these reationships: either as stand-aone estimation methods or within a framework of computer codes. Voume 4 is devoted to the turbuence in mutiphase fows. Nucear therma hydrauics is the science providing knowedge about the physica processes occurring during the transferring the fission heat reeased in structura materias due to nucear reactions into its environment. Aong its way to the environment the therma energy is organized to provide usefu mechanica work or usefu heat or both. Voume 5 is devoted to nucear therma hydrauics. In

9 XII Summary a way this is the most essentia appication of the mutiphase fuid dynamics in anayzing steady and transient processes in nucear power pants. Voume 5 can be summarized as foows: Chapter 1 contains introductory information about the heat reease in the reactor core, the therma power and therma power density in the fue, structures, and moderator, the infuence of the therma power density on the cooant temperature, and the spatia distribution of the therma power density. Finay, some measures are introduced for equaizing the spatia distribution of the therma power density. Chapter gives the methods for describing the steady and the transient temperature fieds in the fue eements. Some information is provided regarding infuence of cadding oxidation, hydrogen diffusion, and corrosion product deposition on the temperature fieds. Didacticay nucear therma hydrauics needs introductions at different eves of compexity, introducing step-by-step new features after the previous ones have been ceary presented. The foowing two chapters serve this purpose. Chapter describes mathematicay the simpe steady boiing fow in a pipe. The steady mass-, momentum-, and energy-conservation equations are soved at different eves of compexity by removing, one after the other, simpifying assumptions. First the idea of mechanica and thermodynamic equiibrium is introduced. Then the assumption of mechanica equiibrium is reaxed. Then the assumption of thermodynamic equiibrium is reaxed in addition. In a cases comparison with experimenta data gives the evidence of the eve of adequacy of the different eve of modeing compexity. The engineering reaxation methods are considered, foowed by the more sophisticated boundary ayer treatment without and with variabe effective bubbe size. Then an introduction to the saturated fowboiing heat transfer is given and the accuracy of the methods is demonstrated by comparison with experiments. The hybrid method of combining the asymptotic method with boundary ayer treatment aowing for variabe effective bubbe size is aso presented. Finay, the idea of using separated momentum equations and bubbe dynamics is introduced and again its adequacy is demonstrated by comparison with experiments. Whie Chap. essentiay deas with the so-caed two-fuid mode, Chap. demonstrates the rea cases where a three-fuid mode is mandatory. Chapter is an introduction to the simpe steady three-fuid boiing fow in a pipe. The fow regime transition from sug to churn turbuent fow is considered in addition to the aready-avaiabe information from Chap.. The idea of the redistribution of the iquid between fim and dropets is presented at two eves of compexity: the instantaneous and the transient iquid redistribution in fim and dropets. The transient redistribution is in fact the introduction of the ideas of dropet entrainment and deposition. The idea for the description of the mechanica interaction of the veocity fieds is again presented at two eves of compexity: by using drift fux correations and by using separated momentum equations defining the forces among the fieds. The next step of the sophistication is then introduced by using modes for the dynamic evoution of the mean dropet size consisting of modes for the dropet size stabiity imit, for dropet production rate due to fragmentation, for duration of the fragmentation, and for coision and coaescence of dropets.

10 Summary XIII Then the heat and mass transfer mechanisms in the fim fow with dropet oading are introduced. Finay, comparisons with experimenta data demonstrate the success of the different ideas and modes. To my view the reader wi not understand the materia of the foowing chapters if Chaps. and 4 are not we understood. Chapter 5 describes the most powerfu methods for describing the core therma hydrauics these days. First an introduction of the design of the reactor pressure vesses for pressurized- and boiing-water reactors is given. Then by using a arge number of experimenta data sets for steady fows in heated bundes the accuracy of the modern methods is demonstrated. The experiments gathered for comparison are the NUPEC experiment, the SIEMENS void data for the ATRIUM 10 fue bunde, the FRIGG experiment, and the THTF experiments: high pressure and ow mass fow. Methods for prediction of the pressure drop for boiing fow in bundes are presented and compared with data. Then by using experimenta data sets for transient fows in heated bundes the accuracy of the modern methods is demonstrated. The experiments gathered for comparison are the NUPEC transients in a channe simuating one subchanne of a pressurized-water reactor fue assemby and the NUPEC transients in a pressurized-water reactor 5 5 fue assemby. Actuay avoiding a boiing crisis is the main target of a proper core design. That is why the methods for anayzing whether the critica heat fux is reached in the cores cooed by steadystate fows are presented in detai at different compexity eves: initia 0D guess and D critica heat fux anaysis. Severa uncertainties of the physica modes are identified during this process and discussed in detai. New ideas for future progress in this fied are presented: arge-scae turbuence modeing in bundes, fineresoution anaysis, etc. Finay, an exampe is given of the most compex case subject in nucear therma hydrauics: the anaysis of the therma processes in a core of a boiing-water reactor using the methods presented in this monograph. The stronger the driving forces for fow processes, the more stabe are the resuting phenomena and vice versa. Many of the processes in nucear therma hydrauics are associated with ow driving forces and tend to instabiity. This chapter presents a noninear stabiity anaysis on some prominent exampes in nucear therma hydrauics: fow boiing and condensation stabiity anaysis. After a stateof-the-art review the AREVA boiing stabiity data for the ATRIUM 10B fue bunde are compared with state-of-the-art predictions using the methods presented in this monograph. The cassica boiing instabiity anaysis is accompished with the sedom-presented fow condensation stabiity anaysis in a compex system of emergency condensers consisting of a arge number of 1D condensing pipes submerged into a D poo. Condensation at the high-pressure side eads to a fow patterns for neary horizonta pipes with a their instabiities. It is couped with the D boiing of the secondary poo side. The compex picture is very informative for what can be expected and what has to be avoided in such designs. Chapter 7 is devoted to critica mutiphase fow. It starts with the mathematica definition of the criticaity condition, with the appropriate design of a numerica grid structure and numerica iteration strategy. Then the methods used in modern design are presented, starting from the simpe modes and graduay increasing the compexity. First the singe-phase critica fow in a pipe is considered for the case with no-friction energy dissipation and constant cross-section. Then the genera

11 XIV Summary case is presented for a perfect gas. Then the same ideas are extended to simpe two-phase cases for pipes and nozzes: subcooed critica mass fow rate in short pipes, orifices, and nozzes; frozen homogeneous nondeveoped fow; inhomogeneous deveoped fow without mass exchange; equiibrium homogeneous fow; equiibrium inhomogeneous fow; inhomogeneous deveoping fow in short pipes and nuzzes with infinitey fast heat exchange and with imited interfacia mass transfer. Then the recent state of the knowedge for describing critica fow is presented by considering physica detais ike: bubbe origination; bubbe fragmentation; bubbe coaescence; dropet origination. Exampes foow for appication of the theory of the critica fow in rea-scae anaysis: bow-down of a cosed pipe and bow-down of a vesse. Chapter 8 is devoted to the basics of designing of steam generators. Chapter 9 is devoted to the basics of designing of moisture separation. First the importance of knowing the characteristic spectra of the moisture is underined for proper anaysis. Then some simpe methods for computation of the efficiency of the separation are given for cycone-type and vane-type separators. Different ideas based on different compexity are presented for description of the veocity fied: the Kreith and Sonju soution for the decay of turbuent swir in a pipe, the potentia gas fow in vanes; description of the trajectory of partices in a known continuum fied; the computationa fuid dynamics (CFD) anayses of cycones; the CFD anayses of vane separators. Then severa experiments are coected from the iterature for boiing-water reactor cycones, pressurized-water reactor steamgenerator cycones, other cycone types, and vane dryers. In severa cases the success of different methods is demonstrated by comparison with data. The nucear power pant consists not ony of arge and sma components but aso by a forest of interconnected pipes. Chapter 10 is devoted to the estimation of the accuracy of modeing of transient processes in pipe networks by using a the methods presented in this monograph. First some basic definitions are introduced of pipes, axes in space, knots, diameters of pipe sections, reductions, ebows, creating a ibrary of pipes, creating a subsystem network, and discretization of pipe networks for numerica treatment. Then seven interesting experiments are simuated and a comparison with measurements is presented in order to derive concusions about the accuracy of the methods derived in this monograph. Not ony are the main systems of interest for the practicing engineer. He or she wi have to hande probems in rea ife in the so-caed auxiiary systems. As one exampe of such a system the high-pressure reduction station is anayzed in Chapter 11. A singe high-pressure pipe break is anayzed and the consequences of such an event are discussed. As a second exampe for processes in auxiiary systems an anaysis of the physica and chemica processes of radioysis gas production, air absorption, diffusion-controed gas reease, and transport in the cooant ceaning system of the research reactor FRM II is given. The evoution of the safety phiosophy in the ast 0 years has ed to the introduction of so-caed passive safety systems. Such exampes are so-caed emergency condensers. Chapter 1 gives first a simpe mathematica iustration of the operation of the system. Then the performance of the condenser as a function of the water eve and pressure are anayzed with the methods introduced in this monograph. The important question of the condensate remova is discussed.

12 Summary XV Chapter 1 is devoted to the core degradation during so-caed severe accidents. Chapter 14 is devoted to met-water interaction, which is an important part of modern nucear reactor therma hydrauics. Chapter 15 is devoted to the cooabiity of ayers of moten nucear reactor materia. Such physics is important for designing stabiization of spread met in reactor compartments. After defining the probem with its boundary conditions and some simpifying assumptions the system of differentia equations describing the process is presented: mass and energy conservation. The foowing effects are taken into account: moten stee dropped in the met or originating inside the met; gas reease from a subayer; the viscous ayer; crust formation; buoyancy-driven convection; fim boiing; heat conduction through the structures; oxide crust formation on coder heat-conducting structures. The existence of a metaic ayer is aso considered. Some test cases are presented to make easy the appication of the presented modes: oxide over meta and oxide beside meta. A simpe mode for gravitationa fooding of a hot soid horizonta surface by water eading to a hyperboic system is aso presented. Chapter 16 is devoted to the so-caed externa cooing of reactor vesses during a severe accident. It is a technoogy aowing the arrest of the met inside the vesse if some initia conditions are fufied. First the state of the art is presented. Then a brief description of the phenomenoogy eading to met in the ower head is discussed: dry core meting scenario, met reocation, wa attack, focusing effect. A brief mathematica mode description is given appropriate for a set of mode assumptions. The mode describes: met poo behavior, two-dimensiona heat conduction through the vesse wa, tota heat fow from the poos into the vesse wa, vesse wa abation, heat fuxes, crust formation, and buoyancy-driven convection. A soution agorithm is provided for a set of boundary conditions adequate for rea situations. A summary of the state of the art regarding the critica heat fux for externay fowing ower head geometry is provided. For severa practica appications different effects are demonstrated: the effect of vesse diameter, of the ower head radius, of the reocation time, of the mass of the interna structures. Varying some important parameters characterizing the process the difference between highpowered pressurized- and boiing-water reactor vesse behavior is demonstrated. Severa modern aspects of the severe accident anaysis cannot be understood if the engineer does not have accurate information on the materia properties for the participating structura materias in soid, in iquid, and in some cases in gaseous states. Chapter 17 contains vauabe sets of thermophysica and transport properties for severe accident anaysis for the foowing materias: uranium dioxide, zirconium dioxide, stainess stee, zirconium, auminum, auminum oxide, siicon dioxide, iron oxide, moybdenum, boron oxide, reactor corium, sodium, ead, bismuth, and ead bismuth eutectic aoy. The emphasis is on the compete and consistent thermodynamic sets of anaytica approximations appropriate for computationa anaysis. December 9, 010 Herzogenaurach Nikoay Ivanov Koev

13 Nomencature Latin A cross-section, m² A surface vector a speed of sound, m/s a w a σ surface of the fied wetting the wa w per unit fow voume max Vo beonging to contro voume Vo (oca voume interface area density of the structure w), m 1 surface of the veocity fied contacting the neighboring fieds per unit fow voume max Vo beonging to contro voume Vo (oca voume = 1 interface area density of the veocity fied ), m 1 a tota surface of the veocity fied per unit fow voume max Vo beonging to contro voume Vo (oca voume interface area density of the = 1 veocity fied ), m 1 Cu i Courant criterion corresponding to each eigenvaue, dimensioness C i mass concentration of the inert component i in the veocity fied c coefficients, dimensioness C m mass concentration of the component m in the veocity fied, dimensioness C i mass concentration of the component i in the veocity fied, dimensioness c p specific heat at constant pressure, J/ ( kgk ) vm c virtua mass force coefficient, dimensioness d c drag force coefficient, dimensioness L c ift force coefficient, dimensioness D hy hydrauic diameter (four times cross-sectiona area / perimeter), m D E diameter of the entrained dropets, m D d size of the bubbes produced after one nuceation cyce on the soid structure, bubbe departure diameter, m = 1

14 XVIII Nomencature D 1dm size of bubbes produced after one nuceation cyce on the inert soid partices of fied m =, D ch critica size for homogeneous nuceation, m D cd critica size in presence of dissoved gases, m D most probabe partice size, m D characteristic ength of the veocity fied, partice size in case of fragmented fied, m D coefficient of moecuar diffusion for species i into the fied, m /s coefficient of turbuent diffusion, m /s tota diffusion coefficient, m /s DC right-hand side of the nonconservative conservation equation for the inert i t D i * D i i component, kg / ( sm ) D diffusivity, m /s d tota differentia E tota energy, J e specific interna energy, J/kg F, f (... function of (... f force per unit fow voume, N/m f fraction of entrained met or water in the detonation theory F w surfaces separating the veocity fied from the neighboring structure within Vo, m F σ surfaces separating the veocity fied from the neighboring veocity fied within Vo, m F surface defining the contro voume Vo, m f im frequency of the nucei generated from one activated seed on the partice beonging to the donor veocity fied m, s 1 f w frequency of the bubbe generation from one activated seed on the channe wa, s 1 f, coa coaescence frequency, s 1 g acceeration due to gravity, m/s H height, m h specific enthapy, J/kg h i eigenvectors corresponding to each eigenvaue I unit matrix, dimensioness i unit vector aong the x-axis J matrix, Jacobian j unit vector aong the y-axis k unit vector aong the k-axis k ce number

15 Nomencature XIX k kinetic energy of turbuent pusation, m /s T k i coefficient of thermodiffusion, dimensioness p k i coefficient of barodiffusion, dimensioness L ength, m M i kg-moe mass of the species i, kg/mo m tota mass, kg n ΔV unit vector pointing aong ΔV m, dimensioness n unit vector pointing outward from the contro voume Vo, dimensioness n unit surface vector pointing outward from the contro voume Vo e n unit interface vector pointing outward from the veocity fied σ n i number of the partice from species i per unit fow voume, kin, n w n h n number of partices of fied i per unit fow voume, partice number density of the veocity fied, m n coa number of partices disappearing due to coaescence per unit time and unit voume, m n partice production rate due to nuceation during evaporation or conden- sation, 1/ ( m s ) number of the activated seeds on unit area of the wa, m number of the nucei generated by homogeneous nuceation in the donor m veocity fied per unit time and unit voume of the fow, 1/ ( m s ) n dis, number of the nucei generated from dissoved gases in the donor veocity fied per unit time and unit voume of the fow, 1/ ( m s ) n sp, number of partices of the veocity fied arising due to hydrodynamic disintegration per unit time and unit voume of the fow, 1/ ( m s ) P probabiity P irreversiby dissipated power from the viscous forces due to deformation of the oca voume and time-averaged veocities in space, W / kg Per perimeter, m p i = 1: partia pressure inside the veocity fied =, : pressure of the veocity fied p pressure, Pa q therma power per unit fow voume introduced into the fuid, W/m q σ = 1,,. Therma power per unit fow voume introduced from the interface into the veocity fied, W/m q w σ therma power per unit fow voume introduced from the structure interface into the veocity fied, W/m

16 XX Nomencature R mean radius of the interface curvature, m r(x,y,z) position vector, m R (with indexes) gas constant, J/(kg K) s arc ength vector, m S tota entropy, J/K s specific entropy, J/(kg K) t Sc turbuent Schmidt number, dimensioness tn Sc turbuent Schmidt number for partice diffusion, dimensioness T temperature, K T temperature of the veocity fied, K T shear stress tensor, N/m t unit tangent vector U dependent variabes vector Vo contro voume, m 1/ Vo size of the contro voume, m Vo voume avaiabe for the fied inside the contro voume, max Vo voume avaiabe for the fow inside the contro voume, = 1 V instantaneous fuid veocity with components, u, v, w in r, θ, and z direction, m/s τ ϑ τ τ V instantaneous fied veocity with components, u, v, w in r, θ, and z directions, m/s V time-averaged veocity, m/s V pusation component of the instantaneous veocity fied, m/s ΔV m V m V, veocity difference, disperse phase, continuous phase m carrying, m/s δ iv τ diffusion veocity, m/s τ V σ interface veocity vector, m/s τ ϑ τ τ V γ instantaneous vector with components, u γ r, vγθ, wγ z in r, θ, and z directions, m/s v specific voume, m /kg x mass fraction, dimensioness y distance between the bottom of the pipe and the center of mass of the iquid, m vector product m m

17 Nomencature XXI Greek α α i the same as part of γ Vo v avaiabe to the veocity fied, oca instantaneous voume fraction of the veocity fied, dimensioness α in the case of gas mixtures; in the case of mixtures consisting of iquid and macroscopic soid partices, the part of γ Vo avaiabe to the inert component i of the veocity fied, oca instantaneous v voume fraction of the inert component i of the veocity fied, dimensioness α,max 0.6, imit for the cosest possibe packing of partices, dimensioness γ v the part of dvo avaiabe for the fow, voumetric porosity, dimensioness γ surface permeabiity, dimensioness γ directiona surface permeabiity with components γ r, γθ, γ z, dimensioness Δ finite difference δ sma deviation with respect to a given vaue δ = 1 for continuous fied; = 0 for disperse fied, dimensioness ε partia differentia dissipation rate for kinetic energy from turbuent fuctuation, power irreversiby dissipated by the viscous forces due to turbuent fuctuations, W/kg η dynamic viscosity, kg/(m s) θ θ -coordinate in the cyindrica or spherica coordinate systems, rad κ = 0 for Cartesian coordinates; = 1 for cyindrica coordinates κ isentropic exponent κ curvature of the surface of the veocity fied, m λ therma conductivity, W/(m K) λ eigenvaue τ μ oca voume-averaged mass transferred into the veocity fied per unit time and unit mixture fow voume, oca voume-averaged instantaneous mass source density of the veocity fied, kg / ( m s ) τ μ, kg / ( m s ) μ time average of μ w mass transport from exterior source into the veocity fied, kg / ( m s ) τ μ i oca voume-averaged inert mass from species i transferred into the veocity fied per unit time and unit mixture fow voume, oca voumeaveraged instantaneous mass source density of the inert component i of the veocity fied, kg / ( m s )

18 XXII Nomencature τ μ i time average of i μ, kg / ( m s ) τ μ im oca voume-averaged instantaneous mass source density of the inert component i of the veocity fied due to mass transfer from fied m, kg / ( m s ) τ μ im time average of im μ, kg / ( m s ) τ μ im oca voume-averaged instantaneous mass source density of the inert component i of the veocity fied due to mass transfer from fied into veocity fied m, kg / ( m s ) μ, kg / ( m s ) τ μ im time average of im ν kinematic viscosity, m /s t ν coefficient of turbuent kinematic viscosity, m /s tn ν coefficient of turbuent partice diffusion, m /s ξ ange between n σ and ΔV m, rad ρ density, kg/m ρ instantaneous density, density; without indexes, mixture density, kg/m ρ instantaneous fied density, kg/m ρ instantaneous inert component density of the veocity fied, kg/m i ρ intrinsic oca voume-averaged phase density, kg/m ( ρ w) entrainment mass fow rate, kg / ( m s ) ( ρ w) deposition mass fow rate, kg / ( m s ) ( τ ) e ρv oca intrinsic surface mass fow rate, kg / ( m s ) σ, σ 1 surface tension between phases 1 and, N/m τ time, s ϕ ange giving the projection of the position of the surface point in the pane norma to ΔV m, rad mσ χ the product of the effective heat transfer coefficient and the interfacia area density, ( ) W / m K. The subscript denotes inside the veocity fied. The superscript mσ denotes ocation at the interface σ dividing fied m from fied. The superscript is ony used if the interfacia heat transfer is associated with mass transfer. If there is heat transfer ony, the inearized interaction coefficient is assigned the subscript m ony, indicating the interface at which the heat transfer takes pace.

19 Nomencature XXIII Subscripts c continuous d disperse m from to m or acting on m w region outside of the fow e entrances and exits for contro voume Vo veocity fied, intrinsic fied average i inert components inside the fied, noncondensabe gases in the gas fied = 1, or microscopic partices in water in fied or i corresponding to the eigenvaue λ i in Chapter 4 M noninert component m mixture of entrained cooant and entrained met debris that is in therma and mechanica equiibrium behind the shock front m from m into im from im into i max maximum number of points n inert component 0 beginning of the time step E entrainment coa coaescence sp spitting, fragmentation σ interface τ od time eve τ +Δ τ new time eve * initia 0 reference conditions p, v, s at constant p, v, s, respectivey L eft R right 1 vapor or in front of the shock wave water or behind the shock wave met 4 entrained cooant behind the front entrained cooant 5 micropartices after the therma interaction entrained met Superscripts time fuctuation ' saturated steam " saturated iquid "' saturated soid phase A air d drag e heterogeneous

20 XXIV Nomencature i component (either gas or soid partices) of the veocity fied i max maximum for the number of the components inside the veocity fied L ift intrinsic fied average e intrinsic surface average σ averaged over the surface of the sphere m component n norma n od iteration n+1 new iteration t turbuent, tangentia vm virtua mass τ tempora, instantaneous Operators averaging sign divergence gradient n norma component of the gradient t tangentia component of the gradient surface gradient operator, 1/m e Lapacian oca voume average oca intrinsic voume average oca intrinsic surface average Nomencature required for coordinate transformations ( xyz,, ) coordinates of a Cartesian, eft-oriented coordinate system (Eucidean x ( i = 1,,) : space). Another notation which is simutaneousy used is i x, x, x 1 V cs g ( ξ, η, ζ ) coordinates of the curviinear coordinate system caed transformed coordinate system. Another notation which is simutaneousy used is ξ i ( i = 1,,) : 1 ξ, ξ, ξ the veocity of the curviinear coordinate system Jacobian determinant or Jacobian of the coordinate transformation x = f ( ξ, η, ζ ), y = g( ξ, η, ζ ), z = h( ξ, η, ζ )

21 Nomencature XXV a ij ij a eements of the Jacobian determinant eements of the determinant transferring the partia derivatives with respect to the transformed coordinates into partia derivatives with respect to the physica coordinates. The second superscript indicates the Cartesian components of the contravariant vectors (,, ) a1 a a covariant base vectors of the curviinear coordinate system tangent vectors to the three curviinear coordinate ines represented by ξ, η, ζ 1 (,, ) ( ) a a a contravariant base vectors, norma to a coordinate surface on which the coordinates ξ, η, and ζ are constant, respectivey covariant metric tensor (symmetric) contravariant metric tensor (symmetric) g ij ij g 1 (,, ) e e e unit vectors norma to a coordinate surface on which the coordinates ξ, η, and ζ are constant, respectivey i V i = a V, contravariant components of the vector V V = a V, covariant components of the vector V i ( ξ, η, ζ ) i γ γ γ permeabiities of coordinate surfaces on which the coordinates ξ, η, and ζ are constant, respectivey Greek Α, α Apha Β, β Beta Γ, γ Gamma Δ, δ Deta Ε, ε Epsion Ζ, ζ Zeta Η, η Eta Θ, ϑ Theta Ι, ι Iota Κ, κ Kappa Λ, λ Lambda Μ, μ Mu Ν, ν Nu Ξ, ξ Xi Ο, ο Omikron Π, π Pi Ρ, ρ Rho Σ, σ Sigma Τ, τ Tau Φ, ϕ Phi Χ, χ Chi ϒ, υ Ypsion Ψ, ψ Psi Ω, ω Omega

22 Tabe of Contents 1 Heat reease in the reactor core Therma power and therma power density Therma power density and fue materia Therma power density and moderator temperature Spatia distribution of the therma power density Equaizing of the spatia distribution of the therma power density Nomencature... 1 References Temperature inside the fue eements Steady-state temperature fied Transient temperature fied Infuence of the cadding oxidation, hydrogen diffusion, and corrosion product deposition Cadding oxidation Hydrogen diffusion Deposition Nomencature References... 4 The simpe steady boiing fow in a pipe Mass conservation Mixture momentum equation Energy conservation The idea of mechanica and thermodynamic equiibrium Reaxing the assumption of mechanica equiibrium Reaxing the assumption of thermodynamic equiibrium The reaxation method The boundary ayer treatment The boundary ayer treatment with considered variabe effective bubbe size Saturated fow boiing heat transfer Combining the asymptotic method with boundary ayer treatment aowed for variabe effective bubbe size Separated momentum equations and bubbe dynamics Nomencature References... 8 Appendix.1 Sani s (1960) data for boiing fow in a pipe... 85

23 XXVIII Tabe of Contents 4 The simpe steady three-fuid boiing fow in a pipe Fow regime transition from sug to churn turbuent fow Instantaneous iquid redistribution in fim and dropets Reaxing the assumption for instantaneous iquid redistribution in fim and dropets, entrainment, and deposition Drift fux correations Separated momentum equation Dynamic evoution of the mean dropet size Dropet size stabiity imit Dropet production rate due to fragmentation Duration of the fragmentation Coision and coaescence Heat transfer Mass transfer Comparison with experiments Nomencature References Core therma hydrauics Reactor pressure vesses Steady-state fow in heated rod bundes The NUPEC experiment The SIEMENS void data for the ATRIUM 10 fue bunde The FRIGG experiments The THTF experiments: high pressure and ow mass fow Pressure drop for boiing fow in bundes Transient boiing The NUPEC transients in a channe simuating one subchanne of a PWR fue assemby The NUPEC transients in PWR 5 5 fue assemby Steady-state critica heat fux Initia zero-dimensiona guess Three-dimensiona CHF anaysis Uncertainties Outook toward arge-scae turbuence modeing in bundes Outook toward fine-resoution anaysis Core anaysis Nomencature References Appendix 5.1 Some reevant constitutive reationships addressed in this anaysis Fow boiing and condensation stabiity anaysis State of the art AREVA boiing stabiity data for the ATRIUM 10B fue bunde Fow condensation stabiity... 0 References... 11

24 Tabe of Contents XXIX 7 Critica mutiphase fow Definition of the criticaity condition Grid structure Iteration strategy Singe phase fow in pipe No friction energy dissipation, constant cross section Genera case, perfect gas Simpe two phase cases for pipes and nozzes Subcooed critica mass fow rate in short pipes, orifices and nozzes Frozen homogeneous non-deveoped fow Non-homogeneous deveoped fow without mass exchange Equiibrium homogeneous fow Equiibrium non-homogeneous fow Inhomogeneous deveoping fow in short pipes and nuzzes with infinitey fast heat exchange and with imited interfacia mass transfer Recent state of the knowedge for describing critica fow Bubbes origination Bubbe fragmentation Bubbe coaescences Dropets origination Exampes for appication of the theory of the critica fow Bow down from initiay cosed pipe Bow down from initiay cosed vesse Nomencature... 9 References Steam generators Introduction Some popuar designs of steam generators U-tube type Once through type Other design types Frequent probems, sound design practices Anaytica toos Some preiminary remarks on the physica probem to be soved Some simpe conservation principes Three-dimensiona anaysis Vaidation exampes Benchmark for heat exchanger design with compex computer codes Benchmark for once through steam generator design with compex computer codes Three-dimensiona benchmarks comparison with predictions of oder computer codes... 4

25 XXX Tabe of Contents 8.6 Primary circuits of PWRs up to Primary circuits of modern PWRs Appendix 1 Some usefu geometrica reations in preparing geometrica data for U-tube steam generator anaysis References Moisture separation Introduction Moisture characteristics Simpe engineering methods for computation of the efficiency of the separation Cycone separators Vane separators Veocity fied modeing in separators Kreith and Sonju soution for the decay of turbuent swir in pipes Potentia gas fow in vanes Trajectory of partices in a known continuum fied Computationa fuid dynamics anayses of cycones Computationa fuid dynamics anayses of vane separators Experiments BWR cycones, PWR steam generator cycones Other cycone types Vane dryers Moisture separation in NPP with PWRs anayzed by three-fuid modes Separation efficiency of the specific cycone design Efficiency of the specific vane separator design Uniformity of the fow passing the vane separators Efficiency of the condensate remova ocay and integray Nomencature References Pipe networks Some basic definitions Pipes Axis in the space Diameters of pipe sections Reductions Ebows Creating a ibrary of pipes Sub system network Discretization of pipes Knots The 198-Interatome experiments Experiment Experiment

26 Tabe of Contents XXXI 10.. Experiment Experiment Experiment Experiment Experiment References Some auxiiary systems High pressure reduction station Gas reease in research reactors piping Soubiity of O, N and H under 1 bar pressure Some genera remarks on the gas reease- and absorption dynamics Gas reease in the siphon safety pipe Radioysis gases: generation, absorption and reease Mixing in the water poo Computationa anayses References Emergency condensers Introduction Simpe mathematica iustration of the operation of the system Performance of the condenser as a function of the water eve and pressure Condensate remova Air-cooed condenser, steam reheater Heat exchanger power Intensifying heat transfer by fins Heat transfer at finned tubes Heat conduction through finned pipe Condensation inside a pipe Nomencature References Core degradation Processes during the core degradation depending on the structure temperature Anaytica toos for estimation of the core degradation Met-cooant interaction Met-cooant interaction anaysis for the boiing water reactor KARENA Interaction inside the guide tubes Met-reocation through the ower core grid Side met-reocation through the core barre Late water injection... 51

27 XXXII Tabe of Contents 14. Pressure increase due to the vapor generation at the surface of the met poo Conditions for water penetration into met Vesse integrity during the core reocation phase References Cooabiity of ayers of moten reactor materia Introduction Probem definition System of differentia equations describing the process Simpifying assumptions Mass conservation Gas reease and gas voume faction Viscous ayer Crust formation Met energy conservation Buoyancy driven convection Fim boiing Heat conducting structures Heat conduction through the structures Boundary conditions Oxide crust formation on coder heat conducting structures Meta ayer Test case Oxide over meta Oxide besides meta Gravitationa fooding of hot soid horizonta surface by water Simpifying assumptions Conservation of mass and momentum, scaing Eigen vaues, eigen vectors and canonica forms Steady state Nomencature Nomencature to Sect References Externa cooing of reactor vesses during severe accident Introduction State of the art Dry core meting scenario, met reocation, wa attack, focusing effect Mode assumptions and brief mode description Moten poo behavior Two dimensiona heat conduction through the vesse wa Boundary conditions Tota heat fow from the poos into the vesse wa Vesse wa abation

28 Tabe of Contents XXXIII Heat fuxes and crust formation Buoyancy convection Critica heat fux Appication exampes of the mode The effect of vesse diameter The effect of the ower head radius The effect of the reocation time The effect of the mass of the interna structures Some important parameters characterizing the process Nomencature References Appendix 1: Some geometrica reations Thermo-physica properties for severe accident anaysis Introduction Summary of the properties at the meting ine at atmospheric pressure Approximation of the iquid state of mets Nomencature References Uranium dioxide caoric and transport properties Soid Liquid Vapor References Zirconium dioxide Soid Liquid References Stainess stee Soid Liquid Vapor References Zirconium Soid Liquid References Auminum Soid Liquid References

29 XXXIV Tabe of Contents 17.7 Auminum oxide, A O Soid Liquid References Siicon dioxide Soid Liquid References Iron oxide Soid Liquid... 7 References Moybdenum Soid Liquid... 7 References Boron oxide Soid Liquid References Reactor corium Liquid Soid References Sodium Some basic characteristics Liquid Vapor References Appendix Lead, bismuth and ead-bismuth eutectic aoy References Index