GOTHIC Code Improvement for Phenomena Involving Post Critical Heat Flux Conditions for In-Vessel Retention

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U

NIVERSITÀ DI

P

ISA

Dottorato di Ricerca in Ingegneria Industriale Curriculum in Ingegneria Nucleare e Sicurezza Industriale

Ciclo XXXI

GOTHIC Code Improvement for Phenomena

Involving Post Critical Heat Flux Conditions

for In-Vessel Retention

Author Alexandru POP Supervisor(s)

Prof. Dr. Walter AMBROSINI Prof. Dr. Nicola FORGIONE Dr. Alessandro PETRUZZI

Coordinator of the PhD Program Prof. Giovanni Mengali

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Abstract

GOTHIC is an integrated, general purpose thermal hydraulic (TH) software package for design, licensing, safety and operating analysis of Nuclear Power Plant containments, conﬁnement buildings and system components. It bridges the gap between the lumped parameter codes frequently used for containment analysis (like the MELCOR, MAAP, COCOSYS and ASTEC codes) and Computational Fluid Dynamics (CFD) codes.

Within a single nodalization model, GOTHIC can include regions treated in both conventional lumped parameter mode and regions with three-dimensional flows in complex geometries. Although it does not include the capability to model the details of the boundary layers as in most CFD codes, through the use of standard wall functions for heat and momentum transfer, it can give good estimates of the three dimensional (3D) flows and distribution with significantly lower computational cost than typical CFD codes. Additionally, it includes the capability to model multiphase flows situations including dropwise and wall condensation, pool surfaces and sprays without special user supplied models. The 3D capabilities of GOTHIC in simulating basic flows, and in detail, hydrogen flows for containment analysis have been investigated extensively, simulating tests in facilities like PANDA, CSTF, BFMC or CVTR.

The heat transfer correlations built into GOTHIC cover the portion of the boiling curve which spans single phase heat transfer to pre-CHF (critical heat flux) heat transfer. The boiling curve is truncated to exclude post-CHF heat transfer as it has not been adequately verified and was considered by the developers to have little application in general containment analysis. As such, one area that the code is not currently qualified is for In-Vessel Corium Retention analysis, where water enters in contact with the high temperature pressure vessel walls, requiring the modelling of CHF heat transfer. Depending on the heat transfer flux, different amounts of steam may be produced as water enters in contact with the high temperature RPV surface; this will have a direct impact on containment pressure as well as on the integrity of the vessel wall.

The performed doctoral research had the goal of creating an external function/subroutine that resolves this GOTHIC code limitation, enabling it to account for CHF phenomena. Being able to model CHF would represent a significant improvement in simulating the external Pressure Vessel Cooling of different reactor types, as well as other types of severe accidents. With proper heat transfer models and qualification, the modelling capabilities of the GOTHIC code can be expanded to address additional phenomena than envisaged to date.

The developed subroutine and its implementation were made without having access to the GOTHIC source code and was based on the 2006 CHF Look-up Tables by Groeneveld, as one of the most widely used methods for the prediction of CHF. A modifier coefficient was applied to the Look-up Table in order account for different surface angles. Based on experimental data from the ULPU facility, installed at the University of California, Santa Barbara in USA, the subroutine was qualified for CHF situations occurring due to In-Vessel Retention (IVR) accident management procedures using data from 9 different full-scale experiments.

In order to better analyze the performance of the CHF subroutine, the model was applied to the real containment model of Atucha-I, for two severe accident scenarios (LBLOCA and SBO). Compared to a previous analysis of the containment without the CHF model, which used very conservative boundary conditions to model the vapor generation due to the postulated IVR procedure, current results show that the postulated procedure of IVR would be successful with a much lower risk to the Containment integrity due to its lower pressurization (notably lower pressure than the initial analysis, while successfully transferring the imposed decay heat).

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Table of Figures:

Figure 2.1: Example of a Boiling Curve as a function of heat flux and heat structure superheating

temperature. ... 6

Figure 2.2: Schematic of in-vessel retention phenomenology (Ref. [8]) ... 7

Figure 2.3: Example of key geometric features of cavity flooding and venting paths. Illustration of the two-phase boundary layer on the lower head. Thermal insulation not shown (Ref. [8]). ... 8

Figure 3.1: SCCRED Flow Chart ... 24

Figure 3.2: The SCCRED Qualification procedure flow chart ... 26

Figure 3.3: Vertical cross-sections of the Atucha I NPP containment. ... 32

Figure 3.4: 3D CAD view of the containment (dome and annular areas excluded). ... 33

Figure 3.5: Free Volume versus elevation curve. ... 33

Figure 3.6: Volume of concrete structures versus elevation curve. ... 33

Figure 3.7: The blockages (grey) used for geometrically representing the SG-A nodalization (left) and Geometrical shape correspondence of SG-A in the containment (right). ... 35

Figure 3.8: Modeling difficulties in GOTHIC for annular spaces. Left – would fail to account for NC. Middle – only X direction grid lines created in order to account for NC. Right –fully accounting for NC only in a small region of the annular space. ... 35

Figure 3.9: GOTHIC representation of the detailed nodalization (on the left are the junctions in green labels, on the right are the thermal conductors in red labels). ... 38

Figure 3.10: Detailed nodalization sketch level 0. ... 38

Figure 3.11: Detailed nodalization sketch level 1. ... 39

Figure 3.12: Detailed nodalization sketch level 2 example. ... 39

Figure 3.13: Example of a level 3 sketch (SG-A room, from -4.5 to -3.8 m). ... 40

Figure 3.14: Comparison of the free volume vs. elevation of the nodalizations and the RDS ... 41

Figure 3.15: Steady State Achievement (pressure in the cavity volume) ... 42

Figure 3.16: Steady State Achievement (temperature in the SG rooms) ... 43

Figure 3.17: Pressure and Steam Mass in the Dome for the LBLOCA Scenario (from 0 s to 5,000 s) ... 48

Figure 3.18: Pressure and Steam Mass in the Dome for the LBLOCA Scenario (from 0 s to 5,000 s) ... 49

Figure 3.19: Pressure and Steam Mass in the Dome for the LBLOCA Scenario (from 0 s to 100,000 s) .... 49

Figure 3.20: Pressure and Steam Mass in the Dome for the LBLOCA Scenario (from 0 s to 200,000 s) .... 50

Figure 3.21: Temperature and Vapor Phase Energy in the Dome for the LBLOCA Scenario (from 0 s to 5,000 s) ... 50

Figure 3.24: Liquid Level and Total Liquid Mass for the LBLOCA scenario (from 0 s to 1,500 s) ... 51

Figure 3.25: Liquid Level and Total Liquid Mass for the LBLOCA scenario (from 0 s to 200,000 s) ... 51

Figure 3.26: Time Evolution of H2 Iso-Concentration Surfaces for the LBLOCA scenario ... 52

Figure 3.27: The influence of the Drop Breakup Model and the Drop Diameter on the Pressure in the Dome, (short term effect) ... 52 Figure 3.28: The influence of the Drop Breakup Model and the Drop Diameter on the Temperature in the

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Figure 3.29: The influence of the Drop Breakup Model and the Drop Diameter on the Pressure in the

Dome, (long term effect) ... 53

Figure 3.30: The influence of the Drop Breakup Model and the Drop Diameter on the Temperature in the Dome, (long term effect) ... 53

Figure 3.31: Pressure values from sensitivity analysis compared to the Reference EMs ... 54

Figure 3.32: Temperature values from sensitivity analysis compared to the Reference Ems... 56

Figure 3.33: Pressure and Steam Mass in the Dome for the SBO Scenario (from 0 s to 100,000 s) ... 57

Figure 3.34: Temperature and Vapor Phase Energy in the Dome for the SBO Scenario (from 0 s to 100,000 s) ... 58

Figure 3.35: Pressure and Steam Mass in the Dome for the SBO Scenario (from 0 s to 200,000 s) ... 58

Figure 3.36: Temperature and Vapor Phase Energy in the Dome for the SBO Scenario (from 0 s to 200,000 s) ... 58

Figure 3.37: Time Evolution of H2 Iso-Concentration Surfaces for the SBO scenario ... 59

Figure 3.38: The effect of droplets on the pressure for the SBO case ... 59

Figure 3.39: The effect of droplets on the temperature for the SBO case ... 59

Figure 3.40: Pressure values from the sensitivity analysis compared to the Reference EMs ... 60

Figure 3.41: Temperature values from the sensitivity analysis compared to the Reference EMs ... 61

Figure 3.42: SBO Pressure results, difference between lumped and coarse calculations ... 63

Figure 3.43: SBO Liquid Level results, difference between lumped and coarse calculations ... 64

Figure 3.44: Model A (left) and Model B (right) ... 64

Figure 3.45: Model A Results (left), Model B results with Liquid Volume Fraction initialized at 80% (center), ... 65

Figure 3.46: Sketch of the model ... 66

Figure 3.47: GOTHIC 8.0 pressure drops when level crosses a horizontal grid line ... 66

Figure 3.48: GOTHIC 8.1 pressure drops when level crosses a horizontal grid line ... 66

Figure 4.1: Initial schematic for the full boiling curve implementation in GOTHIC ... 68

Figure 4.2: RELAP5-3D©_{wall heat transfer flow chart (Ref. [37]). ... 69}

Figure 4.3: Critical Heat Flux Subroutine implementation flow chart (second implementation). ... 71

Figure 4.4: Used GOTHIC nodalization for the code-to-code comparison. ... 72

Figure 4.5: Cavity Volume created with blockages in order to replicate the real hardware of the Atucha-1 NPP. ... 73

Figure 4.6: RPV Thermal Conductor meshing ... 74

Figure 4.7: GOTHIC vs RELAP5 code to code comparison volume liquid fraction ... 75

Figure 4.8: GOTHIC vs RELAP5 code to code comparison Heat Flux... 75

Figure 4.9: ULPU Configuration I (figure based on Ref. [8]). The heater blocks extend over the region -30° to +30°. ... 77

Figure 4.10: Schematic of Configurations II and III in ULPU. The heater blocks extend over the region 0° to 90° (Ref. [8]). ... 77

Figure 4.11: ULPU Configuration V Facility Schematic (Ref. [9])... 78

Figure 4.12: Surface aging effect (Configuration II) (Ref. [8]). ... 78

Figure 4.13: ULPU Configuration III CHF results compared to the paper author’s proposed CHF correlation (Ref. [8]). ... 79

Figure 4.14: Exit from riser in the ULPU Configuration V (Ref. [9]). ... 80

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Figure 4.16: Flow rate as function of input power (data at different power levels) (Ref. [9]). ... 81

Figure 4.17: Flow rate as function of input power at the burnout step, sorted by baffle positions (Ref. [9]) ... 81

Figure 4.18: Flow rate vs. Critical Heat Flux (Ref. [9]). ... 82

Figure 4.19: ULPU-III Simplified nodalization Sketch (left), ULPU Configuration III nodalization diagram as seen in GOTHIC (right) ... 83

Figure 4.20: Heat Flux transferred for GOTHIC ULPU Configuration III nodalization results, compared to the experimentally derived CHF ... 85

Figure 4.21: Heat Flux transferred for GOTHIC ULPU Configuration III at 90°, comparison between Direct and Film Heat Transfer Model ... 85

Figure 4.22: Simplified nodalization Sketch (left), ULPU Configuration III for subroutine implementation nodalization diagram as seen in GOTHIC (right) ... 87

Figure 4.23: CHF Subroutine based on Groeneveld table (with no surface angle information) at 45° ... 88

Figure 4.26: CHF Subroutine based on MELCOR correlation at 90°... 89

Figure 4.27: CHF Subroutine based on MELCOR correlation at 45°... 90

Figure 4.28: ULPU-V Simplified nodalization Sketch (left), ULPU Configuration V nodalization diagram as seen in GOTHIC (right) ... 91

Figure 4.29: Heat Flux transferred for GOTHIC ULPU Configuration V nodalization results, compared to the experimentally derived CHF, for a 15, 43 and 67° surface angle ... 92

Figure 4.30: Heat Flux transferred for GOTHIC ULPU Configuration V nodalization results, compared to the experimentally derived CHF, for a 15° surface angle (better view of oscillations) ... 93

Figure 4.31: Heat Flux transferred for GOTHIC ULPU Configuration V nodalization results, compared to the experimentally derived CHF, for a 71° surface angle ... 93

Figure 4.32: Heat Flux transferred for GOTHIC ULPU Configuration V nodalization results, compared to the experimentally derived CHF, for an 82° surface angle ... 93

Figure 4.33: Simplified nodalization Sketch (left), ULPU Configuration V for subroutine implementation nodalization diagram as seen in GOTHIC (right) ... 94

Figure 4.34: GOTHIC Results for Configuration III, using the CHF subroutine with the angular modification factor ... 96

Figure 4.35: GOTHIC Results for Configuration V, using the CHF subroutine with the angular modification factor, for a 15, 43 and 67° surface angle ... 96

Figure 4.36: GOTHIC Results for Configuration V, using the CHF subroutine with the angular modification factor, for a 71° surface angle ... 97

Figure 4.37: GOTHIC Results for Configuration V, using the CHF subroutine with the angular modification factor, for an 82° surface angle ... 97

Figure 4.38: CHF Subroutine without Heat Feedback function (left), and with (right) ... 98

Figure 4.39: Heat Flux transferred to the fictitious volume (qB1) and transferred to the real volume (qB2) – w/o feedback function ... 99

Figure 4.40: HS A Temperature at side A (imposed heat) – w/o feedback function ... 99

Figure 4.41: Heat Flux transferred to the fictitious volume (qB1) and transferred to the real volume (qB2) – with feedback function ... 100

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Figure 4.43: Critical Heat Flux Subroutine implementation flow chart (second implementation). ... 101

Figure 5.1: Left figure – bottom filler positioning inside the RPV; right figure – subdivision of the RPV heat structure ... 104

Figure 5.2: Nodalization sketch differences between the Atucha-I containment nodalization without the CHF subroutine (left) and with the subroutine (right) ... 104

Figure 5.3: Better view of the differences, left picture represents the 1s, or the Cavity Volume that is connected to the nodalization, while the right picture represents control volume 44s, the copy of the Cavity Volume, DLL Cavity ... 105

Figure 5.4: Side view representation of the Cavity Control Volume, grey areas denote blockages, black areas denote free volume ... 105

Figure 5.5: LBLOCA Pressure in the Dome of the Containment ... 108

Figure 5.6: LBLOCA Temperature in the Dome of the Containment ... 109

Figure 5.7: LBLOCA HS Bottom Heat Flux transferred to the liquid phase ... 109

Figure 5.8: LBLOCA HS Second Heat Flux transferred to the liquid phase (0 to 100000 s) ... 109

Figure 5.9: LBLOCA HS Second Heat Flux transferred to the liquid phase (0 to 100 s) ... 110

Figure 5.10: LBLOCA HS Corium Heat Flux transferred to the liquid phase (0 to 100 000 s) ... 110

Figure 5.11: LBLOCA HS Corium Heat Flux transferred to the liquid phase (0 to 100 s) ... 110

Figure 5.12: SBO Pressure in the Dome of the Containment ... 111

Figure 5.13: SBO Temperature in the Dome of the Containment ... 112

Figure 5.14: SBO HS Bottom Heat Flux transferred to the liquid phase (0 to 100 000 s) ... 112

Figure 5.15: SBO HS Bottom Heat Flux transferred to the liquid phase (17 000 to 19 000 s) ... 112

Figure 5.16: SBO HS Second Heat Flux transferred to the liquid phase ... 113

Figure 5.17: SBO HS Corium Heat Flux transferred to the liquid phase ... 113

Figure 5.18: SBO Temperature profile of the RPV Corium heat structure ... 113

Figure 5.19: SBO Temperature profile of the RPV Corium heat structure (focus on carbon steel region) ... 114

Figure 5.20: LBLOCA Sensitivity Heat Flux for the RPV Corium Heat Structure (focus on GOTHIC results) ... 115

Figure 5.21: LBLOCA Sensitivity Heat Flux for the RPV Corium Heat Structure (focus on CHF subroutine results) ... 116

Figure 5.22: LBLOCA Sensitivity Heat Flux for the RPV Corium Heat Structure (0 to 20 000s) ... 116

Figure 5.23: LBLOCA Sensitivity Pressure in the Containment Dome ... 116

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List of Tables:

Table 2.1. RELAP5 CHF Table lookup multipliers ... 19

Table 3.1: Closure equations/models in GOTHIC ... 29

Table 3.2: Coarse simplification decision matrix ... 37

Table 3.3: Demonstration of geometrical fidelity: acceptable errors for RCS ... 41

Table 3.4: Geometrical fidelity demonstration, comparison with RDS ... 41

Table 3.5: Inherent drift of thermal hydraulic parameters... 43

Table 3.6: RTAs corresponding to the first Ph. W. for the LBLOCA transient ... 46

Table 3.7: Ph. W, phenomena and RTAs for the Atucha I Containment response - LBLOCA transient ... 47

Table 3.8: CNA-I LBLOCA – list of sensitivities ... 55

Table 3.9: List of sensitivities performed in addition to the set performed for the LBLOCA scenario ... 60

Table 4.1: RPV Thermal Conductor meshing ... 74

Table 4.2: ULPU Configuration III Critical Heat Flux ... 79

Table 4.3: GOTHIC ULPU Configuration III Control Volumes ... 82

Table 4.4: GOTHIC ULPU Configuration III Flow Paths ... 83

Table 4.5: GOTHIC ULPU Configuration III Thermal Conductor Properties ... 84

Table 4.6: GOTHIC ULPU Configuration III Control Variables ... 84

Table 4.7: GOTHIC ULPU Configuration III with DLL additional Control Volume ... 86

Table 4.8: GOTHIC ULPU Configuration III with DLL additional Flow Paths ... 86

Table 4.9: GOTHIC ULPU Configuration III with DLL additional Thermal Conductor Properties ... 86

Table 4.10: GOTHIC ULPU Configuration III with DLL additional Control Variables ... 87

Table 4.11: GOTHIC ULPU Configuration V Thermal Conductor Properties ... 90

Table 4.12: GOTHIC ULPU Configuration V Control Variables ... 90

Table 4.13: GOTHIC ULPU Configuration V Control Volumes ... 91

Table 4.14: GOTHIC ULPU Configuration V Flow Paths... 92

Table 4.15: GOTHIC ULPU Configuration V with DLL additional Control Volume ... 94

Table 4.16: GOTHIC ULPU Configuration V with DLL additional Flow Paths... 95

Table 4.17: GOTHIC ULPU Configuration V with DLL additional Thermal Conductor Properties ... 95

Table 4.18: CHF Subroutine Results after implementing the angular modification factor ... 97

Table 5.1: GOTHIC Atucha I CHF subroutine additional Boundary Conditions ... 106

Table 5.2: GOTHIC Atucha I CHF subroutine additional Control Variables ... 107

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List of acronyms

3D Three Dimensional

ABWR Advanced Boiling Water Reactor AFD Annular Film Dryout

BC Boundary Condition

BE Best Estimate

BEPU Best Estimate Plus Uncertainty

BO Burnout

BT Boiling Transition BWR Boiling Water Reactor CAD Computer Aided Design CFD Computational Fluid Dynamics CHF Critical Heat Flux

CNA-I Nuclear Power Plant Atucha-I CPU Central Processing Unit

CSNI Committee on the Safety of Nuclear Installations CV Control Variable

CVM Code Validation Matrix DLL Dynamic Link Library

DLM-FM Diffusion Layer Model with Film roughening and Mist generation DNB Departure from Nuclear Boiling

DO Dryout

EH Engineering Handbook

EM Evaluation Model

EPRI Electric Power Research Institute EXE Executable program for windows FOM Figures of Merit

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HS Heat Structure

IAEA International Atomic Energy Agency

IRIS International Reactor Innovative and Secure ITF Integral Test Facilities

LBLOCA Large Break Loss of Coolant Accident LCC Local Conditions Correlation

LFD Liquid Film Dryout

NA-SA Nucleoeléctrica Argentina NC Natural Circulation NEA Nuclear Energy Agency NPP Nuclear Power Plant

PAR Passive Auto-Catalytic Recombiner PE Portable Executable

Ph. W. Phenomenological Window PRV Pressurizer Relief Valve PWR Pressurized Water Reactor QR Qualification Report RB Reactor Building RCS Reactor Cooling System RDS Reference Data Set

RIA Reactivity Induced Accident RPV Reactor Pressure Vessel

RTA Relevant Thermal-Hydraulic Aspects

SA Severe Accident

SAM Severe Accident Management SBLOCA Small Break Loss of Coolant Accident SBO Station Black-Out

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SETF Separate Effect Test Facility

SG Steam Generator

SS Steady State

TC Thermal Conductor (Heat Structure)

TH Thermal Hydraulic

UCC Upstream Conditions Correlation V &V Verification and Validation

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Nomenclature*

αf Liquid phase void fraction (-)

α_g Vapor/gas phase void fraction (-) cpl Heat Capacity of Liquid (J/kg K)

D or Dh Hydraulic diameter (m)

Drod Diameter of a heated rod (m)

Fnu Non-uniform axial flux factor (-)

G Mass Flux (kg/m2_s)

g Acceleration of gravity (m/s2₎

Gr Grashof number

h_f Liquid specific enthalpy (J/kg)

h_fs Liquid specific enthalpy at saturation conditions (J/kg) hfg Latent heat (J/kg)

h_g Vapor specific enthalpy (J/kg) HTC Heat Transfer Coefficient (W/m2_K)

Kbf Bundle Factor (-)

Kcw Cold Wall Factor (-)

Khl Heated Length Factor (-)

Khor Horizontal flow factor (-)

Khy Hydraulic diameter factor (m)

Knu Axial heat flux distribution factor (-)

Kp Grid Pressure Loss Coefficient (-)

Krp Radial Power Distribution Factor (-)

Ksp Grid Spacer Factor (-)

Kver Vertical flow factor (-)

L Length (m)

P Pressure (Pa)

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Psat Pressure at saturation conditions (Pa)

q or q’’ _{Heat Flux (W/m}2₎

qCHF or Qchf Critical Heat Flux (W/m2)

ρ_f Density of liquid phase (kg/m3₎

ρg Density of vapor/gas phase (kg/m3)

ρ_l Density of liquid (kg/m3₎

ρ_v Density of vapor (kg/m3₎

Re Reynolds number

RPF Radial Power Factor (-)

Tavg Average temperature, (Tsurf+Tsat)/2, (K)

Tf Fluid Temperature (K)

Tmax Maximum fluid temperature in the room

Tmin Minimum initial temperature in the room

Tsat Saturation Temperature (K)

Tw or TS Wall Temperature (K)

V Volume (m3₎

v_f Liquid phase velocity (m/s) vg Vapor/gas phase velocity (m/s)

WA Wetted Area (m2)

Xcrit Local thermodynamic quality at critical conditions (-)

Xin Local thermodynamic quality at inlet (-)

𝑋𝑒 Thermodynamic equilibrium quality (-) 𝜎 Surface tension of water (N/m) 𝜇 Dynamic viscosity of liquid (kg/m s)

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1 Introduction and Motivation

GOTHIC is an integrated, general purpose thermal hydraulic (TH) software package for design, licensing, safety and operating analysis of Nuclear Power Plant containments, conﬁnement buildings and system components (Ref. [1]). It bridges the gap between the lumped parameter codes frequently used for containment analysis (like the MELCOR, MAAP, COCOSYS and ASTEC codes) and Computational Fluid Dynamics (CFD) codes.

While GOTHIC lacks the capability to model the “in-vessel” degraded core and vessel breach phenomena available in MAAP or MELCOR, it can utilize the output from such codes as input for a more detailed containment response modelling, in order to provide greater insight into the subtle phenomena that may significantly impact a containment (e.g. Hydrogen Distribution). Within a single model, GOTHIC can include regions treated in conventional lumped parameter mode and regions with three-dimensional flows in complex geometries. Although it does not include the capability to model the details of the boundary layers as in most CFD codes, through the use of standard wall functions for heat and momentum transfer, it can give good estimates of the three dimensional (3D) flows and distribution with significantly lower computational time than typical CFD codes. Additionally, it includes the capability to model multiphase flow situations including dropwise and wall condensation, pool surfaces and sprays without special user supplied models. The 3D capabilities of GOTHIC in simulating basic flows and, in detail, hydrogen flows for containment analysis have been investigated extensively, simulating tests in facilities like PANDA, CSTF, BFMC or CVTR,(see Andreani and Paladino, 2010); (Papini et al., 2015) and (Fernández-Cosials et al., 2015 for more details). In addition, GOTHIC has been used also for Generation III+ reactors, such as the AP600 and AP1000 (Hung et al., 2015), the ABWR (Chen et al., 2012), or the IRIS reactor (Papini et al., 2010).

The heat transfer correlations built into GOTHIC cover the portion of the boiling curve which spans single phase heat transfer up to pre-CHF (critical heat flux) heat transfer. The boiling curve is truncated to exclude post-CHF heat transfer as it has not been adequately verified and was considered by the developers to have little application in general containment analysis (Ref. [2]). As such, one area that the code is not currently qualified for is In-Vessel Corium Retention analysis, where water enters in contact with the high temperature pressure vessel walls, requiring the modelling of CHF heat transfer. The present doctoral research is aimed to create an external function that solves this limitation, accounting for CHF phenomena. Being able to model CHF would be very useful in order to be able to simulate the external Pressure Vessel Cooling of different reactor types, as well as other types of severe accidents.

The GOTHIC 8.0 code was used to perform a containment safety analysis for the Atucha-I NPP (CNA-I) (Ref. [3]). The CNA-I has a full pressure containment constituted by a steel sphere enveloped by a concrete shell and an annular gap of air in between (Zarate and Valle Cepero, 2001). The target of the analysis is the evaluation of a possible new procedure of an external Reactor Pressure Vessel (RPV) cooling, by ﬂooding the reactor cavity, in order to preserve its integrity in case of a severe accident (SA) situation. As a consequence of this new procedure, there will be an additional production of steam in the reactor cavity, which may cause the pressurization of the containment above the safety limit.

The accident management procedure consisting in the external RPV cooling is to be applied when speciﬁc Severe Accident conditions take place, namely the core melting and relocation in the lower plenum. Two diﬀerent accident scenarios have been analyzed. The Large Break Loss of Coolant Accident (LBLOCA)

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scenario is considered without the actuation of the active Emergency Core Cooling System, while the Station Black Out (SBO) scenario is considered to occur without breaks due to creep. The main focus of the analysis is to evaluate the Thermal Hydraulic consequences of the emergency procedure in the containment and not the Severe Accident evolution of the LBLOCA or SBO scenarios. The following two implicit assumptions are considered:

• The integrity of the containment is assumed for the whole duration of the scenarios;

• The integrity of the vessel is assumed during the application of the accident management procedure. For the initial Analysis of the CNA-I, as the GOTHIC code is not fully qualified for Critical Heat Flux phenomena, the additional production of steam was imposed as a steam injection boundary condition. This condition was very conservative, based on the latent heat of the water and the decay heat of the reactor, by assuming that a large proportion of the decay heat is released to the water, not taking into account CHF phenomena, as these conditions would produce the maximum amount of containment pressurization.

The present research is therefore focused towards implementing a CHF external function inside of GOTHIC, to enable it to properly analyze the two postulated Atucha-1 in-vessel retention scenarios without excessively conservative boundary conditions.

The developed subroutine and its implementation were to be performed without having access to the GOTHIC source code. The subroutine is required to pass through a qualification process by using experimental data related to CHF occurring in In-Vessel Retention situations.

In order to better analyze the performance of the CHF subroutine, the model was applied to the real containment model of Atucha-I, for the two severe accident scenarios (LBLOCA and SBO).

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2 Adopted Methodology

The first step in improving GOTHIC is to identify the exact code limitations regarding two-phase heat transfer. In respect to condensation phenomena, the code was extensively qualified in a high number of tests (Ref. [1]). Concerning boiling phenomena, from the code user and technical manuals it was determined that the heat transfer correlations built into GOTHIC have “the boiling curve truncated to exclude post-Critical Heat Flux heat transfer since these models have little application in general containment analysis” and have not been adequately verified (Ref. [2]). Boiling heat transfer is using a very simplistic model for film boiling that “is not intended to give accurate results for the transition, film boiling and rewet phase of boiling heat transfer” (Ref. [2]). The code was qualified for nucleate and subcooled boiling based on on the following data:

1) FLECHT SEASET Natural Circulation Tests (Ref. [4]): performed on a full height 1/307 volume scale facility of a 4 loop PWR; specific information regarding the tests used for the qualification, the model used, and the results could not be obtained (not found);

2) seven analyses performed in GOTHIC to simulate tests performed by Helge Christensen (Ref. [5]); the testing facility was used to achieve subcooled boiling by having a well-controlled inlet flow (pressure and temperature) enter a vertical rectangular channel which was electrically heated; the system was then able to measure the void fraction along the heated channel with the use of a gamma densitometer; the heated section of the channel was 1.11 cm by 4.44 cm and 127 cm long.

Post CHF Heat transfer actually may occurs in several instances during a severe accident while considering containment safety analysis, for example:

 Water from sprays enters in contact with high temperature surfaces (e.g. the Reactor Pressure Vessel Upper Head, Leg piping);

 Saturated water from a Large Break Loss Of Coolant Accident (LBLOCA) coming in contact with the heated surface of the RPV;

 External Reactor Pressure Vessel cooling;

In Chapter 4.4 of the thesis, where the qualification of the developed sub-routine function is achieved, these limitations are more precisely presented and quantified.

The proposed improvement to the code is meant to enable GOTHIC to be used in dry-out conditions and is possible either through the modification of the source code (unfortunately not available) or through the development of subroutines to be implemented as a Dynamic Link Library so that the boiling curve can be coupled with the code. Implementing the CHF subroutines can enable GOTHIC to predict Critical Heat Flux scenarios such as:

● Possibility of studying In Vessel Retention (Thermal Hydraulic aspect):

 e.g. permitting the code application to the AP1000 reactors and to the Atucha-I NPP; ● Enabling GOTHIC to be partly used as a system code (if properly qualified) :

 Simulating primary system even during dry-out conditions (LBLOCA, SBO scenarios). The initial sub-routine improvement was meant to have various user-selectable options to facilitate its

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situations. Due to the difficulty in qualifying each model and the limited resources available, the focus of the PhD research was dedicated in improving the GOTHIC code for In-Vessel retention CHF situations. In the future, the developed sub-routine may be improved by performing qualification tests so that it can be also applied to in-core CHF situations.

As GOTHIC has full 3D capabilities, with turbulence models (k-ε), the implementation of the post-CHF model in the code, brings a significant improvement in the way severe accidents are modeled, offering the possibility for CFD-similar codes of correctly modeling boiling phenomena, while having relatively low computing requirements. The Atucha-1 analysis is to be the first time a fully 3D code is used in modeling In Vessel Retention (and not only lumped parameter codes).

After the GOTHIC code limitations were determined, the second step was to perform a literature study and find possible improvements. The Groeneveld 2006 CHF Look-up Table (Ref. [6]) was found to be adequate, as it was extensively qualified for use in very diverse situations and the fact several thermal-hydraulic codes make use of it.

Initially the idea was to have the full boiling curve implemented in the code, with all the heat structure calculations performed within the DLL, using boundary conditions to simulate the appropriate amount of steam generated, liquid removed and heating of each phase. This was later determined to be unfeasible without access to source code, because momentum equations and additional evaporation computed by the code could not be successfully compensated for. Evaporation and condensation of droplet fields is very well documented in both the technical manual and the qualification report of GOTHIC, but the evaporation of a heated pool is not well documented. The only information supplied on this aspect was regarding the qualification of GOTHIC, which used experimental data from four tests performed by Bagaasen at the Grout-Pilot-Scale facility (Ref. [7]), which was not sufficient for compensating for this phenomena occurring at other thermodynamic ranges.

Because of this problem, the third step was a work-around designed such that the natural convection boiling and the nucleate boiling phases (pre-CHF) use GOTHIC’s own Chen correlation, while the limitation in the maximum transferred Heat Flux to the liquid phase at the CHF value determined through the sub-routine.

The fourth step consisted in the Dynamic Link Library sub-routine development, which was performed in the C programming language. This sub-routine would receive required information supplied by GOTHIC from a certain cell and heat-structure from a nodalization, would compare the heat flux of that heat structure to the calculated CHF value, limiting the GOTHIC calculated Heat-Flux as required. This would be performed at each time-step.

The fifth step was performed but in the end, it was not used. It consisted of a simple code-to-code comparison between GOTHIC and RELAP5-3D©_{for a model of the Atucha-I Cavity room volume being filled}

with liquid while the Reactor Pressure Vessel simulated heat structure has a high imposed heat flux. This step was meant to provide a code-to-code qualification of the developed DLL subroutine. It was not used for qualification process as the RELAP5-3D©_{results varied greatly based on different heat structure}

options used (more specifically the “reflood” option). It was determined that both RELAP5©_{and TRACE}©

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if the flow is horizontal or vertical). Different surface angles are needed because for In-Vessel retention, the corium settles at the bottom of the vessel (which is an angled/curved surface) and CHF varies significantly in relation to the heated surface angle (Ref. [8], [9], [10]).

The sixth step consisted of the qualification of the sub-routine based on experimental tests. Because the Look-up Table determined maximum heat flux is for an 8 mm tube with vertical flow, the determined values were adjusted and qualified for Reactor Pressure Vessel type structures, and for different surface angles. The experimental facility chosen for this scope is ULPU (Ref. [8]), which is a facility used to perform full-scale simulations of the boiling crisis phenomenon on the hemispherical lower head of a reactor pressure vessel (AP1000) submerged in water. In order to properly qualify the CHF subroutines, the qualification process was comprised of two different facility configurations, Configuration III and Configuration V, with three experiments analyzed in the former configuration and five experiments in the latter configuration. For the qualification purpose, two GOTHIC nodalizations were developed, one for each analyzed configuration. It should be noted that the Groeneveld 2006 CHF Look-up Tables consist of different experimental data for different diameter tubes, the data is normalized to 8 mm tube diameters; the use of the Table for different diameters can be accounted for through the adoption of a diameter correction factor, enabling its usage for high hydraulic diameter values similar to Cavity-RPV situations. The effect of diameter becomes negligible for tubes with a diameter greater than 25 mm (Ref. [11]). During this step, for comparison reasons, the CHF correlation used in MELCOR (version 1.8.6) was also tested, finding that it offered quite conservative results.

The final step was the application of the DLL CHF subroutine to a real containment model (Atucha I NPP) and implementing the accident management procedure consisting in the external RPV cooling following two postulated accidents (LBLOCA and SBO), both consisting in core melting and relocation to the lower plenum. This way it can be shown that the developed sub-routine code improvement is properly working and can be successfully used in a full-scale analysis. The real containment model was previously run with a very conservative assumption (for the containment pressure) that all the decay heat is successfully transferred to the water (at saturation temperature) and produces steam, with only a very limited occurrence of CHF (few tens of seconds) due to the high surface area of the Atucha-1 RPV. To effectively demonstrate that the sub-routine is functional even for prolonged periods, a sensitivity with excessively conservative conditions (fictitious) was also performed.

2.1 Critical Heat Flux Definition

Critical heat flux (CHF) describes the thermal limit of nucleate boiling heat transfer, where phase change occurs during heating (with bubbles forming on a metal surface used to heat water), which suddenly decreases the efficiency of heat transfer, thus causing localized overheating of the surface.

When liquid coolant undergoes a change in phase due to the absorption of heat from a heated solid surface, a higher transfer rate occurs. The more efficient heat transfer from the heated surface (in the form of heat of vaporization plus sensible heat) and the motions of the bubbles (bubble-driven turbulence and convection) leads to rapid mixing of the fluid. Therefore, boiling heat transfer has played an important role in industrial heat transfer processes such as macroscopic heat transfer exchangers in nuclear and fossil power plants, and in microscopic heat transfer devices such as heat pipes and micro-channels for cooling electronic chips.

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The use of boiling is limited by a condition called critical heat flux (CHF), which is also called a boiling crisis or departure from nucleate boiling (DNB), liquid film dry-out (LFD), annular film dryout (AFD), dryout (DO), burnout (BO), boiling crisis, boiling transition (BT), etc., depending on the specific mode of its occurrence. The most serious problem is that the boiling limitation can cause the physical burnout of the materials of a heated surface due to the suddenly inefficient heat transfer through a vapor film formed across the surface resulting from the replacement of liquid by vapor adjacent to the heated surface.

Consequently, the occurrence of CHF with an imposed heat flux boundary condition is accompanied by an inordinate increase in the surface temperature for a surface-heat-flux-controlled system. Otherwise, an inordinate decrease of the heat transfer rate occurs for a surface-temperature-controlled system. This can be explained with Newton's law of cooling:

𝑞 = 𝐻𝑇𝐶 ∙ (𝑇_𝑤− 𝑇_𝑓) where,

q Heat Flux

HTC Heat Transfer Coefficient Tw Wall Temperature

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The Critical Heat Flux is an important point on the boiling curve (Figure 2.1) and it may be desirable to operate a boiling process near this point. However, one could become cautious of dissipating heat in excess of this amount.

2.2 Importance of CHF in nuclear safety

Critical Heat Flux models are used frequently in performing nuclear reactor analyses, both for heat transfer inside the core, between the rods and the coolant, but also outside the core, in the case of certain nuclear containment analyses. Several new reactor types (e.g. AP 1000 reactor) consider using In Vessel Retention (IVR) as an accident management procedure, preventing the RPV failure and the spilling of highly radioactive materials on the containment floor, thus preventing any risk to the integrity of the last safety barrier to the environment.

The effectiveness of In-Vessel Retention of molten core debris by means of external reactor vessel flooding has been thoroughly documented, reviewed as a part of licensing certification, and accepted by the US Nuclear Regulatory Commission for the Westinghouse’s AP600 (Advanced Passive Light Water Reactor) design. A successful In-Vessel Retention would terminate a severe accident, passively, with the core in a stable, coolable configuration (within the lower head), thus avoiding the largely uncertain accident evolution with the molten debris on the containment floor. This passive plant design has been upgraded by Westinghouse to the AP1000, a 1000 MWe plant very similar to the AP600.

In the management of severe accidents, the relocation of molten corium could be stopped in the region of the lower head of a reactor pressure vessel, by external flooding (Figure 2.2). For this accident management procedure to be successful, it is necessary that the thermal load created by natural convection of the corium inside the RPV be below what could cause a critical heat flux on the outside walls of the RPV. Figure 2.3 depicts the reactor-vessel-cavity configuration in the AP600 reactor design with the key features of accident management procedure.

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Figure 2.3: Example of key geometric features of cavity flooding and venting paths. Illustration of the two-phase boundary layer on the lower head. Thermal insulation not shown (Ref. [8]).

2.3 CHF Literature study

To this date, the knowledge of the physical nature of CHF and the mechanism of a boiling crisis is reasonably known, the prediction of this phenomena is still difficult. A large number of empirical correlations have been developed based on CHF experimental data; each of which apply for various geometries and particular parameter ranges. The boiling curve for fast transients (for example RIA, Reactivity Induced Accident) is drastically different from the one for slow transients (Ref. [12]). Differences in water composition and surface roughness influence the nucleation points and can be the cause for more than a 300% difference in the CHF value (Ref. [13]).

In order to successfully implement Critical Heat Flux limitations inside of GOTHIC, a study of the methods for predicting CHF used in different codes was carried out. These methods can be categorized into:

 Empirical Correlations (e.g., MELCOR)  Look-up Tables (e.g., RELAP)

 Mechanistic Models (e.g., CFD)

Regarding CHF prediction, correlations are classified in two categories: upstream conditions correlation (UCC) and local conditions correlation (LCC). For the UCC the CHF is a function of five known independent variables:

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The LCC CHF correlation is an explicit function of local quality, based on four independent variables, the length L is eliminated, and it can be evaluated by using the direct substitution method or the heat balance method (Ref. [14]):

𝑞_𝐶𝐻𝐹= 𝑓(𝐺, 𝐷, 𝑃, 𝑥_{𝑐𝑟𝑖𝑡}) (2) where,

q

CHF Critical heat flux, W/m2

G

Mass Flux, kg/m2_s

D

Hydraulic diameter, m

P

Pressure, Pa

L

Length, m

x

in Local quality at the inlet

x

crit Local quality at critical conditions

The first performed experiments with non-uniform heat flux distribution were conducted on vertical round tubes with upwards flow. For determining the limitations, the form of the axial profile heat flux is a sine wave:

𝑞(𝑧) = 𝑞𝑚𝑎𝑥∙ sin (𝜋𝑧_𝐿) = 𝑞𝑚𝑎𝑥∙ 𝑓(𝑧) (3)

In commercial nuclear reactor cores, the conditions are for rod bundles, in which the diameter of the rods is small and are spaced from one another by the usage of grid spacers. The CHF phenomenon is different in this case, as compared to round tubes or annuli due, for example, to the cross flow between subchannels, the effect of grid spacers, the effect of cold walls.

The effect of non-uniform heating on CHF can be predicted based on one of the following hypotheses (Ref. [15]):

 Local conditions

 The F-factor

 The boiling length average heat flux

In the next section a short comparison between the available hypotheses for predicting CHF (Ref. [15]) is performed.



Local conditions

With this approach, it is assumed that the CHF is controlled only by the local heat flux and the local quality, without the need for upstream information. The linear critical heat flux as a function of quality can be formulated as:

𝑞𝐶𝐻𝐹(𝑧) = 𝐴 − 𝐵𝑥(𝑧) = 𝐴 −

𝐵

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The energy balance equation, when a rod bundle is assumed to behave as a one-dimensional single channel is given by:

ℎ(𝑧_𝐶𝐻𝐹) − ℎ_𝑓 =𝑁𝜋𝐷𝑟𝑜𝑑

𝐺𝐴 ∫ 𝑞(𝑧)𝑑𝑧

𝑍𝐶𝐻𝐹

0

− ∆ℎ_𝑖𝑛 (5)

where N is the number of heated rods, Drod is the diameter of a heated rod, q(z) is the heat flux, and G is

the mass flux. The average thermodynamic quality in this case is given by:

𝑥(𝑧_𝐶𝐻𝐹) = 𝑥_𝑖𝑛+𝑁𝜋𝐷𝑟𝑜𝑑

𝐺𝐴 ∫ 𝑞(𝑧)𝑑𝑧

𝑍𝐶𝐻𝐹

0 (6)



Peak heat flux method

This method is described by Collier and Thome (Ref. [16]). When the heat flux, q(z) and the enthalpy h(z) related by equation 5 satisfies equation 4, the critical heat flux is reached at the axial distance z from the inlet. The peak heat flux at this condition can be derived from the previous equations and is given by:

𝑞𝐶𝐻𝐹(𝑧) = 𝐴 + 𝐵∆ℎ𝑖𝑛/ℎ𝑓𝑔 𝑓(𝑧) + (𝐵𝜋𝐷𝑟𝑜𝑑𝐶/ℎ𝑓𝑔)∫ 𝑓(𝑧)𝑑𝑧₀𝑧 (7) where C is: 𝐶 = { 1 𝐺_𝑠𝑢𝑏𝐴_𝑠𝑢𝑏 𝑓𝑜𝑟 𝑎 𝑠𝑢𝑏𝑐ℎ𝑎𝑛𝑛𝑒𝑙 𝑁 𝐺𝐴 𝑓𝑜𝑟 𝑎𝑣𝑒𝑟𝑎𝑔𝑒𝑑 𝑓𝑙𝑜𝑤 (8)

The minimum value of qCHF(z) can be determined when the derivative of equation 7 is equal to zero, or

specifically: 𝑑 𝑑𝑧(𝑓(𝑧) + 𝐵𝜋𝐷𝑟𝑜𝑑𝐶 ℎ𝑓𝑔 ∫ 𝑓(𝑧)𝑑𝑧 𝑧 0 )=0 (9)

The constants A and B are taken from an adequate CHF correlation of a uniformly heated tube (e.g. from Look-up Tables).



Kirby correlation

By using non-uniform heating data exclusively, Kirby (Ref. [17]) optimized the constants of equation 4 as: 𝑞_𝐶𝐻𝐹 = 𝑌₁𝐺𝑌2𝐷𝑌3− 𝑌

4𝐺𝑌5𝐷𝑌6𝑋(𝑧) (10)

Where the constants Y1, Y2, Y3, Y4, Y5 and Y6 are functions of system pressure.



Macbeth correlation

By assuming that a channel behaves as a one dimensional, with average properties, Macbeth (Ref. [18]) has proposed a correlation as:

𝑞_𝐶𝐻𝐹∙ 10−6₌𝐴 + 𝐵∆ℎ𝑖𝑛

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

Barnett correlation

Adjusting a correlation derived for annulus situations with the help of correction factors, was found to predict a wide range of rod bundle data with remarkable accuracy. The Barnett correlation (Ref. [19]) has the same for as Equation 11 and may be used as:

𝑞_𝐶𝐻𝐹∙ 10−6₌𝐴(ℎ𝑓𝑔/649) + 𝐵∆ℎ𝑖𝑛

𝐶 + 𝑧 (12)



Hench-Levy correlation

For the design purpose of BWR type reactors, limit lines known as Janssen-Levy limit lines, have been developed to provide the CHF value as linear functions of mass flux and critical quality. These were subsequently replaced by the Hench-Levy limit lines (Ref. [20]), and are based on imperial units:

For 1000 psia 𝑞_𝐶𝐻𝐹 = { 1.0 1.9 − 3.3𝑥(𝑧) − 0.7𝑡𝑎𝑛ℎ2_(3𝐺) 0.6 − 0.7𝑥(𝑧) − 0.09𝑡𝑎𝑛ℎ2_(2𝐺) , 𝑥(𝑧) < 𝑥𝑙𝑖𝑚1 , 𝑥_{𝑙𝑖𝑚1}< 𝑥(𝑧) < 𝑥_{𝑙𝑖𝑚2} , 𝑥(𝑧) > 𝑥_{𝑙𝑖𝑚2} (13)

Where xlim1=0.273-0.212tanh2(3G), xlim2=0.5-0.269tanh2(3G)+0.0346tanh2(2G).

For pressures other than 1000 psia, the critical heat flux is given by:

𝑞_𝐶𝐻𝐹 = 𝑞₁₀₀₀(1.1 − 0.1 (𝑝 − 600 400 )

1.25

) (14)

Where p is the pressure in psia, G in Mlb/h ft2_{and q}

CHF in MBtu/h ft2.



Condie and Bengston correlation

Condie and Begston proposed an empirical correlation based on 5200 experiments on bundle CHF data (Ref. [15], [21], [22]):

𝑞_𝐶𝐻𝐹= 25.487 ∙ (𝐺/1356)

(0.1775∙ln(𝑥+1))

(𝑥 + 1)3.3906_{∙ 0.5356 ∙ 𝑝}0.3234_{∙ 𝑅𝑃𝐹}1.053 (15)

Where RPF is the maximum radial power factor for the bundle.



Bowring correlation

A mixed flow correlation that can be applied to all types of nuclear fuel was proposed by Bowring (Ref. [23]):

𝑞_𝐶𝐻𝐹 =𝐴 + 𝐵 ∙ ∆ℎ𝑖𝑛

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Where the constants A, B and C depend on pressure, mass flux, hydraulic diameter, equivalent heated diameter and shape profile of the heat flux. Y is the ratio of average heat flux from entry to position z, which is determined as:

𝑌 = 1 𝑧𝐶𝐻𝐹 ∫ 𝑞(𝑧) 𝑞𝐶𝐻𝐹𝑑𝑧 𝑧𝐶𝐻𝐹 0 (17)



EPRI correlation

Reddy and Fighetti, from EPRI, (Ref. [24]) developed a widely used general correlation which is applicable to both PWR and BWR conditions:

𝑞_𝐶𝐻𝐹 = 𝐴0− 𝑥𝑖𝑛

𝐶0∙ 𝐹𝑠∙ 𝐹𝑁𝑈+ (𝑥(𝑧) − 𝑥_𝑞(𝑧) 𝑖𝑛) (18)

where the constants A0 and C0 are functions of pressure and mass flux, and Fs is a grid spacer factor given

by:

𝐹_𝑠 = 1.3 − 0.3 ∙ 𝐾_𝑝 (19)

where Kp is a grid pressure loss coefficient. Fnu is a non-uniform axial flux factor:

𝐹_𝑛𝑢= 1 +𝑌 − 1

1 + 𝐺 (20)



Tong method

A consequence of the inconsistency of the local conditions’ hypothesis for non-uniform heat flux distributions is that the local critical heat flux depends on the heat flux profile upstream of the point considered. To take into account for the effect of the upstream flux profile on the local critical heat flux, Tong (Ref. [25]) proposed a semi-empirical method by defining a factor:

𝐹 = [𝑞𝐶𝐻𝐹(𝑧)]𝑢

[𝑞_𝐶𝐻𝐹(𝑧)]_𝑛𝑢 (21)

where [𝑞𝐶𝐻𝐹(𝑧)]𝑢 is the critical heat flux in case of uniform heat flux profile, while [𝑞𝐶𝐻𝐹(𝑧)]𝑛𝑢 is the

critical heat flux in non-uniform axial heat flux distribution. By assuming that CHF occurs at a limiting value of superheat in the liquid sublayer, the developed correlation for the factor F will be:

𝐹 = 𝐶 1 − exp (−𝐶𝑧_𝐶𝐻𝐹) ∫ 𝑞(𝑧) 𝑞(𝑧_𝐶𝐻𝐹)exp[−𝐶(𝑧𝐶𝐻𝐹− 𝑧)] 𝑑𝑧 𝑧𝐶𝐻𝐹 0 (22)

where C is a function of both local steam quality x(zCHF) and mass flux G, given by:

𝐶 = 17.323(1 − 𝑥(𝑧𝐶𝐻𝐹))

7.9

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

Smolin method

Based on the Tong shape factor equation (eq. 21), Smolin and Polyakov (Ref. [26]) have proposed slightly revised forms of the F-factor:

𝐹 =𝐶 ∫ 𝑞(𝑧) exp[−𝐶(𝑧𝐶𝐻𝐹− 𝑧)] 𝑑𝑧 𝑧𝐶𝐻𝐹 0 𝑞(𝑧_𝐶𝐻𝐹) (24) where C=1/40D.



Lin method

The form proposed by Lin et al. (Ref. [27]) is:

𝐹 =𝐶 ∫ 𝑞(𝑧) exp[−𝐶(𝑧𝐶𝐻𝐹− 𝑧)] 𝑑𝑧

𝑧𝐶𝐻𝐹

𝑧𝑜𝑛𝑏

𝑞(𝑧_𝐶𝐻𝐹)(1 − exp(−𝐶𝑧_𝐶𝐻𝐹)) (25)

where C is a function of both local steam quality x(zCHF) and mass flux G, given by:

𝐶 = 5.906(1 − 𝑥(𝑧𝐶𝐻𝐹))

4.31

(𝐺/1356)0.478 (26)

and zonb is the height at which the onset of nucleate boiling occurs.



Groeneveld method (1986)

The boiling length average heat flux hypothesis is a tentative modification of the local condition hypothesis to take the history effect into account. In the Groeneveld method (Ref. [28]), the predicted CHF values, at uniform heat flux for a tube diameter of 8 mm, are obtained by linear interpolation between data provided in a look-up table as a function of pressure, mass flux and thermodynamic quality. In other flow situations, including flow in rod bundles, the CHF is calculated based on correction coefficients:

𝑞𝐶𝐻𝐹= 𝑞𝑇𝑎𝑏𝑙𝑒∙ 𝐾ℎ𝑦∙ 𝐾𝑏𝑓∙ 𝐾𝑠𝑝∙ 𝐾ℎ𝑙∙ 𝐾𝑛𝑢∙ 𝐾ℎ𝑜𝑟∙ 𝐾𝑣𝑒𝑟 (27)

where Khy, Kbf, Ksp, Khl, Knu, Khor, Kver are the hydraulic diameter factor, the bundle factor, the grid spacer

factor, the heated length factor, the axial heat flux distribution factor, the horizontal flow factor and the vertical flow factor, respectively. If comparing with other models, the F-factor adopted by Groeneveld et al., which is equivalent to the Bowring’s Y factor over the boiling length, can be expressed as:

𝐹 = 1 𝐾𝑛𝑢= 𝑌 = ∫𝑧𝑐ℎ𝑓𝑞(𝑧)𝑑𝑧 𝑧𝑠𝑎𝑡 𝑞(𝑧_𝑐ℎ𝑓)(𝑧𝑐ℎ𝑓− 𝑧𝑠𝑎𝑡) (28)



Lee method

This method is based on the 1995 Look-up Table by Groeneveld (Ref. [6]) and on the measured bundle CHF data. Lee (Ref. [29]) has proposed a set of correction factors in order to extend the application of the 1995 Look-up Table to other flow conditions. The proposed correction factors are slightly different from the ones originally proposed by Groeneveld:

𝑞𝐶𝐻𝐹 = 𝑞𝑇𝑎𝑏𝑙𝑒∙ 𝐾ℎ𝑦∙ 𝐾𝑏𝑓∙ 𝐾𝑠𝑝∙ 𝐾ℎ𝑙∙ 𝐾𝑛𝑢∙ 𝐾𝑐𝑤∙ 𝐾𝑟𝑝 (29)

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To take into consideration the effect of non-uniform axial heat flux, a new form of correction factor was also proposed:

𝐹 = 1

𝐾𝑛𝑢 = 𝐶3[1 + (𝑌 − 1) exp(2.66𝑥(𝑧))] (30)

Where C3=0.728 for the direct substitution method.

2.4 Pre-CHF correlations

As previously stated, the initial endeavor for the GOTHIC code improvement was meant to have the full boiling curve implemented. This meant the implementation of pre-CHF correlations as well; therefore, a short study of the pre-CHF correlations was also performed.

2.4.1 The Chen pre-CHF correlation

In regard to the pre-CHF boiling correlations in codes, one of the most used correlations is Chen (Ref. [30]). This correlation is used in both GOTHIC and RELAP, with some small differences, as GOTHIC has three phases and is based on internal energy as opposed to enthalpy. In GOTHIC, the Chen correlation assumes that the total heat transfer rate is given by the sum of forced convection and nucleate boiling:

𝐻_{𝑐𝑜𝑛𝑣}= 𝜆_𝑅𝐻_𝑠𝑝𝑙+ 𝐻_𝑛𝑏 (31)

where Hspl is the single-phase liquid heat transfer coefficient and λRis a factor given by

𝜆𝑅 = 𝑀𝑎𝑥 {_{2.35 ∙ (𝑥} 1

𝑚+ 0.213)0.736 (32)

with xm being the Martinelli quality factor, based on the lesser flowing quality is

𝑥_𝑚 = 𝑀𝑖𝑛 ([ 𝑥𝑓 1 − 𝑥_𝑓] 0.9 ∙ [_𝜌𝜌𝑙 𝑣] 0.5 ∙ [𝜇_𝜇𝑣 𝑙] 0.1 , 100) (33)

where 𝜇 is the viscosity (of the liquid and vapor phases, respectively), and the flow quality xf is defined in

respect to the three phases that exist in GOTHIC (vapor, liquid and droplets) as 𝑥_𝑓 = |𝛼𝑣𝑢𝑣𝜌𝑣|

|𝛼𝑣𝑢𝑣𝜌𝑣| + |𝛼𝑙𝑢𝑙𝜌𝑙| + |𝛼𝑑𝑢𝑑𝜌𝑑| (34)

The nucleate boiling coefficient is

𝐻_𝑛𝑏 = 0.00122𝜆_𝑣𝜆_𝑆 𝑘𝑙 0.79_𝑐 𝑝𝑓0.45𝜌𝑓0.49 𝜎_𝑓0.5_𝜇 𝑓 0.29_ℎ 𝑓𝑔0.24𝜌𝑔0.24(𝑇𝑤 − 𝑇_𝑓)0.24(𝑃_𝑤− 𝑃_𝑠𝑎𝑡)0.75 ₍₃₅₎

where 𝜆𝑆 is the suppression factor, Pw is the saturation pressure at Tw and Psat is the saturation pressure

at Tf. The nucleate boiling coefficient is ramped to zero for 𝛼𝑙<0.0001 according to:

𝜆_𝑣 = 𝛼𝑙− 0.0001

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The suppression factor is given by 𝜆𝑆= { (1 + 0.12𝑅𝑒′1.14₎−1_{, 𝑅𝑒′ < 32.5} (1 + 0.42𝑅𝑒′0.78)−1, 32.5 < 𝑅𝑒′ < 50.9 0.1, 𝑅𝑒′ > 50.9 (37) Where, 𝑅𝑒′ = 0.0001𝑅𝑒_𝑡𝑝 (38)

Retp is the two-phase Reynolds number given by:

𝑅𝑒_𝑡𝑝= 𝑅𝑒_𝑙𝜆1.25_𝑅 ₍₃₉₎

Because the derivative of the heat transfer coefficient with respect to temperature is needed for the numerical solution, it is useful to replace the pressure difference factor in Eq 35 with an equivalent temperature difference factor. The substitution is based on the Clausius-Clapeyron equation

(𝑑𝑃 𝑑𝑇)_𝑠𝑎𝑡=

ℎ_𝑓𝑔𝜌_𝑓𝑔

𝑇_𝑓 (40)

For a finite temperature difference, the pressure difference can be approximated as (Ref. [31]) 𝑃_𝑤− 𝑃_𝑠𝑎𝑡=ℎ𝑓𝑔𝜌𝑓𝑔

𝑇_𝑓 (𝑇𝑓− 𝑇𝑓)

𝑁

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where the exponent N has been curve fit to a set of ΔPw=Pw-Psat versus ΔTw=Tw-Tf points obtained from

the water/steam property functions. The functional form of N is 𝑁 = 1.0306 log(𝑃_𝑠𝑎𝑡)−0.017_{+ 0.196}∆𝑇𝑤− 5

95 log (𝑃𝑠𝑎𝑡)−1.087 (42)

with the saturation pressure, Psat, in psia and the wall temperature, ΔTw, in °F.

The wall source terms in the nucleate boiling regime are:

𝑄_𝑤𝑙 = 𝜆_𝑤𝑙𝐻_{𝑐𝑜𝑛𝑣𝑙}𝐴_𝑐𝑛(𝑇_𝑤− 𝑇_𝑙) = 𝑄_𝑛𝑏+ 𝑄_𝑠𝑝𝑙 (43)

and

𝑄_𝑤𝑣= 𝜆_𝑤𝑣𝐻_𝑠𝑝𝑣𝐴_𝑐𝑛(𝑇𝑤− 𝑇𝑣) (44)

where Qnb is the portion of the total wall heat attributable to the generation of steam (Hnb) and Qspl is the

remaining convective heat transfer to the liquid phase. Acn is the conductor surface area.

It should be taken into consideration that even though GOTHIC has three phases, heat structures can only transfer heat to either the vapor phase or the liquid phase; i.e. there is no heat transfer from a heat structure to the droplet phase.

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2.4.2 The Roshenow pre-CHF correlation

The MELCOR code, which is a 1D Thermal Hydraulics code usually used to perform Severe Accident analysis of light water reactor nuclear power plants, uses a modification of the Roshenow (Ref. [32]) correlation (1952), in order to model nucleate boiling. The used MELCOR correlation is in the form:

[𝑐𝑝𝑙(𝑇𝑠− 𝑇𝑠𝑎𝑡) ℎ_𝑓𝑔 ] = 𝐶𝑠𝑓[ 𝑞_𝑛𝑏′′ 𝜇 ∙ ℎ_𝑓𝑔∙ ( 𝜎 𝑔(𝜌_𝑙− 𝜌_𝑣)) 1/2 ] 𝑛 𝑃𝑟𝑚 ₍₄₅₎ where,

𝑞_𝑛𝑏′′ Nucleate boiling heat flux, W/m2

𝑐𝑝𝑙 Heat Capacity of Liquid at Tsat, J/(kg∙K)

T

s Temperature of surface, K

T

sat Saturation temperature, K

C

sf Constant determined empirically for different surfaces and fluids (default =0.013) 𝜇 Dynamic viscosity of liquid at Tavg, kg/(m∙s)

h

fg Latent heat, J/kg

𝜎 Surface tension at Tavg, N/m

g Acceleration of gravity, m/s2

𝜌_𝑙 Density of liquid at Tsat, kg/m3

𝜌_𝑣 Density of vapor at Tsat, kg/m3

n Constant (default=0.33)

Pr Prandtl number for liquid in boundary volume m Constant (default=1.0)

T

avg Average temperature, (Tsurf+Tsat)/2, K

The surface tension of water is given as a function of temperature by:

𝜎 = 0.2358(1 − 0.625𝑇_𝑅)𝑇_𝑅1.256_{+ 𝑐} ₍₄₆₎

where

T

R 1-T/647.3

𝑇 Temperature, K

c

Constant (default =0.0)

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The constants Csf, m, n in Eq. 45 and c in Eq. 46 are implemented in the MELCOR code as sensitivity

coefficient arrays.

The values in Rohsenow equation can be used for any geometry since it is found that the rate of heat transfer during nucleate boiling is essentially independent of the geometry and orientation of the heated surface.  The correlation is applicable to relatively clean and smooth surfaces.  

2.5 Determination of Critical Heat Flux in Thermal Hydraulic Codes

In this sub-chapter a short review of the way that Critical Heat Flux is computed in different thermal-hydraulics codes. This will be used as a basis for the development of the initial sub-routine to be implemented in GOTHIC.

2.5.1 RELAP5-3D

©

_{, TRACE}

©

For determining the value at which Critical Heat Flux occurs, thermal hydraulic codes widely use look-up tables. The most comprehensive table is “The 2006 CHF look-up table” D.C. Groeneveld et. al. (Ref. [6]), itself being based on previous iterations, “1986 AECL-UO Critical Heat Flux Lookup Table” (Ref. [28]), and “The 1995 look-up table for critical heat ﬂux in tubes” (Ref. [33]), representing the completion of work performed over more than 30 years of development. The table was supported by various institutions like CENG Grenoble, University of Ottawa, IPPE-Obninsk and AECL-Chalk River. The 2006 CHF look-up table is created using extensive analysis of experimentally obtained data from different sources: The 1995 Look up-Table (Groeneveld et al. 1996), Smolin et al. (1962), Alekseev et al. (1964), Griffel (1965), Judd et al. (1966), Zenkevich et al. (1969), Becker et al. (1971), Zenkevich (1971), Zenkevich (1974), Smolin et al. (1979), Borodin (1983), Groeneveld (1985), Soderquist (1994). Even though the experiments were performed for vertical water-cooled tubes of various internal diameters, the data was normalized for an 8 mm diameter tube. The effect of tube diameter on CHF was accounted for using the diameter correction factor:

𝐶𝐻𝐹_𝐷/ 𝐶𝐻𝐹_{𝐷=8𝑚𝑚}= (𝐷/8)−1/2 ₍₁₎

for the range 3<D<25 mm. Outside this range the diameter effect appears to be absent (Ref. [11]). In order to predict the CHF occurring at different hydraulic diameters, the same correction factor should be applied.

The 2006 CHF look-up table is based on three main parameters: ● Pressure;

● Mass flux;

● Thermodynamic quality.

Groeneveld table by default, but have the 2006 CHF look-up table as a selectable development option (not qualified). As another example, the TRACE©_{V5.0 code uses the 1995 CHF Groeneveld table.}

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In RELAP5, as the geometry used for populating the table is an 8 mm diameter tube, the CHF table values are adjusted in order to enable its usage for different heat structure geometries (mostly qualified for nuclear core similar geometries):

● Standard Geometry;

● Parallel Plates (narrow and wide gap);

 Based also on Gambill-Weatherhead correlation (Ref. [34]) ● In line rods;

 Different for parallel flow and cross-flows ● CANDU Bundle.

eight different multipliers (Table 2.1) such as axial power, core rod grid spacers (pressure losses and distance from grid spacer) and heated length in order to adjust the precision of their model. The diameter correction factor (k1, “hydraulic factor”) that RELAP5 uses is slightly different from the one used to compile the 2006 CHF Look-up table, becoming constant for all hydraulic diameters above 16 mm (instead of 25 mm). This is likely because the CHF model used by RELAP is meant to be mainly used for in-core situations and would not be fully adequate at modelling the heat transfer in conditions such as the RPV lower hemispherical region located inside of a reactor cavity.