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116 nuclear ThermalHydraulic Phenomena THP are identified in the present paper, following documents issued during the last three decades by the Committee on the Safety of Nuclear Installations of Nuclear Energy Agency of the Organization for Economic Cooperation and Development (OECD/NEA/CSNI) and by the International Atomic Energy Agency (IAEA). The derived THP list includes consideration of experiments performed in Separate Effect Test (SET) and Integral Effect Test (IET) facilities relevant to reactor coolant system and containment of Water Cooled Nuclear Reactors (WCNR). We consider a dozen WCNR types: Pressurized Water Reactors (PWR), Boiling Water Reactors (BWR), Russian reactors (VVER440, VVER1000 and RBMK), pressure tube heavy water reactors by Canada (CANDU) and India (PHWR) and socalled ‘advanced’ reactors (e.g. AP1000 and APR1400 designed in US and Korea, respectively).
We envisage a variety of applications for the THP list. Four of the phenomena are helpful to characterize the current state of art in nuclear thermalhydraulics: Counter Current Flow Limitation (CCFL), Critical Heat Flux (CHF), reflood and TwoPhase Critical Flow (TPCF). Furthermore, the THP identification contributes to addressing the scaling issue, performing uncertainty evaluations, developing constitutive equations and ‘special models’ in codes and prioritizing the research.
phenomena in nuclear thermalhydraulics, experiments, scaling, numerical codes, nuclear reactor safety
When characterizing indispensable nuclear technology areas or disciplines that need specific development to make possible the exploitation of fission energy, one may converge on the following shortlist (not in the order of importance):
• Nuclear Thermal Hydraulics (NTH),
• Radioprotection,
• Neutron Physics,
• Structural Mechanics.
NTH is on the focus in the present paper. Thermal Hydraulic Phenomena (THP) and computer codes are key elements of NTH. The former constitutes the basis of the empirical evidence; the latter is the repository of modeling expertise and competence. The overall NTH implies a universe of knowledge as discussed in [1].
The scope here narrows down to transient NTH and to fundamentals, i.e. the THP, important for applications to the safety of WaterCooled Nuclear Reactors (WCNR).
Any transient part of the Design Basis Accident (DBA) envelope in WCNR is at the origin of an accident scenario (AS). Phenomenological Windows (Ph.W) allow subdividing the time evolution of any AS; then, THP characterize Ph.W. Physical Parameters, with proper Ranges (PP&R), are part of modeling and constitute the solution of numerical code calculations. Therefore, one may depict the logical frame:
(WCNR+DBA)$\rightarrow$AS$\rightarrow$Ph.W$\rightarrow$THP$\rightarrow$PP&R← (codecalculationsresults) ← modeling
Properly scaled Separate Effect Tests (SET) and Integral Effect Tests (IET) constitute the experimental database. Whereas integral test facilities, at the origin of IET, are usually designed to follow the performance of a reference reactor system in various offnormal conditions or accident transients, SET focus on the behavior of a single component, or on the features of one or a limited number of THP.
Already in the year 1987, the OECD/NEA Committee on the Safety of Nuclear Installations (CSNI) published a document that systematically identifies a set of THP and tests detected from IET. Those experiments and phenomena provide the best basis for the assessment of thermal hydraulic codes, [2]. A couple dozen reports, in forthcoming two decades, constitute the bases for the identification and the characterization of 116 THP, [3]. The description of individual phenomena and the connection between THP and AS are the topic of Chapters 6 and 15 of [1].
A twotier objective for the present paper is the use of THP for moving the frontiers of NTH. The two tiers are: (a) to propose multiple roadmaps for exploiting the knowledge associated with phenomena, see also [4], and (b) to provide a view of current modeling capabilities. We achieve the latter objective by considering four THP: Counter Current Flow Limitation (CCFL), Critical Heat Flux (CHF), reflood and TwoPhase Critical Flow (TPCF).
Background information about NTH phenomena, the list of 116 THP, envisaged applications of phenomena and selected modeling limitations constitute the content of the paper.
In 1943, soon after the proof of the reactor chain control and sustainability, e.g. [5], the endeavor started for the design of WCNR. Thermal hydraulics (TH) had a key role since the beginning, Figure 1 (e.g. Chapt. 2 of [1]). A number of design situations required specific research and the need to consider NTH appeared at the early stage of this period.
Two main breakthrough events provided impulse and directions to the development of NTH: 1) the introduction by US Atomic Energy Commission (US AEC) of Interim Acceptance Criteria (IAC) for the design of Emergency Core Cooling Systems (ECCS) in 1971 and consequent public rulemaking hearings in USA in 197273, 2) Three Mile Island (TMI2) accident in 1979. The former brought to the design, construction and operation of experimental facilities (e.g. LOFT and Semiscale) initially aimed at addressing Large Break Loss Of Coolant Accident (LBLOCA) scenarios; the IAC also pushed the development of analytical work for numerical computer codes. The latter shifted the attention from LBLOCA to Small Break LOCA (SBLOCA). Specific experimental programs started (BETHSY, LOBI, etc., reported in Figure 1). Gathering of experimental Data Base (DB) for Accident Scenarios (AS) started.
The need to validate computer codes and the complexity of AS at the basis of the validation brought to the decision to identify and to characterize phenomenological windows and phenomena. The related processes of code validation and phenomena identification started by CSNI in the early 1980’s, [3].
Figure 1. Outline of the history of THP, modified from [1]
The Integral Test Facilities (ITF: this acronym is used interchangeably with IET defined in the abstract), the Separate Effect Test Facilities (SETF, or SET), the Computer Code Validation Matrices (CCVM), the State of Art Report (SOAR) for Thermal hydraulics of ECCS (TECC) and, later on, the identification of Containment phenomena, constituted milestone products from CSNI activities. In 1989, US Nuclear Regulatory Commission (US NRC) introduced the Phenomena Identification and Ranking Table (PIRT) when proposing the Code Scaling Applicability and Uncertainty (CSAU), [6]. A parallel investigation within the International Atomic Energy Agency (IAEA) focused on ‘advanced’ (or ‘new’) reactor designs. The reader may find details on the topics above and the full reference to THP in [3]. A perspective for future use of THP might outcome from the CSNI Specialists Meeting scheduled by the end of 2021.
Two complementary visions for THP derive from Figure 2 and 3 (see e.g. [7]).
On the one hand, phenomena are prerequisite for developing the Partial Differential Equations (PDE) at the bases of system thermal hydraulics and Computational Fluid Dynamic (CFD) codes: this is specifically true in relation to the constitutive equations (embedded into the PDE) and to the needs for validation. The predicted WCNR performance depends upon THP modeling and knowledge, Figure 2.
Figure 2. Role of THP in NTH
On the other hand, a (as far as possible) systematic analysis of a few hundred experiments in SETF and ITF, a variety of code calculation results and recorded data from nuclear reactor transient situations (e.g. including occurred accidents) brought to the identification and characterization of THP, Figure 3.
Figure 3. Derivation of THP, modified from [3]
The Design Basis Accident (DBA, also called by IAEA, Design Basis Conditions, DBC) framework is relevant for the identification and characterization of THP, as well as the processes for code development and scaling (see also Figure 2), Verification and Validation (V&V) and uncertainty evaluation.
The experiments or scaled accident scenarios, performed in a couple dozen ITF and in a thousand SETF brought to the selection of two lists of qualified facilities. BWR, mostly those equipped with jet pumps, and PWR with either UTubes Steam Generators (UTSG) or Once Through Steam Generators (OTSG), were at the center of attention.
An evaluation followed for each experiment of the scaling rationale, the quality of instrumentation and the applicability (to BWR and PWR conditions) of parameter ranges. The impact of experimental data upon the development and the V&V processes of codes paved the way for the selection of tests in the matrices, i.e. the CCVM. Deep reviews by scientists, managers and representatives of research, industry and regulatory body organizations, allowed the finalization of reports [8, 9].
Later on, different groups of scientists developed new and reduced scope CCVM by considering the following reactors, components and specific accident scenarios (the list of references in [3] cites individual related reports):
(a) PWRtype, Russian design reactors (VVER440 and VVER1000) equipped with Horizontal Steam Generators (HOSG).
(b) Reactors designs involving the use of natural circulation for cooling (noticeably AP600 and SBWR) where a tight interaction between reactor coolant system and containment occurs following an accident.
(c) Canadian Deuterium (natural) Uranium (CANDU) reactors with horizontalchannel, core, also called Pressurized Heavy Water Reactor (PHWR).
(d) BWRtype, Russian design reactors (RBMK) and Atuchatype reactors, or PHWR with verticalchannels and reactor pressure vessel: expected THP added by the authors of [3].
(e) Watercooled Small Modular Reactor (SMR) designs with heat exchangers into the reactor pressure vessel.
(f) Containment systems, distinguishing between full pressure (in PWR) and pressure suppression system (in BWR); in the last case, an early CSNI report included suitable information.
(g) Accident Management (AM) scenarios including the thermal hydraulic conditions and parameter ranges expected in ‘Beyond’ DBA (BDBA, recently characterized as Design Extension Conditions, region A, DECA by IAEA), before loss of core structural integrity.
Information from ‘all’ (as far as possible) international institutions reports, OECD/NEA and IAEA, dealing with thermal hydraulic phenomena was gathered; the list of 116 THP in alphabetic order was issued, [3].
As taken from [3] and related to [1], “… 47 accident scenarios (AS), calculated in relation to 13 watercooled nuclear reactors (WCNR), discussed in 68 reference documents (RD), utilizing 15 generalized (thermal hydraulic) parameters (GP), have been ‘aposteriori’ crosslinked with 116 phenomena (THP) in order to prove the origin of phenomena. The crosslink process also shows the direct connection between phenomena and nuclear reactor safety”.
Table 1 deals with the list of 116 THP in alphabetic order (parts 1 and 2). Phenomena associated with a number in the first column form the list of selected THP. Additional information in [3], not part of the present table, allows further characterization for each phenomenon, e.g. which category a) to g) is concerned, crossconnection of THP, etc. Phenomena without a corresponding number constitute typical alternative identification (sometimes synonymous) of THP part of the list. The last row of Table 1 (part 2) includes acronyms in the table.
The description of each THP ([1], Chapter 6), beyond the scope here, includes information about modeling capabilities and adequacy of experimental database. The use of selected THP for characterizing scaling capabilities of numerical codes and current state of knowledge in NTH can be found in documents cited in [3].
Table 1. The list of 116 THP, part 1
No 
THP Identification 
1 
Accumulator behavior 
2 
Asymmetric loop behavior 
3 
Asymmetry due to the presence of a dam 
4 
Behavior of check valves 

Behavior of containment emergency systems (e.g. PCCS) 
5 
Behavior of core makeup tanks 
6 
Behavior of density locks 
7 
Behavior of emergency heat exchangers including PRHR and IC 
8 
Behavior of large pools of liquid 
9 
Blowdown 
10 
Boiler condenser mode (of NC) 

Boiloff 
11 
Boron mixing and transport 
12 
CCF/CCFLChannel inlet orifice 
13 
CCF/CCFLDowncomer 
14 
CCF/CCFLHL & CL (including connection with RPV) 
15 
CCF/CCFLSG tubes 
16 
CCF/CCFLSurgeline 
17 
CCF/CCFLUTP 

Centrifugal pump 
18 
Channel and bypass axial flow and void distribution 
19 
Collapsed level behavior in downcomer 
20 
Condensation due to heat removal 
21 
Condensation due to pressurization 
22 
Condensation in stratified conditionsHorizontal Pipes 
23 
Condensation in stratified conditionsPRZ 
24 
Condensation in stratified conditionsSGPS 
25 
Condensation in stratified conditionsSGSS & BWRPSP 
26 
Containment emergency systems including passive cooling 

Containment pressure and temperature 
27 
Containment pump performance including sump clogging 
28 
Core thermalhydraulics 

Core wide void and flow distribution 
29 
CRGT flashing 
30 
Critical and supercritical flow in discharge pipes 

Critical flow 

Critical Power Ratio 

Deentrainment 

Depressurization 
31 
ECC bypass/downcomer penetration 

ECC mixing and condensation 
32 
Entrainment/DeentrainmentCore 
33 
Entrainment/DeentrainmentDowncomer 
34 
Entrainment/DeentrainmentHot leg with ECCI 
35 
Entrainment/DeentrainmentSG mixing chamber 
36 
Entrainment/DeentrainmentSG tubes 
37 
Entrainment/DeentrainmentUP 
38 
Evaporation due to depressurization (e.g. geom. discontinuities) 
39 
Evaporation due to heat input 
40 
Flow through openings 
41 
Global multiD fluid temperature, void and flow distributionCore 
42 
Global multiD fluid temperature, void and flow distributionDC 
43 
Global multiD fluid temperature, void and flow distributionSG SS 
44 
Global multiD fluid temperature, void and flow distributionUP 
45 
Gravity driven reflood 
46 
Horizontal heated channel HT 
47 
HT [NCO, FCO, SNB, SANB, CHF, postCHF]Core, SG, structures 
48 
HT [radiation]core 
49 
HT [condensation]SG structures 
50 
HT condensation in containment structures., w w/o noncondensable 
51 
Impeller pump behavior 

Instability (in boiling channels) 
52 
Interfacial friction in horizontal flow 
53 
Interfacial friction in vertical flow 
54 
Intermittent 2phase NC 
55 
Internal pump behavior (specific geometry) 
56 
Jet pump behavior 
57 
Liquid accumulation in horizontal SG tubes 

Liquid carryover 
58 
Liquid temperature stratification 
59 
LiquidVapor mixing with condensationCore 
60 
LiquidVapor mixing with condensationDowncomer 
61 
LiquidVapor mixing with condensationECCI in HL and CL 
62 
LiquidVapor mixing with condensationLower plenum 
63 
LiquidVapor mixing with condensationSG mixing chamber 
64 
LiquidVapor mixing with condensationUP 
65 
Loop seal filling and clearance (or clearing) 
66 
LP entrainment 
67 
LP flashing 

Mixture level & entrainmentCore, downcomer and SG SS 
68 
NC, 1phase & 2phasePS & SS 
69 
NC core and downcomer 
70 
NC core bypass, hot and cold bundles 
71 
NC core, gap, downcomer, dummy elements 
72 
NC core, vent valves, downcomer 
73 
NC with horizontal SG 
74 
NC RPV and containment & various system configurations 
75 
Natural convection and H2 distribution 
76 
Non condensable gas effect including condensation HT in RCS 

Nuclear fuel behavior 
77 
Nuclear thermalhydraulics feedback and spatial effect 

Nuclear thermalhydraulics instabilities 
78 
Parallel channel effects and instabilities PCEI 
79 
Phase separation at branches (including effect on TPCF) 
80 
Phase separation/vertical flow with and w/o mixture levelCore 
81 
Phase separation/vertical flow with and w/o mixture level – DC 
82 
Phase separation/vertical flow w w/o mixture levelPipes & Plena 
83 
Pool formation in UP 
84 
Pressure drops at geometric discontinuities, including containment 
85 
Pressure wave propagation including CIWH 
86 
Pressuretemperature increase & boiling due to energy/mass input 
87 
PRZ thermalhydraulics 
88 
QF propagation/rewetFuel rods 
89 
QF propagation/rewetChannel walls, Water rods 
90 
Refill including loop refill in PWRO 
91 
Reflood 
92 
Reflux condenser mode and CCFL 

Return to Nucleate Boiling (RNB) 
93 
Separator behavior (flooding, steam penetration, liquid carryover) 
94 
SG siphon draining (SG interaction with ESF, including gravity driven) 
95 
Spray effectsContainment 
96 
Spray effectsCore (including cooling and distribution) 
97 
Spray effectsOTSG SS 
98 
Spray effectsPRZ 
99 
Steam binding (liquid carryover, etc.) 
100 
Steam dryer behavior 
101 
Steam line dynamics 
102 
Stratification in horizontal flowPipes (in 1phase & 2phase) 
103 
Stratification of boron 
104 
Structural heat and heat losses 
105 
Surgeline hydraulics 
106 
Superheating in OTSG SS 
107 
Thermalhydraulics – Nuclear fuel feedback 
108 
Thermalhydraulics of horizontal SG, PS and SS 
109 
Thermalhydraulics of OTSG, PS and SS 
110 
TPCFBreaks 
111 
TPCFPipes 
112 
TPCFValves 
113 
Tracking of noncondensable gases 

Valve leak flow (construction, operation, maintenance) 

Vapor (or steam) carryunder 

Vapor pull through 
114 
Ventilation blower characteristics 
115 
Void collapse and temperature distribution during pressurization 
116 
Wall to fluid friction 

Water accumulation in horizontal SG tubes 
Acronyms in the table: BWR = Boiling Water Reactor; CCF = Counter Current Flow; CCFL = CCF Limitation; CIWH = Condensation Induced Water Hammer; CHF = Critical Heat Flux; CL = Cold Leg; CRGT = Control Rod Guide Tube; DC = Downcomer; ECC = Emergency Core Cooling; ECCI = ECC Injection; ESF = Engineered Safety Features; FCO = Forced Convection; HL = Hot Leg; HT = Heat Transfer; IC = Isolation Condenser; LP = Lower Plenum; NC = Natural Circulation; NCO = Natural Convection; OTSG = Once Through Steam Generator; PCCS = Passive Containment Cooling System; PCEI = Parallel Channel Effects and Instability; PRHR = Passive Residual Heat Removal; PRZ = Pressurizer; PS = Primary Side; PSP = Pressure Suppression Pool; PWR = Pressurized Water Reactor; PWRO = PWR equipped with OTSG; QF = Quench Front; RCS = Reactor Coolant System; RNB = Return to Nucleate Boiling; RPV = Reactor Pressure Vessel; SG= Steam Generator; SANB = Saturate Nucleate Boiling; SNB = Subcooled Nucleate Boiling; SS = Secondary Side; TPCF = TwoPhase Critical Flow; UP = Upper Plenum; UTP = Upper Tie Plate 
Although THP identified in [8, 9] already found applications e.g. in the areas of V&V and scaling, the issue here is to prepare the basis for a systematic use of the 116 THP list, see also [3].
The first (hidden) step, expected from a newly formed (possibly CSNI) group of experts, is to amend and finally accept the list, [3], and the descriptions in [1], Chapter 6. The notes below aim at supporting in this endeavor.
Figure 4 provides a summary view for the origins of THP, as of today the situation and the areas of NTH for possible applications. Table 2 gives a guidance related to the last item.
Figure 4. Perspective use of THP: Topics
Let us stress again, Figure 4, that phenomena derive from experiments and code applications (DBA) and are the synthesis of 68 RD issued by NEA/CSNI and IAEA. The NTH areas for applications are scaling, [10], V&V [1], uncertainty [6, 11, 12], new experiments and models. The Best Estimate Plus Uncertainty (BEPU) framework, [13], includes all those areas. Furthermore, scaling, V&V, uncertainty and BEPU constitute ‘procedures in NTH’, [4].
Table 2. Perspective use of THP, a systematic approach
No 
THP 
CONSTITUTIVE LAW / LAWS (applicable) 
SPECIAL MODEL need 
SCALING 
UNCERTAINTY PARAMETERS relevant 
MECHANISTIC MODELING crosslink 
PRIORITY (for new research) 

SET/IET data 
Constitutive laws 

1 
Accumulator behavior 







… 
… 







… 
… 







… 
… 







116 
Wall to fluid friction 







The filling of Table 2, i.e. adding the 116 THP in the 2^{nd} column, aims at a comprehensive and common understanding of current system thermal hydraulics, [4]: this would lead to the closure of a 40years long process.
Starting from the third column of the table (i.e. counting from left to right), the following activities are relevant in relation to each phenomenon and an assigned numerical code (when needed):
Research priority is an outcome from filling the columns 3 to 8: selected prioritized THP may enter the process proposed in [18].
Understanding the current inadequacies in the application of models and numerical codes to the calculation of THP is essential for any decision step in previous section (i.e. filling the Table 2). However, a systematic overview of predictive capabilities for each phenomenon is beyond the scope (for this paper or for any other paper).
Hereafter, snapshot information related to four THP provide an idea of amount of errors in predictions and of challenges in modeling: we do not have the objective of summarizing the knowledge available from hundreds or thousands papers dealing with each concerned phenomenon.
5.1 CCFL, THP 14 in Table 1
CCFL may occur at any geometric discontinuity and even in horizontal pipes (typically, nonfully developed flow conditions) free of obstacles.
It is unavoidably a transient phenomenon, e.g. flooding conditions change with time and creation of a pool of liquid downstream flooding or CCFL occurrence. However, SETF experiments and model developments make use of the ‘quasisteady’ condition hypotheses.
Wallis and Kutateladze pioneered investigations in the area, early in the 1960’s, bringing to the wellknown formulations:
$j_{g}^{* 1 / 2}+m_{w a} j_{f}^{* 1 / 2}=C_{w a}$ (1)
$K_{g}^{1 / 2}+m_{k u} K_{f}^{1 / 2}=C_{k u}$ (2)
where, the flooding parameters are, respectively,
$j_{k}^{*}=j_{k}\left[\frac{\rho_{k}}{g d\left(\rho_{f}\rho_{g}\right)}\right]^{1 / 2}$ (3)
and
$K_{k}=j_{k}\left[\frac{\rho_{k}^{2}}{g \sigma\left(\rho_{f}\rho_{g}\right)}\right]^{1 / 4}$ (4)
Later on, it was found, [19], that a forcemomentum balance applied to a control volume in vertical flowreversal conditions leads to eq. (1) if
$\tau_{i}=\frac{1}{2} c_{i}\left(\rho_{g}^{1 / 2} w_{g i}+\rho_{f}^{1 / 2} w_{f}\right)^{2}$ (5)
The complexity and the difficult predictability of the flooding phenomenon derive from Figure 5, [20] and [21], top and bottom diagram, respectively.
Figure 5. Flooding and CCFL ( $j_{f}^{*}=0$) . Top: spread of experimental and calculated data in a vertical pipe, adapted from [20]. Bottom: HLRPV geometry, adapted from [21]
From the right side of bottom diagram in Figure 5, one may derive the impact of CCFL prediction at the HLRPV connection upon the safety analysis of the LBLOCA in Atucha2 reactor, [22]. If calculated $j_{g}^{*}$ remains below 0.60, reflood occurs timely (i.e. no safety concern) following ECC injection; otherwise, if calculated $j_{g}^{*}>$ 0.75, rod surface temperature in the core may overpass safety limits. The spread associated with the knowledge of CCFL (dotted lines in the diagram) imposed a specific uncertainty investigation to prove suitable core cooling conditions.
5.2 CHF, THP 47 in Table 1
CHF is at the center of attention of NTH scientists because of the need to ensure core operation in nucleate boiling heat transfer regime and, at the same time, to allow the maximum value for linear power during operation of WCNR.
We note the publication of hundreds correlations and the availability of data from thousands experiments, see e.g. [23]: errors derived from the application of any correlation in predicting an enough large number of CHF experimental datapoints are barely below 20%, on average.
Therefore, Kirillov and Groeneveld (and coworkers) during a lifelong engagement, almost simultaneously, launched an empirical approach socalled LookUp Tables (LUT). Measured CHF datapoints fill thousands virtual cubes in a threedimensional space, where independent variables are local quality, mass flow and pressure (further details in [23]). Nowadays, almost all thermal hydraulic system codes adopt the LUT approach for predicting CHF for WCNR safety analyses.
System thermal hydraulic codedevelopers who are members of the FONESYS, [24], decided to compare CHF code predictions referring to an assigned fuel assembly. Different codes simulated an imposed flowdecrease transient starting from nominal operation. Figure 6 shows key results.
Figure 6. CHF calculations by different codes of a virtual core channel. Top: DNBR along channel axis at nominal conditions. Bottom: CHF occurrence following a decreaseofflow transient
We can summarize the outcomes related either to initial steady state (top) or to the flowdecrease transient (bottom):
(a) As expected, the entire channel is in nucleate boiling conditions; however, the minimum value of the DNBR is not the same for all calculations (i.e. in the range 1.4 to 2.0, approximately).
(b) DNBR values have larger differences in the bottom one meter of the channel.
(c) Initial clad temperatures differ for about 10 K at the beginning of the transient (bottom diagram); a portion of this difference depends upon the concerned axial location, i.e. the place where CHF occurs first.
(d) Times of CHF occurrence differ for about 40 seconds in a transient where the ‘latest’ calculation predicts dryout at about 80 s.
(e) Difference in rod surface temperature after the CHF occurrence is due also to postCHF model.
5.3 Reflood, THP 91 in Table 1
The modern history of reflood modeling started in 1968 with the (wellknown) Yamanouchi milestonepaper. He derived a twodimensional equation for conduction heat transfer in the clad:
$\frac{\partial^{2} T}{\partial y^{2}}+\frac{\partial^{2} T}{\partial z^{2}}=\rho_{c l} c_{p}^{c l} \kappa_{c l} U \frac{\partial T}{\partial z}$ (6)
The Yamanouchi idea consists in establishing the link between the Quench Front (QF) velocity (U) and the time derivative in the conduction equation (i.e., $\frac{\partial T}{\partial t}=U \frac{\partial T}{\partial z}$ ) solved within a domain of the clad downstream the QF. The introduction of subsequent assumptions brought to the relationship:
$\check{\mathrm{h}}_{r e}=\left(\rho_{c l} c_{p}^{c l} U\right)^{2} \frac{\delta}{2 \kappa}\left\{\left[\frac{2\left(T_{ws}T_{M F B}\right)}{T_{M F B}T_{s a t}}+1\right]1\right\}$ (7)
Current codes for nuclear reactor safety analyses make use of the structure of the above formula. This is at the origin of large discrepancies among predictions: following the BEMUSE project, [12], reflood is at the center of attention for derivation of uncertain parameters and ranges [15, 16].
Figure 7 deals with two complex reflood aspects: the ‘sametime’ or the homogeneous reflood and the multiple (apparently random) values of $T_{M F B}$.
Figure 7. Reflood. Top: QF advancement vs time, ‘sametime’ reflood. Bottom: (T_{QF} $\approx$) T_{MFB }vs T_{P}, several T_{MFB }values for the same T_{P}, modified from [26]
Predicted ‘uniform’ advancements of QF from bottomtotop and from toptobottom of channel (full lines) leads to gathering of two fronts at about middle of channel height at around 250 s, while experimental data (‘+’ points) exhibit a ‘sametime reflood’ at around 220 s, top diagram. Similar experimental and corresponding calculated data are reported in [25] for high velocity reflood. ‘Sametime’ reflood, or homogeneous reflood, is inconsistent with the derivation of eq. (7): a different modeling approach is necessary.
The bottom diagram shows not widely disseminated reflood data for nuclear fuel, measured in Halden nuclear reactor, [26]. Two issues are concerned, widely debated in scientific literature:
(a) The equivalence between T_{QF}, or the temperature when a steep change occurs in the time trend cladtemperature vs time, and T_{MFB}.
(b) The widely spread values for T_{QF}, whatever is the value of the maximum temperature experienced at the same location during the early period of the transient.
5.4 TPCF, THP 111 in Table 1
When Adm. Rickover took the decision to use water as coolantmoderator of nuclear reactors, TwoPhase Critical Flow (TPCF) became of technological interest. Papers summarizing related investigations appeared early in the 1950’s.
Moody and Fauske are the pioneers who published reference TPCF models in the early 1960’s. In order to introduce to the complexity of the issue, we introduce a seedinformation of the Moody model. Moody considered the perfect gas theory and proposed the following energy balance in a control volume including the region from a highpressure reservoir and a hypothetic break connecting with the lowpressure environment:
$h_{0}\left(x_{0}, p_{0}\right)=x\left[h_{g}(p)+\frac{w_{g}^{2}}{2}\right]$
$+(1x)\left[h_{f}(p)+\frac{w_{f}^{2}}{2}\right]$ (8)
Moody obtained one equation in four unknowns: w_{g}, w_{f}, x and p, all at the break location. Following a change of variables, (he introduced twophase mass flux, $\Gamma$, void fraction, $\alpha$, and slip ratio, S = w_{g}/w_{f}, so the unknowns become S, $\alpha$, p, G), he proposed three additional equations:
$s_{0}=$ const. $\quad(9), \quad\left[\frac{\partial \Gamma}{\partial S}\right]_{p}=0 \quad(10), \quad\left[\frac{\partial \Gamma}{\partial p}\right]_{S}$=0 (9)
The Eq. (9) ‘neglects’ friction and wall heat transfer in a highly changing velocities and temperature condition; Eq. (11) ‘recalls’ the perfect gas theory for critical flow; Eq. (10) is a mathematical condition (not having any connection with physics). Other hidden or evident approximations (incomplete list) are:
• No consideration of momentum balance, e.g. pressure drop due to acceleration, interfacial drag, etc.
• Velocity in reservoir assumed negligible.
• Quality in reservoir not affected by TPCF and related depressurization.
• Use of state equations in nonequilibrium conditions.
However, Moody model produced results that compete (nowadays) with hundreds recent models, in terms of accuracy of predictions for newly available experimental data. Notwithstanding efforts made by hundreds of scientists who published papers and correlations, errors in the comparison between measured and predicted values of TPCF are large and strongly depending upon x_{0}: the largest value, of the order of 30% of measured values (also affected by uncertainties), occurs when $x_{0} \approx 0$.
Here we limit ourselves to report, Figure 8, the spread of data resulting from TPCF predictions to show the slow progress made in the area during 45 years. The upper and low diagrams report data from 1976, [27], already elaborated in 1980, [28], and 2020, [29], respectively. Both vertical axes report nondimensional TPCF within the same range of values. However, horizontal axes are different: we use x_{0} and transient time in top and bottom diagrams with $x_{0} \approx 0$. at t=0, for the data in the bottom diagrams.
Figure 8. TPCF. Top: spread of models results in 1976, adapted from [27]. Bottom: spread of code results in 2020, adapted from [29]
A longlasting and widerange investigation performed by hundreds of scientists brought to the identification and characterization of 116 THP within nuclear thermal hydraulics technology. Snapshot notes dealing with CCFL, CHF, reflood and TPCF (four of THP) confirm the complexity of the phenomena and the inaccuracies and the challenges in modeling.
The formulation of constitutive laws and the special models that are part of a numerical code should correspond with each individual phenomenon. However, this is not an objective pursued when developing a numerical code.
Procedures like accuracy quantification, V & V, scaling and uncertainty take benefit from the identification and characterization of the 116 THP. For instance, experimental databases have been associated with selected phenomena with the aim to demonstrate their suitability for code validation.
The ‘perspective THP table’ (Table 2 in the text) involving experimental database, scaling, uncertainty, constitutive laws, mechanistic modeling and prioritization of research is the main outcome from the present investigation: the final step of a multi decade international research and the initial step for progressing in the area.
The authors thank many researchers who contributed to identify and characterize the list of THP: only a few of them are coauthors in reported references.
c 
friction factor 
C 
constant based on system parameters () 
c_{p} 
specific heat, J. kg^{1}. K^{1} 
d 
geometric dimension, m 
g 
gravitational acceleration, m.s^{2} 
h 
enthalpy, J. kg^{1} 
$\check{\mathrm{h}}$ 
heat transfer coefficient, W.m^{2}. K^{1} 
j 
superficial velocity, m.s^{1} 
m 
multiplier for CCFL parameter () 
p 
pressure, N.m^{2} 
q” 
heat flux, w.m^{2} 
s 
entropy, J. kg^{1}. K^{1} 
S 
slip ratio, dimensionless 
t 
time, s 
T 
temperature, K 
U 
quench front velocity, m.s^{1} 
w 
fluid velocity, m.s^{1} 
x 
quality, dimensionless 
y 
geometry coordinate 
z 
geometry coordinate 
Greek symbols 

$\Gamma$ 
mass flux, kg.m^{2}. s^{1} 
$\delta$ 
clad thickness, m 
$\mathcal{E}$ 
spread ( $\frac{\max \min }{\min }$ ) 
$\boldsymbol{K}$ 
thermal conductivity, W.m^{1}. K^{1} 
$\boldsymbol{\rho}$ 
density, kg.m^{3} 
$\sigma$ 
surface tension N.m^{1} 
$\tau$ 
shear stress N.m^{2} 
Subscripts 

cl 
clad material 
f 
(saturated) liquid 
g 
(saturated) vapor 
i 
interface 
k 
f or g 
ku 
Kutateladze 
max 
maximum 
MFB 
minimum film boiling 
P 
peak (maximum during a transient) 
QF 
quench front ≈ MFB 
re 
Reflood 
sat 
Saturation 
wa 
Wallis 
ws 
wallsteam region 
0 
upstream reservoir condition 
Superscripts 

cl 
clad material 
* 
nondimensional 
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