Parametric MELCOR 2.2 sensitivity and uncertainty study with a focus on aerosols based on Ph ´ ebus test FPT1

The Ph ´ ebus test FPT-1 was carried out to investigate the fission product release, transportation, and distribution in the reactor coolant system and the simulated containment during a severe accident. The test was divided into three phases: the core degradation, aerosol transport, and containment phenomena. In this work, a parametric sensitivity and uncertainty analysis of the two first phases, i.e., core degradation and aerosol transport, is conducted with the focus on aerosol parameters. The study was done in a frame of WP4 of the European MUSA project whose main objective is to develop a methodology for severe accident uncertainty analysis. Due to the wide scope of the MUSA project, this study focuses not only on the UQ analysis but also on the parametric dependency and data interpretation. The main subject was investigating the uncertainty of the MELCOR 2.2 aerosol models and the relation between the chosen parameters as well as evaluating the usability of essential analytical tools available for such analyses. The calculations were carried out using the integral severe accident code MELCOR2.2 and the generic input deck of the FPT-1 experiment. The uncertainty quantification tool DAKOTA was used as a statistical tool. It was coupled with MELCOR within the SNAP environment. Fifteen uncertain parameters related with the aerosol release and transport were investigated. Two figures of merit were chosen to be the amount of deposition of Cs in the reactor coolant circuit, and the mass of air-borne aerosol in the containment. Preliminary analysis showed a significant correlation between the aerosol dynamic and agglomeration shape factors, density and minimum aerosol diameter with both FOMs. In addition to the standard uncertainty analysis, five MELCOR/DAKOTA UQ sensitivity calculations with modified parameters were performed and analyzed. Additionally, the impact of the parameters was investigated by excluding the most correlated ones from the analysis. The results confirm the strong correlation of shape factors and the importance of their ranges for overall UQ results. The authors show that by coupling the chosen analytical tools together, it is possible to carry out uncertainty quantification of severe accident analysis. Based on the study performed, the authors underline the need for further investigation of parameter distributions, the effect of the chosen number of samples on the results, and a global time-dependent UQ analysis.


Introduction
Due to their complexity and limited experience, nuclear severe accidents (SA) are one of the most challenging parts of nuclear safety to predict and evaluate.At the same time, maintaining a high level of nuclear safety requires wide knowledge in that field which is constantly used for licensing and updates of the accident guidelines.In the broad perspective of nuclear safety, not everything can be predicted, and severe accidents are one of the areas where scientists try to predict unpredictable and unexpected events.Many codes, which aims to simulate severe accident phenomenology, are burdened with relatively high uncertainty caused by the interdisciplinary nature of the events.The SA codes besides thermo-hydraulic need to calculate: material properties, chemical reactions, mechanistic strength, radioactivity, and many possible interactions between mentioned fields.This interdisciplinary nature needs to meet the computational performance limitations, leading to model simplifications and assumptions necessary to provide reliable data within a reasonable time, which is the main challenge in SA calculations.Due to the abovementioned simplifications, the SA codes often rely on the parametric models using constant values, which are variables in reality.The aerosol dynamics and related parameters could be an example of such generalization, for instance, shape factors or turbulence dissipation, which in MELCOR are constant for the whole transient; in reality, they depend on dynamically changing particles and atmosphere characteristics.Without a deep understanding of the models and involved phenomenology, this kind of simplification could lead to the uncertainty evolution.Years of experience show that, in general, SA codes provide a satisfying level of accuracy; however, each nuclear installation's uniqueness and scenario make this accuracy more difficult to achieve.Oftentimes, do conservative approaches need to be used to keep an additional safety margin.The uncertainty of the constant parameters is one of the subjects of the presented investigations.Moreover, all parameters are related to each other.The authors aim to visualize some of the relations between investigated variables, enhance understanding of parameters correlations, and show a possible way to interpret the results.
The Uncertainty Analysis (UA) is widely used to improve the level of understanding of the particular phenomena or model, leading to increased awareness of potential results variation.The UA is especially useful in the area where experimental data or experience is limited, and the models rely mostly on the assumptions or approximation made based on similar or scaled experiments.The SAs analysis is an example of the engineering field where UA could improve future codes, models, and user performance.This improvement could be done by simply increasing the knowledge about how uncertain SA codes are, showing ways to decrease this uncertainty and understand their origin.The UQ is also used for preliminary evaluation of new models for instance Accident Tolerant Fuels (ATF) (Feng et al., 2020).In the area of thermal-hydraulics or nuclear data, UA has grown for years.Good examples are thermohydraulic system codes like RELAP5 (Chang et al., 2020), Sub-channel analyses (Wang et al., 2013), or reactor physics (Hursin et al., 2015).In the case of SA, the UA were less popular, which was caused mainly by limited computational capacity.Nowadays, due to technical improvements and code development, the interest of UA has increased in the area of SA, which is proven by the number of related projects led by institutions like U.S. NRC (Mattie et al., 2016), IAEA (IAEA and New CRP, 1033), and MUSA project in the frame of which this study was done (MUSA Project, 2020).
The MUSA project focuses on the management and uncertainties of severe accidents.The main goal of the project is to develop a methodology for severe accident Uncertainty Quantification (UQ).To do that, a few factors need to be investigated, such as: tools, approach to implementation of uncertain parameters, statistical analysis, and correct definition of the input decks, etc.The authors assess the suitability of the available tools for UQ of the SA analysis.As computation tools, MEL-COR2.2coupled with DAKOTA, were chosen.Coupling of MELCOR and DOKOTA was done within SNAP environment, which allows the user to efficiently manage the whole UQ process and generate reports with the main results.The work presented here is a first approach to the UQ analysis in the MUSA project.The calculations were performed based on the Phébus FPT-1 experiment, which is a base for numerous similar investigations (Elsalamouny and Kaliatka, 2021) (Foad et al., 2020).
Because the Phébus experiments provided crucial data widely used to improve modelling and code development (Herranz, 2007) (Darnowski et al., 2020), it was chosen as a base case for the MUSA project.The FPT1 test was performed in 1996; it was composed of 18 irradiated and 2 fresh pressurized water reactor (PWR) fuel rods with high burnup.Poison material was Ag-In-Cd (AIC).The simulated scenario was a LOCA in a low-pressure steam environment (Jacquemain et al., 2000), consisting of three experimental phases: core degradation until 18 000 s, aerosol phase after the release from the core was finished between 18 000 s and 30 000 s, and the containment phase starting when the containment was isolated at 30 000 s.In this investigation, the two first phases, i.e., the core degradation and aerosol phase, were addressed.
For base UQ analysis, fifteen uncertain parameters and two figures of merit (FOM) related to the aerosol phenomenology were investigated.Because the project aims to develop a methodology for the future UQ analysis in the SA field in addition to standard UQ analysis, the authors perform a sensitivity study on the UQ settings to bring a more comprehensive perspective for MELCOR 2.2 UQ aerosol analysis.The main motivation was to understand the importance of the often unknown parameter ranges and their relation to final results variations.To do so, five of the UQ analyses were performed and analyzed, studying each of the aforementioned factors.The other goal was to study the usability and robustness of the tools for further analysis of more complex input decks and to highlight potential improvements' needs, directions, and usefulness.

Input deck
The input deck used for this study is based on Sandia National Laboratories (SNL) MELCOR 1.8.6 input deck converted to the MELCOR2.2 using SNAP to ensure best practice and to use the latest models in MELCOR.Investigation of the differences between MELCOR 1.8.6 and MELCOR 2.2 was not a goal of the study.The input deck is similar to the model described in (Humphries et al., 2015) and it contains all FPT1 elements crucial for the results, namely core, hot leg, steam generator, The nodalization of the input deck is shown in Fig. 1, containing 281 Control Functions (CF), 31 Control Volumes Hydrodynamics (CVH), 72 Heat Structures (HS), and 29 Flow Paths (FL).The core is divided by 2 rings horizontally and axially into 31 levels.Each of the core axial levels is connected with a separate Heat Structure (HS); the rest of the CVHs are arranged similarly.Uranium and zircaloy are distributed between both core rings while control rod material is placed only in the first ring.Supporting steel and zircaloy are located in rings 1 and 2 in the bottom part of the core where the support plate is located.
The CORSOR-Booth model for high burn-up fuel is used in the presented input deck to describe fission product release from the fuel.This model is based on the Booth diffusion model, and it calculates total mass release rate as a combination of diffusion and gas-phase mass transport (evaporation) rates.In this model, the release rate is calculated for Cs, and the other radionuclide (RN) classes are released by multiplying the Cs release rate by pre-defined scaling factors (Gauntt, 2010) (Humphries, 2018) (Humphries et al., 2019a).The CORSOR-Booth model approach binds all releases with Cs releases which clearly simplifies the calculation, but at the same time, increases uncertainty.Default values for MELCOR 2. X were used to define CORSOR-Booth class scaling factors, except for class 7 assigned to Mo, set up to 0.2 following the guidelines of the code developers (Gauntt, 2010).Calculations in this work were done with the chemisorption model activated, which describes the absorption of CsOH on stainless steel surfaces.The initial inventory is modelled by implementing CsM and CsI classes via the card RN1_FPN.The mass of CsI and CsM classes were pre-calculated by authors based on the available initial mass of Cs, I, and Mo and manually implemented as initial inventory.Used speciation masses are presented in Table 1.Gap inventory was defined as follows: Xe class and Cs class 1%, Ba class 0.0001%, I2 class 1.7%, Te class 0.01% of total inventory.The MELCOR hygroscopic model is turn on in all invastigated cases.
The control rod Ag-In-Cd (AIC) mass in MELCOR is generally implemented in the input deck via the COR package; and thereby, it is not transferred to the RN package.Because of that, the control rod (CR) materials are considered as structural materials and are not available to release as aerosols or vapors.In this work, to let AIC materials be released and investigate them as aerosols, additional RN classes with corresponding parameters containing the control rod materials were introduced.The values for the necessary sensitivity coefficients for AIC materials along with RN class mapping were determined based on data in (Darnowski et al., 2020).A precise description of the FPT-1 experiment based on which input deck and scenario sequence was developed can be found in FPT1 final reports (Jacquemain et al., 2000).

Uncertainty analysis tools
In reference to the analytical tool used in this study, it should be noted that MELCOR SNAP DAKOTA coupling, is an efficient setup; however, it has numerous limitations which could bring extensive difficulties for more complex input decks.SNAP interface helps to efficiently define uncertain parameters as well as their ranges and distributions, and at the end, could deliver a report with the main parameters.However, if even one calculation crashes, the whole batch is denied, which makes the finalization of the UQ study more difficult.
Origin of the crashes were very different and difficult to clearly define as it is combination of many things.This issue is under continuous investigation, currently we believe that correlation of some uncertain parameters gives set of parameters which makes calculation difficult to convergence however it needs to be confirmed.On top of that, limited flexibility or accessibility of features, like time-dependent correlation coefficients that highlight statistical values in the figures, etc. Makes it less attractive.Due to these limitations, the authors decided to use MELCOR SNAP DAKOTA coupling only for sampling and running the calculations, and the whole analysis and figures were done by in-house Python scripts.Using scripts instead of coupling will be extended also to sampling and calculation runs in further studies due to the aforementioned higher flexibility and potentially more efficient management of crashed calculations.

Uncertainty analysis methodology
Uncertainty analysis aims to identify and characterize essential input parameters and methodology to quantify the global impact of these uncertainties on selected output parameters called Figure of Merits (FOM).There are different ways of treating this type of analysis, and many methods have been developed for uncertainty quantification.Two main groups briefly described below based on (IAEA, 2008) could be highlighted: "propagation of input uncertainties" and "extrapolation of output uncertainty".The first one relies on choosing uncertain parameters and performing a certain number of calculations, varying these parameters in the given ranges and probability distributions.The second one focuses on comparing the results with available data and estimating the uncertainty based on the comparison outcomes.In this study, propagation of input uncertainty is used.
At the beginning of the UQ calculations, the number of parameters needs to be set up.The most important ones are the number of tasks, i.e., how many calculations are carried out, sampling methodology, probability and confidence level, statistical and sensitivity tools, and uncertainty parameters, along with their ranges and distributions (IAEA, 2008) (Frey et al., 2006).
Due to fact that the aim of this study is to develop a methodology for the UQ analysis of SA, the commonly accepted and widely used tools were chosen.Thus, DAKOTA tool and its statistical approach, along with the Wilks formula, was used.The Wilks formula specifies the needed minimum number of samples to ensure that the requested confidence level will be achieved.The BEMUSE project noted that the number of calculations impacts the sensitivity analysis presented by correlation coefficients results, and in that sense, it could be interesting to investigate similar UQ runs with an increased number of runs (Glaeser et al. (NEA, 2011).However, it was decided that this aspect will be investigated in a separate study.The uncertain parameters are sampled randomly using the Monte Carlo method.Models and methods are described in detail in (KeithEldred Michael, 2020).
The probability and confidence levels are both set at 95%.The confidence level says that 95% of obtained results are true, and the probability level says there is a 95% chance of getting this result.This 95/95 approach is widely used and accepted.One of the reasons for this wide use is because the regulatory body accepts it as a licensing limit which makes it relevant for the nuclear industry (IAEA, 2008).
To determine uncertainty, the distribution of the chosen uncertain parameters needs to be defined.Empirical distribution models for nuclear systems in the SA domain usually do not exist or are questionable.Due to the lack of empirical data, analytical distributions are used.Cumulative Distribution Function (CDF) or its first derivative Probability Density Function (PDF) is the way to describe how chosen parameters are distributed within given ranges.The PDF measures how likely the parameters are to have a specific value.The type of PDFs depends on analyzed variables and available data.In the case that expert judgment is used, or only boundary conditions are known, uniform distribution is mainly used.If boundary levels and some select values are known, a more precise distribution could be used, such as triangular, normal, or lognormal.The most important part is to fit available data and knowledge to the certain PDF types in the best possible way for a particular analysis.In a nuclear and thermal-hydraulic study, precise values of many parameters are unknown and uniform distribution is widely used (IAEA, 2008).However, along with the progress of the UQ investigation, distributions could and should be adopted to mirror the reality as much as possible.
As analytical tools, the simple statistical parameters indicated in Table 2 are used to evaluate the statistical significance.The standard, Pearson and Spearman coefficients, supported by partial coefficients are chosen for sensitivity analysis.The Spearman correlation gives the linear correlation between the FOM and the variable while the Pearson correlation gives the monotonic, not necessarily linear, correlation between the FOM and the variable.Partial correlation coefficients are similar to standard (Pearson or Spearman) coefficients, with the difference that the partial correlation coefficients between two variables measure their correlation while adjusting for the effects of the other variables.
In this work, fifteen uncertain parameters and their ranges were defined and described in Table 3.The authors focus on MELCOR 2.2 aerosol parameters and decide to use uniform distribution as a starting point of the investigation due to the lack of data to justify any particular distribution for the uncertain parameters.Using uniform distribution lets to see if the results concentration occurs at any particular values range of the parameters.This information could be used for PDF updates in the second iteration of the study.
The two FOMs have been chosen for this study.
• Cesium retention in the circuit [% of Cs released from the core retained in the RCS] -FOM1.The variable gives the sum of Cs masses that were already released from the fuel in all locations, excluding the containment CVH.• Aerosol amount in the containment's atmosphere -FOM2.The variable gives the total mass of aerosol (radioactive plus nonradioactive) in the gas phase in the containment.

Uncertain parameters
Fifteen uncertain parameters described below have been chosen for further investigation.The literature study was done to learn about possible boundary conditions for uncertainty analysis and the importance of particular parameters.If there was no direct indication about boundaries for uncertainty analysis, engineering judgment was used based on a literature study and general knowledge.

Fuel release
The CORSOR-Booth model for high burn-up fuel, as already mentioned, calculates the release rate for Cs while releases of the rest of the radionuclide classes are done with the help of scaling factors.In this study, the scaling factors for the radionuclide classes Cs, Ba, Te, Mo, CsM and CsI were chosen as uncertain parameters as the elements in those radionuclide classes have a high share in a mass of the aerosols in the containment (Humphries et al., 2015).The scaling factors of mentioned parameters will be investigated in ranges estimated with engineering judgment, supported by several references (Mattie et al., 2016) (Darnowski et al., 2020) (Nichenko et al., 2021).All proposed parameters and their values are shown in Table 3.

Aerosol dynamics in MELCOR
In MELCOR, the processes related to aerosol phenomenology are calculated based on MAEROS computer code.Due to the high complexity and dynamic conditions associated with the aerosols and atmosphere characteristics during the accident, simplified models using constant parameters were implemented into MELCOR (Humphries et al., 2019b) (Ross et al., 2014).In this paper, a number of uncertain parameters were listed and preliminarily investigated to understand the level of uncertainty and strength of the relations between them.The detailed equations and description of the parameters can be found in the MELCOR reference manual (Humphries et al., 2019b).
• DMIN -lower bound aerosol diameter.Using DMIN as an uncertain parameter is motivated by diameters observed in experimental data (KissaneOctober 2008) and discussion in (Schikarski and Schöck, 1984).The default value equal to 1. E− 07 m was used as a lower boundary and a relatively high value of 2. E− 06 m as an upper boundary.
• RHONOM -nominal density of aerosols.RHONOM value of 1000 kg/ m 3 could look underestimated as aerosols often have a much higher density than 1000 kg/m 3 (Powers et al., 1996).However, this default value comes from assumptions that aerosols are wet, and 1000 kg/m 3 sounds more realistic in that case.Nevertheless, for this UQ investigation, a value of 5000 kg/m 3 was chosen as a max value, while 1000 kg/m 3 was used as a minimum.• CHI -dynamic aerosol shape factor.The dynamic shape factor describes the movement of a particle in a gas stream.It is 1 for a spherical particle, and the more the motion of a particle differs from that of a spherical particle due to its shape, the larger the dynamic shape factor.
Based on the literature study, the authors decided to start from default MELCOR value, 1, to a reasonably high value of 4 (Hans-Josef Allelein.Ari Auvinen et al., 2009) (Williams et al., 1987).
• GAMMA -agglomeration aerosol shape factor.The agglomeration shape factor describes the tendency of a particle to agglomerate with other particles.Similar to the dynamic shape factor, a value larger than 1 reflects a higher tendency for a particle to form agglomerates than a spherical particle.
Particles usually are not spherical, and the shape factors (CHI and GAMMA) aim to reflect that geometrical diversity.For a highly irregular particle, the coefficients should be much greater than one, and usually highly unequal (Hans-Josef Allelein.Ari Auvinen et al., 2009) (Williams et al., 1987).
As mentioned in (Ross et al., 2014), varying CHI and GAMMA parameters together with RHONOM could cause unphysical results.Due to that, calculations should be done cautiously.This potential effect is also evaluated in the presented study.
Based on the literature study, the authors decided to use the same range for both shape factors, thus GAMMA range is set up to 1-4.
FSLIP -particle slip coefficient.Is used in the Cunningham slip correction factor, to reduce the Stokes drag force which is used for small particles or at low gas pressures.In MELCOR Cunningham slip correction factor is defined based on the empirical correlation defined by Davies in 1949 (Humphries et al., 2019a) (Davies, 2053) in which particle slip coefficient value is equal to 1.257.Supported by general knowledge and references, authors decided to use slip coefficient as an uncertain parameter with values between 1.1 and 1.3 (Hans-Josef Allelein.Ari Auvinen et al., 2009) (Rader, 1990).

M. Malicki and T. Lind
• TURBDS -turbulence dissipation rate.TURBDS describes turbulence kinetic energy transfer into heat and was pointed out as one of the main uncertain parameters, along with dynamic (CHI) and agglomeration (GAMMA) shape factors (Humphries et al., 2019b) (Powers et al., 1996) (Schikarski and Schöck, 1984).The range of TURBDS is defined as ± 50% of the default value.Believing that the default value is the most expected one, and taking into account the significant parameter variation reported in (Powers et al., 1996), the authors decided to use values 0.0005-0.0015m 2 /s 3 .The proposed range is within value 0.0002-0.002m 2 /s 3 reported in (Powers et al., 1996) as reasonable for a low level of turbulence.• TKGOP -ratio of the gas particle thermal conductivity respectively k gas and k p .MELCOR is using k gas /k p ratio as a TKGOP parameter, which is set to 0.05 by default (Humphries et al., 2019b).The parameter is used as a constant for all aerosols and for the whole duration of the calculation, independently of the composition and temperature of the atmosphere.It means that it needs to be averaged between all potential aerosol/atmosphere combinations which could appear during calculation.Based on data about the large variety of the thermal conductivity of the particles and the atmosphere (Powers et al., 1996), the authors decided to use a wide range of the TKGOP parameter from 0.0002 to 0.055.
• FTHERM -thermal accommodation coefficient.The thermal accommodation coefficient is the parameter that strongly depends on the structure of gas molecules, and the physical and mechanical conditions of the structure.In the case of a gas, it should be less than 1.0 (Powers et al., 1996) (Song and Yovanovich, 1987) (Sipkens and Daun, 2017).In MELCOR, it is introduced as a "constant associated with the thermal accommodation coefficients", which by default, equals to 2.25 (Humphries et al., 2019b).Due to lack of expertise, ±0.5 is used as an uncertainty range.• DELDIF -diffusion boundary layer thickness.DELDIF is the definition of the boundary layer for Brownian diffusion of aerosols.MELCOR treats this parameter as a constant and, based on discussion in (Powers et al., 1996), the authors decided to include this parameter in the investigation with an order of magnitude variation from the default value 1.0E-6 -1.0E-4 m.

Uncertainty calculation cases
The base case results (BC) in which all uncertain parameters are sampled in the whole proposed renges (see, Table 4) show a more substantial influence of the CHI and GAMMA factors than the rest of the uncertain parameters, respectively, for FOM1 and FOM2.

Table 3
The uncertain parameters with the ranges of their values.To better understand the relationship between the parameters, the authors performed multiple UQ analyses on the uncertain parameters set, as presented in Table 4.All cases are a variation of the base case.
-CHI case: In the base case, it was observed that changing the dynamic shape factor CHI in the range 1-4 had a dominant effect on the uncertainty of FOM1.Therefore, the CHI was altered to have a smaller range.Based on the base case, this range was updated to 1-2.All the other uncertain parameters were unchanged.-CHI and GAMMA case: As both the dynamic (CHI) and agglomeration (GAMMA) shape factors were observed to have a dominant effect on the uncertainty, both were given a reduced range of 1-2.-The Constant RHO + CHI and GAMMA: As already mentioned, varying CHI and GAMMA together with RHONOM could cause unphysical results.To study this effect the density (RHO) was given a constant value of 2000 kg/m 3 .Shape factors were varied in range 1-2.-Constant CHI and GAMMA: The final calculation is the same as BC, but with CHI and GAMMA set up as a constant evaluated based on the previous cases, respectively 1.1 and 1.5.

Results analysis overview
The results discussion is done for two FOMs separately as they represent different phenomena; thus, the parameters' impact is different.To improve the results presentation, only the most relevant parameters are presented due to the high total number of the parameters.The performed analysis contains three parts: statistical parameters, correlation coefficients, and time-dependent evolution of FOMs extended by time-depended correlation coefficient analyses.Statistical analysis shows how obtained results are spread at the end of the transient and helps to compare values with experimental data showing overall uncertainty.Sensitivity analysis based correlation coefficients give information about the relevance of a particular parameter and its impact on the abovementioned uncertainty.
The significance of the parameters and their ranges could also be evaluated via scatter plots which are presented for FOM1 in Figs. 2 and  3, and by correlation coefficients in Fig. 4. The scatter plot depicts the distribution and the relation between the FOM and the given parameter value, showing a level of dependency between those two values.An example of a relatively strong dependence is presented in Fig. 2 and a much weaker dependence in Fig. 3, which could be estimated with the help of linear regression printed in the abovementioned figures as a solid red line and correlation parameter printed in the figures legends.In the scatter plots, the experimental value is printed by an orange dashed line as a reference point, which helps highlight the difference between the cases more clearly.However, to investigate how strong the link is between a particular parameter and FOM, the Pearson and Spearman correlation coefficients are used.These two coefficients are a widely used tool for evaluating parameters' dependency, which can also show if the parameter is positively or negatively correlated.Because parameters affect each other, an investigation of multiple parameters without proper understanding of the relation between the parameters could lead to a misleading results interpretation.Standard correlation coefficients (Pearson and Spearman) give a correlation between FOM and particular parameters however affected by the influence of the rest of the parameters included in the investigation.While partial correlation coefficients should give a correlation between FOM and a particular parameter, limiting the impact of the other parameters.
Time-dependent FOM evolution shows the progression of uncertainty in time, which gives additional information about uncertainty propagation and how essential parameters are during the transient.
During the analysis, the authors decided to investigate correlation coefficients in a time-dependent manner.This way of analyzing the results let to investigate changes in the importance of the parameters during the transient.In this paper, time-dependent correlation coefficients are calculated for 300 points between 0.0s and 30000s of the transient.

FOM1 analysis
The results of the first FOM, Cs retention in the reactor coolant system (RCS), analysis shows that the retention is strongly, almost linearly correlated with the dynamic shape (CHI) factor, Fig. 2. In the BC case, correlation is seen as a high value, 0.948, of the Pearson correlation coefficient.The rest of the selected uncertain parameters show only limited correlation with the FOM1 as the highest Pearson correlation coefficient in BC for particle density is 0.077, Fig. 3.
The statistics in Table 5 helps to understand the general trend of uncertainty evolution along the sensitivity cases.This analysis could be done by investigating the most relevant parameters for UQ study, which are a minimum and maximum value and coefficient of variance that provides information about the results dispersion.Table 5 shows relatively high uncertainty for the BC, illustrated by a wide spread of the results (min and max value) and a high coefficient of variance as well as mean value, which is far from the experimental results.When the dynamic shape factor was limited to 1-2, the results showed a significant decrease of variation compared to BC.In the case where both shape factors were adjusted to a smaller range of values, no significant difference occurred.The same results were produced when the particle density was excluded, as min and max values, as well as coefficient of variance, do not vary between these cases.However, the mean value still differs from the experimental value.Additionally, in the case where both shape factors were excluded from the uncertain parameter list, variance of the results decreased even more, and the mean value was much closer to the experiment.This behavior confirms that CHI is the most impactful parameter for Cs retention in the RCS at the end of the transient.
Using scatter plots can help to evaluate implemented distributions and recognize the behavior of the results dependently on the parameter sampling.Linear regression, commonly used in such plots, indicates the strength of correlation and its sign.Linear regression also helps to understand the importance of the parameter distribution and underlines the potential deviations of the results for specific values.Here, in Fig. 3, the three parameters which show the strongest change between the sensitivity calculations are compared.The left side in Fig. 3 shows the BC calculation results for RHO, FTHERM and DMIN and the right side, respectively, shows a case "Constant CHI and GAMMA".By the changed linear regression slope, the graphs show a stronger dependency of the FOM1 on the other uncertain parameters when the shape factors are constant.It also could be seen by increased values of Pearson coefficient and R 2 parameter included in the legend in Fig. 3.The highest correlation increase is visible in the case of the particle density RHO.
In general, obtained results underestimate the experimental data, but it should be noted that in all scatter plots, the value of FOM1 differs greatly from the experiment.This is due not only the uncertain aerosol parameters selected for this investigation but also due to other features of our base case analysis.Analyzing Figs. 2 and 3 in light of the experimental data, one can see that CHI parameter in values 1-1.3 gives much steeper correlation with FOM1 than in the rest of the CHI value range, and the FOM1 values are closer to the values from the experiment.The analysis of the density parameter shows that more expected results are obtained when RHO >3000 kg/m3 is used.The DMIN parameter's behavior shows a slight difference from the other parameters.In DMIN ranges of 0.2E-6 -0.25 E-6m and 1.25E-6 -2.0 E-6m, the FOM1 results are closer to the experimental values.However, even though the results are closer to the experiment, it should be noted that in all these cases, the FOM1 value is dispersed more, thus the uncertainty in that range is higher.
In the BEMUSE project (Glaeser et al. (NEA, 2011), it was observed that the number of calculations is of decisive importance for sensitivity analysis, and increasing the number of runs could help to highlight additional dependency of less relevant parameters.In this investigation, this aspect of uncertainty quantification was not addressed.Consequently, it is recommended to carry out a separate study focusing on parameter range, distribution and the number of calculations.Different correlation coefficients are used to evaluate dependency between the parameter and the FOM.Fig. 4 presents the evolution of the importance of the chosen parameters in the different sensitivity calculations.The low value of the correlation coefficient indicates that, in this calculation, the parameter is irrelevant or of low relevance for FOM results.When the value of the partial coefficient is similar to the standard one, it can be assumed that the parameter is not strongly affected by the rest of the investigated parameters.However, if for a particular parameter the partial coefficient is significantly higher than the standard one, it usually means that together with the chosen uncertain parameters, the parameter shows only low significance but depending on the definition of the rest of the parameters, it could be more significant.Results in Fig. 4 show that in the BC, standard coefficients for RHO, DMIN, TKGOP and FTHERM are low while partial coefficients are visibly higher, whereas for CHI, all the coefficients have similarly high values.Adaptation of the range of values of the shape factors to 1-2 brings unification between partial coefficients for RHO and DMIN parameters.In the case where RHO was excluded, the DMIN partial coefficients decrease, and the Pearson partial coefficient changes sign similarly to GAMMA, which proves complexity of parameters dependency.Calculations with constant shape factors show a high value of investigated correlation coefficients between FOM1 and RHO, as it was in the case of CHI in BC.Also, an increase in the importance of the DMIN, TKGOP, FTHERM is visible.However, greater differences between standard and partial coefficients for these parameters suggests that they are dependent on other parameters, but not as strong as it was in the case of RHO in BC.This strong dependency confirms that these parameters (CHI, GAMMA, RHO) should be used carefully and with an understanding of their interrelationships.
One of the most basic ways to investigate FOMs is using timedependent figures, which show FOM evolution in time.It helps to understand the importance of dynamically changing parameters, and uncertainty propagation.In the case of Cs retention in RCS, high dispersion of the results in the early phase of core degradation, Fig. 5, could be more related to the uncertainty of the release than with aerosol dynamics.In Fig. 5, the BC and cases with constant CHI and GAMMA are presented as the most representative cases.Note that the experimental value is shown as constant in Fig. 5 even though, in reality, the retention in the circuit changed over time.However, the change was not measured, and thereby, only a total retention fraction is used as the FOM1.
The BC shows high dispersion in the early core degradation phase (5000-10000s) when the first relatively small release of fission products and structural materials takes place.From major aerosol release which starts at 10 000 the results' dispersion decreases.The dispersion remains between 18 and 47% of Cs retention until the end of the transient.The case with constant shape factors gives similar dispersion in the core degradation phase; however, after 10000s, it decreases to 39.9-51.6%.The figures confirm the importance of the shape factors for Cs retention in the whole transient.In both cases, the early core degradation phase during the first aerosol release shows relatively high results dispersion independently of the used parameters, which is also the case for the rest  of the UQ calculations.The difference in all sensitivity calculations varies around 40%; however, the highest deviation is in BC ~50%.
Due to the aforementioned correlation differences, additional correlation coefficient analyses were done.To understand the dynamics of the parameter relations and final results, time-dependent correlations for the most influential parameters were calculated and presented in Figs. 6 and 7.This data demonstrates how the particular parameter correlation coefficient is changing with the transient progression, and it could help to understand which parameters are more relevant for the core degradation phase.In Fig. 6, the CHI correlation shows lower importance in the early core degradation phase, and it increases when the main aerosol release takes place.This confirms that CHI is a significant parameter as soon as the main aerosol release and transport start.Based on Fig. 7, the correlation of the shown parameters RHO and DMIN significantly changes during the transient once the main aerosol release starts.The greatest effect was seen for the RHO parameter (particle density), which in the BC, decreases in importance when the main aerosol transport phase starts whereas in the constant CHI and GAMMA calculation, the importance of particle density increases during this time.In addition, particle density (RHO) shows a decrease of the Pearson coefficient and an increase of the Partial Pearson coefficient in the base case.The results are consistent with Fig. 4, which shows a high difference between the standard and partial coefficients.However, when GAMMA and CHI are constant, the Pearson correlation coefficient is increasing as well as the Partial Pearson.This difference confirms the importance of the RHO parameter alone and an influence of the shape factors on RHO significance.

FOM2 analysis
The total mass of the air-borne aerosols in the containment at the end of the transient is the FOM2.FOM2 uncertainty and parameter dependency is different from FOM1.The most important parameter in this case is the aerosol agglomeration shape factor GAMMA, which is then followed by RHO, CHI and DMIN, see Fig. 8, Figs. 9 and 10.
Based on the statistical analysis of the UQ runs, as seen in Table 6, the importance of the GAMMA should be noted.In the case where only the CHI range was updated to 1-2, no changes in the statistics are shown.However, when GAMMA was adopted as well, the results showed a dispersion decrease, which could be seen by the min and max value as well as the coefficient of variance.In the case when RHO or CHI and GAMMA were excluded, uncertainty slightly decreased, but the change was not as significant as in the case where GAMMA was modified.This behavior confirms that GAMMA is the parameter which brings the highest dispersion in the results for FOM2.
A comparison of the CHI and GAMMA scatter plots presented in Fig. 8 for BC and the case with CHI and GAMMA set to range 1-2 help to visualize importance of the GAMMA range for the FOM2 analysis.Fig. 8 shows a significant difference in the distribution of the results in function of the GAMMA parameter.In the BC calculation (left side of Fig. 8), the concentration of the low results is noticeable along with the whole CHI range and from ~2 to 4 in the GAMMA range.This behavior confirms the strong influence of the GAMMA, especially for the values above 2.Because of this correlation, the GAMMA parameter, along with CHI, were updated to the range of 1-2, as seen on the right side of Fig. 8.The update of the GAMMA causes a lack of result concentration in the scatter plot of CHI and a smooth result distribution of the GAMMA parameter, consequently, changing their importance as presented by a linear regression slope.Based on the Pearson and R2 parameters given in the legend boxes, after the update, the CHI relevance increased while GAMMA slightly decreased.This behavior highlights the importance of the correct parameter range implementation or distribution, which will be further investigated.
The most affected parameter along the performed UQ calculations was DMIN, whose evolution is presented in Fig. 9.In the case of BC, the accumulation of the results with the lower values along the whole DMIN range was noted, similar to the case of the CHI discussed above.The concentration phenomena disappeared, as in the case of CHI when GAMMA was limited to 1-2.Additionally, the DMIN presented a much higher and clearer correlation in the case where shape factors were constant.This behavior shows a bond between the DMIN and shape factors and confirms their significance for the parameters setup.
To investigate correlations' coefficients with a broader perspective, the results for the most critical parameters are presented in Fig. 10.As discussed in the FOM1 analysis, a high difference between the standard and partial coefficients could suggest a stronger dependency of a  particular parameter than with others.The case where only the CHI range was adapted to 1-2 is excluded as it does not vary significantly from the case when both shape factors were updated.In the BC, similar to the FOM1, some of the parameters show low standard coefficients while their partial coefficients are much higher, which is the case for RHO, CHI, DMIN and TURBDS.GAMMA gives the highest correlation as expected; however, the different coefficients are not as equal as they were in the case of CHI in FOM1.This could suggest that GAMMA is the most significant parameter but less linearly than CHI was for FOM1 because the Pearson and Spearman coefficients vary more.This statement is confirmed by previously discussed scatter plots Fig. 8 which show the non-linear behavior of GAMMA for the values above 2.An adaptation of the shape factors with the range 1-2 unifies the results by limiting the GAMMA significance of the RHO, CHI, DMIN increase, and partial coefficients of TURBDS and DELDIF.This confirms the high GAMMA values' great impact on all results and correlations between parameters.Excluding the RHO from UQ parameters causes changes in the coefficients of TURBDS, FTHERM and DELDIF, which shows a strong correlation between these parameters.In the case where shape factors were constant, the importance of the RHO was negligible, while DMIN presents a high independent importance as all coefficients are relatively equal.The TURBDS parameter's significance also increases, but the ratio between partial and standard coefficients is still high.The discussed results confirm the importance of the shape factors and the dependency of the DMIN on them.
A time-dependent evolution of the FOM2 is presented in Fig. 11.It shows the result dispersion during the transient and compares it with the experimental data.The BC gives the highest dispersion in the whole transient.An adaptation of the CHI range to 1-2 divides the values in the aerosol phase (18000-30000s) into two parts, highlighting the one with higher deposition and, consequently, lower aerosol mass at the end of the transient caused by high GAMMA values.Excluding RHO from the UQ parameters and adapting the CHI and GAMMA ranges to 1-2 causes a decrease in the result dispersion.Constant RHO caused slightly decreased result variation in the core degradation phase.The disappearance of the results with high deposition in the aerosol phase is related to the limited GAMMA range.In the case where shape factors were constant, the result dispersion decreased.However, the maximum obtained value decreased; even though the final results seem to be closer to the experiment data, the overall transient is far from the expected values.It indicates the need for time-dependent analysis to understand the origin and level of the uncertainty fully.This behavior confirms the relevance of the RHO and shape factor parameters regarding the uncertainty and final results.
To further investigate the importance of the parameters, similarly to FOM1, the time-dependent coefficient graphs are analyzed in Fig. 12.Based on the analyses discussed above, the RHO and DMIN parameters were chosen to compare the case with a CHI and GAMMA range of 1-2 and for the one without shape factors.The results only partially confirm the importance of the RHO and DMIN parameters in the core degradation phase.Fig. 12 shows high correlations only for the beginning of the core degradation phase, which decrease when the main releases occurred, around 11000s and 16000s; however, their significance increased when CHI and GAMMA were constant.In Fig. 13, the timedependent Pearson coefficient is presented for CHI and GAMMA, representing the case where their ranges were set to 1-2.The results show that the CHI significance decreases in the middle of the degradation phase, after which it grows back, achieving its high value when the aerosol mass peaks.On the other hand, GAMMA increases its importance within time.This confirms the GAMMA's significance in the end phase and highlights the CHI impact on the maximum mass of the transported aerosols.These results are in line with the analysis from the FOM1, in which CHI was the leading parameter.High CHI values decrease Cs retention in RCS, thus, the mass of aerosols transported to the containment should increase, which can be seen in Fig. 11 along sensitivity cases.When CHI was limited to 1-2 or set as constant, the maximum mass of the aerosols decreased.Also, based on the time-dependent correlation analysis, other parameters related to release scaling factors significantly impact the FOM2 around the peak time.However, as this paper focuses on aerosol parameters, release scaling factors will be investigated in a separate study.
These results confirm the importance of understanding the process of the uncertainty evolution during the transient.Also, it highlights how different tools could help interpret the results and improve knowledge  For FOM2, the air-borne aerosol concentration in the containment, the results were on average lower when the CHI range was reduced to 1-2.Also, in that case, the aerosol phase showed two focused areas: the first one scattered around experimental values, and the second one much lower than expected.These lower results were correlated with values of the aerosol agglomeration shape factor (GAMMA) parameters above 2.When GAMMA and CHI were set to 1-2, and the particle density (RHO) was constant, the data were dispersed more equally and closer to the experiment, which was confirmed by statistics.In the case of constant CHI and GAMMA, the dispersion was similar to the case where their ranges were adapted to ranges 1-2; however, the dispersion was smoother and more accurate, considering experimental values.The obtained results could suggest that the range and uniform distributions, especially for GAMMA, obscure the importance of other parameters and strongly impact the final results.
When lower, and probably more realistic values for GAMMA were used, the importance of FOM1-RHO and FOM2-DMIN increased.In the case where CHI and GAMMA were constant, it was confirmed that RHO and DMIN were more relevant for FOM1 and FOM2, respectively.Also, RHO for FOM2 loses its importance when the impact of the shape factors is limited.In general, CHI and GAMMA indeed show high correlations with FOMs for the whole transient, and they are the most influential uncertain parameters.Also, the strong link between shape factors and the RHO (particle density) was confirmed and presented as a variation of time-dependent correlation coefficients.
From a methodology point of view, the scatter plots help the most to identify a potential adaptation of PDF distribution or sampling, which is especially crucial for such uncertain phenomenology as a nuclear severe accident.Also, working with the experimental data, like those from FPT-1 experiment, gives an opportunity to investigate the dynamics of uncertainty and uncertainty propagation.It would be interesting to investigate scatter plots at the peak of aerosol mass concentration which, compared to the end state data, could bring more insights about parameters relevant to aerosol transportation within RCS and in the containment.Merging the linear regression of scatter plots with standard and partial correlation coefficients helps to identify potential dependency in the parameter setup, as it was in the case of GAMMAdependency with CHI and DMIN in the FOM2 analysis.A timedependent analysis of both FOMs and correlation coefficients, done by presenting the result dispersion and highlighting the importance of the parameters during the transient, led to a better understanding of the origin of uncertainty and the consequences of parameter variation for the whole analysis.
Based on the study performed, the authors also highlight the need for the further investigation of the effect of parameter distribution and the number of samples on the results, and would recommend a global timedependent UQ analysis.
It needs to be noted that in this study, the FOMs were selected as the values at the end of the transient.The time-dependent analysis should be performed in a broader then in this paper range to understand fully, as only preliminarily explored here, the effect of the parameters range and distribution.

Disclaimer
This paper reflects only the author's view, and the European Commission is not responsible for any use that may be made of the information it contains.

Fig. 2 .
Fig. 2. Scatter plot of the most correlated with FOM1 (percentage of released Cs retained in RCS) parameter, CHI.

Fig. 3 .
Fig. 3. FOM1 (percentage of released Cs retained in RCS), scatter plot of the parameters, parameters that show the strongest change between the sensitivity calculations, are compared, BC on the left side, case Constant CHI, and GAMMA on the right.

Fig. 4 .
Fig. 4. Plots of correlation coefficient of the most impacting parameters for FOM1 (percentage of released Cs retained in RCS).

Fig. 5 .
Fig. 5. Plots of FOM1 (percentage of released Cs retained in RCS) for BC, left side and Constant CHI and GAMMA on the right side.

Fig. 6 .
Fig. 6.Time dependent Pearson coefficient for CHI parameter in FOM1 (percentage of released Cs retained in RCS).

Fig. 7 .
Fig. 7. Time-dependent partial Pearson correlation coefficient for FOM1 (percentage of released Cs retained in RCS) of the most important parameters in BC, on left and case Constant CHI and GAMMA on the right.

Fig. 8 .
Fig. 8. FOM2 (aerosol amount in the containment's atmosphere) scatter plot for CHI and GAMMA parameters, BC on the left, CHI and GAMMA 1-2 on the right.

Fig. 12 .
Fig. 12.Time dependent Pearson coefficient for FOM2 (aerosol amount in the containment's atmosphere) of RHO and DMIN parameters for cases CHI and GAMMA 1-2, on the left, and Constant CHI and GAMMA, on the right.

Fig. 13 .
Fig. 13.CHI and GAMMA time-dependent coefficients for FOM2 (aerosol amount in the containment's atmosphere) in case CHI and GAMMA 1-2.

Table 1
Proposed Cs speciation implemented into investigated input deck.

Table 2
Uncertainty methodology summary.

Table 4
Uncertain parameters and the ranges of their values in the five different sensitivity calculations.

Table 5
Statistical values for performed UQ sensitivity cases on FOM1 (percentage of released Cs retained in RCS).
M.Malicki and T. Lind