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Reactor design requires safety studies to ensure that the reactors will behave appropriately under incidental or accidental situations. Safety studies often involve multiphysics simulations where several branches of reactor physics are necessary to model a given phenomenon. In those situations, it has been observed that the neutron transport part is still a bottleneck in terms of computational times, with more than 80% of the total time. In the case of hexagonal lattice reactors, transport solvers usually invert the discretised Boltzmann equation by discretising the regular hexagon into lozenges or triangles. In this work, we seek to reduce the computational burden of the neutron transport solver by designing a numerical spatial discretisation scheme that would be more appropriate for honeycomb meshes. In our past research efforts, we have set up interesting discretisation schemes in the finite element setting in 2D, and we wish to extend them to 3D geometries that are prisms with a hexagonal base. In 3D, a rigorous method was derived to shrink the tensor product between 2D and 1D bases to minimum terms. We have applied these functions successfully on a reactor benchmark—Takeda Model 4—to compare and contrast the numerical results in a physical setting. [ABSTRACT FROM AUTHOR]

Within the framework of pressurized water reactor severe accidents, the ICE research program is devoted to studies on a variety of fundamental mechanisms involved in Fuel-Coolant Interaction (FCI). Among them, the oxidation of liquid corium is expected to impact the premixing/fragmentation stage, thus possibly influencing the subsequent steam explosion. As a first step towards the exhaustive description of oxidation phenomena occurring in typical FCI layouts, the 1-D oxygen diffusion in liquid (over)stoichiometric spherical corium droplets is investigated in this article. A mesoscopic description is adopted, based on the evolution of oxygen concentration c O , depending on space, time, corium composition and temperature. The underlying physical model is coupled with CALPHAD-based thermodynamic calculations. An asymptotic analytical solution is first determined, assuming that the apparent diffusion coefficient D O does not depend on c O . Then, the general variational form of governing equations is derived, leading to the implementation of a related numerical approach using the finite-element method, in COMSOL® software. This numerical approach is first benchmarked with the analytical calculation, and then extrapolated to the more general case where D O varies with c O . Finally, some case studies representative of FCI issues are selected, in order to determine typical oxygen diffusion transient times within a variety of liquid corium droplets. [ABSTRACT FROM AUTHOR]

This work focuses on the k -eigenvalue problem of the neutron transport equation to solve for the largest eigenvalue (k eff) and the corresponding eigenmode, called the fundamental mode. This problem is usually solved using the power iteration method. However, the convergence of this algorithm can be very slow, especially if the dominance ratio is high as is the case in reactor physics applications. Thus, the power iteration method has to be accelerated in some ways to improve its convergence. One such acceleration is the Chebyshev acceleration method which has been widely applied to legacy codes. In recent years, nonlinear methods have been applied to solve the k -eigenvalue problem. Nevertheless, they are often compared to the unaccelerated power iteration. Past works have compared and contrasted the benefits of the Anderson acceleration method for the neutron transport equation, and the analysis is improved in this work by studying the impact of preconditioning acceleration techniques on the overall performance. In this work, the transport operator is inverted using the source iteration method with the discrete ordinates (S n) method. Therefore, keeping in mind the need for improving the power iteration algorithm, we also focus on a different type of acceleration which is the diffusion synthetic acceleration (DSA) usually employed for improving the inner iterations. In this research effort, DSA is applied to the power iteration. Besides, Anderson and Chebyshev accelerations are combined with this external DSA scheme to evaluate the computational gain. All these methods have been applied to a benchmark case of a small pressurised water reactor, KAIST 3A. • The k -eigenvalue problem for the transport equation is solved with the power iteration. • Its convergence is usually very slow and has to be improved with accelerations. • We propose to apply the Anderson acceleration as a substitute for the legacy Chebyshev method. • We also propose to apply a diffusion synthetic acceleration to the power iteration. • These accelerations have been applied and analysed on the KAIST 3A heterogeneous core benchmark. [ABSTRACT FROM AUTHOR]

The modeling of transient phase segregation phenomena in corium multicomponent thermodynamic systems is complicated especially in the liquid miscibility gap which can lead to phase separation (stratification of the liquid phases for corium) and is of tremendous importance. The modeling of the associated phase separation transients in corium stratification is limited by a deficiency in the interfacial energy data, which are difficult to measure experimentally for these high-temperature systems. The limitations of summarizing the interfacial energies as constant coefficients were observed in a recent work on simulation of the stratification kinetics of a corium pool and the necessity to improve the estimation of interfacial energies [ 1 , 2 ]. This work tests the efficacy of a thermodynamic approach to evaluate the interfacial tensions, initially proposed for metallic systems by Kaptay [3]. This method is based on the Butler equation and makes use of existing thermodynamic databases for multicomponent systems constructed by the CALPHAD method. In practice, the application of this method to corium systems was performed using the open-source code openIEC [4]. To do so, this code was modified to enable required features, both functionally and methodically. In particular, the OpenCalphad Gibbs energy minimizer has been interfaced in this framework and benchmarked against previous results. Then, interfacial energies were calculated for two relevant corium systems (U-O and U-O-Zr-Fe) using two different existing databases (TAF-ID and NUCLEA). The first results obtained were analyzed in terms of general trends and compared to existing theoretical aspects. Large differences were observed depending on the thermodynamic database that was used. [ABSTRACT FROM AUTHOR]

Abstract: The issue of the integration scheme along a trajectory which appears for all tracking-based transport methods is discussed from the point of view of the method of characteristics. The analogy with the discrete ordinates method in slab geometry is highlighted along with the practical limitation in transposing high-order S N schemes to a trajectory-based method. We derived an example of such a transposition starting from the linear characteristic scheme. This new scheme is compared with the standard flat-source approximation of the step characteristic scheme and with the diamond differencing scheme. The numerical study covers a 1D analytical case, 2D one-group critical and fixed-source benchmarks and finally a realistic multigroup calculation on a BWR-MOX assembly. [Copyright &y& Elsevier]

AbstractThis paper describes the implementation of efficient and portable vectorized sweep kernels as part of the resolution of the neutron transport equation on three-dimensional Cartesian grids using the discrete ordinates ($${{\mathcal S}_n}$$Sn) method for the angular variable and the diamond differencing (DD) scheme for the spatial discretization. Vectorization is set up along the directions within the same octant and is independent of the spatial discretization order; therefore, the extension of this technique to high-order DD or discontinuous Galerkin schemes is immediate. Our implementation is written in C++17 and relies on the Kokkos performance portability framework. This library allows one to express shared-memory parallelism (including vectorization) in a machine-independent way and supports many backends including CUDA and OpenMP. Our vectorization procedure relies on the portable single instruction multiple data types provided by Kokkos. The method has been implemented for DD schemes up to order 2 and yields promising results on CPUs supporting standard vector instructions. [ABSTRACT FROM AUTHOR]

The finite element method stands out as a powerful tool for modelling engineering problems. They are particularly well suited thanks to adaptive discretization techniques involving mesh size (h) or polynomial degree (p) or a combination of both (hp). In the case of p -adaptiveness, high-order function bases are required on the elements. For polygonal elements in 2D, Wachspress proved that, in the general case, it is not possible to construct a set of Lagrange finite elements with polynomial functions. Instead, this can be achieved through a popular set of basis functions which are generalized barycentric coordinates. One such family of functions is the Wachspress functions which are well suited for strictly convex polygons. While recent work has led to the construction of quadratic approximations from first-order Wachspress functions, there exists no approach for generalizing to higher orders for any strictly convex polygon. This work provides a general method to develop k -order Wachspress bases for any m -face polygon, where k < m. This method relies on symbolic computation for the assembly of a linear system whose numerical solution gives the values of the unknown coefficients of the Wachspress functions, and can be viewed as a generalization of Dasgupta's GADJ algorithm. This work originates from the development of a high-order Discontinuous Galerkin (DG) scheme for the solution of a linear transport equation over honeycomb meshes. As such, the regular hexagon is the strictly convex polygon taken as an example to describe the proposed method. We ensure that the functions obtained enforce the constraints required for their application in a finite element approach. Furthermore, we have successfully applied these functions to a finite element discretization of the Poisson equation and provided a test case for the linear transport equation on honeycomb meshes. In addition, further results are provided for an irregular pentagon to show the generality of the approach. • The finite element method is a powerful tool for engineering problems. • In 2D, it is not possible to construct Lagrange elements with polynomial functions. • No previous work for higher orders for any strictly convex polygons exists. • This method relies on symbolic computation to obtain high-order bases. • It can be viewed as a generalization of Dasgupta's GADJ algorithm. [ABSTRACT FROM AUTHOR]

Considering transient processes where liquid/solid phase change occurs, this paper focuses on the associated modeling and numerical treatment in the frame of "Computational Fluid Dynamics" simulations. While being of importance in many industrial applications involving solidification and melting of mixed materials, including power and manufacturing engineering, the first application of this work pertains to the analysis of severe accidents in a nuclear reactor. Indeed, in this context, the molten core materials (a.k.a. corium) can form a high-temperature multiphase liquid pool at the boundary of which fusion and solidification phenomena are of prime importance. In this context, even if materials at play are treated as pure components, it is mandatory to distinguish two different phase change temperatures with a solid fusion temperature and a liquid solidification temperature. Accordingly, in the frame of a sharp interface representation, the paper introduces non-classical heterogeneous conditions at the liquid/solid boundary in such a way that both moving interface (through Stefan conditions associated with fusion or solidification) and static interface (imposing heat flux continuity) are supported at the same time on different spatial locations along this boundary. Within a monolithic resolution of Navier–Stokes and heat conduction equations, this interface is explicitly tracked with combined Front-Tracking and VOF methods. In order to ensure zero velocity in the solid phase, an Immersed Boundary Method and a direct forcing penalization are also introduced. The main relevant features of this combination of numerical methods are discussed along with their implementation in the TrioCFD code taking advantage of the pre-existing code capabilities. Numerical simulations including both verification tests and a case of interest for our industrial application are reported and demonstrate the applicability of the proposed triptych model+methods+code to treat such problems. The numerical tools and the simulation code developed in this work could be used not only in the several accident context but also to simulate melting, solidification and fusion processes occurring in aerodynamics, hypersonic reentry vehicles and laser applications to cite but a few. [ABSTRACT FROM AUTHOR]

• Rayleigh-Bénard-Marangoni (RBM) natural convections are analyzed using TrioCFD. • Influences of RBM effects on thermal behavior of thin metallic layer are studied. • Various thermal and mechanical boundary conditions are investigated. • The results may lead to a better evaluation of the focusing effect in IVR strategy. Within the context of nuclear power safety, the In-Vessel Retention (IVR) strategy consists in sustaining the integrity of the reactor pressure vessel by External Reactor Vessel Cooling (ERVC). However, due to the ablation of the vessel wall and the thermochemical effects, a thin metallic layer may be formed on top of the corium pool and lead to the vessel failure. In this article, numerical simulations of the thermal–hydraulic behavior of the thin metallic layer are carried out to reduce the uncertainties in the heat flux evaluation. Specifically, the Rayleigh-Bénard and Bénard-Marangoni convections are considered in the heat transfer models of the metallic layer. The TrioCFD code is validated for the Bénard-Marangoni convection, then applied into the numerical simulations of the metallic layer with different top boundary conditions: pure radiative heat transfer, and radiative heat transfer plus positive or negative Marangoni effects. The temperature profile, mean temperature and focusing effect of the metallic layer are evaluated and compared for various cases. Results show that the Marangoni effects have to be considered for small metallic layer thicknesses. Furthermore, the layer can be divided into two regions, of which the heat flux correlations can be evaluated independently, giving a better estimation of the heat flux profile. [ABSTRACT FROM AUTHOR]

Thermodynamic data are needed for all kinds of simulations of materials processes. Thermodynamics determines the set of stable phases and also provides chemical potentials, compositions and driving forces for nucleation of new phases and phase transformations. Software to simulate materials properties needs accurate and consistent thermodynamic data to predict metastable states that occur during phase transformations. Due to long calculation times thermodynamic data are frequently pre-calculated into “lookup tables” to speed up calculations. This creates additional uncertainties as data must be interpolated or extrapolated and conditions may differ from those assumed for creating the lookup table. Speed and accuracy requires that thermodynamic software is fully parallelized and the OpenCalphad (OC) software is the first thermodynamic software supporting this feature. This paper gives a brief introduction to computational thermodynamics and introduces the basic features of the OC software and presents four different application examples to demonstrate its versatility. [ABSTRACT FROM AUTHOR]

• Experimental studies of the corium melt oxidation kinetics were carried out. • Various melt structures had been studied: one-, two- and three-liquid. • Melt structures had been studied both in the presence and absence of oxide crusts. • Air, argon-oxygen mixture and steam were used as oxidant. • Melt oxidation control processes had been identified. Prototypic oxidic-metallic molten corium oxidation has been studied in the conditions applicable for a severe accident at the NPP with PWR. Suboxidized oxidic melt and different pool configurations have been examined – one-liquid oxidic, two-liquid oxidic-metallic with surface and bottom positions of metallic liquid, and three-liquid with one oxidic and two metallic layers. The sensitivity of oxidation rate to the presence of oxidic surface and intermediate crusts is discussed. Produced experimental data for different pool structures and oxidants have been summarized. [ABSTRACT FROM AUTHOR]

• Original results on molten steel penetration through a corium crust. • Two penetration mechanisms are identified: dissolution process and penetration through cracks. • A deceleration of penetration is observed with increasing exposure time. • Dissolution process leads to molten steel enrichment in uranium and zirconium and these elements tend to reach an equilibrium value. • A two-step dissolution mechanism is proposed. In the framework of the European IVMR project, dedicated to the assessment of In-Vessel Retention (IVR) strategy for high power reactors, VITI-CORMET tests, performed in VITI facility (CEA-Cadarache/PLINIUS Severe Accident Platform), intend to study the interaction between a molten 304L stainless steel droplet and suboxidized solid corium crust, referred to as C-70 (( U 0.54 , Zr 0.46 ) O 1.74 ). In the present paper, original experimental results on molten steel penetration through a corium crust are presented. Identification of penetration mechanisms is performed by SEM/EDX analyses, and has shown that molten steel can penetrate through pre-existing cracks or by a dissolution process. The latter is found to be the dominant penetration mechanism and involves material transport from the crust to molten steel. Penetration kinetics is studied by measuring the average penetration length at different exposure times. It is found that the penetration by dissolution decelerates with time, while U and Zr contents tend to reach an equilibrium value within the metallic droplet. [ABSTRACT FROM AUTHOR]