This paper presents an experimental study for the transient growth of an ice layer in a square channel under laminar flow conditions and a mixed convection heat transfer regime. The ice layer was grown from a cold plate located at the bottom of the channel, capable of reaching temperatures between 0 and −20 °C. The onset of ice formation was marked by a sudden sharp increase of the cold plate temperature followed by a rapid spreading of the ice over the cold plate surface. This was attributed to subcooling effects within the thermal boundary layer of the flow. The flow field was measured using particle image velocimetry (PIV) and the ice profiles were measured at several instances of time after the onset of freezing by a visual tracing of the solid–liquid interface. In addition, a parametric study was performed regarding the effect of the cold plate temperature and the flow rate on the ice growth rate. Suitable approximations to the experimental boundary conditions were found after a detailed analysis of the cold plate's transient temperature response, which could be readily implemented in numerical software. An important novelty of the present work is the measurement of the transient ice development of the ice-layer near the inlet of the channel, in addition to the centre of the channel where the flow is more developed. As such, a comprehensive and well-described experimental data set was generated for transient freezing in laminar internal flow. With this approach, a very good agreement was obtained between the experimental results and numerical simulations which were included to indicate the suitability of the current experimental campaign for numerical benchmarking purposes.

This work presents two color LIF temperature measurements for the transient freezing in a square channel under laminar flow conditions. This is the first time non-intrusive temperature measurements were performed within the thermal boundary layer during the transient growth of an ice layer in internal flow. A combination of a local outlier factor algorithm and a smoothing operation was used to remove the top to bottom striations and reduce the other measurement noise. The temperature uncertainty in our measurements was between σ=0.3∘C and σ=0.5∘C. For the largest temperature difference between the bulk and the melting point of 14.6 °C, good results were obtained. As such, the current campaign demonstrates the potential of LIF as a non-intrusive temperature measurement technique for solid–liquid phase change experiments. However, some artefacts were present within the vicinity of the ice-layer due to the scattering of the laser light, especially near the inlet of the channel where the ice-layer is curved instead of flat. LIF measurements taken within a short time span prior to the onset of ice freezing showed approximately 2 °C of subcooling, consistent with previous findings. In addition, an anomalous behavior within the thermal boundary layer was observed, with a much smaller temperature gradient within the first few mm above the cold plate and a point of inflection in the temperature profile. The anomalous temperature behavior is possibly attributed to enhanced natural convection as a result of the subcooling at the cold plate surface.

We present a finite volume adaptive mesh refinement method for solid-liquid phase change problems with convection. The refinement criterion consisted of three different error estimators for the solid-liquid interface, the flow field, and the temperature field respectively. For the solid-liquid interface, the cells undergoing phase change were refined based on the maximum difference in the liquid fraction over the cell faces. For the flow field and the temperature field, an error indicator was used based on the cell residual method. To maintain a high parallelization efficiency, a dynamic load balancing procedure was used. The adaptive mesh refinement strategy was verified through three different test cases, these are the gallium melting in both 2D and 3D cavities, and the molten salt reactor freeze valve. For all three cases, very good agreement was obtained between the adaptive mesh results and the reference solutions. In addition, more accurate results were obtained with the adaptive meshes compared to static meshes with a similar amount of mesh cells. This illustrated the potential of the current approach for developing computationally efficient numerical methods for solid-liquid phase change problems.

Five Euratom projects launched since 2017 in support of the development of ESNII/Generation-IV reactor systems are briefly presented in the paper in terms of key objectives, results, and recommendations for the future. These projects focus on various aspects of the following ESNII/Generation-IV systems: Sodium Fast Reactor, Gas Cooled Fast Reactor, Supercritical Water Cooled Reactor, and Molten Salt Fast Reactor. The paper does not consider EU projects focused on the Gen-IV reactor technologies based on the use of heavy metals as a coolant because these projects are reviewed in a different paper.

We present a discontinuous Galerkin method for melting/solidification problems based on the “linearized enthalpy approach,” which is derived from the conservative form of the energy transport equation and does not depend on the use of a so-called mushy zone. We use the symmetric interior penalty method and the Lax–Friedrichs flux to discretize diffusive and convective terms, respectively. Time is discretized with a second-order implicit backward differentiation formula, and two outer iterations with second-order extrapolation predictors are used for the coupling of the momentum and energy. The numerical method was validated with three different benchmark cases, i.e., the one-dimensional Stefan problem, octadecane melting in a square cavity and gallium melting in a rectangular cavity. The performance of the method was quantified based on the L ² norm error and the number of iterations needed to convergence the energy equation at each time step. For all three validation cases, a mesh convergence rate of approximately O(h) was obtained, which is below the expected accuracy of the numerical method. Only for the gallium melting case, the use of a higher-order method proved to be beneficial. The results from the present numerical campaign demonstrate the promise of the discontinuous Galerkin finite element method for modeling certain solid–liquid phase change problems where large gradients in the flow field are present or the phase change is highly localized, however, further enhancement of the method is needed to fully benefit from the use of a higher-order numerical method when solving solid–liquid phase change problems.

We present an approach to build a reduced-order model for nonlinear, time-dependent, parametrized partial differential equations in a nonintrusive manner. The approach is based on combining proper orthogonal decomposition (POD) with a Smolyak hierarchical interpolation model for the POD coefficients. The sampling of the high-fidelity model to generate the snapshots is based on a locally adaptive sparse grid method. The novelty of the work is in the adaptive sampling of time, which is treated as an additional parameter. The goal is to have a robust and efficient sampling strategy that minimizes the risk of overlooking important dynamics of the system while disregarding snapshots at times when the dynamics are not contributing to the construction of the reduced model. The developed algorithm was tested on three numerical tests. The first was an advection problem parametrized with a five-dimensional space. The second was a lid-driven cavity test, and the last was a neutron diffusion problem in a subcritical nuclear reactor with 11 parameters. In all tests, the algorithm was able to detect and include more snapshots in important transient windows, which produced accurate and efficient representations of the high-fidelity models.

A method based on ultrasonic wave propagation is applied for the determination of the viscosity of low viscous liquids. A waveguide is used to remotely transmit the ultrasonic waves from a shear piezoelectric transducer into the liquid. At the solid-liquid interface, a guided wave mode, the shear mode, is used to extract the liquid viscosity. The energy of the reflected ultrasonic wave depends upon its operating frequency, the physical properties of the liquid (viscosity and density), and the waveguide (density and shear modulus). The results show that the attenuation of the waves, and thus the viscosity of the liquid, can be retrieved using this method. Measurements on water, ethanol, and mixtures of water/glycerol illustrate that the method can monitor changes in attenuation due to the viscosity of the liquid. The range of viscosities measured was between 0.8 and 60 mPa s. Compared to literature values, the relative error for these measurements was lower than 12% while the uncertainty in the measurements was lower than 5%. Besides its ability to measure low viscosities, this method offers advantages such as the capability to perform in-situ measurements of liquids in harsh environments, the omission of mechanical parts, and the possibility to handle small volumes of liquid. These features make this method suitable for low viscous liquids that are radioactive, corrosive and at high temperature.

In this paper, we present a reduced-order modeling approach to study the Molten Salt Fast Reactor (MSFR). Our approach is nonintrusive and based on the proper orthogonal decomposition method. We include adaptivity in selecting the sampling points both in time and parameter space. Steady-state and transient analysis were both performed using the developed models. In the steady-state analysis, we capture the effect of 30 model parameters on the spatial distributions of fission power and temperature, and on the multiplication factor. The dimensionality of the fission power was reduced from the 104288 nominal dimensions in the physical space to 10 dimensions in the reduced space, whereas the temperature was reduced from 220972 dimensions to 3. The reduced model was then used for uncertainty and sensitivity study of the maximum temperature in the reactor and the multiplication factor. In the transient analysis, the reduced model captured the effect of perturbations in the flow rate of salt in the intermediate circuit on the fission power density and temperature. The reduced models were successfully tested on a set of points that were not part of the snapshots used during the construction stage.

We use a novel nonintrusive adaptive Reduced Order Modeling method to build a reduced model for a molten salt reactor system. Our approach is based on Proper Orthogonal Decomposition combined with locally adaptive sparse grids. Our reduced model captures the effect of 27 model parameters on k_eff of the system and the spatial distribution of the neutron flux and salt temperature. The reduced model was tested on 1000 random points. The maximum error in multiplication factor was found to be less than 50 pcm and the maximum L₂ error in the flux and temperature were less than 1%. Using 472 snapshots, the reduced model was able to simulate any point within the defined range faster than the high-fidelity model by a factor of 5×10⁶. We then employ the reduced model for uncertainty and sensitivity analysis of the selected parameters on k_eff and the maximum temperature of the system.

Accurate methods to solve the Reynolds-Averaged Navier-Stokes (RANS) equations coupled to turbulence models are still of great interest, as this is often the only computationally feasible approach to simulate complex turbulent flows in large engineering applications. In this work, we present a novel discontinuous Galerkin (DG) solver for the RANS equations coupled to the k−ϵ model (in logarithmic form, to ensure positivity of the turbulence quantities). We investigate the possibility of modeling walls with a wall function approach in combination with DG. The solver features an algebraic pressure correction scheme to solve the coupled RANS system, implicit backward differentiation formulae for time discretization, and adopts the Symmetric Interior Penalty method and the Lax-Friedrichs flux to discretize diffusive and convective terms respectively. We pay special attention to the choice of polynomial order for any transported scalar quantity and show it has to be the same as the pressure order to avoid numerical instability. A manufactured solution is used to verify that the solution converges with the expected order of accuracy in space and time. We then simulate a stationary flow over a backward-facing step and a Von Kármán vortex street in the wake of a square cylinder to validate our approach.

Performing accurate numerical simulations of molten salt reactors is challenging, especially in case of fast-spectrum designs, due to the unique physics phenomena characterizing these systems. The limitations of codes traditionally used in the nuclear community often require the development of novel high-fidelity multi-physics tools to advance the design of these innovative reactors. In this work, we present the most recent code developed at Delft University of Technology for multi-physics simulations of liquid-fueled fast reactors. The coupling is realized between an incompressible RANS model and an S_N neutron transport solver. The models are implemented in two in-house codes, based on the discontinuous Galerkin Finite Element discretization, which guarantees high-quality of the solution. We report and discuss the results of preliminary simulations of the Molten Salt Fast Reactor at steady-state and during a Total Loss of Power transient. Results prove our code has capabilities for steady-state and transient analysis of non-moderated liquid-fueled reactors.

In reactor core nodal analysis, the dependence of few-group, homogenized cross sections on the local physical conditions (i.e., the thermal-hydraulic state and material composition) is commonly represented via multivariate interpolation in parameterized libraries. In this paper, we propose a novel approach to model the spectral effects of changes in the moderator density and in the concentrations of diluted boron and xenon. This method is based on the spectral rehomogenization technique developed at Framatome and TU Delft to account for neighbor effects on the nodal cross sections. We compute on the fly the variation in the infinite-medium energy-collapsing spectrum from a nominal state to a perturbed condition (i.e., with different values of the aforementioned state parameters). The dependence of the microscopic and macroscopic cross sections on these three variables is thus resolved without the standard multidimensional interpolation. This strategy reduces substantially the computational burden of the lattice calculation, the cross-section library memory requirements, and the run time of the on-line cross-section reconstruction. The proposed approach is applied to a pressurized-water-reactor UO₂ fuel assembly at zero burn-up, covering a wide range of the values of the water density and of the concentrations of boron and xenon. Both normal and abnormal operating conditions are considered. We show that, in most cases, cross-section changes are predicted with an accuracy comparable to that of traditional interpolation. Higher errors (but reasonably small compared to the range of accuracy of nodal computational tools) are only found at very low moderator densities, typical of accidental conditions. As further validation of the methodology, we simulate a heterogeneous multiassembly configuration. With this benchmark problem, we prove that the method can reconstruct the spectrum variation between the real environment in a perturbed state and the infinite lattice in the nominal one, thus modeling simultaneously the non-separable spectral effects of local physical conditions and internodal neutron leakage.

The core of a Liquid Metal Fast Breeder Reactor (LMFBR) consists of cylindrical fuel rods that are wrapped by a helicoidally-wound wire spacer to enhance mixing and to prevent damage by fretting. It is known that the liquid metal close to the rod is forced to follow the wires, and that liquid metal further away from the rod crosses the wires (called: migratory flow). This work aims at gaining more insight into the physics behind migratory flow and to provide a model for its bending angle. To this purpose, the flow field in a 7-rods, wire-wrapped, hexagonal bundle with water is studied within the Reynolds number range of 4990–16330 by using Particle Image Velocimetry (PIV). Refraction of the light is minimized by using Fluorinated Ethylene Propylene (FEP), which is a refractive index-matching (RIM) material. These measurements confirm that liquid near the rod follows the helicoid path and bends cross-wise with respect to the wire further away from the rod. A theoretical model for the bending angle of the flow is derived from the Euler equations and shows that the bending is primarily caused by the pressure gradient field induced by the wire. The model shows a very good correspondence with the experimentally obtained PIV data. These findings improve our understanding of the physics at play in rod bundle flows with wrapped wires and can be of assistance in developing practical correlations for frictional pressure losses and heat transfer in such bundles.

The core of a Liquid Metal Fast Reactor (LMFR) has wires wrapped helicoidally around the fuel pins. This solution prevents damage of the cladding by fretting, and pushes the coolant flow through the gaps between the pins, thus, enhancing the heat exchange. The wires make the flow through the core very complex to understand and model. The goal of this study is, thus, twofold: it aims to obtain insight into the physics of the flow field near the wire and to provide high-fidelity data for benchmarking numerical simulations. Water at room temperature flowing inside a 7 rods, wire-wrapped hexagonal bundle is studied within a Reynolds number range of 4000−14000 using the Particle Image Velocimetry (PIV) technique. The measured quantities are the axial and lateral velocity components, and their root mean square. The measurement area is close to the wire around the central rod. Fluorinated ethylene propylene (FEP), which is a refractive index-matching (RIM) material, provides optical access to the measured region without disturbance of the light beam. The flow direction below the wire follows the wrapping path, as expected. However, if the flow is measured at the front of the wire, the streamlines bend in the opposite direction. The Euler equations applied to the streamlines show that this effect is caused by the pressure gradient across the wire. These findings deepens the physical understanding of the rod bundle flow in the presence of wire spacers, improving the LMFR designs. Moreover, the generated data will support validation of numerical codes for Gen-IV nuclear reactors.

Large-scale complex systems require high fidelity models to capture the dynamics of the system accurately. The complexity of these models, however, renders their use to be expensive for applications relying on repeated evaluations, such as control, optimization, and uncertainty quantification. Proper Orthogonal Decomposition (POD) is a powerful Reduced Order Modelling (ROM) technique developed to reduce the computational burden of high fidelity models. In cases where the model is inaccessible, POD can be used in a nonintrusive manner. The accuracy and efficiency of the nonintrusive reduced model are highly dependent on the sampling scheme, especially for high dimensional problems. To that end, we study integrating the locally adaptive sparse grids with the POD method to develop a novel nonintrusive POD-based reduced order model. In our proposed approach, the locally adaptive sparse grid is used to adaptively control the sampling scheme for the POD snapshots, and the hierarchical interpolant is used as a surrogate model for the POD coefficients. An approach to efficiently update the surpluses of the sparse grids with each POD snapshots update is also introduced. The robustness and efficiency of the locally adaptive algorithm are increased by introducing a greediness parameter, and a strategy to validate the reduced model after convergence. The proposed validation algorithm can also enrich the reduced model around regions of detected discrepancies. Three numerical test cases are presented to demonstrate the potential of the proposed POD-Adaptive algorithm. The first is a nuclear reactor point kinetics, the second is a general diffusion problem, and the last is a variation of the analytical Morris function. The results show that the developed algorithm reduced the number of model evaluations compared to the classical sparse grid approach. The built reduced models captured the dynamics of the reference systems with the desired tolerances. The non-intrusiveness and simplicity of the method provide great potential for a wide range of practical large scale applications.

Gap vortex streets characterise many industrial applications involving rod bundle flows, such as heat exchangers and nuclear reactors. These structures, known as gap vortex streets, may excite the structural components of the bundle to resonance, leading to fretting and fatigue. This work aims to measure these coherent structures and the resulting displacement and oscillation frequency of the neighbouring rod, to provide unique data for fluid-structure interaction studies and to develop a general correlation for estimating the coherent structure's wavelength. A water loop was built to host a hexagonal rod bundle. Fluorinated Ethylene Prophylene (FEP), a refractive index matching (RIM) material, was used to have undisturbed optical access in the area around the central rod. The flow was measured with Laser Doppler Anemometry (LDA) to detect coherent structures, while the vibrations were measured with a high speed camera. A new correlation for estimating the wavelength of the coherent structures is derived with dimensional analysis based on experimental evidence. The correlation is tested on different geometries: rectangular channels with single or half-rods, and two rod bundles, within the pitch-to-diameter ratio (P/D) range 1.02–1.2. Moreover fluctuations in the flow, given by the detected coherent structures, govern the structural response of the rod. The rod is excited to resonance if these fluctuations match twice the natural frequency of the rod.

This paper focuses on the freeze-plug, a key safety component of the Molten Salt Fast Reactor, one of the Gen. IV nuclear reactors that must excel in safety, reliability, and sustainability. The freeze-plug is a valve made of frozen fuel salt, designed to melt when an event requiring the core drainage occurs. Melting and draining must be passive, relying on decay heat and gravity, and must occur before the reactor incurs structural damage. In this work, we preliminarily investigate the freeze-plug melting behavior, assessing the influence of various design configurations and parameters (e.g., sub-cooling, recess depth). We used COMSOL Multiphysics® to simulate melting, adopting an apparent heat capacity method. Results show that single-plug designs generally outperform multi-plug ones, where melting is inhibited by the formation of a frozen layer on top of the metal grate hosting the plugs. The layer thickness strongly depends on sub-cooling and recess depth. For multi-plug designs, the P/D ratio has a negligible influence on melting and can therefore be chosen to optimize the draining time. The absence of significant mixing in the pipe region above the plug leads to acceptable melting times (i.e., <1000 s) only for distances from the core up to 0.1 m, considered insufficient to host all the cooling equipment on the outside of the draining pipe and to protect the plug from possible large temperature oscillations in the core. Consequently, we conclude that the current freeze-plug design based only on decay heat to melt is likely to be unfeasible. A design improvement, preserving passivity and studied within the SAMOFAR project (http://samofar.eu/), consists in accelerating melting via heat stored in steel masses adjacent to the draining pipe.

We propose a two-dimensional (2-D) modal approach for spatial rehomogenization of nodal cross sections in light water reactor analysis. This algorithm aims to synthesize the variation in the 2-D intranodal distributions of the few-group flux and directional net currents between the core environment and the infinite-lattice approximation. Assembly discontinuity factors are also corrected. The method is validated on a broad set of pressurized-water-reactor benchmark problems. Its accuracy is assessed on both nodal quantities and the reconstructed pin-by-pin flux and power distributions. We show that the errors in the effective multiplication factor and assembly-averaged fission power significantly decrease compared to the calculation with infinite-medium homogenization parameters. In most cases, an improvement is also found at the pin level. A thorough discussion follows, which addresses the use of the 2-D neutron current information to compute the transverse-leakage distribution for the transverse-integrated nodal equations, the potential dual application of the method for rehomogenization and dehomogenization, and the quantification of the contributions of various environmental effects (spatial, spectral, and cross energy-space) to homogenization errors.

Numerical simulations of fast MSRs constitute a challenging task. In fact, classical codes employed in reactor physics cannot be used, and new dedicated multi-physics tools must be developed, to capture the unique features of these systems: the strong coupling between neutronics and thermal-hydraulics due to the use of a liquid fuel, the effects on reactor kinetics induced by the precursors drift, the internal heat generation, and the shape of the core having no fuel pins as a repeated structure. In this work, we present a novel multi-physics tool being developed at TU Delft. The coupling is realized between an S_N radiation transport code (PHANTOM-S_N) and a RANS solver (DGFlows). Both in-house tools are based on a Discontinuous Galerkin Finite Element space discretization, characterized by local conservation, high-order accuracy, and allowing for high geometric flexibility. Implicit discretization in time is performed adopting Backward Differentiation Formulae. Cross sections are computed on an element base, starting from the local average temperature and a set of libraries generated at reference temperatures with Monte Carlo or deterministic codes. Comparison of the results obtained performing a suitable numerical benchmark created at LPSC/CNRS/Grenoble with those available in literature shows that the multi-physics tool is able to capture the unique phenomena characterizing fast liquid-fueled systems.