Lattice Boltzmann Method: Nuclear Reactor: Comparison
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Nuclear engineering requires computationally efficient methods to simulate different components and systems of plants. The Lattice Boltzmann Method (LBM), a numerical method with a mesoscopic approach to Computational Fluid Dynamic (CFD) derived from the Boltzmann equation and the Maxwell–Boltzmann distribution, can be an adequate option.

  • nuclear engineering
  • neutron
  • neutronics
  • heat
  • numerical methods
  • LBM
  • Boltzmann

1. Introduction

A nuclear (fission) reactor is a thermodynamic system that uses a great amount of energy generated by nuclear reactions of heavy radioactive materials, such as uranium, in a controlled and self-sustained nuclear chain reaction. One of the common outcomes is the use of the electricity generated by means of steam turbine generators, fueled by a working fluid heated by means of the energy obtained by the nuclear chain reaction.

The heat removal from the core of the nuclear reactor, where the nuclear fuel is “burned”, is not only the main operating principle of the circuit, but also the most relevant safety issue in this kind of system in which the core temperature can typically rise to hundreds of degrees (°C), and the capability to predict the behavior of the system is fundamental [1][17]. Heat extraction from the core uses the working fluid as a coolant, which can be either single-phase or multi-phase, where phenomena, such as boiling and nucleation, occur. Some commonly used working fluids are water, helium, or liquid metals, such as sodium, lead, or lead-bismuth eutectic. The working fluid removes heat from the core mainly by convection, natural or forced, albeit in some cases conduction is also relevant, e.g., if a liquid metal is used as the coolant. Commonly, the fluid is in a turbulent regime to enhance heat transfer coefficients, which means a flow characterized by high Reynolds number (Re). The interaction between the fluid and the nuclear fuel in such complex geometries (e.g., arrays of fuel bars in a coaxial cylindrical vessel, or fuel spheres in a pebble bed) determines the temperature gradients in the system.

Neutron emission from the atomic nuclei and the subsequent kinetics in the surrounding medium is a fundamental part of a nuclear reactor system. The neutron transport is a complex process that includes the emission, scattering, and absorption, leading to flux patterns in space which determines the probability to find a neutron in a certain place and with a specific energy. As known, neutron kinetics can be (at least in principle) characterized by solving the neutron transport integro-differential equations, or by the simplified neutron diffusion approximation.

Computer modeling and simulation plays an important role in nuclear power research and development in several fields, such as “nuclear safety research, optimization of technical and economic parameters, planning and support of reactor experiments, research and design of new devices and technologies, design and development of simulators for operating personnel training” [2][18]. For this reason, modern, accurate, and efficient numerical techniques are required to design, study, and simulate the physical phenomena that occur in complicate environments, such as nuclear power plants, and reducing the need of expensive and time-consuming experimental investigations: “one important part of nuclear reactor simulation is the benchmarking process is used to demonstrate reliability and repeatability in the simulation of real cases, for which data are well documented (from reactor operation or experiments)” [3][19].

2. Discussion on Lattice Boltzmann Method

Some works in the CFD field related with nuclear engineering do not need a relevant modification of the traditional LBM scheme [4][5][6][46,47,49]; however, to enhance the applications in CFD problems, it is common to adapt the LBM with high discretization schemes in 3D, such as D3Q27, and with turbulence models, LES being the most basic turbulence model [7][8][9][10][11][12][37,38,39,48,51,53]. More sophisticated turbulence models, such as VLES or LES–WALE [13][14][15][16][17][44,52,54,55,58], are also used, leading to better results than LES, and that of the use of the LBM (only without taking into account turbulence). The inclusion of thermal field models is relevant [18][19][12][16][20][42,43,53,55,56] to upgrade the classical isothermal LBM to a numerical method capable of simulating the heat transfer mechanisms present in the reactor (in particular, the convective heat transfer). The most commonly used is based on the secondary distribution approach. The inclusion of species transport is used to simulate chemical reactions, such as corrosion and deposition [21][22][23][24][25][40,41,45,59,60].

The LBM treats the complicated neutrons transport phenomena as simple linear calculations, and the use of angular discretization schemes leads to convergence. In early works, the solutions were obtained for isotropic scattering medium and with domain dimensions significantly larger than the mean free path. The multigroup solution was obtained and tested with some 1D, 2D, and 3D simulations, for both simple and complex geometries. The use of primitive cells is a commonly adopted strategy in both CFD and neutronic problems. It is possible to enhance the efficiency of the model for complex geometries using FV–LBM or mesh refinement schemes; thus, reducing the computational cost.

Recent research not only suggests the possibility to use the LBM in the nuclear engineering field, by the direct comparison of benchmark results obtained by other numerical methods, but also by the recovering of the macroscopic equations, such the Navier–Stokes equations (including the energy equation), the neutron transport equation and the neutron diffusion equation from the basic Lattice Boltzmann equation, thanks to a strong theoretical basis derived mostly from the Chapman–Enskog expansion. The LBM can, thus, be considered more than a computational automata, involving some physical principles that emerge from the statistical mechanics.

It is noted that each evolutionary step in the implementation of the LBM drove (and will drive) to validated engineering tools with flexible features as a single framework to solve fluid dynamics, convective, conductive, and radiative heat transfer, neutron transport, and more complex microphysics phenomena, such as chemical reactions and species transport. This shows that the LBM has an intrinsic capability to integrate powerful multi-physics simulation frameworks. This capability is enhanced with the integration of hybrid models, such as the Finite Volume method to adjust unstructured meshes [26][27][75,79], or the Boundary Immersed Method to calculate fluid-solid interactions [28][61].

To date, a complete multi-physics simulation of a nuclear reactor, including NT, heat transfer (with the integration of radiative, convective, and conductive mechanisms), CFD and transport-deposition mechanisms for a secondary chemical species has not yet been done, but the LBM is a candidate for a possible comprehensive framework to study the nuclear reactor as a whole. A great advance is presented in [29][84], coupling neutron transport and heat transfer in a single framework. Using a single framework, the LBM could overcome, partially, the limitation related to the use of several interfaces “Multi-physics assessment is subject to the use of several methods and, just as importantly, several interfaces between physical domains. For this reason, only limited credit may be claimed in safety analysis” [30][21].

Some commercial software implementing LBM are available, as well as some open-source libraries (see Appendix A for further details), but up to now, the general trend of research is to use and test self-developed codes, proving relatively simple implementation, but also the customization possibilities of this numerical method. To access all of the potential of the method, it is worth mentioning the possibility of implementing the LBM in parallel computing platforms (GPUs and clusters [19][31][32][43,81,82]) that not only enhance the spatial and temporal resolution of the results, but also optimize the computational cost with the inclusion of advanced mesh refinement schemes [33][34][35][76,77,78].

The application of the LBM in NE problems could have many more applications, such as the inclusion of multi-phase (liquid-gas) CFD simulations of BWR, or of the jet breakup (relevant after a core disruptive accident of a sodium-cooled fast reactor) [36][37][85,86], or some other topics (such as in [38][87]) concerning the development of a hybrid framework that couples microscale models and the LBM to simulate flows around reactive blocks (both chemical and nuclear) or a reactive flow: here, two different hybrid models were compared, i.e., the mean-field deterministic equation and the reaction kinetics coupled to LBM for the transport process, and the microscale kMC (kinetic Monte Carlo) coupled to LBM. “Apparently, the coupled kMC–LBM framework can fulfill the requirements of a unified multi-scale reactive flow simulator” [38][87].

Another possible application of the LBM regards the simulation of nanofluids to enhance the heat transfer properties (e.g., [39][40][41][42][88,89,90,91]), with the great advantage being that the simulation uses a distribution function for the fluid, a second one for the nanoparticles, and a third one for the thermal field. All three lattices are overlapped, with simple rules interacting between them, making it unnecessary to define complex geometrical conditions or interfaces between the nanoparticles and the main fluid.

Moreover, there are some characteristics of the Lattice Boltzmann Method that justify the use of the LBM over other numerical techniques, or even work together in a hybrid scheme:

  • The algorithmic implementation is simple.

  • The meshing technique is time-efficient, generating a structured lattice with relatively easy refinement schemes.

  • LBM is highly parallelizable; the collision term is local and then the calculations can be easily distributed between different cores or GPUs. The communication for the streaming process only uses neighbor nodes, and this leads to a highly parallelizable interface.

  • The multiphysics coupling is reduced to a single framework. The communication between a series of overlapping lattices, fluid lattice, thermal lattice, and neutronic lattice is done by a simple local scheme before the collision step, then the parallelization property is conserved.

  • Directly linked with the previous point is the possible inclusion of a multiphase flow using an additional distribution function, also applicable for nanofluids.

  • The thermal-hydraulic and neutronic results are validated by comparison with benchmarks.

  • The results are accurate for the standard methods.

  • The scheme is highly flexible and adaptable with hybrid schemes, using standard methods and turbulence models.

  • The use of the common framework for neutronics proposed in [29][84] enables the use of the LBM as a solver for the Sn, Sp3, or P1 approximations, maintaining the parallelization properties of the method.

  • The use of Sn–LBM, the spatial-angular parallelization mode, and the multi-GPU technique can speed up the NT–LBM, solving the neutron transport problem accurately and effectively without affecting the convergence characteristics [32][82].

3. Conclusions

CFD problems include the mixing process of the coolant, corrosion and deposition of impurities, flow acoustic resonances, hot and cold plume temperature gradients, and coolant with low Prandtl number (liquid–metal). Neutronic problems include neutron flux calculations and the calculation of the Keff eigenvalue (criticality problem).

In CFD problems, it is possible to adapt the LBM with turbulence models, high discretization schemes in three-dimensional lattices, thermal models, and mesh refinement strategies.

In neutronic problems, the LBM is adapted to solve the NT and/or ND equations with the inclusion of an angular discretization scheme, such as the DOM technique, which takes into account scattering effects. A common framework to implement Sn–LBM, Sp3–LBM, and P1–LBM models was developed.

The validation of the LBM results is commonly made by the comparison with experimental measurements of derived macroscopic variables (as friction factor, or velocity field for CFD) or with a direct comparison of the solutions of well-established benchmarks with other numerical methods, such as FVM (for CFD) or DOM (for neutronics).

The interest in applying the LBM to NE problems is increasing, and the related research items show this tendency, with an increase in the number of publications in the last years. Furthermore, the perspective to develop a multi-physics platform based on the LBM is foreseeable in the literature, and may include:

  • Two-dimensional (2D) and 3D high discretization schemes.
  • Turbulence models.
  • Thermal field models.
  • Radiative heat transfer.
  • Neutrons transport models.
  • Mesh refinement.
  • Species transport.

LBM is rapidly approaching to a mature development, and the potential of this numerical method is a strong reason to conduct a better-directed effort to validate its use as a highly accurate multiphysics simulation tool for the nuclear engineering field. It is fundamental to point out that theoretical research in LBM is going on, and is aiming (and succeeding) at resolving issues related to stability, speed, and model expansions.

Finally, it is worthy to remember that to access all of the potential of this numerical method, it is possible to exploit the intrinsic parallelization features of the LBM, using, for example, computing platforms, such as GPUs and clusters. This will not only enhance the spatial and temporal resolution of the results, but will also improve the computational effort. It is important to consider that the flexibility of the LBM opens the opportunity to develop hybrid methods to study complex processes, including microphysics (as deposition, oxidation, and chemical reactions).Tyhe complete paper can be found here:[43] 

Bocanegra Cifuentes, J.A.; Borelli, D.; Cammi, A.; Lomonaco, G.; Misale, M. Lattice Boltzmann Method Applied to Nuclear Reactors—A Systematic Literature Review. Sustainability 2020, 12, 7835. https://doi.org/10.3390/su12187835 [431]

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