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Nuclear reactor modeling has been shifting, over the last decades, towards fullcore multiphysics analysis due to the everincreasing safety requirements and complexity of the designs of innovative systems. This is particularly true for liquidfuel reactor concepts such as the Molten Salt Fast Reactor (MSFR), given their strong intrinsic coupling between thermalhydraulics, neutronics and fuel chemistry. In the MSFR, fission products (FPs) are originated within the liquid fuel and are carried by the fuel flow all over the reactor core and through pumping and heat exchange systems. Some of FP species, in the form of solid precipitates, can represent a major design and safety challenge, e.g., due to deposition on solid boundaries, and their distribution in the core is relevant to the design and safety analysis of the reactor. In this regard it is essential, both for the design and the safety assessment of the reactor, the capability to model the transport of solid FPs and their deposition to the boundary (e.g., wall or heat exchanger structures). To this aim, in this study, models of transport of solid FPs in the MSFR are developed and verified. An Eulerian singlephase transport model is developed and integrated in a consolidated multiphysics model of the MSFR based on the opensource CFD library OpenFOAM. In particular, general mixedtype deposition boundary conditions are considered, to possibly describe different kinds of particlewall interaction mechanisms. For verification purposes, analytical solutions for simple case studies are derived ad hoc based on the extension of the classic Graetz problem to linear decay, distributed source terms and mixedtype boundary conditions. The results show excellent agreement between the two models, and highlight the effects of decay and deposition phenomena of various intensity. The resulting approach constitutes a computationally efficient tool to extend the capabilities of CFDbased multiphysics MSFR calculations towards the simulation of solid fission products transport.
Liquidfuel reactor concepts have gained renewed interest over the last years. Among them, the Molten Salt Reactor (MSR) and, in particular, the fastspectrum MSFR (Molten Salt Fast Reactor) has obtained a prominent role thanks to the selection as one of the Generation IV reference technologies (
In the MSFR, fission products (FPs) that are originated within the fuel as the result of fission reactions are not retained by solid fuel elements and are therefore free to be carried by the liquid fuel flow within the primary circuit. The presence of mobile FPs represents a major design and safety challenge, as some of them are not expected to form stable compounds with the constituents of the fuel salt mixture (
A preliminary implementation is here described and verified against analytical solutions for simplified test cases. The analytical solutions are derived from the wellknown
The paper is organized as follows. The adopted multiphysics approach is briefly described in Section 2.1. In Section 2.2, the proposed FPs transport model implemented in the OpenFOAM solver is described in detail. The analytical solution against which the OpenFOAM FPs transport model is verified is presented in Section 2.3.
The OpenFOAM library, based on standard finitevolume methods for CFD calculations, is used to develop the numerical solver used in the present work. Originally developed for the transient analysis of the MSFR (
Continuity, momentum and energy (in temperature form) conservation equations are expressed in a singlephase incompressible formulation:
Pressurevelocity coupling is performed through the standard SIMPLE/PISO algorithms (
The multigroup diffusion model is adopted for neutron flux calculations (
Due to the circulating nature of the fuel, transport equations are formulated also for delayed neutron and decay heat precursors. The transport equation for the concentration of delayed neutron precursors of the
A discussion on the diffusion coefficient
Group constants are adjusted as functions of local temperature around reference values to account for Doppler and fuel density effects. For a generic neutron reaction
Similarly to other transported scalar quantities, each fission product specie is modelled as a continuous scalar concentration field subject to advection, dispersion and decay mechanisms:
As previously mentioned, this modeling choice is motivated by the need to limit the overall complexity and computational requirements of the MSFR, and to easily integrate such models in stateoftheart MSFR codes. The singlephase Eulerian approach can still represent a valid approximation, provided that some conditions are met. Theoretical and experimental analysis has suggested that Fick’s diffusion law only applies when inertial effects are negligible, and that particles inertia plays an increasingly dominant role in transport mechanisms as particles size increases (
As regards the particle diffusivity, it is commonly assumed that the diffusivity coefficient
Besides particle transport in the bulk flow, transport mechanisms which lead to deposition need to be addressed separately. First of all, when particlewall interaction in the boundary layer is considered, a variable diffusion coefficient can be introduced to model hydrodynamic interactions between particles and solid walls (
Moreover, to model deposition mechanisms and formulate appropriate boundary conditions for the particle concentration field, one possible approach is the inclusion in the transport equation of a particlewall interaction forcing term based on an interaction potential energy
When the energy potential
Decoupling of the transport and deposition problems: solutions for the wall region (
The coefficient
In the general threedimensional case, the wall boundary condition can be written as
In this Section the results of the verification of the implemented FP transport model are described. A simple test case with an analytical solution has been chosen. A twodimensional channel between parallel plates is considered (
Parallel plates geometry: with respect to the flow, the
The problem here considered resembles the wellknown Graetz problem, for which different solutions are available in the literature. An exhaustive treatment of the Graetz theory applied to particle transport problems is given by
Analytical solutions of the momentum equation can be found only for simple steadystate fullydeveloped laminar flow problems. In such a case, the well known parabolic solution reads
The boundary value problem then becomes, in explicit cartesian coordinates,
Longitudinal diffusion is neglected to allow for separation of variables. This assumption is reasonable in all cases where diffusion is negligible compared to advection, i.e., if
For the following discussion, it is convenient to assume that the distributed source can be expressed as
On the other hand, when a distributed source is present, the inlet contribution is forgotten as the fullydeveloped concentration profile is attained and therefore a more meaningful choice should be based on the relative intensity of generation and removal mechanisms. When radioactive decay is dominant, a good definition reads
When decay is negligible, the reference concentration
These values are useful to identify correct scaling with respect to the dominant removal mechanisms. Similar expressions are easily found when solving for the centerline concentration in fullydeveloped profiles with uniform source.
Solutions of the boundary value problem
The functions
The eigenvalue problem is therefore stated as
It is easily verified that
whose general solution (back in terms of
The separated
To obtain this last form,
In this Section, results of the verification of the implemented models against the analytical solutions are presented. In Section 2.3.3 it is shown that, if the effect of desorption is negligible compared to decay, the
Dependence of the 1st eigenvalue
Dependence of the 2nd eigenvalue
Dependence of the 5th eigenvalue
Dependence of the 10th eigenvalue
In the following, results from the comparison between the OpenFOAM model described in Section 2.2 and the corresponding analytical solutions are discussed. To highlight the role of decay and deposition phenomena, the selected parameters are
Values of
Case n°  1  2  3  4  5  6  7  8  9 


0.1  0.1  0.1  1.0  1.0  1.0  10  10  10 

0.1  1.0  10  0.1  1.0  10  0.1  1.0  10 
Model parameters used in all verification cases.






500  1  500  0  0.025 
To simplify the analysis, the inlet concentration is set to zero (
Therefore in dimensionless form the solution does not depend on the source term average value, but only on its shape. For the present analysis, a cosineshaped source term has been selected to resemble the typical shape of fission rate profiles in simplified reactor geometries such as the one here considered:
Concentration profiles obtained for the nine test cases (
Comparison of concentration profiles obtained with the proposed transport model implemented in OpenFOAM (
Comparison of concentration profiles obtained with the proposed transport model implemented in OpenFOAM (
Comparison of concentration profiles obtained with the proposed transport model implemented in OpenFOAM (
Results show excellent agreement between the proposed transport model and the analytical solutions, proving a successful verification of the implemented transport models in OpenFOAM. The influence of decay is evidenced from the decrease in concentration profiles from
We finally report here some brief computational information regarding the verification. All simulations were performed on a structured orthogonal mesh, constituted by 5 × 10^{4} hexahedral volumes, with respectively 500 and 100 divisions on the longitudinal and transversal directions. All 9 cases shown similar convergence behavior, with approximately 5 × 10^{4} pseudotransient iterations needed to ensure tight convergence for the particle concentration field. Computational times are comparable among all cases, showing no significant dependence on the physical parameters within the selected range (
Computational times for the verification cases. 5 × 10^{4} iterations are performed on 5 × 10^{4} volumes with eight CPUs.
Case n°  1  2  3  4  5  6  7  8  9 

Time (s)  2037  2684  2102  2312  2003  2636  2000  2036  2119 
In this paper, the preliminary development of transport models for the solid fission products in the MSFR was discussed. Simplified transport models based on a Eulerian singlephase framework were implemented in consolidated MSFR multiphysics simulation tools based on the opensource finitevolume library OpenFOAM. The resulting model has been verified against analytical solutions for simplified test cases, based on an extended version of the Graetz problem. Results show the good agreement between the two models, proving the correct implementation of the transport and deposition mechanisms considered and the capability of OpenFOAM in treating coupled deposition and decay phenomena of different relative intensities, albeit on a simplified reference problem. The proposed approach constitutes a computationally efficient framework to extend the capabilities of CFDbased multiphysics MSFR calculations towards the simulation of solid fission products transport.
The analytical model used for verification has been developed specifically for this application. The simultaneous presence of distributed internal generation, radioactive decay and mixed deposition boundary condition in a Graetz problem represents an original contribution, and the resulting analytical model could therefore constitute a useful benchmark for future developments of the FPs migration model and for other similar MSFR applications. Future work will include the analysis of the integration of FPs transport in a full reactor simulation. In this regard, turbulent transport in more complex geometries may lead to large concentration gradients close to walls, with the need for mesh refinement. The extension to reactor simulation might therefore lead to numerical and/or computational issues to be addressed. Among other possible limitations of the current modelling approach, we highlight the adoption pure concentrationdriven diffusive transport. As already mentioned, such approximation is strictly valid only for particles of very small size. Further study will be needed to assess the validity of such assumption for MSFR applications and to possibly extend the methodology to more advanced particle transport models.
The datasets presented in this study can be found in online repositories. The names of the repository/repositories and accession number(s) can be found in the article/Supplementary Material. The data that support the findings of this study are openly available in Zenodo at
All authors contributed to conception and design of the study. AD performed the theoretical and technical analysis and wrote the first draft of the manuscript. SL, FG and AC supervised the analysis. All authors contributed to manuscript revision, read, and approved the submitted version.
This project has received funding from the Euratom research and training programme 2014–2018 under grant agreement no. 847527.
The content of this paper does not reflect the official opinion of the European Union. Responsibility for the information and/or views expressed therein lies entirely with the authors.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The Supplementary Material for this article can be found online at:
reference
particle (vol.) concentration
nondim. particle (vol.) concentration
nondim. deposited particle (surf.) conc.
nondim. inlet particle (vol.) concentration
ref. particle (vol.) concentration
deposited particle (surf.) concentration
specific heat capacity
inlet particle (vol.) concentration
laminar particle diffusivity
particle diameter
turbulent particle diffusivity
effective particle diffusivity
bulk effective particle diffusivity
Damköhler number
channel aspect ratio
particle diffusivity correction factor
gravitational acceleration
channel halfwidth
Boltzmann constant
effective multiplication factor
channel length
confluent hypergeometric function
pressure
Péclet number
Prandtl number
turbulent Prandtl number
vol. energy source
Reynolds number
particle (vol.) source
average particle (vol.) source
nondim. particle (vol.) source
unitaverage particle (vol.) source
Schmidt number
turbulent Schmidt number
Sherwood number
temperature
reference temperature (buoyancy)
reference temperature (crosssections)
longitudinal velocity comp.
velocity
nondim. longitudinal velocity comp.
longitudinal velocity comp.
velocity
maximum profile velocity
avg.
longitudinal coordinate
nondim. longitudinal coord.
transversal coord.
nondim. transversal coord.
fission yield of transported fission product
laminar thermal diffusivity
turbulent thermal diffusivity
effective thermal diffusivity
total delayed neutrons fraction
(square root of)
vol. thermal expansion coeff.
desorption number
desorption rate constant
wall region thickness
deposition velocity
particle decay constant
fluid density
reference
avg.
laminar kinematic viscosity
effective kinematic viscosity
turbulent kinematic viscosity
particlewall interaction potential