# Coupled neutronics and thermal-hydraulics using TRUST-NK for high performance computing in molten salt reactor simulation

Ayoub Ouazzani, Yannick Gorsse, Antoine Gerschenfeld

► To cite this version:

Ayoub Ouazzani, Yannick Gorsse, Antoine Gerschenfeld. Coupled neutronics and thermal-hydraulics using TRUST-NK for high performance computing in molten salt reactor simulation. EPJ Web of Conferences, 2024, EPJ Web Of Conferences: SNA + MC 2024, 302, pp.03007. 10.1051/epj-conf/202430203007.cea-05047484

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# Coupled neutronics and thermal-hydraulics using TRUST-NK for high performance computing in molten salt reactor simulation

Ayoub Ouazzani<sup>1,3,*</sup>, Yannick Gorsse<sup>2,**</sup>, and Antoine Gerschenfeld<sup>2,***</sup>

**Abstract.** Molten Salt Reactors display a strong coupling between neutronics and thermal-hydraulics for which various simulation tools have been developed. In this paper we describe TRUST-NK, a neutronics code based on TRUST, CEA's High Performance Computing (HPC) platform. TRUST-NK provides a multi-group diffusion solver with transport of delayed neutron precursors, for both steady-state and transient simulations. Being based on TRUST allows TRUST-NK to take advantage of the platform's HPC capabilities and to easily couple to CFD solvers provided by other derivatives of TRUST. After a brief description of the code, we present an example of application to a simplified MSR model.

## 1 Introduction

The Molten Salt Reactor (MSR) is a nuclear reactor concept in which the fissile material is dissolved in a molten salt at high temperature. The nuclear fuel is thus liquid and travels through the primary circuit. The use of a liquid fuel offers potentially considerable advantages in terms of intrinsic safety and operability, due to a strong and fast-acting density effect, but also, depending on the chosen fuel cycle, of waste management, resource management and economic competitiveness.

In the recent years, there have been many developments in the multi-physics modeling and simulation of MSRs. Many developed tools are based on OpenFOAM®. For example Gen-Foam enables the coupling of a multi-group diffusion solver for neutronics with thermal-hydraulics and thermal-mechanics solvers [1]. Other applications couple OpenFOAM® CFD solvers to a Monte Carlo code [2, 3]. Several studies concern the Molten Salt Fast Reactor (MSFR) [4], a reference design for MSRs. [5] provides a useful survey of multi-physics codes developed within the SAMOFAR project and compare them on a numerical benchmark. More recently, using a new coupling scheme, [6] coupled APOLLO3® with TRUST/TrioCFD, each providing a great number of choices of deterministic neutronics solvers and CFD solvers respectively.

<footnote n="1"><sup>1</sup>CEA DES/ISAS/DM2S/STMF/LMSF, Saclay, 91191 Gif-sur-Yvette, France</footnote>

<footnote n="2"><sup>2</sup>CEA DES/ISAS/DM2S/STMF/LMEC, Saclay, 91191 Gif-sur-Yvette, France</footnote>

<footnote n="3"><sup>3</sup>Institut Polytechnique de Paris, Route de Saclay, 91128 Palaiseau, France</footnote>

<footnote>*e-mail: ayoub.ouazzani@cea.fr</footnote>

<footnote>**e-mail: yannick.gorsse@cea.fr</footnote>

<footnote>***e-mail: antoine.gerschenfeld@cea.fr</footnote>
In this work, we present TRUST-NK, a new neutronics module based on the TRUST thermal-hydraulics platform [7]. In Sec.2 we provide a brief description of the code, the models implemented and their resolution schemes. In Sec.3 we provide an example of application to a MSR toy model.

<table><caption>Table 1: Symbols description.</caption><thead><tr><th>Symbol</th><th>Description</th><th>Unit</th></tr></thead><tbody><tr><td>φ</td><td>neutron flux</td><td>n.m<sup>-2</sup>.s<sup>-1</sup></td></tr><tr><td>k<sub>eff</sub></td><td>neutron multiplication factor</td><td>-</td></tr><tr><td>c</td><td>DNPs concentration</td><td>m<sup>-3</sup></td></tr><tr><td>ν</td><td>neutron velocity</td><td>m.s<sup>-1</sup></td></tr><tr><td>D</td><td>diffusion coefficient</td><td>m</td></tr><tr><td>Σ</td><td>cross-section</td><td>m<sup>-1</sup></td></tr><tr><td>ν</td><td>neutron production per fission</td><td>-</td></tr><tr><td>χ</td><td>neutron energy spectrum</td><td>-</td></tr><tr><td>β</td><td>delayed neutrons fraction</td><td>-</td></tr><tr><td>λ</td><td>DNPs decay constant</td><td>s<sup>-1</sup></td></tr><tr><td>q<sub>f</sub></td><td>power released per fission</td><td>W</td></tr><tr><td>P<sub>0</sub></td><td>total power</td><td>W</td></tr><tr><td>u</td><td>velocity</td><td>m.s<sup>-1</sup></td></tr><tr><td>T</td><td>temperature</td><td>K</td></tr><tr><td>ν<sup>l</sup></td><td>kinematic viscosity</td><td>m<sup>2</sup>.s<sup>-1</sup></td></tr><tr><td>Sc</td><td>Schmidt number</td><td>-</td></tr></tbody></table>

<table><caption>Table 2: Indices description.</caption><thead><tr><th>Index</th><th>Description</th></tr></thead><tbody><tr><td>g</td><td>neutron energy group</td></tr><tr><td>l</td><td>DNPs group</td></tr><tr><td>f</td><td>fission</td></tr><tr><td>r</td><td>removal</td></tr><tr><td>s</td><td>scattering</td></tr><tr><td>d</td><td>delayed</td></tr><tr><td>T</td><td>turbulent</td></tr></tbody></table>

<table><caption>Table 3: Operators abbreviations.</caption><thead><tr><th>Notation</th><th>Definition</th></tr></thead><tbody><tr><td>Pφ</td><td>∫dx Σ<sub>g</sub> q<sub>f</sub><sup>g</sup>Σ<sub>f</sub><sup>g</sup>φ<sup>g</sup></td></tr><tr><td>Lφ</td><td>-∇·(D<sup>g</sup>∇φ<sup>g</sup>) + Σ<sub>f</sub><sup>g</sup>φ<sup>g</sup> - Σ<sub>g'≠g</sub> Σ<sub>s</sub><sup>g'→g</sup>φ<sup>g'</sup></td></tr><tr><td>Fφ</td><td>χ<sub>f</sub><sup>g</sup>(1 - β) Σ<sub>g'</sub> ν<sub>g'</sub>Σ<sub>f</sub><sup>g'</sup>φ<sup>g'</sup></td></tr><tr><td>S<sub>d</sub>c</td><td>Σ<sub>l</sub>χ<sub>l</sub><sup>g</sup>λ<sub>l</sub>c<sub>l</sub></td></tr><tr><td>Ac</td><td>∇·(u<sub>c</sub>l) + λ<sub>l</sub>c<sub>l</sub> - ∇·([ν<sub>l</sub> / Sc + ν<sub>T</sub> / Sc<sub>T</sub>] ∇c<sub>l</sub>)</td></tr><tr><td>S<sub>f</sub>φ</td><td>Σ<sub>g</sub>β<sub>l</sub>ν<sub>g</sub>Σ<sub>f</sub><sup>g</sup>φ<sup>g</sup></td></tr></tbody></table>

# 2 Description of the code

## 2.1 The TRUST platform and TRUST-NK

TRIO_U Software for Thermal hydraulics (TRUST) is an open source thermohydraulic software package for CFD simulations developed at CEA<fnref n="1" /> [7]. It makes use of an object-oriented, inherently parallel C++ program design for High Performance Computing (HPC) simulations. It was originally made for incompressible single-phase and low Mach number flows, but today it allows the simulation of real compressible multi-phase flows. TRUST also provides a basis on which extensions implementing specific models can be developed.

<footnote n="1">¹Commissariat à l'énergie atomique et aux énergies alternatives, 91191 Gif-sur-Yvette, France.</footnote>
TRUST-NK is one such extension, and it implements a multi-group diffusion solver with transport of delayed neutrons precursors (DNPs) for both stationary and transient simulations. Its integration within the TRUST platform allows for direct internal coupling with any thermal-hydraulic (TH) model found in TRUST or its extensions. In particular all turbulence models implemented in TrioCFD [8], another TRUST-based code, are available. In comparison to more conventional approaches, this solution streamlines the coupling procedure and minimizes the computational problems that are sometimes associated with external code couplings, in addition to providing a better user experience. It also enables TRUST-NK to use the HPC capabilities of TRUST, making possible its use for computationally intensive nuclear reactor simulations.

The verification and validation of TRUST-NK is a work in progress. A verification against the numerical benchmark of [5], dedicated to multi-physics codes for MSR simulation, has been performed. But this will be left for a future publication.

## 2.2 The diffusion and DNP equations

*Time-dependent equations* TRUST-NK can solve the following multi-group diffusion equations with transport of DNPs:

$$ \frac{1}{v^g} \frac{\partial \phi^g}{\partial t} - \nabla \cdot (D^g \nabla \phi^g) = -\sum_r \phi^g + \sum_{g' \neq g} \sum_s \phi^{g' \to g} \phi^{g'} + \chi_f^g (1 - \beta) \sum_{g'} \nu_{g'} \sum_f \phi^{g' \to g} + \sum_l \chi_l^g \lambda_l c_l, \quad (1) $$

$$ \frac{\partial c_l}{\partial t} + \nabla \cdot (\mathbf{u} c_l) = \nabla \cdot \left( \frac{v^l}{\mathrm{Sc}} \nabla c_l \right) - \lambda_l c_l + \sum_g \beta_l \nu_g \sum_f \phi^g. \quad (2) $$

The cross-sections and diffusion coefficients are parameterized with temperature, which is provided by the TH problem.

When a turbulent flow model is used for the thermal-hydraulics, we get additional terms. For example if we are solving Reynolds-Averaged Navier-Stokes (RANS) equations, the unknowns are the *average* flux and DNP concentrations, and we get an additional turbulent diffusion term $\nabla \cdot \left( \frac{v^l}{\mathrm{Sc}_T} \nabla c_l \right)$ in the DNP equation, with $v_T^l(x)$ the turbulent kinematic viscosity provided by the turbulence model.<fnref n="2" />

*Steady-state equations* TRUST-NK also solves steady-state problems for a fixed power $P\phi = P_0$, introducing the multiplication factor $k_{\text{eff}}$ via $\nu_g \leftarrow \nu_g/k_{\text{eff}}$. For example:

$$ -\nabla \cdot (D^g \nabla \phi^g) = -\sum_r \phi^g + \sum_{g' \neq g} \sum_s \phi^{g' \to g} \phi^{g'} + \frac{1}{k_{\text{eff}}} \chi_f^g (1 - \beta) \sum_{g'} \nu_{g'} \sum_f \phi^{g' \to g} + \sum_l \chi_l^g \lambda_l c_l, \quad (3) $$

$$ \nabla \cdot (\mathbf{u} c_l) = \nabla \cdot \left( \left[ \frac{v^l}{\mathrm{Sc}} + \frac{v_T^l}{\mathrm{Sc}_T} \right] \nabla c_l \right) - \lambda_l c_l + \frac{1}{k_{\text{eff}}} \sum_g \beta_l \nu_g \sum_f \phi^g, \quad (4) $$

TRUST-NK also allows the simulation of the transport of decay heat precursors (DHPs), modeled by an equation similar to Eq.(2), and can solve the steady-state problem for a fixed total power $P\phi + P_{\text{DHP}} = P_0$.

*Boundary conditions* Concerning flux boundary conditions (BCs), TRUST-NK allows for: imposed flux, vacuum BC and symmetry BC. Albedo BCs are not yet implemented. But an albedo can be modeled in a simulation by adding a solid reflector around the salt subdomain.

<footnote n="2">²As is standard, the diffusion equation is not modified. This corresponds to the approximation that $\langle \Sigma_i(x)\phi(x) \rangle \approx \Sigma_i(\langle T \rangle(x))\langle \phi \rangle(x)$, where we used Reynolds-averaging.</footnote>
*Delayed neutrons fraction* TRUST-NK can give an estimation of the *circulating* delayed neutrons fraction $\beta_{\text{eff}}$ at any given time. In principle, this requires solving an adjoint coupled problem, introducing adjoint DNP concentrations [9]. But in TRUST-NK we use Eq.(5) in which the reaction rates are weighted with the *static* adjoint flux $\phi_0^*$, computed once for the initial static salt:

$$ \beta_{\text{eff},l} = \frac{\int dx \sum_g \phi_0^{*g} \chi_l^g \lambda_l c_l}{\sum_{l'} \int dx \sum_g \phi_0^{*g} \chi_{l'}^{g} \lambda_{l'} c_{l'} + \int dx \sum_g \phi_0^{*g} \chi_f^g (1 - \beta) \sum_{g'} \nu_{g'} \sum_f^{g'} \phi_{g'}^*} \quad (5) $$

The estimation is a good one to the extent that the shape of the adjoint flux does not vary significantly with the distribution of DNPs. This approximation has been observed to be a good one, but its rigorous assessment is left for future work.

## 2.3 Numerical resolution method

*Discretization* The diffusion equations in TRUST-NK can be discretized by one of the discretizations of the “PolyMAC” family. These are finite volume methods, integrated in TRUST, which generalize the MAC discretization [10] to general polyhedral meshes. On the other hand, the possible discretization methods of the TH and DNP equations are provided by TRUST/TrioCFD. Typically, as is the case for the RANS simulation below, we use the “VEF” method (cf. TrioCFD’s documentation [8] for a description).

*Fields interpolation between meshes* Different meshes for diffusion and TH/DNP equations will require, for example, an interpolation of temperature on the diffusion mesh, and an interpolation of nuclear power on the TH mesh. For efficient data exchange, TRUST-NK uses the MEDCoupling library [11] which can manage arbitrary intersections of meshes across processors. Standard interpolation schemes are used. For example, for overlapping meshes, an interpolation on cells of a cell-located intensive field is done by computing the average of the field values on intersecting cells weighted by the volumes of the intersections. For other cases and more details see the MEDCoupling documentation [11].

*Coupling schemes* Now, we describe the resolution strategies for the coupled problems, without DHPs for better clarity.

### 2.3.1 Steady-state: a modified eigenvalue problem

The steady-state problem on $(k_{\text{eff}}, \phi, c, \mathbf{u}, T)^3$ can be written thus:

$$ \left\{ \begin{array}{l} \mathcal{L}(T)\phi = \frac{1}{k_{\text{eff}}} \mathcal{F}(T)\phi + S_d(T)c, \\ \mathcal{A}(\mathbf{u}, T)c = \frac{1}{k_{\text{eff}}} S_f(T)\phi, \\ \mathcal{P}\phi = P_0, \\ \text{steady-state TH equations on } \mathbf{u} \text{ and } T, \\ \text{boundary conditions.} \end{array} \right. \qquad (6) $$

To solve this, we follow [6] by performing a pseudo-transient of the TH and DNP equations while regularly updating the values of $k_{\text{eff}}$ and $\phi$, as we will now describe. An initialization is first performed to obtain a first guess $(k_{\text{eff},0}, \phi_0, c_0, \mathbf{u}_0, T_0)$. We repeat the following for each $n$, until a convergence criterion is satisfied:

<footnote n="3">³In the following, it is implied that with the velocity **u** come pressure and other variables depending on the turbulence model used.</footnote>
a$_{n+1}$) We obtain c$_{n+1}$, u$_{n+1}$, T$_{n+1}$ by integrating over a duration d$_{tupdate}$ the TH and DNP equa-
tions starting from c$_{n}$, u$_{n}$, T$_{n}$, and with flux and multiplication factor fixed at φ$_{n}$ and
k$_{eff,n}$. (This is done numerically, possibly with several time steps.)

b$_{n+1}$) $\phi_{n+1}$ and $k_{\text{eff},n+1}$ are computed as $\phi_{n+1}^{\infty}$ and $\delta k_{\text{eff},n+1}^{\infty} k_{\text{eff},n}$, the convergent values of the following modified power iteration algorithm, where $\phi_{n+1}^{0} = \phi_{n}$:

$$
\left\{
\begin{array}{ll}
\text{solve for } \tilde{\phi}_{n+1}^{i+1}: & \mathcal{L}(T_{n+1})\tilde{\phi}_{n+1}^{i+1} = \frac{1}{k_{\text{eff},n}}\mathcal{F}(T_{n+1})\phi_{n+1}^{i} + S_d c_{n+1}, \\
\text{set:} & \delta k_{\text{eff},n+1}^{i+1} = \frac{\mathcal{P}\tilde{\phi}_{n+1}^{i+1}}{P_0}, \quad \phi_{n+1}^{i+1} = \frac{\tilde{\phi}_{n+1}^{i+1}}{\delta k_{\text{eff},n+1}^{i+1}}.
\end{array}
\right.
\qquad (7)
$$

The problem itself and the resolution algorithm are not standard. So we do not have yet any
strong theoretical guarantees. But it is easy to check that *if* the computation converges to
some $(k_{\text{eff},\infty}, \phi_{\infty}, c_{\infty}, \mathbf{u}_{\infty}, T_{\infty})$, these are solutions of problem (6).

2.3.2 Transient

For transient simulations, the diffusion equation is solved via an implicit Euler scheme. The assembled linear system can be solved with any method found in the PETSc library, thanks to the integration within TRUST. The TH and DNP equations can be solved with any combination of time scheme and linear system resolution method provided by TRUST and TrioCFD. Since the coupled equations are solved separately, it is in principle necessary to perform fixed-point iterations between equations within a time step. But reaching convergence generally takes only few iterations.

3 Application to a simplified MSR

3.1 Description of the model

In this section, we provide an example of application of TRUST-NK in the study of a sim-
plified MSR core. This toy model is designed to display a phenomenology richer than that
of the square cavity of the numerical benchmark of [5] (henceforth the CNRS benchmark),
in particular by displaying turbulence related phenomena, while still being relatively simple
and computationally affordable. It is not designed to describe a specific realizable reactor.
Rather, we want to use it to study generic physical phenomena which can be observed in
various MSR concepts, in particular in pool-type MSRs (e.g. turbulence-related power fluc-
tuations). For this reason, we do not pick a specific salt composition, but instead we choose
freely and directly thermal-hydraulic and neutronics parameters, either to get target physical
behavior or to simplify simulations, as will be explained.

*Geometry* The geometry corresponds to that of an MSFR-like reactor, truncated so as not to model the exterior part of the fuel circuit which contains the heat-exchanger. The inlet and outlet of the core are thus located at the sides of its top and bottom parts. The shape thus resembles that of a soda can as shown in Fig.1, where we also give its dimensions.

*Thermal-hydraulic properties* We choose fluid properties similar to those of the CNRS benchmark, with the exception of the kinematic viscosity which we take to be 4 × 10⁻⁵ m²s⁻¹ instead of 2.5 × 10⁻² to obtain a turbulent flow regime. (The value is still higher than realistic values: e.g. 2 × 10⁻⁶ m²s⁻¹ for the MSFR). The TH parameters are given in Tab.4. We choose
a nominal power of $P_0 = 1$ GWth. Given this value, we choose a flow rate of 4.25 m³/s to obtain a nominal temperature range $T_{max} - T_{min}$ of no more than 100 K, consistent with design choices in various MSR concepts. This flow rate, combined to a diameter of 2 m gives a Reynolds of ~ 70k. We neglect molecular diffusion compared to turbulent diffusion of DNPs. Concerning turbulent diffusion, there is so far no experimental data on turbulent Prandtl number and DNP's turbulent Schmidt number for MSRs. We follow [12] in taking, as an example, a value of 0.85 for both of them. A sensitivity analysis on these parameters could be considered for specific quantities of interest.

<figure><img src="image_2.png" /><figcaption>Figure 1: Illustration of the geometry of the can-shaped core (left; distances are in m) and the domain decomposition used (right).</figcaption></figure>

<table><caption>Table 4: Thermal-hydraulic parameters.</caption><thead><tr><td>Parameter</td><td>Value</td><td>Unit</td></tr></thead><tbody><tr><td>Density</td><td>2500</td><td>kg.m<sup>-3</sup></td></tr><tr><td>Kinematic viscosity</td><td>4 × 10<sup>-5</sup></td><td>m<sup>2</sup>.s<sup>-1</sup></td></tr><tr><td>Specific heat capacity</td><td>3075</td><td>J.K<sup>-1</sup>.kg<sup>-1</sup></td></tr><tr><td>Thermal conductivity</td><td>0.5</td><td>W.m<sup>-1</sup>.K<sup>-1</sup></td></tr><tr><td>Thermal expansion coefficient</td><td>2 × 10<sup>-4</sup></td><td>K<sup>-1</sup></td></tr><tr><td>Turbulent Prandtl number</td><td>0.85</td><td>-</td></tr><tr><td>Turbulent Schmidt number</td><td>0.85</td><td>-</td></tr></tbody></table>

*Nuclear properties* We use the same condensed cross-sections, diffusion coefficients, delayed neutron fraction and decay constants of the six energy groups and eight DNP groups used in the CNRS benchmark [5]. Since our goal is not realistic core design, we do not fine-tune the core parameters in order to make it critical on average. Instead, for Large Eddy Simulations (LES), to make the average thermal-hydraulic state as close as possible to the chosen operating point, the fission terms are divided with an adequate $k_{eff}$.<fnref n="4" />

*Neutronics boundary conditions* For simplicity, we do not add a reflector and impose vacuum boundary conditions everywhere.

<footnote n="4">⁴In the case of a laminar or unsteady RANS simulation, this is the $k_{eff}$ obtained by solving the steady-state problem at the nominal state. In the case of LES, since we cannot obtain a TH steady-state, we keep updating the $k_{eff}$ until it oscillates around a stable value, and arbitrarily choose a time at which to start the time resolution of the neutronics. This introduces a small, but acceptable, deviation of the average power from the target value. A better method, which has yet to be implemented, is to update the $k_{eff}$ using the average temperature field.</footnote>
*Inlet/outlet boundary conditions* At the outlet, we impose open boundary conditions with constant external values of: pressure, temperature and DNP concentration. This is adequate as long as we do not have important influx coming from the outlet. In that case the exact choice of constant values is not important. The inlet temperature is set to 900 °C.

We *do not* model, DNP re-circulation, from the outlet to the inlet.<fnref n="5" /> This makes for an underestimation of β<sub>eff</sub>. However the obtained value, as we will see, is still realistic (at least close to that of the MSFR).

For the velocity inlet, a modeling choice had to be made. To have (statistical) invariance under rotation around the z-axis makes for a conceptually simpler model and a lower cost of pre-calculation. For this reason, we impose that: on each vertical section, the velocity profile at the inlet is that of a fully developed 2D channel flow. For example, for RANS simulations using the *k*–*ε* model, we impose profiles of *u*, *k*, *ε* obtained once from a purely hydraulic pre-computation on a long 2D pipe. The 2D flow rate is chosen so as to obtain the target flow rate in the core. For LES, we add, on top of the average profile, fluctuations generated by a generalisation of TrioCFD’s synthetic turbulence BC. This will be detailed in future work.

## 3.2 Results of coupled RANS simulations

*Description of the computation* As an example, we show here results of a computation solving the steady-state problem coupling the diffusion and DNP equations to RANS equations. The base equations are the incompressible Navier-Stokes equations with a Boussinesq approximation for buoyancy. The turbulence closure model used is a *k*–*ε* model with wall treatment (cf. [13] §10.4, §11.7 and [8]). The TH and DNP equations are discretized over a uniform tetrahedral mesh with mesh size ~ 2 cm. The diffusion equation is discretized over a coarser mesh with mesh size 4–5 cm. The TH mesh has ~ 5 × 10⁶ elements. We use an implicit Euler scheme for both the TH and neutronics equations. The computation was run on 320 cpus for about an hour. The domain decomposition is obtained using the METIS library [14] integrated in TRUST. The decomposition is shown Fig.1.

### Results

<figure><img src="image_3.png" /></figure>

Figure 2: Temperature in °C (left) and velocity streamlines on top of a velocity magnitude map in m/s (right).

<footnote n="5">⁵This is would require using an external code, on top of TRUST-NK, to impose an inlet DNP concentration as a function of the outlet concentration.</footnote>
<figure><img src="image_4.png" /><figcaption>Figure 3: Total flux in n.m⁻².s⁻¹ (left) and delayed neutrons production in m⁻³.s⁻¹ (right).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 4: Effective multiplication factor (left) and delayed neutron fraction estimation (right).</figcaption></figure>

Fig.2 shows the temperature and velocity on the x = 0 section. We have a higher temperature on the low-velocity recirculation regions identified by the streamline loops. In particular, the salt reaches 990 °C at the top of the core, whereas it exits the core at ~ 930 °C, corresponding to a ΔT consistent with the chosen parameters of power, flow rate and heat capacity.

Fig.3 shows the total flux and delayed neutron production rate. The latter, similarly to the temperature, is higher in the low-velocity regions, and its peak is shifted towards the top of the core. However, given that it represent only a small fraction of the total neutron production, a shift of the total flux is not qualitatively noticeable. This is consistent with the value of β<sub>eff</sub> = 125 pcm ~ 0.1% obtained, as shown in Fig.4. This value is similar to that of the MSFR (see e.g. [3]), showing that the loss of DNPs is not too unrealistic. Fig.5 shows examples of DNP concentrations of four different groups. As can be expected, the smaller the decay constant (the longer the decay period), the farther the concentration peak is from the core.

In addition to probing the average phenomenology of the core, a RANS simulation helps estimate the mesh size required for LES to resolve enough turbulent motion. The order of magnitude of the mesh size needed to resolve ~ 80% of the turbulent spectrum, is given by the so-called Taylor micro-scale $\lambda_T$ (cf. [13] §7.3 for a rigorous definition of $\lambda_T$), which can be estimated as $\lambda_T = \sqrt{10vk/\epsilon}$. Alternatively, as is done in [3], it can be estimated as a
fraction $L_T/10$ of the turbulent lengthscale $L_T := k^{3/2}/\epsilon$ (cf. [13] §13.2). Fig.6 shows $\lambda_T$ and $L_T/10$. We observe that, except for the thin stripes starting at the inlet edges, in most of the core, the two estimations agree on a value no less than ~ 1 cm.

<figure><img src="image_6.png" /><figcaption>Figure 5: DNP concentration in $m^{-3}$ for four groups of different decay constants $\lambda$.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 6: Taylor micro-scale $\lambda_T = \sqrt{10vk/\epsilon}$ (left) and $L_T/10$ (right) in m.</figcaption></figure>
# 4 Conclusion

We have briefly described the models and some aspects of the numerical schemes of TRUST-NK, and showed how it can be used for multi-physics core analysis of MSRs. We have presented coupled RANS simulations only, because the realization of a genuine coupled LES simulation is a work in progress. Recent elements suggest that it is computationally within reach for the can-shaped toy model and we expect to have exploitable results soon.

As previously mentioned, there is room for improvement for TRUST-NK in the possibility of adding a solver for the real adjoint problem with adjoint DNP transport, or in the implementation of albedo boundary conditions. But, more importantly, there is also the possibility of implementing an $SP_N$ solver, since the structure of the $SP_N$ equations can also be handled by TRUST's toolbox. This would be an interesting upgrade since it would add the possibility of probing transport effects in MSRs with only a moderate additional computational cost.

# Authors' contributions

**Yannick Gorsse:** Software conception and development. **Antoine Gershenfeld:** Software development. **Ayoub Ouazzani:** Test case conception and simulation, writing.

# Acknowledgements

We would like to thank Elie Saikali and Adrien Bruneton for their help on the development of TRUST-NK. We would also like to thank André Bergeron, Nathan Greiner and Grégoire Allaire for their valuable contribution on the toy model conception and simulation.

# References

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