© B. Babcsány et al., Published by EDP Sciences, 2025

1. Introduction

The supercritical water-cooled reactor (SCWR) technology is one of the six types of nuclear energy systems selected by the Generation IV International Forum (GIF) worthy for further research and development (R&D) to meet the specific technology goals of Generation IV reactors within the areas of sustainability, economics, safety and reliability, proliferation resistance and physical protection. The coolant of the SCWRs is light water at supercritical pressure, i.e. at a pressure higher than 22.1 MPa. The advantage of SCWRs compared to conventional water-cooled reactors lies in the planned once-through steam cycle with higher steam enthalpy at the turbine inlet, which leads to increased efficiency and decreased capital and fuel costs [1]. The aim of reducing the investment costs and risks, the better integrability to smaller electricity grids, and the possibility of providing alternative product streams have also placed small modular reactors in the focus of recent R&D activities [2].

To take advantage of the combined benefits of the SCWR and SMR technologies, an international project called Joint European Canadian Chinese development of Small Modular Reactor Technology (ECC-SMART) was launched in 2020 with funding from an EU/EURATOM/H2020 grant [3]. The Institute of Nuclear Techniques of the Budapest University of Technology and Economics (BME NTI) is actively involved in the development of a supercritical water-cooled small modular reactor (SCW-SMR) concept as a consortium member in the framework of the ECC-SMART project. The primary goal of the ECC-SMART project is to evaluate the feasibility and identify the safety features of an intrinsically and passively safe small modular reactor cooled by supercritical water [3]. The main objectives of the project include defining the design requirements for the future SCW-SMR technology, developing the pre-licensing study and guidelines for the demonstration of the safety in the further development stages of the SCW-SMR concept, including the methodologies and tools to be used, identifying the key obstacles for the future SMR licencing and proposing a strategy for this process [3]. Research within the ECC-SMART project is organised into field-specific work packages (WPs). BME leads the reactor physics work package (WP4) and significantly contributes to the thermal hydraulics research conducted under WP3.

The SCW-SMR pre-conceptual design, currently under development within ECC-SMART and detailed in Section 2, was initially proposed by Schulenberg and Otic in [4]. This design heavily incorporates lessons learned from previous projects focused on designing a High-Performance Light Water Reactor (HPLWR) [5]. In addition to the material challenges that the SCWRs face – such as high temperatures, irradiation and corrosive environments – the hot-channel factor is another concern. Besides core loading pattern optimisations, to limit local enthalpy rises, thus coolant and cladding temperature peaks within the core, compared to a single heat-up step of conventional nuclear reactors, Oka proposed first a two-pass core concept [6], which was later adopted in the HPLWR design but with an additional heat-up step, resulting in three heat-up stages. In the HPLWR design, this three-stage heat-up arrangement significantly reduced the maximum cladding temperature; however, calculations indicated that the cladding was still being overheated in regions with the highest local power density [4]. Consequently, it was recommended to increase the number of heat-up stages further to improve mixing and prevent the formation of hot spots. Hence, in the SCW-SMR concept under development within the ECC-SMART project – similar to the design applied in supercritical coal-fired power plants – the number of heat-up steps has been increased to seven [4]. Generally, a higher number of heat-up steps leads to a lower peak coolant temperature at a given average core outlet temperature. This results in less stringent material requirements, although it also increases the complexity of the core design [1].

To support the complex pre-conceptual design of the SCW-SMR, assembly and core models were developed, coupled and applied by the ECC-SMART consortium members for multipurpose modelling and design work using various tools, including but not limited to the Apros thermal hydraulics system code [7], the Ansys CFX CFD software package [8], the Serpent 2 Monte Carlo code [9], and deterministic reactor physics codes, such as SPNDYN [10]. A summary of the activities performed by BME NTI has recently been published in [11]. The coupling of reactor physics and thermal hydraulics calculations was essential in the case of the pre-conceptual design work of the SCW-SMR concept. The temperature and density distributions of the moderator and coolant, and the temperature distributions of the fuel and other structural materials are strongly influenced by the power distribution within the core, while the neutronics behaviour – mainly due to the large coolant temperature and density changes and inhomogeneous fuel temperature distribution – strongly depends on the thermal hydraulics state of the core. In addition, the thermophysical properties of supercritical water exhibit rapid fluctuations in the pseudocritical region, which significantly impacts the power distribution in SCWRs. This paper presents research focusing on establishing the methodology of online coupling between the Apros thermal hydraulics system code and the SPNDYN in-house deterministic reactor physics code and its testing. The coupled code system was used for modelling the nominal, equilibrium behaviour of the SCW-SMR assuming two different core layouts at a beginning of cycle (BoC) reactor state. One of the core loading patterns modelled and suggested for the first cycle of the supercritical water-cooled small modular reactor consists of solely 5.0 at.% U²³⁵ enriched fuel assemblies, while the other is an optimised core map with three differently enriched UO₂ assemblies with 5.5 wt.%, 7.8 wt.% and 10 wt.% U²³⁵ enrichment, respectively. In the case of SPNDYN, a two-group diffusion model was used with parametrised group constants based on a Serpent 2 group constant database. To verify the coupled code system, the results obtained were compared to reference Apros-Serpent 2 calculations using an offline coupling method (see details in [12, 13]).

For the design and analysis of SCWRs, the coupling of thermal hydraulics and reactor physics calculations is not unprecedented. Over the last few decades, advanced SCWR concepts have been developed primarily in Japan, Canada and the European Union, leading the SCWR-related research within the GIF according to the cooperation agreements.

The Japan SCWR concepts developed at the University of Tokyo include thermal and fast spectrum reactors, specifically the Super Light Water Reactor (LWR) and Super-Fast LWR concepts [6]. For the design of these reactors, coupled core design calculations at fuel assembly level were carried out using a single-channel thermal hydraulics code (SPROD) and a three-dimensional finite difference reactor physics code, SRAC. The peak cladding temperature prediction within the fuel assemblies was based on a pin power reconstruction method and sub channel analysis [6]. Fuel behaviour studies were conducted using the FEMAXI-6 code, while for transient and accident simulations, various versions of an in-house code, named SPRAT, were developed and applied in the design work, along with a code named SCRELA for reflooding analysis. In these simulations, the average and hot fuel channels were differentiated in the nodalisation of the Super LWR/Super-Fast LWR models [6].

Canadian SCWR research activities focus on developing a CANDU-SCWR with a pressurised tube design. For coupled core calculations, recent Canadian research involved the development of a coupled neutronics and thermal hydraulics scheme based on DRAGON5, DONJON5 and CATHENA in evaluating the safety of the latest CANDU-SCWR design [14]. DRAGON5 and DONJON5 are deterministic codes; for the DONJON5 diffusion calculations, the group constants are generated and parametrised using the DRAGON5 lattice code. The diffusion calculations of the full-core simulation code DONJON5 rely on finite-difference and finite-element numerical techniques available in TRIVAC5 [15]. To model the CANDU-SCWR in CATHENA, an 84-channel thermal hydraulics model was developed, taking advantage of the quarter symmetry of the reactor core, hence providing a uniquely detailed simulation tool [16].

The European research on developing an HPLWR design began in 2000 [5]. In the deterministic core design activities, the MULTICELL transport code was applied for group constant database generation for KARATE diffusion calculations, coupled with SPROD, a one-dimensional, parallel channel thermal hydraulics code [5]. For transient simulations necessitating reactor physics and thermal hydraulics coupling – such as reactivity-initiated accident (RIA) simulations – the ATHLET/KIKO3D and SMABRE/TRAB-3D code systems were employed [5]. KIKO3D and TRAB-3D are three-dimensional nodal diffusion codes developed in Hungary and Finland, respectively. KIKO3D was coupled to ATHLET, while TRAB-3D was coupled to SMABRE to accurately model the thermal hydraulics behaviour of the HPLWR system [5]. To provide input parameters for the neutron kinetics calculations, KIKO3D was extended with subroutines that calculate the parameterised response matrices as functions of coolant density, fuel temperature, burnup, and concentrations of xenon and samarium based on the cross sections obtained from MULTICELL [17]. To address asymmetric transients, the ATHLET core nodalisation was optimised based on the locations of the fuel assemblies within the core, resulting in a definition of 16 super-channels and an option to consider one hot channel in addition to the average channel for each heat-up step [18]. For TRAB-3D calculations, the parameterised cross sections and relevant subroutines from KIKO3D were reimplemented and used [19]. In SMABRE, the most detailed model nodalisation features separate flow channels for all 156 fuel assemblies and corresponding moderator channels within the core [19].

Due to the seven coolant heat-up stages in the SCW-SMR design, suggested to reduce further the peak coolant temperatures, a sophisticated thermal hydraulics system model was necessary. In the Apros model of the SCW-SMR developed by BME and presented in detail in Section 3.1, in each heat-up stage, the hot, cold and average coolant channels are treated separately, corresponding to five-five fuel assemblies with the highest and lowest powers, and to the rest of the assemblies, respectively, resulting in a total of 21 coolant channels. The axial power distributions, as the heat sources within the coolant channels, are supplied by reactor physics calculations. As an offline option, coupled equilibrium state calculations can be performed iteratively using Apros and a full-core Serpent 2 SCW-SMR model (see Sect. 3.3) that can provide a reference for online coupled, deterministic equilibrium state core calculations. While the offline coupled Apros and Serpent 2 simulations can provide reliable results (see also [12, 13]), such simulations are quite resource-intensive and, thus, computationally expensive for coupled core design activities. Besides, our future aim is to contribute to the safety evaluation of the SCW-SMR design by developing a coupled thermal hydraulics and reactor physics tool that is able to perform transient simulations within acceptable time and with relatively cheap computational resources. Therefore, we decided to implement online coupling between Apros and SPNDYN (see in detail in Sect. 3.2). SPNDYN has two different finite element modules: one is based on a continuous Galerkin, while the other is based on a hybrid finite element method [10]. In the present work, the continuous Galerkin finite element (CGFEM) module, which does not have any geometric modelling restrictions, was used for two-group, steady-state diffusion calculations. The geometric model of the reactor can be set up, and meshes can be generated in Gmsh, an open-source three-dimensional finite element mesh generator with a built-in computer-aided design (CAD) engine and post-processor [20]. SPNDYN also allows for higher-order transport calculations (the so-called SP₃ method), along with transient simulations having a direct time-integration scheme implemented in the code; however, applying these code capabilities was not within the scope of the present research. By coupling and verifying the Apros-SPNDYN code system, its applicability for SCWR design activities was shown, and we have also taken our first step in the direction of having a modelling tool for RIA simulations of the SCW-SMR core concept.

As for the structure of this paper, Section 2 presents the pre-conceptual design of the SCW-SMR, while Section 3 first covers the developed thermal hydraulics and reactor physics models along with their coupling methodology, based on a Python communication and control module written for this purpose and on the built-in Python module of Apros. The methods used to generate the mesh and parametrised group constants required as inputs to SPNDYN are then presented in detail. In Section 4, sensitivity analyses about mesh refinement and Apros runtime are presented, and the coupled calculation results are discussed and compared with references obtained with the Apros-Serpent 2 code system. We conclude our paper with lessons learned and an outlook on our future plans for further method and code development.

A part of this work was carried out in the frame of an MSc project [21], and preliminary results have been presented in [22]. This paper is related to applying the SPNDYN finite element reactor physics code developed within the framework of the PhD research conducted by B. Babcsány and shortlisted to receive the PhD Award of the High Scientific Council of the European Nuclear Society in 2024.

2. The pre-conceptual SCW-SMR design suggested in the ECC-SMART project

Based on the experiences gained with the HPLWR design, an innovative concept of a supercritical water-cooled small modular reactor was suggested in the framework of the ECC-SMART project with more favourable safety characteristics compared to the HPLWR [4]. Smaller peak cladding temperatures were achieved with an increased number of heat-up stages along with coolant mixing between the stages in the spiral returning channels situated around the core, while passive residual heat removal by natural convection was enabled by a core design with horizontal fuel assemblies, where the coolant flows only horizontally or upwards. The SCW-SMR core configuration has 400 horizontally aligned fuel assemblies arranged in a 20 × 20 square lattice (see Fig. 1). The fuel assemblies are distributed over the seven heat-up stages, where the number of assemblies per stage from the bottom to the top of the core is increasing to compensate for the additional pressure drop within the core caused by the decreasing density and, hence, increasing flow velocity along the coolant flow path. The first three stages at the bottom have only two rows of fuel assemblies each, the fourth and fifth stages have three rows of fuel assemblies each, while the sixth and seventh stages, located at the top of the core, each have four rows of fuel assemblies.

Fig. 1. Top and side views of the reactor pressure vessel (RPV) internals of the SCW-SMR design (with the top cover (left), front plate (center) and outer cylinder removed (right)) [4].

The fuel assembly design is taken from the HPLWR: each fuel assembly contains 40 fuel rods with helical spacers (wire wraps) to improve coolant mixing within the fuel assembly and an internal water channel (moderator box) to compensate for the insufficient moderation of the coolant (see Fig. 2). The fuel rod outer diameter is 8 mm, while its active length is 1680 mm. The moderator box and the assembly have a closed, three-layer outer wall (see also Fig. 2) with a thin layer of stainless steel at the coolant side to ensure corrosion resistance, an insulation layer of yttria-stabilised tetragonal zirconia polycrystal (YTZP) in the middle to minimise heat loss to the moderator, and a thick layer of Zircaloy-4 on the moderator side to minimise neutron absorption and provide structural stability.

Fig. 2. Cross section of the SCW-SMR assembly (left), and the dimensions of the three-layer walls of the assembly and moderator boxes (central and right): black – stainless steel liner, yellow – insulation, grey – Zircaloy-4 [23, 24].

The SCW-SMR concept has a thermal power of 290 MW, an outlet pressure of 25 MPa and a total coolant mass flow of 145 kg/s. The cooling and the moderation are provided by the same medium (light water) flowing multiple times through the core. This leads to a complex flow path in the reactor vessel shown in Figure 3. (Fig. 3 also shows the numbering applied in the following flow path description). The feed water enters the reactor vessel at the inlet nozzles at a temperature of 280°C (1), and then the majority of it flows downwards through the down comer (2) into the lower plenum (3). However, the reactor flange contains some orifices (4), hence providing a bypass route directly to the upper plenum (5). From the lower plenum, the feed water, with a primary function of providing moderation, flows upwards in two parallel paths. The main part enters the core and flows through the inter-assembly space (upward moderator channel) (6), while the rest flows through the reflector space formed on the sides of the core (7). These paths mix again above the core (8), from where the moderator continues towards the upper plenum (9) and mixes with the bypass as well. Thereafter, the flow turns downwards towards the moderator channel inlet plenum (10) and enters the moderator boxes of all seven stages simultaneously from the side of the core. The moderator is collected in the outlet moderator plenum on the opposite side (11). Then, it continues downwards, passing through two channels (12) under the core before returning to the inlet side again (13). In this region, the temperature of the water reaches approx. 310°C. From now on, as coolant, its primary function is to provide adequate heat removal and cooling of the fuel rods while flowing through all seven stages, one after the other. The coolant first passes through the fuel assemblies of the first heat-up stage from the inlet coolant plenum (13) to the outlet (14). From there, the coolant flows through the spiral flow channels (15) – formed around the core and connecting the stage outlets with the inlet of the next stage around the reflector space – to reach the second stage inlet plenum. The mixing plena and the spiral channels ensure the proper coolant mixing between the stages to avoid the formation of hot spots and ensure the same flowing direction at all stages. Finally, when the coolant reaches the top of the core at the temperature of around 500°C, the gas-like supercritical water is collected in the annular steam dome (16) above the core, from where it exits the reactor pressure vessel through the two hot leg nozzles (17).

Fig. 3. Vertical cross sections of the SCW-SMR RPV and its internals showing the coolant flow path [23].

The design of the reactor control and protection system is still under development. Still, the concept is that the control and safety rods or plates would be inserted into the core from above into the gaps between the assemblies. For reloading and -shuffling, the entire core drum would be removed by crane to the spent fuel pool, tilted by 90°, and then, by removing the front plate, the fuel assemblies would be accessible [4].

In the next section, the Apros, SPNDYN and Serpent 2 models developed for the coupled equilibrium state simulation of the above-presented SCW-SMR system are described in detail.

3. The coupled SCW-SMR model development

For thermal hydraulics analyses of the SCW-SMR pre-conceptual design, the Apros code (version 6.11SR1), developed by the Finnish VTT Research Centre and the Fortum Oyj energy company, already adapted for supercritical water analyses and successfully used in the HPLWR projects (see [25, 26]), has been applied by BME and coupled to an in-house, finite element-based reactor physics code SPNDYN in the framework of the ECC-SMART project. To verify the coupling methodology and the results obtained for coupled equilibrium state simulations for the SCW-SMR, Apros-Serpent 2 offline coupled calculation results were applied as a reference. This section details the models developed in the three codes and the developed and applied coupling methodologies.

3.1. The Apros model of the SCW-SMR

The main diagram of the Apros model developed for steady-state and transient analyses of the SCW-SMR pre-conceptual RPV design presented in Section 2 is shown in Figure 4. Data used for building up the system model were derived from an SCW-SMR CAD model and related descriptions provided by the concept developers (see [4, 23]). Although a detailed presentation of the developed system model and its main components are provided in [12, 13], to ease the understanding of the models and their coupling, we briefly give an overview in this paper, as well. We must emphasise that the replication of a thermal hydraulics system code model would necessitate the provision of an extensive database, including secondary parameters, such as hydraulic diameters, flow areas, and pipe lengths initially derived from the detailed, three-dimensional CAD model of the SCW-SMR design, which would be beyond the scope and purpose of this paper; however, any additional data requests from readers are welcomed by the authors.

Fig. 4. The Apros model of the SCW-SMR (the numbering corresponds to that given in Fig. 3) [12].

The Apros model incorporates the entire moderator and coolant flow paths through the RPV, including the reflector and bypass flows, takes into account all passive and active heat structures (except the outer vessel wall), and is, therefore, able to describe the heat transfer between the internal volumes separated by walls. This is a key phenomenon which needs to be considered in the thermal hydraulics model of the SCW-SMR concept, as a complex flow path has been designed in which the water passes through the core 2+7 times while heating up by more than 200°C. Therefore, a wide range of temperatures and states of media are in contact through the walls. The outer RPV walls are neglected in this model version because their thermal equilibrium is established very slowly, and they barely have any significance in the equilibrium state core calculations.

To model the individual stages, the so-called User Component (UC) feature of Apros was applied, and UC instances have been set up to model the seven heat-up stages (indicated by boxes with ‘SC Stage’ label in Fig. 4). The seven UC instances have the same internal structure, but are parametrised individually according to the number of fuel assemblies and other unique characteristics of the actual heat-up stage. The User Components defined for the heat-up stages include the coolant and moderator channels of the assemblies, their inlet and outlet plena, the inter-assembly space, and all associated heat structures. The symbol of the heat-up stage User Component and its internal substructure is presented in Figures 5 and 6. As shown in Figure 6. Parts B and C, three-three parallel coolant and moderator channels are present in the UCs of each heat-up stage, representing the five assemblies with the highest powers, the five assemblies with the lowest powers, and the remaining (N-10) assemblies within a stage, respectively, where N is the total number of fuel assemblies in the given stage. The three-three coolant and moderator channels share common inlet and outlet coolant and moderator plena, and their central, heated section is divided into 20 axial nodes. One-one additional axial nodes were used to model the unheated inlet and outlet channel sections. The heat source in the form of axial power profiles can be set independently for the three types of coolant channels in each stage. A total of 21 (7 × 3) power profiles are provided by a preceding reactor physics calculation (either Serpent 2 or SPNDYN). The steady-state temperature and density profiles, calculated with the received power profiles, are the primary outputs of the Apros calculations required by the consecutive reactor physics simulation.

Fig. 5. Symbol of the heat-up stage User Component in the Apros model [12].

Fig. 6. Substructure of the heat-up stage User Component in the Apros model (the numbering corresponds to that given in Fig. 3) – (A) upward moderator channel; (B) horizontal moderator channels with inlet and outlet plena; (C) horizontal coolant channels with inlet and outlet plena [12].

The exact methodology of such couplings and data transfer are described in Sections 3.2 and 3.3. The vertical moderator channel (the flow area between the fuel assemblies) is represented by one node per stage (see part A of Fig. 6). Although representation of each fuel assembly with separate coolant and moderator channels in the Apros model would have been a modelling-intensive approach and, considering the one-dimensional nature of Apros, might have also been unnecessarily detailed, applying three channels (hot, cold, average) rather than two (hot, average; as it was applied in an SCW-SMR RELAP5 model developed by Chaaraoui et al. [27]) enables a more detailed, interpolation-based consideration of thermal hydraulics properties and their spatial distribution in the reactor physics calculations.

The rest of the RPV model (inlet and outlet nozzles, down comer, lower and upper plenum, reflector space, spiral channels and the outlet steam dome) is modelled separately, as shown in Figure 4. The built-in Apros components, such as DESIGN_REACTOR, DESIGN_REACTOR_HS, NODE, PIPE, BRANCH and HEAT_STRUCTURE_X have been used when the SCW-SMR model was set up.

In the frame of the ECC-SMART project, a comprehensive benchmark activity has also been performed, mainly focusing on the heat transfer phenomenon under supercritical conditions [28]. Based on this experience, the Apros model of the SCW-SMR concept uses the Bishop correlation [29] for heat transfer calculations. The pressure loss coefficient settings of the stages, which also consider the effect of the wire wraps, are based on fuel assembly CFD analyses, also performed at BME using Ansys CFX in the framework of ECC-SMART (see details in [30, 31]). Finally, in addition to the RPV model described above, the primary and safety system models are also being developed to enable transient simulations. For the equilibrium state calculations presented in this paper, however, appropriate boundary conditions have been defined for the inlet temperature, inlet mass flow rate, and outlet pressure boundary conditions based on the design values.

3.2. The SPNDYN model of the SCW-SMR

The SPNDYN in-house finite element reactor physics code was coupled to Apros for equilibrium state deterministic core calculations. In the research presented in this paper, only the steady-state continuous Galerkin finite element module of SPNDYN was used and run in diffusion mode to verify the applicability of the coupled code system for SCW-SMR design purposes. The geometric model of the SCW-SMR core and the finite element meshes were created in Gmsh. We applied two-group diffusion approximation and assembly-wise homogenisation in the core. The reflector region was modelled with homogeneous material regions surrounding the core differentiated by their temperatures and densities. The parametrised group constant database was generated using the Serpent 2 Monte Carlo code and parametrised by non-linear least squares data fitting for two different core layouts at the BoC reactor state suggested for the first cycle of the SCW-SMR. One of the core loading patterns consists of solely 5.0 at.% U²³⁵ UO₂ enriched fuel assemblies (Case A), while the other is an optimised pattern with three differently enriched UO₂ assemblies (Case B).

3.2.1. The SCW-SMR Gmsh model

The 3D SCW-SMR model shown in Figure 7. was created in Gmsh, code version 4.9.5. Only half of the SCW-SMR geometry is modelled to take advantage of the vertical symmetry of the core. The core consists of an assembly lattice with 10 × 20 rectangular fuel assemblies with a lattice pitch of 9.5 cm/9.93 cm (Case A/Case B). The active length of each fuel assembly is 168 cm, which is axially divided into 20 layers, each representing a homogeneous material region characterised by separate, parametrised group constants, taking into account two neutron energy groups. The head and foot ends of the assemblies are represented by a 16 cm homogeneous reflector layer along the x-axis of the model, which is divided vertically according to the seven heat-up stages. The spiral channels on the sides of the core and the reflector regions within were modelled with a 35 cm thick homogeneous reflector region along the y-axis of the model, and finally, the upper and lower plena were also included in the reflector region and modelled with a 30 cm homogeneous layer below and above the core. On the outer surface of the reflector, vacuum boundary condition was defined in both Case A and Case B. Additionally, the normal flux derivative was set to zero on the symmetry plane, serving as the corresponding boundary condition. Unstructured meshes were generated using first-order (4-node) tetrahedral elements with first-order (3-node) triangular boundary elements. Before selecting the appropriate mesh for the coupled reactor physics and thermal hydraulics simulations, a mesh sensitivity analysis was conducted, presented in detail in Section 4.

Fig. 7. The Gmsh model of the SCW-SMR core (different colours indicate different material regions). The symmetry plane is located at the front side of the model.

3.2.2. Group constant database generation and parametrisation

Group constant databases were generated using the Serpent 2 Monte Carlo transport code, version 2.1.31 and the ENDF/B-VII.1 nuclear data library. Databases were developed separately for the two core configurations and the reflector regions, which was identically modelled in both cases.

Core region – Case A: in the preliminary core concept, homogeneously enriched UO₂ fuel (5.0 at.%) was considered; therefore, the group constant generation performed for the core region in Case A was based on a detailed, two-dimensional (axially homogeneous), 5.0 at.% enriched fuel assembly model (see Fig. 2). Here, infinite flux spectra were used for homogenisation and collapsing, and no critical spectrum correction was applied. The intermediate multi-group and final few-group structures were the default, 70-group and two-group structures of Serpent 2 (defaultmg and default2). To parametrise the group constant functions (here denoted by y) within the core, the function given in equation (1) was applied, depending on the coolant temperature (T_cool) and density (ρ_cool), the fuel temperature (T_fuel), and the temperatures of the moderator flowing in the moderator boxes (T_mod) and between the fuel assemblies (T_interass).

$$\begin{array}{cc}\hfill y& =a+b·T_{\mathrm{fuel}}+c·T_{\mathrm{fuel}}^2+d·T_{\mathrm{cool}}+e·T_{\mathrm{cool}}^2\hfill \\ \hfill & +f·T_{\mathrm{mod}}+g·T_{\mathrm{interass}}+h·{\rho }_{\mathrm{cool}}.\hfill \end{array}$$(1)

Core region – Case B: the layout of a further optimised SCW-SMR core design – shown in Figure 8 – contained three differently enriched UO₂ assemblies with 5.5 wt.%, 7.8 wt.% and 10 wt.% U²³⁵ enrichment, respectively. In this case, the group constant generation performed for the core region was based on three detailed, two-dimensional (axially homogeneous) fuel assembly models that solely differed in their U²³⁵ enrichment; hence, the isotopic composition of the material regions defined for the fuel in the Serpent 2 models. Interested readers can find further details on the fuel geometry and material composition of the structural elements of the SCW-SMR core in [32], which summarises the latest results of the reactor physics work package of ECC-SAMRT (WP4). Similar to Case A, infinite flux spectra were used for homogenisation and collapsing, and no critical spectrum correction was applied. The intermediate multi-group and final few-group structures were the default, 70-group and two-group structures of Serpent 2 (defaultmg and default2). To parametrise the group constant functions, the function given in equation (1) was applied in this case, as well.

Fig. 8. The optimised core layout considered in Case B.

Relevant temperature and density combinations were considered a priori and used to define the physical properties of the material domains specified in the Serpent 2 fuel assembly model. The fuel temperatures are considered from 600 K to 1500 K with a step of 300 K, motivated by the fact that the nuclear data are provided in the applied nuclear data library for these temperature values. These fuel temperatures have been paired with six different coolant temperatures, as shown in Figure 9.

Fig. 9. Coolant density as a function of pressure and temperature and the selected calculation points [21].

The coolant temperatures were chosen in the range of 575 K to 800 K, depending on the data availability in the thermal scattering library. The corresponding coolant densities were calculated at the nominal pressure of 25 MPa. Although the pressure drop in the core is not significant in the equilibrium state, the densities are sensitive to the pressure change when the pseudocritical transition occurs; therefore, additional calculation points were taken at 650 K and 680 K water temperatures, as shown in Figure 9. The densities of these additional points were calculated at pressures of 24 MPa and 26 MPa, respectively. The temperature of the moderator flowing in the moderator boxes within the fuel assemblies does not change significantly; therefore, only two temperature values, 575 K and 600 K, were considered, with densities calculated at 25 MPa. For the same reason, only two temperature values, 550 K and 600 K, with densities calculated at 25 MPa, were considered in the case of the moderator flowing in the space among the fuel assemblies (vertical moderator channel). The fuel and coolant temperature combinations, the coolant and moderator box temperature combinations and the moderator box and inter-assembly moderator temperature combinations are shown in Figure 10.

Fig. 10. The fuel and coolant temperature combinations (upper left), the coolant and moderator box temperature combinations (upper right) and the moderator box and inter-assembly moderator temperature combinations (lower) taken into account while preparing the group constant database [21].

Reflector region: the Serpent 2 model used to generate the database for the reflector region is a simplified representation of the SCW-SMR core. It consists of a semi-infinite lattice of fuel assemblies surrounded by a 20 cm thick layer of water with vacuum boundary condition defined at the outer surface (see Fig. 11). While the homogenised group constants were generated only for the reflector region, modelling the assembly lattice was necessary to obtain a more realistic flux spectrum and spatial distribution in the reflector required for the group constant generation. The intermediate multi-group and final few-group structures were the default, 70-group and two-group structures of Serpent 2 (defaultmg and default2).

Fig. 11. Front view of the Serpent 2 model used to generate the group constant database for the reflector region (denoted by green colour) [21].

The range of water temperatures and the densities in the reflector regions is somewhat broader, so it was decided to generate the group constants assuming 575 K, 600 K, 650 K, 680 K and 750 K reflector water temperatures with corresponding densities at 25 MPa nominal pressure. The fuel assemblies in this model are considered with the following parameters: the fuel temperature was 900 K, the coolant temperature was 680 K, the coolant density was 0.15077 g/cm³, the temperature of the moderator flowing between the assemblies was 550 K, while the moderator temperature in the moderator box was 575 K.

The function given in equation (2) was applied to parametrise the group constant functions (here denoted by y) within the reflector, which depend on the reflector temperature (T_ref) and density values (ρ_ref).

$$\begin{array}{c}\hfill y=a+b·T_{\mathrm{ref}}+c·T_{\mathrm{ref}}^2+d·{\rho }_{\mathrm{ref}}+e·{\rho }_{\mathrm{ref}}^2.\end{array}$$(2)

In total, 59 different fuel assembly states and five different reflector states were modelled, and two-group constants for diffusion calculations were generated, which formed the basis for the non-linear least squares data fitting performed in MATLAB using the lsqcurvefit function of the Optimisation Toolbox. The coefficients of the fitted curves are summarised in Appendix A (see Tabs. A.1–A.8) for the two different core configurations, including the differently enriched fuel assemblies, and the reflector regions (see Tabs. A.9 and A.10), respectively.

3.2.3. Consistent mapping of the Apros and SPNDYN regions

The Apros model of the SCW-SMR core requires three axial power profiles in each stage to start the calculation: the averaged axial power profile of the five hottest, the five coldest and the remaining fuel assemblies within a stage. Hence, the power calculation module of SPNDYN has been extended to include a function to find, calculate and normalise the required power profiles for each heat-up stage. Since SPNDYN applies the same axial nodalisation to the fuel assemblies as defined in the Apros model, the correspondence between the heated channels and the provided power profiles is straightforward. In the current version of the SPNDYN model, the positions of the hot, cold and average assemblies were determined a priori for both core loading patterns as results of non-coupled, steady-state SPNDYN calculations and were fixed throughout the coupled calculation process, as shown in Figure 12. The fixing of the positions are supported by previously performed SCW-SMR coupled computations, which showed insignificant changes in their positions [33].

Fig. 12. The arrangement of the hot (orange), cold (blue) and average (green) fuel assemblies in the SCW-SMR SPNDYN model for the core loading pattern considered in Case A (left) and in Case B (right).

Since the reflector model of SPNDYN is simplified compared to the Apros model, an approximate correspondence was taken into account in this case. The temperatures of the upper and lower reflector regions in SPNDYN were approximated by the temperature of the moderator flowing between the fuel assemblies in the first and seventh stages, respectively. The reflector positioned at the side of the reactor was assigned the parameters of the fifth stage inter-assembly moderator, while for the reflector regions at the foot and head of the fuel assemblies, the temperature and density of the unheated part of the average coolant channels at each heat-up stage were considered. Two approximations are introduced using these correspondences, one resulting from neglecting the distinction between the three different channel types and the other from neglecting the structural components of the head and foot parts of the fuel assemblies and the moderator box located there.

3.2.4. The methodology applied for coupling the Apros and SPNDYN codes

The coupling between Apros and SPNDYN was realised by developing a Transmission Control Protocol/Internet Protocol (TPC/IP) based communication module in each code. The socket and pickle Python libraries were used in SPNDYN, which was originally also written in Python. SPNDYN is the server programme during data communication, while Apros is the client. The two programs do not necessarily have to be installed on the same computer, but this was the case in this study. In Apros, a plugin for Python integration is available since code version 6.09. The TPC/IP-based communication module was also written in Apros using the Simantics Constraint Language (SCL) module.

The coupled simulation is started in SPNDYN as the server program; then, the Apros run should also be started. SPNDYN waits to receive the temperature and density distributions returned by Apros in the form of Python arrays at the end of its first simulation using a first guess for the axial power distributions. SPNDYN then recalculates the group constants based on the received data and using the parametrised group constant functions, and then performs a steady-state diffusion calculation. The convergence criteria for the steady-state SPNDYN diffusion calculations were predefined for both the effective multiplication factor and the finite-element node-wise flux distributions. In this study, tolerances of 1e-6 and 1e-8 were applied, respectively, defined as the relative difference between successive source iteration steps. For the inner iteration, solved using the Generalized Minimal RESidual (GMRES) method implemented in SciPy, the convergence criterion was set to 1e-7. The axial power distributions of the hot, cold and average assemblies are derived from the two-group flux distribution, normalised and returned as Python arrays to Apros, the client, which receives the data, overwrites the power profiles of the hot, cold and average channels using an SCL script, and starts the next thermal hydraulics simulation. SPNDYN waits for Apros to finish the simulation and receives the temperature and density distributions returned by Apros, and so on. The convergence of the coupled calculation was monitored through the relative differences in nodal temperatures and powers at each iteration step.

SPNDYN and Apros should perform steady-state calculations in each iteration step during the coupled simulations. Unfortunately, Apros can only perform dynamic (transient) calculations. To overcome this problem, the coupling is structured so that no communication occurs until the Apros transient simulation reaches a specific stopping condition, which was set to be a particular simulation time (called Apros runtime) that elapsed since the last successful communication between the codes. It must be stressed that this condition is based on the simulation time, which describes the length of the transient simulated by Apros and not the actual running time of the code.

3.3. The Serpent 2 models of the SCW-SMR core applied for coupled code system verification

To verify the Apros-SPNDYN coupled code system, Apros-Serpent 2 offline coupled calculations were performed to provide a reference for the two different core configurations. In this section, a brief overview of the applied Serpent 2 models and their offline coupling to Apros are presented. Interested readers can find further information about the core configuration optimisation performed for the SCW-SMR with Serpent 2 in [32], providing details about the developed Serpent 2 models, along with specified geometric and physical parameters, as well as materials present in the core. For the reference calculations, similar to the group constant generation, the Serpent 2 code version 2.1.31 and the ENDF/B-VII.1 nuclear data library were used.

The fuel assembly models are built up from fuel pins with a pellet diameter of 6.9 mm, a gas gap width of 0.05 mm, a cladding width of 0.5 mm, and an active length of 1680 mm (see Fig. 13). The fuel pin pitch is 9.5 mm. The layered fuel assembly walls have an inner width of 67.5 mm, followed by a 0.4 mm 310S stainless steel liner, a 4 mm YTZP insulation and a 2.5 mm Zircaloy layer, where the radii of the rounded corners from the inner to the outer surfaces are 2.5 mm, 2.9 mm, 6.9 mm, and 9.4 mm, respectively. The layered moderator box walls have an inner width of 10.05 mm, followed by a 1.0 mm Zircaloy layer, a 2 mm YTZP insulation and a 0.4 mm 310S stainless steel liner, where the radii of the rounded corners from the inner to the outer surfaces are 0.7 mm, 1.7 mm, 3.7 mm, and 4.1 mm, respectively. The fuel assembly lattice pitch differs in the two core configurations: 95 mm was applied in Case A, while the core optimisation reported in [32] resulted in a fuel assembly lattice pitch of 99.3 mm (81.3 mm outer fuel assembly width and 18 mm water gap), which is applied in Case B. In both calculation cases, for verifying the Apros-SPNDYN coupled code system, the non-active parts of the fuel assemblies, as well as other structural elements surrounding the core, were neglected to obtain a simplified Serpent 2 model that geometrically equivalent to the Gmsh model developed for the SPNDYN runs. The core, consisting of a fuel assembly lattice with 20 × 20, thus 400 fuel assemblies, is surrounded by water reflector layers according to the SPNDYN geometric model: the head and foot ends of the assemblies are represented by a 16 cm water layer, the spiral channels on the sides of the core and the reflector regions within were modelled with a 35 cm thick water layer, and finally, the upper and lower plena were modelled with a 30 cm homogeneous water layer below and above the core. Figure 14 presents the Serpent 2 model used for the Apros-Serpent 2 coupled calculations. For model reproduction purposes, the isotopic composition of the materials and their corresponding densities are provided in Appendix B (Tabs. B.1–B.4). for both core configurations.

Fig. 13. Front and lateral views of the SCW-SMR reference fuel assembly model developed in Serpent 2 [32].

Fig. 14. The front and side views of the Serpent 2 model applied for providing reference results for the Apros-SPNDYN coupled code verification.

The coupling established between Apros and Serpent 2 is a semi-automated and weak coupling method (see details in [12, 13]), initially established in the framework of an MSc project [33]. Using an initial guess for the temperature and density distribution of the different Serpent 2 model components, Serpent 2 computes the detailed axial assembly power distributions characterising the core. For this purpose, neutron detectors with response number of -8 (macroscopic total fission energy production cross section) were used with an evenly spaced axial mesh of 20 bins defined for the fuel region assembly-by-assembly. Then, using MATLAB/Python scripts (due to the evolved modelling process, in Case A and B different tools were applied), the average axial power distribution of the five hottest, the five coldest, and the remaining N-10 fuel assemblies at each heat-up stage are calculated, and a Simantics Constraint Language (SCL) script is generated that sets the obtained power profiles in the corresponding Apros modules (hot, cold and average coolant channels in each heat-up stage). Using the derived power profiles, an Apros simulation is run, and new moderator and coolant temperature and density axial distributions, and fuel temperature axial distributions are obtained for the hot, cold and average channels, which are used by MATLAB/Python script (Case A/Case B) to update the Serpent 2 models. In the Serpent 2 model used in Case A, the temperature and density distributions between the fuel assemblies are interpolated based on the assembly power distribution of the previous iteration step, while in Case B, solely the hot, cold and average channels are differentiated in the Serpent 2 model. Similar to the Apros-SPNDYN runs, the convergence of the coupled calculation was monitored using the relative differences in nodal temperatures and powers at each iteration step.

The temperature of each of the remaining regions and structural materials (gas gap, cladding, the three layers of the assembly and the moderator box walls) is updated by a core-wise average since the neutron cross sections of these materials do not change significantly within their core-wise temperature range. Similar to the SPNDYN model, the temperatures of the upper and lower reflector regions were approximated by the temperature of the moderator flowing between the fuel assemblies in the first and seventh stages, respectively. The reflector positioned at the side of the reactor was assigned the parameters of the fifth stage inter-assembly moderator, while for the reflector regions at the foot and head of the fuel assemblies, the temperature and density of the unheated part of the average coolant channels at each heat-up stage were considered.

It is important to highlight that due to the high number of material regions with various temperatures and densities in the Serpent 2 model, memory usage optimisation was necessary. We applied 0.05 as the fractional reconstruction tolerance of the energy grid using the ‘set egrid’, and the temperature values were rounded to integers to reduce the memory need of the Doppler broadening pre-processor of Serpent 2 initiated using the ‘tmp’ entry of the material definition cards.

4. Results

The results of the mesh and Apros runtime sensitivity analyses performed with the core loading pattern considering solely 5.0 at.% enriched UO₂ fuel assemblies and the final coupled calculations for the two different core configurations, along with their comparison to reference Serpent 2 calculations, are presented in detail below.

4.1. Sensitivity analyses performed for the 5% enriched UO₂ core

The coupled calculations were first started with a sensitivity analyses focusing on two main parameters: the number of elements in the mesh generated in Gmsh for the developed three-dimensional core model, and the Apros runtime defined below. However, other sources of uncertainty must also be considered when performing detailed sensitivity and uncertainty analyses. These include uncertainties in nuclear data, the generation of diffusion group constants, and potential errors arising from modelling approximations (spatial, angular, and energy discretization) as well as from the applied numerical methods. In the present work, we focus specifically on discretization errors in the SPNDYN finite element calculations. Nevertheless, we plan to extend our research in the future to cover a full-scope sensitivity analysis and uncertainty propagation study.

4.1.1. Mesh sensitivity study

The mesh sensitivity analysis aimed to optimize the discretization error of the finite element solution of SPNDYN while considering the computational cost of the simulations. To achieve this, four unstructured tetrahedral meshes were created, each with a different number of elements: 130969 (A), 499619 (B), 1237255 (C), and 2150139 (D).

By refining the mesh and increasing the number of nodes, we can minimize the discretization error of the finite element solution in relation to the effective multiplication factor and the power distribution. This is especially important in areas where the flux gradient is significant, such as towards the boundaries of the core. When the discretization error is reduced to its minimum, the results are considered mesh-independent, meaning their accuracy does not rely on the density of the mesh. However, it is essential to note that a finer mesh increases the computational costs of the simulation. Therefore, optimizing the achievable accuracy while considering the available computational resources is crucial.

The analysis focused on the power profiles calculated by SPNDYN, as they significantly impact the results of the coupled simulations. The effective multiplication factor was also examined; however, the accuracy of reactivity estimation will be more critical for transient calculations. The analysis utilized an estimated temperature profile in SPNDYN while disregarding its coupling with Apros. The results obtained are presented in Figures 15–17. The profiles of successive stages are plotted one after the other, forming a total of 7 × 20 = 140 nodes (the non-active part of the assemblies, the inlet and outlet plena, and the spiral channels are excluded from the plots).

Fig. 15. The obtained effective multiplication factor in function of the number of elements in the finite element mesh [21].

Fig. 16. The average power profiles obtained for the hot (upper left), cold (upper right) and the average (lower center) fuel assemblies [21].

Fig. 17. The average power profile differences in the case of the hot, cold and average channels obtained between the meshes A and B (upper left), B and C (upper right), and C and D (lower center) [21].

The most significant differences in power profiles occur in the cold assemblies, specifically at the first and last (20th) nodes of the assemblies in each stage. In contrast, the variations in other regions of the profiles are minimal. In the case of the effective multiplication factor, there is a significant improvement during the first step of the sensitivity analysis (see Fig. 15), and the value continues to decrease as the number of elements increases. Unfortunately, the computer used for the calculations could not exceed 2150139 elements due to memory limitations; hence, mesh-independent results were not obtained, but the deviation between the k_eff values obtained with meshes C and D were almost negligible, approx. 2.5 pcm.

Considering that our results converged with decreasing discretization error and also taking into account the available computational resources, we performed a simple optimization to balance discretization-induced errors against runtime performance. For the coupled core calculations in both Case A and Case B, we employed a mesh consisting of 499 619 elements. This mesh enables relatively fast SPNDYN calculations with low memory demand while keeping the discretization error within 5–6%.

4.1.2. Apros runtime sensitivity analysis

Each time Apros receives a new power profile, a new transient is initiated, and, depending on the variation of the profile, the effects on the system may be more or less pronounced. Then, depending on the length of the simulated transient (runtime), the system may or may not reach a steady state during the iteration step. If the stopping condition is not set correctly, the data sent to SPNDYN will not be related to the steady-state condition of the reactor. Therefore, the reactor physics calculation will start with group constants calculated from the temperatures of a transient state of the system. This will probably not prevent the coupled calculation from converging, but it will require more iteration steps and a longer time. On the other hand, longer transients unnecessarily increase the computational demand of Apros. Finding the right setting for the parameter Apros runtime is essential to optimise the coupled computation.

For this purpose, the temperature distributions of the fuel, coolant and moderator were analysed using different runtime settings in Apros. The subject of this analysis was only the first step of the coupled calculations, the results of which are shown in Figures 18–20. A coarse mesh of 22115 nodes was used in this study. The profiles of successive stages are plotted one after the other, forming a total of 7 × 20 = 140 nodes (the non-active part of the assemblies, the inlet and outlet plena, and the spiral channels are excluded from the plots). In this case, as a result of the representation, step changes in the coolant temperature are observed at each stage boundary. It should also be noted that in the moderator’s case, the flows are parallel in all the 400 boxes with the same inlet temperature regardless of the seven-stage assignment of the coolant channels; however, the same plotting approach is applied as in the case of the coolant and fuel temperature distributions.

Fig. 18. The fuel temperature profiles obtained in the runtime sensitivity analysis for the hot (upper left), cold (upper right) and the average (lower center) channels [21].

Fig. 19. The coolant temperature profiles obtained in the runtime sensitivity analysis for the hot (upper left), cold (upper right) and the average (lower center) channels [21].

Fig. 20. The moderator temperature profiles obtained in the runtime sensitivity analysis for the hot (upper left), cold (upper right) and the average (lower center) channels [21].

As expected, the fuel temperature was the fastest parameter to converge, as it directly depends on the power distribution. Due to the heat transfer from the fuel rods, the coolant temperatures take longer to converge. Nevertheless, it can be considered converged after 80 s. The moderator box temperature was the slowest parameter to converge because several heat conduction and transfer processes delay the effect of the new power profile. Based on these results, Apros needs to simulate at least 320 s between two steps of the reactor physics calculation to reach a good equilibrium state in the case of the SCW-SMR model.

4.2. Equilibrium state core calculations

With the coupled code system Apros-SPNDYN, finalised based on the results of the sensitivity analyses (see Sect. 4.1), steady-state simulations were performed until the whole system reached equilibrium considering two different core layouts.

4.2.1. Results obtained for the 5% UO₂ core

The results obtained assuming only 5.0 at.% enriched UO₂ assemblies in the core loading pattern are compared in Figures 21–28 with a reference coupled simulation performed with Apros and Serpent 2.

Fig. 21. The nodal power distribution obtained for the 5% UO2 core.

Fig. 22. The differences of the nodal power distribution obtained for the 5% UO2 core.

Fig. 23. The temperature profile of the coolant channels obtained for the 5% UO2 core.

Fig. 24. The density profile of the coolant channels obtained for the 5% UO2 core.

Fig. 25. The temperature profile in the inter-assembly space obtained for the 5% UO2 core.

Fig. 26. The temperature profile of the moderator boxes obtained for the 5% UO2 core.

Fig. 27. The radially averaged fuel temperature profiles obtained for the 5% UO2 core.

Fig. 28. The cladding temperature profiles obtained for the 5% UO2 core.

As seen, the usual cosine-like power profile is formed both axially within each stage and in the vertical direction thanks to the homogeneous 5% enrichment used in this simulation (see Fig. 21). However, the cosine curves are slightly shifted in both directions, horizontally towards the inlet side and vertically downwards, which can be explained by the temperatures of the moderator and the coolant, which are lower in these directions; hence, these regions can provide better moderation.

The two calculations show good agreement. The effective multiplication factor obtained with Serpent 2 was 1.16720 ± 4 pcm, while that obtained with SPNDYN was 1.17442. The largest absolute deviations of the examined physical quantities are observed in the hot channels of heat-up stages two to four, while the largest relative deviations are typically observed at the edges of the active core, i.e. in the cold channels of heat-up stages one and seven (see Fig. 22). The latter remains mostly below ±10%, but even the maximum does not exceed ±21%. The observed differences can be attributed to several possible sources, namely the accuracy of the transport approximation used (diffusion) compared to the Monte Carlo simulation, differences in the way the temperature and density distributions are taken into account in the SPNDYN (three-channel) and Serpent 2 models (interpolation), and, finally, the fixing of the position of the hot and cold assemblies in the SPNDYN model (see Fig. 12 and the related considerations in Sect. 3.2.3).

As a result of the axially cosinusoidal power distribution, the rise in the coolant temperature describes successive S-shaped curves (see Fig. 23), which flatten out around heat-up stages two to four despite the high local power. This apparent contradiction is caused by the supercritical transition (see Fig. 24), a second-order phase transition that consumes much energy, similar to boiling at subcritical pressures.

At the design power of 290 MW and an inlet mass flow rate of 145 kg/s, an RPV outlet temperature of about 520°C was achieved, higher than the target of 500°C, for which a mass flow rate of about 150 kg/s would be required. A significant temperature difference of over 50°C is observed between the three channels, particularly in the upper stages. The design concept has already undergone several improvements during the project to reduce this high temperature peak (see the results obtained for Case B in Sect. 4.2.2 and approaches presented in [13]). The comparison of the two models shows that the temperatures are consistent with the power profiles, and significant deviations from each other are only found in the middle stages. The most considerable temperature difference is about 5°C at the outlet of the hot channel of the fourth stage.

The moderator regions are heated by the coolant through the walls. The slight deviation of the coolant temperature results in a practically negligible variation of the temperature of both the moderator flowing in the inter-assembly space (see Fig. 25) and in the moderator boxes (see Fig. 26). In the inter-assembly space, due to the higher upper coolant temperatures and the increasing number of assemblies per stage, the moderator temperature rises progressively as it moves upwards, reaching a temperature of over 296°C at the outlet. The flow in the moderator boxes is parallel, i.e. the inlet temperature is almost the same for all the 400 boxes. Nevertheless, the first value per stage shown in Figure 25 increases upwards. This is a result of the significant heat transfer along the inlet section of the box, which is not yet inside the active core, especially in the cylindrical part of the box where the wall does not have an internal insulating layer. The temperature of the coolant surrounding the box is higher in the upper stages, resulting in higher moderator temperature rises in both the inlet and active sections. The maximum outlet temperature observed is approximately 315°C.

The radially averaged fuel temperature (see Fig. 27) and the cladding temperature (see Fig. 28) are essentially determined by the combined effects of the power profiles and the continuously increasing coolant temperature. Again, significant differences between the models are only observed in the hot channels of heat-up stages two to four, with a global maximum of 49°C and 18°C in stage four. As a result of the quite uneven power distribution of the three channel types, the cladding temperature has a strong variation, ultimately leading to a maximum of over 650°C.

4.2.2. Results obtained for the UO₂ core with three differently enriched fuel assemblies

The results obtained for the UO₂ core with three differently enriched fuel assemblies are compared in Figures 29–36 with a reference coupled simulation performed with Apros and Serpent 2.

Fig. 29. The nodal power distribution obtained for the UO2 core with three differently enriched fuel assemblies.

Fig. 30. The differences of the nodal power distribution obtained for the UO2 core with three differently enriched fuel assemblies.

Fig. 31. The temperature profile of the coolant channels obtained for the UO2 core with three differently enriched fuel assemblies.

Fig. 32. The density profile of the coolant channels obtained for the UO2 core with three differently enriched fuel assemblies.

Fig. 33. The temperature profile in the inter-assembly space obtained for the UO2 core with three differently enriched fuel assemblies.

Fig. 34. The temperature profile of the moderator boxes obtained for the UO2 core with three differently enriched fuel assemblies.

Fig. 35. The radially averaged fuel temperature profiles obtained for the UO2 core with three differently enriched fuel assemblies.

Fig. 36. The cladding temperature profiles obtained for the UO2 core with three differently enriched fuel assemblies.

Even when considering a more heterogeneous core loading pattern with significantly varying fuel enrichments (5.5 wt.%, 7.8 wt.% and 10 wt.%), there is a good agreement between the reference Apros-Serpent 2 and the obtained Apros-SPNDYN results. As for the effective multiplication factors, the reference taken from the Apros-Serpent 2 run was 1.24719 ± 2 pcm, while that obtained from the Apros-SPNDYN run was 1.25308, showing a deviation of less than 500 pcm.

The largest absolute deviations of the examined physical quantities were observed in heat-up stage one (hot and average channels) and heat-up stage four (cold channel), while the largest relative deviations typically occurred at the edges of the active core, particularly in the cold channels of each heat-up stage, as well as in the first and last (20th) fuel assembly nodes (see Fig. 30). The largest absolute fuel assembly node power difference is approx. 5.5 kW in the cold channel of heat-up stage four. The larger deviations obtained at the core boundaries – specifically in the first and last nodes of the channels, as well as generally in the cold channels typically located at the periphery of the core – can be attributed to the different accuracy of the transport approximation used (diffusion vs. Monte Carlo transport). Additionally, and besides the finite element mesh density, fixing the position of the hot, cold and average assemblies in the SPNDYN model at the start of the coupled calculation versus the dynamic positioning used in the Apros – Serpent 2 coupled calculations, where the positions are determined at the beginning of each iteration step, can also contribute to the obtained differences. In Case B, the reference Apros-Serpent 2 and the performed Apros-SPNDYN coupled calculations both considered only three channel types (hot, cold, average) without interpolating the thermal hydraulics parameters; therefore, additional deviations cannot result from this aspect.

Similarly to the 5% UO₂ core, as a result of the axially cosinusoidal power distribution, the rise in the coolant temperature describes successive S-shaped curves (see Fig. 31), which flatten out around heat-up stages two to four due to the supercritical phase transition (see Fig. 32). Higher differences between the results of the two calculations are observed in the coolant temperatures (over 50°C) of the hot/cold and average channels in the upper stages, especially in heat-up stages five to seven. Generally, good agreement is observed between the obtained and reference coolant temperature distributions (less than 2°C); however, in heat-up stage five, the cold channel coolant temperature profiles differ with a maximum difference of 3–4°C.

Deviations between the obtained and reference coolant temperature profiles (Fig. 31) results in a slight variation of the temperature of both the moderator flowing in the inter-assembly space (see Fig. 33) and in the moderator boxes (see Fig. 34). As it can be seen in Figure 33, the moderator temperature rises monotonously as it moves upwards reaching an upper plenum temperature of almost 300°C, while by flowing through the moderator boxes, its temperature reaches around 301–317°C, depending on the channel type and heat-up stage.

The agreement obtained between the reference Apros – Serpent 2 and Apros – SPNDYN fuel and cladding temperature profiles can also be considered good. However, the radially averaged fuel temperature profiles (see Fig. 35) show higher differences in heat-up stage one and the cold channels of the heat-up stages, where higher deviation in the power profiles were also observed. At the same time, larger differences were obtained between the cladding temperature profiles (see Fig. 36) in those heat-up stages, where the coolant temperature profiles showed higher deviations (in heat-up stages four and five). The highest deviations in the radially averaged fuel temperature profiles of the cold, hot and average channels were approx. 36°C (heat-up stage four), 15°C (heat-up stage one) and 28°C (heat-up stage one), respectively. As a result of the quite uneven power distribution of the three channel types, the cladding temperature has a substantial variation, ultimately leading to a maximum of a little bit over 600°C. This maximum is more than 50°C lower than that obtained in Case A (Fig. 28), which demonstrates well the advantage of the optimised core loading pattern over the homogeneous one from the viewpoint of thermal hydraulics. The highest deviations in the cladding temperature profiles of the cold, hot and average channels were approx. 8°C (heat-up stage four), 3°C (heat-up stage six) and 3°C (heat-up stage one), respectively.

Finally, although a quantitative comparison of total running time between the coupled approaches would not be informative – since the Apros-Serpent 2 coupling requires human interaction, and the calculations were performed on computers with markedly different performance characteristics – we consider it important to emphasize that the same Apros model was applied in both cases. Thus, the differences in runtime primarily stem from the use of different reactor physics codes and from the degree of automation in the coupling methods. As expected, the Serpent 2 runs required significantly more time than the SPNDYN diffusion runs due to the Monte Carlo approach, while the fully automated coupling further reduced the overall computational effort. Consequently, the application of lower-order deterministic transport approximations for SCW-SMR core analysis and design – as is common practice in operating nuclear power plants – is strongly recommended for future research.

5. Conclusion

In the framework of the research presented in this paper, a coupled Apros-SPNDYN simulation framework was developed and applied to analyse the equilibrium state behaviour of the SCW-SMR pre-conceptual design as part of the ECC-SMART project. The coupling was achieved through a Python-based communication module, enabling iterative thermal hydraulics and reactor physics calculations.

The key outcome of the research performed is that we successfully implemented a coupled modelling framework, allowing equilibrium state calculations for the SCW-SMR concept. Detailed sensitivity analysis was performed on mesh density and the Apros runtime parameter, which optimised the computational accuracy and efficiency of the coupled simulations. Coupled equilibrium state calculations were performed using the Apros-SPNDYN coupled code system assuming two different BoC core loading patterns (Case A: homogeneous 5.0 at.% enriched UO₂ fuel assemblies, Case B: an optimised core loading pattern with three differently enriched UO₂ fuel assemblies (5.5 wt.%, 7.8 wt.%, and 10 wt.%)). Compared to Case A, the optimised core layout applied in Case B demonstrated improved thermal performance, reduced peak cladding temperatures, and higher BoC excess reactivity, thus allowing longer fuel cycle length, addressing key design challenges. We verified the Apros-SPNDYN code system by comparing the simulation results obtained for the SCW-SMR concept using the two different core loading patterns against reference results from Apros-Serpent coupled calculations and demonstrated a good agreement. The obtained differences can be mainly attributed to the lower accuracy of the diffusion approximation used by SPNDYN for the presented SCW-SMR analysis compared to a Monte Carlo transport solution. In Case A, generally, higher absolute differences were only observed in the central part of the core (hot channels of stages two to four), while in Case B, higher deviations were observed in heat-up stages one and at the boundaries of the core with steeper flux gradients due to the positioning of the high-enriched fuel assemblies.

We also identified key areas for further improvement, including the improvement of the representation of the reflector region, potentially applying denser finite element mesh due to the more heterogeneous enrichment map, incorporating additional SCW-SMR design modifications, and the extension of the modelling framework for transient analysis, enabling the simulation of reactivity-initiated accidents (RIA) and other dynamic scenarios relevant to safety evaluation.

This research represents a major step forward in SCW-SMR coupled thermal hydraulics and reactor physics modelling and simulation, providing a verified toolset for equilibrium state analysis and laying the groundwork for future transient and safety assessments of Generation IV supercritical water reactor designs.

Acknowledgments

The design work carried out by Prof. Thomas Schulenberg and Ivan Otic (KIT) and the developed CAD model of the SCW-SMR reactor vessel, which was shared with the project partners, made it possible to carry out the present work. We would also like to thank our project partners, especially Prof. Walter Ambrosini (University of Pisa), for their excellent cooperation in the thermal hydraulics modelling and coupled computations. The authors are grateful to Viktor Holl and András Csige, former colleagues at BME NTI, for the effective assistance provided at the beginning of the coupling of Apros and SPNDYN.

Funding

The ECC-SMART project has received funding from the Euratom Research and Training Programme 2014-2018 under Grant Agreement No. 945234. The research presented in this paper was partly funded by the Sustainable Development and Technologies National Programme of the Hungarian Academy of Sciences (FFT NP FTA). This research was partially funded by the National Research, Development, and Innovation Fund of Hungary under Grant TKP2021-NVA-02. The authors would like to thank the High Scientific Council of the European Nuclear Society for funding the Article Processing Charges (APCs).

Conflicts of interest

The authors declare that they have no competing interests to report.

Data availability statement

This article has no associated data generated and/or analysed.

Author contribution statement

Conceptualization, B. Babcsány and T. Varju; Methodology, B. Babcsány and T. Varju; Software, B. Babcsány, Z. Bertesina and T. Varju; Validation, Zs. Várkonyi, Z. Bertesina, P. Mészáros, Cs. Antók; Formal Analysis, B. Babcsány and T. Varju; Investigation, B. Babcsány and T. Varju; Resources, B. Babcsány and T. Varju; Data Curation, B. Babcsány, Z. Bertesina, Cs. Antók and T. Varju; Writing – B. Babcsány, T. Varju, Cs. Antók; Writing – Review & Editing, B. Babcsány; Visualization, B. Babcsány and T. Varju; Supervision, B. Babcsány and T. Varju; Project Administration, B. Babcsány and T. Varju; Funding Acquisition, B. Babcsány and T. Varju.

References

Appendix

Coefficients of the group constant functions

Appendix

Isotopic composition of the materials defined in the Serpent 2 models

The gas gap was modelled as pure ⁴He (ZAID: 2004, atomic fraction: 1.0) with mass density of 1.785E−04 g/cm³, while the coolant and moderator regions were considered with the following isotopic composition (ZAID and atomic fractions) at temperatures and densities taken from the Apros run:

1001 6.66667E−01

8016 3.33200E−01

8017 1.33333E−04

The isotopic concentration of the rest of the materials are provided in Tables B.1–B.4.

All Tables

All Figures

	Fig. 1. Top and side views of the reactor pressure vessel (RPV) internals of the SCW-SMR design (with the top cover (left), front plate (center) and outer cylinder removed (right)) [4].
In the text

	Fig. 2. Cross section of the SCW-SMR assembly (left), and the dimensions of the three-layer walls of the assembly and moderator boxes (central and right): black – stainless steel liner, yellow – insulation, grey – Zircaloy-4 [23, 24].
In the text

	Fig. 3. Vertical cross sections of the SCW-SMR RPV and its internals showing the coolant flow path [23].
In the text

	Fig. 4. The Apros model of the SCW-SMR (the numbering corresponds to that given in Fig. 3) [12].
In the text

	Fig. 5. Symbol of the heat-up stage User Component in the Apros model [12].
In the text

	Fig. 6. Substructure of the heat-up stage User Component in the Apros model (the numbering corresponds to that given in Fig. 3) – (A) upward moderator channel; (B) horizontal moderator channels with inlet and outlet plena; (C) horizontal coolant channels with inlet and outlet plena [12].
In the text

	Fig. 7. The Gmsh model of the SCW-SMR core (different colours indicate different material regions). The symmetry plane is located at the front side of the model.
In the text

	Fig. 8. The optimised core layout considered in Case B.
In the text

	Fig. 9. Coolant density as a function of pressure and temperature and the selected calculation points [21].
In the text

	Fig. 10. The fuel and coolant temperature combinations (upper left), the coolant and moderator box temperature combinations (upper right) and the moderator box and inter-assembly moderator temperature combinations (lower) taken into account while preparing the group constant database [21].
In the text

	Fig. 11. Front view of the Serpent 2 model used to generate the group constant database for the reflector region (denoted by green colour) [21].
In the text

	Fig. 12. The arrangement of the hot (orange), cold (blue) and average (green) fuel assemblies in the SCW-SMR SPNDYN model for the core loading pattern considered in Case A (left) and in Case B (right).
In the text

	Fig. 13. Front and lateral views of the SCW-SMR reference fuel assembly model developed in Serpent 2 [32].
In the text

	Fig. 14. The front and side views of the Serpent 2 model applied for providing reference results for the Apros-SPNDYN coupled code verification.
In the text

	Fig. 15. The obtained effective multiplication factor in function of the number of elements in the finite element mesh [21].
In the text

	Fig. 16. The average power profiles obtained for the hot (upper left), cold (upper right) and the average (lower center) fuel assemblies [21].
In the text

	Fig. 17. The average power profile differences in the case of the hot, cold and average channels obtained between the meshes A and B (upper left), B and C (upper right), and C and D (lower center) [21].
In the text

	Fig. 18. The fuel temperature profiles obtained in the runtime sensitivity analysis for the hot (upper left), cold (upper right) and the average (lower center) channels [21].
In the text

	Fig. 19. The coolant temperature profiles obtained in the runtime sensitivity analysis for the hot (upper left), cold (upper right) and the average (lower center) channels [21].
In the text

	Fig. 20. The moderator temperature profiles obtained in the runtime sensitivity analysis for the hot (upper left), cold (upper right) and the average (lower center) channels [21].
In the text

	Fig. 21. The nodal power distribution obtained for the 5% UO2 core.
In the text

	Fig. 22. The differences of the nodal power distribution obtained for the 5% UO2 core.
In the text

	Fig. 23. The temperature profile of the coolant channels obtained for the 5% UO2 core.
In the text

	Fig. 24. The density profile of the coolant channels obtained for the 5% UO2 core.
In the text

	Fig. 25. The temperature profile in the inter-assembly space obtained for the 5% UO2 core.
In the text

	Fig. 26. The temperature profile of the moderator boxes obtained for the 5% UO2 core.
In the text

	Fig. 27. The radially averaged fuel temperature profiles obtained for the 5% UO2 core.
In the text

	Fig. 28. The cladding temperature profiles obtained for the 5% UO2 core.
In the text

	Fig. 29. The nodal power distribution obtained for the UO2 core with three differently enriched fuel assemblies.
In the text

	Fig. 30. The differences of the nodal power distribution obtained for the UO2 core with three differently enriched fuel assemblies.
In the text

	Fig. 31. The temperature profile of the coolant channels obtained for the UO2 core with three differently enriched fuel assemblies.
In the text

	Fig. 32. The density profile of the coolant channels obtained for the UO2 core with three differently enriched fuel assemblies.
In the text

	Fig. 33. The temperature profile in the inter-assembly space obtained for the UO2 core with three differently enriched fuel assemblies.
In the text

	Fig. 34. The temperature profile of the moderator boxes obtained for the UO2 core with three differently enriched fuel assemblies.
In the text

	Fig. 35. The radially averaged fuel temperature profiles obtained for the UO2 core with three differently enriched fuel assemblies.
In the text

	Fig. 36. The cladding temperature profiles obtained for the UO2 core with three differently enriched fuel assemblies.
In the text