DYN3D and CTF Coupling within a Multiscale and Multiphysics Software Development (Part II)

Abstract

Traditionally, the complex coupled physical phenomena in nuclear reactors has resulted in them being treated separately or, at most, simplistically coupled in between within nuclear codes. Currently, coupling software environments are allowing different types of coupling, modularizing the nuclear codes or multi-physics. Several multiscale and multi-physics software developments for LWR are incorporating these to deliver improved or full coupled reactor physics at the fuel pin level. An alternative multiscale and multi-physics nuclear software development between NURESIM and CASL is being created for the UK. The coupling between DYN3D nodal code and CTF subchannel code can be used to deliver improved coupled reactor physics at the fuel pin level. In the current journal article, the second part of the DYN3D and CTF coupling was carried out to analyse a parallel two-way coupling between these codes and, hence, the outer iterations necessary for convergence to deliver verified improved coupled reactor physics at the fuel pin level. This final verification shows that the DYN3D and CTF coupling delivers improved effective multiplication factors, fission, and feedback distributions due to the presence of crossflow and turbulent mixing.

1. Introduction

Nuclear technology development aims to both improve the existing nuclear reactors, as well as construct new nuclear reactors through innovation resulting in more efficient nuclear reactors [1]. It is usually performed by stages [2,3]. Initially, the requirements analysis is performed to understand the demands of a country. Then, the design is drafted to acknowledge all the components, such as reactor cores, pressurizers, steam generators, turbines, and condensers necessary to address the requirements analysis. Then, the improvement or construction of the nuclear reactor takes place according to the design. Then, testing is performed to guarantee the safety of the nuclear reactor improvement or construction. Then, operation takes place with the nuclear reactor operating along the safety range determined during testing. Finally, maintenance is provided to ensure the correct operation of the nuclear reactor under operation.

Nuclear software development aims to both improve the available nuclear codes, as well as deliver new nuclear codes through innovation resulting in more adequate representations of nuclear reactors. It usually occurs by stages. Initially, the requirements analysis is performed to understand the demands of the industry, the nuclear regulator, and academia [4,5]. Then, the design is drafted to acknowledge all the physical and mathematical models, as well as the possible code structure necessary to address the requirements analysis. Then, implementation and integration are conducted to create a nuclear code according to the design. Then, validation and verification are performed to proof the functionality of the nuclear code created during implementation and integration. Then, deployment takes place with the nuclear code functioning within the range of the validation and verification. Finally, maintenance is provided to ensure the correct functioning of the nuclear code within the deployment.

Coupling is present in both nuclear technology and in nuclear software development, it describes the interdependence, coordination, or information flow either between components within a nuclear reactor [6,7] or between physical and mathematical models or code structure within a nuclear code [8,9,10]. In nuclear technology development the components, such as reactor cores, pressurizers, steam generators, turbines, and condensers are drafted, assembled, safety proven, and operate together. In nuclear software development, the physical and mathematical models, such as the neutronics, thermal hydraulics, and thermo-mechanics, as well as the possible code structure, such as the modularity are drafted, coded, functionally proven, and function together.

Currently, the coupling within a nuclear software development is of interest. The coupling is regarded either as strong or weak [11] depending on the level of interdependence, coordination, and information flow between physical and mathematical models. The coupling is regarded either as tight or loose [12] depending on the level of interdependence, coordination, and information flow between the code structure. The complex coupling phenomena resulting from the non-linearity existing in both the physical and mathematical models, such as the neutronics, thermal hydraulics, and thermo-mechanics, has led to either fully treating them in separate nuclear codes or at most simplistically treating them in different modules within certain nuclear codes. The coupling between nuclear codes or modules can be present in several forms depending on the level of interdependence and coordination between them [13]. In serial coupling, internal libraries are used to merge a nuclear code or module with another becoming a single nuclear code, where extensive modifications are required in the former. In parallel coupling, either external coupling scripts or internal libraries are only used to exchange data between a nuclear code or module with another remaining separate nuclear codes, where mild modifications are required in the former. The coupling between nuclear codes or modules can occur in several ways depending on the information flow between them [14]. In one-way coupling, the merge or exchange of data between nuclear codes or modules takes place only from one to another. In two-way coupling the merge or exchange of data between nuclear codes or modules data takes place both from one to another and vice versa.

Coupling software environments allow to establish either serial or parallel coupling, as well as either one-way and two-way coupling between nuclear codes or modules by offering either a simplified coupling interface or a fully coupled software framework. SALOME (Simulation Numerique par Architecture Logicielle en Open Source et a Methodologie d’Evolution), is a coupling software environment [15,16] which offers a simplified coupling interface to transfer multi-physics and multiscale, modularizing the nuclear codes. It was created in the 2000s by OC (Open Cascade, Guyancourt, France), EDF (Electricite de France, Paris, France) and CEA (Commissariat à l’énergie atomique et aux énergies alternatives, Paris, France) with several versions adapted to different requirements being available. VERA (Virtual Environment for Reactor Applications) is a coupling software environment [17,18] which also offers a simplified coupling interface to transfer multi-physics and multiscale modularizing of the nuclear codes. It was created in the 2000s by ORNL (Oak Ridge National Laboratory, Oak Ridge, TN, USA) with one version created according to specific requirements being available. MOOSE (Multiphysics Object Oriented Simulation Environment) is a coupling software environment [19,20,21] which offers a fully coupled software framework to solve coupled multi-physics and multiscale, modularizing the coupled multi-physics rather than the nuclear codes. It was created in the 2000s by INL (Idaho National Laboratory, Idaho Falls, ID, USA) with several versions adapted to different requirements being available. All the coupling software environments execute the corresponding coupling, mapping between meshes. They allow parallelization across multiple cores within a computational cluster to reduce the simulation times. They include a graphical user interface to provide visualization. They may include additional modules for other specific tasks. Several multiscale and multi-physics software developments for LWR (Light Water Reactor) are being created which incorporate the mentioned coupling software environments to deliver either improved or full coupled reactor physics at the fuel pin level.

NURESIM (Nuclear Reactor Simulator) is a multiscale and multi-physics nuclear software development [22,23] created to deliver in LWR both improved coupled reactor physics and, hence, simplified neutron diffusion and full mixing fluid dynamics at the fuel pin level, as well as simplified coupled reactor physics and, hence, simplified neutron diffusion and non-mixing fluid dynamics at the fuel assembly level. It uses a derivative of the SALOME coupling software environment to couple the nuclear codes. It is being created by EURATOM (European Atomic Energy Community, Brussels, Belgium). Simplified coupled reactor physics at the fuel assembly are delivered by SALOME after transferring via itself the homogenized fuel assembly cross sections from spectral codes to the nodal codes. Improved coupled reactor physics at the fuel pin level are delivered by SALOME [24,25,26,27,28] either after transferring via itself the homogenized fuel pin cross sections from spectral codes to the nodal codes or after performing fuel pin power reconstruction in the nodal codes. Additionally, after transferring via itself the fuel pin power distributions from the nodal codes to the subchannel codes and the fuel cells feedback distributions from the subchannel codes to the nodal codes. Finally, after transferring via itself the fuel parameters from the nodal codes to the fuel performance codes and vice versa. In NURESIM, full coupled reactor physics are not present, which does not result in the most adequate representation of an LWR.

CASL (Consortium for Advanced Simulation of LWRs) is a multiscale and multi-physics nuclear software development [29,30] created to deliver in LWR through its advanced component full coupled reactor physics and, hence, full neutron transport and full mixing fluid dynamics at the fuel pin level, as well as through its baseline component simplified coupled reactor physics at the fuel pin level and, hence, simplified neutron diffusion and non-mixing fluid dynamics at the fuel assembly level. It uses a combination between VERA and a derivative of the MOOSE coupling software environments to couple the nuclear codes. It is being created by USDE (United States Department of Energy, Washington, DC, USA). Simplified coupled reactor physics at the fuel assembly level are delivered by VERA after transferring via itself the homogenized fuel assembly cross sections from spectral codes to the nodal codes. Full coupled reactor physics at the fuel pin level are delivered by VERA [31,32,33,34] after transferring via itself the cross sections from spectral codes to transport codes. Additionally, after transferring via itself the fuel pin power distributions from the transport codes to the subchannel codes and the fuel cells feedback distributions from the subchannel codes to the transport codes. Finally, after transferring via MOOSE the fuel parameters from the transport codes to the fuel performance codes and vice versa. In CASL, full coupled reactor physics at the fuel pin level are extended to all the reactor core, which results in the most computationally expensive representation of an LWR.

An alternative for the UK (United Kingdom) is a multiscale and multi-physics nuclear software development [35] between NURESIM and CASL created to deliver in LWR both improved coupled reactor physics and hence simplified neutron diffusion and full mixing fluid dynamics at the fuel pin level and full coupled reactor physics and, hence, full neutron transport and full mixing fluid dynamics at the fuel pin level, as well as simplified coupled reactor physics and, hence, simplified neutron diffusion and non-mixing fluid dynamics at the fuel assembly level. It uses a customized coupling software environment to couple the nuclear codes. It is being created by UOL (University of Liverpool). Simplified coupled reactor physics at the fuel assembly are delivered by the customized coupling software environment after transferring via itself the homogenized fuel assembly cross sections from the spectral code SCALE [36,37,38] to the nodal code DYN3D [39,40,41]. Improved coupled reactor physics at the fuel pin level are delivered by the customized coupling software environment either after transferring via itself the homogenized fuel pin cross sections from SCALE to DYN3D or after performing fuel pin power reconstruction in DYN3D. Additionally, after transferring via itself the fuel pin power distributions from DYN3D to the subchannel code CTF [42,43,44,45] and the fuel cells feedback distributions from CTF to DYN3D. Finally, after transferring via itself the fuel parameters from DYN3D to the fuel performance code ENIGMA [46] and vice versa. Full coupled reactor physics at the fuel pin level are delivered by the customized coupling software environment after transferring via itself the cross sections from SCALE to the transport code LOTUS [47,48,49]. Additionally, after transferring via itself the fuel pin power distributions from LOTUS to CTF and the fuel cells feedback distributions from CTF to LOTUS. Finally, after transferring via itself the fuel parameters from LOTUS to ENIGMA and vice versa. In this multiscale and multi-physics software development between NURESIM and CASL full coupled reactor physics at the fuel pin level are present but are only extended to the hottest fuel assemblies while simplified coupled reactor physics at the fuel assembly level are extended to all the reactor core, which results in a more reliable but less computationally expensive representation of an LWR. The customized coupling software environment offers a simplified coupling interface to transfer multi-physics and multiscale, modularizing the nuclear codes. It is currently being created by UOL. It executes the corresponding coupling mapping between meshes. It will eventually allow parallelization across multiple cores within a computational cluster. It includes additional modules for specific tasks. Both a one-way and two-way coupling between any two of the mentioned nuclear codes can be performed within the customized coupling software environment as observed in Figure 1.

At present, the aim is the coupling between the nodal code DYN3D and the subchannel code CTF to deliver improved coupled reactor physics at the fuel pin level within the mentioned multiscale and multi-physics nuclear software development. Several objectives were previously carried out to achieve this aim. The first objective [42] was comprehended by CTF and FLOCAL (DYN3D thermal hydraulics module) thermal hydraulics validations and verifications which were carried to analyse the fluid dynamics in both nuclear codes and, hence, the accuracy and mixing methods available to deliver full fluid dynamics at the fuel pin level. Fluid dynamics distributions were obtained in either CTF or FLOCAL through their corresponding modules and convergence criteria. CTF was seen to be very accurate as opposed to other subchannel and system codes. CTF was also seen to include different mixing methods, as opposed to FLOCAL where no crossflow is available. The second objective [45] was comprehended by the first part of the DYN3D and CTF coupling verification which was carried out to analyse an external one-way coupling between both nuclear codes and, hence, the inner coupling iterations contained in an outer coupling iteration to deliver partially verified improved coupled reactor physics at the fuel pin level. Feedback distributions were obtained in the DYN3D and CTF coupling through external coupling scripts, the converged last iteration DYN3D fission power distributions, as well as the CTF modules and convergence criteria. This external one-way coupling analysis determined through the thermal hydraulics when should the DYN3D and CTF coupling be applied instead of DYN3D to deliver improved coupled reactor physics at the fuel pin level. The DYN3D and CTF coupling was seen to deliver more homogeneous feedback distributions as opposed to DYN3D although both were also observed to deliver similar average feedback values.

The current objective is comprehended by the second part of the DYN3D and CTF coupling which has been carried out to analyse a parallel two-way coupling between these nuclear codes and, hence, the outer coupling iterations necessary for convergence to deliver verified improved coupled reactor physics at the fuel pin level. Fission power distributions have been obtained in the DYN3D and CTF coupling through the customized coupling software environment modules, the multiple iterations CTF feedback distributions, as well as the DYN3D and CTF modules and customized coupling software environment convergence criteria. Feedback distributions have been obtained in the DYN3D and CTF coupling through the customized coupling software environment modules, the multiple iterations DYN3D fission power distributions, as well as the DYN3D and CTF modules and customized coupling software environment convergence criteria. This parallel two-way coupling analysis determines through both the neutronics and thermal hydraulics when the DYN3D and CTF coupling should be applied instead of DYN3D to deliver improved coupled reactor physics at the fuel pin level. This third article, hence, comprehends the DYN3D and CTF coupling outer iterations convergence verification to deliver verified improved coupled reactor physics at the fuel pin level, while the LOTUS and CTF coupling convergence verification to deliver verified full coupled reactor physics at the fuel pin level will be covered in a separate journal article.

This journal article is arranged into several sections which are divided into subsections. Initially, the codes used in the coupling outer iterations convergence verification consisting of DYN3D and CTF were described, including main concepts, versions, and approaches. Following, the modules modified or created for the coupling outer iterations, convergence verification within either DYN3D, or the customized coupling software environment were explained, including the main concepts and approach. Then, the specifications used in the coupling outer iterations convergence verification conformed by the modified KAIST (Korean Advanced Institute of Science and Technology) benchmark [50] were presented including the different geometries, materials, and boundary conditions. Additionally, the models simulated in the coupling outer iterations convergence verification were described, including the meshes, methods, correlations, and property tables.

Following, the results and analysis acquired through the modified KAIST benchmark in the DYN3D and CTF coupling outer iterations convergence verification were discussed conformed by DYN3D to DYN3D and CTF coupling comparisons. The tests included provide results for the effective multiplication factor, fission power, and feedback in both 17 × 17 fuel assemblies containing fuel pins, as well as 3 × 3 mini-cores containing the previously mentioned fuel assemblies. The mentioned magnitudes were selected to complete the DYN3D and CTF coupling from a coupled reactor physics perspective.

Finally, the conclusions obtained for the DYN3D and CTF coupling outer iterations convergence verification were outlined to summarize the last objective in the aim of delivering the DYN3D and CTF coupling within the multiscale and multi-physics nuclear software development, which was completed by verifying the DYN3D and CTF coupling outer iterations convergence. Additionally, future work is mentioned to present the last aim of delivering the LOTUS and CTF coupling within the multiscale and multi-physics nuclear software development.

2. Codes within the Verification

Previously it was discussed that the DYN3D and CTF coupling outer iterations convergence verification was carried out using both DYN3D and CTF. Hence, the codes are described in the following subsections.

2.1. DYN3D Nodal Code

DYN3D is a nodal code [51,52] created to deliver neutron diffusion and non-mixing fluid dynamics and hence simplified coupled reactor physics in both LWR and VVER (square and hexagonal geometries). It delivers simplified coupled reactor physics at the fuel assembly level after performing fuel assembly homogenisation. It delivers simplified coupled reactor physics at the fuel pin level either after performing fuel pin homogenization or after performing fuel pin power reconstruction. It was created in the 1990s by FZD (Forschung Zentrum Dresden, Dresden, Germany) using FORTRAN 90 (Formula Translator). DYN3D-MG is a recent version of DYN3D updated through several features. It includes multi-energy group neutron diffusion, reactivity calculation through inverse point kinetics, the Pernica DNB correlation and boron calculation through the particle in cell method to increase the fidelity. It was updated in recent years by HZDR (Helmholtz Zentrum Dresden Rossendorf, Dresden, Germany) also using FORTRAN 90. DYN3D-MG (MOD) is the present version of DYN3D-MG updated through a new feature necessary in this journal article. It allows the de-coupling within DYN3D, as well as external feedback importation to increase the fidelity. It was updated by UOL also using FORTRAN 90. DYN3D is structured into several neutronics modules which define the NK (Neutron Kinetics) code and several thermal hydraulics modules which define the FLOCAL (Thermal Hydraulics) code all of which are strongly and tightly coupled in between. A reactor core or parts of it are modelled through a set of nodes which correspond to channels.

The neutronics modules apply the multi-energy group neutron diffusion equations to the set of nodes in conjunction with cross sections tables, as well as nodal expansion methods that are used to provide a solution to the mentioned equations. The SP3 and HEXNEM1-2 nodal expansion methods can be used to integrate the mentioned equations in either square or hexagonal geometries. Albedos are applied to the neutron currents to address neutrons leakage within the reactor core. ADF (Assembly Discontinuity Factors) can be applied to the neutron fluxes to address the errors in the cross sections resulting from fuel assembly homogenization. Fuel pin power reconstruction can be applied to the neutron fluxes to increase fidelity. A control rod model can be applied to the fuel assemblies to replace at the corresponding nodes the cross sections without control rods by the cross sections with control rods.

The thermal hydraulics modules apply the two fluid non-mixing fluid dynamics equations to the set of nodes in conjunction with thermal and mechanical property tables, as well as heat and mass transfer models that are used to provide a solution to the mentioned equations. A flow regime map is used to integrate the mentioned equations over the nodes depending on the type of boiling. Constitutive relations are applied to the mentioned equations to treat the different phases within a channel according to the flow regime. The boiling model is applied to the channels to determine the evaporation and condensation rates. Heat transfer correlations are used in the channels to obtain heat transfer between phases. Several friction and form models are applied to both the subchannels and fuel rods to provide the pressure losses. A fuel rod model is applied to the fuel assemblies to determine heat transfer between the solids and fluids.

In DYN3D, the mentioned equations are formulated using finite differences and solved by using numerical methods through an implicit method. DYN3D performs iterations between the neutronics and thermal hydraulics until a convergence criterion is met in the case of the steady state which consists of small absolute and relative differences between both the fission power and feedback distributions or until a certain time is achieved in the case of the transient state.

2.2. CTF Subchannel Code

COBRA-TF is a subchannel code [53,54] created to deliver full mixing fluid dynamics in LWR (square geometry). It delivers full mixing fluid dynamics at the fuel pin level after providing fuel pin power distributions. It was created in the 1980s by PNL (Pacific Northwest laboratories, Richland, DC, USA) using FORTRAN 77. CTF is the present version of COBRA-TF updated through several features. It improves the void drift, turbulent mixing, heating models, the computational efficiency, and the user friendliness to increase the fidelity. It was updated in recent years by both PSU (Pennsylvania State University, State College, PA, USA) and NCSU (North Carolina State University, Raleigh, NC, USA) using FORTRAN 90. CTF is structured into several thermal hydraulics modules which define the code all of which are strongly and tightly coupled in between. A reactor core or parts of it are modelled through a matrix of mesh cells that correspond to subchannels.

The thermal hydraulics modules apply the two fluid three-flow field full mixing fluid dynamics equations to the matrix of mesh cells in conjunction with thermal and mechanical property tables, as well as heat and mass transfer models that are used to provide a solution to the mentioned equations. A flow regime map is used to integrate the mentioned equations over the matrix of mesh cells depending on the type of boiling. Macro mesh cell closure terms are applied to the mentioned equations to treat the different phases between subchannels according to the flow regime. Several friction and form models are applied to both the subchannels and fuel rods to provide the pressure losses. Inter-cell models or crossflow is used in the subchannels to obtain heat and mass transfer between them. Turbulent mixing and void drift are used in the subchannels to improve heat and mass transfer between them. Micro-mesh cell closure terms are applied to the mentioned equations to treat the different phases within a subchannel according to the flow regime. The boiling model is applied to the subchannels to determine the evaporation and condensation rates. A droplet model is applied to the subchannels to determine the entrainment and de-entrainment rates. Inter-phase models are used in the subchannels to obtain heat and mass transfer between phases. A fuel rod model is applied to the fuel pins for heat transfer between the solids and fluids.

In CTF, the mentioned equations are formulated using finite differences and solved by using numerical methods through a semi-implicit method known as SIMPLE (Semi Implicit Method for Pressure Linked Equations). CTF performs iterations within the thermal hydraulics until a convergence criterion is met in the case of the steady state which consists of small absolute and relative differences within the feedback distributions or until a certain time is achieved in the case of the transient state.

3. Modules within the Verification

Previously, it was discussed that the DYN3D and CTF coupling outer iterations convergence verification was carried out modifying or creating several modules either within DYN3D or the customized coupling software environment. Hence, the modules are described in the following subsection.

DYN3D and CTF Coupling Modules

Several modules have been both modified within DYN3D via DYN3D-MG (MOD), as well as created within the customized coupling software environment to deliver the coupling between DYN3D and CTF in LWR (square geometries). They deliver improved coupled reactor physics at the fuel pin level after transferring both the fission power distributions from DYN3D-MG(MOD) to CTF, as well as the feedback distributions from CTF to DYN3D-MG(MOD). These were recently created by UOL in the case of DYN3D-MG(MOD) using FORTRAN 90 while in the case of the customized coupling software environment using PYTHON. Such modules are in the case of DYN3D-MG (MOD) strongly and tightly coupled in between while these are in the case of the customized coupling software environment strongly but loosely coupled in between. A reactor core or parts of it can be interpreted by the modules.

The modified modules in DYN3D-MG (MOD) apply the de-coupling within DYN3D between NK and FLOCAL as well as the external feedback importation from CTF into DYN3D through several changes to improve the solution in the steady state. A CTF logical parameter for control and allocatable vectors for the fuel temperature, moderator temperature, moderator density, and boron concentration are defined in the constant power/feedback parameter definitions module. The “CTF FEEDBACK CALCULATION” character flag for selection in the input is checked to enable the decoupling of NK from FLOCAL, activate the CTF logical parameter, allocate the vectors according to the number of nodes, as well as import the feedback, which is defined in the neutron diffusion constant feedback calculation module. The CTF logical parameter allows to decouple NK from FLOCAL, which is defined in the neutron diffusion steady state calculation module. It also allows to associate the allocated vectors to the internal variables, which is defined in both the thermal hydraulics feedback preparation and the neutron diffusion constant feedback calculation modules. The de-coupling within DYN3D between NK and FLOCAL, as well as the external feedback importation into DYN3D can be observed in Figure 2.

The created modules within the customized coupling software environment apply the coupling between DYN3D and CTF through several features to improve the solution in the steady state. A loop is executed which calls DYN3D-MG (MOD), as well as several exportation, relaxation, and importation modules, and CTF, as well as several other exportation, relaxation, and importation modules until a convergence criterion is met which consists of small absolute and relative differences between the effective multiplication factor values, as well as the power and feedback distributions, which is defined in the main DYN3D and CTF coupling module. The inequalities that either the absolute or the relative differences of the mentioned magnitudes must satisfy simultaneously for a certain iteration n to achieve convergence is given by the following expressions.

$$|{\mathrm{k}}_{\mathrm{eff}}^{\mathrm{n}}-{\mathrm{k}}_{\mathrm{eff}}^{\mathrm{n}-1}|\le \max \left({\mathrm{R}}_{{\mathrm{k}}_{\mathrm{eff}}}{\ast \mathrm{k}}_{\mathrm{eff}}^{\mathrm{n}-1},{\mathrm{A}}_{{\mathrm{k}}_{\mathrm{eff}}}\right)$$

(1)

$$\max \left(\left|{\mathrm{q}}^{\mathrm{n}}-{\mathrm{q}}^{\mathrm{n}-1}\right|\right)\le \max \left({\mathrm{R}}_{\mathrm{q}}{\ast \mathrm{q}}^{\mathrm{n}-1},{\mathrm{A}}_{\mathrm{q}}\right)$$

(2)

$$\max \left(\left|{\mathsf{\rho }}_{\mathrm{m}}^{\mathrm{n}}-{\mathsf{\rho }}_{\mathrm{m}}^{\mathrm{n}-1}\right|\right)\le \max \left({\mathrm{R}}_{{\mathsf{\rho }}_{\mathrm{m}}}{\ast \mathsf{\rho }}_{\mathrm{m}}^{\mathrm{n}-1},{\mathrm{A}}_{{\mathsf{\rho }}_{\mathrm{m}}}\right)$$

(3)

$$\max \left(\left|{\mathrm{T}}_{\mathrm{m}}^{\mathrm{n}}-{\mathrm{T}}_{\mathrm{m}}^{\mathrm{n}-1}\right|\right)\le \max \left({\mathrm{R}}_{{\mathrm{T}}_{\mathrm{m}}}{\ast \mathrm{T}}_{\mathrm{m}}^{\mathrm{n}-1},{\mathrm{A}}_{{\mathrm{T}}_{\mathrm{m}}}\right)$$

(4)

$$\max \left(\left|{\mathrm{T}}_{\mathrm{f}}^{\mathrm{n}}-{\mathrm{T}}_{\mathrm{f}}^{\mathrm{n}-1}\right|\right)\le \max \left({\mathrm{R}}_{{\mathrm{T}}_{\mathrm{f}}}{\ast \mathrm{T}}_{\mathrm{f}}^{\mathrm{n}-1},{\mathrm{A}}_{{\mathrm{T}}_{\mathrm{f}}}\right)$$

(5)

In the mentioned expressions, ${\mathrm{k}}_{\mathrm{eff}}$ defines the effective multiplication factor, $\mathrm{q}$ defines the fission power distribution, ${\mathsf{\rho }}_{\mathrm{m}}$ defines the moderator density distribution, ${\mathrm{T}}_{\mathrm{m}}$ defines the moderator temperature distribution, and ${\mathrm{T}}_{\mathrm{f}}$ defines the fuel temperature distribution. Finally, $\mathrm{R}$ defines the relative convergence tolerance for the corresponding magnitude, while $\mathrm{A}$ defines the absolute convergence tolerance for the corresponding magnitude. The results exportation from DYN3D-MG (MOD) to the customized coupling software environment is performed to store both the effective multiplication factor and the fission power distributions with the latter being reshaped into a full 3D representation to improve data handling, which is defined in the DYN3D powers exportation module. Under relaxation within the customized coupling software environment is applied to the fission power distributions with the under-relaxation parameter taking values below 1 to increase the stability and above 0.1 to avoid false convergence, which is defined in the DYN3D powers under relaxation module. The traditional under relaxation method is applied to the fission power for a certain iteration n and node i and is given by the following expression.

$${\mathrm{q}}_{{\mathrm{rel}}_{\mathrm{i}}}^{\mathrm{n}}=\mathsf{\theta }{\mathrm{q}}_{\mathrm{DYN}3{\mathrm{D}}_{\mathrm{i}}}^{\mathrm{n}}+\left(1-\mathsf{\theta }\right){\mathrm{q}}_{{\mathrm{rel}}_{\mathrm{i}}}^{\mathrm{n}-1}$$

(6)

In the mentioned expression, ${\mathrm{q}}_{\mathrm{rel}}$ defines the under relaxed fission power, ${\mathrm{q}}_{\mathrm{DYN}3\mathrm{D}}$ defines the resulting fission power from DYN3D, and $\mathsf{\theta }$ defines the under-relaxation factor. The under-relaxed fission power distributions importation from the customized coupling software environment to CTF is performed to interpolate from node centre values to the node face values, renormalise these by their corresponding average values, reformat these according to the card groups in CTF and, finally, retrieve them into CTF, which is defined in the CTF powers importation module. The results exportation from CTF to the customized coupling software environment is performed to store the fuel temperature, the moderator temperature, the moderator density, and the boron concentration feedback distributions with these being reshaped into a full 3D representation to improve data handling, which is defined in the CTF feedback exportation module. Under relaxation within the customized coupling software environment is applied to the fuel temperature, the moderator temperature, the moderator density, and the boron concentration feedback distributions to increase the stability with the under-relaxation parameter taking values below 1 to increase the stability and above 0.1 to avoid false convergence, which is defined in the CTF feedback under relaxation module. The traditional under relaxation method is applied to the feedback distributions including moderator density and temperature, as well as fuel temperature for a certain iteration n and node I and is given by the following expressions.

$${\mathsf{\rho }}_{{\mathrm{m}}_{{\mathrm{rel}}_{\mathrm{i}}}}^{\mathrm{n}}=\mathsf{\theta }{\mathsf{\rho }}_{{\mathrm{m}}_{{\mathrm{CTF}}_{\mathrm{i}}}}^{\mathrm{n}}+\left(1-\mathsf{\theta }\right){\mathsf{\rho }}_{{\mathrm{m}}_{{\mathrm{rel}}_{\mathrm{i}}}}^{\mathrm{n}-1}$$

(7)

$${\mathrm{T}}_{{\mathrm{m}}_{{\mathrm{rel}}_{\mathrm{i}}}}^{\mathrm{n}}=\mathsf{\theta }{\mathrm{T}}_{{\mathrm{m}}_{{\mathrm{CTF}}_{\mathrm{i}}}}^{\mathrm{n}}+\left(1-\mathsf{\theta }\right){\mathrm{T}}_{{\mathrm{m}}_{{\mathrm{rel}}_{\mathrm{i}}}}^{\mathrm{n}-1}$$

(8)

$${\mathrm{T}}_{{\mathrm{f}}_{{\mathrm{rel}}_{\mathrm{i}}}}^{\mathrm{n}}=\mathsf{\theta }{\mathrm{T}}_{{\mathrm{f}}_{{\mathrm{CTF}}_{\mathrm{i}}}}^{\mathrm{n}}+\left(1-\mathsf{\theta }\right){\mathrm{T}}_{{\mathrm{f}}_{{\mathrm{rel}}_{\mathrm{i}}}}^{\mathrm{n}-1}$$

(9)

In the mentioned expressions, ${\mathsf{\rho }}_{{\mathrm{m}}_{\mathrm{rel}}}$ defines the under relaxed moderator density, ${\mathsf{\rho }}_{{\mathrm{m}}_{\mathrm{CTF}}}$ defines the resulting moderator density from CTF, ${\mathrm{T}}_{{\mathrm{m}}_{\mathrm{rel}}}$ defines the under relaxed moderator temperature, ${\mathrm{T}}_{{\mathrm{m}}_{\mathrm{CTF}}}$ defines the resulting moderator temperature from CTF, ${\mathrm{T}}_{{\mathrm{f}}_{\mathrm{rel}}}$ defines the under relaxed fuel temperature, and ${\mathrm{T}}_{{\mathrm{f}}_{\mathrm{CTF}}}$ defines the resulting fuel temperature from CTF. The under relaxed moderator density, moderator temperature, fuel temperature, and boron concentration feedback distributions importation from the customized coupling software environment to DYN3D-MG(MOD) is performed to reformat these according to the modified modules in DYN3D-MG(MOD) and, finally, retrieve them into DYN3D-MG(MOD) which is defined in the DYN3D feedback importation module. The coupling between DYN3D and CTF can be observed in Figure 3.

4. Specifications within the Verification

Previously it was discussed that the DYN3D and CTF coupling outer iterations convergence verification was carried out using the modified KAIST benchmark. Hence, the specifications are described in the following subsection.

Modified KAIST Benchmark

The modified KAIST benchmark is an improved version of the KAIST benchmark [50] for PWR coupled reactor physics simulations. No experimental data or other code results are available. Tests carried out in the simulations include: steady state 17 × 17 fuel assemblies containing fuel pins and guide tubes, as well as burnable absorber pins with different fission power and feedback distributions. Steady state 3 × 3 mini-cores containing the previously mentioned 17 × 17 fuel assemblies also with different fission power and feedback distributions.

The modified KAIST benchmark includes a multi parameter exercise for the 17 × 17 fuel assemblies consisting of seven coupling tests based on a full power PWR. Each of these tests includes a single parameter change applied to either the total power, the axial albedos, the inlet boron concentration, the inlet temperature, the inlet mass flux, or the outlet pressure. The modified KAIST benchmark also includes a full reactor start up exercise for the 3 × 3 mini-cores consisting of three coupling tests based on a full power PWR. Each of these tests represents a different operation stage including the cold zero, the hot zero, and the full power. In the first part of the DYN3D and CTF coupling, high total powers were used leading to hot PWR behaviour where two phase phenomena is largely present. In the second part of the DYN3D and CTF coupling, standard total powers have been used leading to typical PWR behaviour where two phase phenomena is less present. All the data for the tests has been presented

Specifications include the geometry, materials, and initial and boundary conditions. The geometries for both the 17 × 17 fuel assemblies with or without burnable absorber fuel pins, as well as for the 3 × 3 mini-cores with or without burnable absorber fuel assembly are described as observed in

Table 1. The 17 × 17 fuel assembly and 3 × 3 mini-core geometries from the modified KAIST benchmark.

Type	UOX-2 (CR) 17 × 17 Assembly	UOX-2 (BA-16) 17 × 17 Assembly
Channel Width (m)	0.2142	0.2142
Cell Width (m)	0.0126	0.0126
Axial Length (Active) (m)	3.658	3.658
Number of Fuel Pins	264	248
Number of Burnable Absorber Pins	0	16
Number of Guide Tubes	25	25
Type	Homogeneous 3 × 3 Mini-Core	Heterogeneous 3 × 3 Mini-Core
Channel Width (m)	0.6426	0.6426
Cell Width (m)	0.0126	0.0126
Axial Length (Active) (m)	3.658	3.658
Number of Fuel Pins	2376	2360
Number of Burnable Absorber Pins	0	16
Number of Guide Tubes	225	225

.

The geometries for the fuel pins or burnable absorber pins, as well as for the guide tubes are described as observed in

Table 2. The fuel/burnable absorber in and guide tube geometries from the modified KAIST benchmark.

Type	Fuel/Burnable Absorber Pin
Fuel Pin Diameter (m)	0.0082
Gap Thickness (m)	0.000085
Clad Thickness (m)	0.00057
Clad Diameter (m)	0.0095
Type	Guide Tube
Clad Thickness (m)	0.000405
Guide Tube Diameter (m)	0.01224

.

The materials for the fuel pins or burnable absorber pins, as well as for the guide tubes, are described as observed in

Table 3. The fuel/burnable absorber in and guide tube materials from the modified KAIST benchmark.

Neutron Energy Groups (eV)	Group 0 ≡ (0.62506, 2231300) Group 1 ≡ (0.000014, 0.62506)
Fuel Pin Composition	UO2 (3.3% 235U, 96.7% 238U)
Burnable Absorber Pin Composition	UO2 (0.711% 235U, 90.289% 238U) + Gd2O3 (9.0%)
Clad Composition	Zircalloy (97.91% Zr, 1.59% Sn, 0.5% Fe)
Fuel Density (kg/m3)	10,040
Fuel Specific Heat (J/kg K)	${\mathrm{c}}_{{\mathrm{p}}_{\mathrm{fuel}}}=\frac{8.5013 {10}^8{\mathrm{e}}^{\frac{535.285}{\mathrm{T}}}}{{\mathrm{T}}^2{\left({\mathrm{e}}^{\frac{535.285}{\mathrm{T}}}-1\right)}^2 }+0.0243\mathrm{T}+\frac{1.6587 {10}^{12}}{{\mathrm{T}}^2}{\mathrm{e}}^{-\frac{18968}{\mathrm{T}}}$	(10)
Fuel Thermal Conductivity (W/m K)	${\mathrm{k}}_{\mathrm{fuel}}=\max \left(\frac{2335}{464+\mathrm{T}}, 1.1038\right)+7.027 {10}^{-3} {10}^{-3}{ \mathrm{e}}^{1.867 {10}^{-3} \mathrm{T}}$	(11)
Gap Gas	He
Gap Heat Conductance (kJ/m2 K)	5678
Clad Density (kg/m3)	6400
Clad Specific Heat (J/kg K)	${\mathrm{c}}_{{\mathrm{p}}_{\mathrm{clad}}}=252.54+0.11474\mathrm{T}$	(12)
Clad Thermal Conductivity (W/m K)	${\mathrm{k}}_{\mathrm{clad}}=7.51+2.09 {10}^{-2}\mathrm{T}-1.45 {10}^{-5}{\mathrm{T}}^2+7.67 {10}^{-9}{\mathrm{T}}^3$	(13)

.

The initial and boundary conditions for the 17 × 17 fuel assemblies multi-parameter exercise are described as observed in

Table 4. The 17 × 17 fuel assemblies initial and boundary conditions.

Case	Axial Albedos	Boric Acid Concentration (ppm)	Power (MW)	Inlet Temperature (C)	Mass Flux (kg/m2 s)	Outlet Pressure (Bar)
Full Power	0.5	800	17.308	293.30	2889.33	155
High Power	0.5	800	22.308	293.30	2889.33	155
High Albedo	0.75	800	17.308	293.30	2889.33	155
Zero Boron	0.5	0	17.308	293.30	2889.33	155
High Temperature	0.5	800	17.308	303.30	2889.33	155
Low Flux	0.5	800	17.308	293.30	2476.58	155
Low Pressure	0.5	800	17.308	293.30	2889.33	145

.

The initial and boundary conditions for the 3 × 3 mini-cores full reactor start up exercise are described as observed in

Table 5. The 3 × 3 mini-cores initial and boundary conditions.

Case	Axial Albedos	Boric Acid Concentration (ppm)	Power (MW)	Inlet Temperature (C)	Mass Flux (kg/m2 s)	Outlet Pressure (Bar)
Cold Zero Power	0.5	800	0	25.30	2889.33	155
Hot Zero Power	0.5	800	0	293.30	2889.33	155
Full Power	0.5	800	155.772	293.30	2889.33	155

.

5. Models within the Verification

Previously it was discussed that the DYN3D and CTF coupling outer iterations convergence verification was carried by simulating the modified KAIST benchmark. Hence, the models are described in the following subsection.

Modified KAIST Benchmark

Different models between the 17 × 17 fuel assemblies and the 3 × 3 mini-cores in both DYN3D and CTF include the meshes. In the case of DYN3D and the UOX-2 (CR) fuel assembly, 264 fuel pins as well as 25 guide tubes are included into 289 fuel cells (fuel pin centred system) all of which are composed by 36 uniform axial node layers. In the case of CTF and the UOX-2 (CR) fuel assembly, 264 fuel pins, as well as 25 guide tubes are included in 324 subchannels (subchannel centred system) all of which are connected in between by 612 gaps and composed by 36 uniform axial node layers. In the case of DYN3D and the UOX-2 (BA-16) fuel assembly, 248 fuel pins as well as 16 burnable absorber pins, and 25 guide tubes are included into 289 fuel cells (fuel pin centred system) all of which are composed by 36 uniform axial node layers. In the case of CTF and the UOX-2 (BA-16) fuel assembly, 248 fuel pins as well as 16 burnable absorber pins and 25 guide tubes are included in 324 subchannels (subchannel centred system), all of which are connected in between by 612 gaps and composed by 36 uniform axial node layers. In the case of DYN3D and the homogeneous mini-core, 2376 fuel pins, as well 225 guide tubes, are included in 2601 fuel cells (fuel pin centred system), all of which are composed by 36 uniform axial node layers. In the case of CTF and the homogeneous mini-core, 2376 fuel pins as well as 225 guide tubes are included in 2704 subchannels (subchannel centred system) all of which are connected in between by 5304 gaps and composed by 36 uniform axial node layers. In the case of DYN3D and the heterogeneous mini-core, 2360 fuel pins as well 16 burnable absorbers and 225 guide tubes are included in 2601 fuel cells (fuel pin centred system), all of which are composed by 36 uniform axial node layers. In the case of CTF and the heterogeneous mini-core, 2360 fuel pins, as well as 16 burnable absorber pins and 225 guide tubes are included in 2704 subchannels (subchannel centred system), all of which are connected in between by 5304 gaps and composed by 36 uniform axial node layers. The fuel pin centred system mesh in DYN3D and the subchannel centred system mesh in CTF for both the fuel assemblies and mini-cores can be observed in Figure 4.

Common models between the 17 × 17 fuel assemblies and the 3 × 3 mini-cores in both DYN3D and CTF include the cross sections tables as well as nodal expansion methods and the thermal and mechanical property tables, as well as the heat and mass transfer models. In the case of the neutronics in DYN3D, the neutron diffusion of two energy groups has been modelled as it is available. Homogenised fuel pin cross sections for the fuel and burnable absorber pins, as well as the guide tube, have been produced using SCALE-POLARIS [55]. The SP3 nodal expansion method is modelled as it is the one available for cartesian geometries [56]. Full reflective boundary conditions are only modelled radially while different partial reflective boundary conditions are modelled axially according to the specifications to provide a more complete representation of the system. No ADF have been included for simplicity. No fuel pin power reconstruction is performed, as fuel pin discretization is used instead. No control rods are used for simplicity. In the case of the thermal hydraulics in DYN3D, no crossflow or turbulent mixing has been modelled, as it is not available. Material properties for the fuel and burnable absorber pins, as well as the guide tubes, have been used through the average of the distributions provided in the modified KAIST benchmark. Water properties for the moderator have been modelled using the IFC-67 approach out of several available [57]. Nucleate boiling has been modelled using the Rassokhin and Borishaskji correlation it being the one available [58]. Departure from nucleate boiling has been modelled using the Bezrukov and Astakhov (OKB-2) correlation, it being the most accurate out of several available [59]. No entrainment has been modelled, as it is not available. Friction pressure losses have been modelled using the Filonenko and Osmachkin one- and two-phase multiplier correlation, it being the one available [60]. No spacer grids pressure losses have been modelled for simplicity. In the case of the thermal hydraulics in CTF, crossflow and turbulent mixing have been modelled in the case of the former via the gaps and in the case of the latter using the Rogers and Rosehart correlation as it determines the two-phase mixing coefficient according to the corresponding subchannel [61]. Material properties for the fuel and burnable absorber pins have been modelled through tables of the distributions provided in the modified KAIST benchmark. Water properties have been modelled using the IAPWS approach out of several available [62]. Nucleate boiling has been modelled using the Thom correlation with it remaining valid along a wide range of pressures [63]. Departure from nucleate boiling has been modelled using the W-3 correlation it being widely used to analyse PWR. Entrainment has been modelled using the CTF method [64]. Friction pressure losses have been modelled using the McAdam’s one phase and two-phase multipliers correlation it being widely used to analyse PWR [65]. No spacer grids pressure losses have been modelled for simplicity.

In addition, another common model between the 17 × 17 fuel assemblies and the 3 × 3 mini-cores in both DYN3D and the DYN3D and CTF coupling includes the under relaxation. In both cases, it has been modelled using the traditional under relaxation method with an under-relaxation factor value of 0.5 improving the stability while avoiding false convergence.

6. Results and Analysis

Results for the coupled reactor physics in DYN3D and the DYN3D and CTF coupling were acquired by simulating the modified KAIST benchmark. DYN3D to DYN3D and CTF coupling comparisons within the DYN3D and CTF coupling outer iterations convergence verification in the steady state are discussed for the effective multiplication factor and the fission power in the following subsections while these are discussed for the feedback in the appendixes.

6.1. Fuel Assemblies

DYN3D to DYN3D and CTF coupling comparisons within the modified multi-parameter exercise for the UOX-2 (CR) and the UOX-2 (BA-16) fuel assemblies are presented for the effective multiplication factor and the fission power distributions from both physical and convergence perspectives while the fluid density feedback, fluid temperature feedback, and fuel temperature feedback are presented from both physical and convergence perspectives in Appendix A. The final iteration effective multiplication factor value for both the UOX-2 (CR) and the UOX-2 (BA-16) fuel assemblies is given for all the tests to show the similarities and differences between the DYN3D and the DYN3D and CTF coupling values. Such values are described as observed in

Table 6. UOX-2 (CR) and UOX-2 (BA-16) fuel assemblies effective multiplication factor values.

	UOX-2 (CR) Fuel Assembly			UOX-2 (BA-16) Fuel Assembly
	Effective Multiplication Factor		Reactivity Difference (pcm)	Effective Multiplication Factor		Reactivity Difference (pcm)
	DYN3D	DYN3D and CTF Coupling		DYN3D	DYN3D and CTF Coupling
Full Power	1.23923	1.23733	−124	0.99991	1.00044	53
High Power	1.23387	1.23097	−191	0.99057	0.99365	313
High Albedo	1.24018	1.23855	−106	1.00069	1.00147	78
Zero Boron	1.33922	1.33532	−218	1.06322	1.06290	−28
High Temperature	1.23635	1.23445	−124	0.99570	0.99735	166
Low Flux	1.23796	1.23594	−132	0.99709	0.99875	167
Low Pressure	1.23897	1.23715	−119	0.99913	1.00022	109

.

DYN3D derives the effective multiplication factor from fast energy group fission and removal cross sections, as well as leakage contributions and from thermal energy group scattering and absorption cross sections, as well as leakage contributions by solving the two energy groups neutron diffusion equations where the reactivity is updated through feedback coefficients by applying cross sections interpolation.

In DYN3D, as well as in the DYN3D and CTF coupling, a decrease in the effective multiplication factor was seen in all the tests in the UOX-2 (BA-16) fuel assembly when compared to the UOX-2 (CR) fuel assembly. This decrease in the effective multiplication factor occurred due to the higher neutron absorption cross section in the burnable absorber fuel pins, which resulted in less thermal neutrons and, therefore, lower reactivities according to the two energy groups neutron diffusion equations.

In DYN3D, as well as in the DYN3D and CTF coupling, a decrease in the effective multiplication factor was seen in the high power, high temperature, low flow, and low-pressure tests when compared to the full power test in both the UOX-2 (CR) and in the UOX-2 (BA-16) fuel assemblies. This decrease in the effective multiplication factor occurred due to several reasons: In the high-power test, this occurred due to the higher fuel and moderator temperature, as well as higher moderator density feedback coefficients in the fuel pin cells which resulted in less thermal neutrons and therefore lower reactivities according to the cross sections interpolation. In the high-temperature test, this occurred due to the higher moderator temperature feedback coefficient in the fuel pin cells which resulted in less thermal neutrons and, therefore, lower reactivities according to the cross sections interpolation. In the low flow test, this occurred due to the higher moderator density feedback coefficient in the fuel pin cells which resulted in less thermal neutrons and, therefore, lower reactivities according to the cross sections interpolation. In the low-pressure test, this occurred due to the higher moderator density feedback coefficient in the fuel pin cells which resulted in less thermal neutrons and, therefore, lower reactivities according to the cross sections interpolation. In DYN3D, as well as in the DYN3D and CTF coupling, an increase in the effective multiplication factor was seen in the high albedo, and zero boron tests when compared to the full power test in both the UOX-2 (CR) and in the UOX-2 (BA-16) fuel assemblies. This increase in the effective multiplication factor occurred due to several reasons: In the high albedo test, this occurred due to the lower neutron leakage in the fuel pins, which resulted in more neutrons and, therefore, higher reactivities according to the two energy groups neutron diffusion equations. In the zero-boron test, this occurred due to the lower boron feedback coefficient in the fuel pin cells, which resulted in more thermal neutrons and therefore higher reactivities according to the cross-sections interpolation.

Between DYN3D and the DYN3D and CTF coupling, differences in the effective multiplication factor values, as well as reactivity difference in both the UOX-2 (CR) and in the UOX-2 (BA-16) fuel assemblies are seen to be present. These differences in the effective multiplication factor as well as reactivity difference occurred due to the different fuel temperature feedback coefficients resulting from different fuel rod models, as well as the different moderator temperature and density feedback coefficients resulting from either the presence or absence of crossflow and turbulent mixing and the different boiling and inter-phase models. The achieved effective multiplication factor, as well as reactivity difference values in both the UOX-2 (CR) and UOX-2 (BA-16) fuel assemblies are compatible with each other due to either the similarity or low difference between them.

All iterations reactivity differences for both the UOX-2 (CR) and the UOX-2 (BA-16) fuel assemblies are given for all the tests to show the convergence of the DYN3D and CTF coupling values. Such differences are represented in Figure 5.

The DYN3D and CTF coupling achieves the convergence of the reactivity via the under relaxation of both the power and feedback distributions until it achieves the steady state.

In the DYN3D and CTF coupling, the convergence of the reactivity was seen with higher iteration number in all the tests in both the UOX-2 (CR) and the UOX-2 (BA-16) fuel assemblies. The convergence of the reactivity occurred between 8 and 9 iterations due to several reasons: the presence of standard or mildly hot boundary conditions, which resulted in a faster or slower convergence rate and, therefore, less or more required iterations according to the convergence criteria. The reasonable under relaxation factor, which results in higher stability and, therefore, in part more required iterations according to the convergence criteria. In the DYN3D and CTF coupling, a slower convergence of the reactivity was seen with higher iteration number in all the tests in the UOX-2 (BA-16) fuel assembly when compared to the UOX-2 (CR) fuel assembly. This slower convergence of the reactivity occurred due to the higher neutron absorption cross section in the burnable absorber fuel pins, which resulted in more heterogeneity and, therefore, more required iterations according to the convergence criteria.

The final iteration average axial fission power distribution peak for both the UOX-2 (CR) and the UOX-2 (BA-16) fuel assemblies are given for all the tests to show the similarities and differences between the DYN3D and the DYN3D and CTF coupling distributions. Such distributions peaks are represented in Figure 6.

DYN3D derives the fission power distribution from fast and thermal energy groups fission reaction rate contributions by solving the two energy groups neutron diffusion equations where the former are affected by the feedback.

In DYN3D, as well as in the DYN3D and CTF coupling, an increase in asymmetry in the average axial fission power distributions was seen in all the tests in the UOX-2 (BA-16) fuel assembly when compared to the UOX-2 (CR) fuel assembly. This increase in asymmetry in the average axial fission power distributions occurred due to the higher neutron absorption cross section in the burnable absorber fuel pins, which resulted in less thermal neutrons and, therefore, lower fission reaction rates in the former, leading to the use of remaining neutrons in the fuel pins to preserve the same total power as when there are only fuel pins, which resulted, therefore, in higher fission reaction rates in the latter according to the two energy groups neutron diffusion equations.

In DYN3D, as well as in the DYN3D and CTF coupling, an increase in asymmetry in the average axial fission power distributions was seen in the high-power, high-albedo, zero-boron, high-temperature, low-flow, and low-pressure tests when compared to the full power test in both the UOX-2 (CR) and in the UOX-2 (BA-16) fuel assemblies. This increase in asymmetry in the average axial fission power distributions occurred due to different reasons: In the high-power test, this occurred due to the higher fuel and moderator temperature, as well as higher moderator density feedback coefficients in the fuel pin cells, which resulted in less thermal neutrons and, therefore, lower fission reaction rates at the top layers of the fuel pin cells, leading to the use of remaining neutrons to preserve the total power, which therefore resulted in higher fission reaction rates at the bottom layers of the fuel pin cells according to the previously mentioned equations. In the high-albedo test, this occurred due to the lower neutron leakage in the fuel pin cells, which resulted in more neutrons and, therefore, higher fission reaction rates in the fuel pin cells according to the previously mentioned equations. In the zero-boron test, this occurred due to the lower boron feedback coefficient in the fuel pin cells, which resulted in more thermal neutrons and, therefore, higher fission reaction rates in the fuel pin cells according to the previously mentioned equations. In the high temperature test, this occurred due to the higher moderator temperature coefficient in the fuel pin cells, which resulted in less thermal neutrons and, therefore, lower fission reaction rates at the top layers of the fuel pins, leading to the use of remaining neutrons to preserve the total power, which, therefore, resulted in higher fission reaction rates at the bottom layers of the fuel pin cells according to the previously mentioned equations. In the low flow test, this occurred due to the higher moderator density feedback coefficient in the fuel pin cells, which resulted in less thermal neutrons and, therefore, lower fission reaction rates at the top layers of the fuel pins, leading to the use of remaining neutrons to preserve the total power, which, therefore, resulted in higher fission reaction rates at the bottom layers of the fuel pin cells according to the previously mentioned equations. In the low-pressure test, this occurred due to the higher moderator density feedback coefficient in the fuel pin cells, which resulted in less thermal neutrons and, therefore, lower fission reaction rates at the top layers of the fuel pins, leading to the use of remaining neutrons to preserve the total power, which therefore resulted in higher fission reaction rates at the bottom layers of the fuel pin cells according to the previously mentioned equations.

Between DYN3D and the DYN3D and CTF coupling, differences in the average axial fission power distributions in both the UOX-2 (CR) and in the UOX-2 (BA-16) fuel assemblies are seen to be present. These differences in the average axial fission power distributions occurred due to the different fuel temperature feedback coefficients resulting from different fuel rod models, as well as the different moderator temperature and density feedback coefficients resulting from either the presence or absence of crossflow and turbulent mixing and the different boiling and inter-phase models. The achieved average axial fission power distributions in the UOX-2 (CR) fuel assembly are compatible with each other due to the similarity between them. The achieved average axial fission power distributions in the UOX-2 (BA-16) fuel assembly are less compatible with each other due to the differences between them.

The final iteration axial fission power distribution peak for central, side, and corner fuel pins, and average between fuel pins, as well as the final iteration transversal fission power distribution for all the fuel pins at the average axial node layer are given for both the UOX-2 (CR) and UOX-2 (BA-16) fuel assemblies full power test to show the similarities and differences between the DYN3D and the DYN3D and CTF coupling distributions. Such distributions are represented in Figure 7 and Figure 8.

In DYN3D, as well as in the DYN3D and CTF coupling, an increase in the axial fission power distribution, as well as an increase in the transversal fission power distribution at the average axial node layer was seen in the central fuel pin cells when compared to the side and corner fuel pin cells in the full power test in the UOX-2 (CR) fuel assembly. This increase in the fission power distribution occurred due to the lower neutron leakage in the central fuel pin cells, which resulted in more neutrons and, therefore, higher fission reaction rates in the former according to the two energy groups neutron diffusion equations. In DYN3D, as well as in the DYN3D and CTF coupling, a decrease in the axial fission power distribution, as well as a decrease in the transversal fission power at the average axial node layer, was seen in the central fuel pin cells when compared to the side and corner fuel pin cells in the full power test in the UOX-2 (BA-16) fuel assembly. This decrease in the fission power distribution occurred due to the higher neutron absorption cross section in the burnable absorber fuel pins, which resulted in less neutrons and, therefore, lower fission reaction rates in the central fuel pins according to the two energy groups neutron diffusion equations.

The relative difference between the former is given for both the UOX-2 (CR) and UOX-2 (BA-16) fuel assemblies full power test to show the differences between the DYN3D and the DYN3D and CTF coupling distributions. Such relative difference between distributions is represented in Figure 9.

Between DYN3D and the DYN3D and CTF coupling, differences in the axial fission power distribution and in the transversal fission power distributions at the average axial node layer, as well as in the relative difference between the previously mentioned in the full power test either the UOX-2 (CR) or UOX-2 (BA-16) fuel assemblies full power test were seen to be present. These differences in the axial and transversal fission power distributions occurred due to the different fuel temperature feedback coefficients resulting from different fuel rod models, as well as the different moderator temperature and density feedback coefficients resulting from either the presence or absence of crossflow and turbulent mixing and the different boiling and inter-phase models.

All iterations maximum fission power values for both the UOX-2 (CR) and the UOX-2 (BA-16) fuel assemblies are given for all the tests to show the convergence of the DYN3D and CTF coupling distributions. Such values are represented in Figure 10.

The DYN3D and CTF coupling achieves the convergence of the power distributions via their own under relaxation until these achieve the steady state.

In the DYN3D and CTF coupling, the convergence of the maximum fission power values was seen with higher iteration number in in all the tests in both the UOX-2 (CR) and the UOX-2 (BA-16) fuel assemblies. This convergence of the maximum fission power values occurred between 6 and 9 iterations due to several reasons: the presence of standard or mildly hot boundary conditions, which resulted on a faster or slower convergence rate and, therefore, less or more required iterations according to the convergence criteria. The reasonable under relaxation factor, which resulted on higher stability and, therefore, more required iterations according to the convergence criteria. In the DYN3D and CTF coupling, a slower convergence of the maximum fission power values was seen with higher iteration number in all the tests in the UOX-2 (BA-16) fuel assembly when compared to the UOX-2 (CR) fuel assembly. This slower convergence of the maximum fission power values occurred due to the higher neutron absorption cross section in the burnable absorber fuel pins, which resulted in more heterogeneity and, therefore, more required iterations according to the convergence criteria.

6.2. Mini-Cores

DYN3D to DYN3D and CTF coupling comparisons within the full reactor start up exercise for the homogeneous and heterogeneous mini-cores are presented for the effective multiplication factor and the fission power distributions from both physical and convergence perspectives while the fluid density feedback, fluid temperature feedback and fuel temperature feedback are presented from both physical and convergence perspectives in Appendix B. The final iteration effective multiplication factor for both the homogeneous and heterogeneous mini-cores is given for all the tests to show the similarities and differences between the DYN3D and the DYN3D and CTF coupling values. Such values are described as observed in

Table 7. Homogeneous and heterogeneous mini-cores effective multiplication factor values.

	Homogeneous Mini-Core			Heterogeneous Mini-Core
	Effective Multiplication Factor		Reactivity Difference (pcm)	Effective Multiplication Factor		Reactivity Difference (pcm)
	DYN3D	DYN3D and CTF Coupling		DYN3D	DYN3D and CTF Coupling
Cold Zero Power	1.26175	1.26175	0	1.24026	1.24026	0
Hot Zero Power	1.25553	1.25552	−1	1.23292	1.23291	−1
Full Power	1.23924	1.23737	−122	1.21578	1.21394	−125

.

In DYN3D, as well as in the DYN3D and CTF coupling, a mild decrease in the effective multiplication factor was seen in all the tests in the heterogeneous mini-core when compared to the homogeneous mini-core. This mild decrease in the effective multiplication factor occurred due to the compensation between the higher neutron absorption cross-section in the central fuel assembly, which resulted in less thermal neutrons and, therefore, lower reactivity in the former with the lower neutron absorption cross section in the side and corner fuel assemblies, which resulted in more thermal neutrons and therefore higher reactivity in the latter according to the two energy groups neutron diffusion equations.

In DYN3D, as well as in the DYN3D and CTF coupling, a decrease in the effective multiplication factor was seen in the hot zero power and full power tests when compared to the cold zero power test in both the homogeneous and heterogeneous mini-cores. This decrease in the effective multiplication factor occurred due to several reasons: In the hot zero power test, this occurred due to the higher moderator temperature feedback coefficient in the fuel assemblies, which resulted in less thermal neutrons and, therefore, lower reactivities according to the cross sections interpolation. In the full power test, this occurred due to the higher fuel and moderator temperature, as well as higher moderator density feedback coefficients in the fuel assemblies, which resulted in less thermal neutrons and, therefore, lower reactivities according to the cross sections interpolation.

Between DYN3D and the DYN3D and CTF coupling, differences in the effective multiplication factor values, as well as reactivity difference in both the homogeneous and heterogeneous mini-cores are seen to be present. These differences in the effective multiplication factor, as well as reactivity difference occurred due to the different fuel temperature feedback coefficients resulting from different fuel rod models, as well as the different moderator temperature and density feedback coefficients resulting from either the presence or absence of crossflow and turbulent mixing and the different boiling and inter-phase models. The achieved effective multiplication factor, as well as reactivity difference values in both the homogeneous and heterogeneous mini-cores are compatible with each other due to either the similarity or low difference between them.

All iterations reactivity differences for both the homogeneous and heterogeneous mini-cores are given for all the tests to show the convergence of the DYN3D and CTF coupling values. Such differences are represented in Figure 11.

In the DYN3D and CTF coupling, the convergence of the reactivity was seen with higher iteration number in all the tests in both the homogeneous and heterogeneous mini-cores. The convergence of the reactivity occurred between 3 and 9 due to several reasons: the presence of cold or standard boundary conditions, which resulted on a very fast or fast convergence rate and therefore less required iterations according to the convergence criteria. The reasonable under relaxation factor, which resulted on higher stability and, therefore, in part more required iterations according to the convergence criteria. In the DYN3D and CTF coupling, a mildly slower convergence of the reactivity was seen with higher iteration number mainly in the full power case in the heterogeneous mini-core when compared to the homogeneous mini-core. This mildly slower convergence of the reactivity occurred due to the compensation between the higher neutron absorption cross section in the central fuel assembly with the lower neutron absorption cross section in the side and corner fuel assemblies, which resulted only in slightly more heterogeneity and therefore similar required iterations according to the convergence criteria.

The final iteration average axial fission power distribution peak for both the homogeneous and heterogeneous mini-cores are given for all the tests to show the similarities and differences between the DYN3D and the DYN3D and CTF coupling distributions. Such distributions peaks are represented in Figure 12.

In DYN3D, as well as in the DYN3D and CTF coupling, the preservation of symmetry in the average axial fission power distributions was seen in all the tests in the heterogeneous mini-core when compared to the homogeneous mini-core. This preservation of symmetry in the average axial fission power distributions occurred due to the compensation between the higher neutron absorption cross section in the central fuel assembly, which resulted in less thermal neutrons and, therefore, lower fission reaction rates in the former with the lower neutron absorption cross section in the side and corner fuel assemblies, which resulted in more thermal neutrons and, therefore, higher fission reaction rates in the latter according to the two energy groups neutron diffusion equations.

In DYN3D, as well as in the DYN3D and CTF coupling, an increase in asymmetry in the average axial fission power distributions was only in the full power test when compared to the cold zero power test in both the homogeneous and heterogeneous mini-cores. This increase in asymmetry in the average axial fission power distributions occurred due to the higher fuel and moderator temperature, as well as higher moderator density feedback coefficients in the fuel assemblies, which resulted in less thermal neutrons and, therefore, lower fission reaction rates at the top layers of the fuel assemblies, leading to the use of remaining neutrons to preserve the total power, which, therefore, resulted in higher fission reaction rates at the bottom layers of the fuel assemblies according to the previously mentioned equations.

Between DYN3D and the DYN3D and CTF coupling, differences in the average axial fission power distributions in both the homogeneous and heterogeneous mini-cores are seen to be present. These differences in the average axial fission power distributions occurred due to the different fuel temperature feedback coefficients resulting from different fuel rod models, as well as the different moderator temperature and density feedback coefficients resulting from either the presence or absence of crossflow and turbulent mixing and the different boiling and inter-phase models. The achieved average axial fission power distributions in both the homogeneous and heterogeneous mini-cores are compatible with each other due to the similarity between them.

The final iteration axial fission power distribution peak for central, side, and corner fuel assemblies and average between fuel assemblies, as well as the final iteration transversal fission power distribution for all the fuel assemblies at the average axial node layer are given for both the homogeneous and heterogeneous mini-cores full power test to show the similarities and differences between the DYN3D and the DYN3D and CTF coupling distributions. Such distributions are represented in Figure 13 and Figure 14.

In DYN3D, as well as in the DYN3D and CTF coupling, an equivalent axial fission power distribution, as well as an equivalent transversal fission power distribution at the average axial node layer, was seen in the central fuel assembly when compared to the side and corner fuel assemblies in full power test in the homogeneous mini-core. This equivalence in the fission power distribution occurred due to the equivalent neutron absorption cross section in all the fuel assemblies, which resulted in an equivalent number of neutrons with no inter assembly power redistribution and, therefore, equivalent fission reaction rates in the central, side, and corner fuel assemblies according to the two energy groups neutron diffusion equations. In DYN3D, as well as in the DYN3D and CTF coupling, a decrease in the axial fission power distribution, as well as a decrease in the transversal fission power distribution at the average axial node layer was seen central fuel assembly when compared to the side and corner fuel assemblies in full power test in the heterogeneous mini-core. This decrease in the axial fission power distribution occurred due to the higher neutron absorption cross section in the central fuel assembly which resulted in a lower number of neutrons with inter-assembly power redistribution and, therefore, lower fission reaction rates in the central fuel assembly than in the side and corner fuel assemblies according to the two energy groups neutron diffusion equations.

The relative difference between the former are given for both the homogeneous and heterogeneous mini-cores full power test to show the differences between the DYN3D and the DYN3D and CTF coupling distributions. Such relative differences between distributions are represented in Figure 15.

Between DYN3D and the DYN3D and CTF coupling, differences in the axial fission power distribution and in the transversal fission power distributions at the average axial node layer, as well as in the relative difference between the previously mentioned in the full power test in either the homogeneous or heterogeneous mini-core were seen to be present. These differences in the axial and transversal fission power distributions occurred due to the different fuel temperature feedback coefficients resulting from different fuel rod models, as well as the different moderator temperature and density feedback coefficients resulting from either the presence or absence of crossflow and turbulent mixing and the different boiling and inter-phase models.

All iterations maximum fission power values for both the homogeneous and the heterogeneous mini-cores are given for all the tests to show the convergence of the DYN3D and CTF coupling distributions. Such values are represented in Figure 16.

The DYN3D and CTF coupling achieves the convergence of the fission power distributions via their own under relaxation until these achieve the steady state.

In the DYN3D and CTF coupling, the convergence of the maximum fission power values was seen with higher iteration number in all the tests in both the homogeneous and heterogeneous mini-cores. This convergence of the maximum fission power values occurred between 1 and 7 iterations due to several reasons: the presence of cold or standard boundary conditions, which resulted on a faster convergence rate and, therefore, less required iterations according to the convergence criteria. The reasonable under relaxation factor, which resulted on higher stability and therefore in part more required iterations according to the convergence criteria. In the DYN3D and CTF coupling, a mildly slower convergence of the maximum fission power values was seen with higher iteration number in the full power case in the heterogeneous mini-core when compared to the homogeneous mini-core. This mildly slower convergence of the maximum fission power values occurred due to the compensation between the higher neutron absorption cross section in the central fuel assembly with the lower neutron absorption cross section in the side and corner fuel assemblies, which resulted only in slightly more heterogeneity and, therefore, similar required iterations according to the convergence criteria.

7. Conclusions

Finally, the last objective in the aim of delivering the coupling between DYN3D and CTF within the multiscale and multi-physics nuclear software development has been completed through the analysis of the parallel two ways coupling between them. The coupling outer iterations convergence has been verified delivering improved coupled reactor physics at the fuel pin level. In this parallel two ways coupling both converged fission power and feedback distributions were obtained through the customized coupling software environment modules, the multiple iterations CTF feedback and DYN3D fission power distributions, as well as the DYN3D and CTF modules and customized coupling software environment convergence criteria.

Comparing the improved coupled reactor physics at the fuel pin level for both the fuel assemblies and mini-cores delivered by the DYN3D and CTF coupling to the simplified coupled reactor physics at the fuel pin level delivered by DYN3D. In the case of the fuel assemblies, the DYN3D and CTF coupling differs from DYN3D through the effective multiplication factor and reactivity difference values, as well as through the fission power and feedback distributions and maximum and minimum values due to the different fuel temperature feedback coefficients resulting from different fuel rod models, as well as the different moderator temperature and density feedback coefficients resulting from either the presence or absence of crossflow and turbulent mixing and the different boiling and inter-phase models. In the case of the mini-cores the DYN3D and CTF coupling differs less from DYN3D through the effective multiplication factor and reactivity difference values as well as through the fission power and feedback distributions and maximum and minimum values due to the compensation between the former reasons for the fuel assemblies with power redistribution between the fuel assemblies.

In general, the DYN3D and CTF coupling provides improved coupled reactor physics over DYN3D due to the presence in the former as opposed to in the latter of crossflow and turbulent mixing. However, the DYN3D and CTF coupling is more computationally expensive compared to DYN3D which is less computationally expensive as the full simulation time in the latter ranges from around 2 min for the fuel assemblies to 20 min for the mini-cores, while the full simulation time in the former ranges from 1 h for the fuel assemblies to 10 h for the mini-cores.

8. Future Work

The final aim and objective is comprehended by the LOTUS and CTF coupling verification which will be carried out to analyse a parallel two-way coupling between these codes to deliver verified full coupled reactor physics at the fuel pin level. Fission power distributions will be obtained in the LOTUS and CTF coupling through the customized coupling software environment modules, the multiple iterations CTF feedback distributions, as well as the LOTUS and CTF modules and customized coupling software environment convergence criteria. Feedback distributions will be obtained in the LOTUS and CTF coupling through the customized coupling software environment modules, the multiple iterations LOTUS fission power distributions, as well as the LOTUS and CTF modules and customized coupling software environment convergence criteria. This parallel two-way coupling analysis will determine through both the neutronics and thermal hydraulics when should the LOTUS and CTF coupling be used instead of the DYN3D and CTF coupling to deliver full coupled reactor physics at the fuel pin level.