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A coupled neutronics thermal-hydraulics code NODAL3 has been developed based on the few-group neutron diffusion equation in 3-dimensional geometry for typical PWR static and transient analyses. The spatial variables are treated by using a polynomial nodal method while for the neutron dynamic solver the adiabatic and improved quasistatic methods are adopted. In this paper we report the benchmark calculation results of the code against the OECD/NEA CRP PWR rod ejection cases. The objective of this work is to determine the accuracy of NODAL3 code in analysing the reactivity initiated accident due to the control rod ejection. The NEACRP PWR rod ejection cases are chosen since many organizations participated in the NEA project using various methods as well as approximations, so that, in addition to the reference solutions, the calculation results of NODAL3 code can also be compared to other codes’ results. The transient parameters to be verified are time of power peak, power peak, final power, final average Doppler temperature, maximum fuel temperature, and final coolant temperature. The results of NODAL3 code agree well with the PHANTHER reference solutions in 1993 and 1997 (revised). Comparison with other validated codes, DYN3D/R and ANCK, shows also a satisfactory agreement.

The National Nuclear Energy Agency of Indonesia (BATAN) has been operating three research reactors with the nominal thermal power of 100 kW, 2 MW, and 30 MW, respectively, for nuclear science, technology, and engineering research and development (R&D). The oldest 100 kW reactor has been operated since 1965. To the present date there is no nuclear power plant (NPP) due to strong dependency of the national primary energy on the fossil fuels [

The development of analytical tools in the agency was initiated 2 decades ago by the development of the 2-dimensional (2D) and 3-dimensional (3D) multigroup neutron diffusion codes for various types of research reactors, namely, the BATAN-2DIFF and BATAN-3DIFF codes, respectively [

In the last several years, a 3D coupled neutronics and thermal-hydraulic calculation code, MTR-DYN, had been developed for safety analysis of a material testing research reactor (MTR) [

Based on the experiences in developing the static and transient calculation codes for research reactors, the development of the incore fuel management and 3D transient analysis codes is carried out for typical pressurized water reactors (PWR). PWR-type reactor is chosen based on the guidance of the national research programs in 2010–2014 [

As is well known, the PWR core dimension is considerably much larger compared to one of the research reactors, so that the neutron diffusion problem in PWRs is commonly solved by modern nodal methods [

The NODAL3 code has been verified with the steady state light water reactor (LWR) benchmarks, such as IAEA-2D, KOERBERG, BIBLIS, and IAEA-3D, and very satisfactory results were obtained [

This paper is organized as follows. In Section

NODAL3 code consists of three modules; the first module deals with the nodal equation for the steady state problems; the second module deals with the thermal-hydraulics model of a typical PWR fuel pin; and the third module is the time-dependent solver for the reactor dynamics. In the first module, the few-group neutron diffusion equation in 3D Cartesian geometry is discretized spatially using the polynomial nodal method (PNM). A coarse mesh finite difference (CFMD) formulation is used to determine the node-averaged neutron fluxes and the eigenvalue, while the PNM method is used to estimate the accurate coupling between adjacent nodes in the core. Quadratic polynomial expansion for the transverse-integrated flux is adopted [

In the second module, that is, the thermal-hydraulic module, the heat conduction problem in the fuel rods is discretized in time and space using the conventional finite-difference method. Heat conduction is considered only in the radial direction. Fluid dynamic of the cooling water is modelled under a single-phase flow condition. The mass flow rate in each cooling channel is assumed to be known and specified by the code user. As a result, only the mass continuity and energy conservation equations are to be solved. These are discretized in space and time using finite-difference method and implicit scheme, respectively [

In the third module, two time-dependent reactor dynamics models are available, that is, the adiabatic method (AM) and improved quasistatic methods (IQSM). These two methods are selected since they have a high accuracy [^{−1}); ^{−2} s^{−1}); ^{−1}); ^{−1}); ^{−3}).

In the AM, firstly, the difference between the neutron spectra of delayed neutrons and the ones of prompt neutrons is neglected. In other words, the delayed neutrons from their precursors are assumed to be born at the same time with the prompt neutrons. Secondly, all time derivatives of the amplitude and shape functions are neglected. Thus, (

In the IQSM, the time derivative of the shape function is approximated with the backward finite difference scheme:

In addition to the three modules, what is not less important is the time-step adjustment. Different time steps for amplitude function (^{−5} seconds. On the other hand,

Control rod ejection event can occur as a consequence of the rupture of the control rod drive mechanism (CRDM) in a PWR. This event is followed by a significant localized perturbation of the neutronics and thermal-hydraulics core parameters. Therefore, the PWR rod ejection benchmark prepared by the OECD/NEA can be used to evaluate the accuracy of a coupled neutronics thermal-hydraulics code in analysing the transient characteristic of a PWR [

The transient events in the benchmark are initiated by a rapid ejection of control rod (CR) at HZP (hot zero power, 2775 W) and HFP (hot full power, 2775 MW) conditions. The core configuration and operational data, such as geometry and neutron cross sections, are derived from a real PWR. To allow the problem of a single rod ejection, a CR is added in the center of the core. As shown in [

There are 157 fuel assemblies, where 49 assemblies are with CR, with a radial dimension of 21.606 cm

2 layers for the upper and lower axial reflector with thickness of 30 cm each;

16 layers, from bottom to top, with heights of 7.7 cm, 11.0 cm, 15.0 cm, 30.0 cm (10 layers), 12.8 cm (2 layers), and 8.0 cm.

The radial (a) and axial (b) PWR benchmark core configuration [

The four selected cases are described in Table

Operational data for the A1, A2, B1, and B2 benchmark cores.

Case name | Core condition | Number of ejected CR | Initial position of CR (in steps)/number | |
---|---|---|---|---|

Ejected | Not ejected | |||

A1 | HZP | 1 (central) | 0/1 | 228/40 |

A2 | HFP | 1 (central) | 100/1 | 200/40 |

B1 | HZP | 4 (peripheral) | 0/4 | 0/5 |

B2 | HFP | 4 (peripheral) | 150/4 | 150/1 |

The initial control rod positions for case A1 (circled position is ejected).

The initial control rod positions for case A2 (circled position is ejected).

The initial control rod positions for case B1 (circled position is ejected).

The initial control rod positions for case B2 (circled position is ejected).

Table

The calculation results of NODAL3 code for steady sate and transient problems.

Parameter | Case/core condition | |||
---|---|---|---|---|

A1/HZP | A2/HFP | B1/HZP | B2/HFP | |

Critical boron concentration, ppm | ||||

PANTHER (1993) | 567.7 | 1,160.6 | 1,254.6 | 1,189.4 |

PANTHER (1997) | 561.2 | 1,156.6 | 1,248.0 | 1,183.8 |

NODAL3 (adiabatic) | 563.0 | 1,151.7 | 1,253.0 | 1,179.3 |

NODAL3 (quasistatic) | 563.0 | 1,151.7 | 1,253.0 | 1,179.3 |

Time of power peak, s | ||||

PANTHER (1993) | 0.560 | 0.100 | 0.517 | 0.120 |

PANTHER (1997) | 0.538 | 0.095 | 0.523 | 0.100 |

NODAL3 (adiabatic) | 0.566 | 0.099 | 0.501 | 0.126 |

NODAL3 (quasistatic) | 0.579 | 0.100 | 0.507 | 0.121 |

Power peak | ||||

PANTHER (1993) | 1.18 | 1.080 | 2.44 | 1.063 |

PANTHER (1997) | 1.27 | 1.083 | 2.32 | 1.064 |

NODAL3 (adiabatic) | 1.17 | 1.076 | 2.73 | 1.055 |

NODAL3 (quasistatic) | 1.15 | 1.076 | 2.67 | 1.055 |

Final power (at 5 s) | ||||

PANTHER (1993) | 0.196 | 1.035 | 0.32 | 1.038 |

PANTHER (1997) | 0.197 | 1.036 | 0.32 | 1.039 |

NODAL3 (adiabatic) | 0.190 | 1.032 | 0.31 | 1.033 |

NODAL3 (quasistatic) | 0.190 | 1.032 | 0.31 | 1.033 |

Final average Doppler temperature (at 5 s), °C | ||||

PANTHER (1993) | 324.30 | 554.60 | 349.9 | 552.0 |

PANTHER (1997) | 324.90 | 555.20 | 350.0 | 552.4 |

NODAL3 (adiabatic) | 323.26 | 555.71 | 349.1 | 552.2 |

NODAL3 (quasistatic) | 323.17 | 555.83 | 348.9 | 552.5 |

Maximum fuel temperature (at 5 s), °C | ||||

PANTHER (1993) | 673.3 | 1,691.8 | 559.8 | 1,558.1 |

PANTHER (1997) | 679.3 | 1,679.6 | 559.7 | 1,576.1 |

NODAL3 (adiabatic) | 659.6 | 1,699.1 | 554.9 | 1,598.0 |

NODAL3 (quasistatic) | 658.5 | 1,699.0 | 554.1 | 1,598.0 |

Final coolant outlet temperature (at 5 s), °C | ||||

PANTHER (1993) | 293.1 | 324.6 | 297.6 | 324.5 |

PANTHER (1997) | 293.2 | 324.9 | 297.7 | 324.8 |

NODAL3 (adiabatic) | 308.7 | 335.1 | 303.1 | 334.6 |

NODAL3 (quasistatic) | 308.7 | 335.1 | 303.0 | 334.6 |

The behaviour of reactor power and average Doppler temperature are shown in Figures

Comparison between NODAL3 and references results for case A1.

The maximum deviations of the fuel temperature parameters obtained by NODAL3 are lower compared to the final power parameters, since they are only 0.53% (

Comparison between NODAL3 and references results for case A2.

Comparison between NODAL3 and references results for case B1.

Comparison between NODAL3 and references results for case B2.

A comparison with the codes that have been validated for the same benchmark, DYN3D/R and ANCK codes, has been carried out as shown in Tables

Maximum deviations of DYN3D/R and NODAL3 compared to the reference solution in 1993 [

Parameter | Maximum deviation (%) |
---|---|

Critical boron concentration, ppm | |

DYN3D/R | 0.86 |

NODAL3 (adiabatic) | 0.85 |

NODAL3 (quasistatic) | 0.85 |

Time of power peak, s | |

DYN3D/R | 12.50 |

NODAL3 (adiabatic) | 5.00 |

NODAL3 (quasistatic) | 3.39 |

Power peak | |

DYN3D/R | 20.32 |

NODAL3 (adiabatic) | 11.89 |

NODAL3 (quasistatic) | 9.43 |

Final power (at 5 s) | |

DYN3D/R | 4.65 |

NODAL3 (adiabatic) | 3.13 |

NODAL3 (quasistatic) | 3.13 |

Final average Doppler temperature (at 5 s), °C | |

DYN3D/R | 1.29 |

NODAL3 (adiabatic) | 0.32 |

NODAL3 (quasistatic) | 0.35 |

Maximum fuel temperature (at 5 s), °C | |

DYN3D/R | 3.53 |

NODAL3 (adiabatic) | 2.56 |

NODAL3 (quasistatic) | 2.56 |

Final coolant outlet temperature (at 5 s), °C | |

DYN3D/R | 0.24 |

NODAL3 (adiabatic) | 5.32 |

NODAL3 (quasistatic) | 5.32 |

Maximum deviations of ANCK and NODAL3 compared to the reference solution in 1997 [

Parameter | Maximum deviation (%) |
---|---|

Critical boron concentration, ppm | |

ANCK | 0.75 |

NODAL3 (adiabatic) | 0.42 |

NODAL3 (quasistatic) | 0.42 |

Time of power peak, s | |

ANCK | 5.26 |

NODAL3 (adiabatic) | 26.00 |

NODAL3 (quasistatic) | 21.00 |

Power peak | |

ANCK | 5.17 |

NODAL3 (adiabatic) | 17.67 |

NODAL3 (quasistatic) | 15.09 |

Final power (at 5 s) | |

ANCK | 2.54 |

NODAL3 (adiabatic) | 3.55 |

NODAL3 (quasistatic) | 3.55 |

Final average Doppler temperature (at 5 s), °C | |

ANCK | Not available |

NODAL3 (adiabatic) | 0.51 |

NODAL3 (quasistatic) | 0.53 |

Maximum fuel temperature (at 5 s), °C | |

ANCK | 1.43 |

NODAL3 (adiabatic) | 2.90 |

NODAL3 (quasistatic) | 3.06 |

Final coolant outlet temperature (at 5 s), °C | |

ANCK | Not available |

NODAL3 (adiabatic) | 5.29 |

NODAL3 (quasistatic) | 5.29 |

Furthermore, compared to the ANCK code, the maximum deviations of NODAL3 code are higher in the range of 1.02%–15.74%, except for the critical boron concentration parameter which is lower by 0.32%. The relatively large differences occur in two parameters, the time of power peak and the power peak, with the maximum deviations, that is, 15.74% and 12.50%, respectively. The difference of 15.74% is equivalent to Δ

A coupled neutronics thermal-hydraulics code, NODAL3, based on the few-group neutron diffusion equation in 3-dimensional geometry using the polynomial nodal method, has been verified with the OECD/NEACRP PWR rod ejection benchmark. The results of NODAL3 code show very good agreement with the PHANTHER reference solutions in 1993 and 1997 (revised).

As future works, the sensitivity analysis is needed to be carried out to analyse the cause of relatively higher deviation for the power peak parameters. In addition, the code will be verified to the PWR benchmark of uncontrolled withdrawal control rod to elaborate the accuracy in fast reactivity insertion.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work has been partially supported by the Finance Ministry of Indonesia for fiscal years of 2010–2012. The authors express a special gratitude to Dr. Kunihiko Nabeshima (JAEA) for providing an important reference for this paper.