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1. Test ID: AER-DYN-004
2. a, Solution Submitted by: Ulrich Rohde, Forschungszentrum Rossendorf,
Institute of Safety Research, Germany
Date: 31.07.2000
b, Reviewed by: P. Siltanen, Fortum Engineering Ltd
Date: 27.10.2000
c, Accepted by: (name)
Date:
3. Code or Program Applied: DYN3D/H1.1
4. Short description of the Code:
The code DYN3D/H1.1 consists of the 3 - dimensional neutron kinetic model and the thermohydraulic module FLOCAL. The neutron kinetics of DYN3D/H1.1 is calculated by using a nodal expansion method (NEM) for hexagonal geometry. The developed method solves the neutron diffusion equation for two energy groups. Stationary state and transient behaviour can be calculated. The code FLOCAL consisting of a two-phase coolant flow model, a fuel rod model and a heat transfer regime map up to superheated steam is coupled with neutron kinetics by the neutron physical constants. One coolant channel per fuel assembly and additional hot channels can be considered.
5. Known Approximations:
Neutron Kinetics
Neutron diffusion theory
Two group theory
Nodewise homogenised cross sections
Thermal Hydraulics
One-dimensional four equation model for two phase coolant flow (momentum equation,energy equation and mass balance of mixture, mass balance of the vapour phase)
Constitutive equations for pressure drop, evaporation and condensation, phase slip
Heat transfer regime map for fuel to coolant heat transfer
Radial heat conduction equation in fuel pin
Feedback
Calculation of neutron cross section by using libraries or input data
6. Mathematical Model:
Neutron Kinetics
The 3-dimensional neutron kinetic model is based on the solution of the 3-dimensional 2group neutron diffusion equation by a nodal expansion method which is specific for the geometry of fuel assemblies [1,2]. It is assumed that the macroscopic cross sections are spatially constant in a node being a part of the hexagonal fuel assembly. The stationary diffusion equation in the node is solved by factorising the space dependency of neutron fluxes in the radial plane and the axial direction. A 2-dimensional diffusion equation in the radial plane and a 1-dimensional equation in axial direction are obtained. The two equations are coupled by the transversal bucklings. In the hexagonal plane the fluxes are expanded by using Bessel functions being the solutions of the Helmholtz equation. The low order coefficients are expressed by the node averaged fluxes and the incoming partial currents averaged over the interface of the hexagon. In this way, the outgoing partial currents at the interfaces are given by the node fluxes and the incoming partial currents. The matrix elements of these relations depend on the transversal buckling and the eigenvalue keff. The 1-dimensional equation in axial direction is solved by a polynomial expansion up to the fourth order. The outgoing partial currents in axial direction are given by the averaged fluxes, incoming partial currents and higher order coefficients. The equations for the 3rd and 4th order polynomials are obtained by Galerkin weighting. The outgoing partial currents at a node interface are the incoming currents in the neighbouring nodes. The steady state diffusion equation is solved by an inner and outer iteration process. The outer iterations are the fission source iterations accelerated by a Chebychev extrapolation scheme. A small number of inner iterations (3 - 5) are sufficient for the convergence. During the outer iteration process the matrix elements are recalculated few times (3-5).
Concerning the time integration over the neutronic time step an implicit difference scheme with exponential transformation is used. The exponents in each node are calculated from the previous time step or during the iteration process. For the calculation of matrix elements describing the relation between partial currents and averaged fluxes it is assumed that the time behaviour of the neutron fluxes in the nodes is exponential and the local variation of the source of delayed neutrons is proportional to the source of prompt neutrons. These assumptions allow the same treatment of diffusion equation in the nodes as in the steady state. In the iteration process we have to solve an inhomogeneous problem. Similar methods as used for the steady state are applied.
Thermal Hydraulics
The thermohydraulic model of the reactor core and the fuel rod model are implemented in the module FLOCAL [3] being a part of DYN3D. The reactor core is modelled by parallel cooling channels which can describe one or more fuel elements. The parallel channels are coupled hydraulically by the condition of equal pressure drop over all core channels. Additionally, so called hot channels can be considered for the investigation of hot spots and uncertainties in power density, coolant temperature or mass flow rate. Thermohydraulic boundary conditions for the core like coolant inlet temperature, pressure, coolant mass flow rate or pressure drop must be given as input for DYN3D.
Mixing of coolant from different loops before entering the core can be modelled by applying several options. Homogeneous mixing can be assumed for each reactor type and number of loops. For VVER-440 type reactors, a mixing model for the downcomer and the lower plenum is implemented in the code. The model is based on the analytical solution of the Navier-Stokes equations in the potential flow approximation in three-dimensional cylindrical geometry and the diffusion equation for heat transport or soluble poison. Turbulent Peclet numbers for the downcomer and the lower plenum are parameters of the model, which are used for a best fit adaptation to experimental results. The mixing model represents an interface between the cold legs of the primary loop and the core inlet.
The module FLOCAL comprises
- a one- or two-phase coolant flow model on the basis of four differential balance equations for mass, energy and momentum of the two-phase mixture and the mass balance for the vapour phase allowing the description of thermodynamic non-equilibrium between the phases,
- a heat transfer regime map from one-phase liquid up to post-critical heat transfer regimes and superheated steam,
- a fuel rod model for the calculation of fuel and cladding temperatures and the determination of some parameters for fuel rod failure estimation.
The two-phase flow model is closed by constitutive laws for heat, mass and momentum transfer, e.g. vapour generation at the heated walls, condensation in the subcooled liquid, phase slip ratio, pressure drop at single flow resistance's and due to friction along the flow channels as well as heat transfer correlations. Different packages of water and steam thermophysical properties presentation can be used.
The heat transfer regime map which is implemented in FLOCAL ranges from one-phase liquid convection up to superheated steam. The occurrence of heat transfer crisis is stated by different correlation's for the critical heat flux. The transient boiling region is described by the KIRCHNER and GRIFFITH interpolation for the heat flux. In the stable post-crisis region for inverted annular or dispersed flow the GROENEVELD - DELORME or a modified BROMLEY correlation's are used. After full evaporation of coolant, heat transfer to superheated steam is estimated by a forced convection correlation [3,4].
Special emphasis is put on adopting the model to the conditions of RIA accidents where a combination of high heat fluxes with high degree of coolant subcooling is typical and thermodynamic non-equilibrium effects are important. A correlation for the Leidenfrost or minimum stable film boiling point TMSFB takes into account the influence of fluid subcooling. In the post-crisis region, a special non-equilibrium correction to the heat transfer coefficient due to subcooled liquid is used.
For the estimation of fuel and cladding temperatures, the heat conduction equation in one-dimensional radial geometry is solved. In the gas gap between fuel and cladding the heat transfer components due to conduction in the gas, radiation and fuel- cladding contact are considered. The idea of the model is that the parameters for the stationary reference state (geometrical gap width , gas pressure and composition) are given and have to be obtained from detailed fuel rod behaviour codes. The changes during the instationary process are estimated by the model.
A thermomechanical model of the fuel and cladding behaviour is implemented into the code. The aim of this model is the estimation of gas gap conductance behaviour for a realistic temperature calculation being important especially for RIA calculations. The thermomechanical model is based on following assumptions:
- one-dimensional modelling of mechanics in radial direction,
- simple superposition of radial thermal, elastic and plastic deformations without axial coupling,
- elastic deformation of the fuel is taken into account only in the case of fuel-cladding contact, plastic deformations of the fuel are not considered,
- cladding is described in the thin shell approximation.
A deterministic model of fuel rod failure during accidents is not included in FLOCAL, but some parameters for the diagnostics of possible fuel rod failure are given, that is:
- fuel enthalpy for each axial node of the rod,
- cladding oxide thickness,
- signalisation of possible cladding rupture, when the cladding stress is positive (inner pressure is larger than outer pressure) and exceeds the yield point.
For testing and validation of the fuel rod and heat transfer model in the code FLOCAL experiments on fuel rod behaviour under RIA conditions were calculated [4].
For the solution of the 4th dynamic benchmark of AER, a special boron transport model based on the particle-in-cell (PIC) method was developed and used [5]. This method is free of any numerical diffusion and allows a correct numerical description of the moving boron dilution front, what was very important for the 4th benchmark.
7. Features of Techniques Used:
special boron transport model based on a PIC method to avoid numerical diffusion
standard DYN3D models for heat transfer, subcooled boiling and two-phase friction without modifications
the unheated lower and upper sections of the core are included in the coolant channels and treated as axial reflectors in neutronic calculations
8. Computer, Operational System: SUN workstation, Solaris
9. References:
[1] U. GRUNDMANN and U. ROHDE, "DYN3D/M2 - a Code for Calculation of ReactivityTransients in Cores with Hexagonal Geometry", Proc. of IAEA Technical Committee Meeting on Reactivity Initiated Accidents, Vienna (1989)
Rep. FZR 93-01, Research Centre Rossendorf (1993)
[2] U. GRUNDMANN, "A NEM for Solving Time-Dependent 3-Dimensional Diffusion Equation for Hexagonal Geometry", Proc. International Conference on the Physics of Reactors PHYSOR'90, Marseille (1990)
[3] U. ROHDE, "Modelling of Fuel Rod Behaviour and Heat Transfer in the Code FLOCAL for Reactivity Accident Analysis of Reactor Cores", 1st Baltic Heat Transfer Conference, Gotemborg, (1991), published in: Transport Processes in Engineering, 2: Elsevier Publ., Amsterdam, (1992)
[4] U. ROHDE, The modelling of fuel rod behaviour under RIA conditions in the code DYN3D, 8th Symposium of AER, Bystrice nad Pernstejnem (CZ), September 21-25, 1998, Proc. Pp. 721 - 740
[5] U. ROHDE, D. LUCAS, Solution of the 4th AER Dynamic Benchmark by Use of the Code DYN3D with a Particle-in-Cell Method for the Description of Boron Transport, 7th Symposium of AER, Hrnitz near Zittau (Germany), September 23-26, 1997, Proc. pp. 387-398
10. Results:
The requested results:
key parameters
time functions
axial distributions
power distributions
hot channel time functions
hot channel axial distributions
are included in the ASCII-file FZR_B4.TXT. The comparisons with other solutions are presented in the specification document DYN-004.
11. Comparison to Recommended Solution:
No reference solution does exist so far. The comparisons with solutions of other nodal codes are presented in the specification document DYN-004.
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