Zircaloy Loss-of-Coolant Accident Thermal Creep and Hoppe Irradiation Creep

Calculates the Erbacher secondary thermal creep under loss-of-coolant accident conditions, the Limback-Andersson primary thermal creep, and the Hoope irradiation creep for Zircaloy cladding. This material must be run in conjunction with ComputeMultipleInelasticStress.

Description

LOCA range thermal creep and Hoppe irradiation creep are both calculated in this single class, and thermal creep transitions automatically from the standard temperature Limback-Andersson model under normal operating conditions to the LOCA secondary thermal creep model in the LOCA temperature range. The LOCA thermal creep model requires the fraction of zirconium in the beta phase; this fraction of beta phase is computed by the separate material ZrPhase. This material, which must be run in conjunction with ComputeMultipleInelasticStress calculates the inelastic creep strain, the elastic strain, and the resulting stress for zircaloy materials.

warning:ZrPhase Required in the LOCA Regime Model

The material ZrPhase must be included in the input file when running LOCA simulations.

The contributions to creep from irradiation, primary, and thermal secondary creep are summed at each iteration.

Thermal Creep

The LOCA thermal creep model implemented in Bison includes both primary and secondary creep. The model will transition automatically to the LOCA thermal creep model in higher temperatures if the simulation is started at a lower temperature, below the start of the LOCA thermal creep temperature range. The default end of the standard Limback-Andersson model is set to 700K, and the start of the LOCA thermal creep range is set to 900K. Between these two temperature bounds, a linear interpolation is used to average the contributions from both the Limback Andersson Matsuo and Erbacher LOCA thermal secondary creep models.

LOCA Thermal Creep (High Temperature Accident Conditions)

During a loss of coolant accident, or LOCA, outward creep deformation of the cladding tube under the effect of internal pressurization and high temperature drives cladding ballooning and eventual failure due to burst.

For LOCA analysis, the large creep deformation of the cladding is defined by a strain rate correlation in the form of a Norton power equation (Van Uffelen et al., 2008; Neitzel and Rosinger, 1980; Erbacher et al., 1982): (1) where (s) is the effective creep strain rate, A (MPas) the strength coefficient, Q (J/mol) the activation energy for the creep deformation, T (K) the temperature, (MPa) the effective (von Mises) stress, and n (dimensionless) is the stress exponent. The components of the strain tensor are then updated at each time step based on the effective strain increment and a flow rule. The material parameters, shown in Table 1, are used in the model were obtained from tension tests on Zircaloy-4 tubes (Neitzel and Rosinger, 1980; Erbacher et al., 1982).

Table 1: Material Parameters used to Calculate the Creep in Zircaloy-4 (Erbacher et al., 1982; Markiewicz and Erbacher, 1988)

Zircaloy Phase PresentEffective Creep Rate ()A ()Q ()n ()
Pure phaseall creep rates
Mixed: 50% phase, 50% phase
Mixed: 50% phase, 50% phaseLinear interpolation of ln (A)Linear interpolationLinear interpolation
Pure phaseall creep rates

In the mixed phase () region, interpolations are made to calculate the Norton material parameters. Depending on the strain rate, different approaches are adopted from Neitzel and Rosinger (1980):

For s:

  • linear interpolation of ln(A), n, and Q is made between the values for pure and the equally mixed phases of (),

  • and between and pure phase.

For s, it is assumed that the values of ln(A), n, and Q vary linearly between the values for pure and pure phase.

Figure 1: Effective creep strain rate of Zircaloy-4 as a function of temperature for different values of the effective stress. The approximate temperature regions corresponding to the different crystallographic phases of the material are highlighted.

To perform the interpolation, the fraction of each phase calculated is in the separate material model ZrPhase. The effective creep strain rate as a function of temperature for different stress values is illustrated in Figure 1.

When running a simulation where the temperature in the cladding increases from normal operating conditions (about 600K) up to LOCA temperatures (about 900K), the effective creep strain rate is linearly interpolated between the Limback thermal creep model (Matsuo, 1987) and the LOCA thermal creep model. There are a number of regression tests that demonstrate the LOCA behavior and the transition between normal operation secondary thermal creep and LOCA creep.

Limback Thermal Creep (Standard Operating Conditions)

The Limback-Andersson model includes both primary and secondary creep; primary creep can be important as part of power changes when the load on the cladding changes relatively suddenly.

Limback-Andersson Secondary Thermal Creep

Secondary thermal creep rate in the Limback-Andersson model is given as the Matsuo (1987) model where the creep rate (hr) is (2) where the constants , , and are shown in the table below for the different cladding materials, is the temperature (K), = 65 (dimensionless), = 8.314 (J/mol/K), = 0.56 (dimensionless), = 1.4 10 ((n/cm)), and = 1.3 (dimensionless).

Based on the Limback model, a new model for ZIRLO was developed by adjusting some parameters to fit data on ZIRLO material using (Foster et al., 2008; Quecedo et al., 2009; Seok et al., 2011).

Table 2: Standard Thermal Creep Zircaloy Material Constants

Clad TypeA (K/MPa/hr)Q (kJ/mol)n
stress relief annealed (Zr2 or Zr4)
recrystallization annealed (Zr2)
partially recrystallization annealed (Zr2)
stress relief annealed ZIRLO

Note that is a function of effective stress: (3)

Primary Creep from Limback-Andersson

The primary thermal creep rate is calculated as a non zero value when the secondary thermal creep rate is greater than zero while the primary creep strain is below the saturation value. Within these bounds, the primary thermal creep rate is calculated as (4) where = 52 (dimensionless) and is a time constant type variable defined as: (5) where is the saturated primary creep strain and is the steady state creep rate: the sum of the secondary thermal and irradiation creep rates.

The primary saturated strain, , can be determined by either the Matsuo model or Limback's modified Matsuo model, (Matsuo, 1987). The Limback modified model, given below, is used as the default method to calculate primary thermal creep strain. (6)

Table 3: Parameters for the Eqn Eq. 6

Model ParameterParameter Value
(dimensionless)
(dimensionless)

Both primary creep strain and secondary thermal creep strain are saved as independent material properties, primary_creep_strain and thermal_secondary_creep_strain; these material properties can be saved to the output file through the use of AuxKernels to individually examine these components of the creep strain.

Hoppe Irradiation Creep

Irradiation-induced creep of cladding materials is based on an empirical model developed by Hoppe (1991) that relates the creep rate to the current fast neutron flux and stress. The specific relation implemented is: (7) where is the effective irradiation creep rate (1/s), is the fast neutron flux , is the effective (Mises) stress (MPa), and , , and are material constants. The material constants , , and are shown in the table for different cladding materials. Note that the original Hoppe formulation is given in terms of circumferential stress, whereas the relation implemented in Bison assumes an effective (von Mises) stress.

Table 4: Irradiation Creep Zircaloy Material Constants

Clad Type
stress relief annealed (Zr2 or Zr4)
recrystallization annealed (Zr2)
partially recrystallization annealed (Zr2)
stress relief annealed ZIRLO

The constants used in the irradiation creep model depend on the material selected as an input parameter.

Total Zircaloy Creep Strain

Total creep strain is the combination of the primary and secondary creep strains: (8)

Example Input Syntax


[./zry_thermal_creep]
  type = ZryCreepLOCAErbacherLimbackHoppeUpdate
  block = 1
  temperature = temp
  model_primary_creep = false
  model_irradiation_creep = false
[../]
(test/tests/tensor_mechanics/zry_creep/operating_to_loca_creep_1_tm.i)

ZryCreepLOCAErbacherLimbackHoppeUpdate must be run in conjunction with the inelastic strain return mapping stress calculator as shown below:


[./stress]
  type = ComputeMultipleInelasticStress
  tangent_operator = elastic
  inelastic_models = 'zry_thermal_creep'
  block = 1
[../]
(test/tests/tensor_mechanics/zry_creep/operating_to_loca_creep_1_tm.i)

The material ZrPhase must also be included in the input file when running simulations with ZryCreepLOCAErbacherLimbackHoppeUpdate, such as


[./phase]
  type = ZrPhase
  block = 1
  temperature = temp
  numerical_method = 2
[../]
(test/tests/tensor_mechanics/zry_creep/operating_to_loca_creep_1_tm.i)

Input Parameters

  • zircaloy_material_typestress_relief_annealedType of zircaloy material properties to use in calculating creep. Choices are: stress_relief_annealed recrystalization_annealed partial_recrystallization_annealed zirlo

    Default:stress_relief_annealed

    C++ Type:MooseEnum

    Description:Type of zircaloy material properties to use in calculating creep. Choices are: stress_relief_annealed recrystalization_annealed partial_recrystallization_annealed zirlo

  • acceptable_multiplier10Factor applied to relative and absolute tolerance for acceptable convergence if iterations are no longer making progress

    Default:10

    C++ Type:double

    Description:Factor applied to relative and absolute tolerance for acceptable convergence if iterations are no longer making progress

  • fast_neutron_fluxThe fast neutron flux

    C++ Type:std::vector

    Description:The fast neutron flux

  • temperature_loca_creep_begin900The lower limit of the temperature range (in K) in which the thermal creep model applies

    Default:900

    C++ Type:double

    Description:The lower limit of the temperature range (in K) in which the thermal creep model applies

  • temperatureThe coupled temperature (K)

    C++ Type:std::vector

    Description:The coupled temperature (K)

  • max_creep_increment0.001Maximum creep strain increment allowed by accuracy time step criterion

    Default:0.001

    C++ Type:double

    Description:Maximum creep strain increment allowed by accuracy time step criterion

  • boundaryThe list of boundary IDs from the mesh where this boundary condition applies

    C++ Type:std::vector

    Description:The list of boundary IDs from the mesh where this boundary condition applies

  • fast_neutron_fluenceThe fast neutron fluence

    C++ Type:std::vector

    Description:The fast neutron fluence

  • absolute_tolerance1e-11Absolute convergence tolerance for Newton iteration

    Default:1e-11

    C++ Type:double

    Description:Absolute convergence tolerance for Newton iteration

  • temperature_standard_thermal_creep_end700The upper limit of temperature (in K) where the standard thermal creep model for normal operating temperature no longer applies

    Default:700

    C++ Type:double

    Description:The upper limit of temperature (in K) where the standard thermal creep model for normal operating temperature no longer applies

  • model_primary_creepTrueSet true to activate primary creep

    Default:True

    C++ Type:bool

    Description:Set true to activate primary creep

  • relative_tolerance1e-08Relative convergence tolerance for Newton iteration

    Default:1e-08

    C++ Type:double

    Description:Relative convergence tolerance for Newton iteration

  • max_inelastic_increment0.0001The maximum inelastic strain increment allowed in a time step

    Default:0.0001

    C++ Type:double

    Description:The maximum inelastic strain increment allowed in a time step

  • base_nameOptional parameter that defines a prefix for all material properties related to this stress update model. This allows for multiple models of the same type to be used without naming conflicts.

    C++ Type:std::string

    Description:Optional parameter that defines a prefix for all material properties related to this stress update model. This allows for multiple models of the same type to be used without naming conflicts.

  • initial_fast_fluence0The initial fast neutron fluence

    Default:0

    C++ Type:double

    Description:The initial fast neutron fluence

  • model_thermal_creepTrueSet true to activate steady state thermal creep

    Default:True

    C++ Type:bool

    Description:Set true to activate steady state thermal creep

  • model_irradiation_creepTrueSet true to activate irradiation induced creep

    Default:True

    C++ Type:bool

    Description:Set true to activate irradiation induced creep

  • outputThe reporting postprocessor to use for the max_iterations value.

    C++ Type:PostprocessorName

    Description:The reporting postprocessor to use for the max_iterations value.

  • max_its30Maximum number of Newton iterations

    Default:30

    C++ Type:unsigned int

    Description:Maximum number of Newton iterations

  • blockThe list of block ids (SubdomainID) that this object will be applied

    C++ Type:std::vector

    Description:The list of block ids (SubdomainID) that this object will be applied

Optional Parameters

  • effective_inelastic_strain_nameeffective_creep_strainName of the material property that stores the effective inelastic strain

    Default:effective_creep_strain

    C++ Type:std::string

    Description:Name of the material property that stores the effective inelastic strain

  • enableTrueSet the enabled status of the MooseObject.

    Default:True

    C++ Type:bool

    Description:Set the enabled status of the MooseObject.

  • use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

    Default:False

    C++ Type:bool

    Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

  • control_tagsAdds user-defined labels for accessing object parameters via control logic.

    C++ Type:std::vector

    Description:Adds user-defined labels for accessing object parameters via control logic.

  • seed0The seed for the master random number generator

    Default:0

    C++ Type:unsigned int

    Description:The seed for the master random number generator

  • implicitTrueDetermines whether this object is calculated using an implicit or explicit form

    Default:True

    C++ Type:bool

    Description:Determines whether this object is calculated using an implicit or explicit form

  • constant_onNONEWhen ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeSubdomainProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped

    Default:NONE

    C++ Type:MooseEnum

    Description:When ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeSubdomainProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped

Advanced Parameters

  • creeprate_scale_factor1scaling factor for total creep rate. Used for calibration and sensitivity studies

    Default:1

    C++ Type:double

    Description:scaling factor for total creep rate. Used for calibration and sensitivity studies

Advanced: Scaling Factors Parameters

  • internal_solve_output_onon_errorWhen to output internal Newton solve information

    Default:on_error

    C++ Type:MooseEnum

    Description:When to output internal Newton solve information

  • internal_solve_full_iteration_historyFalseSet true to output full internal Newton iteration history at times determined by `internal_solve_output_on`. If false, only a summary is output.

    Default:False

    C++ Type:bool

    Description:Set true to output full internal Newton iteration history at times determined by `internal_solve_output_on`. If false, only a summary is output.

Debug Parameters

  • output_propertiesList of material properties, from this material, to output (outputs must also be defined to an output type)

    C++ Type:std::vector

    Description:List of material properties, from this material, to output (outputs must also be defined to an output type)

  • outputsnone Vector of output names were you would like to restrict the output of variables(s) associated with this object

    Default:none

    C++ Type:std::vector

    Description:Vector of output names were you would like to restrict the output of variables(s) associated with this object

Outputs Parameters

Input Files

References

  1. F. J. Erbacher, H. J. Neitzel, H. Rosinger, H. Schmidt, and K. Wiehr. Burst criterion of Zircaloy fuel claddings in a loss-of-coolant accident. In Zirconium in the Nuclear Industry, Fifth Conference, ASTM STP 754, D.G. Franklin Ed., 271–283. American Society for Testing and Materials, 1982.[BibTeX]
  2. J.P. Foster, H.K. Yueh, and R.J. Comstock. Zirloâ„¢ cladding improvement. Journal of ASTM International, 2008.[BibTeX]
  3. N. E. Hoppe. Engineering model for zircaloy creep and growth. In Proceedings of the ANS-ENS International Topical Meeting on LWR Fuel Performance, 157–172. Avignon, France, April 21-24, 1991.[BibTeX]
  4. M. E. Markiewicz and F.J. Erbacher. Experiments on ballooning in pressurized and transiently heated Zircaloy-4 tubes. Technical Report KfK 4343, Kernforschungszentrum Karlsruhe GmbH (Germany, Kernforschungszentrum Karlsruhe, Germany, 1988.[BibTeX]
  5. Y. Matsuo. Thermal creep of zircaloy-4 cladding under internal pressure. Journal of Nuclear Science and Technology, 24(2):111–119, February 1987.[BibTeX]
  6. H. J. Neitzel and H. Rosinger. The development of a burst criterion for zircaloy fuel cladding under loca conditions. Technical Report KfK 4343, Kernforschungszentrum Karlsruhe GmbH (Germany, Kernforschungszentrum Karlsruhe, Germany, 1980.[BibTeX]
  7. M. Quecedo, M. Lloret, J.M. Conde, C. Alejano, J.A. Gago, and F.J. Fernandez. Results of thermal creep test on highly irradiated zirloâ„¢. Nuclear Engineering and Technology, 41(2):179–186, 2009.[BibTeX]
  8. C.S. Seok, B. Marple, Y.J. Song, S. Gollapudi, I. Charit, and K.L. Murty. High temperature deformation characteristics of zirloâ„¢ tubing via ring-creep and burst tests. Nuclear Engineering and Design, 241:599–602, 2011.[BibTeX]
  9. P. Van Uffelen, C. GyÅ‘ri, A. Schubert, J. van de Laar, Z. Hózer, and G. Spykman. Extending the application range of a fuel performance code from normal operating to design basis accident conditions. Journal of Nuclear Materials, 383:137–143, 2008.[BibTeX]