Zirconium Phase Transformation

Computes the volume fraction of beta phase for Zr-based cladding materials as a function of temperature and time

Description

The ZrPhase model computes the volume fraction of phase for Zr-based cladding materials as a function of temperature and time.

Under extreme in-service conditions, e.g., during a postulated loss-of-coolant accident (LOCA), fuel cladding will be subjected to a rapid increase in temperature (up to 1000-1500K), which involves time-dependent phase transformation of Zr alloy from hexagonal (-phase) to cubic (-phase) crystal structure. Modeling the kinetics of crystallographic phase transformation is pivotal for the assessment of the mechanical properties essential for fuel rod integrity (deformation and burst) during a postulated LOCA.

The crucial parameter for the transformation kinetics is the evolution of the volume fraction of the new phase as a function of time and temperature. A model is available in Bison for calculation of the volume fraction of the favoured phase in Zircaloy-4 as a function of time and temperature during phase transformation in non- isothermal conditions. The model is based on Massih (2009), Massih and Jernkvist (2009), and Massih (2011). The phase transformation rate is expressed by (1) where is the volume fraction of -phase, (s) the time, (/) the steady- state or equilibrium value of , and (s) the rate parameter. The -phase equilibrium fraction is represented by a sigmoid function of temperature (2) where and are material specific parameters related to the center and span of the mixed-phase temperature region, respectively. For Zircaloy-4, (K) and (K) (Massih, 2009) are used, with being the hydrogen concentration in the range wppm (weight parts per million hydrogen). The rate parameter is expressed in the form (3) where is a kinetic prefactor, an effective activation energy, the Boltzmann constant, and a constant. For Zircaloy-4, (s) and (K) (Massih, 2009; Massih, 2011) are used, where (Ks) is the heat rate in the range Ks. The transformation is purely diffusion controlled, while the transformation is partly martensitic. This is represented by the constant given in the form (Massih, 2011): (4) The starting temperatures for the onset of and phase transformations are calculated as (in Kelvin) (Massih, 2009) (5)

(6) for wppm. The -phase volume fraction as a function of time is calculated by numerical integration of Eq. 1. As default option, this is accomplished using the second order Adams-Moulton (AM2) method. The backward Euler method is also available. The calculated volume fractions of phase as a function of temperature at equilibrium and for temperature variation rates of 10 Ks are shown in Figure 1.

Figure 1: Calculated volume fraction of phase as a function of temperature. Equilibrium conditions (slow temperature variation) and temperature variation rates of 10 Ks are considered.

Example Input Syntax


[./phase]
  type = ZrPhase
  block = 1
  temperature = temp
  numerical_method = 2
[../]
(test/tests/phase_transition_zircaloy/test_zrphase.i)

Input Parameters

  • temperatureCoupled temperature

    C++ Type:std::vector

    Description:Coupled temperature

Required Parameters

  • 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

  • computeTrueWhen false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the Material via MaterialPropertyInterface::getMaterial(). Non-computed Materials are not sorted for dependencies.

    Default:True

    C++ Type:bool

    Description:When false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the Material via MaterialPropertyInterface::getMaterial(). Non-computed Materials are not sorted for dependencies.

  • numerical_method2Select numerical method to solve the equation for beta phase fraction

    Default:2

    C++ Type:short

    Description:Select numerical method to solve the equation for beta phase fraction

  • 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

  • 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

  • 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. A.R. Massih. Transformation kinetics of zirconium alloys under non-isothermal conditions. Journal of Nuclear Materials, 384:330–335, 2009.[BibTeX]
  2. A.R. Massih. Evaluation of loss-of-coolant accident simulation tests with the fuel rod analysis code FRAPTRAN-1.4. Technical Report TR11-008V1, Quantum Technologies AB, 2011.[BibTeX]
  3. Ali R Massih and Lars Olof Jernkvist. Transformation kinetics of alloys under non-isothermal conditions. Modelling and Simulation in Materials Science and Engineering, 17(5):055002, 2009.[BibTeX]