# Thermal Properties for UOvar element = document.getElementById("moose-equation-45c0641c-1df6-4133-aa0d-51a6f7263565");katex.render("_2", element, {displayMode:false,throwOnError:false}); Fuel

Model that computes specific heat and thermal conductivity for oxide fuel.

## Description

The ThermalFuel model computes specific heat and thermal conductivity for oxide fuel. A number of correlations are available.

## UO2 Thermal Properties

Five empirical models are available in Bison to compute UO thermal conductivity and its dependence on temperature, porosity, burnup, and, for four of the models, Gadolinia content. Choices for UO fuel include models referred to as Fink-Lucuta (Fink, 2000; Lucuta et al., 1996), Halden (Lanning et al., 2005), NFIR (A. Marion (NEI) letter dated June 13, 2006 to H. N. Berkow (USNRC/NRR), 2006; Lyon, 2015), MATPRO (Allison et al., 1993), and modified NFI (Ohira and Itagaki, 1997) (modifications are described in Lanning et al. (2005)). The Halden, MATPRO, NFIR and modified NFI models can account for Gadolinia content.

Empirical fits for the temperature dependent specific heat of UO accompany both the Fink- Lucuta and MATPRO conductivity models.

For the most part, the thermal conductivity of urania is represented as the sum of a lattice vibration (phonon) and an electronic (electron hole pair effect) term or for unirradiated material at 95% theoretical density (TD) (1) The first term in Eq. 1 is typically inversely proportional to the sum of temperature and burnup dependent functions, while the second term, usually an exponential function of inverse temperature, is inversely proportional to temperature or temperature squared. For example, (2) (3) where , , and are constants, is burnup, is temperature, and , , , and are functions of burnup or temperature. While each of the thermal conductivity models has the basic form given by Eq. 2 and Eq. 3, each has their own specific set of constants and perhaps additional corrections that account for effects of dissolved fission products, precipitated fission products, porosity, deviation from stoichiometry, and radiation damage. In general, the final conductivity corrected for these effects is given as (4) where is the dissolved fission products correction, is the precipitated fission products correction, is the porosity correction, is the deviation from stoichiometry (1.0 for uranium fuel but 1 if Gadolinia is present), and is the radiation damage correction.

### Fink-Lucuta

In the Fink-Lucuta model, the temperature-dependence of unirradiated material is defined using the equation suggested by Fink (2000). This relationship is then modified to account for the effects of irradiation, porosity and burnup using a series of multipliers, as outlined in detail by Lucuta et al. (1996). The Fink equation is (5) where is the temperature in K divided by 1000. Eq. 5 is multiplied by the following factor to obtain 100% TD thermal conductivity (6) Eq. 6 is then corrected per Eq. 4 as perscribed by Lucuta with the following correction factors.

The dissolved fission products correction is given as (7) and the precipitated fission products correction is calculated as (8) where is the burnup in atomic percent and is the temperature in K. The porosity correction is determined with (9) where is the porosity.

Finally the radiation damage correction factor is given by (10) where is the temperature in K, and bu is the burnup in atomic percent. The Fink-Lucuta model is valid from 298 to 3120 K (Carbajo et al., 2001).

### MATPRO

The MATPRO model (Allison et al., 1993) is based on an equation proposed by Ohira and Itagaki (1997). The thermal conductivity for 95% theoretical density is given as (11) where the reciprocal expression and term6 correspond to k and k, respectively. The terms for Eq. 11 are given in Table 1.

Table 1: Parameters used in the MATPRO Thermal Fuel Model (Allison et al., 1993)

Model ExpressionParameter Value

In Table 1 is temperature in K, is burnup in MWd/kgU, and is the Gadolinia concentration in weight percent. Eq. 11 is multiplied by the appropriate factor to return the thermal conductivity to 100% TD and then multiplied by a density correction factor (similar to Eq. 9 but written in terms of percent TD) to provide a thermal conductivity representative of the material of interest (12) where D is the fractional TD. The multiplier 1.0789 is the inverse of the density correction factor evaluated at 0.95 TD. The MATPRO correlation is valid over the following ranges (Lanning et al., 2005) (13) Figure 1 compares the two models as a function of temperature and burnup for fully dense UO.

Figure 1: A comparison of the Fink-Lucuta and MATPRO empirical models for the thermal conductivity of full density UO, as a function of temperature and burnup.

### Halden

The Halden model has the same form as Eq. 11. However, the terms are different, and different temperature and burnup units are used. For 95% TD fuel, the terms for Eq. 11 are given in Table 2.

Table 2: Parameters used in the Halden Thermal Fuel Model (Lanning et al., 2005)

Model ExpressionParameter Value

where is the Gadolinia concentration in weight percent, is the deviation from stoichiometry, i.e. (2 - oxygen/metal ratio), is the burnup in MWd/kgUO2, and is the temperature in C. Eq. 12 is used to compute the thermal conductivity at the TD of interest.

The Halden UO correlation is valid over the following ranges (Lanning et al., 2005) (14)

Figure 2 compares the the Fink-Lucuta and Halden models as a function of temperature and burnup for 95% theoretical density UO.

Figure 2: A comparison of the Fink-Lucuta and Halden empirical models for the thermal conductivity of 95% theoretical density UO, as a function of temperature and burnup.

### NFIR

The NFIR correlation also has the general form of Eq. 1. However, the NFIR model contains a temperature dependent thermal recovery function that accounts for self-annealing of defects in the fuel as it heats up. The ultimate effect of the self- annealing is a slight increase of the thermal conductivity over a range of temperatures up to 1200 K. As a result of this formulation, two components of are used, one at the start of thermal recovery and one at the end of thermal recovery. The thermal recovery function is used to interpolate between these two values to compute k. Thus (15) where is the thermal recovery function, is temperature in C, is the phonon contribution at the start of thermal recovery, and is the phonon contribution at the end of thermal recovery. This model has a gadolinium correction associated with it. Gadolinium is reported as weight percent with the default set to zero. The individual terms are (16) (17)

(18)

(19)

(20)

(21)

(22)

(23) where is burnup in MWd/kgU. Eq. 15 is then multiplied by a temperature dependent density correction factor to give (24) where is the fractional density. Figure 3 compares the the Fink-Lucuta and NFIR models as a function of temperature and burnup for 95% theoretical density UO.

Figure 3: A comparison of the Fink-Lucuta and NFIR empirical models for the thermal conductivity of 95% theoretical density UO, as a function of temperature and burnup.

### Modified NFI

The modified NFI model is also of the form of Eq. 11. Terms are defined in Table 3.

Table 3: Parameters used in the Modified NFI Thermal Fuel Model

Model ExpressionParameter Value

where is the Gd concentration in weight percent, is the temperature in K, is the burnup in MWd/kgU. Again, Eq. 12 is used to convert to the TD of interest. The modified NFI model is valid over the following ranges (Lanning et al., 2005) (25)

Figure 4 compares the Fink-Lucuta and NFI modified models as a function of temperature and burnup for 95% theoretical density UO.

Figure 4: A comparison of the Fink-Lucuta and modified NFI empirical models for the thermal conductivity of 95% theoretical density UO, as a function of temperature and burnup.

## MOX Thermal Properties

Three models are available to compute MOX thermal properties. For these models, thermal conductivity of unirradiated material is first defined. In general, these relationships are then multiplied by correction factors to account for effects of irradiation, burnup, MOX content, and porosity. The corrections factors used in Bison have been developed by Lucuta et al. (1996) and are recommended by Carbajo et al. (2001).

### Duriez-Ronchi

The first model is recommended by Carbajo et al. (2001) and is a combination of Duriez et al. (2000) and Ronchi et al. (1999) models. In this first model, thermal conductivity of unirradiated MOX is given by: (26) where is the thermal conductivity (), is a deviation from stoichiometry (unitless), and the following expressions define the remaining terms: (27) This model provides temperature and deviation from stoechiometry. It is valid from 700 to 3100 K, x less than 0.05, and plutonium concentration between 3 weight percent and 15 wt. %. According to Carbajo et al. (2001), thermal conductivity does not depend on Pu concentration in this range. Thus this model is valid essentially for thermal reactor MOX.

### Amaya

The second model available in Bison has been proposed by Amaya et al. (2011). Unlike the previous model, Amaya provides a plutonium concentration dependence. It starts from pure UO thermal conductivity and applies corrections to account for Pu content. Unirradiated MOX thermal conductivity is given by: (28) where is the MOX unirradiated thermal conductivity (), is the UO unirradiated thermal conductivity (), is the temperature (K), is the plutonium concentration (weight percent), and the following constants are used: (29)

Figure 5: Unirradiated thermal conductivities for UO and MOX from different models implemented in Bison.

Bison uses Fink model to compute unirradiated UO thermal conductivity. Amaya model's coefficients have been fitted in the temperature range from 400 K to 1500 K and the plutonium concentration up to 30 weight percent (Amaya et al., 2011). Figure 5 shows a comparison of the computed thermal conductivities for the Fink-Lucuta (for reference), Fink-Amaya, and Duriez-Ronchi models for unirradiated MOX at 95% theoretical density.

### Halden

The Halden correlation discussed in the previous section for uranium fuel is also applicable, with one change, to MOX fuel. Reduction in thermal conductivity due to the presence of mixed oxides is accounted for by multiplying the k term in Eq. 1 by 0.92. This is consistent with the statement above regarding the lack of dependence of MOX thermal conductivity on Pu concentration. The k part of the equation is unchanged and Eq. 12 is used to account for the TD of interest.

The Halden MOX correlation is valid over the following ranges (Lanning et al., 2005) (30) Figure 6 is a comparison of the Fink-Lucuta (for reference) urania correlation and the Fink-Amaya, Duriez-Ronchi, and Halden correlations for MOX for unirradiated 95% theoretical density MOX fuel with 0.07% Pu concentration.

Figure 6: Unirradiated thermal conductivities for UO (for reference) and MOX from different models implemented in Bison. Results are for 95% theoretical density and Pu concentraton of 7 weight percent. Correction factors appropriate for each correlation have been applied.

## Thermal Conductivity of Crumbled Fuel

When simulating Loss of Coolant Accident (LOCA) conditions some portions of the fuel column may collapse into a crumbled mixture of fuel fragments and gas. In these regions the effective fuel thermal conductivity of the mixture is calculated by the model proposed by Chiew and Glandt (1983).

The correlation is given by: (31) where is the reduced thermal polarizability, is the thermal conductivity of the fuel determined by one of the models above, is the packing fraction of the crumbled fuel, and is a function of and defined later. The reduced thermal polarizability is given by: (32) where is the thermal conductivity of the gas surrounding the crumbled fuel particles. The function is approximated by: (33) where Jernkvist and Massih (2015) used best fit approximations to the tabulated values of Chiew and Glandt (1983) to obtain: (34)

(35)

## Example Input Syntax

[./fuel_thermal]
type = ThermalFuel
block = 2
model = 2
initial_porosity = 0.0
temp = temp
burnup_function = burnup
[../]
(test/tests/fission_gas_behavior_sifgrs/sifgrs_athermal_release.i)

## Input Parameters

• modelThermal conductivity models: 0==Duriez (MOX), 1==Amaya (MOX), 2==Fink-Lucuta (UO2), 3==Halden (UO2 (w/wo Gd) or MOX), 4==NFIR (UO2 (w/wo Gd)), or 5 == Modified NFI (UO2 w/wo Gd)

C++ Type:int

Description:Thermal conductivity models: 0==Duriez (MOX), 1==Amaya (MOX), 2==Fink-Lucuta (UO2), 3==Halden (UO2 (w/wo Gd) or MOX), 4==NFIR (UO2 (w/wo Gd)), or 5 == Modified NFI (UO2 w/wo Gd)

### Required Parameters

• 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.

• tempCoupled Temperature

C++ Type:std::vector

Description:Coupled Temperature

• initial_porosity0.05Initial porosity. Must be between 0.0 and 1.0

Default:0.05

C++ Type:double

Description:Initial porosity. Must be between 0.0 and 1.0

• cp_scalef1scaling factor for fuel specific heat

Default:1

C++ Type:double

Description:scaling factor for fuel specific heat

• porosity_materialFalseWhether a material property for porosity is supplied

Default:False

C++ Type:bool

Description:Whether a material property for porosity is supplied

• gap_thermal_conductivityThe layered average thermal conductivity across the gas gap

C++ Type:std::vector

Description:The layered average thermal conductivity across the gas gap

• Pu_content0Weight fraction of Pu in MOX fuel (typically ~0.07)

Default:0

C++ Type:double

Description:Weight fraction of Pu in MOX fuel (typically ~0.07)

• thcond_scalef1scaling factor for fuel thermal conductivity

Default:1

C++ Type:double

Description:scaling factor for fuel thermal conductivity

• axial_relocation_objectName of the AxialRelocationUserObject that determines whether the fuel has crumbled.

C++ Type:UserObjectName

Description:Name of the AxialRelocationUserObject that determines whether the fuel has crumbled.

• Gd_content0Weight fraction of gadolinium in fuel

Default:0

C++ Type:double

Description:Weight fraction of gadolinium in fuel

• porosityCoupled Porosity

C++ Type:std::vector

Description:Coupled Porosity

• 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

• oxy_to_metal_ratio2Deviation from stoichiometry

Default:2

C++ Type:double

Description:Deviation from stoichiometry

• 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

• burnup_functionBurnup function

C++ Type:FunctionName

Description:Burnup function

• burnupCoupled Burnup Rate

C++ Type:std::vector

Description:Coupled Burnup Rate

### 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

## References

1. C. M. Allison, G. A. Berna, R. Chambers, E. W. Coryell, K. L. Davis, D. L. Hagrman, D. T. Hagrman, N. L. Hampton, J. K. Hohorst, R. E. Mason, M. L. McComas, K. A. McNeil, R. L. Miller, C. S. Olsen, G. A. Reymann, and L. J. Siefken. SCDAP/RELAP5/MOD3.1 code manual, volume IV: MATPROâ€“A library of materials properties for light-water-reactor accident analysis. Technical Report NUREG/CR-6150, EGG-2720, Idaho National Engineering Laboratory, 1993.[BibTeX]
2. M. Amaya, J. Nakamura, F. Nagase, and T. Fuketa. Thermal conductivity evaluation of high burnup mixed-oxide (mox) fuel pellet. Journal of Nuclear Materials, 414:303â€“308, 2011.[BibTeX]
3. J. Carbajo, L. Gradyon, S. Popov, and V. Ivanov. A review of the thermophysical properties of mox and uo2 fuels. Journal of Nuclear Materials, 299:181â€“198, 2001.[BibTeX]
4. Y. C. Chiew and E. D. Glandt. The effect of structure on the conductivity of a dispersion. Journal of Colloid and Interface Science, 91(1):90â€“104, 1983.[BibTeX]
5. C. Duriez, J.-P. Alessandri, T. Gervais, and Y. Philipponneau. Thermal conductivity of hypostoichiometriclow pu content mixed oxide. Journal of Nuclear Materials, 277:143â€“158, 2000.[BibTeX]
6. J. K. Fink. Thermophysical properties of uranium dioxide. Journal of Nuclear Materials, 279(1):1â€“18, 2000.[BibTeX]
7. L. O. Jernkvist and A. Massih. Model for axial relocation of fragmented and pulverized fuel pellets in distending fuel rods and its effects on fuel rod heat load. Technical Report SSM-2015:37, Str\r al sÃ¤kerhets myndigheten, 2015.[BibTeX]
8. D. D. Lanning, C. E. Beyer, and K. J. Geelhood. Frapcon-3 updates, including mixed-oxide fuel properties. Technical Report NUREG/CR-6534, Vol. 4 PNNL-11513, Pacific Northwest National Laboratory, 2005.[BibTeX]
9. P. G. Lucuta, H. J. Matzke, and I. J. Hastings. A pragmatic approach to modelling thermal conductivity of irradiated UO$_2$ fuel: review and recommendations. Journal of Nuclear Materials, 232:166â€“180, 1996.[BibTeX]
10. W. F. Lyon. Summary report: gd thermal conductivity model updates. Technical Report ANA-P1400138-TN03 Rev. 2, Anatech Corp., 2015.[BibTeX]
11. K. Ohira and N. Itagaki. Thermal conductivity measurements of high burnup UO$_2$ pellet and a benchmark calculation of fuel center temperature. In Proceedings of the American Nuclear Society Meeting on Light Water Reactor Fuel Performance, 541. Portland, Oregon, Mar 2 to Mar 6, 1997.[BibTeX]
12. C. Ronchi, M. Sheindlin, M. Musella, and G.J. Hyland. Thermal conductivity of uranium dioxide up to 2900 k from simultaneous measurement of the heat capacity and thermal diffusivity. Journal of Applied Physics, 85:776â€“789, 1999.[BibTeX]
13. A. Marion (NEI) letter dated June 13, 2006 to H. N. Berkow (USNRC/NRR). Safety Evaluation by the Office of Nuclear Reactor Regulation of Electric Power Research Institute (EPRI) Topical Report TR-1002865, "Topical Report on Reactivity Initiated Accidents: Bases for RIA Fuel rod Failures and Core Coolability Criteria". http://pbadupws.nrc.gov/docs/ML0616/ML061650107.pdf, 2006.[BibTeX]