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THERMAL_CHARACTERISTIC_CURVES¶
This option specifies the thermal characteristic curves (e.g. thermal conductivity and associated parameters) associated with a material property. This expands thermal conductivity as a function of both temperature and saturation (with the exception of the CONSTANT, DEFAULT, and LINEAR thermal conductivity functions).
The legacy input method of specifying thermal conductivity by MATERIAL_PROPERTY (i.e. with THERMAL_CONDUCTIVITY_DRY and THERMAL_CONDUCTIVTY_WET) is backwards-compatible, where parameters are adapted to the DEFAULT thermal characteristic curve and functions are numbered in material sequence. However, THERMAL_CHARACTERISTIC_CURVES cannot be combined with the legacy convention in the same input file.
Required Blocks and Cards:¶
- THERMAL_CONDUCTIVITY_FUNCTION <string>
Opens a thermal conductivity block, where <string> indicates the type of thermal conductivity function to be employed.
Supported THERMAL_CONDUCTIVITY_FUNCTIONs (along with their required cards):
CONSTANT
CONSTANT_THERMAL_CONDUCTIVITY
DEFAULT
THERMAL_CONDUCTIVITY_DRY
THERMAL_CONDUCTIVITY_WET
LINEAR
THERMAL_CONDUCTIVITY_DRY
THERMAL_CONDUCTIVITY_WET
POWER
THERMAL_CONDUCTIVITY_DRY
THERMAL_CONDUCTIVITY_WET
REFERENCE_TEMPERATURE
EXPONENT
CUBIC_POLYNOMIAL
THERMAL_CONDUCTIVITY_DRY
THERMAL_CONDUCTIVITY_WET
REFERENCE_TEMPERATURE
CUBIC_POLYNOMIAL_COEFFICIENTS
LINEAR_RESISTIVITY
THERMAL_CONDUCTIVITY_DRY
THERMAL_CONDUCTIVITY_WET
REFERENCE_TEMPERATURE
LINEAR_RESISTIVITY_COEFFICIENTS
FROZEN
THERMAL_CONDUCTIVITY_DRY
THERMAL_CONDUCTIVITY_WET
KERSTEN_EXPONENT
THERMAL_CONDUCTIVITY_FROZEN
KERSTEN_EXPONENT_FROZEN
ICE_MODEL
COMPOSITE
COMPOSITE_X
COMPOSITE_Y
COMPOSITE_Z
ASM_AXIAL (Assembly axial model)
THERMAL_CONDUCTIVITY_WATER
THERMAL_CONDUCTIVITY_SOLID
POROSITY_ASSEMBLY
ASM_RADIAL (Assembly radial model)
THERMAL_CONDUCTIVITY_WATER
THERMAL_CONDUCTIVITY_SOLID
POROSITY_ASSEMBLY
THERMAL_CONDUCTIVITY_DRY
ASM_DRY_COEFFICIENT
ASM_DRY_EXPONENT
ASM_WATER_FILLED (Standalone model for water-filled assembly)
THERMAL_CONDUCTIVITY_WATER
THERMAL_CONDUCTIVITY_SOLID
POROSITY_ASSEMBLY
THERMAL_CONDUCTIVITY_DRY (optional)
ASM_DRY (Standalone model for dry assembly)
THERMAL_CONDUCTIVITY_DRY
ASM_DRY_COEFFICIENT
ASM_DRY_EXPONENT
THERMAL_CONDUCTIVITY_WET (optional)
Thermal Characteristic Curves Parameter Definitions¶
- CONSTANT_THERMAL_CONDUCTIVITY <float>
Thermal conductivity of porous medium that does not depend on temperature or saturation [W/m-K].
- THERMAL_CONDUCTIVTY_WET <float>
Thermal conductivity of the wet porous medium (\(s_l=1\)) [W/m-K].
- THERMAL_CONDUCTIVITY_DRY <float>
Thermal conductivity of the dry porous medium (\(s_l=0\)) [W/m-K].
In the DEFAULT model, effective thermal conductivity (\(\kappa_T\)) at the given liquid saturation is computed as \(\kappa_T(s_l)=\kappa_T^{dry} + \sqrt{s_l}(\kappa_T^{wet} - \kappa_T^{dry})\) [W/m-K]. [1]
In the LINEAR model, effective thermal conductivity (\(\kappa_T\)) at the given liquid saturation is computed as \(\kappa_T(s_l)=\kappa_T^{dry} + \saturation_l(\kappa_T^{wet} - \kappa_T^{dry})\) [W/m-K].
- REFERENCE_TEMPERATURE <float>
This temperature is subtracted from the actual temperature before the calculation (useful for conversion from Celsius to Kelvin, or to shift the zero a polynomial) [°C].
- EXPONENT <float>
In the POWER model, this is the exponent of temperature, called \(\gamma\) [-].
Thermal conductivity for the POWER model is computed as \(\kappa_T(s_l,T)=\kappa_T(s_l)[(T-T_{ref})/300]^\gamma\) [W/m-K].
The saturation dependence of the POWER model comes from the DEFAULT model, and when using the default \(T_{ref}=-273.15\:°C\), THERMAL_CONDUCTIVITY_WET and THERMAL_CONDUCTIVITY_DRY are at 26.85 °C.
- CUBIC_POLYNOMIAL_COEFFICIENTS <float> <float> <float>
Coefficients of a cubic polynomial expression for the temperature-dependence, called \(\beta_i\).
Thermal conductivity for the CUBIC_POLYNOMIAL model is computed as \(\kappa_T(s_l,T)=\kappa_T(s_l)[1 + \beta_1 (T-T_{ref}) + \beta_2 (T-T_{ref})^2 + \beta_3 (T-T_{ref})^3]\) [W/m-K].
The saturation dependence of the CUBIC_POLYNOMIAL model comes from the DEFAULT model, and when using the default \(T_{ref}=0\:°C\), THERMAL_CONDUCTIVITY_WET and THERMAL_CONDUCTIVITY_DRY are at 0 °C.
- LINEAR_RESISTIVITY_COEFFICIENTS <float> <float>
Coefficients of a linear inverse conductivity (i.e., resistivity), called \(a_i\).
Thermal conductivity for the LINEAR_RESISTIVITY model is computed as \(\kappa_T(s_l,T)=\kappa_T(s_l)/[a_1 + a_2 (T - T_{ref})]\) [W/m-K], with the default \(T_{ref}=0\:°C\).
The saturation dependence of the LINEAR_RESISTIVITY model comes from the DEFAULT model, and when using the default \(T_{ref}=0\:°C\), THERMAL_CONDUCTIVITY_WET and THERMAL_CONDUCTIVITY_DRY are at 0 °C. Typically \(a_1=1\).
Note: this function also implements porosity dependence. See Porosity-Dependent Thermal Conductivity Definition for additional parameters needed to turn on this behavior.
- KERSTEN_EXPONENT <float>
In TH mode, this is the exponent (\(\alpha_{u}\) [-]) of liquid saturation used to derive the Kersten number for unfrozen soil: \(Ke_{u}=s^{\alpha_{u}}_{l}\) (see Ice Model).
Outside of TH mode, only the dry and wet components of the ice model are utilized for FROZEN.
- THERMAL_CONDUCTIVITY_FROZEN <float>
In the FROZEN model, this is the thermal conductivity of frozen soil [W/m-K] (see Ice Model).
When this parameter is specified in TH mode, the FREEZING option (see SIMULATION Options) automatically becomes active.
- KERSTEN_EXPONENT_FROZEN <float>
In the FROZEN model, this is the exponent (\(\alpha_{f}\) [-]) of ice saturation used to derive the Kersten number for frozen soil: \(Ke_{f}=s^{\alpha_{f}}_{i}\) (see Ice Model).
This parameter must be specified with THERMAL_CONDUCTIVITY_FROZEN.
- ICE_MODEL
- Specifies the ice model for the FROZEN model. Options include:
PAINTER_EXPLICIT [2]
PAINTER_KARRA_IMPLICIT [3]
PAINTER_KARRA_EXPLICIT [3]
PAINTER_KARRA_EXPLICIT_NOCRYO [3]
DALL_AMICO [4,5]
This parameter must be specified with THERMAL_CONDUCTIVITY_FROZEN.
Composite Function¶
In the COMPOSITE function, the following parameters are used to employ previously-defined thermal characteristic curves along certain principal axes (see example). Anisotropy ratios can also be specified if needed.
- COMPOSITE_X <string>
Name of the thermal characteristic curve governing conduction in the X direction.
- COMPOSITE_Y <string>
Name of the thermal characteristic curve governing conduction in the Y direction.
- COMPOSITE_Z <string>
Name of the thermal characteristic curve governing conduction in the Z direction.
Assembly Models¶
Models are available to describe the conduction of heat in spent nuclear fuel assemblies along both radial and axial directions.
The radial model (ASM_RADIAL) takes the form of the DEFAULT curve, albeit with a temperature-dependent dry component and a special wet component: \(\kappa_{radial}(s_l,T)=\kappa_{d}(T)+[\kappa_{w}^{\prime}-\kappa_{d}(T)\sqrt{s_{l}}]\) [W/m-K].
- The dry thermal conductivity of the radial model takes the form of a power law with temperature: \(\kappa_{d}(T)=\kappa_{d}^{0}+\alpha T^{\beta}\) [W/m-K] for \(T\ge\:0 °C\).[6]
- The wet thermal conductivity of the radial model takes into account the porosity of the assembly \((\Phi)\) and the thermal conductivities of its solid constituents and contained water (\(\kappa_{s}\) and \(\kappa_{l}\)): \(\kappa_{w}^{\prime}=\kappa_{l}\Bigg[1-\sqrt{1-\Phi}+\frac{\sqrt{1-\Phi}}{1+(\frac{\kappa_{l}}{\kappa_{s}}-1)\sqrt{1-\Phi}}\Bigg]\) [W/m-K].[7]
This model can be used on its own with the ASM_WATER_FILLED function.
A constant \(\kappa_{d}\) may be specified to use the saturation dependence of the DEFAULT model.
The axial model (ASM_AXIAL) assumes parallel conduction between solid constituents in the assembly and the surrounding water. When applied to an unsaturated system, it assumes that the thermal conductivity of gas is negligible. It differs from the DEFAULT curve by having linear saturation dependence and by using the thermal conductivities of assembly solids and water (as opposed to dry and wet components): \(\kappa_{axial}(s_{l})=(1-\Phi)\kappa_{s}+\Phi \saturation_{l}\kappa_{l}\) [W/m-K].
- THERMAL_CONDUCTIVITY_WATER <float>
The thermal conductivity of water (\(\kappa_{l}\) [W/m-K]) contained in the assembly.
- THERMAL_CONDUCTIVITY_SOLID <float>
The thermal conductivity of the solid components in the assembly, including rods and baskets (\(\kappa_{s}\) [W/m-K]).
- POROSITY_ASSEMBLY <float>
The porosity of the assembly \((\Phi)\), or the ratio of the volume of void to the total volume.
- THERMAL_CONDUCTIVITY_DRY <float>
For the radial assembly model, the dry thermal conductivity is applied as the zero-order term describing the baseline thermal conductivity of the dry assembly at 0 °C (\(\kappa_{d}^{0}\) [W/m-K]).
- ASM_DRY_COEFFICIENT <float>
For the dry state of the radial assembly model, this is the coefficient for the temperature-dependent term (\(\alpha\)).
- ASM_DRY_EXPONENT <float>
For the dry state of the radial assembly model, this is the exponent of temperature in the temperature-dependent term (\(\beta\)). Both \(\alpha\) and \(\beta\) must be fitted to align with the units of \(\kappa_{d}^{0}\).
Optional Blocks and Cards:¶
Thermal Conductivity Anisotropy Parameter Definitions¶
The following parameters are used to impart a direction-dependent treatment of thermal conductivity for thermal characteristic curves that employ \(\kappa_T(s_l)\) from the DEFAULT function. The following inputs are ratios that determine what fraction of the user-input values (THERMAL_CONDUCTIVITY_DRY or THERMAL_CONDUCTIVITY_WET) comprise particular components of the thermal conductivity tensor.
- ANISOTROPY_RATIO_X <float>
The ratio applied to user-input thermal conductivity to derive the \(\kappa_{xx}\) component of the thermal conductivity tensor. Requires additional input of Y and Z ratios.
- ANISOTROPY_RATIO_Y <float>
The ratio applied to user-input thermal conductivity to derive the \(\kappa_{yy}\) component of the thermal conductivity tensor. Requires additional input of X and Z ratios.
- ANISOTROPY_RATIO_Z <float>
The ratio applied to user-input thermal conductivity to derive the \(\kappa_{zz}\) component of the thermal conductivity tensor. Requires additional input of X and Y ratios.
Porosity-Dependent Thermal Conductivity Definition¶
The LINEAR_RESISTIVITY thermal conductivity model has three optional parameters. If these parameters are not specified, the model has no variation with porosity (see LINEAR_RESISTIVITY_COEFFICIENTS in Thermal Characteristic Curves Parameter Definitions). All three optional parameters presented below must be specified to turn on this behavior.
- REFERENCE_POROSITY <float>
Maximum porosity expected (\(\porosity_{\mathrm{ref}}\)). Value used to normalize porosity values between 0 and 1. If porosity goes above REFERENCE_POROSITY, the normalized value is capped at 1.0.
- POROSITY_EXPONENT <float>
A dimensionless exponent applied to the solid fraction (\(\xi\)).
- INITIAL_LINEAR_COEFFICIENTS <float> <float>
Two coefficients (\(b_1\) and \(b_2\)) used to express linear temperature dependence of the thermal conductivity of the pore space (\(b_1 + b_2 T\)).
The expression was developed for reconsolidation of granular salt with air-filled porosity (Bollingerfehr et al., 2012; Table B.4, Saltgrus), \(\kappa_T(s_l,T,\porosity) = \kappa_T(s_l,T) (1-\frac{\porosity}{\porosity_{\mathrm{ref}}})^\xi + (\frac{\porosity}{\porosity_{\mathrm{ref}}}) \cdot (b_1 + b_2 T)\).
\(\kappa_T(s_l,T)\) is the LINEAR_RESISTIVITY function presented previously (associated with LINEAR_RESISTIVITY_COEFFICIENTS without porosity variation).
At \(\porosity=0\), the thermal conductivity is equal to the LINEAR_RESISTIVITY model without porosity variation. At \(\porosity\ge\porosity_\mathrm{ref}\) the thermal conductivity has a linear dependence on temperature, given by \(b_1\) and \(b_2\). At porosities between these two endmembers, the porosity is interpolated as a combination of the two.
Test Thermal Characteristic Curve¶
- TEST
- Including this keyword will produce output (.dat file) for a thermal characteristic curve that includes:
temperature [C] \((T)\),
liquid saturation [-] \((s_l)\),
porosity [-] \((\porosity)\)
thermal conductivity [W/m*K] \((\kappa_T)\),
derivative of thermal conductivity with respect to liquid saturation \((\frac{\partial \kappa_T}{\partial \saturation_l})\),
derivative of thermal conductivity with respect to temperature \((\frac{\partial \kappa_T}{\partial T})\),
numerical approximation to (e.),
numerical approximation to (f.), and
numerical approximation to derivative of thermal conductivity with respect to porosity \((\frac{\partial \kappa_T}{\partial \porosity})\)
- When the FROZEN model is in use with FREEZING active, there are additional parameters in the output:
ice saturation \((s_i)\)
\(\frac{\partial \kappa_T}{\partial \saturation_i}\)
numerical approximation to \(\frac{\partial \kappa_T}{\partial \saturation_i}\)
Examples¶
Material with thermal characteristic curve named “cct_power”¶
MATERIAL_PROPERTY soil ID 1 CHARACTERISTIC_CURVES cc1 POROSITY 0.000001 TORTUOSITY 1.0 ROCK_DENSITY 2650.0 kg/m^3 THERMAL_CHARACTERISTIC_CURVES cct_power HEAT_CAPACITY 830.0 J/kg-C PERMEABILITY PERM_ISO 1.d-12 / / THERMAL_CHARACTERISTIC_CURVES cct_constant THERMAL_CONDUCTIVITY_FUNCTION CONSTANT CONSTANT_THERMAL_CONDUCTIVITY 5.5000D+0 W/m-C END TEST END THERMAL_CHARACTERISTIC_CURVES cct_default THERMAL_CONDUCTIVITY_FUNCTION DEFAULT THERMAL_CONDUCTIVITY_DRY 5.5000D+0 W/m-C THERMAL_CONDUCTIVITY_WET 7.0000D+0 W/m-C END TEST END THERMAL_CHARACTERISTIC_CURVES cct_power THERMAL_CONDUCTIVITY_FUNCTION POWER THERMAL_CONDUCTIVITY_DRY 5.5000D+0 W/m-C THERMAL_CONDUCTIVITY_WET 7.0000D+0 W/m-C #REFERENCE_TEMPERATURE -273.15 ! default value EXPONENT -1.18D+0 END TEST END THERMAL_CHARACTERISTIC_CURVES cct_cubic_polynomial THERMAL_CONDUCTIVITY_FUNCTION CUBIC_POLYNOMIAL THERMAL_CONDUCTIVITY_DRY 5.5000D+0 W/m-C THERMAL_CONDUCTIVITY_WET 7.0000D+0 W/m-C #REFERENCE_TEMPERATURE 0.d0 ! default value CUBIC_POLYNOMIAL_COEFFICIENTS -4.53398D-3 1.41580D-5 -1.94840D-8 END TEST END THERMAL_CHARACTERISTIC_CURVES cct_linear_resistivity THERMAL_CONDUCTIVITY_FUNCTION LINEAR_RESISTIVITY THERMAL_CONDUCTIVITY_DRY 5.5000D+0 W/m-C THERMAL_CONDUCTIVITY_WET 7.0000D+0 W/m-C #REFERENCE_TEMPERATURE 0.d0 ! default value LINEAR_RESISTIVITY_COEFFICIENTS 1.0d0 5.038D-3 END TEST END THERMAL_CHARACTERISTIC_CURVES cct_frozen THERMAL_CONDUCTIVITY_FUNCTION FROZEN THERMAL_CONDUCTIVITY_DRY 0.2500D+0 W/m-C THERMAL_CONDUCTIVITY_WET 1.3000D+0 W/m-C KERSTEN_EXPONENT 0.45 #THERMAL_CONDUCTIVITY_FROZEN 2.3500D+0 W/m-C #KERSTEN_EXPONENT_FROZEN 0.95 #ICE_MODEL PAINTER_EXPLICIT END TEST END
Material with composite thermal characteristic curve named “cct_composite”¶
MATERIAL_PROPERTY wp ID 1 CHARACTERISTIC_CURVES cc_wp POROSITY 0.50 TORTUOSITY 1.0 ROCK_DENSITY 5000.0 kg/m^3 THERMAL_CHARACTERISTIC_CURVES cct_composite HEAT_CAPACITY 450.0 J/kg-C PERMEABILITY PERM_ISO 1.d-16 / / THERMAL_CHARACTERISTIC_CURVES cct_axial THERMAL_CONDUCTIVITY_FUNCTION ASM_AXIAL THERMAL_CONDUCTIVITY_WATER 1.7200D+0 W/m-C THERMAL_CONDUCTIVITY_SOLID 1.6700D+1 W/m-C POROSITY_ASSEMBLY 5.0000D-1 END TEST END THERMAL_CHARACTERISTIC_CURVES cct_radial THERMAL_CONDUCTIVITY_FUNCTION ASM_RADIAL THERMAL_CONDUCTIVITY_DRY 0.1430D+0 W/m-C THERMAL_CONDUCTIVITY_WATER 1.7200D+0 W/m-C THERMAL_CONDUCTIVITY_SOLID 1.6700D+1 W/m-C ASM_DRY_COEFFICIENT 3.8300D-5 ASM_DRY_EXPONENT 1.6700D+0 POROSITY_ASSEMBLY 5.0000D-1 END TEST END THERMAL_CHARACTERISTIC_CURVES cct_composite THERMAL_CONDUCTIVITY_FUNCTION COMPOSITE COMPOSITE_X cct_radial COMPOSITE_Y cct_radial COMPOSITE_Z cct_axial END END
Material with anisotropic thermal conductivity¶
MATERIAL_PROPERTY soil ID 1 CHARACTERISTIC_CURVES cc1 POROSITY 0.25 TORTUOSITY 0.5 ROCK_DENSITY 2650.0 kg/m^3 THERMAL_CHARACTERISTIC_CURVES cct_linear_resistivity HEAT_CAPACITY 830.0 J/kg-C PERMEABILITY PERM_ISO 1.d-12 / / THERMAL_CHARACTERISTIC_CURVES cct_linear_resistivity THERMAL_CONDUCTIVITY_FUNCTION LINEAR_RESISTIVITY THERMAL_CONDUCTIVITY_DRY 5.5000D+0 W/m-C THERMAL_CONDUCTIVITY_WET 7.0000D+0 W/m-C #REFERENCE_TEMPERATURE 0.d0 ! default value LINEAR_RESISTIVITY_COEFFICIENTS 1.0d0 5.038D-3 ANISOTROPY_RATIO_X 1.0000D+0 ANISOTROPY_RATIO_Y 0.8000D+0 ANISOTROPY_RATIO_Z 0.5000D+0 END TEST END
References¶
Somerton, W.H., J.A. Keese, and S.L. Chu (1974). Thermal behavior of unconsolidated oil sands. Society of Petroleum Engineers Journal 14(5), 513-521. https://doi.org/10.2118/4506-PA
Painter, S.L. (2011). Three-phase numerical model of water migration in partially frozen geological media: model formulation, validation, and applications. Computational Geosciences 15, 69–85. https://doi.org/10.1007/s10596-010-9197-z
Painter, S.L., and S. Karra (2014). Constitutive model for unfrozen water content in subfreezing unsaturated soils. Vadose Zone 13(4), 1-8. https://doi.org/10.2136/vzj2013.04.0071
Dall’Amico, M. (2010). Coupled water and heat transfer in permafrost modeling. Ph.D. thesis, Institute of Civil and Environmental Engineering, Universita’ degli Studi di Trento, Trento, Italy. http://eprints-phd.biblio.unitn.it/335/
Dall’Amico, M., S. Endrizzi, S. Gruber, and R. Rigon (2011). A robust and energy-conserving model of freezing variably-saturated soil. The Cryosphere 5(2), 469-484. https://doi.org/10.5194/tc-5-469-2011
TRW Environmental Safety Systems (1996). Spent nuclear fuel effective thermal conductivity report. U.S. Department of Energy, Yucca Mountain Site Characterization Project Office, Las Vegas, NV. MOL.19961202.0030. https://doi.org/10.2172/778872
Cheng, P., and C.-T. Hsu (1999). The effective stagnant thermal conductivity of porous media with periodic structures. Journal of Porous Media 2(1), 19-38. https://doi.org/10.1615/JPorMedia.v2.i1.20
Bollingerfehr, W., W. Filbert, S. Dorr, P. Herold, C. Lerch, P. Burgwinkel, F. Charlier, B. Thomauske, G. Bracke, R. Kliger (2012). Endlangerauslegung und -optimierung. GRS-281, ISBN 978-3-939355-57-1, Gesellschaft für Anlagen- und Reaktorsicherheit (GRS) gGmbH.
