Conductivity

Methods

Thermal conductivity is defined as

GeoParams.MaterialParameters.Conductivity.ConstantConductivity Type

ConstantConductivity(k=3.0W/m/K)

Set a constant conductivity

$$k=cst$$

where $k$ is the thermal conductivity [$W/m/K$].

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GeoParams.MaterialParameters.Conductivity.T_Conductivity_Whittington Type

T_Conductivity_Whittington()

Sets a temperature-dependent conductivity following the parameterization of _Whittington, A.G., Hofmeister, A.M., Nabelek, P.I., 2009. Temperature-dependent thermal diffusivity of the Earth’s crust and implications for magmatism. Nature 458, 319–321. https://doi.org/10.1038/nature07818._ Their parameterization is originally given for the thermal diffusivity, together with a parameterization for thermal conductivity, which allows us to compute

$$Cp=a+bT-c/T^2$$$$\kappa =d/T-e\text{ if }T<=846K$$$$\kappa =f-g\ast T\text{ if }T>846K$$$$\rho =2700kg/m^3$$$$k=\kappa \rho Cp$$

where $Cp$ is the heat capacity [$J/mol/K$], and $a,b,c$ are parameters that dependent on the temperature T:

• a = 199.50 J/mol/K if T<= 846 K

• a = 199.50 J/mol/K if T> 846 K

• b = 0.0857J/mol/K^2 if T<= 846 K

• b = 0.0323J/mol/K^2 if T> 846 K

• c = 5e6J/mol*K if T<= 846 K

• c = 47.9e-6J/mol*K if T> 846 K

• d = 576.3m^2/s*K

• e = 0.062m^2/s

• f = 0.732m^2/s

• g = 0.000135m^2/s/K

This looks like:

Example

julia> using GLMakie, GeoParams
julia> p=T_Conductivity_Whittington();
julia> T,k,plt = PlotConductivity(p)

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GeoParams.MaterialParameters.Conductivity.T_Conductivity_Whittington_parameterised Type

T_Conductivity_Whittington_parameterised()

Sets a temperature-dependent conductivity that is parameterization after Whittington, et al. 2009

The original parameterization involves quite a few parameters; this is a polynomial fit that is roughly valid from 0-1000 Celsius

$$k[W/m/K]=-2{10}^{-9}(T-Ts{)}^3+6{10}^{-6}(T-Ts{)}^2-0.0062(T-Ts)+4$$$$Ts=273.15K$$

where T[K] is the temperature in Kelvin (or the nondimensional equivalent of it).

The comparison of this parameterisation vs. the original one is:

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GeoParams.MaterialParameters.Conductivity.TP_Conductivity Type

TP_Conductivity()

Sets a temperature (and pressure)-dependent conductivity parameterization as described in Gerya, Numerical Geodynamics (2nd edition, Table 21.2). The general for

$$k=(a_k+\frac{b_k}{T+c_k})(1+d_{k}P)$$

where $k$ is the conductivity [$W/K/m$], and $a_k,b_k,c_k,d_k$ are parameters that dependent on the temperature T and pressure P:

• $a_k$ = 1.18Watt/K/m

• $b_k$ = 474Watt/m

• $c_k$ = 77K

• $d_k$ = 0/MPa

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GeoParams.MaterialParameters.Conductivity.Set_TP_Conductivity Function

Set_TP_Conductivity["Name of temperature(-pressure) dependent conductivity"]

This is a dictionary with pre-defined laws:

• "UpperCrust"

• "LowerCrust"

• "OceanicCrust"

• "Mantle"

Example

julia> k=Set_TP_Conductivity["Mantle"]
T/P dependent conductivity: k = (0.73 W K⁻¹ m⁻¹ + 1293 W m⁻¹/(T + 77 K))*(1 + 4.0e-5 MPa⁻¹*P)

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Computational routines

To compute, use this:

GeoParams.MaterialParameters.Conductivity.compute_conductivity Function

k = compute_conductivity(P, T, s:<AbstractConductivity)

Returns the thermal conductivity k at any temperature T and pressure P using any of the parameterizations implemented.

Currently available:

• ConstantConductivity

• T_Conductivity_Whittington

• TP_Conductivity

Example

Using dimensional units

julia> T  = (250:100:1250)*K;
julia> cp = T_HeatCapacity_Whittington()
julia> Cp = ComputeHeatCapacity(0,T,cp)

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compute_conductivity!(K::AbstractArray{<:AbstractFloat}, Phases::AbstractArray{<:Integer}, P::AbstractArray{<:AbstractFloat},Temp::AbstractArray{<:AbstractFloat}, MatParam::AbstractArray{<:AbstractMaterialParamsStruct})

In-place computation of conductivity K for the whole domain and all phases, in case a vector with phase properties MatParam is provided, along with P and Temp arrays. This assumes that the Phase of every point is specified as an Integer in the Phases array.

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compute_conductivity!(k::AbstractArray{T,N}, PhaseRatios::AbstractArray{T, M}, P::AbstractArray{<:AbstractFloat,N},T::AbstractArray{<:AbstractFloat,N}, MatParam::AbstractArray{<:AbstractMaterialParamsStruct})

In-place computation of conductivity k for the whole domain and all phases, in case a vector with phase properties MatParam is provided, along with P and T arrays. This assumes that the PhaseRatio of every point is specified as an Integer in the PhaseRatios array, which has one dimension more than the data arrays (and has a phase fraction between 0-1)

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GeoParams.MaterialParameters.Conductivity.compute_conductivity! Function

compute_conductivity(k_array::AbstractArray{<:AbstractFloat,N},P::AbstractArray{<:AbstractFloat,N},T::AbstractArray{<:AbstractFloat,N}, s::ConstantConductivity) where N

In-place routine to compute constant conductivity

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