[cig-commits] [commit] doc_updates: Draft of thermoelastics background completed (da7b46c)
cig_noreply at geodynamics.org
cig_noreply at geodynamics.org
Tue Dec 30 12:59:08 PST 2014
Repository : https://github.com/geodynamics/burnman
On branch : doc_updates
Link : https://github.com/geodynamics/burnman/compare/b409e7148867eb5ae2d2f67417ffcff0212aa52c...da7b46ce4b4de76165deef26fe01df3b079425c2
>---------------------------------------------------------------
commit da7b46ce4b4de76165deef26fe01df3b079425c2
Author: Bob Myhill <myhill.bob at gmail.com>
Date: Tue Dec 30 20:58:50 2014 +0000
Draft of thermoelastics background completed
>---------------------------------------------------------------
da7b46ce4b4de76165deef26fe01df3b079425c2
sphinx/background_thermoelastics.txt | 53 +++++++++++++++++++++++++++++-------
sphinx/ref.bib | 28 +++++++++++++++++++
2 files changed, 71 insertions(+), 10 deletions(-)
diff --git a/sphinx/background_thermoelastics.txt b/sphinx/background_thermoelastics.txt
index 9c3f56b..48bf9e8 100644
--- a/sphinx/background_thermoelastics.txt
+++ b/sphinx/background_thermoelastics.txt
@@ -65,7 +65,7 @@ experimental fits or EoS modeling).
Modified Tait (isothermal)
^^^^^^^^^^^^^^^^^^^^^^^^^^
-The Modified Tait equation of state was developed by :cite:`HC1974`
+The Modified Tait equation of state was developed by :cite:`HC1974`. It has the considerable benefit of allowing volume to be expressed as a function of pressure. It performs very well to pressures and temperatures relevant to the deep Earth :cite:`HP2011`.
.. math::
\frac{V_{P, T}}{V_{1 bar, 298 K}} &= 1 - a(1-(1+bP)^{-c}), \\
@@ -75,8 +75,40 @@ The Modified Tait equation of state was developed by :cite:`HC1974`
:label: mtait
-Mie-Grüneisen-Debye (thermal)
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+Mie-Grüneisen-Debye (thermal correction to Birch-Murnaghan)
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+The Debye model for the Helmholtz free energy can be written as follows :cite:`Matas2007`
+
+.. math::
+ \mathcal{F} &= \frac{9nRT}{V}\frac{1}{x^3} \int_{0}^{x} \xi^2 \ln (1-e^{-\xi}) d\xi, \\
+ x &= \theta / T, \\
+ \theta &= \theta_0 \exp \left( \frac{\gamma_0-\gamma}{q_0} \right), \\
+ \gamma &= \gamma_0 \left( \frac{V}{V_0} \right)^{q_0}
+
+where :math:`\theta` is the Debye temperature and :math:`\gamma` is the Grüneisen parameter.
+
+Using thermodynamic relations we can derive equations for the thermal pressure and bulk modulus
+
+.. math::
+ P_{th}(V,T) &= - \frac{\partial \mathcal{F(V, T)}}{\partial V}, \\
+ &= \frac{3 n \gamma R T}{V} D(x), \\
+ K_{th}(V,T) &= -V \frac{\partial P(V, T)}{\partial V}, \\
+ &= \frac{3 n \gamma R T}{V} \gamma \left[ (1-q_0 - 3 \gamma) D(x) + 3\gamma \frac{x}{e^x - 1} \right], \\
+ D(x) &= \frac{3}{x^3} \int_{0}^{x} \frac{\xi^3}{e^{\xi} - 1} d\xi
+
+The thermal shear correction used in BurnMan was developed by :cite:`HS1998`
+
+.. math::
+ G_{th}(V,T) &= \frac{3}{5} \left[ K_{th} (V, T) - 2\frac{3nRT}{V}\gamma D(x) \right]
+
+
+The total pressure, bulk and shear moduli can be calculated from the following sums
+
+.. math::
+ P(V, T) &= P_{\textrm{ref}}(V, T_0) + P_{th}(V, T) - P_{th}(V, T_0), \\
+ K(V, T) &= K_{\textrm{ref}}(V, T_0) + K_{th}(V, T) - K_{th}(V, T_0), \\
+ G(V, T) &= G_{\textrm{ref}}(V, T_0) + G_{th}(V, T) - G_{th}(V, T_0)
HP2011 (thermal correction to Modified Tait)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
@@ -84,7 +116,7 @@ HP2011 (thermal correction to Modified Tait)
The thermal pressure can be incorporated into the Modified Tait equation of state, replacing :math:`P` with :math:`P-P_{\textrm{thermal}}` in Equation :eq:`mtait` :cite:`HP2011`. Thermal pressure here is calculated using a Mie-Grüneisen equation of state and an Einstein model for heat capacity, even though the Einstein model is not actually used for the heat capacity when calculating the enthalpy and entropy (see following section).
.. math::
- P_{\textrm{thermal}} &= \frac{\alpha_0 K_0 E_{\textrm{th}}}{C_{V0}}, \\
+ P_{\textrm{th}} &= \frac{\alpha_0 K_0 E_{\textrm{th}}}{C_{V0}}, \\
C_{V0} &= 3 n R \frac{(\frac{\Theta}{T})^2\exp(\frac{\Theta}{T})}{(\exp(\frac{\Theta}{T})-1)^2}, \\
E_{\textrm{th}} &= \int_{T0}^T \frac{1}{C_{V0}} dT
@@ -217,14 +249,15 @@ EoS, we refer readers to :cite:`Stixrude2005`.
| | | | |
+--------------+------------------+-----------------------------------+-------------------------+
-CORK (for fluids, thermal)
+Compensated-Redlich-Kwong (for fluids, thermal)
^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-A Murnaghan EoS is used to approximate the change in volume with pressure and temperature
+The CORK equation of state :cite:`HP1991` is a simple virial-type extension to the modified Redlich-Kwong (MRK) equation of state. It was designed to compensate for the tendency of the MRK equation of state to overestimate volumes at high pressures and accommodate the volume behaviour of coexisting gas and liquid phases along the saturation curve.
.. math::
- V_{P, T} = V_{1 bar, T} \left( 1- \frac{K_0'P}{K_0'P + K_0(1-\frac{dK_0}{dT}(T-298)} \right)^{\frac{1}{K_0'}}
+ V &= \frac{RT}{P} + c_1 - \frac{c_0 R T^{0.5}}{(RT + c_1 P)(RT + 2 c_1 P)} + c_2 P^{0.5} + c_3 P, \\
+ c_0 &= c_{0,0} T_c^{2.5}/P_c + c_{0,1} T_c^{1.5}/P_c T, \\
+ c_1 &= c_{1,0} T_c/P_c, \\
+ c_2 &= c_{2,0} T_c/P_c^{1.5} + c_{2,1}/P_c^{1.5} T, \\
+ c_3 &= c_{3,0} T_c/P_c^2 + c_{3,1}/P_c^2 T
-.. math::
- V_{1 bar, T} = (1+\alpha(T-298))
diff --git a/sphinx/ref.bib b/sphinx/ref.bib
index 94a6748..db50ae5 100644
--- a/sphinx/ref.bib
+++ b/sphinx/ref.bib
@@ -909,3 +909,31 @@ year = {2011}
volume = "25",
year = "2007"
}
+
+ at ARTICLE{HS1998,
+ author = {{Hama}, J. and {Suito}, K.},
+ title = "{High-temperature equation of state of CaSiO $_{3}$ perovskite and its implications for the lower mantle}",
+ journal = {Physics of the Earth and Planetary Interiors},
+ year = 1998,
+ month = feb,
+ volume = 105,
+ pages = {33-46},
+ doi = {10.1016/S0031-9201(97)00074-5},
+ adsurl = {http://adsabs.harvard.edu/abs/1998PEPI..105...33H},
+ adsnote = {Provided by the SAO/NASA Astrophysics Data System}
+}
+
+ at article{HP1991,
+ year={1991},
+ author = "Holland, Tim and Powell, Roger",
+ doi = "10.1007/BF00306484",
+ issn = "0010-7999",
+ journal = "Contributions to Mineralogy and Petrology",
+ language = "English",
+ number = "2",
+ pages = "265–273",
+ publisher = "Springer-Verlag",
+ title = "{A Compensated-Redlich-Kwong (CORK) equation for volumes and fugacities of CO2 and H2O in the range 1 bar to 50 kbar and 100–1600°C}",
+ url = "http://dx.doi.org/10.1007/BF00306484",
+ volume = "109"
+}
\ No newline at end of file
More information about the CIG-COMMITS
mailing list