[cig-commits] [commit] doc_updates: Draft of thermodynamics background completed (c2bb5f2)

[cig_noreply at geodynamics.org]

Repository : https://github.com/geodynamics/burnman

On branch  : doc_updates
Link       : https://github.com/geodynamics/burnman/compare/da7b46ce4b4de76165deef26fe01df3b079425c2...c2bb5f2f08c50589c2e81d3d54ad91228051efab

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commit c2bb5f2f08c50589c2e81d3d54ad91228051efab
Author: Bob Myhill <myhill.bob at gmail.com>
Date:   Tue Dec 30 22:17:40 2014 +0000

    Draft of thermodynamics background completed


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c2bb5f2f08c50589c2e81d3d54ad91228051efab
 sphinx/background_thermodynamics.txt | 30 +++++++++++++++++++++++++++---
 1 file changed, 27 insertions(+), 3 deletions(-)

diff --git a/sphinx/background_thermodynamics.txt b/sphinx/background_thermodynamics.txt
index 4cbf1e0..2785564 100644
--- a/sphinx/background_thermodynamics.txt
+++ b/sphinx/background_thermodynamics.txt
@@ -19,12 +19,36 @@ HP2011
 ^^^^^^
 
 .. math::
-    \mathcal{G}(P,T) = \mathcal{H}_{\textrm{1 bar, T}} - T\mathcal{S}_{\textrm{1 bar, T}} + \int_{\textrm{1 bar}}^P V(P,T)~dP
+    \mathcal{G}(P,T) &= \mathcal{H}_{\textrm{1 bar, T}} - T\mathcal{S}_{\textrm{1 bar, T}} + \int_{\textrm{1 bar}}^P V(P,T)~dP, \\
+    \mathcal{H}_{\textrm{1 bar, T}} &= \Delta_f\mathcal{H}_{\textrm{1 bar, 298 K}} + \int_{298}^T C_P~dT, \\
+    \mathcal{S}_{\textrm{1 bar, T}} &= \mathcal{S}_{\textrm{1 bar, 298 K}} + \int_{298}^T \frac{C_P}{T}~dT, \\
+    \int_{\textrm{1 bar}}^P V(P,T)~dP &= P V_0 \left( 1 - a + \left( a \frac{(1-b P_{th})^{1-c} - (1 + b(P-P_{th}))^{1-c}}{b (c-1) P} \right) \right)
     :label: gibbs_hp2011  
 
+
+The heat capacity at one bar is given by an empirical polynomial fit to experimental data
+
+.. math::
+    C_p = a + bT + cT^{-2} + dT^{-0.5}
+
+The entropy at high pressure and temperature can be calculated by differentiating the expression for :math:`\mathcal{G}` with respect to temperature
+
 .. math::
-    \mathcal{G}(P,T) = \left( \Delta_f\mathcal{H}_{\textrm{1 bar, 298 K}} + \int_{298}^T C_P~dT \right) - T\left(\mathcal{S}_{\textrm{1 bar, 298 K}} + \int_{298}^T \frac{C_P}{T}~dT \right) + \int_{\textrm{1 bar}}^P V(P,T)~dP
-    :label: gibbs_hp2011_2   
+    \mathcal{S}(P,T) &= S_{\textrm{1 bar, T}} + \frac{\partial  \int V dP }{\partial T}, \\
+    \frac{\partial  \int V dP }{\partial T} &= V_0 \alpha_0 K_0 a \frac{C_{V0}(T)}{C_{V0}(T_\textrm{{ref}})} ((1+b(P-P_{th}))^{-c} - (1-bP_{th})^{-c} )
+
+Finally, the enthalpy at high pressure and temperature can be calculated
+
+.. math::
+    \mathcal{H}(P,T) = \mathcal{G}(P,T) + T\mathcal{S}(P,T) 
 
 SLB2005
 ^^^^^^^
+
+The Debye model yields the Helmholtz free energy and entropy due to lattice vibrations
+
+.. math::
+    \mathcal{G} &= \mathcal{F} + PV, \\
+    \mathcal{F} &= nRT \left(3 \ln( 1 - e^{-\frac{\theta}{T}}) - \int_{0}^{\frac{\theta}{T}}\frac{\tau^3}{(e^{\tau}-1)}d\tau \right), \\	
+    \mathcal{S} &= nR \left(4 \int_{0}^{\frac{\theta}{T}}\frac{\tau^3}{(e^{\tau}-1)}d\tau - 3 \ln(1-e^{-\frac{\theta}{T}}) \right), \\
+    \mathcal{H} &= \mathcal{G} + T\mathcal{S}