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21 | 21 |
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22 | 22 | .. _PowerMG: https://www.osti.gov/biblio/1762624 |
23 | 23 |
|
24 | | -.. _SimpleMACAW: https://www.osti.gov/biblio/2479474 |
25 | | - |
26 | | -.. _MACAW EOS: https://pubs.aip.org/aip/jap/article/134/12/125102/2912256 |
27 | 24 |
|
28 | 25 | EOS Models |
29 | 26 | =========== |
@@ -1323,56 +1320,6 @@ This constructor also optionally accepts `MeanAtomicProperties` for |
1323 | 1320 | the atomic mass and number as a final optional parameter. |
1324 | 1321 |
|
1325 | 1322 |
|
1326 | | -Simple MACAW |
1327 | | -```````````` |
1328 | | -The `Simple MACAW EOS <_SimpleMACAW>`_ is a simplified version of the `MACAW EOS <MACAW EOS>`_ |
1329 | | -and is thermodynamically consistent. It is constructed from a the Helmholtz |
1330 | | -free energy using a Murnaghan cold curve and a simplified thermal component |
1331 | | -from the MACAW EOS. |
1332 | | - |
1333 | | -Fundamentally, the equation of state can be written in Mie-Gruneisen form (with constant Gruneisen parameter) as: |
1334 | | - |
1335 | | -.. math:: |
1336 | | -
|
1337 | | - P(v, e) = P_{\text{cold}}(v) + \Gamma_c \rho (e - e_{\text{cold}}(v)) |
1338 | | -
|
1339 | | -where the cold curves are given by: |
1340 | | - |
1341 | | -.. math:: |
1342 | | -
|
1343 | | - e_{\text{cold}}(v) = A v_0 \Big[ \Big( \frac{v}{v_0} \Big)^{-B} + \Big( \frac{v}{v_0} \Big) B - (B+1) \Big] |
1344 | | -
|
1345 | | -and |
1346 | | - |
1347 | | -.. math:: |
1348 | | -
|
1349 | | - p_{\text{cold}}(v) := -e'_{\text{cold}}(v) = AB \Big( \Big( \frac{v}{v_0} \Big)^{-(B+1)} - 1 \Big) |
1350 | | -
|
1351 | | -The specific heat capacity at constant volume for this model is given by, |
1352 | | - |
1353 | | -.. math:: |
1354 | | -
|
1355 | | - C_v(v, \tau) = C^\infty_v \frac{\tau^2 + 2\tau}{(\tau + 1)^2} \quad \text{where } \tau = \frac{T}{\theta(v)} \quad \text{ and } \quad \theta(v) := T_0 \Big( \frac{v}{v_c} \Big)^{-\Gamma_c} |
1356 | | -
|
1357 | | -Note it obeys the expected physical behavior; that, :math:`\lim_{T\to 0^+} C_v(v,\tau(v,T)) = 0` and |
1358 | | -:math:`\lim_{T\to\infty} C_v(v,\tau(v,T) = C^\infty_v < \infty` (Dulong-Petit law). |
1359 | | - |
1360 | | -The constructor for the Simple MACAW EOS is |
1361 | | - |
1362 | | -.. code-block:: cpp |
1363 | | -
|
1364 | | - SimpleMACAW(const Real A, const Real B, const Real Cvinf, const Real v0, |
1365 | | - const Real T0, const Real Gc) |
1366 | | -
|
1367 | | -where ``A`` is :math:`A`, ``B`` is :math:`B`, ``Cvinf`` is :math:`C^\infty_v`, |
1368 | | -``v0`` is :math:`v_0`, ``T0`` is :math:`T_0`, ``Gc`` is :math:`\Gamma_c`. |
1369 | | - |
1370 | | -In order to maintain thermodynamic stability, a sufficient set of constraints |
1371 | | -is given by :math:`A > 0`, :math:`B > 0`, :math:`C^\infty_v > 0`, :math:`v_0 > |
1372 | | -0`, :math:`T_0 > 0`, and :math:`\Gamma_c \in (0,1]`. One can still select |
1373 | | -:math:`\Gamma_c > 1`, just note that the isothermal bulk modulus can be |
1374 | | -negative (the isentropic bulk modulus will still be positive though). |
1375 | | - |
1376 | 1323 | JWL EOS |
1377 | 1324 | `````````` |
1378 | 1325 |
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