Merge pull request #92 from usnistgov/assocmix

Add association for mixtures

doc/source/models/CPA.ipynb

@@ -7,8 +7,64 @@
    "source": [
     "# Cubic Plus Association (CPA)\n",
     "\n",
-    "The combination of a cubic EOS with association"
+    "The combination of a cubic EOS with association with the association term. The sum of the terms goes like:\n",
+    "$$\n",
+    "\\alpha^{\\rm r} = \\alpha^{\\rm r}_{\\rm cub} + \\alpha^{\\rm r}_{\\rm assoc}\n",
+    "$$\n",
+    "\n",
+    "## Cubic part\n",
+    "\n",
+    "The residual contribution to $\\alpha$ is expressed as the sum :\n",
+    "$$\n",
+    "\\alpha^{\\rm r}_{\\rm cub,rep} +\\alpha^{\\rm r}_{\\rm cub,att}\n",
+    "$$ where the cubic parts come from\n",
+    "\n",
+    "The repulsive part of the cubic EOS contribution:\n",
+    "$$\n",
+    "\\alpha^{\\rm r}_{\\rm cub,rep} = -\\ln(1 - b_{\\rm mix}\\rho) \n",
+    "$$\n",
+    "The attractive part of the cubic EOS contribution:\n",
+    "$$\n",
+    "\\alpha^{\\rm r}_{\\rm cub,att} = -\\frac{a_{\\rm mix}}{RT}\\dfrac{\\ln\\left(\\frac{\\Delta_1 b_{\\rm mix}\\rho + 1}{\\Delta_2b_{\\rm mix}\\rho + 1}\\right)}{b_{\\rm mix}\\cdot(\\Delta_1 - \\Delta_2)}\n",
+    "$$\n",
+    "with the coefficients depending on the cubic type:\n",
+    "\n",
+    "SRK: $\\Delta_1=1$, $\\Delta_2=0$\n",
+    "\n",
+    "PR: $\\Delta_1=1+\\sqrt{2}$, $\\Delta_2=1-\\sqrt{2}$\n",
+    "\n",
+    "The mixture models used for the $a_{\\rm mix}$ and $b_{\\rm mix}$ are the classical ones:\n",
+    "\n",
+    "$$\n",
+    "a_{\\rm mix} = \\sum_i\\sum_jx_ix_j(1-k_{ij})a_{ij}(T)\n",
+    "$$\n",
+    "with x the mole fraction, $k_{ij}$ a weighting parameter\n",
+    "$$\n",
+    "a_{ij}(T) = \\sqrt{a_ia_j}\n",
+    "$$\n",
+    "and\n",
+    "$$\n",
+    "a_{i}(T) = a_{0i}\\left[1+c_{1i}(1-\\sqrt{T/T_{{\\rm crit},i}})\\right]^2\n",
+    "$$\n",
+    "and for $b$:\n",
+    "$$\n",
+    "b_{\\rm mix} = \\sum_ix_ib_i\n",
+    "$$\n",
+    "so there are three cubic parameters per fluid that need to be obtained though fitting: $b_{i}$, $a_{0i}$, $c_{1i}$. The value of $a_{\\rm ij}$ depends on temperature while $b_{\\rm mix}$ does not.\n",
+    "\n",
+    "## Association part\n",
+    "\n",
+    "For the association, one must have a solid understanding of the association approach that is being applied. To this end, a short discussion of the general approach is required. \n",
+    "\n"
    ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "2180ef67",
+   "metadata": {},
+   "outputs": [],
+   "source": []
   }
  ],
  "metadata": {
@@ -27,7 +83,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.10.4"
+   "version": "3.10.13"
   }
  },
  "nbformat": 4,