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"# Cubic Plus Association (CPA)\n",
"\n",
"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",
"$$\n",
"\\alpha^{\\rm r}_{\\rm assoc} = \\alpha^{\\rm r}_{\\rm cub,rep} +\\alpha^{\\rm r}_{\\rm cub,att}\n",
"$$ where the cubic parts come from two separate contributions. The repulsive part of the cubic EOS contribution is:\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 $a_{ij}$ depends on temperature while $b_{\\rm mix}$ does not.\n",
"\n",
"## Association part\n",
"\n",
"The residual contribution of the association to the Helmholtz energy $\\alpha^{\\rm r}_{\\rm assoc}$ is formulated as:\n",
"$$\n",
"\\alpha^{\\rm r}_{\\rm assoc} = \\sum_{i} x_i \\sum_{A_i} \\left(\\ln(Y_{A_i}) - \\frac{Y_{A_i}}{2} \\right) + \\frac{M_i}{2},\n",
"$$\n",
"where $x_i$ is the mole fraction of molecule $i$, $A_i$ is a site of type $A$ on molecule $i$ that can interact with other site types (e.g., type $B$). $Y_{A_i}$ then is the monomer mole fraction of site $A_i$ and $M_i$ is the count of association sites of molecule $i$.\n",
"The monomer mole fraction $Y_{A_i}$ can be calculated with:\n",
"$$\n",
"Y_{A_i} = \\frac{1}{1+\\rho N_{\\rm A} \\sum_j \\sum_{B_j} x_j Y_{B_j} \\Delta_{A_i B_j}}.\n",
"$$\n",
"$\\rho$ is the molar density of the mixture, $N_{\\rm A}$ is the Avogadro constant, $j$ is the index for a molecule that can be the same as molecule $i$. $B_j$ is a site of type $B$ on molecule $j$ and $\\Delta_{A_i B_j}$ is the interaction strength between the sites. The calculation of $\\Delta_{A_i B_j}$, $Y_{A_i}$ and $Y_{B_j}$ is explained in more detail in the documentation about association.\n",
"\n",
"The important thing to understand is whether a site is an electron acceptor (positively charged) or electron donor (negatively charged) and how many sites are present on a molecule. For example, water is usually considered a molecule with two electron acceptor (labeled as \"H\" in teqp) and two electron donor (labeled as \"e\" in teqp) sites. The site types have to be specified. "
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"import teqp, numpy as np"
]
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"D: [[0 2]\n",
" [2 0]]\n",
"∆: [[1.24389806e-27 1.24389806e-27]\n",
" [1.24389806e-27 1.24389806e-27]]\n",
"X_A: [0.54879025 0.54879025]\n",
"siteid->(component, name): [[0, [0, 'H']], [1, [0, 'e']]]\n",
"(component, name)->siteid: [[[0, 'H'], 0], [[0, 'e'], 1]]\n",
"multiplicities: [2 2]\n",
"pressure in Pa: 84414.03871039051\n"
]
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"source": [
"water = {\n",
" \"a0i / Pa m^6/mol^2\": 0.12277,\n",
" \"bi / m^3/mol\": 0.0000145,\n",
" \"c1\": 0.6736,\n",
" \"Tc / K\": 647.13,\n",
" \"epsABi / J/mol\": 16655.0,\n",
" \"betaABi\": 0.0692,\n",
"# here the site types are declared. \"e\" for electron donor, \"H\" for electron acceptor\n",
" \"sites\": [\"e\",\"e\",\"H\",\"H\"]\n",
"}\n",
"\n",
"jCPA = {\n",
" \"cubic\": \"SRK\",\n",
" \"radial_dist\": \"CS\",\n",
"# \"combining\": \"CR1\", # No other option is implemented yet\n",
" \"pures\": [water],\n",
" \"R_gas / J/mol/K\": 8.31446261815324\n",
"}\n",
"\n",
"model = teqp.make_model({\"kind\": \"CPA\", \"model\": jCPA, \"validate\": False}, False)\n",
"\n",
"T = 303.15 # K\n",
"rhomolar = 1000 # mol/m^3\n",
"molefracs = np.array([1.0])\n",
"\n",
"# Note: passing data back and forth in JSON format is done for convenience and flexibility, not speed\n",
"res = model.get_assoc_calcs(T, rhomolar, molefracs)\n",
"\n",
"print('D:', np.array(res['D']))\n",
"print('∆:', np.array(res['Delta']))\n",
"print('X_A:', np.array(res['X_A']))\n",
"print('siteid->(component, name):', res['to_CompSite'])\n",
"print('(component, name)->siteid:', res['to_siteid'])\n",
"print('multiplicities:', np.array(res['counts']))\n",
"\n",
"\n",
"print('pressure in Pa: ',T*rhomolar*model.get_R(molefracs) * ( 1+ model.get_Ar01(T,rhomolar,molefracs)))"
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