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388080231e
- remove stray AC_MSG_RESULT from DUMUX_CHECK_QUAD - slight reformulations of the usage message of test_2p - replace tabulators with four spaces - remove trailing whitespaceformating
506 lines
17 KiB
C++
506 lines
17 KiB
C++
// -*- mode: C++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
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// vi: set et ts=4 sw=4 sts=4:
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/*****************************************************************************
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* Copyright (C) 2010-2011 by Andreas Lauser *
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* Institute for Modelling Hydraulic and Environmental Systems *
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* University of Stuttgart, Germany *
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* email: <givenname>.<name>@iws.uni-stuttgart.de *
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* *
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* This program is free software: you can redistribute it and/or modify *
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* it under the terms of the GNU General Public License as published by *
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* the Free Software Foundation, either version 2 of the License, or *
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* (at your option) any later version. *
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* *
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* This program is distributed in the hope that it will be useful, *
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* but WITHOUT ANY WARRANTY; without even the implied warranty of *
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *
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* GNU General Public License for more details. *
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* *
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* You should have received a copy of the GNU General Public License *
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* along with this program. If not, see <http://www.gnu.org/licenses/>. *
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*****************************************************************************/
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/*!
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* \file
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*
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* \brief Implements the Peng-Robinson equation of state for liquids
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* and gases.
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*
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* See:
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*
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* D.-Y. Peng, D.B. Robinson: A new two-constant equation of state,
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* Industrial & Engineering Chemistry Fundamentals, 1976, 15 (1),
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* pp. 59–64
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*
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* R. Reid, et al.: The Properties of Gases and Liquids,
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* 4th edition, McGraw-Hill, 1987, pp. 42-44, 82
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*/
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#ifndef DUMUX_PENG_ROBINSON_HH
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#define DUMUX_PENG_ROBINSON_HH
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#include <dumux/material/fluidstates/temperatureoverlayfluidstate.hh>
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#include <dumux/material/idealgas.hh>
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#include <dumux/common/exceptions.hh>
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#include <dumux/common/math.hh>
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#include <dumux/common/tabulated2dfunction.hh>
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#include <iostream>
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namespace Dumux
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{
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/*!
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* \brief Implements the Peng-Robinson equation of state for liquids
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* and gases.
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*
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* See:
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*
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* D.-Y. Peng, D.B. Robinson: A new two-constant equation of state,
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* Industrial & Engineering Chemistry Fundamentals, 1976, 15 (1),
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* pp. 59–64
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*
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* R. Reid, et al.: The Properties of Gases and Liquids,
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* 4th edition, McGraw-Hill, 1987, pp. 42-44, 82
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*/
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template <class Scalar>
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class PengRobinson
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{
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//! The ideal gas constant [Pa * m^3/mol/K]
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static constexpr Scalar R = Dumux::Constants<Scalar>::R;
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PengRobinson()
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{ }
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public:
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static void init(Scalar aMin, Scalar aMax, int na,
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Scalar bMin, Scalar bMax, int nb)
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{
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// resize the tabulation for the critical points
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criticalTemperature_.resize(aMin, aMax, na, bMin, bMax, nb);
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criticalPressure_.resize(aMin, aMax, na, bMin, bMax, nb);
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criticalMolarVolume_.resize(aMin, aMax, na, bMin, bMax, nb);
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Scalar VmCrit, pCrit, TCrit;
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for (int i = 0; i < na; ++i) {
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Scalar a = criticalTemperature_.iToX(i);
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for (int j = 0; j < nb; ++j) {
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Scalar b = criticalTemperature_.jToY(j);
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findCriticalPoint_(TCrit, pCrit, VmCrit, a, b);
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criticalTemperature_.setSamplePoint(i, j, TCrit);
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criticalPressure_.setSamplePoint(i, j, pCrit);
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criticalMolarVolume_.setSamplePoint(i, j, VmCrit);
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}
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}
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};
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/*!
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* \brief Predicts the vapor pressure for the temperature given in
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* setTP().
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*
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* Initially, the vapor pressure is roughly estimated by using the
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* Ambrose-Walton method, then the Newton method is used to make
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* difference between the gas and liquid phase fugacity zero.
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*/
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template <class Params>
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static Scalar computeVaporPressure(const Params ¶ms, Scalar T)
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{
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typedef typename Params::Component Component;
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if (T >= Component::criticalTemperature())
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return Component::criticalPressure();
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// initial guess of the vapor pressure
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Scalar Vm[3];
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const Scalar eps = Component::criticalPressure()*1e-10;
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// use the Ambrose-Walton method to get an initial guess of
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// the vapor pressure
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Scalar pVap = ambroseWalton_(params, T);
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// Newton-Raphson method
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for (int i = 0; i < 5; ++i) {
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// calculate the molar densities
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int numSol = molarVolumes(Vm, params, T, pVap);
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assert(numSol == 3);
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Scalar f = fugacityDifference_(params, T, pVap, Vm[0], Vm[2]);
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Scalar df_dp =
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fugacityDifference_(params, T, pVap + eps, Vm[0], Vm[2])
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-
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fugacityDifference_(params, T, pVap - eps, Vm[0], Vm[2]);
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df_dp /= 2*eps;
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Scalar delta = f/df_dp;
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pVap = pVap - delta;
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if (std::abs(delta/pVap) < 1e-10)
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break;
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}
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return pVap;
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}
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/*!
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* \brief Computes molar volumes where the Peng-Robinson EOS is
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* true.
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*/
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template <class FluidState, class Params>
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static Scalar computeMolarVolume(const FluidState &fs,
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Params ¶ms,
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int phaseIdx,
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bool isGasPhase)
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{
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Valgrind::CheckDefined(fs.temperature(phaseIdx));
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Valgrind::CheckDefined(fs.pressure(phaseIdx));
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Scalar Vm = 0;
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Valgrind::SetUndefined(Vm);
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Scalar T = fs.temperature(phaseIdx);
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Scalar p = fs.pressure(phaseIdx);
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Scalar a = params.a(phaseIdx); // "attractive factor"
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Scalar b = params.b(phaseIdx); // "co-volume"
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Scalar RT= R*T;
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Scalar Astar = a*p/(RT*RT);
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Scalar Bstar = b*p/RT;
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Scalar a1 = 1.0;
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Scalar a2 = - (1 - Bstar);
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Scalar a3 = Astar - Bstar*(3*Bstar + 2);
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Scalar a4 = Bstar*(- Astar + Bstar*(1 + Bstar));
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// ignore the first two results if the smallest
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// compressibility factor is <= 0.0. (this means that if we
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// would get negative molar volumes for the liquid phase, we
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// consider the liquid phase non-existant.)
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Scalar Z[3] = {0.0,0.0,0.0};
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int numSol = invertCubicPolynomial(Z, a1, a2, a3, a4);
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if (numSol == 3) {
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// the EOS has three intersections with the pressure,
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// i.e. the molar volume of gas is the largest one and the
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// molar volume of liquid is the smallest one
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if (isGasPhase)
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Vm = Z[2]*RT/p;
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else
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Vm = Z[0]*RT/p;
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}
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else if (numSol == 1) {
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// the EOS only has one intersection with the pressure,
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// for the other phase, we take the extremum of the EOS
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// with the largest distance from the intersection.
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Scalar VmCubic = Z[0]*RT/p;
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if (T > criticalTemperature_(a, b)) {
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// if the EOS does not exhibit any extrema, the fluid
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// is critical...
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Vm = VmCubic;
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handleCriticalFluid_(Vm, fs, params, phaseIdx, isGasPhase);
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}
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else {
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// find the extrema (if they are present)
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Scalar Vmin, Vmax, pmin, pmax;
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if (findExtrema_(Vmin, Vmax,
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pmin, pmax,
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a, b, T))
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{
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if (isGasPhase)
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Vm = std::max(Vmax, VmCubic);
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else
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Vm = std::min(Vmin, VmCubic);
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}
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else
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Vm = VmCubic;
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}
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}
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Valgrind::CheckDefined(Vm);
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assert(std::isfinite(Vm) && Vm > 0);
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return Vm;
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}
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/*!
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* \brief Returns the fugacity coefficient for a given pressure
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* and molar volume.
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*
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* This is the same value as computeFugacity() because the mole
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* fraction of a component in a pure fluid is obviously always
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* 100%.
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*
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* \param params Parameters
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*/
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template <class Params>
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static Scalar computeFugacityCoeffient(const Params ¶ms)
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{
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Scalar T = params.temperature();
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Scalar p = params.pressure();
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Scalar Vm = params.molarVolume();
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Scalar RT = R*T;
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Scalar Z = p*Vm/RT;
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Scalar Bstar = p*params.b() / RT;
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Scalar tmp =
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(Vm + params.b()*(1 + std::sqrt(2))) /
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(Vm + params.b()*(1 - std::sqrt(2)));
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Scalar expo = - params.a()/(RT * 2 * params.b() * std::sqrt(2));
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Scalar fugCoeff =
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std::exp(Z - 1) / (Z - Bstar) *
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std::pow(tmp, expo);
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return fugCoeff;
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}
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/*!
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* \brief Returns the fugacity coefficient for a given pressure
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* and molar volume.
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*
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* This is the fugacity coefficient times the pressure. The mole
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* fraction of a component in a pure fluid is obviously always
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* 100%, so it is not required.
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*
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* \param params Parameters
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*/
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template <class Params>
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static Scalar computeFugacity(const Params ¶ms)
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{ return params.pressure()*computeFugacityCoeff(params); }
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protected:
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template <class FluidState, class Params>
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static void handleCriticalFluid_(Scalar &Vm,
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const FluidState &fs,
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const Params ¶ms,
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int phaseIdx,
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bool isGasPhase)
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{
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Scalar Vcrit = criticalMolarVolume_(params.a(phaseIdx), params.b(phaseIdx));
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if (isGasPhase)
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Vm = std::max(Vm, Vcrit);
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else
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Vm = std::min(Vm, Vcrit);
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}
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static void findCriticalPoint_(Scalar &Tcrit,
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Scalar &pcrit,
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Scalar &Vcrit,
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Scalar a,
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Scalar b)
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{
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Scalar minVm;
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Scalar maxVm;
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Scalar minP(0);
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Scalar maxP(1e100);
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// first, we need to find an isotherm where the EOS exhibits
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// a maximum and a minimum
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Scalar Tmin = 250; // [K]
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for (int i = 0; i < 30; ++i) {
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bool hasExtrema = findExtrema_(minVm, maxVm, minP, maxP, a, b, Tmin);
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if (hasExtrema)
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break;
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Tmin /= 2;
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};
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Scalar T = Tmin;
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// Newton's method: Start at minimum temperature and minimize
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// the "gap" between the extrema of the EOS
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for (int i = 0; i < 25; ++i) {
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// calculate function we would like to minimize
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Scalar f = maxVm - minVm;
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// check if we're converged
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if (f < 1e-10) {
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Tcrit = T;
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pcrit = (minP + maxP)/2;
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Vcrit = (maxVm + minVm)/2;
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return;
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}
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// backward differences. Forward differences are not
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// robust, since the isotherm could be critical if an
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// epsilon was added to the temperature. (this is case
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// rarely happens, though)
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const Scalar eps = - 1e-8;
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bool DUMUX_UNUSED hasExtrema = findExtrema_(minVm, maxVm, minP, maxP, a, b, T + eps);
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assert(hasExtrema);
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Scalar fStar = maxVm - minVm;
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// derivative of the difference between the maximum's
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// molar volume and the minimum's molar volume regarding
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// temperature
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Scalar fPrime = (fStar - f)/eps;
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// update value for the current iteration
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Scalar delta = f/fPrime;
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if (delta > 0)
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delta = -10;
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// line search (we have to make sure that both extrema
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// still exist after the update)
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for (int j = 0; ; ++j) {
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if (j >= 20) {
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DUNE_THROW(NumericalProblem,
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"Could not determine the critical point for a=" << a << ", b=" << b);
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}
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if (findExtrema_(minVm, maxVm, minP, maxP, a, b, T - delta)) {
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// if the isotherm for T - delta exhibits two
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// extrema the update is finished
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T -= delta;
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break;
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}
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else
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delta /= 2;
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};
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}
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};
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// find the two molar volumes where the EOS exhibits extrema and
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// which are larger than the covolume of the phase
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static bool findExtrema_(Scalar &Vmin,
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Scalar &Vmax,
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Scalar &pMin,
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Scalar &pMax,
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Scalar a,
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Scalar b,
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Scalar T)
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{
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Scalar u = 2;
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Scalar w = -1;
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Scalar RT = R*T;
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// calculate coefficients of the 4th order polynominal in
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// monomial basis
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Scalar a1 = RT;
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Scalar a2 = 2*RT*u*b - 2*a;
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Scalar a3 = 2*RT*w*b*b + RT*u*u*b*b + 4*a*b - u*a*b;
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Scalar a4 = 2*RT*u*w*b*b*b + 2*u*a*b*b - 2*a*b*b;
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Scalar a5 = RT*w*w*b*b*b*b - u*a*b*b*b;
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// Newton method to find first root
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// if the values which we got on Vmin and Vmax are usefull, we
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// will reuse them as initial value, else we will start 10%
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// above the covolume
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Scalar V = b*1.1;
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Scalar delta = 1.0;
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for (int i = 0; std::abs(delta) > 1e-9; ++i) {
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Scalar f = a5 + V*(a4 + V*(a3 + V*(a2 + V*a1)));
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Scalar fPrime = a4 + V*(2*a3 + V*(3*a2 + V*4*a1));
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if (std::abs(fPrime) < 1e-20) {
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// give up if the derivative is zero
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Vmin = 0;
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Vmax = 0;
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return false;
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}
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delta = f/fPrime;
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V -= delta;
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if (i > 20) {
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// give up after 20 iterations...
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Vmin = 0;
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Vmax = 0;
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return false;
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}
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}
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// polynomial division
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Scalar b1 = a1;
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Scalar b2 = a2 + V*b1;
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Scalar b3 = a3 + V*b2;
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Scalar b4 = a4 + V*b3;
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// invert resulting cubic polynomial analytically
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Scalar allV[4];
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allV[0] = V;
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int numSol = 1 + Dumux::invertCubicPolynomial(allV + 1, b1, b2, b3, b4);
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// sort all roots of the derivative
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std::sort(allV + 0, allV + numSol);
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// check whether the result is physical
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if (allV[numSol - 2] < b) {
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// the second largest extremum is smaller than the phase's
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// covolume which is physically impossible
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Vmin = Vmax = 0;
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return false;
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}
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// it seems that everything is okay...
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Vmin = allV[numSol - 2];
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Vmax = allV[numSol - 1];
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return true;
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}
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/*!
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* \brief The Ambrose-Walton method to estimate the vapor
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* pressure.
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*
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* \return Vapor pressure estimate in bar
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*
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* See:
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*
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* D. Ambrose, J. Walton: "Vapor Pressures up to Their Critical
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* Temperatures of Normal Alkanes and 1-Alkanols", Pure
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* Appl. Chem., 61, 1395-1403, 1989
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*/
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template <class Params>
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static Scalar ambroseWalton_(const Params ¶ms, Scalar T)
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{
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typedef typename Params::Component Component;
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Scalar Tr = T / Component::criticalTemperature();
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Scalar tau = 1 - Tr;
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Scalar omega = Component::acentricFactor();
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Scalar f0 = (tau*(-5.97616 + std::sqrt(tau)*(1.29874 - tau*0.60394)) - 1.06841*std::pow(tau, 5))/Tr;
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Scalar f1 = (tau*(-5.03365 + std::sqrt(tau)*(1.11505 - tau*5.41217)) - 7.46628*std::pow(tau, 5))/Tr;
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Scalar f2 = (tau*(-0.64771 + std::sqrt(tau)*(2.41539 - tau*4.26979)) + 3.25259*std::pow(tau, 5))/Tr;
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return Component::criticalPressure()*std::exp(f0 + omega * (f1 + omega*f2));
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}
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/*!
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* \brief Returns the difference between the liquid and the gas phase
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* fugacities in [bar]
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*
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* \param params Parameters
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* \param T Temperature [K]
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* \param p Pressure [bar]
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* \param VmLiquid Molar volume of the liquid phase [cm^3/mol]
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* \param VmGas Molar volume of the gas phase [cm^3/mol]
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*/
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template <class Params>
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static Scalar fugacityDifference_(const Params ¶ms,
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Scalar T,
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Scalar p,
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Scalar VmLiquid,
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Scalar VmGas)
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{ return fugacity(params, T, p, VmLiquid) - fugacity(params, T, p, VmGas); }
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static Tabulated2DFunction<Scalar> criticalTemperature_;
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static Tabulated2DFunction<Scalar> criticalPressure_;
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static Tabulated2DFunction<Scalar> criticalMolarVolume_;
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};
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template <class Scalar>
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Tabulated2DFunction<Scalar> PengRobinson<Scalar>::criticalTemperature_;
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template <class Scalar>
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Tabulated2DFunction<Scalar> PengRobinson<Scalar>::criticalPressure_;
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template <class Scalar>
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Tabulated2DFunction<Scalar> PengRobinson<Scalar>::criticalMolarVolume_;
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} // end namepace
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#endif
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