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/* ASCEND modelling environment |
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Copyright (C) 2008 Carnegie Mellon University |
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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, or (at your option) |
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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, write to the Free Software |
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Foundation, Inc., 59 Temple Place - Suite 330, |
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Boston, MA 02111-1307, USA. |
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*//** @file |
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Implementation of the reduced molar Helmholtz free energy equation of state. |
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|
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For nomenclature see Tillner-Roth, Harms-Watzenberg and Baehr, Eine neue |
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Fundamentalgleichung f端r Ammoniak. |
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|
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John Pye, 29 Jul 2008. |
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*/ |
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|
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#include <math.h> |
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|
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#include "helmholtz.h" |
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#include "ideal_impl.h" |
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|
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#ifdef TEST |
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#include <assert.h> |
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#include <stdlib.h> |
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#include <stdio.h> |
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#endif |
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|
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/* forward decls */ |
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|
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static double helm_resid(double tau, double delta, const HelmholtzData *data); |
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static double helm_resid_del(double tau, double delta, const HelmholtzData *data); |
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static double helm_resid_tau(double tau, double delta, const HelmholtzData *data); |
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static double helm_resid_deltau(double tau, double delta, const HelmholtzData *data); |
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static double helm_resid_deldel(double tau, double delta, const HelmholtzData *data); |
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|
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/** |
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Function to calculate pressure from Helmholtz free energy EOS, given temperature |
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and mass density. |
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|
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@param T temperature in K |
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@param rho mass density in kg/m続 |
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@return pressure in Pa??? |
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*/ |
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double helmholtz_p(double T, double rho, const HelmholtzData *data){ |
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|
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double tau = data->T_star / T; |
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double delta = rho / data->rho_star; |
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|
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#ifdef TEST |
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assert(data->rho_star!=0); |
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assert(T!=0); |
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assert(!isnan(tau)); |
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assert(!isnan(delta)); |
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assert(!isnan(data->R)); |
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|
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//fprintf(stderr,"p calc: T = %f\n",T); |
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//fprintf(stderr,"p calc: tau = %f\n",tau); |
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//fprintf(stderr,"p calc: rho = %f\n",rho); |
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//fprintf(stderr,"p calc: delta = %f\n",delta); |
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//fprintf(stderr,"p calc: R*T*rho = %f\n",data->R * T * rho); |
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|
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//fprintf(stderr,"T = %f\n", T); |
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//fprintf(stderr,"rhob = %f, rhob* = %f, delta = %f\n", rho/data->M, data->rho_star/data->M, delta); |
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#endif |
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|
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return data->R * T * rho * (1 + delta * helm_resid_del(tau,delta,data)); |
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} |
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|
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/** |
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Function to calculate internal energy from Helmholtz free energy EOS, given |
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temperature and mass density. |
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|
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@param T temperature in K |
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@param rho mass density in kg/m続 |
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@return internal energy in ??? |
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*/ |
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double helmholtz_u(double T, double rho, const HelmholtzData *data){ |
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|
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double tau = data->T_star / T; |
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double delta = rho / data->rho_star; |
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|
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#ifdef TEST |
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assert(data->rho_star!=0); |
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assert(T!=0); |
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assert(!isnan(tau)); |
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assert(!isnan(delta)); |
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assert(!isnan(data->R)); |
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#endif |
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|
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#ifdef TEST |
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fprintf(stderr,"ideal_tau = %f\n",helm_ideal_tau(tau,delta,data->ideal)); |
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fprintf(stderr,"resid_tau = %f\n",helm_resid_tau(tau,delta,data)); |
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fprintf(stderr,"R T = %f\n",data->R * data->T_star); |
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#endif |
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|
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return data->R * data->T_star * (helm_ideal_tau(tau,delta,data->ideal) + helm_resid_tau(tau,delta,data)); |
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} |
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|
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/** |
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Function to calculate enthalpy from Helmholtz free energy EOS, given |
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temperature and mass density. |
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|
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@param T temperature in K |
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@param rho mass density in kg/m続 |
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@return enthalpy in J/kg |
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*/ |
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double helmholtz_h(double T, double rho, const HelmholtzData *data){ |
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|
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double tau = data->T_star / T; |
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double delta = rho / data->rho_star; |
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|
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#ifdef TEST |
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assert(data->rho_star!=0); |
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assert(T!=0); |
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assert(!isnan(tau)); |
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assert(!isnan(delta)); |
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assert(!isnan(data->R)); |
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#endif |
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|
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return data->R * T * (1 + tau * (helm_ideal_tau(tau,delta,data->ideal) + helm_resid_tau(tau,delta,data)) + delta*helm_resid_del(tau,delta,data)); |
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} |
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|
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/** |
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Function to calculate entropy from Helmholtz free energy EOS, given |
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temperature and mass density. |
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|
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@param T temperature in K |
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@param rho mass density in kg/m続 |
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@return entropy in J/kgK |
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*/ |
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double helmholtz_s(double T, double rho, const HelmholtzData *data){ |
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|
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double tau = data->T_star / T; |
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double delta = rho / data->rho_star; |
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|
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#ifdef TEST |
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assert(data->rho_star!=0); |
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assert(T!=0); |
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assert(!isnan(tau)); |
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assert(!isnan(delta)); |
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assert(!isnan(data->R)); |
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|
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fprintf(stderr,"helm_ideal_tau = %f\n",helm_ideal_tau(tau,delta,data->ideal)); |
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fprintf(stderr,"helm_resid_tau = %f\n",helm_resid_tau(tau,delta,data)); |
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fprintf(stderr,"helm_ideal = %f\n",helm_ideal(tau,delta,data->ideal)); |
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fprintf(stderr,"helm_resid = %f\n",helm_resid(tau,delta,data)); |
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#endif |
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return data->R * ( |
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tau * (helm_ideal_tau(tau,delta,data->ideal) + helm_resid_tau(tau,delta,data)) |
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- helm_ideal(tau,delta,data->ideal) - helm_resid(tau,delta,data) |
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); |
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} |
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|
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/** |
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Calculation zero-pressure specific heat capacity |
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*/ |
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double helmholtz_cp0(double T, const HelmholtzData *data){ |
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double val = helm_cp0(T,data->ideal); |
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#if 0 |
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fprintf(stderr,"val = %f\n",val); |
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#endif |
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return val; |
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} |
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|
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/*--------------------------------------------- |
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UTILITY FUNCTION(S) |
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*/ |
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|
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/* ipow: public domain by Mark Stephen with suggestions by Keiichi Nakasato */ |
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static double ipow(double x, int n){ |
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double t = 1.0; |
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|
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if(!n)return 1.0; /* At the top. x^0 = 1 */ |
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|
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if (n < 0){ |
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n = -n; |
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x = 1.0/x; /* error if x == 0. Good */ |
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} /* ZTC/SC returns inf, which is even better */ |
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|
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if (x == 0.0)return 0.0; |
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|
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do{ |
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if(n & 1)t *= x; |
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n /= 2; /* KN prefers if (n/=2) x*=x; This avoids an */ |
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x *= x; /* unnecessary but benign multiplication on */ |
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}while(n); /* the last pass, but the comparison is always |
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true _except_ on the last pass. */ |
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|
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return t; |
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} |
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|
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/** |
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Residual part of helmholtz function. |
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*/ |
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double helm_resid(double tau, double delta, const HelmholtzData *data){ |
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#if 0 |
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double sum, res = 0; |
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unsigned n, i; |
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const HelmholtzPowTerm *pt; |
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const HelmholtzExpTerm *et; |
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|
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n = data->np; |
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pt = &(data->pt[0]); |
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|
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/* power terms */ |
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sum = 0; |
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unsigned oldl; |
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for(i=0; i<n; ++i){ |
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sum += pt->a * pow(tau, pt->t) * ipow(delta, pt->d); |
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oldl = pt->l; |
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++pt; |
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if(i+1==n || oldl != pt->l){ |
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if(oldl == 0){ |
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res += sum; |
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}else{ |
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res += sum * exp(-ipow(delta,pt->l)); |
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} |
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sum = 0; |
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} |
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} |
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|
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/* now the exponential terms */ |
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n = data->ne; |
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et = &(data->et[0]); |
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for(i=0; i< n; ++i){ |
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fprintf(stderr,"i = %d, a = %e, t = %f, d = %d, phi = %d, beta = %d, gamma = %f\n",i+1, et->a, et->t, et->d, et->phi, et->beta, et->gamma); |
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|
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double e1 = -et->phi * delta*delta |
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+ 2 * et->phi * delta |
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- et->beta * tau * tau |
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+ 2 * et->beta * et->gamma * tau |
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- et->phi |
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- et->beta * et->gamma * et->gamma; |
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sum = et->a * pow(tau,et->t) * ipow(delta,et->d) * exp(e1); |
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//fprintf(stderr,"sum = %f\n",sum); |
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res += sum; |
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++et; |
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} |
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|
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return res; |
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} |
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|
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#else |
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double sum; |
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double res = 0; |
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double delX; |
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unsigned l; |
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unsigned n, i; |
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const HelmholtzPowTerm *pt; |
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const HelmholtzExpTerm *et; |
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|
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n = data->np; |
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pt = &(data->pt[0]); |
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|
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delX = 1; |
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|
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l = 0; |
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sum = 0; |
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for(i=0; i<n; ++i){ |
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//fprintf(stderr,"i = %d, a = %e, t = %f, d = %d, l = %d\n",i+1, pt->a, pt->t, pt->d, pt->l); |
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sum += pt->a * pow(tau, pt->t) * ipow(delta, pt->d); |
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++pt; |
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//fprintf(stderr,"l = %d\n",l); |
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if(i+1==n || l != pt->l){ |
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if(l==0){ |
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//fprintf(stderr,"Adding non-exp term\n"); |
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res += sum; |
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}else{ |
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//fprintf(stderr,"Adding exp term with l = %d, delX = %e\n",l,delX); |
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res += sum * exp(-delX); |
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} |
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/* set l to new value */ |
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if(i+1!=n){ |
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l = pt->l; |
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//fprintf(stderr,"New l = %d\n",l); |
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delX = ipow(delta,l); |
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sum = 0; |
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} |
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} |
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} |
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|
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/* now the exponential terms */ |
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n = data->ne; |
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et = &(data->et[0]); |
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for(i=0; i< n; ++i){ |
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#ifdef TEST |
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fprintf(stderr,"i = %d, a = %e, t = %f, d = %d, phi = %d, beta = %d, gamma = %f\n",i+1, et->a, et->t, et->d, et->phi, et->beta, et->gamma); |
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#endif |
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|
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double e1 = -et->phi * delta*delta |
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+ 2 * et->phi * delta |
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- et->beta * tau * tau |
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+ 2 * et->beta * et->gamma * tau |
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- et->phi |
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- et->beta * et->gamma * et->gamma; |
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sum = et->a * pow(tau,et->t) * ipow(delta,et->d) * exp(e1); |
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//fprintf(stderr,"sum = %f\n",sum); |
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res += sum; |
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++et; |
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} |
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|
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return res; |
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} |
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#endif |
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|
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/** |
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Derivative of the helmholtz residual function with respect to |
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delta. |
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*/ |
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double helm_resid_del(double tau,double delta, const HelmholtzData *data){ |
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double sum, res = 0; |
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double dell, ldell; |
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unsigned n, i; |
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const HelmholtzPowTerm *pt; |
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const HelmholtzExpTerm *et; |
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|
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n = data->np; |
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pt = &(data->pt[0]); |
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|
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sum = 0; |
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dell = ipow(delta,pt->l); |
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ldell = pt->l * dell; |
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unsigned oldl; |
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for(i=0; i<n; ++i){ |
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sum += pt->a * pow(tau, pt->t) * ipow(delta, pt->d - 1) * (pt->d - ldell); |
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oldl = pt->l; |
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++pt; |
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if(i+1==n || oldl != pt->l){ |
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if(oldl == 0){ |
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res += sum; |
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}else{ |
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res += sum * exp(-dell); |
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} |
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sum = 0; |
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dell = ipow(delta,pt->l); |
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ldell = pt->l*dell; |
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} |
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} |
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|
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#if 1 |
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/* now the exponential terms */ |
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n = data->ne; |
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et = &(data->et[0]); |
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for(i=0; i< n; ++i){ |
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//fprintf(stderr,"i = %d, a = %e, t = %f, d = %d, phi = %d, beta = %d, gamma = %f\n",i+1, et->a, et->t, et->d, et->phi, et->beta, et->gamma); |
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|
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double del2 = delta*delta; |
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double tau2 = tau*tau; |
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double gam2 = et->gamma * et->gamma; |
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double e1 = -et->phi * del2 |
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+ 2 * et->phi * delta |
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- et->beta * tau2 |
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+ 2 * et->beta * et->gamma * tau |
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- et->phi |
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- et->beta * gam2; |
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sum = -et->a * pow(tau,et->t) * ipow(delta,et->d-1) |
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* (2 * et->phi * del2 - 2 * et->phi * delta - et->d) |
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* exp(e1); |
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//fprintf(stderr,"sum = %f\n",sum); |
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res += sum; |
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++et; |
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} |
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#endif |
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|
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return res; |
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} |
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|
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/** |
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Derivative of the helmholtz residual function with respect to |
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tau. |
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*/ |
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double helm_resid_tau(double tau,double delta,const HelmholtzData *data){ |
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|
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double sum; |
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double res = 0; |
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double delX; |
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unsigned l; |
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unsigned n, i; |
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const HelmholtzPowTerm *pt; |
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const HelmholtzExpTerm *et; |
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|
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n = data->np; |
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pt = &(data->pt[0]); |
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|
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delX = 1; |
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|
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l = 0; |
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sum = 0; |
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for(i=0; i<n; ++i){ |
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if(pt->t){ |
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//fprintf(stderr,"i = %d, a = %e, t = %f, d = %d, l = %d\n",i+1, pt->a, pt->t, pt->d, pt->l); |
| 402 |
sum += pt->a * pow(tau, pt->t - 1) * ipow(delta, pt->d) * pt->t; |
| 403 |
} |
| 404 |
++pt; |
| 405 |
//fprintf(stderr,"l = %d\n",l); |
| 406 |
if(i+1==n || l != pt->l){ |
| 407 |
if(l==0){ |
| 408 |
//fprintf(stderr,"Adding non-exp term\n"); |
| 409 |
res += sum; |
| 410 |
}else{ |
| 411 |
//fprintf(stderr,"Adding exp term with l = %d, delX = %e\n",l,delX); |
| 412 |
res += sum * exp(-delX); |
| 413 |
} |
| 414 |
/* set l to new value */ |
| 415 |
if(i+1!=n){ |
| 416 |
l = pt->l; |
| 417 |
//fprintf(stderr,"New l = %d\n",l); |
| 418 |
delX = ipow(delta,l); |
| 419 |
sum = 0; |
| 420 |
} |
| 421 |
} |
| 422 |
} |
| 423 |
|
| 424 |
#if 1 |
| 425 |
/* now the exponential terms */ |
| 426 |
n = data->ne; |
| 427 |
et = &(data->et[0]); |
| 428 |
for(i=0; i< n; ++i){ |
| 429 |
//fprintf(stderr,"i = %d, a = %e, t = %f, d = %d, phi = %d, beta = %d, gamma = %f\n",i+1, et->a, et->t, et->d, et->phi, et->beta, et->gamma); |
| 430 |
|
| 431 |
double tau2 = tau*tau; |
| 432 |
double del2 = delta*delta; |
| 433 |
double gam2 = et->gamma * et->gamma; |
| 434 |
double e1 = -et->phi * del2 |
| 435 |
+ 2 * et->phi * delta |
| 436 |
- et->beta * tau2 |
| 437 |
+ 2 * et->beta * et->gamma * tau |
| 438 |
- et->phi |
| 439 |
- et->beta * gam2; |
| 440 |
sum = -et->a * pow(tau,et->t - 1) * ipow(delta,et->d) |
| 441 |
* (2 * et->beta * tau2 - 2 * et->beta * et->gamma * tau - et->t) |
| 442 |
* exp(e1); |
| 443 |
//fprintf(stderr,"sum = %f\n",sum); |
| 444 |
res += sum; |
| 445 |
++et; |
| 446 |
} |
| 447 |
#endif |
| 448 |
|
| 449 |
return res; |
| 450 |
} |
| 451 |
|
| 452 |
|
| 453 |
|
| 454 |
/** |
| 455 |
Mixed derivative of the helmholtz residual function with respect to |
| 456 |
delta and tau |
| 457 |
*/ |
| 458 |
double helm_resid_deltau(double tau,double delta,const HelmholtzData *data){ |
| 459 |
|
| 460 |
double sum; |
| 461 |
double phir = 0; |
| 462 |
unsigned i; |
| 463 |
double XdelX; |
| 464 |
|
| 465 |
const HelmholtzPowTerm *pt = &(data->pt[0]); |
| 466 |
|
| 467 |
for(i=0; i<5; ++i){ |
| 468 |
phir += pt->a * pow(tau, pt->t - 1) * ipow(delta, pt->d - 1) * pt->d * pt->t; |
| 469 |
++pt; |
| 470 |
} |
| 471 |
|
| 472 |
sum = 0; |
| 473 |
XdelX = delta; |
| 474 |
for(i=5; i<10; ++i){ |
| 475 |
sum += pt->a * pow(tau, pt->t - 1) * ipow(delta, pt->d - 1) * pt->t *(pt->d - XdelX); |
| 476 |
++pt; |
| 477 |
} |
| 478 |
phir += exp(-delta) * sum; |
| 479 |
|
| 480 |
sum = 0; |
| 481 |
XdelX = 2*delta*delta; |
| 482 |
for(i=10; i<17; ++i){ |
| 483 |
sum += pt->a * pow(tau, pt->t - 1) * ipow(delta, pt->d - 1) * pt->t *(pt->d - XdelX); |
| 484 |
++pt; |
| 485 |
} |
| 486 |
phir += exp(-delta*delta) * sum; |
| 487 |
|
| 488 |
sum = 0; |
| 489 |
XdelX = 3*delta*delta*delta; |
| 490 |
for(i=17; i<21; ++i){ |
| 491 |
sum += pt->a * pow(tau, pt->t - 1) * ipow(delta, pt->d - 1) * pt->t *(pt->d - XdelX); |
| 492 |
++pt; |
| 493 |
} |
| 494 |
phir += exp(-delta*delta*delta) * sum; |
| 495 |
|
| 496 |
return phir; |
| 497 |
} |
| 498 |
|
| 499 |
#define SQ(X) ((X)*(X)) |
| 500 |
|
| 501 |
/** |
| 502 |
Second derivative of helmholtz residual function with respect to |
| 503 |
delta (twice). |
| 504 |
*/ |
| 505 |
double helm_resid_deldel(double tau,double delta,const HelmholtzData *data){ |
| 506 |
|
| 507 |
double sum; |
| 508 |
double phir = 0; |
| 509 |
unsigned i; |
| 510 |
unsigned X; |
| 511 |
double XdelX; |
| 512 |
|
| 513 |
const HelmholtzPowTerm *pt = &(data->pt[0]); |
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|
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for(i=0; i<5; ++i){ |
| 516 |
phir += pt->a * pow(tau, pt->t) * ipow(delta, pt->d - 2) * (SQ(pt->d) - X); |
| 517 |
++pt; |
| 518 |
} |
| 519 |
|
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sum = 0; |
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X = 1; |
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XdelX = delta; |
| 523 |
for(i=5; i<10; ++i){ |
| 524 |
sum += pt->a * pow(tau, pt->t) * ipow(delta, pt->d - 2) * (SQ(XdelX) - X*XdelX - 2*pt->d*XdelX + XdelX + SQ(pt->d) - pt->d); |
| 525 |
++pt; |
| 526 |
} |
| 527 |
phir += exp(-delta) * sum; |
| 528 |
|
| 529 |
sum = 0; |
| 530 |
X = 2; |
| 531 |
XdelX = 2*delta*delta; |
| 532 |
for(i=10; i<17; ++i){ |
| 533 |
sum += pt->a * pow(tau, pt->t) * ipow(delta, pt->d - 2) * (SQ(XdelX) - X*XdelX - 2*pt->d*XdelX + XdelX + SQ(pt->d) - pt->d); |
| 534 |
++pt; |
| 535 |
} |
| 536 |
phir += exp(-delta*delta) * sum; |
| 537 |
|
| 538 |
sum = 0; |
| 539 |
X = 3; |
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XdelX = 3*delta*delta*delta; |
| 541 |
for(i=17; i<21; ++i){ |
| 542 |
sum += pt->a * pow(tau, pt->t) * ipow(delta, pt->d - 2) * (SQ(XdelX) - X*XdelX - 2*pt->d*XdelX + XdelX + SQ(pt->d) - pt->d); |
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++pt; |
| 544 |
} |
| 545 |
phir += exp(-delta*delta*delta) * sum; |
| 546 |
|
| 547 |
return phir; |
| 548 |
} |
| 549 |
|
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