doublereal WaterProps::ADebye(doublereal T, doublereal P_input, int ifunc)
{
doublereal psat = satPressure(T);
doublereal P;
if (psat > P_input) {
P = psat;
} else {
P = P_input;
}
doublereal epsRelWater = relEpsilon(T, P, 0);
doublereal epsilon = epsilon_0 * epsRelWater;
doublereal dw = density_IAPWS(T, P);
doublereal tmp = sqrt(2.0 * Avogadro * dw / 1000.);
doublereal tmp2 = ElectronCharge * ElectronCharge * Avogadro /
(epsilon * GasConstant * T);
doublereal tmp3 = tmp2 * sqrt(tmp2);
doublereal A_Debye = tmp * tmp3 / (8.0 * Pi);
// dAdT = - 3/2 Ad/T + 1/2 Ad/dw d(dw)/dT - 3/2 Ad/eps d(eps)/dT
// dAdT = - 3/2 Ad/T - 1/2 Ad/Vw d(Vw)/dT - 3/2 Ad/eps d(eps)/dT
if (ifunc == 1 || ifunc == 2) {
doublereal dAdT = - 1.5 * A_Debye / T;
doublereal depsRelWaterdT = relEpsilon(T, P, 1);
dAdT -= A_Debye * (1.5 * depsRelWaterdT / epsRelWater);
// calculate d(lnV)/dT _constantP, i.e., the cte
doublereal cte = coeffThermalExp_IAPWS(T, P);
doublereal contrib2 = - A_Debye * (0.5 * cte);
dAdT += contrib2;
if (ifunc == 1) {
return dAdT;
}
if (ifunc == 2) {
// Get the second derivative of the dielectric constant wrt T
// -> we will take each of the terms in dAdT and differentiate
// it again.
doublereal d2AdT2 = 1.5 / T * (A_Debye/T - dAdT);
doublereal d2epsRelWaterdT2 = relEpsilon(T, P, 2);
d2AdT2 += 1.5 * (- dAdT * depsRelWaterdT / epsRelWater
- A_Debye / epsRelWater *
(d2epsRelWaterdT2 - depsRelWaterdT * depsRelWaterdT / epsRelWater));
doublereal deltaT = -0.1;
doublereal Tdel = T + deltaT;
doublereal cte_del = coeffThermalExp_IAPWS(Tdel, P);
doublereal dctedT = (cte_del - cte) / Tdel;
doublereal contrib3 = 0.5 * (-(dAdT * cte) -(A_Debye * dctedT));
d2AdT2 += contrib3;
return d2AdT2;
}
}
// A_Debye = (1/(8 Pi)) sqrt(2 Na dw / 1000)
// (e e/(epsilon R T))^3/2
//
// dAdP = + 1/2 Ad/dw d(dw)/dP - 3/2 Ad/eps d(eps)/dP
// dAdP = - 1/2 Ad/Vw d(Vw)/dP - 3/2 Ad/eps d(eps)/dP
// dAdP = + 1/2 Ad * kappa - 3/2 Ad/eps d(eps)/dP
//
// where kappa = - 1/Vw d(Vw)/dP_T (isothermal compressibility)
if (ifunc == 3) {
doublereal dAdP = 0.0;
doublereal depsRelWaterdP = relEpsilon(T, P, 3);
dAdP -= A_Debye * (1.5 * depsRelWaterdP / epsRelWater);
doublereal kappa = isothermalCompressibility_IAPWS(T,P);
dAdP += A_Debye * (0.5 * kappa);
return dAdP;
}
return A_Debye;
}
/**
* ADebye calculates the value of A_Debye as a function
* of temperature and pressure according to relations
* that take into account the temperature and pressure
* dependence of the water density and dieletric constant.
*
* A_Debye -> this expression appears on the top of the
* ln actCoeff term in the general Debye-Huckel
* expression
* It depends on temperature. And, therefore,
* most be recalculated whenever T or P changes.
*
* A_Debye = (1/(8 Pi)) sqrt(2 Na dw / 1000)
* (e e/(epsilon R T))^3/2
*
* Units = sqrt(kg/gmol) ~ sqrt(1/I)
*
* Nominal value = 1.172576 sqrt(kg/gmol)
* based on:
* epsilon/epsilon_0 = 78.54
* (water at 25C)
* epsilon_0 = 8.854187817E-12 C2 N-1 m-2
* e = 1.60217653E-19 C
* F = 9.6485309E7 C kmol-1
* R = 8.314472E3 kg m2 s-2 kmol-1 K-1
* T = 298.15 K
* B_Debye = 3.28640E9 sqrt(kg/gmol)/m
* Na = 6.0221415E26
*
* ifunc = 0 return value
* ifunc = 1 return temperature derivative
* ifunc = 2 return temperature second derivative
* ifunc = 3 return pressure first derivative
*
* Verification:
* With the epsRelWater value from the BP relation,
* and the water density from the WaterDens function,
* The A_Debye computed with this function agrees with
* the Pitzer table p. 99 to 4 significant digits at 25C.
* and 20C. (Aphi = ADebye/3)
*
* (statically defined within the object)
*/
double WaterProps::ADebye(double T, double P_input, int ifunc) {
const double e = 1.60217653E-19;
const double epsilon0 = 8.854187817E-12;
const double R = 8.314472E3;
double psat = satPressure(T);
double P;
if (psat > P_input) {
//printf("ADebye WARNING: p_input < psat: %g %g\n",
// P_input, psat);
P = psat;
} else {
P = P_input;
}
double epsRelWater = relEpsilon(T, P, 0);
//printf("releps calc = %g, compare to 78.38\n", epsRelWater);
//double B_Debye = 3.28640E9;
const double Na = 6.0221415E26;
double epsilon = epsilon0 * epsRelWater;
double dw = density_IAPWS(T, P);
double tmp = sqrt( 2.0 * Na * dw / 1000.);
double tmp2 = e * e * Na / (epsilon * R * T);
double tmp3 = tmp2 * sqrt(tmp2);
double A_Debye = tmp * tmp3 / (8.0 * Pi);
/*
* dAdT = - 3/2 Ad/T + 1/2 Ad/dw d(dw)/dT - 3/2 Ad/eps d(eps)/dT
*
* dAdT = - 3/2 Ad/T - 1/2 Ad/Vw d(Vw)/dT - 3/2 Ad/eps d(eps)/dT
*/
if (ifunc == 1 || ifunc == 2) {
double dAdT = - 1.5 * A_Debye / T;
double depsRelWaterdT = relEpsilon(T, P, 1);
dAdT -= A_Debye * (1.5 * depsRelWaterdT / epsRelWater);
//int methodD = 1;
//double ddwdT = density_T_new(T, P, 1);
// double contrib1 = A_Debye * (0.5 * ddwdT / dw);
/*
* calculate d(lnV)/dT _constantP, i.e., the cte
*/
double cte = coeffThermalExp_IAPWS(T, P);
double contrib2 = - A_Debye * (0.5 * cte);
//dAdT += A_Debye * (0.5 * ddwdT / dw);
dAdT += contrib2;
#ifdef DEBUG_HKM
//printf("dAdT = %g, contrib1 = %g, contrib2 = %g\n",
// dAdT, contrib1, contrib2);
#endif
if (ifunc == 1) {
return dAdT;
}
if (ifunc == 2) {
/*
* Get the second derivative of the dielectric constant wrt T
* -> we will take each of the terms in dAdT and differentiate
* it again.
*/
double d2AdT2 = 1.5 / T * (A_Debye/T - dAdT);
double d2epsRelWaterdT2 = relEpsilon(T, P, 2);
//double dT = -0.01;
//double TT = T + dT;
//double depsRelWaterdTdel = relEpsilon(TT, P, 1);
//double d2alt = (depsRelWaterdTdel- depsRelWaterdT ) / dT;
//printf("diff %g %g\n",d2epsRelWaterdT2, d2alt);
// HKM -> checks out, i.e., they are the same.
d2AdT2 += 1.5 * (- dAdT * depsRelWaterdT / epsRelWater
- A_Debye / epsRelWater *
(d2epsRelWaterdT2 - depsRelWaterdT * depsRelWaterdT / epsRelWater));
double deltaT = -0.1;
double Tdel = T + deltaT;
double cte_del = coeffThermalExp_IAPWS(Tdel, P);
double dctedT = (cte_del - cte) / Tdel;
//double d2dwdT2 = density_T_new(T, P, 2);
double contrib3 = 0.5 * ( -(dAdT * cte) -(A_Debye * dctedT));
d2AdT2 += contrib3;
return d2AdT2;
}
}
/*
* A_Debye = (1/(8 Pi)) sqrt(2 Na dw / 1000)
* (e e/(epsilon R T))^3/2
*
* dAdP = + 1/2 Ad/dw d(dw)/dP - 3/2 Ad/eps d(eps)/dP
*
* dAdP = - 1/2 Ad/Vw d(Vw)/dP - 3/2 Ad/eps d(eps)/dP
*
* dAdP = + 1/2 Ad * kappa - 3/2 Ad/eps d(eps)/dP
*
* where kappa = - 1/Vw d(Vw)/dP_T (isothermal compressibility)
*/
if (ifunc == 3) {
double dAdP = 0.0;
double depsRelWaterdP = relEpsilon(T, P, 3);
dAdP -= A_Debye * (1.5 * depsRelWaterdP / epsRelWater);
double kappa = isothermalCompressibility_IAPWS(T,P);
//double ddwdP = density_T_new(T, P, 3);
dAdP += A_Debye * (0.5 * kappa);
return dAdP;
}
return A_Debye;
}
