void
GeneralVaporMixtureFluidProperties::e_from_p_T(Real p,
Real T,
std::vector<Real> x,
Real & e,
Real & de_dp,
Real & de_dT,
std::vector<Real> & de_dx) const
{
const Real x_primary = primaryMassFraction(x);
Real ep, dep_dp, dep_dT;
_fp_primary->e_from_p_T(p, T, ep, dep_dp, dep_dT);
e = x_primary * ep;
de_dp = x_primary * dep_dp;
de_dT = x_primary * dep_dT;
de_dx.resize(_n_secondary_vapors);
for (unsigned int i = 0; i < _n_secondary_vapors; ++i)
{
Real ei, dei_dp, dei_dT;
_fp_secondary[i]->e_from_p_T(p, T, ei, dei_dp, dei_dT);
e += x[i] * ei;
de_dp += x[i] * dei_dp;
de_dT += x[i] * dei_dT;
de_dx[i] = ei - ep;
}
}
Real
GeneralVaporMixtureFluidProperties::k_from_p_T(Real p, Real T, std::vector<Real> x) const
{
const Real x_primary = primaryMassFraction(x);
Real M_primary = _fp_primary->molarMass();
Real sum = x_primary / M_primary;
for (unsigned int i = 0; i < _n_secondary_vapors; ++i)
sum += x[i] / _fp_secondary[i]->molarMass();
Real M_star = 1. / sum;
Real vp = _fp_primary->v_from_p_T(p, T);
Real ep = _fp_primary->e_from_p_T(p, T);
Real k = x_primary * M_star / M_primary * _fp_primary->k_from_v_e(vp, ep);
for (unsigned int i = 0; i < _n_secondary_vapors; ++i)
{
Real vi = _fp_secondary[i]->v_from_p_T(p, T);
Real ei = _fp_secondary[i]->e_from_p_T(p, T);
Real Mi = _fp_secondary[i]->molarMass();
k += x[i] * M_star / Mi * _fp_secondary[i]->k_from_v_e(vi, ei);
}
return k;
}
void
GeneralVaporMixtureFluidProperties::v_from_p_T(Real p,
Real T,
std::vector<Real> x,
Real & v,
Real & dv_dp,
Real & dv_dT,
std::vector<Real> & dv_dx) const
{
const Real x_primary = primaryMassFraction(x);
Real vp, dvp_dp, dvp_dT;
_fp_primary->v_from_p_T(p, T, vp, dvp_dp, dvp_dT);
v = x_primary * vp;
dv_dp = x_primary * dvp_dp;
dv_dT = x_primary * dvp_dT;
dv_dx.resize(_n_secondary_vapors);
for (unsigned int i = 0; i < _n_secondary_vapors; ++i)
{
Real vi, dvi_dp, dvi_dT;
_fp_secondary[i]->v_from_p_T(p, T, vi, dvi_dp, dvi_dT);
v += x[i] * vi;
dv_dp += x[i] * dvi_dp;
dv_dT += x[i] * dvi_dT;
dv_dx[i] = vi - vp;
}
}
Real
IdealRealGasMixtureFluidProperties::k_from_T_v(Real T, Real v, const std::vector<Real> & x) const
{
const Real x_primary = primaryMassFraction(x);
Real M_primary = _fp_primary->molarMass();
Real sum = x_primary / M_primary;
for (unsigned int i = 0; i < _n_secondary_vapors; i++)
sum += x[i] / _fp_secondary[i]->molarMass();
Real M_star = 1. / sum;
Real vp = v / x_primary;
Real ep = _fp_primary->e_from_T_v(T, vp);
Real k = x_primary * M_star / M_primary * _fp_primary->k_from_v_e(vp, ep);
for (unsigned int i = 0; i < _n_secondary_vapors; i++)
{
Real vi = v / x[i];
Real ei = _fp_secondary[i]->e_from_T_v(T, vp);
Real Mi = _fp_secondary[i]->molarMass();
k += x[i] * M_star / Mi * _fp_secondary[i]->k_from_v_e(vi, ei);
}
return k;
}
void
IdealRealGasMixtureFluidProperties::p_T_from_v_e(
Real v, Real e, const std::vector<Real> & x, Real & p, Real & T) const
{
Real v_primary = v / primaryMassFraction(x);
static const Real vc = 1. / _fp_primary->criticalDensity();
static const Real ec = _fp_primary->criticalInternalEnergy();
// Initial estimate of a bracketing interval for the temperature
Real lower_temperature, upper_temperature;
if (v_primary > vc)
{
Real e_sat_primary = _fp_primary->e_spndl_from_v(v_primary);
lower_temperature = _fp_primary->T_from_v_e(v_primary, e_sat_primary);
}
else
lower_temperature = _fp_primary->T_from_v_e(v_primary, ec);
upper_temperature = _T_mix_max;
// Use BrentsMethod to find temperature
auto energy_diff = [&v, &e, &x, this](Real T) { return this->e_from_T_v(T, v, x) - e; };
BrentsMethod::bracket(energy_diff, lower_temperature, upper_temperature);
T = BrentsMethod::root(energy_diff, lower_temperature, upper_temperature);
p = p_from_T_v(T, v, x);
}
Real
GeneralVaporMixtureFluidProperties::e_from_p_T(Real p, Real T, std::vector<Real> x) const
{
const Real x_primary = primaryMassFraction(x);
Real e = x_primary * _fp_primary->e_from_p_T(p, T);
for (unsigned int i = 0; i < _n_secondary_vapors; ++i)
e += x[i] * _fp_secondary[i]->e_from_p_T(p, T);
return e;
}
Real
IdealRealGasMixtureFluidProperties::cv_from_T_v(Real T, Real v, const std::vector<Real> & x) const
{
const Real x_primary = primaryMassFraction(x);
Real cv = x_primary * _fp_primary->cv_from_T_v(T, v / x_primary);
for (unsigned int i = 0; i < _n_secondary_vapors; i++)
cv += x[i] * _fp_secondary[i]->cv_from_T_v(T, v / x[i]);
return cv;
}
void
IdealRealGasMixtureFluidProperties::e_from_T_v(Real T,
Real v,
const std::vector<Real> & x,
Real & e,
Real & de_dT,
Real & de_dv,
std::vector<Real> & de_dx) const
{
Real e_primary, de_dT_primary, de_dv_primary, de_dx_primary, dxi_dx_primary;
Real de_dT_sec, dxj_dxi, dx_primary_dxi;
std::vector<Real> e_sec, de_dv_sec;
const Real x_primary = primaryMassFraction(x);
de_dx.resize(_n_secondary_vapors);
e_sec.resize(_n_secondary_vapors);
de_dv_sec.resize(_n_secondary_vapors);
_fp_primary->e_from_T_v(T, v / x_primary, e_primary, de_dT_primary, de_dv_primary);
e = x_primary * e_primary;
de_dT = x_primary * de_dT_primary;
de_dv = de_dv_primary;
de_dx_primary = e_primary - x_primary * de_dv_primary * v / (x_primary * x_primary);
// get the partial pressures and their derivatives first
for (unsigned int i = 0; i < _n_secondary_vapors; i++)
{
_fp_secondary[i]->e_from_T_v(T, v / x[i], e_sec[i], de_dT_sec, de_dv_sec[i]);
e += x[i] * e_sec[i];
de_dT += x[i] * de_dT_sec;
de_dv += de_dv_sec[i];
dxi_dx_primary = -x[i] / (1. - x_primary);
de_dx_primary +=
dxi_dx_primary * e_sec[i] - x[i] * de_dv_sec[i] * v / (x[i] * x[i]) * dxi_dx_primary;
}
// get the composition dependent derivatives of the secondary vapors
for (unsigned int i = 0; i < _n_secondary_vapors; i++)
{
de_dx[i] = e_sec[i] - x[i] * de_dv_sec[i] * v / (x[i] * x[i]);
for (unsigned int j = 0; j < _n_secondary_vapors; j++)
{
if (j == i)
continue;
dxj_dxi = -x[j] / (1. - x[i]);
de_dx[i] += dxj_dxi * e_sec[j] - x[j] * de_dv_sec[j] * v / (x[j] * x[j]) * dxj_dxi;
}
dx_primary_dxi = -x_primary / (1. - x[i]);
de_dx[i] += dx_primary_dxi * e_primary -
x_primary * de_dv_primary * v / (x_primary * x_primary) * dx_primary_dxi;
}
}
Real
GeneralVaporMixtureFluidProperties::cv_from_p_T(Real p, Real T, std::vector<Real> x) const
{
const Real x_primary = primaryMassFraction(x);
Real vp = _fp_primary->v_from_p_T(p, T);
Real ep = _fp_primary->e_from_p_T(p, T);
Real cv = x_primary * _fp_primary->cv_from_v_e(vp, ep);
for (unsigned int i = 0; i < _n_secondary_vapors; ++i)
{
Real vi = _fp_secondary[i]->v_from_p_T(p, T);
Real ei = _fp_secondary[i]->e_from_p_T(p, T);
cv += x[i] * _fp_secondary[i]->cv_from_v_e(vi, ei);
}
return cv;
}
Real
IdealRealGasMixtureFluidProperties::cp_from_T_v(Real T, Real v, const std::vector<Real> & x) const
{
Real p, dp_dT, dp_dv;
Real h, dh_dT, dh_dv;
p_from_T_v(T, v, x, p, dp_dT, dp_dv);
const Real x_primary = primaryMassFraction(x);
_fp_primary->h_from_T_v(T, v / x_primary, h, dh_dT, dh_dv);
Real cp = x_primary * (dh_dT - dh_dv * dp_dT / dp_dv);
for (unsigned int i = 0; i < _n_secondary_vapors; i++)
{
_fp_secondary[i]->h_from_T_v(T, v / x[i], h, dh_dT, dh_dv);
cp += x[i] * (dh_dT - dh_dv * dp_dT / dp_dv);
}
return cp;
}
void
IdealRealGasMixtureFluidProperties::s_from_T_v(
Real T, Real v, const std::vector<Real> & x, Real & s, Real & ds_dT, Real & ds_dv) const
{
Real s_primary, ds_dT_primary, ds_dv_primary;
Real s_sec, ds_dT_sec, ds_dv_sec;
const Real x_primary = primaryMassFraction(x);
_fp_primary->s_from_T_v(T, v / x_primary, s_primary, ds_dT_primary, ds_dv_primary);
s = x_primary * s_primary;
ds_dT = x_primary * ds_dT_primary;
ds_dv = ds_dv_primary;
for (unsigned int i = 0; i < _n_secondary_vapors; i++)
{
_fp_secondary[i]->s_from_T_v(T, v / x[i], s_sec, ds_dT_sec, ds_dv_sec);
s += x[i] * s_sec;
ds_dT += x[i] * ds_dT_sec;
ds_dv += ds_dv_sec;
}
}
void
IdealRealGasMixtureFluidProperties::p_from_T_v(
Real T, Real v, const std::vector<Real> & x, Real & p, Real & dp_dT, Real & dp_dv) const
{
Real p_primary, dp_dT_primary, dp_dv_primary;
Real p_sec, dp_dT_sec, dp_dv_sec;
const Real x_primary = primaryMassFraction(x);
_fp_primary->p_from_T_v(T, v / x_primary, p_primary, dp_dT_primary, dp_dv_primary);
p = p_primary;
dp_dT = dp_dT_primary;
dp_dv = dp_dv_primary / x_primary;
for (unsigned int i = 0; i < _n_secondary_vapors; i++)
{
_fp_secondary[i]->p_from_T_v(T, v / x[i], p_sec, dp_dT_sec, dp_dv_sec);
p += p_sec;
dp_dT += dp_dT_sec;
dp_dv += dp_dv_sec / x[i];
}
}
Real
IdealRealGasMixtureFluidProperties::v_from_p_T(Real p, Real T, const std::vector<Real> & x) const
{
const Real x_primary = primaryMassFraction(x);
Real M_primary = _fp_primary->molarMass();
Real sum = x_primary / M_primary;
for (unsigned int i = 0; i < _n_secondary_vapors; i++)
sum += x[i] / _fp_secondary[i]->molarMass();
Real M_star = 1. / sum;
Real v_ideal = R_molar * T / (M_star * p);
// check range of validity for primary (condensable) component
static const Real Tc = _fp_primary->criticalTemperature();
static const Real vc = 1. / _fp_primary->criticalDensity();
Real v_spndl, e_spndl;
if (T < Tc)
_fp_primary->v_e_spndl_from_T(T, v_spndl, e_spndl);
else
v_spndl = vc;
Real lower_spec_volume = v_spndl * x_primary;
Real upper_spec_volume = v_ideal; // p*v/(RT) <= 1
// Initial estimate of a bracketing interval for the temperature
Real p_max = p_from_T_v(T, lower_spec_volume, x);
if (p > p_max || upper_spec_volume < lower_spec_volume)
return getNaN();
// Use BrentsMethod to find temperature
auto pressure_diff = [&T, &p, &x, this](Real v) { return this->p_from_T_v(T, v, x) - p; };
BrentsMethod::bracket(pressure_diff, lower_spec_volume, upper_spec_volume);
Real v = BrentsMethod::root(pressure_diff, lower_spec_volume, upper_spec_volume);
return v;
}
Real
GeneralVaporMixtureFluidProperties::c_from_p_T(Real p, Real T, std::vector<Real> x) const
{
const Real x_primary = primaryMassFraction(x);
Real vp, dvp_dp_T, dvp_dT_p;
_fp_primary->v_from_p_T(p, T, vp, dvp_dp_T, dvp_dT_p);
Real v = x_primary * vp;
Real dv_dp_T = x_primary * dvp_dp_T;
Real dv_dT_p = x_primary * dvp_dT_p;
Real sp, dsp_dp_T, dsp_dT_p;
_fp_primary->s_from_p_T(p, T, sp, dsp_dp_T, dsp_dT_p);
Real ds_dp_T = x_primary * dsp_dp_T;
Real ds_dT_p = x_primary * dsp_dT_p;
for (unsigned int i = 0; i < _n_secondary_vapors; ++i)
{
Real vi, dvi_dp_T, dvi_dT_p;
_fp_secondary[i]->v_from_p_T(p, T, vi, dvi_dp_T, dvi_dT_p);
v += x[i] * vi;
dv_dp_T += x[i] * dvi_dp_T;
dv_dT_p += x[i] * dvi_dT_p;
Real si, dsi_dp_T, dsi_dT_p;
_fp_secondary[i]->s_from_p_T(p, T, si, dsi_dp_T, dsi_dT_p);
ds_dp_T += x[i] * dsi_dp_T;
ds_dT_p += x[i] * dsi_dT_p;
}
Real dv_dp_s = dv_dp_T - dv_dT_p * ds_dp_T / ds_dT_p;
if (dv_dp_s >= 0)
mooseWarning(name(), ":c_from_p_T(), dv_dp_s = ", dv_dp_s, ". Should be negative.");
return v * std::sqrt(-1. / dv_dp_s);
}
Real
IdealRealGasMixtureFluidProperties::xs_prim_from_p_T(Real p,
Real T,
const std::vector<Real> & x) const
{
Real T_c = _fp_primary->criticalTemperature();
Real xs;
if (T > T_c)
// return 1. to indicate that no water would condense for
// given (p,T)
xs = 1.;
else
{
Real pp_sat = _fp_primary->pp_sat_from_p_T(p, T);
if (pp_sat < 0.)
{
// return 1. to indicate that no water would condense for
// given (p,T)
xs = 1.;
return xs;
}
Real v_primary = _fp_primary->v_from_p_T(pp_sat, T);
Real pp_sat_secondary = p - pp_sat;
Real v_secondary;
if (_n_secondary_vapors == 1)
v_secondary = _fp_secondary[0]->v_from_p_T(pp_sat_secondary, T);
else
{
std::vector<Real> x_sec(_n_secondary_vapors);
const Real x_primary = primaryMassFraction(x);
Real sum = 0.;
for (unsigned int i = 0; i < _n_secondary_vapors; i++)
{
x_sec[i] = x[i] / (1. - x_primary);
sum += x_sec[i] / _fp_secondary[i]->molarMass();
}
Real M_star = 1. / sum;
v_secondary = R_molar * T / (M_star * pp_sat_secondary);
int it = 0;
double f = 1., df_dvs, pp_sec, p_sec, dp_dT_sec, dp_dv_sec, dp_dT, dp_dv;
double tol_p = 1.e-8;
while (std::fabs(f / pp_sat_secondary) > tol_p)
{
pp_sec = 0.;
dp_dT = 0.;
dp_dv = 0.;
for (unsigned int i = 0; i < _n_secondary_vapors; i++)
{
_fp_secondary[i]->p_from_T_v(T, v_secondary / x_sec[i], p_sec, dp_dT_sec, dp_dv_sec);
pp_sec += p_sec;
dp_dT += dp_dT_sec;
dp_dv += dp_dv_sec / x_sec[i];
}
f = pp_sec - pp_sat_secondary;
df_dvs = dp_dv;
v_secondary -= f / df_dvs;
if (it++ > 15)
return getNaN();
}
}
xs = v_secondary / (v_primary + v_secondary);
}
return xs;
}