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Added additional derivatives of alpha0 (xi,tau; xi,delta; xi,xj)

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This commit is contained in:
Ian Bell
2016-08-25 23:23:26 -06:00
parent 36787f9d8e
commit 952ec9486e
2 changed files with 148 additions and 2 deletions
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147
src/Backends/Helmholtz/MixtureDerivatives.cpp
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@@ -738,7 +738,7 @@ CoolPropDbl MixtureDerivatives::dalpha0_dxi(HelmholtzEOSMixtureBackend &HEOS, st
double delta_oi = HEOS.delta()*rhor/rhoci;
double Rratioi = 1;//HEOS.gas_constant()/HEOS.components[i].EOS().R_u;
double term = Rratioi*HEOS.components[i].EOS().alpha0.base(tau_oi, delta_oi) + log(HEOS.mole_fractions[i]);
double term = Rratioi*HEOS.components[i].EOS().alpha0.base(tau_oi, delta_oi) + log(HEOS.mole_fractions[i]) + 1;
std::size_t kmax = HEOS.mole_fractions.size();
if (xN_flag == XN_DEPENDENT){ kmax--; }
@@ -753,7 +753,147 @@ CoolPropDbl MixtureDerivatives::dalpha0_dxi(HelmholtzEOSMixtureBackend &HEOS, st
double Rratiok = 1;//HEOS.gas_constant()/HEOS.components[k].EOS().R_u;
double dalpha0_ok_dxi = HEOS.components[k].EOS().alpha0.dTau(tau_ok, delta_ok)*dtauok_dxi + HEOS.components[k].EOS().alpha0.dDelta(tau_ok, delta_ok)*ddeltaok_dxi;
term += xk*(Rratiok*dalpha0_ok_dxi + 1/xk*Kronecker_delta(k,i));
term += xk*(Rratiok*dalpha0_ok_dxi);
}
return term;
}
CoolPropDbl MixtureDerivatives::d2alpha0_dxi_dDelta(HelmholtzEOSMixtureBackend &HEOS, std::size_t i, x_N_dependency_flag xN_flag)
{
// Reducing values are constant for all components under consideration
double Tr = HEOS.T_reducing();
double rhor = HEOS.rhomolar_reducing();
// Values for the i-th component
double Tci = HEOS.get_fluid_constant(i, iT_critical);
double rhoci = HEOS.get_fluid_constant(i, irhomolar_critical);
double tau_oi = HEOS.tau()*Tci/Tr;
double delta_oi = HEOS.delta()*rhor/rhoci;
double Rratioi = 1;//HEOS.gas_constant()/HEOS.components[i].EOS().R_u;
double term = rhor/rhoci*Rratioi*HEOS.components[i].EOS().alpha0.dDelta(tau_oi, delta_oi);
std::size_t kmax = HEOS.mole_fractions.size();
if (xN_flag == XN_DEPENDENT){ kmax--; }
for (std::size_t k = 0; k < kmax; ++k){
double xk = HEOS.mole_fractions[k];
double Tck = HEOS.get_fluid_constant(k, iT_critical);
double rhock = HEOS.get_fluid_constant(k, irhomolar_critical);
double tau_ok = HEOS.tau()*Tck / Tr;
double delta_ok = HEOS.delta()*rhor / rhock;
double dtauok_dxi = -tau_ok / Tr*HEOS.Reducing->dTrdxi__constxj(HEOS.mole_fractions, i, xN_flag); // (Gernert, supp, B.19)
double drhor_dxi = HEOS.Reducing->drhormolardxi__constxj(HEOS.mole_fractions, i, xN_flag);
double ddeltaok_dxi = delta_ok/rhor*drhor_dxi; // (Gernert, supp. B.20)
double Rratiok = 1;//HEOS.gas_constant()/HEOS.components[k].EOS().R_u;
double dalpha0ok_ddeltaok = HEOS.components[k].EOS().alpha0.dDelta(tau_ok, delta_ok);
double d_dalpha0ok_ddeltaok_dxi = HEOS.components[k].EOS().alpha0.dDelta_dTau(tau_ok, delta_ok)*dtauok_dxi + HEOS.components[k].EOS().alpha0.dDelta2(tau_ok, delta_ok)*ddeltaok_dxi;
term += xk/rhock*(rhor*d_dalpha0ok_ddeltaok_dxi + drhor_dxi*dalpha0ok_ddeltaok);
}
return term;
}
CoolPropDbl MixtureDerivatives::d2alpha0_dxi_dTau(HelmholtzEOSMixtureBackend &HEOS, std::size_t i, x_N_dependency_flag xN_flag)
{
// Reducing values are constant for all components under consideration
double Tr = HEOS.T_reducing();
double rhor = HEOS.rhomolar_reducing();
// Values for the i-th component
double Tci = HEOS.get_fluid_constant(i, iT_critical);
double rhoci = HEOS.get_fluid_constant(i, irhomolar_critical);
double tau_oi = HEOS.tau()*Tci/Tr;
double delta_oi = HEOS.delta()*rhor/rhoci;
double Rratioi = 1;//HEOS.gas_constant()/HEOS.components[i].EOS().R_u;
double term = Tci/Tr*Rratioi*HEOS.components[i].EOS().alpha0.dTau(tau_oi, delta_oi);
std::size_t kmax = HEOS.mole_fractions.size();
if (xN_flag == XN_DEPENDENT){ kmax--; }
for (std::size_t k = 0; k < kmax; ++k){
double xk = HEOS.mole_fractions[k];
double Tck = HEOS.get_fluid_constant(k, iT_critical);
double rhock = HEOS.get_fluid_constant(k, irhomolar_critical);
double tau_ok = HEOS.tau()*Tck / Tr;
double delta_ok = HEOS.delta()*rhor / rhock;
double dTr_dxi = HEOS.Reducing->dTrdxi__constxj(HEOS.mole_fractions, i, xN_flag);
double dtauok_dxi = -tau_ok/Tr*dTr_dxi; // (Gernert, supp, B.19)
double drhor_dxi = HEOS.Reducing->drhormolardxi__constxj(HEOS.mole_fractions, i, xN_flag);
double ddeltaok_dxi = delta_ok/rhor*drhor_dxi; // (Gernert, supp. B.20)
double Rratiok = 1;//HEOS.gas_constant()/HEOS.components[k].EOS().R_u;
double dalpha0ok_dtauok = HEOS.components[k].EOS().alpha0.dTau(tau_ok, delta_ok);
double d_dalpha0ok_dTauok_dxi = HEOS.components[k].EOS().alpha0.dTau2(tau_ok, delta_ok)*dtauok_dxi + HEOS.components[k].EOS().alpha0.dDelta_dTau(tau_ok, delta_ok)*ddeltaok_dxi;
term += xk*Tck*(1/Tr*d_dalpha0ok_dTauok_dxi + -1/POW2(Tr)*dTr_dxi*dalpha0ok_dtauok);
}
return term;
}
CoolPropDbl MixtureDerivatives::d2alpha0_dxidxj(HelmholtzEOSMixtureBackend &HEOS, std::size_t i, std::size_t j, x_N_dependency_flag xN_flag)
{
// Reducing values are constant for all components under consideration
double Tr = HEOS.T_reducing();
double rhor = HEOS.rhomolar_reducing();
// Values for the i-th component
double Tci = HEOS.get_fluid_constant(i, iT_critical);
double rhoci = HEOS.get_fluid_constant(i, irhomolar_critical);
double tau_oi = HEOS.tau()*Tci/Tr;
double delta_oi = HEOS.delta()*rhor/rhoci;
double dTr_dxi = HEOS.Reducing->dTrdxi__constxj(HEOS.mole_fractions, i, xN_flag);
double drhor_dxi = HEOS.Reducing->drhormolardxi__constxj(HEOS.mole_fractions, i, xN_flag);
// Values for the j-th component
double Tcj = HEOS.get_fluid_constant(j, iT_critical);
double rhocj = HEOS.get_fluid_constant(j, irhomolar_critical);
double tau_oj = HEOS.tau()*Tcj/Tr;
double delta_oj = HEOS.delta()*rhor/rhocj;
double dTr_dxj = HEOS.Reducing->dTrdxi__constxj(HEOS.mole_fractions, j, xN_flag);
double drhor_dxj = HEOS.Reducing->drhormolardxi__constxj(HEOS.mole_fractions, j, xN_flag);
// Cross-terms with both i & j
double dtauoi_dxj = -tau_oi/Tr*dTr_dxj; // (Gernert, supp, B.19)
double ddeltaoi_dxj = delta_oi/rhor*drhor_dxj; // (Gernert, supp. B.20)
double dtauoj_dxi = -tau_oj/Tr*dTr_dxi; // (Gernert, supp, B.19)
double ddeltaoj_dxi = delta_oj/rhor*drhor_dxi; // (Gernert, supp. B.20)
double d2Tr_dxidxj = HEOS.Reducing->d2Trdxidxj(HEOS.mole_fractions, i, j, xN_flag);
double d2rhor_dxidxj = HEOS.Reducing->d2rhormolardxidxj(HEOS.mole_fractions, i, j, xN_flag);
double Rratioi = 1;//HEOS.gas_constant()/HEOS.components[i].EOS().R_u;
double d_dalpha0oi_dxj = HEOS.components[i].EOS().alpha0.dTau(tau_oi, delta_oi)*dtauoi_dxj + HEOS.components[i].EOS().alpha0.dDelta(tau_oi, delta_oi)*ddeltaoi_dxj;
double d_dalpha0oj_dxi = HEOS.components[j].EOS().alpha0.dTau(tau_oj, delta_oj)*dtauoj_dxi + HEOS.components[j].EOS().alpha0.dDelta(tau_oj, delta_oj)*ddeltaoj_dxi;
double xi = HEOS.mole_fractions[i];
double term = d_dalpha0oi_dxj + d_dalpha0oj_dxi + Kronecker_delta(i,j)/xi;
std::size_t kmax = HEOS.mole_fractions.size();
if (xN_flag == XN_DEPENDENT){ kmax--; }
for (std::size_t k = 0; k < kmax; ++k){
// Values for the k-th component
double xk = HEOS.mole_fractions[k];
double Tck = HEOS.get_fluid_constant(k, iT_critical);
double rhock = HEOS.get_fluid_constant(k, irhomolar_critical);
double tau_ok = HEOS.tau()*Tck / Tr;
double delta_ok = HEOS.delta()*rhor / rhock;
double dtauok_dxj = -tau_ok/Tr*dTr_dxj; // (Gernert, supp, B.19)
double ddeltaok_dxj = delta_ok/rhor*drhor_dxj; // (Gernert, supp. B.20)
double dtauok_dxi = -tau_ok/Tr*dTr_dxi; // (Gernert, supp, B.19)
double ddeltaok_dxi = delta_ok/rhor*drhor_dxi; // (Gernert, supp. B.20)
double dalpha0ok_dtauok = HEOS.components[k].EOS().alpha0.dTau(tau_ok, delta_ok);
double d2tauok_dxidxj = -Tck*HEOS.tau()*(POW2(Tr)*d2Tr_dxidxj-dTr_dxi*(2*Tr*dTr_dxj))/POW4(Tr);
double d_dalpha0ok_dtauok_dxj = HEOS.components[k].EOS().alpha0.dTau2(tau_ok, delta_ok)*dtauok_dxj + HEOS.components[k].EOS().alpha0.dDelta_dTau(tau_ok, delta_ok)*ddeltaok_dxj;
double dalpha0ok_ddeltaok = HEOS.components[k].EOS().alpha0.dDelta(tau_ok, delta_ok);
double d2deltaok_dxidxj = HEOS.delta()/rhock*d2rhor_dxidxj;
double d_dalpha0ok_ddeltaok_dxj = HEOS.components[k].EOS().alpha0.dDelta_dTau(tau_ok, delta_ok)*dtauok_dxj + HEOS.components[k].EOS().alpha0.dDelta2(tau_ok, delta_ok)*ddeltaok_dxj;
term += xk*(dalpha0ok_dtauok*d2tauok_dxidxj + d_dalpha0ok_dtauok_dxj*dtauok_dxi
+dalpha0ok_ddeltaok*d2deltaok_dxidxj + d_dalpha0ok_ddeltaok_dxj*ddeltaok_dxi);
}
return term;
}
@@ -1103,6 +1243,9 @@ public:
//two_comp("d_ndalphardni_dxj__constT_V_xi", MD::d_ndalphardni_dxj__constT_V_xi, MD::ndalphar_dni__constT_V_nj);
one_comp("dalpha0_dxi",MD::dalpha0_dxi, MD::alpha0);
one("d2alpha0_dxi_dDelta", MD::d2alpha0_dxi_dDelta, MD::dalpha0_dxi, DELTA);
one("d2alpha0_dxi_dTau", MD::d2alpha0_dxi_dTau, MD::dalpha0_dxi, TAU);
two_comp("d2alpha0_dxidxj",MD::d2alpha0_dxidxj, MD::dalpha0_dxi);
one_comp("dalphar_dxi",MD::dalphar_dxi, MD::alphar);
two_comp("d2alphardxidxj",MD::d2alphardxidxj, MD::dalphar_dxi);

3
src/Backends/Helmholtz/MixtureDerivatives.h
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@@ -668,6 +668,9 @@ class MixtureDerivatives{
static CoolPropDbl dalpha0_dxi(HelmholtzEOSMixtureBackend &HEOS, std::size_t i, x_N_dependency_flag xN_flag);
static CoolPropDbl d2alpha0_dxi_dTau(HelmholtzEOSMixtureBackend &HEOS, std::size_t i, x_N_dependency_flag xN_flag);
static CoolPropDbl d2alpha0_dxi_dDelta(HelmholtzEOSMixtureBackend &HEOS, std::size_t i, x_N_dependency_flag xN_flag);
static CoolPropDbl d2alpha0_dxidxj(HelmholtzEOSMixtureBackend &HEOS, std::size_t i, std::size_t j, x_N_dependency_flag xN_flag);
/// ****************************************************************************
/// ****************************************************************************