#include <derivs.hpp>

Static Public Member Functions
static auto	get_splus (const Model &model, const Scalar &T, const VectorType &rhovec)
	Calculate the residual entropy ( \(s^+ = -s^r/R\)) from derivatives of alphar.

static auto	get_pr (const Model &model, const Scalar &T, const VectorType &rhovec)
	Calculate the residual pressure from derivatives of alphar.

static auto	get_Ar00 (const Model &model, const Scalar &T, const VectorType &rhovec)

static auto	get_Ar10 (const Model &model, const Scalar &T, const VectorType &rhovec)

static auto	get_Ar01 (const Model &model, const Scalar &T, const VectorType &rhovec)

static auto	get_Psir (const Model &model, const Scalar &T, const VectorType &rhovec)
	Calculate \(\Psi^r=a^r \rho\).

static auto	get_dPsirdT_constrhovec (const Model &model, const Scalar &T, const VectorType &rhovec)
	Calculate derivative \(\Psi^r=a^r \rho\) w.r.t. T at constant molar concentrations.

static auto	build_Psir_Hessian_autodiff (const Model &model, const Scalar &T, const VectorType &rho)
	Calculate the Hessian of \(\Psi^r = a^r \rho\) w.r.t. the molar concentrations.

static auto	build_Psir_fgradHessian_autodiff (const Model &model, const Scalar &T, const VectorType &rho)
	Calculate the function value, gradient, and Hessian of \(Psi^r = a^r\rho\) w.r.t. the molar concentrations.

static auto	build_Psi_Hessian_autodiff (const Model &model, const Scalar &T, const VectorType &rho)
	Calculate the Hessian of \(\Psi = a \rho\) w.r.t. the molar concentrations.

static auto	build_Psir_gradient_autodiff (const Model &model, const Scalar &T, const VectorType &rho)
	Gradient of Psir = ar*rho w.r.t. the molar concentrations.

static auto	build_Psir_gradient_complex_step (const Model &model, const Scalar &T, const VectorType &rho)
	Gradient of Psir = ar*rho w.r.t. the molar concentrations.

template<ADBackends be = ADBackends::autodiff>
static auto	build_Psir_gradient (const Model &model, const Scalar &T, const VectorType &rho)

static auto	get_chempotVLE_autodiff (const Model &model, const Scalar &T, const VectorType &rho)
	Calculate the chemical potential of each component.

template<ADBackends be = ADBackends::autodiff>
static auto	get_fugacity_coefficients (const Model &model, const Scalar &T, const VectorType &rhovec)
	Calculate the fugacity coefficient of each component.

template<ADBackends be = ADBackends::autodiff>
static auto	get_ln_fugacity_coefficients (const Model &model, const Scalar &T, const VectorType &rhovec)
	Calculate the natural logarithm of fugacity coefficient of each component.

template<ADBackends be = ADBackends::autodiff>
static auto	get_ln_fugacity_coefficients_Trhomolefracs (const Model &model, const Scalar &T, const Scalar &rhotot, const VectorType &molefrac)

template<ADBackends be = ADBackends::autodiff>
static auto	get_ln_fugacity_coefficients1 (const Model &model, const Scalar &T, const VectorType &rhovec)

template<ADBackends be = ADBackends::autodiff>
static auto	get_ln_fugacity_coefficients2 (const Model &model, const Scalar &T, const VectorType &rhovec)

static Eigen::ArrayXd	build_d2PsirdTdrhoi_autodiff (const Model &model, const Scalar &T, const VectorType &rho)

static Eigen::ArrayXd	build_d2alphardrhodxi_constT (const Model &model, const Scalar &T, const Scalar &rhomolar, const VectorType &molefrac)

template<ADBackends be = ADBackends::autodiff>
static auto	get_d_ln_fugacity_coefficients_dT_constrhovec (const Model &model, const Scalar &T, const VectorType &rhovec)
	Calculate the derivative of the natural logarithm of fugacity coefficient of each component w.r.t. temperature at constant mole concentrations (implying constant mole fractions and density)

template<ADBackends be = ADBackends::autodiff>
static auto	get_lnZ_Z_dZdrho (const Model &model, const Scalar &T, const VectorType &rhovec)
	Calculate ln(Z), Z, and dZ/drho at constant temperature and mole fractions.

template<ADBackends be = ADBackends::autodiff>
static auto	get_d_ln_fugacity_coefficients_drho_constTmolefracs (const Model &model, const Scalar &T, const VectorType &rhovec)
	Calculate the derivative of the natural logarithm of fugacity coefficient of each component w.r.t. molar density at constant temperature and mole fractions.

template<ADBackends be = ADBackends::autodiff>
static auto	get_d_ln_fugacity_coefficients_dv_constTmolefracs (const Model &model, const Scalar &T, const VectorType &rhovec)
	Calculate the derivative of the natural logarithm of fugacity coefficient of each component w.r.t. molar volume at constant temperature and mole fractions.

template<ADBackends be = ADBackends::autodiff>
static auto	get_d_ln_fugacity_coefficients_dmolefracs_constTrho (const Model &model, const Scalar &T, const VectorType &rhovec)
	Calculate the derivative of the natural logarithm of fugacity coefficient of each component w.r.t. mole fraction of each component, at constant temperature and molar density.

template<ADBackends be = ADBackends::autodiff>
static auto	get_d_ln_fugacity_coefficients_dmolefracs_constTrho1 (const Model &model, const Scalar &T, const VectorType &rhovec)

static auto	get_dchempotdT_autodiff (const Model &model, const Scalar &T, const VectorType &rhovec)
	Calculate the temperature derivative of the chemical potential of each component.

static auto	get_dpdT_constrhovec (const Model &model, const Scalar &T, const VectorType &rhovec)
	Calculate the temperature derivative of the pressure at constant molar concentrations.

static auto	get_dpdrhovec_constT (const Model &model, const Scalar &T, const VectorType &rhovec)
	Calculate the molar concentration derivatives of the pressure at constant temperature.

static auto	get_partial_molar_volumes (const Model &model, const Scalar &T, const VectorType &rhovec)
	Calculate the partial molar volumes of each component.

static VectorType	get_Psir_sigma_derivs (const Model &model, const Scalar &T, const VectorType &rhovec, const VectorType &v)

Detailed Description

template<typename Model, typename Scalar = double, typename VectorType = Eigen::ArrayXd>
struct teqp::IsochoricDerivatives< Model, Scalar, VectorType >

In the isochoric formalism, the fugacity coefficient array can be obtained by the gradient of the residual Helmholtz energy density (which is a scalar) and the compressibility factor \(Z\) (which is also a scalar) in terms of the temperature \(T\) and the molar concentration vector \(\vec\rho\):

\[ \ln\vec\phi = \frac{1}{RT}\frac{\partial \Psi^r}{d\vec\rho} - \ln(Z) \]

Easy: temperature derivative at constant molar concentrations (implying constant volume and molar composition)

\[ \deriv{ \ln\vec\phi}{T}{\vec\rho} = \frac{1}{RT}\frac{\partial^2 \Psi^r}{\partial \vec\rho\partial T} + \frac{-1}{RT^2}\deriv{\Psi^r}{\vec\rho}{T} - \frac{1}{Z}\deriv{Z}{T}{\vec\rho} \]

Medium: molar density derivative at constant temperature and mole fractions

\[ \deriv{ \ln\vec\phi}{\rho}{T,\vec x} = \frac{1}{RT}\frac{\partial^2 \Psi^r}{\partial \vec\rho\partial \rho} - \frac{1}{Z}\deriv{Z}{\rho}{T,\vec x} \]

\[ Z = 1+\rho\deriv{\alpha^r}{\rho}{T} \]

\[ \deriv{Z}{\rho}{T,\vec x} = \rho\deriv{^2\alpha^r}{\rho^2}{T} + \deriv{\alpha^r}{\rho}{T} \]

Back to basics, for a quantity \(\chi\) that is a function of \(T\) and \(\vec\rho\), and then the derivative taken w.r.t. density at constant temperature and mole fractions:

\[ \deriv{\chi}{\rho}{T, \vec x} = \deriv{\chi}{T}{\vec \rho}\cancelto{0}{\deriv{T}{\rho}{T}} + \sum_i\deriv{\chi}{\rho_i}{T, \rho_{j\neq i}}\deriv{\rho_i}{\rho}{T,\vec x} \]

with \(\rho_i =x_i\rho\)

\[ \deriv{\rho_i}{\rho}{T, \vec x} = x_i \]

thus

\[ \deriv{\chi}{\rho}{T, \vec x} = \sum_i\deriv{\chi}{\rho_i}{T, \rho_{j\neq i}}x_i \]

and following the pattern yields

\[ \frac{\partial^2 \Psi^r}{\partial \vec\rho\partial \rho} = \sum_i\deriv{\frac{\partial \Psi^r}{d\vec\rho} }{\rho_i}{T, \rho_{j\neq i}}x_i \]

where the big thing is the Hessian of the residual Hessian matrix of the residual Helmholtz energy density. This uses terms that are already developed.

Medium+: Volume derivative, based on the density derivative

\[ \deriv{ \ln\vec\phi}{v}{T,\vec x} = \deriv{ \ln\vec\phi}{\rho}{T,\vec x}\deriv{ \rho}{v}{} \]

\[ \deriv{\rho}{v}{} = -1/v^2 = -\rho^2 \]

Hard: mole fraction derivatives (this results in a matrix rather than a vector)

\[ \deriv{ \ln\vec\phi}{\vec x}{T,\rho} = ? \]

The first term is conceptually tricky. Again, considering a generic quantity \(\chi\)

\[ \deriv{\chi}{x_i}{T, \rho, x_{j\neq i}} = \deriv{\chi}{T}{\vec \rho}\cancelto{0}{\deriv{T}{x_i}{T,\rho,x_{j\neq i}}} + \sum_i\deriv{\chi}{\rho_i}{T, \rho_{j\neq i}}\deriv{\rho_i}{x_i}{T,\rho, x_{j\neq i}} \]

yields

\[ \deriv{\chi}{x_i}{T, \rho, x_{j\neq i}} = \rho \sum_i\deriv{\chi}{\rho_i}{T, \rho_{j\neq i}} \]

so the first part becomes

\[ \deriv{\frac{\partial \Psi^r}{d\vec\rho}}{x_i}{T, \rho, x_{j\neq i}} = \rho \sum_i\deriv{\frac{\partial \Psi^r}{d\vec\rho}}{\rho_i}{T, \rho_{j\neq i}} \]

or

\[ \deriv{^2\partial \Psi^r}{\vec\rho \partial \vec x}{T, \rho} = \rho H(\Psi^r) \]

which is somewhat surprising because the order of derivatives with respect to composition and density doesn't matter, as the Hessian is symmetric

The second part, from derivatives of \(\ln Z\), with \(Z\) given by

\[ Z = 1+\rho\deriv{\alpha^r}{\rho}{T, \vec x} \]

yields

\[ \deriv{\ln Z}{x_i}{T,\rho,x_{k \neq j}} = \frac{1}{Z}\deriv{Z}{x_i}{T,\rho,x_{k \neq i}} \]

which results in a vector because you have

\[ \deriv{Z}{x_i}{T,\rho,x_{k \neq i}} = \rho \deriv{^2\alpha^r}{\rho\partial x_i}{T} \]

Definition at line 949 of file derivs.hpp.

Member Function Documentation

◆ build_d2alphardrhodxi_constT()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

static Eigen::ArrayXd teqp::IsochoricDerivatives< Model, Scalar, VectorType >::build_d2alphardrhodxi_constT	(	const Model &	model,
		const Scalar &	T,
		const Scalar &	rhomolar,
		const VectorType &	molefrac )

inlinestatic

Definition at line 1274 of file derivs.hpp.

◆ build_d2PsirdTdrhoi_autodiff()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

static Eigen::ArrayXd teqp::IsochoricDerivatives< Model, Scalar, VectorType >::build_d2PsirdTdrhoi_autodiff	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rho )

inlinestatic

Definition at line 1256 of file derivs.hpp.

◆ build_Psi_Hessian_autodiff()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::build_Psi_Hessian_autodiff	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rho )

inlinestatic

Calculate the Hessian of \(\Psi = a \rho\) w.r.t. the molar concentrations.

Uses autodiff derivatives to calculate second partial derivatives

Definition at line 1068 of file derivs.hpp.

◆ build_Psir_fgradHessian_autodiff()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::build_Psir_fgradHessian_autodiff	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rho )

inlinestatic

Calculate the function value, gradient, and Hessian of \(Psi^r = a^r\rho\) w.r.t. the molar concentrations.

Uses autodiff to calculate the derivatives

Definition at line 1043 of file derivs.hpp.

◆ build_Psir_gradient()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

template<ADBackends be = ADBackends::autodiff>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::build_Psir_gradient	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rho )

inlinestatic

Definition at line 1165 of file derivs.hpp.

◆ build_Psir_gradient_autodiff()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::build_Psir_gradient_autodiff	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rho )

inlinestatic

Gradient of Psir = ar*rho w.r.t. the molar concentrations.

Uses autodiff to calculate derivatives

Definition at line 1107 of file derivs.hpp.

◆ build_Psir_gradient_complex_step()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::build_Psir_gradient_complex_step	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rho )

inlinestatic

Gradient of Psir = ar*rho w.r.t. the molar concentrations.

Uses complex step to calculate derivatives

Definition at line 1144 of file derivs.hpp.

◆ build_Psir_Hessian_autodiff()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::build_Psir_Hessian_autodiff	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rho )

inlinestatic

Calculate the Hessian of \(\Psi^r = a^r \rho\) w.r.t. the molar concentrations.

Requires the use of autodiff derivatives to calculate second partial derivatives

Definition at line 1023 of file derivs.hpp.

◆ get_Ar00()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::get_Ar00	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rhovec )

inlinestatic

Definition at line 972 of file derivs.hpp.

◆ get_Ar01()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::get_Ar01	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rhovec )

inlinestatic

Definition at line 984 of file derivs.hpp.

◆ get_Ar10()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::get_Ar10	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rhovec )

inlinestatic

Definition at line 978 of file derivs.hpp.

◆ get_chempotVLE_autodiff()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::get_chempotVLE_autodiff	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rho )

inlinestatic

Calculate the chemical potential of each component.

Uses autodiff to calculate derivatives See Eq. 5 of https://doi.org/10.1002/aic.16730, but the rho in the denominator should be a rhoref (taken to be 1)

Note: : Some contributions to the ideal gas part are missing (reference state and cp0), but are not relevant to phase equilibria

Definition at line 1188 of file derivs.hpp.

◆ get_d_ln_fugacity_coefficients_dmolefracs_constTrho()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

template<ADBackends be = ADBackends::autodiff>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::get_d_ln_fugacity_coefficients_dmolefracs_constTrho	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rhovec )

inlinestatic

Calculate the derivative of the natural logarithm of fugacity coefficient of each component w.r.t. mole fraction of each component, at constant temperature and molar density.

\[ \deriv{ \ln\vec\phi}{\vec x}{T, \rho} \]

Definition at line 1367 of file derivs.hpp.

◆ get_d_ln_fugacity_coefficients_dmolefracs_constTrho1()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

template<ADBackends be = ADBackends::autodiff>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::get_d_ln_fugacity_coefficients_dmolefracs_constTrho1	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rhovec )

inlinestatic

Definition at line 1388 of file derivs.hpp.

◆ get_d_ln_fugacity_coefficients_drho_constTmolefracs()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

template<ADBackends be = ADBackends::autodiff>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::get_d_ln_fugacity_coefficients_drho_constTmolefracs	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rhovec )

inlinestatic

Calculate the derivative of the natural logarithm of fugacity coefficient of each component w.r.t. molar density at constant temperature and mole fractions.

\[ \deriv{ \ln\vec\phi}{\rho}{T,\vec x} = \frac{1}{RT}H(\Psi_r)\vec x - \frac{1}{Z}\deriv{Z}{\rho}{T,\vec x} \]

Definition at line 1336 of file derivs.hpp.

◆ get_d_ln_fugacity_coefficients_dT_constrhovec()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

template<ADBackends be = ADBackends::autodiff>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::get_d_ln_fugacity_coefficients_dT_constrhovec	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rhovec )

inlinestatic

Calculate the derivative of the natural logarithm of fugacity coefficient of each component w.r.t. temperature at constant mole concentrations (implying constant mole fractions and density)

Uses autodiff to calculate derivatives by default

\[ \deriv{ \ln\vec\phi}{T}{\vec \rho} \]

Definition at line 1299 of file derivs.hpp.

◆ get_d_ln_fugacity_coefficients_dv_constTmolefracs()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

template<ADBackends be = ADBackends::autodiff>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::get_d_ln_fugacity_coefficients_dv_constTmolefracs	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rhovec )

inlinestatic

Calculate the derivative of the natural logarithm of fugacity coefficient of each component w.r.t. molar volume at constant temperature and mole fractions.

\[ \deriv{ \ln\vec\phi}{v}{T,\vec x} = \deriv{ \ln\vec\phi}{\rho}{T,\vec x}\deriv{ \rho}{v}{} \]

Definition at line 1353 of file derivs.hpp.

◆ get_dchempotdT_autodiff()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::get_dchempotdT_autodiff	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rhovec )

inlinestatic

Calculate the temperature derivative of the chemical potential of each component.

Note: : Some contributions to the ideal gas part are missing (reference state and cp0), but are not relevant to phase equilibria

Definition at line 1403 of file derivs.hpp.

◆ get_dpdrhovec_constT()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::get_dpdrhovec_constT	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rhovec )

inlinestatic

Calculate the molar concentration derivatives of the pressure at constant temperature.

Definition at line 1423 of file derivs.hpp.

◆ get_dpdT_constrhovec()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::get_dpdT_constrhovec	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rhovec )

inlinestatic

Calculate the temperature derivative of the pressure at constant molar concentrations.

Definition at line 1413 of file derivs.hpp.

◆ get_dPsirdT_constrhovec()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::get_dPsirdT_constrhovec	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rhovec )

inlinestatic

Calculate derivative \(\Psi^r=a^r \rho\) w.r.t. T at constant molar concentrations.

Definition at line 1010 of file derivs.hpp.

◆ get_fugacity_coefficients()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

template<ADBackends be = ADBackends::autodiff>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::get_fugacity_coefficients	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rhovec )

inlinestatic

Calculate the fugacity coefficient of each component.

Uses autodiff to calculate derivatives

Definition at line 1201 of file derivs.hpp.

◆ get_ln_fugacity_coefficients()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

template<ADBackends be = ADBackends::autodiff>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::get_ln_fugacity_coefficients	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rhovec )

inlinestatic

Calculate the natural logarithm of fugacity coefficient of each component.

Uses autodiff to calculate derivatives by default

Definition at line 1212 of file derivs.hpp.

◆ get_ln_fugacity_coefficients1()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

template<ADBackends be = ADBackends::autodiff>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::get_ln_fugacity_coefficients1	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rhovec )

inlinestatic

Definition at line 1237 of file derivs.hpp.

◆ get_ln_fugacity_coefficients2()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

template<ADBackends be = ADBackends::autodiff>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::get_ln_fugacity_coefficients2	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rhovec )

inlinestatic

Definition at line 1248 of file derivs.hpp.

◆ get_ln_fugacity_coefficients_Trhomolefracs()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

template<ADBackends be = ADBackends::autodiff>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::get_ln_fugacity_coefficients_Trhomolefracs	(	const Model &	model,
		const Scalar &	T,
		const Scalar &	rhotot,
		const VectorType &	molefrac )

inlinestatic

Definition at line 1225 of file derivs.hpp.

◆ get_lnZ_Z_dZdrho()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

template<ADBackends be = ADBackends::autodiff>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::get_lnZ_Z_dZdrho	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rhovec )

inlinestatic

Calculate ln(Z), Z, and dZ/drho at constant temperature and mole fractions.

Uses autodiff to calculate derivatives by default

Definition at line 1317 of file derivs.hpp.

◆ get_partial_molar_volumes()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::get_partial_molar_volumes	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rhovec )

inlinestatic

Calculate the partial molar volumes of each component.

\[ \hat v_i = \left(\frac{\partial V}{\partial n_i}\right)_{T,V,n_{j \neq i}} \]

Eq 7.32 from GERG-2004

\[ \hat v_i = \deriv{V}{n_i}{T,p,n_j} = \displaystyle\frac{-n\deriv{p}{n_i}{T,V,n_j}}{n\deriv{p}{V}{T,\vec{n}}} \]

Total differential of a variable \(Y\) that is a function of \(T\), \(\vec{\rho}\):

\[ \mathrm{d} Y = \deriv{Y}{T}{\vec{\rho}} \mathrm{d} T + \sum_k \deriv{Y}{\rho_k}{T,\rho_{j\neq i}} \mathrm{d} \rho_k \]

so

\[ \deriv{Y}{n_i}{T,V,n_{j\neq i}} = \deriv{Y}{T}{\vec{\rho}} \cancelto{0}{\deriv{T}{n_i}{T,V,n_j}} + \sum_k \deriv{Y}{\rho_k}{T,\rho_{j\neq i}} \deriv{\rho_k}{n_i}{T,V,n_j} \]

\[ \deriv{\rho_k}{n_i}{T,V,n_j} = \frac{\delta_{ik}}{V} \]

because \(\rho_k = n_k/V\).

Thus in the isochoric framework, the partial molar volume is given by

\[ \hat v_i = \displaystyle\frac{-\frac{n}{V}\deriv{p}{\rho_i}{T,\rho_j}}{n\deriv{p}{V}{T,\vec{n}}} \]

The final formulation includes this term in the numerator (see the supporting info from isochoric paper):

\[ \deriv{p}{\rho_i}{T,\rho_j} = RT + \sum_k\rho_k\deriv{^2\Psi^{\rm r}}{\rho_k\partial \rho_i}{T,\rho_{j\neq k}} \]

and the denominator is given by (Eq. 7.62 of GERG-2004)

\[ n\deriv{p}{V}{T,\vec{n}} = -\rho^2 RT\left(1+2\delta\deriv{\alpha^{\rm r}}{\delta}{\tau} + \delta^2\deriv{^2\alpha^{\rm r}}{\delta^2}{\tau} \right) \]

and finally

\[ \hat v_i = \displaystyle\frac{-\rho\deriv{p}{\rho_i}{T,\rho_j}}{n\deriv{p}{V}{T,\vec{n}}} \]

Definition at line 1474 of file derivs.hpp.

◆ get_pr()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::get_pr	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rhovec )

inlinestatic

Calculate the residual pressure from derivatives of alphar.

Definition at line 963 of file derivs.hpp.

◆ get_Psir()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::get_Psir	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rhovec )

inlinestatic

Calculate \(\Psi^r=a^r \rho\).

Definition at line 1001 of file derivs.hpp.

◆ get_Psir_sigma_derivs()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

static VectorType teqp::IsochoricDerivatives< Model, Scalar, VectorType >::get_Psir_sigma_derivs	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rhovec,
		const VectorType &	v )

inlinestatic

Definition at line 1488 of file derivs.hpp.

◆ get_splus()

template<typename Model , typename Scalar = double, typename VectorType = Eigen::ArrayXd>

static auto teqp::IsochoricDerivatives< Model, Scalar, VectorType >::get_splus	(	const Model &	model,
		const Scalar &	T,
		const VectorType &	rhovec )

inlinestatic

Calculate the residual entropy ( \(s^+ = -s^r/R\)) from derivatives of alphar.

Definition at line 954 of file derivs.hpp.

The documentation for this struct was generated from the following file:

include/teqp/derivs.hpp

Static Public Member Functions

Detailed Description

Member Function Documentation

◆ build_d2alphardrhodxi_constT()

◆ build_d2PsirdTdrhoi_autodiff()

◆ build_Psi_Hessian_autodiff()

◆ build_Psir_fgradHessian_autodiff()

◆ build_Psir_gradient()

◆ build_Psir_gradient_autodiff()

◆ build_Psir_gradient_complex_step()

◆ build_Psir_Hessian_autodiff()

◆ get_Ar00()

◆ get_Ar01()

◆ get_Ar10()

◆ get_chempotVLE_autodiff()

◆ get_d_ln_fugacity_coefficients_dmolefracs_constTrho()

◆ get_d_ln_fugacity_coefficients_dmolefracs_constTrho1()

◆ get_d_ln_fugacity_coefficients_drho_constTmolefracs()

◆ get_d_ln_fugacity_coefficients_dT_constrhovec()

◆ get_d_ln_fugacity_coefficients_dv_constTmolefracs()

◆ get_dchempotdT_autodiff()

◆ get_dpdrhovec_constT()

◆ get_dpdT_constrhovec()

◆ get_dPsirdT_constrhovec()

◆ get_fugacity_coefficients()

◆ get_ln_fugacity_coefficients()

◆ get_ln_fugacity_coefficients1()

◆ get_ln_fugacity_coefficients2()

◆ get_ln_fugacity_coefficients_Trhomolefracs()

◆ get_lnZ_Z_dZdrho()

◆ get_partial_molar_volumes()

◆ get_pr()

◆ get_Psir()

◆ get_Psir_sigma_derivs()

◆ get_splus()