Module that defines the basics of a residual Helmholtz energy.

All the residual properties that are calculated in this library are based on residual Helmholtz Equations of State. Following the book by Michelsen and Mollerup.

In this library up to second derivatives of residual Helmholtz energy are used. Because they’re the fundamentals for phase equilibria calculation.

Note

Later on, third derivative with respect to volume will be included since it’s importance on calculation of critical points.

Properties

Available properties:

pressure(n, V, T)
fugacity(n, V, T)
fugacity(n, P, T, root=[vapor, liquid, stable])
volume

Calculate thermodynamic properties using Helmholtz energy as a basis. All the routines in this module work with the logic:

call foo(x, V, T, [dfoodv, dfoodT, ...])

Where the user can call the routine of the desired property. And include as optional values the desired derivatives of said properties.

Uses

- yaeos__constants
- yaeos__models_base

Interfaces

public interface size

private pure function size_ar_model(eos)

Get the size of the model.

Arguments

Type	Intent	Optional	Attributes		Name
class(ArModel),	intent(in)			::	eos

Return Value integer

Abstract Interfaces

abstract interface

private subroutine abs_residual_helmholtz(self, n, v, t, Ar, ArV, ArT, ArTV, ArV2, ArT2, Arn, ArVn, ArTn, Arn2)

Residual Helmholtz model generic interface.

This interface represents how an Ar model should be implemented. By our standard, a Resiudal Helmholtz model takes as input:

The mixture’s number of moles vector.
Volume, by default in liters.
Temperature, by default in Kelvin.

All the output arguments are optional. While this keeps a long signature for the implementation, this is done this way to take advantage of any inner optimizations to calculate derivatives inside the procedure.

Once the model is implemented, the signature can be short like model%residual_helmholtz(n, v, t, ArT2=dArdT2)

Arguments

Type	Intent	Optional		Name
class(ArModel),	intent(in)		::	self	ArModel
real(kind=pr),	intent(in)		::	n(:)	Moles vector
real(kind=pr),	intent(in)		::	v	Volume [L]
real(kind=pr),	intent(in)		::	t	Temperature [K]
real(kind=pr),	intent(out),	optional	::	Ar	Residual Helmoltz energy
real(kind=pr),	intent(out),	optional	::	ArV	$\frac{dAr}{dV}$
real(kind=pr),	intent(out),	optional	::	ArT	$\frac{dAr}{dT}$
real(kind=pr),	intent(out),	optional	::	ArTV	$\frac{d^2Ar}{dTV}$
real(kind=pr),	intent(out),	optional	::	ArV2	$\frac{d^2Ar}{dV^2}$
real(kind=pr),	intent(out),	optional	::	ArT2	$\frac{d^2Ar}{dT^2}$
real(kind=pr),	intent(out),	optional	::	Arn(size(n))	$\frac{dAr}{dn_i}$
real(kind=pr),	intent(out),	optional	::	ArVn(size(n))	$\frac{d^2Ar}{dVn_i}$
real(kind=pr),	intent(out),	optional	::	ArTn(size(n))	$\frac{d^2Ar}{dTn_i}$
real(kind=pr),	intent(out),	optional	::	Arn2(size(n),size(n))	$\frac{d^2Ar}{dn_{ij}}$

abstract interface

private function abs_volume_initializer(self, n, p, t)

Function that provides an initializer value for the liquid-root of newton solver of volume. In the case the model will use the volume_michelsen routine this value should provide the co-volume of the model.

Arguments

Type	Intent		Name
class(ArModel),	intent(in)	::	self	Ar Model
real(kind=pr),	intent(in)	::	n(:)	Moles vector
real(kind=pr),	intent(in)	::	p	Pressure [bar]
real(kind=pr),	intent(in)	::	t	Temperature [K]

Return Value real(kind=pr)

Initial volume [L]

Derived Types

type, public, abstract, extends(BaseModel) :: ArModel

Abstract residual Helmholtz model.

Components

Type	Visibility	Attributes		Name		Initial
type(Substances),	public		::	components			Substances contained in the module
character(len=:),	public,	allocatable	::	name			Name of the model

Type-Bound Procedures

procedure, public :: Cp_residual_vt
procedure, public :: Cv_residual_vt
procedure, public :: Psat_pure
procedure, public :: enthalpy_residual_vt
procedure, public :: entropy_residual_vt
procedure(abs_volume_initializer), public, deferred :: get_v0
procedure, public :: gibbs_residual_vt
procedure, public :: internal_energy_residual_vt
procedure, public :: lnfug_vt
procedure, public :: lnphi_pt
procedure, public :: lnphi_vt
procedure, public :: pressure
procedure(abs_residual_helmholtz), public, deferred :: residual_helmholtz
procedure, public :: volume

Functions

private function Psat_pure(eos, ncomp, T)

Calculation of saturation pressure of a pure component using the secant method.

Arguments

Type	Intent		Name
class(ArModel),	intent(in)	::	eos	Model that will be used
integer,	intent(in)	::	ncomp	Number of component in the mixture from which the saturation pressure will be calculated
real(kind=pr),	intent(in)	::	T	Temperature [K]

Return Value real(kind=pr)

private pure function size_ar_model(eos)

Get the size of the model.

Arguments

Type	Intent	Optional	Attributes		Name
class(ArModel),	intent(in)			::	eos

Return Value integer

Subroutines

public subroutine volume(eos, n, P, T, V, root_type)

Volume solver routine for residual Helmholtz models.

Arguments

Type	Intent		Name
class(ArModel),	intent(in)	::	eos	Model
real(kind=pr),	intent(in)	::	n(:)	Moles number vector
real(kind=pr),	intent(in)	::	P	Pressure [bar]
real(kind=pr),	intent(in)	::	T	Temperature [K]
real(kind=pr),	intent(out)	::	V	Volume [L]
character(len=*),	intent(in)	::	root_type	Desired root-type to solve. Options are: `["liquid", "vapor", "stable"]`

private subroutine Cp_residual_vt(eos, n, V, T, Cp)

Calculate residual heat capacity pressure constant given V and T.

Arguments

Type	Intent		Name
class(ArModel),	intent(in)	::	eos	Model
real(kind=pr),	intent(in)	::	n(:)	Moles number vector
real(kind=pr),	intent(in)	::	V	Volume [L]
real(kind=pr),	intent(in)	::	T	Temperature [K]
real(kind=pr),	intent(out)	::	Cp	heat capacity P constant [bar L / K]

private subroutine Cv_residual_vt(eos, n, V, T, Cv)

Calculate residual heat capacity volume constant given V and T.

Arguments

Type	Intent		Name
class(ArModel),	intent(in)	::	eos	Model
real(kind=pr),	intent(in)	::	n(:)	Moles number vector
real(kind=pr),	intent(in)	::	V	Volume [L]
real(kind=pr),	intent(in)	::	T	Temperature [K]
real(kind=pr),	intent(out)	::	Cv	heat capacity V constant [bar L / K]

private subroutine enthalpy_residual_vt(eos, n, V, T, Hr, HrV, HrT, Hrn)

Calculate residual enthalpy given volume and temperature.

Arguments

Type	Intent	Optional		Name
class(ArModel),	intent(in)		::	eos	Model
real(kind=pr),	intent(in)		::	n(:)	Moles number vector
real(kind=pr),	intent(in)		::	V	Volume [L]
real(kind=pr),	intent(in)		::	T	Temperature [K]
real(kind=pr),	intent(out),	optional	::	Hr	Residual enthalpy [bar L]
real(kind=pr),	intent(out),	optional	::	HrV	$\frac{dH^r}}{dV}$
real(kind=pr),	intent(out),	optional	::	HrT	$\frac{dH^r}}{dT}$
real(kind=pr),	intent(out),	optional	::	Hrn(size(n))	$\frac{dH^r}}{dn}$

private subroutine entropy_residual_vt(eos, n, V, T, Sr, SrV, SrT, Srn)

Calculate residual entropy given volume and temperature.

Arguments

Type	Intent	Optional		Name
class(ArModel),	intent(in)		::	eos	Model
real(kind=pr),	intent(in)		::	n(:)	Moles number vector
real(kind=pr),	intent(in)		::	V	Volume [L]
real(kind=pr),	intent(in)		::	T	Temperature [K]
real(kind=pr),	intent(out),	optional	::	Sr	Entropy [bar L / K]
real(kind=pr),	intent(out),	optional	::	SrV	$\frac{dS^r}}{dV}$
real(kind=pr),	intent(out),	optional	::	SrT	$\frac{dS^r}}{dT}$
real(kind=pr),	intent(out),	optional	::	Srn(size(n))	$\frac{dS^r}}{dn}$

private subroutine gibbs_residual_vt(eos, n, V, T, Gr, GrV, GrT, Grn)

Calculate residual Gibbs energy given volume and temperature.

Arguments

Type	Intent	Optional		Name
class(ArModel),	intent(in)		::	eos	Model
real(kind=pr),	intent(in)		::	n(:)	Moles number vector
real(kind=pr),	intent(in)		::	V	Volume [L]
real(kind=pr),	intent(in)		::	T	Temperature [K]
real(kind=pr),	intent(out),	optional	::	Gr	Gibbs energy [bar L]
real(kind=pr),	intent(out),	optional	::	GrV	$\frac{dG^r}}{dV}$
real(kind=pr),	intent(out),	optional	::	GrT	$\frac{dG^r}}{dT}$
real(kind=pr),	intent(out),	optional	::	Grn(size(n))	$\frac{dG^r}}{dn}$

private subroutine internal_energy_residual_vt(eos, n, V, T, Ur, UrV, UrT, Urn)

Calculate residual internal energy given volume and temperature.

Arguments

Type	Intent	Optional		Name
class(ArModel),	intent(in)		::	eos	Model
real(kind=pr),	intent(in)		::	n(:)	Moles number vector
real(kind=pr),	intent(in)		::	V	Volume [L]
real(kind=pr),	intent(in)		::	T	Temperature [K]
real(kind=pr),	intent(out),	optional	::	Ur	Internal energy [bar L]
real(kind=pr),	intent(out),	optional	::	UrV	$\frac{dU^r}}{dV}$
real(kind=pr),	intent(out),	optional	::	UrT	$\frac{dU^r}}{dT}$
real(kind=pr),	intent(out),	optional	::	Urn(size(n))	$\frac{dU^r}}{dn}$

private subroutine lnfug_vt(eos, n, V, T, P, lnf, dlnfdV, dlnfdT, dlnfdn, dPdV, dPdT, dPdn)

Calculate natural logarithm of fugacity given volume and temperature.

Arguments

Type	Intent	Optional		Name
class(ArModel)			::	eos	Model
real(kind=pr),	intent(in)		::	n(:)	Mixture mole numbers
real(kind=pr),	intent(in)		::	V	Volume [L]
real(kind=pr),	intent(in)		::	T	Temperature [K]
real(kind=pr),	intent(out),	optional	::	P	Pressure [bar]
real(kind=pr),	intent(out),	optional	::	lnf(size(n))	$\ln(\f_i)$ vector
real(kind=pr),	intent(out),	optional	::	dlnfdV(size(n))	$ln(f_i)$ Volume derivative
real(kind=pr),	intent(out),	optional	::	dlnfdT(size(n))	$ln(f_i)$ Temp derivative
real(kind=pr),	intent(out),	optional	::	dlnfdn(size(n),size(n))	$ln(f_i)$ compositional derivative
real(kind=pr),	intent(out),	optional	::	dPdV	$\frac{dP}{dV}$
real(kind=pr),	intent(out),	optional	::	dPdT	$\frac{dP}{dT}$
real(kind=pr),	intent(out),	optional	::	dPdn(:)	$\frac{dP}{dn_i}$

private subroutine lnphi_pt(eos, n, P, T, V, root_type, lnPhi, dlnPhidP, dlnPhidT, dlnPhidn, dPdV, dPdT, dPdn)

Calculate natural logarithm of fugacity given pressure and temperature.

Arguments

Type	Intent	Optional		Name
class(ArModel),	intent(in)		::	eos	Model
real(kind=pr),	intent(in)		::	n(:)	Mixture mole numbers
real(kind=pr),	intent(in)		::	P	Pressure [bar]
real(kind=pr),	intent(in)		::	T	Temperature [K]
real(kind=pr),	intent(out),	optional	::	V	Volume [L]
character(len=*),	intent(in)		::	root_type	Type of root desired [“liquid”, “vapor”, “stable”]
real(kind=pr),	intent(out),	optional	::	lnPhi(size(n))	$\ln(phi)$ vector
real(kind=pr),	intent(out),	optional	::	dlnPhidP(size(n))	ln(phi) Presssure derivative
real(kind=pr),	intent(out),	optional	::	dlnPhidT(size(n))	ln(phi) Temperature derivative
real(kind=pr),	intent(out),	optional	::	dlnPhidn(size(n),size(n))	ln(phi) compositional derivative
real(kind=pr),	intent(out),	optional	::	dPdV	$\frac{dP}{dV}$
real(kind=pr),	intent(out),	optional	::	dPdT	$\frac{dP}{dT}$
real(kind=pr),	intent(out),	optional	::	dPdn(size(n))	$\frac{dP}{dn_i}$

private subroutine lnphi_vt(eos, n, V, T, P, lnPhi, dlnPhidP, dlnPhidT, dlnPhidn, dPdV, dPdT, dPdn)

Calculate natural logarithm of fugacity coefficent.

Arguments

Type	Intent	Optional		Name
class(ArModel)			::	eos	Model
real(kind=pr),	intent(in)		::	n(:)	Mixture mole numbers
real(kind=pr),	intent(in)		::	V	Volume [L]
real(kind=pr),	intent(in)		::	T	Temperature [K]
real(kind=pr),	intent(out),	optional	::	P	Pressure [bar]
real(kind=pr),	intent(out),	optional	::	lnPhi(size(n))	$\ln(\phi_i)$ vector
real(kind=pr),	intent(out),	optional	::	dlnPhidP(size(n))	$ln(phi_i)$ Presssure derivative
real(kind=pr),	intent(out),	optional	::	dlnPhidT(size(n))	$ln(phi_i)$ Temp derivative
real(kind=pr),	intent(out),	optional	::	dlnPhidn(size(n),size(n))	$ln(phi_i)$ compositional derivative
real(kind=pr),	intent(out),	optional	::	dPdV	$\frac{dP}{dV}$
real(kind=pr),	intent(out),	optional	::	dPdT	$\frac{dP}{dT}$
real(kind=pr),	intent(out),	optional	::	dPdn(:)	$\frac{dP}{dn_i}$

private subroutine pressure(eos, n, V, T, P, dPdV, dPdT, dPdn)

Calculate pressure.

Arguments

Type	Intent	Optional		Name
class(ArModel),	intent(in)		::	eos	Model
real(kind=pr),	intent(in)		::	n(:)	Moles number vector
real(kind=pr),	intent(in)		::	V	Volume [L]
real(kind=pr),	intent(in)		::	T	Temperature [K]
real(kind=pr),	intent(out)		::	P	Pressure [bar]
real(kind=pr),	intent(out),	optional	::	dPdV	$\frac{dP}}{dV}$
real(kind=pr),	intent(out),	optional	::	dPdT	$\frac{dP}}{dT}$
real(kind=pr),	intent(out),	optional	::	dPdn(:)	$\frac{dP}}{dn_i}$

yaeos__models_ar Module

Contents

Interfaces

Abstract Interfaces

Derived Types

Functions

Subroutines

Module that defines the basics of a residual Helmholtz energy.

Properties

Available properties:

Uses

Interfaces

public interface size

private pure function size_ar_model(eos)

Arguments

Return Value integer

Abstract Interfaces

abstract interface

private subroutine abs_residual_helmholtz(self, n, v, t, Ar, ArV, ArT, ArTV, ArV2, ArT2, Arn, ArVn, ArTn, Arn2)

Arguments

abstract interface

private function abs_volume_initializer(self, n, p, t)

Arguments

Return Value real(kind=pr)

Derived Types

type, public, abstract, extends(BaseModel) :: ArModel

Components

Type-Bound Procedures

Functions

private function Psat_pure(eos, ncomp, T)

Arguments

Return Value real(kind=pr)

private pure function size_ar_model(eos)

Arguments

Return Value integer

Subroutines

public subroutine volume(eos, n, P, T, V, root_type)

Arguments

private subroutine Cp_residual_vt(eos, n, V, T, Cp)

Arguments

private subroutine Cv_residual_vt(eos, n, V, T, Cv)

Arguments

private subroutine enthalpy_residual_vt(eos, n, V, T, Hr, HrV, HrT, Hrn)

Arguments

private subroutine entropy_residual_vt(eos, n, V, T, Sr, SrV, SrT, Srn)

Arguments

private subroutine gibbs_residual_vt(eos, n, V, T, Gr, GrV, GrT, Grn)

Arguments

private subroutine internal_energy_residual_vt(eos, n, V, T, Ur, UrV, UrT, Urn)

Arguments

private subroutine lnfug_vt(eos, n, V, T, P, lnf, dlnfdV, dlnfdT, dlnfdn, dPdV, dPdT, dPdn)

Arguments

private subroutine lnphi_pt(eos, n, P, T, V, root_type, lnPhi, dlnPhidP, dlnPhidT, dlnPhidn, dPdV, dPdT, dPdn)

Arguments

private subroutine lnphi_vt(eos, n, V, T, P, lnPhi, dlnPhidP, dlnPhidT, dlnPhidn, dPdV, dPdT, dPdn)

Arguments

private subroutine pressure(eos, n, V, T, P, dPdV, dPdT, dPdn)

Arguments