yaeos__equilibria_critical

Uses

Variables

Type	Visibility	Attributes		Name		Initial
type(CPSpecs),	public,	parameter	::	spec_CP	=	CPSpecs()	Specification variables for a critical point or critical line calculation.

Derived Types

type, public :: CriticalLine

This derived type is used to store a critical line between two fluids. The critical line is calculated using the critical_line function. It uses the continuation method.

Components

Type	Visibility	Attributes		Name
type(EquilibriumState),	public		::	CEP	Critical End Point
real(kind=pr),	public,	allocatable	::	P(:)	Pressure [bar]
real(kind=pr),	public,	allocatable	::	T(:)	Temperature [K]
real(kind=pr),	public,	allocatable	::	V(:)	Volume [L/mol]
real(kind=pr),	public,	allocatable	::	a(:)	Molar fraction of the second fluid
integer,	public,	allocatable	::	iters(:)	Iterations needed for this point
integer,	public,	allocatable	::	ns(:)	Specified variable
real(kind=pr),	public,	allocatable	::	z0(:)	Molar fractions of the first fluid
real(kind=pr),	public,	allocatable	::	zi(:)	Molar fractions of the second fluid

type, private :: CPSpecs

Enumerator to handle the possible specifications for a critical point.

Components

Type	Visibility		Name		Initial
integer,	public	::	P	=	4	Specify $P$
integer,	public	::	T	=	3	Specify $T$
integer,	public	::	V	=	2	Specify $V$
integer,	public	::	a	=	1	Specify $\alpha$

Functions

public function F_critical(model, X, ns, S, z0, zi, u)

Function that should be equal to zero at a critical point is found. The second criticality condition is calculated as a numerical derivative with eps=1e-4.

Arguments

Type	Intent		Name
class(ArModel),	intent(in)	::	model	Equation of state model
real(kind=pr),	intent(in)	::	X(4)	Vector of variables
integer,	intent(in)	::	ns	Position of the specification variable
real(kind=pr),	intent(in)	::	S	Specification variable value
real(kind=pr),	intent(in)	::	z0(:)	Molar fractions of the first fluid
real(kind=pr),	intent(in)	::	zi(:)	Molar fractions of the second fluid
real(kind=pr),	intent(inout)	::	u(:)	Eigen-vector

Return Value real(kind=pr), (4)

public function critical_line(model, a0, z0, zi, ns0, S0, dS0, v0, t0, p0, max_points, maxP, first_point, stability_analysis)

Calculates the critical line between two mixtures using the continuation method. The two mixtures compositions are restricted to the relation between them, by a parameter $\alpha$ , which represents the molar fraction of the second fluid with respect to the whole mixture.

Arguments

Type	Intent	Optional		Name
class(ArModel),	intent(in)		::	model	Equation of state model
real(kind=pr),	intent(in)		::	a0	Initial $\alpha$ value
real(kind=pr),	intent(in)		::	z0(:)	Molar fractions of the first fluid
real(kind=pr),	intent(in)		::	zi(:)	Molar fractions of the second fluid
integer,	intent(in)		::	ns0	Position of the specification variable
real(kind=pr),	intent(in)		::	S0	Specified value
real(kind=pr),	intent(in)		::	dS0	Initial step size
real(kind=pr),	intent(in),	optional	::	v0	Initial volume [L/mol]
real(kind=pr),	intent(in),	optional	::	t0	Initial temperature [K]
real(kind=pr),	intent(in),	optional	::	p0	Initial pressure [bar]
integer,	intent(in),	optional	::	max_points	Maximum number of points
real(kind=pr),	intent(in),	optional	::	maxP	Maximum pressure
type(EquilibriumState),	intent(in),	optional	::	first_point
logical,		optional	::	stability_analysis

Return Value type(CriticalLine)

public function critical_point(model, z0, zi, spec, S, max_iters, V0, T0, a0, P0)

Calculates a single critical point of a mixture using a Newton-Raphson method. It is possible to specify different variables to be fixed with the spec argument, the spec_CP variable helps when selecting the specified variable.

Arguments

Type	Intent	Optional		Name
class(ArModel),	intent(in)		::	model	Equation of state model
real(kind=pr),	intent(in)		::	z0(:)	Molar fractions of the first fluid
real(kind=pr),	intent(in)		::	zi(:)	Molar fractions of the second fluid
integer,	intent(in)		::	spec	Specification `[1:"z", 2:"V", 3:"T", 4:"P"]`
real(kind=pr),	intent(in)		::	S	Specification value
integer,	intent(in)		::	max_iters	Maxiumum number of iterations
real(kind=pr),	intent(in),	optional	::	V0	Initial volume [L/mol].
real(kind=pr),	intent(in),	optional	::	T0	Initial temperature [K].
real(kind=pr),	intent(in),	optional	::	a0	Initial $\alpha$ value
real(kind=pr),	intent(in),	optional	::	P0	Initial Pressure [bar]

Return Value type(EquilibriumState)

public function df_critical(model, X, ns, S, z0, zi, u)

Calculates the Jacobian of the critical point function F_critical.

Arguments

Type	Intent		Name
class(ArModel),	intent(in)	::	model	Equation of state model
real(kind=pr),	intent(in)	::	X(4)	Vector of variables
integer,	intent(in)	::	ns	Position of the specification variable
real(kind=pr),	intent(in)	::	S	Specification variable value
real(kind=pr),	intent(in)	::	z0(:)	Molar fractions of the first fluid
real(kind=pr),	intent(in)	::	zi(:)	Molar fractions of the second fluid
real(kind=pr),	intent(inout)	::	u(:)	Eigen-vector

Return Value real(kind=pr), (4,4)

Jacobian of the critical point function

public function lambda1(model, X, s, z0, zi, u, u_new, P)

Calculation of the first restriction of a critical point.

Arguments

Type	Intent	Optional		Name
class(ArModel),	intent(in)		::	model
real(kind=pr),	intent(in)		::	X(4)	Vector of variables
real(kind=pr),	intent(in)		::	s	Distance between the two fluids compositions to the main composition
real(kind=pr),	intent(in)		::	z0(:)	Molar fractions of the first fluid
real(kind=pr),	intent(in)		::	zi(:)	Molar fractions of the second fluid
real(kind=pr),	intent(in)		::	u(:)	Eigen-vector that defines the direction between the two compositions $n_i = z_i + s u_i \sqrt{z_i}$
real(kind=pr),	intent(out),	optional	::	u_new(:)	Eigen-vector corresponding to the smallest eigenvalue of the matrix $M_{ij} = \sqrt{z_i z_j} \frac{\partial \ln f_i}{\partial n_j}$
real(kind=pr),	intent(out),	optional	::	P	Pressure of the system [bar]

Return Value real(kind=pr)

private function F_cep(model, nc, X, z0, zi, u)

Arguments

Type	Intent		Name
class(ArModel),	intent(in)	::	model	Equation of state model
integer,	intent(in)	::	nc
real(kind=pr),	intent(in)	::	X(nc+4)	Vector of variables
real(kind=pr),	intent(in)	::	z0(nc)	Molar fractions of the first fluid
real(kind=pr),	intent(in)	::	zi(nc)	Molar fractions of the second fluid
real(kind=pr),	intent(inout)	::	u(nc)	Eigen-vector

Return Value real(kind=pr), (nc+4)

private function df_cep(model, nc, X, z0, zi, u)

Calculates the Jacobian of the critical point function F_critical.

Arguments

Type	Intent		Name
class(ArModel),	intent(in)	::	model	Equation of state model
integer,	intent(in)	::	nc
real(kind=pr),	intent(in)	::	X(nc+4)	Vector of variables
real(kind=pr),	intent(in)	::	z0(nc)	Molar fractions of the first fluid
real(kind=pr),	intent(in)	::	zi(nc)	Molar fractions of the second fluid
real(kind=pr),	intent(inout)	::	u(nc)	Eigen-vector

Return Value real(kind=pr), (nc+4,nc+4)

Jacobian of the critical point function

private function get_a(X)

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=pr),	intent(in)			::	X

Return Value real(kind=pr)

private function set_a(a)

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=pr),	intent(in)			::	a

Return Value real(kind=pr)

Subroutines

private subroutine look_for_cep(model, z0, zi, Pc, Vc, Tc, a, u, found, CEP)

Arguments

Type	Intent		Name
class(ArModel),	intent(in)	::	model	Equation of state model
real(kind=pr),	intent(in)	::	z0(:)	Molar fractions of the first fluid
real(kind=pr),	intent(in)	::	zi(:)	Molar fractions of the second fluid
real(kind=pr),	intent(in)	::	Pc	Pressure [bar]
real(kind=pr),	intent(in)	::	Vc	Volume [L/mol]
real(kind=pr),	intent(in)	::	Tc	Temperature [K]
real(kind=pr),	intent(in)	::	a	Molar fraction of the second fluid
real(kind=pr),	intent(inout)	::	u(:)	Eigen-vector
logical,	intent(out)	::	found	Found a Critical End Point
type(EquilibriumState),	intent(out)	::	CEP	Critical End Point

private subroutine stability_check(model, z0, zi, Pc, Vc, Tc, a, unstable, y_other, V_other)

Check the stability of a point in the critical line. The stability is determined by tpd analysis.

Arguments

Type	Intent		Name
class(ArModel),	intent(in)	::	model	Equation of state model
real(kind=pr),	intent(in)	::	z0(:)	Molar fractions of the first fluid
real(kind=pr),	intent(in)	::	zi(:)	Molar fractions of the second fluid
real(kind=pr),	intent(in)	::	Pc	Pressure [bar]
real(kind=pr),	intent(in)	::	Vc	Volume [L/mol]
real(kind=pr),	intent(in)	::	Tc	Temperature [K]
real(kind=pr),	intent(in)	::	a	Molar fraction of the second fluid
logical,	intent(out)	::	unstable	Stability of the point)
real(kind=pr),	intent(out)	::	y_other(:)	Molar fractions of the second fluid
real(kind=pr),	intent(out)	::	V_other	Volume [L/mol]

yaeos__equilibria_critical Module

Contents

Variables

Derived Types

Functions

Subroutines

Uses

Variables

Derived Types

type, public :: CriticalLine

Components

type, private :: CPSpecs

Components

Functions

public function F_critical(model, X, ns, S, z0, zi, u)

Arguments

Return Value real(kind=pr), (4)

public function critical_line(model, a0, z0, zi, ns0, S0, dS0, v0, t0, p0, max_points, maxP, first_point, stability_analysis)

Arguments

Return Value type(CriticalLine)

public function critical_point(model, z0, zi, spec, S, max_iters, V0, T0, a0, P0)

Arguments

Return Value type(EquilibriumState)

public function df_critical(model, X, ns, S, z0, zi, u)

Arguments

Return Value real(kind=pr), (4,4)

public function lambda1(model, X, s, z0, zi, u, u_new, P)

Arguments

Return Value real(kind=pr)

private function F_cep(model, nc, X, z0, zi, u)

Arguments

Return Value real(kind=pr), (nc+4)

private function df_cep(model, nc, X, z0, zi, u)

Arguments

Return Value real(kind=pr), (nc+4,nc+4)

private function get_a(X)

Arguments

Return Value real(kind=pr)

private function set_a(a)

Arguments

Return Value real(kind=pr)

Subroutines

private subroutine look_for_cep(model, z0, zi, Pc, Vc, Tc, a, u, found, CEP)

Arguments

private subroutine stability_check(model, z0, zi, Pc, Vc, Tc, a, unstable, y_other, V_other)

Arguments