The implementation employs a method of corresponding states, using the Takahashi [48] approach for binary diffusion coefficients (using mixture averaging rules for the mixture properties), the Lucas method for the viscosity, and a method from Ely and Hanley for the thermal conductivity. More...

#include <HighPressureGasTransport.h>

Inheritance diagram for HighPressureGasTransport:

[legend]

Detailed Description

The implementation employs a method of corresponding states, using the Takahashi [48] approach for binary diffusion coefficients (using mixture averaging rules for the mixture properties), the Lucas method for the viscosity, and a method from Ely and Hanley for the thermal conductivity.

All methods are described in Poling et al. [38] (viscosity in Ch. 9, thermal conductivity in Ch. 10, and diffusion coefficients in Ch. 11).

Definition at line 151 of file HighPressureGasTransport.h.

Public Member Functions
string	transportModel () const override
	Identifies the model represented by this Transport object.

void	init (ThermoPhase *thermo, int mode=0) override
	Initialize a transport manager.

double	thermalConductivity () override
	Returns the mixture high-pressure thermal conductivity [W/m/K] using a method of corresponding states by Ely and Hanley.

double	viscosity () override
	Returns the mixture high-pressure viscosity [Pa·s] using the Lucas method.

Public Member Functions inherited from HighPressureGasTransportBase
void	getBinaryDiffCoeffs (const size_t ld, double *const d) override
	Computes the matrix of binary diffusion coefficients [m²/s] using the Takahashi correction factor.

void	getMixDiffCoeffs (double *const d) override
	Returns the mixture-averaged diffusion coefficients [m²/s].

void	getMixDiffCoeffsMole (double *const d) override
	Returns the mixture-averaged diffusion coefficients [m²/s].

void	getMixDiffCoeffsMass (double *const d) override
	Returns the mixture-averaged diffusion coefficients [m²/s].

virtual void	updateCorrectionFactors ()
	Updates the matrix of species-pair Takahashi correction factors for use in computing the binary diffusion coefficients, see takahashiCorrectionFactor()

Public Member Functions inherited from MixTransport
	MixTransport ()=default
	Default constructor.

string	transportModel () const override
	Identifies the model represented by this Transport object.

void	getThermalDiffCoeffs (double *const dt) override
	Return the thermal diffusion coefficients [kg/m/s].

double	thermalConductivity () override
	Returns the mixture thermal conductivity [W/m/K].

void	getMobilities (double *const mobil) override
	Get the electrical mobilities [m²/V/s].

void	update_T () override
	Update the internal parameters whenever the temperature has changed.

void	update_C () override
	Update the internal parameters whenever the concentrations have changed.

void	getSpeciesFluxes (size_t ndim, const double const grad_T, size_t ldx, const double const grad_X, size_t ldf, double *const fluxes) override
	Get the species diffusive mass fluxes [kg/m²/s] with respect to the mass averaged velocity, given the gradients in mole fraction and temperature.

void	init (ThermoPhase *thermo, int mode=0) override
	Initialize a transport manager.

Public Member Functions inherited from GasTransport
double	viscosity () override
	Get the viscosity [Pa·s] of the mixture.

void	getSpeciesViscosities (double *const visc) override
	Get the pure-species viscosities [Pa·s].

void	getBinaryDiffCoeffs (const size_t ld, double *const d) override
	Returns the matrix of binary diffusion coefficients [m²/s].

void	getMixDiffCoeffs (double *const d) override
	Returns the Mixture-averaged diffusion coefficients [m²/s].

void	getMixDiffCoeffsMole (double *const d) override
	Returns the mixture-averaged diffusion coefficients [m²/s].

void	getMixDiffCoeffsMass (double *const d) override
	Returns the mixture-averaged diffusion coefficients [m²/s].

void	getViscosityPolynomial (size_t i, double *coeffs) const override
	Return the polynomial fits to the viscosity of species `i`.

void	getConductivityPolynomial (size_t i, double *coeffs) const override
	Return the temperature fits of the heat conductivity of species `i`.

void	getBinDiffusivityPolynomial (size_t i, size_t j, double *coeffs) const override
	Return the polynomial fits to the binary diffusivity of species pair (i, j)

void	getCollisionIntegralPolynomial (size_t i, size_t j, double astar_coeffs, double bstar_coeffs, double *cstar_coeffs) const override
	Return the polynomial fits to the collision integral of species pair (i, j)

void	setViscosityPolynomial (size_t i, double *coeffs) override
	Modify the polynomial fits to the viscosity of species `i`

void	setConductivityPolynomial (size_t i, double *coeffs) override
	Modify the temperature fits of the heat conductivity of species `i`

void	setBinDiffusivityPolynomial (size_t i, size_t j, double *coeffs) override
	Modify the polynomial fits to the binary diffusivity of species pair (i, j)

void	setCollisionIntegralPolynomial (size_t i, size_t j, double astar_coeffs, double bstar_coeffs, double *cstar_coeffs, bool actualT) override
	Modify the polynomial fits to the collision integral of species pair (i, j)

void	init (ThermoPhase *thermo, int mode=0) override
	Initialize a transport manager.

bool	CKMode () const override
	Boolean indicating the form of the transport properties polynomial fits.

void	invalidateCache () override
	Invalidate any cached values which are normally updated only when a change in state is detected.

Public Member Functions inherited from Transport
	Transport ()=default
	Constructor.

	Transport (const Transport &)=delete

Transport &	operator= (const Transport &)=delete

virtual string	transportModel () const
	Identifies the model represented by this Transport object.

ThermoPhase &	thermo ()
	Phase object.

void	checkSpeciesIndex (size_t k) const
	Check that the specified species index is in range.

void	checkSpeciesArraySize (size_t kk) const
	Check that an array size is at least m_nsp.

virtual void	getSpeciesFluxes (size_t ndim, const double const grad_T, size_t ldx, const double const grad_X, size_t ldf, double *const fluxes)
	Get the species diffusive mass fluxes [kg/m²/s] with respect to the specified solution averaged velocity, given the mole fraction and temperature gradients.

virtual void	getMolarFluxes (const double const state1, const double const state2, const double delta, double *const cfluxes)
	Get the molar fluxes [kmol/m²/s], given the thermodynamic state at two nearby points.

virtual void	getMassFluxes (const double state1, const double state2, double delta, double *mfluxes)
	Get the mass fluxes [kg/m²/s], given the thermodynamic state at two nearby points.

virtual void	getThermalDiffCoeffs (double *const dt)
	Return a vector of thermal diffusion coefficients [kg/m/s].

virtual void	getBinaryDiffCoeffs (const size_t ld, double *const d)
	Returns the matrix of binary diffusion coefficients [m²/s].

virtual void	getMultiDiffCoeffs (const size_t ld, double *const d)
	Return the multicomponent diffusion coefficients [m²/s].

virtual void	getMixDiffCoeffs (double *const d)
	Return a vector of mixture averaged diffusion coefficients [m²/s].

virtual void	getMixDiffCoeffsMole (double *const d)
	Returns a vector of mixture averaged diffusion coefficients [m²/s].

virtual void	getMixDiffCoeffsMass (double *const d)
	Returns a vector of mixture averaged diffusion coefficients [m²/s].

virtual void	getViscosityPolynomial (size_t i, double *coeffs) const
	Return the polynomial fits to the viscosity of species `i`.

virtual void	getConductivityPolynomial (size_t i, double *coeffs) const
	Return the temperature fits of the heat conductivity of species `i`.

virtual void	getBinDiffusivityPolynomial (size_t i, size_t j, double *coeffs) const
	Return the polynomial fits to the binary diffusivity of species pair (i, j)

virtual void	getCollisionIntegralPolynomial (size_t i, size_t j, double astar_coeffs, double bstar_coeffs, double *cstar_coeffs) const
	Return the polynomial fits to the collision integral of species pair (i, j)

virtual void	setViscosityPolynomial (size_t i, double *coeffs)
	Modify the polynomial fits to the viscosity of species `i`

virtual void	setConductivityPolynomial (size_t i, double *coeffs)
	Modify the temperature fits of the heat conductivity of species `i`

virtual void	setBinDiffusivityPolynomial (size_t i, size_t j, double *coeffs)
	Modify the polynomial fits to the binary diffusivity of species pair (i, j)

virtual void	setCollisionIntegralPolynomial (size_t i, size_t j, double astar_coeffs, double bstar_coeffs, double *cstar_coeffs, bool flag)
	Modify the polynomial fits to the collision integral of species pair (i, j)

AnyMap	parameters () const
	Return the parameters for a phase definition which are needed to reconstruct an identical object using the newTransport() function.

AnyMap	fittingErrors () const
	Get error metrics about any functional fits calculated for pure species transport properties.

virtual void	invalidateCache ()
	Invalidate any cached values which are normally updated only when a change in state is detected.

virtual double	bulkViscosity ()
	The bulk viscosity [Pa·s].

virtual double	electricalConductivity ()
	Get the electrical conductivity [siemens/m].

Protected Member Functions
	HighPressureGasTransport ()=default
	default constructor

double	elyHanleyDilutePureSpeciesViscosity (double V, double Tc, double Vc, double Zc, double acentric_factor, double mw)
	Get the viscosity [Pa·s] of a pure species using the method of Ely and Hanley.

double	thetaShapeFactor (double Tr, double Vr, double acentric_factor)
	Returns the theta shape factor of Leach and Leland for a pure species.

double	phiShapeFactor (double Tr, double Vr, double Zc, double acentric_factor)
	Returns the phi shape factor of Leach and Leland for a pure species.

double	elyHanleyDiluteReferenceViscosity (double T0)
	Returns the viscosity [Pa·s] for the reference fluid (methane) for low pressures.

double	elyHanleyReferenceThermalConductivity (double rho0, double T0)
	Returns the thermal conductivity [W/m/K] of the reference fluid of methane.

void	computeMixtureParameters ()
	Computes the composition-dependent values of parameters that are needed for the Lucas viscosity model.

double	lowPressureNondimensionalViscosity (double Tr, double FP, double FQ)
	Returns the non-dimensional low-pressure mixture viscosity in using the Lucas method.

double	highPressureNondimensionalViscosity (double Tr, double Pr, double FP_low, double FQ_low, double P_vap, double P_crit)
	Returns the non-dimensional high-pressure mixture viscosity in using the Lucas method.

double	quantumCorrectionFactor (double Q, double Tr, double MW)
	Calculates quantum correction term of the Lucas method for a species based on the reduced temperature(Tr) and molecular weight(MW), used in viscosity calculation.

double	polarityCorrectionFactor (double mu_r, double Tr, double Z_c)
	Returns the polarity correction term for a species based on reduced temperature, reduced dipole moment, and critical compressibility.

Protected Member Functions inherited from HighPressureGasTransportBase
	HighPressureGasTransportBase ()=default
	default constructor

void	getTransportData () override
	Obtain required parameters from the 'critical-parameters' species input section, and checks the critical-properties.yaml file if an acentric factor is not specified.

void	initializeCriticalProperties ()
	Computes and stores the estimate of the critical properties for each species.

double	Tcrit_i (size_t i)
	Returns the stored value of the critical temperature for species 'i'.

double	Pcrit_i (size_t i)
	Returns the stored value of the critical pressure for species 'i'.

double	Vcrit_i (size_t i)
	Returns the stored value of the critical volume for species 'i'.

double	Zcrit_i (size_t i)
	Returns the stored value of the critical compressibility for species 'i'.

Protected Member Functions inherited from MixTransport
void	updateCond_T ()
	Update the temperature dependent parts of the species thermal conductivities.

Protected Member Functions inherited from GasTransport
virtual void	update_T ()

virtual void	update_C ()=0

virtual void	updateViscosity_T ()
	Update the temperature-dependent viscosity terms.

virtual void	updateSpeciesViscosities ()
	Update the pure-species viscosities.

virtual void	updateDiff_T ()
	Update the binary diffusion coefficients.

virtual void	setupCollisionParameters ()
	Setup parameters for a new kinetic-theory-based transport manager for low-density gases.

void	setupCollisionIntegral ()
	Setup range for polynomial fits to collision integrals of Monchick & Mason [31].

void	makePolarCorrections (size_t i, size_t j, double &f_eps, double &f_sigma)
	Corrections for polar-nonpolar binary diffusion coefficients.

void	fitCollisionIntegrals (MMCollisionInt &integrals)
	Generate polynomial fits to collision integrals.

virtual void	fitProperties (MMCollisionInt &integrals)
	Generate polynomial fits to the viscosity $ \eta $ and conductivity $ \lambda $.

virtual void	fitDiffCoeffs (MMCollisionInt &integrals)
	Generate polynomial fits to the binary diffusion coefficients.

void	getBinDiffCorrection (double t, MMCollisionInt &integrals, size_t k, size_t j, double xk, double xj, double &fkj, double &fjk)
	Second-order correction to the binary diffusion coefficients.

Private Attributes
Reference fluid properties
These are the properties of the reference fluid, which is methane in this case. These are used by the thermalConductivity() method.
const double	m_ref_MW = 16.04
	Molecular weight [kg/kmol].

const double	m_ref_Tc = 190.4
	Critical temperature [K].

const double	m_ref_Vc = 0.0986
	Critical volume [m^3/kmol].

const double	m_ref_Zc = 0.288
	Critical compressibility [unitless].

const double	m_ref_rhoc = 0.1628
	Critical density [g/cm^3].

const double	m_ref_acentric_factor = 0.011
	Acentric factor [unitless].

Lucas method viscosity parameters
These are the parameters that are needed to calculate the viscosity using the Lucas method.
double	m_FQ_mix_o
	Quantum correction factor.

double	m_FP_mix_o
	Polarity correction factor.

double	m_Tr_mix
	Reduced temperature.

double	m_Pr_mix
	Reduced pressure.

double	m_Pc_mix
	Critical pressure.

double	m_Tc_mix
	Critical temperature.

double	m_MW_mix
	Molecular weight.

double	m_P_vap_mix
	Vapor pressure.

Friends
class	TransportFactory

Additional Inherited Members
Protected Attributes inherited from HighPressureGasTransportBase
vector< double >	m_Tcrit
	Critical temperature [K] of each species.

vector< double >	m_Pcrit
	Critical pressure [Pa] of each species.

vector< double >	m_Vcrit
	Critical volume [m³/kmol] of each species.

vector< double >	m_Zcrit
	Critical compressibility [unitless] of each species.

DenseMatrix	m_P_corr_ij
	Matrix of Takahashi binary diffusion coefficient corrections.

Protected Attributes inherited from MixTransport
vector< double >	m_cond
	vector of species thermal conductivities [W/m/K]

double	m_lambda = 0.0
	Internal storage for the calculated mixture thermal conductivity [W/m/K].

bool	m_spcond_ok = false
	Update boolean for the species thermal conductivities.

bool	m_condmix_ok = false
	Update boolean for the mixture rule for the mixture thermal conductivity.

Protected Attributes inherited from GasTransport
vector< double >	m_molefracs
	Vector of species mole fractions.

double	m_viscmix = 0.0
	Internal storage for the viscosity of the mixture [Pa·s].

bool	m_visc_ok = false
	Update boolean for mixture rule for the mixture viscosity.

bool	m_viscwt_ok = false
	Update boolean for the weighting factors for the mixture viscosity.

bool	m_spvisc_ok = false
	Update boolean for the species viscosities.

bool	m_bindiff_ok = false
	Update boolean for the binary diffusivities at unit pressure.

int	m_mode = 0
	Type of the polynomial fits to temperature.

DenseMatrix	m_phi
	Viscosity weighting function. size = m_nsp * m_nsp.

vector< double >	m_spwork
	work space length = m_nsp

vector< double >	m_visc
	vector of species viscosities [Pa·s].

vector< vector< double > >	m_visccoeffs
	Polynomial fits to the viscosity of each species.

vector< double >	m_mw
	Local copy of the species molecular weights.

DenseMatrix	m_wratjk
	Holds square roots of molecular weight ratios.

DenseMatrix	m_wratkj1
	Holds square roots of molecular weight ratios.

vector< double >	m_sqvisc
	vector of square root of species viscosities.

vector< double >	m_polytempvec
	Powers of the ln temperature, up to fourth order.

double	m_temp = -1.0
	Current value of the temperature [K] at which the properties in this object are calculated.

double	m_kbt = 0.0
	Current value of Boltzmann constant times the temperature [J].

double	m_sqrt_t = 0.0
	current value of temperature to 1/2 power

double	m_logt = 0.0
	Current value of the log of the temperature.

double	m_t14 = 0.0
	Current value of temperature to 1/4 power.

vector< vector< double > >	m_diffcoeffs
	Polynomial fits to the binary diffusivity of each species.

DenseMatrix	m_bdiff
	Matrix of binary diffusion coefficients at the reference pressure and the current temperature Size is m_nsp x m_nsp.

vector< vector< double > >	m_condcoeffs
	temperature fits of the heat conduction

vector< vector< int > >	m_poly
	Indices for the (i,j) interaction in collision integral fits.

vector< vector< double > >	m_omega22_poly
	Fit for omega22 collision integral.

vector< vector< int > >	m_star_poly_uses_actualT
	Flag to indicate for which (i,j) interaction pairs the actual temperature is used instead of the reduced temperature.

vector< vector< double > >	m_astar_poly
	Fit for astar collision integral.

vector< vector< double > >	m_bstar_poly
	Fit for bstar collision integral.

vector< vector< double > >	m_cstar_poly
	Fit for cstar collision integral.

vector< double >	m_zrot
	Rotational relaxation number for each species.

vector< double >	m_crot
	Dimensionless rotational heat capacity of each species.

vector< bool >	m_polar
	Vector of booleans indicating whether a species is a polar molecule.

vector< double >	m_alpha
	Polarizability [m³] of each species in the phase.

vector< double >	m_eps
	Lennard-Jones well-depth [J] of the species in the current phase.

vector< double >	m_sigma
	Lennard-Jones diameter [m] of the species in the current phase.

DenseMatrix	m_reducedMass
	This is the reduced mass [kg] of the interaction between species i and j.

DenseMatrix	m_diam
	hard-sphere diameter [m] for (i,j) collision

DenseMatrix	m_epsilon
	The effective well depth [J] for (i,j) collisions.

DenseMatrix	m_dipole
	The effective dipole moment [Coulomb·m] for (i,j) collisions.

DenseMatrix	m_delta
	Reduced dipole moment of the interaction between two species.

vector< double >	m_w_ac
	Pitzer acentric factor [dimensionless].

vector< double >	m_disp
	Dispersion coefficient normalized by the square of the elementary charge [m⁵].

vector< double >	m_quad_polar
	Quadrupole polarizability.

Protected Attributes inherited from Transport
ThermoPhase *	m_thermo
	pointer to the object representing the phase

size_t	m_nsp = 0
	Number of species in the phase.

AnyMap	m_fittingErrors
	Maximum errors associated with fitting pure species transport properties.

Constructor & Destructor Documentation

◆ HighPressureGasTransport()

HighPressureGasTransport ( )

protecteddefault

default constructor

Member Function Documentation

◆ transportModel()

string transportModel ( ) const

inlineoverridevirtual

Identifies the model represented by this Transport object.

Each derived class should override this method to return a meaningful identifier.

Since: New in Cantera 3.0. The name returned by this method corresponds to the canonical name used in the YAML input format.

Reimplemented from Transport.

Definition at line 158 of file HighPressureGasTransport.h.

◆ init()

void init	(	ThermoPhase *	thermo,
		int	mode = `0`
	)

overridevirtual

Initialize a transport manager.

This routine sets up a transport manager. It calculates the collision integrals and populates species-dependent data structures.

Parameters

thermo	Pointer to the ThermoPhase object
mode	Chemkin compatible mode or not. This alters the specification of the collision integrals. defaults to no.

Reimplemented from GasTransport.

Definition at line 343 of file HighPressureGasTransport.cpp.

◆ thermalConductivity()

double thermalConductivity ( )

overridevirtual

Returns the mixture high-pressure thermal conductivity [W/m/K] using a method of corresponding states by Ely and Hanley.

This method for computing the thermal conductivity is described in [9] . There are references to earlier work in [8] in the description of the model for thermal conductivity. The equations referenced in this description primarily are from [9] , and equations that come from [8] are marked as such. The equations referenced here are the same as the ones from the papers from Ely and Hanley.

The thermal conductivity is assumed to have two components: a translational/collisional part and an internal part that is related to the transfer of energy to internal degrees of freedom.

\[ \lambda_{mix}(\rho, T) = \lambda^{''}_{mix}(T) + \lambda^{'}_{mix}(\rho, T) \]

The first term on the right-hand side is the internal part and the second term is the translational part. The internal part is only a function of temperature, and the translational part is a function of both temperature and density.

The internal component of the thermal conductivity is defined by Equation 2 in [9] :

\[ \lambda^{''}_{mix}(T) = \sum_i \sum_j X_i X_j \lambda^{''}_{ij}(T) \]

The mixing rule for combining pure-species internal thermal conductivity components is given by Equation 3 in [9] :

\[ (\lambda^{''}_{ij})^{-1} = 2[(\lambda^{''}_{i})^{-1} + (\lambda^{''}_{j})^{-1}] \]

The pure species internal thermal conductivity is given by Equation 1 in [9] :

\[ \lambda^{''}_{i} = \frac{\eta_i^*}{MW_i}f_{int} \left(C_{p,i}^0 - 2.5R \right) \]

Where $ \eta_i^* $ is the referred to as the dilute gas viscosity and is the component that is temperature dependent described in [8] (see elyHanleyDilutePureSpeciesViscosity() ), $ MW_i $ is the molecular weight of species i, $ f_{int} $ is a constant and is 1.32, $ C_{p,i}^0 $ is the ideal gas heat capacity of species i, and R is the gas constant (J/kmol/K).

For the translational component of the thermal conductivity, Equation 5 in [9] is used:

\[ \lambda^{'}_{mix}(\rho, T) = \lambda^{'}_{0}(\rho_0, T_0) F_{\lambda} \]

Where $ \lambda^{'}_{0}(\rho_0, T_0) $ is the internal component of the thermal conductivity of a reference fluid that is at a reference temperature and density, $ T_0 $ and $ \rho_0 $. These reference conditions are not the same as the conditions that the mixture is at ( $ T $ and $ \rho $). The subscript 0 refers to the reference fluid. The term $ F_{\lambda} $ is defined by Equation 6 in [9] :

\[ F_{\lambda} = \left( \frac{MW_0}{MW_m} \right)^{1/2} f_{m,0}^{1/2} h_{m,0}^{-2/3} \]

Where $ MW_0 $ is the molecular weight of the reference fluid, $ MW_m $ is the molecular weight of the mixture.

Note: : $ MW_m $ is an effective mixture molecular weight and should not be confused with a mole-weighted average of the molecular weights of the species in the mixture.

The terms $ f_{m,0} $ and $ h_{m,0} $ are called reducing ratios. The $ h_{m,0} $ quantity is defined by Equation 9 in [9] :

\[ h_{m,0} = \sum_i \sum_j X_i X_j h_{ij} \]

with $ h_{ij} $ defined by Equation 11 in [9] :

\[ h_{ij} = \frac{1}{8} \left( h_i^{1/3} + h_j^{1/3} \right)^3 (1 - l_{ij}) \]

The variable $ l_{ij} $ is a binary interaction parameter and is assumed to be zero as is done by Ely and Hanley. The pure species reducing ratio $ h_i $ is defined by Equation 13 in [9] :

\[ h_i = \frac{V_c^i}{V_c^0} \phi_i(T_R^i, V_R^i, \omega_i) \]

Where $ \phi_i $ is a shape factor of Leach and Leland (see phiShapeFactor()), $ T_R^i $ is the reduced temperature of species i, $ V_R^i $ is the reduced volume of species i, and $ \omega_i $ is the acentric factor of species i.

At this point, the value of $ h_{m,0} $ can be computed. This leaves the other reducing ratio, $ f_{m,0} $, to be computed. The value of that reducing ratio is defined by Equation 8 in [9] :

\[ f_{m,0} = \frac{1}{h_{m,0}} \sum_i \sum_j X_i X_j f_{ij} h_{ij} \]

The value of $ h_{ij} $ is the same as the one defined earlier. The combining rule for $ f_{ij} $ is given by Equation 10 in [9] :

\[ f_{ij} = (f_i f_j)^{1/2} (1-\kappa_{ij}) \]

$ \kappa_{ij} $ is binary interaction parameters and is assumed to be zero as was done in the work of Ely and Hanley. The species-specific value of $ f_i $ is defined by Equation 12 in [9] :

\[ f_i = \frac{T_c^i}{T_c^0} \theta_i(T_R^i, V_R^i, \omega_i) \]

Where $ \theta_i $ is a shape factor of Leach and Leland (see thetaShapeFactor()), $ T_c^i $ is the critical temperature of species i, and $ T_c^0 $ is the critical temperature of the reference fluid.

The value of $ h_{m,0} $ can be computed from the equation that defined the value of $ f_{m,0} h_{m,0} $. The value of $ h_{m,0} $ must be computed first and then the value of $ f_{m,0} $ can be computed.

The expression for $ MW_m $ is somewhat complex, but keep in mind that all terms except $ MW_m $ are known at this point and so simple algebra can be used to isolate the molecular weight of the mixture. The expression for the mixture molecular weight is given by Equation 14 in [9] :

\[ MW_m^{1/2} f_{m,0}^{1/2} h_{m,0}^{-4/3} = \sum_i \sum_j X_i X_j MW_{ij}^{-1/2} f_{ij}^{1/2} h_{ij}^{-4/3} \]

where the mixing rule for the molecular weight of the mixture is given by Equation 15 in [9] :

\[ MW_{ij} = \frac{1}{2} (MW_i^{-1} + MW_j^{-1}) \]

For clarity, if we assume the right-hand-side of the molecular weight mixture equation is A, then the mixture molecular weight is given by:

\[ MW_m = A^{-2} f_{m,0} h_{m,0}^{-8/3} \]

All of the above descriptions allow for the calculation of $ F_{\lambda} $ in the expression for the mixture value of the translational component of the thermal conductivity. The reference fluid value of the translational component of the thermal conductivity ( $ \lambda^{'}_{0}(\rho_0, T_0) $ ) is still needed, which requires the the temperature and density of the reference fluid.

The reference fluid density is computed using Equation 7 in [9] :

\[ \rho_0 = \rho h_{m,0} \]

The reference fluid temperature is computed using Equation 7 in [9] :

\[ T_0 = \frac{T}{f_{m,0}} \]

The reference fluid translational component of thermal conductivity can now be computed using Equation 18 in [9]. See elyHanleyReferenceThermalConductivity().

Note: The correlations for the reference fluid viscosity return a value of micro-grams/cm/s and the parts of the thermal conductivity that utilize the correlation fit parameters give values in units of mW/m/K. Appropriate conversions were made in the relevant functions: elyHanleyDiluteReferenceViscosity() and elyHanleyReferenceThermalConductivity()

Reimplemented from Transport.

Definition at line 350 of file HighPressureGasTransport.cpp.

◆ viscosity()

double viscosity ( )

overridevirtual

Returns the mixture high-pressure viscosity [Pa·s] using the Lucas method.

This uses the approach described in chapter 9-7. In this method, the mixture viscosity at high pressure is computed using the pure-fluid high pressure relation of Lucas with the difference being that mixture values of the model parameters are used. These mixture values are computed using the mixing rules described in see computeMixtureParameters() .

The mixture pseudo-critical temperature and pressure are calculated using Equations 9-5.18 and 9-5.19. The mixture molecular weight is computed using Equation 9-5.20. The mixture values of the low-pressure polarity and quantum correction factors are computed using Equations 9-5.21 and 9-5.22.

Reimplemented from GasTransport.

Definition at line 555 of file HighPressureGasTransport.cpp.

◆ elyHanleyDilutePureSpeciesViscosity()

double elyHanleyDilutePureSpeciesViscosity	(	double	V,
		double	Tc,
		double	Vc,
		double	Zc,
		double	acentric_factor,
		double	mw
	)

protected

Get the viscosity [Pa·s] of a pure species using the method of Ely and Hanley.

Uses the method outlined in [8] to compute the viscosity of a pure species.

The primary equation is given by Equation 1 in [8] :

\[ \eta_i(\rho,T) = \eta_0(\rho_0, T_0) F_{\eta} \]

Where $ F_{\eta} $ is defined by Equation 2 in [8] :

\[ F_{\eta} = \left( \frac{MW_i}{MW_0} \right)^{1/2} f_{i,0}^{1/2} h_{i,0}^{-2/3} \]

The 0 subscripts refer to the reference fluid, which is methane in this case, and the i subscripts refer to the species of interest. No mixing rules need to be used here because this is for a pure species. As such, the terms $ f_{i,0} $ and $ h_{i,0} $ have simpler expressions. The value of $ f_{i,0} $ is given by Equation 7 in [8] :

\[ f_{i,0} = \frac{T_c^i}{T_c^0} \theta_i(T_R^i, V_R^i, \omega_i) \]

and the value of $ h_{i,0} $ is given by Equation 8 in [8] :

\[ h_{i,0} = \frac{V_c^i}{V_c^0} \phi_i(T_R^i, V_R^i, Z_{c,i}, \omega_i) \]

Where $ \theta_i $ and $ \phi_i $ are shape factors of Leach and Leland ( see thetaShapeFactor() and phiShapeFactor() ), $ T_c^i $ is the critical temperature of species i, $ T_c^0 $ is the critical temperature of the reference fluid, $ V_c^i $ is the critical volume of species i, $ V_c^0 $ is the critical volume of the reference fluid, $ Z_{c,i} $ is the critical compressibility of species i, and $ \omega_i $ is the acentric factor of species i.

Parameters

V	Molar volume of the species
Tc	Critical temperature of the species
Vc	Critical volume of the species
Zc	Critical compressibility of the species
acentric_factor	Acentric factor of the species
mw	Molecular weight of the species

Definition at line 456 of file HighPressureGasTransport.cpp.

◆ thetaShapeFactor()

double thetaShapeFactor	(	double	Tr,
		double	Vr,
		double	acentric_factor
	)

protected

Returns the theta shape factor of Leach and Leland for a pure species.

The shape factor $ \theta_i $ is defined by Equation 11 in [8] as follows:

\[ \theta_i(T_R^i, V_R^i, \omega_i) = 1 + (\omega_i - \omega_0)F(T_R^i, V_R^i)] \]

Where $ \omega_i $ is the acentric factor of species i, $ \omega_0 $ is the acentric factor of the reference fluid, and $ F(T_R^i, V_R^i) $ is a function of the reduced temperature and reduced volume of species i. The function $ F(T_R^i, V_R^i) $ is defined by Equation 13 in [8] :

\[ F(T_R^i, V_R^i) = a_1 + b_1 ln(T_+^i) + (c_1 + d_1/T_+^i) (V_+^i - 0.5) \]

Where $ T_+^i $ and $ V_+^i $ are limited values of the reduced temperature and pressure. The limited temperature is defined by Equation 15 in [8] :

\[ T_+^i = \text{min}(2, \text{max}(T_R^i, 0.5)) \]

and the limited pressure is defined by Equation 16 in [8] :

\[ V_i^+ = \text{min}(2, \text{max}(V_R^i, 0.5)) \]

The values of $ a_1 $, $ b_1 $, $ c_1 $, and $ d_1 $ are given in Table II of [8]. They are:

\[ a_1 = 0.090569, b_1 = -0.862762, c_1 = 0.316636, d_1 = -0.465684 \]

Parameters

Tr	Reduced temperature
Vr	Reduced volume
acentric_factor	Acentric factor

Definition at line 478 of file HighPressureGasTransport.cpp.

◆ phiShapeFactor()

double phiShapeFactor	(	double	Tr,
		double	Vr,
		double	Zc,
		double	acentric_factor
	)

protected

Returns the phi shape factor of Leach and Leland for a pure species.

The shape factor $ \phi_i $ is defined by Equation 12 in [8] as follows:

\[ \phi_i(T_R^i, V_R^i, \omega_i, Z_c^i) = \frac{Z_c^0}{Z_c^i} [1 + (\omega_i - \omega_0)G(T_R^i, V_R^i)] \]

Where $ Z_c^0 $ is the critical compressibility of the reference fluid, $ Z_c^i $ is the critical compressibility of species i, $ \omega_0 $ is the acentric factor of the reference fluid, and $ G(T_R^i, V_R^i) $ is a function of the reduced temperature and reduced volume of species i. The function $ G(T_R^i, V_R^i) $ is defined by Equation 14 in [8] :

\[ G(T_R^i, V_R^i) = a_2(V_+^i + b_2) + c_2(V_+^i + d_2)ln(T_+^i) \]

Where $ T_+^i $ and $ V_+^i $ are limited values of the reduced temperature and pressure. The limited temperature is defined by Equation 15 in [8] :

\[ T_+^i = \text{min}(2, \text{max}(T_R^i, 0.5)) \]

and the limited pressure is defined by Equation 16 in [8] :

\[ V_i^+ = \text{min}(2, \text{max}(V_R^i, 0.5)) \]

The values of $ a_2 $, $ b_2 $, $ c_2 $, and $ d_2 $ are given in Table II of [8]. They are:

\[ a_2 = 0.394901, b_2 = -1.023545, c_2 = -0.932813, d_2 = -0.754639 \]

Parameters

Tr	Reduced temperature
Vr	Reduced volume
Zc	Critical compressibility
acentric_factor	Acentric factor

Definition at line 489 of file HighPressureGasTransport.cpp.

◆ elyHanleyDiluteReferenceViscosity()

double elyHanleyDiluteReferenceViscosity ( double T0 )

protected

Returns the viscosity [Pa·s] for the reference fluid (methane) for low pressures.

Computes the temperature dependent (referred to as the dilute viscosity in the reference) component only (eta_0) from the expression in Table III in [8] . Prevents inputs larger than 10,000 Kelvin by just returning the value at 10,000 Kelvin.

Parameters

T0	Temperature of the reference fluid

Definition at line 500 of file HighPressureGasTransport.cpp.

◆ elyHanleyReferenceThermalConductivity()

double elyHanleyReferenceThermalConductivity	(	double	rho0,
		double	T0
	)

protected

Returns the thermal conductivity [W/m/K] of the reference fluid of methane.

Computes the temperature and density dependent reference fluid thermal conductivity from the expression in Table I in [9] .

The reference fluid translational component of thermal conductivity is computed using Equation 18 in [9] :

\[ \lambda^{'}_{0}(\rho_0, T_0) = \frac{15R}{4MW_0} \eta_0^* + \lambda_0^{(1)}\rho_0 + \Delta\lambda_0(\rho_0, T_0) \]

Where $ \eta_0^* $ is the dilute reference gas viscosity, $ \lambda_0^{(1)} $ is the first density correction to the thermal conductivity, and $ \Delta\lambda_0 $ is the high-density contribution. Ely and Hanley provide a correlation equation to solve for this quantity for the methane reference fluid. The details of the correlation and the parameters are shown in Table I of [9].

For completeness, the relations are reproduced here.

\[ \lambda_0^*(T_0) = \frac{15R}{4MW_0} \eta_0^* \]

\[ \eta_0^* = \sum_{n=1}^{9} C_n T_0^{(n-4)/3} \]

\[ \lambda_0^{(1)} = b_1 + b_2[b_3 - ln(T_0 / b_4)]^2 \]

\[ \Delta\lambda_0 = \text{exp}[a_1 + a_2/T_0] (exp[(a_3 + a_4/T_0^{3/4})\rho_0^{0.1} + (\rho_0 / \rho_0,c - 1)\rho_0^{0.5} (a_5 + a_6/T_0 + a_7/T_0^2) ] - 1) \]

The constants a, b, and C are not related to any ones used in the previous equations, their values are defined in Table I of [9].

Parameters

rho0	Density of the reference fluid
T0	Temperature of the reference fluid

Definition at line 521 of file HighPressureGasTransport.cpp.

◆ computeMixtureParameters()

void computeMixtureParameters ( )

protected

Computes the composition-dependent values of parameters that are needed for the Lucas viscosity model.

The equations for the mixing rules defined on page 9.23 of [38] for the Lucas method's composition dependent parameters. The primary mixing rules are defined below, and the reduced properties are just the properties divided by the pseudo-critical mixture properties defined below.

Note: Equation numbers are from [38]

\[ T_{\text{c,m}} = \sum_i X_i T_{\text{c,i}} \quad \text{( Equation 9-5.18)} \]

Where $ T_{\text{c,i}} $ is the critical temperature of species i, and $ X_i $ is the mole fraction of species i.

\[ P_{\text{c,m}} = R T_{\text{c,m}} \frac{\sum_i X_i Z_{\text{c,i}}}{\sum_i X_i V_{\text{c,i}}} \quad \text{(Equation 9-5.19)} \]

Where $ Z_{\text{c,i}} $ is the critical compressibility of species i, and $ V_{\text{c,i}} $ is the critical volume of species i.

\[ M_m = \sum X_i M_i \quad \text{(Equation 9-5.20)} \]

Where $ M_i $ is the molecular weight of species i.

\[ F_{P,m}^{\text{o}} = \sum X_i F_{P,i}^{\text{o}} \quad \text{(Equation 9-5.21)} \]

Where $ F_{P,i}^{\text{o}} $ is the low-pressure polarity correction factor of species i from equation 9-4.18.

\[ F_{Q,m}^{\text{o}} = \left ( \sum X_i F_{Q,i}^{\text{o}} \right ) A \quad \text{(Equation 9-5.22)} \]

Where $ F_{Q,i}^{\text{o}} $ is the low-pressure quantum correction factor of species i from equation 9-4.19, and A is defined below.

\[ A = 1 - 0.01 \left ( \frac{M_H}{M_L} \right )^{0.87} \quad \text{(Equation 9-5.23)} \]

For $ \frac{M_H}{M_L} > 9 $ and $ 0.05 < X_H < 0.7 $, otherwise A = 1. In the above equation, $M_H$ and $M_L$ are the molecular weights of the heaviest and lightest components in the mixture, and $ X_H $ is the mole fraction of the heaviest component.

While it isn't returned as a parameter, the species-specific reduced dipole moment ( $ \mu_r $) is used to compute the mixture polarity correction factor. It is defined as:

\[ \mu_r = 52.46 \frac{\mu^2 P_{\text{c,i}}}{T_{\text{c,i}}} \quad \text{(Equation 9-4.17)} \]

Definition at line 573 of file HighPressureGasTransport.cpp.

◆ lowPressureNondimensionalViscosity()

double lowPressureNondimensionalViscosity	(	double	Tr,
		double	FP,
		double	FQ
	)

protected

Returns the non-dimensional low-pressure mixture viscosity in using the Lucas method.

\[ \eta \xi = F_P^{\text{o}} F_Q^{\text{o}} [0.807 T_r^{0.618} - 0.357 e^{-0.449 T_r} + 0.340e^{-4.058 T_r} + 0.018] \quad \text{(Equation 9-4.16)} \]

This function is structured such that it can be used for pure species or mixtures, with the only difference being the values that are passed to the function (pure values versus mixture values).

For the definition of the mixture rules, see computeMixtureParameters() .

Parameters

Tr	Reduced temperature [unitless]
FP	Polarity correction factor [unitless]
FQ	Quantum correction factor [unitless]

Definition at line 675 of file HighPressureGasTransport.cpp.

◆ highPressureNondimensionalViscosity()

double highPressureNondimensionalViscosity	(	double	Tr,
		double	Pr,
		double	FP_low,
		double	FQ_low,
		double	P_vap,
		double	P_crit
	)

protected

Returns the non-dimensional high-pressure mixture viscosity in using the Lucas method.

\[ \eta \xi = Z_2 F_P F_Q \quad \text{(Equation 9-6.12)} \]

This returns the value of η*ξ (by multiplying both sides of 9-6.12 by ξ and returning the right-side of the resulting equation).

This function is structured such that it can be used for pure species or mixtures, with the only difference being the values that are passed to the function (pure values versus mixture values).

For the definition of the mixture rules, see computeMixtureParameters() .

Parameters

Tr	Reduced temperature [unitless]
Pr	Reduced pressure [unitless]
FP_low	Low-pressure polarity correction factor [unitless]
FQ_low	Low-pressure quantum correction factor [unitless]
P_vap	Vapor pressure [Pa]
P_crit	Critical pressure [Pa]

Definition at line 683 of file HighPressureGasTransport.cpp.

◆ quantumCorrectionFactor()

double quantumCorrectionFactor	(	double	Q,
		double	Tr,
		double	MW
	)

protected

Calculates quantum correction term of the Lucas method for a species based on the reduced temperature(Tr) and molecular weight(MW), used in viscosity calculation.

\[ F_{Q}^{\text{o}} = 1.22 Q^{0.15} {1 + 0.00385[ (T_r - 12)^2 ]^{\frac{1}{MW}} \text{sign} (T_r - 12 )} \quad \text{(Equation 9-4.19)} \]

Parameters

Q	Species-specific constant
Tr	Reduced temperature [unitless]
MW	Molecular weight [kg/kmol]

Definition at line 753 of file HighPressureGasTransport.cpp.

◆ polarityCorrectionFactor()

double polarityCorrectionFactor	(	double	mu_r,
		double	Tr,
		double	Z_c
	)

protected

Returns the polarity correction term for a species based on reduced temperature, reduced dipole moment, and critical compressibility.

Used in the calculation of viscosity.

Calculates polarity correction term of the Lucas method for a species based on the reduced temperature(Tr) and molecular weight(MW). Equation 9.4.18.

\[ \begin{equation} F_P^0 = \begin{cases} 1 & 0 \leq \mu_r < 0.022 \\ 1 + 30.55(0.292 - Z_c)^{1.72} & 0.022 \leq \mu_r < 0.075 \\ 1 + 30.55(0.292 - Z_c)^{1.72} \times 0.96 + 0.1(T_r - 0.7) & 0.075 \leq \mu_r \end{cases} \end{equation} \]

Note: The original description in Poling(2001) neglects to mention what happens when the quantity raised to the 1.72 power goes negative. That is an undefined operation that generates real and imaginary numbers. For now, only positive values are allowed.

Parameters

mu_r	Species Reduced dipole moment
Tr	Reduced temperature
Z_c	Species Critical compressibility

Definition at line 760 of file HighPressureGasTransport.cpp.

Friends And Related Symbol Documentation

◆ TransportFactory

friend class TransportFactory

friend

Definition at line 376 of file HighPressureGasTransport.h.

Member Data Documentation

◆ m_ref_MW

const double m_ref_MW = 16.04

private

Molecular weight [kg/kmol].

Definition at line 784 of file HighPressureGasTransport.h.

◆ m_ref_Tc

const double m_ref_Tc = 190.4

private

Critical temperature [K].

Definition at line 785 of file HighPressureGasTransport.h.

◆ m_ref_Vc

const double m_ref_Vc = 0.0986

private

Critical volume [m^3/kmol].

Definition at line 786 of file HighPressureGasTransport.h.

◆ m_ref_Zc

const double m_ref_Zc = 0.288

private

Critical compressibility [unitless].

Definition at line 787 of file HighPressureGasTransport.h.

◆ m_ref_rhoc

const double m_ref_rhoc = 0.1628

private

Critical density [g/cm^3].

Definition at line 788 of file HighPressureGasTransport.h.

◆ m_ref_acentric_factor

const double m_ref_acentric_factor = 0.011

private

Acentric factor [unitless].

Definition at line 789 of file HighPressureGasTransport.h.

◆ m_FQ_mix_o

double m_FQ_mix_o

private

Quantum correction factor.

Definition at line 798 of file HighPressureGasTransport.h.

◆ m_FP_mix_o

double m_FP_mix_o

private

Polarity correction factor.

Definition at line 799 of file HighPressureGasTransport.h.

◆ m_Tr_mix

double m_Tr_mix

private

Reduced temperature.

Definition at line 800 of file HighPressureGasTransport.h.

◆ m_Pr_mix

double m_Pr_mix

private

Reduced pressure.

Definition at line 801 of file HighPressureGasTransport.h.

◆ m_Pc_mix

double m_Pc_mix

private

Critical pressure.

Definition at line 802 of file HighPressureGasTransport.h.

◆ m_Tc_mix

double m_Tc_mix

private

Critical temperature.

Definition at line 803 of file HighPressureGasTransport.h.

◆ m_MW_mix

double m_MW_mix

private

Molecular weight.

Definition at line 804 of file HighPressureGasTransport.h.

◆ m_P_vap_mix

double m_P_vap_mix

private

Vapor pressure.

Definition at line 805 of file HighPressureGasTransport.h.

The documentation for this class was generated from the following files:

Detailed Description

Public Member Functions

Protected Member Functions

Private Attributes

Friends

Additional Inherited Members

Constructor & Destructor Documentation

◆ HighPressureGasTransport()

Member Function Documentation

◆ transportModel()

◆ init()

◆ thermalConductivity()

◆ viscosity()

◆ elyHanleyDilutePureSpeciesViscosity()

◆ thetaShapeFactor()

◆ phiShapeFactor()

◆ elyHanleyDiluteReferenceViscosity()

◆ elyHanleyReferenceThermalConductivity()

◆ computeMixtureParameters()

◆ lowPressureNondimensionalViscosity()

◆ highPressureNondimensionalViscosity()

◆ quantumCorrectionFactor()

◆ polarityCorrectionFactor()

Friends And Related Symbol Documentation

◆ TransportFactory

Member Data Documentation

◆ m_ref_MW

◆ m_ref_Tc

◆ m_ref_Vc

◆ m_ref_Zc

◆ m_ref_rhoc

◆ m_ref_acentric_factor

◆ m_FQ_mix_o

◆ m_FP_mix_o

◆ m_Tr_mix

◆ m_Pr_mix

◆ m_Pc_mix

◆ m_Tc_mix

◆ m_MW_mix

◆ m_P_vap_mix