# Pure Component Helmholtz EoS¶

The Helmholtz Equation of State (EoS) classes serve as a common core for pure component property packages where very accurate and thermodynamically consistent pure component properties are required. This contains general information. Thermodynamic properties for all Helmholtz EoS packages are calculated by the core class only the parameters differ between specific component implementation. Specific implementations may also contain additional properties such as viscosity and thermal conductivity. For specific property packages details see the pages below.

The basic Helmholtz EoS is described “Revised Release on the IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use.”. The Helmholtz EoS as used in the IAPWS-95 contains non-analytic terms to improve accuracy near the critical point. These terms, however cause a singularity at the critical point and can causes computational difficulty, so the non-analytic where omitted in the IDAES implementation.

The IDAES implementation is of the Helmholtz EoS makes use of external function for many of the properties. Solving the VLE and changing state variables require solution of non-linear equations with multiple solutions, so solving then externally provides a method of decomposition where it can be guaranteed that the nonlinear equations associated with the Helmholtz EoS are solved correctly. The external functions provide first and second derivatives, and are compatible with advanced optimization solvers. Phase change does cause problems to be non-smooth, but, as a practical matter, problem using the IDAES implementation of the Helmholtz EoS, still seem to solve well even with phase change.

## Units¶

The iapws95 property module uses SI units (m, kg, s, J, mol) for all public variables and expressions. Temperature is in K. Note that this means molecular weight is in the unusual unit of kg/mol.

A few expressions intended used internally and all external function calls use units of kg, kJ, kPa, and K. These generally are not needed by the end user.

## Phase Presentation¶

The property package wrapper can present fluid phase information to the IDAES framework in different ways. For specifics on how to set the options see a specific implementation page.

The PhaseType.MIX option causes to modeling framework to view water and steam as a single mixed liquid and vapor phase. This generally reduces model complexity. Phase equilibrium is still calculated and vapor_frac and individual phase properties are available, just are they would be with the two-phase presentation. The mixed-phase presentation can be used with most standard unit models that do not provide phase separation. If phase separation is required, either use the two-phase presentation or create a custom model.

The PhaseType.LG option appears to the IDAES framework to be two phases “Vap” and “Liq”. This option requires one of two unit model options to be set. You can use the total material balance option for unit models, to specify that only one material balance equation should be written not one per phase. The other possible option is to specify has_phase_equlibrium=True. This will write a material balance per phase, but will add a phase generation term to the model. For Helmholtz EoS packages, it is generally recommended that specifying total material balances is best because it results in a problem with fewer variables, and phase equilibrium is always calculated by the property package.

There are two single phase options PhaseType.L and PhaseType.G; these present a single phase “Liq” or “Vap” to the framework. The vapor fraction will also always return 0 or 1 as appropriate. These options can be used when the phase of a fluid is know for certain to only be liquid or only be vapor. For the temperature-pressure-vapor fraction formulation, this eliminates the complementarity constraint, but for the enthalpy-pressure formulation, where the vapor fraction is always calculated, the single phase options probably do not provide any real benefit.

## State Variables¶

There is a choice of state variables, pressure-enthalpy and temperature-pressure-vapor fraction. In general the enthalpy-pressure form is preferable. Both the pressure and enthalpy variables are smooth and sufficient to define the fluid state. For systems where two-phases may be present, it is expected that pressure-enthalpy is the best choice of state variables.

The temperature-pressure-vapor fraction form is more convenient, since temperature is directly measurable and more familiar than enthalpy. Complementarity constraints are used to deal with the vapor fraction variable, but the additional complimentary constraints may make the problem less robust. Temperature-pressure is often a good choice of state variables where there is only a single known phase.

### Pressure-Enthalpy Formulation¶

The advantage of this choice of state variables is that it is more robust when phase changes occur, and is especially useful when it is not known if a phase change will occur. The disadvantage of this choice of state variables is that for equations like heat transfer equations that are highly dependent on temperature, a model could be harder to solve near regions with phase change. Temperature is a non-smooth function with non-smoothness when transitioning from the single-phase to the two-phase region. Temperature also has a zero derivative with respect to enthalpy in the two-phase region, so near the two-phase region solving a constraint that specifies a specific temperature may be difficult.

The variables for this form are flow_mol (mol/s), pressure (Pa), and enth_mol (J/mol).

Since temperature and vapor fraction are not state variables in this formulation, they are provided by expressions, and cannot be fixed. For example, to set a temperature to a specific value, a constraint could be added which says the temperature expression equals a fixed value.

These expressions are specific to the P-H formulation:

temperature

Expression that calculates temperature by calling an ExternalFunction of enthalpy and pressure. This expression is non-smooth in the transition from single-phase to two-phase and has a zero derivative with respect to enthalpy in the two-phase region.

vapor_frac

Expression that calculates vapor fraction by calling an ExternalFunction of enthalpy and pressure. This expression is non-smooth in the transition from single-phase to two-phase and has a zero derivative with respect to enthalpy in the single-phase region, where the value is 0 (liquid) or 1 (vapor).

### Temperature-Pressure-Vapor Fraction¶

This formulation uses temperature (K), pressure (Pa), and vapor fraction as state variables. When a single phase option is given, the vapor fraction is fixed to the appropriate value and the complementarity constraint is deactivated.

A complementarity constraint is required for the T-P-x formulation when two-phases may be present. First, two expressions are defined below where $$P^-$$ is pressure under saturation pressure and $$P^+$$ is pressure over saturation pressure. The $$\max()$$ function is provided as an IDAES utility which provides a smooth max expression.

$P^- = \max(0, P_{\text{sat}} - P)$
$P^+ = \max(0, P - P_{\text{sat}})$

With the “pressure over” and “pressure under” expressions a complementarity constraint can be written. If the pressure under saturation is more than zero, only vapor exists. If the pressure over saturation is greater than zero only a liquid exists. If both are about zero two phases can exist. The saturation pressure function maxes out at the critical pressure and any temperature above the critical temperature will yield a saturation pressure that is the critical pressure, so supercritical fluids will be classified as liquids as is the convention for this property package.

$0 = xP^+ - (1 - x)P^-$

Assuming the vapor fraction ($$x$$) is positive and noting that only one of $$P^+$$ and $$P^-$$ can be nonzero (approximately), the complementarity equation above requires $$x$$ to be 0 when $$P^+$$ is not zero (liquid) or $$x$$ to be 1 when $$P^-$$ is not zero (vapor). When both $$P^+$$ and $$P^-$$ are about 0, the complementarity constraint says nothing about x, but it basically reduces another constraint, that $$P=P_{\text{sat}}$$. When two phases are present $$x$$ is found by the unit model energy balance, where the temperature will be $$T_{\text{sat}}$$ (because $$P=P_{\text{sat}}$$).

An alternative approach is sometimes useful to simplify the problem when it is certain that there are two phases. The complementarity constraint can be deactivated and a $$P=P_{\text{sat}}$$ or $$T=T_{\text{sat}}$$ constraint can be added.

Using the T-P-x formulation requires better initial guesses than the P-H form. It is not strictly necessary but it is best to try to get an initial guess that is in the correct phase region for the expected result model.

### Expressions¶

Unless otherwise noted, the property expressions are common to both the T-P-x and P-H formulations. For phase specific properties, valid phase indexes are "Liq" and "Vap". Even when using the mixed phase version of the property package, both liquid and vapor properties are available.

Expression

Description

mw

Molecular weight (kg/mol)

tau

Critical temperature divided by temperature (unitless)

temperature

Temperature (K) if PH form

temperature_red

Reduced temperature, temperature divided by critical temperature (unitless)

temperature_sat

Saturation temperature (K)

tau_sat

Critical temperature divided by saturation temperature (unitless)

pressure_sat

Saturation pressure (Pa)

dens_mass_phase[phase]

Density phase (kg/m3)

dens_phase_red[phase]

Phase reduced density ($$\delta$$), mass density divided by critical density (unitless)

dens_mass

Total mixed phase mass density (kg/m3)

dens_mol

Total mixed phase mole density (kg/m3)

flow_vol

Total volumetric flow rate (m3/s)

enth_mass

Mass enthalpy (J/kg)

enth_mol_sat_phase[phase]

Saturation enthalpy of phase, enthalpy at P and Tsat (J/mol)

enth_mol

Molar enthalpy (J/mol) if TPx form

enth_mol_phase[phase]

Molar enthalpy of phase (J/mol)

energy_internal_mol

molar internal energy (J/mol)

energy_internal_mol_phase[phase]

Molar internal energy of phase (J/mol)

entr_mol_phase

Molar entropy of phase (J/mol/K)

entr_mol

Total mixed phase entropy (J/mol/K)

cp_mol_phase[phase]

Constant pressure molar heat capacity of phase (J/mol/K)

cv_mol_phase[phase]

Constant pressure volume heat capacity of phase (J/mol/K)

cp_mol

Total mixed phase constant pressure heat capacity (J/mol/K)

cv_mol

Total mixed phase constant volume heat capacity (J/mol/K)

heat_capacity_ratio

cp_mol/cv_mol

speed_sound_phase[phase]

Speed of sound in phase (m/s)

dens_mol_phase[phase]

Mole density of phase (mol/m3)

vapor_frac

Vapor fraction, if PH form

phase_frac[phase]

Phase fraction

flow_mol_comp["H2O"]

Same as total flow since only water (mol/s)

P_under_sat

Pressure under saturation pressure (kPa)

P_over_sat

Pressure over saturation pressure (kPa)

### ExternalFunctions¶

This provides a list of ExternalFuctions available in the wrapper. These functions do not use SI units and are not usually called directly. If these functions are needed, they should be used with caution. Some of these are used in the property expressions, some are just provided to allow easier testing with a Python framework.

All of these functions provide first and second derivative and are generally suited to optimization (including the ones that return derivatives of Helmholtz free energy). Some functions may have non-smoothness at phase transitions. The delta_vap and delta_liq functions return the same values in the critical region. They will also return real values when a phase doesn’t exist, but those values do not necessarily have physical meaning.

There are a few variables that are common to a lot of these functions, so they are summarized here $$\tau$$ is the critical temperature divided by the temperature $$\frac{T_c}{T}$$, $$\delta$$ is density divided by the critical density $$\frac{\rho}{\rho_c}$$, and $$\phi$$ is Helmholtz free energy divided by the ideal gas constant and temperature $$\frac{f}{RT}$$.

Object

C Function

Returns

Arguments

func_p

p

pressure (kPa)

$$\delta, \tau$$

func_p_stau

p_stau

pressure (kPa)

s (kJ/kg/K), $$\tau$$

func_u

u

internal energy (kJ/kg)

$$\delta, \tau$$

func_s

s

entropy (kJ/K/kg)

$$\delta, \tau$$

func_h

h

enthalpy (kJ/kg)

$$\delta, \tau$$

func_hvpt

hvpt

vapor enthalpy (kJ/kg)

P (kPa), $$\tau$$

func_hlpt

hlpt

liquid enthalpy (kJ/kg)

P (kPa), $$\tau$$

func_svpt

svpt

vapor entropy (kJ/kg/K)

P (kPa), $$\tau$$

func_slpt

slpt

liquid entropy (kJ/kg/K)

P (kPa), $$\tau$$

func_uvpt

uvpt

vapor internal energy (kJ/kg)

P (kPa), $$\tau$$

func_ulpt

ulpt

liquid internal energy (kJ/kg)

P (kPa), $$\tau$$

func_tau

tau

$$\tau$$ (unitless)

h (kJ/kg), P (kPa)

func_tau_sp

tau_sp

$$\tau$$ (unitless)

s (kJ/kg/K), P (kPa)

func_tau_up

tau_up

$$\tau$$ (unitless)

u (kJ/kg), P (kPa)

func_vf

vf

vapor fraction (unitless)

h (kJ/kg), P (kPa)

func_vfs

vfs

vapor fraction (unitless)

s (kJ/kg/K), P (kPa)

func_vfu

vfu

vapor fraction (unitless)

u (kJ/kg), P (kPa)

func_g

g

Gibbs free energy (kJ/kg)

$$\delta, \tau$$

func_f

f

Helmholtz free energy (kJ/kg)

$$\delta, \tau$$

func_cv

cv

const. volume heat capacity (kJ/K/kg)

$$\delta, \tau$$

func_cp

cp

const. pressure heat capacity (kJ/K/kg)

$$\delta, \tau$$

func_w

w

speed of sound (m/s)

$$\delta, \tau$$

func_delta_liq

delta_liq

liquid $$\delta$$ (unitless)

P (kPa), $$\tau$$

func_delta_vap

delta_vap

vapor $$\delta$$ (unitless)

P (kPa), $$\tau$$

func_delta_sat_l

delta_sat_l

sat. liquid $$\delta$$ (unitless)

$$\tau$$

func_delta_sat_v

delta_sat_v

sat. vapor $$\delta$$ (unitless)

$$\tau$$

func_p_sat

p_sat

sat. pressure (kPa)

$$\tau$$

func_tau_sat

tau_sat

sat. $$\tau$$ (unitless)

P (kPa)

func_phi0

phi0

$$\phi$$ idaes gas part (unitless)

$$\delta, \tau$$

func_phi0_delta

phi0_delta

$$\frac{\partial \phi_0}{\partial \delta}$$

$$\delta$$

func_phi0_delta2

phi0_delta2

$$\frac{\partial^2 \phi_0}{\partial \delta^2}$$

$$\delta$$

func_phi0_tau

phi0_tau

$$\frac{\partial \phi_0}{\partial \tau}$$

$$\tau$$

func_phi0_tau2

phi0_tau2

$$\frac{\partial^2 \phi_0}{\partial \tau^2}$$

$$\tau$$

func_phir

phir

$$\phi$$ real gas part (unitless)

$$\delta, \tau$$

func_phir_delta

phir_delta

$$\frac{\partial \phi_r}{\partial \delta}$$

$$\delta, \tau$$

func_phir_delta2

phir_delta2

$$\frac{\partial^2 \phi_r}{\partial \delta^2}$$

$$\delta, \tau$$

func_phir_tau

phir_tau

$$\frac{\partial \phi_r}{\partial \tau}$$

$$\delta, \tau$$

func_phir_tau2

phir_tau2

$$\frac{\partial^2 \phi_r}{\partial \tau^2}$$

$$\delta, \tau$$

func_phir_delta_tau

phir_delta_tau

$$\frac{\partial^2 \phi_r}{\partial \delta \partial \tau}$$

$$\delta, \tau$$

## Initialization¶

The IAPWS-95 property functions do provide initialization functions for general compatibility with the IDAES framework, but as long as the state variables are specified to some reasonable value, initialization is not required. All required solves are handled by external functions.

## References¶

Although a general Helmholtz EoS was developed, the equations where taken from the IAPWS-95 standard. For specific parameter sources see specific implementation documentation.

International Association for the Properties of Water and Steam (2016). IAPWS R6-95 (2016), “Revised Release on the IAPWS Formulation 1995 for the Properties of Ordinary Water Substance for General Scientific Use,” URL: http://iapws.org/relguide/IAPWS95-2016.pdf

Wagner, W., A. Pruss (2002). “The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use.” J. Phys. Chem. Ref. Data, 31, 387-535.

Akasaka, R. (2008). “A Reliable and Useful Method to Determine the Saturation State from Helmholtz Energy Equations of State.” Journal of Thermal Science and Technology, 3(3), 442-451.