.AixLib.Media.Refrigerants.Interfaces.PartialHybridTwoPhaseMediumFormula

Information

This package provides the implementation of a refrigerant modelling approach using a hybrid approach. The hybrid approach is developed by Sangi et al. and consists of both the Helmholtz equation of state and fitted formula for thermodynamic state properties at bubble or dew line (e.g. p_sat or h_l,sat) and thermodynamic state properties depending on two independent state properties (e.g. T_ph or T_ps). In the following, the basic formulas of the hybrid approach are given.

The Helmholtz equation of state

The Helmholtz equation of state (EoS) allows the accurate description of fluids' thermodynamic behaviour and uses the Helmholtz energy as fundamental thermodynamic relation with temperature and density as independent variables. Furthermore, the EoS allows determining all thermodynamic state properties from its partial derivatives and its general formula is given below:

Calculation procedure of dimensionless Helmholtz energy

As it can be seen, the general formula of the EoS can be divided in two part: The ideal gas part (left summand) and the residual part (right summand). Both parts' formulas are given below:

Calculation procedure of dimensionless ideal gas Helmholtz energy

Calculation procedure of dimensionless residual Helmholtz energy

Both, the ideal gas part and the residual part can be divided in three subparts (i.e. the summations) that contain different coefficients (e.g. nL, l_i, p_i or e_i). These coefficients are fitting coefficients and must be obtained during a fitting procedure. While the fitting procedure, the general formula of the EoS is fitted to external data (e.g. obtained from measurements or external media libraries) and the fitting coefficients are determined. Finally, the formulas obtained during the fitting procedure are implemented in an explicit form.

For further information of the EoS and its partial derivatives, please read the paper " HelmholtzMedia - A fluid properties library" by Thorade and Saadat as well as the paper " Partial derivatives of thermodynamic state properties for dynamic simulation" by Thorade and Saadat.

Fitted formulas

Fitted formulas allow to reduce the overall computing time of the refrigerant model. Therefore, both thermodynamic state properties at bubble and dew line and thermodynamic state properties depending on two independent state properties are expresses as fitted formulas. The fitted formulas' approaches implemented in this package are developed by Sangi et al. within their "Fast_Propane" model and given below:

"Formulas for calculating saturation properties" cellspacing="0" cellpadding="2" border="1" width="80%" style= "border-collapse:collapse;">

Saturation pressure
Saturation temperature
Bubble density
Dew density
Bubble Enthalpy
Dew Enthalpy
Bubble Entropy
Dew Entropy

"Formulas for calculating thermodynamic properties at superheated and supercooled regime" cellspacing="0" cellpadding="3" border="1" width="80%" style= "border-collapse:collapse;">

Temperature_ph	First Input
Temperature_ph	Second Input
Temperature_ps	First Input
Temperature_ps	Second Input
Density_pT	First Input
Density_pT	Second Input
Functional approach

As it can be seen, the fitted formulas consist basically of the coefficients e_i, c_i as well as of the parameters Mean_i and Std_i. These coefficients are the fitting coefficients and must be obtained during a fitting procedure. While the fitting procedure, the formulas presented above are fitted to external data (e.g. obtained from measurements or external media libraries) and the fitting coefficients are determined. Finally, the formulas obtained during the fitting procedure are implemented in an explicit form.

For further information of the hybrid approach, please read the paper "A Medium Model for the Refrigerant Propane for Fast and Accurate Dynamic Simulations" by Sangi et al..

Smooth transition

To ensure a smooth transition between different regions (e.g. from supercooled region to two-phase region) and, therefore, to avoid discontinuities as far as possible, Sangi et al. implemented functions for a smooth transition between the regions. An example (i.e. specificEnthalpy_ps) of these functions is given below:

"Calculation procedures to avoid numerical instability at phase change" cellspacing="0" cellpadding="2" border="1" width="80%" style= "border-collapse:collapse;">

From supercooled region to bubble line and vice versa
From dew line to superheated region and vice versa
From bubble or dew line to two-phase region and vice versa

Assumptions and limitations

Two limitations are known for this package:

The modelling approach implemented in this package is a hybrid approach and, therefore, is based on the Helmholtz equation of state as well as on fitted formula. Hence, the refrigerant model is just valid within the valid range of the fitted formula.
It may be possible to have discontinuities when moving from one region to another (e.g. from supercooled region to two-phase region). However, functions are implemented to reach a smooth transition between the regions and to avoid these discontinuities as far as possible. (Sangi et al., 2014)

Typical use and important parameters

The refrigerant models provided in this package are typically used for heat pumps and refrigerating machines. However, it is just a partial package and, hence, it must be completed before usage. In order to allow an easy completion of the package, a template is provided in AixLib.Media.Refrigerants.Interfaces.TemplateHybridTwoPhaseMediumFormula.

References

Thorade, Matthis; Saadat, Ali (2012): HelmholtzMedia - A fluid properties library. In: Proceedings of the 9th International Modelica Conference; September 3-5; 2012; Munich; Germany. Linköping University Electronic Press, S. 63–70.

Thorade, Matthis; Saadat, Ali (2013): Partial derivatives of thermodynamic state properties for dynamic simulation. In: Environmental earth sciences 70 (8), S. 3497–3503.

Sangi, Roozbeh; Jahangiri, Pooyan; Klasing, Freerk; Streblow, Rita; Müller, Dirk (2014): A Medium Model for the Refrigerant Propane for Fast and Accurate Dynamic Simulations. In: The 10th International Modelica Conference. Lund, Sweden, March 10-12, 2014: Linköping University Electronic Press (Linköping Electronic Conference Proceedings), S. 1271–1275

Klasing,Freerk: A New Design for Direct Exchange Geothermal Heat Pumps - Modeling, Simulation and Exergy Analysis. Master thesis

Name	Description
BaseProperties	Base properties of refrigerant
ThermodynamicState	Thermodynamic state
SmoothTransition	Record that contains ranges to calculate a smooth transition between different regions
f_Idg	Dimensionless Helmholtz energy (Ideal gas contribution alpha_0)
f_Res	Dimensionless Helmholtz energy (Residual part alpha_r)
t_fIdg_t	Short form for tau*(dalpha_0/dtau)_delta=const
tt_fIdg_tt	Short form for tautau(ddalpha_0/(dtau*dtau))_delta=const
t_fRes_t	Short form for tau*(dalpha_r/dtau)_delta=const
tt_fRes_tt	Short form for tautau(ddalpha_r/(dtau*dtau))_delta=const
d_fRes_d	Short form for delta*(dalpha_r/(ddelta))_tau=const
dd_fRes_dd	Short form for deltadelta(ddalpha_r/(ddeltadelta))_tau=const
td_fRes_td	Short form for taudelta(ddalpha_r/(dtau*ddelta))
ttt_fIdg_ttt	Short form for tautautau(dddalpha_0/(dtaudtau*dtau))_delta=const
ttt_fRes_ttt	Short form for tautautau(dddalpha_r/(dtaudtau*dtau))_delta=const
ddd_fRes_ddd	Short form for deltadeltadelta* (dddalpha_r/(ddeltaddeltaddelta))_tau=const
tdd_fRes_tdd	Short form for taudeltadelta(dddalpha_r/(dtauddelta*ddelta))
ttd_fRes_ttd	Short form for tautaudelta(dddalpha_r/(dtaudtau*ddelta))
setSmoothState	Return thermodynamic state so that it smoothly approximates: if x > 0 then state_a else state_b
setDewState	Return thermodynamic state of refrigerant on the dew line
setBubbleState	Return thermodynamic state of refrigerant on the bubble line
setState_dTX	Return thermodynamic state of refrigerant as function of d and T
setState_pTX	Return thermodynamic state of refrigerant as function of p and T
setState_phX	Return thermodynamic state of refrigerant as function of p and h
setState_psX	Return thermodynamic state of refrigerant as function of p and s
pressure	Pressure of refrigerant
temperature	Temperature of refrigerant
density	Density of refrigerant
specificEnthalpy	Specific enthalpy of refrigerant
specificInternalEnergy	Specific internal energy of refrigerant
specificGibbsEnergy	Specific Gibbs energy of refrigerant
specificHelmholtzEnergy	Specific Helmholtz energy of refrigerant
specificEntropy	Specific entropy of refrigerant
specificHeatCapacityCp	Specific heat capacity at constant pressure of refrigerant
specificHeatCapacityCv	Specific heat capacity at constant volume of refrigerant
velocityOfSound	Velocity of sound of refrigerant
isobaricExpansionCoefficient	Isobaric expansion coefficient beta of refrigerant
isentropicExponent	Isentropic exponent defined as -v/p*(dp/dv)_s=const
isentropicEnthalpy	Isentropic enthalpy calculated by downstream pressure and reference state
isothermalCompressibility	Isothermal compressibility factor of refrigerant
isothermalThrottlingCoefficient	Isothermal throttling coefficient of refrigerant
jouleThomsonCoefficient	Joule-Thomson coefficient of refrigerant
pressure_dT	Computes pressure as a function of density and temperature
density_ph	Computes density as a function of pressure and enthalpy
density_ps	Computes density as a function of pressure and entropy
specificEnthalpy_pT	Computes specific enthalpy as a function of pressure and temperature
specificEnthalpy_dT	Computes specific enthalpy as a function of density and temperature
specificEnthalpy_ps	Computes specific enthalpy as a function of pressure and entropy
pressure_derd_T	Calculates pressure derivative (dp/dd)_T=const
pressure_derT_d	Calculates pressure derivative (dp/dT)_d=const
specificEnthalpy_derT_d	Calculates enthalpy derivative (dh/dT)_d=const
specificEnthalpy_derd_T	Calculates enthalpy derivative (dh/dd)_T=const
specificEntropy_derd_T	Calculates derivative (ds/dd)_T=const.
specificEntropy_derT_d	Calculates derivative (ds/dT)_d=const.
specificInternalEnergy_derT_d	Calculated derivative (du/dT)_d=const.
specificInternalEnergy_derd_T	Calculated derivative (du/dd)_T=const.
density_derp_T	Calculates the derivative (dd/dp)_T=const.
density_derT_p	Calculates the derivative (dd/dT)_p=const.
temperature_derp_h	Calculates temperature derivative (dT/dp)_h=const.
temperature_derp_s	Calculates temperature derivative (dT/dp)_s=const.
temperature_ders_p	Calculates temperature derivative (dT/ds)@p=const.
temperature_derh_p	Calculates temperature derivative (dT/dh)_p=const.
density_derp_h	Calculates density derivative (dd/dp)_h=const.
density_derh_p	Calculates density derivative (dd/dh)_p=const.
density_derp_s	Calculates density derivative (dd/dp)_s=const.
density_ders_p	Calculates density derivative (dd/ds)_p=const.
specificEnthalpy_derp_T	Calculates derivative (dh/dp)_T=const.
specificEnthalpy_derT_p	Calculates derivative (dh/dT)_p=const.
specificEnthalpy_ders_p	Calculates derivative (dh/ds)_p=const.
specificEnthalpy_derp_s	Calculates derivative (dh/dp)_s=const.
saturationPressure_derT	Calculates derivative (dp/dT)_saturation
saturationTemperature_derp	Calculates derivative (dT/dp)_saturation
dBubbleDensity_dPressure	Calculates bubble point density derivative
dDewDensity_dPressure	Calculates dew point density derivative
dBubbleEnthalpy_dPressure	Calculates bubble point enthalpy derivative
dDewEnthalpy_dPressure	Calculates dew point enthalpy derivative
dBubbleEntropy_dPressure	Calculates bubble point entropy derivative
dDewEntropy_dPressure	Calculates dew point entropy derivative
dBubbleDensity_dTemperature	Calculates bubble point density derivative
dDewDensity_dTemperature	Calculates dew point debsity derivative
dBubbleEnthalpy_dTemperature	Calculates bubble point enthalpy derivative
dDewEnthalpy_dTemperature	Calculates dew point enthalpy derivative
dBubbleInternalEnergy_dTemperature	Calculates bubble point internal energy derivative
dDewInternalEnergy_dTemperature	Calculates dew point internal energy derivative
f_Idg_der	Calculates time derivative of f_Idg
f_Res_der	Calculates time derivative of f_Res
t_fIdg_t_der	Calculates time derivative of t_fIdg_t
tt_fIdg_tt_der	Calculates time derivative of tt_fIdg_tt
t_fRes_t_der	Calculates time derivative of t_fRes_t
tt_fRes_tt_der	Calculates time derivative of tt_fRes_tt
d_fRes_d_der	Calculates time derivative of d_fRes_d
dd_fRes_dd_der	Calculates time derivative of dd_fRes_dd
td_fRes_td_der	Calculates time derivative of td_fRes_td
setState_dTX_der	Calculates time derivative of the thermodynamic state record calculated by d and T
setState_pTX_der	Calculates time derivative of the thermodynamic state record calculated by p and T
setState_phX_der	Calculates time derivative of the thermodynamic state record calculated by p and h
setState_psX_der	Calculates time derivative of the thermodynamic state record calculated by p and s
pressure_der	Calculates time derivative of pressure calculated by thermodynamic state record
temperature_der	Calculates time derivative of temperature calculated by thermodynamic state record
density_der	Calculates time derivative of density calculated by thermodynamic state record
specificEnthalpy_der	Calculates time derivative of specific enthalpy calculated by thermodynamic state record
pressure_dT_der	Calculates time derivative of pressure_dT
temperature_ph_der	Calculates time derivative of temperature_ph
temperature_ps_der	Calculates time derivative of temperature_ps
density_ph_der	Calculates time derivative of density_ph
density_pT_der	Calculates time derivative of density_pT
density_ps_der	Calculates time derivative of density_ps
specificEnthalpy_pT_der	Calculates time derivative of specificEnthalpy_pT
specificEnthalpy_dT_der	Calculates time derivative of specificEnthalpy_dT
specificEnthalpy_ps_der	Calculates time derivative of specificEnthalpy_ps