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. psat or hl,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:
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:
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, li, pi or ei). 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:
|
Saturation pressure |
|
|
Saturation temperature |
|
|
Bubble density |
|
|
Dew density |
|
|
Bubble Enthalpy |
|
|
Dew Enthalpy |
|
|
Bubble Entropy |
|
|
Dew Entropy |
|
|
Temperature_ph |
First Input |
|
|
Second Input |
|
|
|
Temperature_ps |
First Input |
|
|
Second Input |
|
|
|
Density_pT |
First Input |
|
|
Second Input |
|
|
|
Functional approach |
|
|
As it can be seen, the fitted formulas consist basically of the coefficients ei, ci as well as of the parameters Meani and Stdi. 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:
|
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 |
|
Two limitations are known for this package:
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.
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 tau*tau*(ddalpha_0/(dtau*dtau))_delta=const |
| t_fRes_t | Short form for tau*(dalpha_r/dtau)_delta=const |
| tt_fRes_tt | Short form for tau*tau*(ddalpha_r/(dtau*dtau))_delta=const |
| d_fRes_d | Short form for delta*(dalpha_r/(ddelta))_tau=const |
| dd_fRes_dd | Short form for delta*delta(ddalpha_r/(ddelta*delta))_tau=const |
| td_fRes_td | Short form for tau*delta*(ddalpha_r/(dtau*ddelta)) |
| ttt_fIdg_ttt | Short form for tau*tau*tau*(dddalpha_0/(dtau*dtau*dtau))_delta=const |
| ttt_fRes_ttt | Short form for tau*tau*tau*(dddalpha_r/(dtau*dtau*dtau))_delta=const |
| ddd_fRes_ddd | Short form for delta*delta*delta* (dddalpha_r/(ddelta*ddelta*ddelta))_tau=const |
| tdd_fRes_tdd | Short form for tau*delta*delta*(dddalpha_r/(dtau*ddelta*ddelta)) |
| ttd_fRes_ttd | Short form for tau*tau*delta*(dddalpha_r/(dtau*dtau*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 |
