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HelmholtzMedia – A Fluid Properties Library

This document presents an in-depth exploration of the Helmholtz energy equation of state (EoS) and its applications in fluid properties, including organic Rankine cycles, power cycles, and refrigeration cycles. Key topics include vapor-liquid equilibrium (VLE), iterative calculation methods, and the validation of properties derived from the Helmholtz EoS. The framework also describes essential components such as heat exchangers, pumps, turbines, and valves. Insights on the implementation and future outlook of this research are provided, enhancing understanding of its significance in geoscience technologies.

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HelmholtzMedia – A Fluid Properties Library

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  1. HelmholtzMedia – A Fluid Properties Library Matthis Thorade, Ali Saadat Helmholtz Centre Potsdam GFZ German Research Centre for Geosciences Section 4.1 Reservoir Technologies

  2. Motivation • Organic Rankine Cycle • Power cycles • Refrigeration cycles • Components: HE, PHE, Pumps, Turbines, Valves, etc.

  3. Outline • Helmholtz energy equation of state (EoS) • Vapour-Liquid-Equilibrium (VLE) • Iterative Procedures • Additional Properties • Implementation & Validation • Outlook & Summary

  4. Equations of State (EoS)

  5. Helmholtz EoS: functional form logarithmic: ≈ 2 terms polynomial: ≈ 2 terms Planck-Einstein: ≈ 4 terms polynomial: ≈4…10 terms BWR: Benedict-Webb-Rubin: ≈ 10…50 Gaussian Bell ≈ 2…4 terms Benedict, M.; Webb, G. B. & Rubin, L. C. (1940), 'An Empirical Equation for Thermodynamic Properties of Light Hydrocarbons and Their Mixtures I. Methane, Ethane, Propane and n-Butane', The Journal of Chemical Physics8 (4) , 334-345 . Setzmann, U. & Wagner, W. (1991), 'A New Equation of State and Tables of Thermodynamic Properties for Methane Covering the Range from the Melting Line to 625 K at Pressures up to 100 MPa', Journal of Physical and Chemical Reference Data20 (6) , 1061-1155 .

  6. Thermodynamic State Properties from EoS … Span, R. (2000), Multiparameter equations of state: an accurate source of thermodynamic property data , Springer Verlag .

  7. Derivatives of Thermodynamic State Properties and and and . . . and and Thorade, M. & Saadat, A. (2012), 'Partial derivatives of thermodynamic state properties for dynamic simulation', submitted to: Environmental Earth Sciences(GeoEn Special Issue).

  8. Further Partial Derivatives Tummescheit, H. (2002), 'Design and Implementation of Object-Oriented Model Libraries using Modelica', PhD thesis, Lund University. Thorade, M. & Saadat, A. (2012), 'Partial derivatives of thermodynamic state properties for dynamic simulation',

  9. Outline • Helmholtz energy equation of state (EoS) • Vapour-Liquid-Equilibrium (VLE) • Iterative Procedures • Additional Properties • Implementation & Validation • Outlook & Summary

  10. Vapour-Liquid-Equilibrium (VLE) • Thermal equilibrium: • Mechanical equilibrium: • Diffusional equilibrium: • Have to be solved simultaneously to find the VLE from the Helmholtz EoS • For a given temperature T, solve mechanical and diffusional equilibrium: and 2 unknowns, 2 equations

  11. 2D-Newton setSat_T: after 4 iterations initial guess better guess RES(initial guess) Jacobian matrix Press, W.; Teukolsky, S.; Vetterling, W. & Flannery, B. (2007), Numerical Recipes: The Art of Scientific Computing , Cambridge University Press .

  12. VLE initial guess

  13. VLE ancillary equations • dewDensity_T • bubbleDensity_T • saturationPressure_T • saturationTemperature_d (iterative inversion using Ridders'method) • Boundaries: • saturationTemperature_p(iterative inversion using Newton's method) Ridders, C. (1979), 'A new algorithm for computing a single root of a real continuous function', IEEE Transactions onCircuits and Systems, 26(11) , 979 - 980 . Press, W.; Teukolsky, S.; Vetterling, W. & Flannery, B. (2007), Numerical Recipes: The Art of Scientific Computing , Cambridge University Press .

  14. VLE by pressure and density • setSat_d: 2D-Newton • If d<d_crit: vapor side, find T and liq.d • If d>d_crit: liquid side, find T and vap.d • RES_p = p(T,liq.d) – p(T,vap.d) = 0 • RES_g = g(T,liq.d) – g(T,vap.d) = 0 • setSat_p: 3D-Newton • Find T, liq.d and vap.d such that • RES_p = p(T,liq.d) – p = 0 • RES_p = p(T,vap.d) – p = 0 • RES_g = g(T,liq.d) – g(T,vap.d) = 0

  15. Outline • Helmholtz energy equation of state (EoS) • Vapour-Liquid-Equilibrium (VLE) • Iterative Procedures • Additional Properties • Implementation & Validation • Outlook & Summary

  16. Property functions and frequency of use + Iterative algorithms !!! Wagner, W. et.al.: The IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam In: Journal of Engineering for Gas Turbines and Power , Vol. 122 , Nr. 1 ASME (2000) , S. 150-184 .

  17. setStatefunctions • setState_pT • Find d such that RES_p=p(d,T)-p=0 • setState_ph • Find d and T such that RES_p=p(d,T)-p=0 and RES_h=h(d,T)-h=0 • setState_ps • Find d and T such that RES_p=p(d,T)-p=0 and RES_s=s(d,T)-s=0 • setState_pd • Find T such that RES_p=p(d,T)-p=0 • setState_Ts • Find d such that RES_s=s(d,T)-s=0

  18. setState_ph • Input p and h • Calculate VLE from ancillary functions • sat.Tsat:=saturationTemperature_p • sat.vap.d:=dewDensity_T; and sat.liq.d:=bubbleDensity_T; • sat.vap.h:=h(Tsat,vap.d);and sat.liq.h:=h(Tsat,liq.d); • If necessary, calculate VLE from EoS • Determine region and set boundaries • If single-phase, start Newton from saturation values (± offset), • check boundaries About 4 iterations, maximum 10 About 10 timesslowerthansetState_dT

  19. Outline • Helmholtz energy equation of state (EoS) • Vapour-Liquid-Equilibrium (VLE) • Iterative Procedures • Additional Properties • Implementation & Validation • Outlook & Summary

  20. Additional properties • Surface tension • Viscosity • Thermal conductivity

  21. Outline • Helmholtz energy equation of state (EoS) • Vapour-Liquid-Equilibrium (VLE) • Iterative Procedures • Additional Properties • Implementation & Validation • Outlook & Summary

  22. Implementation • Based on Modelica.Media • extends PartialTwoPhaseMedium • All functions available, same input & output • Add entropy s to ThermodynamicState • Add states liq and vap to SaturationProperties • annotation(inverse=xyz) • annotation(derivative=xyz_der) Elmqvist, H.; Tummescheit, H. & Otter, M. (2003), Object-oriented modeling of thermo-fluid systems, in 'Proceedings of the 3rd International Modelica Conference' , pp. 269--286 .

  23. Validation • 6 fluids: n-Butane, Isobutane, Isopentane, Propane, R134a, Ethanol • Possible working fluids for geothermal ORC • Manually validated by comparing values to values calculated from RefProp • d, T, p, h, u, s (ThermodynamicState) • Tsat, psat, dliq, dvap (SaturationProperties) • cp, beta, kappa, speed of sound • About 20 points, including (d,T)=(0,Tmax), (d,T)=(dmax,Tmax), (d,T)=… • Pretty-print coefficient matrix using MSL function • Validate analytical derivatives by comparing them to numerical derivatives

  24. Stability • Single-phase: • T=ramp.y from T_trip to T_max • p=sine.y from 0 to p_max • state:=setState_pTis a valid single-phase state • Call all other setState functions using values from state • Two-phase: • T=ramp.y from T_trip to T_crit • d=d_crit • state:=setState_dTis a valid two-phase state • Call all other setState functions using values from state • Ancillary functions and setSat functions • T=ramp.y from T_trip to T_crit • sat:=setSat_T • Call all other setSatfunctions using values from sat

  25. Outline • Helmholtz energy equation of state (EoS) • Vapour-Liquid-Equilibrium (VLE) • Iterative Procedures • Additional Properties • Implementation & Validation • Outlook & Summary

  26. Known issues = Outlook • BaseProperties currently fixed to (p,h) • Numerical Jacobians when running Modelica.Media.Examples.Test.MediaTestModels • EoS always slower than table or spline based media • Functional form: Hyperbolic & non-analytical terms • Mixtures, e.g. GERG2008

  27. Summary • HelmholtzMedia is a library for the calculation of fluid properties for pure fluids that can be described by the Helmholtz energy EoS • 6 fluids implemented, about 1 day for each additional fluid • Additional properties: Surface tension, viscosity, thermal conductivity • Implemented in Modelica, released under Modelica license

  28. Grant 03G0767A

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