Environmental Earth Sciences

, Volume 70, Issue 8, pp 3497–3503 | Cite as

Partial derivatives of thermodynamic state properties for dynamic simulation

Special Issue


The thermodynamic behaviour of fluids can be accurately described by equations of state (EoS) in terms of the Helmholtz energy, with temperature and density as independent variables. The known properties in dynamic simulations of power or refrigeration cycles are usually different from temperature and density. Partial derivatives of state properties with respect to the known properties of the simulation have to be transformed into partial derivatives with respect to the independent variables of the EoS. This transformation is demonstrated step by step for the single-phase region, along the saturation line and within the two-phase region.


Thermodynamic properties Helmholtz energy Partial derivatives Dynamic simulation 


  1. Akasaka R (2008) A reliable and useful method to determine the saturation state from Helmholtz energy equations of state. J Therm Sci Technol 3(3):442–451. doi:10.1299/jtst.3.442 CrossRefGoogle Scholar
  2. Arnas AÖ (2000) On the physical interpretation of the mathematics of thermodynamics. Int J Therm Sci 39(5):551–555. doi:10.1016/S1290-0729(00)00249-0 CrossRefGoogle Scholar
  3. Baehr HD (1998) Thermodynamische Fundamentalgleichungen und charakteristische Funktionen. Forschung im Ingenieurwesen 64(1):35–43. doi:10.1007/PL00010764 CrossRefGoogle Scholar
  4. Baehr HD, Tillner-Roth R (1995) Thermodynamic properties of environmentally acceptable refrigerants. Springer, BerlinGoogle Scholar
  5. Bauer O (1999) Modelling of two-phase flows with Modelica. Master’s Thesis, Department of Automatic Control, Lund University, SwedenGoogle Scholar
  6. Bridgman PW (1914) A complete collection of thermodynamic formulas. Phys Rev 3(4):273–281. doi:10.1103/PhysRev.3.273 CrossRefGoogle Scholar
  7. Carroll B (1965) On the use of Jacobians in thermodynamics. J Chem Educ 42(4):218. doi:10.1021/ed042p218 CrossRefGoogle Scholar
  8. Crawford FH (1949) Jacobian methods in thermodynamics. Am J Phys 17(1):1–5. doi:10.1119/1.1989489 CrossRefGoogle Scholar
  9. Gibbs JW (1873) A method of geometrical representation of the thermodynamic properties of substances by means of surfaces. Trans Conn Acad Arts Sci 2:382–404Google Scholar
  10. Hakala RW (1964) A method for relating thermodynamic first derivatives. J Chem Educ 41(2):99. doi:10.1021/ed041p99 CrossRefGoogle Scholar
  11. Iglesias-Silva GA, Bonilla-Petriciolet A, Eubank PT, Holste JC, Hall KR (2003) An algebraic method that includes Gibbs minimization for performing phase equilibrium calculations for any number of components or phases. Fluid Phase Equilib 210(2):229–245. doi:10.1016/S0378-3812(03)00171-7 CrossRefGoogle Scholar
  12. Lemmon EW, Jacobsen RT (2005) A new functional form and new fitting techniques for equations of state with application to pentafluoroethane (HFC-125). J Phys Chem Ref Data 34(1):69–108. doi:10.1063/1.1797813 CrossRefGoogle Scholar
  13. Lemmon EW, Jacobsen RT, Penoncello SG, Friend DG (2000) Thermodynamic properties of air and mixtures of nitrogen, argon, and oxygen from 60 to 2000 K at pressures to 2000 MPa. J Phys Chem Ref Data 29(3):331–385. doi:10.1063/1.1285884 CrossRefGoogle Scholar
  14. Lemmon EW, Huber ML, McLinden MO (2010) NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP, 9th edn. National Institute of Standards and Technology, Standard Reference Data Program, GaithersburgGoogle Scholar
  15. O’Connell J, Haile J (2005) Thermodynamics: fundamentals for applications. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  16. Richter C (2008) Proposal of new object-oriented equation-based model libraries for thermodynamic systems. Dissertation, TU BraunschweigGoogle Scholar
  17. Shaw AN (1935) The derivation of thermodynamical relations for a simple system. Philos Trans R Soc Lond Ser A. Math Phys Sci 234(740):299–328. doi:10.1098/rsta.1935.0009 CrossRefGoogle Scholar
  18. Somerton CW, Arnas AÖ (1985) On the use of Jacobians to reduce thermodynamic derivatives. Int J Mech Eng Educ 13(1):9–18Google Scholar
  19. Span R (2000) Multiparameter equations of state: an accurate source of thermodynamic property data. Springer, Berlin CrossRefGoogle Scholar
  20. Span R, Wagner W, Lemmon EW, Jacobsen RT (2001) Multiparameter equations of state—recent trends and future challenges. Fluid Phase Equilib 183–184(1–2):1–20. doi:10.1016/S0378-3812(01)00416-2 CrossRefGoogle Scholar
  21. Thorade M, Saadat A (2012) HelmholtzMedia—a fluid properties library. In: Proceedings of the 9th International Modelica Conference. doi:10.3384/ecp1207663
  22. Tummescheit H (2002) Design and implementation of object-oriented model libraries using Modelica. Dissertation, Lund UniversityGoogle Scholar
  23. Wagner W, Kretzschmar H (2008) International steam tables: properties of water and steam based on the industrial formulation IAPWS-IF97. Springer, Berlin. doi:10.1007/978-3-540-74234-0 CrossRefGoogle Scholar
  24. Wagner W, Pruß A (2002) The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use. J Phys Chem Ref Data 31(2):387–535. doi:10.1063/1.1461829 CrossRefGoogle Scholar
  25. Wagner W, Cooper JR, Dittmann A, Kijima J, Kretzschmar HJ, Kruse A, Mareš R, Oguchi K, Sato H, Stöcker I, Šifner O, Takaishi Y, Tanishita I, Trübenbach J, Willkommen T (2000) The IAPWS industrial formulation 1997 for the thermodynamic properties of water and steam. J Eng Gas Turbines Power 122(1):150–184. doi:10.1115/1.483186 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Helmholtz Centre Potsdam, GFZ German Research Centre for GeosciencesPotsdamGermany

Personalised recommendations