A method for the accurate and smooth approximation of standard thermodynamic functions

Abstract

A method is proposed for the calculation of approximations of standard thermodynamic functions. The method is consistent with the physical properties of standard thermodynamic functions. This means that the approximation functions are, in contrast to the hitherto used approximations, continuous and smooth in every temperature interval in which no phase transformations take place. The calculation algorithm was implemented by the SmoothSTF program in the C++ language which is part of this paper. New version program summaryProgram title: SmoothSTFCatalogue identifier: AENH_v1_1Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENH_v1_1.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 3807No. of bytes in distributed program, including test data, etc.: 138212Distribution format: tar.gzProgramming language: C++.Computer: Any computer with a gcc compiler.Operating system: Debian GNU Linux 6.0. The program can be run in operating systems in which the gcc compiler can be installed.RAM: 256 Mbytes are sufficient for the table of standard thermodynamic functions with 500 linesClassification: 4.9.Catalogue identifier of previous version: AENH_v1_0Journal reference of previous version: Comput. Phys. Comm. 184 (2013) 194Does the new version supersede the previous version?: YesNature of problem: Standard thermodynamic functions (STF) of individual substances are given by thermal capacity at constant pressure, entropy and enthalpy. STF are continuous and smooth in every temperature interval in which no phase transformations take place. The temperature dependence of STF as expressed by the table of its values is for further application approximated by temperature functions. In the paper, a method is proposed for calculating approximation functions which, in contrast to the hitherto used approximations, are continuous and smooth in every temperature interval.Solution method: The approximation functions are determined by coefficients that are calculated by the least squares method coupled with meeting the conditions set (calculation of minimum with equality constraints). To calculate the coefficients the values of STF derivatives with respect to temperature must be available in addition to the table of STF values. The values of the derivatives are established using cubic splines and the derivative of the Lagrange interpolation polynomial.Reasons for new version: There are two small mistakes in the original program, which show up for only some input data and do not have any effect on the sample input and output data for the test run. These mistakes do not require any changes to be made in the text of the program summary and the accompanying paper.Summary of revisions: One mistake appeared in the case when the partial interval was defined by two neighbouring tabular temperatures in the given table of standard thermodynamic functions. The other mistake occurred when the calculated value of some coefficient of the approximation function was lower than δA (i.e. the prescribed error caused by rounding the approximation coefficients).Restrictions: The program was tested and is used on the assumption that the upper bound of the temperature interval in which approximation is being performed does not exceed 50 kK. This value is sufficient in the thermal plasma field. For temperatures higher than 50 kK the calculation might entail difficulties, which could however be removed by calculating with higher than double precision.Additional comments: The program package includes a README file and input and output files for a test suite.Running time: A few seconds, if the approximation is being determined in the maximum interval, i.e. from ambient temperature to 50 kK.