Fast Calculation of Thermophysical Properties in Extensive Process Simulations with the Spline-Based Table Look-Up Method (SBTL)

Typ: Fortschritt-Berichte VDI
Erscheinungsdatum: 03.07.2018
Reihe: 6
Band Nummer: 618
Autor: Dipl.-Ing. (FH) Matthias Kunick
Ort: Görlitz
ISBN: 978-3-18-361806-4
ISSN: 0178-9414
Erscheinungsjahr: 2018
Anzahl Seiten: 172
Anzahl Abbildungen: 80
Anzahl Tabellen: 58
Produktart: Buch (paperback, DINA5)

Produktbeschreibung

The presented Spline-Based Table Look-Up Method (SBTL) is intended to be used for fast and accurate property calculations in computationally extensive process simulations, such as Computational Fluid Dynamics (CFD), heat-cycle calculations, simulations of non-stationary processes, and real-time process optimizations, where conventional multiparameter equations of state may be unsuitable because of their computing time consumption. Through the use of the SBTL method, the results of existing property formulations are accurately reproduced at high computational speed. SBTL property functions, their first derivatives, and inverse functions are continuous and numerically consistent with each other. The developed algorithms are successfully applied in commercial and non-commercial software products for CFD, heat-cycle calculations, and simulations of non-stationary processes. The International Association for the Properties of Water and Steam has adopted the “Guideline on the Fast Calculation of Steam and Water Properties with the Spline-Based Table Look-Up Method (SBTL)”.

Contents

List of Symbols and Nomenclature ……………………………………………………………………….. VIII
1 Introduction …………………………………………………………………………………………………………… 1
2 Property Calculation Algorithms for Numerical Process Simulations ………………………. 4
2.1 Simple Thermal Equations of State ……………………………………………………………………… 4
2.1.1 Ideal-Gas Model ………………………………………………………………………………………… 4
2.1.2 Cubic Equations of State …………………………………………………………………………….. 6
2.2 Fundamental Equations of State ………………………………………………………………………… 10
2.2.1 Reference Equations of State ……………………………………………………………………… 11
2.2.2 Short Fundamental Equations of State for Industrial Applications ………………….. 14
2.2.3 Fast Fundamental Equations for Separate Regions ……………………………………….. 14
2.3 Backward Equations ………………………………………………………………………………………… 18
2.4 Table Look-Up Methods ………………………………………………………………………………….. 19
2.4.1 One-Dimensional Functions ………………………………………………………………………. 20
2.4.1.1 Local Polynomial Interpolation …………………………………………………………. 21
2.4.1.2 Tabular Taylor Series Expansion Method (TTSE) ……………………………….. 22
2.4.1.3 Spline Interpolation and Approximation Algorithms ……………………………. 23
2.4.2 Two-Dimensional Functions ……………………………………………………………………… 27
2.4.2.1 Local Polynomial Interpolation …………………………………………………………. 28
2.4.2.2 Tabular Taylor Series Expansion Method (TTSE) ……………………………….. 29
2.4.2.3 Spline Interpolation and Approximation Algorithms ……………………………. 31
2.4.3 Thermodynamic Consistency of Table Look-up Methods ……………………………… 37
2.5 Computing Time ………………………………………………………………………………………………38
2.5.1 Computing Times of Various Operations …………………………………………………….. 38
2.5.2 Computationally Efficient Implementation of Property Formulations ……………… 41
2.5.3 Computing-Time Comparisons ………………………………………………………………….. 44
2.6 Conclusions for the Development of a Fast and Accurate Property Calculation
Method for Extensive Process Simulations ………………………………………………………… 48
3 The Spline-Based Table Look-Up Method (SBTL) ………………………………………………… 50
3.1 One-Dimensional Spline Functions …………………………………………………………………….50
3.1.1 Spline Functions ………………………………………………………………………………………. 50
3.1.2 Transformations ……………………………………………………………………………………….. 53
3.1.3 Inverse Spline Functions …………………………………………………………………………… 54
3.1.4 Derivatives ………………………………………………………………………………………………. 55
3.2 Two-Dimensional Spline Functions …………………………………………………………………… 56
3.2.1 Spline Functions ………………………………………………………………………………………. 56

3.2.2 Transformations ……………………………………………………………………………………….. 63
3.2.3 Inverse Spline Functions …………………………………………………………………………… 65
3.2.4 Derivatives ………………………………………………………………………………………………. 67
3.2.5 Calculations in the Two-Phase Region ………………………………………………………… 68
4 FluidSplines – Software for Generating SBTL Property Functions ………………………… 71
4.1 Basic Structure ……………………………………………………………………………………………….. 71
4.2 Generation of One-Dimensional SBTL Property Functions ………………………………….. 74
4.3 Generation of Two-Dimensional SBTL Property Functions………………………………….. 75
5 SBTL Property Functions Based on IAPWS-IF97 for Water and Steam ………………… 79
5.1 Spline Functions of (v,u) and Inverse Functions ………………………………………………….. 79
5.1.1 Range of Validity ……………………………………………………………………………………… 79
5.1.2 Spline Functions for the Single-Phase Region ……………………………………………… 80
5.1.3 Calculations in the Two-Phase Region ………………………………………………………… 81
5.1.4 Derivatives ………………………………………………………………………………………………. 81
5.1.5 Deviations from IAPWS-IF97 ……………………………………………………………………. 81
5.1.6 Numerical Consistency at Region Boundaries ……………………………………………… 83
5.2 Spline Functions of (p,h) and Inverse Functions ………………………………………………….. 84
5.2.1 Range of Validity ……………………………………………………………………………………… 84
5.2.2 Spline Functions for the Single-Phase Region ……………………………………………… 85
5.2.3 Calculations in the Two-Phase Region ………………………………………………………… 86
5.2.4 Derivatives ………………………………………………………………………………………………. 86
5.2.5 Deviations from IAPWS-IF97 ……………………………………………………………………. 86
5.2.6 Numerical Consistency at Region Boundaries ……………………………………………… 88
5.3 Spline Functions for the Metastable-Vapor Region ……………………………………………… 89
5.3.1 Spline Functions of (v,u) …………………………………………………………………………… 89
5.3.2 Spline Functions of (p,h) …………………………………………………………………………… 89
5.3.3 Deviations from IAPWS-IF97 ……………………………………………………………………. 89
5.4 Computing-Time Comparisons …………………………………………………………………………. 92
6 SBTL Property Functions Based on IAPWS-95 for Water and Steam ……………………. 95
6.1 Spline Functions of (v,u) ………………………………………………………………………………….. 95
6.1.1 Range of Validity ……………………………………………………………………………………… 95
6.1.2 Spline Functions for the Single-Phase Region ……………………………………………… 96
6.1.3 Deviations from IAPWS-95 ………………………………………………………………………. 96
6.2 Spline Functions of (p,h) ………………………………………………………………………………….. 98
6.2.1 Range of Validity ……………………………………………………………………………………… 98
6.2.2 Spline Functions for the Single-Phase Region ……………………………………………… 99

Keywords: Fast Calculation, thermophysical, process Stimulation, Spline-Based Table Look-up Method, Thermophysical Properties, Spline-Interpolation, Table Look-Up Method, Equation of State, Process Simulations, Computational Fluid Dynamics, Heat-Cycle Simulations, Transient Processes, Fast Calculation, thermophysical, process Stimulation, Spline-Based Table Look-up Method, Thermophysical Properties, Spline-Interpolation, Table Look-Up Method, Equation of State, Process Simulations, Computational Fluid Dynamics, Heat-Cycle Simulations, Transient Processes

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