Zittau/Goerlitz University of Applied Sciences Department of Technical Thermodynamics Germany. K. Knobloch, H.-J. Kretzschmar, K. Miyagawa, W.

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1 Content Zittau/Goerlitz University of Applied Sciences Department of Technical Thermodynamics Germany University of Applied Sciences K. Knobloch, H.-J. Kretzschmar, K. Miyagawa, W. Wagner Supplementary Backward Equations for the Industrial Formulation IAPWS-IF of Water and Steam for Fast Calculations of Heat Cycles, Boilers, and Steam Turbines Contents Structure of IAPWS-IF and Supplementary Backward Equations Backward and Boundary Equations for Functions of (p,h) Backward and Boundary Equations for Functions of (h,s) Computing Times in Comparison with IAPWS-IF Fundamental Equations 2

2 IF IAPWS Industrial Formulation 19 for the Thermodynamic Properties of Water and Steam IAPWS-IF p / MPa Result of international surveys in industry, organized by IAPWS p B23 ( T ) f 3 ( vt, ) 1 2 ( ph, ) g1 ( p, T ) g 2 ( p, T) ( ps, ) 1 ( p, h) T 2 ( p, h) ( hs, ) 1 ( p, s) T 2 ( p, s) ( pt, ) hs, T T ( hs, ) ( ) IAPWS-IF c p T sat sat ( T ) ( p) ( hs, ) g ( p T) 5, T / K 3

3 Requirements on Backward Equations 1. Extremely high numerical consistency Deviation between the backward equation and the relating fundamental equation Example: Backward equations T(p,h) T = T- T(p,h) where h(p,t) derived from the fundamental equation g(p,t) Corresponds to iteration accuracy otherwise used in numerical calculations of process modeling Determined by IAPWS based on an international survey in industry Problem: The numerical consistency is more than one magnitude higher then accuracy of the properties themselfs 2. Calculation of the backward equations should be much faster than the corresponding iterations of the fundamental equations

4 Supplementary Backward Equations for IAPWS-IF p / MPa p B23 ( T ) 1 f 3 ( vt, ) 05 3 (, ) 2 g1 ( p, T) T3 ( p, h) g 2 ( p, T) T1 ( p, h) v T 2 ( p, h) 3 ( p, h) T1 ( p, s) T 2 ( p, s T ( ) ) 3 p, s p1 ( hs, ) v ( ) p 2 ( hs, ) 3 p, s p ( hs) v pt 05 IAPWS-IF-S05 c IAPWS-IF 3 IAPWS-IF-S IAPWS-IF-Srev IAPWS-IF-S 3, sat sat p T T ( T ) ( p) ( h s ) sat, g ( p T) 5, T / K 5

5 Suppl. Releases Supplementary Release Equations Status IAPWS-IF-S: Supplementary Release on Backward Equations for Pressure as a Function of Enthalpy and Entropy p(h,s) to the IAPWS Industrial Formulation 19 for the Thermodynamic Properties of Water and Steam. p 1 (h,s) p 2 (h,s) adopted in 20 IAPWS-IF-Srev: Revised Supplementary Release on Backward Equations for the Functions T(p,h), v(p,h) and T(p,s), v(p,s) for Region 3 of the IAPWS Industrial Formulation 19 for the Thermodynamic Properties of Water and Steam. IAPWS-IF-S: Supplementary Release on Backward Equations p(h,s) for Region 3, Equations as a Function of h and s for the Region Boundaries, and an Equation T sat (h,s) for Region of the IAPWS Industrial Formulation 19 for the Thermodynamic Properties of Water and Steam. IAPWS-IF-S05: Supplementary Release on Backward Equations for Specific Volume as a Function of Pressure and Temperature v(p,t) for Region 3 of the IAPWS Industrial Formulation 19 for the Thermodynamic Properties of Water and Steam. T 3,v 3 (p,h) T 3,v 3 (p,s) p 3sat (h) p 3sat (s) p 3 (h,s) T sat (h,s) h'(s), h"(s) h B13 (s) T B23 (h,s) adopted in 20 revised in 20 adopted in 20 v 3 (p,t) adopted in 2005 Further information concerning supplementary releases or other releases issued by IAPWS can be obtained from 6

6 ph Backward and Boundary Equations for Functions of Pressure and Enthalpy (p,h) v p / MPa ( p, h) h T b ( p) v 3b ( p, h) B23 ( p) p 2bc ( h) MPa 0 C 350 C T 1 ( p, h) T ( p, h) c p 3sat T 3b ( p, h) 3b ( h) 2c MPa ( T2c p, h) 2b T 2b ( p, h ) 2a T sat ( T 2a ( ) 800 C IAPWS-IF IAPWS-IF-Srev 10 MPa 5 Regions 1, 2 T (p,h) g (p,t) Region3 T (p,h), v (p,h) f (T,v) h / kj/kg 1 7

7 ph Backward Equations T(p,h) and v(p,h) Structure N Ii Ji = N Ii vph (, ) p h n * i + a + b * * i = 1 * i v i = 1 p * h * T( p, h) p h T p h = n + a + b Numerical consistency J i Equation N a b T tol mk T max mk T1 ( p, h) T2a ( p, h) T2b ( p, h) T2c ( p, h) T ( p, h) T3b ( p, h) Equation N a b v/v tol % v/v max % v ( p, h ) b v ( p, h )

8 Backward Equations for Functions of Enthalpy and Entropy (h,s) Regions 1, 2 p (h,s) h /kjkg p p ( hs) 1, 1 ( hs) 3b, p ( hs), p ( hs) 2c, s c 3b 5.85 kj kg 1 K 1 c ( h s) sat, s ''( K) c T p ( hs) 2b, 2b h 2ab p 2a ( s) ( hs) 2a, 5 IAPWS-IF-S IAPWS-IF-S s / kj kg 1 K 1 T (p,h) g (T,p) Region 3 p (h,s), T (p,h), v (p,s) f (T,v) Region Tsat ( h, s) sat p ( Tsat ) h h x = h h 9

9 Backward Equations p(h,s) and T sat (h,s) N Ii p(h,s) h s = n i +a +b * * * p i=1 h s J i c Structure 36 i T sat(h,s) h s = n * i * * T h s i=1 I J i Numerical consistency Equation p/p tol % p/p max % p1 ( hs, ) p 2.5MPa p > 2.5MPa 15 kpa 1 kpa T tol mk T max mk p2a ( h, s) p2b ( h, s) p2c ( hs, ) v/v tol % v/v max % p ( h, s) p3b ( h, s) Equation T tol mk T max mk p/p tol % p/p max % x tol x max T h s s kj kg K x x 10 6 sat (, ) -1-1 s > 5.85 kj kg K x x

10 Boundary Equations for Functions of Enthalpy and Entropy (h,s) h /kjkg h B13 ( s) T ( h s) B23, 3b c h ( s) 2c h h 2b 2ab 2c3b ( s) ( s) 2a h ( s) 1 IAPWS-IF-S s / kj kg 1 K 1 11

11 CT Computing Time in Comparison with IAPWS-IF Fundamental Equations Computing Time Ratio (CTR) Compting time of fundamental eq. CTR = Computing time of backward eq. CTR Backward Function Region (p,h) (p,s) (h,s) (p,t) 1 Liquid Vapor Critical and Supercritical Two-Phase Calculations of heat cycles, boilers and steam turbines may be times faster when using the backward and boundary equations 12

12 Conclusions - Backward and boundary equations for the functions of (p,h), (p,s), (h,s) and (p,t) have been developed. - The equations were adopted as supplements to the Industrial Formulation IAPWS-IF. - Their numerical consistencies are sufficient for most applications in heatcycle, boiler, and steam-turbine calculations. - Using the equations, the properties as functions of (p,t), (p,h), (p,s), and (h,s) including determination of the region can be calculated without iterations. - Resulting, process calculations will be between 2 and 3 times faster when using the supplementary backward and boundary equations. - For applications where the demands on numerical consistency are extremely high, the equations can be used for calculating very accurate starting values in iterations. 13

13 Working Group Industrial Requirements and Solutions (IRS) Task Group on Supplementary Backward Equations for IAPWS-IF and Contributors J. R. Cooper A. Dittmann D. G. Friend A. H. Harvey K. Knobloch H.-J. Kretzschmar (chair) R. Mareš R. Span I. Stöcker W. Wagner Evaluation Task Group J. Gallagher K. Miyagawa (chair) N. Okita I. Weber Working Group Thermophysical Properties of Water and Steam (TPWS) 15

14 ps Backward and Boundary Equations for Functions of Pressure and Entropy (p,s) p / MPa MPa 350 C 0 C 1 T v T 1 ( p, s) ( p, s) ( p, s) T 3b v 3b s c ( p, s) ( p, s) c T B23 ( p) 3b p 3sat ( s) 2c T 2c ( p, s) T 2b ( p, s) kj kg K 2b MPa 800 C 10 MPa IAPWS-IF IAPWS-IF-Srev 2000 C Regions 1, 2 T (p,s) g (p,t) T sat ( p) 2a T 2a ( p, s) 5 Region3 T (p,s), v (p,s) f (T,v) s / kj kg 1 K 1 16

15 Boundary Equation h 2ab (s) " h ab ( s) exp 5.21kJ kg K s = n i kj kg i = 1 s 9.2 kj kg K I = J = i I i J i i " h 5 2ab, Lemmon( s) n i = 1 m 1kJkg i = 1 i s 1 1 5kJkg K m = , m = , m = , m = , m = Compting time of fundamental eq. CTR = Computing time of backward eq. 17

16 Computing Time in Comparison with IAPWS-IF Fundamental Equations Computing Time Ratio (CTR) Compting time of fundamental eq. CTR = Computing time of backward eq. Backward Equations Function Reg. Backward Equation(s) CTR (p,h) 1 T ( p, h) p h 6 2 T (, ) 3 T3 ( p, h) & v3 ( p, h) 16 (p,s) 1 T ( p, s) p s 7 2 T (, ) 3 T3 ( p, s) & v3 ( p, s) 18 (h,s) 1 p1 ( h, s) & T1 ( p, h) 23 2 p2 ( h, s) & T2 ( p, h) 38 (p,t) 3 v ( p, T) 3 p3 ( h, s) & T3 ( p, h) & v3 ( p, s) 10 h h Tsat ( h, s) & psat ( T) & x = 11 h h

17 Computing Time in Comparison with IAPWS-IF Fundamental Equations Boundary Equations Funct. Bound. Reg.-Reg. Bound. Eq. CTR (p,h) x = 0 x = p3sat ( h) 12 (p,s) x = 0 x = p3sat ( s) 9 (h,s) x = h 1 ( s) h ( s) x = h ( ) 3 - h ( ) 90 2ab s, h 2c3b ( s) 20 2c3b s 60 B13 s K 1-3 h ( ) pb23 ( T ) 2-3 TB23 ( h, s), p ( hs, ) 2c 20 Calculation of backward functions including determination of region boundaries times faster than iteration of fundamental equations Calculations of heat cycles, boilers and steam turbines may be times faster when using the backward and boudary equations 19