Numerical Consistency of IAPWS-IF97 Backward Equations

IAPWS-IF97 provides backward equations for calculation of properties without iteration.   The backward equations reduce calculation time, but sacrifice accuracy.    IAPWS-IF97 calls the accuracy as "Numerical Consistency from Basic Equation".   IAPSW-IF97 made the backward equations under the limit of certain numerical consistency from the basic equations.   For the numerical consistency of each backward equation. please refer to IAPWS-IF97 documents.       

IAPWS provides backward equations for getting temperature by pressure and enthalpy given or by pressure and entropy given in Region 1, 2 and 3, i.e. T(p,h) and T(p,s).   By using these backward equations and basic equations, other properties by pressure and enthalpy or by pressure by entropy given can be looked up.   For example, h(p,s) can be made from h(p, T(p,s)) using the backward equation T(p,s) and the basic equation h(p,T).

ENGSoft tested the numerical consistency of the IAPWS-IF97 backward equations, but decided not to use the backward equations because the numerical inconsistency is considerable even though the time for calculation is reduced.

Exceptionally the functions of ESStb_PbyHS and ESStb_TbyHS use the IAPWS-IF97 backward equations in order to reduce calculation time.

 

Instead, ENGSoft developed sub-routines to get T(p,h) and T(p,s) values from the basic equations by its own iteration method for Region 1, 2 and 3 as well as Region 5.   Please note that IAPWS-IF97 does not provide backward equations for Region 5, but ES_StableIF97 provides T(p,h) and T(p,s) values by its own sub-routines.

ENGSoft tested consistency of all IAPWS-IF97 backward equations.  Herein the test data of the Table 5 of IAPWS-IF97 for Region 1 is presented as shown in the table below.    According to ENGSoft's investigation, the backward equations for other regions show similar numerical inconsistency  

In the consistency verification tables below, the column "IAPWS-IF97 Backware Eq." shows the temperatures calculated by relevant IAPWS-IF97 backward equations, given the same pressure and enthalpy(or entropy) with the Table 5 data.   And the column "ES_StableIF97" shows the temperature calculated by ENGSoft's iteration subroutines using the IAPWS-IF97 basic equations, given the same pressure and enthalpy(or entropy) with the Table 5 data.

As seen in the tables below, the numerical consistency of IAPWS-IF97 backward equations from the basic equations are not satisfactory, while those of ES_StableIF97 show good consistency.

Numerical Consistency Verification Table for IAPWS-IF97 Backward Equation T(p,h) for Region 1 :

Case 1 for T(p,h)

IAPWS-IF97 Table 5 Data

IAPWS-IF97 Backward Eq.

ES_StableIF97

Pressure Given

MPa

3

3

3

Enthalpy Given)

kJ/kg

115.331273

115.331273

115.331273

Temperature Output

K

300

300.01783 (0.01783)*

300.00000 (0.00000)*

 

Case 2 for T(p,h)

IAPWS-IF97 Table 5 Data

IAPWS-IF97 Backward Eq.

ES_StableIF97

Pressure Given

MPa

80

80

80

Enthalpy Given

kJ/kg

184.142828

184.142828

184.142828

Temperature Output

K

300

299.98103 (0.01897)*

300.00000 (0.00000)*

 

Case 3 for T(p,h)

IAPWS-IF97 Table 5 Data

IAPWS-IF97 Backward Eq.

ES_StableIF97

Pressure Given

MPa

3

3

3

Enthalpy Given

kJ/kg

2549.91145

2549.91145

2549.91145

Temperature Output

K

500

499.34232 (0.65768)*

500.00000 (0.00000)*

 

Numerical Consistency Verification Table for IAPWS-IF97 Backward Equation T(p,s) for Region 1 :

Case 1 for T(p,s)

IAPWS-IF97 Table 5 Data

IAPWS-IF97 Backward Eq. (13)

ES_StableIF97

Pressure Given

MPa

3

3

3

Entropy Given

kJ/kg/K

0.392294792

0.392294792

0.392294792

Temperature Output

K

300

299.99870 (0.00130)*

300.00000 (0.00000)*

 

Case 2 for T(p,s)

IAPWS-IF97 Table 5 Data

IAPWS-IF97 Backward Eq.

ES_StableIF97

Pressure Given

MPa

80

80

80

Entropy Given

kJ/kg/K

0.368563852

0.368563852

0.368563852

Temperature Output

K

300

299.99034 (0.00966)*

300.00000 (0.00000)*

 

Case 3 for T(p,s)

IAPWS-IF97 Table 5 Data

IAPWS-IF97 Backward Eq.

ES_StableIF97

Pressure Given

MPa

3

3

3

Entropy Given

kJ/kg/K

2.58041912

2.58041912

2.58041912

Temperature Output

K

500

499.99132 (0.00868)*

500.00000 (0.00000)*

* The figures in parenthesis show numerical inconsistency.


Copyright (c) : ENGSoft Inc., Seoul, Korea, All right reserved since 2008.  / www.engsoft.co.kr