Numerical Consistency of IAPWSIF97 Backward Equations
IAPWSIF97 provides backward equations for calculation of properties without iteration. The backward equations reduce calculation time, but sacrifice accuracy. IAPWSIF97 calls the accuracy as "Numerical Consistency from Basic Equation". IAPSWIF97 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 IAPWSIF97 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 IAPWSIF97 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 IAPWSIF97 backward equations in order to reduce calculation time.
Instead, ENGSoft developed subroutines 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 IAPWSIF97 does not provide backward equations for Region 5, but ES_StableIF97 provides T(p,h) and T(p,s) values by its own subroutines.
ENGSoft tested consistency of all IAPWSIF97 backward equations. Herein the test data of the Table 5 of IAPWSIF97 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 "IAPWSIF97 Backware Eq." shows the temperatures calculated by relevant IAPWSIF97 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 IAPWSIF97 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 IAPWSIF97 backward equations from the basic equations are not satisfactory, while those of ES_StableIF97 show good consistency.
Numerical Consistency Verification Table for IAPWSIF97 Backward Equation T(p,h) for Region 1 :
Case 1 for T(p,h) 
IAPWSIF97 Table 5 Data 
IAPWSIF97 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) 
IAPWSIF97 Table 5 Data 
IAPWSIF97 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) 
IAPWSIF97 Table 5 Data 
IAPWSIF97 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 IAPWSIF97 Backward Equation T(p,s) for Region 1 :
Case 1 for T(p,s) 
IAPWSIF97 Table 5 Data 
IAPWSIF97 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) 
IAPWSIF97 Table 5 Data 
IAPWSIF97 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) 
IAPWSIF97 Table 5 Data 
IAPWSIF97 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.
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