Debye-Hückel model does not satisfy consistency relationship when using variable A_Debye and "dilute" approximation

[speth]

Problem description

When using the the dilute limit form of the Debye-Hückel model with A_Debye as a function of T and P, the following thermodynamic identity is not satisfied: $$\left. \frac{\partial s}{\partial P}\right|_T = - \left. \frac{\partial v}{\partial T}\right|_P$$

Steps to reproduce

Create a phase definition by adding the following phase and species definitions to test/data/debye-huckel-all.yaml:

phases:
- name: debye-huckel-dilute-IAPWS
  elements: [O, H, C, E, Fe, Si, N, Na, Cl]
  species:
  species:
  - water_IAPWS: [H2O(L)]
  - species_waterSolution: [Na+, Cl-, H+, OH-, NaCl(aq), NaOH(aq), SiO2(aq),
      NaH3SiO4(aq), H3SiO4-]
  thermo: Debye-Huckel
  activity-data:
    model: dilute-limit
    A_Debye: variable
    B_Debye: 3.28640E9 kg^0.5/gmol^0.5/m
    max-ionic-strength: 3.0
    use-Helgeson-fixed-form: true
    default-ionic-radius: 3.5 angstrom
  state: {T: 300.0 K, P: 1.01325e+05 Pa, molalities: {Na+: 9.3549, Cl-: 9.3549, H+: 1.0499e-08,
      OH-: 1.3765e-06, NaCl(aq): 0.98492, NaOH(aq): 3.8836e-06, NaH3SiO4(aq): 6.8798e-05,
      SiO2(aq): 3.0179e-05, H3SiO4-: 1.0231e-06}}

water_IAPWS:
- name: H2O(L)
  composition: {H: 2, O: 1}
  equation-of-state:
    model: liquid-water-IAPWS95

Add a test case to consistency.cpp / consistency-cases.yaml to test the same states that are used for the other Debye-Huckel tests, then run the gtest test suite.

Behavior

In addition to the errors related to calculations of cp noted in #1500, which occur for any "form" of the model, in the dilute limit the following errors are also noted:

[ RUN      ] DebyeHuckelDilute_IAPWS/TestConsistency.dsdP_const_T_eq_minus_dV_dT_const_P/0
The difference between dsdP and -dvdT is 3.3643129666229918e-07, which exceeds tol, where
dsdP evaluates to -8.6653262251322138e-06,
-dvdT evaluates to -9.001757521794513e-06, and
tol evaluates to 9.001757521794514e-11.
[  FAILED  ] DebyeHuckelDilute_IAPWS/TestConsistency.dsdP_const_T_eq_minus_dV_dT_const_P/0

[ RUN      ] DebyeHuckelDilute_IAPWS/TestConsistency.dsdP_const_T_eq_minus_dV_dT_const_P/1
The difference between dsdP and -dvdT is 1.1736104129706076e-07, which exceeds tol, where
dsdP evaluates to -6.2996960828039867e-06,
-dvdT evaluates to -6.4170571241010475e-06, and
tol evaluates to 6.4170571241010481e-11.
[  FAILED  ] DebyeHuckelDilute_IAPWS/TestConsistency.dsdP_const_T_eq_minus_dV_dT_const_P/1

[ RUN      ] DebyeHuckelDilute_IAPWS/TestConsistency.dsdP_const_T_eq_minus_dV_dT_const_P/2
The difference between dsdP and -dvdT is 4.1357386968356715e-07, which exceeds tol, where
dsdP evaluates to -1.2342163995588498e-05,
-dvdT evaluates to -1.2755737865272065e-05, and
tol evaluates to 1.2755737865272067e-10.
[  FAILED  ] DebyeHuckelDilute_IAPWS/TestConsistency.dsdP_const_T_eq_minus_dV_dT_const_P/2

[ RUN      ] DebyeHuckelDilute_IAPWS/TestConsistency.dsdP_const_T_eq_minus_dV_dT_const_P/3
The difference between dsdP and -dvdT is 4.1030475777195397e-07, which exceeds tol, where
dsdP evaluates to -1.2281317817166268e-05,
-dvdT evaluates to -1.2691622574938222e-05, and
tol evaluates to 1.2691622574938223e-10.
[  FAILED  ] DebyeHuckelDilute_IAPWS/TestConsistency.dsdP_const_T_eq_minus_dV_dT_const_P/3\

This test succeeds with a relative tolerance of 1e-5 for all of the other variants of the Debye-Huckel model.

System information

• Cantera version: main at 13cb334
• OS: macOS 13.3

Additional context