Debye-Hückel cp calculations incorrect when using variable A_Debye

[speth]

Problem description

When using the mode where A_Debye is a function of T and P, calculations of the constant pressure heat capacity do not satisfy thermodynamic consistency tests.

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-B-dot-ak-IAPWS
  elements: [O, H, C, E, Fe, Si, N, Na, Cl]
  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: B-dot-with-variable-a
    A_Debye: variable
    B_Debye: 3.28640E9 kg^0.5/gmol^0.5/m
    max-ionic-strength: 50.0
    B-dot: 0.041
    default-ionic-radius: 4.0 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

The following test output (slightly simplified) shows that the cp calculations are off:

[ RUN      ] DebyeHuckel_b_dot_ak_IAPWS/TestConsistency.cp_eq_dhdT/0
The difference between cp_fd and cp_mid is 605.84843225777149, which exceeds abstol, where
cp_fd evaluates to 3377833.5181276002,
cp_mid evaluates to 3378439.366559858, and
abstol evaluates to 3.3784393665598578.
[  FAILED  ] DebyeHuckel_b_dot_ak_IAPWS/TestConsistency.cp_eq_dhdT/0

[ RUN      ] DebyeHuckel_b_dot_ak_IAPWS/TestConsistency.cp_eq_dhdT/1
The difference between cp_fd and cp_mid is 261.74634717032313, which exceeds abstol, where
cp_fd evaluates to 2365772.22687006,
cp_mid evaluates to 2366033.9732172303, and
abstol evaluates to 2.3660339732172302.
[  FAILED  ] DebyeHuckel_b_dot_ak_IAPWS/TestConsistency.cp_eq_dhdT/1

[ RUN      ] DebyeHuckel_b_dot_ak_IAPWS/TestConsistency.cp_eq_dhdT/2
The difference between cp_fd and cp_mid is 543.4203562955372, which exceeds abstol, where
cp_fd evaluates to 3378542.5882879645,
cp_mid evaluates to 3379086.00864426, and
abstol evaluates to 3.37908600864426.
[  FAILED  ] DebyeHuckel_b_dot_ak_IAPWS/TestConsistency.cp_eq_dhdT/2

[ RUN      ] DebyeHuckel_b_dot_ak_IAPWS/TestConsistency.cp_eq_dhdT/3
The difference between cp_fd and cp_mid is 536.87257880996913, which exceeds abstol, where
cp_fd evaluates to 3378463.8793207705,
cp_mid evaluates to 3379000.7518995805, and
abstol evaluates to 3.3790007518995804.
[  FAILED  ] DebyeHuckel_b_dot_ak_IAPWS/TestConsistency.cp_eq_dhdT/3

[ RUN      ] DebyeHuckel_b_dot_ak_IAPWS/TestConsistency.cp_eq_dsdT_const_p_times_T/0
The difference between cp_fd and cp_mid is 605.77241257159039, which exceeds abstol, where
cp_fd evaluates to 3377833.9096641899,
cp_mid evaluates to 3378439.6820767615, and
abstol evaluates to 3.3784396820767615.
[  FAILED  ] DebyeHuckel_b_dot_ak_IAPWS/TestConsistency.cp_eq_dsdT_const_p_times_T/0

[ RUN      ] DebyeHuckel_b_dot_ak_IAPWS/TestConsistency.cp_eq_dsdT_const_p_times_T/1
The difference between cp_fd and cp_mid is 261.7126909350045, which exceeds abstol, where
cp_fd evaluates to 2365772.3556048786,
cp_mid evaluates to 2366034.0682958136, and
abstol evaluates to 2.3660340682958134.
[  FAILED  ] DebyeHuckel_b_dot_ak_IAPWS/TestConsistency.cp_eq_dsdT_const_p_times_T/1

[ RUN      ] DebyeHuckel_b_dot_ak_IAPWS/TestConsistency.cp_eq_dsdT_const_p_times_T/2
The difference between cp_fd and cp_mid is 543.39873323589563, which exceeds abstol, where
cp_fd evaluates to 3378543.1893083248,
cp_mid evaluates to 3379086.5880415607, and
abstol evaluates to 3.3790865880415604.
[  FAILED  ] DebyeHuckel_b_dot_ak_IAPWS/TestConsistency.cp_eq_dsdT_const_p_times_T/2

[ RUN      ] DebyeHuckel_b_dot_ak_IAPWS/TestConsistency.cp_eq_dsdT_const_p_times_T/3
The difference between cp_fd and cp_mid is 536.85167412599549, which exceeds abstol, where
cp_fd evaluates to 3378464.4802999701,
cp_mid evaluates to 3379001.3319740961, and
abstol evaluates to 3.3790013319740959.
[  FAILED  ] DebyeHuckel_b_dot_ak_IAPWS/TestConsistency.cp_eq_dsdT_const_p_times_T/3

System information

• Cantera version: main at 13cb334
• OS: macOS 13.3, though presumably this applies universally