{
  "version": 1,
  "topic": "TH09",
  "title": "Calibrating heavy end PVT parameters without hiding residuals",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Synthetic heavy-end calibration residuals",
    "premise": "Default supported gamma heavy-end model: T 350 K, M 200 g/mol, SG 0.85, alpha 2; six synthetic vapor-fraction targets at 1–6 MPa. Full inputs and fitted outputs are downloadable.",
    "formula": "residual_i = predicted vapor mole fraction_i − supplied target_i",
    "variables": "Targets are synthetic, not laboratory measurements. The fitted interaction is a model parameter, not a measured fluid property.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Pressure (MPa(a)) = 4",
        "explanation": "Default supported gamma heavy-end model: T 350 K, M 200 g/mol, SG 0.85, alpha 2; six synthetic vapor-fraction targets at 1–6 MPa. Full inputs and fitted outputs are downloadable."
      },
      {
        "label": "Select the relation",
        "calculation": "residual_i = predicted vapor mole fraction_i − supplied target_i",
        "explanation": "Targets are synthetic, not laboratory measurements. The fitted interaction is a model parameter, not a measured fluid property."
      },
      {
        "label": "Substitute the values",
        "calculation": "Fitted methane / heavy-cut interaction parameter = 0.0703360405 dimensionless; Vapor mole-fraction RMSE = 0.0000121872986 dimensionless; Observation count = 6 dimensionless; Vapor-fraction sensitivity norm = 0.202330764 dimensionless",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Observed vapor mole fraction = 0.729672772; Fitted vapor mole fraction = 0.729663175; Residual = -0.00000959662336",
        "explanation": "Small training residuals verify this synthetic fitting path. Held-out states and independent PVT properties are required for prediction assessment."
      }
    ],
    "result": "Observed vapor mole fraction = 0.729672772; Fitted vapor mole fraction = 0.729663175; Residual = -0.00000959662336",
    "interpretation": "Small training residuals verify this synthetic fitting path. Held-out states and independent PVT properties are required for prediction assessment.",
    "control": "Pressure (MPa(a))",
    "columns": [
      "Pressure (MPa(a))",
      "Observed vapor mole fraction",
      "Fitted vapor mole fraction",
      "Residual"
    ],
    "rows": [
      [
        1,
        0.8772817998873333,
        0.8772704370607538,
        -0.000011362826579541796
      ],
      [
        2,
        0.8228104123626551,
        0.8227937422683129,
        -0.000016670094342141617
      ],
      [
        3,
        0.7734980165387924,
        0.7734825311673061,
        -0.00001548537148621243
      ],
      [
        4,
        0.7296727716500049,
        0.729663175026642,
        -0.000009596623362861934
      ],
      [
        5,
        0.6903038635788712,
        0.6903036749087409,
        -1.8867013029666424e-7
      ],
      [
        6,
        0.654311854809398,
        0.6543241936950557,
        0.000012338885657658949
      ]
    ],
    "baseline": 3,
    "rowNotes": [
      "Pressure (MPa(a)) = 1. Observed vapor mole fraction: 0.8772818; Fitted vapor mole fraction: 0.877270437; Residual: -0.0000113628266.",
      "Pressure (MPa(a)) = 2. Observed vapor mole fraction: 0.822810412; Fitted vapor mole fraction: 0.822793742; Residual: -0.0000166700943.",
      "Pressure (MPa(a)) = 3. Observed vapor mole fraction: 0.773498017; Fitted vapor mole fraction: 0.773482531; Residual: -0.0000154853715.",
      "Pressure (MPa(a)) = 4. Observed vapor mole fraction: 0.729672772; Fitted vapor mole fraction: 0.729663175; Residual: -0.00000959662336.",
      "Pressure (MPa(a)) = 5. Observed vapor mole fraction: 0.690303864; Fitted vapor mole fraction: 0.690303675; Residual: -1.8867013e-7.",
      "Pressure (MPa(a)) = 6. Observed vapor mole fraction: 0.654311855; Fitted vapor mole fraction: 0.654324194; Residual: 0.0000123388857."
    ],
    "provenance": "Actual supported bounded calibration run with its raw target rows preserved in the downloadable record.",
    "questions": [
      {
        "question": "Reproduce the baseline Observed vapor mole fraction without rounding intermediate values.",
        "answer": "Fitted methane / heavy-cut interaction parameter = 0.0703360405 dimensionless; Vapor mole-fraction RMSE = 0.0000121872986 dimensionless; Observation count = 6 dimensionless; Vapor-fraction sensitivity norm = 0.202330764 dimensionless. Observed vapor mole fraction = 0.729672772. Small training residuals verify this synthetic fitting path. Held-out states and independent PVT properties are required for prediction assessment."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Pressure (MPa(a)) changes from 1 to 6. Other controls remain as stated in the premise. Observed vapor mole fraction changes from 0.8772818 to 0.654311855."
      },
      {
        "question": "What would be needed to apply this example to a different physical system?",
        "answer": "Replace the synthetic controls with compatible documented inputs and check the relation's assumptions. Targets are synthetic, not laboratory measurements. The fitted interaction is a model parameter, not a measured fluid property. Small training residuals verify this synthetic fitting path. Held-out states and independent PVT properties are required for prediction assessment."
      }
    ]
  },
  "recorded": {
    "inputs": {
      "T": 350,
      "M": 200,
      "SG": 0.85,
      "z1": 0.45,
      "z2": 0.15,
      "z3": 0.15,
      "z4": 0.1,
      "z5": 0.05,
      "z6": 0.03,
      "z7": 0.07,
      "alpha": 2
    },
    "result": {
      "outputs": [
        {
          "label": "Fitted methane / heavy-cut interaction parameter",
          "value": 0.07033604047644901,
          "unit": "dimensionless"
        },
        {
          "label": "Vapor mole-fraction RMSE",
          "value": 0.000012187298572544832,
          "unit": "dimensionless"
        },
        {
          "label": "Observation count",
          "value": 6,
          "unit": "dimensionless"
        },
        {
          "label": "Vapor-fraction sensitivity norm",
          "value": 0.202330763754408,
          "unit": "dimensionless"
        }
      ],
      "columns": [
        "Pressure (MPa(a))",
        "Observed vapor mole fraction",
        "Fitted vapor mole fraction",
        "Residual"
      ],
      "rows": [
        [
          1,
          0.8772817998873333,
          0.8772704370607538,
          -0.000011362826579541796
        ],
        [
          2,
          0.8228104123626551,
          0.8227937422683129,
          -0.000016670094342141617
        ],
        [
          3,
          0.7734980165387924,
          0.7734825311673061,
          -0.00001548537148621243
        ],
        [
          4,
          0.7296727716500049,
          0.729663175026642,
          -0.000009596623362861934
        ],
        [
          5,
          0.6903038635788712,
          0.6903036749087409,
          -1.8867013029666424e-7
        ],
        [
          6,
          0.654311854809398,
          0.6543241936950557,
          0.000012338885657658949
        ]
      ],
      "chart": {
        "xLabel": "Pressure (MPa(a))",
        "yLabel": "Vapor mole fraction",
        "series": [
          {
            "name": "Observed",
            "x": [
              1,
              2,
              3,
              4,
              5,
              6
            ],
            "y": [
              0.8772817998873333,
              0.8228104123626551,
              0.7734980165387924,
              0.7296727716500049,
              0.6903038635788712,
              0.654311854809398
            ],
            "mode": "markers"
          },
          {
            "name": "Fitted",
            "x": [
              1,
              2,
              3,
              4,
              5,
              6
            ],
            "y": [
              0.8772704370607538,
              0.8227937422683129,
              0.7734825311673061,
              0.729663175026642,
              0.6903036749087409,
              0.6543241936950557
            ],
            "mode": "lines+markers"
          }
        ]
      },
      "warnings": [
        "Only one common methane/heavy-cut kij is fitted at fixed temperature and feed. All other pairs remain zero. This does not calibrate cut distributions, volume shifts or calorics, and is not proof of a unique or experimentally accurate fluid model. Use independent holdout data before prediction; no extrapolation accuracy is claimed."
      ]
    }
  }
}