{
  "version": 1,
  "topic": "PV08",
  "title": "Explore gas Z correlations over reduced pressure",
  "createdAt": "2026-10-24",
  "case": {
    "title": "DAK and Hall-Yarborough Gas Z-Factor Solver",
    "premise": "Fixed controls: Pseudo-reduced absolute temperature = 1.5 dimensionless.",
    "formula": "Solve rho_r * Z_DAK(rho_r, T_pr) = 0.27 * P_pr / T_pr; Z = 0.27 * P_pr / (rho_r * T_pr).; Solve the Hall-Yarborough residual F(y) = 0 for 0 < y < 1; Z = A * P_pr / y; t = 1/T_pr.; Z_DAK = 1 + c1*rho_r + c2*rho_r^2 - c3*rho_r^5 + c4; coefficients A1 to A11 are listed below.; F(y) = -A*P_pr + (y+y^2+y^3-y^4)/(1-y)^3 - B*y^2 + C*y^D; B = 14.76t-9.76t^2+4.58t^3; C = 90.7t-242.2t^2+42.4t^3; D = 2.18+2.82t.",
    "variables": "Pseudo-reduced pressure is varied within the supported input range. This comparison deliberately supports only 0.2 <= P_pr <= 15 and 1.2 <= T_pr <= 3. The software envelope is narrower than the cited DAK range; near-critical, subcritical and higher-pressure extensions are not implemented. Bounds are enforced, not treated as extrapolation warnings.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Pseudo-reduced pressure (dimensionless) = 3",
        "explanation": "Fixed controls: Pseudo-reduced absolute temperature = 1.5 dimensionless."
      },
      {
        "label": "Select the relation",
        "calculation": "Solve rho_r * Z_DAK(rho_r, T_pr) = 0.27 * P_pr / T_pr; Z = 0.27 * P_pr / (rho_r * T_pr).; Solve the Hall-Yarborough residual F(y) = 0 for 0 < y < 1; Z = A * P_pr / y; t = 1/T_pr.; Z_DAK = 1 + c1*rho_r + c2*rho_r^2 - c3*rho_r^5 + c4; coefficients A1 to A11 are listed below.; F(y) = -A*P_pr + (y+y^2+y^3-y^4)/(1-y)^3 - B*y^2 + C*y^D; B = 14.76t-9.76t^2+4.58t^3; C = 90.7t-242.2t^2+42.4t^3; D = 2.18+2.82t.",
        "explanation": "Pseudo-reduced pressure is varied within the supported input range. This comparison deliberately supports only 0.2 <= P_pr <= 15 and 1.2 <= T_pr <= 3. The software envelope is narrower than the cited DAK range; near-critical, subcritical and higher-pressure extensions are not implemented. Bounds are enforced, not treated as extrapolation warnings."
      },
      {
        "label": "Substitute the values",
        "calculation": "P_pr = 3 dimensionless; with the fixed controls above, the workflow evaluates DAK Z = 0.776127593 dimensionless; Hall-Yarborough Z = 0.774828273 dimensionless",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "DAK Z (dimensionless) = 0.776127593; Hall-Yarborough Z (dimensionless) = 0.774828273",
        "explanation": "Single gas-phase correlation evaluation only. No dew-point or phase-equilibrium test, composition adjustment, sour-gas correction, density in mass units, gas compressibility derivative or fluid flash is performed. Passing an envelope check does not establish that a supplied fluid is in one phase. Brackets, a bounded iteration count and final residual tests prevent an unverified number being returned. The chart shows correlation sensitivity within the selected interval, not measured error bars. Correlation difference and solver residual are different diagnostics and must not be presented as guaranteed accuracy."
      }
    ],
    "result": "DAK Z (dimensionless) = 0.776127593; Hall-Yarborough Z (dimensionless) = 0.774828273",
    "interpretation": "Single gas-phase correlation evaluation only. No dew-point or phase-equilibrium test, composition adjustment, sour-gas correction, density in mass units, gas compressibility derivative or fluid flash is performed. Passing an envelope check does not establish that a supplied fluid is in one phase. Brackets, a bounded iteration count and final residual tests prevent an unverified number being returned. The chart shows correlation sensitivity within the selected interval, not measured error bars. Correlation difference and solver residual are different diagnostics and must not be presented as guaranteed accuracy.",
    "control": "Pseudo-reduced pressure (dimensionless)",
    "columns": [
      "Pseudo-reduced pressure (dimensionless)",
      "DAK Z (dimensionless)",
      "Hall-Yarborough Z (dimensionless)"
    ],
    "rows": [
      [
        0.5,
        0.9509373234549884,
        0.9496854734591493
      ],
      [
        1,
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      ],
      [
        3,
        0.7761275930721209,
        0.7748282730532944
      ],
      [
        5,
        0.8091313605975782,
        0.8068393899298351
      ],
      [
        10,
        1.1300196263479692,
        1.1337877036829704
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Pseudo-reduced pressure (dimensionless) = 0.5. DAK Z (dimensionless): 0.950937323; Hall-Yarborough Z (dimensionless): 0.949685473.",
      "Pseudo-reduced pressure (dimensionless) = 1. DAK Z (dimensionless): 0.90340059; Hall-Yarborough Z (dimensionless): 0.901817933.",
      "Pseudo-reduced pressure (dimensionless) = 3. DAK Z (dimensionless): 0.776127593; Hall-Yarborough Z (dimensionless): 0.774828273.",
      "Pseudo-reduced pressure (dimensionless) = 5. DAK Z (dimensionless): 0.809131361; Hall-Yarborough Z (dimensionless): 0.80683939.",
      "Pseudo-reduced pressure (dimensionless) = 10. DAK Z (dimensionless): 1.13001963; Hall-Yarborough Z (dimensionless): 1.1337877."
    ],
    "provenance": "Original recorded synthetic cases solved with the site's gas-z-comparison engine. The full inputs, outputs and diagnostic rows are in the downloadable JSON. These are discrete solved states; the lesson does not interpolate a new solve.",
    "questions": [
      {
        "question": "Reproduce the baseline DAK Z (dimensionless) without rounding intermediate values.",
        "answer": "P_pr = 3 dimensionless; with the fixed controls above, the workflow evaluates DAK Z = 0.776127593 dimensionless; Hall-Yarborough Z = 0.774828273 dimensionless. DAK Z (dimensionless) = 0.776127593. Single gas-phase correlation evaluation only. No dew-point or phase-equilibrium test, composition adjustment, sour-gas correction, density in mass units, gas compressibility derivative or fluid flash is performed. Passing an envelope check does not establish that a supplied fluid is in one phase. Brackets, a bounded iteration count and final residual tests prevent an unverified number being returned. The chart shows correlation sensitivity within the selected interval, not measured error bars. Correlation difference and solver residual are different diagnostics and must not be presented as guaranteed accuracy."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Pseudo-reduced pressure (dimensionless) changes from 0.5 to 10. Other controls remain as stated in the premise. DAK Z (dimensionless) changes from 0.950937323 to 1.13001963."
      },
      {
        "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. Pseudo-reduced pressure is varied within the supported input range. This comparison deliberately supports only 0.2 <= P_pr <= 15 and 1.2 <= T_pr <= 3. The software envelope is narrower than the cited DAK range; near-critical, subcritical and higher-pressure extensions are not implemented. Bounds are enforced, not treated as extrapolation warnings. Single gas-phase correlation evaluation only. No dew-point or phase-equilibrium test, composition adjustment, sour-gas correction, density in mass units, gas compressibility derivative or fluid flash is performed. Passing an envelope check does not establish that a supplied fluid is in one phase. Brackets, a bounded iteration count and final residual tests prevent an unverified number being returned. The chart shows correlation sensitivity within the selected interval, not measured error bars. Correlation difference and solver residual are different diagnostics and must not be presented as guaranteed accuracy."
      }
    ]
  },
  "recorded": {
    "kind": "workflow",
    "workflow": "gas-z-comparison",
    "parameter": "P_pr",
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      {
        "inputs": {
          "P_pr": 0.5,
          "T_pr": 1.5
        },
        "result": {
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            {
              "label": "DAK Z",
              "value": 0.9509373234549884,
              "unit": "dimensionless"
            },
            {
              "label": "Hall-Yarborough Z",
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            },
            {
              "label": "HY minus DAK / DAK",
              "value": -0.1316437965954235,
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            },
            {
              "label": "DAK relative residual",
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              "unit": "dimensionless"
            },
            {
              "label": "HY relative residual",
              "value": 5.834797932986854e-13,
              "unit": "dimensionless"
            },
            {
              "label": "DAK function evaluations",
              "value": 51,
              "unit": "count"
            },
            {
              "label": "HY function evaluations",
              "value": 49,
              "unit": "count"
            }
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          "columns": [
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          "chart": {
            "xLabel": "Pseudo-reduced pressure",
            "yLabel": "Z",
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      },
      {
        "inputs": {
          "P_pr": 1,
          "T_pr": 1.5
        },
        "result": {
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      },
      {
        "inputs": {
          "P_pr": 3,
          "T_pr": 1.5
        },
        "result": {
          "outputs": [
            {
              "label": "DAK Z",
              "value": 0.7761275930721209,
              "unit": "dimensionless"
            },
            {
              "label": "Hall-Yarborough Z",
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            {
              "label": "HY minus DAK / DAK",
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            {
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            {
              "label": "DAK function evaluations",
              "value": 51,
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