{
  "version": 1,
  "topic": "PR05",
  "title": "Mixture density and gas volume basis in flowline calculations",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Local no-slip mixture density",
    "premise": "Liquid rate 0.001 m³/s, liquid density 850 kg/m³ and gas density 20 kg/m³. Actual gas rate varies.",
    "formula": "ρm = (ρgQg + ρlQl)/(Qg + Ql)",
    "variables": "Rates and densities are local and compatible. The gas fraction is a no-slip flowing fraction, not measured holdup.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Actual gas rate (m³/s) = 0.001",
        "explanation": "Liquid rate 0.001 m³/s, liquid density 850 kg/m³ and gas density 20 kg/m³. Actual gas rate varies."
      },
      {
        "label": "Select the relation",
        "calculation": "ρm = (ρgQg + ρlQl)/(Qg + Ql)",
        "explanation": "Rates and densities are local and compatible. The gas fraction is a no-slip flowing fraction, not measured holdup."
      },
      {
        "label": "Substitute the values",
        "calculation": "(20 × 0.001 + 850 × 0.001)/(0.001 + 0.001) = 435 kg/m³; gas fraction = 0.5",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "No-slip mixture density (kg/m³) = 435; Gas flowing fraction = 0.5",
        "explanation": "A large gas volume share need not imply a large gas mass share."
      }
    ],
    "result": "No-slip mixture density (kg/m³) = 435; Gas flowing fraction = 0.5",
    "interpretation": "A large gas volume share need not imply a large gas mass share.",
    "control": "Actual gas rate (m³/s)",
    "columns": [
      "Actual gas rate (m³/s)",
      "No-slip mixture density (kg/m³)",
      "Gas flowing fraction"
    ],
    "rows": [
      [
        0.0001,
        774.5454545454545,
        0.09090909090909091
      ],
      [
        0.0005,
        573.3333333333333,
        0.3333333333333333
      ],
      [
        0.001,
        435,
        0.5
      ],
      [
        0.005,
        158.33333333333331,
        0.8333333333333334
      ],
      [
        0.01,
        95.45454545454547,
        0.9090909090909092
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Actual gas rate (m³/s) = 0.0001. No-slip mixture density (kg/m³): 774.545455; Gas flowing fraction: 0.0909090909.",
      "Actual gas rate (m³/s) = 0.0005. No-slip mixture density (kg/m³): 573.333333; Gas flowing fraction: 0.333333333.",
      "Actual gas rate (m³/s) = 0.001. No-slip mixture density (kg/m³): 435; Gas flowing fraction: 0.5.",
      "Actual gas rate (m³/s) = 0.005. No-slip mixture density (kg/m³): 158.333333; Gas flowing fraction: 0.833333333.",
      "Actual gas rate (m³/s) = 0.01. No-slip mixture density (kg/m³): 95.4545455; Gas flowing fraction: 0.909090909."
    ],
    "provenance": "Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.",
    "questions": [
      {
        "question": "Reproduce the baseline No-slip mixture density (kg/m³) without rounding intermediate values.",
        "answer": "(20 × 0.001 + 850 × 0.001)/(0.001 + 0.001) = 435 kg/m³; gas fraction = 0.5. No-slip mixture density (kg/m³) = 435. A large gas volume share need not imply a large gas mass share."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Actual gas rate (m³/s) changes from 0.0001 to 0.01. Other controls remain as stated in the premise. No-slip mixture density (kg/m³) changes from 774.545455 to 95.4545455."
      },
      {
        "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. Rates and densities are local and compatible. The gas fraction is a no-slip flowing fraction, not measured holdup. A large gas volume share need not imply a large gas mass share."
      }
    ]
  },
  "recorded": null
}