{
  "version": 1,
  "topic": "PR04",
  "title": "Actual pipe velocity versus an erosional screening limit",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Actual speed beside an empirical screen",
    "premise": "Actual mixture Q = 500 US gpm, mixture density 12.2916130645 lb/ft³, assumed field C = 100. Pipe ID varies.",
    "formula": "vactual = Q/(2.448 D²); vscreen = C/√ρ",
    "variables": "Both speeds in ft/s. The density-based screen is not a guarantee against erosion or corrosion.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Pipe ID (in) = 4",
        "explanation": "Actual mixture Q = 500 US gpm, mixture density 12.2916130645 lb/ft³, assumed field C = 100. Pipe ID varies."
      },
      {
        "label": "Select the relation",
        "calculation": "vactual = Q/(2.448 D²); vscreen = C/√ρ",
        "explanation": "Both speeds in ft/s. The density-based screen is not a guarantee against erosion or corrosion."
      },
      {
        "label": "Substitute the values",
        "calculation": "v = 500/(2.448 × 4²) ft/s; screen = 100/√12.2916130645 ft/s; ratio = v/screen",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Actual velocity (ft/s) = 12.7655229; Assumed screen (ft/s) = 28.5230235; Velocity/screen ratio = 0.447551532",
        "explanation": "Diameter changes actual speed; the fixed-density empirical screen remains constant."
      }
    ],
    "result": "Actual velocity (ft/s) = 12.7655229; Assumed screen (ft/s) = 28.5230235; Velocity/screen ratio = 0.447551532",
    "interpretation": "Diameter changes actual speed; the fixed-density empirical screen remains constant.",
    "control": "Pipe ID (in)",
    "columns": [
      "Pipe ID (in)",
      "Actual velocity (ft/s)",
      "Assumed screen (ft/s)",
      "Velocity/screen ratio"
    ],
    "rows": [
      [
        2,
        51.06209150326797,
        28.523023504926556,
        1.7902061292502325
      ],
      [
        3,
        22.69426289034132,
        28.523023504926556,
        0.795647168555659
      ],
      [
        4,
        12.765522875816993,
        28.523023504926556,
        0.4475515323125581
      ],
      [
        5,
        8.169934640522877,
        28.523023504926556,
        0.28643298068003725
      ],
      [
        6,
        5.67356572258533,
        28.523023504926556,
        0.19891179213891474
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Pipe ID (in) = 2. Actual velocity (ft/s): 51.0620915; Assumed screen (ft/s): 28.5230235; Velocity/screen ratio: 1.79020613.",
      "Pipe ID (in) = 3. Actual velocity (ft/s): 22.6942629; Assumed screen (ft/s): 28.5230235; Velocity/screen ratio: 0.795647169.",
      "Pipe ID (in) = 4. Actual velocity (ft/s): 12.7655229; Assumed screen (ft/s): 28.5230235; Velocity/screen ratio: 0.447551532.",
      "Pipe ID (in) = 5. Actual velocity (ft/s): 8.16993464; Assumed screen (ft/s): 28.5230235; Velocity/screen ratio: 0.286432981.",
      "Pipe ID (in) = 6. Actual velocity (ft/s): 5.67356572; Assumed screen (ft/s): 28.5230235; Velocity/screen ratio: 0.198911792."
    ],
    "provenance": "Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.",
    "questions": [
      {
        "question": "Reproduce the baseline Actual velocity (ft/s) without rounding intermediate values.",
        "answer": "v = 500/(2.448 × 4²) ft/s; screen = 100/√12.2916130645 ft/s; ratio = v/screen. Actual velocity (ft/s) = 12.7655229. Diameter changes actual speed; the fixed-density empirical screen remains constant."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Pipe ID (in) changes from 2 to 6. Other controls remain as stated in the premise. Actual velocity (ft/s) changes from 51.0620915 to 5.67356572."
      },
      {
        "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. Both speeds in ft/s. The density-based screen is not a guarantee against erosion or corrosion. Diameter changes actual speed; the fixed-density empirical screen remains constant."
      }
    ]
  },
  "recorded": null
}