{
  "version": 1,
  "topic": "DH02",
  "title": "Why annular velocity changes when the hole gets wider",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Continuity through changing hole sizes",
    "premise": "Actual rate 600 US gpm, pipe OD 5 in, hole diameter varies. All local volume assumptions remain fixed.",
    "formula": "v = Q/[2.448(Dh² − Dp²)] ft/s; ft/min = 60v",
    "variables": "This average speed is not a local velocity field or a cuttings-transport criterion.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Hole diameter (in) = 8.5",
        "explanation": "Actual rate 600 US gpm, pipe OD 5 in, hole diameter varies. All local volume assumptions remain fixed."
      },
      {
        "label": "Select the relation",
        "calculation": "v = Q/[2.448(Dh² − Dp²)] ft/s; ft/min = 60v",
        "explanation": "This average speed is not a local velocity field or a cuttings-transport criterion."
      },
      {
        "label": "Substitute the values",
        "calculation": "600/[2.448 × (8.5² − 5²)] = 5.18726009 ft/s; multiply by 60 for ft/min",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Average velocity (ft/s) = 5.18726009; Average velocity (ft/min) = 311.235605",
        "explanation": "The wider local flow area lowers average speed at fixed actual Q."
      }
    ],
    "result": "Average velocity (ft/s) = 5.18726009; Average velocity (ft/min) = 311.235605",
    "interpretation": "The wider local flow area lowers average speed at fixed actual Q.",
    "control": "Hole diameter (in)",
    "columns": [
      "Hole diameter (in)",
      "Average velocity (ft/s)",
      "Average velocity (ft/min)"
    ],
    "rows": [
      [
        6.5,
        14.208581983518044,
        852.5149190110826
      ],
      [
        7.5,
        7.8431372549019605,
        470.5882352941176
      ],
      [
        8.5,
        5.187260089220874,
        311.2356053532524
      ],
      [
        10,
        3.2679738562091503,
        196.078431372549
      ],
      [
        12.25,
        1.9598044115197302,
        117.58826469118381
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Hole diameter (in) = 6.5. Average velocity (ft/s): 14.208582; Average velocity (ft/min): 852.514919.",
      "Hole diameter (in) = 7.5. Average velocity (ft/s): 7.84313725; Average velocity (ft/min): 470.588235.",
      "Hole diameter (in) = 8.5. Average velocity (ft/s): 5.18726009; Average velocity (ft/min): 311.235605.",
      "Hole diameter (in) = 10. Average velocity (ft/s): 3.26797386; Average velocity (ft/min): 196.078431.",
      "Hole diameter (in) = 12.25. Average velocity (ft/s): 1.95980441; Average velocity (ft/min): 117.588265."
    ],
    "provenance": "Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.",
    "questions": [
      {
        "question": "Reproduce the baseline Average velocity (ft/s) without rounding intermediate values.",
        "answer": "600/[2.448 × (8.5² − 5²)] = 5.18726009 ft/s; multiply by 60 for ft/min. Average velocity (ft/s) = 5.18726009. The wider local flow area lowers average speed at fixed actual Q."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Hole diameter (in) changes from 6.5 to 12.25. Other controls remain as stated in the premise. Average velocity (ft/s) changes from 14.208582 to 1.95980441."
      },
      {
        "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. This average speed is not a local velocity field or a cuttings-transport criterion. The wider local flow area lowers average speed at fixed actual Q."
      }
    ]
  },
  "recorded": null
}