{
  "version": 1,
  "topic": "PR08",
  "title": "Explore how pipe diameter changes velocity and friction screening",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Diameter, velocity and turbulent liquid friction",
    "premise": "Water-like liquid: Q = 0.01 m³/s, ρ = 1,000 kg/m³, μ = 0.001 Pa·s, absolute roughness 0.00001 m, L = 100 m.",
    "formula": "v = 4Q/(πD²); Re = ρvD/μ; Δp = fD(L/D)ρv²/2",
    "variables": "Haaland fD is recalculated with Re and ε/D for every diameter, all inside its turbulent domain.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Pipe ID (m) = 0.1",
        "explanation": "Water-like liquid: Q = 0.01 m³/s, ρ = 1,000 kg/m³, μ = 0.001 Pa·s, absolute roughness 0.00001 m, L = 100 m."
      },
      {
        "label": "Select the relation",
        "calculation": "v = 4Q/(πD²); Re = ρvD/μ; Δp = fD(L/D)ρv²/2",
        "explanation": "Haaland fD is recalculated with Re and ε/D for every diameter, all inside its turbulent domain."
      },
      {
        "label": "Substitute the values",
        "calculation": "At D = 0.1 m: v = 4 × 0.01/(π × 0.1²); Re = 1,000v × 0.1/0.001; evaluate Haaland at ε/D = 0.0001, then Darcy–Weisbach",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Velocity (m/s) = 1.27323954; Darcy friction loss (kPa) = 14.1634002; Reynolds number = 127323.954",
        "explanation": "Changing diameter alters speed, Re, relative roughness and the computed friction factor together."
      }
    ],
    "result": "Velocity (m/s) = 1.27323954; Darcy friction loss (kPa) = 14.1634002; Reynolds number = 127323.954",
    "interpretation": "Changing diameter alters speed, Re, relative roughness and the computed friction factor together.",
    "control": "Pipe ID (m)",
    "columns": [
      "Pipe ID (m)",
      "Velocity (m/s)",
      "Darcy friction loss (kPa)",
      "Reynolds number"
    ],
    "rows": [
      [
        0.05,
        5.09295817894065,
        424.5210774226862,
        254647.90894703253
      ],
      [
        0.075,
        2.263536968418067,
        57.5191728643961,
        169765.272631355
      ],
      [
        0.1,
        1.2732395447351625,
        14.163400177375795,
        127323.95447351626
      ],
      [
        0.125,
        0.8148733086305042,
        4.809432073394143,
        101859.16357881302
      ],
      [
        0.15,
        0.5658842421045167,
        1.9967344562222584,
        84882.6363156775
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Pipe ID (m) = 0.05. Velocity (m/s): 5.09295818; Darcy friction loss (kPa): 424.521077; Reynolds number: 254647.909.",
      "Pipe ID (m) = 0.075. Velocity (m/s): 2.26353697; Darcy friction loss (kPa): 57.5191729; Reynolds number: 169765.273.",
      "Pipe ID (m) = 0.1. Velocity (m/s): 1.27323954; Darcy friction loss (kPa): 14.1634002; Reynolds number: 127323.954.",
      "Pipe ID (m) = 0.125. Velocity (m/s): 0.814873309; Darcy friction loss (kPa): 4.80943207; Reynolds number: 101859.164.",
      "Pipe ID (m) = 0.15. Velocity (m/s): 0.565884242; Darcy friction loss (kPa): 1.99673446; Reynolds number: 84882.6363."
    ],
    "provenance": "Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.",
    "questions": [
      {
        "question": "Reproduce the baseline Velocity (m/s) without rounding intermediate values.",
        "answer": "At D = 0.1 m: v = 4 × 0.01/(π × 0.1²); Re = 1,000v × 0.1/0.001; evaluate Haaland at ε/D = 0.0001, then Darcy–Weisbach. Velocity (m/s) = 1.27323954. Changing diameter alters speed, Re, relative roughness and the computed friction factor together."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Pipe ID (m) changes from 0.05 to 0.15. Other controls remain as stated in the premise. Velocity (m/s) changes from 5.09295818 to 0.565884242."
      },
      {
        "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. Haaland fD is recalculated with Re and ε/D for every diameter, all inside its turbulent domain. Changing diameter alters speed, Re, relative roughness and the computed friction factor together."
      }
    ]
  },
  "recorded": null
}