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    "SI Hazen–Williams k=10.67",
    "Circular full-flow water pipe",
    "Empirical C; not Darcy f, not Colebrook, not Manning n",
    "Not Q=Av identities",
    "Not US customary gpm/in"
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  "limitations": [
    "SI Hazen–Williams: hf = 10.67 L Q^{1.852} / (C^{1.852} D^{4.87})",
    "Circular full-flow water pipe; empirical C; k=10.67; not US gpm/in",
    "Not Darcy–Weisbach, not Colebrook, not Manning, not Q=Av"
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  "boundary_conditions": [
    "missing C, L, Q, D, or hf/S when required → MISSING_REQUIRED_INPUT",
    "C, L, Q, D, hf, or S ≤ 0 → VALUE_MUST_BE_POSITIVE",
    "hf and S together → MIXED_INPUT_ENCODING",
    "f, Re, n, v, ρ, R, or C_d → MIXED_INPUT_ENCODING",
    "k given and k ≠ 10.67 → INVALID_MODE",
    "darcy, colebrook, or manning as mode → INVALID_MODE"
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  "sources": [
    {
      "title": "ISO 80000-4 — Mechanics",
      "section": "Volume flow rate and hydraulic head",
      "url": "https://www.iso.org/standard/31895.html",
      "supports": "Q is volume flow; hf is a length of hydraulic head",
      "kind": "standard"
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    {
      "title": "Williams, G.S. and Hazen, A. Hydraulic Tables",
      "section": "Empirical pipe formula",
      "supports": "Hazen–Williams C correlation for water pipes; SI form uses k=10.67 with Q in m³/s and D in m",
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        "L": 100,
        "Q": 0.02,
        "D": 0.2
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        "C": 120,
        "L": 100,
        "D": 0.2,
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        "v": 0.6366197723675813,
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        "A": 0.031415926535897934,
        "S": 0.0027227779496920654,
        "k": 10.67,
        "model": "hazen_williams"
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      "tol": 1e-12
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        "L": 100,
        "D": 0.2,
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        "model": "hazen_williams"
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        "hf": 0.27227779496920657,
        "Q": 0.020000000000000004,
        "v": 0.6366197723675814,
        "C": 120,
        "L": 100,
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        "A": 0.031415926535897934,
        "S": 0.0027227779496920654,
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        "model": "hazen_williams"
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        "L": 100,
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        "A": 0.031415926535897934,
        "S": 0.0027227779496920654,
        "k": 10.67,
        "model": "hazen_williams"
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        "C": 119.99999999999999,
        "L": 100,
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        "A": 0.031415926535897934,
        "S": 0.0027227779496920654,
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        "model": "hazen_williams"
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        "A": 0.031415926535897934,
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        "f": 0.02
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      "code": "MIXED_INPUT_ENCODING",
      "error": "Hazen–Williams does not take Darcy f. That is /calc/physics/darcy-weisbach"
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    {
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        "n": 0.013
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      "ok": true,
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        "Q": 0.02,
        "D": 0.2,
        "v": 2
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      "expectError": "MIXED_INPUT_ENCODING",
      "code": "MIXED_INPUT_ENCODING",
      "error": "Hazen–Williams does not take v. That is Q=Av. Give Q"
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        "Q": 0.02,
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        "mode": "darcy"
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        "Q": 0.02,
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        "k": 1.318
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