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  "failed": 0,
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  "assumptions": [
    "Circular-arc bend: h = K v² / (2g); K = 0.21 (θ/90)/√(R/D); Idelchik 90° B1 scaled linearly in θ; K is reported, not an input",
    "Require 0 < θ ≤ 180° and R/D ≥ 1; default g=9.80665 m/s²; default ρ=1000 kg/m³; incompressible",
    "Not minor-loss K given, not mitered R/D<1, not Darcy f, not orifice C_d, not Q=Av, not gradual contraction α"
  ],
  "limitations": [
    "Circular-arc bend: h = K v² / (2g); K = 0.21 (θ/90)/√(R/D); Idelchik 90° B1 scaled linearly in θ; K is reported, not an input",
    "Require 0 < θ ≤ 180° and R/D ≥ 1; default g=9.80665; default ρ=1000 for Δp; incompressible",
    "Not minor-loss K given, not mitered R/D<1, not Darcy f, not orifice C_d, not Q=Av, not gradual contraction α"
  ],
  "boundary_conditions": [
    "missing v_m_s, h, θ, or rd when required → MISSING_REQUIRED_INPUT",
    "v_m_s ≤ 0, h ≤ 0, θ ≤ 0, or rd ≤ 0 → VALUE_MUST_BE_POSITIVE",
    "rd < 1 or θ > 180° → VALUE_OUT_OF_RANGE",
    "K, k, f/Re, Q, C_d, v1/v2, alpha_deg → MIXED_INPUT_ENCODING",
    "minor, mitered, darcy, or orifice as mode → INVALID_MODE"
  ],
  "sources": [
    {
      "title": "ISO 80000-4 — Mechanics",
      "section": "Length, speed, pressure, plane angle",
      "url": "https://www.iso.org/standard/31895.html",
      "supports": "h is a head; v is a speed; Δp is a pressure; θ is a plane angle",
      "kind": "standard"
    },
    {
      "title": "Idelchik — Handbook of Hydraulic Resistance",
      "section": "Circular-arc bends, B1 at 90°",
      "supports": "ζ_90 = 0.21 / √(R/D) for a circular-arc of circular section in the quadratic region; this page scales linearly with θ/90°",
      "kind": "reference"
    }
  ],
  "methods": [],
  "fingerprint": "sha256:1d126b6c47b6a579fb15719ef325004c92359151293d5a62d37a69327ef4701e",
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      "ok": true,
      "kind": "golden",
      "inputs": {
        "v_m_s": 2,
        "theta_deg": 90,
        "rd": 2
      },
      "expect": {
        "h": 0.030284026461467467,
        "v": 2,
        "K": 0.14849242404917495,
        "rd": 2,
        "dP": 296.9848480983499,
        "mode": "head",
        "model": "circular_arc_bend"
      },
      "expectWarning": null,
      "actual": {
        "mode": "head",
        "h": 0.030284026461467467,
        "v": 2,
        "K": 0.14849242404917495,
        "theta": 1.5707963267948966,
        "theta_deg": 90,
        "rd": 2,
        "g": 9.80665,
        "rho": 1000,
        "dP": 296.9848480983499,
        "model": "circular_arc_bend"
      },
      "tol": 1e-12
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      "kind": "golden",
      "inputs": {
        "mode": "pressure",
        "v_m_s": 2,
        "theta_deg": 90,
        "rd": 2
      },
      "expect": {
        "dP": 296.9848480983499,
        "model": "circular_arc_bend"
      },
      "expectWarning": null,
      "actual": {
        "mode": "pressure",
        "h": 0.030284026461467467,
        "v": 2,
        "K": 0.14849242404917495,
        "theta": 1.5707963267948966,
        "theta_deg": 90,
        "rd": 2,
        "g": 9.80665,
        "rho": 1000,
        "dP": 296.9848480983499,
        "model": "circular_arc_bend"
      },
      "tol": 1e-12
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      "ok": true,
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      "inputs": {
        "mode": "velocity",
        "h": 0.030284026461467467,
        "theta_deg": 90,
        "rd": 2
      },
      "expect": {
        "v": 2,
        "model": "circular_arc_bend"
      },
      "expectWarning": null,
      "actual": {
        "mode": "velocity",
        "h": 0.030284026461467467,
        "v": 2,
        "K": 0.14849242404917495,
        "theta": 1.5707963267948966,
        "theta_deg": 90,
        "rd": 2,
        "g": 9.80665,
        "rho": 1000,
        "dP": 296.9848480983499,
        "model": "circular_arc_bend"
      },
      "tol": 1e-12
    },
    {
      "id": "boundary-pb-theta",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "mode": "angle",
        "v_m_s": 2,
        "h": 0.030284026461467467,
        "rd": 2
      },
      "expect": {
        "theta_deg": 90,
        "model": "circular_arc_bend"
      },
      "expectWarning": null,
      "actual": {
        "mode": "angle",
        "h": 0.030284026461467467,
        "v": 2,
        "K": 0.14849242404917495,
        "theta": 1.5707963267948966,
        "theta_deg": 90,
        "rd": 2,
        "g": 9.80665,
        "rho": 1000,
        "dP": 296.9848480983499,
        "model": "circular_arc_bend"
      },
      "tol": 1e-9
    },
    {
      "id": "boundary-pb-missing",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "v_m_s": 2,
        "theta_deg": 90
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      "expectError": "MISSING_REQUIRED_INPUT",
      "code": "MISSING_REQUIRED_INPUT",
      "error": "rd is required"
    },
    {
      "id": "boundary-pb-K",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "v_m_s": 2,
        "theta_deg": 90,
        "rd": 2,
        "K": 0.15
      },
      "expectError": "MIXED_INPUT_ENCODING",
      "code": "MIXED_INPUT_ENCODING",
      "error": "Pipe bend does not take minor-loss K. That is /calc/physics/minor-loss. K=0.21 (θ/90)/√(R/D) is reported"
    },
    {
      "id": "boundary-pb-Q",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "v_m_s": 2,
        "theta_deg": 90,
        "rd": 2,
        "Q": 1
      },
      "expectError": "MIXED_INPUT_ENCODING",
      "code": "MIXED_INPUT_ENCODING",
      "error": "Pipe bend does not take Q or A. That is /calc/physics/flow-rate. Give v_m_s, θ, and R/D"
    },
    {
      "id": "boundary-pb-mode",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "v_m_s": 2,
        "theta_deg": 90,
        "rd": 2,
        "mode": "minor"
      },
      "expectError": "INVALID_MODE",
      "code": "INVALID_MODE",
      "error": "mode must be head|pressure|velocity|angle. Not minor-loss, mitered, Darcy, or Q=Av"
    },
    {
      "id": "boundary-pb-zero",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "v_m_s": 0,
        "theta_deg": 90,
        "rd": 2
      },
      "expectError": "VALUE_MUST_BE_POSITIVE",
      "code": "VALUE_MUST_BE_POSITIVE",
      "error": "v_m_s must be greater than zero"
    },
    {
      "id": "boundary-pb-rd",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "v_m_s": 2,
        "theta_deg": 90,
        "rd": 0.5
      },
      "expectError": "VALUE_OUT_OF_RANGE",
      "code": "VALUE_OUT_OF_RANGE",
      "error": "R/D < 1 is mitered or sharp, not a circular-arc. That is not this page"
    },
    {
      "id": "boundary-pb-wide",
      "ok": true,
      "kind": "boundary",
      "inputs": {
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        "theta_deg": 181,
        "rd": 2
      },
      "expectError": "VALUE_OUT_OF_RANGE",
      "code": "VALUE_OUT_OF_RANGE",
      "error": "Deflection > 180° is not a single circular-arc. That is not this page"
    },
    {
      "id": "property-pb-h",
      "ok": true,
      "kind": "property",
      "detail": "h=K v²/(2g)",
      "expect": 0.030284026461467467,
      "actual": 0.030284026461467467
    },
    {
      "id": "property-pb-K",
      "ok": true,
      "kind": "property",
      "detail": "K=0.21 (θ/90)/√(R/D)",
      "expect": 0.14849242404917495,
      "actual": 0.14849242404917495
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    {
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      "kind": "property",
      "detail": "circular_arc_bend",
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