{
 "about": {
  "what": "The three scale-up rules for geometrically similar vessels at the same vvm: constant ungassed power per volume (N2 = N1·(D1/D2)^(2/3)), constant tip speed (N2 = N1·D1/D2) and constant kLa (the coalescing correlation with Hughmark's gassed power inverted by Brent's method), each with its power, P/V, vs, kLa and range flags.",
  "oracle": "hand-worked in Python; roots by scipy.optimize.brentq",
  "tolerance": "within 1e-9 relative; the constant-kLa speed within 1e-6 relative",
  "versions": {
   "python": "3.12.14",
   "scipy": "1.17.0",
   "numpy": "2.3.5"
  },
  "recordedOn": "2026-10-07",
  "script": "pipeline/bioreactor-simulator/reference.py"
 },
 "cases": [
  {
   "small": {
    "tankD": 0.22,
    "impellerD": 0.088,
    "impellers": 1,
    "Np": 5.0,
    "volumeL": 7.0,
    "rpm": 700.0,
    "vvm": 1.0,
    "medium": "coalescing",
    "tipSpeed": 3.22536845769,
    "PgPerV": 2925.1721107,
    "kLaPerH": 126.253141038
   },
   "volumeRatio": 10.0,
   "large": {
    "tankD": 0.473975631807,
    "impellerD": 0.189590252723,
    "impellers": 1,
    "Np": 5.0,
    "volumeL": 70.0
   },
   "rules": {
    "constant-power-per-volume": {
     "rpm": 419.638975223,
     "tipSpeed": 4.16572355107,
     "P0W": 419.009369126,
     "PgW": 189.63396723,
     "PgPerV": 2709.05667472,
     "vs": 0.00661218323254,
     "kLaPerH": 179.711357607,
     "vanTRietInRange": true,
     "hughmarkInRange": true
    },
    "constant-tip-speed": {
     "rpm": 324.911218353,
     "tipSpeed": 3.22536845769,
     "P0W": 194.486920891,
     "PgW": 91.4638470247,
     "PgPerV": 1306.62638607,
     "vs": 0.00661218323254,
     "kLaPerH": 134.248319618,
     "vanTRietInRange": true,
     "hughmarkInRange": true
    },
    "constant-kla": {
     "rpm": 307.873914641,
     "tipSpeed": 3.05624046551,
     "P0W": 165.468359822,
     "PgW": 78.4481664626,
     "PgPerV": 1120.68809232,
     "vs": 0.00661218323254,
     "kLaPerH": 126.253141038,
     "vanTRietInRange": true,
     "hughmarkInRange": true
    }
   }
  },
  {
   "small": {
    "tankD": 0.22,
    "impellerD": 0.088,
    "impellers": 1,
    "Np": 5.0,
    "volumeL": 7.0,
    "rpm": 600.0,
    "vvm": 0.5,
    "medium": "coalescing",
    "tipSpeed": 2.76460153516,
    "PgPerV": 2241.8707596,
    "kLaPerH": 80.2621618553
   },
   "volumeRatio": 100.0,
   "large": {
    "tankD": 1.02114954339,
    "impellerD": 0.408459817358,
    "impellers": 1,
    "Np": 5.0,
    "volumeL": 700.0
   },
   "rules": {
    "constant-power-per-volume": {
     "rpm": 215.628819828,
     "tipSpeed": 4.61163330594,
     "P0W": 2638.659584,
     "PgW": 1345.99034898,
     "PgPerV": 1922.84335568,
     "vs": 0.00712275846651,
     "kLaPerH": 162.621334226,
     "vanTRietInRange": true,
     "hughmarkInRange": true
    },
    "constant-tip-speed": {
     "rpm": 129.266081402,
     "tipSpeed": 2.76460153516,
     "P0W": 568.481974295,
     "PgW": 313.118416916,
     "PgPerV": 447.312024166,
     "vs": 0.00712275846651,
     "kLaPerH": 90.7494886501,
     "vanTRietInRange": false,
     "hughmarkInRange": true
    },
    "constant-kla": {
     "rpm": 116.064915081,
     "tipSpeed": 2.48226943164,
     "P0W": 411.496192605,
     "PgW": 230.343138134,
     "PgPerV": 329.061625906,
     "vs": 0.00712275846651,
     "kLaPerH": 80.2621618555,
     "vanTRietInRange": false,
     "hughmarkInRange": true
    }
   }
  },
  {
   "small": {
    "tankD": 0.22,
    "impellerD": 0.088,
    "impellers": 1,
    "Np": 5.0,
    "volumeL": 7.0,
    "rpm": 800.0,
    "vvm": 1.0,
    "medium": "non-coalescing",
    "tipSpeed": 3.68613538021,
    "PgPerV": 4279.84925936,
    "kLaPerH": 788.378436093
   },
   "volumeRatio": 50.0,
   "large": {
    "tankD": 0.810486929701,
    "impellerD": 0.32419477188,
    "impellers": 1,
    "Np": 5.0,
    "volumeL": 350.0
   },
   "rules": {
    "constant-power-per-volume": {
     "rpm": 335.383040132,
     "tipSpeed": 5.69305954709,
     "P0W": 3127.3002477,
     "PgW": 1314.81358859,
     "PgPerV": 3756.6102531,
     "vs": 0.0113066742827,
     "kLaPerH": 934.021278721,
     "vanTRietInRange": true,
     "hughmarkInRange": true
    },
    "constant-tip-speed": {
     "rpm": 217.153409328,
     "tipSpeed": 3.68613538021,
     "P0W": 848.879888475,
     "PgW": 380.94038451,
     "PgPerV": 1088.4010986,
     "vs": 0.0113066742827,
     "kLaPerH": 392.42055959,
     "vanTRietInRange": true,
     "hughmarkInRange": true
    },
    "constant-kla": {
     "rpm": 308.061781222,
     "tipSpeed": 5.22928668064,
     "P0W": 2423.5933624,
     "PgW": 1032.02412646,
     "PgPerV": 2948.64036131,
     "vs": 0.0113066742827,
     "kLaPerH": 788.378436094,
     "vanTRietInRange": true,
     "hughmarkInRange": true
    }
   }
  }
 ]
}