{ "about": { "what": "Oxygen solubility (Benson and Krause, with Sander's Henry's law constant as the cross-check), the exhaust gas balance, and Van 't Riet kLa with Hughmark's gassed power worked by hand at ten points, two outside the ranges.", "bensonKrause": "USGS OWQ Technical Memorandum 2011.03 (13 July 2011), attachment eq. 7 (DO° at 1 atm, zero salinity), eqs. 9 to 11 (the pressure factor with the water vapour and θ0 terms); read 6 October 2026; valid 0 to 40 °C and 0.5 to 1.1 atm. Enriched gas by y_O2/0.2095 (from memory).", "sander": "Sander 2023, Atmos Chem Phys 23:10901, the Burkholder et al. 2019 row (Hcp 1.3e-5 mol m⁻³ Pa⁻¹ at 298.15 K, d ln H/d(1/T) 1500 K); read 6 October 2026; the cross-check only", "exhaust": "y_out = (n·y_in − OUR·V)/(n − OUR·V + CER·V), n = p·(vvm·V)/(R·T); inlet CO2 and water vapour not counted (dry basis)", "vanTRiet": { "coalescing": "0.026·(P/V)^0.4·vs^0.5, as quoted by Yawalkar et al. 2002, Can J Chem Eng 80:840-848, Table 2 (read 6 October 2026)", "nonCoalescing": "0.002·(P/V)^0.7·vs^0.2, from memory", "range": "500 to 10,000 W/m³, 0.002 to 2.6 m³ (coalescing) or 4.4 m³: from memory" }, "hughmark": "Pg/P0 = 0.1·(Q/(N·V))^−0.25·(N²D⁴/(g·w·V^(2/3)))^−0.2, w = D/5, g = 9.81 (Yawalkar et al. 2002 eq. 2, Hughmark 1980); its range from Yawalkar Table 3: T 0.21 to 3.33 m, D/T 0.33 to 0.576, H/T 0.75 to 1.87, vs at most 0.053 m/s, Pg/P0 0.31 to 0.8", "power": "P0 = impellers·Np·ρ·N³·D⁵, ρ 1,000 kg/m³, Np 5 for a Rushton turbine (from memory)", "tolerance": "C* and kLa within 1e-9 relative; the range flags exact", "versions": { "python": "3.12.14", "scipy": "1.17.0", "numpy": "2.3.5" }, "recordedOn": "2026-10-07", "script": "pipeline/bioreactor-simulator/reference.py" }, "solubility": [ { "tempC": 20.0, "pressureBar": 1.01325, "yO2": 0.2095, "do0": 9.09242604289, "pressureFactor": 1.0, "cstarAir": 9.09242604289, "cstar": 9.09242604289, "sanderCstar": 9.39929057847 }, { "tempC": 25.0, "pressureBar": 1.01325, "yO2": 0.2095, "do0": 8.26345669782, "pressureFactor": 1.0, "cstarAir": 8.26345669782, "cstar": 8.26345669782, "sanderCstar": 8.55407870779 }, { "tempC": 30.0, "pressureBar": 1.01325, "yO2": 0.2095, "do0": 7.55879604783, "pressureFactor": 1.0, "cstarAir": 7.55879604783, "cstar": 7.55879604783, "sanderCstar": 7.78666023655 }, { "tempC": 37.0, "pressureBar": 1.01325, "yO2": 0.2095, "do0": 6.72699264344, "pressureFactor": 1.0, "cstarAir": 6.72699264344, "cstar": 6.72699264344, "sanderCstar": 6.81811094096 }, { "tempC": 37.0, "pressureBar": 1.1, "yO2": 0.2095, "do0": 6.72699264344, "pressureFactor": 1.09121782521, "cstarAir": 7.34061428255, "cstar": 7.34061428255, "sanderCstar": 7.44038548914 }, { "tempC": 30.0, "pressureBar": 0.8, "yO2": 0.2095, "do0": 7.55879604783, "pressureFactor": 0.780439607039, "cstarAir": 5.89918381725, "cstar": 5.89918381725, "sanderCstar": 6.07624374514 }, { "tempC": 37.0, "pressureBar": 1.01325, "yO2": 0.4, "do0": 6.72699264344, "pressureFactor": 1.0, "cstarAir": 6.72699264344, "cstar": 12.8439000352, "sanderCstar": 13.0178729183 }, { "tempC": 37.0, "pressureBar": 1.01325, "yO2": 1.0, "do0": 6.72699264344, "pressureFactor": 1.0, "cstarAir": 6.72699264344, "cstar": 32.109750088, "sanderCstar": 32.5446822958 }, { "tempC": 25.0, "pressureBar": 1.05, "yO2": 0.35, "do0": 8.26345669782, "pressureFactor": 1.03741506612, "cstarAir": 8.57263447656, "cstar": 14.3218237079, "sanderCstar": 14.8258701743 } ], "exhaust": [ { "ourMmolPerLh": 100.0, "cerMmolPerLh": 105.0, "vvm": 1.0, "volumeL": 10.0, "tempC": 37.0, "pressureBar": 1.01325, "gasMolPerH": 23.5755531999, "yO2out": 0.166729572935, "yCO2out": 0.0444434037635 }, { "ourMmolPerLh": 50.0, "cerMmolPerLh": 26.0, "vvm": 0.5, "volumeL": 6.0, "tempC": 37.0, "pressureBar": 1.01325, "gasMolPerH": 7.07266595998, "yO2out": 0.170555706602, "yCO2out": 0.0225151567273 }, { "ourMmolPerLh": 20.0, "cerMmolPerLh": 20.0, "vvm": 1.5, "volumeL": 8.0, "tempC": 37.0, "pressureBar": 1.01325, "gasMolPerH": 28.2906638399, "yO2out": 0.20384442398, "yCO2out": 0.0056555760199 } ], "kla": [ { "tankD": 0.22, "impellerD": 0.088, "impellers": 1, "Np": 5.0, "volumeL": 7.0, "rpm": 500, "vvm": 1.0, "medium": "coalescing", "P0W": 15.2700207407, "gassedRatio": 0.51397847975, "PgW": 7.84846204607, "PgPerV": 1121.20886372, "vs": 0.00306910358579, "liquidHeight": 0.184146215148, "kLaPerS": 0.0238975693312, "kLaPerH": 86.0312495925, "vanTRietInRange": true, "hughmarkInRange": true }, { "tankD": 0.22, "impellerD": 0.088, "impellers": 1, "Np": 5.0, "volumeL": 7.0, "rpm": 700, "vvm": 1.0, "medium": "coalescing", "P0W": 41.9009369126, "gassedRatio": 0.488681310817, "PgW": 20.4762047749, "PgPerV": 2925.1721107, "vs": 0.00306910358579, "liquidHeight": 0.184146215148, "kLaPerS": 0.035070316955, "kLaPerH": 126.253141038, "vanTRietInRange": true, "hughmarkInRange": true }, { "tankD": 0.22, "impellerD": 0.088, "impellers": 1, "Np": 5.0, "volumeL": 7.0, "rpm": 900, "vvm": 1.0, "medium": "coalescing", "P0W": 89.05476096, "gassedRatio": 0.470602315527, "PgW": 41.9093767165, "PgPerV": 5987.05381664, "vs": 0.00306910358579, "liquidHeight": 0.184146215148, "kLaPerS": 0.0467051085803, "kLaPerH": 168.138390889, "vanTRietInRange": true, "hughmarkInRange": true }, { "tankD": 0.22, "impellerD": 0.088, "impellers": 1, "Np": 5.0, "volumeL": 7.0, "rpm": 900, "vvm": 1.5, "medium": "coalescing", "P0W": 89.05476096, "gassedRatio": 0.425237195213, "PgW": 37.869396771, "PgPerV": 5409.91382443, "vs": 0.00460365537869, "liquidHeight": 0.184146215148, "kLaPerS": 0.054928898446, "kLaPerH": 197.744034406, "vanTRietInRange": true, "hughmarkInRange": true }, { "tankD": 0.22, "impellerD": 0.088, "impellers": 1, "Np": 5.0, "volumeL": 9.0, "rpm": 800, "vvm": 0.8, "medium": "coalescing", "P0W": 62.5460049541, "gassedRatio": 0.523729500318, "PgW": 32.7571879215, "PgPerV": 3639.68754683, "vs": 0.00315679225967, "liquidHeight": 0.236759419476, "kLaPerS": 0.0388170020557, "kLaPerH": 139.741207401, "vanTRietInRange": true, "hughmarkInRange": true }, { "tankD": 0.22, "impellerD": 0.088, "impellers": 1, "Np": 5.0, "volumeL": 7.0, "rpm": 700, "vvm": 1.0, "medium": "non-coalescing", "P0W": 41.9009369126, "gassedRatio": 0.488681310817, "PgW": 20.4762047749, "PgPerV": 2925.1721107, "vs": 0.00306910358579, "liquidHeight": 0.184146215148, "kLaPerS": 0.167779270537, "kLaPerH": 604.005373935, "vanTRietInRange": true, "hughmarkInRange": true }, { "tankD": 0.22, "impellerD": 0.088, "impellers": 1, "Np": 5.0, "volumeL": 11.0, "rpm": 1000, "vvm": 1.2, "medium": "non-coalescing", "P0W": 122.160165926, "gassedRatio": 0.470075637812, "PgW": 57.4245179129, "PgPerV": 5220.41071935, "vs": 0.00578745247607, "liquidHeight": 0.289372623803, "kLaPerS": 0.285706916555, "kLaPerH": 1028.5448996, "vanTRietInRange": true, "hughmarkInRange": true }, { "tankD": 1.2, "impellerD": 0.45, "impellers": 1, "Np": 5.0, "volumeL": 1500.0, "rpm": 150, "vvm": 0.5, "medium": "coalescing", "P0W": 1441.62597656, "gassedRatio": 0.562594441663, "PgW": 811.050761372, "PgPerV": 540.700507581, "vs": 0.0110524266036, "liquidHeight": 1.32629119243, "kLaPerS": 0.0338754348352, "kLaPerH": 121.951565407, "vanTRietInRange": true, "hughmarkInRange": true }, { "tankD": 0.22, "impellerD": 0.088, "impellers": 1, "Np": 5.0, "volumeL": 7.0, "rpm": 1400, "vvm": 1.0, "medium": "coalescing", "P0W": 335.207495301, "gassedRatio": 0.440424257443, "PgW": 147.633512207, "PgPerV": 21090.5017439, "vs": 0.00306910358579, "liquidHeight": 0.184146215148, "kLaPerS": 0.0772883233362, "kLaPerH": 278.23796401, "vanTRietInRange": false, "hughmarkInRange": true }, { "tankD": 0.15, "impellerD": 0.06, "impellers": 1, "Np": 5.0, "volumeL": 2.5, "rpm": 300, "vvm": 1.0, "medium": "coalescing", "P0W": 0.486, "gassedRatio": 0.608695414588, "PgW": 0.29582597149, "PgPerV": 118.330388596, "vs": 0.00235785100877, "liquidHeight": 0.141471060526, "kLaPerS": 0.00852059736831, "kLaPerH": 30.6741505259, "vanTRietInRange": false, "hughmarkInRange": false } ] }