DahnekeCondensationEvaporation

{
  "reference": {
    "citation": "Dahneke, B. (1983) Simple Kinetic Theory of Brownian Diffusion in Vapors and Aerosols, in Theory of Dispersed Multiphase Flow, Academic Press, pp. 97–133. Eqs. 2.1–2.3, 3.6, 5.4–5.8, 6.2. Migrated from Aerosol.jl src/dahneke_brownian_diffusion.jl:157 @ commit c8c640bb636b22f2425460e9ed5cba50793bfee1 (esm-ivb).",
    "notes": "source_ref: Aerosol.jl@c8c640bb:src/dahneke_brownian_diffusion.jl:157\n\nUpstream DahnekeCondensationEvaporation is an MTK @component that composes two subsystems — DahnekeMassTransportCorrection and DahnekeHeatTransportCorrection — and adds four algebraic equations. All variables are determined by explicit algebraic equations with no time derivatives; all are purely observed.\n\nThis ESM flattens both subsystems into a single model. The constant k_B = 1.380649×10⁻²³ J/K and π = 3.14159265358979 are folded as floating-point literals. 8/π = 2.5464790894703255 and 4π = 12.566370614359172 are precomputed.\n\nEquation chain (inlined from DahnekeMassTransportCorrection, with T = T_inf, r shared):\n  c_bar_v = sqrt(8 k_B T_inf / (π m_v))              [Eq. 6.2: mean thermal velocity of vapor]\n  ell_D   = 2 D_v / c_bar_v                            [Eq. 3.6: diffusional mean-free-path]\n  Kn_D    = ell_D / r                                   [diffusion Knudsen number]\n  β       = (Kn_D+1) / (1 + 2 Kn_D(Kn_D+1)/δ_m)       [Eq. 5.5: mass transport correction]\n\nEquation chain (inlined from DahnekeHeatTransportCorrection, with r shared):\n  Kn_K    = 2 κ / (c_bar_gas n_gas m_gas c_v r)         [Eq. 5.7: heat transfer Knudsen number]\n  β_q     = (Kn_K+1) / (1 + 2 Kn_K(Kn_K+1)/α)          [Eq. 5.7: heat transport correction]\n\nMain equations:\n  I_M     = 4π r D_v (n_inf - n_s)                      [Eq. 2.1: Maxwell molecular growth rate]\n  I       = β I_M                                        [Eq. 5.4: corrected growth rate]\n  Q_M     = 4π r κ (T_inf - T_o)                        [Eq. 2.2: Maxwell heat conduction rate]\n  Q       = β_q Q_M                                     [Eq. 5.6: corrected heat conduction rate]\n\nThe conservation equation β I_M = β_q Q_M / L_vap (Eq. 5.8) couples mass and heat transfer at steady state. It is NOT enforced as an equation in this model — T_inf, T_o, n_inf, n_s are all independent input parameters. L_vap is included as a parameter for reference.\n\nLimiting cases for β (Eq. 5.5):\n  Kn_D → 0 (continuum): β → 1, I → I_M (Maxwell limit)\n  Kn_D → ∞ (free-molecular): β → δ_m c_bar_v r / (4 D_v)\n\nLimiting cases for β_q (Eq. 5.7):\n  Kn_K → 0 (continuum): β_q → 1, Q → Q_M\n  Kn_K → ∞ (free-molecular): β_q → α / (2 Kn_K)\n\nDefault parameters correspond to water vapor in air at 298.15 K with a 100 nm droplet: I_M = 2.513×10¹⁰ s⁻¹, β = 0.444 (strong transition-regime correction), Q_M = 1.634×10⁻⁷ W, β_q = 0.345.\n\nTopological evaluation order: {T_inf,m_v} → c_bar_v → ell_D → Kn_D → β; {κ,c_bar_gas,n_gas,m_gas,c_v,r} → Kn_K → β_q; {r,D_v,n_inf,n_s} → I_M → I; {r,κ,T_inf,T_o} → Q_M → Q. The dependency graph is acyclic.\n\nTolerance rationale (model-level rel=1e-9): all equations are purely algebraic compositions of *, /, sqrt, + in float64. Round-off is ~1e-15 relative; rel=1e-9 leaves six orders of headroom — consistent with capillary_penetration.esm and coagulation_rate.esm from the same Dahneke (1983) source."
  },
  "tolerance": {
    "rel": 1e-09
  },
  "variables": {
    "T_inf": {
      "type": "parameter",
      "units": "K",
      "default": 298.15,
      "description": "Gas temperature at infinity (far from the droplet). Default 298.15 K (25 °C). Also used as the vapor-phase temperature for the mass transport correction (T in DahnekeMassTransportCorrection)."
    },
    "T_o": {
      "type": "parameter",
      "units": "K",
      "default": 293.15,
      "description": "Droplet surface temperature. Default 293.15 K (20 °C), 5 K cooler than T_inf — a typical evaporative cooling offset."
    },
    "n_inf": {
      "type": "parameter",
      "units": "m^-3",
      "default": 1e+21,
      "description": "Vapor number concentration at infinity (far from the droplet). Default 1×10²¹ m⁻³ (~40 µg/m³ for water at 298 K)."
    },
    "n_s": {
      "type": "parameter",
      "units": "m^-3",
      "default": 0.0,
      "description": "Saturation vapor number concentration at the droplet surface. Default 0 (maximum driving force for condensation). Set equal to n_inf for no net flux; set greater than n_inf for evaporation."
    },
    "L_vap": {
      "type": "parameter",
      "units": "J",
      "default": 7.48e-20,
      "description": "Molecular latent heat of vaporization (energy released per molecule condensed). Default 7.48×10⁻²⁰ J (≈ 45 kJ/mol for water). Used in the conservation equation β I_M = β_q Q_M / L_vap (Eq. 5.8), which links mass and heat transport at steady state. Not used in any computed variable; included for reference."
    },
    "r": {
      "type": "parameter",
      "units": "m",
      "default": 1e-07,
      "description": "Droplet (or particle) radius. Default 100 nm — transition-regime aerosol where both β and β_q differ significantly from 1. Shared between the mass transport and heat transport subsystems."
    },
    "D_v": {
      "type": "parameter",
      "units": "m^2/s",
      "default": 2e-05,
      "description": "Diffusion coefficient of vapor molecules in the carrier gas. Default 2.0×10⁻⁵ m²/s (air at 298 K)."
    },
    "κ": {
      "type": "parameter",
      "units": "W/(m*K)",
      "default": 0.026,
      "description": "Thermal conductivity of the carrier gas. Default 0.026 W/(m·K) (air at 298 K)."
    },
    "m_v": {
      "type": "parameter",
      "units": "kg",
      "default": 4.81e-26,
      "description": "Mass of a diffusing vapor molecule. Default 4.81×10⁻²⁶ kg (~29 g/mol, dry air average; replace with the actual vapor molecule mass, e.g. 2.99×10⁻²⁶ kg for water). Determines the mean thermal velocity c_bar_v."
    },
    "δ_m": {
      "type": "parameter",
      "units": "1",
      "default": 1.0,
      "description": "Condensation-evaporation sticking coefficient (mass accommodation coefficient). Dimensionless; range (0, 1]. Default 1.0 (perfect sticking; every molecule that strikes the surface is absorbed). Smaller δ_m suppresses β and thus I in the transition/free-molecular regime."
    },
    "n_gas": {
      "type": "parameter",
      "units": "m^-3",
      "default": 2.46e+25,
      "description": "Number concentration of the carrier gas molecules. Default 2.46×10²⁵ m⁻³ (air at 298 K, 1 atm). Used only in Kn_K."
    },
    "c_bar_gas": {
      "type": "parameter",
      "units": "m/s",
      "default": 500.0,
      "description": "Mean thermal velocity of carrier gas molecules. Default 500 m/s (air at 298 K). Used only in Kn_K."
    },
    "c_v": {
      "type": "parameter",
      "units": "J/(kg*K)",
      "default": 718.0,
      "description": "Specific heat at constant volume of the carrier gas per unit mass. Default 718 J/(kg·K) (air). Used only in Kn_K."
    },
    "m_gas": {
      "type": "parameter",
      "units": "kg",
      "default": 4.81e-26,
      "description": "Mass of a carrier gas molecule. Default 4.81×10⁻²⁶ kg (~29 g/mol, dry air). Used only in Kn_K."
    },
    "α": {
      "type": "parameter",
      "units": "1",
      "default": 1.0,
      "description": "Thermal accommodation coefficient: probability that a gas molecule striking the droplet surface reaches thermal equilibrium with it. Dimensionless; range (0, 1]. Default 1.0 (perfect accommodation). Smaller α suppresses β_q and thus Q in the transition/free-molecular regime."
    },
    "c_bar_v": {
      "type": "unknown",
      "units": "m/s",
      "description": "Mean thermal velocity of vapor molecules (Eq. 6.2): c_bar_v = sqrt(8 k_B T_inf / (π m_v)). k_B = 1.380649×10⁻²³ J/K and π = 3.14159265358979 folded as literals; 8/π = 2.5464790894703255 precomputed."
    },
    "ell_D": {
      "type": "unknown",
      "units": "m",
      "description": "Diffusional mean-free-path of vapor molecules (Eq. 3.6): ell_D = 2 D_v / c_bar_v. Characteristic length for the mass transport correction; Kn_D = ell_D / r."
    },
    "Kn_D": {
      "type": "unknown",
      "units": "1",
      "description": "Diffusion Knudsen number for mass transport: Kn_D = ell_D / r. Dimensionless; Kn_D ≪ 1: continuum (β → 1); Kn_D ≫ 1: free-molecular (β → δ_m c_bar_v r / (4 D_v))."
    },
    "β": {
      "type": "unknown",
      "units": "1",
      "description": "Mass transport correction factor (Eq. 5.5): β = (Kn_D+1) / (1 + 2 Kn_D(Kn_D+1)/δ_m). Dimensionless; ranges from 1 (continuum) to 0 (free-molecular). Multiplies the Maxwell rate I_M to give the actual rate I."
    },
    "Kn_K": {
      "type": "unknown",
      "units": "1",
      "description": "Heat transfer Knudsen number (Eq. 5.7): Kn_K = 2 κ / (c_bar_gas n_gas m_gas c_v r). Dimensionless; the product n_gas m_gas c_v is the volumetric heat capacity of the carrier gas [J/(m³·K)]. Kn_K ≪ 1: continuum (β_q → 1); Kn_K ≫ 1: free-molecular (β_q → α/(2 Kn_K))."
    },
    "β_q": {
      "type": "unknown",
      "units": "1",
      "description": "Heat transport correction factor (Eq. 5.7): β_q = (Kn_K+1) / (1 + 2 Kn_K(Kn_K+1)/α). Dimensionless; parallel to β. Multiplies the Maxwell heat conduction rate Q_M to give the actual rate Q."
    },
    "I_M": {
      "type": "unknown",
      "units": "1/s",
      "description": "Maxwell (continuum) molecular growth rate (Eq. 2.1): I_M = 4π r D_v (n_inf - n_s). Rate at which vapor molecules condense in the continuum limit. 4π = 12.566370614359172 folded as literal."
    },
    "I": {
      "type": "unknown",
      "units": "1/s",
      "description": "Transition-regime corrected molecular growth rate (Eq. 5.4): I = β I_M. The actual condensation/evaporation rate accounting for non-continuum effects. I < I_M for Kn_D > 0."
    },
    "Q_M": {
      "type": "unknown",
      "units": "W",
      "description": "Maxwell (continuum) heat conduction rate (Eq. 2.2): Q_M = 4π r κ (T_inf - T_o). Heat flux from the gas to the droplet surface in the continuum limit. Positive for T_inf > T_o (droplet cooler than ambient)."
    },
    "Q": {
      "type": "unknown",
      "units": "W",
      "description": "Transition-regime corrected heat conduction rate (Eq. 5.6): Q = β_q Q_M. The actual heat flux accounting for non-continuum effects. Q < Q_M for Kn_K > 0."
    }
  },
  "equations": [
    {
      "lhs": "c_bar_v",
      "rhs": {
        "op": "sqrt",
        "args": [
          {
            "op": "/",
            "args": [
              {
                "op": "*",
                "args": [
                  2.5464790894703255,
                  1.380649e-23,
                  "T_inf"
                ]
              },
              "m_v"
            ]
          }
        ]
      }
    },
    {
      "lhs": "ell_D",
      "rhs": {
        "op": "/",
        "args": [
          {
            "op": "*",
            "args": [
              2.0,
              "D_v"
            ]
          },
          "c_bar_v"
        ]
      }
    },
    {
      "lhs": "Kn_D",
      "rhs": {
        "op": "/",
        "args": [
          "ell_D",
          "r"
        ]
      }
    },
    {
      "lhs": "β",
      "rhs": {
        "op": "/",
        "args": [
          {
            "op": "+",
            "args": [
              "Kn_D",
              1.0
            ]
          },
          {
            "op": "+",
            "args": [
              1.0,
              {
                "op": "/",
                "args": [
                  {
                    "op": "*",
                    "args": [
                      2.0,
                      "Kn_D",
                      {
                        "op": "+",
                        "args": [
                          "Kn_D",
                          1.0
                        ]
                      }
                    ]
                  },
                  "δ_m"
                ]
              }
            ]
          }
        ]
      }
    },
    {
      "lhs": "Kn_K",
      "rhs": {
        "op": "/",
        "args": [
          {
            "op": "*",
            "args": [
              2.0,
              "κ"
            ]
          },
          {
            "op": "*",
            "args": [
              "c_bar_gas",
              "n_gas",
              "m_gas",
              "c_v",
              "r"
            ]
          }
        ]
      }
    },
    {
      "lhs": "β_q",
      "rhs": {
        "op": "/",
        "args": [
          {
            "op": "+",
            "args": [
              "Kn_K",
              1.0
            ]
          },
          {
            "op": "+",
            "args": [
              1.0,
              {
                "op": "/",
                "args": [
                  {
                    "op": "*",
                    "args": [
                      2.0,
                      "Kn_K",
                      {
                        "op": "+",
                        "args": [
                          "Kn_K",
                          1.0
                        ]
                      }
                    ]
                  },
                  "α"
                ]
              }
            ]
          }
        ]
      }
    },
    {
      "lhs": "I_M",
      "rhs": {
        "op": "*",
        "args": [
          12.566370614359172,
          "r",
          "D_v",
          {
            "op": "-",
            "args": [
              "n_inf",
              "n_s"
            ]
          }
        ]
      }
    },
    {
      "lhs": "I",
      "rhs": {
        "op": "*",
        "args": [
          "β",
          "I_M"
        ]
      }
    },
    {
      "lhs": "Q_M",
      "rhs": {
        "op": "*",
        "args": [
          12.566370614359172,
          "r",
          "κ",
          {
            "op": "-",
            "args": [
              "T_inf",
              "T_o"
            ]
          }
        ]
      }
    },
    {
      "lhs": "Q",
      "rhs": {
        "op": "*",
        "args": [
          "β_q",
          "Q_M"
        ]
      }
    }
  ],
  "tests": [
    {
      "id": "default_conditions",
      "description": "Default parameters: r=100 nm, T_inf=298.15 K, T_o=293.15 K, n_inf=1e21 m⁻³, n_s=0, D_v=2e-5 m²/s, κ=0.026 W/(m·K), m_v=4.81e-26 kg, δ_m=1, n_gas=2.46e25 m⁻³, c_bar_gas=500 m/s, c_v=718 J/(kg·K), m_gas=4.81e-26 kg, α=1. Verifies the full equation chain: vapor thermal velocity, diffusional mean-free-path, both Knudsen numbers, both correction factors, Maxwell rates, and corrected rates.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "c_bar_v",
          "time": 0.0,
          "expected": 466.8276456276172,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "ell_D",
          "time": 0.0,
          "expected": 8.568472834598901e-08,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Kn_D",
          "time": 0.0,
          "expected": 0.8568472834598901,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "β",
          "time": 0.0,
          "expected": 0.4440020569957207,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Kn_K",
          "time": 0.0,
          "expected": 1.2241332983231208,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "β_q",
          "time": 0.0,
          "expected": 0.3450798590680775,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "I_M",
          "time": 0.0,
          "expected": 25132741228.71835,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "I",
          "time": 0.0,
          "expected": 11158988803.492104,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Q_M",
          "time": 0.0,
          "expected": 1.6336281798666924e-07,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Q",
          "time": 0.0,
          "expected": 5.637321820780381e-08,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    },
    {
      "id": "continuum_limit",
      "description": "Large droplet (r=1 µm): Kn_D = 0.0857 ≪ 1 and Kn_K = 0.122 ≪ 1, both correction factors near 1. β = 0.9154, β_q = 0.8805. I ≈ I_M and Q ≈ Q_M to within ~15%, confirming the continuum-limit approach. I_M is 10× larger due to 10× larger r.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "r": 1e-06
      },
      "assertions": [
        {
          "variable": "Kn_D",
          "time": 0.0,
          "expected": 0.08568472834598902,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "β",
          "time": 0.0,
          "expected": 0.9153760779725606,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Kn_K",
          "time": 0.0,
          "expected": 0.12241332983231208,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "β_q",
          "time": 0.0,
          "expected": 0.8804645668444205,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "I_M",
          "time": 0.0,
          "expected": 251327412287.18344,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "I",
          "time": 0.0,
          "expected": 230059100946.43472,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Q_M",
          "time": 0.0,
          "expected": 1.6336281798666921e-06,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Q",
          "time": 0.0,
          "expected": 1.4383517277711663e-06,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    },
    {
      "id": "deep_continuum_limit",
      "description": "Very large droplet (r=100 µm): Kn_D = 8.57×10⁻⁴ ≪ 1 and Kn_K = 1.22×10⁻³ ≪ 1. β = 0.9991, β_q = 0.9988 — within 0.1% of unity. Confirms the continuum limit β → 1, β_q → 1 as r → ∞.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "r": 0.0001
      },
      "assertions": [
        {
          "variable": "Kn_D",
          "time": 0.0,
          "expected": 0.0008568472834598901,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "β",
          "time": 0.0,
          "expected": 0.9991431539725586,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Kn_K",
          "time": 0.0,
          "expected": 0.0012241332983231206,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "β_q",
          "time": 0.0,
          "expected": 0.9987758703614391,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    },
    {
      "id": "free_molecular_limit",
      "description": "Very small droplet (r=1 nm): Kn_D = 85.7 ≫ 1, β = 5.835×10⁻³. Asymptotic formula β → δ_m c_bar_v r / (4 D_v) = 5.835×10⁻³ (0.01% agreement). Similarly Kn_K = 122 ≫ 1, β_q = 4.084×10⁻³. Both mass and heat fluxes are strongly suppressed relative to the continuum limit.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "r": 1e-09
      },
      "assertions": [
        {
          "variable": "Kn_D",
          "time": 0.0,
          "expected": 85.684728345989,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "β",
          "time": 0.0,
          "expected": 0.0058349527795214465,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Kn_K",
          "time": 0.0,
          "expected": 122.41332983231206,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "β_q",
          "time": 0.0,
          "expected": 0.004084387321960984,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "I_M",
          "time": 0.0,
          "expected": 251327412.28718346,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "I",
          "time": 0.0,
          "expected": 1466483.5828950338,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    },
    {
      "id": "low_sticking_coefficient",
      "description": "Reduced sticking coefficient δ_m=0.1 at default r=100 nm. β drops from 0.444 (δ_m=1) to 0.0566 — a 7.8× reduction. Q_M and β_q are unchanged (heat subsystem is independent of δ_m). Confirms that poor mass accommodation strongly suppresses I in the transition regime.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "δ_m": 0.1
      },
      "assertions": [
        {
          "variable": "β",
          "time": 0.0,
          "expected": 0.05657550850298151,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "I",
          "time": 0.0,
          "expected": 1421897615.088589,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "β_q",
          "time": 0.0,
          "expected": 0.3450798590680775,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Q",
          "time": 0.0,
          "expected": 5.637321820780381e-08,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    },
    {
      "id": "low_thermal_accommodation",
      "description": "Reduced thermal accommodation α=0.1 at default r=100 nm. β_q drops from 0.345 (α=1) to 0.0401 — an 8.6× reduction. β and I are unchanged (mass subsystem is independent of α). Confirms that poor thermal accommodation strongly suppresses Q.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "α": 0.1
      },
      "assertions": [
        {
          "variable": "β",
          "time": 0.0,
          "expected": 0.4440020569957207,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "I",
          "time": 0.0,
          "expected": 11158988803.492104,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "β_q",
          "time": 0.0,
          "expected": 0.04010864741742192,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Q",
          "time": 0.0,
          "expected": 6.552261667743789e-09,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    },
    {
      "id": "vapor_flux_linearity",
      "description": "Verify I_M ∝ (n_inf - n_s): set n_s = 5×10²⁰ m⁻³ (half of n_inf). I_M should halve relative to the default (n_s=0) case. I follows proportionally since β is unchanged. Exercises the (n_inf - n_s) driving force term in Eq. 2.1.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "n_s": 5e+20
      },
      "assertions": [
        {
          "variable": "I_M",
          "time": 0.0,
          "expected": 12566370614.359175,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "I",
          "time": 0.0,
          "expected": 5579494401.746052,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    },
    {
      "id": "heat_flux_temperature_difference",
      "description": "Larger temperature difference: T_inf=300 K, T_o=280 K (ΔT=20 K vs default 5 K). Q_M scales linearly: 4× larger ΔT gives 4× larger Q_M and Q. β_q is unchanged (depends only on r, κ, gas properties). Exercises the (T_inf - T_o) driving force in Eq. 2.2.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "T_inf": 300.0,
        "T_o": 280.0
      },
      "assertions": [
        {
          "variable": "Q_M",
          "time": 0.0,
          "expected": 6.53451271946677e-07,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Q",
          "time": 0.0,
          "expected": 2.2549287283121525e-07,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    }
  ],
  "analyses": [
    {
      "id": "beta_vs_radius",
      "description": "Mass transport correction factor β as a function of droplet radius r (log sweep 1 nm to 100 µm) at default T_inf, D_v, m_v, δ_m=1. Shows the transition from the free-molecular regime (β ≪ 1, r ≪ ell_D) through the transition regime (r ~ ell_D ≈ 86 nm) to the continuum limit (β → 1, r ≫ ell_D).",
      "initial_state": {},
      "parameters": {},
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "r",
            "range": {
              "start": 1e-09,
              "stop": 0.0001,
              "count": 61,
              "scale": "log"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "beta_vs_r",
          "type": "line",
          "description": "Mass transport correction factor β vs droplet radius r. Approaches 1 in the continuum regime (r ≫ ell_D ≈ 86 nm) and decreases toward 0 in the free-molecular regime (r ≪ ell_D).",
          "x": {
            "variable": "r",
            "label": "Droplet radius r (m)"
          },
          "y": {
            "variable": "β",
            "label": "Mass correction factor β"
          }
        }
      ]
    },
    {
      "id": "beta_q_vs_radius",
      "description": "Heat transport correction factor β_q as a function of droplet radius r (log sweep 1 nm to 100 µm) at default κ, n_gas, c_bar_gas, c_v, m_gas, α=1. Parallel to the mass correction sweep; shows the analogous transition for heat transport.",
      "initial_state": {},
      "parameters": {},
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "r",
            "range": {
              "start": 1e-09,
              "stop": 0.0001,
              "count": 61,
              "scale": "log"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "beta_q_vs_r",
          "type": "line",
          "description": "Heat transport correction factor β_q vs droplet radius r. Approaches 1 in the continuum regime and decreases in the free-molecular regime.",
          "x": {
            "variable": "r",
            "label": "Droplet radius r (m)"
          },
          "y": {
            "variable": "β_q",
            "label": "Heat correction factor β_q"
          }
        }
      ]
    },
    {
      "id": "correction_vs_sticking",
      "description": "Mass correction factor β vs sticking coefficient δ_m (log sweep 0.01 to 1.0) at default r=100 nm. Shows that poor accommodation (δ_m → 0) suppresses the mass transfer rate, particularly in the transition regime where Kn_D ≈ 0.86.",
      "initial_state": {},
      "parameters": {},
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "δ_m",
            "range": {
              "start": 0.01,
              "stop": 1.0,
              "count": 50,
              "scale": "log"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "beta_vs_delta_m",
          "type": "line",
          "description": "Mass correction factor β vs sticking coefficient δ_m at r=100 nm. β → 0 as δ_m → 0 (no sticking, no mass transfer) and reaches 0.444 at δ_m=1 (perfect sticking, transition regime).",
          "x": {
            "variable": "δ_m",
            "label": "Sticking coefficient δ_m"
          },
          "y": {
            "variable": "β",
            "label": "Mass correction factor β"
          }
        }
      ]
    }
  ]
}