DahnekeMassTransportCorrection

{
  "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. 3.6, 5.5, 6.2. Migrated from Aerosol.jl src/dahneke_brownian_diffusion.jl:33 @ c8c640bb (esm-xn1).",
    "notes": "source_ref: Aerosol.jl@c8c640bb:src/dahneke_brownian_diffusion.jl:33\n\nUpstream DahnekeMassTransportCorrection is an MTK @component with 1 constant (k_B=1.380649e-23 J/K), 5 parameters (T, D_v, m_v, r, δ_m), and 4 algebraic variables determined by explicit equations with no time derivative — all purely observed.\n\nConstant k_B is folded as the floating-point literal 1.380649e-23, and π is folded as 3.14159265358979.\n\nEquation chain:\n  c_bar = sqrt(8 k_B T / (π m_v))              [mean thermal velocity, Eq. 6.2]\n  ℓ_D   = 2 D_v / c_bar                         [diffusional mean-free-path, Eq. 3.6]\n  Kn_D  = ℓ_D / r                               [diffusion Knudsen number]\n  β     = (Kn_D+1) / (2 Kn_D(Kn_D+1)/δ_m + 1) [mass transport correction, Eq. 5.5]\n\nLimiting cases:\n  Kn_D → 0: β → 1  (continuum limit, Maxwell's result)\n  Kn_D → ∞: β → δ_m c̄ r / (4 D_v)  (free-molecular limit)\n\nThe correction factor β relates the actual transport rate I to the continuum result I_M:\n  I = β I_M = β (4π r D_v (n_∞ - n_s))\n\nTopological evaluation order: T, m_v → c_bar → ℓ_D → Kn_D → β. The dependency graph is acyclic.\n\nTolerance rationale (model-level rel=1e-9): purely algebraic system with sqrt, *, /, and + in float64. Round-off is ~1e-15 relative; rel=1e-9 leaves six orders of headroom above machine epsilon."
  },
  "tolerance": {
    "rel": 1e-09
  },
  "variables": {
    "T": {
      "type": "parameter",
      "units": "K",
      "default": 298.15,
      "description": "Temperature. Default 298.15 K (25 °C), standard atmospheric."
    },
    "D_v": {
      "type": "parameter",
      "units": "m^2/s",
      "default": 2e-05,
      "description": "Diffusion coefficient of vapor molecules. Default 2×10⁻⁵ m²/s (water vapor in air near 298 K)."
    },
    "m_v": {
      "type": "parameter",
      "units": "kg",
      "default": 4.81e-26,
      "description": "Mass of diffusing vapor molecule. Default 4.81×10⁻²⁶ kg (water molecule, M=18 g/mol)."
    },
    "r": {
      "type": "parameter",
      "units": "m",
      "default": 1e-07,
      "description": "Sphere (droplet/particle) radius. Default 100 nm — transition-regime aerosol."
    },
    "δ_m": {
      "type": "parameter",
      "units": "1",
      "default": 1.0,
      "description": "Condensation-evaporation (sticking) coefficient. Dimensionless; range (0, 1]. Default 1.0 (perfect accommodation). Smaller δ_m reduces β, reflecting partial reflection of vapor molecules at the particle surface."
    },
    "c_bar": {
      "type": "unknown",
      "units": "m/s",
      "description": "Mean thermal velocity of vapor molecules (Eq. 6.2): c̄ = sqrt(8 k_B T / (π m_v)). k_B = 1.380649×10⁻²³ J/K and π = 3.14159265358979 folded as literals."
    },
    "ℓ_D": {
      "type": "unknown",
      "units": "m",
      "description": "Diffusional mean-free-path of vapor molecules (Eq. 3.6): ℓ_D = 2 D_v / c̄. Characterises the length scale below which diffusive transport becomes ballistic."
    },
    "Kn_D": {
      "type": "unknown",
      "units": "1",
      "description": "Diffusion Knudsen number: Kn_D = ℓ_D / r. Dimensionless; Kn_D ≪ 1 means continuum diffusion dominates (β → 1); Kn_D ≫ 1 means free-molecular transport (β → δ_m c̄ r / (4 D_v))."
    },
    "β": {
      "type": "unknown",
      "units": "1",
      "description": "Mass transport correction factor (Eq. 5.5): β = (Kn_D+1) / (2 Kn_D(Kn_D+1)/δ_m + 1). Dimensionless; β = 1 in the continuum limit and β → δ_m c̄ r / (4 D_v) in the free-molecular limit. Multiplies the Maxwell (continuum) rate I_M to give the actual rate I = β I_M."
    }
  },
  "equations": [
    {
      "lhs": "c_bar",
      "rhs": {
        "op": "sqrt",
        "args": [
          {
            "op": "/",
            "args": [
              {
                "op": "*",
                "args": [
                  8,
                  1.380649e-23,
                  "T"
                ]
              },
              {
                "op": "*",
                "args": [
                  3.14159265358979,
                  "m_v"
                ]
              }
            ]
          }
        ]
      }
    },
    {
      "lhs": "ℓ_D",
      "rhs": {
        "op": "/",
        "args": [
          {
            "op": "*",
            "args": [
              2,
              "D_v"
            ]
          },
          "c_bar"
        ]
      }
    },
    {
      "lhs": "Kn_D",
      "rhs": {
        "op": "/",
        "args": [
          "ℓ_D",
          "r"
        ]
      }
    },
    {
      "lhs": "β",
      "rhs": {
        "op": "/",
        "args": [
          {
            "op": "+",
            "args": [
              "Kn_D",
              1
            ]
          },
          {
            "op": "+",
            "args": [
              1,
              {
                "op": "/",
                "args": [
                  {
                    "op": "*",
                    "args": [
                      2,
                      "Kn_D",
                      {
                        "op": "+",
                        "args": [
                          "Kn_D",
                          1
                        ]
                      }
                    ]
                  },
                  "δ_m"
                ]
              }
            ]
          }
        ]
      }
    }
  ],
  "tests": [
    {
      "id": "default_conditions",
      "description": "Default parameters: T=298.15 K, D_v=2e-5 m²/s, m_v=4.81e-26 kg, r=100 nm, δ_m=1.0. Checks the full equation chain: c̄=466.83 m/s, ℓ_D=85.68 nm, Kn_D=0.857 (transition regime), β=0.444.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "c_bar",
          "time": 0.0,
          "expected": 466.8276456276172,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "ℓ_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
          }
        }
      ]
    },
    {
      "id": "continuum_limit",
      "description": "Large particle (r=100 µm): Kn_D = 8.57×10⁻⁴ ≪ 1, firmly continuum. β → 1 (within 0.1% of 1). Confirms the continuum limit of Eq. 5.5.",
      "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
          }
        }
      ]
    },
    {
      "id": "free_molecular_limit",
      "description": "Very small particle (r=0.1 nm): Kn_D = 856.8 ≫ 1, deep free-molecular regime. β ≈ δ_m c̄ r / (4 D_v) = 5.84×10⁻⁴ (asymptotic formula accurate to 0.001% at this Kn_D).",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "r": 1e-10
      },
      "assertions": [
        {
          "variable": "Kn_D",
          "time": 0.0,
          "expected": 856.84728345989,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "β",
          "time": 0.0,
          "expected": 0.0005835341600963319,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    },
    {
      "id": "partial_accommodation",
      "description": "Partial sticking coefficient δ_m=0.5 at default r=100 nm: β drops from 0.444 (δ_m=1) to 0.252, reflecting that half of vapor molecules are reflected rather than accommodated. Confirms δ_m enters only the denominator of Eq. 5.5.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "δ_m": 0.5
      },
      "assertions": [
        {
          "variable": "Kn_D",
          "time": 0.0,
          "expected": 0.8568472834598901,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "β",
          "time": 0.0,
          "expected": 0.2521472618237901,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    },
    {
      "id": "temperature_200K",
      "description": "Low temperature T=200 K: c̄ drops to 382.3 m/s, ℓ_D increases to 104.6 nm, Kn_D increases to 1.046 (more free-molecular), β falls to 0.387. Confirms the temperature dependence enters via c̄ ∝ √T and the resulting change in ℓ_D and Kn_D.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "T": 200.0
      },
      "assertions": [
        {
          "variable": "c_bar",
          "time": 0.0,
          "expected": 382.343893295089,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Kn_D",
          "time": 0.0,
          "expected": 1.0461786025997393,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "β",
          "time": 0.0,
          "expected": 0.38743575347283044,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    },
    {
      "id": "temperature_400K",
      "description": "High temperature T=400 K: c̄ rises to 540.7 m/s, ℓ_D decreases to 73.98 nm, Kn_D decreases to 0.740 (more continuum), β rises to 0.487. Confirms the temperature-β relationship reversal: higher T → shorter ℓ_D → smaller Kn_D → β closer to 1.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "T": 400.0
      },
      "assertions": [
        {
          "variable": "c_bar",
          "time": 0.0,
          "expected": 540.7159193884463,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Kn_D",
          "time": 0.0,
          "expected": 0.739759984230542,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "β",
          "time": 0.0,
          "expected": 0.4867809997545371,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    }
  ],
  "analyses": [
    {
      "id": "beta_vs_radius",
      "description": "Correction factor β as a function of particle radius r (log sweep 0.1 nm to 10 µm) at T=298 K, D_v=2e-5 m²/s, m_v=4.81e-26 kg, δ_m=1. Shows the transition from the continuum limit (β→1 for large r) to the free-molecular limit (β→0 for small r), with the transition regime near r ~ ℓ_D ≈ 86 nm.",
      "initial_state": {},
      "parameters": {},
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "r",
            "range": {
              "start": 1e-10,
              "stop": 1e-05,
              "count": 61,
              "scale": "log"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "beta_vs_r",
          "type": "line",
          "description": "β vs particle radius r. Shows the three regimes: free-molecular (small r, β→0), transition (r ~ ℓ_D), and continuum (large r, β→1).",
          "x": {
            "variable": "r",
            "label": "Particle radius r (m)"
          },
          "y": {
            "variable": "β",
            "label": "Mass transport correction factor β"
          }
        }
      ]
    },
    {
      "id": "beta_vs_delta_m",
      "description": "Correction factor β as a function of sticking coefficient δ_m (linear sweep 0.01 to 1.0) at default r=100 nm. Illustrates how partial accommodation (δ_m < 1) suppresses β in the transition regime, with β → 0 as δ_m → 0 (no accommodation).",
      "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": "linear"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "beta_vs_delta_m_plot",
          "type": "line",
          "description": "β vs sticking coefficient δ_m for r=100 nm (transition regime). β increases monotonically with δ_m from near zero (specular reflection) to 0.444 (perfect accommodation).",
          "x": {
            "variable": "δ_m",
            "label": "Sticking coefficient δ_m"
          },
          "y": {
            "variable": "β",
            "label": "Mass transport correction factor β"
          }
        }
      ]
    }
  ]
}