MassTransferCoefficient

{
  "reference": {
    "citation": "Seinfeld, J. H. and Pandis, S. N. (2006) Atmospheric Chemistry and Physics, 2nd Edition, Chapter 12, Eq. 12.116 — combined gas-phase and interfacial mass transfer coefficient. Migrated from Aerosol.jl src/mass_transfer.jl `MassTransferCoefficient` (esm-f9f).",
    "notes": "source_ref: Aerosol.jl@FWrvp:src/mass_transfer.jl:231\n\nUpstream `MassTransferCoefficient` is an MTK `@component` with 1 constant (R=8.314 J/(mol·K)), 5 parameters (R_p, D_g, α, M_A, T), and 3 algebraic variables (tau_g, tau_i, k_mt). All three variables are determined by explicit algebraic equations with no time derivative — purely observed.\n\nEquations (S&P 2006, Eq. 12.116):\n  tau_g = R_p² / (3 D_g)                        [gas-phase diffusion resistance, s]\n  tau_i = (R_p / 3α) × √(2π M_A / (R T))        [interfacial resistance, s]\n  k_mt  = 1 / (tau_g + tau_i)                    [mass transfer coefficient, s⁻¹]\n\nThe constant R = 8.314 J/(mol·K) and π = 3.141592653589793 are folded as floating-point literals into the tau_i expression.\n\nVariable renaming for ASCII compatibility: Julia `τ_g` → ESM `tau_g`; Julia `τ_i` → ESM `tau_i`. The parameter `α` retains Unicode (matching uptake_coeff.esm convention for the accommodation coefficient).\n\nPhysics: tau_g = R_p²/(3 D_g) is the characteristic time for a gas-phase molecule to diffuse through a shell of radius R_p; it scales as R_p² and is independent of molecular properties. tau_i = (R_p/3α) √(2πM_A/(RT)) is derived from the mean molecular speed c̄ = √(8RT/(πM_A)) via the free-molecular flux formula; it scales as R_p/α and increases with molecular weight (heavier molecules are slower). In the continuum limit (R_p large), tau_g ≫ tau_i and gas-phase diffusion limits; in the free-molecular limit (R_p small), tau_i can dominate if α is small. The interfacial term vanishes as α → 1 and R_p → 0 (fully accommodating small particles), leaving the free-molecular flux as the only limit.\n\nTopological substitution order: tau_g and tau_i each depend only on parameters and constants, so they resolve independently before k_mt. The dependency graph is acyclic — no SimulationError.\n\nTolerance rationale (model-level rel=1e-9): purely algebraic system with no ODE integration. All expressions are finite compositions of *, /, ^(2), sqrt in float64; round-off is ~1e-15 relative. rel=1e-9 leaves six orders of magnitude of headroom."
  },
  "tolerance": {
    "rel": 1e-09
  },
  "variables": {
    "R_p": {
      "type": "parameter",
      "units": "m",
      "default": 1e-05,
      "description": "Particle or droplet radius. Default 10 µm is typical for cloud/fog droplets. tau_g scales as R_p² and tau_i scales as R_p, so both resistances increase with particle size but gas-phase diffusion grows faster for large particles."
    },
    "D_g": {
      "type": "parameter",
      "units": "m^2/s",
      "default": 2e-05,
      "description": "Gas-phase molecular diffusivity of the absorbing species. Default 2×10⁻⁵ m²/s is representative of mid-weight atmospheric trace gases. Larger D_g reduces the gas-phase diffusion resistance tau_g."
    },
    "α": {
      "type": "parameter",
      "units": "1",
      "default": 1.0,
      "description": "Mass accommodation coefficient: probability that a molecule striking the particle surface is taken up. Dimensionless; range (0, 1]. Default 1.0 (perfect accommodation). Smaller α increases the interfacial resistance tau_i."
    },
    "M_A": {
      "type": "parameter",
      "units": "kg/mol",
      "default": 0.029,
      "description": "Molar mass of the gas-phase species. Default 0.029 kg/mol is representative of dry air. Heavier molecules have lower mean thermal speed, which increases the interfacial resistance tau_i as √M_A."
    },
    "T": {
      "type": "parameter",
      "units": "K",
      "default": 298.15,
      "description": "Temperature. Used in the interfacial resistance via the mean molecular speed factor √(M_A/(RT)). Higher temperature increases mean molecular speed, reducing tau_i."
    },
    "tau_g": {
      "type": "unknown",
      "units": "s",
      "description": "Gas-phase diffusion resistance: tau_g = R_p² / (3 D_g). Time scale for diffusion of the gas-phase species through a hemispherical shell of thickness R_p. Scales as R_p²/D_g; dominates for large particles or slowly diffusing species."
    },
    "tau_i": {
      "type": "unknown",
      "units": "s",
      "description": "Interfacial (accommodation) resistance: tau_i = (R_p / (3α)) × √(2π M_A / (R T)). Time scale associated with the finite probability of surface accommodation; derived from the mean free-molecular flux. Scales as R_p / (α √T) and increases with molecular weight. Negligible for α ≈ 1 and large R_p."
    },
    "k_mt": {
      "type": "unknown",
      "units": "s^-1",
      "description": "Combined mass transfer coefficient (Eq. 12.116): k_mt = 1 / (tau_g + tau_i). Reciprocal of the total resistance to mass transfer between the gas phase and the particle surface. Represents the rate constant for transfer of a gaseous species to or from the particle. High k_mt means fast transfer; low k_mt means slow transfer limited by diffusion or accommodation."
    }
  },
  "equations": [
    {
      "lhs": "tau_g",
      "rhs": {
        "op": "/",
        "args": [
          {
            "op": "^",
            "args": [
              "R_p",
              2.0
            ]
          },
          {
            "op": "*",
            "args": [
              3.0,
              "D_g"
            ]
          }
        ]
      }
    },
    {
      "lhs": "tau_i",
      "rhs": {
        "op": "*",
        "args": [
          {
            "op": "/",
            "args": [
              "R_p",
              {
                "op": "*",
                "args": [
                  3.0,
                  "α"
                ]
              }
            ]
          },
          {
            "op": "sqrt",
            "args": [
              {
                "op": "/",
                "args": [
                  {
                    "op": "*",
                    "args": [
                      2.0,
                      3.141592653589793,
                      "M_A"
                    ]
                  },
                  {
                    "op": "*",
                    "args": [
                      8.314,
                      "T"
                    ]
                  }
                ]
              }
            ]
          }
        ]
      }
    },
    {
      "lhs": "k_mt",
      "rhs": {
        "op": "/",
        "args": [
          1.0,
          {
            "op": "+",
            "args": [
              "tau_g",
              "tau_i"
            ]
          }
        ]
      }
    }
  ],
  "tests": [
    {
      "id": "default_conditions",
      "description": "Default parameters: R_p=10 µm, D_g=2×10⁻⁵ m²/s, α=1, M_A=0.029 kg/mol, T=298.15 K. tau_g = (1e-5)²/(3×2e-5) = 1e-10/6e-5 ≈ 1.6667×10⁻⁶ s dominates; tau_i = (1e-5/3)×√(2π×0.029/(8.314×298.15)) ≈ 2.858×10⁻⁸ s is ~60× smaller. k_mt = 1/(tau_g + tau_i) ≈ 5.899×10⁵ s⁻¹. Anchors all three computed quantities at the declared defaults.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "tau_g",
          "time": 0.0,
          "expected": 1.666666666666667e-06,
          "tolerance": {
            "rel": 1e-12
          }
        },
        {
          "variable": "tau_i",
          "time": 0.0,
          "expected": 2.857888372453761e-08,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "k_mt",
          "time": 0.0,
          "expected": 589885.0463103912,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    },
    {
      "id": "large_particle",
      "description": "Large particle R_p = 100 µm = 10⁻⁴ m (10× the default). tau_g scales as R_p² → 100× larger (≈ 1.667×10⁻⁴ s). tau_i scales as R_p → 10× larger (≈ 2.858×10⁻⁷ s). Gas-phase diffusion completely dominates: k_mt ≈ 5.990×10³ s⁻¹, about 100× smaller than the default. Exercises R_p² scaling in tau_g.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "R_p": 0.0001
      },
      "assertions": [
        {
          "variable": "tau_g",
          "time": 0.0,
          "expected": 0.0001666666666666666,
          "tolerance": {
            "rel": 1e-12
          }
        },
        {
          "variable": "tau_i",
          "time": 0.0,
          "expected": 2.857888372453761e-07,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "k_mt",
          "time": 0.0,
          "expected": 5989.729213515929,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    },
    {
      "id": "low_accommodation",
      "description": "Reduced accommodation coefficient α = 0.1 (10% sticking probability). tau_g unchanged; tau_i scales as 1/α → 10× larger (≈ 2.858×10⁻⁷ s), now comparable to tau_g. k_mt drops from ≈5.899×10⁵ to ≈5.122×10⁵ s⁻¹. Isolates the 1/α scaling in the interfacial resistance.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "α": 0.1
      },
      "assertions": [
        {
          "variable": "tau_i",
          "time": 0.0,
          "expected": 2.85788837245376e-07,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "k_mt",
          "time": 0.0,
          "expected": 512175.564562853,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    },
    {
      "id": "hot_temperature",
      "description": "High temperature T = 320 K. tau_i ∝ 1/√T decreases slightly (faster molecules → lower interfacial resistance). k_mt increases marginally from ≈5.899×10⁵ to ≈5.902×10⁵ s⁻¹ (sub-1% change since tau_g dominates). Exercises T wiring in tau_i via the thermal speed factor √(1/(RT)).",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "T": 320.0
      },
      "assertions": [
        {
          "variable": "tau_i",
          "time": 0.0,
          "expected": 2.758593313298192e-08,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "k_mt",
          "time": 0.0,
          "expected": 590230.7602291733,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    },
    {
      "id": "heavy_molecule",
      "description": "Heavy molecule M_A = 0.1 kg/mol (≈3.4× the default, analogous to SO₂ at 64 g/mol or roughly this value). tau_i ∝ √M_A increases by √(0.1/0.029) ≈ 1.856×: tau_i ≈ 5.307×10⁻⁸ s. k_mt drops from ≈5.899×10⁵ to ≈5.815×10⁵ s⁻¹. Exercises √M_A wiring in the interfacial term.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "M_A": 0.1
      },
      "assertions": [
        {
          "variable": "tau_i",
          "time": 0.0,
          "expected": 5.306965477950655e-08,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "k_mt",
          "time": 0.0,
          "expected": 581484.4912730998,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    },
    {
      "id": "perfect_accommodation_small_particle",
      "description": "Small particle R_p = 1 nm = 10⁻⁹ m with perfect accommodation α = 1. tau_g = (1e-9)²/(6e-5) ≈ 1.667×10⁻¹⁴ s (negligible); tau_i = (1e-9/3)×√(2π×0.029/(8.314×298.15)) ≈ 2.858×10⁻¹² s. k_mt ≈ 1/tau_i ≈ 3.498×10¹¹ s⁻¹ — extremely fast transfer at nanometer scale. Confirms that tau_g ≪ tau_i in the free-molecular limit for small, well-accommodated particles.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "R_p": 1e-09
      },
      "assertions": [
        {
          "variable": "tau_g",
          "time": 0.0,
          "expected": 1.666666666666666e-14,
          "tolerance": {
            "rel": 1e-12
          }
        },
        {
          "variable": "tau_i",
          "time": 0.0,
          "expected": 2.857888372453761e-12,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "k_mt",
          "time": 0.0,
          "expected": 347879927985.6495,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    }
  ],
  "analyses": [
    {
      "id": "k_mt_vs_particle_radius",
      "description": "Sweep particle radius R_p from 1 nm to 1 mm log-spaced at default conditions (D_g=2×10⁻⁵, α=1, M_A=0.029, T=298.15 K). For small R_p, tau_i dominates (proportional to R_p) and k_mt ∝ 1/R_p — the free-molecular regime. For large R_p, tau_g dominates (proportional to R_p²) and k_mt ∝ 1/R_p² — the continuum diffusion regime. The crossover occurs around R_p where tau_g ≈ tau_i, i.e., R_p ≈ D_g × √(2πM_A/(RT)) / α.",
      "initial_state": {},
      "parameters": {},
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "R_p",
            "range": {
              "start": 1e-09,
              "stop": 0.001,
              "count": 61,
              "scale": "log"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "k_mt_vs_rp",
          "type": "line",
          "description": "Mass transfer coefficient k_mt as a function of particle radius R_p. Shows transition from free-molecular regime (k_mt ∝ 1/R_p for small R_p, interfacial-limited) to continuum regime (k_mt ∝ 1/R_p² for large R_p, diffusion-limited).",
          "x": {
            "variable": "R_p",
            "label": "Particle radius R_p (m)"
          },
          "y": {
            "variable": "k_mt",
            "label": "Mass transfer coefficient k_mt (s⁻¹)"
          }
        }
      ]
    },
    {
      "id": "k_mt_vs_accommodation",
      "description": "Sweep accommodation coefficient α from 0.001 to 1.0 log-spaced at default R_p = 10 µm. For α near 1, tau_i ≪ tau_g and k_mt is insensitive to α (diffusion-limited). As α decreases, tau_i grows as 1/α and eventually dominates, making k_mt ∝ α (accommodation-limited). The crossover is around α where tau_i ≈ tau_g.",
      "initial_state": {},
      "parameters": {},
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "α",
            "range": {
              "start": 0.001,
              "stop": 1.0,
              "count": 51,
              "scale": "log"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "k_mt_vs_alpha",
          "type": "line",
          "description": "Mass transfer coefficient k_mt vs. accommodation coefficient α. Shows plateau at diffusion-limited value for α near 1 and linear decrease k_mt ∝ α in the accommodation-limited regime for small α.",
          "x": {
            "variable": "α",
            "label": "Accommodation coefficient α"
          },
          "y": {
            "variable": "k_mt",
            "label": "Mass transfer coefficient k_mt (s⁻¹)"
          }
        }
      ]
    }
  ]
}