MassTransfer

{
  "reference": {
    "citation": "Seinfeld, J. H. and Pandis, S. N. (2006) Atmospheric Chemistry and Physics, 2nd Edition, Chapter 12 — transition-regime mass transfer. Eq. 12.24 (mean molecular speed), Eq. 12.12 (continuum flux), Eq. 12.31 (Knudsen number), Eq. 12.43 (Fuchs-Sutugin correction). Migrated from Aerosol.jl src/mass_transfer.jl `MassTransfer` (esm-44h).",
    "notes": "source_ref: Aerosol.jl@YeW6d:src/mass_transfer.jl:319\n\nUpstream `MassTransfer` is an MTK `@component` with 3 constants (R=8.314, k_B=1.380649e-23, N_Av=6.02214076e23), 10 parameters (T, P, M_A, M_air, μ, R_p, D_g, α, c_inf, c_s), and 7 algebraic variables (c_bar, ρ_air, λ, Kn, f_FS, J_c, J). It returns `System(eqs, t; name)` with no subsystems — a flat system with no `couple` call. The migration-tracker `coupling.couple` feature tag and `depends_on: MeanMolecularSpeed, Knudsen, FuchsSutugin` describe a conceptual relationship, not an implementation dependency; the source equations use only the component's own parameters and constants. The component is therefore migrated as a standalone flat model, following the same rig convention as wet_deposition_flux.esm.\n\nAll 7 source `@variables` are purely algebraic (no time derivative appears on any left-hand side in the `eqs` array), so each is migrated as `type: observed` with an inline `expression`. The `equations` array is empty; the ESS runner evaluates observed values by direct expression substitution in topological order.\n\nConstant folding:\n- R = 8.314 J/(mol·K) — appears in ρ_air and λ. Folded as the numeric literal 8.314.\n- k_B = 1.380649e-23 J/K, N_Av = 6.02214076e23 mol⁻¹ — appear only in c_bar as the product k_B·N_Av. They are NOT folded to R (8.31446261815324) because the upstream uses k_B and N_Av separately with their own values and the product k_B·N_Av ≠ R (upstream R=8.314 is slightly different from k_B·N_Av=8.31446261815324). Both constants are retained as their exact upstream literal values in the c_bar expression.\n- π is inlined as the Float64 literal 3.141592653589793 (Julia `pi`).\n\nDynamic viscosity μ is a `@parameter` in the upstream (default 1.81e-5 Pa·s), not a `@constant`, so it is migrated as a parameter.\n\nEquation mapping (one observed expression per source equation):\n  - c_bar ~ sqrt(8·k_B·T·N_Av/(π·M_A))  → Eq. 12.24 — mean molecular speed of species A\n  - ρ_air ~ P·M_air/(R·T)                → ideal gas law for air density\n  - λ ~ 2·μ/(ρ_air·sqrt(8·R·T/(π·M_air))) → mean free path of air (μ is viscosity, ρ_air is computed above)\n  - Kn ~ λ/R_p                            → Eq. 12.31 — Knudsen number\n  - f_FS ~ 0.75·α·(1+Kn)/(Kn²+Kn+0.283·Kn·α+0.75·α) → Eq. 12.43 — Fuchs-Sutugin correction\n  - J_c ~ 4·π·R_p·D_g·(c_inf-c_s)        → Eq. 12.12 — continuum-regime molar flow rate\n  - J ~ f_FS·J_c                           → transition-regime corrected molar flow rate\n\nTopological order of the observed chain: c_bar ← {T, M_A}; rho_air ← {P, M_air, T}; λ ← {μ, rho_air, T, M_air}; Kn ← {λ, R_p}; f_FS ← {α, Kn}; J_c ← {R_p, D_g, c_inf, c_s}; J ← {f_FS, J_c}. Note: c_bar does not feed into λ — the λ formula uses sqrt(8·R·T/(π·M_air)) for the mean speed of air molecules, not c_bar (which uses M_A for species A). These coincide when M_A = M_air (the default case) but are distinct in general. The density variable is spelled rho_air (ASCII) rather than ρ_air (Unicode) following the sibling mean_free_path.esm convention.\n\nRegime notes: For Kn ≪ 1 (large particles, continuum regime), f_FS → 1 and J → J_c. For Kn ≫ 1 (small particles, kinetic regime), f_FS → 0 and J ≪ J_c. At the default R_p=100 nm, Kn ≈ 0.65 (transition regime); f_FS ≈ 0.615. At R_p=10 μm (continuum), Kn ≈ 0.0065 and f_FS ≈ 0.995. At R_p=10 nm (kinetic), Kn ≈ 6.5 and f_FS ≈ 0.109.\n\nTolerance rationale (model-level rel=1e-9): purely algebraic system with no ODE integration — each observed variable is evaluated by Float64 arithmetic (sqrt, *, /, +, -). Float64 round-off is ~1e-15 relative; rel=1e-9 leaves six orders of magnitude headroom, matching the sibling mean_molecular_speed.esm."
  },
  "tolerance": {
    "rel": 1e-09
  },
  "variables": {
    "T": {
      "type": "parameter",
      "units": "K",
      "default": 298.15,
      "description": "Temperature of the gas. Affects mean molecular speed (∝ √T), air density (∝ 1/T), mean free path (∝ √T/ρ_air ∝ T^(3/2)/P), and Knudsen number."
    },
    "P": {
      "type": "parameter",
      "units": "Pa",
      "default": 101325.0,
      "description": "Total air pressure. Affects air density (∝ P) and mean free path (∝ 1/P via ρ_air)."
    },
    "M_A": {
      "type": "parameter",
      "units": "kg/mol",
      "default": 0.029,
      "description": "Molar mass of the diffusing species A. Affects mean molecular speed c_bar only (∝ M_A^{-1/2}). Default 0.029 kg/mol is dry air."
    },
    "M_air": {
      "type": "parameter",
      "units": "kg/mol",
      "default": 0.029,
      "description": "Molar mass of air. Used in the ideal-gas air density and in the mean-free-path expression for the thermal speed of air molecules. Independent of M_A."
    },
    "μ": {
      "type": "parameter",
      "units": "Pa*s",
      "default": 1.81e-05,
      "description": "Dynamic viscosity of air. Used in the mean free path λ = 2μ/(ρ_air·c̄_air). Typical range 1.7–1.95×10⁻⁵ Pa·s over 250–350 K."
    },
    "R_p": {
      "type": "parameter",
      "units": "m",
      "default": 1e-07,
      "description": "Particle radius. Controls the Knudsen number Kn = λ/R_p and the continuum flux J_c = 4πR_p D_g Δc. Default 100 nm is a typical accumulation-mode aerosol."
    },
    "D_g": {
      "type": "parameter",
      "units": "m^2/s",
      "default": 2e-05,
      "description": "Gas-phase diffusivity of species A in air. Controls the continuum-regime flux J_c (∝ D_g)."
    },
    "α": {
      "type": "parameter",
      "units": "1",
      "default": 1.0,
      "description": "Mass accommodation (sticking) coefficient. Range (0, 1]. α=1 means every collision results in uptake; smaller α reduces the Fuchs-Sutugin correction and the transition-regime flux."
    },
    "c_inf": {
      "type": "parameter",
      "units": "mol/m^3",
      "default": 1e-06,
      "description": "Bulk gas-phase molar concentration of species A far from the particle."
    },
    "c_s": {
      "type": "parameter",
      "units": "mol/m^3",
      "default": 0.0,
      "description": "Molar concentration of species A at the particle surface. The driving force for mass transfer is c_inf - c_s."
    },
    "c_bar": {
      "type": "unknown",
      "units": "m/s",
      "description": "Mean molecular speed of gas species A: c̄ = √(8k_B N_Av T/(π M_A)) = √(8RT/(π M_A)) using the literal k_B and N_Av values. Identical to the standalone MeanMolecularSpeed component (esm-1i9). Positive-definite for T > 0."
    },
    "rho_air": {
      "type": "unknown",
      "units": "kg/m^3",
      "description": "Air density from the ideal gas law: rho_air = P·M_air/(R·T), with R=8.314 J/(mol·K) (upstream @constant value, upstream variable name ρ_air). At standard conditions (T=298.15 K, P=101325 Pa, M_air=0.029 kg/mol), rho_air ≈ 1.185 kg/m³."
    },
    "λ": {
      "type": "unknown",
      "units": "m",
      "description": "Mean free path of air molecules: λ = 2μ/(rho_air·c̄_air) where c̄_air = √(8RT/(π M_air)) is the mean speed of air molecules (using M_air, distinct from c_bar which uses M_A). At standard conditions, λ ≈ 65 nm."
    },
    "Kn": {
      "type": "unknown",
      "units": "1",
      "description": "Knudsen number: Kn = λ/R_p (Eq. 12.31). Kn ≪ 1 → continuum regime; Kn ≫ 1 → kinetic regime; Kn ~ 1 → transition regime. At 100 nm radius and standard conditions, Kn ≈ 0.65."
    },
    "f_FS": {
      "type": "unknown",
      "units": "1",
      "description": "Fuchs-Sutugin transition-regime correction factor (Eq. 12.43): f_FS = 0.75α(1+Kn)/(Kn²+Kn+0.283Kn·α+0.75α). In the continuum limit (Kn→0), f_FS→1. In the kinetic limit (Kn→∞), f_FS→0. At Kn=1 and α=1, f_FS≈0.494."
    },
    "J_c": {
      "type": "unknown",
      "units": "mol/s",
      "description": "Continuum-regime molar flow rate to a spherical particle (Eq. 12.12): J_c = 4πR_p D_g (c_inf - c_s). Positive when c_inf > c_s (net condensation). Independent of temperature and viscosity."
    },
    "J": {
      "type": "unknown",
      "units": "mol/s",
      "description": "Transition-regime corrected molar flow rate: J = f_FS·J_c. Equals J_c in the continuum limit (f_FS→1) and approaches 0 in the kinetic limit (f_FS→0). The physically realised flux for particle growth/evaporation across all Knudsen number regimes."
    }
  },
  "equations": [
    {
      "lhs": "c_bar",
      "rhs": {
        "op": "sqrt",
        "args": [
          {
            "op": "/",
            "args": [
              {
                "op": "*",
                "args": [
                  8,
                  1.380649e-23,
                  6.02214076e+23,
                  "T"
                ]
              },
              {
                "op": "*",
                "args": [
                  3.141592653589793,
                  "M_A"
                ]
              }
            ]
          }
        ]
      }
    },
    {
      "lhs": "rho_air",
      "rhs": {
        "op": "/",
        "args": [
          {
            "op": "*",
            "args": [
              "P",
              "M_air"
            ]
          },
          {
            "op": "*",
            "args": [
              8.314,
              "T"
            ]
          }
        ]
      }
    },
    {
      "lhs": "λ",
      "rhs": {
        "op": "/",
        "args": [
          {
            "op": "*",
            "args": [
              2,
              "μ"
            ]
          },
          {
            "op": "*",
            "args": [
              "rho_air",
              {
                "op": "sqrt",
                "args": [
                  {
                    "op": "/",
                    "args": [
                      {
                        "op": "*",
                        "args": [
                          8,
                          8.314,
                          "T"
                        ]
                      },
                      {
                        "op": "*",
                        "args": [
                          3.141592653589793,
                          "M_air"
                        ]
                      }
                    ]
                  }
                ]
              }
            ]
          }
        ]
      }
    },
    {
      "lhs": "Kn",
      "rhs": {
        "op": "/",
        "args": [
          "λ",
          "R_p"
        ]
      }
    },
    {
      "lhs": "f_FS",
      "rhs": {
        "op": "/",
        "args": [
          {
            "op": "*",
            "args": [
              0.75,
              "α",
              {
                "op": "+",
                "args": [
                  1,
                  "Kn"
                ]
              }
            ]
          },
          {
            "op": "+",
            "args": [
              {
                "op": "^",
                "args": [
                  "Kn",
                  2
                ]
              },
              "Kn",
              {
                "op": "*",
                "args": [
                  0.283,
                  "Kn",
                  "α"
                ]
              },
              {
                "op": "*",
                "args": [
                  0.75,
                  "α"
                ]
              }
            ]
          }
        ]
      }
    },
    {
      "lhs": "J_c",
      "rhs": {
        "op": "*",
        "args": [
          4,
          3.141592653589793,
          "R_p",
          "D_g",
          {
            "op": "-",
            "args": [
              "c_inf",
              "c_s"
            ]
          }
        ]
      }
    },
    {
      "lhs": "J",
      "rhs": {
        "op": "*",
        "args": [
          "f_FS",
          "J_c"
        ]
      }
    }
  ],
  "tests": [
    {
      "id": "default_parameters",
      "description": "Default parameters: T=298.15 K, P=101325 Pa, M_A=M_air=0.029 kg/mol, μ=1.81e-5 Pa·s, R_p=100 nm, D_g=2e-5 m²/s, α=1, c_inf=1e-6 mol/m³, c_s=0. Kn≈0.655 — transition regime. Anchors all seven observed variables at standard atmospheric conditions.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "c_bar",
          "time": 0.0,
          "expected": 466.5579107928379,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "rho_air",
          "time": 0.0,
          "expected": 1.1854132477839954,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "λ",
          "time": 0.0,
          "expected": 6.545537554702463e-08,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Kn",
          "time": 0.0,
          "expected": 0.6545537554702464,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "f_FS",
          "time": 0.0,
          "expected": 0.6148523302653344,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "J_c",
          "time": 0.0,
          "expected": 2.5132741228718345e-17,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "J",
          "time": 0.0,
          "expected": 1.5452924510433117e-17,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    },
    {
      "id": "continuum_regime_large_particle",
      "description": "Large particle R_p=10 μm: Kn=λ/R_p≈0.0065 ≪ 1 (continuum regime). Fuchs-Sutugin correction f_FS→1 and J≈J_c. Exercises the continuum limit of the correction factor. J/J_c > 0.99.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "R_p": 1e-05
      },
      "assertions": [
        {
          "variable": "Kn",
          "time": 0.0,
          "expected": 0.0065455375547024625,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "f_FS",
          "time": 0.0,
          "expected": 0.9953435842005406,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "J_c",
          "time": 0.0,
          "expected": 2.513274122871835e-15,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "J",
          "time": 0.0,
          "expected": 2.501571273537722e-15,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    },
    {
      "id": "kinetic_regime_small_particle",
      "description": "Small particle R_p=10 nm: Kn=λ/R_p≈6.55 ≫ 1 (kinetic regime). Fuchs-Sutugin correction f_FS≈0.109 and J≈0.109·J_c. J/J_c < 0.15 as expected for the kinetic limit.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "R_p": 1e-08
      },
      "assertions": [
        {
          "variable": "Kn",
          "time": 0.0,
          "expected": 6.545537554702463,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "f_FS",
          "time": 0.0,
          "expected": 0.10884664234173425,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "J_c",
          "time": 0.0,
          "expected": 2.5132741228718344e-18,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "J",
          "time": 0.0,
          "expected": 2.735614495589664e-19,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    },
    {
      "id": "partial_accommodation",
      "description": "Partial accommodation α=0.1: the Fuchs-Sutugin factor is reduced by ~83% compared to α=1 at the same Kn≈0.655 (f_FS drops from 0.615 to 0.105). Exercises the α dependence of the correction factor; J follows proportionally.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "α": 0.1
      },
      "assertions": [
        {
          "variable": "Kn",
          "time": 0.0,
          "expected": 0.6545537554702464,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "f_FS",
          "time": 0.0,
          "expected": 0.10547352931380244,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "J_c",
          "time": 0.0,
          "expected": 2.5132741228718345e-17,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "J",
          "time": 0.0,
          "expected": 2.6508389187234356e-18,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    },
    {
      "id": "hot_atmosphere_350K",
      "description": "High temperature T=350 K: c_bar increases as √T (≈505.5 m/s), ρ_air decreases (≈1.010 kg/m³), λ increases (≈70.9 nm), Kn rises (≈0.709), and f_FS drops slightly (≈0.593). J_c is unchanged (no T dependence). Exercises the full temperature dependence of the thermodynamic chain.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "T": 350.0
      },
      "assertions": [
        {
          "variable": "c_bar",
          "time": 0.0,
          "expected": 505.5011790653896,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "rho_air",
          "time": 0.0,
          "expected": 1.0098027423622806,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "λ",
          "time": 0.0,
          "expected": 7.091889077384121e-08,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "Kn",
          "time": 0.0,
          "expected": 0.7091889077384121,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "f_FS",
          "time": 0.0,
          "expected": 0.5926895669140232,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "J_c",
          "time": 0.0,
          "expected": 2.5132741228718345e-17,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "J",
          "time": 0.0,
          "expected": 1.4895913514211292e-17,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    },
    {
      "id": "heavy_species_SO2",
      "description": "Heavy diffusing species (M_A=0.064 kg/mol, representative of SO₂). c_bar decreases as M_A^{-1/2} (≈314.1 m/s vs 466.6 m/s for air). λ, Kn, f_FS, J_c, and J are unchanged because they depend on M_air (not M_A). Tests that c_bar varies independently of the mass-transfer chain.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "M_A": 0.064
      },
      "assertions": [
        {
          "variable": "c_bar",
          "time": 0.0,
          "expected": 314.06140521147375,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "λ",
          "time": 0.0,
          "expected": 6.545537554702463e-08,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "J",
          "time": 0.0,
          "expected": 1.5452924510433117e-17,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    },
    {
      "id": "nonzero_surface_concentration",
      "description": "Surface concentration c_s=0.5e-6 mol/m³ (half of c_inf). The continuum flux J_c = 4πR_p D_g (c_inf-c_s) is halved relative to the default (c_s=0) case, and J follows proportionally. Tests that the (c_inf-c_s) driving force is correctly captured.",
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_overrides": {
        "c_s": 5e-07
      },
      "assertions": [
        {
          "variable": "Kn",
          "time": 0.0,
          "expected": 0.6545537554702464,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "f_FS",
          "time": 0.0,
          "expected": 0.6148523302653344,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "J_c",
          "time": 0.0,
          "expected": 1.2566370614359173e-17,
          "tolerance": {
            "rel": 1e-09
          }
        },
        {
          "variable": "J",
          "time": 0.0,
          "expected": 7.726462255216558e-18,
          "tolerance": {
            "rel": 1e-09
          }
        }
      ]
    }
  ],
  "analyses": [
    {
      "id": "flux_vs_particle_radius",
      "description": "Transition-regime corrected flux J and Fuchs-Sutugin factor f_FS as a function of particle radius R_p, swept log-spaced from 1 nm to 10 μm. Traces the crossover from the kinetic regime (Kn≫1, f_FS→0, J≪J_c) through the transition regime to the continuum regime (Kn≪1, f_FS→1, J≈J_c). The classical two-regime behaviour is captured by the Eq. 12.43 correction.",
      "initial_state": {},
      "parameters": {},
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "R_p",
            "range": {
              "start": 1e-09,
              "stop": 1e-05,
              "count": 41,
              "scale": "log"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "fFS_vs_Rp",
          "type": "line",
          "description": "Fuchs-Sutugin correction factor f_FS vs particle radius R_p. Approaches 1 in the continuum limit (R_p≫λ) and 0 in the kinetic limit (R_p≪λ).",
          "x": {
            "variable": "R_p",
            "label": "Particle radius R_p (m)"
          },
          "y": {
            "variable": "f_FS",
            "label": "Fuchs-Sutugin factor f_FS"
          }
        },
        {
          "id": "J_vs_Rp",
          "type": "line",
          "description": "Transition-regime corrected molar flow rate J vs particle radius R_p. Scales as R_p in the continuum limit (J≈J_c∝R_p) and is suppressed by f_FS in the kinetic limit.",
          "x": {
            "variable": "R_p",
            "label": "Particle radius R_p (m)"
          },
          "y": {
            "variable": "J",
            "label": "Molar flow rate J (mol/s)"
          }
        }
      ]
    },
    {
      "id": "flux_vs_accommodation_coefficient",
      "description": "Transition-regime flux J as a function of the mass accommodation coefficient α, swept from 0.001 to 1 (log scale) at the default particle radius (100 nm, transition regime). Shows how incomplete accommodation suppresses the flux: at α=1 (perfect accommodation) J is highest; at small α the Fuchs-Sutugin factor collapses and J decreases substantially.",
      "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": 31,
              "scale": "log"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "J_vs_alpha",
          "type": "line",
          "description": "Transition-regime molar flow rate J vs accommodation coefficient α at R_p=100 nm (transition regime). Higher α → higher f_FS → higher J.",
          "x": {
            "variable": "α",
            "label": "Accommodation coefficient α"
          },
          "y": {
            "variable": "J",
            "label": "Molar flow rate J (mol/s)"
          }
        }
      ]
    },
    {
      "id": "flux_over_radius_and_accommodation",
      "description": "Two-dimensional sweep of the transition-regime flux J over particle radius R_p (1 nm to 10 μm, log) and accommodation coefficient α (0.01 to 1, log). Surfaces the joint dependence: the Fuchs-Sutugin regime boundary shifts with both parameters, and partial accommodation becomes increasingly important in the kinetic regime.",
      "initial_state": {},
      "parameters": {},
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "R_p",
            "range": {
              "start": 1e-09,
              "stop": 1e-05,
              "count": 20,
              "scale": "log"
            }
          },
          {
            "parameter": "α",
            "range": {
              "start": 0.01,
              "stop": 1.0,
              "count": 10,
              "scale": "log"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "J_heatmap",
          "type": "heatmap",
          "description": "Molar flow rate J (mol/s) over particle radius R_p and accommodation coefficient α.",
          "x": {
            "variable": "R_p",
            "label": "Particle radius R_p (m)"
          },
          "y": {
            "variable": "α",
            "label": "Accommodation coefficient α"
          },
          "value": {
            "variable": "J",
            "reduce": "final"
          }
        }
      ]
    }
  ]
}