Henry’s Law Phase Transfer

Henry’s Law Phase Transfer#

Gas-condensed phase mass transfer governed by Henry’s Law equilibrium and Fuchs-Sutugin transition-regime kinetics.

Physics#

For gas species A partitioning into a condensed phase:

\[\text{A(gas)} \rightleftharpoons \text{A(condensed)}\]

The condensation rate describes how fast gas molecules transfer into particles, the evaporation rate describes the reverse, and Henry’s Law defines the equilibrium ratio.

Henry’s Law Constant#

Temperature-dependent:

\[H(T) = H_\text{ref} \cdot \exp\!\left(C \left(\frac{1}{T} - \frac{1}{T_0}\right)\right)\]

Symbol	Description	Units
\(H_\text{ref}\)	Reference HLC at \(T_0\)	mol m⁻³ Pa⁻¹
\(C\)	Temperature dependence parameter	K
\(T_0\)	Reference temperature (default 298.15)	K

HLC is evaluated once per time step and stored as a state parameter.

Condensation Rate#

Mean molecular speed#

\[\bar{c} = \sqrt{\frac{8 R T}{\pi M_w}}\]

Mean free path#

\[\lambda = \frac{3 D_g}{\bar{c}}\]

Knudsen number#

\[\text{Kn} = \frac{\lambda}{r_\text{eff}}\]

Characterizes the gas-particle interaction regime (Kn ≪ 1 → continuum, Kn ≫ 1 → free molecular).

Fuchs-Sutugin correction#

\[f(\text{Kn}) = \frac{1 + \text{Kn}}{1 + \frac{2 \text{Kn}(1 + \text{Kn})}{\alpha}}\]

Interpolates between the continuum (\(f \to 1\)) and free-molecular (\(f \to \alpha / (2 \text{Kn})\)) limits.

Derivative with respect to Kn:

\[\frac{df}{d\text{Kn}} = \frac{\alpha - 2 - 4\text{Kn} - 2\text{Kn}^2}{\alpha \cdot D^2}\]

where \(D = 1 + 2\text{Kn}(1 + \text{Kn})/\alpha\).

Condensation rate constant#

\[k_c = 4\pi \, r_\text{eff} \, N \, D_g \, f(\text{Kn})\]

Units: s⁻¹. First-order rate constant for gas-to-condensed transfer.

Partial derivatives:

\[\frac{\partial k_c}{\partial r_\text{eff}} = 4\pi N D_g \left[ f(\text{Kn}) - \text{Kn} \cdot \frac{df}{d\text{Kn}} \right]\]

\[\frac{\partial k_c}{\partial N} = \frac{k_c}{N}\]

Evaporation Rate#

\[k_e = \frac{k_c}{H \cdot R \cdot T}\]

Partial derivatives — obtained by dividing \(k_c\) partials by the same constant factor:

\[\frac{\partial k_e}{\partial r_\text{eff}} = \frac{1}{HRT} \frac{\partial k_c}{\partial r_\text{eff}},\qquad \frac{\partial k_e}{\partial N} = \frac{k_e}{N}\]

Phase Volume Fraction#

For multi-phase modes, the exposed surface area of phase \(p\) is proportional to its volume fraction:

\[\varphi_p = \frac{V_\text{phase}}{V_\text{total}}\]

where

\[V_\text{phase} = \sum_i [\text{species}_{p,i}] \cdot \frac{M_{w,i}}{\rho_i}, \qquad V_\text{total} = \sum_q \sum_i [\text{species}_{q,i}] \cdot \frac{M_{w,i}}{\rho_i}\]

Solvent Volume Fraction#

Converts between concentration bases:

\[f_v = [\text{solvent}] \cdot \frac{M_{w,\text{solvent}}}{\rho_\text{solvent}}\]

Net Transfer Rate#

\[R_\text{net} = \varphi_p \, k_c \, [\text{A}]_\text{gas} - \varphi_p \, k_e \, \frac{[\text{A}]_\text{aq}}{f_v}\]

The ODE right-hand sides:

\[\frac{d[\text{A}]_\text{gas}}{dt} = -R_\text{net}, \qquad \frac{d[\text{A}]_\text{aq}}{dt} = +R_\text{net}\]

When multiple condensed-phase instances exist (e.g., multiple aerosol modes containing the same phase), each contributes its own \(R_\text{net}\) and contributions are summed.

Jacobian#

The MICM Rosenbrock solver stores −J (negative Jacobian). All values below show the mathematical derivative and the stored value.

Direct entries#

Entry	Mathematical \(J\)	Stored \(-J\)
\(J[\text{gas}, \text{gas}]\)	\(-\varphi_p k_c\)	\(+\varphi_p k_c\)
\(J[\text{gas}, \text{aq}]\)	\(+\varphi_p k_e / f_v\)	\(-\varphi_p k_e / f_v\)
\(J[\text{gas}, \text{solvent}]\)	\(-\varphi_p k_e [\text{A}]_\text{aq} / (f_v \cdot [\text{solvent}])\)	\(+\varphi_p k_e [\text{A}]_\text{aq} / (f_v \cdot [\text{solvent}])\)
\(J[\text{aq}, x]\)	\(-J[\text{gas}, x]\)	\(-(-J[\text{gas}, x])\)

The antisymmetry \(J[\text{aq}, x] = -J[\text{gas}, x]\) is a direct consequence of mass conservation.

Indirect entries (chain rule)#

When an aerosol property \(P\) depends on state variable \(y_j\), additional terms appear:

Through \(r_\text{eff}\):

\[\text{stored } {-J[\text{gas}, y_j]} \mathrel{+}= \varphi_p \left( \frac{\partial k_c}{\partial r} [\text{A}]_\text{gas} - \frac{\partial k_e}{\partial r} \frac{[\text{A}]_\text{aq}}{f_v} \right) \frac{\partial r}{\partial y_j}\]

Through \(N\):

\[\text{stored } {-J[\text{gas}, y_j]} \mathrel{+}= \varphi_p \left( \frac{\partial k_c}{\partial N} [\text{A}]_\text{gas} - \frac{\partial k_e}{\partial N} \frac{[\text{A}]_\text{aq}}{f_v} \right) \frac{\partial N}{\partial y_j}\]

Through \(\varphi_p\):

\[\text{stored } {-J[\text{gas}, y_j]} \mathrel{+}= R \cdot \frac{\partial \varphi_p}{\partial y_j}\]

where \(R = k_c [\text{A}]_\text{gas} - k_e [\text{A}]_\text{aq}/f_v\) is the un-scaled net rate.

Aerosol Property Derivatives#

Effective radius#

SingleMomentMode / UniformSection: \(\partial r / \partial y_j = 0\) (parameterized).
TwoMomentMode:

\[\frac{\partial r_\text{eff}}{\partial [\text{species}_{p,i}]} = \frac{r_\text{eff} \cdot M_{w,i}}{3 \, V_\text{total} \, \rho_i}, \qquad \frac{\partial r_\text{eff}}{\partial N} = -\frac{r_\text{eff}}{3 N}\]

Number concentration#

SingleMomentMode / UniformSection: \(N = V_\text{total}/V_\text{single}\), so

\[\frac{\partial N}{\partial [\text{species}_{p,i}]} = \frac{M_{w,i}}{\rho_i \cdot V_\text{single}}\]
TwoMomentMode: \(N\) is a prognostic variable, so \(\partial N / \partial N_\text{var} = 1\).

Phase volume fraction#

For a mode/section with a single phase, \(\varphi_p = 1\) and all partials are zero. For multi-phase:

\[\frac{\partial \varphi_p}{\partial [\text{species}_{p,i}]} = \frac{M_{w,i}}{\rho_i} \cdot \frac{1 - \varphi_p}{V_\text{total}}\]

\[\frac{\partial \varphi_p}{\partial [\text{species}_{q,i}]} = -\frac{M_{w,i}}{\rho_i} \cdot \frac{\varphi_p}{V_\text{total}} \qquad (q \neq p)\]