Back to parent page - Gas-aerosol Exchange

Mass transfer rates¶

Mass transfer to/from a single particle¶

We start by assuming the ambient air contains a semi-volatile trace gas denoted by \(L\). The net mass exchange between the trace gas and a single particle can be quantified using the mass transfer rate \(\widetilde{J}_{L}\) (unit: kmol\(s^{-1}\)) where positive and negative rates correspond to condensation and evaporation, respectively. \(\widetilde{J}_{L}\) can be expressed following Fuchs and Sutugin (1971) as

\[ \begin{equation} \label{eqn:mass_transfer_rate_single_particle} \widetilde{J}_{L} = 2 \pi D_{p} \mathbb{D}_{g,L} (\widetilde{c}_{L} - \widetilde{c}_{L,sat}) f(K_{n,L}, \alpha_{L}) \end{equation} \]

(see also Eq. (12.12) and (12.43) in Seinfeld and Pandis (2016)), where

\(D_{p}\) is the diameter (unit: m) of the aerosol particle,
\(\mathbb{D}_{g,L}\) is the gas diffusivity (unit: m\(^2\)s\(^{-1}\), see expression below).
\(\widetilde{c}_{L}\) is the ambient concentration of gas \(L\) (kmol per m\(^{3}\) air),
\(\widetilde{c}_{L,sat}\) is the saturation concentration of gas \(L\) just above the surface of the aerosol particle. The value of \(\widetilde{c}_{L,sat}\) depends on \(T\), \(p\) and the chemical species. In the real world, \(\widetilde{c}_{L,sat}\) also depends on the size and composition of the aerosol particle.
\(f(K_{n,L}, \alpha_{L})\) denotes a unitless function of the Knudsen number \(K_{n}\) and the accommodation coefficient \(\alpha\). The Knudsen number is defined as the ratio between the mean free path of gas \(L\) in air and the diameter of the particle. \(\alpha_{L}\) (also unitless) is the probability of sticking when a vapor molecule encounters the surface of a particle (cf. Chapter 12.1.3 of Seinfeld and Pandis (2016)).

The calculation of the gas diffusivity follows Fuller et al., 1966.

\[ \begin{equation} \label{gasd} \mathbb{D}_{g,L} = \frac{10^{-7}T^{1.75} \sqrt{\frac{1}{M_{w,L}}+\frac{1}{M_{w,air}}}} {p \left[ (\mathbb{V}_{D,L})^\frac{1}{3}+ (\mathbb{V}_{D,air})^\frac{1}{3} \right]^2} \end{equation} \]

Note that here the unit of \(p\) is \(\rm atm\), i.e., the standard atmosphere defined as 101325Pa. \(\mathbb{V}_{D,L}\) is the molecular diffusion volume of \(L\), a constant for a given species determined by regression analysis of experimental data (Fuller et al., 1966; Poling et al., 2001; Tang et al., 2015). \(M_{w,L}\) and \(M_{w,air}\) are molecular weights of gas \(L\) and the air, respectively (unit: \(\rm g \cdot mol^{-1}\)). The values of these physical constants are listed in Table~1.

The expression of \(f(K_{n,L}, \alpha_{L})\) follows Fuchs and Sutugin (1971),

\[ \begin{equation} \label{fka} f(K_{n,L}, \alpha_{L}) = \frac{0.75 \alpha_{L} (1+K_{n,L})} {0.75 \alpha_{L} + K_{n,L} (1+K_{n,L}+0.283 \alpha_{L})} \end{equation} \]

where the Knudson number is defined as

\[ \begin{equation} K_{n,L} = \frac{2\lambda_{L}}{D_{p}} \end{equation} \]

with \(\lambda_{L} = \frac{3\mathbb{D}_{g,L}}{v_{L}} \label{mfp}\) (see Eq. (9-15) in Seinfeld and Pandis, 2016)). \(v_{L} = \sqrt{\frac{8RT}{\pi M_{w,L}}}\) is the mean molecular speed (\(\rm m~s^{-1}\)) of gas \(L\).

Table 1: Constants used in MAM's gas-aerosol exchange parameterization.

Constant	Value	Reference
\(\mathbb{V}_{D,H_2SO_4}\)	42.88
\(\mathbb{V}_{D,air}\)	20.1
\(M_{w,H_2SO_4}\)	98.0783997 g mol\(^{-1}\)
\(M_{w,air}\)	28.966 g mol\(^{-1}\)
\(\alpha_{H_2SO_4}\)	0.65	\cite{poschl-1998-jpca} \

Mass transfer to/from a log-normal mode¶

Recall that MAM uses a collection of log-normal functions ("modes") to describe the size distribution of the aerosol particles. The governing equations are written in the form of time-evolution equations of the aerosol number concentrations of the various modes and the aerosol mass concentrations of the chemical species in each mode. Below we explain the derivation of the equations related to the condensation/evaporation of chemical gases.

Mass transfer rate expressed with concentrations per unit volume of air¶

Let \(n_i(\ln D_p)\) denote the size distribution of aerosol particles in mode \(i\) (unit: m\(^{-3}\)). The net mass transfer rate to/from a mode \(i\), in the unit of mol species \(L\) per \(\rm m^3 s^{-1}\) of air, can be obtained by integrating the mass transfer rate \(\widetilde{J}_{L}\) of a single particle over the mode's size distribution, i.e.,

\[ \begin{eqnarray}\label{eqn:mass_transfer_rate_integral} J_{L,i} &=& \int_{-\infty}^{\infty} \widetilde{J}_{L} n_{i}(\ln D_{p}) d\ln D_{p} \end{eqnarray} \]

Elements on the right-hand side of the expression of \(\widetilde{J}_{L}\) that depend on particle size include (i) \(D_p\), (ii) the saturation gas concentration \(\widetilde{c}_{L,sat}\), and (iii) \(f(K_{n,L}, \alpha_{L})\).

MAM uses the simplifying assumption that \(\widetilde{c}_{L,sat}\) is independent of \(D_p\) within each mode although its value can differ from mode to mode. This helps to simplify the expression of \(J_{L,i}\) into

\[ \begin{equation}\label{eqn:mass_transfer_rate_mode_i} J_{L,i} = (\widetilde{c}_{L} - \widetilde{c}_{L,sat,i})\, C_{L,i} \end{equation} \]

where

\[ \begin{equation} \label{kmt} C_{L,i} = 2 \pi \,\mathbb{D}_{g,L} \int_{- \infty}^{\infty} f(K_{n,L}, \alpha_{L}) D_{p}n_{i}(\ln D_{p}) d\ln D_{p} . \end{equation} \]

\(C_{L,i}\) is referred to as the mass transfer coefficient (unit: s\(^{-1}\)) of mode \(i\). It is worth noting that, as a chemical species condenses or evaporates, particle sizes will change, hence \(C_{L,i}\) will also evolve. We also note that \(D_p\) in the equations above should be interpreted as the wet diameter of particles or modes if the particles are soluble.

Mass transfer rate expressed with molar mixing ratios¶

MAM's aerosol microphysics parameterizations use molar mixing ratios to describe the concentrations of aerosol and gases, i.e., kmol kmol\(^{-1}\)s\(^{-1}\) for mass concentrations and kmol\(^{-1}\)s\(^{-1}\) for number concentrations. Let us use the following notation:

\(q_{v,L}\): molar mixing ratio of gas species \(L\), unit: kmol (kmol-air)\(^{-1}\);
\(q_{L,sat,i}\): saturation molar mixing ratio of gas species \(L\) for aerosol mode \(i\), unit: kmol~(kmol-air)\(^{-1}\);
\(q_{m,L,i}\): molar mixing ratio of species \(L\) in mode \(i\), unit: kmol (kmol-air\(^{-1}\));
\(q_{n,i}\): molar mixing ratio of particle number of mode \(i\), unit: (kmol-air)\(^{-1}\),
\(c_{air}\): concentration of dry air in the unit of kmol m\(^{-3}\),
\(N_i\): number of particles of mode \(i\) per unit volume of air (unit: m\(^{-3}\)).

By definition, we have

\[ \begin{eqnarray} N_i &=& c_{air}\,q_{n,i} \label{eqn:Ni} \,,\\ \widetilde{c}_{L} &=& c_{air}\,q_{v,L} \,,\\ \widetilde{c}_{L,sat,i} &=& c_{air}\,q_{L,sat,i}\,. \end{eqnarray} \]

Dividing the mass transfer rate \(J_{L,i}\) by \(c_{air}\) gives

\[ \begin{equation}\label{eqn:molar_mass_transfer_rate_mode_i} \frac{J_{L,i}}{c_{air}} = (q_{L} - q_{L,sat,i})\, C_{L,i} \,. \end{equation} \]

This is the rate of change in the molar mixing ratio of aerosol mass in mode \(i\) due to the condensation/evaporation of species \(L\).

To help illustrate the nonlinearity of MAM's governing equations presented in a separate section, we now express \(C_{L,i}\) in terms of the aerosol mixing ratios predicted by MAM. Plugging the expression of \(n_i\left(\ln D_p\right)\) (see the section on the modal approach) into the expression of \(C_{L,i}\) gives

\[ \begin{eqnarray} \label{kmt_detailed} C_{L,i} &=& 2 \pi \,\mathbb{D}_{g,L} \int_{- \infty}^{\infty} f(K_{n,L}, \alpha_{L}) % D_{p}\frac{N_{i}}{\sqrt{2\pi}\ln \sigma_{g,i}} \ \exp \left [ - \frac{(\ln D_{p} - \ln D_{gn,w,i})^2} {2\ln^2\sigma_{g,i}} \right ] % d\ln D_{p} \\ % &=& 2\pi c_{air}\,q_{n,i}\,\mathbb{D}_{g,L} \int_{- \infty}^{\infty} f(K_{n,L}, \alpha_{L}) % \frac{D_{p}}{\sqrt{2\pi}\ln \sigma_{g,i}} \! \exp\! \left [ - \frac{(\ln D_{p} - \ln D_{gn,w,i})^2} {2\ln^2\sigma_{g,i}} \right ] % d\ln D_{p} %\nonumber \label{eqn:kmt_expressed_in_q} \end{eqnarray} \]

The geometric mean wet diameter \(D_{gn,w,i}\) is linked to the aerosol mass and number mixing ratios by

\[ \begin{eqnarray} D_{gn,w,i}^3 &=& \frac{6\rho_{air}}{\pi c_{air}\,q_{n,i} \exp \left ( \frac{9}{2} \ln^2 \sigma_{g,i} \right )} \left[ \sum\limits_{SP} \left( q_{m,SP,i}\frac{ M_{w,SP}} {\rho_{SP}}\right) + q_{m,H_2O,i} \frac{M_{w,H_2O}} {\rho_{H_2O}} \right] \label{eqn:D_gn_w_i_and_qm_qn} \end{eqnarray} \]

where \(SP\) refers to any of the aerosol species in mode \(i\). For MAM4, SP can be SO_4, POA, SOA, BC, SS, DST and MOA. The subscript H\(_2\)O in the expression of \(D_{gn,w,i}^3\) refers to aerosol water (see documentation on water uptake). \(\rho_{air}\) is the density of air (kg m\(^{-3}\)). \(\rho_{SP}\) and \(\rho_{H_2O}\) are the densities of the aerosol species (kg m\(^{-3}\)) and liquid water, respectively.