# A Multi-Scale Approach for Modelling Airborne Transport of Mucosalivary Fluid
## 2 Governing Equations
### PSI-PBE Coupling  
One of the main novelty of this work is related to the coupling between PSI-Cell method and PBE. Precisely, PBE is solved in the droplet's inner part in order to model nucleation and growth processes of the NaCl crystals. PBE solution enables the possibility to predict droplet nuclei generation phenomenon. In this context, PBE is reformulated ignoring spatial terms, i.e., both convective and diffusive terms are removed. This is possible due to the homogeneous spatial condition of the considered system. Hence, PBE collapses in an ordinary differential equation which requires limited computational resources for its solution. Moreover, a similar formulation is also attractive from the numerical point of view because of its increased numerical stability. PBE reads as follows (28):  
\[\frac{\partial N_{j}}{\partial t}+\mathbf{\
abla}\cdot\big{(}N_{j}\mathbf{u}_{p} \big{)}-\mathbf{\
abla}\cdot\big{(}D_{t}\mathbf{\
abla}N_{j}\big{)}=-\sum_{j}\frac{ \partial\big{[}G_{j}N_{j}\big{]}}{\partial r_{j}}+B\prod_{j}\delta\big{(}r_{ j}-r_{\beta}\big{)}+h \tag{3}\] \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Nu** & **Sh** \\ \hline [PERSON] (23) & \(2+0.6\text{Re}_{d}^{0.6}\text{Pr}^{\frac{1}{3}}\) & \(2+0.6\text{Re}_{d}^{0.6}\text{Sc}^{\frac{1}{3}}\) \\ \hline Clift (24) & \((1+\text{Re}\cdot\text{Pr})^{\frac{1}{3}}\cdot\text{max}\Big{[}1,\text{Re}_{d} ^{0.77}\Big{]}\) & \((1+\text{Re}\cdot\text{Sh})^{\frac{1}{3}}\cdot\text{max}\Big{[}1,\text{Re}_{d} ^{0.77}\Big{]}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Correlations for Nusselt and Sherwood numbers. where \(N_{j}\) is density number of \(i\)-th chemical specie, \(D_{t}\) is the local turbulent diffusivity, \(G_{j}\) is the growth rate, \(r_{j}\) is the particle internal coordinate, \(r_{j0}\) is the particle internal coordinate for a crystal nucleus. Furthermore, \(\delta\) is the Dirac function, \(B\) is the nucleation rate, and h is the creation or destruction of particles due to aggregation, agglomeration, and breakage. The interest of this paper is limited to crystallization phenomena issues of NaCl. Consequently, PBE needs to include only \(B\) and \(G\) terms. Moreover, the convective and diffusive terms for the particles disappear because we assume each droplet to be well mixed and tracked independently in the Lagrangian frame. It is feasible to re-write PBE in a semi-discrete form integrating Equation (3) over \(r\):  
\[\frac{df_{j}}{dt}=-\frac{1}{\Delta r}\left[G_{j+1/2}\left(f_{j}+\frac{\Delta r }{2}(f_{r})_{j}\right)-G_{j-1/2}\left(f_{j-1}+\frac{\Delta r}{2}(f_{r})_{j-1} \right)\right] \tag{4}\]  
where \(f_{j}\) is the particle-averaged population density. It is also important to remark that in Equation (4) nucleation term is included averaging the nucleation rate within discrete linear computational cells built inside the droplets. Starting from [PERSON] et al. [28], it is possible to evaluate particle-averaged crystal mass:  
\[N_{w,j}=\frac{1}{4}\rho_{c}k_{\mathrm{p}}f_{j}\left(r_{j+1/2}^{4}-r_{j-1/2}^{ 4}\right) \tag{5}\]  
which, in turns, allows to calculate the total mass related to crystallization process as:  
\[m_{cr,i}=\pi\frac{D_{p,i}^{3}}{6}\sum_{j}N_{w,j} \tag{6}\]  
and the radius of the (dry) solid part of the droplets, \(r_{N}\):  
\[r_{N}=\frac{\sum_{j}N_{w,j}r_{j}^{4}}{\sum_{j}N_{w,j}r_{j}^{3}}, \tag{7}\]  
actually, combining Equations (5) and (6), semi-discrete PBE can be formulated as:  
\[ \begin{split}\frac{dN_{w,j}}{dt}&=-\frac{\rho k_{ \mathrm{p}}}{4\Delta r}\left(r_{j+1/2}^{4}-r_{j-1/2}^{4}\right)\max(\mathrm{ sign}\Delta c,0)\times\\ &\left[G_{j+1/2}\left(f_{j}+\frac{\Delta r}{2}(f_{r})_{j}\right) -G_{j-1/2}\left(f_{j-1}+\frac{\Delta r}{2}(f_{r})_{j-1}\right)+\underbrace{B }_{j=0}\right].\end{split}  \tag{8}\]  
In the above equation \(\Delta c=c-c^{*}\) is supersaturation, while \(c\) and \(c^{*}\) are the NaCl concentration and its solubility in pure water.