Introduction & Starting point
Air quality in underground stations is a growing concern in cities with an urban transportation network. In France, continuous and open access measurement campaigns are carried out at the SNCF or at the RATP, giving access to data on the concentration of PM_{2.5} and PM_{10}.
By using the available data, the correlation between rail traffic (continuous lines) and PM_{10} (dotted line). The figure below represents an average week in the StMichel NotreDame underground station (Paris). It can then be assumed that the majority of the amount of aerosol in suspension comes from braking and resuspension.
In this figure, it can be seen that the maximum PM_{10} are concomitant with morning and evening traffic peaks, on the other hand, weekend concentrations decrease with rail traffic.
By observing the PM_{2.5}/PM_{10} ration over one week at Magenta station (Paris) as shown in Figure 2 below from [Fortain 2007], we can see that it is between ~0.25 and 0.35 during the day, while it reaches 1 during the night. The PM_{2.5} being included in the PM_{10}, this means that the larger particles gradually disappear from the aerosol during the night time: only the finer ones remain.
This can be explained by the deposition rate of the particles. Indeed, it is more important for large diameter particles. Figure 3 below shows the deposition rate of particles as a function of their diameter based on [Lai et Nazaroff 2000]. Not surprisingly, gravity deposition is predominant for On particles larger than 1[µm] in diameter. It should be noted that the finest particles, with a diameter of less than 0.01[µm] also settle more significantly: this is related to the Brownian motion which diffuses very small particles onto surfaces. In the intermediate diameter range, the particles are too large to be subjected to the Brownian motion of the gas in which they evolve (air) and do not have enough mass for gravity sedimentation: the natural convection movements of the air are sufficient to keep them in suspension.
These observations have led to the development of successive models for predicting air quality in underground stations. The following sections describe the method developed in [Walther et Bogdan 2017a] and [Walther et al. 2017b], available in the section Publications of this site.
Equation
Based on the previous finding and the homogeneous mixing hypothesis, a differential system is established that considers the concentrations of two classes of particles C_a for the PM_{2.5} and C_b for the PM_{2.510}, the sum of the two being equal to the concentration of the PM_{10}. This makes it possible to capture the slower and faster dynamics of the aerosol particles:
\frac{d C_a}{dt} = \alpha_a N^2(t) + \tau (C_a^\text{ext}C_a)  \delta_a C_a
\frac{d C_b}{dt} = \alpha_b N^2(t) + \tau (C_b^\text{ext}C_b)  \delta_b C_b
Where N is the number of trains per unit of time, \alpha_a,\alpha_b are the terms of emission of the PM_{2.5} and PM_{2.510} and \delta_a,\delta_b the deposition rates calculated according to the [Lai et Nazaroff 2000].
The air exchange rate \tau is broken down into a natural ventilation rate \tau_0 and a piston effect ventilation \beta \times N, where \beta [] is the volume of outside air caused by the departure and the arrival of the train, divided by the volume of the station.
In the literature review by[Nicholson 1988], the resuspension rate follows speed with a power law with an exponent ranging from 1 to 6. The apparent issue term \alpha is thus assumed to vary with the square of the number of trains N^2 : Indeed, if trains are discrete events, they are considered to lead to an increase in the average speed at the station. The experimental air velocity measurement campaign conducted at StMichel NotreDame station in February 2017 corroborates this hypothesis (see Figure 4 below).
Identification of model parameters
With the differential system in place, it remains to identify the unknown parameters of the model. There are six of them:

 the piston effect \beta whose order of magnitude can be determined by measurement or simulation (see our research on this subject)
 the rate of air renewal by natural or mechanical ventilation \tau_0
 the apparent source terms \alpha_a,\alpha_b which give the proportions of fine and larger particles emitted by braking and resuspension
 the deposition constants \delta_a,\delta_b which are actually identified from equivalent aerosol diameters d_a,d_b : Indeed, the approach developed here considers that the total concentration of PM_{10} is that of an aerosol composed of only two types of particles, in different concentrations.
The detailed identification of these parameters makes it possible to find fairly accurately the concentrations measured as a function of rail traffic, as shown in the figure below Figure 5, where measurements and model are compared.
The PM_{2.5} are also simulated by the model (see green curve in Figure 5) but without experimental confrontation, as these measurements are not available for the SaintMichel station at the time of preparing this document (this paragraph will have to be updated).
The PM_{2.5}/PM_{10} ratio obtained, presented in Figure 6, is very similar to the one of the experimental campaign of [Fortain 2007] (see Figure 2).
Pending experimental results, this finding is considered to support the results obtained from the model. On this basis, it is possible to simulate the evolution of the PM_{10} concentration as a function of ventilation and filtration installation, by integrating these terms into the model with the identified parameters.
Further development
The model is being improved, depending on the availability of measured data. Two major axes are being explored: the addition of the quantity deposited on surfaces and the generalization of a model to n classes.
Closing equation
The addition of the equation on the surfaces closes the differential system previously presented as in [Qian et al. 2008]. This results in:
V \times \frac{\partial C}{\partial t} = a N + Q_{\text{v}} (C^{\text{ext}}  C)  \delta V C + \rho S_{\text{r}} L
S_{\text{d}} \times \frac{dL}{dt} = \delta V C  \rho S_{\text{r}} L
In the first equation, V is the volume of the station, Q_v is the external air flow rate, \delta is the deposition rate, \rho is the resuspension constant, S_r the surface accessible for resuspension and L the quantity of particles deposited per unit area [\mu g/m^2]. In the second equation, S_d is the surface accessible for deposition.
It should be noted that these two equations are coupled by the cross terms \pm \delta V C and \pm \rho S_r L. These respectively represent the quantity of particles that passes from the air "compartment" to the surface "compartment" by deposition and the quantity resuspended from the surfaces to the air volume.
Model with n classes
With a reasoning similar to that used for two particle size classes, it is possible to extend the model to n classes: this would represent the evolution of an aerosol by size class, as shown in the following figure.
It becomes possible to define sources from their « emission spectrum » by size class. An example is given below:
Among the scientific locks to be removed are the quantity of particles initially deposited, the differentiation of the resuspension term by size class (in other words: the "resuspension spectrum" for which there is not yet a unified theory) and finally the experimental comparison: external concentrations by size class are rarely available and the quantity of particles deposited on surfaces is difficult to evaluate.
References
[Fortain 2007] Caractérisation des particules en gares souterraines. PhD thesis, University of La Rochelle
[Lai et Nazaroff 2000] Lai, A. C. K. and Nazaroff, W. W. (2000). Modeling indoor particle deposition from turbulent flow onto smooth surfaces. Journal of Aerosol Science, 31(4):463–476.
[Nicholson 1988] Nicholson, K. W. (1988). A review of particle resuspension. Atmospheric Environment, 22.
[Qian et al. 2008] Qian, J., Ferro, A.R., Fowler, K.R., 2008. Estimating the resuspension rate and residence time of indoor particles. J. Air Waste Manage. Assoc. 58 (4).
[Walther et Bogdan 2017a] Walther, E., & Bogdan, M. (2017). A novel approach for the modelling of air quality dynamics in underground railway stations. Transportation Research Part D: Transport and Environment, 56, 3342.
[Walther et al. 2017b] Walther, E., Bogdan, M., & Cohen, R. (2017). Modelling of airborne particulate matter concentration in underground stations using a two sizeclass conservation model. Science of The Total Environment, 607, 13131319.