## 2 Materials and Methods
### Estimation of Ventilation Rates  
The ventilation rate is a magnitude that expresses the net volumetric flow of reposition air \(Q\) that enters a room. Said flow is usually expressed as Air Changes per Hour (ACH) by referring it to the volume of the room:  
\[\text{ACH}=\frac{Q}{V} \tag{1}\]  
where \(\text{ACH}\) is conventionally expressed in \(\text{h}^{-1}\). ACH is an IAQ indicator, as it is one of the main parameters that regulates the concentration of airborne contaminants. Assuming that the reposition air that enters the room is cleaner than the air inside, higher ventilation rates allow for remotion and dilution of contaminants. Consequently, the ventilation rate is one of the main engineering parameters for the design of both natural and mechanical ventilation systems. For example, the ANSI/ASHRAE 62.1-2019 standard [30] stipulates a minimum ventilation rate for occupied spaces according to their intended use, surface area and number of occupants. For the classroom previously described, the standard yields a minimum ventilation rate of 3.8 \(\text{h}^{-1}\). Additionally, in the context of the SARS-CoV-2 pandemic, several recommendations have been made regarding ventilation rates. For example, the World Health Organization (WHO) has recommended a minimum ventilation rate of 10 L/s per person [31]. This implies 6.4 \(\text{h}^{-1}\) for the selected classroom. One relatively simple way of estimating the ventilation rate in a certain environment is by using fully-mixed box models. These are based on performing a mass balance to an airborne tracer by assuming that the air in the room is perfectly and instantly mixed. Hence, the tracer concentration is supposed to be uniform. Several researchers have applied box models for studying IAQ. There seems to be consensus that \(\text{CO}_{2}\) is a suitable tracer for studying ventilation, particularly in schools [5]. It has the advantage of being an inert gas that is relatively easy and inexpensive to measure. Furthermore, its generation is guaranteed in any occupied environment, as it is produced by humans when they breathe. Nonetheless, \(\text{CO}_{2}\) poses some disadvantages. There is a significant concentration of it in outside air, which can have slight variations. Additionally, determining the human generation rate of \(\text{CO}_{2}\) is not straightforward. The mass balance for \(\text{CO}_{2}\) in a certain room can be expressed as  
\[V\frac{dC(t)}{dt}=Q(C_{\text{ext}}-C(t))+nG \tag{2}\]  
where \(V\) is the room's volume, \(C\) is the volumetric concentration of \(\text{CO}_{2}\), \(t\) is the time, \(Q\) is the volumetric flow of outside air, \(C_{\text{ext}}\) is the concentration of \(\text{CO}_{2}\) in the air that enters the room, \(n\) is the number of persons in the room and \(G\) is the volumetric generation rate of \(\text{CO}_{2}\) of one individual. Assuming that \(C\) is the only time function, Equation (2) can be solved to obtain  
\[C(t)=C_{\text{ext}}+\frac{nG}{Q}-\left(C_{\text{ext}}-C_{\text{o}}+\frac{nG}{ Q}\right)e^{-\frac{Q}{V}t} \tag{3}\]  
where \(C_{\text{o}}\) is an initial condition for \(C\). A concentration time series could be fitted via a non-linear least-squares technique in order to determine the model parameters. Although box models are a compelling option for determining ventilation rates, they present some difficulties. Firstly, they rely on the perfect mix hypothesis, which could be inappropriate. There is little information regarding this concern. Quantitative criteria is mentioned in the ASTM D6249 and E741 standards [32; 33], which establish that the concentration difference between measurement points should be below 10% of the difference between the room's average concentration and the outdoors concentration. Secondly, although model parameters can be obtained by fitting concentration measurements, in practice this is quite difficult. Some systems may exhibit rapid dynamics, which permit getting just some measurements before there are changes in the model parameters.