COVID-19: Analysis of models used to assess risk for infection in a classroom

Dr. Neil Canter, Contributing Editor | TLT Tech Beat September 2021

A recent study shows what’s important in reducing transmission in the classroom.


Two models, the Wells-Riley equation and computational fluid dynamics, have been used to better understand COVID-19 transmission in a classroom.
Case studies where no ventilation was present and where an HVAC system that had a particulate removal efficiency of 72% were evaluated.
Use of masks and ventilation were found to be very important in reducing COVID-19 transmission. Social distancing from an infected individual was not considered to be a significant issue, as the probability of infection in a non-ventilated classroom was with individuals 4.5 meters from an infected individual instead of the recommended two meters. 

Note: COVID-19 continues to impact how the lubricant industry is conducting business. This month’s Tech Beat presents articles from two recent research papers that will hopefully provide further insight into measures that need to be taken to minimize exposure and help you, our readers, continue to deal with the challenges of living and working with COVID-19. Both articles evaluate the effect of COVID-19 exposure in a classroom environment. The first article discusses the accuracy of two models in evaluating COVID-19 transmission. The second article uses carbon dioxide concentration as an indication of the potential degree of risk of COVID-19 infection. Use of masks, proper ventilation and reducing physical activity and vocalization indoors are all measures that will reduce the risk of COVID-19 infection.

Airborne transmissions have been identified as one of the main mechanisms for how COVID-19 can be transmitted from one individual to another. An important source of transmission is small respiratory droplets that emerge as aerosols from an infected individual into an indoor environment.

Effective ventilation is a key parameter in minimizing the spread of the virus that causes COVID-19, SARS-CoV-2 (see Figure 1). For an indoor environment such as a classroom, heating, ventilation and air conditioning (HVAC) systems are keys for preventing infection because they have a strong influence on how rapidly air moves in an indoor environment where infection is much more likely compared to the outdoors.

Figure 1. The use of two models to evaluate two case studies involving classrooms shows that ventilation and wearing masks are the two most important factors in minimizing transmission of COVID-19. Figure courtesy of STLE.

In a previous TLT article,1 researchers recognized the importance in having the air conditioning system in an indoor environment not spread aerosol particles containing SARS-CoV-2. They developed a heated air filter that, when incorporated into an air conditioner, was found to inactivate 99.8% of the SARS-CoV-2 particles that passed through it when heated to 200 C. The filter was prepared from highly porous nickel foam that exhibited sufficiently high surface area to use van der Waals forces to capture SARS-CoV-2 particles. The researchers who developed this concept also found it to be effective against bacteria causing diseases such as anthrax. Using a similar set up, 99.9% of the bacterium that causes anthrax (Bacillus anthracis) were killed when passed through a nickel foam heated air conditioning filter.

Researchers have been working with various models to provide an estimate for the risk of airborne exposure to COVID-19 in indoor environments. One of the leading mathematical models is derived from using the Wells-Riley equation, which evaluates the risk of infection in an indoor environment. Michael Kinzel, assistant professor of mechanical and aerospace engineering at the University of Central Florida in Orlando, Fla., says, “The Wells-Riley equation provides an almost analytical solution to fluid dynamics by determining the probability of an individual becoming infected. Parameters such as exposure time, ventilation rate, room volume and quanta (virus released) are part of the mathematical calculation. This equation assumes an underlying probability of quanta that an individual exhales, leading to a build-up of viral particle concentration in an indoor environment. The equation then converts this information into a probability of inhalation of the viral particles.”

The Wells-Riley equation by default assumes a perfectly mixed environment, according to Kinzel. The mathematical model assumes that the quanta are uniformly dispersed throughout an indoor environment. Unfortunately, that is not the case, as dead zones will occur in indoor environments where there is poor mixing efficiency, leading to areas with either high or low concentrations of quanta.

To compensate for this issue, computational fluid dynamics (CFD) models are used to provide more detailed investigation about virus transmission. Kinzel says, “CFD has the ability to solve aerodynamic problems through organizing them into many mini problems. This approach has the potential to complement findings from the Wells-Riley equation, which may allow for a better understanding of COVID-19 transmission in a specific environment.”

The researchers have now utilized both models to determine how they compare in studying transmission in a classroom.

Infection probability
The classroom used in the study was configured to have a floor area of 66 square meters, with rectangular dimensions of 9.5 by 7 meters, with a height of 2.7 meters. For ventilation, a single centrally located inlet is placed with a single, corner-located return supply. The air flow is comparable to a typical HVAC system with commercial-grade venting. 

A teacher and nine students were placed in the classroom. Kinzel says, “This means that there are nine potential transmission routes that could infect a specific individual in the classroom, leading to a total of 90 possible transmission paths. We chose to use short class durations, which better reflects actual conditions in a classroom. This timeline also reduces the probability of transmission.”

The researchers evaluated both the Wells-Riley equation and the CFD models in two case studies. The first one involves a classroom with no ventilation, which means that the HVAC system is not operating, no fans are present to move the air, and the windows are closed. Ventilation using an HVAC system is added for the second case study and includes filters that exhibit a particulate removal efficiency of 72%.

The two models produced similar results in predicting infection probability in both case studies though the Wells-Riley equation under predicted infection probability by about 29% in the ventilated scenario. Both models indicate that the presence of ventilation and filtration will reduce infection probability by a significant percentage. The Wells-Riley equation predicted a 50% reduction, while CFD predicted a 40% reduction. Ventilation produces a steady air flow that guides aerosol viral particles through the filtration system, leading to the removal of a significant percentage. In the unventilated case study, aerosol viral particles tend to remain in the air space above the people in the classroom.

Masks were included in both case studies, and the researchers assumed that they had a particulate removal efficiency of 44%. They proved to be beneficial in the absence and presence of ventilation. The researchers believe the use of masks directed air flow of aerosols containing viral particles vertically above the individuals present in the classroom, reducing infection probability.

The one parameter that both models showed does not appear to be a factor is social distancing. Kinzel says, “Aerosols move around the classroom with air mixing, and we do not see a drop off in infection probability as the distance between individuals increases.” In the unventilated case study, the researchers found that the maximum infection probability occurred when individuals are 4.5 meters apart instead of the recommended two-meter social distancing. The models predict that the probability of infection at six meters is similar to that of 4.5 meters.

In comparing the two models, the researchers determined that the Wells-Riley equation could be more effective if it incorporated some elements of the CFD. Kinzel says, “Future work will involve a more extensive analysis of how parameters, such as air patterns, room geometry and ventilation parameters, influence the accuracy of the models. The objective is to develop corrective techniques that improve the results produced from the models.”

Another approach taken by the researchers is to study how the composition of human saliva can affect the formation of aerosol droplets exhaled by an individual. Kinzel says, “We found that if the saliva produced by an individual can be changed, then there is a possibility of increasing the diameter of aerosol droplets exhaled, reducing the possibility of virus transmission.”

The model analysis showing the importance of ventilation and the use of masks in a classroom can clearly be applied to individuals working in offices and laboratories. Further information on this research can be found in a recent article2 or by contacting Kinzel at

1. Canter, N. (2020), “Use of heated air filtration to inactivate COVID-19,” TLT, 76 (9), pp. 14-16. Available here.
2. Foster, A. and Kinzel, M. (2021), “Estimating COVID-19 exposure in a classroom setting: A comparison between mathematical and numerical models,” Physics of Fluids, 33, 021904.
Neil Canter heads his own consulting company, Chemical Solutions, in Willow Grove, Pa. Ideas for Tech Beat can be submitted to him at