Airborne transmission is one of the main modes for the transmission of highly infectious diseases such as COVID-19. Pathogen laden aerosols from an infected person can be transported by air to a susceptible population. A widely used model for airborne transmission considers the indoor space to be well-mixed such that the pathogen concentration is spatially homogeneous. Other models that employ computational fluid dynamics (CFD) allow tracking the spatiotemporal variation of infection probability in indoor spaces but are computationally expensive. Here, we compare the predictions of a well-mixed continuously stirred tank reactor (CSTR) model for indoor transmission with CFD for airflow, along with the Lagrangian tracking of aerosol particles. Of particular interest is the ventilation using ceiling fans, which are common in South East Asia. It is found that the behavior of particles at the walls plays an important role. Two limiting cases are studied: all particles reaching the wall get trapped vs all particles being reflected from the solid boundaries. We propose a modification to the CSTR equation to include the wall effect, and it matches the CFD data closely.

## I. INTRODUCTION

Indoor air quality is closely related to the health of the occupants. This is particularly true for the airborne transmission of infectious diseases, such as tuberculosis, Mers-CoV, SARS-CoV, influenza, etc. The Covid-19 pandemic highlighted the role of indoor ventilation in mitigating the risk of airborne infection. In particular, it was shown that the risk of infection is higher in poorly ventilated indoor spaces, where pathogen carrying micrometer-sized droplets or aerosols can persist for long periods of time. Models to predict the infection probability due to airborne transmission in indoor spaces account for ventilation to evaluate the concentration of infectious pathogen.

Modeling the air in an indoor space as a well-mixed reactor greatly simplifies the study of indoor airborne transmission.^{1–3} In such models, droplets and aerosol particles are generated by people in the indoor space and are distributed throughout the spatial volume. The well-mixed continuously stirred tank reactor (CSTR) model considers the pathogen concentration to be homogeneous everywhere in the space irrespective of the type of airflow pattern, ventilation strategy, and location of sources and fresh air inflow. The model assumes a constant production rate of droplets, removal rate due to ventilation, settling and filtration of droplets, as well as deactivation of pathogen. It can be used to estimate the steady state pathogen concentration in the indoor space, as well as the kinetics of concentration change for specific situations.^{4}

To determine the probability of infection in an indoor space, Wells^{5} and Riley^{6} combined the well-mixed model with the “quanta” of infection. In the Wells–Riley model, a “quantum” of infection is the dose of infectious pathogen required to infect 63.2% of the population.^{5} Quanta is related to the pathogen load, pathogen infectivity, and pulmonary ventilation rates.^{7–9} A recent study^{4} has used this definition of pathogen quanta and the well-mixed model for airborne transmission to calculate safety guidelines for indoor spaces.

Another way of studying pathogen transmission in an indoor space is using computational fluid dynamics (CFD). Using CFD, turbulent airflow in an indoor space can be simulated and the location of source of infectious aerosol can be specified. Each droplet can be tracked through space and time using Lagrangian approach. Alternately, the Euler framework can be used to compute the spatiotemporal variation of particle concentration in the room.^{10} Either way, CFD models for airborne transmission are much more computationally demanding when compared to simple well-mixed models.

A compilation of CFD applied to indoor airflow and comparison with well-mixed models can be found in the works of Nielsen^{11} and others.^{12} Recently, studies have presented simulations of airflow fields and pathogen trajectories in various enclosed spaces, like washrooms,^{13} restaurants,^{14} buses,^{15} airliner cabins,^{16,17} classrooms,^{18} hospital rooms,^{19} and offices,^{20} including the effect of walking,^{21} etc. Studies have also predicted the local infection probability in indoor spaces by coupling the Wells–Riley equation with CFD.^{22–25} It can be inferred from these studies that the infection probability is strongly dependent on the air circulation in the room, the type of ventilation strategy,^{26} and the location of source.^{27} A significant difference has been reported^{28} between CFD and well-mixed model in the case of forced ventilation. Another study^{29} has shown that changing ventilation scenarios results in differences in the tracer gas concentration with respect to a well-mixed reactor model.

Since infection risk is directly linked to pathogen concentration, developing an understanding of ventilation and pathogen distribution in closed spaces is important to decide safety guidelines. We are particularly interested in the airflow set up by ceiling fans. Ceiling fans are widely used in equatorial countries like India. They are expected to promote the mixing of air, as in the case of a well-mixed CSTR. The majority of studies focus on thermal comfort by ceiling fan^{30} and the use of ceiling fans as assisted devices in ventilation.^{31} However, the effect of ceiling fan on airborne transmission has not been investigated directly.^{30}

The objective of this study is to compare results from simple CSTR models with CFD simulations for a realistic classroom with four ceiling fans. The classroom has two exhaust fans that throw out the used air, and the door is kept open to draw in fresh air. We consider a lecture of 50 minutes duration and simulate the spatiotemporal buildup of infectious aerosol exhaled by the instructor delivering the lecture. The CFD simulations calculate the airflow in the room and track the aerosol particles in space and time. The concentration of the aerosol particles is converted to an equivalent infection causing quanta and compared with the predictions of the CSTR model. The relative quanta exposure (defined subsequently) assesses the infectivity of air in different parts of the room to assess the validity of the well-mixed assumption.

The paper is organized as follows: the details of the CSTR model and CFD methodology are presented in Secs. II and III, respectively. This is followed by the CFD results in Sec. IV, and their comparison with the CSTR model in terms of the residence time of air in the room and the spatiotemporal variation of the number of particles. We study the behavior of the aerosol particles at the wall by simulating two limiting cases: the one where all the particles hitting a solid boundary reflect back and re-enter in the airflow, and second, where all the particles reaching the walls get trapped there and are not part of the airflow simulation subsequently. We find large differences between the two cases, which raises question about the validity of the well-mixed CSTR model. Conclusions are presented in Sec. V.

## II. CSTR MODEL OF INDOOR AIRBORNE TRANSMISSION

Indoor spaces are often modeled as well-mixed continuously stirred tank reactors (CSTRs) to study the airborne transmission of pathogens.^{1–3} Ventilation in indoor spaces typically involves turbulent airflows, resulting in mixing.^{4} Forced convection induced by the presence of exhaust fans, ceiling fans, HVAC systems, etc., increases the turbulence and mixing of air in the space. Ventilation systems are defined in terms of air changes per hour (ACH), which is a ratio of the volumetric rate of air removal through vents or ducts to the volume of the enclosed space. Thus, the inverse of the ACH provides a timescale, *τ*, that characterizes the average time required to vent the air from the entire volume.

*N*(

*t*) is the number of the droplets at time

*t*,

*V*is the volume of an individual droplet,

_{d}*t*is the exposure time, and

*P*is the production rate of droplets by exhalation. The removal rate of droplets is given by $ \lambda = \tau \u2212 1 = Q / V$. It is the inverse of ACH timescale,

*τ*, due to ventilation, where

*V*is the volume of the room and

*Q*is the volumetric flow rate of air moving out of the room. On integrating the above equation, we get

*N*is the number of droplets in the indoor space after steady state has been reached and is given as $ P / \lambda V d$. Here, we have neglected the settling of particles, the effect of evaporation, and other particle dynamics like diffusion, etc. Diffusion due to Brownian motion is neglected for particles much larger than 0.1

_{s}*μ*m, which is the case for the micrometer-sized particles considered in the current study. For such particles, the settling velocity has been found to be negligible compared to the air velocity.

^{4}Also, the evaporation timescale of such droplets is found to be less than a second,

^{4}which is much smaller than the ventilation timescale,

*τ*. Hence, it is assumed that the droplets reach equilibrium size as soon as they are injected into the air.

*K*, is proposed in the recently modified version of the CSTR model to take into account the non-ideality of mixing.

^{28}The mixing factor is taken as the ratio of the ACH timescale to the average removal timescale of non-ideal flow. A value of

*K*= 1 represents perfectly mixed, while

*K*< 1 corresponds to poorly-mixed scenario. It is found that the exponential buildup of particle numbers given in Eq. (2) gets modified as

*K*can be obtained from CFD simulations.

^{28}

Bazant and Bush^{4} estimate the safety exposure time for indoor spaces by assuming the removal rate of pathogen as the ACH timescale. However, in our previous study,^{13} we found the removal timescale to be much larger than the ACH timescale when there is insufficient mixing. The removal timescale was found to agree with the ACH timescale only upon increasing levels of turbulence in the airflow. This further stresses upon the importance of mixing factor in CSTR equation. The actual safe exposure time will be different in non-ideal situations from the estimates from guideline calculators.^{4}

## III. SIMULATION METHODOLOGY

### A. Classroom geometry and computational mesh

We use the three-dimensional geometry of a classroom with realistic dimensions for this study; see Fig. 1. There are two exhaust fans that draw air from the indoor space and expel it out of the room. Fresh air enters through the door. There are four ceiling fans that promote the mixing of air in the classroom. This is expected to result in a well-mixed scenario, and the results presented below corroborate this fact. We focus on a comparison of the CFD results with the well-mixed model based on the volume of the entire room. Therefore, to reduce computational complexity, we do not include mannekins or furniture in the room.

In the computational model, the door and exhaust fans are modeled as a pressure inlet and velocity outlet boundaries. A more realistic simulation of the exhaust fans will need to consider the motion of the fan blades. Here, we are only concerned about the airflow exiting the room due to the exhaust fans. Therefore, for simplicity, we have modeled the exhaust fans as circular velocity outlet boundaries providing the volumetric flow rate of the fan. The air delivery rate of exhaust fans is 18.75 m^{3}/min (information provided by the fan manufacturer), and the diameter of the exhaust fan is 0.3 m, which gives the velocity of air as 4.4 m/s. For the dimensions of the classroom, the two fans can vent out an equivalent volume of air in 370 s. This gives ACH as 9.7, which means that the fans will remove the used air in the room on an average 9.7 times in an hour.

The four identical ceiling fans are modeled as 2D surfaces and created as part of the 3D fluid domain. Each ceiling fan is assigned a boundary condition. Based on the air delivery value of 270 m^{3}/min and sweep of 1.4 m, the velocity at the fan is calculated as 2.92 m/s. An axial pressure jump of 5.52 Pa, computed based on this value of velocity and density of the air, is applied at the 2D surfaces representing the ceiling fans. Our aim is to study the mixing of air in the room due to the ceiling fans. The mixing of air influences the airborne transmission of particles and the infection risk. The simple approach of modeling the ceiling fans as 2D surfaces with a pressure jump generates enough mixing to achieve close to well-mixed scenario; details are presented in Sec. IV. Additional complexity of modeling the rotation of the blades may further enhance the mixing of air in the room, bringing the results even closer to the well-mixed model. We believe our current 2D modeling of the ceiling fans is adequate to address the objective of the paper, i.e., do ceiling fans create a well-mixed scenario in an indoor space?

The computational mesh consists of 9.68 × 10^{5} polyhedral cell elements generated using Ansys FLUENT,^{32} with finer elements near the ceiling fans and exhaust fans [Fig. 1(b)]. The average wall-normal spacing corresponds to $ y + = 175$ along the walls. The airflow in the classroom is not wall-dominated, and the focus is on the airflow generated by the ceiling and exhaust fans. In such cases, enhanced wall treatment allows us to place the first cell next to the wall in the turbulent layer. The effect of grid refinement is discussed later in the section.

### B. Airflow and particle dynamics

We use commercial CFD solver, Ansys FLUENT 2022 R1,^{32} to solve airflow and particle dynamics via an Eulerian–Lagrangian approach. This is similar to earlier works on indoor airflow simulation.^{22,26,27} The Eulerian framework treats air as the continuum phase, and the Lagrangian framework tracks the movement of discrete particles. We assume a one-way coupling, viz. the airflow determines the particle trajectories, while the particles have no effect on the airflow. This is reasonable since the volume fraction of particles in the air is low.^{33}

We solve the continuity and momentum equations for incompressible air using the Reynolds-averaged Navier–Stokes (RANS) approach. We have chosen the realizable *k*-*ε* model^{34} to close the turbulence related terms in the RANS equation, as the model works well for high Reynolds number turbulent flows. Several recent studies of indoor airflow are based on variants of the *k*–*ε* model.^{35,36} We found realizable *k*–*ε* model to work well for ventilation in a public washroom.^{13} No slip condition is applied at the walls, with zero gauge pressure at the open door. Turbulent intensity at the inlet door is taken as 5%, which signifies a medium level of turbulence. Turbulent viscosity at the inlet door is taken to be 10 times the dynamic viscosity, following similar methodology used in indoor airflow studies.^{13} The SIMPLE algorithm is used for pressure–velocity coupling, with a second-order upwind scheme for the discretization of pressure and convective terms of momentum equation, as well as the turbulence transport equations. Hybrid initialization is used, and iterative convergence to steady state is obtained when scaled residuals drop below $ 10 \u2212 4$ for the continuity, momentum, and turbulence variables.

*t*is measured from the point of fresh air entry. In the current Eulerian framework, the equation is integrated over the steady state velocity field

*u*, with a zero value specified at the inlet door boundary. The air residence time at a point in the room is the amount of time a packet of air has spent in the domain till it reaches the point, starting from the door inlet, where the value of the $\varphi $ is zero. It is higher for streamlines that are trapped in dead air zones or recirculating zones. On the other hand, the low air residence time at a point means that the used air is quickly replaced by fresh air. A comparison of the air residence time with the ACH timescale gives a measure of how well-mixed the air in the room is, i.e., air residence time close to ACH timescale means that there are no dead zones with trapped air. This is elaborated further in Sec. IV.

_{j}^{22}The velocity of each particle $ v \u2192$ is calculated by balancing lift, drag, and gravitational forces as follows:

*m*is the mass of the particle and $ g \u2192$ is the acceleration due to gravity. Inter-particle interactions are neglected for a low number density of particles in the flow.

^{32}

We use mono-dispersed particles of diameter 2.5 *μ*, with a velocity of 0.1 m/s. Exhalation data^{4,37} shows a large fraction of aerosol in the range below 5 *μ*m; we use 2.5 *μ*m as a representative size of the air-borne particles. Our previous work on washroom ventilation^{13} shows that particles of diameter between 1 and 5 *μ*m follow the airflow. Both mono-dispersed and poly-dispersed particles (following the Rosin–Rammler distribution) below 5 *μ*m give essentially the same number of particles in the domain as a function of time.^{13} We note that the critical diameter of particles (obtained by comparing settling vs evacuation time scales) for the current scenario is 5 *μ*m,^{4} which implies that droplets below this size remain air-borne for long duration and are the focus of the current study.

The Reynolds number for 2.5 *μ*m particles is much less than unity. The drag force is thus computed using the spherical drag model, where the drag coefficient depends inversely on Reynolds number. Saffman's model is used to compute lift force.^{32} The effect of turbulent dispersion is modeled using a discrete random walk model. The particles can either reflect at a solid wall or get trapped by the wall. The effect of the wall boundary condition is discussed in detail in Sec. IV. The escape boundary condition is used for the particles at the door and exhaust fans.

Particles are injected at the instructor location (shown in Fig. 1). This is based on a higher respiratory rate of the instructor (speaking loudly) than students (passive listeners). We run the particle tracking simulations over a period of 3000 s, which is comparable to the duration of a typical lecture. Based on the ACH timescale, *τ* = 370 s, we expect 99.3% of the steady state particle concentration in the classroom to be reached within 5*τ* = 1850 s. We have chosen the production rate of droplets *P* to be 5.7 $ \xd7 10 \u2212 12$ m^{3}/s in Eq. (2). This is much higher than the estimate given for a single sneeze,^{38} which translates to a production rate of $ O ( 10 \u2212 15 )$ m^{3}/s for a droplet diameter of 2.5 *μ*m. A higher production rate is used to achieve statistical consistency.

The time step of particle injection is taken as 0.05 s, which is equivalent to 60 000 parcels injected (one per time step) over the entire duration of the simulation. Each parcel has 34 898 particles. The number of parcels and particles is found to be large enough to get statistically converged results. Results obtained with double the number of parcels and particles give proportionally higher values of particle numbers in the domain. The same is true for the CSTR results, where the particle concentration and the total number of particles in the room are proportional to the production rate *P*. Since we are interested in the ratio of the concentration of particles between CFD and well-mixed models, the exact value of the production rate is not important for the results presented in the current study, as long as they are equal in the two models.

### C. Grid convergence

Grid convergence is studied using four meshes with different levels of refinement. The baseline mesh of 968 k points is described above. Two coarser meshes of 481 k and 138 k and a finer mesh of 4 $ \xd7 10 6$ cells are used in the computation. The airflow in the classroom is dominated by the ceiling fans. Figure 2 shows the velocity profile in planes passing through the center of ceiling fans. The results obtained from the four meshes are close to each other. Peak velocity magnitude is under the ceiling fans, and lower velocity is found in the regions in between the fans. There are small variations between the four solutions; for example, between the baseline and fine mesh in the range *x* = 2.5–4 m in Fig. 2(a). This corresponds to the recirculation region between the two ceiling fans, where the velocity magnitude is low. Small deviations (less than 3% of the peak velocity induced by the ceiling fans) in this region has only a small effect on the overall mixing.

The average residence time of air in the room computed using the baseline and the fine meshes are within 3% of each other (see Table I and the associated text in Sec. IV B). The average air residence time and its comparison with ACH time quantifies the mixing in the room, as noted earlier in Sec. III B. It therefore has a direct effect on the particle buildup in the room over time. The difference between the steady state number of particles in the room computed on the two finer meshes is less than 4% (see Fig. 7 in Sec. IV). We thus consider the baseline grid resolution to be adequate for quantifying the level of mixing in the room and its effect on the particle concentration in the room and the resulting infectivity of air (in Sec. IV C).

. | ART . | K
. |
---|---|---|

Grid sizes | ||

138k | 454.8 s | 0.82 |

481k | 402.4 s | 0.92 |

968k | 403.5 s | 0.92 |

4M | 412.7 s | 0.89 |

Turbulence models | ||

Standard k–ε | 389.2 s | 0.95 |

Realizable k–ε | 403.5 s | 0.92 |

RNG k–ε | 442.6 s | 0.84 |

. | ART . | K
. |
---|---|---|

Grid sizes | ||

138k | 454.8 s | 0.82 |

481k | 402.4 s | 0.92 |

968k | 403.5 s | 0.92 |

4M | 412.7 s | 0.89 |

Turbulence models | ||

Standard k–ε | 389.2 s | 0.95 |

Realizable k–ε | 403.5 s | 0.92 |

RNG k–ε | 442.6 s | 0.84 |

## IV. RESULTS

### A. Airflow simulation results

Figure 3 shows the variation of airflow velocity magnitude in different horizontal, longitudinal, and lateral plane sections in the classroom. In plane velocity, vectors are also plotted to visualize the airflow circulation. The cross-sectional planes are chosen to bring out the salient features of the airflow circulation in the room and its effect on the air residence time. The ceiling fans generate a strong airflow in the downward direction, as seen at planes corresponding to *x* = 1.78 and *y* = 6.57 m (axes shown in Fig. 1), passing through the center of the fans. The air stream impinges on the floor and turns upward between the ceiling fans and near the side walls of the room. The four streams are visible under the four ceiling fans as high velocity regions in the *z* = 1.6 plane. These are not perfectly aligned with the ceiling fan positions, possibly because of the asymmetry in the room geometry (door inlet and exhaust fan outlets). The air stream entering through the door is visible in the vector plot in the horizontal plane.

Figure 4 plots the contours of turbulent kinetic energy in the representative *x*, *y*, and *z* planes. There is a significant generation of turbulence in the air streams produced by the ceiling fans. As expected, the regions of high TKE match with the high downward velocity under the ceiling fans. A high level of turbulence is also observed in regions where the downward air streams impinge on the floor; see, for example, the vertical plane at *x* = 5.62 m. The effect of the airflow impinging the floor on the particle dynamics is discussed in the subsequent section.

Figure 5 shows the variation of air residence time (ART) in different horizontal and vertical planes through the classroom. Here, the air residence time obtained from Eq. (4) is normalized by the ACH timescale of 370 s (computed in Sec. III A). The plots show that the normalized air residence time is in the range of 0.9–1.2 in the majority of the domain. The only exception is near the door inlet, where fresh air enters the classroom and where the age of air is low. The effect of the fresh air stream entering through the door can be seen in *z* = 1.6 and *x* = 0.8 m sections, as it penetrates into the room and gets mixed with the older air elsewhere.

Overall, the velocity vectors in all the planes show a significant cross-circulation of air in the domain, as generated by the four ceiling fans; see Figs. 3–5. The high level of turbulence generated by the ceiling fans also aid in mixing. The relatively uniform yellow and green color in Fig. 5 indicates that well-mixedness of air in the room has been mostly achieved due to the cross-circulation of air. The ceiling fans are thus successful in homogenizing the air in the classroom. Therefore, the indoor space can be approximated to be well-mixed, like in a CSTR model.

### B. Particle tracking results

Figure 6 shows the spatial distribution of particles in the domain at different instances of time. On injection from the instructor location (marked in Fig. 1), the particles follow the airflow streamlines and spread in the room. A fraction of the particles exits the room through the exhaust fans, and the rest accumulate in the domain. Note that the simulation uses reflect boundary conditions for the particles at the wall. The number of particles increase over the time span shown in the figure. The particles are colored as per the time they have spent in the room post-injection, i.e., the particle residence time. We see more blue and green particles (low particle residence time) in Fig. 6(a) and 6(b). Additional yellow and red particles are present at later instants in Figs. 6(c) and 6(d). The particles appear to be uniformly mixed in the volume of air in the room in terms of the spatial distribution. The only exception is the region near the door, where fresh air entering through the door dilutes the particle concentration and we find fewer particles in this region; see, for example, in 200 s.

Figure 7 plots the total number of particles present in the classroom as a function of time. After the airflow achieves steady state, injection is initiated and time is set to *t* = 0. The CFD results show an exponential buildup of particle number in the domain as anticipated by the CSTR model; equation (2) is also plotted for reference. The steady state particle number obtained from CFD is slightly higher than the CSTR value. From Eq. (2), the steady state number of particles, *N _{s}*, is given by P/ $ \lambda V d$. For production rate,

*P*= 5.7 $ \xd7 10 \u2212 12$ m

^{3}/s, removal rate,

*λ*= 1/

*τ*= 1/370 s

^{−1}, and volume of one droplet of diameter 2.5

*μ*,

*V*, we estimate

_{d}*N*as 26 $ \xd7 10 7$ for the analytical CSTR model.

_{s}The difference in the steady state value can be explained in terms of the air residence time. The volume-averaged ART obtained from CFD is 403.5 s, as opposed to the ACH timescale 1/*λ* = 370 s. Under the CSTR assumption, the air in the room is well-mixed and the particles are uniformly distributed in the room. The exhaust fans expel air equivalent to the volume of the room once every 370 s. This can be interpreted as the mean age of air if the room is a perfect CSTR. A higher average ART in the CFD simulation retains the air longer on average and results in a higher buildup of particles in the room. It is an indicator of spatial inhomogeneity, with possible pockets of trapped air in the corners. Interestingly, the CFD steady state particle number is close to the modified CSTR curve computed using a mixing factor of *K* = 0.92 = 370/403.5 obtained as the ratio of the ACH timescale to the volume-averaged ART from CFD simulation. Note that the value of *K* depends on the specific geometry of the room and the type of ventilation.

Figure 7 also shows the effect of grid refinement on the particle buildup in the domain. Results obtained using the 481k and 4 M grids are very close (the time-averaged steady state values are within 4%) to that of the baseline 968k grid discussed above. The 138k coarse mesh results show a larger deviation, with a higher number of particles in the classroom. The volume-averaged ART and the corresponding mixing factor values obtained from different grids are listed in Table I. Once again, the 138k grids have a much higher average ART, while the values for the other three grid are comparable (within 3% of each other).

We note that the grids mentioned above have a first cell height of 17 mm adjacent to the solid boundaries. Comparable results are obtained when the boundary layer is refined successively to obtain the first cell height of 8.5 and 5 mm. In all the cases, the first grid point next to the wall is placed in the turbulent flow region. Much finer meshes, with the first cell placed in the viscous sublayer, show the accumulation of particles in the low momentum near-wall regions. This leads to a high concentration of particles at corners very near the floor and ceiling. Excessive clustering of particles can violate the assumptions of the DPM method and needs further investigation. For the purpose of the present study, we proceed with the results obtained on the baseline 968 k grid (with first cell height of 17 mm).

Table I also presents the data obtained using different turbulence models. The volume-averaged ART for the standard and realizable k–*ε* models are found to be comparable. The RNG k–*ε* model gives a relatively high value of the volume-averaged ART. The corresponding mixing factor K values are also reported in the table.

### C. Estimation of Relative Quanta Exposure (RQE)

^{5}and Riley

^{6}proposed a model for estimating the probability of infection via airborne particles. They assumed that the infection probability,

*IP*, was governed by a Poisson process and depended upon the total infectious quanta inhaled,

*q*

_{T.}^{7}

*C*, RNA copies/m

_{v}^{3}, and pathogen strain infectivity

*C*, quanta/RNA copies, are additionally required to quantitatively define one quantum.

_{i}^{9}The amount of quanta present per unit volume of air in the room is

*V*is the volume of the domain. RH is the relative humidity and

*θ*is the diffusivity of water in air. The factor $ ( 1 \u2212 R H ) / \theta $ takes care of the change in droplet size due to evaporation.

^{39}The total quanta inhaled by a person depends on the infectious quanta present in the room, $ q \u2032 ( t )$, the pulmonary ventilation rate of susceptible,

*Q*, and the total time of exposure,

_{b}*τ*. The total amount of quanta inhaled throughout the initial transient buildup period up to steady state is

_{e}The dose of infection or quanta is determined by converting the number density of droplets in air, *N*(*t*)/*V*, to the quantity of pathogen using clinically obtained parameters, *C _{v}* and

*C*. The volume of air a healthy individual inhales at any instant of time is $ Q b d t$. The total volume of droplets inhaled over exposure time

_{i}*τ*is $ \u222b 0 \tau V d$ N(t)

_{e}*Q*dt $ / V$. Infectivity,

_{b}*C*, gives the infectious dose of pathogen per RNA copies. Pathogen load,

_{i}*C*gives the number of RNA copies present in the unit volume of sputum collected from the mouth. Thus, $ C i C v$ gives quanta/m

_{v,}^{3}of pathogen present in bioaerosols. The volume of droplets inhaled can thus be converted to the dose of pathogen using these parameters.

The infection probability is proportional to the amount of infection quanta inhaled. Higher the inhaled infection quanta, higher is the infection risk. The CSTR model provides an estimate of the number of particles in the room and, therefore, the amount of quanta a person is exposed to, assuming the entire indoor space to be homogeneous.

To study the spatial variation in quanta in the classroom as estimated from CFD simulation, we have divided the classroom into 20 sub-domains or boxes each at four layers of varying heights; see Fig. 8. The first level is $ 0 \u2013 1$ m, which is the area near the floor. The next level is the breathing zone, $ 1 \u2013 2$ m, which includes the sitting and standing breathing heights for an average human. The third level is the region up to the height of ceiling fans, $ 2 \u2013 2.84$ m, and the final level is above the ceiling fan 2.84–3.66 m.

*i*is used to denote the boxes, while subscript CSTR refers to the entire space. The values in the numerator are taken from CFD simulation, while the denominator is from the CSTR model.

Figure 9 shows RQE for the different boxes to estimate the spatial inhomogeneity in infection probability, with respect to the well-mixed assumption of CSTR. In the breathing zone, *z* = 1–2 m, RQE is between 0.8 and 1.2 (within 20% of the CSTR estimate) for the majority of the boxes. Only exceptions are box A1 and A3. Outdoor air entering the room through the door dilutes the number concentration of the infectious particles in air and results in a lower RQE in box A1. Box A3, on the other hand, is close to the location of particle injection (see Fig. 1) and has a higher particle number in the CFD solution and hence a higher RQE.

The lower level of boxes and those up to the ceiling fan in the figure show similar trend as the breathing zone, discussed above. By comparison, the region above the ceiling fans has relatively higher values of RQE. There are several regions (for example, boxes E1, D2, A3, A4, and others) where the relative infection probability is appreciably higher (more than 30%) than that estimated by the well-mixed CSTR model for the classroom. This is possibly because of dead zones at the corners of the ceiling. The region near the ceiling is possibly of lower concern as it is much higher than the breathing level of most humans.

The RQE computed using the fine grid solution shows similar trends as the baseline grid results in Fig. 9. The values are within ±0.2 of the reference CSTR value for the majority of the classroom. The only exceptions are in the vicinity of the door entry, the injection location, and some boxes near the ceiling.

### D. Trap boundary condition at walls

In this section, we study the effect of the boundary condition applied to particles as they interact with the walls. The particles can either reflect or get trapped at the walls. Previous results generated with the reflect boundary condition are consistent with the CSTR framework. Here, we present results from DPM simulations with the trap boundary condition. The simulation is performed using identical the steady state airflow field solution and identical particle injection parameters as reported above. The only difference is that particles are taken out of the simulation as soon as they hit the walls.^{32} This can be seen as a limiting case where no particles reflect back. In reality, a fraction of the particles may be trapped by the wall, depending on their velocity, size, and other characteristics. The transition between trap and reflect conditions is discussed below.

Figure 10 shows that the total number of particles in the domain obtained using the trap boundary condition is much lower than the reflect case presented earlier. A large fraction of particles gets trapped at the floor, ceiling, and other walls of the room. The steady state particle numbers are found to be ten times lower with the trap boundary condition, and the transient buildup to steady state is faster (within 100 s) than the reflect case (for the 2.5 *μ*m particles considered in this work). The ceiling fans have a significant effect in pushing the particles to the floor and walls. This reduces the total number of particles in the room, as well as in each of the boxes identified in Fig. 8. The RQE values presented in Fig. 11 are lower than those in Fig. 9 by an order of magnitude. The data also shows spatial homogeneity across the room, except for an increase toward the right side of the room, possibly due to the effect of the exhaust fans placed near boxes B4 and C4. Additionally, higher RQE is observed at the location of particle injection (box A3) and because of fresh air coming in through the door (box A1).

Air follows a no-slip condition at the walls, which implies that the wall-normal velocity of air in the grid layer adjacent to the wall is close to zero. The CFD simulations, however, predict a high value of TKE adjacent to the floor due to the significant amount of turbulence generated by the ceiling fans; see Fig. 5. The effect of turbulent dispersion on particle motion is modeled via the random walk model. The particle trajectories in the near-wall region are, therefore, governed by the turbulence in the airflow. A high turbulent velocity component normal to the wall tends to make a large number of particles impinge on the wall and get trapped there. The number of particles in the domain, under the trap boundary condition, is thus significantly lower than that obtained using the reflect boundary condition for the particles at the walls. The RQE results bring out this effect quantitatively, and the results are relatively invariant to the turbulence model used in the airflow simulation. The RQE predicted by the different turbulence models is below 0.1 for most of the boxes in the breathing zone (Fig. 11). The results can, however, be limited by the accuracy of the random walk model, which needs further investigation.

The Stokes number of spherical particles in the boundary layer grid cell next to a solid wall is a good way of estimating if the particles will stick to a boundary wall or get reflected back into the flow.^{40} According to the elasto-hydrodynamic lubrication theory, the critical Stokes number is in the range $ 8 \u2013 15$, below which the particles are trapped at the wall. We estimate that the Stokes number for 2.5 *μ*m particles impinging on the walls (other than the floor) is below 1. At the floor surface, a small fraction of particles (less than 10%), have a Stokes number greater than the critical range of 15. Only this fraction of particles are expected to reflect back into the airflow. The majority of particles are thus expected to stick to the walls.

The Weber number of particles in the exhalation plume is at most of the order of 1, suggesting that surface tension effects dominate over inertia for the particles. If the impaction energy of droplets is small, the particles will not deform upon hitting the walls to produce a splashing event. The critical Weber number for transition from sticking to splashing regime is 80.^{41}

*λ*as the removal rate of air through the exhaust fans. We propose to use an additional

*λ*to account for the particle removal rate at the walls. Thus,

_{w}*N*= P/ $ \lambda total V d$. As expected, an increase in

_{s}*λ*decreases the steady state buildup in the room, resulting in lower infection probability and lower RQE, as reported above. Also, the exponential time constant given by

*τ*= 1/

*λ*

_{total}is reduced by the particles getting trapped at the wall. This decreases the time required to reach steady state, as evident clearly from Fig. 10. The exponential buildup curve from the CFD simulation with the trap boundary condition reaches steady state much faster than the CSTR model or the CFD simulation with the reflect boundary condition. The removal rate through walls,

*λ*, is estimated by taking the slope of fraction of particles escaping from all walls.

_{w}*λ*is estimated as 0.03 s

_{w}^{−1}in this study. Finally, the CSTR model prediction using

*λ*

_{total}is shown to overlap with the CFD result obtained using the trap boundary condition for the particles at the walls.

## V. CONCLUSION

In this paper, we study airborne transmission in a classroom setting, where the airflow is dominated by ceiling fans that are common in South-east Asia. Specifically, we look at the air circulation, air residence time, and concentration of aerosol particles in the classroom. We use computational fluid dynamics, along with Lagrangian tracking, to predict the airflow in the room and the spatiotemporal variation of aerosol particles. The results are compared with the predictions of the well-mixed CSTR model that is commonly used for indoor ventilation. CSTR model assumes that the air in the room is well-mixed, such that the concentration of pathogen in the room is spatially homogeneous. The buildup of pathogen concentration in time is governed by the ACH timescale, given in terms of the ventilation rate and the volume of the room.

The CFD simulations show that the ceiling fans generate large scale mixing and high level of turbulence in the air. This leads to close to a well-mixed scenario, where the average residence time of air computed from CFD is within 10% of the ACH timescale given by the well-mixed model. Aerosol particles injected at the instructor location are uniformly distributed throughout the room, except for the locations of injection and the inflow of fresh air. Interestingly, the number of aerosol particles in the room and its concentration largely depends on the interaction of the particles with the walls. In the scenario, where every particle reaching the walls, floor, and ceiling get trapped, the particle concentration is significantly lower than when the particles reflect back from the walls and re-enter the airflow. This results in an order of magnitude decrease in the infectivity of air, defined in terms of a relative quanta exposure of a susceptible individual in the room. The trap boundary condition is justified in terms of the particle Stokes number, while the reflect case is close to the well-mixed model that is widely used for calculating infection probability in indoor spaces. Thus, the CFD method predicts a drastic reduction in the infection level in the presence of ceiling fans. It is important to include this effect in ventilation calculations. We propose a correction to the well-mixed model to account for the same.

## ACKNOWLEDGMENTS

We would like to thank Dr. Ravichandran of the Center of Climate Studies at IIT Bombay for his fruitful discussions. We also acknowledge the contribution of Abhijit Patil from Ansys in generating some of the computational meshes used in the study. Deep Narayan Singh Baudh and Lagoon Biswal of the Department of Aerospace Engineering, IIT Bombay, assisted in checking for grid sensitivity and the grid refinement studies.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Archita Mullick:** formal analysis (equal); investigation (equal); methodology (equal); and writing—original draft (lead). **Guruswamy Kumaraswamy:** conceptualization (equal); methodology (equal); supervision (equal); and writing—review and editing (equal). **Sarika Mehra:** conceptualization (equal); methodology (equal); supervision (equal); and writing—review and editing (supporting). **Janani Murallidharan:** methodology (equal) and writing—review and editing (supporting). **Vivek Kumar:** methodology (equal) and writing—review and editing (supporting). **Krishnendu Sinha:** conceptualization (equal); project administration (equal); supervision (equal); and writing—review and editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.

## REFERENCES

*Airborne Contagion and Air Hygiene*

*ANSYS FLUENT Theory Guide 15*

*Physics of COVID-19 Transmission*

*ESAIM: Proceedings*(EDP Sciences, 2008), Vol. 23, pp.