2.1. Global ventilation models

Prior to detailing the modelling framework of the present study, a brief overview of two pertinent models is provided. First, we describe the Wells–Riley model. The spatiotemporal generalisation of this model implemented by Lau et al. [Reference Lau, Griffiths, English and Kaouri23] is then described.

The Wells–Riley model and its variants [Reference Gammaitoni and Nucci15, Reference Miller28] assume a well-mixed room, meaning the aerosol concentration is spatially uniform and depends only on time, $t$ . The aerosol concentration in the room, $\mathcal{C}(t)$ , is thus governed by the ODE

(2.1) \begin{equation} \frac {\mathrm{d} \mathcal{C}}{\mathrm{d} t} = \frac {R}{V} - (\lambda + \beta + \sigma ) \mathcal{C}. \end{equation}

This closely matches the formulation used by Miller [Reference Miller28]. The first term on the right-hand side describes aerosol production as a global source, with $R$ denoting the rate at which aerosols are produced and $V$ being the room volume. The second term on the right-hand side describes first-order removal of aerosols through a global sink, where each coefficient is the rate at which aerosols are removed by ventilation ( $\lambda$ ), biological deactivation ( $\beta$ ) and gravitational settling ( $\sigma$ ).

Table 1.

Parameters and their values

Table 1. Long description

A table with 12 rows and 4 columns. The columns are labeled Parameter, Symbol, Value, and Source. The table lists various parameters related to a room and airflow study, including room dimensions, airflow speed, aerosol emission rate, virus deactivation rate, gravitational settling rate, air exchange rate, turbulent eddy diffusion coefficient, breathing rate, infectivity constant, location of purifier center, and radius of purifier. Each row provides the symbol, value, and source for each parameter. For example, Room length is listed with a value of 8 meters from Lau et al. [23], Room width is also 8 meters from the same source, and Room height is 3 meters. The table includes specific values for rates and coefficients, such as the aerosol emission rate of 5 aerosols per second and the virus deactivation rate of 1.7 times 10 to the power of -4 per second.

In addition to (2.1), Wells–Riley models relate the aerosol concentration to the probability that a susceptible person becomes infected. Following Riley et al. [Reference Riley, Murphy and Riley32], the infection risk can be related to the concentration according to

(2.2) \begin{equation} P(t) = 1 - \exp \left [ - I \int _0^t \eta \mathcal{C}(\tau ) \, \mathrm{d} \tau \right ], \end{equation}

where $I$ describes the infectivity of the disease and $\eta$ is a breathing rate (see Table 1 for units and values).

In order to capture the spatiotemporal evolution of aerosol concentration and infection risk, Lau et al. [Reference Lau, Griffiths, English and Kaouri23] generalised (2.1) to account for spatial effects: advection of aerosols around a recirculating AC flow and diffusion due to turbulent eddies. This results in an ADR equation governing the 2D concentration around the recirculating loop, $\mathscr{C}$ . This is an areal concentration (in aerosols/m $^2$ ), with different units to the volumetric concentration $\mathcal{C}$ (in aerosols/m $^3$ ) introduced above. The production term is modified so that an infected person is modelled as a point source at location $(x_0,y_0)$ , i.e. $R/V \rightarrow R \delta (x-x_0) \delta (y-y_0)$ . As in (2.1), the removal terms are treated as global sinks.

The Lau et al. [Reference Lau, Griffiths, English and Kaouri23] model was validated against two superspreader outbreaks: one in a courtroom in Switzerland [Reference Vernez, Schwarz, Sauvain, Petignat and Suarez35] and one in a restaurant in Guangzhou, China [Reference Li25, Reference Lu26]. In both cases, the results agreed well with real-world observations. For both scenarios, the transport of aerosols was dominated by a recirculating airflow generated by AC units that were ineffective at removing aerosols.

In this paper, we introduce a local treatment of ventilation in addition to the global sink used in both Wells–Riley models (2.1) and Lau et al. [Reference Lau, Griffiths, English and Kaouri23]. The modelling framework detailed below is a generalisation of the Lau et al. [Reference Lau, Griffiths, English and Kaouri23] framework. The assumptions that recover the Lau et al. [Reference Lau, Griffiths, English and Kaouri23] model are highlighted after the current model is described.

2.2. Spatiotemporal model with local ventilation

Consider a three-dimensional (3D) room of volume $V$ , with dimensions $L_x$ , $L_y$ , $L_z$ , as depicted in Figure 1(a) (values given in Table 1). Following Lau et al. [Reference Lau, Griffiths, English and Kaouri23], we assume a recirculating flow produced by a single AC vent along the top corner and introduce the arclength coordinate $\xi$ , which follows the recirculating loop. This assumption is consistent with the real-life case studies [Reference Li25, Reference Vernez, Schwarz, Sauvain, Petignat and Suarez35] with which Lau et al. [Reference Lau, Griffiths, English and Kaouri23] validated their model. Moreover, van Hooff et al. [Reference van Hooff, Blocken, Defraeye, Carmeliet and Van Heijst20] measure the flow generated by a similar AC configuration and observe a recirculating flow throughout the majority of the room. Additional flow features are also reported by van Hooff et al. [Reference van Hooff, Blocken, Defraeye, Carmeliet and Van Heijst20], such as a fast-moving jet near the ceiling and a counter-rotating loop near the top corner opposite the AC unit, both of which are confined to the top 15% of the room ( $z \gt 2.55$ m here). These flow features are neglected from the current modelling framework since we assume that aerosols are produced at head height ( $\approx 1.5$ m, half the height of the room), so these near-ceiling structures are assumed to have a negligible effect on aerosol transport.

Table 2.

Parameters for two purifiers, with $\lambda = 2\times 10^{-4}$ s $^{-1}$ (0.7 ACH)

Table 2. Long description

A table comparing parameters for two purifiers. The table has 11 rows and 5 columns. The columns are labeled Parameter, Symbol, Weak purifier, Strong purifier, and Source. The rows list various parameters such as Clean air delivery rate, Flow-rate, Air velocity into purifier, Purifier air exchange rate, Purifier ACH, Total air exchange rate, and Equivalent global ACH. Each row provides specific values for the weak and strong purifiers, along with the source of the data. For example, the Clean air delivery rate (CADR) is 140 cubic meters per hour for the weak purifier and 1000 cubic meters per hour for the strong purifier. The Flow-rate (Q) is 0.039 cubic meters per second for the weak purifier and 0.28 cubic meters per second for the strong purifier. The Air velocity into purifier (vp) is 0.04 meters per second for the weak purifier and 0.3 meters per second for the strong purifier. The Purifier air exchange rate (λp) is 2.0 times 10 to the power of negative 4 per second for the weak purifier and 1.5 times 10 to the power of negative 3 per second for the strong purifier. The Purifier ACH is 0.7 ACH for the weak purifier and 5.3 ACH for the strong purifier. The Total air exchange rate (λtot) is 4.0 times 10 to the power of negative 4 per second for the weak purifier and 1.7 times 10 to the power of negative 3 per second for the strong purifier. The Equivalent global ACH is 1.4 ACH for the weak purifier and 6.0 ACH for the strong purifier.

* Weak purifier: Dbouk et al. [Reference Dbouk, Roger and Drikakis11], Strong purifier: Kähler et al. [Reference Kähler, Fuchs and Hain21].

Figure 1.

A 3D room with dimensions $L_x$ , $L_y$ , $L_z$ is shown in (a). A cylindrical local ventilation system crosses the two recirculating layers and the arclength coordinate $\xi$ follows the recirculating loop. The computational domain $(\xi ,y)$ is shown in (b).

The distance between the recirculation layers is taken to be $L_z/2$ , and the total arclength is assumed to be $2L_x$ (as in Lau et al. [Reference Lau, Griffiths, English and Kaouri23]). The height and arclength of the recirculating loop were determined by inspection of the recirculating flow reported by van Hooff et al. [Reference van Hooff, Blocken, Defraeye, Carmeliet and Van Heijst20].

The local ventilation system employed here is inspired by a common purifier design – a cylindrical device that draws air in through the top and expels air radially outwards near the bottom [Reference Dbouk, Roger and Drikakis11]. Inside the device, air passes through a filter which traps entrained particles. The filter efficacy $\zeta$ is the proportion of particles that are trapped by the filter, and hence removed from the room. We will assume that no aerosols that pass through the purifier re-enter the room, so $\zeta = 1$ . This assumption is common in purifier modelling [Reference Burgmann and Janoske7, Reference Dbouk, Roger and Drikakis11].

We will denote the radius of this device by $r$ and assume that it extends over both recirculating layers: air is drawn into an inlet in the upper layer and expelled from an outlet in the lower layer.

Figure 1(b) shows the computational domain $(\xi ,y)$ . The left/right halves correspond to the upper/lower layers and the purifier inlet and outlet appear as circles removed from the domain, with boundaries

(2.3a) \begin{equation} \partial _{\mathrm{in}} = \{(\xi ,y) \,:\, |(\xi ,y) - (x_p,y_p)| = r\}, \end{equation}

(2.3b) \begin{equation} \partial _{\mathrm{out}} = \{ (\xi ,y) \,:\, |(\xi ,y) - (2L_x-x_p,y_p)| = r\}, \end{equation}

where $(x_p,y_p)$ denotes the location of the centre of the purifier in the $(x,y)$ -plane, and we will assume throughout this work that the purifier is in the centre of the room.

In this quasi-3D model, the device extends the entire height of the room, which is significantly taller than real-life purifiers. This simplification allows for a comparison with the model of Lau et al. [Reference Lau, Griffiths, English and Kaouri23] while still offering useful and practical insights into the local effects of air purifiers.

Consider a single infectious individual at $\boldsymbol{x}_0 = (x_0,y_0)$ and talking continuously. Talking produces around 10 times as many aerosols as breathing [Reference Asadi, Wexler, Cappa, Barreda, Bouvier and Ristenpart1]: a reasonable worst-case scenario for an asymptomatic person. The infectious source will be placed in the top recirculating loop ( $\xi = x_0$ ), meaning the infectious person is standing (e.g. a presenting teacher or lecturer). Let $\mathscr{C}\kern1pt(\xi ,y,t)$ denote the concentration of aerosols (in aerosols/m $^2$ ) around the 2D recirculating loop depicted in Figure 1(b). We assume, as in Lau et al. [Reference Lau, Griffiths, English and Kaouri23], that $\mathscr{C}\kern1pt(\xi ,y,t)$ is governed by the ADR equation

(2.4) \begin{equation} \frac {\partial \mathscr{C}}{\partial t} + \nabla \cdot ( \boldsymbol{v} \mathscr{C}\kern1.8pt ) - \nabla \cdot ( K \nabla \mathscr{C}\kern1.9pt) = R \delta (\xi - x_0) \delta (y - y_0) - (\lambda + \beta + \sigma ) \mathscr{C}, \end{equation}

where $\boldsymbol{v} = (u,v)$ is a vector field describing the airflow around the recirculating loop; $K$ is the eddy diffusion coefficient; $R$ is the (constant) aerosol production rate and $\lambda$ , $\beta$ , $\sigma$ are global removal rates due to ventilation, biological deactivation and gravitational settling, respectively. Parameter values are given in Table 1; more details are provided in Lau et al. [Reference Lau, Griffiths, English and Kaouri23].

Following Lau et al. [Reference Lau, Griffiths, English and Kaouri23], we model the inbuilt ventilation with the global removal term $\lambda$ and set $\lambda = 2 \times 10^{-4}$ s $^{-1}$ ; the ‘poor ventilation’ scenario of Lau et al. [Reference Lau, Griffiths, English and Kaouri23] (from classroom data, Guo et al., [Reference Guo, Morawska, He and Gilbert17]). This reflects a broken or poorly maintained AC system that moves air around but is ineffective at removing aerosols [Reference Lu26] with an air exchange rate of 0.7 air changes per hour (ACH). ACH will be given to 1 decimal place throughout this paper.

Gravitational settling is treated here as a global sink. It is assumed that aerosols land on the ground with a given rate $\sigma$ and do not return to the air thereafter (i.e. they are removed from the room). Following Lau et al. [Reference Lau, Griffiths, English and Kaouri23], the effect of gravity in transporting aerosols from the top layer of the recirculating loop to the bottom is neglected, since this process occurs over much longer time-scales than the transport of the aerosols along the recirculating loop.

Aerosol production begins at $t = 0$ , so we set the initial condition $\mathscr{C}\kern1pt(\xi ,y,0) = 0$ . Periodic conditions across the left and right boundaries ( $\xi = 0, 2L_x$ ) complete the recirculating loop,

(2.5a) \begin{equation} \mathscr{C}\kern1pt(0,y,t) = \mathscr{C}\kern1pt(2L_x,y,t), \end{equation}

(2.5b) \begin{equation} \frac {\partial \mathscr{C}}{\partial \xi } (0,y,t) = \frac {\partial \mathscr{C}}{\partial \xi } (2L_x,y,t), \end{equation}

and there is no flux through the walls at $y = 0, L_y$ ,

(2.5c) \begin{equation} \frac {\partial \mathscr{C}}{\partial y}(\xi ,0,t) = \frac {\partial \mathscr{C}}{\partial y}(\xi ,L_y,t) = 0. \end{equation}

At the purifier inlet, we assume that aerosols are removed solely through advection (and cannot escape through diffusion). We therefore set zero diffusive flux here

(2.6a) \begin{align} \boldsymbol{\hat {n}} \cdot (K \nabla \mathscr{C}\kern1.9pt) = 0 \; \; \mathrm{on} \; \; \partial _{\mathrm{in}}, \end{align}

which is also known as the Danckwerts condition [Reference Danckwerts10]. The assumption that $\zeta = 1$ (the purifier filter traps all aerosols that pass through the device) means that no aerosols enter the domain through the purifier outlet. We thus set zero total (diffusive and advective) flux there,

(2.6b) \begin{equation} \boldsymbol{\hat {n}} \cdot ( \boldsymbol{v} \mathscr{C} - K \nabla \mathscr{C}\kern1.9pt) = 0 \; \; \mathrm{on} \; \; \partial _{\mathrm{out}}. \end{equation}

Here, $\boldsymbol{\hat {n}}$ denotes the unit vector normal to each boundary, directed out of the domain.

We extend the formula for the diffusion coefficient, $K$ , from Foat et al. [Reference Foat, Drodge, Nally and Parker14]. The turbulent eddies generated by the AC unit and purifier have different characteristic lengths (due to different vent sizes) and different characteristic velocities, hence the turbulent dissipation associated with each will be different. Letting $K_{ac}$ and $K_p$ denote the turbulent diffusion coefficients associated with the AC unit and purifier (respectively), the formula from Foat et al. [Reference Foat, Drodge, Nally and Parker14] gives

\begin{equation*} K_{ac} = c \lambda \sqrt [3]{\frac {V^2}{2}}, \qquad K_p = c \lambda _p \sqrt [3]{\frac {V^2}{2}}, \end{equation*}

where $\lambda _p$ is the air exchange rate of the purifier (discussed below), and we have assumed that there is a single vent of each size. Here, $c = c_v/ \sqrt [3]{c_{\varepsilon }}$ , where $c_v$ is the von Karman constant and $c_{\varepsilon }$ is the constant of proportionality in Taylor’s dissipation law. Following the rationale of Bodin [Reference Bodin5] and Karlsson et al. [Reference Karlsson, Sjöstedt and Håkansson22], we assume $c_{\varepsilon } = c_v^3$ , giving $c = 1$ (as in Lau et al. [Reference Lau, Griffiths, English and Kaouri23]).

The total eddy diffusion coefficient throughout the room is then assumed to be the sum of $K_{ac}$ and $K_p$ , giving

(2.7) \begin{equation} K = (\lambda + \lambda _p) \sqrt [3]{\frac {V^2}{2}}, \end{equation}

Several alternative combinations of $K_{ac}$ and $K_p$ were explored and found to produce similar results to those presented below. The expression (2.7) was chosen due to its mathematical simplicity and also because it produces the same value of $K$ for equivalent global and local ventilation levels, making direct comparison straightforward. However, by assuming a constant $K$ throughout the room, the model does not account for the increased levels of diffusion anticipated near the purifier. The exact form that a spatially varying $K$ would take has not been determined in the literature, to the best of our knowledge. To incorporate this feature into the model would necessitate a detailed study of turbulent eddy diffusion, which is outside the scope of this work.

Due to the inherent 3-dimensionality of turbulence, it is possible for turbulent eddies to transport aerosols between the upper and lower layers of the recirculating loop. This effect is neglected here under the assumption that these layers are sufficiently separated to evolve independently. This is a reasonable assumption for small $K$ .

Following Lau et al. [Reference Lau, Griffiths, English and Kaouri23], we assume that the majority of aerosols remain within the recirculating loop and consider the height-averaged concentration over the loop height ( $L_z/2$ ). This concentration, in aerosols/m $^3$ , is given by

(2.8) \begin{equation} C(x,y,t) = \frac {\mathscr{C}\kern1pt(\xi = x,y,t) + \mathscr{C}\kern1pt(\xi = 2L_x - x,y,t)}{L_z/2}. \end{equation}

Here, we have divided the recirculating loop into an upper layer, $\xi \in [0, L_x]$ , and a lower layer, $\xi \in [L_x, 2 L_x]$ , as depicted in Figure 1, to determine the height-averaged concentration, $C$ . This represents the concentration of particles (in aerosols/m $^3$ ) at a given $(x,y) \in [0, L_x] \times [0, L_y]$ in the 3D room, whereas $\mathscr{C}$ represents the concentration (in aerosols/m $^2$ ) at a given $(\xi ,y) \in [0, 2 L_x] \times [0, L_y]$ on the 2D recirculating loop.

The Lau et al. [Reference Lau, Griffiths, English and Kaouri23] model is a special case of the model described above and is recovered by removing the purifier (and hence the internal boundaries $\partial _{\mathrm{in}}$ and $\partial _{\mathrm{out}}$ ) and assuming a constant, unidirectional flow around the recirculating loop, $\boldsymbol{v} = (u_0, 0)$ . All other modelling assumptions are the same.

2.3. Infection risk

The risk of infection to a susceptible person at any $(x,y)$ is calculated using

(2.9) \begin{equation} P(x,y,t) = 1 - \mathrm{exp} \left [ - I \int _0^t \eta C(x,y,\tau ) \, \mathrm{d} \tau \right ]\!, \end{equation}

where $I$ is the infectivity constant of the virus and $\eta$ is the breathing rate (see Table 1). This is a spatiotemporal version of (2.2), the infection risk used in Wells–Riley models and the same as that used in the Lau et al. [Reference Lau, Griffiths, English and Kaouri23] model.

2.4. Airflow

In Lau et al. [Reference Lau, Griffiths, English and Kaouri23], aerosols are advected around the recirculating loop at constant speed $u_0$ (Table 1). Here, we assume that the recirculating loop remains coherent in the presence of a purifier, but we do not assume constant velocity. We assume that air enters and leaves the purifier with a specified constant speed $v_p$ , and this leads to a modified $\boldsymbol{v} = (u,v)$ such that

(2.10a) \begin{equation} \boldsymbol{v} \cdot \boldsymbol{\hat {n}} = v_p \; \; \mathrm{on} \; \; \partial _{\mathrm{in}}, \end{equation}

(2.10b) \begin{equation} \boldsymbol{v} \cdot \boldsymbol{\hat {n}} = - v_p \; \; \mathrm{on} \; \; \partial _{\mathrm{out}}. \end{equation}

We also require that $\boldsymbol{v}$ satisfies the periodic boundary condition,

(2.11) \begin{equation} \boldsymbol{v}(0,y) = \boldsymbol{v}(2 L_x,y). \end{equation}

The velocity vector field that satisfies (2.10) and (2.11) is determined using the Shear Stress Transport (SST) turbulent flow solver [Reference Menter27] in COMSOL Multiphysics $^{\circledR }$ (see COMSOL Inc., [8]). A turbulent flow solver is used since $Re \gt 10,000$ and hence numerical solutions to the Navier–Stokes equations were found to be sensitive to the mesh size. The resulting 2D flow in the $(\xi ,y)$ -plane does not incorporate the diffusive properties of a true 3D turbulent flow; the spreading of aerosols by small-scale turbulent eddies is accounted for by the eddy diffusion coefficient $K$ , as discussed above.

Imposing no-slip and no-penetration conditions at the walls,

(2.12) \begin{equation} \boldsymbol{v}(\xi ,0) = \boldsymbol{v}(\xi ,L_y) = (0,0), \end{equation}

and setting a suitable pressure gradient over the periodic boundaries to ensure

(2.13) \begin{equation} \max _y u (\xi = 0, y) = u_0, \end{equation}

we run the SST solver until a steady state is reached. This steady velocity, $\boldsymbol{v}$ , is then used in the ADR equation (2.4). The results are compared against those of Lau et al. [Reference Lau, Griffiths, English and Kaouri23] for: (i) no purifier and (ii) a switched-off purifier ( $v_p = 0$ ). There is good agreement in both cases provided (2.13) is satisfied. Several turbulent models were compared, and all resulted in a similar concentration distribution $\mathscr{C}$ . The details of the SST flow solver used here are provided in Appendix A.

The steady-state assumption is consistent with Hooff et al. [Reference van Hooff, Blocken, Defraeye, Carmeliet and Van Heijst20] and realistic for many locations (e.g. classrooms and offices) where the AC is managed automatically and is therefore expected to have settled into a steady state prior to use. Note that during the development of this steady recirculating flow, we would anticipate complex transient effects such as multiple vortices of different sizes.

Two purifier settings are considered based on the clean air delivery rate (CADR) of purifiers used in experimental and computational studies. We define a weak purifier with a CADR of 140 m $^{3}$ h $^{-1}$ , representative of devices for small spaces such as domestic rooms [Reference Dbouk, Roger and Drikakis11] and individual offices; and a strong purifier with a CADR of 1,000 m $^{3}$ h $^{-1}$ , representative of devices for larger spaces such as classrooms [Reference Kähler, Fuchs and Hain21] and open-plan offices.

Let $Q$ denote the flow rate through the device in m $^3$ s $^{-1}$ . In general, the CADR (stated in m $^3$ h $^{-1}$ by convention) is given by $\zeta Q$ (for filter efficacy $\zeta$ ). Here, the assumption $\zeta = 1$ means the CADR, and flow rate $Q$ are equivalent. For a cylindrical purifier (circumference $2 \pi r$ ) with an inlet half the height of the room ( $L_z/2$ ),

(2.14) \begin{equation} Q = \pi r L_z v_p. \end{equation}

Using $v_p = Q/\pi r L_z$ (see Table 2) as the air speed at the inlet and outlet boundaries in (2.10), we use the SST solver to determine the velocity field $\boldsymbol{v}$ for each purifier.

Each purifier is compared with an equivalent increase in the global removal term $\lambda$ . The air exchange rate associated with each purifier is given by $\lambda _p = Q/V$ , which we add to the global removal rate of the AC unit (0.7 ACH) to determine a total air exchange rate, $\lambda _{\mathrm{tot}} = \lambda + \lambda _p$ (see Table 2). The weak purifier doubles the total air exchange to 1.4 ACH, less than half the recommended ventilation for classrooms – 3 ACH for 30 occupants [3]. The total air exchange rate with the strong purifier is 6 ACH, exceeding this recommendation and also sufficient to meet the guidelines for times of heightened infection risk, provided the number of occupants is halved [4].

The streamlines of the airflow $\boldsymbol{v}$ are shown in Figure 2 for both purifiers. The flow is broadly unidirectional for the weak purifier (Figure 2 a). Regions in which the streamlines are directed into or out of the purifier are shaded and take up a greater area for the strong purifier (Figure 2 b). In both cases, these regions meet at the periodic boundary.

Figure 2.

The airflow streamlines in the $(\xi ,y)$ -plane are shown for (a) the weak purifier ( $v_p = 0.04$ ) and (b) the strong purifier ( $v_p = 0.3$ ). The shaded regions indicate streamlines that pass through the purifier inlet (left) and the purifier outlet (right).

2.5. Terminology: global and local ventilation models

To improve clarity in the discussions below, we here define shorthand labels that refer to the models explored. There are three features that all models compared have in common: (i) the computational domain, which is depicted in Figure 1(b); (ii) the SST solver is used to determine the airflow $\boldsymbol{v}$ ; and (iii) the global ventilation rate $\lambda$ is non-zero.

The term global ventilation model here refers to cases in which there are no local ventilation effects. To ensure the computational domain is consistent, the purifier is present but switched off, $v_p = 0$ . This global ventilation model is subtly different to the model of Lau et al. [Reference Lau, Griffiths, English and Kaouri23] due to the existence of the boundaries $\partial _{\mathrm{in}}$ , $\partial _{\mathrm{out}}$ and the non-uniform vector field $\boldsymbol{v}$ . Aside from these minor differences, the underlying modelling assumptions are the same. This model was compared with results reported in Lau et al. [Reference Lau, Griffiths, English and Kaouri23], and the contours of aerosol concentration $C$ were found to be virtually indistinguishable except in small regions near to the switched-off purifier where differences remained minimal.

We consider three global ventilation levels, which will be referred to by the ACH: The control case 0.7 ACH, in which the only ventilation is provided by the ineffective AC unit; 1.4 ACH, equivalent to the total air exchange rate for the weak purifier; and 6 ACH, equivalent to the total air exchange rate for the strong purifier.

The term local ventilation model refers to the cases in which the purifier is switched on, i.e. $v_p \neq 0$ . In these cases, the global ventilation rate is that of the control case, 0.7 ACH, so we are considering the combined ventilation of the ineffective AC unit (treated globally) and the purifier (treated locally). The two local ventilation cases explored are referred to by the purifier strength: The weak purifier (CADR of 140 m $^3$ h $^{-1}$ ) and the strong purifier (CADR of 1,000 m $^3$ h $^{-1}$ ).

2.6. Computational speed

The model was solved in COMSOL Multiphysics $^{\circledR }$ on a triangular mesh using the predefined ‘Finer’ mesh, with a maximum element size of $0.224$ m. The automatic ‘Fluid dynamics’ calibration resolved the boundary layers with smaller elements near to boundaries. The point source was approximated by uniform production within a small circle, radius $r_s = 2.5$ mm. Consequently, a finer mesh was produced around $\boldsymbol{x}_0 = (x_0, y_0)$ . To ensure the symmetry of the problem was numerically realisable, a symmetric mesh about the centreline $y_0 = L_y/2$ was used. For scenarios in which $y_0 \neq L_y/2$ , this resulted in an additional region of finer meshing near $(x_0, L_y - y_0)$ , resulting in slightly more mesh elements than when $y_0 = L_y/2$ . Each mesh contained approximately $21,000$ nodes (though the precise number differed for each $\boldsymbol{x}_0$ ). Both the flow field and the ADR equation were solved on the same mesh.

The mesh independency was explored as follows: all parameters were fixed, and the problem (both airflow and ADR equation) was solved using the automatic resolution settings ranging from ‘Coarser’ to ‘Extremely Fine’. For direct comparison, each solution was interpolated onto a fine uniform grid (using cubic splines), and the relative error between the ‘Extremely Fine’ mesh (around 90,000 nodes) and each of the other meshes was calculated at each grid point, then integrated over the domain (using Simpson’s rule). The ‘Finer’ mesh ensured a discrepancy of the order of $10^{-3}$ and resulted in plots that were visually indistinguishable from those for the ‘Extremely Fine’ mesh.

Simulations were performed on a Lenovo IdeaPad Flex 5 laptop, with a 1.3 GHz 4-core Intel Core i7-1065G7 processor and 8 GB of RAM. For each airflow simulation (switched-off purifier, weak purifier and strong purifier) to reach a steady state, computation takes approximately 10 minutes. For the ADR equation (2.4), an event lasting 4 hours takes around 5 minutes to run, including calculation of the infection risk (2.9). At these computational speeds, advice and guidance can be quickly updated with new information during a fast-changing epidemic.