Reducing airborne infection risk of COVID-19 by locating air cleaners at proper positions indoor: Analysis with a simple model.

Portable air cleaners (PACs) can remove airborne SARS-CoV-2 exhaled by COVID-19 infectors indoor. However, effectively locating PAC to reduce the infection risk is still poorly understood. Here, we propose a simple model by regressing an equation of seven similarity criteria based on CFD-modeled results of a scenario matrix of 128 cases for office rooms. The model can calculate the mean droplet nucleus concentration with very low computing costs. Combining this model with the Wells-Riley equation, we estimate the airborne infection risk when a PAC is located in different positions. The two similarity criteria, B p + and G p + , are critical for characterizing the effect of the position and airflow rate of PAC on the infection risk. An infection probability of less than 10% requires B p   +  to be larger than 144 and G p   +  to be larger than 0.001. These criteria imply that locating PAC in the center of the room is optimal under the premise that the airflow rate of PAC is greater than a certain level. The model provides an easy-to-use approach for real-time risk control strategy decisions. Furthermore, the placement strategies offer timely guidelines for precautions against the prolonged COVID-19 pandemic and common infectious respiratory diseases.

Introduction

Convincing evidence has shown that human exhaled droplets are closely related to the transmission of infectious respiratory diseases in indoor environments [[1], [2], [3]]. World Health Organization (WHO) pointed that the droplet nucleus evaporated from respiratory droplets exhaled by the coronavirus disease 2019 (COVID-19) infected patients can contain severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) [4]. The airborne infection cannot be ignored, particularly in crowded spaces with insufficient ventilation. Under the current severe situation of the COVID-19 pandemic, portable air cleaners (PACs) have been suggested as a supplement to remove airborne SARS-CoV-2 [5]. An analysis of viral RNA from air samples taken in households inhabited by a COVID-19 positive patient was carried out using real-time RT-PCR method [6]. The study clarified a PAC with HEPA filter can remove airborne SARS-CoV-2 by effectively decreasing the concentration of virus attached droplet nuclei, thus further reduce the risk of airborne infection.

Previous studies have concluded that the indoor position of PAC plays an important role in air cleaning efficacy. Chen et al. [2] studied the purification effect of PAC on cigarette smoke and cough pollutants. They concluded that the position of PAC affects the transport of particles. Kashkooli et al. [7] simulated the airflow field and obtained an optimal placement plan for the PAC. Novoselac and Sigel [8] found that PAC in different locations can produce up to three times the difference in air cleaning efficacy. In addition, factors such as the air outlet angle, spatial scales, and particle sizes have also been demonstrated to have different effects on the concentration distribution of particle-phase indoor pollutants [9,10]. Therefore, it is necessary to investigate the effect of the indoor position of PAC on reducing the airborne infection risk of COVID-19 due to exhaled droplet nuclei from infected subjects.

Computational fluid dynamics (CFD) is a powerful tool capable of modeling air pollutant distribution indoors; thus, CFD may be an optional approach to study the effect of the indoor position of PAC on removing human exhaled droplet nucleus [2,7,8]. However, the position of the PAC affects the indoor airflow distribution and results in different pollutant distributions. To investigate the human exhaled droplet nucleus distribution caused by the different positions of PAC, the turbulent non-isothermal airflow and dispersion of different sizes of droplet nuclei should have to be incorporated in the CFD simulation for each case, a high cost of simulation. Most importantly, such a complicated and time-consuming simulation cannot meet decision-making needs in actual engineering.

Here, we established a simple model to quantify the relationship between the airborne infection risk of COVID-19 and the indoor position of PAC. The simple model is a regressed equation based on seven criteria deduced with similarity analysis. The regression data were obtained by CFD modeling of a scenario matrix of 128 cases. Then, the airborne infection risk of COVID-19 was estimated using the Wells–Riley (W–R) equation under the scenario of office rooms, commonly seen working postures, and exposure time.

Methodology

Fig. 1 shows the process of methodology development. In the first step, the input data are prepared for predictive equation regression. We performed similarity analysis to deduce the similarity criteria. We then modeled the mean concentration of the droplet nucleus using the CFD approach. We employed a multiple linear regression (MLR) approach to regress the predictive equation in the second step. In the third step, we estimated the airborne infection risk of COVID-19 by combining the regressed predictive equation with the W–R equation.

Fig. 1

Overall methodology flowchart.

Similarity analysis

Dimensional similarity analysis has been introduced in detail in previous textbooks [11]. The core idea is the dimensional homogeneous method, which is to establish several similar criteria in a large number of parameters through a dimensional matrix. Thus, new dimensionless parameters are deduced to have the simplest possible mathematical form with clear physical meaning. The method needs to meet two requirements: the basic physical quantities must have dimensions, but the dimensions must not be repeated in the same group; the dimensions contained in basic physical quantities can include all dimensions of all the related parameters.

Here, we focus mainly on office rooms. The office is a crowded and inadequately ventilated space, where airborne transmission in COVID-19 outbreaks mainly occurs [4]. Furthermore, as office-based workers represent half the world's population and typically spend an average of over 8 h on weekdays [12,13], we believe it is suitable and applicable to demonstrate the key idea of this study by focusing on office rooms.

Upward ventilation systems (ceiling air supply and ceiling air return flow pattern) and prolonged sitting are the most common in office spaces [12,14]. Therefore, such scenarios were incorporated into our analysis, as shown in Fig. 2 (a). For such scenarios, the key factors influencing the airflow distribution may be few because people, facilities, inlets, and ventilation outlets are fixed at certain locations. The popular PACs in the market have typical ejection directions of air current from the inlet and outlet, as shown in Fig. 2 (b). In addition, the differences between horizontal and vertical ejections are generally insignificant [2].

Fig. 2

Schematic of the studied indoor environment (a) office room. (b) portable air cleaner (PAC). (c) the relative position of the PAC and the source (infector). (d) meshing diagram in three typical cut planes of case-setting.

The parameters for the dimensional similarity analysis and their units are listed in Table 1 . Seven similarity criteria were deduced using similarity analysis approach (Table 2 ). Considering the airborne infection risk due to inhalation of the droplet nucleus, the mean droplet nucleus concentration in the breathing zone, z = 1.2 m [10], is our target for regression.

Table 1

Parameters for similarity analysis and their range of values.

Parameter	Description	Unit	Setting range
ρa	Density of air	kg/m3	1.205
cp	Specific heat of air	J/kg∙K	1005
ν	Kinetic viscosity of air	m2/s	15.06e-6
g	Acceleration of gravity	m/s2	9.8
b	Length of room	m	5–12
A	Floor area	m2	20–72
V	Volume of room	m3	2.8A–3.2A
uv	Speed of air supply	m/s	0–0.3
Gv	Volume of air supply	m3/s	0–0.375
tv	Temperature of air supply	K	292–296
up	Air speed of PAC	m/s	1–9
Gp	Air flow rate of PAC	m3/s	0.06–0.54
hp	Height of PAC	m	0.6–1.5
Xp	Crosswise relative distance of PAC	–	0.1–0.5
Yp	Lengthwise relative distance of PAC	–	0.1–0.5
Q	Heat gain of room	J/s	770–2670
hs	Height of the source	m	1.2
us	Expiratory speed	m/s	1.945
ts	Expiratory temperature	K	308
Xs	Crosswise relative position of the source	–	0.2–0.8
Ys	Lengthwise relative position of the source	–	0.2–0.8

Table 2

Deduced similarity criteria.

No.	Similarity criteria	Description	Physical significance	Min	Max
п1	V+	VA^1..5	Represents the influence of the room shape	0.377	0.630
п2	Bp+	u_pAX_pY_pG_p	Represents the position of PAC relative to the room	1.85	300
п3	Bs+	h_sAX_sY_sV	Represents the position of emitting source relative to the room	0.0160	0.224
п4	Gv+	G_su_sgV	Represents the influence of the mechanical air supply volume rate	0	5.00e-5
п5	Gp+	G_pu_pgV	Represents the influence of airflow rate of PAC	2.66e-5	8.86e-3
п6	Q+	Qc_pρ(t_s−t_v)(G_v+G_p)	Represents the relationship of heat gain and volume of total air supply	0.0981	0.883
п7	H+	g(h_p−h_s)^3v^2	Represents the influence of height difference between the position of PAC and the emitting source	−8.70e9	1.10e9

Fig. 1(c) is a schematic diagram of the x-y plane corresponding to Fig. 1(a). We defined the crosswise relative distance X and the lengthwise relative distance Y to generalize the object's distance to the origin. An object's X or Y is defined as the ratio of its x- or y-coordinates to the corresponding coordinate value of the room. Fig. 1(c) shows the definition of the sample case, where (X , Y ) and (X , Y ) are (0.4, 0.3) and (0.5, 0.2), respectively. Set the range of X and Y to 0.1–0.5, and the range of X and Y to 0.2–0.8, which can cover the scenarios of concern. Two similarity criteria, B and B , were then deduced to reflect the relative position of the PAC and the source to the room (Table 2).

To perform MLR, each similarity criterion affecting the droplet nucleus concentration distribution should cover at least two values in all cases. Therefore, at least 27 CFD cases are required, as we have seven similar criteria. Thus, we performed MLR based on the scenario matrix of the 128 CFD modeled cases to obtain the correlation equation between the mean droplet nucleus concentration in the breathing zone () and the other seven similar criteria. Because of the changes in the parameter settings in different cases, the similarity criteria will also change accordingly (Table 2), indicating that the results obtained by the above regression analysis method apply to all office cases where the seven similarity criteria are within the actual applications of these ranges.

CFD modeling

Airflow model

The renormalization group (RNG) k–ε turbulence model was adopted to model the turbulent airflow indoors as the model has been verified to be suitable for indoor airflow modeling [15]. ANSYS Fluent 17.0 (ANSYS, 2017) was used for the CFD modeling.

Drift flux model for droplet nuclei dispersion

Previous studies have validated the accuracy of the drift flux model in modeling airborne particle dispersion in indoor environments. The drift flux model is a Eulerian method that integrates the gravitational settling effects of particles into the concentration equation. The governing equation is as follows [16]:where , and are the air density, velocity vector, and concentration of the particles, respectively. is the turbulent diffusivity of . is the sum of the molecular and turbulent dynamic viscosities. is the settling velocity of particles.

The field measurements indicated that the size of the human exhaled droplet nucleus containing SARS-CoV-2 is mainly distributed in the range of 0.25–1 μm [17], corresponding to the initial droplet size range of approximately 1–5 μm [18]. Chen and Zhao [19] investigated the dispersion characteristics of human exhaled droplets in ventilation rooms. They found that the transient process from droplets to droplet nuclei due to evaporation at the mouth can be neglected in the modeling when the initial diameter is less than 100 μm. Therefore, the virus-laden droplets can be directly calculated as droplet nuclei. This study determined the droplet nuclei 0.2–10 μm for CFD simulation, which is widely adopted for airborne viruses in transmission models [3,20,21].

The settling velocity of the droplet nucleus derived by equaling the fluid drag force on the droplet nuclei with the gravitational force can be expressed as [22]:where and are the densities of the droplet nucleus and ambient air, respectively, is the gravitational acceleration. is the drag coefficient. The settling velocity always has the same direction as gravitation, that is, perpendicularly downward.

The above equations were discretized into algebraic equations by the finite volume method using the second-order upwind scheme. The model was fulfilled with a user-defined function (UDF) in ANSYS Fluent 17.0 (ANSYS, 2017). The pressure-velocity coupling was solved using the SIMPLE algorithm.

Boundary conditions

For airflow, all variables were defined at the supply inlet of the ventilation system and the PAC. Mass flow boundaries were specified at the outlet to ensure that the outflow mass flow rates correspond with the inflow mass flow rates. Standard logarithmic law wall functions were adopted. The walls were assumed to be stationary and adiabatic because the simulation object of this research studies the office located in the inner zone.

For droplet nucleus dispersion, You et al. [23] established an empirical equation to determine indoor particle deposition velocities, which can be used to calculate the boundary conditions for the drift-flux model easily. Combined with the particle size range studied in this study, the deposition velocity is expressed as follows: Where

, , are the deposition velocities at the upward-facing horizontal surface (floor), vertical walls, and the downward-facing horizontal surface (ceiling), respectively. θ is the inclination angle, which is 0°, 90°, 180° at floor, vertical walls and ceiling, respectively. is the friction velocity of the wall.

is the shear stress at the walls.

The particle size of the droplet nucleus was divided into four segments between 0.2 and 10 μm; size distribution meets Weibull's law of probability frequency f [24]. The mass flow of particulate matter at the outlet of the PAC had a linear relationship with the inlet, and the ratio was (1-η). η is the size-dependent filtration efficiency, which has been reported by Ref. [25]. The values of η and f are listed in Table 3 . Another UDF was integrated into the model to fulfill the above boundary conditions using the ANSYS Fluent 17 program (ANSYS, 2017).

Table 3

The values of size-dependent filtration efficiency (η) and frequency (f).

Size (μm)	η	fd
0.2	70%	1.27e-12
0.8	87%	2.08e-8
5	95%	0.00775
10	90%	0.992

For each case, the total concentration of the four size segments was normalized by the source concentration:where is the normalized total mean droplet nucleus concentration in the office room's breathing zone (z = 1.2 m). is the mean droplet nucleus concentration at z = 1.2 m for each size segment. is the concentration at the mouth of the source for each size segment.

A grid independence test was conducted by calculating the same mode with finer grids until the simulation changed little. To determine the relative test error, the grid convergence index (GCI) was calculated using the Richardson extrapolation method [26]:where  = 3, p = 2, r is the ratio of fine to the coarse grid, is defined asWhere is defined as:Where u is the velocity magnitude.

We set three groups of grid numbers: 782,217,1,532,423, and 3,910,099. The GCI values of the tested group grids were all less than 10%. Then, CFD cases were simulated based on a grid number of 1,500,000 to balance sufficient accuracy and calculation speed. The grid diagram at three most representative locations is shown in Fig. 2 (d).

Validation of the model

The numerical model used in this research was validated using different sets of experiments. Zhao et al. [16] conducted a full-scale experiment. They verified that the results of the numerical model were consistent with the measurement results for ultrafine particles. Zhou and Zhao [27] further compared the numerical model for particle dispersion with measurements to verify the model, covering the particle size from fine (sub-micron) to the coarse (micron) range. Hence, we tend not to repeat the validation here because the model has been fully verified several times. We are confident that it is capable of analyzing the scenarios in this study.

Multiple linear regression (MLR)

MLR has been widely and successfully applied in predictive indoor air pollution concentrations [[28], [29], [30], [31]]. When using MLR, the input variables should be independent of each other; there is no multicollinearity. Data analysis of the scenario matrix of the 128 CFD simulation cases was performed using SPSS (Statistical Package for Social Science, version 22).

The results were checked for multicollinearity by examining the predictor variables' variance inflation factors (VIFs) [32,33]. The Durbin Watson statistic was also used to check if the model did not have any order autocorrelation problem. The residuals (or error) were normally distributed with zero mean and constant variance to ensure the adequacy of the statistical model [34]. The reliability of the regression equation was further verified by the coefficient of multiple determinations (R2).

Wells–Riley (W–R) equation for airborne infection risk assessment

We established an association between the airborne infection probability and droplet nucleus concentration based on a revised W–R equation [35,36]. The difference in quantum concentration between locations essentially reflects the dilution capacity of office room ventilation and PAC's airflow rate for droplet nuclei in different locations, which is exactly the normalized concentration mentioned above. The W–R equation is modified as:where is the infection probability, is the quantum generation rate produced by a source (h−1), set as 14 h−1 based on our previous study [37]. This value is also verified by real cases and estimation of COVID-19 viral emission [38]; is the droplet nucleus concentration at a certain location normalized by the source concentration, and t is the exposure time (h), set as 8 h to represent the most popular working times [12].

Results and discussions

Regression equation and verification

The correlating equation between and the seven similar criteria is:

The result of the regression model fitted to 128 sets of data yielded a strong coefficient of determination value (R2) of 0.63.

The logic to verify the equation is that if the results calculated by the equation are consistent with the simulated results with CFD, the simple equation can be considered feasible. Therefore, we simulated three new cases other than the 128 cases to verify the correlating equation by comparing the CFD simulated mean droplet nucleus concentration with the values calculated using Eqs. (10), (11), respectively: The parameter settings of the three verification cases (the parameter settings vary in a wide range compared to those 128 cases) and comparisons are listed in Table 4 .

Table 4

Parameters of the verification cases and the comparing results.

Parameter	Description	Unit	Case 1	Case 2	Case 3
b	Length of room	m	5	8	12
A	Floor area	m2	20	40	72
V	Volume of room	m3	56	120	230.4
uv	Speed of air supply	m/s	0	0.12	0.3
Gv	Volume of air supply	m3/s	0	0.15	0.375
up	Air speed of PAC	m/s	2	4.5	9
Gp	Air flow rate of PAC	m3/s	0.12	0.27	0.54
hp	Height of PAC	m	0.6	1	1.5
Xp	Crosswise relative distance of PAC	–	0.1	0.3	0.5
Yp	Lengthwise relative distance of PAC	–	0.2	0.3	0.5
Q	Heat gain of room	J/s	770	2010	2670
Xs	Crosswise relative position of a person	–	0.32	0.2	0.7
Ys	Lengthwise relative position of a person	–	0.4	0.2	0.6

The relative errors are less than 15%, indicating good agreement and acceptable accuracy from engineering applications. It takes a few seconds to obtain the results with the simple model. In contrast, CFD usually takes several hours to calculate a case.

Airborne infection risk corresponding to the position of PAC

A total of 20 typical cases representing the influence of the similarity criteria for airborne infection risk analysis is listed in Table 5 . Droplet nucleus dispersion and infection probability vary widely from case to case. The impact of the position of PAC is closely related to G   as the airflow mainly determines the droplet nucleus motion, and G   represents the degree of effect of the PAC air flow rate relative to the room volume. When G is small, PAC has relatively little effect on the airflow field and vortices near the wall and human body, regardless of location. Hence, the droplet nuclei accumulate in the breathing zone, resulting in a high risk of infection, as in cases 1–4. The infection probabilities of cases 1–4 were larger than 20%, even as high as 44% in case 1. By contrast, the G in case 5–8 is larger than 0.001, the infection probability is reduced to less than 20%, and the G in case 9 is as high as 0.0024, corresponding to a reduction in the infection probability to less than 10%. This is because the operation of a higher airflow rate of PAC results in faster indoor air velocity and enhances convection in the office room, reducing the vortices around the human body and more effectively preventing droplet nuclei from being trapped in the breathing zone.

Table 5

The and infection probability of 20 typical cases.

Case No.	Image 1	Bp+	Gp+	C‾^+	Infection probability
1	Image 2	3	0.0007	0.0052	44.2%
2	Image 3	3	0.0001	0.0045	39.5%
3	Image 4	17	0.0001	0.0022	22.3%
4	Image 5	83	0.0001	0.0022	21.4%
5	Image 6	7	0.0018	0.0016	16.7%
6	Image 7	19	0.0010	0.0017	17.4%
7	Image 8	30	0.0018	0.0011	11.7%
8	Image 9	60	0.0022	0.0011	11.5%
9	Image 10	72	0.0022	0.0009	9.7%
10	Image 11	83	0.0010	0.0025	24.1%
11	Image 12	144	0.0010	0.0006	7.0%
12	Image 13	146	0.0006	0.0014	14.3%
13	Image 14	300	0.0002	0.0012	12.6%
14	Image 15	83	0.0007	0.0026	25.2%
15	Image 16	80	0.0018	0.0012	12.3%
16	Image 17	146	0.0010	0.0010	10.3%
17	Image 18	144	0.0010	0.0009	9.5%
18	Image 19	1	0.0015	0.0007	7.6%
19	Image 20	300	0.0022	0.0006	6.2%
20	Image 21	300	0.0022	0.0005	5.9%

It should be noted that blindly increasing G   does not always reduce the probability of infection. For example, the G of case 10 was higher than 0.001, but the risk was still as high as 24%. This implies that the airflow rate of PAC does not entirely characterize its effectiveness in controlling droplet nucleus exposure.

We found that increasing B   can reduce infection risk by comparing cases 1 and 11, cases 2 and 3, and cases 5 and 7. According to the definition of this similarity criterion in Section 2.1, it can be expected that the larger the room area relative to the area of the PAC outlet, the larger the B , when will naturally decrease because of the larger room volume. Meanwhile, the larger the B , the closer the PAC is to the center of the room when the room size is constant. This placement also enhances the mixing effect and reduces airborne infection risk. For example, B in case 11 is as high as 144, and the corresponding infection probability drops below 10%.

However, B and G   did not increase infinitely at the same time by definition. When the room size, airflow rate, and position of the PAC are fixed, B   increases. At the same time, G   decreases, and the effect of reducing the risk of infection may be weakened. For example, the risk in case 12 was doubled compared to case 11; or although the B in case 13 was as high as 300, the infection probability was still higher than 10% because G in case 13 was only 0.0002. Further observation of cases 14–20 shows that when B is larger than 144 and G is larger than 0.001, and the infection probability can be reduced to less than 10%. Considering that the other five similar criteria in these 20 typical cases have changed in relatively large ranges, we can confirm the credibility of the criterion.

The above criterion implies that locating the PAC in the center of the room is optimal for reducing the airborne infection risk of COVID-19. The optimal placement was also related to the airflow rate of the PAC. When operating at a low flow rate, the PAC has little effect on the airflow field. The infection probability is as high as 20%–40% no matter where it is placed. Combining the definition of G and PAC's common air supply speed in the market, for example, 4.5 m/s, G   larger than 0.001 means that the PAC provides an air change rate of more than 7 h−1 to the room. On this premise, locating the PAC close to the center of the room may reduce the infection probability to less than 10%. In addition, these two criteria are significantly affected by the room size. We suggest that the room area be at least 500 times larger than the PAC outlet area.

The practical implication for further application

The simple model is efficient for predicting the mean droplet nucleus concentration and assessing infection risk in confined spaces without time-consuming and complicated CFD simulations when the position of PAC is arbitrarily changed. The model is a set of algebraic equations that require simple input parameters. As epidemic prevention and control work continues to deepen, current strict precautions have ensured that COVID-19 infectors are unlikely to enter public spaces. However, prolonged pandemic periods have caused a shortage of medical resources worldwide, especially in temporary hospitals or home isolation scenarios, where the existing ventilation systems provide ventilation rates far below the standard recommended by the WHO [39]. The simple equation can maximize its effectiveness when applied to reducing risk with very few costs on the premise that PAC serves as a useful supplement. In addition, droplet nuclei in the size range are the pathogen carriers for most respiratory diseases, that is, the influenza A virus, SARS-CoV-1, and norovirus [20,40,41], implying that the simple model is not only for COVID-19 precaution but also makes indoor real-time risk monitoring possible for pervasive airborne transmission in daily life.

For other types of indoor environments, the similar process may be repeated to obtain a similar correlation equation of similarity criteria. Then, the model can be extended to analyze the effectiveness of PAC in controlling pathogen-laden particle dispersion for other potential patient scenarios. For example, if the ventilation system is changed to underfloor or side air supply, the cases would accordingly change; thus, we may obtain a new simplified regressed correlating equation based on new CFD modeled results. In addition, it is difficult to describe an object's position using a certain parameter involved into the similarity analysis, the Bp + we deduced is only applicable to square rooms. However, the office room shown in Fig. 2 is the most common design in actual engineering, and rectangular buildings have the lowest life-cycle costs [42,43]. Thus, the scenario we chose represents the most commonly observed case in real office rooms. The guidance of PAC placement may be extended for the actual application.

Conclusions

We established a simple model to investigate the impact of the indoor position of PAC on reducing the airborne infection risk of COVID-19 in typical poorly ventilated scenarios. The following conclusions were drawn.

A simple equation correlating seven similarity criteria is obtained to predict the mean droplet nucleus concentration, further estimating the airborne risk of COVID-19 infection using the W–R equation. The simple equations can be applied in actual engineering to avoid case-by-case complicated CFD simulations.

An infection probability of less than 10% requires locating the PAC in the center of the room (corresponding to B  + larger than 144) and enough air flow rate of the PAC (corresponding to G   larger than 0.001)

CRediT authorship contribution statement

Hui Dai: Writing – original draft. Bin Zhao: Writing – review & editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.