Recent studies have shown that the effectiveness of the face masks depends not only on the mask material but also on their fit on faces. The mask porosity and fit dictate the amount of filtered flow and perimeter leakage. Lower porosity is usually associated with better filtration; however, lower porosity results in higher perimeter leakage. The resulting leakage jets generated from different types of faces and different mask porosities are of particular interest. Direct numerical simulations of the flow dynamics of respiratory events while wearing a face mask can be used to quantify the distribution of the perimeter leaks. Here, we present a novel model for porous membranes (i.e., masks) and use it to study the leakage pattern of a fabric face mask on a realistic face obtained from a population study. The reduction in perimeter leakage with higher porosities indicates that there would be an optimal porosity such that the total leakage and maximum leakage velocities are reduced. The current model can be used to inform the quantification of face mask effectiveness and guide future mask designs that reduce or redirect the leakage jets to limit the dispersion of respiratory aerosols.

Due to the continuing COVID-19 pandemic (caused by the SARS-CoV-2 coronavirus), the use of face masks has been adopted by the general public, albeit with some initial pushback. To ensure that N95 respirators and surgical masks are available to those in high-risk situations such as medical personnel and infected patients, the World Health Organization (WHO) and Centers for Disease Control and Prevention (CDC) recommended the use of fabric face masks, and it has been considered the status quo in personal protection equipment (PPE). The CDC has even made a tutorial and pattern design available for the general public to make their own face masks at home. The use of face masks and social distancing have been the primary response to mitigating droplet and airborne transmission.

Morawska and Milton, with the support of over 200 other clinicians, infectious disease experts, engineers, and aerosol scientists, urged public health authorities to acknowledge the potential for airborne transmission.1,2 Airborne transmission happens when a susceptible person inhales microscopic bio-aerosols in the air, which are generated from a respiratory event such as a cough, sneeze, or even just talking.3,4 Aerosols are a suspension of solid or liquid particles in a gas, which are made up of droplets smaller than 5μm and the remains of evaporated infectious droplets referred to as droplet nuclei. While larger droplets (100μm) reach the ground within a second, aerosols can linger in the air for hours.5–7 

Sneezing, coughing, talking, and even just breathing can spread the virus in the environment by an infected person. The size of the particles depends on the environment's temperature and humidity and can change as they evaporate. The larger droplets (>5μm) can easily be stopped/filtered by face masks. Smaller droplets (<5μm), as previously mentioned, are slower to settle and remain infectious for hours.7 Therefore, smaller droplets can transmit the virus over larger distances than larger droplets. The amount of particles is also an important factor in determining the dispersion of the virus. The previous studies8,9 show that the size ranges of particle diameters generated by coughing and talking are similar (0.5μm<d<1000μm). However, sneezing produces significantly more particles than coughing, and coughing more so than talking.10 Still, significantly more research is required to quantify the distribution of particle sizes during different respiratory events.11 In addition, it should be noted that the amount or concentration of the virus is not the same across all particle sizes. For example, it has been shown that particles smaller than 1μm contain the majority of the influenza virus.12 However, the percentage of virus found at each particle size is still not well documented for the novel SARS-CoV-2 coronavirus, and the available literature does not agree.13,14

The evaporation rate of the different-sized droplets also affects the transmission of an airborne virus. Small particles of less than 1μm in the expiratory flow evaporate fast into aerosols, and these droplet nuclei remain suspended as passive tracers in the expiratory flow.15 The droplet diameter reduces as the droplet evaporates, and therefore, the settling time of the droplet changes.9 Studies have offered various ways of modeling the heat and mass transfer of virus-laden droplets,9,16–20 a critical component to determine the droplet size, settling time, viral concentration, droplet concentration, and viability. However, due to the separations between the time scales of respiratory events O(0.1 s) and the evaporation of the droplets O(1 s), these effects are not considered here, and instead, the droplets are treated as passive tracers.

Experimental studies, on both humans and manikins, show that the mask usage can limit the transmission of various infections to and from the wearer.21–29 However, the effectiveness of the outward protection of masks remains a topic of contention. Outward protection refers to the effectiveness of a face mask when the wearer is infectious. Air leakage around the mask perimeter where it does not make a seal with the face has been observed, therefore reducing the effectiveness of the mask.5,30,31 Perimeter leakage can be significantly impacted by facial dimensions and features.30–33 Proper fitting of N95 respirators has always been stressed for the efficacious filter of all contaminants. However, surgical masks are primarily designed for the outward protection of droplets; therefore, the fit is much looser. Homemade masks made from cotton or similar fabrics are likely to have an even looser fit with more leakage. In a study comparing the effectiveness of different face masks, homemade masks were shown to be half as effective as surgical masks and 50 times less effective than an FFP2 respirator (a European standard medical grade “filtering face piece,” similar to N95 respirators). These effects were even more pronounced among the children subjects, likely due to an inferior fit on their smaller faces.29 

Determining mask effectiveness depends not only on the mask fit but also on the mask material properties. Filtration efficiency (FE) is a widely accepted and used parameter for determining how much of the virus-laden aerosols are captured or blocked by the mask. However, the data on the filtration efficiency of different mask materials vary widely from source to source. Studies34–36 do agree that typical fabric masks have a filtration efficiency of less than 40%. In comparison, surgical masks and respirators can achieve much higher filtration efficiencies of up to 95% for the well-fitted N95 respirators. This number comes with another caveat, and that is, filtration efficiency is dependent on the aerosol particle sizes. For example, the N95 respirator's FE=95% is based on particles of 0.3μm.

Computer simulations have the potential to systematically estimate the location and amount of leakage for different facial structures. Headform models provide a high fidelity representation of the head anatomy based on 3D scanning of different subjects ranging in age, weight, gender, etc.37,38 These models can account for deformation of the face due to contact pressure, which is important to accurately predict the seal made between the perimeter of the mask and the face. The dispersion of virus-laden aerosols into the environment is highly dependent on the gap between the face and mask, which allows flow to leak out. Tang et al. studied the jet created from coughing and the effect of wearing surgical masks or N95 respirators.39 They found that a surgical mask effectively blocks the forward momentum of the cough jet, but the loose fit of the mask allows air leakage around the perimeter of the mask, especially through the top and sides.40 Oestenstad and collaborators used a fluorescent tracer to identify leak location and shape on subjects wearing half-mask respirators.32,33 They tested the effect of gender, race, respirator brand, and facial dimensions. Facial dimensions were found to be highly correlated with leak location, while the other categorical analysis showed weaker correlations. Hui et al. used smoke visualization to look at the flow after a coughing event of a subject wearing no mask, a surgical mask, and an N95 respirator while lying in a hospital bed.41 The N95 respirators proved to be the most effective at stopping the cough jet, while the surgical mask showed significant leakage around the perimeter. Lei and Yang used a headform model combined with a finite element computational fluid dynamics (CFD) model to study the leakage locations of an N95 respirator and found that most leakage appears along the top perimeter of the mask near the nose.38 More recently, Verma et al. experimentally investigated the effect of different mask types by visualizing the respiratory jets and observed leakage through the perimeter of the mask.42 The study reports that both the material and fit have an important impact on the mask's effectiveness, with all masks tested showing leakage from the top of the mask due to poor fitting. Dbouk and Drikakis43 employed multiphase CFD to study the transmission of droplets and the droplet dynamics and determined that mask efficiency drops with consecutive coughing cycles. This was also verified by the experimental studies of Morris et al.44 Arumuru et al.45 provide PIV results of sneezing into a mask which show the peripheral gap leakages and also state that a surgical mask cannot block the sneeze completely. Shah et al.46 also used particle image velocimetry (PIV) to study the exhaled flow through a face mask and discuss the reduction in mask efficiency due to the peripheral gap leakages. All of the studies so far have focused on the difference between an N95 respirator and a surgical mask. No particular focus on the effect of homemade masks and the subject's facial structure has been found.

In earlier studies by the authors,30,31 a headform model was used to predict leakage locations and the gap open area between the mask and the face. The effect of gender, age, weight, and height was studied. Similar to the previous studies, it was observed that facial dimensions had the most significant impact on predicting the leakage. In this continuation, we focus on the flow physics of respiratory events and the effect of mask fitting based on the headform model and results from part Solano et al.30 

We consider a cough jet originating from the mouth of a headform model wearing a face mask as depicted in Fig. 1. The particular face shown in Fig. 1(a) is that of an adult male wearing a mask that follows the CDC guidelines. This face was previously identified as a characteristic face based on the peripheral gap sizes and location by the clustering study.30 The mask is a rectangle cloth mask (9×5.5in.2), and its deployment on the face is modeled using a minimum energy method. The mask consisting of a membrane (stretching and no bending) with a border (stretching and bending) that is initially placed in front of the face with elastic bands (stretching only) wrapped around the ears but with zero tension. The resting length of the band gradually decreases from the initial length to its final value during the initial transient phases. The intermediate quasi-static equilibrium position of the mask is calculated iteratively until the mask rests in the final configuration on the face. The contact between the face and the mask is treated as a soft contact. For a complete description of the mask deployment model, refer to Solano et al.30 

FIG. 1.

(a) The placement of the mask on the face calculated from the mask deployment model.30 (b) Domain center-plane schematic [inset: velocity profile of a cough jet adapted from Gupta et al., Indoor Air 19, 517–525 (2009).47].

FIG. 1.

(a) The placement of the mask on the face calculated from the mask deployment model.30 (b) Domain center-plane schematic [inset: velocity profile of a cough jet adapted from Gupta et al., Indoor Air 19, 517–525 (2009).47].

Close modal

The cough is modeled as a jet with a horizontal velocity U0(t) originating from a circular opening at the mouth with diameter d0=2.2cm as shown in Fig. 1(b). The time evolution of the velocity of a cough has been studied and reported in literature.47–50 The peak velocity of a cough varies in the literature between 5m/s and over 20m/s with an average of around 10 m/s. The cough velocity depends on the opening of the mouth, which can also vary from person to person. Gupta et al. collected data from men and women to determine the flow rate of a single cough as well as the mouth opening.47 The inset in Fig. 1(b) shows the evolution of the cough jet velocity over time adapted from Gupta et al., and the chosen mouth diameter is based on the average male mouth opening.47 

The flow is modeled by numerical simulations of the Navier–Stokes equations of continuity and momentum:

·u=0,
(1)
ut+·(uu)=1ρp+ν2uf,
(2)

where u(x,t) is the velocity field, p(x,t) is the pressure, and f(x,t) is a forcing term added to enforce the effect of a porous mask. Here, x=(x,y,z) is the Cartesian coordinate vector. The sharp interface immersed boundary condition is employed to represent the face.51 The fluid forcing function is written as a diffusive interface force

f(x,t)=SF(s,t)δ(xX(s,t))ds,
(3)

where the position of the membrane, i.e., the face mask, is denoted by X(s,t)=(X(s,t),Y(s,t),Z(s,t)) and s is the mask's reference coordinate in the Lagrangian reference frame. The function F(s,t) represents the immersed boundary force, and δ is the three-dimensional delta function.

The mask is a permeable material that provides resistance to the cough jet and is modeled as a porous, rigid membrane. The Darcy law for porous media is used to model the porosity of the mask. Due to the mask being inherently a thin structure, it is considered a membrane with no transverse porosity. The Darcy relation can be integrated through the thickness of the mask to reach

[p]n=μkv=ckv,
(4)

where v(s,t) is the through mask or slip velocity and [p]=p+p is the pressure jump over the mask. The Lagrangian coordinate s describes the position of the mask points, and n is the vector normal to mask surface. The mask's effect permeability coefficient is denoted by k; however, for simplicity, we adopt the parameter ck=μ/k known as the effective flow resistance coefficient. It is important to note that higher values of ck correspond to less porous masks, i.e., ck is inversely proportional to mask porosity.

The interaction force at time t is calculated by Goldstein's feedback law52 as

F(s,t)=c10t(vUib)dt+c2(vUib),
(5)

where c1=106 and c2=103 are large penalty constants chosen to enforce the immersed boundary condition. The fluid velocity at the mask (Uib) is evaluated at the center of the triangular elements comprising the mask mesh structure using a smoothed delta function

Uib(s,t)=Vu(x,t)δ(xX(s,t)dS.
(6)

The Darcy law, Eq. (4), can be expressed in terms of the interaction force as

[p]=(F·n)n=ckv.
(7)

We can split the interaction force [Eq. (5)] into the normal and tangential components, F=Fn+Fτ, and write the normal component of the force at time step i as

Fni=(F·n)n+Fpi11+(c1dt+c2)ck1,
(8)

where dt is the time step. The new variables F and Fp are defined as

F=c10tUibdtc2Uib,
(9)
Fp=c1ck10tdt(F·n)ndt.
(10)

The coefficients c1 and c2 are chosen to be sufficiently large so that there is no implicit porosity (other than that imposed by the coefficient ck) for the face mask.

The governing equations (1) and (2) are discretized on a Cartesian grid using a finite volume approach.51,53,54 A second-order Crank–Nicolson scheme is used for implicitly advancing the diffusion and advection terms. A finite volume approach with a four-step fractional-step time integration scheme proposed by Choi and Moin55 is used to advance the momentum equation. The conditions at the domain boundaries are Neumman boundary conditions for the pressure and velocities. The cough jet is imposed as a Dirichlet type boundary condition.

The computational domain is 25×30×30cm3 in the x, y, and z directions. Special consideration is given to the grid refinement between the mask and the face. A mesh resolution with dx=0.06cm was chosen to have sufficient resolution in the area between the mouth and the mask, where the jet is generated and impinges on the mask. The maximum Reynolds number of the simulation based on the peak cough jet velocity and mouth opening diameter is Re=U0D/ν2.4×104, in the same range as some experimental studies of similar problems.45,56 In this work, we do not use any explicit model for the subgird-scale fluctuations and instead implicit large eddy simulations are employed.57 The time step used for the simulations depends on the porosity ck. With higher porosity (i.e., lower ck), the interaction force f is lower, and therefore, the constraint on the time step can be relaxed. The time step and final domain and mesh sizes are selected from the convergence study. The smallest time step used was dt=2.5×105s for ck=2×104Pas/m, and the largest time step was dt=5×105s.

Our primary interest is to quantify the fitted face masks effectiveness given different Darcy coefficients (ck) and filtration efficiencies (FEs). The flow rates through the mask and from the perimeter gaps cannot be directly used to compute the mask's effectiveness since the through-mask flow is filtered and the leaked flow is unfiltered. The extent of the filtration depends on the mask's filtration efficiency (FE), which is still an active research topic. The effective FE for common fabric face mask (cotton, polyesters, and blends) is reported to be less than 40%.34–36,58 To account for FE in our characterization of mask effectiveness, we define the unfiltered volume flow rate as

ρ=qm(1FE)+qg,
(11)

where qm is the through-mask flow rate and qg is the total sum of all the peripheral gap-leak flow rates.

We also define the fitted filtration efficiency (FFE) for convenience as

FFE=1ρqm+qg.
(12)

The fitted filtration efficiency is a standard way of comparing the mask's nominal effectiveness (FE) to the real effectiveness of the mask while it is deployed on a face. As an example, an N95 respirator has a nominal FE=95%; however, an incorrectly sized N95 respirator can reduce FFE.59 

As previously discussed in Sec. I B, the filtration efficiency is dependent on particle size and mask material. The smaller particles with dp<5μm are especially important since these aerosols can reach deep into the alveolar region of the lungs, increasing the likelihood of transmission,60 and are also more likely to leak out unfiltered. It is also generally believed that escaped droplets >5μm settle within 6 feet from the person who produces them due to gravity.1,10Figure 2 shows FE for various fabric face masks as a function of particle diameter dp, as reported in the literature.35,36,58 While there is some variation in the data, the general trend is consistent between different studies. For clarity, in our analysis, we will use a second-order rational polynomial curve fit to represent these data as shown in Fig. 2. From Fig. 2, we see that the most common fabric face masks can filter 100% of particles larger than 5μm. We note that the viral load of the SARS-CoV-2 coronavirus, i.e., RNAcopies/ml, is not necessarily constant across the range of size of particles produced during a cough.13,14 However, there is not sufficient data available on this subject to take into account in our calculations.

FIG. 2.

Filtration efficiency of different fabric masks as a function of particle size. A second-order rational polynomial is fitted to represent the mean of the data for each particle size.

FIG. 2.

Filtration efficiency of different fabric masks as a function of particle size. A second-order rational polynomial is fitted to represent the mean of the data for each particle size.

Close modal

The time evolution of a cough while wearing a face mask is illustrated in Fig. 3 (Multimedia view) by the iso-surface contours of the velocity magnitude. Initially, there is a flow that slips through mask, as time progresses, two leakage jets begin to form at the top of the mask. The through-mask flow starts as a uniform stream at the location where the cough jet impinges on the mask. While the through-mask flow is concentrated at the impingement location, i.e., the mouth, there is also some leaked flow in the region between the nose and the chin. Note that at τ=t/tp=1.67, a secondary stream forms closer to the chin. As the cough jet impinges on the inside of the mask, flow spreads radially away from the original site of impingement. The flow continues to develop as a channel-like flow between the face and the mask toward the mask edges. At the chin, the mask comes in contact with the face creating another impingement site and initiating a secondary stream through the mask. This secondary stream that develops later on, while not directly discussed, is also seen in the experimental results of Verma et al.42 for the less porous face masks they tested. As the cough velocity increases, so does the flow through the mask until the cough velocity reaches its maximum when τ = 1. For τ>1, the through-mask flow begins to reduce as the pressure inside drops due to decreasing cough velocity. However, the gap flow is not as significantly affected by the initial pressure drop as is evident by the still growing plumes at the top of the mask.

FIG. 3.

Time evolution of the flow during a coughing episode for a male face with a porous masks (ck=33.3Pas/m). Front and side views of the iso-surface contours of velocity magnitude over time τ nondimensionalized by the peak coughing velocity time. Multimedia view: https://doi.org/10.1063/5.0086320.1

FIG. 3.

Time evolution of the flow during a coughing episode for a male face with a porous masks (ck=33.3Pas/m). Front and side views of the iso-surface contours of velocity magnitude over time τ nondimensionalized by the peak coughing velocity time. Multimedia view: https://doi.org/10.1063/5.0086320.1

Close modal

The flow can be described by three leading “streams”: through-mask stream and two plumes at the top of the mask and on either side of the nose. The two plumes at the top of the mask are not symmetrical. In the present case, the plume on the right side of the face (the reader's left) is larger than the plume on the left side. This asymmetry is due to a slight asymmetry in the facial structure itself. The asymmetry is amplified after the mask deployment, which results in a tighter fit on the left side of the face than on the right side. However, the facial and mask fit asymmetry is almost imperceivable to the eye, but it is made obvious by the flow. For this particular case, the only unfiltered leakage observed is through the top as previously mentioned. However, for different face shapes, leakage through the bottom and sides of the mask is also possible.30 From the author's previous mask deployment study,30 this face and mask combination was one of the best fitting face masks. The gaps around the sides and bottom of the face are very small, and from a quantitative analysis of the potential flow leakage, it was deemed one of the more effective mask fittings. However, as shown in Fig. 3, the unfiltered leakage flow is dispersed straight upwards where environmental cross-flows can potentially further aid in the dispersion of the aerosols.

The time history of the volume flow rates (q) is illustrated in Fig. 4(a). An initial peak is observed in the through-mask flow (qm) at τp0.4. The end of the peak coincides with the emergence of the gap leakage flow as shown in Fig. 3. It effectively captures the capacitance effect of the space between the mask and the face at the beginning of the cough. The through-mask flow rate increases to its maximum at τ = 1, which coincides with the maximum cough velocity and then decreases. The gap flow rate (qg) slowly ramps up between 0<τ<0.5 and then continues to steadily increase up to τ = 2.

FIG. 4.

(a) Volumetric flow rate through the mask (qm; blue square symbols) and through the peripheral gaps (qg; red cross symbols). Also, the total volumetric flow rate (qm+qg)equals the cough flow rate. (b) Unfiltered volume flow rate for different mask filtration efficiencies FE=10%,30%,50%. All quantities shown over the nondimensional time τ for a male face with mask porosity mask ck=33.3Pas/m.

FIG. 4.

(a) Volumetric flow rate through the mask (qm; blue square symbols) and through the peripheral gaps (qg; red cross symbols). Also, the total volumetric flow rate (qm+qg)equals the cough flow rate. (b) Unfiltered volume flow rate for different mask filtration efficiencies FE=10%,30%,50%. All quantities shown over the nondimensional time τ for a male face with mask porosity mask ck=33.3Pas/m.

Close modal

The unfiltered flow is shown in Fig. 4(b) by ρ, defined by Eq. (11), for different mask filtration efficiencies. The curves for the different FE illustrate ρ for the typical range of FE of common fabric face masks. At τ = 1 (i.e., time of maximum coughing velocity), there is the highest potential for dispersion/transmission of virus-laden aerosols (highest unfiltered flow leaked, ρ). It is also at τ = 1 where we see the largest difference in unfiltered flow for the different values of FE since it is when the difference between the through-mask flow and gap flows is the largest. For the remaining discussion, we will focus on the worst-case scenario, maximum viral dispersion, which occurs at τ = 1.

The flow at τ = 1, for different porosities, is shown in Fig. 5 (Multimedia view) as iso-surface contours of the velocity magnitude. The mask porosity increases from left to right, as shown by the arrow. It is immediately evident that less porous masks result in more flow leakage through the gaps (compare ck=2×105Pas/m with ck=102Pas/m). This trend was also observed in experiments.42,44,46 For the first case shown ck=2×105Pas/m (leftmost in Fig. 5), there is virtually no through-mask flow. All of the cough flow is redirected out through the peripheral gaps. The flow can be observed to be leaking through the mask and the top, sides, and bottom of the mask. This is unlike the case explored in Sec. III A, in which we only observed leakage through the mask and top of the mask. Although the mask leaves very small gaps at the sides and bottom, the high pressure caused by an effectively rigid wall results in leakage through all gap openings. This type of leakage has been shown for very high filtering efficiency masks such as the N95 respirator. This is similar to multiple PIV and Schlieren studies41,42,45,46 of a fitted N95 respirator showing that although the through-mask flow is significantly reduced, peripheral gap leakage still occurs especially through the top edge of the mask.41,42 This is especially true for looser fitting masks such as surgical masks and raises the question: If gaps are unavoidable, should face masks be designed to direct the leakage in the least dangerous direction?

FIG. 5.

Time evolution of the flow during a coughing episode for a male face with a porous masks with different porosity levels. The figure shows the front views of the iso-surface contours of velocity magnitude at τ = 1. Multimedia view: https://doi.org/10.1063/5.0086320.2

FIG. 5.

Time evolution of the flow during a coughing episode for a male face with a porous masks with different porosity levels. The figure shows the front views of the iso-surface contours of velocity magnitude at τ = 1. Multimedia view: https://doi.org/10.1063/5.0086320.2

Close modal

The previous section discussed the secondary flow stream through the mask caused by the impingement of the internal flow at the chin. Here, we see this secondary stream is more pronounced for the cases of ck=104 and ck=103Pas/m. As the mask porosity increases, the pressure gradient across the mask and the velocity of the flow impinging at the chin decreases; consequently, the secondary chin stream is reduced. Although the pressure gradient decreases, the mask's pressure becomes more uniform than localized at the impingement sites. Therefore, the secondary stream reduction is accompanied by more uniform and distributed flow through the mask and an overall increase in the through-mask flow qm. This observation is in agreement with the experimental visualization results of Verma et al.42 in which only the less porous masks exhibited the secondary stream near the chin and the most porous face mask they tested showed a more spatially uniform through-mask flow.

Figure 6 shows the maximum volume flow rates through the mask and peripheral gaps for all the different mask porosities tested (33.3Pas/m<ck<2×104Pas/m). As expected, we see that as the mask's porosity increases (decreasing ck), qm increases while qg decreases. The system changes from gap-dominated leaks to mask-dominated leaks at ck500Pas/m, as seen by the inversion of the flow rate curves. The maximum unfiltered flow rate ρmax is also shown in Fig. 6 for three different effective FEs. The chosen effective FE corresponds to the lowest (FE=1.3%), highest (FE=15.5%), and average (FE=34%) effective overall face mask FE of the 38 fabric masks tested by Zangmeister et al.36 The trade-off between the through-mask flow and through-gap flow, as ck changes, results in an apparent optimal ck for which the unfiltered flow rate ρmax is lowest, i.e., most effective not perfectly fitted face mask. For example, the lowest unfiltered flow rate for a face mask with FE=34% is seen at ck1=102m/Pas. One could say that face masks with higher FE benefit from higher mask porosities. However, the dependence or coupling between ck and FE is not well characterized and cannot be ignored. What is true is that the optimal value for ck shifts to more porous masks as FE increases. That is, the optimal ck is lower for masks with higher FE, and the optimal ck is higher for masks with lower FE (ck,optFE1).

FIG. 6.

Left axis: The effect of mask porosity on the maximum volume flow rate through the mask (+) and through the peripheral gaps (□). Right axis: the unfiltered flow rate ρ for different filtration efficiencies (FE).

FIG. 6.

Left axis: The effect of mask porosity on the maximum volume flow rate through the mask (+) and through the peripheral gaps (□). Right axis: the unfiltered flow rate ρ for different filtration efficiencies (FE).

Close modal

The relation between ck and FE is not clear due to the complex nature of FE, which depends on porosity, weave pattern, thread/fiber size, and the electro-static properties of the material, among other things. Recently, ck and effective FE for different fabric face masks, surgical masks, and respirators have been measured.36,61,62 The data collected by these authors can be used to determine that there is effectively no clear relationship between porosity and filtration efficiency, except for a slight inverse relation in fabric masks. The maximum fitted filtration efficiency (FFE) [Eq. (12)] is shown in Fig. 7 as a function of ck and FE. From the data collected by Zangmeister et al., Duncan et al., and Konda et al., we can visualize where the different mask materials would lie on the FFE contour map shown by the regions shown in the dashed and dotted curves in Fig. 7. It is important to note that we refer to the mask material and not the mask itself since the mask design simulated here is not the same. The purpose of overlaying the different mask material regions in Fig. 7 is to illustrate the overall effectiveness of the generic face mask design modeled here if it were to be made of these different materials. We see that most face masks have a porosity of 104m/Pas<ck1<102m/Pas. In this range of ck, the unfiltered flow through the peripheral gaps is anywhere between 40% and 80% of the total cough. Note that N95/KF95 respirators only reach their capacities of filtering 95% of particles (dp>0.3μm) if fitted properly, i.e., fitted tightly on the face such that the peripheral gaps are sealed. However, here, the effective filtration efficiency, FFE30%, is significantly smaller than the nominal FE. The drastic reduction in filtration is primarily due to the large gaps around the nose, allowing unfiltered flow to leak out. The mask design deployed on the face in the present study is clearly not an N95 respirator; its fit on the face is more similar to a surgical or fabric face mask. And although the mask fits snugly on this face leaving very small or no gaps on the sides and chin, the larger gaps near the nose allow much flow to leak out unfiltered, thereby reducing the FFE. This is true for all mask materials and shows that the fit of a face mask is equally if not more important than the FE.

FIG. 7.

The fitted filtration efficiency (FFE) for aerosols (<5μm) as a function of porosity and nominal mask filtration efficiency. Symbols represent different face masks materials.

FIG. 7.

The fitted filtration efficiency (FFE) for aerosols (<5μm) as a function of porosity and nominal mask filtration efficiency. Symbols represent different face masks materials.

Close modal

All fabric face masks have an effective FE<40%; however, the FFE is less than half the nominal FE. The FFE for surgical masks shown in Fig. 7 is comparable to the experimental results in literature46,63 (FFE=12%36%). Shah et al.46 reported reduced FFE for surgical masks and N95 respirators of 12% and 46%, respectively. Note that the FFE they reported for N95 respirators is equally as low as that presented here because they created artificial gaps (3 mm) to account for a “loose-fitting” N95 respirator similar to what is studied here. And while adding another mask can increase the FE, it should be noted that double masking may not really increase the wearers' protection significantly. Double masking usually refers to wearing a two masks on top of each other, cloth on top of cloth or cloth on top of surgical, and has become a more and more popular recommendation in lieu of more effective respirators if those aren't available. While the FE of multi-layered masks is linear with respect to the number of layers, the porosity is inversely proportional to the number of layers (more layers, less porous). An average surgical mask with FE=45% offers an FFE=20%. Adding a fabric mask of equal porosity and FE=10% would see the combination move up and to the left of the surgical mask in Fig. 7 resulting in negligible, if any, increase in protection. This could, in turn, give the wearer a false sense of security when choosing to double mask. The effects of double masking need to be further explored to accurately quantify its benefit. However, there is evidence that double masking does not provide a significant enhancement in protection without proper mask fit.63 

So far, we have discussed the overall effectiveness of face masks based on a nominal or average FE. However, as discussed in Sec. II D, the face mask FE is not the same across all particle sizes (dp). Figure 8 shows the FFE as a function of ck and dp where the relation between FE and dp is formed based on the fitted rational polynomial curve described in Sec. II D. The FE(dp) is concave with the lowest FE observed for particles with 200nm<dp<400nm. The effect of this is also apparent in Fig. 8 where the FFE is lowest for the same range of dp. Recall that the porosity observed for real face mask was 104m/Pas<ck1<102m/Pas. In this porosity range, less than 50% of particles are filtered out, even for those with the largest dp. The number of smaller (dp500nm) unfiltered particles is a combination of through-mask leakage and peripheral gap leakage. The majority of the large (dp>5×103nm) particles that remain unfiltered are due to the leakage through the peripheral gaps.

FIG. 8.

The fitted filtration efficiency (FFE) as a function of porosity and particle size.

FIG. 8.

The fitted filtration efficiency (FFE) as a function of porosity and particle size.

Close modal

It should be noted that we are calculating the fitted filtration efficiency of aerosols; hence, it should correctly be referred to as the “FFE of (nanoscale) aerosols.” Larger particles have inertia and might, upon impinging on the mask, be filtered from the internal flow entirely before leaking through the peripheral gaps. Dbouk and Drikakis43 modeled the transmission of droplets during a mild coughing incident in which the droplet size distribution generated by the cough approximately followed a normal Gaussian distribution centered at dp70μm. The study showed that a significant number of droplets stick to the mask. We have already discussed the fact that most face masks have a FE=100% for droplets with dp>5μm. The distribution of particles used by Dbouk et al. is not very different from the distribution of particles generated during a coughing episode reported by other researchers,9 which show that the majority of particles are larger than 5μm. These studies show that the real overall filtration efficiency of a mask will be higher than the filtration efficiency of aerosols reported in this and other studies. Despite this, we follow the norm to report the FFE of aerosols since small leaked aerosol particles can be transmitted through the environmental airflow and induce disease transmission over a longer distance.

At the start of the pandemic, it was debated that the primary driver of transmission was droplets (dp>5μm), particles that (a) settle to the ground faster and (b) can be filtered by common fabric face masks (assuming no peripheral gaps). Recently, scientists have shown that aerosols (dp<5μm) are also a significant transmission source. Aerosols are arguably more dangerous since they linger in the air longer, can be transmitted in the flow over larger distances, and are more easily inhaled by susceptible people. While it is unclear how much of the virus is contained in the small aerosol particles vs the larger droplets, from Fig. 8, we see that most aerosols will go unfiltered by fabric face masks.

Here, we used an immersed-boundary and diffuse interface model for porous membranes to study the flow physics of face masks. We focused on the fluid dynamics behind a coughing person wearing a generic face mask. The face is not a generic manikin face; instead, it is a 3D headform model constructed using a systematic principle component analysis (PCA) method, from 100 male and 100 female subjects.37 The mask deployment on the face was modeled using a minimum energy method to ensure an accurate and realistic fit.30 A cough jet was generated at the mouth with a peak velocity of 10 m/s, and the flow distribution for different mask porosities was studied.

We found that the flow through the mask increases with an increase in mask porosity. On the other hand, the gap leakage is inversely proportional to the mask's porosity. The through-mask leakage behaves very differently for lower mask porosity (higher ck) than for higher mask porosity (lower ck). The leakage from the low porosity face mask is characterized by a primary uniform stream where the cough jet initially impinges on the mask and a secondary stream at the chin. The secondary stream is due to the increased pressure where the mask and the face come in contact at the chin. The through-mask flow rate increases as the mask porosity increases. Due to the increase in mask porosity, the pressure gradient across the mask not only drops but also becomes more uniform over the surface of the mask leading to a more uniform and distributed flow through the face mask.

The nominal FE of a face mask can be significantly higher than the practical FFE if there is considerable unfiltered leakage. This is primarily due to the gaps between the face and the mask. Note that here the FFE is specifically the fitted filtration efficiency of (0<dp<5μm) aerosols, which is of interest when studying airborne transmission, but does not account for the larger droplets that would stick to the inside of the mask increasing the overall effectiveness of the mask at blocking virus-laden particles. We can conclude that the effectiveness of face masks against airborne transmission is significantly reduced (as low as FFE = 5%) with loose fitting masks, even N95 respirators can have their effectiveness reduced by half. We also showed that double masking might not significantly improve the mask's outward protection. While it can theoretically increase the overall FE, the added layer(s) also decrease the porosity. In loose-fitting face masks, the increase of porosity, albeit with an accompanying larger FE, will have an adverse effect of higher unfiltered flow through the peripheral gaps.

In the case presented in this study, the mask fits well on the male's face leaving little to no gaps around the sides and bottom of the mask. However, large gaps around the nose (something that can be mitigated with a bending metal clip) resulted in an avenue for unfiltered flow leakage. Previous studies have shown the reduction in the effectiveness (FFE) of N95 respirators if they are fitted incorrectly; here, we verify that a good fit on the face, that is, minimizing the gaps, is imperative for an effective face mask. Moreover, if gaps are unavoidable, should we consider mask designs that intentionally redirect the unfiltered flow in a more benign direction? This would depend on the environmental flow, which is characterized by the type of ventilation system used. The main ventilation design categories are (1) mixed ventilation39,64 where the inflow of air is through a vent close to the ceiling of the room pushing air out at high velocities, in turn, promoting mixing of the air in the room; (2) displacement ventilation39,64 in which the inflow of air is provided at the floor level at lower velocities and the air in the room is divided into the “polluted” air at the top and “clean” air at the bottom part of the room. In a displacement ventilation system, redirecting the unfiltered leakage flow upwards would be beneficial to essentially remove the virus-laden aerosol from the room.39,64 In a mixed ventilation system, on the other hand, redirecting the unfiltered flow upwards enhances mixing and dispersion of the aerosols through the room, especially in the breathing region. One possible solution would be to redirect the unfiltered flow downward and toward the chest of the mask wearer to increase the probability of the aerosols sticking to the outer surface of the subject and not dispersed into the environment. However, the problem can be alleviated largely by better mask designs such that the gaps are reduced as much as possible, and for the unavoidable gaps, the leaked flow is minimal.

We wish to acknowledge the support of our collaborators Dr. Kenneth Breuer and his group for their suggestions, and providing the initial range of Darcy coefficients from their experiments used in this study. This study was made possible due to the computational resources provided by NSF's XSEDE systems through the Award No. TG-MCH220004. This work was supported by National Science Foundation Grant Nos. CBET-2034992 and CEBT-2034983.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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