Quantifying the effect of a mask on expiratory flows

Face masks are used widely to mitigate the spread of infectious diseases. While their main purpose is to filter pathogenic droplets, masks also represent a porous barrier to exhaled and inhaled air flow. In this study, we characterize the aerodynamic effect of the presence of a mask by tracking the air exhaled by a person through a mask, using both infrared imaging and particle image velocimetry performed on illuminated fog droplets surrounding a subject. We show how a mask confines the exhaled flows within tens of centimeters in front of a person breathing or speaking. In addition, we show that the tissue of common surgical face masks has a low permeability, which efficiently transforms the jetlike flows of exhalation produced during breathing or speaking into quasivertical buoyancy-driven flows. Therefore, wearing a mask offers a strong mitigation of direct transport of infectious material in addition to providing a filtering function. By comparing results on human subjects and model experiments, we propose a model to rationalize how a mask changes the air flow, and thus we provide quantitative insights that are useful for descriptions of disease transmission.

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Figure 1

Typical air flows during four distinct exhalations without and with a face mask. (a) Visualization of the exhaled ${\mathrm{CO}}_2$ without wearing a mask (false colors). From left to right: (i, blue) soft respiration from the nose, (ii, red) mild respiration with an open mouth, (iii, green) heavy respiration, and (iv, orange) blowing. The scale bar represents 10 cm. (b) Visualization of the exhaled ${\mathrm{CO}}_2$ for the four kinds of exhalations (i)–(iv) when the subject wears a mask (false colors). While exhaled flow is contained within 1 cm from the mask in the case of soft respirations (i) and (ii), it reaches distances of the order of 10 cm for heavy respiration (iii) and (iv) as ${\mathrm{CO}}_2$ jets emanate from the mask. The scale bar represents 10 cm. (c) Measurements of the expiratory flow-rate $Q(t$) during those four kinds of respirations. Depending on the kind of respiration, the expiratory flow-rate $Q$ reaches a maximum at approximately 15 L/min (i, blue data), 40 L/min (ii, red), or 200 L/min (iii–iv, green and orange). The horizontal dashed line delimits exhalations ($Q>0$) and inhalations ($Q<0$).

Figure 2

Mask wearing induces localized buoyant flows during respiration. (a)–(c) Snapshots of normal breathing, 1, 2, and 3 s after exhalation. The exhaled air is detected as a dark cloud in a bright fog illuminated by a laser sheet. The dashed line is a guide for the eyes delimiting the approximate area affected by the ascending exhaled air. (d)–(f) Measurements of the velocity field of (a)-(c). The black arrows (and the color map) show the velocity direction and the magnitude of the flow, according to the PIV analysis. See the movie and the processed PIV results in supplementary movie S7.

Figure 3

Trajectory of a controlled jet of cold air. (a) Sketch of the experiment: a constant flow rate $Q$ of cold air (at the temperature $T^{-}$) is imposed for 3 s from a pipe of diameter $2a$ leading to an initial velocity $V_0$. The pipe could be open or covered by a 3-ply mask (blue layers) sealed over the end of the tube, as seen in Fig. S7(a). Depending on the imposed flow rate $Q$, the trajectory of the jet is linear ($y=0$, light purple) or quadratic ($y=\kappa x^2$, in red). (b) Trajectories $y\left(x\right)$ of cold air jets emanating from a pipe covered by a mask as a function of the flow rate $Q$ ($2\le Q\le 80$ L/min). The solid lines represent fits of the trajectories approximating the flow, leading to a quadratic shape $y=\kappa x^2$ for low flow rates ($Q\le 20$ L/min) and straight trajectories ($y=0$) for $Q=50$ and 80 L/min. (c) Values of the radius of curvature ${\kappa }^{-1}$ of the jet as a function of the flow rate $Q$ with (filled squares) and without (open circles) a mask covering the pipe. The lines represent the equation ${\kappa }^{-1}=\alpha Q^2$ obtained after fitting the data without a mask (dashed line) and with a mask (solid line). The parameter $\alpha$ decreases by an order of magnitude when the pipe is covered by the mask. (d) Values of the radius of curvature ${\kappa }^{-1}$ of the jet as a function of the measured initial velocity of the jet $V_0$ with (filled squares) and without (open circles) a mask covering the pipe. All data collapse on the solid line of equation ${\kappa }^{-1}=\beta {V_0}^2$.

Figure 4

Permeability of a 3-ply mask. The pressure drop $\mathrm{\Delta }P$ across a 3-ply mask varies with the flow-rate $Q$ following the Forchheimer equation (solid line). At low flow-rates, $\mathrm{\Delta }P$ increases linearly with $Q$ (dashed line) whereas a quadratic response (dotted line) is noticeable for increasing flow-rates. Standard deviations are within 5% of the averaged values represented by the red circles. The inset represents the sketch of the experiment. The flow-rate $Q$ is imposed through the 3-ply mask (blue layers) covering a pipe of radius $a$, and the upstream pressure $P_0+\mathrm{\Delta }P$ is measured.

Figure 5

Extent of the exhaled air breathed out through a mask. (a) Trajectories $y\left(x\right)$ of the front of the air jets exhaled without (open symbols) and with a mask (filled symbols). The trajectories are detected for four kinds of respirations: soft breathing from the nose (blue circles), soft breathing with an open mouth (red triangles), heavy breathing (green squares), and blowing (orange downward triangles). (b) Horizontal distance $x$ reached by the front of the exhaled air cloud as a function of time $t$ for the four kinds of respirations with and without the mask. Masked exhaled flows (filled symbols) are slower than an equivalent unmasked flow (open symbols) and reach a saturation $x_s$ as shown by the solid line. (c) Horizontal projection of the initial velocity with a mask (filled symbols) and without a mask (open symbols) for the four kinds of respirations. The mask leads to a substantial decrease of the initial velocity. (d) Radius of curvature ${\kappa }^{-1}$ for masked flows emanating from a human (colored filled symbols) as a function of the inertial-buoyant distance $\frac{\rho V_{0,x}^2}{\mathrm{\Delta }\rho g}$. The data on human breathing are in relatively good agreement with the measurement with the radius of curvature ${\kappa }^{-1}$ on the model experiment reported in Fig. 3. The solid line represents a fit of the data obtained with the model experiment (black symbols) of equation ${\kappa }^{-1}\approx \frac{1}{2}\frac{\rho V_{0,x}^2}{\mathrm{\Delta }\rho g}$, lower than the expected theoretical scaling law by a factor $\approx 4$. The dashed vertical lines represent the inertial-buoyant distance for unmasked respirations from the measured values of $V_{0,x}$ in Fig. 5. The expected gain in terms of radius of curvature is around two orders of magnitude.