Methods

The surgical mask creates additional resistance to airflow compared to only an N95 FFR. As a porous medium, the ease of movement of the fluid through the N95 can be modeled as permeability “k” in Darcy’s law. ^{Reference Whitaker5} Darcy’s law states that the flow rate through the porous medium is proportional to the permeability and the pressure drop across this medium:

(1) $$Q = {{ - kA} \over {\mu L}}\Delta p\comma $$

where Q is the volumetric flow rate (analogous to minute ventilation), A is the cross-sectional area, µ is fluid viscosity, L is the length of the porous medium, and Δp is the pressure drop. ^{Reference Whitaker5,Reference Nishiyama and Yokoyama6} This law states that for the same amount of pressure drop, the flow rate permeating the porous medium decreases as the length of the medium increases. According to Eq. (1), if a surgical mask is being used to cover a N95 FFR, the length of the porous medium (ie, thickness of both masks) is increased and therefore a lower air flow rate penetrates through the masks for the same pressure drop across the masks. To maintain a normal minute ventilation, the breathing pressure (pressure drop, Δp) must therefore increase as the resistance increases.

Considering Eq. (1), if Q_B is the normal volumetric flow rate passing through a mask, then the pressure drop across this mask is defined as follows:

(2) $$\Delta p = {{ - {Q_B}\mu } \over A}{L \over k} = - {Q_B}\mu \ R$$

where R is defined as flow resistance ( $${L \over {kA}}$$ ), analogous to a resistor in conduction of electricity. ^{Reference Hagen7}

For multiple masks (ie, a surgical mask over a N95 FFR), the total pressure drop across the masks is equal to the sum of the pressure drop across each mask individually.

Therefore, the total pressure drop can be defined as follows:

(3) $$\Delta {p_{tot}} = - {Q_B}\mu \ {R_{eq}} = - {Q_B}\mu \ \sum\nolimits_{i \\comma = 1}^2 {{R_i} = - {Q_B}\mu } \\comma ({R_1} + {R_2})$$

where R _eq is the equivalent flow resistance of the 2 masks and R ₁ and R ₂ are the flow resistances for the N95 FFR and the surgical mask, respectively. Equivalent flow resistance is calculated by the summation of resistances, similar to resistors in series in an electrical circuit. ^{Reference Hagen7} From Eq. (3), because the breathing flow rate is unchanged and the flow resistance has increased by the amount of R ₂, the total pressure drop across the 2 masks increases as compared to only using an N95 FFR. This additional resistance from the addition of the surgical mask, in turn, creates increased breathing pressures within the mask and airway relative to atmospheric (room air) pressure when the user maintains normal minute ventilation.

Thus, the respiratory cycle pressures will necessarily be more negative on inspiration and more positive on expiration to overcome the increased resistance of the combined masks in an attempt to maintain normal airflow or minute ventilation. As the pressure drop increases across the masks, the same atmospheric-to-airway pressure drop applies to the N95 FFR edge-to-face seal. As a result, as higher pressure differentials pulsate across a pliable mechanical seal, such as the N95 FFR edge-to-face seal, leakage can incrementally occur.

When the face and N95 FFR edge meet to form a seal, any separation between the mating surfaces increases leakage substantially. Doubling seal-surface separation can increase leakage by a factor of 8. ^{Reference Lorenz and Persson8} This is shown by approximating the critical constriction at a seal interface as a pore with a rectangular cross section with a long width and relatively small height. Assuming incompressible Newtonian fluid, and u ₁ as the average height separating the mating surfaces, the volume flow per unit time, Q_l (leakage flow rate), through the critical constriction is given by the following equation (Poiseuille flow):

(4) $${Q_l} \propto {{u_1^3} \over \eta }\Delta p$$

where η is the fluid viscosity, and Δp is the pressure differential across the masks as shown in Eq. (3). ^{Reference Lorenz and Persson8}