Representation of effective facial mask design as a novel dimensionless number—Implications for COVID 19 pandemic

Abstract

A dimensionless number Nr for the effective design of facial masks was derived and compared with other dimensionless numbers of fluid mechanics. Nr is found to closely resemble Euler’s number (Eu). Nr is equal to the logarithmic function of the ratio of inertial force (Fi) of the air to the pressure force (Fp) of the air through the porous membrane of the mask. Nr is then introduced as a novel dimensionless number (Habib number) Ha in which the air flow through a facial mask is derived with parameters for an effective barrier from the COVID-19 virus (SARS COV 2). The introduction of Ha was not only for a comparison reason with other dimensionless numbers of fluid mechanics but also the number Ha is an essential extension of an early work on “Fluid mechanics of facial masks as personal protection equipment (PPE) of COVID-19 virus” [Rev. Sci. Instrum. 92, 074101 (2021)], in which the air flow through the mask is in its optimum design conditions to shield against the COVID-19 virus. As a result, an optimum Nr of expressing the flow of the O2 and N2 gases through the porous membrane was determined (Nr = NO2 = NN2 = Ha = −4.4). This was obtained when the N95 mask with specifications of a = 20 nm, l = 30 µm, and ε = 30% was used, with respect to the pressure gradient of the human lungs during exhaled and inhaled conditions, PAverage = 20 mm Hg (g cm−1 S−2), and to the size of the COVID-19 virus of about 125 nm (0.125 µm). In addition, a range of values of Nr was analyzed with respect to the optimum (Ha) value of Nr. On the one hand, when the range value of Nr falls between 0 ≥ Nr ≥ Ha, the mask has less resistance than Nr < −4.4, but not necessarily its optimum design conditions. On the other hand, when Nr = 0, the flow through the mask has no resistance at all, as if the mask does not exist.