Numerical Modeling of Face Shield Protection against a Sneeze

mathematics Article Numerical Modeling of Face Shield Protection against a Sneeze Ainara Ugarte-Anero 1, Unai Fernandez-Gamiz 1,* , Iñigo Aramendia 1, Ekaitz Zulueta 2 and Jose Manuel Lopez-Guede 2   Citation: Ugarte-Anero, A.; Fernandez-Gamiz, U.; Aramendia, I.; Zulueta, E.; Lopez-Guede, J.M. Numerical Modeling of Face Shield Protection against a Sneeze. Mathematics 2021,9, 1582. https:// doi.org/10.3390/math9131582 Academic Editor: Michael Booty Received: 6 June 2021 Accepted: 1 July 2021 Published: 5 July 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1Nuclear Engineering and Fluid Mechanics Department, University of the Basque Country, UPV/EHU, Nieves Cano 12, Vitoria-Gasteiz, 01006 Araba, Spain; [email protected] (A.U.-A.); [email protected] (I.A.) 2System Engineering and Automation Control Department, University of the Basque Country, UPV/EHU, Nieves Cano 12, Vitoria-Gasteiz, 01006 Araba, Spain; [email protected] (E.Z.); [email protected] (J.M.L.-G.) *Correspondence: [email protected] Abstract: The protection provided by wearing masks has been a guideline worldwide to prevent the risk of COVID-19 infection. The current work presents an investigation that analyzes the effectiveness of face shields as personal protective equipment. To that end, a multiphase computational fluid dynamic study based on Eulerian–Lagrangian techniques was defined to simulate the spread of the droplets produced by a sneeze. Different scenarios were evaluated where the relative humidity, ambient temperature, evaporation, mass transfer, break up, and turbulent dispersion were taken into account. The saliva that the human body generates was modeled as a saline solution of 8.8 g per 100 mL. In addition, the influence of the wind speed was studied with a soft breeze of 7 km/h and a moderate wind of 14 km/h. The results indicate that the face shield does not provide accurate protection, because only the person who is sneezed on is protected. Moreover, with a wind of 14 km/h, none of the droplets exhaled into the environment hit the face shield, instead, they were deposited onto the neck and face of the wearer. In the presence of an airflow, the droplets exhaled into the environment exceeded the safe distance marked by the WHO. Relative humidity and ambient temperature play an important role in the lifetime of the droplets. Keywords: COVID-19 protection; face shield; sneeze; droplet evaporation; relative humidity; environment temperature; computational fluid dynamics (CFD) 1. Introduction Currently, we are experiencing a pandemic caused by the new coronavirus SARS-CoV2, which causes the disease known as COVID-19. The WHO defines the term pandemic as “the worldwide spread of a new disease”. The expansion and spread of COVID-19 have changed our lifestyle habits. The strategies for combating such a phenomenon are rather limited, and prevention is the best way to control and to reduce the transmission of COVID-19. This is not the first time the world has had to deal with a global situation due to the spread of an infectious disease. During human history there have been several global pandemics such as the Black death, the Spanish flu, and smallpox, among others. According to Saunders-Hastings et al. [ 1 ], advances in the medical sector change when it comes to dealing with a pandemic. Verma et al. [ 2 ] showed that social distance and hand washing are essential to combat COVID-19, all accompanied by the use of masks. When coughing, sneezing, or even talking, dozens of droplets are exhaled into the environment with a high chance of infecting another human being. These droplets are defined as aerosols, which may be in a liquid or gaseous state. The study of Zhu et al. [ 3 ] noted that 6.7 mg of saliva is expelled with each individual sneeze. The evaporation of these particles causes the reduction of the particle diameter, and this phenomenon depends on the relative humidity and the environmental temperature and flows, according to Wang et al. [ 4 ]. Redow et al. [ 5 ] observed that when the droplets Mathematics 2021,9, 1582. https://doi.org/10.3390/math9131582 https://www.mdpi.com/journal/mathematics Mathematics 2021,9, 1582 2 of 15 leave our mouths their temperature immediately drops, acquiring the temperature of the environment. The study developed by Li et al. [ 6 ] showed that once exhaled droplets evaporate and their diameter decreases, there is an increase in the concentration of pathogens per unit volume, leading to an increase in the risk of infection. Wei et al. [ 7 ] concluded that in particles below 50 µ m, evaporation is most significant. The research carried out by Mowraska et al. [ 8 ] claimed that the most important factor to consider in the contagion of a virus is the particle diameter. Xie et al. [ 9 ] stressed that the size of a droplet depends on two factors: evaporation and the movement of the droplets. The computational fluid dynamics (CFD) numerical model provided by Chillón et al. [ 10 ] indicated that the movement of particles with a diameter of less than 30 µ m is governed by the Brownian effect. On the other hand, average particles up to 80 µ m are subject to major forces, while larger diameter droplets are affected by gravity. According to Redow et al. [ 5 ], droplets with a diameter of 10 µ m evaporate at about 550 milliseconds when relative humidity is 80% (very wet environment). They also stated that the same particle, under conditions of 50% relative humidity, evaporates in 300 milliseconds, and in 20% relative humidity in 250 milliseconds. These results are in agreement with the work of Wang et al. [ 4 ], assuring that relative humidity is an important factor in evaporation. However, the evaporation of these aerosols does not remove the chance of infection, as shown in the work of Van Doremalen et al. [ 11 ]. They determined that SARS-CoV-2 has a half-life of about 1 h in the environment whereas if those aerosols fall on plastic or stainless steel, the half-life will be 6.8 h and 5.6 h, respectively. Avoiding close contact and maintaining a social distance of 1.83 m (6 feet) are two of the main guidelines to prevent infection. However, Feng et al. [ 12 ] observed that larger distances need to be considered due to ventilation conditions or external winds. The results of Li et al. [ 13 ] showed that with a wind speed of 2 m/s, a particle of 100 µ m could reach up to 6.6 m. The computational model by Dbouk et al. [ 14 ] also concluded that with a wind speed between 4 km/h and 15 km/h, the droplets of saliva exhaled into the environment could traverse 6 m. They also noted that particles decrease their concentration in the wind direction. On the other hand, in an enclosed space, that is, when air speed is 0 m/s and when it is not possible to keep the social distance, it is essential to use ventilation strategies to reduce the chance of infection. Sen et al. [ 15 ] demonstrated this in their CFD study inside an elevator. When there is a vent, the droplets fall to the ground without impacting any person. In contrast, aerosols would reach the person and could infect them. However, maintaining a safe social distance without any measure of individual protection is not enough, and, therefore, wearing masks has also been recommended. Principally, there are two types of masks: masks that cover only the mouth, nose, and chin and face shields, which cover the whole face (eyes, nose, and mouth). Regarding the masks that cover the mouth, nose, and chin, there are several types and they offer different functions; for example, surgical masks last 4 h and prevent the exhaled air from being dispersed, whereas the personal protective equipment (PPE) that have three levels of filtering facepiece respirators (FFP1, FFP2, FFP3) vary in their effectiveness at filtration. They filter the inhaled air and protect against the inhalation of aerosols. The face shields, however, are plastic screens with the purpose of covering our faces and acting as a protective barrier against aerosols. Akhtar et al. [ 16 ] performed an experimental study to prove the effectiveness of wearing a mask when sneezing. Five different masks were evaluated: surgical, cloth, cloth PM 2.5, wetted cloth PM 2.5, and N95. They showed that the aerosols came out into the environment while wearing all masks except for the N95 mask. The mathematical model by Arumuru et al. [ 17 ] agrees with Akhtar et al. [ 16 ] in stating that the N95 masks completely prevent leakage of droplets in the direction of advancement, although they observed that the droplets disperse to the environment between the holes of the mask and the nose. The mask recommended by Arumuru et al. [ 17 ] that has a lower leakage ratio is the commercial five-layered mask. It can be concluded that when wearing both masks combined, the risk of infection can be reduced. Recently, Salimnia et al. [ 18 ] undertook a study on the protection offered by the two types of masks and the protection offered by both combined, Mathematics 2021,9, 1582 3 of 15 and no improvements were noted when combining them. Arumuru et al. [17] also agrees with Salimnia et al. [ 18 ], arguing that the combination of the two masks is not feasible. In contrast, the computational model by Akagi et al. [ 19 ] investigated the protection offered by face shields, concluding that this type of mask is not a good protection tool, as 4.4% of the droplets exhaled into the environment enter the gap between the human and the face shield. Wendling et al. [ 20 ] carried out an experimental work to compare the barrier performance of face masks and face shields. The experimental data showed that, in a conversation between two people, a mask does not protect the human who sneezes, but the one who is sneezed on is protected. Moreover, when a face shield is used, the protection is better since it reduces the number of particles. Besides, when the human who sneezes is the one who is protected, there is hardly any difference between the two masks. Several numerical works for multiphase flows coupled with incompressible fluid motion can be found in the literature, such as the convergence analysis of a fully discrete finite difference scheme for the Cahn–Hilliard–Hele–Shaw equation presented by Chen et al. [ 21 ] and the error analysis of a mixed finite element method proposed by Feng and Prohl [ 22 ]. In that field, Yan et al. [ 23 ] showed that a second-order energy-stable scheme for the Cahn–Hilliard–Hele–Shaw equation was able to produce accurate long-time numerical results with a reasonable computational cost. The coupling of the Cahn–Hilliard equation to the Navier–Stokes equation of fluid flow gained the attention of Diegel et al. [ 24 ], with a convergence analysis and error estimates for a second-order accurate finite element method. In the current work, a CFD numerical model is presented with the aim of analyzing the spread of a sneeze containing viral droplets and evaluating the effectiveness of a face shield as a personal protective device. The human who wears the face shield has a height of 1.8 m and is placed at a social distance of 1.5 m from another human. Two scenarios were studied. In the first scenario, the mouths of both individuals are situated at the same height, while in the second scenario the mouths are located at a height difference of 0.2 m. The evaporation of aerosols was checked by modifying the ambient temperature and relative humidity. Two different temperatures were applied, 25 ◦ C and 15 ◦ C, and relative humidity (RH) values of 40% and 60% were studied for each case. In addition, two different wind speeds, 7 km/h and 14 km/h, were added to the numerical model to verify the social distance, with a favorable wind in the same direction as the sneeze. 2. Materials and Methods 2.1. Validation For the validation of our numerical model, the evaporation of a single droplet over time was numerically simulated. Three different diameters were evaluated: 1 µ m, 10 µ m, and 100 µ m. The temperature of the environment and the droplet were T e = 293.15 K and T d = 310.15 K, respectively. Figure 1shows the results with four different relative humidity ratios (0%, 20%, 60%, and 80%), the same method used by Redow et al. [ 5 ], Mowraska et al. [8], and Li et al. [13]. In addition, the study of Hamey [ 25 ] was used to validate the distance that a single droplet can traverse. The study consisted of a free fall water droplet into a wet space of relative humidity RH = 70%, T e = 293 K, and T d = 289 K, and droplet dimeter of 110 µ m or 115 µm, as shown in Figure 2. Mathematics 2021,9, 1582 4 of 15 Figure 1. Evaporation of a single pure water droplet under different environment conditions, environment temperature of 293.15 K, particle temperature 310.15 K, and different droplet sizes (1 µm, 10 µm, and 100 µm). Figure 2. Freely falling water droplets based on the study of Hamey (1982). Particle sizes of 110 µ m and 115 µ m, environment temperature 293 K, particle temperature 289 K, and 70% relative humidity. Mathematics 2021,9, 1582 5 of 15 2.2. Computational Domain and Initial Conditions CFD techniques, based on numerical methods and algorithms, allow for the analysis of complex problems under a wide range of conditions and parameters. In the current work, an individual was placed in an open three-dimensional domain. The domain size was 3 m × 2 m × 2.5 m (X, Y, Z), but the walls were simulated as a symmetry plane. The human had a height of 1.8 m and the height of its mouth was located at 1.6 m. Figure 3 shows the geometry of the subject and the face shield that it is wearing that covers its whole face. Table 1shows the main dimensions of the face shield. Figure 3. Human geometry: (a) Human body (1.8 m tall); (b) Face shield in detail. Table 1. Face shield dimensions. Dimension Value Height 250 mm Length 196 mm Area 660 mm2 Another human has been placed in front of the first at a distance of 1.5 m. Only the mouth has been modeled to simulate the sneezing by means of a surface injector, see Figure 4 . The aerosols and air jet are exhaled into the environment at a temperature of 36 ◦ C, the average temperature of the human body. According to Carpenter et al. [ 26 ] saliva is the most characteristic fluid of the body, having to perform the function of taste and being composed of different elements, giving rise to a composition that is difficult to recreate. Nicas et al. [ 27 ] summarizes the composition of saliva in water and non-volatile solids. The concentration of ions and cations is 150 mM, which in terms of mass equals 8.8 g/L. In the current study, saliva was modeled as a saline mixture in which it only has influence in reducing the saturation pressure of the water. The mass of each sneeze was 6.7 mg, based on the research of Zhu et al. [ 3 ] showing the average mass of aerosols poured into the environment in each sneeze. The mouth geometry was defined as: DM = 40 mm and dm = 20 mm, as shown in Figure 4c, in order to represent the mouth shape when sneezing. A sneezing velocity condition of 16 m/s with a duration of 400 milliseconds were used. In the first scenario, the mouths of the two individuals were at the same height. With the aim of comparing the effect of the people’s height, a second case was modeled with a difference between the height of the two mouths of 20 cm. For each height of mouth, two different temperatures were applied, one warmer at 25 ◦ C and another cooler at 15 ◦ C. Since the average suitable relative humidity range is 40–60%, each ambient temperature was simulated first with 40% and then with 60% relative humidity. To model an open space, wind factor was included, giving a comparison between a soft breeze of 7 km/h Mathematics 2021,9, 1582 6 of 15 and a moderate wind of 14 km/h. The wind had the same direction as the jet exhaled by the subject. Figure 4. Position of the mouth at 1.5 m: ( a ) The same height as the human with the face shield, 1.6 m; ( b ) A difference of height of 20 cm; ( c ) Geometry of the mouth (DM = 40 mm and dm = 20 mm) to simulate a sneeze. The discretization of the computational domain was made by means of a trimmer mesh. The domain was composed of 6.7 × 10 6 (6,755,678) cells. The most important zone of this study was the face shield, where the droplets were expected to impact, therefore, a finer mesh was made near it using a volume control (VC). The same volume control was also used in the direction of sneezing. In addition, a second volume control was used above and below the jet. Figure 5a illustrates the mesh used when the mouths were at the same height and Figure 5b show the meshes used when the difference between the height of the two mouths was 20 cm. Figure 5. Mesh distribution, the block ( ) indicates the mouth of the individual that has the virus and sneezes: ( a ) The mesh in the domain when the mouths were at the same height; ( b ) The mesh in the domain when the mouths were at different heights (20 cm difference). 2.3. Numerical Setup The fundamental laws that govern the mechanics of fluids are the conservation of mass, linear momentum, and energy, see Equations (1)–(3). ∂ρ ∂t+∇·(ρv)=0 (1) ∂(ρv) ∂t+∇·(ρv⊗v)=∇·(pI)+∇·T+fb(2) ∂(ρE) ∂t+∇·(ρEv)=fb·v+∇·(v·σ)− ∇·q+SE(3) Mathematics 2021,9, 1582 7 of 15 where ρ is the density, that is, the mass per unit volume, vis the continuum velocity, ⊗ denotes the outer product, f b is the resultant of the body forces per unit volume acting on the continuum, σ is the stress tensor, pis the pressure, Tis the viscous stress tensor, Eis the total energy per unit mass, qis the heat flux, and SEis an energy source per unit volume. The governing equations for the continuous phase, composed of dry air and water vapor, were expressed in Eulerian form, whereas the Lagrangian description was used to solve the dispersed saliva droplets as they crossed the computational domain. This homogeneous composition has, in all cases, the same temperature, pressure, and velocity. This is a non-reacting species; after determining the properties of each species, the properties of the mixture are calculated as a mass function of the mixture’s components. To that end, the mass-weighted mixture method proposed by Busco et al. [28] was used, see Equation (4). φmix = N=2 ∑ i=1 φiYi. (4) where Y i is the mass fraction of air and water vapor and φi is the property values of mixture component. Nis the total number of components in the mixture; in this case, N= 2. Relative humidity depends on the initial air and water vapor mass that is present in the environment and the temperature of the airflow. The equilibrium pressure is the most significant parameter that affects it. To calculate the density and viscosity of air, vapor, and liquid water the equations of Kukkonen et al. [ 29 ] were used. These properties were updated as the temperature was varied. As was mentioned previously in this section, saliva is composed mainly of water and non-volatile solids. However, in order to simplify the mathematical model, an assumption was made to consider saliva as a saline solution. This factor will only influence the equilibrium pressure of the water since dissolved inorganic salts help to reduce the saturation pressure of the water. Xie et al. [ 9 ] showed that the saturation pressure of these droplets can be calculated using Raoult´s law. Pva,s=XdPva(Tw)(5) where Pva,s is the saturation pressure of the droplet in the saline mixture, Pva is the equilibrium pressure of the water at a specific temperature (Tw) , and Xd is the mole fraction of the droplet, calculated as shown in Equation (6). Xd= 1+6imsMw πρLMs(dp)3!−1 (6) where M w and M s are the molecular weights of water and of solute, respectively; m s the mass of the solute in the droplet; dp is the diameter of the particle (droplet), and the ion factor “i” is equal to 2, in this case for the NaCl. Droplets, which cross the computational domain, were modeled by Lagrangian equations. In order to calculate the interactions between the mixture of dry air and water vapor and the droplets, the two-way coupling model was used. When droplets are exhaled into the environment, evaporation begins due to a temperature difference. Therefore, a quasisteady evaporation model, which allows droplets to lose mass through evaporation, was introduced and determined by Sherwood ´ s number. Equation (7) expresses the rate of change of droplet mass due to evaporation [28]. . mp=g∗×Asln(1+B)(7) where Bis the Spalding transfer number, g∗ is the mass transfer conductance, and As is the droplet surface area. However, this phenomenon is not the only one that occurs, the droplets may also be distorted and break up under the action of non-uniform surface forces. This behavior was taken into account using the Taylor analogy breakup (TAB) model. Mathematics 2021,9, 1582 8 of 15 Taylor analogy represents the distortion of a droplet as in a damping spring mass system; it reflects only the basic mode of oscillation of the droplet. The distance that these droplets can achieve is affected by the dispersion and turbulence they suffer. This effect was introduced in the present work with the addition of the Reynolds-averaged Navier–Stokes (RANS) equations with a kω Shear Stress Transport (SST) turbulence model, developed by Menter [ 30 ]. The transport equations for the kinetic energy kand the specific dissipation rate ωare shown in Equations (8) and (9). ∂ ∂t(ρk)+∇·(ρkv)=∇·[(µ+σkµt)∇k]+Pk−ρβ∗fβ∗(ωk−ω0k0)+Sk(8) ∂ ∂t(ρω)+∇·(ρωv)=∇·[(µ+σωµt)∇ω]+Pω−ρβ∗fβ∗ω2−ω2 0+Sω(9) where v is the mean velocity, µ is the dynamic viscosity, σk , σω , P k ,and P ω are Production Terms, f β* is the free-shear modification factor, S k and S ω are the user-specified source terms, and k0and ω0are the ambient turbulence values that counteract turbulence decay. Considering experimental data shown by Xie et al. [ 31 ] and subsequently validated by Dbouk et al. [ 14 ], the initial size distribution of the droplets representing the sneezing was modeled using the Rosin–Rammler distribution. Equation (10) shows the expression to define the probability density functions f , also known as the Weibull distribution. A minimum diameter of 10 µm and a maximum diameter of 300 µm were defined. f=n dp dp dp!n−1 e−(d dp)n ,n=8 (10) where dpis the mean diameter and is equal to 80 µm. The equation of conservation of linear momentum for a material droplet of mass m p is given by Equation (11). mp dvp dt =Fs+Fb(11) where v p denotes the instantaneous particle velocity, F s is the resultant of the forces that act on the surface of the particle, and Fbis the resultant of the body forces. Spherical particles were assumed and the energy that these spherical particles release is implemented with the correlation of Ranz–Marshall. This correlation is suitable for spherical particles up to Re ≈ 5000 and was applied in the evolution of the mass of salivaonly droplet particles due to evaporation. This correlation defines the coefficient of heat transfer as a derivative correlation as a function of the Nusselt number. The droplets injected into the computational domain were under the influence of the drag force, defined by Equation (12). Fd=1 2CdρAp|vs|vs(12) were C d is the drag coefficient of the droplet, ρ is the density of the continuous phase, A p is the projected area of the droplet, and v s is the droplet slip velocity (v s =v − v p ) with v being the instantaneous velocity of the continuous phase. The Schiller–Naumann model was used to calculate this force, the same method employed by Wang et al. [ 4 ]. According with Karunarathne et al. [ 32 ], this model was used for modelling of the drag between fluid phases in a multiphase flow. Equation (13) shows the expression to calculate the drag coefficient Cd. Cd   24 Re ,Re ≤1 24 Re 1+0.15Re0.687, 1 <Re ≤1000 0.44, Re >1000 (13) where Re is the Reynolds number. Mathematics 2021,9, 1582 9 of 15 The influence of the gravity force was taken into account as was the turbulent particle dispersion with the exact eddy interaction time. In this work, the commercial CFD code STAR-CCM + v.14.02 (Siemens, London, UK) [ 33 ] was used to define and solve the numerical model of aerosols produced by sneezing. A personal server-clustered parallel computer with Intel Xeon © E5-2609 v2 CPU @ 2.5 GHz (16 cores) and 45 GB RAM were used to run all the simulations. Each simulation took about 35 h of computation, with a total time of approximately 24 days. 3. Results 3.1. Effect of Relative Humidity and Environment Temperature in the Droplets’ Evaporation In this section, the influence of relative humidity and ambient temperature are analyzed. To that end, the quantity of particles exhaled and their corresponding diameter were observed. Firstly, a comparison between t = 0.4 s, when the sneeze is finished, and t = 2.5 s , when most of the droplets have impacted the person, is presented at Te= 15 ◦C and RH = 40% . Secondly, a comparison is made to evaluate the influence of relative humidity at t = 2.5 s in the following scenarios: T e = 15 ◦ C and RH = 40% and T e = 15 ◦ C and RH = 60%. Finally, a comparison between T e = 15 ◦ C and RH = 40% and T e = 25 ◦ C and RH = 40% is presented to see the difference in ambient temperature. The evolution of the droplets exhaled to the environment was tracked at t = 0.4 s and t = 2.5 s and are shown in Figure 6, with a droplet size ranging from 19 µ m to 96 µ m diameter. At t = 2.5 s, a maximum droplet size of 92 µ m and a minimum droplet size of 19 µ m were observed. At t = 0.4 s, in contrast, there were no droplets of 19 µ m diameter, but 461 droplets of 29 µ m were observed. In that timeframe, the evaporation process takes place, and the 29 µm droplets manage to evaporate down to a diameter of 19 µm. Figure 6. The distribution of the droplets size at t = 0.4 s and t = 2.5 s when the environment temperature was 15 ◦ C and the relative humidity was 40%.