A physicist view of COVID-19 airborne infection through convective airflow in indoor spaces
AA physicist view of COVID-19 airborne infection through convective airflow in indoor spaces
Luis A. Anchordoqui and Eugene M. Chudnovsky
Physics Department, Herbert H. Lehman College and Graduate School, The City University of New York250 Bedford Park Boulevard West, Bronx, New York 10468-1589, USA (Dated: March 2020)
General Idea:
Naturally produced droplets from humans (such as those produced by breathing, talking,sneezing, and coughing) include several types of cells (e.g., epithelial cells and cells of the immunesystem), physiological electrolytes contained in mucous and saliva (e.g. Na + , K + , Cl − ), as well as,potentially, several infectious agents (e.g. bacteria, fungi, and viruses). In response to the novelcoronavirus SARS-CoV-2 epidemic, which has become a major public health issue worldwide, weprovide a concise overview of airborne germ transmission as seen from a physics perspective. We alsostudy whether coronavirus aerosols can travel far from the immediate neighborhood and get airbornewith the convective currents developed within confined spaces. Methodology:
Methods of fluid dynamics are utilized to analyze the behavior of various-size airbornedroplets containing the virus.
Study Findings:
We show that existing vortices in the air can make a location far away from the sourceof the virus be more dangerous than a nearby (e.g., 6 feet away) location.
Practical Implications : Our study reveals that it seems reasonable to adopt additional infection-controlmeasures to the recommended 6 feet social distancing. We provide a recommendation that could helpto slow down the spread of the virus.
I. INTRODUCTION
The recent outbreak of the respiratory disease identi-fied as COVID-19 is caused by the severe acute respira-tory syndrome coronavirus 2, shortened to SARS-CoV-2 [1–3]. The outbreak, first reported in December 2019,has rapidly evolved into a global pandemic. Indeed,COVID-19 is spreading across the globe with a speed andstrength that laid bare the limits of our understandingof the transmission pathways and the associated factorsthat are key to the spread of such diseases. In particular,the virus can spread from seemingly healthy carriers orpeople who had not yet developed symptoms [4]. Over-all, this has transformed the face of healthcare aroundthe world.To understand and prevent the spread of a virus likeSARS-CoV-2, it is important to estimate the probabil-ity of airborne transmission as aerosolization with par-ticles potentially containing the virus. There have beenreports favoring the possibility of creating coronavirusaerosols [5]. Thus far no aerosolized coronavirus parti-cles have been found in hospital searches within the mostpublic areas, but evidence has been detected in rooms ofSARS-CoV-2 patients [6–9]. In this paper we provide anoverview on the possible threat of SARS-CoV-2 airborneinfection from a physics point of view, focusing attentionon the e ff ect of convection currents in indoor spaces.The layout of the paper is as follows. In Sec. II wefirst provide a concise discussion of the motion of thevirus in suspended aerosols. After that, using the Sim-Scale program [10] we study the convective airflow ina meeting room and o ffi ce space. We show that ex-isting vortices in the air can make a location far awayfrom the source of the virus be more dangerous than anearby (e.g., 6 feet away) location. In Sec. III we presentour conclusions. Throughout we adopt the convention of the World Health Organization to nickname particlesthat are (cid:38) µ m diameter as droplets and those (cid:46) µ mas aerosols or droplet nuclei [11]. II. MODELING THE EFFECT OF CONVECTIONCURRENTS IN THE TRANSMISSION OF SARS-COV-2
In the presence of air resistance, compact heavy objectsfall to the ground quickly, while light objects exhibitBrownian motion and follow the pattern of turbulentconvection of the air. For aerosol particles containingthe virus, the boundary between these two behaviorsdepends on the size of the particle. We begin with asimple question: how long does a virus float in the airunder the influence of gravity? To answer this query wemodel the virus as a sphere of radius r ∼
90 nm and mass m ∼ . × − kg [12], and we assume that this sphericalparticle is suspended in a viscous fluid (the air) feelingthe Earth’s gravitational field. Herein, gravity tends tomake the particles settle, while di ff usion and convectionact to homogenize them, driving them into regions ofsmaller concentration. On the one hand, the convectionmechanism provides particle macro-mixing within thefluid through the tendency of hotter and consequentlyless dense material to rise, and colder, denser materialto sink under the influence of gravity. On the otherhand, the di ff usion mechanism acts on the scale of anindividual particle (micro-mixing) slowly and randomlymoving through the media.Under the action of gravity, the virus acquires a down-ward terminal speed that follows from Stokes law and isgiven by v down = µ mg , (1) a r X i v : . [ phy s i c s . pop - ph ] A ug TABLE I: Evaporation time of water droplets.Droplet diameter ( µ m) Evaporation time (s)2000 6601000 165500 41200 6 . .
750 0 . where g (cid:39) . / s is the acceleration due to gravity and µ = πη r , (2)is the virus mobility in the fluid, and where η = . × − kg / (ms) is the dynamic viscosity of air [13]. Sub-stituting (2) into (1) we find that the downward termi-nal speed of the virus in dry air is indeed negligible, v down ∼ × − m / s. It is therefore clear that gravityplays no role in the motion of an isolated virus throughthe air. Rather it follows a convection pattern in a man-ner similar to how smelly substances move through theair. The survival probability of the virus in the dry air isthen given by the likelihood of survival outside its natu-ral environment. The half-life of SARS-CoV-2 in aerosolshas been found to be about 1.1 hours [5].We have seen that the coronavirus can go airbornestaying suspended in the air. However, the virus istransmitted through respiratory droplets and dropletnuclei produced mostly while sneezing and coughing.Then to ascertain whether airborne transmissible SARS-CoV-2 can survive and stay infectious in aerosols wemust double-check that the falling time of a droplet or adroplet nuclei from a height of about two meters is largerthan its evaporation time scale. To this end, we assumethat the drops are also spherical and hence the mass canbe simply estimated as m = π r ρ , (3)where ρ =
997 kg / m is the density of water and r thedroplet / aerosol radius. For large droplets whose diam-eters (cid:38) µ m, the e ff ect of air resistance is negligi-ble and so the falling time can be directly estimated us-ing Newton’s equations for gravitational settling. Forsmaller droplets whose diameters < µ m, the fallingtimes must instead be determined using the downwardterminal speed given in (1) to account for the air resis-tance upon the falling droplets. It is now an instructiveand straightforward exercise to show that the time forfalling 2 m in saturated air is 0.6 s for droplets with r > µ m, 6.0 s for those of r ∼ µ m, 600 s (about10 minutes) for those of r ∼ µ m, and 60,000 s (about16.6 hours) for those of r ∼ . µ m. The droplet evap-oration time scale, as computed by Wells using droplet evaporation data collected by Whytlaw-Gray and Pat-terson, is shown in Table I [14]. The assumption of purewater droplets in unsaturated air at 18 ◦ C was used forthe evaporation calculations, such that the theoreticaldroplets are capable of complete evaporation. By di-rect comparison of the droplet evaporation and fallingtimes we can conclude that somewhere between 100 and200 µ m lies the droplet size (i.e. the diameter) whichidentifies droplets of mouth spray that reach the groundwithin the life of the droplet as against droplets thatevaporate and remain in the air as droplet-nuclei withattached SARS-CoV-2 infection. Several investigationshave been carried out to continue improving the preci-sion of Wells analyses and to study the various externalenvironmental (such as temperature and humidity) fac-tors that may alter his estimates; see e.g. [15].The sizes of the droplets and droplet-nuclei producedby sneezing and coughing were studied by the micro-scopic measurement of 12,000 droplet stain-marks foundon slides exposed directly to mouth-spray, and of 21,000stain-containing droplet-nuclei recovered from the airon to oiled slides exposed in the slit sampler [16]. Fromthis data sample it was found that the original diame-ters of respiratory droplets ranged from 1 to 2000 µ mand that 95% were between 2 and 100 µ m and that themost common were between 4 and 8 µ m. Similar re-sults were reported in [17]. This suggest that, in princi-ple, droplet-spray could drive direct airborne infectionof SARS-CoV-2. The transmission of the COVID-19 dis-ease, however, still depends on the infectious virus loadcarried by the droplets, which must be determined ex-perimentally. The number of virions needed for infectionis yet unknown, but we can use other viral transmission(e.g., influenza [18–20]) for a template. The spread of asneeze in the air has been studied by ultrafast imaging atMIT [21–23]. It was found that even the largest dropletsfrom a sneeze can float in the air for up to 10 minutes,which allows them to reach the far end of a large room.This points towards convection in the air being the pri-mary mechanism of the spread of the infection.From the physics point of view, we cannot find a goodjustification for a stationary 6-feet separation in a situa-tion when people spend long time together in a room.Small droplets or aerosols containing the virus move inthe air via convection. The convection pattern in a roomcan be very complex; see Fig. 1. It depends on the lo-cation of air conditioners, radiators, windows, and allitems in the room, as well as on people producing vor-tices by moving around. The existing vortices in theair can make a location far away from the source of thevirus more dangerous than the location 6 feet away. Thisapplies to meeting rooms, o ffi ce spaces, supermarkets,department stores, etc. The airflow pattern should bestudied for all such facilities to avoid the spread of infec-tion to large distances from a single infected person. Thesafest rooms must be those equipped with the air suckingventilator at the top, like hospital surgery rooms [24].By all means, re-configuring the ventilation of public FIG. 1: Visual representation of airflow streamlines in a meeting room (left) and o ffi ce space (right) colored to velocity magnitudefrom low (blue) to high (red). The convection pattern in the meeting room demonstrates how the infection can be persistentlycarried by the airflow between two chairs separated by 6 feet. The convection pattern in the o ffi ce space illustrates how theinfection can be taken by the airflow from one cubicle to the other. Simulation by SimScale [10]. and private facilities cannot be done within the timescaleof the pandemic. The question is what to do now if wewant to slow its pace. The answer is very simple. Peoplemust be required to wear face masks in public spaces toprevent the virus from becoming airborne in the firstplace. III. CONCLUSIONS
Airborne SARS-CoV-2 virus spreads from infectedindividuals and accumulates in confined spaces whereit can linger in the air in the aerosol form for hours. Theinhaled virus load depends on the virus concentration inthe air and the time of exposure. Concentration can varyfrom one spot to another. It depends on the location ofthe spreaders and the pattern of the airflow. The latteris determined by many factors, such as the location ofdoors and windows, ventilators, heaters, movement ofpeople, etc. Common central air conditioning systemthat is cooling the indoor air but is not exchanging itwith the outside air and is not filtering the virus helpsto spread it across the airconditioned space. It caneasily take the virus to distant locations and make it accumulate in the least expected places. Computerstudies of the airflow in public spaces are importantfor mitigating this problem. In the long run businessand educational facilities should consider redesigningventilation and air conditioning systems to e ff ectivelyreduce concentration of the aerosol virus in the air. Funding / Support:
The theoretical and computationaltechniques and resources used in this research weresupported by the U.S. National Science Foundation,NSF Grant PHY-1620661 (L.A.A.), and the U.S. De-partment of Energy, O ffi ce of Science, DOE Grant DE-FG02-93ER45487 (E.C.). Role of the Funder / Sponsor:
The sponsors had norole in the preparation, review or approval of themanuscript and decision to submit the manuscript forpublication. Any opinions, findings, and conclusions orrecommendations expressed in this article are those ofthe authors and do not necessarily reflect the views ofthe NSF or DOE.
Conflict of Interest Disclosures:
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