Necessity of ventilation for mitigating virus transmission quantified simply
NNecessity of ventilation for mitigating virus transmission quantified simply
Eric G. Blackman a,b, ∗ , Gourab Ghoshal a a Department of Physics and Astronomy, University of Rochester, Rochester, NY, 14627, USA b Laboratory for Laser Energetics, University of Rochester, Rochester NY, 14623, USA
SummaryBackground
To mitigate the SARS-CoV-2 pandemic, officials have employed social distancing and stay-at-home measures, with increased attention to room ventilation emerging only more recently. Effective dis-tancing practices for open spaces can be ineffective for poorly ventilated spaces, both of which are commonlyfilled with turbulent air. This is typical for indoor spaces that use mixing ventilation. While turbulenceinitially reduces the risk of infection near a virion-source, it eventually increases the exposure risk for alloccupants in a space without ventilation. To complement detailed models aimed at precision, minimalistframeworks are useful to facilitate order of magnitude estimates for how much ventilation provides safety,particularly when circumstances require practical decisions with limited options.
Method
Applying basic principles of transport and diffusion, we estimate the time-scale for virions injectedinto a room of turbulent air to infect an occupant, distinguishing cases of low vs. high initial virion massloads and virion-destroying vs. virion-reflecting walls. We consider the effect of an open window as a proxyfor ventilation.
Findings
When the airflow is dominated by isotropic turbulence, the minimum area needed to ensure safetydepends only on the ratio of total viral load to threshold load for infection.
Interpretation
The minimalist estimates here convey simply that the equivalent of ventilation by modestsized open window in classrooms and workplaces significantly improves safety.
Keywords: elsarticle.cls , L A TEX, Elsevier, template
Introduction
The SARS CoV-2 virus, first reported in 2019 [1, 2] has since spread to at least 213 countries andterritories leading to an unprecedented global pandemic [3]. The lack of therapeutics and vaccines have ledpublic health officials to employ non-pharmaceutical interventions focused on physical distancing measuresand masks, but comparatively much less on ventilation. With the resumption of visits to offices, bars,restaurants, salons and universities, where people are expected to be in close proximity in for prolongedperiods of time, ventilation and HVAC filtering are safety precautions that must be prioritized [4, 5].Airborne viral particles like SARS-CoV-2, which cause Covid-19, get trapped on moisture droplets thatcarry the virions [6]. Interior air is commonly, if not unavoidably, turbulent. This keeps droplets airbonemuch longer than their free-fall time [7]. The droplets form a spectrum of sizes, and those below 5 µ m canremain airborne in aerosols for at least 3 hours [8]. Since virions are attached to these aerosol droplets,basic principles of turbulent diffusion and transport or aerosols [9, 10, 11, 12, 13, 14, 15, 16] become directlyapplicable to guiding HVAC issues and the efficacy of masks [17]. Forced central air/heating in HVACsystems that use mixing ventilation without sufficient replenishment of fresh air exacerbates the danger ofairborne virions. The rapid spread of aerosol droplets in room of turbulent air is exemplified by spraying ∗ Corresponding author
Email addresses: [email protected] (Eric G. Blackman), [email protected] (Gourab Ghoshal)
Preprint submitted to Journal of L A TEX Templates August 20, 2020 a r X i v : . [ phy s i c s . s o c - ph ] A ug njected mass of virions M mass of virions to infect one person M c N = M/M c rate of virions encountering a face ˙ M h density of virions uniformly spread over room ρ room radius R eddy size l turbulent eddy air speed v l eddy turnover time t l = l/v l eddy diffusivity ν T = v l l/ x ≥ l v dif( x ) = ν T /x mean air flow v p face width h face area A h = h required time for infection t c window area W time for load to diffuse via window t saf Table 1: list of variables a scented aerosol and measuring how quickly a person on the other side of the room can detect it. Thebenefits of physical distancing are diminished by prolonged exposure to viral filled turbulent air in a closedroom because virions are transported throughout by turbulent diffusion of the host droplets. Without atightly sealed mask and proper eye protection, accumulated indoor-exposure is likely.Virions can also be transported by HVAC systems between rooms. People staying home may be exposedto the virus in poorly ventilated apartment buildings with forced circulating air. Staying at home in isolatedhouses is not equivalent to staying home within a poorly ventilated apartment building. Evidence for suchnon-local transport of viruses has been found in restaurants in China [18, 19], a call-center in South Korea[20], and a choir-setting in Washington state [21]. The study of airbone disease transmission and ventilationhas a empirical and computational history [22, 23, 24, 25].Much about the transmission modes of SARS CoV-2 remains unknown, such as the viral load requiredto cause infection [26]. Nevertheless, it is crucial to use basic principles that we do know to inform policychoices to reduce the risk of transmission [4]. As such, here we use employ basic concepts of turbulenttransport to provide minimalist order-of-magnitude estimates to quantify the efficacy of ventilation.Specifically, we estimate the time scale for an individual to be infected in an enclosed space of arbitrarysize, subjected to an injection of virions. We distinguish between virion-absorbing vs. virion-reflectingwalls and cases in which the injected mass of virions is sufficient vs. insufficient to infect a person overone diffusion crossing time from the virial injector to the room boundary. We consider the role of an openwindow as a proxy for ventilation, and estimate the typical cross-section needed for safety. We also discussthe role of a systematic drift velocity in the room. Since the threshold viral load for severity of infectionis unknown, despite the different loads produced by talking, breathing and other modes [27, 28, 29], wepresent our results simply in terms of a dimensionless quantity–the virion mass required to infect a singleindividual. Our results demonstrate simply the importance of ventilation and filtering, complementing, butin agreement with, more detailed computational [7, 18, 30] and empirical efforts [31].2 esearch in Context
Evidence before this study
Coughing, sneezing and breathing release moisture droplets onto which SARS-CoV-2 bind. Droplets of sizebelow 5 µm in aerosols remain viable and airborne with a half life of three hours [8]. Interior air is commonlyturbulent and diffuses aerosols throughout a room and building over much shorter time scales, spreadingthe infectious virions over distances much larger than the 6ft recommended physical distancing separation. Added value of this study
We quantify the mitigating role of ventilation using simple from simple considerations of fluid transportand diffusion, by comparing the load of virions encountering a persons face in a room with and without awindow. Specifically, this becomes a determination of the size of the window or filter needed to maintainsafety. The results help guide practical decisions about safety and interior ventilation in spaces where peoplespend extended periods, such as classrooms and office buildings.
Implications of all the available evidence
Proper ventilation mitigates the airborne spread of SARS-CoV-2 and needs to be viewed along with masksand physical distancing as the third pillar of prevention, We show its efficacy quantitatively in a simplebroadly accessible way.
Results
While progressively smaller droplets take longer and longer to fall out of the air [31] after the initialload (cough, breathing, or sneeze), we are mainly concerned with particles that remain suspended in the airafter it is well mixed, namely these ≤ µ that can remain airborne for 3hr [8]. Such particles are equivalentcarriers of virions for present purposes.We assume that a combined mass M of such viron-loaded aerosol droplets (VLADs) are injected in aroom of radius R , and that the room has steady turbulent air of local eddy air speed v l and eddy scale l << R . We define t c as the time scale for a single person to get infected and M c is critical mass of VLADsneeded to infect one person. These two quantities are related by t c = M c / ˙ M h , (1)where ˙ M h is the rate of air mass encountering a human face of diameter h . We assume that the thresholdviral load for infection is small compared to the total viral content in the room. In Table 1, we list thevariables used in the following calculation results. Large viral injection mass, no ventilation
We first consider the case of a closed room (no ventilation), for which the injected mass load is sufficientlyhigh that a single passing of the diffusion front after the viral injection is sufficient to cause infection. Forthis case, the time scale to infect any individual is bounded by the diffusion time from the injection point tothe room boundary. Assuming isotropic turbulence, the eddy diffusivity is ν T = v l × l/
3. The critical timefor infection is therefore t c ≤ R /ν T = 3 t l (cid:0) Rl (cid:1) ≤ . (cid:0) t l . (cid:1) (cid:0) R (cid:1) (cid:0) l . (cid:1) − min . (2)Here t l = l/v l is the eddy turnover time and its scaling of 0 .
75s was estimated using an eddy scale of l = 0 .
75m corresponding to injection by a room fan at a flow speed 1m / s.3 mall viral injection mass, no ventilation The likely more commonly important case is when the initial injection of viral mass is small enough sothat the room is fully mixed with VLADs before any individual is infected. First, we consider two sub-caseswithout ventilation: (i) completely absorbing walls (say if the walls are infused with anti-viral material suchas copper fibers) and (ii) completely reflecting walls. For the former, sub-case virions are removed uponcontact with the wall and no one in the room is infected.For the latter sub-case, the virion-carrying droplets remain airborne. Then, given that the accumulatedmass of virions on a single person is ˙ M h (cid:39) ρv dif A h , (3)where ρ = 3 M πR , (4)and v dif is the diffusion speed, which is simply the turbulent diffusion coefficient ν T divided by the netdisplacement from the source to point of measurement. Given that we are in the well-mixed regime, therelevant displacement is just the eddy scale, since any new virion-mass accumulates via neighboring eddies.Therefore, v dif = 13 v l , (5)Combining Eqns. (3), (4), and (5) into Eq. (1) yields, for the critical infection time, t c = 13 . (cid:0) t l .
75 s (cid:1) (cid:16)
M/M c (cid:17) − (cid:0) R (cid:1) (cid:0) l . (cid:1) − (cid:0) h . (cid:1) − min , (6)where we have used fiducial values to facilitate estimates. Note, that while M c is unknown, in all cases itenters as as fraction of the input viral mass. In the upper panel of Fig. 1 we plot Eq. (6) as a function of theroom radius R and injected load N = M/M c . By definition, to prevent infection an occupant must spend aduration t ≤ t c in the room. Role of an open window
To quantify the effect of ventilation, we consider window of open area
W << R , that allows exchange ofinterior and exterior air. If pressure and temperature equilibrate everywhere, turbulence facilitates mixinginterior VLAD filled air with virus-free air. VLADs leave the window with a mass loss rate˙ M W = ρv dif W. (7)From the time of VLAD injection in the room, it takes a single diffusion time from the source to the windowfor exterior air to interact with the viral air. This has no effect for the case when the viral mass exceeds thecritical threshold for a single infection. Nor does it have any bearing for absorbing walls, where the viralload is absorbed over one diffusion crossing time.The ventilation is however, very influential for the reflecting wall case of Eq. (6), which now requiresmodification. For this case, the VLAD mass lost over a time t >> R /ν T can be estimated as M L ( t ) = ˙ M W t .Dividing this by M and using equations (4) and (7) gives the fraction of VLAD mass lost from the room as M L ( t ) /M (cid:39) π (cid:18) v l tR (cid:19) (cid:18) WR (cid:19) . (8)Safety would be achieved once M L ≥ M (1 − M c /M ), corresponding to ≥
99% replacement of the room airwhen M c le . M . (If using a ventilation system measured in conventional ventilation units of air changesper hour (ACH) [ ? ], for M c /M < .
01 this would require > ACH to achieve this replacement in 1 hoursince each ACH replaces only 63 .
2% of the air.)Solving Eq. (8) for the associated t = t saf gives, t saf = 4 πR v l W . (9)4ow we set Eq. (6) equal to Eq. (9) to get the minimum open window area W c ensuring t saf < t c , so thatenough virions are removed before anyone gets infected. This gives W c ≥ (cid:0) h . (cid:1) (cid:16) M/M c (cid:17) m . (10)This is independent of the room size because the same viral mass in a larger room reduces the viral density.Therefore the flux of mass incident on a face, as well as on the surface of a window is reduced by the samefactor. In the lower panel of Fig. 1b we show t c /t saf as a function of W and N . The intersection of thecurved surface with the plane is equation (10).Given that ρ is uniform in the room for the fully mixed case by diffusive action of turbulence, the ratioof mass flux through a window to that encountering a person’s face is approximately W/h . Increasing thisratio enhances the probability for viral mass encountering the window. Our example shows that the strategyfor safety is to increase this ratio such that the leftover virial density in the room is insufficient to producea critical infectious load for individuals during their period of stay.What is the role of masks in the context of the above estimates? They reduce the surface area of theface by at least a factor of 2 since droplets must slide around the smaller open area around the edges of themask. A factor of ≥ h has the equivalent effect of reducing the required size of the windowarea for safety by that same factor as seen from Eq. 10.A generalization of interest is the case in which there is a systematic bulk flow in the from one side tothe other, for example from a pressure gradient or fan. In this case, the ratio of the flux through the windowto that onto a face depends on the angles of the bulk flow to the window, θ w and to the face, θ f . If thebulk flow is aimed toward the window, then only the latter angle enters and this ratio of fluxes becomes ( v T + v P ) W ( v T − v P cos θ f ) h where cos θ f is the angle between the flow and the normal to the face. If | v P cos θ f | >> v T ,then this ratio becomes simply − W/ ( h cos θ f ) highlighting that simply having ones back to the flow is thesafer orientation, as the flux ratio is negative when the flow passes from back to front. In practice detailedmodeling of flow around the head must be considered to assess the influence of boundary layers. Discussion
Turbulence is hard to avoid in interior spaces and when the viral mass injected into a turbulent roomis sufficiently high, a person can be infected over one turbulent diffusion time. as per equation (2). Morelikely are spaces such as school classrooms or work places where occupants remain in the room for extendedperiods after which virions are fully mixed are only exposed to a critical load over longer times. For reflectingwalls and no open windows, the time scale for infection is given by Eq. (6) and illustrated in the upper panelof Fig. 1. Spending more time than this in such an unventilated room would lead to infection, independentof physical distancing.To provide simple practical information and intuition the role of ventilation, we estimated the minimumsize of an open window needed to mitigate infection for a one-time viral load for the case that would otherwisecause infection on the time scale of equation (6). The critical window size is given by Eq. (10) and dependson one unknown, the critical VLAD mass for infection M c . The lower panel of Fig. 1 exemplifies the effect,showing that a window area ≥ m is enough to prevent occupants being infected for a viral load thatcould potentially infect 50 people. Open windows are therefore extremely helpful. These conclusions areconsistent with, but complementary to empirical approaches such as those of of [31]. Masks reduce the areaof the face and thus decrease the required open window area to obtain the same protection.In addition to windows, retrofitting HVAC systems with UVC (ultra violet c) or other anti-viral filterswill certainly also reduce exposure, particularly from air passing between rooms in building complexes anddepending on the efficiency of the filter, there is a correspondence in efficacy to a window area of a particularsize. In recognize that the efficacy of ventilation can be thouht of as air passing through a window or filterof a specific area, note that interior walls themselves provide a useful large surface area. If they were imbuedwith anti-viral materials and constructed to mitigate boundary layer effects, rooms could potentially be safeafter one turbulent diffusion time following any VLAD injection. Such measures are also effective for other5 igure 1: Critical time for infection and condition for safety. Upper panel : the time t c for a non-ventilated room withreflecting walls according to Eq. (6) plotted as a function of the number of people that can be infected by the injected loadand room radius R . Infection is prevented by spending a time t ≤ t c . Lower panel : mitigation by introducing ventilation viaan open window of area W . The planar surface is the critical surface t c /t saf = 1 above which the time scale for infection t c exceeds the time scale t saf for which enough virons have diffused through the window. Points on the curved surface abovethe plane indicate safe occupancy duration. The line of intersection between the surfaces shows the window area needed as afunction of the viral mass injected into the room according to Eq. (10). airborne illnesses such as the flu and offer the economic benefits of reducing sick days, even non-pandemiccircumstances.Precision is not required for our key results and message, but its further pursuit involves more complexnumerical modeling. These can include turbulent eddy and droplet size spectra with size-dependent airbornesurvival times; different droplet-eddy coupling times as a function of droplet and eddy sizes and humidity;repeated, continuous, or time-dependent injection of VLADs from different room-locations and sources[28, 29]; temperature gradients [32]; room geometry, inhomogeneous turbulence; and wall boundary layereffects [30]. Acknowledgments
EB acknowldges support from NSF Grant AST-1813298 and Department of Energy grants DE-SC0020432and DE-SC0020434. GG acknowledges support from NSF Grant IIS-2029095. We thank W.J. Forrest andJ.A. Tarduno for related discussions. None of these funding sources had any involvement or influence in anyaspect of the study.
Author Contributions
Authors Blackman and Ghoshal jointly conceived the idea for the study. Author Blackman carried out thebasic calculations and drafted the initial version of the manuscript based on joint discussions. Ghoshal furthercontributed to writing the manuscript, and the interpretation and elucidation of the results, Correspondingauthor Blackman confirms full access to everything in this study and had final responsibility for the decisionto submit for publication. 6 onflicts of Interest
We have no conflicts of interest to report.
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