ma hema ics
A icle
Nume ical Modeling o Face Shield P o ec ion agains a Sneeze
Aina a Uga e-Ane o 1, Unai Fe nandez-Gamiz 1,* , Iñigo A amendia 1, Ekai z Zulue a 2
and Jose Manuel Lopez-Guede 2
Ci a ion: Uga e-Ane o, A.;
Fe nandez-Gamiz, U.; A amendia, I.;
Zulue a, E.; Lopez-Guede, J.M.
Nume ical Modeling o Face Shield
P o ec ion agains a Sneeze.
Ma hema ics 2021,9, 1582. h ps://
doi.o g/10.3390/ma h9131582
Academic Edi o : Michael Boo y
Recei ed: 6 June 2021
Accep ed: 1 July 2021
Published: 5 July 2021
Publishe ’s No e: MDPI s ays neu al
wi h ega d o ju isdic ional claims in
published maps and ins i u ional a il-
ia ions.
Copy igh : © 2021 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
1Nuclea Enginee ing and Fluid Mechanics Depa men , Uni e si y o he Basque Coun y, UPV/EHU,
Nie es Cano 12, Vi o ia-Gas eiz, 01006 A aba, Spain; [email p o ec ed] (A.U.-A.);
[email p o ec ed] (I.A.)
2Sys em Enginee ing and Au oma ion Con ol Depa men , Uni e si y o he Basque Coun y, UPV/EHU,
Nie es Cano 12, Vi o ia-Gas eiz, 01006 A aba, Spain; [email p o ec ed] (E.Z.);
[email p o ec ed] (J.M.L.-G.)
*Co espondence: [email p o ec ed]
Abs ac :
The p o ec ion p o ided by wea ing masks has been a guideline wo ldwide o p e en he
isk o COVID-19 in ec ion. The cu en wo k p esen s an in es iga ion ha analyzes he e ec i eness
o ace shields as pe sonal p o ec i e equipmen . To ha end, a mul iphase compu a ional luid
dynamic s udy based on Eule ian–Lag angian echniques was de ined o simula e he sp ead o
he d ople s p oduced by a sneeze. Di e en scena ios we e e alua ed whe e he ela i e humidi y,
ambien empe a u e, e apo a ion, mass ans e , b eak up, and u bulen dispe sion we e aken
in o accoun . The sali a ha he human body gene a es was modeled as a saline solu ion o 8.8 g
pe 100 mL. In addi ion, he in luence o he wind speed was s udied wi h a so b eeze o 7 km/h
and a mode a e wind o 14 km/h. The esul s indica e ha he ace shield does no p o ide accu a e
p o ec ion, because only he pe son who is sneezed on is p o ec ed. Mo eo e , wi h a wind o
14 km/h, none o he d ople s exhaled in o he en i onmen hi he ace shield, ins ead, hey we e
deposi ed on o he neck and ace o he wea e . In he p esence o an ai low, he d ople s exhaled in o
he en i onmen exceeded he sa e dis ance ma ked by he WHO. Rela i e humidi y and ambien
empe a u e play an impo an ole in he li e ime o he d ople s.
Keywo ds:
COVID-19 p o ec ion; ace shield; sneeze; d ople e apo a ion; ela i e humidi y; en i-
onmen empe a u e; compu a ional luid dynamics (CFD)
1. In oduc ion
Cu en ly, we a e expe iencing a pandemic caused by he new co ona i us SARS-CoV-
2, which causes he disease known as COVID-19. The WHO de ines he e m pandemic
as “ he wo ldwide sp ead o a new disease”. The expansion and sp ead o COVID-19
ha e changed ou li es yle habi s. The s a egies o comba ing such a phenomenon a e
a he limi ed, and p e en ion is he bes way o con ol and o educe he ansmission
o COVID-19. This is no he i s ime he wo ld has had o deal wi h a global si ua ion
due o he sp ead o an in ec ious disease. Du ing human his o y he e ha e been se e al
global pandemics such as he Black dea h, he Spanish lu, and smallpox, among o he s.
Acco ding o Saunde s-Has ings e al. [
1
], ad ances in he medical sec o change when i
comes o dealing wi h a pandemic. Ve ma e al. [
2
] showed ha social dis ance and hand
washing a e essen ial o comba COVID-19, all accompanied by he use o masks. When
coughing, sneezing, o e en alking, dozens o d ople s a e exhaled in o he en i onmen
wi h a high chance o in ec ing ano he human being. These d ople s a e de ined as ae osols,
which may be in a liquid o gaseous s a e. The s udy o Zhu e al. [
3
] no ed ha 6.7 mg o
sali a is expelled wi h each indi idual sneeze.
The e apo a ion o hese pa icles causes he educ ion o he pa icle diame e , and
his phenomenon depends on he ela i e humidi y and he en i onmen al empe a u e
and lows, acco ding o Wang e al. [
4
]. Redow e al. [
5
] obse ed ha when he d ople s
Ma hema ics 2021,9, 1582. h ps://doi.o g/10.3390/ma h9131582 h ps://www.mdpi.com/jou nal/ma hema ics
Ma hema ics 2021,9, 1582 2 o 15
lea e ou mou hs hei empe a u e immedia ely d ops, acqui ing he empe a u e o he
en i onmen . The s udy de eloped by Li e al. [
6
] showed ha once exhaled d ople s e ap-
o a e and hei diame e dec eases, he e is an inc ease in he concen a ion o pa hogens
pe uni olume, leading o an inc ease in he isk o in ec ion. Wei e al. [
7
] concluded
ha in pa icles below 50
µ
m, e apo a ion is mos signi ican . The esea ch ca ied ou
by Mow aska e al. [
8
] claimed ha he mos impo an ac o o conside in he con a-
gion o a i us is he pa icle diame e . Xie e al. [
9
] s essed ha he size o a d ople
depends on wo ac o s: e apo a ion and he mo emen o he d ople s. The compu a ional
luid dynamics (CFD) nume ical model p o ided by Chillón e al. [
10
] indica ed ha he
mo emen o pa icles wi h a diame e o less han 30
µ
m is go e ned by he B ownian
e ec . On he o he hand, a e age pa icles up o 80
µ
m a e subjec o majo o ces, while
la ge diame e d ople s a e a ec ed by g a i y. Acco ding o Redow e al. [
5
], d ople s
wi h a diame e o 10
µ
m e apo a e a abou 550 milliseconds when ela i e humidi y is
80% ( e y we en i onmen ). They also s a ed ha he same pa icle, unde condi ions o
50% ela i e humidi y, e apo a es in 300 milliseconds, and in 20% ela i e humidi y in
250 milliseconds. These esul s a e in ag eemen wi h he wo k o Wang e al. [
4
], assu ing
ha ela i e humidi y is an impo an ac o in e apo a ion. Howe e , he e apo a ion
o hese ae osols does no emo e he chance o in ec ion, as shown in he wo k o Van
Do emalen e al. [
11
]. They de e mined ha SARS-CoV-2 has a hal -li e o abou 1 h in he
en i onmen whe eas i hose ae osols all on plas ic o s ainless s eel, he hal -li e will be
6.8 h and 5.6 h, espec i ely.
A oiding close con ac and main aining a social dis ance o 1.83 m (6 ee ) a e wo o
he main guidelines o p e en in ec ion. Howe e , Feng e al. [
12
] obse ed ha la ge
dis ances need o be conside ed due o en ila ion condi ions o ex e nal winds. The esul s
o Li e al. [
13
] showed ha wi h a wind speed o 2 m/s, a pa icle o 100
µ
m could each
up o 6.6 m. The compu a ional model by Dbouk e al. [
14
] also concluded ha wi h a wind
speed be ween 4 km/h and 15 km/h, he d ople s o sali a exhaled in o he en i onmen
could a e se 6 m. They also no ed ha pa icles dec ease hei concen a ion in he wind
di ec ion. On he o he hand, in an enclosed space, ha is, when ai speed is 0 m/s and
when i is no possible o keep he social dis ance, i is essen ial o use en ila ion s a egies
o educe he chance o in ec ion. Sen e al. [
15
] demons a ed his in hei CFD s udy inside
an ele a o . When he e is a en , he d ople s all o he g ound wi hou impac ing any
pe son. In con as , ae osols would each he pe son and could in ec hem.
Howe e , main aining a sa e social dis ance wi hou any measu e o indi idual p o ec-
ion is no enough, and, he e o e, wea ing masks has also been ecommended. P incipally,
he e a e wo ypes o masks: masks ha co e only he mou h, nose, and chin and ace
shields, which co e he whole ace (eyes, nose, and mou h). Rega ding he masks ha
co e he mou h, nose, and chin, he e a e se e al ypes and hey o e di e en unc ions;
o example, su gical masks las 4 h and p e en he exhaled ai om being dispe sed,
whe eas he pe sonal p o ec i e equipmen (PPE) ha ha e h ee le els o il e ing acepiece
espi a o s (FFP1, FFP2, FFP3) a y in hei e ec i eness a il a ion. They il e he inhaled
ai and p o ec agains he inhala ion o ae osols. The ace shields, howe e , a e plas ic
sc eens wi h he pu pose o co e ing ou aces and ac ing as a p o ec i e ba ie agains
ae osols. Akh a e al. [
16
] pe o med an expe imen al s udy o p o e he e ec i eness o
wea ing a mask when sneezing. Fi e di e en masks we e e alua ed: su gical, clo h, clo h
PM 2.5, we ed clo h PM 2.5, and N95. They showed ha he ae osols came ou in o he
en i onmen while wea ing all masks excep o he N95 mask. The ma hema ical model by
A umu u e al. [
17
] ag ees wi h Akh a e al. [
16
] in s a ing ha he N95 masks comple ely
p e en leakage o d ople s in he di ec ion o ad ancemen , al hough hey obse ed ha
he d ople s dispe se o he en i onmen be ween he holes o he mask and he nose. The
mask ecommended by A umu u e al. [
17
] ha has a lowe leakage a io is he comme cial
i e-laye ed mask. I can be concluded ha when wea ing bo h masks combined, he
isk o in ec ion can be educed. Recen ly, Salimnia e al. [
18
] unde ook a s udy on he
p o ec ion o e ed by he wo ypes o masks and he p o ec ion o e ed by bo h combined,
Ma hema ics 2021,9, 1582 3 o 15
and no imp o emen s we e no ed when combining hem. A umu u e al. [17] also ag ees
wi h Salimnia e al. [
18
], a guing ha he combina ion o he wo masks is no easible. In
con as , he compu a ional model by Akagi e al. [
19
] in es iga ed he p o ec ion o e ed
by ace shields, concluding ha his ype o mask is no a good p o ec ion ool, as 4.4%
o he d ople s exhaled in o he en i onmen en e he gap be ween he human and he
ace shield. Wendling e al. [
20
] ca ied ou an expe imen al wo k o compa e he ba ie
pe o mance o ace masks and ace shields. The expe imen al da a showed ha , in a
con e sa ion be ween wo people, a mask does no p o ec he human who sneezes, bu he
one who is sneezed on is p o ec ed. Mo eo e , when a ace shield is used, he p o ec ion is
be e since i educes he numbe o pa icles. Besides, when he human who sneezes is
he one who is p o ec ed, he e is ha dly any di e ence be ween he wo masks.
Se e al nume ical wo ks o mul iphase lows coupled wi h incomp essible luid
mo ion can be ound in he li e a u e, such as he con e gence analysis o a ully disc e e
ini e di e ence scheme o he Cahn–Hillia d–Hele–Shaw equa ion p esen ed by Chen
e al. [
21
] and he e o analysis o a mixed ini e elemen me hod p oposed by Feng and
P ohl [
22
]. In ha ield, Yan e al. [
23
] showed ha a second-o de ene gy-s able scheme o
he Cahn–Hillia d–Hele–Shaw equa ion was able o p oduce accu a e long- ime nume ical
esul s wi h a easonable compu a ional cos . The coupling o he Cahn–Hillia d equa ion
o he Na ie –S okes equa ion o luid low gained he a en ion o Diegel e al. [
24
], wi h a
con e gence analysis and e o es ima es o a second-o de accu a e ini e elemen me hod.
In he cu en wo k, a CFD nume ical model is p esen ed wi h he aim o analyzing
he sp ead o a sneeze con aining i al d ople s and e alua ing he e ec i eness o a ace
shield as a pe sonal p o ec i e de ice. The human who wea s he ace shield has a heigh
o 1.8 m and is placed a a social dis ance o 1.5 m om ano he human. Two scena ios we e
s udied. In he i s scena io, he mou hs o bo h indi iduals a e si ua ed a he same heigh ,
while in he second scena io he mou hs a e loca ed a a heigh di e ence o 0.2 m. The
e apo a ion o ae osols was checked by modi ying he ambien empe a u e and ela i e
humidi y. Two di e en empe a u es we e applied, 25
◦
C and 15
◦
C, and ela i e humidi y
(RH) alues o 40% and 60% we e s udied o each case. In addi ion, wo di e en wind
speeds, 7 km/h and 14 km/h, we e added o he nume ical model o e i y he social
dis ance, wi h a a o able wind in he same di ec ion as he sneeze.
2. Ma e ials and Me hods
2.1. Valida ion
Fo he alida ion o ou nume ical model, he e apo a ion o a single d ople o e
ime was nume ically simula ed. Th ee di e en diame e s we e e alua ed: 1
µ
m, 10
µ
m,
and 100
µ
m. The empe a u e o he en i onmen and he d ople we e T
e
= 293.15 K
and T
d
= 310.15 K, espec i ely. Figu e 1shows he esul s wi h ou di e en ela i e
humidi y a ios (0%, 20%, 60%, and 80%), he same me hod used by Redow e al. [
5
],
Mow aska e al. [8], and Li e al. [13].
In addi ion, he s udy o Hamey [
25
] was used o alida e he dis ance ha a single
d ople can a e se. The s udy consis ed o a ee all wa e d ople in o a we space o
ela i e humidi y RH = 70%, T
e
= 293 K, and T
d
= 289 K, and d ople dime e o 110
µ
m o
115 µm, as shown in Figu e 2.
Ma hema ics 2021,9, 1582 4 o 15
Figu e 1.
E apo a ion o a single pu e wa e d ople unde di e en en i onmen condi ions,
en i onmen empe a u e o 293.15 K, pa icle empe a u e 310.15 K, and di e en d ople sizes
(1 µm, 10 µm, and 100 µm).
Figu e 2.
F eely alling wa e d ople s based on he s udy o Hamey (1982). Pa icle sizes o 110
µ
m
and 115
µ
m, en i onmen empe a u e 293 K, pa icle empe a u e 289 K, and 70% ela i e humidi y.
Ma hema ics 2021,9, 1582 5 o 15
2.2. Compu a ional Domain and Ini ial Condi ions
CFD echniques, based on nume ical me hods and algo i hms, allow o he analysis
o complex p oblems unde a wide ange o condi ions and pa ame e s. In he cu en
wo k, an indi idual was placed in an open h ee-dimensional domain. The domain size
was 3 m
×
2 m
×
2.5 m (X, Y, Z), bu he walls we e simula ed as a symme y plane. The
human had a heigh o 1.8 m and he heigh o i s mou h was loca ed a 1.6 m.
Figu e 3
shows he geome y o he subjec and he ace shield ha i is wea ing ha co e s i s
whole ace. Table 1shows he main dimensions o he ace shield.
Figu e 3. Human geome y: (a) Human body (1.8 m all); (b) Face shield in de ail.
Table 1. Face shield dimensions.
Dimension Value
Heigh 250 mm
Leng h 196 mm
A ea 660 mm2
Ano he human has been placed in on o he i s a a dis ance o 1.5 m. Only
he mou h has been modeled o simula e he sneezing by means o a su ace injec o , see
Figu e 4
. The ae osols and ai je a e exhaled in o he en i onmen a a empe a u e o
36
◦
C, he a e age empe a u e o he human body. Acco ding o Ca pen e e al. [
26
]
sali a is he mos cha ac e is ic luid o he body, ha ing o pe o m he unc ion o as e
and being composed o di e en elemen s, gi ing ise o a composi ion ha is di icul o
ec ea e. Nicas e al. [
27
] summa izes he composi ion o sali a in wa e and non- ola ile
solids. The concen a ion o ions and ca ions is 150 mM, which in e ms o mass equals
8.8 g/L. In he cu en s udy, sali a was modeled as a saline mix u e in which i only has
in luence in educing he sa u a ion p essu e o he wa e . The mass o each sneeze was
6.7 mg, based on he esea ch o Zhu e al. [
3
] showing he a e age mass o ae osols pou ed
in o he en i onmen in each sneeze. The mou h geome y was de ined as: DM = 40 mm
and dm = 20 mm, as shown in Figu e 4c, in o de o ep esen he mou h shape when
sneezing. A sneezing eloci y condi ion o 16 m/s wi h a du a ion o 400 milliseconds
we e used. In he i s scena io, he mou hs o he wo indi iduals we e a he same heigh .
Wi h he aim o compa ing he e ec o he people’s heigh , a second case was modeled
wi h a di e ence be ween he heigh o he wo mou hs o 20 cm. Fo each heigh o mou h,
wo di e en empe a u es we e applied, one wa me a 25
◦
C and ano he coole a 15
◦
C.
Since he a e age sui able ela i e humidi y ange is 40–60%, each ambien empe a u e
was simula ed i s wi h 40% and hen wi h 60% ela i e humidi y. To model an open
space, wind ac o was included, gi ing a compa ison be ween a so b eeze o 7 km/h
Ma hema ics 2021,9, 1582 6 o 15
and a mode a e wind o 14 km/h. The wind had he same di ec ion as he je exhaled by
he subjec .
Figu e 4.
Posi ion o he mou h a 1.5 m: (
a
) The same heigh as he human wi h he ace shield,
1.6 m; (
b
) A di e ence o heigh o 20 cm; (
c
) Geome y o he mou h (DM = 40 mm and dm = 20 mm)
o simula e a sneeze.
The disc e iza ion o he compu a ional domain was made by means o a imme
mesh. The domain was composed o 6.7
×
10
6
(6,755,678) cells. The mos impo an zone
o his s udy was he ace shield, whe e he d ople s we e expec ed o impac , he e o e,
a ine mesh was made nea i using a olume con ol (VC). The same olume con ol was
also used in he di ec ion o sneezing. In addi ion, a second olume con ol was used abo e
and below he je . Figu e 5a illus a es he mesh used when he mou hs we e a he same
heigh and Figu e 5b show he meshes used when he di e ence be ween he heigh o he
wo mou hs was 20 cm.
Figu e 5.
Mesh dis ibu ion, he block ( ) indica es he mou h o he indi idual ha has he i us
and sneezes: (
a
) The mesh in he domain when he mou hs we e a he same heigh ; (
b
) The mesh in
he domain when he mou hs we e a di e en heigh s (20 cm di e ence).
2.3. Nume ical Se up
The undamen al laws ha go e n he mechanics o luids a e he conse a ion o
mass, linea momen um, and ene gy, see Equa ions (1)–(3).
∂ρ
∂ +∇·(ρ )=0 (1)
∂(ρ )
∂ +∇·(ρ ⊗ )=∇·(pI)+∇·T+ b(2)
∂(ρE)
∂ +∇·(ρE )= b· +∇·( ·σ)− ∇·q+SE(3)
Ma hema ics 2021,9, 1582 7 o 15
whe e
ρ
is he densi y, ha is, he mass pe uni olume, is he con inuum eloci y,
⊗
deno es he ou e p oduc ,
b
is he esul an o he body o ces pe uni olume ac ing on
he con inuum,
σ
is he s ess enso , pis he p essu e, Tis he iscous s ess enso , Eis he
o al ene gy pe uni mass, qis he hea lux, and SEis an ene gy sou ce pe uni olume.
The go e ning equa ions o he con inuous phase, composed o d y ai and wa e
apo , we e exp essed in Eule ian o m, whe eas he Lag angian desc ip ion was used o
sol e he dispe sed sali a d ople s as hey c ossed he compu a ional domain. This homo-
geneous composi ion has, in all cases, he same empe a u e, p essu e, and eloci y. This is
a non- eac ing species; a e de e mining he p ope ies o each species, he p ope ies o
he mix u e a e calcula ed as a mass unc ion o he mix u e’s componen s. To ha end, he
mass-weigh ed mix u e me hod p oposed by Busco e al. [28] was used, see Equa ion (4).
φmix =
N=2
∑
i=1
φiYi. (4)
whe e Y
i
is he mass ac ion o ai and wa e apo and
φi
is he p ope y alues o mix u e
componen . Nis he o al numbe o componen s in he mix u e; in his case, N= 2.
Rela i e humidi y depends on he ini ial ai and wa e apo mass ha is p esen in
he en i onmen and he empe a u e o he ai low. The equilib ium p essu e is he mos
signi ican pa ame e ha a ec s i . To calcula e he densi y and iscosi y o ai , apo ,
and liquid wa e he equa ions o Kukkonen e al. [
29
] we e used. These p ope ies we e
upda ed as he empe a u e was a ied. As was men ioned p e iously in his sec ion, sali a
is composed mainly o wa e and non- ola ile solids. Howe e , in o de o simpli y he
ma hema ical model, an assump ion was made o conside sali a as a saline solu ion. This
ac o will only in luence he equilib ium p essu e o he wa e since dissol ed ino ganic
sal s help o educe he sa u a ion p essu e o he wa e . Xie e al. [
9
] showed ha he
sa u a ion p essu e o hese d ople s can be calcula ed using Raoul ´s law.
P a,s=XdP a(Tw)(5)
whe e
P a,s
is he sa u a ion p essu e o he d ople in he saline mix u e,
P a
is he equilib-
ium p essu e o he wa e a a speci ic empe a u e
(Tw)
, and
Xd
is he mole ac ion o
he d ople , calcula ed as shown in Equa ion (6).
Xd= 1+6imsMw
πρLMs(dp)3!−1
(6)
whe e M
w
and M
s
a e he molecula weigh s o wa e and o solu e, espec i ely; m
s
he
mass o he solu e in he d ople ;
dp
is he diame e o he pa icle (d ople ), and he ion
ac o “i” is equal o 2, in his case o he NaCl.
D ople s, which c oss he compu a ional domain, we e modeled by Lag angian equa-
ions. In o de o calcula e he in e ac ions be ween he mix u e o d y ai and wa e apo
and he d ople s, he wo-way coupling model was used. When d ople s a e exhaled in o
he en i onmen , e apo a ion begins due o a empe a u e di e ence. The e o e, a quasi-
s eady e apo a ion model, which allows d ople s o lose mass h ough e apo a ion, was
in oduced and de e mined by She wood
´
s numbe . Equa ion (7) exp esses he a e o
change o d ople mass due o e apo a ion [28].
.
mp=g∗×Asln(1+B)(7)
whe e Bis he Spalding ans e numbe ,
g∗
is he mass ans e conduc ance, and
As
is
he d ople su ace a ea. Howe e , his phenomenon is no he only one ha occu s, he
d ople s may also be dis o ed and b eak up unde he ac ion o non-uni o m su ace o ces.
This beha io was aken in o accoun using he Taylo analogy b eakup (TAB) model.
Ma hema ics 2021,9, 1582 8 o 15
Taylo analogy ep esen s he dis o ion o a d ople as in a damping sp ing mass sys em;
i e lec s only he basic mode o oscilla ion o he d ople .
The dis ance ha hese d ople s can achie e is a ec ed by he dispe sion and u bu-
lence hey su e . This e ec was in oduced in he p esen wo k wi h he addi ion o he
Reynolds-a e aged Na ie –S okes (RANS) equa ions wi h a k-
ω
Shea S ess T anspo
(SST) u bulence model, de eloped by Men e [
30
]. The anspo equa ions o he kine ic
ene gy kand he speci ic dissipa ion a e ωa e shown in Equa ions (8) and (9).
∂
∂ (ρk)+∇·(ρk )=∇·[(µ+σkµ )∇k]+Pk−ρβ∗ β∗(ωk−ω0k0)+Sk(8)
∂
∂ (ρω)+∇·(ρω )=∇·[(µ+σωµ )∇ω]+Pω−ρβ∗ β∗ω2−ω2
0+Sω(9)
whe e
is he mean eloci y,
µ
is he dynamic iscosi y,
σk
,
σω
, P
k
,and P
ω
a e P oduc ion
Te ms,
β*
is he ee-shea modi ica ion ac o , S
k
and S
ω
a e he use -speci ied sou ce
e ms, and k0and ω0a e he ambien u bulence alues ha coun e ac u bulence decay.
Conside ing expe imen al da a shown by Xie e al. [
31
] and subsequen ly alida ed
by Dbouk e al. [
14
], he ini ial size dis ibu ion o he d ople s ep esen ing he sneezing
was modeled using he Rosin–Rammle dis ibu ion. Equa ion (10) shows he exp ession
o de ine he p obabili y densi y unc ions
, also known as he Weibull dis ibu ion.
A minimum diame e o 10 µm and a maximum diame e o 300 µm we e de ined.
=n
dp dp
dp!n−1
e−(d
dp)n
,n=8 (10)
whe e dpis he mean diame e and is equal o 80 µm.
The equa ion o conse a ion o linea momen um o a ma e ial d ople o mass m
p
is
gi en by Equa ion (11).
mp
d p
d =Fs+Fb(11)
whe e
p
deno es he ins an aneous pa icle eloci y, F
s
is he esul an o he o ces ha ac
on he su ace o he pa icle, and Fbis he esul an o he body o ces.
Sphe ical pa icles we e assumed and he ene gy ha hese sphe ical pa icles elease
is implemen ed wi h he co ela ion o Ranz–Ma shall. This co ela ion is sui able o
sphe ical pa icles up o Re
≈
5000 and was applied in he e olu ion o he mass o sali a-
only d ople pa icles due o e apo a ion. This co ela ion de ines he coe icien o hea
ans e as a de i a i e co ela ion as a unc ion o he Nussel numbe . The d ople s
injec ed in o he compu a ional domain we e unde he in luence o he d ag o ce, de ined
by Equa ion (12).
Fd=1
2CdρAp| s| s(12)
we e C
d
is he d ag coe icien o he d ople ,
ρ
is he densi y o he con inuous phase, A
p
is he p ojec ed a ea o he d ople , and
s
is he d ople slip eloci y (
s
=
−
p
) wi h
being he ins an aneous eloci y o he con inuous phase.
The Schille –Naumann model was used o calcula e his o ce, he same me hod
employed by Wang e al. [
4
]. Acco ding wi h Ka una a hne e al. [
32
], his model was used
o modelling o he d ag be ween luid phases in a mul iphase low. Equa ion (13) shows
he exp ession o calcula e he d ag coe icien Cd.
Cd
24
Re ,Re ≤1
24
Re 1+0.15Re0.687, 1 <Re ≤1000
0.44, Re >1000
(13)
whe e Re is he Reynolds numbe .
Ma hema ics 2021,9, 1582 9 o 15
The in luence o he g a i y o ce was aken in o accoun as was he u bulen pa icle
dispe sion wi h he exac eddy in e ac ion ime. In his wo k, he comme cial CFD code
STAR-CCM + .14.02 (Siemens, London, UK) [
33
] was used o de ine and sol e he nume i-
cal model o ae osols p oduced by sneezing. A pe sonal se e -clus e ed pa allel compu e
wi h In el Xeon
©
E5-2609 2 CPU @ 2.5 GHz (16 co es) and 45 GB RAM we e used o un
all he simula ions. Each simula ion ook abou 35 h o compu a ion, wi h a o al ime o
app oxima ely 24 days.
3. Resul s
3.1. E ec o Rela i e Humidi y and En i onmen Tempe a u e in he D ople s’ E apo a ion
In his sec ion, he in luence o ela i e humidi y and ambien empe a u e a e an-
alyzed. To ha end, he quan i y o pa icles exhaled and hei co esponding diame e
we e obse ed. Fi s ly, a compa ison be ween = 0.4 s, when he sneeze is inished, and
= 2.5 s
, when mos o he d ople s ha e impac ed he pe son, is p esen ed a
Te= 15 ◦C
and
RH = 40%
. Secondly, a compa ison is made o e alua e he in luence o ela i e hu-
midi y a = 2.5 s in he ollowing scena ios: T
e
= 15
◦
C and RH = 40% and T
e
= 15
◦
C and
RH = 60%. Finally, a compa ison be ween T
e
= 15
◦
C and RH = 40% and T
e
= 25
◦
C and
RH = 40% is p esen ed o see he di e ence in ambien empe a u e.
The e olu ion o he d ople s exhaled o he en i onmen was acked a = 0.4 s
and = 2.5 s and a e shown in Figu e 6, wi h a d ople size anging om 19
µ
m o 96
µ
m
diame e . A = 2.5 s, a maximum d ople size o 92
µ
m and a minimum d ople size o
19
µ
m we e obse ed. A = 0.4 s, in con as , he e we e no d ople s o 19
µ
m diame e ,
bu 461 d ople s o 29
µ
m we e obse ed. In ha ime ame, he e apo a ion p ocess akes
place, and he 29 µm d ople s manage o e apo a e down o a diame e o 19 µm.
Figu e 6.
The dis ibu ion o he d ople s size a = 0.4 s and = 2.5 s when he en i onmen empe a u e was 15
◦
C and he
ela i e humidi y was 40%.
