nanoma e ials
A icle
Nume ical Compa ison o P edic ion Models
o Ae osol Fil a ion E iciency Applied on
a Hollow-Fibe Memb ane Po e S uc u e
Pa el Bulejko 1,2 ID
1Hea T ans e and Fluid Flow Labo a o y, Facul y o Mechanical Enginee ing, B no Uni e si y o
Technology, Technická2, 616 69 B no, Czech Republic; [email p o ec ed]; Tel.: +420-541-144-912
2MemB ain s. .o. Pod Vinicí87, 471 27 S áž pod Ralskem, Czech Republic
Recei ed: 18 May 2018; Accep ed: 15 June 2018; Published: 19 June 2018


Abs ac :
Hollow- ibe memb anes (HFMs) ha e been widely applied o many liquid ea men
applica ions such as was ewa e ea men , memb ane con ac o s/bio eac o s and memb ane
dis illa ion. Despi e he ac ha HFMs a e widely used o gas sepa a ion om gas mix u es, hei use
o mechanical il a ion o ae osols is e y sca ce. In his wo k, we compa ed ma hema ical models
de eloped o he p edic ion o ai il a ion e iciency by applying hem on he s uc u al pa ame e s
o polyp opylene HFMs. These memb anes a e cha ac e is ic o po e diame e s o abou 90 nm
and ha e high solidi y, hus p o iding high po en ial o nanopa icle emo al om ai . A single
ibe /collec o and capilla y po e app oach was chosen o compa e be ween models de eloped o
ib ous il e s and capilla y-po e memb anes (Nuclepo e il e s) based on h ee main mechanisms
occu ing in ae osol il a ion (ine ial impac ion, in e cep ion and di usion). The collec ion e iciency
due o indi idual mechanisms di e s signi ican ly. The di e ences a e caused by he pa ame e s o
which he indi idual models we e de eloped, i.e., gi en alues o go e ning dimensionless numbe s
(Reynolds, S okes and Pecle numbe ) and also gi en alues o il e po osi y and il e ibe diame e .
Some models can be used o p edic he e iciency o HFMs based on assump ions depending on he
condi ions and exac memb ane pa ame e s.
Keywo ds:
hollow- ibe memb ane; ae osol; il a ion e iciency; in e cep ion; ine ial impac ion;
di usion
1. In oduc ion
Ai il a ion is he mos equen ly used me hod o ai bo ne pa icula e ma e mi iga ion [1,2].
Dus , alle gens, mic oo ganisms, welding umes and combus ion-gene a ed pa icles ha e been o
g owing in e es due o associa ed heal h conce ns. I has been ound ha he e is a di ec ela ionship
be ween inc eased concen a ions o ai bo ne pa icles and human heal h diso de s [
3
–
7
]. Wi h
ega d o inc easing nano echnology applica ions, ai bo ne nanopa icles ha e been o g owing
in e es [
8
–
11
] as well as echnologies o hei mi iga ion. This mainly en ails he de elopmen
o a ious il a ion ma e ials based on nano ibe s o a memb ane s uc u e [
12
–
15
]. The o me
has ecen ly been a subjec o many wo ks while he la e was o in e es mainly when dealing
wi h capilla y po e memb anes (CPMs, so called Nuclepo e il e s) used o measu ing wo kplace
exposu e [
16
–
18
]. P edic ing he pe o mance o such il e s/memb anes was o g ea conce n in
e ms o hei pa icle emo al e iciency, including minimum e iciency, mos pene a ing pa icle size
(MPPS) and p essu e d op. The e o e, many wo ks ha e been ca ied ou o de elop ma hema ical
exp essions o calcula e il a ion e iciency in ela ion o pa icle size. Di e en models we e de eloped
o e iciency p edic ions o ib ous il e s and CPMs. While il a ion mechanisms o a CPM ela es
mainly o su ace il a ion and sie ing, ib ous il e s sepa a e pa icles mainly ia ine ial impac ion,
Nanoma e ials 2018,8, 447; doi:10.3390/nano8060447 www.mdpi.com/jou nal/nanoma e ials
Nanoma e ials 2018,8, 447 2 o 24
in e cep ion, B ownian mo ion (di usion), g a i a ional se ling and elec os a ic deposi ion in elec e
il e s [19].
Hollow- ibe memb ane (HFM) is a special ype o memb ane geome y cha ac e is ic o
compac ness: I con ains a la ge su ace a ea in a small olume. Thus a , all applica ions o po ous
HFMs could ha e been ound mos ly in wa e ea men applica ions [
20
–
23
] and non-po ous HFMs
in gas sepa a ion and hea exchange s [
24
–
26
]. Such geome y ensu es a high su ace a ea necessa y
o mass/hea ans e applica ions. HFMs can ha e a symme ic o asymme ic po ous s uc u e
depending on he way o p epa a ion. A symme ic s uc u e o HFM can be achie ed in a numbe o
ways, such as ia d y s e ching o ex uded polyme ic hollow ibe [
27
]. An asymme ic s uc u e is
ob ained when a hin skin laye is coa ed in a HFM su ace (shell side o lumen side) o can be in eg al
i.e., he po e size dec eases in he di ec ion o memb ane su ace [28–30].
The numbe o publica ions conce ning he use o HFMs in gas il a ion applica ions is e y
sca ce. The e ha e ecen ly been only h ee publica ions ocused on he sepa a ion o solid pa icles
om ai using HFMs. Wang e al. [
31
] p epa ed asymme ic poly inylidene luo ide-polye hylene
glycol (PVDF-PEG) HFMs and es ed o ai il a ion pe o mance agains ul a ine polydispe se NaCl
ae osol wi h a geome ic a e age pa icle size o 30 nm. The esul s showed a high il a ion e iciency
o 99.999%. In he o he wo k, Li e al. [
32
] ocused on design and cha ac e iza ion o HFMs based on
poly(e he sul one) p epa ed ia d y-je we spinning. They p epa ed an asymme ic HFM composed
o a ib ous-like po ous subs a e wi h a memb ane sie e-like laye on i s su ace and achie ed an
e iciency o 99.995% when challenged wi h less han 300 nm ammonium sul a e pa icula es. Las ly,
he au ho s o he hi d publica ion [
27
] used symme ic polyp opylene HFMs o sepa a e polydispe se
nanoae osol gene a ed using incense s ick bu ning. They disco e ed ha he HFMs ha e MPPS in
ange o 34–40 nm wi h a MPPS e iciency o 79–87% depending on low a e. Fu he mo e, he esul s
showed e iciency le els highe han 99% o pa icles abo e 60 nm and emain unchanged wi h
he low a e. Po en ial applica ions o HFM as ai il e s a e mainly in low olume applica ions.
Such applica ions include p in ing boa d il e s, mic oelec onics, s e ilized wa e ank en ila ion and
clean ai o sensi i e analy ical o medical de ices. The HFM can p o ide high e iciency due o i s
na ow po e size dis ibu ion and small po e sizes in he ange o 90 nm. Mo eo e , HFMs can p o ide
long se ice li e due o possibili y o simple egene a ion.
In his wo k, we applied ma hema ical models de eloped o p edic ion o ai il a ion e iciency
o ib ous il e s and CPMs on a HFM po e s uc u e. Fo calcula ion, we used pa ame e s o symme ic
polyp opylene HFM p oduced by ZENA Memb anes s. .o. [
33
]. F om he po ous s uc u e o hese
HFMs (Figu e 1), and compa ing wi h a ypical s uc u e o a ib ous il e (see e.g., [
12
,
34
–
38
] and a
CPM (see e.g., [
39
–
43
]), we can see se e al simila i ies bu also se e al main di e ences. Fi s , he HFM
po e s uc u e is composed o longi udinal segmen s ( e e ed o as collec o s) wi h an a e age diame e
o abou 90 nm. These can be conside ed ibe s analogically o ib ous il e s. Second, he po e s uc u e
con ains ellip ical po es ha a e analogical o CPM, which has ci cula po es. Con e sely, HFMs ha e
e y high solidi y (he e 0.48) compa ed o comme cial ib ous il e s, which ypically ha e solidi y
be ween 0.01 and 0.3 [44]. So wi h some assump ions, he models o ib ous il e s and CPMs can be
applied on he HFMs conside ed in his s udy. The e o e, he main e o o his wo k is o compa e
hese models by nume ically applying hem on HFM assuming ha he collec ion mechanisms a e
analogical o hose conside ed in ib ous il e s and CPMs. Based on single ibe heo y de eloped o
ib ous il e s, we de e mined single collec o e iciencies (SCE) based on di e en mechanisms aking
place in ae osol il a ion. The e iciency esul s we e compa ed be ween SCE models o indi idual
mechanisms de eloped by a ious esea che s. The e o e, his wo k can also se e as an o e iew o
ma hema ical models o SCE due o di e en cap u ing mechanisms.
Nanoma e ials 2018,8, 447 3 o 24
Nanoma e ials 2018, 8, x FOR PEER REVIEW 3 o 25
Figu e 1. Polyp opylene HFM po e s uc u e.
2. P edic ion Models o Ai Fil a ion E iciency
Ai il a ion ma e ials o whole ai il a ion uni s a e mos ly e alua ed in e ms o il a ion
e iciency and p essu e d op. The o me desc ibes he abili y o a il e uni o emo e pa icles om
ai s eam while he la e one is ela ed mainly o ene gy equi emen s. The il a ion e iciency η is
de ined as ollows:
up
down
C
C
−=1
η
(1)
whe e Cdown and Cup a e he numbe o pa icles downs eam and ups eam o he il e , espec i ely.
2.1. E iciency P edic ion o Fib ous Fil e s
Non-wo en ib ous il e s a e composed o ibe s, which a e andomly o ien ed e en hough he
o ien a ion is mos ly no mal o he ai low. The diame e o ibe s is mos ly no uni o m and can be
p oduced om a ious mos ly polyme ic ma e ials. The il a ion e iciency o ib ous il e s may be
p edic ed based on se e al pa ame e s and assump ion o an idealized il e s uc u e. The o mula
is as ollows [45]:






−
−−=
)1(
4
exp1 d
Z
απ
αη
η
(2)
whe e α, η , Z and d a e he il e solidi y, SCE, il e hickness and a e age collec o diame e ,
espec i ely. The o al SCE is a sum o con ibu ions om di e en collec ion mechanisms and can
be w i en as ollows:
ADRI )(
η
η
η
η
η
++= (3)
whe e ηI, ηR, ηD and ηA a e he single collec o e iciencies due o ine ial impac ion, in e cep ion,
di usion and adhesion, espec i ely. The il a ion heo y, which is based on h ee main mechanisms,
ine ial impac ion, in e cep ion and di usion (Figu e 2), does no ake in o accoun pa icle- ibe
in e ac ion, i.e., he pa icle ebound and e-en ainmen . The e o e, we used Equa ion (3) o calcula e
he SCE based on collision e iciency (sum o collec ion e iciencies due o impac ion in e cep ion and
di usion) mul iplied by he collec ion e iciency caused by adhesion e ec s [46,47].
Figu e 1. Polyp opylene HFM po e s uc u e.
2. P edic ion Models o Ai Fil a ion E iciency
Ai il a ion ma e ials o whole ai il a ion uni s a e mos ly e alua ed in e ms o il a ion
e iciency and p essu e d op. The o me desc ibes he abili y o a il e uni o emo e pa icles om
ai s eam while he la e one is ela ed mainly o ene gy equi emen s. The il a ion e iciency
η
is
de ined as ollows:
η=1−Cdown
Cup (1)
whe e Cdown and Cup a e he numbe o pa icles downs eam and ups eam o he il e , espec i ely.
2.1. E iciency P edic ion o Fib ous Fil e s
Non-wo en ib ous il e s a e composed o ibe s, which a e andomly o ien ed e en hough he
o ien a ion is mos ly no mal o he ai low. The diame e o ibe s is mos ly no uni o m and can be
p oduced om a ious mos ly polyme ic ma e ials. The il a ion e iciency o ib ous il e s may be
p edic ed based on se e al pa ame e s and assump ion o an idealized il e s uc u e. The o mula is
as ollows [45]:
η=1−exp−4αη Z
π(1−α)d (2)
whe e
α
,
η
,Zand d
a e he il e solidi y, SCE, il e hickness and a e age collec o diame e ,
espec i ely. The o al SCE is a sum o con ibu ions om di e en collec ion mechanisms and can be
w i en as ollows:
η = (ηI+ηR+ηD)ηA(3)
whe e
ηI
,
ηR
,
ηD
and
ηA
a e he single collec o e iciencies due o ine ial impac ion, in e cep ion,
di usion and adhesion, espec i ely. The il a ion heo y, which is based on h ee main mechanisms,
ine ial impac ion, in e cep ion and di usion (Figu e 2), does no ake in o accoun pa icle- ibe
in e ac ion, i.e., he pa icle ebound and e-en ainmen . The e o e, we used Equa ion (3) o calcula e
he SCE based on collision e iciency (sum o collec ion e iciencies due o impac ion in e cep ion and
di usion) mul iplied by he collec ion e iciency caused by adhesion e ec s [46,47].
Nanoma e ials 2018,8, 447 4 o 24
Nanoma e ials 2018, 8, x FOR PEER REVIEW 4 o 25
Figu e 2. Schema ic p esen a ion o indi idual collec ion mechanisms a a single collec o .
2.1.1. SCE Due o B ownian Mo ion
Fil a ion e iciency due o di usion (B ownian mo ion) is a signi ican pa o he o e all
il a ion e iciency. The andomly changing ajec o y o e y small pa icles (Figu e 2) inc eases he
p obabili y o hi ing he collec o and hei cap u e by il e . The go e ning pa ame e o di usion
mechanism is Pecle numbe , which is he a io o con ec ion and di usion anspo a e as ollows:
D
Ud
Pe
= (4)
whe e U is he ace eloci y and D is he di usion coe icien o pa icle calcula ed as ollows:
p
sB
3d
TCk
D
πμ
= (5)
whe e kB, T, µ and dp a e he Bol zmann cons an , absolu e empe a u e, ai dynamic iscosi y and
pa icle diame e , espec i ely and Cs is he Cunningham slip co ec ion ac o :










−++= Kn
KnCs
78.0
exp44.0207.11 (6)
whe e Kn is he Knudsen numbe o pa icle wi h λ as mean ee pa h o gas molecules:
p
2
d
Kn
λ
= (7)
Se e al ela ionships ha e been p oposed o p edic SCE due o di usion (ηD). Fo nanopa icles
ha ha e high di usion coe icien , hence smalle Pecle numbe , Wang e al. [48] ga e he ollowing
ela ionship:
43.0
D84.0 −
=Pe
η
(8)
Equa ion (8) sugges s a lowe dependence o di usion e iciency on he Pecle numbe , hough
i is in good ag eemen wi h expe imen al da a o whole ange o Pecle numbe s. Ano he
ela ionship was p oposed by Ki sch and Fuchs [49]:
3/2
D7.2 −
=Pe
η
(9)
Equa ions (8) and (9) does no include he e ec o low ield dis o ion a he gas- ibe in e ace
and a e independen . The e o e, se e al esea che s p oposed di e en exp essions based on
heo e ical de i a ion o expe imen al da a. S echkina e al. [50] p oposed ollowing ela ionship:
13/23/1
D62.09.2 −−− += PePeKu
η
(10)
while analysis o Pich [51] and Lee and Liu [52] lead o Equa ions (11) and (12), espec i ely:
Figu e 2. Schema ic p esen a ion o indi idual collec ion mechanisms a a single collec o .
2.1.1. SCE Due o B ownian Mo ion
Fil a ion e iciency due o di usion (B ownian mo ion) is a signi ican pa o he o e all il a ion
e iciency. The andomly changing ajec o y o e y small pa icles (Figu e 2) inc eases he p obabili y
o hi ing he collec o and hei cap u e by il e . The go e ning pa ame e o di usion mechanism is
Pecle numbe , which is he a io o con ec ion and di usion anspo a e as ollows:
Pe =Ud
D(4)
whe e Uis he ace eloci y and Dis he di usion coe icien o pa icle calcula ed as ollows:
D=kBTCs
3πµdp(5)
whe e k
B
,T,
µ
and d
p
a e he Bol zmann cons an , absolu e empe a u e, ai dynamic iscosi y and
pa icle diame e , espec i ely and Csis he Cunningham slip co ec ion ac o :
Cs=1+Kn1.207 +0.44 exp−0.78
Kn  (6)
whe e Kn is he Knudsen numbe o pa icle wi h λas mean ee pa h o gas molecules:
Kn =2λ
dp(7)
Se e al ela ionships ha e been p oposed o p edic SCE due o di usion (
ηD
). Fo nanopa icles
ha ha e high di usion coe icien , hence smalle Pecle numbe , Wang e al. [
48
] ga e he ollowing
ela ionship:
ηD=0.84Pe−0.43 (8)
Equa ion (8) sugges s a lowe dependence o di usion e iciency on he Pecle numbe , hough i
is in good ag eemen wi h expe imen al da a o whole ange o Pecle numbe s. Ano he ela ionship
was p oposed by Ki sch and Fuchs [49]:
ηD=2.7Pe−2/3 (9)
Equa ions (8) and (9) does no include he e ec o low ield dis o ion a he gas- ibe in e ace
and a e independen . The e o e, se e al esea che s p oposed di e en exp essions based on heo e ical
de i a ion o expe imen al da a. S echkina e al. [50] p oposed ollowing ela ionship:
ηD=2.9Ku−1/3Pe−2/3 +0.62Pe−1(10)
Nanoma e ials 2018,8, 447 5 o 24
while analysis o Pich [51] and Lee and Liu [52] lead o Equa ions (11) and (12), espec i ely:
ηD=2.27Ku−1/3Pe−2/3(1+0.62KnPe1/3Ku−1/3)(11)
ηD=1.61−α
Ku 1/3
Pe−2/3 (12)
whe e Ku is he Kuwaba a hyd odynamic ac o . The Kuwaba a ac o compensa es he low ield
dis o ion a ound a collec o occu ing due o i s p oximi y o neighbo ing ibe s. The Kuwaba a ac o
is a dimensionless pa ame e and depends only on il e solidi y α o d ≥2µm as ollows:
Ku =−ln α
2+α−α2
4−3
4(13)
As he slip e ec becomes signi ican o il e s wi h ibe diame e smalle han 2
µ
m (which is
ue o HFMs conside ed in his wo k), Ki sch and S echkina [
53
] ecommended adding he Knudsen
numbe o ibe (13) o compensa e o he slip e ec :
Kn =2λ
d
(14)
Thus, o a ibe diame e smalle han 2 mic ons, he ela ionship o he Kuwaba a ac o is:
Ku =2λ
d −ln α
2+α−α2
4−3
4(15)
The same was p oposed o he ela ionships o di usion e iciency, i.e., modi ying using a
co ec ion ac o accoun ing o slip low when he ibe diame e is in he same magni ude as he
mean ee pa h o he gas molecules. Using he wo k o Lee and Liu [
52
] as a basis (Equa ion (12)),
Liu and Rubow [54] co ec ed his model o conside he slip e ec as ollows:
ηD=1.61−α
Ku 1/3
Pe−2/3C1(16)
whe e C1is a cons an calcula ed as ollows:
C1=1+0.388Kn (1−α)Pe
Ku1/3
(17)
Howe e , e iciencies calcula ed using Equa ion (16) migh exceed uni y o e y small pa icles
(low Pecle numbe s). The e o e, Paye e al. [
55
] in oduced ano he co ec ion ac o , o ge he
e iciency o e y small pa icles unde uni y, as ollows:
ηD=1.61−α
Ku 1/3
Pe−2/3C1C2(18)
whe e C2is calcula ed as ollows:
C2=1
1+1.61−α
Ku 1/3Pe−2/3C1
(19)
No e ha he cons an 1.6 in Equa ion (12) and he o he de i ed based on he same cons an may
be subs i u ed wi h a di e en alue (mos ly highe alue o 2.6 o 2.9) o ob ain a be e ag eemen
wi h expe imen al da a. The commonly used single collec o heo y was de eloped o he Kuwaba a

Nanoma e ials 2018,8, 447 6 o 24
cell model [
56
]. This model, howe e , does no conside possible he e ogenei ies o il e s uc u e
(local po osi y a ia ions) ela ed o non-uni o m ibe dis ibu ion o hei size polydispe si y [57].
2.1.2. SCE Due o In e cep ion
In e cep ion occu s when a pa icle ollowing luid s eamline lowing a ound he collec o is in a
dis ance o one pa icle adius om he collec o su ace (Figu e 1). The in e cep ion mechanism is
go e ned by he in e cep ion pa ame e R, which is he a io o pa icle o ibe diame e :
R=dp
d
(20)
In e cep ion e iciency inc eases by inc easing he in e cep ion pa ame e [
58
]. Following his,
he in e cep ion e iciency should be independen o he ai low eloci y, which is ue o mos models
de eloped o SCE due o in e cep ion. Howe e , conside ing he il e ibe s as isola ed cylinde s,
he in e cep ion e iciency ob ained om Lamb’s solu ion o Na ie -S okes equa ions [
59
] is dependen
on Reynolds numbe hence ai low eloci y. Langmui [
60
] de i ed his ela ionship o low Reynolds
numbe s (Re < 1) as ollows:
ηR=2(1+R)ln(1+R)−(1+R) + 1/(1+R)
2(2−ln Re )(21)
wi h Re as ibe Reynolds numbe cha ac e izing low ield a ound a ibe calcula ed as ollows:
Re =d Uρ
µ(22)
whe e
ρ
is he luid densi y. Majo i y o ma hema ical exp essions o in e cep ion e iciency a e based
on he Kuwaba a cell model [
56
] and a e independen o luid eloci y. Ki sch and S echkina ga e a
comple e model o SCE due o in e cep ion as ollows [53]:
ηR=1+R
2Ku "2 ln(1+R)−1+α+1
1+R21−α
2−α
2(1+R)2#(23)
This is he basic o mula o he SCE due o in e cep ion based on he Kuwaba a low ield.
Howe e , i is a a he long and complica ed exp ession, which Lee and Liu educed o ollowing
simple o ms [52]:
ηR=1−α
Ku
R2
1+R(24)
ηR=0.61−α
Ku
R2
1+R(25)
Equa ion (24) is alid o R< 0.2 and
α
< 0.5. Wi h he assump ion ha ibe s a e no o ien ed
pe pendicula o he low di ec ion and o non-uni o m ibe dis ibu ion, Lee and Liu [
52
] modi ied
Equa ion (24) by mul iplying i by a coe icien o 0.6. The in e cep ion e iciency model can hus
be simpli ied e en hough i has se e al limi a ions, mainly small in e cep ion pa ame e and il e
solidi y, he la e o which is no oo es ic i e and can be used o calcula ions in his wo k. Se e al
in es iga o s sugges ed o he co ec ions o Equa ion (23). Fo example, S echkina and Fuchs [
61
]
app oxima ed his ela ionship by omi ing all he e ms con aining he il e solidi y
α
and ob ained
he ollowing equa ion:
ηR=1+R
2Ku "2 ln(1+R)−1+1
(1+R)2#(26)
Nanoma e ials 2018,8, 447 7 o 24
The limi a ions a e he same as o Equa ion (22) i.e., Rand
α
mus be small. Owing o he
omission o il e solidi y, he app oxima ion is less accu a e wi h inc easing solidi y. The e o e, hey
p oposed ano he modi ica ion as ollows:
ηR=2.4α1/3R1.75 (27)
Lee and Gieseke [62] p oposed ano he modi ica ion o Equa ion (24) as ollows:
ηR=1−α
Ku
R2
(1+R)2
3(1−α)
(28)
None o he p edic ion models o in e cep ion e iciency (Equa ions (23)–(28)) conside s he gas
slip e ec . Pich [
63
] p oposed a ela ionship o in e cep ion e iciency, conside ing gas slip, o small
Knudsen numbe s:
ηR=(1+R)−1−(1+R) + 2(1+1.996Kn)(1+R)ln(1+R)
2(−0.75 −0.5 ln α) + 1.996Kn(−0.5 −ln α)(29)
Ano he ela ionship conside ing he gas slip e ec was de eloped by Liu and Rubow [
54
] who
u he modi ied he model o Lee and Liu [
52
] (Equa ion (25)) by mul iplying i by a co ec ion ac o
o he gas slip as ollows:
ηR=0.61−α
Ku
R2
1+R1+1.996Kn
R(30)
2.1.3. SCE Due o Ine ial Impac ion
Ine ial impac ion akes place in highe ai low eloci ies o pa icles wi h a la ge diame e
(mos ly la ge han 1
µ
m depending on condi ions) due o hei highe ine ia, which causes hem
o ollow a di e en ajec o y han ha o ai low s eamlines. The s eamlines nea he collec o
ab up ly changes. The pa icle hus sepa a es om he s eamlines and hi s he collec o . Collec ion
e iciency due o ine ial impac ion depends on S okes’ numbe cha ac e izing he pa icle ine ia,
which is de ined as ollows:
S k =d2
pρpCsU
18µd
(31)
whe e
ρp
is he pa icle densi y. I he S okes’ numbe is highe han uni y, he pa icles sepa a e
om s eamlines and hi he collec o . On he o he hand, o S okes’ numbe lowe han one, he
ine ia e ec will no ake place. Se e al o mulae ha e been de i ed o SCE due o ine ial impac ion.
The mos o en used ela ionship is ha p oposed by S echkina e al. [50]:
ηI=S k
4Ku229.6 −28α0.62R2−27.5R2.8 (32)
o 0.0035 < α< 0.111 and 0.01 < R< 0.4, while o R> 0.4, he ela ionship is modi ied as ollows:
ηI=S k
2Ku2(33)
Landahl and He mann [
64
] p oposed a ela ionship based on expe imen al da a o Re
> 10.
Howe e , as sugges ed by Saleh e al. [
65
], his equa ion may also be used o Re
< 2. The ela ionship
is as ollows:
ηI=S k3
S k3+0.77S k2+0.22 (34)
Nanoma e ials 2018,8, 447 8 o 24
Fuchs ga e ano he ela ionship o impac ion e iciency as ollows [66]:
ηI=S k2
(S k +0.25)2(35)
while Gougeon e al. [
67
] and F iedlande [
68
] p oposed empi ical Equa ions (36) and (37), espec i ely:
ηI=0.039S k3/2 (36)
ηI=0.075S k6/5 (37)
Equa ions (36) and (37) a e alid o 0.0263 < Re
< 0.25 and 0.5 < S k < 4.1 and Re
< 1, 0.8 < S k < 2
and R< 0.2, espec i ely. Zhu e al. [
69
] de i ed a ela ionship wi h no es ic ions conce ning S k, Re
and αas ollows:
ηI=2R(1−α)S k√α+ (1−α)αS k2
Ku (38)
Se e al esea che s p oposed models accoun ing o he e ec o ibe and pa icle Reynolds
numbe on he SCE due o ine ial impac ion. Suneja and Lee [
70
] de i ed a ela ionship o
1 < Re < 60
and 1 < S k < 20 as ollows:
ηI="1+1.53 −0.23 ln Re +0.0167(ln Re )2
S k #−2
(39)
Ilias and Douglas [
71
] heo e ically in es iga ed ine ial ae osol deposi ion on an isola ed cylinde
by sol ing ime dependen Na ie -S okes equa ions. They p oposed a co ela ion o 30 < Re
< 40,000
and 0.07 < S k < 5 as ollows:
ηI=S k3+1.622 ×10−4/S k
1.031S k3+(1.14 +0.04044 ln Re )S k2+0.01479 ln Re +0.2013 (40)
2.1.4. SCE Due o Adhesion
Fo adhesion e iciency, se e al au ho s p oposed empi ical ela ionships o a ying ma e ial
combina ions, wi h di e en anges o Reynolds and S okes numbe s. Based on expe imen al esul s,
an exp ession o adhesion e iciency was p oposed by P ak and Ja oszczyk as ollows [72]:
ηA=190
(RepS k)0.68 +190 (41)
whe e Repis he pa icle Reynolds numbe calcula ed as ollows:
Repp =dpUρp
µ(42)
Re
pp
is no he s anda d luid dynamics Reynolds numbe , i uses he pa icle densi y
ρp
o he
calcula ion [73]. Equa ion (41) was accu a e o 1 < S k < 120 and 0.4 < Re < 5.75.
2.2. E iciency P edic ion o CPM
CPMs a e hin polyca bona e memb anes wi h ci cula po es. The heo e ical p edic ion o
he il a ion e iciency is based on se e al mechanisms simila o ib ous il e s bu wi h physically
di e en meanings (Figu e 3).
Nanoma e ials 2018,8, 447 9 o 24
The heo e ical impac ion e iciency
ηI
o Nuclepo e il e s can be calcula ed using he model
p oposed by Pich [74] as ollows:
ηI=2η0I
1+ξ−η0I2
(1+ξ)2(43)
whe e η0Iand ξa e calcula ed as ollows:
η0I=2S kpξ+25S k2exp−1
S k√ξ−2S k2ξ(44)
ξ=√P
1−√P(45)
whe e Pis he memb ane po osi y and S k is he S okes numbe and is calcula ed as ollows:
S k =d2
pρpCcU
9µdo(46)
wi h Ccas he slip co ec ion ac o and calcula ed as ollows [66]:
Cc=1+2.49 λ
dp
+0.84 λ
dpexp−0.44dp
λ(47)
The di usion e iciency in po es ηDcan be calcula ed as ollows [75]:
ηD=2.56N2/3
D−1.2ND−0.177N4/3
D(48)
i ND< 0.01 o
ηD=1−0.819 exp(−3.657ND)−0.098 exp(−22.305ND)
−0.032 exp(−56.95ND)−0.016 exp(−107.6ND)(49)
i ND> 0.01, whe e NDis:
ND=4ZPD
d2
oU(50)
whe e Dis di usion coe icien calcula ed acco ding o Equa ion (5) and d
o
is he po e diame e .
The in e cep ion e iciency on po e opening
ηR
can be calcula ed using he model sugges ed by
Spu ny e al. [75]:
ηR=Ro(2−Ro)(51)
whe e Rois he in e cep ion pa ame e o capilla y po e il e s calcula ed as ollows:
Ro=dp
do(52)
Nanopa icles can also deposi on he on su ace o Nuclepo e il e s when pa icles a e smalle
han 100 nm and ace eloci y is low. The su ace-di usion e iciency ηDS can be calcula ed using he
exp ession p oposed by Man on [76]:
ηDS =1−exp"−β1δ2/3
1+(β1/β2)δ7/15 #(53)
whe e β2= 4.5 and β1and δa e coe icien s ha a e calcula ed as ollows:
β1=4.57 −6.46P+4.58P2(54)
Nanoma e ials 2018,8, 447 16 o 24
calcula ed as i was assumed ha he esul would be he same o would a y somewhe e in he o de
o 10−70, which is negligible.
Nanoma e ials 2018, 8, x FOR PEER REVIEW 17 o 25
(a)
(b)
65
70
75
80
85
90
95
100
1 10 100 1000
Single collec o e iciency, η (%)
Pa icle diame e , dp(nm)
5 cm/s
10 cm/s
65
70
75
80
85
90
95
100
1101001000
Single collec o e iciency, η (%)
Pa icle diame e , dp(nm)
15 cm/s
20 cm/s
60
70
80
90
100
1 10 100 1000
O e all e iciency, η(%)
Pa icle diame e , dp(nm)
5 cm/s
10 cm/s
15 cm/s
20 cm/s
Figu e 9. Con .

Nanoma e ials 2018,8, 447 17 o 24
Nanoma e ials 2018, 8, x FOR PEER REVIEW 18 o 25
(c)
Figu e 9. Single collec o e iciency (a), o e all il e e iciency (b) and o e all pene a ion (c).
4.2. CPM
The app oach based on memb ane po e size ins ead o memb ane ibe diame e is p esen ed in
his sec ion. Ine ial impac ion is s onge o la ge pa icles a highe eloci ies, which is in
acco dance wi h heo y. Howe e , he model o Pich [74] is less accu a e as i does no conside he
possible sie ing e ec in memb ane il e s i.e., comple e cap u e o pa icles on he memb ane su ace
o pa icles la ge han memb ane po e size. This is ob ious om Figu e 10. The memb ane po e
size conside ed in he calcula ions is 205 nm (Table 1). I ci cula po es a e assumed, which is a
simpli ica ion in he model, we should ob ain 100% e iciency o pa icles abo e 205 nm ega dless
o he ace eloci y. This is no seen o be ue om Figu e 10.
Figu e 10. Impac ion e iciency based on CPM model.
Mo e plausible esul s a e ob ious o in e cep ion e iciency (Figu e 11). The in e cep ion
e iciency inc eases up o a pa icle size o 202 nm wi h an e iciency o 99.97%. Fo a pa icle size o
209 nm (sligh ly la ge han po e size), e iciency is 100% which is easonable. The e o e, he model
0
20
40
60
80
100
0 100 200 300 400 500 600
Impac ion e iciency, ηI(%)
Pa icle diame e , dp(nm)
5 cm/s
10 cm/s
15 cm/s
20 cm/s
Figu e 9. Single collec o e iciency (a), o e all il e e iciency (b) and o e all pene a ion (c).
The main eason o hese esul s is he high solidi y o he HFM s uc u e, which is 0.48, while
mos o he ib ous il e s ha e solidi y be ween 0.01–0.3 [
44
] and mos o he models a e de eloped
o his solidi y ange. Mo eo e , he memb ane collec o diame e is e y small, gi ing a e y dense
s uc u e. I we look a Figu e 4a, we can see collec o diame e s o abou 100 nm in size. The hickness
o he memb ane wall is 36
µ
m. This means ha he e a e abou 360 such laye s in he memb ane wall
c ea ing a dense ne wo k ha is e y ha d o pa icles o pene a e. The e o e, he esul s seems o be
easonable. In p ac ice, his memb ane could se e as an absolu e il e which a e used o ae osols
which mus ha e 100% emo al e iciency. Such ae osols include some adioac i e pa icles, oxic
ae osols and i uses.
4.2. CPM
The app oach based on memb ane po e size ins ead o memb ane ibe diame e is p esen ed in
his sec ion. Ine ial impac ion is s onge o la ge pa icles a highe eloci ies, which is in acco dance
wi h heo y. Howe e , he model o Pich [
74
] is less accu a e as i does no conside he possible sie ing
e ec in memb ane il e s i.e., comple e cap u e o pa icles on he memb ane su ace o pa icles
la ge han memb ane po e size. This is ob ious om Figu e 10. The memb ane po e size conside ed
in he calcula ions is 205 nm (Table 1). I ci cula po es a e assumed, which is a simpli ica ion in he
model, we should ob ain 100% e iciency o pa icles abo e 205 nm ega dless o he ace eloci y.
This is no seen o be ue om Figu e 10.
Mo e plausible esul s a e ob ious o in e cep ion e iciency (Figu e 11). The in e cep ion
e iciency inc eases up o a pa icle size o 202 nm wi h an e iciency o 99.97%. Fo a pa icle
size o 209 nm (sligh ly la ge han po e size), e iciency is 100% which is easonable. The e o e,
he model p oposed by Spu ny e al. [
75
] (Equa ion (51)) seems o be accu a e o he s uc u e o
polyp opylene HFMs.
Nanoma e ials 2018,8, 447 18 o 24
Nanoma e ials 2018, 8, x FOR PEER REVIEW 18 o 25
(c)
Figu e 9. Single collec o e iciency (a), o e all il e e iciency (b) and o e all pene a ion (c).
4.2. CPM
The app oach based on memb ane po e size ins ead o memb ane ibe diame e is p esen ed in
his sec ion. Ine ial impac ion is s onge o la ge pa icles a highe eloci ies, which is in
acco dance wi h heo y. Howe e , he model o Pich [74] is less accu a e as i does no conside he
possible sie ing e ec in memb ane il e s i.e., comple e cap u e o pa icles on he memb ane su ace
o pa icles la ge han memb ane po e size. This is ob ious om Figu e 10. The memb ane po e
size conside ed in he calcula ions is 205 nm (Table 1). I ci cula po es a e assumed, which is a
simpli ica ion in he model, we should ob ain 100% e iciency o pa icles abo e 205 nm ega dless
o he ace eloci y. This is no seen o be ue om Figu e 10.
Figu e 10. Impac ion e iciency based on CPM model.
Mo e plausible esul s a e ob ious o in e cep ion e iciency (Figu e 11). The in e cep ion
e iciency inc eases up o a pa icle size o 202 nm wi h an e iciency o 99.97%. Fo a pa icle size o
209 nm (sligh ly la ge han po e size), e iciency is 100% which is easonable. The e o e, he model
0
20
40
60
80
100
0 100 200 300 400 500 600
Impac ion e iciency, ηI(%)
Pa icle diame e , dp(nm)
5 cm/s
10 cm/s
15 cm/s
20 cm/s
Figu e 10. Impac ion e iciency based on CPM model.
Nanoma e ials 2018, 8, x FOR PEER REVIEW 19 o 25
p oposed by Spu ny e al. [75] (Equa ion (51)) seems o be accu a e o he s uc u e o polyp opylene
HFMs.
Figu e 11. In e cep ion e iciency based on CPM model.
Di usion is an impo an pa o he o e all e iciency. We can dis inguish be ween di usion
cap u e in po es and di usion cap u e on memb ane su ace (Figu e 3). P edic ion models we e
de eloped o bo h (Equa ions (48) and (53)). Figu e 12a shows po e di usion e iciency. To alk abou
di usion cap u e wi hin memb ane po e s uc u e is possible only o pa icles smalle han he
la ges po e size (i.e., smalle han 205 nm). La ge pa icles will only be a subjec o su ace di usion
cap u e (Figu e 12b) which is possible o whole pa icle size ange. F om Figu e 12a, a simila
p oblem o he model o impac ion e iciency is ob ious. While e iciency o impac ion should be
100% o pa icles abo e 205 nm, po e di usion should be equaled o ze o because no pa icle la ge
han 205 nm canno pene a e he po e s uc u e, so he e is no di usion cap u e o hese pa icles.
(a)
0
20
40
60
80
100
070140210
In e cep ion e iciency, ηR(%)
Pa icle diame e , dp(nm)
60
70
80
90
100
0 200 400 600 800 1000
Po e di usion e iciency, ηD(%)
Pa icle diame e , dp(nm)
5 cm/s
10 cm/s
15 cm/s
20 cm/s
Figu e 11. In e cep ion e iciency based on CPM model.
Di usion is an impo an pa o he o e all e iciency. We can dis inguish be ween di usion
cap u e in po es and di usion cap u e on memb ane su ace (Figu e 3). P edic ion models we e
de eloped o bo h (Equa ions (48) and (53)). Figu e 12a shows po e di usion e iciency. To alk abou
di usion cap u e wi hin memb ane po e s uc u e is possible only o pa icles smalle han he la ges
po e size (i.e., smalle han 205 nm). La ge pa icles will only be a subjec o su ace di usion cap u e
(Figu e 12b) which is possible o whole pa icle size ange. F om Figu e 12a, a simila p oblem o he
model o impac ion e iciency is ob ious. While e iciency o impac ion should be 100% o pa icles
abo e 205 nm, po e di usion should be equaled o ze o because no pa icle la ge han 205 nm canno
pene a e he po e s uc u e, so he e is no di usion cap u e o hese pa icles.
Nanoma e ials 2018,8, 447 19 o 24
Nanoma e ials 2018, 8, x FOR PEER REVIEW 19 o 25
p oposed by Spu ny e al. [75] (Equa ion (51)) seems o be accu a e o he s uc u e o polyp opylene
HFMs.
Figu e 11. In e cep ion e iciency based on CPM model.
Di usion is an impo an pa o he o e all e iciency. We can dis inguish be ween di usion
cap u e in po es and di usion cap u e on memb ane su ace (Figu e 3). P edic ion models we e
de eloped o bo h (Equa ions (48) and (53)). Figu e 12a shows po e di usion e iciency. To alk abou
di usion cap u e wi hin memb ane po e s uc u e is possible only o pa icles smalle han he
la ges po e size (i.e., smalle han 205 nm). La ge pa icles will only be a subjec o su ace di usion
cap u e (Figu e 12b) which is possible o whole pa icle size ange. F om Figu e 12a, a simila
p oblem o he model o impac ion e iciency is ob ious. While e iciency o impac ion should be
100% o pa icles abo e 205 nm, po e di usion should be equaled o ze o because no pa icle la ge
han 205 nm canno pene a e he po e s uc u e, so he e is no di usion cap u e o hese pa icles.
(a)
0
20
40
60
80
100
070140210
In e cep ion e iciency, ηR(%)
Pa icle diame e , dp(nm)
60
70
80
90
100
0 200 400 600 800 1000
Po e di usion e iciency, ηD(%)
Pa icle diame e , dp(nm)
5 cm/s
10 cm/s
15 cm/s
20 cm/s
Nanoma e ials 2018, 8, x FOR PEER REVIEW 20 o 25
(b)
Figu e 12. Collec ion e iciency due o di usion in po es (a) and on he memb ane su ace (b).
O e all e iciency is p edic ed based on he models o indi idual mechanisms and calcula ed
using Equa ion (56). Figu e 13 shows 99.997% MPPS (290 nm) e iciency a a eloci y o 5 cm/s. Wi h
inc easing eloci y, he e iciency o MPPS dec eases. Howe e , i is s ill in he ange o 99.7% a a
eloci y o 20 cm/s. MPPS is shi ed o smalle pa icle size wi h eloci y. I is 250, 225 and 202 nm o
10, 15 and 20 cm/s, espec i ely. This model gi es mo e ealis ic esul s compa ed o he model o
ib ous il e s, whe e uncondi ional 100% e iciency was ob ained o all ace eloci ies.
(a)
0
20
40
60
80
100
1 10 100 1000
Su ace di usion e iciency, ηDS (%)
Pa icle diame e , dp(nm)
5 cm/s
10 cm/s
15 cm/s
20 cm/s
99.8
99.85
99.9
99.95
100
0 50 100 150 200 250 300 350 400
O e all e iciency, η(%)
Pa icle diame e , dp(nm)
10 cm/s
5 cm/s
Figu e 12. Collec ion e iciency due o di usion in po es (a) and on he memb ane su ace (b).
O e all e iciency is p edic ed based on he models o indi idual mechanisms and calcula ed
using Equa ion (56). Figu e 13 shows 99.997% MPPS (290 nm) e iciency a a eloci y o 5 cm/s. Wi h
inc easing eloci y, he e iciency o MPPS dec eases. Howe e , i is s ill in he ange o 99.7% a a
eloci y o 20 cm/s. MPPS is shi ed o smalle pa icle size wi h eloci y. I is 250, 225 and 202 nm o
10, 15 and 20 cm/s, espec i ely. This model gi es mo e ealis ic esul s compa ed o he model o
ib ous il e s, whe e uncondi ional 100% e iciency was ob ained o all ace eloci ies.
Nanoma e ials 2018,8, 447 20 o 24
Nanoma e ials 2018, 8, x FOR PEER REVIEW 20 o 25
(b)
Figu e 12. Collec ion e iciency due o di usion in po es (a) and on he memb ane su ace (b).
O e all e iciency is p edic ed based on he models o indi idual mechanisms and calcula ed
using Equa ion (56). Figu e 13 shows 99.997% MPPS (290 nm) e iciency a a eloci y o 5 cm/s. Wi h
inc easing eloci y, he e iciency o MPPS dec eases. Howe e , i is s ill in he ange o 99.7% a a
eloci y o 20 cm/s. MPPS is shi ed o smalle pa icle size wi h eloci y. I is 250, 225 and 202 nm o
10, 15 and 20 cm/s, espec i ely. This model gi es mo e ealis ic esul s compa ed o he model o
ib ous il e s, whe e uncondi ional 100% e iciency was ob ained o all ace eloci ies.
(a)
0
20
40
60
80
100
1 10 100 1000
Su ace di usion e iciency, ηDS (%)
Pa icle diame e , dp(nm)
5 cm/s
10 cm/s
15 cm/s
20 cm/s
99.8
99.85
99.9
99.95
100
0 50 100 150 200 250 300 350 400
O e all e iciency, η(%)
Pa icle diame e , dp(nm)
10 cm/s
5 cm/s
Nanoma e ials 2018, 8, x FOR PEER REVIEW 21 o 25
(b)
Figu e 13. O e all e iciency in ela ion o pa icle size based on CPM model o a eloci y o 5 and
10 cm/s (a) and 15 and 20 cm/s (b)
5. Conclusions
P edic ion models o ai il a ion e iciency o ib ous and memb ane il e s we e nume ically
compa ed by applying an HFM po e s uc u e. Wi h some assump ions, hese models can be used
o p edic ions o he ae osol sepa a ion e iciency o HFMs. Fib ous il e models gi e 100%
e iciency no ma e wha le el o ace eloci y, i.e., ze o pene a ion. This is gi en by e y small
collec o s in memb ane s uc u e simila ly o nano ib ous il e s. Compa ed o nano ib ous il e s,
HFMs ha e a e y high solidi y o 0.48. The HFM s uc u e is e y dense and he calcula ions can
o e es ima e, as mos o he models p edic il e e iciency o solidi y up o 0.3. CPM models p edic
e iciencies ha a e mo e ealis ic. Pene a ion up o 0.00014% was calcula ed o a ace eloci y o 20
cm/s. CPM models seem o gi e mo e plausible esul s o hese HFMs, howe e , an expe imen al
e i ica ion should be app op ia e o compa e accu acy o bo h app oaches. Howe e , his is a he
a sugges ion o ano he s udy, as his e i ica ion would p obably be challenging, conce ning
expe imen al wo k. Thus, i would also be possible o empi ically de elop a new accu a e model o
HFMs.
Supplemen a y Ma e ials: The ollowing a e a ailable online a www.mdpi.com/xxx/s1, Figu e S1: SCE due o
ine ial impac ion based on model o S echkina e al. (a), Landahl and He man (b), Fuchs (c), Gougeon e al. (d),
Suneja and Lee (e), F iedlande ( ), Zhu e al. (g) and Illias and Douglas (h), Figu e S2: Compa ison o impac ion
e iciency based on di e en models and ai low eloci y o 5 cm/s (a) and 20 cm/s (b), Figu e S3: Collec ion
e iciency due o in e cep ion mechanism based on Langmui model o di e en ai low eloci ies (a) and a
compa ison o SCE due o in e cep ion based on models de eloped by a ious esea che s (b), Figu e S4: SCE
due o di usion mechanism based on ma hema ical models de eloped by Paye e al. (a), Ki sch and Fuchs (b),
S echkina e al. (c), Lee and Liu (d), Wang e al. (e) and Pich ( ), Figu e S5; Compa ison o SCE due o di usion
mechanism based on di e en models o an ai low eloci y o 5 cm/s (a) and 20 cm/s (b).
Funding: The wo k was ca ied ou wi hin he amewo k o he p ojec No. LO1202 “NETME Cen e PLUS”
and No. LO1418 “P og essi e de elopmen o Memb ane Inno a ion Cen e” suppo ed by he p og am NPU I
Minis y o Educa ion You h and Spo s o he Czech Republic, using he in as uc u e Memb ane Inno a ion
Cen e.
Con lic s o In e es : The au ho decla es no con lic o in e es .
Re e ences
99
99.2
99.4
99.6
99.8
100
0 50 100 150 200 250 300 350 400
O e all e iciency, η(%)
Pa icle diame e , dp(nm)
15 cm/s
20 cm/s
Figu e 13.
O e all e iciency in ela ion o pa icle size based on CPM model o a eloci y o 5 and
10 cm/s (a) and 15 and 20 cm/s (b).
5. Conclusions
P edic ion models o ai il a ion e iciency o ib ous and memb ane il e s we e nume ically
compa ed by applying an HFM po e s uc u e. Wi h some assump ions, hese models can be used o
p edic ions o he ae osol sepa a ion e iciency o HFMs. Fib ous il e models gi e 100% e iciency
no ma e wha le el o ace eloci y, i.e., ze o pene a ion. This is gi en by e y small collec o s in
memb ane s uc u e simila ly o nano ib ous il e s. Compa ed o nano ib ous il e s, HFMs ha e a e y
high solidi y o 0.48. The HFM s uc u e is e y dense and he calcula ions can o e es ima e, as mos
o he models p edic il e e iciency o solidi y up o 0.3. CPM models p edic e iciencies ha a e
mo e ealis ic. Pene a ion up o 0.00014% was calcula ed o a ace eloci y o 20 cm/s. CPM models
seem o gi e mo e plausible esul s o hese HFMs, howe e , an expe imen al e i ica ion should be
app op ia e o compa e accu acy o bo h app oaches. Howe e , his is a he a sugges ion o ano he
Nanoma e ials 2018,8, 447 21 o 24
s udy, as his e i ica ion would p obably be challenging, conce ning expe imen al wo k. Thus, i
would also be possible o empi ically de elop a new accu a e model o HFMs.
Supplemen a y Ma e ials:
The ollowing a e a ailable online a h p://www.mdpi.com/2079-4991/8/6/447/s1,
Figu e S1: SCE due o ine ial impac ion based on model o S echkina e al. (a), Landahl and He man (b),
Fuchs (c),
Gougeon e al.
(d), Suneja and Lee (e), F iedlande ( ), Zhu e al. (g) and Illias and Douglas (h), Figu e S2:
Compa ison o impac ion e iciency based on di e en models and ai low eloci y o 5 cm/s (a) and
20 cm/s (b)
,
Figu e S3: Collec ion e iciency due o in e cep ion mechanism based on Langmui model o di e en ai low
eloci ies (a) and a compa ison o SCE due o in e cep ion based on models de eloped by a ious esea che s (b),
Figu e S4: SCE due o di usion mechanism based on ma hema ical models de eloped by Paye e al. (a), Ki sch
and Fuchs (b),
S echkina e al.
(c), Lee and Liu (d), Wang e al. (e) and Pich ( ), Figu e S5; Compa ison o SCE due
o di usion mechanism based on di e en models o an ai low eloci y o 5 cm/s (a) and 20 cm/s (b).
Funding:
The wo k was ca ied ou wi hin he amewo k o he p ojec No. LO1202 “NETME Cen e PLUS”
and No. LO1418 “P og essi e de elopmen o Memb ane Inno a ion Cen e” suppo ed by he p og am
NPU I Minis y o Educa ion You h and Spo s o he Czech Republic, using he in as uc u e Memb ane
Inno a ion Cen e.
Con lic s o In e es : The au ho decla es no con lic o in e es .
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