Resea ch A icle Indoo /Ou doo Ai low
and Ai Quali y
E-mail: joaquim.ceb[email p o ec ed]
Simula ing nanopa icle concen a ion using s ochas ic models o
imp o e indoo ai quali y in he indus y
Joaquim Cebolla-Alemany1 (), Ma cel Maca ulla Ma í1, Ma Viana2,3, San iago Gasso-Domingo1, Ve ónica
Mo eno-Ma ín3, Da id Bou4, Vicen a San Félix4, Rubén D. López-Ca eño1
1. Uni e si a Poli ècnica de Ca alunya, Depa men o P ojec and Cons uc ion Enginee ing, G oup o Cons uc ion Resea ch and Inno a ion
(GRIC), Te assa, 08222, Spain
2. Spanish Minis y o Ecological T ansi ion, Pollu ion P e en ion Uni , Mad id, 28071, Spain
3. Ins i u e o En i onmen al Assessmen and Wa e Resea ch (IDAEA-CSIC), Ba celona, 08034, Spain
4. Ins i u e o Ce amic Technology (ITC-AICE), Cas elló de la Plana, 12006, Spain
Abs ac
In indus ial scena ios, nanopa icles a e inciden ally gene a ed in high concen a ions du ing
di e se ma e ial ans o ma ion p ocesses, p esen ing po en ial heal h haza ds o exposed wo ke s.
Consequen ly, as an indoo ai quali y managemen measu e, hei concen a ion is commonly
educed h ough localized o ced en ila ion. Howe e , he con ol o hese sys ems usually elies
on adi ional ule-based algo i hms, which canno deploy e icien con ol s a egies such as
model p edic i e con ol. To sol e his issue, we p opose a no el g ey-box educed o de model
me hod, ne e used be o e o indus ial indoo nanopa icles. This app oach can be deployed in
model p edic i e con ol algo i hms in buildings and does no p esen he da a- eliance and
ans e abili y issues o black-box modeling. To es his model, a da a collec ion campaign was
conduc ed unde eal-wo ld ope a ing condi ions in an indus ial-scale he mal sp aying boo h,
aiming o es he me hod’s iabili y o model calib a ion and alida ion o indoo o al nanopa icle
concen a ion h ough he maximum likelihood me hod, s a is ical alida ion es s, and physical
iabili y assessmen . Resul s o h ee di e en lumped sum models illus a e he e ec i eness o
g ey-box modeling in indus ial scena ios wi h con ined p ocesses and o ced en ila ion sys ems,
handling obse a ions’ noise and backg ound concen a ion luc ua ions, and allowing a pe o mance
compa ison be ween models. Fu he esea ch could be conduc ed o s udy he iabili y o indoo
o al nanopa icle concen a ion educed o de models wi h highe spa ial esolu ion, non-con ined
sou ces, and na u al ai lows.
Keywo ds
educed-o de models
lumped sum model
g ey-box modeling
indus y
HVAC
A icle His o y
Recei ed: 27 Sep embe 2024
Re ised: 18 Decembe 2024
Accep ed: 13 Janua y 2025
© Au ho (s) 2025
1 In oduc ion
In indus ial en i onmen s, one o he pollu an s which is
ecei ing inc easing a en ion om he Eu opean Union
conce ning indoo ai quali y (IAQ) is nanopa icles (NP)
(Council o he Eu opean Union 2024) due o hei p esence
in high concen a ions. These pa icles (also e e ed o as
ul a ine pa icles (S one e al. 2017; Kwon e al. 2020)) a e
suspended in he ai in a solid o liquid s a e and p esen an
ae odynamic diame e below 100 nm. They a e inciden ally
gene a ed (hence named inciden al NPs, INP) a high a es
by p ocesses such as engine combus ion, welding, he mal
sp aying, lase cu ing, g inding, o ce amic ile i ing, among
o he s (Bessa e al. 2020; Sonwani e al. 2021). When inhaled
in high concen a ions, hey a e a po en ial sou ce o di e se
heal h haza ds (Gwinn and Vallya han 2006; Kuma e al.
2010; Nemma e al. 2013). Consequen ly, indus ies usually
ely on localized en ila ion sys ems o elimina e INPs
om he en i onmen (e.g., Jam iska e al. 2003; Hussein
e al. 2015).
Ven ila ion sys ems, howe e , o en ely on adi ional
eac i e ule-based con olle s (Gwe de e al. 2010; Bo odinecs
BUILD SIMUL (2025) 18: 847–862
h ps://doi.o g/10.1007/s12273-025-1245-7
Cebolla-Alemany e al. / Building Simula ion / Vol. 18, No. 4
848
e al. 2022) despi e he ele a ed ene gy consump ion o
indus ial ai ex ac ion sys ems. This app oach bols e s he
deploymen o ene gy-e icien con ol s a egies (Tash oush
e al. 2005; P í a a e al. 2013; Qian e al. 2024). The eme ging
endency o sol e his issue in building ai ex ac ion and
en ila ion sys ems uses model p edic i e con ol (MPC)
o an icipa e and plan he ac ua ion policy based on ene gy
sa ing, ene gy lexibili y and/o IAQ p io i ies (see, e.g.,
S u zenegge e al. 2016; A am e al. 2017; Adegben o e al.
2021; Bi d e al. 2022; Wang e al. 2023; Xia e al. 2023; Yue
e al. 2023; Klepic e al. 2024). MPC can be deployed h ough
IoT-based solu ions employing moni o ed en i onmen al
pa ame e s o adjus en ila ion (Sha ma e al. 2023) and ligh
models o pe o m he necessa y simula ions o ind he
op imal solu ion o each si ua ion. This can be conduc ed
h ough a i icial in elligence (AI) algo i hms p edic ing
IAQ and op imizing ene gy consump ion (Ren and Cao
2020; Suo e al. 2022), compu a ional luid dynamics (CFD)
models, o mul izone models, among o he s (Feng e al. 2019;
Cao e al. 2020).
Rega ding AI me hods, he use o black-box models has
also been add essed in he li e a u e h ough he use o , e.g.,
a i icial neu al ne wo ks (Xie e al. 2009; Pu a e al. 2018;
Gaowa e al. 2024) o sea ch algo i hms (Zhang e al. 2019;
Saini e al. 2022; Al Mindeel e al. 2024). None heless, hese
algo i hms p esen issues o esul s ans e abili y o o he
case s udies, no enhancing he comp ehension o NPs, and
occasionally elying on ela i ely simple scena ios o show
good pe o mance wi h a ewe amoun o da a and dynamic
equi emen s, complica ing hei deploymen in indus ial
buildings (Ren and Cao 2020). Mo eo e , hei da a eliance
migh be c i ical o INP modeling gi en he cos and
accu acy o cu en ins umen a ion.
When compu a ional esou ces and speed a e no
limi ing ac o s, compu a ional luid dynamics me hods
a e equen ly employed o hei high spa ial- empo al
esolu ion, enabling highly accu a e analyses o en ila ion
sys ems (see, e.g., Mousa i 2019; An e al. 2024; Zhang e al.
2024). Howe e , o applica ions equi ing nea eal- ime
da a and op imiza ion, hese models become imp ac ical
(Masoumi-Ve ki e al. 2022; Ta iq e al. 2022; Masoumi-Ve ki
e al. 2023).
The e o e, educed-o de models eme ge as a physics-
based solu ion o deploy MPC in buildings. Pa icula ly,
s ochas ic (also named g ey-box) educed-o de models wi h
lumped sum app oaches ha e been p o en compe i i e
o building simula ions and managemen o empe a u e
(Tha lo and Madsen 2015; B as ein e al. 2018), ene gy (Li
e al. 2021), and ca bon dioxide concen a ion (Maca ulla
e al. 2017, 2018; Wol e al. 2019a, 2019b). Addi ionally, hey
a e compa ible mass-balance models, highligh ed in he
p e ious NP s udies due o hei anspa ency and capabili y
o simpli y eali y wi hou enouncing o include physical
p ocesses (Koi is o e al. 2019). Howe e , li e a u e has no
ye s udied he deploymen o g ey-box models o o he
IAQ pollu an s, including indoo INP.
P e ious app oaches used o indoo NP concen a ion
es ima ions (Pi jola 1999; Asmi e al. 2004; C eely e al. 2005;
Ma qua e al. 2008; Tielemans e al. 2008; Che ie e al.
2011; F ansman e al. 2011; Als up Jensen e al. 2013; Ciuzas
e al. 2015; Koi is o e al. 2015; Ka l e al. 2022; Macko e al.
2023) ely on he de ini ion o con ol bands, mul iplying
ac o s o o he semi-quali a i e analyses, including he
de ini ion o he use o di e en sys em a iables. These
me hods p e en hei applica ion on au oma ed con ol
sys ems o , o example, he de ini ion o necessa y ai low
a es o ex ac ion ime a e inishing an INP emi ing
p ocess o ob ain speci ic concen a ion le els a wo king
condi ions o o clean a pollu ed space, espec i ely. These
si ua ions a e c i ical in, o example, con ined p ocesses wi h
high INP emission a es such as he mal sp aying boo hs,
whe e wo ke s go in and ou o he boo hs while sp aying is
ongoing o check he pieces o mo e hem.
O he explo ed modeling me hods a e es ained o
highly con olled labo a o y condi ions, una ailable in
indus ial se ings, o de ine speci ic pa icle p ocesses such
as o ma ion (Fonseca e al. 2015), elease (Fonseca e al.
2018), deposi ion (He e al. 2005; Smolík e al. 2005), o
coagula ion (Seipenbusch e al. 2008). Despi e he g owing
end in li e a u e o model speci ic p ope ies o NPs (such
as hei s uc u e o p opaga ion h ough di e en ma e ials
o enginee ed o ein o ced nanoma e ials (Kim e al. 2008;
Zhang 2012; Yan and Xie 2013; Odega d e al. 2017)), esea ch
is ye sca ce in he modeling o hei concen a ion in complex
en i onmen s such as indoo indus ial scena ios.
Rega ding en i onmen al NP mass and numbe
concen a ion moni o s, hey a e based on h ee b oad
ca ego ies: ae odynamic classi ica ion, ajec o y obse a ion,
and elec os a ic classi ica ion (Johnson e al. 2020). Wi hin
hese ca ego ies, he used physical p inciples include op ical
de ec ion, condensa ion pa icle coun , scanning mobili y
pa icle size, op ical pa icle coun , di usion size classi ica ion,
and ae osol spec ome y (Ribal a 2019). Op ical de ec o s
use a ligh sou ce o de ec pa icles h ough hei e ec
on ligh sca e ing (McMu y 2000). Condensa ion pa icle
coun e s de ec small pa icles based on liquid condensa ion
on o he pa icles. Scanning mobili y pa icle size s (TSI
2022) a e composed by a condensa ion pa icle coun e
and a di e en ial mobili y analyze ha classi ies pa icles
depending on hei elec ical mobili y (see Ribal a (2019))
and e e ences he ein). Di usion size classi ie minia u es
(Tes o Inc 2024) a e based on he use o a co ona cha ge
o pa icle posi i e unipola cha ging. Finally, wide ange
ae osol spec ome e s combine elec ical mobili y p inciples
Cebolla-Alemany e al. / Building Simula ion / Vol. 18, No. 4
849
wi h op ical ligh sca e ing, using he o me o he smalle
pa icle diame e s.
None heless, he e exis s a ecen conce n in he li e a u e
o de elop new me hods o op imize low-cos and compac
moni o s o ul a ine pa icles, o e coming he challenges o
cu en de ices and enabling deploymen o senso ne wo ks
o di ec ly measu e pa icle concen a ion o IAQ (Nishida
e al. 2020), consequen ly enabling MPC deploymen .
Examples o his end include andem elec os a ic/ine ial
classi ica ion and elec ical de ec ion me hods below 100 nm
(Kwon e al. 2021), NP-imp in ed ma ices o ai and
liquids moni o ing up o 200 nm esembling he molecula ly
imp in ed polyme app oach (De y e al. 2022), bipola
cha ging eaching obse a ion o pa icles be ween 44 nm
and 1.05 μm (Nishida e al. 2020), he combined use o
ae odynamic ae osol classi ie s and di e en ial mobili y
analyze s o he 40–596 nm size ange (Johnson e al. 2020).
Simul aneously, modeling esea ch ies o be e p edic
NP concen a ions in bo h indoo and ou doo en i onmen s.
Fo his, s a is ical analyses, such as Bayesian linea eg ession
models (Has ie e al. 2017; an E p e al. 2019) explo ed
o p edic spa ial a ia ions (Ripley e al. 2022) o a i icial
neu al ne wo ks o size- esol ed numbe concen a ion
(Al-Dabbous e al. 2017). Mechanis ic app oaches ha e
also been s udied by coupling chemical and kine ic models
o es ima e NP g ow h om ola ile o ganic compounds
oxida ion in gas-phase (Li e al. 2024) o pe o ming
simula ions o u bulen anspo o NPs (Raja e al. 2016;
Vanaki e al. 2016; Shimbe g e al. 2022). Howe e , ecen
li e a u e ega ding NP modeling mainly ocuses on small-
scale chemical and medical pe spec i es, aiming o disease
models and quan i ica ion biological esponses (see, e.g.,
Eckmann e al. 2020; Rie be g e al. 2021; Le en i e al. 2023;
Minnema e al. 2023; de Roode e al. 2024), o nanoma e ials
(such as in (Amodeo and Pizzagalli 2021; Choi e al. 2022;
No ozhilo e al. 2023; Spel hann e al. 2023)).
Consequen ly, i can be s a ed ha despi e he
ad ancemen s in NP concen a ion modeling, he e emains
a signi ican esea ch gap in applying i o en i onmen al
NP concen a ion in indus ial scena ios, speci ically in
s ochas ic g ey-box models o eal- ime indus ial indoo
NP concen a ion p edic ion, which would esul c ucial
o ene gy op imiza ion in indus ial se ings con olling
en ila ion sys ems. Consequen ly, he no el y o his
pape lies in he in oduc ion and alida ion o a s ochas ic
g ey-box modeling app oach o accu a ely p edic INP
concen a ions wi hin indus ial en i onmen s. This s udy
aims o demons a e he e icacy and p ac ical applicabili y o
his s a is ical me hod, b idging he gap be ween heo e ical
modeling and eal-wo ld indus ial applica ions in ol ing
NP- ela ed indoo ai quali y. By le e aging he s eng hs
o s ochas ic g ey-box models, we o e a obus solu ion
o model indus ial NPs and manage IAQ, enabling mo e
e icien and in o med con ol s a egies in en i onmen s
whe e adi ional NP modeling me hods all sho .
This pape is o ganized as ollows. Sec ion 2 desc ibes
he me hodology used o model NP and de ails he case s udy
unde in es iga ion. Sec ion 3 p esen s he esul s ob ained
om he simula ions and expe imen al alida ion. Sec ion 4
discusses he implica ions o hese esul s, highligh ing hei
signi icance in he con ex o cu en esea ch and po en ial
applica ions. Finally, Sec ion 5 p o ides conclusions and
sugges s di ec ions o u u e esea ch.
2 Ma e ials and me hods
The p ocess ollowed in his esea ch o apply g ey-box
modeling o INP concen a ion simula ions in indoo
indus ial en i onmen s is conduc ed in h ee s ages:
indus ial da a collec ion in ope a i e condi ions, o al
INP numbe concen a ion modeling, and model alida ion.
Du ing da a collec ion s age, o al nanopa icle numbe
concen a ion is measu ed and si ua ions o INP gene a ion,
o ced en ila ion, and na u al ai low p esence a e moni o ed.
Then, o INP concen a ion modeling, an o dina y di e en ial
equa ion (ODE) is p oposed o desc ibe he de e minis ic
beha io o he sys em, subsequen ly ans o med in o a
s ochas ic di e en ial equa ion (SDE) o conduc pa ame e
calib a ion and model simula ion. Finally, models a e
alida ed h ough signi icance and ue op ima es s o
hei pa ame e s, a easibili y assessmen o he calib a ed
alues, he absolu e esiduals analysis, a whi e noise es ,
and hei ep esen a ion in a Taylo diag am.
2.1 Expe imen al design and da a collec ion
The loca ion unde s udy is a p ecision mechanics indus ial
acili y belonging o he company T.M. Comas, loca ed in
Blanes, Gi ona, Spain. This company specializes in he
epai o high-p ecision componen s o a ious indus ies,
including he pe ochemical, pha maceu ical, and ene gy
sec o s. Se e al ac i i ies ake place in his acili y o epai
pa s, such as welding, sandblas ing, cu ing, o plasma
sp aying. Focusing on he plasma sp aying p ocesses, a
wo kshop o app oxima ely 375 m2 is dedica ed o his
ac i i y. This wo kshop is sepa a ed om he es o he
plan by a wall wi h wo doo s: one o wo ke s and ano he
o o kli ucks, usually kep closed. This a ea con ains
h ee he mal sp aying boo hs o di e en olumes whe e
wo di e en p ocesses a e conduc ed: high- eloci y oxy- uel
coa ing sp aying and a mosphe ic plasma sp aying (APS).
This space was p e iously s udied by Ribal a e al. (2019b),
bu i has since unde gone modi ica ions o isola e he
he mal sp aying p ocesses om he wo ke a ea (WA).
Cebolla-Alemany e al. / Building Simula ion / Vol. 18, No. 4
850
The p esen s udy ocuses on boo h #2, whe e APS
is conduc ed wi h dich omium ioxide (C 2O3) a high
empe a u es (5000–20000 °C) and low sp aying eloci y
(200–500 m/s). The p ocess was ca ied ou in e mi en ly
o wo consecu i e days in July 2022 du ing he epa a ion
o di e en pieces se ies. This boo h has a olume o 102 m3
and a en ila ion sys em sha ed wi h boo h #3 wi h a o al
ai low o 15000 m3/h, spli o no depending i bo h a e
wo king o no in unc ion o clien s’ eques s, ha ing he
ex ac ion low ac i a ion manually con olled by he wo ke
while sp aying and i e mo e minu es o clean he boo h
a e inishing he APS. Du ing he da a collec ion, boo h #3
had no ac i i y, so boo h #2 used he o al en ila ion
capaci y, esul ing in oughly 147 ai changes pe hou . The
ai o he en ila ion is ob ained di ec ly om he WA
h ough wo di e en poin s a bo h sides o he boo h, as
seen in Figu e 1. These poin s a e connec ed h ough an ai
duc o he es o he boo h by i s cen e . Then, he ai low
lea es he boo h h ough he en ila ion sys em on he
opposi e wall.
All boo hs wo k independen ly o he o he s and ollow
i egula and in e mi en ou ines o ganized by he p io i y
o he clien companies, he size and numbe o pieces,
he occupancy o he boo hs, and he p epa a ion o he
(a)
(b)
Fig. 1 (a) The mal sp aying boo h #2 pho og aphy and (b) schema ic
diag am o he he mal sp aying a ea and modeled boo h a T.M.
Comas acili ies wi h dimensions (m)
componen s, among o he c i e ia. In pa allel, he o kli
uck’s access doo opens a ew imes daily o anspo
la ge pieces om o in o he WA, a ec ing i s concen a ion.
The e o e, Booleans desc ibing he ac i a ion o he sp aying
pis ol, mechanical en ila ion, and boo h’s doo opening
we e manually moni o ed du ing he expe imen .
The ins umen s used o collec NP concen a ion
da a we e wo NanoScan SMPS TSI Model 3910 (NS), an
elec ical mobili y spec ome e ha measu es pa icle
numbe concen a ion be ween 100 and 106 pa icles/cm3
and pa icle size dis ibu ion om 10 o 420 nm (TSI 2022).
One was placed inside he boo h and ano he in he WA
in on o he boo h’s doo . To be able o measu e
concen a ions o e he ins umen ’s limi inside he boo h,
he 1:100 dilu ion sys em DIL 550 (Topas 2019) was used.
These ins umen s cap u ed size- esol ed concen a ion
da a wi h 1-minu e ime esolu ion, which was he ea e
con e ed h ough di ec sum in o o al concen a ion da a.
The manu ac u e ’s in e sion algo i hm was used o con e
he moni o ’s ou pu o pa icle size dis ibu ion da a (TSI
2022). The eliabili y o his ins umen o occupa ional
and indoo ai quali y moni o ing was based on exis ing
li e a u e e iews (S abile e al. 2014; Fonseca e al. 2016;
Jø gensen 2019). Pa icle numbe size dis ibu ions desc ibing
obse a ions inside and ou side he boo h du ing he
measu emen s a e ep esen ed in Figu e 2, calcula ed
ollowing he p ocedu e desc ibed in Cebolla-Alemany e al.
(2024) wi h a Pea son coe icien o he i ing o a logno mal
dis ibu ion exceeding 0.996 in bo h cases. Addi ionally,
o al concen a ion da a was a e aged e e y wo minu es o
educe noise be o e being used o model calib a ion and
pa ame e iza ion.
A summa y o he sys em’s mos ele an da a is a ailable
in Table 1, while moni o ed da a (boo h’s concen a ion, WA
concen a ion, and Booleans o gene a ion, en ila ion, and
boo h doo opening) can be seen in Figu e 3.
Fig. 2 Pa icle numbe size dis ibu ions inside (Nin) and ou side
(Nou ) he boo h du ing he measu emen
Cebolla-Alemany e al. / Building Simula ion / Vol. 18, No. 4
851
Table 1 Main ea u es o he case s udy
Pa ame e Value
Boo h’s olume (m3) 102
Boo h’s doo heigh × wid h (m) 2.4×2.2
Ven ila ion low a e (m3/h) 15000
Minimum pa icle size (nm) 10
Maximum pa icle size (nm) 420
P ocess APS
Sp ay ma e ial C 2O3
2.2 Modeling p ocess
G ey-box models in eg a e in o ma ion om obse a ions
wi h physical knowledge o a sys em (Bache and Madsen
2011). The sys em’s physics is ep esen ed h ough a se o
i s -o de s ochas ic di e en ial equa ions consis ing o a
d i e m and a di usion e m. The d i e m desc ibes he
de e minis ic p ocesses go e ning he sys em. Fo his s udy,
h ee models (M1, M2, and M3) desc ibing NP concen a ion
inside he modeled boo h ha e been p oposed and e alua ed.
The i s s udied model (M1) conside s gene a ion, o ced
en ila ion, and di usion phenomena. I is he simples
o he h ee and i is based on well-es ablished li e a u e
(see, e.g., Ribal a e al. (2019b) and e e ences he ein). I s
equa ion is:
in sou ce en ila ion di usion
dd
NJJ J
=+ + (1)
whe e dNin/d is he pa icle numbe concen a ion a ia ion
h ough ime inside he boo h, Jsou ce is he INP gene a ion
e m inside he boo h, J en ila ion is he e m ha ep esen s he
change when mechanical en ila ion o ai ex ac ion is on,
and Jdi usion desc ibes he a ia ion o di usion p ocesses
be ween he boo h and he WA when he doo is open. These
e ms can be u he de eloped, esul ing in Equa ion (2):
()()
ou in en ou in di
in
ddgS N N Q d N N Q
N
V
⋅+⋅-⋅+⋅-⋅
=
(2)
whe e Nin is he numbe concen a ion inside he s udied
boo h, Nou he numbe concen a ion in he WA, S is he
sou ce s eng h o emission a e o he p ocess, V is he
olume o he boo h, Q en is he low a e ha comes in om
he WA and lea es he boo h h ough he en ila ion sys em
when i is on, Qdi is he low a e ha occu s by di usion
be ween he boo h and he WA when he boo h’s doo is
open, g is a Boolean a iable desc ibing when he e is
gene a ion, is ano he Boolean indica ing when mechanical
en ila ion occu s, and d is ano he Boolean ha ac i a es
he doo o he boo h is open.
Equa ion (2) s ems om p e iously used ideal gasses’
mass-balance educed-o de models (Maca ulla e al. 2017,
2018) and simila models ha e al eady been used in li e a u e
o ae osol and o he pollu an s concen a ion, no es ic ed
o NPs (see, o example, Fu aw e al. (1996) and Jensen
e al. (2018)).
(a)
(b)
Fig. 3 (a) Boolean inpu s o he model aining ep esen ing he ins an s o gene a ion (g), en ila ion ( ), and doo opening (d) in he
boo h; and (b) concen a ion da a collec ed inside (Nin) and ou side (Nou ) he boo h
Cebolla-Alemany e al. / Building Simula ion / Vol. 18, No. 4
852
The o he wo models unde s udy (M2 and M3) include
a losses e m (Jlosses) ep esen ing agg ega ion and deposi ion
phenomena, ex ensi ely desc ibed in he li e a u e (see, e.g.,
Jacobson (2005); Hussein e al. (2009); Yu e al. (2013)),
al hough some s udies concluded ha hese losses a e no
signi ican (Ribal a e al. 2019b). Fo hese wo models,
Equa ion (1) changes o Equa ion (3):
in sou ce en ila ion di usion losses
dd
NJJ J J
=+ + +
(3)
The losses e m is de ined in wo di e en ways as shown
in Equa ion (4) and Equa ion (5), esul ing in he i s
model M1 wi hou losses, a second wi h a linea dec ease
o pa icles (M2), and a hi d wi h a quad a ic loss a e
(M3), whe e Kloss is he pa icle loss a e ha includes bo h
deposi ion and agg ega ion o e he measu ed size ange,
simpli ying he agg ega ion and deposi ion componen s
used in Jensen e al. (2018), o example, as i can be
obse ed in Equa ion (6), whe e agg ega ion and deposi ion
phenomena a e a unc ion o he Fuks coagula ion coe icien
Kagg ega ion be ween pa icles (Fuks 1964), k is each possible
deposi ion di ec ion, Ak is he a ailable a ea in each di ec ion
k, and k, is he eloci y o pa icle deposi ion.
loss in
losses KN
JV
⋅-
= (4)
2
loss in
losses KN
JV
⋅-
= (5)
agg ega ion deposi ion agg ega ion in in
11
2kk
k
JJ K NAN
V
⋅⋅--⋅+= å
(6)
Sys em inpu s, ha is, concen a ions inside and ou side
he boo h ( om he concen a ion moni o s) and he
Booleans (moni o ed in he ield), se e o ain and alida e
he models as p e iously desc ibed on g ey-box modeling
li e a u e (see, e.g., Tha lo and Madsen 2015; Maca ulla
e al. 2017, 2018; B as ein e al. 2018; Tugo es e al. 2024).
These inpu s a e ep esen ed in Figu e 3.
These equa ions assume a known backg ound
concen a ion (since se e al non-moni o ed ac i i ies ac as
sou ces in luencing he INP concen a ion o ai coming
om he plan ). O he assump ions a e ha he ai en e ing
he boo h equals he ai lea ing, ha gene a ion and ai lows
a e cons an h ough ime, and he e is a pe ec mix u e o
NPs wi hin he boo h and in he WA be ween imes eps
(Jayjock e al. 2011; Jensen e al. 2018). I is also assumed
ha di usion p e ails o e na u al ai lows. Consequen ly,
when he e is no o ced en ila ion, pa icles always mo e
om zones wi h high concen a ions o zones wi h lowe
alues. This hypo hesis is easonable gi en he absence o
windows and doo s connec ed o he ou doo s in he WA
causing wind ai s eams. Addi ionally, Equa ions (4) and (5)
assume ha losses caused by deposi ion and agg ega ion a e
i e e sible, and no esuspension is con empla ed (Jensen
e al. 2018).
To ans o m an ODE in o an SDE, he di usion
e m ep esen ing measu emen noise and he impac o
unmodeled o unknown dis u bances is inco po a ed. This
e m consis s o a unc ion desc ibing he dis u bances o
he sys em and a Wiene p ocess ep esen ing he whi e
noise. Mo eo e , obse a ions in disc e e ime include a
Gaussian-dis ibu ed measu emen e o . Consequen ly, a
se o i s -o de SDEs has o be de ined. The gene alized
equa ions (Maca ulla e al. 2017, 2018; Tugo es e al. 2024)
o his sys em a e:
()
()
d,,,d,d
X X U θ Gθ W=⋅+⋅ (7)
()
,,,
k
YhXUθ e=+ (8)
whe e X is he ec o o sys em s a es, U is he ec o
ha con ains expe imen al measu emen s, θ is a ec o
wi h he pa ame e s ha ha e o be iden i ied, k
Y is he
INP concen a ion measu emen om inside he boo h
in he ins an k, h(X , U , θ, ) is he ela ionship be ween
measu emen s and he s a e a iables, and e is a ec o
desc ibing he whi e noise om he measu emen s. In his
sys em, he d i e m is (X , U , θ, )·d , whe e (X , U , θ, )
equals Equa ion (1) o M1 and Equa ion (3) o M2 and M3;
he di usion e m is G(θ, ), and dW is a Wiene p ocess.
Rew i ing he s a e-o -space sys em o M1, he esul is:
()
in sou ce en ila ion di usion
dddNJ J J σw=+ + ⋅+⋅ (9)
in,
kk
k
YN e=+
(10)
whe e dw is he Wiene p ocess, σ is he inc emen al a iance
in he Wiene p ocess, in, k
N is he measu ed NP numbe
concen a ion inside he boo h, and ek desc ibes he noise
o he measu emen s. Table 2 summa izes he h ee exposed
models and hei es ima ed pa ame e s (excluding he ini ial
concen a ion o he simula ions): emission a e S, ai low
a es Q en and Qdi , Kloss, σ, and ek.
The Con inuous Time S ochas ic Modeling lib a y o
R (CTSM-R) (Juhl e al. 2016) has been used o he model
calib a ion and simula ion, including he calcula ion o
he boo h’s ini ial concen a ion (N0). This lib a y uses he
maximum likelihood me hod o his pu pose, commonly
employed in he li e a u e (Tha lo and Madsen 2015;
Ande sen e al. 2000), sol ing equa ion sys ems in Table 2
h ough he ga he ed expe imen al da a. This me hod equi es
he de ini ion o an ini ial assump ion o each es ima ed
pa ame e and he lowe and uppe bounds o he sea ch
Cebolla-Alemany e al. / Building Simula ion / Vol. 18, No. 4
853
o he op imal solu ion o he simula ion. The e o e, he
ini ial alues and bounds o each calib a ed pa ame e
can be seen in Table 3. Mo eo e , his lib a y also p o ides
he means o conduc ing he s a is ical es s desc ibed in
Sec ion 2.3.
Table 3 Ini ial alues and bounds
Ini ial
alue Lowe
bound Uppe
bound
N0 (pa icles/cm3) 60000 40000 500000
ek 5 0 100
σ 0.01 0 10
Kloss (cm6/(pa icles·min)) 1000 10 10000
Q en (cm3/min) 108 106 109
Qdi (cm3/min) 106 104 108
S (pa icles/min) 1015 1013 1016
2.3 Model alida ion
The p oposed alida ion p ocess is based on p e ious s udies
(Tha lo and Madsen 2015; B as ein e al. 2018; Wol e al.
2019b; Li e al. 2021). I includes s a is ical es s o assess
pa ame e signi icance, and i calib a ed alues a e ue
op ima. Then, he physical easibili y o pa ame e es ima ions
is analyzed. A e wa ds, esiduals a e s udied o e i y he
whi e noise assump ion in he measu emen s, and absolu e
esiduals a e compa ed wi h he ins umen s’ ole ance.
Finally, he Taylo diag am ep esen a ion comple es he
alida ion h ough a isual compa ison o he models’
pe o mance.
Fi s , a se o s a is ical es s a e used o alida e he
g ey-box models h ough he CTSM-R lib a y (Ande sen e
al. 2000; K is ensen e al. 2004; Bache and Madsen 2011).
The p- alue o each pa ame e in he model, ob ained om
he - es sco es, has o be less han 0.05 o he pa ame e
o be signi ican , assuming a signi icance le el o 95%.
Following, he calcula ion o he de i a i e o he objec i e
unc ion wi h espec o he ini ial condi ions (dF/dPa )
assesses whe he he solu ion ob ained is a ue op imum.
The esul is close o ze o when he alue is he ue op imum.
Then, he penal y unc ion’s de i a i e wi h espec o he
ini ial s a e (dPen/dPa ) is used o know i he pa ame e o
i s ini ial s a e is oo close o one o i s limi s.
Second, absolu e esiduals o he simula ions a e calcula ed
o es i he model’s accu acy alls wi hin he ins umen
ole ance ange. Mo eo e , he one-s ep-ahead esiduals
analysis shows i he whi e noise esiduals assump ion is
ul illed h ough he au oco ela ion unc ion (ACF) esiduals
and he cumula ed pe iodog am (Ande sen e al. 2000;
Bache and Madsen 2011).
Then, he physical easibili y o he es ima ions is s udied
gi en he known sys em’s in o ma ion coming om design
and equipmen speci ica ions.
Finally, he Taylo diag am (Taylo 2001) o all he
models is also calcula ed o compa e hei pe o mance
isually. This g aph simul aneously ep esen s he oo -
mean-squa e e o , he no malized s anda d de ia ion,
and he co ela ion coe icien be ween modeled da a and
obse a ions. Each model is shown as a poin on he diag am
and hus allows o an in ui i e and comp ehensi e
assessmen o mul iple pe o mance me ics in a single plo .
3 Resul s
In his sec ion, p esen ed models a e alida ed and
in e compa ed. Feeding he h ee models in Table 2 wi h
obse a ions in Figu e 3, and gi en he ini ial alues and
bounda y condi ions o he calcula ions shown in Table 3,
he es ima ed pa ame e s, hei s anda d de ia ion (SD),
and he alues om he alida ion p ocess a e shown in
his sec ion.
Table 4 summa izes da a in ol ing M1. Obse ing he
p- alues, all o hem a e below 0.05 and so a e signi ican .
Taking a look a he de i a i e o he objec i e unc ion wi h
espec o he ini ial condi ions and he penal y unc ion’s
de i a i e wi h espec o he ini ial s a e, all pa ame e s
a e nea 0, concluding ha hey a e ue op ima and a e no
Table 2 S udied g ey-box models
Model S a e space ep esen a ion Es ima ed pa ame e s
M1
()()
ou in en ou in di
in
ddd
gS N NQ dN NQ
N σw
V
⋅+⋅ - ⋅ +⋅ - ⋅
=⋅+⋅
in
k
k
YNe=+ S, Q en, Qdi , σ, ek
M2
(
)
(
)
ou in en ou in di loss in
in
ddd
gS N NQ dN NQ KN
N σw
V
⋅+⋅-⋅+⋅-⋅-
=⋅+⋅
⋅
in
k
k
YNe=+ S, Q en, Qdi , Kloss, σ, ek
M3
()()
2
ou in en ou in di loss in
in
ddd
gS N NQ dN NQ KN
N σw
V
⋅+⋅-⋅+⋅-⋅-⋅
=⋅+⋅
in
k
k
YNe=+ S, Q en, Qdi , Kloss, σ, ek
Cebolla-Alemany e al. / Building Simula ion / Vol. 18, No. 4
854
Table 4 Es ima ion esul s o M1 (no losses)
Es ima e S d. e o p- alue dF/dPa dPen/dPa
N0 (pa icles/cm3) 60000 15 0.0000 −1.2362×10−3 −0.0006
ek 26.6 0.1 0.0000 1.8807×10−2 0.0000
σ 0.01 0.00 0.0000 7.0650×10−4 0.0000
Q en (cm3/min) 2.45×108 0.45×108 0.0000 2.9107×10−3 0.0000
Qdi (cm3/min) 1.00×106 7900 0.0000 −9.7668×10−2 0.0000
S (pa icles/min) 1.38×1015 0.25×1015 0.0000 8.0814×10−4 0.0000
limi ed by hei dis ance o he bounda y alues o he
calcula ions, espec i ely. Rega ding i s es ima ions, he
ini ial concen a ion N0 o 60000 pa icles/cm3 is cohe en
wi h he measu emen s, which s a wi h alues be ween
60000 and 65000 pa icles/cm3; he en ila ion low a e Q en
o 2.45 × 108 cm3/min wi h an SD o 0.45 × 108 cm3/min
shows a easonable alue wi hin he sys em design alue o
2.5 × 108 cm3/min, esul ing in an ai change pe hou a e
(ACH) o oughly 144 h−1 conside ing he boo h olume
o 102 m3; he di usi e low a e Qdi h ough he boo h’s
doo shows an ai low speed below 0.1 m/s (ha ing a speed
o 0.007 m/s i conside ing he mos usual si ua ion o jus
hal doo open), con i ming ha he WA, sepa a ed o he es
o he plan by a wall and wo usually closed doo s, does no
con ain majo ai s eams when en ila ion is o inside he
boo hs; he sou ce’s emission a e S o 1.38 × 1015 pa icles/min
is wo o de s o magni udes highe han p e ious analysis
in li e a u e (Ribal a e al. 2019b). Howe e , hese esul s
a e incompa able gi en a subs an ial change in he sys em’s
layou : he enclosu e was di e en in he p e ious se ing, as
was he en ila ion sys em. Addi ionally, used da a in Ribal a
e al. (2019b) o calcula e he emission a e was om he
WA, no om he boo h. Consequen ly, hey de ined hei
pa icle emission a es as anspo ed emission a es o he
WA om he boo hs, so hey did no quan i y he p ocess’
emission, bu he anspo ed INP om he boo h o he
WA. The e o e, since he es o he calcula ed pa ame e s
in M1 a e consis en wi h eali y and design speci ica ions,
he es ima ion is conside ed eliable.
Obse ing M2 in o ma ion in Table 5, p- alues also show
ha all pa ame e s a e signi ican o he sys em, and he
penal y unc ion’s de i a i e wi h espec o he ini ial s a e
con i ms again ha he es ima ions and hei ini ial alues a e
no oo close o hei limi s. Ini ial concen a ion, bo h low
a es, and he emission a e a e he same as he calcula ed
in M1, addi ionally showing simila SDs. The loss a e Kloss
emains a he s a ing alue o 1000 cm6/(pa icles·min),
al hough he de i a i e o he objec i e unc ion wi h espec
o he ini ial condi ions es shows ha he solu ion is a ue
op imum, like all he o he pa ame e s.
In M3 (see Table 6), he p- alues show ha Kloss is no
signi ican , and all he es ima ions a e ue op ima a
Table 5 Es ima ion esul s o M2 (linea losses)
Es ima e S d. e o p- alue dF/dPa dPen/dPa
N0 (pa icles/cm3) 60000 13 0.0000 1.1439×10−3 0.0006
ek 26.6 0.07 0.0000 −4.2010×10−3 0.0000
σ 0.01 0.00 0.0000 −8.6994×10−5 0.0000
Kloss
(cm6/(pa icles·min)) 1000 0.6 0.0000 −1.8664×10−4 0.0000
Q en (cm3/min) 2.45×1080.21×108 0.0000 −9.2949×10−3 0.0000
Qdi (cm3/min) 1.00×106800 0.0000 −1.0066×10−1 0.0000
S (pa icles/min) 1.38×1015 0.12×1015 0.0000 1.1087×10−3 0.0000
Table 6 Es ima ion esul s o M3 (quad a ic losses)
Es ima e S d. e o p- alue dF/dPa dPen/dPa
N0 (pa icles/cm3) 60000 643 0.0000 4.0134×10−3 −0.0006
ek 26.6 0.06 0.0000 −3.4132×10−4 0.0000
σ 0.01 0.00 0.0000 −3.9153×10−7 0.0000
Kloss
(cm6/(pa icles·min)) 11.2 7.4 0.1289 2.3045×10−3 −0.0079
Q en (cm3/min) 9.37×1082.82×108 0.0010 8.9533×10−3 0.0240
Qdi (cm3/min) 1.00×1060.50×106 0.0000 −2.8549×10−2 0.0000
S (pa icles/min) 5.63×1015 1.35×1015 0.0000 2.3030×10−4 0.0003
om hei calcula ion limi s. In his model, he ini ial
concen a ion and he di usion low a e a e simila o he
es ima ions o he o he wo models, bu hei s anda d
e o s inc ease. The sou ce’s emission a e and he en ila ion
low a e oughly quad uple in alue and p esen an SD o
hei o de o magni ude, esul ing in an ex ac ion beyond
he physical capabili ies o he sys em and inc easing he
unce ain y o he model. The loss a e Kloss dec eases wi h
espec o M2 o 11.2 cm6/(pa icles·min) and also ge s a
high SD.
Obse ing he models’ simula ions and hei absolu e
esiduals in Figu e 4, all models exhibi simila beha io s.
Concen a ion apidly ascends and descends when gene a ion
is u ned on and o , and i slowly dec eases pa abolically
when he doo opens (and, he e o e, di usion occu s).
S eady s a es wi h gene a ion and mechanical ex ac ion
ac i a ed p esen a concen a ion o 5.7 × 106 pa icles/cm3.
When he doo is closed and no gene a ion o en ila ion
occu s, M1 and M2 p esen almos iden ical beha io s. Fo
M1, concen a ion emains in a ian in hese si ua ions.
Fo M2, concen a ion sub ly dec eases linea ly. Fo M3,
concen a ion diminishes quad a ically when he doo is
closed and, once i opens, he d op accele a es, as can be
clea ly obse ed a ound 19:00 on July 19. Since all hese
di e ences occu a lowe concen a ions, absolu e esidual
plo s show mino di e ences, ha ing high alues a he
beginning and he end o he simula ions due o he high
o de o magni ude in he pa icle concen a ion a ec ed by
Cebolla-Alemany e al. / Building Simula ion / Vol. 18, No. 4
855
measu emen noise ampli ied due o he dilu ion sys em’s
e ec on he ins umen .
Analyzing ACF esiduals and cumula ed pe iodog ams
(Figu e 5), hey p esen high simila i y among hem. Some
lags o he ACF esiduals a e ou side he con idence in e al
o 95%. None heless, mos o hem a e wi hin and p esen no
pa e n, showing no unde pa ame e iza ion in any model.
Consequen ly, he whi e noise esidual assump ion is e i ied.
Fig. 4 (a), (b), and (c) Simula ions o he boo h’s in e io INP concen a ion ( u quoise) wi h measu ed in e io (black), and ex e io (pink)
concen a ions o M1, M2, and M3, espec i ely; (d), (e), and ( ) absolu e esiduals o he simula ions o M1, M2, and M3, espec i ely
Fig. 5 (a), (b), and (c) Accumula ed unc ion esiduals o M1, M2, and M3, espec i ely; (d), (e), and ( ) cumula ed pe iodog am o
M1, M2, and M3, espec i ely
Cebolla-Alemany e al. / Building Simula ion / Vol. 18, No. 4
862
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