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Op imiza ion ia Mul imodel Simula ion
A New App oach o Op imiza ion o Cyclone Sepa a o Geome ies
Thomas Ba z-Beiels ein, Ma in Zae e e , Quoc Cuong Pham
This is a p e-p in o an a icle published in
S uc u al and Mul idisciplina y Op imiza ion .
The final au hen ica ed e sion is a ailable online a :
h ps://doi.o g/10.1007/s00158-018-1934-2
S uc u al and Mul idisciplina y Op imiza ion manusc ip No.
(will be inse ed by he edi o )
Op imiza ion ia Mul imodel Simula ion
A New App oach o Op imiza ion o Cyclone Sepa a o Geome ies
Thomas Ba z-Beiels ein ·Ma in Zae e e ·Quoc Cuong Pham
Recei ed: da e / Accep ed: da e
Abs ac Inc easing compu a ional powe and he a ail-
abili y o 3D p in e s p o ide new ools o he combina-
ion o modeling and expe imen a ion. Se e al simula-
ion ools can be un independen ly and in pa allel, e.g.,
long unning compu a ional luid dynamics simula ions
can be accompanied by expe imen s wi h 3D p in e s.
Fu he mo e, esul s om analy ical and da a-d i en
models can be inco po a ed. Howe e , he e a e unda-
men al di e ences be ween hese modeling app oaches:
some models, e.g., analy ical models, use domain knowl-
edge, whe eas da a-d i en models do no equi e any in-
o ma ion abou he unde lying p ocesses. A he same
ime, da a-d i en models equi e inpu and ou pu da a,
bu analy ical models do no . Combining esul s om
models wi h di e en inpu -ou pu s uc u es migh im-
p o e and accele a e he op imiza ion p ocess. The op i-
miza ion ia mul imodel simula ion (OMMS) app oach,
which is able o combine esul s om hese di e en
models, is in oduced in his pape .
Using cyclonic dus sepa a o s as a eal-wo ld simu-
la ion p oblem, he easibili y o his app oach is demon-
s a ed and a p oo -o -concep is p esen ed. Cyclones
a e popula de ices used o il e dus om he emi -
Thomas Ba z-Beiels ein (co esponding au ho )
Technische Hochschule K¨oln
S einm¨ulle allee 1, 51643 Gumme sbach, Ge many
Tel.: +49-2261-8196-6391
Fax: +49-2261-8196-6666
E-mail: homas.ba z-beiels ein@ h-koeln.de
ORCID: 0000-0002-5938-5158
Ma in Zae e e
Technische Hochschule K¨oln
S einm¨ulle allee 1, 51643 Gumme sbach, Ge many
ORCID: 0000-0003-2372-2092
Quoc Cuong Pham
Technische Hochschule K¨oln
S einm¨ulle allee 1, 51643 Gumme sbach, Ge many
ed lue gases. They a e applied as p e- il e s in many
indus ial p ocesses including ene gy p oduc ion and
g ain p ocessing acili ies. P os and cons o his mul-
imodel op imiza ion app oach a e discussed and expe-
iences om expe imen s a e p esen ed.
Keywo ds Combined simula ion ·mul imodeling ·
simula ion-based op imiza ion ·me amodel ·mul i-
ideli y op imiza ion ·s acking · esponse su ace
me hodology ·3D p in ing ·compu a ional luid
dynamics
1 In oduc ion
Modeling allows he es ima ion o sys em pe o mance
unde new condi ions as well as he compa ison o di -
e en ope a ing condi ions and pa ame e iza ions, e.g.,
new geome ies. This a icle desc ibes di e en model
ypes, namely analy ical, su oga e, compu a ional luid
dynamics (CFD), and 3D p in ing models. Because e -
e y modeling app oach has i s p os and cons, a combi-
na ion, which uses in o ma ion om se e al models a
he same ime, migh be bene icial. S a ing wi h ma h-
ema ical modeling, we will desc ibe di e en modeling
app oaches i s .
Loosely speaking, ma hema ical modeling is “ he
link be ween ma hema ics and he es o he wo ld”
(Mee schae 2013). Ma hema ical modeling can be pe -
o med using analy ical and nume ical models: Analy i-
cal models a e ma hema ical models ha ha e a closed
o m solu ion, i.e., he solu ion o he equa ions used
o desc ibe changes in a sys em can be exp essed as
a ma hema ical analy ic unc ion. Nelson (1995) e e s
o analy ical models as “ ough-cu models”, i.e., ma h-
ema ically sol able and ypically less de ailed models.
2 Thomas Ba z-Beiels ein e al.
Nume ical (simula ion) models a e ma hema ical mod-
els ha use some so o nume ical ime-s epping me h-
ods such as New on’s me hod o simula e he model’s
beha io o e ime. In con as o analy ical models,
solu ions o simula ion models a e usually p esen ed
as ables o plo s. Simula ion is a widely used me hod
o s udying complex eal-wo ld sys ems, because many
sys ems canno be comple ely desc ibed by analy ical
models and expe imen a ion wi h he eal sys em is in-
easible o expensi e (Law 2007).
Nowadays, CFD simula ion is a well es ablished ech-
nique. I is also used in many s udies, which desc ibe
he opic discussed in his publica ion: he op imiza ion
o cyclone sepa a o geome ies (Ho mann and S ein
2007; Elsayed and Laco 2010).
O e he las decades, su oga e models, also known
as me amodels, gained impo ance (Jin e al. 2001; Ba z-
Beiels ein and Zae e e 2017). They a e build om and
hen used ins ead o he unde lying eal p ocesses o
simula ion models. Popula me amodelling echniques
include eg ession, adial basis unc ions, and K iging
(San ne e al. 2003; Kleijnen 2008).
3D-p in ing is a popula modeling echnique. I is
commonly used o alida e he esul s, e.g., a ce ain
geome y, om CFD simula ions. Recen ly, 3D-p in ing
was in eg a ed in o he op imiza ion ia simula ion loop
(P een and Bull 2014).
Al hough he model based app oach can be consid-
e ed a success s o y, i also causes some p oblems. Se -
e al c i ical issues in simula ion s udies a e ela ed o
e o s (Nelson 1995). These e o s can be due o bias
(e.g., ini ial-condi ion e ec s) o o p oblems wi h he
pseudo andom-numbe gene a o s. I easible, an ana-
ly ical analysis is in many cases p e e able o simula-
ion, because o he lack o sampling e o . Simula-
ion models can also be compu a ionally demanding,
because each simula ion desc ibes only one single se -
ing. The e o e se e al epea s wi h a ying inpu da a
a e necessa y, whe eas an analy ical model allows he
calcula ion o he exac cha ac e is ics o he sys em o
se e al se ings.
Fu he mo e, an inapp op ia e le el o model de-
ail, ailu e o collec adequa e sys em da a, and using
w ong pe o mance indica o s o compa isons a e com-
mon pi alls in bo h analy ical and nume ical simula-
ion s udies (Law 2007).
Many ex books desc ibe me hods o inding he
bes model, bu do no discuss he combina ion o se -
e al models. Nelson (1995) s a ed ha ex books “ end
o gi e he imp ession ha he e is a unique bes model
o any eal o concep ual sys em. This is no co ec .”
Mo e han one ype o model will be used in p ac-
ice. The inc easing compu a ional powe and he a ail-
abili y o 3D p in e s p o ide ools o new modeling
app oaches. Se e al simula ions can be un in pa al-
lel, e.g., long unning CFD simula ions can be accom-
panied by expe imen s wi h 3D p in e s, whe eas he
analy ical model is e alua ed as a baseline. Combina-
ions o he ollowing app oaches a e possible: (i) an-
aly ical models, (ii) nume ical simula ion, (iii) su o-
ga e models, (i ) lab expe imen s, and ( ) ield ex-
pe imen s. The cen al ques ion in his con ex is: A e
he e any bene i s in combining di e en simula ion ap-
p oaches and can he weakness o one app oach be com-
pensa ed by o he app oaches? To answe his ques ion,
an app oach o combining hese he e ogenous esul s
is necessa y. This a icle p esen s a new app oach o
handling se e al simula ion models in pa allel, which
will be e e ed o as op imiza ion ia mul imodel sim-
ula ion (OMMS). The OMMS app oach can be used
as he cen al pa o he well-es ablished op imiza-
ion ia simula ion me hodology (Fu 1994). To exem-
pli y OMMS, a eal-wo ld applica ion is used: cyclone
dus collec o s. This a icle p esen s esul s om an ex-
pe imen al s udy, which can be ega ded as a p oo -
o -concep o OMMS. Fo he expe imen s, we ha e
chosen a combina ion o ou di e en modeling ap-
p oaches:
(M-A) analy ical,
(M-C) CFD simula ion,
(M-S) su oga e (me amodels), and
(M-P) 3D p in ing models.
This pape is s uc u ed as ollows: Sec ion 2 de-
sc ibes ela ed wo k. Cyclone dus abso be s a e b ie ly
desc ibed in Sec ion 3. Sec ion 4 p esen s he OMMS
loop. Sec ion 5 compa es esul s om di e en mod-
eling app oaches. Expe imen al esul s based on hese
modeling app oaches a e p esen ed in Sec ion 6. How
o combine esul s om a ious models ia ensemble
building is shown in Sec ion 7. Finally, Sec ion 8 gi es
a conclusion and an ou look.
2 Rela ed Wo k
The idea o using di e en models wi h di e en es-
olu ions has been discussed in he li e a u e o many
yea s. Zeigle and O en (1986) desc ibe mul iple le els
o model agg ega ion ( esolu ion, abs ac ion). These
le els depend on he objec i es, knowledge, and he
a ailable budge ( esou ces, e.g., ime). Fishwick and
Zeigle (1992) p esen a o malism and a me hodology
o de eloping mul iple, coope a i e models o physi-
cal sys ems om quali a i e physics. Ba zie and Pe y
(1991) desc ibe a wo-le el modeling app oach o de-
eloping simula ion models in he shipbuilding indus-
y. Chaudhu i e al. (2015) desc ibe a lapping wing
OMMS: Op imiza ion ia Mul imodel Simula ion 3
op imiza ion ask. They use mul iple su oga es, mul-
iple in ill c i e ia, and mul iple poin s o he same ex-
pe imen al da a se . Kazemi e al. (2016) use di e en
machine lea ning app oaches o c ea e simple and eli-
able models o p edic ing g anule size dis ibu ions. An
i e a i e p ocedu e assis ed by c oss alida ion was im-
plemen ed o ind ou he bes model among housands.
The cyclone modeling, simula ion, and op imiza ion ap-
p oach p esen ed in ou s udy is ela ed o he wo k
om P een and Bull (2014), who op imized e ical-
axis wind u bines using minia u ized 3D-p in ed wind
u bines.
Yang (2003) s a es ha selec ion o one model can
be be e when he e o s in p edic ion a e small and
ha he model combina ion wo ks be e when he e -
o s a e la ge. Simpson e al. (2012) p esen a hough -
ul e iew o se e al mul imodel app oaches. They s a e
ha “ he use o mul iple su oga es (i.e., a se o su -
oga es and possibly a weigh ed a e age su oga e) is
e y appealing in design op imiza ion due o he ac
ha he bes su oga e may no lead o he bes e-
sul ; and complemen a y because i ing many su o-
ga es and epea ing op imiza ions is cheap compa ed
o cos o simula ion.” They also desc ibe a mul idis-
ciplina y app oach which is no di ec ly compa able o
OMMS, because independen models o di e en sub-
sys ems a e combined a he han in eg a ing se e al
models o he same sys em.
Fu he mo e, co-K iging, which is a popula me hod
ha combines esul s om ine and coa se g ained mod-
els, can be men ioned in his con ex (Fo es e e al.
2007). Typically, co-K iging ies o combine da a om
models which ha e di e en ideli y, e.g., a ine model
ha is expensi e o compu e and a less accu a e, coa se
model, which is cheape o compu e. In con as o
single- ideli y K iging models, co-K iging a emp s o
lea n he co ela ion be ween he coa se and ine model,
hus being able o exploi he la ge amoun o da a
de i ed om he coa se model o imp o e he ep esen-
a ion o he expensi e, ine model. This could be used
o he me a-modeling s ep, especially when di e en
le els o ideli y a e a ailable.
In gene al, he e a e wo op ions o deal wi h mul-
iple models: (i) selec ion o he bes model and (ii)
combina ion o esul s om se e al models. Mos ap-
p oaches y o selec one model, whe eas OMMS com-
bines esul s om se e al models using s acked eg es-
sion (Wolpe 1992; Ba z-Beiels ein 2016). Ou s udy
p esen s an in eg a ed simula ion and expe imen a ion
me hodology on a ious scales (o laye s).
Da
Da
D
D
be
be
he
he
h
h
hh
be
be
Da
Da
D
D
F on & iew Top& iew
Du
Du
Du
Du
hz
hz
Fig. 1 S anda d geome y o he cyclone conside ed in his
s udy. The co esponding geome y pa ame e s, xg, a e de-
sc ibed in Table 1.
3 Cyclone Dus Collec o s
Cyclones a e used in oil and gas, i on and s eel, chem-
ical and ood indus y o il e a maximal amoun o
dus om lue gas (Ho mann and S ein 2007). They
can be applied in ex emely ha sh and demanding en-
i onmen s, bu show a ela i ely low sepa a ion com-
pa ed o elec os a ic dus collec o s. An e icien cy-
clone equi es he op imiza ion o i s geome y pa am-
e e s, which a e shown in Figu e 1. E en wi h oday’s
mode n ools, he complexi y o cyclone beha io is
such ha expe imen al s udies a e necessa y o a solid
unde s anding o he phenomena go e ning hei be-
ha io . The cyclone geome y can be speci ied by he
pa ame e ec o , xg, wi h he ollowing en ies: inle
wid h be, body diame e Da, diame e o he o ex
inde D , diame e o he dus exi Du, o al heigh h,
inle heigh he, o ex inde imme sion h , and cylin-
de heigh hz. In addi ion o hese geome y pa ame-
e s, xg, he speci ica ion o he ope a ing pa ame e s,
xp, is necessa y. The geome y and p ocess pa ame e
se s a e shown in Table 1. We will concen a e in his
s udy on he collec ion e iciency as speci ied in L¨o le
(1988), which will be explained in Sec ion 5.1.
4 Thomas Ba z-Beiels ein e al.
Table 1 Nomencla u e om L¨o le (1988). Values (L, M, S) e e s o he alues o he geome y pa ame e s xg o he L¨o le ,
Muschelknau z E., and S ai mand high e iciency cyclones, espec i ely. The o ex inde imme sion, h , is modi ied o e e y
cyclone geome y. The ype “xp” deno es ope a ing pa ame e s. Pa ame e alues, which depend on o he alues, a e labeled
as “*” in he Type column. Pa ame e s o be op imized a e labeled in he las column.
Pa ame e Uni s Values (L, M, S) Type Desc ip ion Op imized
bemm 12.8; 9.92; 7.97 xginle wid h yes
Damm 80.64; 116.48; 39.97 xgbody diame e yes
D mm 26.88; 29.12; 19.98 xgdiame e o he o ex inde yes
Dumm 26.88; 39.04; 15.04 xgdiame e o he dus exi yes
hmm 160; 160; 160 xg o al heigh o he cyclone yes
hemm 38.4; 29.6; 19.98 xginle heigh yes
h mm 0; 35; 44 xg o ex inde (ou le pipe) imme sion yes
hzmm 44.8; 29.64; 59.95 xgcylinde heigh yes
amm Da/2 * cyclone adius no
imm D /2 * adius o he o ex inde no
himm h−h * heigh o he imagina y cylinde CS no
emm a−be/2 * mean inle pipe adius no
F-Fe/Fi* a io be ween inle and ou le a ea no
Femm2he×be* inle a ea no
Fimm2π× 2
i* ou le a ea no
ems−120 xpinle eloci y no
λg- 0.005 xpload- ee ic ion coe icien no
µPa s 1.8×10−5xp iscosi y no
% kg/m31.2000 xpgas densi y no
%pkg/m32700 xppa icle densi y no
c oh kg/m30.061 xp aw gas concen a ion no
B-B=c oh/ρ * mass load no
ims−1˙
V /(π 2
i) * eloci y o ex inde (ou le pipe) no
( i) ms−1Eq. (1) * adial gas eloci y on he ou le pipe no
ϕi ms−1Eq. (2) * angen ial eloci y a CS no
˙
Vm3/hFe× e* olume ic low a e h ough he cyclone no
λ-λg(1 + 2√B) * wall ic ion ac o ; ic ion coe icien no
4 Op imiza ion ia Mul imodel Simula ion in
he Loop
In he op imiza ion ia simula ion se ing, he goal is
o pe o m uns o he simula ion model in an e icien
manne and o de e mine hose inpu a iables, which
esul in an op imal (o nea op imal) solu ion (Fu 1994).
The OMMS app oach ex ends he s anda d op imiza-
ion ia simula ion se ing by in eg a ing esul s om
se e al model ypes. In con as o ma hema ical mod-
els, which usually equi e some inpu alues only, da a-
d i en models equi e he speci ica ion o inpu and ou -
pu alues. To cla i y he da a low and model building
p ocess in he OMMS app oach, he ollowing model
ca ego ies will be used:
–X-models use inpu pa ame e s, e.g., geome y and
p ocess pa ame e s.
–XY -models use he inpu pa ame e s as well as he
co esponding ou pu alues, e.g., collec ion e iciency.
So, he analy ical (M-A), CFD (M-C), and 3D p in ing
(M-P) models a e conside ed as X-models, whe eas he
su oga e (M-S) models a e XY -models.
The gene al concep o OMMS is illus a ed in Fig-
u e 2. He e, we conside he op imiza ion o he cy-
clone’s geome y pa ame e s, which should be dis in-
guished om he p ocess pa ame e s. I consis s o he
ollowing s eps:
(S-1) Selec an ini ial design. Se = 1, whe e de-
no es he numbe o pa ame e se s. The i s se o
geome y pa ame e s, x( )
g, is gene a ed.
(S-2) Speci y he p ocess pa ame e s xp. They a e no
changed du ing he op imiza ion.
(S-3) Selec X-models (e.g., CFD, analy ical). In ad-
di ion o he geome y and p ocess pa ame e se s,
u he pa ame e s migh be necessa y o each sep-
a a e model. These model speci ic pa ame e s will
be e e ed o as xm. Fo example, he CFD simula-
o equi es he speci ica ion o pa ame e s o hea
ans e , su ace p ope ies, damping, collision, and
adia ion. These pa ame e s a e no used in o he
simula ion models. They a e no changed du ing he
op imiza ion. The se x( )= (x( )
g,xp,xm) will be
used o build he X-models.
(S-4) Build X-models. Fo building hese models no in-
o ma ion abou he dependen (ou pu ) a iables
yis needed. In his s ep, one o se e al models ( 1,
. . ., p) om he se o X-models, which comp e-
hends 3D-p in ed objec s, analy ical model o mu-
las, o CFD simula ion models, a e gene a ed. The
OMMS: Op imiza ion ia Mul imodel Simula ion 5
Legend
(S-3)
Selec X-models
(S-7)
Selec XY-models
(S-1)
Ini ial design
(S-5)
E alua e
X-models
(S-11)
S o e op imized
design
(S-4)
Build X-models
(S-2)
P ocess
pa ame e s
Ex e nal da a
(S-6)
Collec esul s
(S-9)
Op imize on
me amodel
(S10)
Te mina e?
P ocess
Pa allel
P ocesses
Da abase
(S-8)
Build me amodel
Fig. 2 Op imiza ion ia mul imodel simula ion in he loop. Se e al simula ion models a e used in pa allel. Elemen s o he i s
se o models, i.e., du ing s eps (S-3), (S-4), and (S-5), can be one o se e al CFD simula o s, analy ical models, o expe imen s
based on 3D-p in ed objec s. Resul s om hese di e en models a e collec ed and op ionally combined wi h addi ional esul s,
which we e s o ed in a da abase. The second se o models is build du ing s ep (S-8). Models om he second se a e classical
su oga e models, e.g., neu al ne wo ks, linea eg ession models, o K iging models. Because simula ion esul s, i.e., y- alues
a e a ailable a his s age o he mul imodel simula ion p ocess, a b oade se o models can be used han du ing he i s s eps
(S-3) o (S-5). Resul s om hese models can combined in se e al ways. We desc ibe an app oach ha is based on s acked
gene aliza ion (Wolpe 1992). Op imiza ion is pe o med on he s acked model (S-9).
cons uc ion p ocess esul s in se e al models, which
use he same se o pa ame e s x( ).
(S-5) E alua e models. The models a e e alua ed, i.e.,
each model gene a es an ou pu : j:x( )→y( )
j.
No e, some models gene a e a de e minis ic ou pu ,
e.g., CFD models, whe eas o he , e.g., 3D-p in ed
models, gene a e s ochas ic (noisy) ou pu s. The e-
o e, epea s should be conside ed o he s ochas ic
models, o imp o e he quali y o he measu ed al-
ues (Law 2007; Ha ka e al. 2016).
(S-6) Collec esul s. Besides he se o pai s {(x(k), y(k)
j)},
o k= 1, . . . , and j= 1, . . . , p, addi ional esul s
{(x(m), y(m)
l)}, o m= 1, . . . , s and l= 1, . . . , q,
e.g., om his o ical da a o da a om he li e a u e,
can be used in he cons uc ion o he me amodels.
(S-7) Selec XY -models. XY -models use he pa ame-
e se , x(k)= (x(k)
g,xp) as well as he co espond-
ing ou pu alues y(k)
i o model building, wi h k=
1, ..., ( +s). In gene al he numbe o design poin s
(p+s) is equi ed o be la ge enough o allow build-
ing easonable models.
(S-8) Build me amodel. An ensemble engine builds a
me amodel by combining in o ma ion om se e al
models, say Fi. I implemen s s acking me hods.
The me amodel will be e e ed o as F∗. The en-
semble engine selec s se e al models om he huge
a ie y o su oga e models (e.g., andom o es , o
6 Thomas Ba z-Beiels ein e al.
K iging). These se e as basic o le el-0 models.
C oss alida ion is used o build an ensemble model
om he po olio o le el-0 models. The le el-1 ain-
ing algo i hm is ypically a ela i ely simple linea
model. Ins ead o s acking, a weigh ed combina ion
o models Fio co-K iging can be used. I mod-
els o simila ideli y a e combined, we would sug-
ges o employ o s acking. In a mixed case, a co-
K iging model could be in eg a ed in o a s acked
me amodel (i.e., as a single model Fi). O he en-
semble echniques may also be applicable, e.g., bag-
ging o boos ing (Mu phy 2012). Howe e , s ack-
ing is e y e ec i e e en when combining only ew,
s ong lea ne s and i p o ides addi ional in o ma-
ion, e.g., he con ibu ion o each o he combined
models o he ensemble.
(S-9) Op imize on he me amodel. The model F∗is
used as a su oga e o pe o ming he op imiza-
ion s ep. The op imiza ion esul s in a new se
o p omising geome y pa ame e s, which will be
e alua ed in he ollowing s ep. The coun e o he
numbe o pa ame e se s is inc emen ed and he
new design can be e e ed o as x( )
g. Ins ead o in-
c easing by one, se e al new design poin s, e.g.,
om models wi h di e en un imes, can be added
o he pa ame e se .
(S-10) Check he e mina ion c i e ion.
(S-11) S o e he op imized design. Op ionally, i can be
added o a da abase.
5 Modeling App oaches
To exempli y he OMMS app oach, ou di e en mod-
eling app oaches a e desc ibed in he ollowing: analy i-
cal (M-A), su oga e (M-S), CFD simula ion (M-C) and
3D p in ing (M-P) .
5.1 The Analy ical Model (M-A)
A b oad a ie y o analy ical models in ended o p e-
dic cyclone sepa a ion pe o mance exis s in he li e -
a u e (L¨o le 1988; O e camp and Man ha 1998; Ho -
mann and S ein 2007). The analy ical app oach de el-
oped by Ba h (1956) and Muschelknau z (1972) can
be conside ed as s anda d. I will be e e ed o as he
Ba -Muschelknau z me hod o modeling. This me hod
is based on he assump ion ha a pa icle ca ied by
he o ex is in luenced by wo o ces: a cen i ugal o ce
and a low esis ance. They a e exp essed a he ou le
pipe adius iwhe e he highes angen ial eloci y oc-
cu s. Some assump ions can be conside ed easonable
enough o ob ain a good comp omise be ween accu a e
p edic ion and simpli ica ion o he equa ions, e.g., he
pa icles a e sphe ical, he pa icle mo ion is no in lu-
enced by he p esence o neighbo ing pa icles, and he
adial o ce on he pa icle is gi en by S okes’s law.
Based on he geome y and ope a ing pa ame e s
om Table 1, he ollowing calcula ions can be pe -
o med. The equilib ium-o bi model assumes a cylin-
d ical con ol-su ace (CS), which is cons uc ed by ex-
ending he o ex inde wall o he bo om o he cy-
clone. Le hideno e he heigh o he CS. The adial
eloci y a iequals o:
( i) = ˙
V
2π i(h−h ).(1)
Fo a gi en mass load B=c oh/ρ , he wall ic ion
ac o λcan be calcula ed as λ=λg(1 + 2√B). The
co ec ion ac o α o con ac ion is equal o:
α= 1.0−0.54 −0.153
FB
1
3
e.
Using he ou le pipe eloci y i=˙
V
π 2
i,Ba h (1956)
de i ed he ollowing equa ion
ϕi
i
=U=1
Fα · i
e+λ·h
i
= i eπ
αFe+hi eπλ.(2)
These eloci ies a e used o de e mine he collec ion
e iciency. The equilib ium-o bi model is based on a
o ce balance on a pa icle ha is o a ing a adius
i. Small pa icles lea e he cyclone h ough he o ex
inde , whe eas la ge pa icles a e mo ing o he cyclone
wall. The cu size, x50, plays a cen al ole in hese cal-
cula ions. Fo cyclones, pa icles o size x50 ha e a 50-
50 chance o being cap u ed, smalle pa icles a e less
likely o be cap u ed, la ge pa icles a e mo e likely
o be cap u ed. The o ces ac ing on a pa icle o a -
ing on he CS, which is assumed o sepa a e he ou e
egion o downwa d low om he inne egion o up-
wa d low, a e (i) he cen i ugal o ce ac ing ou wa d
wi h a magni ude o πx3ρp 2
ϕi/(6 i) and (ii) he S oke-
sian d ag ac ing inwa d 3πxµ ( i).By equa ing hese
o ces, Ba h (1956) de eloped an analy ical model o
he cu size as ollows:
x50 =s18µ ( i) i
(%p−% ) 2
ϕi
.
The ac ional e iciency cu e assigns an e iciency o
he pa icle diame e . I is desc ibed by
T(x) =
1 + 2
x
x50 3.564
−1.235
,
OMMS: Op imiza ion ia Mul imodel Simula ion 7
Table 2 Pa icle size dis ibu ion able. Values co espond
o he dus used in he 3D p in ing expe imen s.
Pa icle Size x[µm]∆x Mean
˜x
[µm]
∆Qe(x) Cumula i e
0-1 1 0.5 0.1 0.1
1-2.7 1.7 1.85 0.1 0.2
2.7-5.5 2.8 4.1 0.1 0.3
5.5-8.7 3.2 7.1 0.1 0.4
8.7-12.7 4 10.7 0.1 0.5
12.7-16.9 4.2 14.8 0.1 0.6
16.9-21.2 4.2 19 0.1 0.7
21.2-25.4 4.2 23.25 0.1 0.8
25.4-30.8 5.4 28.1 0.1 0.9
30.8-63 31.2 46.9 0.1 1.0
whe e xis he pa icle size. The o e all collec ion e i-
ciency Eis p edic ed acco ding o:
E=Zxmax
xmin
T(x)qe(x)dx ≈
xmax
X
xmin
T( ˜xi)∆Qe(xi),(3)
whe e xmin is he lowe bound o he pa icle size, xmax
is he uppe bound o he pa icle size, ˜xiis he mean
pa icle size in each ac ion, ∆Qe(xi) is he change in
dis ibu ion o pa icle sizes and qe(x) = ∆Qe(xi)/∆xi.
The pa icle size dis ibu ion able, which was used in
ou s udies, is shown in Table 2. Resul s om ou col-
lec ion e iciency Ecalcula ions o models om he
li e a u e and o models used in ou expe imen s a e
shown in Table 3 and Table 5, espec i ely. The co e-
sponding unc ion is a ailable in he R package SPOT
as unCyclone().
5.2 CFD Simula ions (M-C)
Compu a ional Fluid Dynamics simula ions ha e p o-
en o be use ul o s udying he luid and pa icle lows
in cyclones (Hoeks a e al. 1999). They ha e clea
ad an ages o unde s anding he de ails o he low
in cyclones, bu also limi a ions in e ms o modeling
cyclone sepa a ion pe o mance accu a ely (Ho mann
and S ein 2007). Nume ical simula ions a e pe o med
by sol ing he uns eady-s a e, h ee-dimensional Rey-
nolds a e aged Na ie -S okes (RANS) equa ions com-
bined wi h a closu e model o he u bulen s esses
and he la ge eddy simula ion app oach.
The CFD simula ions we e ca ied ou wi h he open
sou ce so wa e OpenFOAM, which has been de eloped
o sol ing nume ical p oblems (Konan and Huckaby
2015). The mesh o hese CFD simula ions consis s o
app oxima ely 30,000 o 50,000 hexahed al cells. The
ansien MPPICFoam sol e was chosen o calcula e
he wo-phase low (Eule -Lag ange). The cyclone sim-
ula ion om he OpenFOAM cyclone u o ial was used
as a basis (OpenFOAM Founda ion 2016). The se ings
o Schemes, Solu ion, anspo P ope ies, and
u bulenceP ope ies we e adap ed o ob ain he same
se up as o he 3D p in ing expe imen s. The se ings
in he kinema icCloudP ope ies ile we e adjus ed
o he cha ac e is ics o he used pa icles. The densi y
o he pa icles was changed o 2,700 kg/m3(as in Ta-
ble 1 abo e). Using he gene alDis ibu ion model,
he pa icle dis ibu ion om Table 2 can be p ecisely
mapped. In he simula ions, 20,000 pa cels ep esen
he en i e y o he pa icles, whe e each pa cel has he
same mass. The amoun o 20,000 pa cels was chosen
o minimize he ep esen ed mass pe pa cel and no
o blow up he calcula ion ime o each imes ep. The
minimiza ion causes a lowe e o when a pa cel escapes
a an ou le .
The hea T ans e ,su aceFilm,damping,s o-
chas icCollision, and adia ion submodels we e le
unchanged a he “o ” s a e. In he expe imen s, a o-
al o 6 g was sp ead o e 10 h us s and he wai ing
ime be ween each h us was app oxima ely 3 seconds.
The simula ion akes only one h us o 0.6 g ins ead
o pe o ming he 10 epe i ions in o de o a oid e y
long simula ion imes. The pa icle eloci y in he sim-
ula ion was se o he same alue as he de e mined
eloci y o he ai a he inle . O e all, a ime ame
o 3 seconds is simula ed. Fo his, a o al calcula ion
ime o app oxima ely 96 hou s (wall clock ime) us-
ing 16 p ocesso co es is equi ed. The simula ion was
con olled by he ime s ep and elaxa ion ac o s and
beha ed ela i ely s able.
A e each expe imen , a ce ain amoun o dus e-
mains in he cyclone. We conside dus as sepa a ed, i
i lea es he cyclone h ough he dus exi (usually a
he bo om o he cyclone). The e alua ion o he sim-
ula ion esul s is shown in column (M-C) in Table 5. I
he emaining dus in he cyclone is also conside ed as
sepa a ed, he collec ion e iciency is inc eased.
5.3 Su oga e Modeling (M-S)
Compu a ional luid dynamics simula ions a e compu-
a ionally expensi e. Da a-d i en models o cyclone sep-
a a o s, which a e subs an ially cheape o e alua e,
can be used ins ead. Well-known app oaches o handle
cos ly objec i e unc ions is o employ esponse su ace
o su oga e models (Jin 2003; Kleijnen 2008). Tha is,
da a-d i en su oga e models may be cons uc ed based
on expe imen al esul s. Then, an op imiza ion algo-
i hm may wo k on he su oga e model ins ead o he
ac ual objec i e unc ion. To exempli y his app oach,
This p ojec has ecei ed unding om he Eu opean Union’s Ho izon 2020
esea ch and inno a ion p og amme unde g an ag eemen No 692286.
