Resea ch A icle - Pee Re iewed Con e ence P oceeding
ESCAPE 35 - Eu opean Symposium on Compu e Aided P ocess Enginee ing
Ghen , Belgium. 6-9 July 2025
Jan F.M. Van Impe, G égoi e Léona d, Sa yajee S. Bhonsale,
Monika E. Polańska, Filip Logis (Eds.)
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Modeling he Impac o Non-Ideal Mixing on Con inuous
C ys alliza ion: A Non-Dimensional App oach
Jan T nkaa, F an išek Š ěpáneka*
a Uni e si y o Chemis y and Technology, Depa men o Chemical Enginee ing, P ague, Czech Republic
* Co esponding Au ho : F an isek.S epanek@ sch .cz
ABSTRACT
Ma hema ical modeling is essen ial o he e ec i e con ol o many chemical enginee ing p o-
cesses, including c ys alliza ion. Howe e , mos exis ing c ys alliza ion models used in indus y
and academia assume ideal mixing. As a esul , he unclea e ec s o impe ec mixing on c ys al-
liza ion, epo ed in expe imen al s udies, emain la gely unexplained. In his wo k we aim o ad-
d ess his gap in unde s anding by examining an isol en c ys alliza ion p ocesses on a gene al
heo e ical le el, using a no el dimensionless model. To add ess he impac o mixing on c ys al-
liza ion, we employ he Engul men model coupled wi h a popula ion balance, and we nondimen-
sionalize he model equa ions. Using his model, we explo e he dependence o he mean pa icle
size on he homogeniza ion a e, ep esen ed by he Damköhle numbe o c ys alliza ion. Mo e-
o e , we s udy he impac o mixing a a ious alues o he model's kine ic pa ame e s o simula e
di e ence in p ope ies o indi idual p oduc s. We show ha we a e able o explain he complex
in e ac ion be ween c ys alliza ion and mixing, p o ing ou model can se e as a ool o achie ing
a be e unde s anding o he p ocesses in ol ed. Finally, due o i s e iciency and educed numbe
o pa ame e s, he model is sui able o di ec i ing o expe imen al da a.
Keywo ds: c ys alliza ion, modeling, mixing, con inuous, non-dimensional
INTRODUCTION
C ys alliza ion is a undamen al echnique used o
sepa a ing and/o pu i ying solids in a ious indus ies,
including pha maceu icals and ood p oduc ion. The p o-
cess condi ions du ing c ys alliza ion signi ican ly a ec
he pa icle size and pu i y o he inal p oduc . Bo h pa -
icle size and shape in luence a ious ma e ial p ope ies,
such as lowabili y, il e abili y, and dissolu ion beha io .
E ec i e con ol o he c ys alliza ion p ocess can en-
hance p oduc quali y and elimina e he need o addi-
ional s eps like comminu ion o g anula ion.
Unlike cooling o e apo a i e c ys alliza ion, eac-
i e and an isol en c ys alliza ion in ol e mixing wo liq-
uids o dis inc composi ions. Supe sa u a ion is induced
h ough mixing, ei he due o he p oduc being syn he-
sized ( eac i e) o he educ ion o solubili y (an isol-
en ). Howe e , mos models cu en ly used in bo h in-
dus y and academia assume ideal mixing and ail o ac-
coun o he sys em's dependence on mixing dynamics,
he eby limi ing ou unde s anding o he p ocess.
Nume ous expe imen al s udies ha e add essed he
impac o mixing on c ys alliza ion. The e ec o inc eas-
ing mixing in ensi y on pa icle size is inconsis en , a y-
ing ac oss di e en subs ances and p oduc s [1]. The
mean c ys al size may inc ease, dec ease, o e en each
a maximum o minimum as he mixing in ensi y a ies.
E en using he same c ys allize ype o he same p od-
uc may lead o quali a i ely di e en ou comes. Addi-
ionally, he a e o homogeniza ion has also been shown
o a ec he wid h o he pa icle size dis ibu ion (PSD),
pa icle shape, and/o he ex en o agglome a ion.
Al hough hese dependencies a e well documen ed,
a sys ema ic app oach o s udying hem has no been es-
ablished. Resea ch, bo h on he modeling and expe i-
men al sides, ends o ocus on a single sys em a a ime.
In con as o he cu en end o de eloping complex
models, we aim o use a ela i ely simple modeling ame-
wo k, keeping he numbe o pa ame e s low and ena-
bling ex ensi e pa ame ic s udies necessa y o ou he-
o e ical esea ch.
The au ho s ha e al eady su icien ly explained he
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mechanism behind he exis ence o a maximum in he de-
pendency o pa icle size on mixing in ensi y o he semi-
ba ch case [2]. We now aim o ex end ou esea ch using
a no el dimensionless model o con inuous c ys alliza-
ion. As in o he a eas o chemical enginee ing, nondi-
mensionaliza ion o e s aluable insigh s by educing he
numbe o pa ame e s and e ealing cha ac e is ic sys-
em p ope ies. I also p o ides sui able scaling o a ia-
bles, imp o ing bo h he p ecision and e iciency o nu-
me ical compu a ions.
MODELING
Cu en Modeling App oaches
The popula ion balance equa ion (PBE) is he s and-
a d app oach o c ys alliza ion modeling. Depending on
he le el o de ail in desc ibing he hyd odynamics in-
ol ed, h ee main modeling app oaches a e commonly
conside ed:
▪ The ideal mixing assump ion is s ill widely used as
i allows o neglec ing mixing in he model
equa ions, hus simpli ying calcula ions. This
app oach assumes an in ini ely as
homogeniza ion a e and is he e o e applicable o
sys ems ha do no exhibi a dependence on he
agi a ion a e.
▪ Models based on compa men aliza ion ep esen
a balanced app oach o accoun o he complexi y
o mixing. These models di ide he s udied olume
in o se e al compa men s and desc ibe he
ma e ial lux be ween hem. Wi hin each
compa men , mixing is ypically assumed o be
in ini ely as . One o he mos popula models in
his ca ego y is he mechanis ic mic omixing
model de eloped by Baldyga and Bou ne, also
known as he Engul men model [3].
▪ CFD-based app oaches, whe e he PBE is
in eg a ed wi hin he CFD amewo k, a e pe haps
he mos igo ous. Howe e , e en CFD canno ully
cap u e mixing p ocesses a he molecula le el
(i.e., mic omixing). Since c ys alliza ion is
a molecula p ocess, hese me hods o en equi e
addi ional mic omixing models. Al e na i ely, hey
assume ha he homogeniza ion p ocess is limi ed
only by mac oscale mixing (i.e., as mic omixing).
Despi e he complexi y and high compu a ional
cos , he imp o emen in accu acy o e he
Engul men model appea s o be limi ed.
Model Desc ip ion
Coupling PBE Wi h he Engul men Model
We ha e based ou app oach on he Engul men
model, as we belie e i p o ides he mos sui able choice
o ou heo e ical s udy. In his Lag angian model, he
sys em olume is disc e ized in o wo well-mixed e-
gions: (1) he mixed zone, en iched wi h he e e ence
compound, and (2) he su ounding bulk luid. Mixing is
desc ibed as he expansion o he mixed zone h ough
he engul men o he bulk, leading o dilu ion o he com-
pound in he mixed zone i no addi ional sou ce is p e-
sen . This app oach was ini ially de eloped in eac o en-
ginee ing bu has since been applied o c ys alliza ion
(e.g., [4]). Al hough he model was o iginally de eloped
based on u bulen low mixing mechanisms, we ind i s
amewo k applicable e en o non- u bulen lows.
Acco ding o he Engul men model [3], he concen-
a ion o a compound in he mixed zone e ol es o e
ime acco ding o Eq. 1 as olume ac ion o he mixed
zone (𝑋) expands. The a e o homogeniza ion is de e -
mined by a single cons an , he mixing ime 𝑡𝑚𝑖𝑥 (Eq. 2).
𝑑𝑐
𝑑𝑡 = 1
𝑋𝑑𝑋
𝑑𝑡 (𝑐𝑏−𝑐)+𝑟 (1)
𝑑𝑋
𝑑𝑡 =1
𝑡𝑚𝑖𝑥 𝑋 (2)
Coupling he engul men model wi h PBE esul s in
Eq. 3, accoun ing only o p ima y nuclea ion and g ow h
(neglec ing agglome a ion and b eakage). The mola bal-
ance is hen desc ibed in Eq. 4. These equa ions a e alid
o bo h eac i e and an isol en c ys alliza ion in ba ch
o in con inuous ubula c ys allize s ope a ing a s eady
s a e, o which ime co esponds o he coo dina e ime.
𝜕𝑓
𝜕𝑡 +𝐺𝜕𝑓
𝜕𝐿 =1
𝑡𝑚𝑖𝑥 (𝑓𝑏−𝑓), 𝑓(0,𝑡)=𝐽
𝐺 (3)
𝑑𝑐
𝑑𝑡 = 1
𝑡𝑚𝑖𝑥 (𝑐𝑏−𝑐)−3𝑘𝑣𝜌𝑐𝑟
𝑀𝐺𝜙2+𝑟 (4)
Model Nondimensionaliza ion
To s udy he e ec o mixing on c ys alliza ion a
a heo e ical le el, we ha e de eloped an e icien
me hod o nondimensionalizing Equa ions 3 and 4. We
begin by in oducing he ollowing dimensionless a ia-
bles:
𝜏 = 𝑡
𝑡0, 𝜆= 𝐿
𝐿0, 𝜒= 𝑐∗
𝑐0, 𝑆 = 𝑐
𝑐0𝜒, 𝑛=𝐿0𝑉0𝑓 (5)
𝐺
=𝐺
𝐺𝑟𝑒𝑓 , 𝐽=𝐽
𝐽𝑟𝑒𝑓 , 𝑟 =𝑡0
𝑐0𝑟, 𝑐0=𝑐𝑟𝑒𝑓 (6)
De ining:
𝑉0=𝑘𝑣𝜌𝑐𝑟𝐿0
3
𝑀𝑐𝑟𝑐 e ,𝐿0=𝑡0𝐺𝑟𝑒𝑓,𝑡0=( 𝑐0𝑀𝑐𝑟
𝑘𝑣𝜌𝑐𝑟𝐽𝑟𝑒𝑓𝐺𝑟𝑒𝑓
3)1
4 (7)
leads o signi ican educ ion in he numbe o model pa-
ame e s. The e e ence alues 𝐽𝑟𝑒𝑓,𝐺𝑟𝑒𝑓,𝑐𝑟𝑒𝑓 can be se
a bi a ily, al hough hey do a ec scaling. The app op i-
a e alue o 𝑐𝑟𝑒𝑓 is he maximal solubili y while he sug-
ges ed choice o 𝐽𝑟𝑒𝑓 and 𝐺𝑟𝑒𝑓 is discussed la e .
Assuming iso he mal condi ions and solubili y as
a sole unc ion o he sol en olume ac ion 𝜑, he inal
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o m o ou non-dimensional model is desc ibed by he
ollowing equa ions:
𝑑𝑆
𝑑𝜏 =𝐷𝑎𝑐𝑟𝑆(𝑆𝑏𝜒𝑏
𝑆𝜒 +(𝜑−𝜑𝑏)
𝜒𝑑𝜒
𝑑𝜑 −1)+1
𝜒(𝑟 −3𝐺
2) (8)
𝜕𝑛
𝜕𝜏 +𝐺
𝜕𝑛
𝜕𝜆 =𝐷𝑎𝑐𝑟(𝑛𝑏−𝑛), 𝑛(0,𝜏)=𝐽
𝐺
(9)
De i a ion o he non-dimensional model leads o
he eme gence o a dimensionless numbe , 𝐷𝑎𝑐𝑟, de ined
as he a io o homogeniza ion a e o c ys alliza ion a e
(Damköhle numbe o c ys alliza ion):
𝐷𝑎𝑐𝑟 =𝑡0
𝑡𝑚𝑖𝑥 =𝑣𝑚𝑖𝑥
𝑣𝑐𝑟 (10)
We u he assume exp essions o g ow h and
c ys alliza ion a es acco ding o Eq. 11. By se ing he
e e ence alues equal o he a e coe icien s, we con-
enien ly educe he numbe o kine ic cons an s om
ou o wo, as shown in Eq. 12.
𝐺 =𝑘𝐺(𝑆−1)𝑔, 𝐽= 𝑘𝐽exp(−𝑗
ln2𝑆) (11)
𝐺
=(𝑆−1)𝑔, 𝐽=exp(−𝑗
ln2𝑆) (12)
The cons an s 𝑗 and 𝑔 oge he wi h he solubili y
da a emain he only p oduc p ope ies needed as inpu
o he model.
Applica ion o Con inuous An isol en P ocess
In his s udy, we ocus on iso he mal an isol en
c ys alliza ion in a ubula de ice de eloped in ou e-
sea ch g oup. The p ocess is schema ically illus a ed in
Figu e 1. A solu ion o candesa an cilexe il, an ac i e
pha maceu ical ing edien (API), in ace one is injec ed
pe pendicula ly o he leng h di ec ion o he ube in o
a s eam o wa e . Mixing o he wo s eams gene a es
supe sa u a ion, which leads o nuclea ion and subse-
quen g ow h o he p oduc pa icles.
The ini ial condi ions o he mixed zone a e gi en by
he p ope ies o he o ganic phase (𝑆(0)=1,𝜑(0)=0.85)
while bulk is ep esen ed by pu e wa e . The olume low
a e a io o o ganic o ino ganic phase is 1:9 (co e-
sponds o 𝑋(0)=0.1). Se ing he bulk a iables 𝑆𝑏, 𝜑𝑏
and 𝑛𝑏 o ze o and omi ing he eac ion e m esul s in
a simpli ied se o equa ions:
𝑑𝑆
𝑑𝜏 =𝐷𝑎𝑐𝑟𝑆(𝜑
𝜒𝑑𝜒
𝑑𝜑 −1)−3𝐺
2
𝜒 (13)
𝜕𝑛
𝜕𝜏 +𝐺
𝜕𝑛
𝜕𝜆 =−𝐷𝑎𝑐𝑟𝑛, 𝑛(0,𝜏)=𝐽
𝐺
(14)
The solubili y da a o candesa an cilexe il used o
ou s udy we e aken om li e a u e [5]. The model o he
con inuous de ice is implemen ed in he Py hon en i on-
men . Eq. 13 is con e ed in o a se o o dina y di e en ial
equa ions (ODEs) using 1D ini e olume me hod wi h
Ko en lux limi e . The se o ODEs is hen in eg a ed nu-
me ically using he Runge–Ku a me hod (RK45). All
simula ion esul s a e s eady wi h espec o he coo di-
na e ime, assuming an in ini ely long ube.
Figu e 1. Schema ic ep esen a ion o he se up and
spa ial disc e iza ion o he con inuous ASP de ice
acco ding o he Engul men model ( he shade o blue
colo ep esen s local concen a ion o API).
RESULTS
The Mixing Impac on he Mean Pa icle Size
As men ioned, ou model akes h ee p oduc -spe-
ci ic inpu s: pa ame e s 𝑗 and 𝑔 and he solubili y unc ion
𝜒(𝜑). In ou s udy, we a y he kine ic pa ame e s in o de
o add ess he in luence o he p oduc p ope ies on he
esul s o ou simula ions. The e ec s o changing he
solubili y cu e a e no p esen ed in his s udy.
As 𝑡0 is a cons an o a gi en subs ance, inc easing
𝐷𝑎𝑐𝑟 has he meaning o inc easing he a e o homoge-
niza ion. We p oceed wi h a sensi i i y analysis o he e -
ec o kine ic cons an s 𝑗 and 𝑔 on he mean pa icle size
(𝜆43) and i s dependence on 𝐷𝑎𝑐𝑟.
The In luence o 𝑔 on 𝜆43(𝐷𝑎𝑐𝑟)
The esul s o a ying 𝑔 along wi h 𝐷𝑎𝑐𝑟 a cons an
alue o 𝑗 a e depic ed in Figu e 2. As shown, all he e-
po ed scena ios o he dependency o he mean pa icle
size on mixing in ensi y a e co e ed by ou model. In
ag eemen wi h he expe imen al s udies, he pa icle
size acco ding o ou model may dec ease, inc ease,
each a maximum o minimum o emain cons an wi h
change in he mixing in ensi y.
All he pa ame ic cu es a e shaped simila ly. A
e y low 𝐷𝑎𝑐𝑟, he pa icle size dec eases while inc eas-
ing he mixing a e. This happens as he esul o mono -
onous inc ease in maximal supe sa u a ion wi h 𝐷𝑎𝑐𝑟
demons a ed on Figu e 3. A highe supe sa u a ion,
mo e nuclei a e o med due o enhanced nuclea ion a e,
esul ing in dec ease in pa icle size. The c ys al size
eaches minimum a 𝐷𝑎𝑐𝑟 = 10−1. In e es ingly, he pa a-
me ic cu es swi ch hei o de sho ly be o e eaching
he minimum. This happens as he maximal supe sa u a-
ion eaches he alue o wo ( he powe unc ion a gu-
men o he g ow h a e eaches one).
Fo mo e in ense mixing, 𝜆43 g ows wi h 𝐷𝑎𝑐𝑟. This
mos likely happens because he inc ease in nuclea ion
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a e diminishes a high supe sa u a ion while he g ow h
a e keeps accele a ing due o he na u e o he espec-
i e equa ions (𝐽 is limi ed unlike 𝐺
). Howe e , he in-
c ease o 𝜆43 is discon inued qui e ab up ly a alues o
𝐷𝑎𝑐𝑟 unique o e e y cu e. The alue o he local maxi-
mum and i s loca ion inc eases wi h highe g ow h a e
exponen .
To unde s and he sudden dec ease, le us desc ibe
he kine ics o c ys alliza ion acco ding o ou model in
he phase space o he non-dimensional concen a ion
(𝑆𝜒) and he sol en olume ac ion (Figu e 4). The ini ial
s a e ma ks he composi ion o he o ganic phase. The
yellow line ep esen s he condi ions close o pe ec
mixing, whe e he mixing and c ys alliza ion e en s a e
sepa a e. The ansi ion om he ini ial s a e o he mixed
s a e along he s aigh line is caused by dilu ion o he
island by engul men o he bulk. Any de ia ion om he
yellow line ep esen s induced c ys alliza ion while mix-
ing. The maximal possible supe sa u a ion is eached a
he yellow line a 𝜑𝑆𝑚𝑎𝑥 = 0.42. Fo 𝜑 >𝜑𝑆𝑚𝑎𝑥 he supe -
sa u a ion always g ows due o mixing while o
𝜑<𝜑𝑆𝑚𝑎𝑥 mixing causes dec ease in 𝑆. I he mixing a e
is oo as ela i e o he c ys alliza ion a e, c ys alliza-
ion is no induced be o e eaching 𝜑𝑆𝑚𝑎𝑥, causing he
c ys alliza ion o begin a signi ican ly lowe supe sa u-
a ion, educing he inal size o he c ys als a high al-
ues o 𝐷𝑎𝑐𝑟. This beha io is encoded in Eq. 13 as he
e m 𝜑
𝜒𝑑𝜒
𝑑𝜑 −1 is posi i e a 𝜑 >𝜑𝑆𝑚𝑎𝑥 and nega i e a
𝜑<𝜑𝑆𝑚𝑎𝑥. The alue o 𝜑𝑆𝑚𝑎𝑥 is he e o e de e mined
only by he shape o he solubili y unc ion.
The In luence o 𝑗 on 𝜆43(𝐷𝑎𝑐𝑟)
Le us now conside he scena io o a ying 𝑗 along
wi h 𝐷𝑎𝑐𝑟 a cons an 𝑔. O e all, inc easing 𝑗 hin e s he
nuclea ion a e and hus p omo es g ow h, causing
a gene al inc ease in pa icle sizes. Inc easing 𝑗 also
leads o la e onse o nuclea ion induced a highe su-
pe sa u a ion, u he a o ing g ow h o e nuclea ion.
The esul s o he simula ions a e shown in Figu e 5.
Fo low alues o 𝑗, he shape o he cu es emains
unal e ed compa ed o he esul s in Figu e 2. Howe e ,
a alues oughly om 10 o 80, he sys em unde goes
a quali a i e change. In his p ocess, he local maximum
anishes and he dec ease in 𝜆43 associa ed wi h ap-
p oaching pe ec mixing is u ned in o an inc ease.
P e iously, o 𝑗=1, he eason o he dec ease in
pa icle size was sudden d op in supe sa u a ion due o
as mixing, esul ing in less p onounced g ow h. How-
e e , nuclea ion a 𝑗 =100 is abou hund ed imes slowe .
A hese condi ions, lowe supe sa u a ion p oduces sig-
ni ican ly smalle numbe o pa icles, esul ing in seem-
ingly pa adoxical inc ease in pa icle size.
Figu e 2. Dependence o he mean non-dimensional
pa icle size on he Damköhle numbe o c ys alliza ion
wi h a ying he g ow h a e exponen .
Figu e 3. E olu ion o he supe sa u a ion o e ime
(maximal supe sa u a ion inc eases wi h 𝐷𝑎𝑐𝑟). The
x-axis is scaled by 𝐷𝑎𝑐𝑟 o be e compa ison.
Figu e 4. Rep esen a ion o he e olu ion o mixing and
c ys alliza ion in he phase space o he non-dimensional
concen a ion and he sol en olume ac ion.
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Figu e 5. Dependence o he mean non-dimensional
pa icle size on he Damköhle numbe o c ys alliza ion
wi h a ying nuclea ion cons an .
Pa ame ic Fi ing
As he de eloped non-dimensional model has only
wo kine ic pa ame e s in con as o ou in he dimen-
sional one, i is easie o i he model o expe imen al
da a. Mo eo e , using he non-dimensional model e-
qui es no knowledge abou he p oduc p ope ies o he
han i s solubili y beha io . To make use o hese ad-
an ages, we u he p esen how o use he model o
pa ame ic i ing.
Le 𝑳 deno e he ec o o 𝑁 measu ed c ys al sizes
a di e en olume low a es 𝑽 ( he mixing a e in ou
sys em is assumed o be dependen only on 𝑉). As e i-
den om Eq. 15, di iding 𝑳 by one o i s elemen s gi es
he same esul s o bo h dimensional and non-dimen-
sional da a:
𝑳
𝐿[0] =𝐿0𝝀
𝐿0𝜆[0] =𝝀
𝜆[0] (15)
We use his iden i y o de ine he objec i e unc ion as
ollows:
𝐹 =∑(𝐿[𝑖]
𝐿[0]−𝜆[𝑖]
𝜆[0])2
𝑁
𝑖=0 (16)
The emaining p oblem o be sol ed is inding he
link be ween he low a es and he co esponding
Damköhle numbe s. Based on he esea ch done on sim-
ila mic omixe s [6], we expec he mixing ime o be in-
e sely p opo ional o he olume low a e o he powe
o 1.5. Thus:
𝐷𝑎𝑐𝑟 =𝑡0
𝑡𝑚𝑖𝑥(𝑉)=𝑡0
𝐾𝑉1.5 =𝑐𝑉1.5 (17)
whe e he coe icien 𝑐 is unknown.
A e applying he same s a egy as o he c ys al
sizes, we ge :
𝑫𝒂𝒄𝒓
𝐷𝑎𝑐𝑟[0] =𝑐𝑽1.5
𝑐𝑉[0]
1.5 =𝑽1.5
𝑉[0]
1.5 (18)
The poin s a which 𝝀 a e o be e alua ed om simula-
ions a e he e o e:
𝑫𝒂𝒄𝒓 =𝐷𝑎𝑐𝑟[0] 𝑽1.5
𝑉[0]
1.5 (19)
whe e 𝐷𝑎𝑐𝑟[0] is unknown and he e o e i is ano he pa-
ame e o be op imized. O e all, inding he kine ic pa-
ame e s p esen s an op imiza ion p oblem:
minimize
𝑔,𝑗,𝐷𝑎𝑐𝑟[0] 𝐹 (20)
We ha e used expe imen al da a measu ed by ou
g oup o es he use o his me hod and alida e ou
model. The bes i o he model o he expe imen al da a
is p esen ed in Figu e 6. In he expe imen s, we ha e
used wo dis inc mixing uni s, he T-junc ion and he
FDmiX mixe . The la e de ice inc eases he homogeni-
za ion a e by passi ely in oducing low oscilla ions a
o he wise lamina condi ions. We ha e used a combina-
ion o di ec g id sea ch and he gene ic algo i hm o
sol e he op imiza ion p oblem.
The model quali a i ely desc ibes he measu ed
ends qui e well. The alue o 𝐷𝑎𝑐𝑟[0] o he FDmiX was
ound a highe alues han o he T-junc ion as ex-
pec ed. As mixing is e y slow in he T-junc ion, he pa -
icle size d ops qui e apidly be o e i le els ou as ex-
plained in desc ibing Figu e 2. Howe e , agg ega ion
may also con ibu e o he la ge size o pa icles meas-
u ed a low mixing a es. On he o he hand, he FDmiX
seems o ope a e a mixing a es close o he local maxi-
mum, which is e y well p edic ed by ou model.
Ongoing CFD analysis o he low in he T-junc ion
has e ealed ha mixing is no inished be o e eaching
he ou le , which ou model is no accoun ing o . Ex-
panding he model o add ess his issue may u he im-
p o e he esul s in he u u e.
Figu e 6. The bes i o he model o ou expe imen al
da a o wo di e en mixe s.
[LAPSE_DoNo Change] Sys Con ol T ans 4:XXXX-YYYY (2025) 6
CONCLUSION
In his wo k, we p esen a simple, ye e icien c ys-
alliza ion model de eloped o s udying he in e ac ion
be ween c ys alliza ion and mixing in ba ch and con inu-
ous ubula c ys allize s. We ha e ound an e icien way
o nondimensionalize he model equa ions, signi ican ly
educing he numbe o pa ame e s. As a esul , a new
non-dimensional numbe has eme ged du ing he p o-
cess – he Damköhle numbe o c ys alliza ion – ep e-
sen ing he a io o homogeniza ion and c ys alliza ion
a es.
We use he model o s udy he p ocess o con inu-
ous an isol en c ys alliza ion by means o pa ame ic
sensi i i y analysis. We we e able o unde co e he
complex in e ac ion be ween nuclea ion, g ow h and
mixing and inc ease he unde s anding o he p ocesses
in ol ed.
In addi ion o i s sui abili y o heo e ical esea ch,
ou model is also con enien o i ing he kine ic pa am-
e e s o expe imen al da a as he educed dimension o
he sea ch space s eamlines sol ing o he op imiza ion
p oblem.
Despi e he simplici y o he model, we ha e shown
i is able o accoun o all he epo ed scena ios o mix-
ing impac on pa icle size and o i measu ed da a su -
icien ly well.
ACKNOWLEDGEMENTS
The inancial suppo is om he p ojec
OP JAK INTER-MICRO ( egis a ion numbe
CZ.02.01.01/00/22_008/0004597), he Speci ic Uni e -
si y Resea ch (MSMT), co- unded by Eu opean Union.
LIST OF SYMBOLS
𝑐∗,𝜒 solubili y [mol m−3], [1]
𝑡, 𝜏 ime [s], [1]
𝑓, 𝑛 popula ion densi y [m−4], [1]
𝐿, 𝜆 c ys al size [m], [1]
𝜙2,
2 second momen [m−1], [1]
𝐽, 𝐽 nuclea ion a e [m−3], [1]
𝐺,𝐺
g ow h a e [m s−1], [1]
𝑟, 𝑟 eac ion a e [mol m−3 s−1], [1]
𝑐 concen a ion [mol m−3]
𝑆 supe sa u a ion [1]
𝑋 island olume ac ion [1]
𝜑 sol en olume ac ion [1]
𝑀 mola mass [kg mol−1]
𝑘𝑣 olume shape ac o [1]
𝜌𝑐𝑟 c ys al densi y [kg m−3]
𝑘𝐺 g ow h a e coe . [m s−1]
𝑘𝐽 nuclea ion a e coe . [m−3]
𝑔 g ow h a e exponen [1]
𝑗 nuclea ion cons an [1]
𝐷𝑎𝑐𝑟 Damköhle numbe [1]
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