In e na ional Jou nal o The mal Sciences 173 (2022) 107349
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Nume ical simula ion o he ansien hea ans e in a blas u nace main
ough du ing i s comple e campaign cycle
P. Ba al a,b,c, L.J. Pé ez-Pé eza,c, P. Quin ela a,b,c,∗
aDepa men o Applied Ma hema ics, Uni e sidade de San iago de Compos ela, 15782 San iago de Compos ela, Spain
bTechnological Ins i u e o Indus ial Ma hema ics (ITMATI), 15782 San iago de Compos ela, Spain
cIns i u o de Ma emá icas (IMAT), Uni e sidade de San iago de Compos ela, 15782 San iago de Compos ela, Spain
ARTICLE INFO
Keywo ds:
S eelmaking
Blas u nace ough
La ge-scale ansien simula ion
Radia i e hea ans e
Global op imiza ion
ABSTRACT
To achie e highe blas u nace (BF) main ough a ailabili y and o minimize he equency o epa a ions is
a key conce n in he s eelmaking indus y. Fo his pu pose, s a egies o assess e ac o y wea a e equi ed,
which is hea ily in luenced by he empe a u e in he e ac o y linings. In his wo k, a ma hema ical model
o assess he ansien beha iou o he empe a u e in a c oss-sec ion o a BF main ough du ing a comple e
campaign is p esen ed. The scope is o in es iga e he e ec ha he cas ing s ops ha e on he empe a u e in
he ough. A sequence o p oblems co esponding o each BF apping and he subsequen s op is de e mined
using p ocess da a o a BF.
The open-sou ce ini e elemen compu ing pla o m FEniCS is employed o sol e he model. The disc e iza-
ion and he nume ical algo i hm ha e been p esen ed and alida ed wi h a manu ac u ed solu ion es in a
p e ious wo k. The nume ical esul s show ha he e ec o he s ops du ing hese campaign cycles is non-
negligible, p e en ing he bulk o he solid laye s om eaching a s eady s a e. Quali a i e ag eemen wi h
empe a u e measu emen s ob ained wi h he mocouples embedded in he ough is obse ed. Since he e is
a signi ican deg ee o unce ain y conce ning he placemen o he de ices, a minimiza ion p oblem o adjus
hei posi ions wi hin he co esponding easible egions is p oposed. A he iden i ied posi ions, good le els o
i be ween he measu ed and he compu ed empe a u es a e achie ed. The ag eemen dec eases owa ds he
end o he campaign cycle, being sugges i e o se e e e ac o y wea , especially a he la e als o he ough.
1. In oduc ion
In he con ex o he in eg a ed i on and s eelmaking p ocess, i
is a majo conce n o he indus y o minimize he wea su e ed
by e ac o y linings. F equen ly, in he expe imen al and nume ical
s udies abou such p ocess, he ocus is pu on he blas u nace (BF),
he me allu gical u nace whe e he i on o e is educed and smel ed
o ob ain ho me al. Howe e , he e ac o y linings used a he main
ough, whose pu pose is o anspo and sepa a e ho me al om slag
a e i s ex ac ion om he BF h ough he aphole, a e subjec o
equen epai s and ope a ion p oblems wi h he consequen cos s o
he companies.
Deg ada ion and wea o he e ac o ies used in he s eelmaking
indus y is a complex phenomenon ha in ol es se e al ac o s. A
b ie summa y o hese was gi en in [1]. Due o he high- empe a u e
en i onmen posing a signi ican challenge o expe imen a ion and i s
c i ical ole in he s eelmaking p ocess, a signi ican body o esea ch
using nume ical expe iences o in es iga e he BF p ocess is a ailable.
∗Co esponding au ho a : Depa men o Applied Ma hema ics, Uni e sidade de San iago de Compos ela, 15782 San iago de Compos ela, Spain.
E-mail add esses: [email p o ec ed] (P. Ba al), [email p o ec ed] (L.J. Pé ez-Pé ez), [email p o ec ed] (P. Quin ela).
Fo ins ance, in [2,3], he ansien low inside he BF hea h was
in es iga ed. In [4], an in e se hea ans e model was de eloped
o es ima e he wea p o ile in he hea h wall using he measu ed
empe a u es by se e al he mocouples. In [5], he low h ough he BF
aphole was modelled a emp ing o a oid splashing due o en ained
ai , which is ela ed o highe deg ees o wea . Mo eo e , in [6], se e al
models designed o p edic he low in he BF hea h and aphole
we e p esen ed, aiming o co e all he a ious aphole condi ions. The
in e es ed eade is e e ed o [7] o a ho ough e iew o he s a e
o nume ical simula ion inside he BF. Simila mul iphysics models,
inco po a ing ei he luid low, he mal adia ion and coupled hea
ans e ha e been de eloped o aluminium u naces (see e.g. [8,9]).
Al hough he a ailable li e a u e ela ed o he main ough is
smalle han ha ela ed o he BF i sel , some e o s ha e been made,
especially in ecen yea s. In [10], se e al s a egies o inc ease he
main ough p oduc i i y we e de ailed. The in ica e low pa e ns in
he ough ha e also been a ma e o esea ch by se e al au ho s. Some
h ps://doi.o g/10.1016/j.ij he malsci.2021.107349
Recei ed 10 June 2021; Recei ed in e ised o m 16 Sep embe 2021; Accep ed 20 Oc obe 2021
In e na ional Jou nal o The mal Sciences 173 (2022) 107349
2
P. Ba al e al.
Nomencla u e
G eek symbols
𝛼𝑏Back acking line sea ch pa ame e
𝛼𝑐Simula ed annealing pa ame e
𝜀Emissi i y
𝜆Pa ame e on ansien empe a u e p o ile
[K/s]
𝛩Blocking ac o
𝛤Bounda y o compu a ional domain
𝛤𝑅Radia ion ca i y
𝛤𝑅In e sec ion be ween he solid subdomain
and he adia ion ca i y
𝛤𝐷
𝑅Slag uppe su ace
𝛺(𝑡)Compu a ional domain a ime 𝑡
𝜔Ke nel o nonlocal adia ion in eg al equa-
ion
𝜈S ep leng h in p ojec ed g adien descen
me hod
𝜌Densi y [kg/m3]
𝜎S e an–Bol zmann cons an [W/(m2K4)]
𝜏ℎMesh o he compu a ional domain
La in symbols
𝑐𝑝Speci ic hea a cons an p essu e [J/(kg
K)]
𝑐𝑛Limi ed s ep size change a io a 𝑛 h ime
s ep
Feasible egion
𝐝Descen di ec ion
ℎHea ans e coe icien [W/(m2K)]
ℎℎ𝑚 Heigh o he ho me al and slag in e ace
[m]
ℎ𝑠𝑙𝑎𝑔 Heigh o he slag ee su ace [m]
ℎ𝐵Heigh o he bo om o he ough channel
bo om [m]
Iden i y ope a o
𝐽Cos unc ional
𝑘The mal conduc i i y [W/(m K)]
𝑘𝑖𝑛 The mal conduc i i y a he insula ion
lining [W/(m K)]
Nonlocal adia ion in eg al ope a o
𝐿The mocouple leng h [m]
𝑀Numbe o uni cycles
𝐧Ou wa d-poin ing uni no mal ec o
𝑁Geome ic dimension
𝑁𝑐Simula ed annealing maximum i e a ions
𝑞𝑟𝑎𝑑 Hea lux due o adia ion [W/m2]
In eg al ope a o o adia i e hea lux
𝑟𝑛No m o he local e o es ima e a he 𝑛 h
ime s ep
𝑅Radiosi y [W/m2]
publica ions ha e elied on physical scale models, while o he au ho s
ha e used CFD o sol e he Na ie –S okes equa ions o cha ac e ize
he mul iphase low in he ough. In [11], se e al expe imen s we e
ca ied ou using a physical model o in es iga e me al sepa a ion
e iciency. To emula e slag and ho me al, oil and wa e we e used,
espec i ely. The a ia ion o se e al p ocess a iables, such as he BF
𝑡Time [s]
𝑡𝑠𝑡𝑒𝑎𝑑𝑦 Requi ed ime o each he s eady s a e [s]
𝑡𝑐Time a which 𝑐 h cas o s op begins [s]
𝑡𝑒𝑛𝑑 Final ime o p oblem [s]
Time in e al [s]
𝑇Tempe a u e [K]
𝑇𝑒𝑥𝑡,𝑐,𝑛𝑐Con ec ion ex e nal empe a u e on 𝛤𝑛𝑐
𝐶[K]
𝑇𝑒𝑥𝑡,𝑟,𝑛𝑐Radia ion ex e nal empe a u e on 𝛤𝑛𝑐
𝐶[K]
𝑇𝑒𝑥𝑡,𝑐,𝑅 Con ec ion ex e nal empe a u e on 𝛤𝑅[K]
𝑇0Ini ial empe a u e [K]
𝑊Weigh unc ional
Subsc ip s
𝑐Index o imes a which BF ac i i y changes
om cas o s op
𝐶Ex e nal bounda ies
𝑐𝑎𝑠𝑡 Cas p oblems
𝑑Index on he mocouples
𝐷Di ichle
𝐹Fluid
𝑚Index on uni cycles
𝑛𝑐Index on con ec ion bounda ies
𝑅Nonlocal adia ion bounda y
𝑆Solid
𝑠𝑡𝑜𝑝 S op p oblems
Supe sc ip s
𝑚Index on uni cycles
𝑛𝑛 h ime s ep
𝑛𝑐Index on con ec ion bounda ies
discha ge a e o ough geome y was in es iga ed. Flow sepa a ion, as
well as i s main s uc u es and cha ac e is ics, we e also expe imen ally
s udied in [12] and [13], desc ibing simila indings.
Conce ning he nume ical simula ion in he main ough, [14]
sol ed he ho me al low o in es iga e he pa e ns esul ing om he
je s eam impingemen , ob aining quali a i ely simila esul s o hose
epo ed in he physical models. In [15], he low s uc u es we e s ud-
ied by sol ing a simila ma hema ical model, also inco po a ing slag as
a di e en low phase. Ho me al losses wi h slag we e assessed, inding
ha geome y changes, such as dec easing he ough bo om slope,
ha e a subs an ial impac on he slag–me al sepa a ion. Mo eo e ,
in [16], he mul iphase u bulen low was sol ed wi h ANSYS CFX. To
alida e he CFD esul s, a 1 ∶ 10 scale model was used o unde ake
ex ensi e expe imen s. Flow modi ie s we e in oduced o con ain he
egion whe e mos o he mixing occu s, achie ing plug-like low in a
g ea e egion o he ough. In [17], he luid low in he ough was
also in es iga ed, bu inco po a ing buoyancy due o he mal e ec s. I
was concluded ha con olling he heigh le el be ween he slag unne
and he i on dam is essen ial o imp o e sepa a ion. In addi ion, in [18],
OpenFOAM was used o sol e he ansien Na ie –S okes equa ions,
also including he conjuga e hea ans e wi h he solid e ac o y. I
was ound ha he shea s ess in he ough wall inc eases wi h highe
aphole inclina ion angles.
Rega ding he he mal esponse o he h ough, in [19], he hea
ans e in a 2D c oss-sec ion was modelled, assuming s eady s a e
condi ions and conside ing a simpli ied ea men o he adia i e
hea ans e . In a p e ious wo k [20], he empe a u e ield in a 3D
domain co esponding o he inal pa o he main ough was s udied,
including he low o ho me al and slag, and he e ac o y co e , also
in a s eady s a e. Mo eo e , in [1], a ansien he mal model o a
In e na ional Jou nal o The mal Sciences 173 (2022) 107349
3
P. Ba al e al.
Fig. 1. Schema ic o a ypical BF main ough.
2D c oss-sec ion o he ough was p oposed, conside ing he ime span
o a single BF apping. The e, he e ec o he he mal adia ion was
add essed by using a nonlocal adia ion bounda y condi ion.
The pu pose o his wo k is o s udy he in luence o he cas ing
p ocess s ops on he empe a u e in he main ough. Usually, hese
s ops a e kep sho . Howe e , a ious ac o s can lead o longe s ops
occu ing. In such case, he bulk o he solids may cool down signi -
ican ly, esul ing in la ge empe a u e swings and enhanced isk o
spalling o he in il a ed e ac o y laye s. Special a en ion is de o ed
o he e olu ion o he posi ion o he so-called c i ical iso he ms in
he e ac o ies, which can be used o diagnose he wea p o iles in he
ough. The e o e, he model de eloped in [1] o he ansien hea ing
du ing a single BF apping is ex ended o he comple e campaign
cycle. To his end, wo di e en submodels a e inco po a ed: one o
hose imes when he BF is no being cas and a di e en one o
imes co esponding o BF appings. Real BF ope a ion da a is used o
de e mine he sequence o p oblems o be sol ed, comp ising mo e han
300 BF appings and he co esponding s ops, which span o a ound
wo mon hs o du a ion. The nume ical esul s a e compa ed wi h
expe imen al measu emen s ob ained wi h he mocouples embedded in
he ough. P o ided ha he e is a subs an ial deg ee o unce ain y
ega ding hei placemen , a minimiza ion p oblem is posed and sol ed.
As a as he au ho s a e awa e, he e is no p e ious li e a u e on he
simula ion o comple e campaigns, which equi es long imescales, no
e i ica ion and calib a ion o hei esul s wi h expe imen al da a.
The ou line o his wo k is as ollows. In Sec ion 2, we de ail he
undamen al cha ac e is ics o he physical p ocess. In Sec ion 3, we
desc ibe he ma hema ical model o he ansien beha iou o he
empe a u e in a 2D c oss-sec ion o he BF ough. In Sec ion 4, we
p esen he compu ed nume ical esul s. Mo eo e , in Sec ion 5, he
me hodology o adjus he posi ion o he he mocouples wi hin hei
easible egions is p oposed. Las ly, in Sec ion 6, we highligh he main
conclusions de i ed om his pape .
2. Physical p oblem
Slag and ho me al a e s a i ied inside he hea h o he BF. When
he ho me al le el is su icien ly high, i s ex ac ion is done h ough
he aphole, an o i ice in he BF wall. The BF d aining p ocess is known
as apping. While he p ocess is ac i e, he luids all in o one o he BF
main oughs, whe e hey sepa a e by densi y di e ence. In Fig. 1, a
schema ic o an emp y ypical main ough is shown be o e he s a o
he cas ing egime. Du ing i s ope a ion, i ills wi h slag and ho me al.
Abo e he ough, a sys em o emo able e ac o y co e s is in-
s alled, which spans o he comple e leng h o he unne , e en hough
in Fig. 1 hey ha e been cu o cla i y. The pu pose o his sys em is
wo- old: i s , i allows a mo e e ec i e aspi a ion o pollu an umes
emi ed du ing he apping p ocess (see [21]); and second, i a oids
Fig. 2. Tempe a u e schema ic a he con ac a ea wi h luids o he wo king lining
du ing a campaign cycle.
excessi e cooling due o adia ion emission om he slag uppe su ace,
which is o key ele ance gi en i s high empe a u e. The ume aspi a-
ion, pe o med nex o he aphole, gene a es a o ced ai ci cula ion
below he co e s, which enhances he cooling o he e ac o ies. The
BF ough design g ea ly a ies among di e en cas houses. In his
wo k, we use a design ha co esponds o a main ough ope a ed by
a s eelmaking company, simila o ha conside ed in [20]. As mos
ough designs, i has h ee dis inc ma e ial laye s, as shown in Fig. 5.
These laye s a e cons uc ed wi h di e en e ac o y ma e ials ha a e
p epa ed o pe o m speci ic unc ions.
The BF apping p ocess is no ully con inuous, as each cas las s o
abou 60-90 minu es wi h i s end being indica ed by BF gas bu s ing
ou h ough he aphole, ime a which i is plugged. A e wa ds, he e
is a s op o allow he liquid le el inside he BF hea h o ise back
(see [22]). Depending on he size o he BF and i s numbe o apholes,
a di e en aphole may be used o he nex cas . La ge mode n BF
designs, wi h up o ou apholes, can be ope a ed wi h wo apholes
being open simul aneously (see [21]). We de ine a uni cycle as a
single cas and he subsequen s op. Due o he se e e deg ada ion
ha sus ains, he wo king lining is eplaced a e se e al uni cycles
a e comple ed o p e en damage on he sa e y and insula ion linings.
We deno e as campaign cycle he se o uni cycles ha each wo king
lining wi hs ands be o e i s eplacemen . Be ween campaign cycles, he
luids in he ough a e d ained and epa a ions a e pe o med. Then,
he main ough is hea ed, and a new campaign cycle begins. In Fig. 2, a
schema ic o he empe a u e alues on he con ac a ea wi h he luids
o he wo king lining is depic ed du ing a campaign cycle. The ime
alues
𝑡𝑐, wi h 𝑐= 0,…,2𝑀− 1, indica e he imes a which appings
In e na ional Jou nal o The mal Sciences 173 (2022) 107349
4
P. Ba al e al.
Fig. 3. Tempe a u e measu emen s in he insula ion lining du ing a unne li e ime
cycle.
a e s a ed and ended, 𝑀being he numbe o appings o uni cycles.
S anda d campaign cycles las up o 2mon hs.
A unne li e ime cycle is de ined as he se o campaign cycles and
he d ains and epai s be ween hem. When i ends, he main ough
is comple ely ebuil . Typically, unne li e ime cycles a e much longe
han campaign cycles and can each se e al yea s o du a ion. In Fig. 3,
empe a u e measu emen s wi hin he insula ion lining a e depic ed
du ing campaign cycles, comp ising a comple e unne li e ime cycle.
I he uni cycle s ops a e kep sho , expe imen al measu emen s show
ha he empe a u e in he bulk o he solids composing he unne
eaches a quasi s eady s a e. Howe e , in eal uni cycles he down ime
o he ough may be a om such si ua ion. This explains he la ge
empe a u e swings ha a e obse ed in Fig. 3 o some campaigns.
These la ge oscilla ions co espond o longe s ops ha can las o up
o se e al days.
3. Ma hema ical model
To assess he cooling ela ed o he longe s ops ha occu s in eal
ope a ion condi ions and how i may a ec he e ac o y ma e ials, we
aim o ob ain he empe a u e ield co esponding o a ull campaign
cycle o a main ough. The e o e, a ime in e al o a ound wo mon hs
o ope a ion needs o be sol ed. Since i would be ex emely di icul o
sol e a de ailed 3D model including he luid low o such ime scales,
we ollow a simpli ied app oach. Hence, we conside only he 2D c oss-
sec ion co esponding o he middle pa o he main ough (see Fig. 5),
si ua ed be o e he slag unne a 𝑥3= 8 m, whe e he coo dina e 𝑥3
deno es he dis ance o he BF. The p oposed model is an ex ension o
ha desc ibed in [1], inco po a ing he model co esponding o s ops.
3.1. Tapping cycles
A ime pe iod = (𝑡0, 𝑡𝑒𝑛𝑑 ]is conside ed, which co esponds o
a comple e campaign cycle. This pe iod can be spli in wo subse s,
𝑠𝑡𝑜𝑝 and 𝑐𝑎𝑠𝑡, which deno e he ime in e al du ing which he e is
no luid discha ge om he BF hea h, and he o al cas ime in e al,
espec i ely. Consequen ly, we ha e ha
= (𝑡0, 𝑡𝑒𝑛𝑑 ] = 𝑠𝑡𝑜𝑝 ∪𝑐𝑎𝑠𝑡.(1)
The se s 𝑐𝑎𝑠𝑡 and 𝑠𝑡𝑜𝑝 may be subdi ided in many smalle ime subin-
e als, which co espond o he BF cas s and s ops du ing each uni
cycle. Fo ins ance, i we conside a campaign cycle composed o 𝑀
cas s, we ind:
𝑐𝑎𝑠𝑡 =
𝑀−1
⋃
𝑚=0
(
𝑡2𝑚,
𝑡2𝑚+1] =
𝑀−1
⋃
𝑚=0
𝑚
𝑐𝑎𝑠𝑡,(2)
Table 1
Example o p o ided BF ac i i y alues.
Regis e index [h] 1 2 3 4 5 6 7
Value 1.0 0.5 0.0 0.1 1.0 0.4 0.0
Fig. 4. Signal indica ing whe he he BF is apping o s opped.
𝑠𝑡𝑜𝑝 =
𝑀−2
⋃
𝑚=0
(
𝑡2𝑚+1,
𝑡2𝑚+2] =
𝑀−2
⋃
𝑚=0
𝑚
𝑠𝑡𝑜𝑝,(3)
whe e we assume ha
𝑡2𝑀−1 =𝑡𝑒𝑛𝑑 .
A a gi en ime 𝑡∈, we sol e a di e en he mal model depending
on whe he he BF is s opped o apping, i.e., i 𝑡belongs o 𝑚
𝑠𝑡𝑜𝑝 o 𝑚
𝑐𝑎𝑠𝑡
o some 𝑚. We deno e he co esponding p oblems as 𝑚
𝑠𝑡𝑜𝑝 and 𝑚
𝑐𝑎𝑠𝑡,
espec i ely. These p oblems a e sol ed sequen ially, meaning ha
he P oblem 𝑚
𝑐𝑎𝑠𝑡 has as ini ial condi ion he inal empe a u e alue
compu ed in 𝑚−1
𝑠𝑡𝑜𝑝 , o 𝑚= 1,…𝑀− 1. Fo 𝑚= 0, which co esponds
o he i s cas , an ini ial s a e o he empe a u e in he ough is
supplied as ini ial condi ion, which is discussed in Sec ion 3.5. The
same applies o 𝑚
𝑠𝑡𝑜𝑝, which uses as ini ial alue he inal empe a u e
in 𝑚
𝑐𝑎𝑠𝑡, o 𝑚= 0,1,…, 𝑀 − 2.
To gene a e he spli ing among 𝑐𝑎𝑠𝑡 and 𝑠𝑡𝑜𝑝 ollowing (2) and (3),
he ime alues
𝑡𝑐a e equi ed, wi h 𝑐= 0,…,2𝑀− 1. To ind hem,
we use ope a ion da a o a BF, supplied by he s eelmaking company
collabo a ing in he esea ch p ojec PID2019-105615RB-I00. The da a
a e composed o alues anging om 0 o 1, which a e equal o he
ac ion o each hou ha he BF was discha ging ma e ial h ough
he aphole. Assuming ha he e is only one cas s a o s op du ing
each hou , a unc ion which is 1 o 𝑡∈𝑐𝑎𝑠𝑡 and 0i 𝑡∈𝑠𝑡𝑜𝑝 is
cons uc ed, simila o he BF ac i i y cu e displayed in Fig. 2. The
unc ion de e mines he alues
𝑡𝑐. Fo ins ance, o he se o alues
ga he ed in Table 1, we econs uc he unc ion shown in Fig. 4. In
pa icula , o his simple example, we ind ha he e a e wo cas s.
The i s one s a s and ends a
𝑡0= 0 h and
𝑡1= 1.5h, espec i ely.
The second cas s a s a
𝑡2= 3.9h and ends a
𝑡3= 5.4h.
3.2. Compu a ional domain
The compu a ional domain is selec ed as he c oss-sec ion depic ed
in Fig. 5. We conside a luid subdomain, 𝛺𝐹, which comp ises he a ea
occupied by he slag and ho me al pool, and a solid subdomain, 𝛺𝑆,
which co esponds o he solid e ac o y ma e ials. The s eel casing is
no included in he compu a ional domain, as i inco po a es se e al
ins and ibs, which a e no compa ible wi h a 2D he mal model.
Ne e heless, hei e ec is conside ed by app op ia ely enhancing he
hea losses h ough he bounda ies.
Du ing he uni cycle s ops, he main ough holds he emaining
liquids om p eceding cas s. This allows o p ese e hea in he ough,
In e na ional Jou nal o The mal Sciences 173 (2022) 107349
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Fig. 5. Compu a ional domains and hei bounda ies.
Fig. 6. F ee su ace posi ions in he luids.
keeping ho me al and slag in a liquid s a e, which helps sepa a ion in
subsequen cas s (see e.g [21] o [23]). The ough is d ained only o
pe o m eme gency epa a ions o a he end o each campaign cycle.
The uppe su ace o he slag and he in e ace sepa a ing ho me al
and slag a e no ully s eady du ing he campaign cycle, as he liquid
le el sligh ly inc eases du ing each cas . The le el changes a e small
and e y di icul o measu e accu a ely; in he apping s eady s a e
i s alue is es ima ed o g ow up o a 15% wi h espec o he le el
du ing he s ops. P o ided ha he scope o his wo k is o assess he
in luence o he s ops be ween consecu i e cas s in he hea ans e
owa ds he solid e ac o y laye s, hese liquid le el a ia ions can be
neglec ed. Hence, we assume ha bo h he slag and ho me al uppe
su aces emain a a cons an heigh du ing he campaign cycle. This
alue is calcula ed assuming ha he luids a e a es du ing he uni
cycle s ops. Consequen ly, he app oxima e heigh s depend only on he
ough design, speci ically on he i on and slag unne heigh , as well as
on he luid densi y alues, as de ailed in [12,17]. Using his app oach,
conside ing he coo dina e axes and o igin depic ed in Fig. 5, we ind
ha he slag uppe su ace is loca ed a ℎ𝑠𝑙𝑎𝑔 = 1.5m and he ho me al
su ace is a ℎℎ𝑚 = 1.15 m, as shown in Fig. 6. The bo om o he luids
is deno ed as ℎ𝐵and he co esponding alue o he selec ed c oss-
sec ion is ℎ𝐵= 1.04 m. No e ha he unne has some inclina ion o
acili a e he sliding o he ho liquids, so ℎ𝐵depends on he coo dina e
𝑥3o he conside ed c oss-sec ion.
Du ing each cas , we ollow he s a egy p oposed in [1] o a single
apping, whe e i was assumed ha a s a iona y empe a u e is eached
swi ly in he luids. To model his phenomenon, he empe a u e in
𝛺𝐹is assumed o ollow a ime-dependen empe a u e p o ile (see
Sec ion 3.4), which depends on he ini ial empe a u e. Thus, he
domain o he P oblem 𝑚
𝑐𝑎𝑠𝑡 is 𝛺𝑐𝑎𝑠𝑡 =𝛺𝑆 o all 𝑚, as depic ed in
Fig. 5(a).
In he p oblem co esponding o uni cycle s ops, 𝑚
𝑠𝑡𝑜𝑝, he aim is
o assess he cooling o he main ough. The comple e c oss-sec ion is
conside ed, including he luids ha a e e ained in he unne du ing
he s ops. Hence, o 0≤𝑚≤𝑀− 2, he domain o P oblem 𝑚
𝑠𝑡𝑜𝑝 is
𝛺𝑠𝑡𝑜𝑝 =𝛺𝐹∪𝛺𝑆∪
𝛤𝐷, as displayed in Fig. 5(b). The bounda y 𝛤𝐷is
de ined as 𝛤𝐷=𝛺𝐹∩𝛺𝑆. Fo he sake o simplici y, we deno e by 𝛺(𝑡)
Table 2
P ope ies o he ma e ials.
𝑘[W/(m K)]𝑐𝑝[J∕(kg K)] 𝜌[kg∕m3]
Wo king lining 2.6 1212 2500
Sa e y lining 1.8 1172 2900
Insula ion lining 𝑘𝑖𝑛(𝑇) 1050 800
Co e e ac o y 2.5 1296 2750
Ho me al 16.5 850 7015
Slag 9.7 807 2600
he compu a ional domain a ime 𝑡∈:
𝛺(𝑡) = {𝛺𝑠𝑡𝑜𝑝,i 𝑡∈𝑠𝑡𝑜𝑝,
𝛺𝑐𝑎𝑠𝑡,i 𝑡∈𝑐𝑎𝑠𝑡.(4)
3.3. Model equa ions
To model he e olu ion o he empe a u e in he ough c oss-
sec ion, bo h o he cas and s op p oblems, we use he ansien hea
equa ion (see [24])
𝜌𝑐𝑝
𝜕𝑇
𝜕𝑡 −di (𝑘(𝑇)∇𝑇)=0,(5)
o 𝑡∈and 𝐱∈𝛺(𝑡), wi h 𝐱= (𝑥1, 𝑥2). The he mal conduc i i y, he
speci ic hea , and he densi y a e deno ed as 𝑘,𝑐𝑝and 𝜌, espec i ely,
and ake he alues ga he ed in Table 2, whe e 𝑘𝑖𝑛 is he ollowing
empe a u e-dependen unc ion (see [1]):
𝑘𝑖𝑛(𝑇) = ⎧
⎪
⎨
⎪
⎩
0.13,i 𝑇≤293,
1
98 000 (9𝑇+ 10 103),i 293 < 𝑇 < 1273,
0.22,i 𝑇≥1273.
(6)
The alues in Table 2 ha e been supplied by he company, in some
cases aken om he da a shee s elabo a ed by he ma e ial man-
u ac u e . The cons an alues a e gi en a he expec ed ope a ion
empe a u e. Only in he case o he insula ion lining, empe a u e-
dependen da a a e a ailable, so he 𝑘𝑖𝑛 unc ion is ob ained by i ing
he alues.
On he bounda y 𝛤𝑅, which co esponds o he adia ion ca i y
(see Fig. 5), an in eg al equa ion is sol ed o model he adia ion hea
exchange. Unde he assump ion ha he adia ion ca i y cons i u es an
opaque, di use and g ey su ace, whe eas he ai wi hin is a anspa en
medium o adia ion, he adiosi y 𝑅on 𝛤𝑅sa is ies, o 𝑡∈, ha (see
e.g. [25]):
(− (1 − 𝜀𝑅(𝐱)))(𝑅(𝑡))(𝐱) = 𝜀𝑅(𝐱)𝜎𝑇 4(𝐱, 𝑡),on 𝛤𝑅,(7)
whe e is he iden i y ope a o and 𝜀𝑅is he emissi i y, conside ed
cons an wi h alue 𝜀𝑅= 0.7. Mo eo e , 𝜎= 5.67𝑒−8 W/(m2K4) is
In e na ional Jou nal o The mal Sciences 173 (2022) 107349
6
P. Ba al e al.
Table 3
Hea ans e coe icien s.
ℎ[W/(m2K)]
ℎ𝑅35
ℎ17
ℎ28
ℎ34
he S e an–Bol zmann cons an and deno es he ollowing in eg al
ope a o
(𝑅(𝑡))(𝐱) = ∫𝛤𝑅
𝜔(𝐱,𝐲)𝑅(𝐲, 𝑡)𝑑𝑠𝐲,(8)
whe e he in eg al ke nel 𝜔, known as iew ac o , is such ha (see
[26])
𝜔(𝐱,𝐲) = 𝐧(𝐱)⋅(𝐲−𝐱)𝐧(𝐲)⋅(𝐱−𝐲)
2|𝐱−𝐲|3𝛩(𝐱,𝐲),wi h 𝐱≠𝐲,(9)
𝐧being he ou wa d-poin ing uni no mal ec o . The isibili y o
blocking ac o 𝛩accoun s o obs acles ha obs uc he iew be ween
poin s and is de ined as
𝛩(𝐱,𝐲) = {0,i 𝐱𝐲 ∩𝛺≠∅,
1,i 𝐱𝐲 ∩𝛺= ∅,(10)
wi h 𝐱𝐲 deno ing he segmen ha connec s he poin s 𝐱and 𝐲.
To e alua e he adia i e hea lux, 𝑞𝑟𝑎𝑑 , which appea s as a e m
in he bounda y condi ion co esponding o 𝛤𝑅o he hea Eq. (5), he
in eg al ope a o is in oduced, which yields 𝑞𝑟𝑎𝑑 (see [1]):
𝑞𝑟𝑎𝑑 (𝐱, 𝑡) = (𝑅(𝑡))(𝐱) ∶= (−)(𝑅(𝑡))(𝐱).(11)
Fo a mo e de ailed desc ip ion o he nonlocal adia ion model, we
e e o [1,27]. A heo e ical analysis o he s eady coupled conduc ion
and nonlocal adia ion can be seen in [28] and he e e ences he ein.
The ansien case is add essed in [29]. No e ha in gene al applica-
ions, he hypo hesis ha a e assumed o desc ibe he mal adia ion
using (7) a e a om ealis ic. This is he case o en i onmen s wi h
combus ion (see e.g. [30]), whe e speci ic me hods ha e o be used o
sol e he adia i e ans e equa ion (see [25,31]).
3.4. Bounda y condi ions
The bounda ies o he domains a e decomposed as displayed in
Fig. 5 o 𝛺𝑐𝑎𝑠𝑡 and 𝛺𝑠𝑡𝑜𝑝. We s a by b ie ly summa izing he bounda y
condi ions used o he cas p oblems, which a e he same al eady p e-
sen ed wi h de ail in [1]. Subsequen ly, he modi ica ions in oduced
o he s op p oblems a e desc ibed.
3.4.1. Cas p oblems
In he ollowing discussion, we de ail he bounda y condi ions
conside ed o each 𝑚
𝑐𝑎𝑠𝑡. The bounda y o i s domain is independen
o 𝑚and is gi en by
𝛤𝑐𝑎𝑠𝑡 =
3
⋃
𝑛𝑐=1
𝛤𝑛𝑐
𝐶∪𝛤𝐷∪
𝛤𝑅,(12)
as depic ed in Fig. 5.
•On he ex e nal bounda ies 𝛤𝑛𝑐
𝐶,1≤𝑛𝑐≤3, mixed con ec ion–
adia ion bounda y condi ions a e conside ed (see [24]):
−𝑘(𝑇)𝜕𝑇
𝜕𝐧=ℎ𝑛𝑐(𝑇−𝑇𝑒𝑥𝑡,𝑐,𝑛𝑐) + 𝜎𝜀𝑛𝑐(𝑇4−𝑇4
𝑒𝑥𝑡,𝑟,𝑛𝑐),(13)
o all 𝑡∈𝑐𝑎𝑠𝑡. The alues o he hea ans e coe icien s,
ℎ𝑛𝑐, a e displayed in Table 3. The con ec ion and adia ion am-
bien empe a u es as well as he emissi i y a e conside ed equal
(𝑇𝑒𝑥𝑡,𝑟,𝑛𝑐=𝑇𝑒𝑥𝑡,𝑐,𝑛𝑐= 293 K and 𝜀𝑛𝑐= 0.7).
Fig. 7. Tempe a u e p o ile 𝑇𝑚
𝐷,𝑐𝑎𝑠𝑡 o a ypical ini ial empe a u e du ing he ough
campaign cycle.
•On 𝛤𝐷=𝛺𝐹∩𝛺𝑆, which is he in e ace be ween he luids and
solids, we se a Di ichle condi ion, using a known empe a u e:
𝑇𝑚
𝑐𝑎𝑠𝑡(𝐱, 𝑡) = 𝑇𝑚
𝐷,𝑐𝑎𝑠𝑡(𝐱, 𝑡),(14)
o all 𝐱∈𝛤𝐷and 𝑡∈𝑚
𝑐𝑎𝑠𝑡. In (14), he empe a u e 𝑇𝑚
𝐷,𝑐𝑎𝑠𝑡
is he p o ile de eloped in [1], sui ably modi ying he ini ial
empe a u e conside ed he ein so ha he empe a u e a he
luids ises om he empe a u e a he ini ial ime o he 𝑚 h cas
p oblem o a s a iona y empe a u e. The s a iona y empe a u e
sa is ies he ollowing exp ession:
𝑇𝐹 ,𝑐𝑎𝑠𝑡(𝑥2) = −1.52𝑒−9𝑥63.46
2+ 1773,(15)
whe e ℎ𝐵≤𝑥2≤ℎ𝑠𝑙𝑎𝑔 , wi h ℎ𝐵= 1.04 m, as de ailed in Sec-
ion 3.2. The empe a u e p o ile (15) is a i ing o he solu ion
compu ed o he s eady s a e in [20] a 𝑥3= 8 and 𝑥1= 1.5
m. No e ha , as desc ibed in [1], any a ia ions o he s a iona y
empe a u e along he ho izon al coo dina e a e neglec ed.
Speci ically, a uni cycle 𝑚, wi h 1≤𝑚≤𝑀− 1, and ime
𝑡∈𝑚
𝑐𝑎𝑠𝑡, we se he slag and ho me al empe a u e o
𝑇𝑚
𝐷,𝑐𝑎𝑠𝑡(𝐱, 𝑡) = ⎧
⎪
⎪
⎨
⎪
⎪
⎩
𝑇𝑚−1
𝑠𝑡𝑜𝑝 (𝐱,
𝑡2𝑚) − 3𝜆2(𝑡−
𝑡2𝑚)2
𝛥𝑇 𝑚(𝐱)
−2𝜆3(𝑡−
𝑡2𝑚)3
(𝛥𝑇 𝑚(𝐱))2,i 𝑡≤𝑡𝑚
𝑠𝑡𝑒𝑎𝑑𝑦(𝐱),
𝑇𝐹 ,𝑐𝑎𝑠𝑡(𝑥2),o he wise,
(16)
whe e
𝑡2𝑚deno es he ime alue co esponding o he end o he
p eceding s op, and 𝑇𝑚−1
𝑠𝑡𝑜𝑝 (𝐱,
𝑡2𝑚)is he compu ed empe a u e o
𝑚−1
𝑠𝑡𝑜𝑝 a i s end ime. Mo eo e , 𝛥𝑇 𝑚(𝐱)s ands o he di e ence
among he s eady empe a u e and he ini ial empe a u e a poin
𝐱, i.e.:
𝛥𝑇 𝑚(𝐱) = 𝑇𝑚−1
𝑠𝑡𝑜𝑝 (𝐱,
𝑡2𝑚) − 𝑇𝐹 ,𝑐𝑎𝑠𝑡(𝑥2).(17)
I 𝑚= 0, co esponding o he i s uni cycle o he ough
campaign, in (16) we conside 𝑇𝑚−1
𝑠𝑡𝑜𝑝 =𝑇0, whe e 𝑇0is he ini ial
empe a u e alue, discussed in Sec ion 3.5. Fu he mo e, he
ime needed o each he s a iona y empe a u e is
𝑡𝑚
𝑠𝑡𝑒𝑎𝑑𝑦(𝐱) =
𝑡2𝑚+
𝑇𝐹 ,𝑐𝑎𝑠𝑡(𝑥2) − 𝑇𝑚−1
𝑠𝑡𝑜𝑝 (𝐱,
𝑡2𝑚)
𝜆.(18)
The pa ame e 𝜆 akes he alue 𝜆= 3∕2 [K/s] in (16) and (18).
This choice gua an ees ha in he mos ex eme scena io (𝑇𝑚−1
𝑠𝑡𝑜𝑝 ≈
293 K), he s a iona y empe a u e is eached in app oxima ely
15 min. Fo a egula cas ing egime, he s ops a e no long
enough o he luids o cool down signi ican ly, and he s eady
s a e is eached much as e , as depic ed in Fig. 7 using a ypical
alue o he ini ial empe a u e 𝑇𝑚−1
𝑠𝑡𝑜𝑝 (𝐱,
𝑡2𝑚), ex ac ed om he
nume ical esul s discussed in Sec ion 4.
In e na ional Jou nal o The mal Sciences 173 (2022) 107349
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P. Ba al e al.
Fig. 8. Tempe a u e con ou s o he ini ial empe a u e 𝑇0.
•On
𝛤𝑅=𝛤𝑅∩𝛺𝑆, a con ec ion con ibu ion (see [24]) due o he
o ced ai low below he co e is added o he adia ion hea lux
𝑞𝑟𝑎𝑑 , ob ained by sol ing (7):
−𝑘𝜕𝑇
𝜕𝐧=ℎ𝑅(𝑇−𝑇𝑒𝑥𝑡,𝑐,𝑅) + 𝑞𝑟𝑎𝑑 ,(19)
o all 𝑡∈𝑐𝑎𝑠𝑡 and whe e he empe a u e o he su oundings is
𝑇𝑒𝑥𝑡,𝑐,𝑅 = 293 K. The hea ans e coe icien , ℎ𝑅, is equal o he
alue indica ed in Table 3. No e ha e en hough he adia ion
con ibu ion is only applied on he pa o he adia ion ca i y
ha belongs o he bounda y o 𝛺𝑐𝑎𝑠𝑡, he in eg al Eq. (7) is sol ed
on he comple e ca i y 𝛤𝑅, as he adiosi y on he slag uppe
su ace, 𝛤𝐷
𝑅, is unknown e en i he empe a u e sa is ies (16).
The e o e, o 𝑚
𝑐𝑎𝑠𝑡,(7) is sol ed o all 𝑡∈𝑚
𝑐𝑎𝑠𝑡 assuming ha
𝑇|𝛤𝐷
𝑅=𝑇𝑚
𝐷,𝑐𝑎𝑠𝑡.
3.4.2. S op p oblems
Fo 𝑚
𝑠𝑡𝑜𝑝, hei domain, 𝛺𝑠𝑡𝑜𝑝, and he e o e hei bounda y, 𝛤𝑠𝑡𝑜𝑝,
a e also independen o 𝑚. In his case, we ha e ha
𝛤𝑠𝑡𝑜𝑝 =
3
⋃
𝑛𝑐=1
𝛤𝑛𝑐
𝐶∪𝛤𝑅,(20)
as displayed in Fig. 5.
On he ex e nal pa o he ough, he bounda y condi ions a e
he same o he cas and he s op p oblems. Thus, o 𝑚
𝑠𝑡𝑜𝑝, on 𝛤𝑛𝑐
𝐶,
wi h 1≤𝑛𝑐≤3, we se (13) o all 𝑡∈𝑠𝑡𝑜𝑝. Howe e , no e
ha he adia ion ca i y bounda y is sligh ly di e en o he s op
p oblems when compa ed o he cas p oblems, as depic ed in Fig. 5,
whe e he bounda y becomes 𝛤𝑅ins ead o
𝛤𝑅, as du ing he s ops he
empe a u e on he luids is no known and he hea equa ion is also
sol ed in 𝛺𝐹.
3.5. Ini ial condi ion
The epa a ions be ween campaign cycles las o a ound 20 days,
as displayed in Fig. 2. Du ing his pe iod, he co e is emo ed, and
he e ac o y laye s cool down as epa a ions a e pe o med. Also, he
wo king lining is hea ed a e being ebuil o p e en damage om he
ini ial he mal shock. The empe a u e s a e a e his p ocess, which
co esponds o he ini ial empe a u e a he s a o a campaign cycle,
is unknown. Ne e heless, an app oxima ed empe a u e ield mus be
supplied as ini ial condi ion a 𝑡=
𝑡0 o he p oposed model. As i
will be de ailed in Sec ion 5, he measu emen s ob ained wi h h ee
he mocouples loca ed in he conside ed c oss-sec ion o he ough a e
a ailable. They show ha he e is a signi ican amoun o esidual hea
in he ough a he s a o he campaign cycles, wi h he bo om being
a a highe empe a u e han he la e als. To ob ain he equi ed ini ial
condi ion, a possibili y would be o conside an a e aged homogeneous
alue using he h ee he mocouple measu emen s in he wo campaign
cycles o which we ha e da a, equal o 322 K. Howe e , i is no
e y ad isable o cap u e a 2D empe a u e ield in he whole c oss-
sec ion only using he measu emen s in hese h ee poin s, gi en ha
he empe a u e in he sa e y lining could be subs an ially la ge han
in he insula ion lining, whe e he he mocouples a e placed. Hence,
o ind a mo e ealis ic ini ial empe a u e we p opose he ollowing
p ocedu e:
(1) A s eady s a e p oblem co esponding o 𝑚
𝑐𝑎𝑠𝑡 is sol ed o ob ain
he s eady s a e empe a u e in he solids. This in ol es sol ing
he hea Eq. (5) in 𝛺𝑐𝑎𝑠𝑡, assuming a s eady s a e as well as
𝑇𝑚
𝑐𝑎𝑠𝑡 =𝑇𝐹 ,𝑐𝑎𝑠𝑡 on 𝛤𝐷, and sa is ying bounda y condi ions (13)
and (19).
(2) A P oblem 𝑚
𝑠𝑡𝑜𝑝 using he s eady s a e empe a u e compu ed in
he p e ious s ep as ini ial condi ion is sol ed, un il eaching a
ime alue 𝑡= 20 days.
Following hese s eps, we ob ain he empe a u e ield depic ed in
Fig. 8. The empe a u e dis ibu ion in he c oss-sec ion shows ha he
esidual hea is concen a ed in he a ea close o he bo om o he
ough, as p edic ed by he he mocouple measu emen s. Howe e , i is
obse ed ha he empe a u es wi hin he sa e y lining a e subs an ially
highe han hose in he insula ion lining, whe e he he mocouples
a e loca ed. The ob ained empe a u e is deno ed as 𝑇0and is used as
ini ial condi ion o he i s cas , i.e., he P oblem 0
𝑐𝑎𝑠𝑡. Mo eo e , as
we a e sol ing a sequence o p oblems al e na ing be ween cas s and
s ops, each o hese needs an ini ial condi ion, which co esponds o
he compu ed solu ion a he end o he p e ious p oblem.
3.6. S ong o mula ion
A summa y o he comple e o mula ion o he wo p oblems ha
a e sequen ially sol ed is p o ided below, including he PDE, he
adia ion in eg al equa ion and he co esponding bounda y condi ions
ha we e discussed p e iously:
P oblem 𝑚
𝑐𝑎𝑠𝑡. Find 𝑇𝑚
𝑐𝑎𝑠𝑡(𝐱, 𝑡)and 𝑅𝑚
𝑐𝑎𝑠𝑡(𝐱, 𝑡)such ha :
𝜌𝑐𝑝
𝜕𝑇 𝑚
𝑐𝑎𝑠𝑡
𝜕𝑡 −di (𝑘(𝑇𝑚
𝑐𝑎𝑠𝑡)∇𝑇𝑚
𝑐𝑎𝑠𝑡)=0,in 𝛺𝑐𝑎𝑠𝑡 ×𝑚
𝑐𝑎𝑠𝑡,(21)
𝑇𝑚
𝑐𝑎𝑠𝑡(
𝑡2𝑚) = {𝑇0,i 𝑚= 0,
𝑇𝑚−1
𝑠𝑡𝑜𝑝 (
𝑡2𝑚),i 𝑚≥1,in 𝛺𝑐𝑎𝑠𝑡,(22)
𝑇𝑚
𝑐𝑎𝑠𝑡 =𝑇𝑚
𝐷,𝑐𝑎𝑠𝑡,on 𝛤𝐷×𝑚
𝑐𝑎𝑠𝑡,(23)
−𝑘(𝑇𝑚
𝑐𝑎𝑠𝑡)𝜕𝑇 𝑚
𝑐𝑎𝑠𝑡
𝜕𝐧=ℎ𝑛𝑐(𝑇𝑚
𝑐𝑎𝑠𝑡 −𝑇𝑒𝑥𝑡,𝑐,𝑛𝑐)
+𝜎𝜀𝑛𝑐((𝑇𝑚
𝑐𝑎𝑠𝑡)4−𝑇4
𝑒𝑥𝑡,𝑟,𝑛𝑐),on 𝛤𝑛𝑐
𝐶×𝑚
𝑐𝑎𝑠𝑡,(24)
−𝑘𝜕𝑇 𝑚
𝑐𝑎𝑠𝑡
𝜕𝐧=ℎ𝑅(𝑇𝑚
𝑐𝑎𝑠𝑡 −𝑇𝑒𝑥𝑡,𝑐,𝑅) + 𝑞𝑟𝑎𝑑 ,on
𝛤𝑅×𝑚
𝑐𝑎𝑠𝑡,(25)
𝑞𝑟𝑎𝑑 =(𝑅𝑚
𝑐𝑎𝑠𝑡),on 𝛤𝑅×𝑚
𝑐𝑎𝑠𝑡,(26)
(− (1 − 𝜀𝑅))(𝑅𝑚
𝑐𝑎𝑠𝑡) = 𝜀𝑅𝜎(𝑇𝑚
𝑐𝑎𝑠𝑡)4,on 𝛤𝑅×𝑚
𝑐𝑎𝑠𝑡,(27)
whe e 1≤𝑛𝑐≤3and 0≤𝑚≤𝑀− 1.
In (22), a he i s cas o he main ough campaign cycle, he
ini ial condi ion, 𝑇0, is ha al eady desc ibed in Sec ion 3.5. In (23),
he empe a u e 𝑇𝑚
𝐷,𝑐𝑎𝑠𝑡 is gi en by (16).
P oblem 𝑚
𝑠𝑡𝑜𝑝. Find 𝑇𝑚
𝑠𝑡𝑜𝑝(𝐱, 𝑡)and 𝑅𝑚
𝑠𝑡𝑜𝑝(𝐱, 𝑡)such ha :
𝜌𝑐𝑝
𝜕𝑇 𝑚
𝑠𝑡𝑜𝑝
𝜕𝑡 −di (𝑘(𝑇𝑚
𝑠𝑡𝑜𝑝)∇𝑇𝑚
𝑠𝑡𝑜𝑝)=0,in 𝛺𝑠𝑡𝑜𝑝 ×𝑚
𝑠𝑡𝑜𝑝,(28)
𝑇𝑚
𝑠𝑡𝑜𝑝(
𝑡2𝑚+1) = 𝑇∗,𝑚
𝑐𝑎𝑠𝑡(
𝑡2𝑚+1),in 𝛺𝑠𝑡𝑜𝑝,(29)
−𝑘(𝑇𝑚
𝑠𝑡𝑜𝑝)
𝜕𝑇 𝑚
𝑠𝑡𝑜𝑝
𝜕𝐧=ℎ𝑛𝑐(𝑇𝑚
𝑠𝑡𝑜𝑝 −𝑇𝑒𝑥𝑡,𝑐,𝑛𝑐)
+𝜎𝜀𝑛𝑐((𝑇𝑚
𝑐𝑎𝑠𝑡)4−𝑇4
𝑒𝑥𝑡,𝑟,𝑛𝑐),on 𝛤𝑛𝑐
𝐶×𝑚
𝑐𝑎𝑠𝑡,(30)
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P. Ba al e al.
−𝑘
𝜕𝑇 𝑚
𝑠𝑡𝑜𝑝
𝜕𝐧=ℎ𝑅(𝑇𝑚
𝑠𝑡𝑜𝑝 −𝑇𝑒𝑥𝑡,𝑐,𝑅) + 𝑞𝑟𝑎𝑑 ,on 𝛤𝑅×𝑚
𝑠𝑡𝑜𝑝,(31)
𝑞𝑟𝑎𝑑 =(𝑅𝑚
𝑠𝑡𝑜𝑝),on 𝛤𝑅×𝑚
𝑠𝑡𝑜𝑝,(32)
(− (1 − 𝜀𝑅))(𝑅𝑚
𝑠𝑡𝑜𝑝) = 𝜀𝑅𝜎(𝑇𝑚
𝑠𝑡𝑜𝑝)4,on 𝛤𝑅×𝑚
𝑠𝑡𝑜𝑝,(33)
whe e 1≤𝑛𝑐≤3and 0≤𝑚≤𝑀− 2. Mo eo e , wi h 𝑇∗,𝑚
𝑐𝑎𝑠𝑡 we deno e
an ex ension o he compu ed empe a u e o he cas p oblem using
(16):
𝑇∗,𝑚
𝑐𝑎𝑠𝑡(𝐱,
𝑡2𝑚+1) = {𝑇𝑚
𝑐𝑎𝑠𝑡(𝐱,
𝑡2𝑚+1),i 𝐱∈𝛺𝑐𝑎𝑠𝑡,
𝑇𝑚
𝐷,𝑐𝑎𝑠𝑡(𝐱,
𝑡2𝑚+1),o he wise.(34)
3.7. Weak o mula ion
The weak o m o bo h 𝑚
𝑐𝑎𝑠𝑡 and 𝑚
𝑠𝑡𝑜𝑝, deno ed as 𝑚
𝑐𝑎𝑠𝑡 and 𝑚
𝑠𝑡𝑜𝑝
espec i ely, is p esen ed below. The weak o mula ion o he p oblems
is he o m ha is sol ed nume ically using a Fini e Elemen Me hod,
and is ob ained analogously o ha p oposed in [1] o a apping
p oblem.
P oblem 𝑚
𝑐𝑎𝑠𝑡. Fo each 𝑡∈𝑚
𝑐𝑎𝑠𝑡, ind 𝑇𝑚
𝑐𝑎𝑠𝑡(𝑡), de ined in 𝛺𝑐𝑎𝑠𝑡,
and 𝑅𝑚
𝑐𝑎𝑠𝑡(𝑡), de ined on 𝛤𝑅, such ha 𝑇𝑚
𝑐𝑎𝑠𝑡(𝑡) = 𝑇𝑚
𝐷,𝑐𝑎𝑠𝑡(𝑡)on 𝛤𝐷and
∫𝛺𝑐𝑎𝑠𝑡
𝜌𝑐𝑝
𝜕𝑇 𝑚
𝑐𝑎𝑠𝑡(𝑡)
𝜕𝑡 𝑣 𝑑𝐴 +∫𝛺𝑐𝑎𝑠𝑡
𝑘(𝑇𝑚
𝑐𝑎𝑠𝑡(𝑡))∇𝑇𝑚
𝑐𝑎𝑠𝑡(𝑡)⋅∇𝑣 𝑑𝐴
+∫
𝛤𝑅
ℎ𝑅𝑇𝑚
𝑐𝑎𝑠𝑡(𝑡)𝑣 𝑑𝑠
+
3
∑
𝑛𝑐=1 ∫𝛤𝑛𝑐
𝐶(ℎ𝑛𝑐𝑇𝑚
𝑐𝑎𝑠𝑡(𝑡) + 𝜎𝜀𝑛𝑐(𝑇𝑚
𝑐𝑎𝑠𝑡(𝑡))4)𝑣 𝑑𝑠 (35)
=
3
∑
𝑛𝑐=1 ∫𝛤𝑛𝑐
𝐶(ℎ𝑛𝑐𝑇𝑒𝑥𝑡,𝑐,𝑛𝑐+𝜎𝜀𝑛𝑐𝑇4
𝑒𝑥𝑡,𝑟,𝑛𝑐)𝑣 𝑑𝑠
+∫
𝛤𝑅
(ℎ𝑅𝑇𝑒𝑥𝑡,𝑅 −(𝑅𝑚
𝑐𝑎𝑠𝑡(𝑡)))𝑣 𝑑𝑠,
o all es unc ions 𝑣de ined in 𝛺𝑐𝑎𝑠𝑡 being su icien ly egula , and
sa is ying:
(− (1 − 𝜀𝑅))(𝑅𝑚
𝑐𝑎𝑠𝑡(𝑡)) = 𝜀𝑅𝜎(𝑇𝑚
𝑐𝑎𝑠𝑡(𝑡))4,on 𝛤𝑅,(36)
o 𝑡∈𝑚
𝑐𝑎𝑠𝑡. Fu he mo e, 𝑇𝑚
𝑐𝑎𝑠𝑡(
𝑡2𝑚) = 𝑇𝑚−1
𝑠𝑡𝑜𝑝 (
𝑡2𝑚) o 1≤𝑚≤𝑀− 1. I
𝑚= 0, hen 𝑇0
𝑐𝑎𝑠𝑡(0) = 𝑇0.
P oblem 𝑚
𝑠𝑡𝑜𝑝. Fo each 𝑡∈𝑚
𝑠𝑡𝑜𝑝, ind 𝑇𝑚
𝑠𝑡𝑜𝑝(𝑡), de ined in 𝛺𝑠𝑡𝑜𝑝,
and 𝑅𝑚
𝑠𝑡𝑜𝑝(𝑡), de ined on 𝛤𝑅, such ha
∫𝛺𝑠𝑡𝑜𝑝
𝜌𝑐𝑝
𝜕𝑇 𝑚
𝑠𝑡𝑜𝑝(𝑡)
𝜕𝑡 𝑣 𝑑𝐴 +∫𝛺𝑠𝑡𝑜𝑝
𝑘(𝑇𝑚
𝑠𝑡𝑜𝑝(𝑡))∇𝑇𝑚
𝑠𝑡𝑜𝑝(𝑡)⋅∇𝑣 𝑑𝐴
+∫𝛤𝑅
ℎ𝑅𝑇𝑚
𝑠𝑡𝑜𝑝(𝑡)𝑣 𝑑𝑠
+
3
∑
𝑛𝑐=1 ∫𝛤𝑛𝑐
𝐶(ℎ𝑛𝑐𝑇𝑚
𝑠𝑡𝑜𝑝(𝑡) + 𝜎𝜀𝑛𝑐(𝑇𝑚
𝑠𝑡𝑜𝑝(𝑡))4)𝑣 𝑑𝑠 (37)
=
3
∑
𝑛𝑐=1 ∫𝛤𝑛𝑐
𝐶(ℎ𝑛𝑐𝑇𝑒𝑥𝑡,𝑐,𝑛𝑐+𝜎𝜀𝑛𝑐𝑇4
𝑒𝑥𝑡,𝑟,𝑛𝑐)𝑣 𝑑𝑠
+∫𝛤𝑅
(ℎ𝑅𝑇𝑒𝑥𝑡,𝑅 −(𝑅𝑚
𝑠𝑡𝑜𝑝(𝑡)))𝑣 𝑑𝑠,
o all es unc ions 𝑣de ined in 𝛺𝑠𝑡𝑜𝑝 being su icien ly egula , and
sa is ying:
(− (1 − 𝜀𝑅))(𝑅𝑚
𝑠𝑡𝑜𝑝(𝑡)) = 𝜀𝑅𝜎(𝑇𝑚
𝑠𝑡𝑜𝑝(𝑡))4,on 𝛤𝑅,(38)
o 𝑡∈𝑚
𝑠𝑡𝑜𝑝 and 0≤𝑚≤𝑀− 2. In addi ion, 𝑇𝑚
𝑠𝑡𝑜𝑝(
𝑡2𝑚+1) = 𝑇∗,𝑚
𝑐𝑎𝑠𝑡(
𝑡2𝑚+1)
(see (34)).
4. Nume ical esul s
In his sec ion, he nume ical esul s conce ning he hea ans-
e in he BF main ough c oss-sec ion a e p esen ed. The model is
Fig. 9. Mesh o he BF main ough c oss-sec ion.
sol ed using he open-sou ce ini e elemen compu ing pla o m FEniCS
(see [32,33]), which allows o sol e PDEs om a high-le el amewo k.
The di e en componen s o FEniCS enable o au oma ically pe o m
he s eps o gene a e e icien code o disc e ize he p oblem using
a ini e elemen me hod, using use -speci ied high le el abs ac ions
simila o he p oblem weak o mula ion as s a ing poin .
A iangula ion 𝜏ℎo he domain 𝛺𝑠𝑡𝑜𝑝 is cons uc ed such ha
𝛺𝑠𝑡𝑜𝑝 = ∪𝐾∈𝜏ℎ𝐾, shown in Fig. 9. The iangula ion is con o mal o he
di e en ma e ials in he BF main ough. Thus, he iangula ion used
o he domain 𝛺𝑐𝑎𝑠𝑡 is a subse o 𝜏ℎand is iden ical o ha p esen ed
in [1], whe e a mesh sensi i i y s udy was ca ied ou . I is o med by
a o al 65 440 iangles whose es ic ions o 𝛤𝑅a e a o al o 400 edges.
F om he weak o mula ions de ailed in Sec ion 3.7, a me hod o
lines leads o he spa ial semi-disc e iza ion, whe eas he adiosi y
in eg al equa ions a e disc e ized using a colloca ion me hod, also
ollowing he p ocedu e desc ibed in [1]. The H211b adap i e s ep
size con olle (see [34]) is used o adjus he s ep size using he
local e o es ima e p o ided by he ESDIRK 3/2a RK scheme, which
simply becomes he di e ence be ween he las wo s ages in i ue
o i s p ope ies (see [35]). No e ha he aim is o sol e a p oblem
wi h e y la ge ime scales (in he o de o mon hs) wi h mos ly
gen le empe a u e changes, bu also in ol ing sho pe iods o sha p
empe a u e change in ime, especially a he beginning o each cas .
The e o e, s ep size adap i i y as well as he use o high-o de ime
disc e iza ion schemes a e essen ial o con ol he commi ed e o
while educing he compu a ional cos .
4.1. Algo i hm
To couple he adiosi y in eg al equa ion wi h he hea equa ion,
we use he seg ega ed app oach desc ibed in [1], which in ol es a
ixed poin algo i hm o each RK s age. I consis s o h ee embedded
loops; an inne one o compu e he adiosi y and empe a u e a a gi en
s age and ime s ep, ano he o he ime s epping and an ou e loop o
ad ance h oughou he sequence o cas and s op p oblems. I ollows
he low cha depic ed in Fig. 10. The ixed poin algo i hm con ol is
based on he ela i e inc emen al esidual o bo h he empe a u e and
adiosi y, which mus all below a gi en h eshold. In addi ion, he s ep
size con ol ejec s and ecompu es he cu en ime s ep i he change
a io is below a gi en alue o 0.9. I he end ime o he co esponding
p oblem is eached, hen he s ep size is adjus ed acco dingly and, i
ha s ep is inally accep ed, he algo i hm ad ances o he subsequen
p oblem. The same pa ame e s o he H211b s ep size con olle o
hose epo ed in [1] a e used in his wo k, wi h he only di e ence
ha a maximum s ep size 𝛥𝑡 = 300 s is se du ing he cas p oblems o
imp o e he s abili y o he algo i hm.
In e na ional Jou nal o The mal Sciences 173 (2022) 107349
9
P. Ba al e al.
Fig. 10. Flow diag am o he algo i hm.
Table 4
S a da e, end da e and o al numbe o cas s (𝑀) o each sol ed campaign cycle.
S a da e End da e Numbe o cas s
Campaign cycle 1 28/05/2018 28/07/2018 347
Campaign cycle 2 21/08/2018 18/10/2018 325
4.2. Simula ion o wo campaign cycles
The p oblem desc ibed in Sec ion 3.6, composed o a sequence o
cas and s op p oblems, is sol ed. This sequence is de e mined by da a
spanning wo di e en campaign cycles o assess whe he he BF is
being cas , as desc ibed in Sec ion 3.1. A summa y o he main cha ac-
e is ics o he conside ed campaign cycles is displayed in Table 4. The
wo sol ed cycles a e consecu i e, wi h he down ime be ween hem
emo ed.
Table 5
Maximum s ep size, numbe o accep ed s eps, numbe o ejec ed s eps and accep ance
a e using he H211b con olle o he conside ed campaign cycles.
Max. s ep size (s) Accep ed s eps Rejec ed s eps Accep ance a e
Campaign 1 1.89𝑒+04 21 509 408 9.81𝑒−01
Campaign 2 9.05𝑒+03 21 171 411 9.81𝑒−01
In Fig. 11, he s ep size alue is plo ed agains he ime alue o he
wo conside ed campaign cycles, om day 10 o day 20. The s ep size
ises quickly om he s a size a each cas o s op p oblem, which is
𝛥𝑡0= 1 s. F equen ly, his i s s ep is ejec ed and ecompu ed wi h
a smalle alue. Fu he mo e, he p esence o long s ops du ing he
campaign cycles is e iden , as hey allow o much la ge s ep sizes
wi hou inc easing he e o beyond he selec ed ole ance alue. In
Table 5, he maximum s ep size, he numbe o accep ed and ejec ed
s eps, and he accep ance a e a e ga he ed o he wo campaign cycles
ha a e sol ed. The accep ance a e is abo e 0.98 in bo h cases, while
he maximum s ep size in each campaign is de e mined by he leng h o
he longes s op du ing he BF ac i i y eco ded du ing he campaign,
which las s o longe han wo days, as shown in Fig. 11.
In Fig. 12(a), he compu ed empe a u e ield 𝑇0
𝑐𝑎𝑠𝑡,ℎ, solu ion o
P oblem 0
𝑐𝑎𝑠𝑡, a e he i s cas o he BF in Campaign cycle 1is
displayed, i.e. a ime
𝑡1. E en hough du ing each 𝑚
𝑐𝑎𝑠𝑡 only he domain
𝛺𝑐𝑎𝑠𝑡 =𝛺𝑆is sol ed, he luids a e also included in he shown con ou
plo s as hei empe a u e is assumed o be known and equal o (16),
as desc ibed in Sec ion 3.4. High empe a u es a e obse ed only in he
zone ha is close o he liquids, as mos o he solid is s ill una ec ed,
showing simila empe a u e alues o hose o he ini ial condi ion (see
Fig. 8). In con as , he pa s o he wo king lining ha a e a ec ed by
he adia ion om he slag su ace show signi ican hea ing, as well as
he e ac o y co e , eaching empe a u es a ound 1100 K.
To showcase he e ec o he sho s ops in he unne ac i i y,
he empe a u e a e he subsequen s op ollowing he i s apping,
𝑇0
𝑠𝑡𝑜𝑝,ℎ(⋅,
𝑡2)is depic ed in Fig. 12(b). App oxima ely hal an hou has
passed since he end o he p eceding cas , and mos o he cooling
akes place in he luids. On he o he hand, he co e and he wo king
lining s ill emain a a high empe a u e.
In Fig. 12(c), he empe a u e con ou s a e app oxima ely 10 days
o cas s and s ops a e shown. Speci ically, he con ou s co espond o
𝑇76
𝑐𝑎𝑠𝑡,ℎ(⋅,
𝑡153). The empe a u e ield is subs an ially di e en o ha dis-
played in Figs. 12(a) and 12(b), as he bulk o he solids displays a much
highe empe a u e. Towa ds he end o he campaign, he empe a u e
is simila , as depic ed in Fig. 12( ) o 𝑡= 57.84 days. Howe e , he
empe a u e ield is a om a s eady s a e, as demons a ed by he
empe a u e con ou s shown in Fig. 12(d). This empe a u e snapsho
co esponds o a ime alue a e a long s op, which las ed o a ound
2.2 days. The co e and he uppe pa o he wo king lining and luids
has subs an ially cooled down. None heless, due o he la ge he mal
ine ia o he ough, he bulk o he solids emains abo e 1000 K.
Fig. 11. Time s ep alues compu ed by he H211b con olle du ing he wo campaign cycles.
In e na ional Jou nal o The mal Sciences 173 (2022) 107349
16
P. Ba al e al.
Fig. 21. Tempe a u e o e ime a he co ec ed he mocouple loca ions.
Despi e he simpli ying assump ions ha we e made, he compu ed
empe a u es show quali a i e ag eemen wi h he measu emen s. Ne -
e heless, as he e is a subs an ial deg ee o unce ain y conce ning
he posi ion o he he mocouples, a echnique o ind a co ec ed
posi ion inside a easible egion was used. In pa icula , he hyb id
GRSA algo i hm was employed o ind he posi ion o each he mo-
couple, minimizing he co esponding cos unc ional. This posi ion is
calcula ed as he one ha yields he closes ime-dependen nume ical
empe a u e p o ile o he measu emen . The compu ed placemen s
a e simila o he wo campaigns and h ee analysed he mocouples.
A hese co ec ed posi ions, much be e le els o ag eemen we e
achie ed among he measu emen s and he compu ed empe a u es.
P o ided ha he model does no accoun o wea o ma e ial de-
e io a ion, he disc epancies among he compu ed empe a u e alues
and he measu emen s, which g ow owa ds he end o he campaigns,
sugges se e e damage and loss o p ope ies in he wo king lining.
The esul s also indica e ha mos o he wea is loca ed in he la e al
sec ions o he unne , as he di e ences a e much la ge o he la e al
he mocouples.
In u u e wo k, u he s udies should be conduc ed o expe imen-
ally alida e he model. Mo e de ailed expe imen al measu emen s o
he ma e ial p ope ies should be ca ied ou o add ess hei empe a-
u e dependency. The model could also be imp o ed by in oducing he
phase change ha akes place in he luids du ing he s ops, especially
du ing hose ha a e p olonged. Using be e he mocouple moni o ing,
bo h conce ning he numbe o de ices and he eliabili y o hei
placemen , an in e se hea ans e model could be de eloped, which
would allow o use he compu ed de ia ion o he nume ical solu ion
om he expe imen al measu emen s o assess he wea p o ile in he
wo king lining.
Decla a ion o compe ing in e es
The au ho s decla e ha hey ha e no known compe ing inan-
cial in e es s o pe sonal ela ionships ha could ha e appea ed o
in luence he wo k epo ed in his pape .
Acknowledgemen
This wo k was pa ially suppo ed by ERDF and Xun a de Galicia
unds unde he ED431C 2017/60 g an , by he Minis e io de Cien-
cia, Inno ación y Uni e sidades h ough he Plan Nacional de I+D+i
(MTM2015-68275-R) and he g an BES-2016-077228, and by he
Agencia Es a al de In es igación h ough p ojec [PID2019-105615RB-
I00/ AEI/10.13039/501100011033].
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