Gas–solid con e sion in luidised bed eac o s
A. G´
omez-Ba eaa,∗, B. Leckne b, D. San anac, P. Olle oa
aBioene gy G oup, Chemical and En i onmen al Enginee ing Depa men , Escuela Supe io de Ingenie os,
Uni e si y o Se ille, Camino de los Descub imien os s/n, 41092 Se ille, Spain
bDepa men o Ene gy and En i onmen , Chalme s Uni e si y o Technology,
S-412 96 G¨o ebo g, Sweden
cThe mal and Fluid Enginee ing Depa men , Uni e sidad Ca los III de Mad id,
A da. de la Uni e sidad 30, 28911 Legan´es Mad id, Spain
Abs ac
A simpli ied model o gas–solid eac ions in luidised bed (FB) is p oposed. Such models al eady exis o ca aly ic gas–solid eac ions (CGSRs),
p o iding gene al desc ip ion o he sys em in e ms o main go e ning pa ame e s. Expansion o his app oach o non-ca aly ic gas–solid eac ions
(NCGSRs) is di icul , because he solid eac an akes pa in he eac ion. The e o e, FB eac o models o NCGSR a e usually de ised only
o speci ic cases, and a gene al analysis has no been p esen ed up o da e. The p esen model allows analysis o di e en ypes o NCGSR in a
gene alised way, handling ca aly ic eac ions as a pa icula , simple , case. I is shown ha he eac o beha iou can be desc ibed by h ee go e ning
dimensionless pa ame e s. Two addi ional pa ame e s, quan i ying he impo ance o di usion e ec s in single pa icles a e also iden i ied, and
hei impac on eac o beha iou is analysed. Possible simpli ica ions a e explo ed. Model limi a ions, ha is, assump ion o iso he mal bed and
pa icle and he occu ence o only one eac ion, a e discussed. Examples a e ou lined o show he applicabili y o he me hod.
Keywo ds: Fluidised beds; Gas–solid eac ions; T anspo p ocesses; Modelling; Non-ca aly ic eac ions; Di usion e ec s
1. In oduc ion
Many impo an p ocesses, in which non-ca aly ic gas–solid
eac ions a e in ol ed, ake place in luidised bed. Typical
applica ions a e ound in me allu gical and he mochemical
con e sion p ocesses. Examples o me allu gical impo ance
comp ise he educ ion o me als oxides (i on, nickel, e c.), oas -
ing o o es o hea y me als in sulphide o m, such as coppe ,
nickel, zinc and lead. The mochemical examples a e combus-
ion, gasi ica ion and py olysis o coal and biomass, including
sulphu cap u e (in-bed desulphu isa ion) using mine al ocks,
such as dolomi e o limes one. To his class o p ocesses belong
also he mal decomposi ion eac ions, luo ina ion o u anium
and plu onium compounds, some g anula ion p ocesses, e c. The
op imisa ion and scale-up o hese p ocesses bene i g ea ly om
modelling o he sys em. De ailed desc ip ion o physical and
Abb e ia ions: BFB, bubbling luidised bed; CGSR, ca aly ic gas solid eac-
ion; FB, luidised bed; FDE, ee o di usion e ec s; NCGSR, non-ca aly ic gas
solid eac ion; SIM, sha p in e ace model; UCM, uni o m con e sion model.
∗Co esponding au ho . Tel.: +34 95 4487223; ax: +34 95 4461775.
E-mail add ess: [email p o ec ed] (A. G´
omez-Ba ea).
chemical p ocesses occu ing inside an FB o NCGSR is, how-
e e , a di icul ask. Simple me hods p o iding app oxima e
solu ions o i s es ima es a e qui e use ul. Fo ins ance, an
app oxima e desc ip ion o gas–solid eac ions in FB can be su -
icien o selec ion o mode o gas–solid con ac , p elimina y
design, and op imal ope a ing condi ions by sensi i i y analysis.
Such models al eady exis o CGSR in FB [1–4]. In his ype o
sys em, he solids a e unchanged as eac ion p oceeds and bed
emo al is no usually unde aken du ing s eady-s a e ope a ion
i he ca alys is no poisoned. In con as , simple models ha e no
been de eloped o NCGSR due o hei complexi y compa ed
o hei ca aly ic coun e pa s. Al hough many FB eac o mod-
els ha e been published, hey a e de ised o speci ic eac ions
only. A gene al amewo k o simpli ied ea men o NCGSR
is no ye a ailable.
The pe o mance o gas–solid eac ions in FB has been
desc ibed by se e al app oaches. Ea ly models ea ed he FB
eac o as i he gas and solids we e mixed, a oiding he mul i-
phase na u e o he bed. These ‘single-phase’ models assumed
ha he eac o pe o mance was de e mined by he esidence
ime o he gas. The b eak h ough caused by he in oduc ion
o he wo-phase heo y p oposed in he ea ly 1950s, allowed
.
1
Nomencla u e
A Non- eac i e componen in he solids (ash o
equi alen )
ATbed c oss sec ion (m2)
Bi Bio numbe
cgas concen a ion (mol m−3)
C eac i e componen in he solids
Dee ec i e di usi i y o he eac an solid pa icle
(m2s−1)
D bed diame e (m)
DapDamk¨
ohle numbe a pa icle scale, de ined in
Eq. (30)
DaRDamk¨
ohle numbe a eac o scale, de ined in Eq.
(22)
DasDamk¨
ohle numbe o he solid eac an , de ined
in Eq. (40)
unc ion
1(xc0,λ) unc ion de ined in Eq. (45)
2(xc0,λ) unc ion de ined in Eq. (48)
F(xc) unc ion exp essing he dependence o dxc/d on
xc o any ηp
Fi(xc) unc ion exp essing he dependence o dxc/d on
xcwhen ηp= 1 (kine ic egime)
F0,F1inle and ou le low a e o solids (kg s−1)
gaccele a ion o g a i y (m s−2)
g(xc) unc ion exp essing he change o e ec i e di u-
si i y wi h xc
G gas eac an
Hheigh o he essel con aining he bed (m)
kn h-o de kine ic coe icien in he kine ics
((− )=kcn) ((kgmol m−3)1−ns−1)
kbcoe icien o in e change be ween bubble and
emulsion (s−1)
kGex e nal mass- ans e coe icien (m s−1)
K kine ic coe icien accoun ing o gas concen a-
ion and empe a u e (s−1)
Lequ equi alen size o solid pa icle (m)
L bed heigh (m)
MThiele module, unc ion o con e sion
Mcmolecula mass o solid eac an (kg kgmol−1)
no de o eac ion
Naconcen a ion e iciency, de ined in Eq. (23)
NTU numbe o ans e uni s, de ined in Eq. (24)
pb(xc) dis ibu ion o con e sion in he bed (mass basis)
p0,p1dis ibu ion o con e sion in he inle and ou le
s eams (mass basis)
Pbp essu e d op ac oss he bed (Pa)
(− ) in insic eac ion a e pe uni o pa icle olume
((− )=kcn) (kgmol m−3s−1)
c,b o e all a e o eac ion in he bed (kg s−1)
(−R) obse ed eac ion a e pe uni o pa icle olume
(kgmol m−3s−1)
R(xc) eac i i y o solid eac an (s−1)
Reppa icle Reynolds numbe
sdummy a iable o in eg a ion
Sh She wood numbe
ime (s)
Tbbed empe a u e (K)
ugas eloci y (m s−1)
Vppa icle olume (m3)
wbmass o A and C in he bed (wb=wA+wc) (kg)
wcmass o solid eac an (C) in he bed (kg)
wTb o al mass o he bed (wTb =wb+wine ) (kg)
xccon e sion o solid eac an in a pa icle
xc,b a e age con e sion o solids in he bed
Xggas con e sion
Yc0 mass ac ion o solid eac an in he eed
Yc,b mass ac ion o solid eac an in he bed
zaxial coo dina e
G eek symbols
αdimensionless pa ame e a eac o le el, de ined
in Eq. (52)
βdimensionless excess o low, de ined in Eq. (25)
δkine ic pa ame e in Eq. (60), also Di ac’s del a
unc ion
εpo osi y
εbbubble ac ion ((m3bubbles) (m−3bed))
ηeex e nal e ec i eness ac o
ηiin e nal e ec i eness ac o
ηppa icle e ec i eness ac o
ηph in e phase e ec i eness ac o
κpa ame e de ined in Eq. (59)
λdimensionless pa ame e de ined in Eq. (41)
νs oichiome ic ac o o he eac ion
ξkine ic pa ame e (see Table 1)
ρdensi y o solid (kg m−3)
τRsolid esidence ime (s)
Θ(xc) unc ion de ined in Eq. (39)
Subsc ip s
b bubble, bed, a e age in he bed
c eac i e componen in he pa icle
c i c i ical
e emulsion
i in apa icle
in inle
ou ou le
p pa icle
eac ion
s su ace
0 ini ial, supe icial
conside a ion o he mul iphase na u e o he FB by means o
a simpli ied desc ip ion o wo phases, in which he solids and
he gas we e dis ibu ed in he bed. The ea e he concep o
‘con ac ime dis ibu ion’ was ecognized as a key ac o o
aking in o accoun he ime o gas con ac wi h he solid eac-
an [2]. May [1], O cu e al. [3] and Da idson and Ha ison
[4] used he wo-phase heo y o luidisa ion o calcula ion o
gas con e sion in a ious iso he mal FB ca aly ic gas eac o s.
2
Table 1
Main models applied o NCGSR kine ics
Name Abb e ia ion Fi(xc)Θ(X) Re e ence
Volume ic model UCM 1 −xc−ln(1 −xc)[22]
G ain model; o sha p in e ace model GM (SIM) (1 −xc)2/3 3(1 −(1 −xc)1/3)[23,24]
Random po e model RPM (1 −xc)(1 −ξln(1 −xc)) (2/ξ)(1 −ξln(1 −xc))1/2 [25]
Simons model SM (1 −xc)(xc+ξ(1 −xc))1/2 2a c gh((1 −ξ)xc+ξ)1/2 [26]
Johnson model JM (1 −xc)2/3eξx2
cNAEF [27]
Du a model DM [1 ±100xξ1ξ2
cexp(−ξ2xc)](1 −xc) NAEF [20]
Ga dne model GM (1 −xc)e
−ξxcNAEF [28]
Cho ne model CM √xc(1 −xc) 2a c gh(√xc)[29]
Modi ied olume ic model MVM ξ1/2
1ξ2(1 −xc)[−ln(1 −xc)] NAEF [30]
T adi ional model TM (1 −xc)ξ(ξ−1)−1[(1 −X)1−ξ−1] [31]
Polynomial model PM
n

i=1
ξixc(1 −xc)iNAEF [32]
The hi d column p esen s Fi(xc), he unc ion modelling he beha iou de ined in Eq. (1). The ou h column is he unc ion de ined in Eq. (39).ξia e kine ic model
pa ame e s; NAEF: no analy ical exp ession ound.
Ex ensi e e iews ha e been published on modelling o FB eac-
o s [2,5–7], whe e he analyses we e pe o med wi h di e en
deg ee o sophis ica ion.
Se e al publica ions ha e su eyed he abili y o FB eac o
models in a a ie y o gas–solid eac ions [5,8–11]. Gene ally,
solu ions o wo-phase models based on iso he mal ca aly ic
sys ems ha e been p esen ed in e ms o wo main dimension-
less g oups: one ep esen ing he dimensionless eac ion a e
and he o he accoun ing o he in e phase mass- ans e esis-
ance [2,5–7]. Analy ical solu ions ha e been epo ed o simple
kine icschemes, such as i s -,second-,e c., o de kine ic(see o
ins ance Table 11.5 in [2]). Solu ions o mo e complex kine ics
ha e been p esen ed o ca aly ic eac ions, based on con en-
ional wo-phase models and Kunii–Le enspiel’s model [12,13].
Expansion o include he mal e ec s has also been unde aken
in ca aly ic sys ems o simple eac ions [7], bu his ex ension
causes di icul ies because o he complexi y o he ea men
e en o he simples eac ion scheme.
In FB ca aly ic sys ems he solids a e unchanged as eac ion
p oceeds (i no ca alys poisoning occu s) and he solids a e only
conside ed as a sink in he e alua ion o he eac ion a e. The
eac ion a e on he ca alys pa icle can be subjec ed o di usion
e ec s, bu hese do no change wi h ime. Fo NCGSR in an FB,
in con as , he solid eac an is cons an ly consumed and solids
make-up is equi ed o s eady-s a e ope a ion. A any ins an ,
he eac o con ains pa icles ha ha e spen di e en leng hs o
ime inside he bed, and, hus, hey ha e a wide bu n-o dis i-
bu ion o pa icle age. Du ing he cou se o eac ion, he solid
eac an con ained in he pa icles is g adually a ec ed, and he
densi y and size o he pa icles change depending on he ope -
a ing condi ions in he bed. This beha iou can change om one
pa icle-size ac ion o ano he depending on he concen a ion
o he solid eac an wi hin he pa icles. A gene al desc ip ion
o he bed should accoun o a ia ion in size and densi y o he
eac ing pa icles [14,15]. In addi ion o he a o emen ioned wo
main dimensionless g oups appea ing in he iso he mal FB ca -
aly ic eac o , a hi d pa ame e aking in o accoun he ela i e
amoun s o gas and solid eac an s ed o he eac o is equi ed
o desc ibe he NCGSR in an FB [9,10].
The abb e ia ion NCGSR ep esen s he e ogeneous eac-
ions whe e he ac i e solid pa icipa es in he eac ion, in
con as o ca aly ic sys ems, which, i no poisoned, emain
unchangeddu ing eac ion. Someca aly ice ec smay exis any-
way caused by he ine ma e ial, o ins ance, mine als in coal o
biomass pa icles in he mochemical p ocesses. Howe e , such
e ec s a e included in he gas–solid kine ics (exp ession dxc/d ,
see Eq. (1)) de e mined in he labo a o y, in his way being an
inpu o he model p esen ed.
F om his discussion i is clea ha an FB eac o model
o NCGSR should conside : (1) con inuous bed emo al; (2)
a ia ion o physical p ope ies and eac ion a e o single pa -
icles as eac ion p oceeds; (3) he dis ibu ion o con e sion
o he pa icles in he bed; (4) a ying di usion ilm and in a-
pa icle mass- ans e limi a ions wi h bu n-o . As a esul , FB
eac o models o ca aly ic eac ions a e no gene ally alid
o NCGSR. The need o all hese (and in some cases o he )
conside a ions is he eason why dedica ed models ha e been
de eloped o NCGSR in FB. Many eac o models exis , bu
hey a e de ised solely o speci ic eac ions. The e iews o
Ya es [16], Do aiswamy and Sha ma [17], and G ace [2] su ey
he mos popula models de eloped un il he end o 1980s.
In he p esen wo k, a me hod is de eloped o he solu ion
o NCGSR in an iso he mal FB, allowing analysis o gene al
NCGSR by a common p ocedu e. In his way, he simple mod-
elling app oach al eady exis ing o ca aly ic sys ems is ex ended
o non-ca aly ic sys ems. The ea men conside s iso he mal
condi ions bo h in he phases and wi hin he eac ing pa icles,
which imposes some limi a ions o he applica ion o he me hod.
Fu he mo e, some NCGSRs imply conside a ion o a ious he -
e ogeneous eac ions, and his could limi he me hod u he .
This and o he limi a ions a e deal wi h a he end o his wo k
whe e ex ension o he me hod and possibili ies o o e come
limi a ions a e discussed.
2. P oblem desc ip ion and defini ions
Fig. 1 illus a es he p oblem deal wi h. The gas eac an G
is in oduced in o he FB eac o as pa o he luidisa ion agen
3
Fig. 1. (a) Model concep showing he hypo hesis assumed in his wo k: he igh -hand d awing o (a) p esen s he basis o he wo-phase model de eloped showing
he esis ance o mass ans e be ween bubble and emulsion. Le -hand d awing o (a) is a zoom o he p ocesses occu ing in a eac ing pa icle (wi h la geome y
o simpli ica ion), including he main esis ances o mass anspo : in he ilm (ex e nal mass esis ance) and wi hin he solid pa icle ( eac ion and in apa icle
esis ances). (b) P ocess scheme showing he con ac pa e n in an FB and popula ion balance de ini ions.
wi h a concen a ion cin. I passes h ough he bed as bubbles
wi h a concen a ion cb, and h ough he well-mixed emulsion
phase wi h a concen a ion ce. The G species is ans e ed om
bubble o emulsion o each he eac ing si es wi hin he eac -
ing pa icles, whe e he eac ion is C(s) + νG(g) →p oduc s.
The esis ances o anspo and eac ion and he main assump-
ions ha ha e been made o de elop he ma hema ical model
a e shown in Fig. 1a. The esis ances a e: bubble o emulsion
esis ance, ex e nal ilm esis ance a ound he solid pa icle ( he
esis ance wi hin he emulsion phase is assumed o be concen-
a ed a ound he pa icles), and in apa icle esis ance. The inle
and ou le s eams o he solids including eac an C a e shown
in Fig. 1b, whe e he bed in en o y, wb, is also speci ied. The
eac o con ains pa icles ha ha e spen di e en imes inside
he bed and, hus, ha e a wide dis ibu ion o con e sion, pb(xc).
This la e is conside ed equal o he dis ibu ion o he ou low
s eam, p1(xc), since pe ec mixing o solid is assumed.
We assume ha a solid pa icle S, is made up o ac i e
solid eac an ma e ial C and non- eac i e solid ma e ial A,
ash o simila . In addi ion, he e could be ine ma e ial, ed
o he sys em o a ious easons ( o ins ance, sand o keep
he bed cons an ). The con e sion o he solid eac an C con-
ained in S a any ins an , xc, is de ined as he ela i e di e ence
be ween he ini ial amoun o C and he ins an aneous one,
xc=(Yc0( 0)−Yc( ))/Yc0( 0). Ycis he mass ac ion o solid eac-
an C in a gi en mass o ma e ial: Yc0 (kg C/kg S a xc0), and
Yc(xc) (kg C/kg S a xc). Following hese de ini ions (1 −Yc0xc)
is he mass ac ion o S in a s eam o con e sion xcand
Yc0/(1 −Yc0xc)iskgCa xc0/kg S a xc. No e he di e ence
be ween Ycand xc:Ycis an in eg al measu e o he amoun o C
con ained in a s eam o ma e ial (o in he bed) con aining a i-
ous compounds (C+A+ine ), whe eas xcis a ma k o he s a e
o con e sion o an indi idual pa icle e e eed o hei ini ial
s a e o con e sion, when i was ed o he eac o , xc0. Speci i-
ca ion o xc o a s eam (o o he bed) has no meaning because,
in he gene al case, in a gi en s eam he e will be pa icles wi h
di e en deg ees o con e sion. Howe e , he a e age o xcin a
s eam (o in he bed) is uniquely ela ed wi h Yc(see Eq. (9)).
The a e o con e sion o a single pa icle due o chemical
eac ion, unde chemical eac ion con ol, may be exp essed as
[18,19]:
dxc
d =K ,eFi(xc) (1)
K ,e is he kine ic coe icien , accoun ing o he concen a ion o
he gaseous eac an and empe a u e in he emulsion, whe e he
eac ion akes place. The unc ion Fi(xc) exp esses he depen-
dence o he con e sion a e on xc.K ,e is e alua ed o he
condi ions in he emulsion, whe e he eac ion akes place. The
a e o eac ion can also be o mula ed as [20,21]:
dxc
d =Mc
ρc0
k(xc)cn
e
ν(2)
kis he kine ic coe icien based on pa icle olume, and ela es
he a e o eac ion pe uni o olume wi h he gas eac an con-
cen a ion, i.e. (− )=kcn
e. This de ini ion is ypical in GSCR
whe e kis a cons an o iso he mal condi ions. In con as , o
iso he mal NCGSR kdepends on con e sion. Wi h he ini ial
ime as a e e ence, Fi(xc=xc0)=1 and k(xc)=k0Fi(xc), so o
xc>xc0 one ob ains om Eqs. (1) and (2):
K ,e=Mc
ρc0
k0cn
e
ν(3)
When di usion plays a ole he con e sion a e o a pa icle
is w i en as
dxc
d =ηp(xc)Fi(xc)K ,e(4)
whe e he pa icle’s e ec i eness ac o ηp(xc) accoun s o he
di usion esis ance a pa icle scale ( he ex e nal gas ilm and
4
he in apa icle esis ance). Some au ho s ha e used exp essions
like Eq. (4) o es ima e he ole o in e nal di usion [20,21].
ηp(xc) is de ined as he a io o he ac ual con e sion a e o a
pa icle o he a e ee o di usion e ec s (FDE):
ηp(xc)=dxc/d
dxc/d |FDE =dxc/d
Fi(xc)K ,e
(5)
Fo mally, he Fi(xc) unc ion should be ee o di usion limi a-
ions, i.e. i should be de e mined in he kine ically con olled
egime. In his wo k Fi(xc) is he a io o a ailable solid su ace
a a ce ain con e sion xc o ha o a e e ence case, xc0.Table 1
p o ides some accep ed models o Fi(xc) used o NCGSR eac-
ions.
F om Eq. (4) a unc ion can be de ined: F(xc)=Fi(xc)ηp(xc),
yielding
dxc
d =K ,eF(xc) (6)
The o e all mass a e o eac ion c,b in he en i e bed is
compu ed by
c,b=1
xc0
wbR(s)pb(s)ds(7)
whe e wbis he mass o S (A + C) in he bed and pb(xc)is he
dis ibu ion o con e sion in he bed (mass basis). The in eg and
wbR(xc)pb(xc) is he a e o eac ion o solid pa icles in he
bed ha ing a con e sion be ween xcand xc+dxc.R(xc)is he
eac i i y exp essed as kg C eac ed/kg S a xcand ime:
R(xc)=Yc0
1−Yc0xc
dxc
d =Yc0K ,e
1−Yc0xc
F(xc) (8)
The solids can accumula e in he bed, depending on he ne
balance be ween he a es o eed, con e sion, and emo al o
solids. A solid pa icle is ed in o he eac o wi h an ini ial
con e sion xc0 and i is emo ed om he bed wi h a con e sion,
xc,b (a e age con e sion o pe ec ly mixed pa icles in he bed).
The ac ion o C in he bed, Yc,b (kg C/kg S in he bed) is [9]:
Yc,b=Yc0(1 −xc,b)
1−Yc0xc,b
(9)
Sol ing o xc,b gi es
xc,b=Yc0 −Yc,b
Yc0(1 −Yc,b)(10)
The bed ma e ial consis s o he solid eac an C and he
ma e ial emaining a e comple ing he eac ion A, ash o any
o he ype o ine componen o igina ing om he eed s eam.
The mass o solid eac an C in he bed is he p oduc o wband
Yc,b ob ained om Eq. (9):
wc=wbYc,b(11)
I ine ma e ial is ed o he bed, o i a ba ch o such ma e ial
is used o ill he bed ini ially ( o ins ance sand as ini ial bu e
in biomass he mochemical con e sion p ocesses), an addi ional
mass balance o his ma e ial has o be o mula ed. The compo-
si ion o he bed a a gi en ime depends on he way o ope a ion.
A de ailed case by case analysis is ou o he scope o he p esen
ea men . In his wo k, a any ins an , he e a e h ee amoun s
o ma e ials in he bed: wc,wA, and wine .wbis he sum o wc
and wA(wb=wc+wA), whe eas wTb includes also he ine
(wTb =wc+wA+wine ), which is known by, o ins ance,
p essu e measu emen s (wTb =Pb(AT/g)). wine , i i exis s,
has o be calcula ed om an addi ional mass balance. The way
o include his addi ional balance in pa allel o he main p oblem
is ou lined in Sec ion 4.3.
3. De elopmen o he model
3.1. Modelling app oach
A ealis ic ep esen a ion o he bed should accoun o he
a ia ion in size and densi y o he eac ing ma e ial, as in he
ea men s by Chen and Saxena [14] and O e u [15]. The
model p oposed aims a simpli ying his gene al ea men . The
me hod is based on wo main s eps:
S ep 1: Applica ion o a luid-dynamic model o a ca -
aly ic sys em. This model is de i ed wi hou conside ing he
non-ca aly ic na u e o he eac ion and he ac ual bu n-o
dis ibu ion in he bed.
S ep 2: Allowance is made o he de ia ion om he ca aly ic
case, conside ing he ex en o con e sion in he FB by a solids
popula ion balance, which is sol ed by a kine ic model o a
single pa icle.
3.2. FB eac o modelling o CGSR
Wi h he assump ions discussed in Fig. 1a mola balances o
he gas in he bubble and emulsion phases lead o
βu0dcb=kbεb(ce−cb)dz(12)
(1 −β)u0(cin −ce)=L
0
kbεb(ce−cb)dz+ν c,b
McAT
(13)
The bounda y condi ions a e
cb(z=0) =cb,in =cin (14)
cou =βcb(z=L )+(1 −β)ce(15)
The gas con e sion Xgand he in e phase e ec i eness ac o
ηph a e de ined by
ηph =ce
cin n
and Xg=1−cou
cin
,(16)
In eg a ing Eqs. (12) and (13), aking Eq. (16) in o accoun ,
gi es [33]:
(1 −Xg/Na)n
Xg/Na=Na
DaR
(17)
whe e he pa ame e s DaRand Naa e de ined in Eqs. (22) and
(23). Combining Eqs. (16) and (17):
Xg=(1 −η1/n
ph )Na(18)
5

DaR=Xg
ηph
(19)
These exp essions can be combined o gi e
DaR
Na=Xg/Na
(1 −Xg/Na)n(20)
(1 −η1/n
ph )
ηph =DaR
Na
(21)
Eqs. (18)–(21) p o ide wo independen ela ionships o ou
quan i ies: DaR,Na,Xgand ηph.DaRis he Damk¨
ohle numbe a
eac o scale, exp essing he ela i e impo ance o gas esidence
ime and eac ion ime:
DaR=ηpDaR,in wi h DaR,in =kcn−1
in
u0/L
(22)
DaRis known o a CGSR i di usion e ec s a e absen , i.e.
ηp= 1 (kine ic egime) because DaR,in is known. He e, we
assume ha he bed heigh L is known o can be de e mined
by he p essu e d op ac oss he bed. In con as , DaR,in is no
known o a NCGSR e en when he pa icles a e in he kine ic
egime, because he concen a ion o he eac ing pa icles in he
bed is unknown. We shall deal wi h his ma e below. Nais he
concen a ion e iciency in he one-dimensional bed, de ined by
Na=cin −cou
cin −ce=1−βexp −NTU
β(23)
NTU is he numbe o ans e uni s and βis he dimensionless
excess gas low:
NTU =kbεb
u0/L
(24)
β=u0−um
u0
(25)
The exp ession o βassumes ha all gas in excess o min-
imum luidisa ion eloci y lows h ough he bed in he o m
o bubbles. This es s on he “ wo-phase heo y” o luidisa-
ion [12]. The e is e idence, howe e , ha he e is a sho -cu
low h ough he bubbles, especially in la ge pa icle sys ems.
This has been quan i ied in se e al models and co ela ions and
depends on he g oups o ub/um ,u0/um and εb. Analyses o he
h ough low in a ious wo-phase models ha e been e iewed in
[4,5,12,34]. The impac o h ough low on he p edic ion o gas
con e sion in simple and dynamic wo-phase low models was
assessed by Mos ou i e al. [11]. A co ec ion o h ough low
could be needed when using βin Eq. (23), especially o la ge
pa icle sys ems. Sensi i i y s udies employing he inal eac o
model a e help ul in iden i ying he need o u he e inemen .
In he cases ou lined in his wo k, he esul s ha e been ound
insensi i e o his pa ame e .
We conclude ha he in e phase e ec i eness ac o ηph is
only a unc ion o he g oup Na/DaR, and Xgis a unc ion o
he wo g oups Naand Na/DaR. The ela ionships needed o
calcula e NTU and βand ela ed pa ame e s depend on he low
pa e n and he pa icle sys em unde conside a ion. Examples o
o mulae use ul o bubbling luidisa ion in lab-scale FB can be
ound in Table 1 [33]. Fu he in o ma ion o o he luidisa ion
sys ems and scales is ound in [12,34].
Fo eac ion o de s o 1, 1/2 and 2 explici solu ions o ηph
a e ound in he li e a u e [2,35]. In a gene al case, o n h-
o de kine ics, explici solu ions o ηph as a unc ion o DaR
and Na, can be ob ained by F ank-Kamene skii’s app oxima ion
(see Appendix A) applied o Eq. (21):
ηph =⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
([{(1 −n)(DaR/Na)}1/n+1]n+n(DaR/Na))−1
wi h 0 <n≤1
2n[(2n)1/n−1+(1 +2n(DaR/Na))1/n]−n
wi h 1 <n≤2.7
(26)
Once ηph is known, he gas con e sion is de e mined by Eq.
(18) o (19).
3.3. Modelling o pa icle kine ics in CGSR
To calcula e he pa icle e ec i eness ac o ηpin he ca aly ic
case, we assume ha ηpdoes no depend on xco , mo e use ul
o he la e expansion o NCGSR, we conside ha xc=xc0. The
es ima ion o ηpis achie ed h ough an in e nal and an ex e nal
e ec i eness ac o , ηiand ηe, so ha ηp=ηiηe. The de ailed
de i a ion o he equa ions needed and he way o ob ain he
necessa y in o ma ion om expe imen s ha e been published in
[33]. The e ec i eness ac o s o iso he mal CGSR a e w i en
as
ηe=cs
cen
(27)
ηi=(−R)
kcn
s=1
Vpkcn
sVp
kcndVp(28)
Unde pseudo-s eady-s a e condi ions, he iso he mal mass-
ans e p oblem o n h-o de kine ics can be exp essed as
ηe=(1 −Dapeηp)n=(1 −Dapeηiηe)n(29)
whe e a second Damk¨
ohle numbe , Dape, ep esen s he a io
o he maximum di usion a e (when cs=ce) o he eac ion a e
con olled by ex e nal di usion (when cs∼0):
Dape =kLequcn
e
kGce
(30)
Taking in o accoun Eqs. (A.1), Eq. (29) can be app oxima ely
sol ed o ηe:
ηe=[({(1 −n)Dapeηi}1/n+1)−n+nDapeηi]−1,0<n<1
2n[(2n)1/n−1+(1 +2nDapeηi)1/n+1]−n,1<n<2.7
(31)
By sol ing he eac ion-di usion p oblem o a eac an pa icle,
an app oxima e solu ion o ηiis o en used:
ηi= anh(Ms)
Ms
(32)
6
Msbeing a con e sion-dependen Thiele module:
Ms=Lequn+1
2
kcn−1
s
De1/2
(33)
I is well know ha Eq. (32) is s ic ly alid o i s -o de
kine ics in a slab, bu ne e heless, he gene alised Thiele mod-
ule de ined in Eq. (33) makes he use o Eq. (32) a easonable
app oxima ion o any geome y (cha ac e ised by Lequ) and
eac ion o de , n[20,24,31,33,34].
In Eqs. (31) and (33) Msand Dape a e e alua ed o su ace
(index ‘s’) and emulsion (index ‘e’) condi ions, espec i ely. The
gas concen a ions in he emulsion and a he su ace di e om
he inle concen a ion, so hey a e no a p io i known. The e o e
Msand Dape should be ela ed o known quan i ies, ha is, hey
should be exp essed as unc ions o quan i ies e alua ed o gas
inle condi ions (index ‘in’). This is done by aking in o accoun
Eqs. (16) and (28), yielding
Dape =Dapin(ηph)(n−1)/n(34)
Ms=Min(ηphηe)(n−1)/2n(35)
He e ηph is calcula ed by Eq. (26), and ηeand ηiby Eqs. (31) and
(32) whe e Msand Dape a e calcula ed by Eqs. (34) and (35).
The e is a loop o ηph and he scheme o solu ion is i e a i e.
F om his ea men i is clea ha he dependence o ηpis in he
o m:
ηp= (ηph,M
in,Da
pin,n) (36)
The algo i hm o solu ion and he g aphical solu ion o his
sys em is p esen ed below in Sec ion 4. The ex ension o accoun
o he e ec o solid con e sion s a e is de eloped in Sec ion
3.5.
3.4. FB eac o modelling o NCGSR
In he model de eloped abo e o CGSR DaR,in was assumed
o be known, gi en by he ba ch o ca alys in he bed and he
p ope ies o he ca alys . In NCGSR, on he o he hand, DaR
is unknown, because nei he he amoun o solid eac an no i s
dis ibu ion o con e sion pb(xc) in he bed a e known. Conse-
quen ly, a solids popula ion balance should be o mula ed and
sol ed. The de ini ion o DaR o NCGSR is be e gi en in he
ollowing e ms:
DaR=∀xc∈VbwbR(xc)pb(xc)dxc
u0ATcinMc/ν = c,b
u0ATcinMc/ν (37)
Fig. 1b shows he main aspec s and he nomencla u e used.
The main condi ions we e al eady discussed in Sec ion 3.2.An
addi ional assump ion is ha all he ines a e e u ned o he
eac o : he e is no ca yo e , and all pa icles lea e wi h he
exi ash discha ge. Also, all pa icles a e assumed o en e wi h
he same con e sion xc0. The ea men ollows he p ocedu e
de eloped by [36]. The p esen app oach, howe e , exp esses
he equa ions in ex en o con e sion ins ead o ime o pa icle
size [37,10]. Following he nomencla u e o Fig. 1a, a popula ion
balance o e he eac o yields he dis ibu ion o he con e sion
o he solids (see Appendix B):
pb(xc)=1
Das
1
F(xc)
1−Yc0xc
1−Yc0xc0
exp −Θ(xc)
λ(38)
Θ(xc) being a unc ion
Θ(xc)=xc
xc0
ds
F(s)(39)
Exp essions o Θ(xc), associa ed wi h well-known kine ic mod-
els a e included in Table 1. The wo dimensionless pa ame e s
Dasand λin Eq. (38) a e de ined as
Das=K ,ewb
F0=K ,eτR(40)
λ=K ,ewb
F1=K ,ewb
F0− c,b
(41)
Dasis he Damk¨
ohle numbe o he solid eac an , exp essing
he a io o esidence ime o solids τR=wb/F0and eac ion
ime 1/K ,e. No e ha Das anges om 0 o 1 ( he Das= 1 case is
when he pa icles a e made up o C en i ely emaining in he bed
jus he ime hey need o eac comple ely). The ela ion be ween
Dasand λis ob ained h ough he no malisa ion equa ion [36]:
1
xc0
pb(s)ds=1 (42)
Eq. (42) is no sa is ied, howe e , wi h he dis ibu ion calcula ed
by Eq. (38) because pb(xc) should include bo h pa icles ha -
ing solid eac an le (C + A) and pa icles comple ely eac ed
ha s ill emain in he bed (only consis ing o A). In Eq.
(42) his second class o solids is no accoun ed o . Ca am
and Amundson [9,10] showed ha o an FB coal gasi ie
(C ≡ca bon + A ≡ash) an ash balance could sol e his appa en
di icul y (equi alen o and eplacing Eq. (42)). The ollowing
ea men uses he same app oach as ha in [10]. An ash (A)
balance o e he sys em yields
1−Yc0xc0
1−Yc0xc,b=F0
F1=Das
λ−1
(43)
The C-concen a ion in he bed can be ob ained by aking in o
accoun all pa icles ha ing a C-concen a ion in he bed, Yc(s)
gi en by Eq. (9). In eg a ion o e he bed using he dis ibu ion
in Eq. (38) gi es
Yc,b=1
xc0
Yc(s)pb(s)ds= 1(xc0,λ)
1/Yc0 −xc0
1
Das
(44)
whe e
1(xc0,λ)=1
xc0
1−s
F(s)exp −Θ(s)
λds(45)
Elimina ion o xc,b be ween Eqs. (43) and (44) yields
Das
λ= 1(xc0,λ)/λ+(1/Yc0 −1)
1/Yc0 −xc0
(46)
7
which is he equi alen o Eq. (42) and eplaces ha equa ion by
accoun ing o he pa icles ha ha e eached comple e con e -
sion and a e s ill in he bed.
An al e na i e equa ion (equi alen o Eq. (46)) is ound by
combina ion o Eqs. (38) and (7) and aking in o accoun Eqs.
(8),(39) and (41) o gi e
Das
λ=1
λ− 2(xc0,λ)Yc0
1−Yc0xc0
(47)
whe e
2(xc0,λ)=1
xc0
exp −Θ(s)
λds(48)
Once Das,xc0 and Yc0 a e gi en, Eq. (46) o (47) p o ide one
equa ion o λ(o Das/λ). The dis ibu ion pb(xc) can be hen
calcula ed by Eq. (38), and he a e age con e sion in he bed,
Eq. (10), is compu ed by
xc,b=1− 1(xc0,λ)
λ(49)
F om Eqs. (40) and (41) c,b becomes
c,b=1−Das
λF0(50)
The o e all mass balance on he solids and gas eac an
and he s oichiome y o he eac ion link he con e sions o
solids and gas. By equalling he a e o disappea ance o solids,
c,b/Mcwi h he a e o consump ion o he gaseous eac an ,
(cin −cou )u0AT/νone ob ains
Xg=1
α1−Das
λ(51)
whe e αis a dimensionless pa ame e de ined by he s oichio-
me ic a io o he eed a es o he eac an gas and he solids:
α=u0ATcinMc
νF0
(52)
Elimina ion o 1(xc0,λ) by combina ion o Eqs. (46) and (49)
enables o ela e xc,b and Das/λ:
xc,b=xc0 +1
Yc0 −xc01−Das
λ(53)
Eq. (53) es ablishes clea ly he bounda y limi s o Das/λwhich
a e ob ained o he limi ing cases xc,b = 0 and 1:
Das
λ∈1−Yc0
1−Yc0xc0
,1
1−Yc0xc0 (54)
Das/λnea ze o means comple e con e sion o solids, whe eas
Das/λclose o uni y s ands o he case o null solid con e sion.
In he pa icula case when all pa icles en e he bed wi h xc0 =0
and Yc0 =1, Das/λis equal o one minus he solid con e sion
ha is a ained in he bed, ha is 1 −xc,b. Thus, wi h xc0 =0
and Yc0 =1,Das/λ anges om 0 o 1. Elimina ion o Das/λby
combina ion o Eqs. (51) and (53) gi es a ela ion be ween Xg
and xc,b:
Xg=xc,b−xc0
α(1/Yc0 −xc0)(55)
The pa ame e Dasshould be e alua ed o emulsion condi ions,
i.e. Das=Das,e, bu he a io Das/λdoes no depend on he e -
e ence si ua ion o which he gas con e sion is e alua ed (see
Eqs. (40) and (41)). The known pa ame e is ac ually Das,in
(e alua ed o he inle condi ions), bu acco ding o Eq. (16)
Das(=Das,e) can be de i ed om Das,in:
Das=Das,inηph (56)
Hence, ηph has o be known o calcula e Das o emulsion con-
di ions. An exp ession o ηph esul s om Eqs. (18) and (51):
ηph =1−1
Naα1−Das
λn
(57)
Eqs. (51) and (53) allow he calcula ion o Xg= (Das/λ,α,
ηph) and xc,b = (Das/λ). Taking in o accoun Eq. (57) gi es
Xg= (Das/λ,α,Na) so he eac o beha iou is go e ned by h ee
pa ame e s: Das/λ,α, and Na.Nacomes om he luid dynamics
(Eq. (23)), whe eas αis ob ained om a ailable inpu s (see Eq.
(52)). The in en o y o he bed, wbis known, o ins ance, om
measu emen s o p essu e d op ac oss he bed (see Sec ion 4.3).
The g oup Das/λis ob ained om Eq. (46) o (47). To apply
hese equa ions, he unc ions 1(o 2) de ined in Eqs. (45) and
(48) equi e he alue o Θ(xc), de ined in Eq. (39). The e o e
F(xc) has o be in eg a ed o all he deg ees o con e sion in he
FB eac o . To unde ake his es ima ion, a kine ic model should
be es ablished in o de o ha e a ailable he exp essions Fi(xc)
and ηp(xc). To sum up: o he es ima ion o Das/λby Eq. (46) o
(47), a kine ic model should be o mula ed i s and hen sol ed
o he condi ions in he eac o .
3.5. Modelling o pa icle kine ics in NCGSR
Fo he non-ca aly ic case he eac ion a e o a pa icle de el-
oped in Sec ion 3.3 has o be expanded o include he e ec
o con e sion. This leads o he solu ion o a ime-dependen
p oblem wi h a mo ing in e ace wi hin a pa icle. The a e o
sh inkage/expansion o a pa icle’s ex e nal su ace is di icul
o gene alise because i depends on he na u e o he NCGSR.
Fo ins ance, o gasi ica ion eac ions, a h eshold o he local
con e sion has been ixed a he ins an when he ash laye o
pa icle peels o [38,39]. This h eshold condi ion allows heo-
e ical compu a ion o he pa icle’s bounda y a any ime. The
h eshold depends on ype o eac o , esis ance o ash, and ope -
a ing condi ions. In a FB, o example, he emo al o an ash
laye may be caused by a i ion. In con as , o eac ions whe e
a solid p oduc is o med, he ela ion be ween he mola ol-
umes o eac an and p oduc is usually employed, oge he wi h
some empi ical pa ame e , o de e mine he a e o change in ol-
ume (sh inkage o expansion) [17]. In gene al, empi ical inpu
is needed a some le el. Excep ions o his a e he well-known
uni o m con e sion model (UCM) and he sha p in e ace model
(SIM) as we shall see la e on, ep esen ing limi ing cases. In si -
8
ua ions whe e hese ex eme cases a e no alid, gene al, bu s ill
simple, models could be applied. In he ollowing such a model
will be o mula ed using Eq. (4) oge he wi h an es ima e o he
e ec i eness ac o o he pa icle ηp(xc).
As in he ca aly ic case, ηp(xc) is composed o an in e nal
and an ex e nal e ec i eness ac o , ηi(xc) and ηe(xc). These a e
de ined o NCGSR in he ollowing way:
ηe=cs
cen
=K ,s
K ,e
,η
i=dxc/d
Fi(xc0)K ,s
(58)
As seen hey depend on con e sion. Eqs. (32) and (33) s ill
apply. Howe e , Msand Dape gi en by Eqs. (34) and (35) ha e
o conside he e ec o con e sion, because k,Lequ,kGand De
depend on i . The change wi h con e sion can be ollowed by he
a ia ion in eac ion a e, Eq. (4), and he change in di usi i y o
he pa icle by g(xc), a unc ion o he local po osi y ε, and De0,
he ini ial e ec i e di usi i y. An empi ical equa ion o g(xc)
is usually accep ed o gas–solid eac ing sys ems [33,40,41]:
g(xc)=De(xc)
De0 =ε(xc)
ε0κ
=1+1−ε0
ε0xcκ
(59)
To es ima e he change o pa icle size wi h xc, an addi ional
ela ion is equi ed. The sh inkage o a pa icle du ing consump-
ion is no included in Fi(xc) ha only measu es he change o
he in e nal su ace. I a de ailed model (in eg a ion inside he
pa icle) is used, i is possible o es ablish he a e o sh inkage as
shown by S ini as and Amundson [38] and Mo ell e al. [39] o
he case o gasi ica ion o coal pa icles. Howe e , he e we ha e
o mula ed he model in e ms o he global pa icle con e sion
xc, and he change in size wi h pa icle con e sion canno be
calcula ed. Ne e heless, his in o ma ion can be p o ided by a
simple empi ical equa ion:
Lequ(xc)=Lequ,0(1 −xc)δ(60)
whe e a judicious choice o he pa ame e δgi es Lequ o any xc.
Only in he limi ing kine ic models, such as UCM and SIM, he
assump ion o an a bi a y alue o δis no necessa y: UCM
implies a cons an pa icle size (δ= 0) and in SIM, δ= 1/3. In
be ween hese wo limi ing si ua ions, he mo e gene al p o-
g essi e con e sion model wi h changes in size and densi y can
be applied by p ope ly choosing a alue o δin he ange o
0–1/3.
Now, wi h he de ini ions gi en in Eqs. (30) and (33), and wi h
Eqs. (4),(59) and (60), he ini ial alue Dapin,0 can be ela ed o
Dapin(xc) and Min,0 o Min(xc)as
Dapin(xc)=Dapin,0[Fi(xc)(1 −xc)(3/2)δ] (61)
Min(xc)=Min,0Fi(xc)
g(xc)1/2
(1 −xc)δ(62)
whe e a co ela ion o he ex e nal di usion coe icien kGo
he ype o Sh ∝Re1/2
pwas used o de i e Eq. (61) [43]. By Eqs.
(31),(33),(61) and (62) he desi ed ela ionships a e de e mined:
Dape(xc)=Dapin,0[(ηph)(n−1)/n(Fi(xc)(1 −xc)(3/2)δ)] (63)
Ms(xc)=Min,0(ηphηe)(n−1)/2nFi(xc)
g(xc)1/2
(1 −xc)δ
(64)
In conclusion, Eqs. (31) and (33) allow calcula ion o ηe
and ηiand so ηp. The quan i ies Msand Dape, appea ing in
hese equa ions, a e calcula ed by Eqs. (63) and (64). F om
his ea men i is clea ha he dependence o ηpis in he
o m
ηp= (ηph,M
in,0,Da
pin,0,F
i(xc),g(xc),δ) (65)
being he NCGSR e sion o Eq. (36).
4. Discussion
4.1. Solu ion o CGSR
The explici solu ion o ηph ound in Eq. (26), i.e.
ηph = (DaR/Na,ηp,n), is displayed in Fig. 2 by solid lines.
I can be demons a ed ha his solu ion includes as pa icu-
la cases published analy ical exp essions, such as epo ed by
[2] o he modi ied O cu model o i e e sible eac ions wi h
n= 1/2, 1 and 2 (symbols in Fig. 2). The di e ence be ween
he solid lines and he symbols is e y small and en i ely asso-
cia ed wi h F ank-Kamene skii’s app oxima ion used o de i e
Eq. (26) om Eq. (21). In ac , Eq. (20) is equi alen o he
solu ions o O cu ’s model, bu he p esen o mula ion p o-
ides an addi ional scheme o simple es ima ion o di usion
limi a ions a he pa icle scale and o expanding his scheme
o NCGSR. The simples case (n= 1) allows a s aigh o -
wa d physical in e p e a ion o he solu ion. In his case he
Fig. 2. Solu ion o gas–solid ca aly ic eac ions in an FB d awn o he in e -
phasic e ec i eness ac o , ηph as a unc ion o DaR,inηp/Na o a ious eac ion
o de s, n(be ween 0.25 and 2), acco ding o Eq. (26). (Solid lines ep esen he
p esen model, whe eas symbol lines a e esul s ob ained om O cu ’s model.)
The pa icle e ec i eness ac o ηpis unknown, so his igu e mus be used in
pa allel wi h Fig. 3 o de e mine i e a i ely ηpand ηph.
9
ion is o en made. I he mal e ec s should be conside ed o no
depends la gely on he ype o eac ion, bu also, on he ope -
a ion condi ions, he la e making i e y di icul o es ablish
gene al guidelines o he applicabili y o he model. Assess-
men o he p esence o hese po en ial he mal g adien s p io
o applica ion o he model is, he e o e, ecommended. This
can be made by es ima ion o he he mal Bio numbe and he
maximum he mal g adien be ween he emulsion and pa icle
om a hea balance o e a eac ing pa icle.
Non-iso he mal analysis a a pa icle scale, in gene al,
equi es wo u he pa ame e s o accoun o he he mal sen-
si i i y o he chemical eac ions ( he A henius pa ame e ) and
o quan i y he he mal e ec s ela i e o he hea conduc ion
( he P a e numbe ). I , in addi ion, he di e ence be ween he
phases is impo an , u he pa ame e s ha e o be conside ed
by o mula ing an ene gy balance o e he eac o . To accoun
o all hese phenomena in a gene alised o mula ion such as
he one p esen ed, complica es he p esen a ion and makes less
meaning ul a comp ehensi e analysis o go e ning pa ame e s.
When mo e han one eac ion occu s, o when he gas eac-
an and he p oduc gas species u he combine homogeneously
o /and eac wi h o he compounds in he gas mix u e, some
ex ensions ha e o be made. An example is gi en o illus-
a e how easonable simpli ica ions can lead o applica ion o
he me hod. Le us conside , o ins ance, simul aneous CO2
and H2O gasi ica ion o cha gene a ed a e de ola ilisa ion o
biomass. The ela i e amoun o H2O and CO2depends on p e-
ious d ying, de ola ilisa ion, and combus ion p ocesses. These
p ocesses occu a much highe a e han he gasi ica ion o he
cha , and so hey can be calcula ed uncoupled o cha gasi ica-
ion. Cha educ ion p ocesses can be simpli ied by conside ing
ha H2O and CO2and H2and CO a e lumped in o he same
pseudo-componen s, R and P, espec i ely. As a esul , he only
he e ogeneous eac ion o be conside ed is cha + R →P. This
eac ion is assumed o occu in he emulsion phase whe e mos
cha pa icles a e ound. This scheme is jus i ied by he simila
s oichiome y o cha -CO2and cha -H2O and by he commonly
assumed equilib ium o he WGSR (wa e –gas shi eac ion).
The ela i e concen a ions o H2O and CO2a e adjus ed o
he local he mal en i onmen a ound he pa icles, because he
WGSR is mo e apid. The me hod o his wo k has been applied
o such a case [44], gi ing close ag eemen wi h esul s om
ad anced models, sho ening conside able he compu a ions,
and mos impo an ly, educing he inpu da a needed o he
calcula ions.
In he me hod wb(=wc+wA) has been conside ed known.
Howe e , when he e is ine ma e ial in he bed, wine , his can
be calcula ed because a p essu e measu emen p o ides he o al
amoun o bed, wTb and hence, also he amoun o ine ma e ial
(wTb =wb+wine ). wine , can be calcula ed once he manne
o ope a ion o he FB sys em is speci ied. As an example, le
us conside an FB ope a ing wi h a cons an solids in en o y: a
con inuous d ainage o bed ma e ial om he sys em is made,
and so, o keep he bed ma e ial cons an , a con inuous make-
up o ine ma e ial is needed. A s eady s a e, a simple mass
balance o e he ine ma e ial yields an addi ional equa ion o
wine . In gene al, he composi ion o he bed has o be known o
he calcula ion o wine , so his equa ion is coupled o he main
p oblem, because o sol e he new equa ion, he composi ion o
he bed has o be known, i.e. Yc,b. The e o e, Yc,b is assumed,
wine is calcula ed, and hen, also wb. I he new Yc,b di e s om
he assumed alue, a new wine is es ima ed and he calcula ion
is epea ed un il con e gence. To sum up, he conside a ion o
ine ma e ial in oduces a second loop, which has o be sol ed
in pa allel wi h he main p oblem. In p ac ise, wo i e a ions a e
o en enough o a ain he solu ion because he main p oblem is
no sensi i e o his loop.
5. Summa y and conclusions
A me hodology is p oposed o e alua ion o gene al
gas–solid eac ions in iso he mal FB. A model is de eloped in
wo s ages. Fi s , a me hod o e alua ion o gas con e sion is o -
mula ed by applying he wo-phase heo y o luidisa ion on FB
ca aly ic eac o s, in which only gas con e sion is conside ed.
A condensed o mula ion is gi en o calcula e gas con e sion
as a unc ion o he go e ning pa ame e s. In a second s age,
he model is ex ended o accoun o non-ca aly ic eac ions by
inco po a ing a ia ion o pa icle p ope ies and eac ion a e
wi h con e sion, as well as he dis ibu ion o he con e sion o
eac ing pa icles in he bed. Th ee g oups go e n he solid and
gas con e sion in he eac o : (1) he a io o eac an gas and
solid eed low a es, α; (2) he concen a ion e iciency in he
en i e bed, Na; and (3) Das/λ, being an indica ion o he solid
con e sion. The g oup Das/λis ob ained om a popula ion bal-
ance aking in o accoun he o e all con ibu ion o all eac ing
pa icles in he bed. A simpli ied kine ic model o a single pa i-
cle is de eloped o cha ac e ise he go e ning pa ame e s a he
pa icle scale. Besides he in insic kine ics, wo pa ame e s a e
iden i ied, quan i ying he di usion e ec s a he pa icle scale:
a gene alised Thiele module Min,0, and a Damk¨
ohle numbe a
a pa icle scale Dapin,0, bo h aking ze o con e sion and gas inle
con e sion as e e ence s a es making hese pa ame e s known
quan i ies. Simpli ica ion is possible o limi ing alues o he
h ee p incipal eac o pa ame e s (α,Na, and Das/λ). The sim-
ples case o NCGSR neglec s he ole o he dis ibu ion o
con e sion in he bed and allows ob aining a apid solu ion o
anykine ics.Examples con i med hegood esul s o he me hod.
Mo eo e , he selec ion o examples allowed iden i ica ion o
he limi ing solu ion de i ed. Discussion is also included on he
applica ion o he model o indus ial FB p ocesses, ocussing
on he unde s anding o he model limi a ions in o de o p o-
ide guidelines o ex ensions. This wo k complemen s exis ing
gene alised FB eac o models o ca aly ic gas–solid eac ions,
he e o e, allowing simila gene alised analysis o non-ca aly ic
gas–solid eac ions.
Acknowledgmen s
The au ho s acknowledge he Eu opean Commission, he
Commission o Science and Technology o Spain, and Jun a
de Andalusia o hei inancial suppo . The i s au ho
acknowledges he The mal and Fluid Enginee ing Depa men
a Uni e si y Ca los III (Mad id) o kindly in i ed him o lec u e
16

pa o he ma e ial o his a icle. Discussions om he lec u ing
sessions made, hope ully, he ac ual e sion o he pape mo e
comp ehensi e and accessible.
Appendix A. F ank-Kamene skii’s app oxima ion
F ank-Kamene skii [45] p oposed an explici solu ion o he
a iable yin a gene al equa ion o he o m
y−(1 −μy)n=0 wi h 0 <n<1 (A.1)
whe e μis a cons an o any alue in he in e al (0–2.7). This
solu ion is
y=⎧
⎪
⎪
⎨
⎪
⎪
⎩
min 1,1
μwi h n=0
[({(1 −n)μ}1/n+1)n+nμ]−1wi h 0 <n<1
2n[(2n)1/n−1+(1 +2nμ)1/n +1]−nwi h 1 <n<2.7
(A.2)
Eqs. (21) and (29) can be exp essed in he o m o Eq. (A.1)
whe e yis ηph o ηe, and he co esponding alues o μa e
DaR/Naand DaRηi. This leads o explici solu ions gi en by
Eqs. (26) and (31) o ηph and ηe, espec i ely.
Appendix B. Fo mula ion o he popula ion balance:
de i a ion o Eq. (38)
Fo bed mass and low a e o solids, as shown in Fig. 1b, an
o e all s eady-s a e mass balance in he bed gi es
F0= c,b+F1(B.1)
c,b is de ined in Eq. (7). Now, by making a balance on con e sion
o he pa icles be ween xcand xc+dxcwe ha e
wbK
d(F(xc)pb(xc))
dxc=F0p0(xc)−F1p1(xc)
−wbR(xc)pb(xc) (B.2)
In a well mixed bed pb(xc)=p1(xc). Sol ing o he dis ibu-
ion o con e sion in he bed pb(xc) yields [10]:
dpb(xc)
dxc+dlnF(xc)
dxc+1
λF(xc)+1
1/Yc0 −xcpb(xc)
=1
Das
p0(xc)
F(xc)(B.3)
whe e Dasand λa e dimensionless pa ame e s de ined in Eqs.
(40) and (41). Eq. (B.3) is in eg a ed using he condi ion pb(0)=0
o ob ain
pb(xc)=1
Das
1/Yc0 −xc
F(xc)xc
0exp −Θ(s)
λp0(s)
1/Yc0 −sds
(B.4)
Assuming ha all he pa icles en e wi h he same con e sion
xc0, he eed dis ibu ion is p0(xc)=δ(xc−xc0) and Eq. (B.4) is
simpli ied o gi e Eq. (38).
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18