Gas–solid conversion in fluidised bed reactors

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). Re e ences [1] W.G. May, Fluidized bed eac o s s udies, Chem. Eng. P og. 55 (1959) 49–56. [2] J.R. G ace, Fluid beds as chemical eac o s, in: D. Gelda (Ed.), Gas Fluidiza ion Technology, John Wiley & Sons L d., 1986, pp. 285–339. [3] J.C. O cu , J.F. Da idson, R.L. Pig o d, Reac ion ime dis ibu ions in luidized bed eac o s, Chem. Eng. P og. Symp. Se . 59 (1962) 1–15. [4] J.F. Da idson, D. Ha ison, Fluidised Pa icles, Camb idge Uni e si y P ess, London, 1963. [5] M. Ho io, C.Y. Wen, An assessmen o luidized-bed modelling, AIChE Symp. Se . 73 (1977) 9–21. [6] W.P.M. an Swaaij, Chemical eac o s, in: J.F. Da idson, R. Cli , D. Ha - ison (Eds.), Fluidiza ion, 2nd edi ion, Academic P ess, London, 1985, pp. 595–629, Chap e 18. [7] J.F. Da idson, D. Ha ison, R.C. Da on, R.D. LaNauze, The wo-phase he- o y o luidiza ion and i s applica ion o chemical eac o s, in: L. Lapidus, N.R. Amundson (Eds.), Chemical Reac o Theo y, A Re iew, P en ice- Hall, Englewood Cli s, NJ, 1977, pp. 583–685. [8] C. Cha a ie, J.R. G ace, Pe o mance analysis o a luidized bed eac o . II. Obse ed eac o beha io compa ed wi h simple wo-phase models, Ind. Eng. Chem. Fund. 14 (1975) 79–86. [9] H.S. Ca am, N.R. Amundson, Fluidized bed gasi ica ion eac o modeling. 1. Model desc ip ion and nume ical esul s o a single bed, Ind. Eng. Chem. P oc. Des. De . 18 (1978) 80–96. [10] H.S. Ca am, N.R. Amundson, Fluidized bed gasi ica ion eac o model- ing. 2. E ec o he esidence ime dis ibu ion and mixing o pa icles. S aged bed modelling, Ind. Eng. Chem. P oc. Des. De . 18 (1978) 96– 102. [11] N. Mos ou i, H.P. Cui, J. Chaouki, A compa ison o wo- and single-phase models o luidized bed eac o s, 2001, Ind. Eng. Chem. Res. 40 (2001) 5526–5532. [12] D. Kunii, O. Le enspiel, Fluidiza ion Enginee ing, 2nd edi ion, Bu e wo h-Heinemann, London, 1991. [13] R.K. I ani, B.D. Kulka ni, L.K. Do aiswamy, Analysis o complex eac ion schemes in a luidized bed. Applica ion o he Kunii-Le enspiel model, Ind. Eng. Chem. P oc. Des. De . 19 (1980) 24–30. [14] T.P. Chen, S.C. Saxena, Ma hema ical modelling o coal combus ion in luidized beds wi h sulphu -emission con ol by limes one o dolomi e, Fuel 56 (1977) 401–413. [15] B.W. O e u , Towa d a gene al desc ip ion o gas–solid eac ions in a luidised bed eac o , Ph.D. Disse a ion, Uni e si y o Pu due, 1980. [16] J.G. Ya es, Fundamen als o Fluidized-bed Chemical P ocesses, Bu e - wo hs, London, 1983. [17] L.K. Do aiswamy, M.M. Sha ma, He e ogeneous Reac ions. Analysis, Examples and Reac o Design, ol. 1, Wiley, New Yo k, 1984. [18] A.B.M. Heesink, J. Klaus, W.P.M. Van Swaaij, In luence o pa icle esi- dence ime dis ibu ion on he eac i i y in luidized bed eac o s, Chem. Eng. Sci. 49 (1994) 2243–2261. [19] L. Liliedahl, K. Sj¨ os ¨ om, Modelling o cha -gas eac ion kine ics, Fuel 76 (1997) 29–37. [20] S. Du a, C.Y. Wen, R.J. Bel , Reac i i y o coal and cha . 1. In ca bon dioxide a mosphe e, Ind. Eng. Chem. P oc. Des. De . 16 (1977) 20–30. [21] H. Huang, A.P. Wa kinson, Coal gasi ica ion in a s i ed bed eac o , Fuel 75 (1997) 1617–1674. [22] T. Adschi i, T. Shi aha, T. Kojima, T. Fu usawa, P edic ion o CO2gasi i- ca ion a e o cha in luidized-bed gasi ie , Fuel 65 (1986) 1688–1693. [23] J. Szekely, J.W. E ans, H.Y. Sohn, Gas–Solid Reac ions, Academic P ess, New Yo k, 1976. [24] F.G. an den Aa sen, Fluidised bed wood gasi ie . Pe o mance and mod- elling, Ph.D. Disse a ion, Twen e Uni e si y Publica ion, Uni e si y o Twen e, 1985. [25] S.K. Bha ia, D.D. Pe lmu e , A andom po e model o luid-solid eac- ions. I. Iso he mal, kine ic con ol, AIChE J. 26 (1980) 379–386. [26] G.A. Simons, The uni ied coal-cha eac ion, Fuel 59 (1980) 143–144. [27] J.L. Johnson, Kine ics o Coal Gasi ica ion, Wiley, New Yo k, 1979. [28] N.C. Ga dne , E. Samuels, K. Wilks, Ca alysed hyd ogasi ica ion o coal cha s, Ad . Chem. Se . 131 (1974) 217–236. 17 [29] E. Cho ne , J.M. Baldo oso, H.T. Ta ki, Kine ic exp essions o coal cha - gas eac ions, Fuel 58 (1979) 395–397. [30] S. Kasaoka, Y. Saka a, C. Tong, Kine ic e alua ion o he eac i i y o a ious coal cha s o gasi ica ion wi h ca bon dioxide in compa ison wi h s eam, In . Chem. Eng. 25 (1985) 160–175. [31] N.M. Lau endeau, He e ogeneous kine ics o coal cha gasi ica ion and combus ion, P og. Ene gy Combus . Sci. 4 (1978) 221–270. [32] A. G´ omez-Ba ea, P. Olle o, R. A jona, Reac ion-di usion model o TGA gasi ica ion expe imen s o es ima ing di usional e ec s, Fuel 84 (2005) 1695–1704. [33] A. G´ omez-Ba ea, P. Olle o, B. Leckne , Mass anspo e ec s du ing mea- su emen s o gas– solid eac ion kine ics in a luidised bed, Chem. Eng. Sci. 62 (2007) 1477–1493. [34] S.O. Oka, Fluidised Bed Combus ion, Ma cel Dekke Inc., New Yo k, 2004. [35] G.F. F omen , K.B. Bischo , Chemical Reac o Analysis and Design, 2nd edi ion, Wiley, New Yo k, 1990. [36] O. Le enspiel, D. Kunii, T. Fi zge ald, The p ocessing o solids o changing size in bubbling luidized beds, Powde Technol. 2 (1968) 87–96. [37] G. Ma b´ an, A.B. Fue es, J.J. Pis, Calcula ion o bu no dis ibu ions o cha pa icles in a bubbling luidized bed eac o by means o ba ch combus ion expe imen s, Chem. Eng. Res. Des. 76 (1998) 864–872. [38] B. S ini as, N.R. Amundson, A single-pa icle cha gasi ica ion model, AIChE J. 26 (1980) 487–496. [39] J.I. Mo ell, N.R. Amundson, S.K. Pa k, Dynamics o a single pa icle du ing cha gasi ica ion, Chem. Eng. Sci. 45 (1990) 387–401. [40] C.Y. Wen, Nonca aly ic he e ogeneous solid luid eac ions, Ind. Eng. Chem. 60 (1968) 34–54. [41] G. B em, J.J.H. B ouwe s, Analy ical solu ions o non-linea con e sion o a po ous solid pa icle in a gas. I. Iso he mal con e sion, Chem. Eng. Sci. 45 (1990) 1905–1913. [42] A. Gomez-Ba ea, B. Leckne , P. Olle o, C. Fe nandez-Baco, L. Sal ado , Reduc ion o physical e ec s du ing eac i i y es s in luidized bed, Ind. Eng. Chem. Res. 45 (2006) 7344–7350. [43] A. G´ omez-Ba ea, Modelling o di usional e ec s du ing gasi ica ion o biomass cha pa icles in luidised-bed, Ph.D. Disse a ion, Uni e si y o Se ille, Spain, 2006. [44] A. G´ omez-Ba ea, B. Leckne , M. Campoy, Con e sion o cha in CFB gasi- ie s, in: E.-U. Ha ge (Ed.), P oceedings o 9 h In e na ional Con e ence on Ci cula ing Fluidized Beds, Hambu g, Ge many, 2008. [45] D.A. F ank-Kamene skii, Di usion and Hea Exchange in Chemical Kine - ics, P ince on Uni e si y P ess, P ince on, 1955. 18