Unsteady-State Modeling of the Fluidized Bed Polyethylene Reactor

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1 Iranan Journal of hmcal Engnrng ol. 6, o. (Wntr, 009, IhE sa rch not nstay-stat Molng of th Fluz B Polythyln actor. assm,. Mostouf,. Sotuh-Gharagh Procss Dsgn an Smulaton sarch ntr, Dpartmnt of hmcal Engnrng, nvrsty of hran, Iran. stract mathmatcal mol s vlop for scrng th ynamc havor of th gas phas thyln ymrzaton ractor. h mol s as on th ynamc two-phas concpt of fluzaton n whch th uls may contan sol partcls an th mulson s capal of contanng mor gas than that of mnmum fluzaton. h fluz ractor s v nto svral sral sctons consstng of ul an mulson phass. Flow of th gas s consr as plug flow through th uls an prfctly mx through th mulson phas. Polymrzaton ractons occur n oth mulson an ul phass. araton of th procss varals as wll as th ymr proprts wr stu as a functon of opratng tm. h hght was controll y th prouct wthrawal rat wth a PID controllr. h rsults of th mol wr compar wth th xprmntal ata an a goo agrmnt was osrv twn th mol prcton an actual ata. h smulaton rsults ncat that a sgnfcant amount of ymr proucton (roughly % taks plac n th uls. Kywors: Fluzaton, Polythyln, Dynamc smulaton,gas-phas ymrzaton. Introucton Gas-phas catalytc fluz- ractors hav n wly us as on of th man mthos for proucng yolfns. Proucton of lnar low nsty ythyln (LLDPE n a gas-phas ractor usng Zglr-atta catalyst s on of such procsss. nrstanng th ynamc havor of ths ractor s crtcal, spcally n th start-up an shut-own of th ractor as wll as n th gra transton procss. hr ar varous mols for th ynamc havor of th ractor an ymr proprts. ho an ay [] wr th frst to consr oth mulson an ul phass n molng th ythyln fluz ractor. hy assum constant ul sz along th an controll th ractor tmpratur wth a PID controllr unr fack control. Mculy t al. [] consr th ymrzaton ractor as a wll-mx ractor to vlop th kntc mol. Mculy t al. [3] compar th twophas an wll mx mols. hy show that th smpl wll mx mol os not ntrouc sgnfcant rror as compar to that of a two-phas mol. Frnans an Lona [,5] mol th orrsponng author: E-mal: mostouf@ut.ac.r 3

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3 assm, Mostouf, Sotuh-Gharagh S j Y (0, j kp =, ( S [ M ] Y (0, j kfh ( j whr S s th numr of typs of actv sts. h kntc rat constants wr thos propos y Kashmshak t al. [9]. h ymr proucton rat p coul thn calculat from: p MW MW (3 n whch an ar th nstantanous rat of racton for monomr an hyrogn otan from Eq. ( an (, rspctvly. al. Elmntary ractons of thyln ymrzaton systm acton Dscrpton * ( j kf ( j (0, j Formaton racton wth co-catalyst ( 0, j M k ( j (, j Intaton racton wth monomrs kp k (r, j M (j (r, j Propagaton ( r, j M kfm k ( j k (, j Q ( r, j ransfr to monomr kfh ( j ( r, j (0, j Q( r, j kh ( j (0, j M (, j khr ( j ( 0, j let3 (, j kfr ( j ( r, j let3 (, j Q( r, j ransfr to hyrogn ransfr to co-catalyst ( r, j kfs ( j (0, j Q( r, j Spontanous transfr ( r, j (0, j (0, j ks ( j ks ( j ks ( j ( j ( j ( j Q ( r, j Dactvaton ractons Iranan Journal of hmcal Engnrng, ol. 6, o. 5

4 naroc Boconvrson of avy yrocarons sng atv onsorta Y(0, j t Y(, j t al. Momnt quatons [M ] k ( j(0, j kh ( j (0, j kh ( j (0, j[let ] r 3 Y(0, j kfh ( j[ ] kfs ( j ks( j r 3 P [M ] k (j(0, j kh (j (0, j kh (j (0, j[let ] [M ]k (jy(0, j Y(0, j Y(, j v p Y(, j t kfm (j[m ] kfr (j[let ] Y(, j kfh (j[ ] kfs (j ks(j 3 [M ] k ( j(0, j kh ( j (0, j kh ( j (0, j[let ] r 3 v p [M ]kp ( j Y(, j Y(0, j Y(0, j Y(, j kfm ( j[m ] kfr ( j[let ] 3 X(n, j t Y(, j kfh ( j[ ] kfs ( j ks( j p Y(n, j (, j kfm ( j[m ] kfr ( j[let ] kfh ( j[ ] kfs ( j ks( j 3 v X(n, j n 0,, v p.3. yroynamcs Most rsarchs hav mploy th smpl two-phas concpt for molng th gasphas thyln ymrzaton (.g., [3, 5, 6]. In th smpl two-phas thory, t s assum that all th gas n xcss of that rqur for mnmum fluzaton passs through th as sol fr uls whl th mulson stays at mnmum fluzaton contons. Exstnc of sol partcls n uls has n shown oth xprmntally [, 3] an thortcally [, 5]. h mulson phas also os not rman at th mnmum fluzaton conton an may contan a hghr amount of gas at hghr gas vlocty [3, 6]. Mostouf t al. [7] show that assumng th smpl two-phas structur of th fluz s woul rsult n unrprctng th prformanc of th ractor. hy conclu that such an ovrsmplfcaton of th flow structur of gas an sols n th fluz ractors coul qut mslang n th prcton of th prformanc of such ractors. Bas on ths facts, th concntraton of partcls n oth mulson an ul phass hav n stmat n th prsnt stuy from th ynamc two-phas concpt of fluzaton propos y u t al. []. ccorng to th ynamc two-phas concpt of fluzaton th uls may contan sol partcls. In a ral fluz, th concntraton of partcls n th mulson phas can lss than that at th mnmum fluzaton, an th uls can contan varous amounts of partcls [, 8]. optng th concpts of th ynamc twophas hyroynamc structur [], Jafar t 6 Iranan Journal of hmcal Engnrng, ol. 6, o.

5 assm, Mostouf, Sotuh-Gharagh al. [0] propos a squntal moular approach to molng fluz ractors. ccorng to thr procur, a fluz ractor shoul v nto a numr of sral sctons n whch ach scton conssts of ul an mulson phass. h flow of gas s consr as plug flow through th uls an s prfctly mx through th mulson phas n ach scton. umr of sctons s trmn from al 3 aftr valuatng th mnsonlss numr J (whch nclus oth kntc an hyroynamc paramtrs fn as: J a 0 mf ( that th numr of sctons n ths cas shoul qual to four an show that thr s a goo agrmnt twn th mol an actual ythyln plant ata. hrfor, th hyroynamc mol of Kashmshak t al. [9] has n mploy n th prsnt stuy. h schmatc of such an arrangmnt s llustrat n Fg.. Equatons us n ths hyroynamc mol ar also gvn n al. al 3. umr of sgmnts [0] J. < J ata numr n ths racton systm coul trmn from: a D g k Pk ( j ( r, j K 0.5 (5 5.6< J <. 0.63< J <5.6 J < h accuracy of Jafar t al. s mol [0] has n confrm y th xprmntal ata rport n th ltratur ovr a w rang of suprfcal gas vlocty. Kashmshak t al. [9] aopt Jafar s mol. [0] an vlop t for a ythyln fluz ractor. hy foun.. Mass alanc quatons Bas on th aov hyroynamc sumol, ynamc molar alancs for th two monomrs (.., thyln an -utn an hyrogn n th nth scton of th ar gvn y th followng quatons: Scton : Bul: [M ] [M ] F [M ] K ([M ] [M ],(n,( out,(,(,(,(,( PLG,( z( z( ( z [M ] t,(,( p,(,(,(,( (6 Emulson: [ M ([ M ],( n ],( [ M ] [ M,( ],( ( F out,( [ M ],( t,( [ M ( p,(,(,(,( K ] ( ( (7 Iranan Journal of hmcal Engnrng, ol. 6, o. 7

6 naroc Boconvrson of avy yrocarons sng atv onsorta al. orrlatons an quatons us n th hyroynamc mol Paramtr Formula frnc Mnmum fluzaton vlocty / mf ( r 9. 5 [9] Bul vlocty 0 ur [0] Bul rs vlocty Emulson vlocty / u r 0.7( g [0] 0 ( n / 3 Bul amtr 0 7( 0 ( 6.8h 0= for Glart B [] Mass transfr coffcnt K K K c c K K.5 c 6.77 D c 5.85 g,( n r u D / g / g 5 / / [] at transfr coffcnt Bul phas fracton Emulson phas porosty Bul phas porosty olum of th mulson phas n ach scton olum of th ul phas n ach scton olum of PF n ach scton olum of ymr n th mulson phas n ach scton olum of ymr n th ul phas n ach scton c c ( n,( n,( n,( n,( n c g ( g 0.53 mf 0. c pg pg k xp 0.6 xp h( ( n h ( n PLG p,( n p,( n g ( k 5.85 / xp,( n,( n,( n g 3 g,( n r h(,( n ( ( n h (,( n ( n u mf 0 mf pg 5/ 0.9 / mf / g / [3] [] [] [] 8 Iranan Journal of hmcal Engnrng, ol. 6, o.

7 assm, Mostouf, Sotuh-Gharagh Sctons, 3, : Bul: [M ] [M ] K ([M ] [M ] ( z z (n,(n,(n,(n,(n,(n,(n,(n p,(n z(n PLG,(n [M ] t,(n,(n,(n (8 Emulson: (,( n,( n t,( n ( n p,( n,( n,( n,( n K ( n ( n (,( n,( n (9 In th aov quatons, th rcton of mass transfr s assum to from ul to mulson phas. It s worth notng that snc th ovrall convrson n th ythyln ractor s low (an n th uls s vn lowr, th concntraton of th monomrs n th ul phas coul consr to constant (or avrag n ach sgmnt whn calculatng th accumulaton trms. h ovrall mass alanc (quaton of contnuty of th ymr n th, consrng constant nsty an porosty, s: ( av F pr F out ( av h( whr av shoul valuat from: av av p h t (0 ( (.5. Enrgy alanc quatons h nrgy alanc quatons ar: Scton : Bul: n ( F t out ( (,( n p,( n,(,( PLG,(,( ( (,(,( rf rf,( p,,(,( rf p ( z ( z (,( p,(,( n,( rf p,(,( z p ( ( z ( z (,(,(,(,( ( rf p,(,( z p. p p,,( ( Iranan Journal of hmcal Engnrng, ol. 6, o. 9

8 naroc Boconvrson of avy yrocarons sng atv onsorta Emulson: ( n F out ( (,( n p,( n,( rf,( rf p, p (,(,(,( n p rf,( ( (,(,( rf p p,,( (3 (,( p,(,( p,(,( ( ( (,(,( t,( (,( rf,( p,( (,( p, Sctons, 3, : Bul: ( [M ] ( [M ],(n rf p,(n,(n rf p,(n n ( ( ( ( p,( n,(n p,,(n rf p,( n,(n p,,(n rf n z z(n z(n z(n,( n p,( n,( n,( n z(n p PLG,(n z ( (,(n,(n,(n,(n (,(n rf,(n p [M ],(n (,(n p, t Emulson: ( n t,( n ( p,( n,( n ( (,( n,( n,( n rf p,( n rf p p, (,( n,( n,( n,( n p,( n p rf ( n,( n,( n (,( n ( p,( n rf,( n ( n,( n ( n p p, (,( n ( p,,( n,( n,( n rf (5 30 Iranan Journal of hmcal Engnrng, ol. 6, o.

9 assm, Mostouf, Sotuh-Gharagh ot that th ovrall rcton of movmnt of ymr partcls n th s th oppost of that for th gasous stram (s Fg.. GS O Pr-Polymr Pr-Polymr S K PF S3 K PF3 S K PF POLYME S K PF POLYME GS FEED Fgur. Schmatc agram of th molng structur of fluz Onc th nrgy alanc quatons ar solv, th tmpratur of th unract gass xt from th top of th ractor coul valuat from: p ( p av (6 ( p p whr p an p ar th molar avrag spcfc hat capacty of th gas n ul an mulson phass n th outlt gasous stram from th top of th ractor, rspctvly..6. Lvl controllr Lvl of th sols n th s controll y th volumtrc rat of th prouct wthrawn Iranan Journal of hmcal Engnrng, ol. 6, o. 3

10 ! $ % # & ' % % $( $ ##* #####$ $( +!!,! +! -$. / $. 0,,$, 0 0,$,$,,$,$ -,,... 0 $*, $,, $ 0, 0,,, $,, $ + +! $ $,, 3 $, $, 3 # $! &% $*!,!,! $ + 5 +!! 6 78!9+9! + & * $$! : #( ' $( # $ ; $$ %! < # $= < $( >! < <

11 assm, Mostouf, Sotuh-Gharagh comparson twn rsults of th smulaton an th actual plant ata for a sampl gra of LLDPE s gvn n al 7. s shown n th tal, thr s a goo agrmnt twn opratng contons (tmpratur an proucton rat prct y th mol an th ral plant ata. ctual wght avrag molcular wghts rport n al 7 wr stmat as on th masur mlt flow nx of th prouct [9]. alculat avrag molcular wghts ar also clos to th ral molcular wghts. Evoluton of th hght vs. tm at th spcf opratng contons s llustrat n Fg.. It can sn n ths fgur that th hght has rach th st pont aftr aout 8 hours an rman stal aftrwars. Paks (or vallys n th hght woul hav a consral ffct on th prformanc of th ractor caus t trmns th rsnc tm of sols an th gas n th ractor. hs ffct s shown an scuss n th followng fgurs. m aftr start -up (hr al 7. omparson twn th smulaton rsults an th xprmntal ata ctual plant ata mpratur ( Smulaton rsults Proucton at (ton/hr ctual plant ata Smulaton rsults vrag Molcular Wght (kg/kgmol ctual plant ata Smulaton rsults B hght (m m (hr. (r. Fgur. B hght vs. tm of opraton Iranan Journal of hmcal Engnrng, ol. 6, o. 33

12 ! #! #% # &% & $% $ $ % &$ &% &# &$ ( ' * * # $ $ # ' ( &$ &# +

13 assm, Mostouf, Sotuh-Gharagh Morovr, t can sn n Fg. that th proucton rat of ymr n th ul phas s lss than that n th mulson phas u to th fact that mulson contans mor catalyst. It coul conclu from Fg. that aout 88% of th prouct s prouc n th mulson phas whl th shar of th uls n th proucton s aroun %. It s worth mntonng that th catalyst s strut among ths phas n almost th sam porton. Fg. confrms that contruton of sol partcls ns th ul phas s not nglgl n proucton of th ymr, as prvously pont out y Kashmshak t al. [9]. h ffrnc twn th tmpraturs of th ul an mulson phass s llustrat n Fg. 5. s xpct, th mulson phas tmpratur s sgnfcantly hghr than th ul phas tmpratur caus of th hghr ymr proucton rat n th mulson phas. Occurrnc of th maxmum n th tmpratur corrspons to th maxmum n th hght, as scuss aov av mpratur (K m (r. (hr. Fgur 5. Bul an mulson phas tmpraturs ontrol of gas-phas ymrzaton ractors s a complcat task caus of th hgh non-lnarty of th systm. In an nustral gas-phas ythyln ractor, controllng th lvl s on of th most crtcal an ffcult tasks. It has n shown n th prvous fgurs that all th procss valus n th ar corrlat to th hght of th sols n th ractor. It s known that whn a larg procss s to controll, th proportonal controllr s sutal. hs s th cas for many lqu lvl control loops. lso, th ffct of proportonal gan s mor sgnfcant than th ntgral an rvatv tm. hrfor, th ffct of K c on th ractor was nvstgat aftr sttng ntgral an rvatv trms nto thr rasonal valus. h trn llustrat n Fg. has Iranan Journal of hmcal Engnrng, ol. 6, o. 35

14 naroc Boconvrson of avy yrocarons sng atv onsorta n achv aftr tunng th control paramtrs n orr to amp th lvl fluctuatons an rach th sr valu. It s mportant to mnton that an mpropr tunng of th lvl controllr woul rsult n an oscllaton pattrn, loosnng th or ncrasng th hght wthout a practcal lmt. Snstvty of th sols lvl n th ractor wth th paramtrs of th controllr s llustrat n Fg. 6. Snstvty of th havor to th valu of th proportonal gan of th controllr s shown n ths fgur whl othr paramtrs of th controllr ar kpt unchang. It s shown that n ths cas, th lvl of th coul proprly controll whn K c s lss than hs fgur shows that slctng an mpropr gan mght la to loosnng th control of th lvl of th. It s thn ncssary to furthr nvstgat th ynamc havor of th mor thoroughly an carryout ynamc optmzaton of th ythyln ractor at ffrnt unstay-stat contons. B ght (m Kc=-0.05 Kc=-0.0 Kc=-0.05 Kc= m (r. m (hr. Fgur 6. araton of th lvl of th vs. tm of opraton at ffrnt proportonal gans. onclusons two-phas mol consstng of 5 ODEs has n vlop to prct th ynamc havor of th gas phas fluz ractor of thyln ymrzaton. h ractor was v nto four sral sctons whr ach scton conssts of ul an mulson phass. Flow of th gas was consr as plug flow through th uls an prfctly mx through th mulson phas. Exstnc of sols n th uls was consr n th prsnt stuy, thus, th ymrzaton ractons occur n oth mulson an ul phass. Dynamc smulaton of th ractor was accomplsh y mans of ntgratng th mol quatons along wth controllng th hght wth a PID controllr.du to hgh non-lnarty of th mol quatons, controllng th gasphas ymrzaton ractor s a ffcult task. ontrollng th hght s an mportant factor n th ractor prformanc. It was shown that paks or vallys of th hght hav a sgnfcant ffct on all of th 36 Iranan Journal of hmcal Engnrng, ol. 6, o.

15 assm, Mostouf, Sotuh-Gharagh ractor varals, spcally thos rlat to th mulson phas u to a hghr contruton of ths phas n th proucton rat. sults of th smulaton wr compar wth th plant ata an a clos agrmnt was osrv twn th mol prcton an ral ata. It has n shown that snc thr s a sgnfcant amount of catalyst partcls n th ul phas, nglctng th proucton rat n ths phas coul la to sgnfcant rror. h propos mol s sutal for smulatng th ynamc havor of th ractor at start-up, shut-own an gra transton. cknowlgmnts h authors gratfully acknowlg all th hlp from Mr.. Kashmshak. h authors woul also lk to thank Mr. S. Mostafav from rak Ptrochmcal omplx for hs commnts an provng usful nformaton n ffrnt stags of th stuy. otaton cross sctonal ara of th ractor, m let 3 trthyl alumnum r rchms numr spcfc hat capacty of componnt, p J/kg.K p, spcfc hat capacty of sol prouct, J/kg.K ul amtr, m p partcl amtr, m D g gas ffuson coffcnt, m /s D t ractor amtr, m E vaton from st pont F out volumtrc prouct rmoval rat, m 3 /s F pr pr-ymr flow rat, kg/s gas thrmal conuctvty, W/m.K k g J kfh kfm k kfr mnsonlss no. transfr to hyrogn rat constant for a ymr chan wth trmnal monomr, m 3 /kmol.s transfr to monomr k rat constant for a ymr chan wth trmnal monomr, m 3 /kmol.s transfr to co-catalyst rat constant for a ymr chan wth trmnal monomr, m 3 /kmol.s kfs spontanous transfr rat constant for a ymr chan wth trmnal monomr, m 3 /kmol.s kh rat constant for r-ntaton y monomr, m 3 /kmol.s kh r rat constant for r-ntaton y cocatalyst, m 3 /kmol.s k rat constant for ntaton y monomr, m 3 /kmol.s kp k propagaton rat constant for a ymr chan wth trmnal monomr ractng wth monomr k,m 3 /kmol.s K c ul to clou mass transfr coffcnt, s - K ul to mulson mass transfr coffcnt, s - K c proportonal gan of th controllr K c clou to mulson mass transfr coffcnt, s - Ks spontanous actvaton rat constant, s - Kf Formaton rat constant, s - h hght, m a ata numr c ul to clou hat transfr coffcnt, W/m 3.K c clou to mulson hat transfr coffcnt, W/m 3.K ul to mulson hat transfr coffcnt, W/m 3.K m numr of monomrs m man monomr molcular wght, kg/kgmol [M ] concntraton of componnt n th ractor, kmol/m 3 [M ] n concntraton of componnt n th nlt gasous stram [M ] total molar monomr concntraton, kmol/m 3 M th th monomr Mw wght avrag molcular wght, kg/kmol MW molcular wght of monomr, kg/kmol Iranan Journal of hmcal Engnrng, ol. 6, o. 37

16 naroc Boconvrson of avy yrocarons sng atv onsorta MW molcular wght of hyrogn, kg/kmol (0,j actv st typ j prouc y formaton racton * ( j potntal actv st typ j (j spontanously actvat st of typ j (0,j actv st of typ j prouc y transfr to hyrogn (r,j lv ymr of lngth r growng on st of typ j wth trmnal monomr S numr of typ of actv sts n numr of mols of ract monomr oun n th ymr n th ractor, kmol OP controllr output P prssur, Pa Q(r,j a ymr of lngth r prouc at st of typ j nstantanous rat of racton for monomr n th ul phas, kmol/m 3.s nstantanous rat of racton for monomr n th mulson phas, kmol/m 3.s nstantanous rat of racton for monomr, kmol/m 3.s nstantanous rat of racton for hyrogn, kmol/m 3.s p proucton rat, kg/m 3.s v volumtrc flow rat of ymr from th ractor, m 3 /s t opratng tm, s tmpratur, K n tmpratur of th nlt gasous stram, K 0 suprfcal gas vlocty, m/s ul vlocty, m/s mulson gas vlocty, m/s mf mnmum fluzaton vlocty, m/s volum, m 3 PLG volum of PF n ach scton, m 3 p volum of ymr ns th ractor, m 3 X(n,j nth momnt of a ymr prouc at st of typ j Y(n,j nth momnt of lv ymr prouc z at st of typ j axal poston, m Grk lttrs hat of racton, KJ/kmol volum fracton of uls n vo fracton av avrag voag of th gas vscosty, Pa.s nsty, kg/m 3 ntgral tm, s rvatv tm, s Suscrpts an suprscrpts thyln -utn 3 hyrogn nrt gas ul phas mulson phas g Gas n f or nlt monomr typ no. j actv st typ no. mf mnmum fluzaton n scton no. ymr rf rfrnc contons frncs. ho, K. Y., ay, W.., h ynamc havour of fluz ractors for sol catalys gas phas olfn ymrzaton, hm. Eng. Sc., 0, 6-79 (985.. Mculy, K. B., Macgrgor, J. F., amlc,. E., kntc mol for nustral gasphas thyln coymrzaton, IhE J., 36, ( Mculy, K. B., alot, J. P., arrs,. J., comparson of two phas an wll-mx mols for fluz ythyln ractors, hm. Eng. Sc., 9, (99.. Frnans, F..., Lona, L. M. F., Fluz ractor an physcalchmcal proprts molng for 38 Iranan Journal of hmcal Engnrng, ol. 6, o.

17 assm, Mostouf, Sotuh-Gharagh ythyln proucton, omp. hm. Eng., 3, S803- S806 ( Frnans, F..., Lona, L. M. F., trognous molng for fluz ymrzaton ractor, hm. Eng. Sc., 56, ( atzantons,., Yannoulaks,., Yagopoulos,., Kparsss,., cnt vlopmnts n molng gas-phas catalyz olfn ymrzaton fluz ractors: h ffct of ul sz varaton on th ractor s prformanc, hm. Eng. Sc., 55, ( hatzoukas,., Prkns, J. D., Pstkopoulos, E.., Kparsss,. Optmal gra transton an slcton of clos-loop controllrs n a gas-phas olfn ymrzaton fluz ractor, hm. Eng. Sc., 58, , ( lzah, M., Mostouf,., Pourmahans, S., Sotuh-Gharagh,., Molng of fluz ractor of thyln ymrzaton, hm. Eng. J., 97, 7-35 ( Kashmshak,., Mostouf, M., Sotuh- Gharagh,., wo-phas molng of th gas phas ythyln fluz ractor, hm. Eng. Sc., 6, ( Jafar,., Sotuh-Gharagh,., Mostouf,., Moular smulaton of fluz ractors, hm. Eng. chnol., 7, 3-, Erratum (00, 7, (00.. u,., Mostouf,., haouk, J., haractrzaton of ynamc gas-sol struton n fluz s, hm. Eng. J., 79, 33-3 (000.. oyag, M., Kunn, D., Importanc of sprs sols n uls for xothrmc ractons n fluz s. hm. Eng. ommun.,, 9-97 ( haouk, J., Gonzalz,., Guy,., Klvana, D., wo-phas mol for a catalytc turulnt fluz- ractor: pplcaton to thyln synthss, hm. Eng. Sc., 5, (999.. Batchlor, G. K., tch, J. M., Expulson of partcls from a uoyant lo n a fluz, J. Flu Mch., 78, 63-8 ( Glrtson, M.., Yats, J. G., h moton of partcls nar a ul n a gas-fluz, J. Flu Mch., 33, ( rahamson,.., Glart, D., Bhavour of gas-fluz s of fn powrs: Part II, oag of th ns phas n ulng s, Powr chnol., 6, 7-55 ( Mostouf,., u,., haouk, J., comparson of two- an sngl-phas mols for fluz ractors, In. Eng. hm. s., 0, ( L, J.., Wn, L. X., Qan, G.., u,. P., Kwauk, M., Schoutn, J.., van n Blk,. M., Structur htrognty, rgm multplcty an nonlnar havor n partcl-flu systms, hm. Eng. Sc., 5, ( Lucas,., rnalos, J., asal, J., Pugjanr, L., Improv quaton for th calculaton of mnmum fluzaton vlocty, In. Eng. hm. Proc. Ds. Dvlop., 5, 6-9 ( Davson, J. F., arrson, D., Fluz Partcls, amrg nvrsty Prss, w York (963.. llgart, K., Wrthr, J., Local ul gas hol-up an xpanson of gas/sol fluz s, Gr. hm. Eng., 9, 5- (986.. Kunn, D., Lvnspl, O. Fluzaton Engnrng, n., Boston, M, Buttrworth-nmann (99. Iranan Journal of hmcal Engnrng, ol. 6, o. 39