EMPIRICAL CORRELATION FOR PREDICTION OF THE ELUTRIATION RATE CONSTANT

Transcription

1 EMPIRICAL CORRELATION FOR PREDICTION OF THE ELUTRIATION RATE CONSTANT by Valentno STOJOVSI and Zvonmr OSTI] Orgnal scentc paper UDC: 53.59: BIBLID: , 7 (003),, In vessels contanng ludzed solds, the gas leavng carres some suspended partcles. Ths lux o solds s called entranment, E, or carryover and the bulk densty o solds on ths leavng gas stream s called the holdup. For desgn we need to know the rate o ths entranment and the sze dstrbuton o these entraned partcles R m n relaton to the sze dstrbuton n the bed, R b, as well as the varaton o both these quanttes wth gas and solds propertes, gas low rate, bed geometry and locaton o the leavng gas stream. Steady-state elutraton experments have been done n a ludsed bed 0, m dameter by,94 m hgh reeboard wth supercal gas veloctes up to 1 m/s usng solds rangng n mean sze rom 0,15 to 0,58 mm and wth partcle densty 660 kg/m 3. When the ne and coarse partcles were mxed, the total entranment lux above the reeboard was ncreased. None o the publshed correlatons or estmatng the elutraton rate constant were useul. A new smple equaton, whch s developed on the base o expermental results and theory o dmensonal analyses, s presented. Introducton A ludzaton vessel usually conssts o two zones, a dense bubblng phase havng a more or less dstnct upper surace separatng t rom a lean or dspersed phase. The secton o the vessel between the surace o the dense phase and the extng gas stream rom the vessel s called reeboard and ts heght s called reeboard heght, H. The purpose o the reeboard s to allow the solds to separate rom the gas stream, and as ts heght s ncreased entranment lessens. A reeboard heght s reached above whch entranment becomes nearly constant. Ths s called the transport dsengagng heght, TDH. Unortunately there are two related but undamentally derent dentons n current use: (1) The TDH s the heght above the surace o the bed requred or the coarses, lung up by the bubbles, to dsengage and all back nto the bed. Above the TDH only nes are ound; 43

2 THERMAL SCIENCE: Vol. 7 (003), No., pp () The TDH s the heght above whch the elutraton rate remans constant or declnes only slghtly. For a reeboard heght less than the TDH, the sze dstrbuton o solds n the reeboard changes wth poston. When the gas stream exts are above the TDH then both the sze dstrbuton and entranment rate become nearly constant. Elutraton reers to the selectve removal o the nes rom mxture and ths may occur ether below or above the TDH. Partcles are splashed nto the reeboard when bubbles burst at the bed surace. The solds trown up nto the reeboard contans the whole spectrum partcles szes present n the bed. Although bubbles clearly play an mportant role n entranment, there s stll no general agreement concernng the mechansms by whch solds are ejected nto and transported throughout the reeboard. The basc ejecton mechansms emphasze the role o the bubble nose (roo) partcles, the partcles n the bubble wake or both nose and wake partcles. The pcture s urther complcated by the reeboard eects, n whch partcles are transported upward as dspersed partcles or as agglomerates and downwards near the wall o the column as a lowng suspenson or n the center as large agglomerates. The term nes s reserved or sze racton o partcles whch have termnal velocty smaller than the supercal gas velocty. Denton o elutraton rate coecent Several models and correlatons or estmatng total entranment lux are reported n the lterature. The total entranment lux can be expressed by the ollowng equaton: ah E E ( E E ) e (1) 0 where E s the entranment lux at a pont h above the bed surace, and E s the elutraton lux (or the entranment lux above TDH). E 0 s the entranment lux o partcles at the bed surace and t can be obtaned by extrapolaton o the entranment lux data n the reeboard to the bed surace. Such a correlaton equaton s gven by Wen and Chen 5 : E AD b r g ( U U ). 5 m m 0 9 [ kg / m s ] (). 5 5 where A s the cross-secton area o column. D b s the bubble dameter at the bed surace and can be calculated usng an approprate equaton at derent low regmes. U m s the mnmum ludzaton velocty. The constant a s ndependent o the bed s composton and can be evaluated rom expermental data or E as a uncton o h. Ths constant representng the characterstcs o the ludzed bed entranment system. Values o a are ranged rom 3.5 to 6.4 m 1, and a 44

3 Stojkovsk, V., ost}, Z.: Emprcal Correlaton or Predcton o the Elutraton... value o 4 m 1 s recommended or a system n whch no normaton on entranment rate s avalable. The total lux o partcles elutrated can be calculated by the summaton o the lux o each partcle elutrated (a component elutraton lux). Thus, E (3) E Expermental evdence shows that only a small partcles whose termnal velocty s less than the gas velocty wll be carred out o the reeboard the reeboard heght s tall enough and no nternals are present n the reeboard. The proportonalty between the elutraton lux o a component and ts concentraton n the bed s well establshed over a very wde range o concentraton and szes. The lux o the partcles elutrated, E, s determnate by both the elutraton rate constant,, and a weght racton o the ne partcles present n the bed, X, as: E X (4) The elutraton rate constant can be regarded as the amount o partcles blown away rom a bed composed o one-szed partcles. s dened mathematcally as: 1 dw X W (5) A dt W For a contnuous eed o solds to a bed at equlbrum, X and W are constant and then elutraton rate constant can be expermentally ndng rom relaton x W x e X A X E (6) where x s weght racton o ne partcles wth sze d concerned nto the entraned lux. Many correlatons publshed (see table on the next page) or determnng the elutraton rate constant are lmted to the expermental condtons employed by each o the nvestgators. Extrapolaton o these correlatons to derent operatng condtons oten leads to very strange results. The purpose o ths study s to provde a new perspectve based on the exstng entranment and elutraton data and to propose a new correlaton whch s more relable than exstng correlatons or the smulaton and modelng o reeboard phenomena n ludzed bed. Dmensonal analyss Followng three equatons may be consdered as the basc equatons or elutraton: 45

4 THERMAL SCIENCE: Vol. 7 (003), No., pp Emprcal correlatons or elutraton rate constant Author Yag-Aoch (1955) Zenz-Wel (1958) gd p t m ( U U ) Re 0. 01Re t t Equaton U U ; 3 10 ru gd pr p gd pr p r U U 4 U ; gd pr p gd pr p Wen-Hashnger (1960) Tanaka et al. (197) ( U U t) r ( U U t) gd p ( U U t) r ( U U t) gd p Re Re t 0. 3 t rp r r rp r r Merrck-Hghley (1974) r U U t A 130 exp U 0. 5 U m U U m 0. 5 Geldart (1979) Colakyan (1979) Bachovchn (1979) r U U 3. 7 exp 54. t U U 331 t U U ( d pm g) r r p d m pm D x d s 0 pm Ln et al. (1980) r U U gd p 1.65 Wen-Chen (198) Colakyan- Levenspel (1984) r ( 1 e) U p s 1 / 4. 7 l( U U t) e 1 gds r ( U U t) d p lrp m m d p r r ( U U t) d p 1. 3 m r U t p 1 or U U t U U U t d p 38 Ds or r ( ). m D ( ). 38 D or r U U t d p s m D s s 46

5 Stojkovsk, V., ost}, Z.: Emprcal Correlaton or Predcton o the Elutraton... or the momentum balance o sngle partcles rp d 3p p U p 6 t = Cd pd p (U p - U ) r 4 - pd 3p 6 ( rp - r ) g (7) or the momentum balance o lud n column r U or elutraton t =- æ U p + m 1 çç r z r rè r ö + r g ø (8) 1 d( X M b ) = X 4 As dt Usng the theory o dmensonal analyss, the ollowng equaton s obtaned: 4 = r (U - U t ) æ r - r (U - U t ) d p (U - U t ) r D ö s ç p,,, ç r gd p m d p ø è (9) where the second and thrd term on the rght sde, denotes Froudes and Reynolds number, respectvely. Expermental The research o partcles entranment was carred out n ludzed beds wth derent constant bed composton. The steady-state ludzed bed was obtaned by a contnuos eedng and dschargng o the materal rom the bed. The cylndrcal column wth a dameter Ds= 00 mm was made o glass and PVC parts. An outlne o the expermental apparatus s shown n g. 1. A gas o-take was constructed lke a concal secton wth a dmenson 00/50/150 and t was equpped at the top o the column. The amount o entranment materal was collected n a lter cloth. Ar was used as ludzng gas. The ar low rate was measured by a orce nserted n 36 mm dameter ppe-lne. Fgure 1. Expermental apparatus 1 an, ppe-lne, 3 orce, 4 dstrbuton plate, 5 concal secton, 6 lter cloth, P1-P pressure taps, V1-V3 varaton o voltage, R rear reductor, EM1, EM electrcal motor, H heght o reeboard, h bed heght 47

6 THERMAL SCIENCE: Vol. 7 (003), No., pp The bed materal was slca sand, whch partcle densty was r p = 660 kg/m 3. Two narrow ractons o a materal were used n the experments: a racton wth equvalent F dameter o partcles d pm mm (approxmately ne F) and a racton wth P d pm mm (approxmately coarse P), as well as ther mxtures: P0/F80 (0% coarse and 80% ne materal), P40/F60, P60/F40, and P80/F0. The man physcal propertes o the bed mxtures are gven n tab. 1. Table 1. The man physcal propertes o bed mxtures [7] Materal F P0/F80 P40/F60 P60/F40 P80/F0 P d pm mm r o kg/m e o U m-p m/s U m-t m/s The experments presented n ths paper were carred out or: the supercal gas velocty was ranged rom U = m/s, the reeboard heght was H = 940 mm, the ludzed bed heght was h =30 mm, and the ludzaton ar temperature ranged rom 0 to 35 C. 48 Fgure. Expermentally obtaned curves o ludzaton or derent mxtures

7 Stojkovsk, V., ost}, Z.: Emprcal Correlaton or Predcton o the Elutraton... Expermentally determnaton on mnmum ludzaton velocty The ludsaton curve o poldsperzve materals has a transton area between the ltraton and ludsaton state o the bed. The apparent, U m-p and the total U m-t mnmum ludsaton velocty can be determnate. The expermentally obtaned ludsaton curves or derent mxtures are presented on g.. The values or apparent and total mnmum ludsaton velocty are gven n tab. 1. The relaton U /U m s termed as a ludsaton number. Expermental results and dscusson Eect o supercal gas velocty on total entranment lux The total entranment lux s the sum o all component luxes and the varaton o E wth U depends thereore on the bed composton as well as on partcle sze. I the supercal gas velocty s ntally below the termnal veloctes o most o the powder and s ncreased to a value above that o most partcles, a larger and larger proporton o the powder becomes avalable or carry-over. The expermental results presented n g. 3 conorms that the supercal gas velocty had a sgncant eect on the total lux o partcles entraned above the TDH. Snce the reeboard heght was always well above the TDH, the ncrease n the lux o partcles entraned above the TDH can be attrbute to an ncrease o burstng bubbles ntensty at the bed surace and reasonly to an ncrease n the lux o partcles ejected rom the bed surace. The expermental data o the total elutraton lux obtaned or derent bed mxtures are grouped when ts present n depence o total number o ludsaton. Ths Fgure 3. Total entranment lux as a uncton o the supercal gas velocty or varous bed materals, bed heght, and reeboard heght 49

8 THERMAL SCIENCE: Vol. 7 (003), No., pp Fgure 4. Elutraton lux n uncton o total number o ludzaton result s presented on g. 4. Ths result can be explaned by the condtons o the ludsed bed at the total mnmum ludsaton velocty. The total mnmum ludzaton velocty s occurred when the all partcles n the bed are n ludsed condton. In that case the segregaton n the bed s exceeded, the orgn o burstng bubbles s appears and the ntensve mxng o the partcles n bed s occurred. These hydrodynamc condtons are smlar or the beds wth varous compostons at the pont o total mnmum ludsaton velocty. Eect o nes n bed on total entranment lux The sze dstrbuton o bed materal had an nluence on the hydrodynamc o the bed and the reeboard. The segregaton wthn the bed had the nluence on the total entranment lux. I the sze dstrbuton s very wde elutraton rates may ncrease due to segregaton. As the concentraton o nes n the bed s ncreased, the heght requred by the coarses or dsengagement ncreases (g. 3). Eect o ludsaton velocty on sze dstrbuton o elutrated partcles The seve anal y ss on en traned ma te ral s done. The cu mu la tve curves or sze dstrbuton o partcles, or a reeboard heght H = 940 mm and var ous su per cal gas ve - loc ty are gven on g. 5. The su per cal gas ve loc ty has a dom nant n lu ence o the sze dstrbuton o elutrated par t cles be cause the n creas ng o the ve loc ty en ables to reach the ter m nal ve loc ty o the wde par t cles rac ton. On the other sde, the lm ted ree board heght and en sur ng the TDH depends o the supercal gas velocty ntensty. 50

9 Stojkovsk, V., ost}, Z.: Emprcal Correlaton or Predcton o the Elutraton... Fgure 5. Cumulatve curves or sze dstrbuton o partcles comprsed n the elutrated materal at reeboard heght H The ex per men tal re sults shows that the rate o the coarse par t cles nto the en traned ma te ral s less than 3%, what means that the ex per men tal con d tons (low and geometrcal) reer or reeboard heght nearest to TDH. The pres ence o coarse par t cles n the elutrated ma te ral s greater or the hgher su per cal gas ve loc ty. Determnaton o the elutraton rate constant The elutraton rate constant or expermentally obtaned results was determned by usng the eq. (6). The temnal velocty or each racton o partcles was obtaned wth Todes' equaton: Ar s Re t log Ar The value o partcles shape actor s = 0.75 was expermentally determnated [6]. 51

10 THERMAL SCIENCE: Vol. 7 (003), No., pp Fgure 6. Elutraton rate constant n dependence o the supercal gas velocty and the ne partcles content n the bed 5

11 Stojkovsk, V., ost}, Z.: Emprcal Correlaton or Predcton o the Elutraton... Eect o ludsaton velocty on the elutraton rate constant The nluence o supercal velocty on the elutraton rate constant or the derent bed composton o ne partcles s gven on g. 6. The ncrease o supercal gas velocty U ncreased the value o elutraton rate constant and had a strong nluence on ts determnaton. On the other sde, the ncrease o the presence o ne partcles n the bed mxture does not have eects o change to the value o the elutraton rate constant. Ths expermental result conrms the physcal meanng o the elutraton rate constant as a coecent o proporton between the rato o the ne partcles concentraton at reeboard heght and the total weght o ne partcles remanng n the bed. Comparson wth the publshed equatons n the lterature The expermentally determned elutraton rate constants, by usng eq. (6), are compared wth equatons publshed n the lterature. The comparatve dagrams are presented on g. 7. Remarkable derences between expermentally obtaned results and predcton o elutraton rate constant by usng the publshed correlaton occurred. Concernng the number o varables and the complexty (and nter-relaton o the mechansms occurrng n, and above the bed), t s hardly surprsng that the numerous publshed correlatons predct wdely derent rates o carry-over (when appled to systems other than those whch they are based on). It s not certan at all that the gas o-take was above the TDH n the used columns. Another problem that complcates the predcton o carry-over rates s that nes may be generated n the bed. The best t to the expermental results are obtaned by usng the Geldarts emprcal correlaton. New emprcal correlaton or the elutraton rate constant Takng nto account the domnant dmensonless number dervate by usng the theory o dmenson analyses, eq. (9), a new orm has been ound to t the expermentally data the best: U t U r U Fr 1 Re where 1 and are constants, and Froudes and Reynolds number s dened wth the (10) relatons: ( U U ) ( U U ) d ; Re gd m t Fr p r t p 53

12 THERMAL SCIENCE: Vol. 7 (003), No., pp Fgure 7. Comparson o the expermentally obtaned elutraton rate constant and the equatons publshed n the lterature The values o the constants 1 and are changed n dependence o the reeboard heghts. On a base o the realzed analyses or the lower reeboard heghts 7, the change o constant values wth the reeboard heght s gven on the dagram on g

13 Stojkovsk, V., ost}, Z.: Emprcal Correlaton or Predcton o the Elutraton... Fgure 8. Values o constant 1 and n dependence o a reeboard heght The t o the expermentally obtaned results wth the new proposed emprcal correlaton s gven on g. 9. The accuracy o the expermental data nterpolaton wth the proposed correlaton s R = The new proposed correlaton s also used or presentaton o the expermental results publshed by Colakyan 6. The dagram presented on g. 10(a) shows the Fgure 9. Comparson between expermental results and proposed correlaton 55

14 THERMAL SCIENCE: Vol. 7 (003), No., pp Fgure 10. Usng the new correlaton or ttng the expermental results rom Colakyan 6 Colakyan s approach or the ttng o oven expermental results, and on g. 10(b) the same expermental results are tted by usng the new proposed correlaton or estmaton the elutraton rate constant. As t can be seen, the new proposed correlaton enable to t the same expermental results wth hgher accuracy (R = 0,94). The applyng o the new proposed emprcal equaton or ttng the expermental results rom Colakyan shows that the values or constant 1 and are much hgher than that gven on g. 8. The comparson o the expermental results wth the publshed equaton, g. 7, shows that the proposed Colakyan equaton predcts the greatest values or the elutraton rate constant than all another equatons. However, the proposed emprcal correlaton gves a better groupng o the expermental results and hgher accuracy or ttng uncton. The denton and determnaton o the values or the constants 1 and are work or urther analyses.n Conclusons Usng a steady-state ludzed bed wth a sucently hgh reeboard, ths experment was carred out n order to nvestgate the eect o the reeboard heght, supercal gas velocty and bed composton on the elutraton rate constant. Wthn the operatng condtons, the ollowng conclusons were drown: A physcal meanng o the elutraton rate constant as a constant o proportonal to the rato o the ne partcles concentraton at the reeboard heght and the total weght o ne partcles remanng n the bed s expermentally obtaned. The eect o supercal gas velocty on was rarely nvestgated. The value o ncreases wth the ncrease U. None o the publshed equatons predcts all the expermental results; though some show the correct trends, other show devatons o several orders o magntude. The elutraton rate constant s aected by Froudes and Reynolds numbers, determned wth the theory o dmensonal analyss. The emprcal correlaton or elutraton rate constant rom a ludsed bed s well correlated wth the developed emprcal equaton. 56

15 Stojkovsk, V., ost}, Z.: Emprcal Correlaton or Predcton o the Elutraton... Nomenclature A s C d D s d p d pm E F Fr g H h 1, I M b R Re t TDH U U m U p U t X b x cross-sectonal area o bed at surace, m drag coecent, nner dameter o column, m artmetcal mean o adjacent seve apertures, m partcle mean sze o powder, m total entranment lux, kg/m s component entranment lux, kg/m s partcle Froude number, acceleraton o gravty, m/s reeboard heght, m bed heght, m elutraton rate constant, kg/m s coecents n eq. (10), upper and down dmenson o partcles racton, mm, mm mass o sold n bed, kg component entranment rate, kg/s partcle Reynolds number, tme, s transport dsengagng heght supercal gas velocty, m/s mnmum ludzaton velocty, m/s velocty o partcles, m/s termnal velocty, m/s equlbrum concentraton o sze n bed, mass racton o sze n elutrated materal, Greek letters m r r p gas vscosty, kg/ms gas densty, kg/m 3 partcle densty, kg/m 3 Reerences 1 Zenz, P. A., Wel, N. A., A Theoretcal-Emprcal Approach to the Mechansm o Partcle Entranment rom Fludzed Bed, AIChE J., 4 (1958), pp Wen, C. Y., Hashnger, R. F., Elutraton o Sold Partcles rom a Dense Phase Fludzed Bed, AIChE J., 6 (1960), pp Tanaka, I., Shnohara, H., Tanaka, Y., Elutraton o Fnes rom Fludzed Bed, J.Chem.Eng.Jpn., 5 (197), pp Ln, L., Sears, J. T., Wen, C. Y., Elutraton and Attrton o Char rom a Large Fludzed Bed, Powder Technology, 7 (1980), pp Wen, C. Y., Chen, L. H., Fludzed Bed Freeboard Phenomena: Entranment and Elutraton, AIChE J., 8 (198) 1, pp Colakyan, M., Levenspel, O., Elutraton rom Fludzed Beds, Powder Technology, 38 (1984), pp Stojkovsk, V., Entranment and Elutraton o Partcle rom Fludzed Bed, M. Sc Thess, Unversty St. rl and Metodj, Faculty o Mechancal Engneerng, Skopje,

16 THERMAL SCIENCE: Vol. 7 (003), No., pp Authors' address: V. Stojkovsk, Z. ost} Unversty St. rl and Metodj, Faculty o Mechancal Engneerng P. O. Box Skopje, R. Macedona Correspondng author (V. Stojkovsk): E-mal: tno@ereb.m.ukm.edu.mk Paper submted: September 9, 003 Paper revsed: September 0, 003 Paper accepted: October 1,