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Computed particle hold-up in a vertical pneumatic conveying line
Computedparticle hold-up in a vertical pneumatic conveyingline KUO-MING LUO1 andDIMITRI GIDASPOW2 Institute ROC Taiwan, Chemistry, 1Department ofApplied Chung Cheng ofTechnology, Taoyuan, Illinois Institute IL60616, USA 2Department ofChemical ofTechnology, Chicago, Engineering, Received forAPT 16October 21December 1990 1990; accepted from drift-flux model were used topredict thesolid Abstract-Theoretical formulas developed hold-up ofgas-solid inavertical line. These formulas ontheir pneumatic conveying physitransportation depend and ofparticles, and theflow rates ofthetwo Forlow calproperties: diameter density, viscosity phases. solid thecalculations from this drift-flux model well with results. Athigh flux, agree very experimental solid flow orlow thesolid distribution intheconveying line isnotuniform. We obtain a rate, gasflow between thetheoretical andexperimental results. Using hydrodynamic larger discrepancy computation and two momentum to the solid two equations including continuity equations equations predict hold-up better results. produces NOMENCLATURE a defined constant, byEq.(29) b defined constant, byEq.(29) formultiparticle gasdragcoefficient CD forsingle particle CDs gasdragcoefficient D diameter oftheconveyer [m] ofparticle diameter [m] dp friction coefficient gas-wall fg friction force[N/m3] fgw gas-wall solid-wall friction coefficient fp totalsolids friction forces duetosolids andsolid-wall friction FS dragforce [N/m'] friction forcebetween solids andwall[N/m3] fsw forces duetogas-wall friction andsolid-wall friction FW totalwallfriction [N/m3] acceleration g gravity [m/s2] H oftube[m] height drift-flux [m/s] molecular ofgas[g/gmol] weight Mg p pressure [N/M21 totalpressure [N/m3] dropperunitlength R universal gasconstant [KgM2/S2 kgmolk] number ofgas Reg Reynolds number ofparticle Rep Reynolds
256 Ug US Ut Vg Vr VS
[m/s] superficial gasvelocity solidvelocity [m/s] superficial terminal velocity [m/s] gasvelocity [m/s] relative velocity [m/s] solidvelocity [m/s] solidflux[kg/m' s]
letters Greek 0 thehorizontal inclination ofthetubeaxisflow fraction cg gasvolume fraction Es solidvolume s] [kg/m ,ug gasviscosity [kg/m3] pg gasdensity [kg/m3] p, soliddensity ofsolid sphericity formomentum transfer fl dragfriction [kg/m2 s] 1.INTRODUCTION ina vertical oftwo-phase flowpatterns lineis Understanding pneumatic conveying a gas-solid forustodesign Several have veryuseful transport system. investigators various ortheoretical studies tounderstand thetwophase performed experimental inpneumatic invertical flowinvolved Literature ongas-solid systems conveying. isabundant Solid intheconveying lineisavery pneumatic conveying [1-4]. hold-up variable forundestanding theflowpattern. Soo et al.[5]measured the important ballprobe. andSaito[3]anda photographic solidfluxes anelectric Konno using tomeasure thesolid volume fraction. andNakamura technique Capes [1]measured concentrations ofa series ofquick-closing valves. Katoet bymeans average particle al.[2]measured thesolid concentration witha subtube thistube bysuddenly sliding to another tube.Luo[6]usedanX-ray densitometer to corresponding transport line.Thistechnique was measure theaverage solidhold-up alongtheconveying further thelocalsolid volume fraction intheradial andusedtomeasure developed direction ina fullydeveloped region. 2.EQUIPMENT 1shows Ascrew a schematic oftheexperimental feeder Figure diagram apparatus. wasinstalled atthebottom tofeedsolids. Thisgives usa homogeneous inletcondition. Amovable densitometer wasinstalled to continuously measure the X-ray intheinletsection. Liftgasflows a rotameter, andthen porosity profile through enters thelift-line andtravels to transport theparticles. Airat room upwards wasusedinthisexperiment astheliftgasandfluidizing gas.Airwas temperature available ata gauge of8.825 atmfroma compressor. Glass beadparticles pressure of520,um diameter a verynarrow sizedistributon withdensities of having particle 2620 were usedinthese studies. kg/m3
257
1.Pneumatic setup. conveying Figure 3.THEORETICAL 3.1.Drift-flux model Thedrift-flux model fortwo-phase flowisbased ontherelative motion ofboth It is validin fullydeveloped flowwithnowalleffects. In two-phase phases. driftflux,Jgs,asthevolumetric which is flow,Wallis [7]defined average velocity, given by andegandesarethegas where velocities, gasandsolid UgandUsarethesuperficial andsolidporosities, which canbeexpressed as: respectively,
arethegasandsolidvelocities, respectively. Substituting Eqs(2) where and Vg V, and(3)into(1),weobtain
258 InEq.(4),( YgV,) canbedefined as - istheslipvelocity, as: Therefore, Eq.(4)canbeexpressed Fora liquid-solid flowsystem, Richardson andZaki[8]assumed therelation between theslipvelocity andterminal ofparticle is velocity Eqs(7)into(6),weget Substituting withtherelation ofEg+ es= 1,itfollows thatEq.(8)isalsoa unique function of n =2.36and thevalue ofnisstillunknown. Wallis However, voidage. [7]assumed 3inhiscalculations tofitexperimental data. Theterminal velocities arefunction ofphysical invarious flowregions. properties ForlowReynolds i.e.Rep<_0.4,thedragcoefficient canberepresented number; by Stokes' lawas theparticle as Where, number, Reynolds Repisdefined Thisleadstotheexpression fortheterminal velocity Intheintermediate thedragcoefficient canbe i.e.,0.4< Rep< 1000, region, as expressed inthisregion is Thustheterminal velocity thedragcoefficient isalmost ForhighReynolds i.e.Rep> 1000, constant numbers, andexpressed as inthisregion andtheterminal is velocity
259 Table 1. ofdrift-flux invarious conditions* Comparison equation
*In inintermediate used general, Eq.(TI) region. From momentum balance inpneumatic ifweneglect themomentum due conveying, tothewallfriction, thenthebuoyant forceisequal tothedragforce. Thisrelation canbeexpressed as ofphysical thefluid-particle friction coefficient where, /3,isalsofunction properties.Ergun as [9]expressed where WenandYu[101 thefriction coefficient as expressed 0, issphericity.
where forsingle andmultiparticle particle CDsandCDarethedragcoefficients systems respectively. Gidaspow [11]suggested using Eq.(17)inhighsolidvolume fraction andEq.(18)inlowsolid volume fraction. Withthedefinition ofdrift-flux andcorrelation offriction thedrift-flux canbeexpressed in coefficient, equation various flowregion asinTable1. 3.2.Hydrodynamic model in two-phase flowcalculations inboth Therearefivevariables i.e.,porosities velocities inbothphases andpressure intheconveying Thesolid phases, system. fraction canbeexpressed as volume
260 Thuswereduce thevariables fromfivetofour,i.e.Vg,V,,P andEg.Continuity forgasandsolid obtain threevariables equations phases namely Vg,Vsandeg.We needonemoreequation todetermine threevariables plusoneadditional equation fordetermination ofpressure. Thecommon are: equations Gasphase continuity Solid phase continuity Mixture momentum Thefourth describes themomentum balances between twophases. This equation momentum balance assumes thatthepressure inbothsolidandgas equation drops flowmodel) andwasproposed fromliterature phases (annular [1].
Intheabove B is theangle between thepipeaxiswitha horizontal line. equations, Forthisvertical to1.Theterm is conveying line,sinB is equal per fsthedragforce unitvolume of particles; forceperunitvolume of particles fswis thefrictional between thesolids andthewall.These termscanbeexpressed asthefollowing equations.
where thefluid-particle friction coefficient inEqs(17)and 8 hasbeendescribed ofthepipeand fp isthesolidfriction coefficient. (18).Disthediameter Many workers havefound to totheparticle and [1-4] proportional velocity fpbeinversely tohavethefollowing form. Inthiscalculation, weusea momenwhere a and bvarywithvarious investigators. tumequation andexpress thesolidfriction coefficient as
261 fromthebottom where isthetotalpressure canbemeasured dropwhich (OPlL)T force to thetopinthepneumatic line.Thetermfgwis thefrictional conveying oftheusual between thegasandthewallwhich canbeexpressed Faningbymeans as: typeequation isa function ofthegasReynolds number friction coefficient, where the fg,gas-wall tobesmooth inour ofthepipe.Thepipeisassumed andtherelative roughness ofthegasReynolds andcomputations. Therefore, experiments fgisonlya function thefriction coefficient canbeobtained from number. ForlowReynolds numbers, as: theHagen-Poiseuille expression thefriction canbeobtained fromtheBlasius formula: ForhighReynolds numbers, thefriction factorcanbeestimated from ForveryhighgasReynolds numbers, Prandtl's expression:
ofgas,pg,isa function thedensity Theairisassumed tobeanidealgas.Thus, as which canbeexpressed ofpressure andT isthe ofair,Ristheuniversal gasconstant Where, weight Mgisthemolecular canbesolved numerical calculausing operating temperature. Equations (21)-(24) linesimultaneously. theconveying tiontoobtainVg,V,,egandP along RESULTS 4.COMPUTATIONAL 2shows the ofhold-up between calculated results fromthedriftFigure comparison andKwauk Wecanexpress dataofWilhelm fluxmodel andtheexperimental [12]. drift-flux as thedimensionless From 1weknow that oftheparticles number. Table where nisa function Reynolds ofReynolds numbers. Wallis n isequal to3.65,2.89and2.33invarious regions fromEq.(T1). Wecanseethat n = 3.Thesolid lineinFig.2iscalculated assumed
262
2.Acomparison oftheoretically calculated from drift-flux model toexperimental dataof Figure Wilhelm andKwauk [12]. thecurve inwhich n = 2.89gives thebestfittoallofexperimental data.Figures 3 and4arethecomparison ofdrift-flux calculation using Eqs(Tl)and(T3)with various solidflux.Thesolidhold-up isfunction ofsuperficial Atlow gasvelocity. solidflowrate(Ws= 13.68 solidvolume fractions s),calculated usingEq. kg/mz withexperimental dataverywell.However, thecalculated results (Tl)agree using
ofexperimental data ofsolids volume fraction with thepredicted 3.Acomparison values from Figure drift flux with (kg/m2s). equations W,=13.68
263
with thepredicted values from ofsolids volume fraction ofexperimental data 4.Acomparison Figure with drift flux (kg/m2s). equations W,=40.20 resultonlyforlargersuperficial gasvelocity Eq.(T3)agreewithexperimental = 40.20 s),using Eq.(T3)isbetter kg/m2 m/s).Athighsolidflux( WS (Ug> 8.717 thanusing results. Eq.(Tl)incalculated animportant factor inpneumatic effects become Ifthewallfriction transporta5 isusedforcalculation. model thatthehydrodynamic Figures tion,itissuggested lineusingthe calculated and6 showthe porosities alongthe conveying Friction model. coefficients, 6 ofEqns(17)and(18),wereusedin hydrodynamic From roleinourcalculation. These coefficients thiscomputation. playanimportant calculation dataandthetheoretical givea Figs5and6weseethattheexperimental 0.1m)oftheconresult when closer (below region Eq.(17)isusedintheentrance Aftergasandsolidsflow of thehighsolidconcentration. linebecause veying
calculated results model for with 5.Acomparison ofexperimental using hydrodynamic porosities Figure inlow solid flux. friction coefficients various
264
6.Acomparison ofexperimental with calculated results model for Figure porosities using hydrodynamic various friction coefficients inhigh solid flux. intothefullydeveloped thanthe0.1mpoint), thesolidconcentraregion (higher tionsareverylow,i.e.theporosities arehigher than0.99.Then, using Eq.(18)in wecanobtain a betteragreement totheexperimental calculations, hydrodynamic data. REFERENCES 1.C.E.Capes and K.Nakamura, Vertical anexperimental with pneumatic conveying: study particles intheintermediate andturbulent flow Can. J.Chem. 1973. regimes. Eng., 51,31-38, 2.K.Kato, Y.Ozawa, H.Endo, M.Hiroyasu and T.Hanzawa, Particles andaxial hold-up pressure invertical reactor In:Fluidization Fundrop pneumatic transport (riser). V,Proc. Fifth Engineering dation K.Ostergaad andA.Sorensen New Conf. of Fluidization, Elsinore, Denmak, (eds). York: 265-272. Fundation, 1986, Engineering 3.H.Konno andS.Saito, J.Chem. Pneumatic ofsolid conveying through straight pipes. Eng. Jpn, 2(2), 211-217, 1969. 4.K.V.S.Reddy andD.C.T.Pei, insolid-gas Particle flow inavertical I&EC Fundynamics pipe. 1969. damentals, 8(3), 490-497, 5.S.L.Soo, G.J.Trezek, R.C.Dimick and G.F.Hohnstriiter, Concertration and mass flow distribuinagas-solid tions I&EC 1964. Fundermentals, suspension. 3(2), 98-106, 6.K.M.Luo, vertical PHD Illinois Institute ofTechnology, Experimental gas-solid Thesis, transport, 1987. Chicago, 7.G.B.Wallis, One-Dimensional Two-Phase Flow. New York: 1969. McGraw-Hill, 8.J.F.Richardson and W.N.Zaki, Sedimentation and Fluidization: I.Trans. Inst. Chem. part Eng., 1954. 32,35-53, 9.S.Ergun, Fluid flow Columns. Chem. 1952. through packed Eng. Prog., 48,89-94, 10.C.Y.Wen and Y.H.Yu, Mechanics offluidization. Chem. Eng. Prog. Series., Symp. 62, 100-111, 1966. 11.D.Gidaspow, Y.C.SeoandB.Ettehadieh, offluidization: and Hydrodynamic experimental theoretical bubble sizes inatwo-dimensional bed with ajet.Chem. Commun., Eng. 22, 253-272, 1983. 12.R.H.Wilhelm andM.Kwauk, Fluidization ofsolid Chem. 44(3), 201-217, particles. Eng. Prog. 1948.
256 Ug US Ut Vg Vr VS
[m/s] superficial gasvelocity solidvelocity [m/s] superficial terminal velocity [m/s] gasvelocity [m/s] relative velocity [m/s] solidvelocity [m/s] solidflux[kg/m' s]
letters Greek 0 thehorizontal inclination ofthetubeaxisflow fraction cg gasvolume fraction Es solidvolume s] [kg/m ,ug gasviscosity [kg/m3] pg gasdensity [kg/m3] p, soliddensity ofsolid sphericity formomentum transfer fl dragfriction [kg/m2 s] 1.INTRODUCTION ina vertical oftwo-phase flowpatterns lineis Understanding pneumatic conveying a gas-solid forustodesign Several have veryuseful transport system. investigators various ortheoretical studies tounderstand thetwophase performed experimental inpneumatic invertical flowinvolved Literature ongas-solid systems conveying. isabundant Solid intheconveying lineisavery pneumatic conveying [1-4]. hold-up variable forundestanding theflowpattern. Soo et al.[5]measured the important ballprobe. andSaito[3]anda photographic solidfluxes anelectric Konno using tomeasure thesolid volume fraction. andNakamura technique Capes [1]measured concentrations ofa series ofquick-closing valves. Katoet bymeans average particle al.[2]measured thesolid concentration witha subtube thistube bysuddenly sliding to another tube.Luo[6]usedanX-ray densitometer to corresponding transport line.Thistechnique was measure theaverage solidhold-up alongtheconveying further thelocalsolid volume fraction intheradial andusedtomeasure developed direction ina fullydeveloped region. 2.EQUIPMENT 1shows Ascrew a schematic oftheexperimental feeder Figure diagram apparatus. wasinstalled atthebottom tofeedsolids. Thisgives usa homogeneous inletcondition. Amovable densitometer wasinstalled to continuously measure the X-ray intheinletsection. Liftgasflows a rotameter, andthen porosity profile through enters thelift-line andtravels to transport theparticles. Airat room upwards wasusedinthisexperiment astheliftgasandfluidizing gas.Airwas temperature available ata gauge of8.825 atmfroma compressor. Glass beadparticles pressure of520,um diameter a verynarrow sizedistributon withdensities of having particle 2620 were usedinthese studies. kg/m3
257
1.Pneumatic setup. conveying Figure 3.THEORETICAL 3.1.Drift-flux model Thedrift-flux model fortwo-phase flowisbased ontherelative motion ofboth It is validin fullydeveloped flowwithnowalleffects. In two-phase phases. driftflux,Jgs,asthevolumetric which is flow,Wallis [7]defined average velocity, given by andegandesarethegas where velocities, gasandsolid UgandUsarethesuperficial andsolidporosities, which canbeexpressed as: respectively,
arethegasandsolidvelocities, respectively. Substituting Eqs(2) where and Vg V, and(3)into(1),weobtain
258 InEq.(4),( YgV,) canbedefined as - istheslipvelocity, as: Therefore, Eq.(4)canbeexpressed Fora liquid-solid flowsystem, Richardson andZaki[8]assumed therelation between theslipvelocity andterminal ofparticle is velocity Eqs(7)into(6),weget Substituting withtherelation ofEg+ es= 1,itfollows thatEq.(8)isalsoa unique function of n =2.36and thevalue ofnisstillunknown. Wallis However, voidage. [7]assumed 3inhiscalculations tofitexperimental data. Theterminal velocities arefunction ofphysical invarious flowregions. properties ForlowReynolds i.e.Rep<_0.4,thedragcoefficient canberepresented number; by Stokes' lawas theparticle as Where, number, Reynolds Repisdefined Thisleadstotheexpression fortheterminal velocity Intheintermediate thedragcoefficient canbe i.e.,0.4< Rep< 1000, region, as expressed inthisregion is Thustheterminal velocity thedragcoefficient isalmost ForhighReynolds i.e.Rep> 1000, constant numbers, andexpressed as inthisregion andtheterminal is velocity
259 Table 1. ofdrift-flux invarious conditions* Comparison equation
*In inintermediate used general, Eq.(TI) region. From momentum balance inpneumatic ifweneglect themomentum due conveying, tothewallfriction, thenthebuoyant forceisequal tothedragforce. Thisrelation canbeexpressed as ofphysical thefluid-particle friction coefficient where, /3,isalsofunction properties.Ergun as [9]expressed where WenandYu[101 thefriction coefficient as expressed 0, issphericity.
where forsingle andmultiparticle particle CDsandCDarethedragcoefficients systems respectively. Gidaspow [11]suggested using Eq.(17)inhighsolidvolume fraction andEq.(18)inlowsolid volume fraction. Withthedefinition ofdrift-flux andcorrelation offriction thedrift-flux canbeexpressed in coefficient, equation various flowregion asinTable1. 3.2.Hydrodynamic model in two-phase flowcalculations inboth Therearefivevariables i.e.,porosities velocities inbothphases andpressure intheconveying Thesolid phases, system. fraction canbeexpressed as volume
260 Thuswereduce thevariables fromfivetofour,i.e.Vg,V,,P andEg.Continuity forgasandsolid obtain threevariables equations phases namely Vg,Vsandeg.We needonemoreequation todetermine threevariables plusoneadditional equation fordetermination ofpressure. Thecommon are: equations Gasphase continuity Solid phase continuity Mixture momentum Thefourth describes themomentum balances between twophases. This equation momentum balance assumes thatthepressure inbothsolidandgas equation drops flowmodel) andwasproposed fromliterature phases (annular [1].
Intheabove B is theangle between thepipeaxiswitha horizontal line. equations, Forthisvertical to1.Theterm is conveying line,sinB is equal per fsthedragforce unitvolume of particles; forceperunitvolume of particles fswis thefrictional between thesolids andthewall.These termscanbeexpressed asthefollowing equations.
where thefluid-particle friction coefficient inEqs(17)and 8 hasbeendescribed ofthepipeand fp isthesolidfriction coefficient. (18).Disthediameter Many workers havefound to totheparticle and [1-4] proportional velocity fpbeinversely tohavethefollowing form. Inthiscalculation, weusea momenwhere a and bvarywithvarious investigators. tumequation andexpress thesolidfriction coefficient as
261 fromthebottom where isthetotalpressure canbemeasured dropwhich (OPlL)T force to thetopinthepneumatic line.Thetermfgwis thefrictional conveying oftheusual between thegasandthewallwhich canbeexpressed Faningbymeans as: typeequation isa function ofthegasReynolds number friction coefficient, where the fg,gas-wall tobesmooth inour ofthepipe.Thepipeisassumed andtherelative roughness ofthegasReynolds andcomputations. Therefore, experiments fgisonlya function thefriction coefficient canbeobtained from number. ForlowReynolds numbers, as: theHagen-Poiseuille expression thefriction canbeobtained fromtheBlasius formula: ForhighReynolds numbers, thefriction factorcanbeestimated from ForveryhighgasReynolds numbers, Prandtl's expression:
ofgas,pg,isa function thedensity Theairisassumed tobeanidealgas.Thus, as which canbeexpressed ofpressure andT isthe ofair,Ristheuniversal gasconstant Where, weight Mgisthemolecular canbesolved numerical calculausing operating temperature. Equations (21)-(24) linesimultaneously. theconveying tiontoobtainVg,V,,egandP along RESULTS 4.COMPUTATIONAL 2shows the ofhold-up between calculated results fromthedriftFigure comparison andKwauk Wecanexpress dataofWilhelm fluxmodel andtheexperimental [12]. drift-flux as thedimensionless From 1weknow that oftheparticles number. Table where nisa function Reynolds ofReynolds numbers. Wallis n isequal to3.65,2.89and2.33invarious regions fromEq.(T1). Wecanseethat n = 3.Thesolid lineinFig.2iscalculated assumed
262
2.Acomparison oftheoretically calculated from drift-flux model toexperimental dataof Figure Wilhelm andKwauk [12]. thecurve inwhich n = 2.89gives thebestfittoallofexperimental data.Figures 3 and4arethecomparison ofdrift-flux calculation using Eqs(Tl)and(T3)with various solidflux.Thesolidhold-up isfunction ofsuperficial Atlow gasvelocity. solidflowrate(Ws= 13.68 solidvolume fractions s),calculated usingEq. kg/mz withexperimental dataverywell.However, thecalculated results (Tl)agree using
ofexperimental data ofsolids volume fraction with thepredicted 3.Acomparison values from Figure drift flux with (kg/m2s). equations W,=13.68
263
with thepredicted values from ofsolids volume fraction ofexperimental data 4.Acomparison Figure with drift flux (kg/m2s). equations W,=40.20 resultonlyforlargersuperficial gasvelocity Eq.(T3)agreewithexperimental = 40.20 s),using Eq.(T3)isbetter kg/m2 m/s).Athighsolidflux( WS (Ug> 8.717 thanusing results. Eq.(Tl)incalculated animportant factor inpneumatic effects become Ifthewallfriction transporta5 isusedforcalculation. model thatthehydrodynamic Figures tion,itissuggested lineusingthe calculated and6 showthe porosities alongthe conveying Friction model. coefficients, 6 ofEqns(17)and(18),wereusedin hydrodynamic From roleinourcalculation. These coefficients thiscomputation. playanimportant calculation dataandthetheoretical givea Figs5and6weseethattheexperimental 0.1m)oftheconresult when closer (below region Eq.(17)isusedintheentrance Aftergasandsolidsflow of thehighsolidconcentration. linebecause veying
calculated results model for with 5.Acomparison ofexperimental using hydrodynamic porosities Figure inlow solid flux. friction coefficients various
264
6.Acomparison ofexperimental with calculated results model for Figure porosities using hydrodynamic various friction coefficients inhigh solid flux. intothefullydeveloped thanthe0.1mpoint), thesolidconcentraregion (higher tionsareverylow,i.e.theporosities arehigher than0.99.Then, using Eq.(18)in wecanobtain a betteragreement totheexperimental calculations, hydrodynamic data. REFERENCES 1.C.E.Capes and K.Nakamura, Vertical anexperimental with pneumatic conveying: study particles intheintermediate andturbulent flow Can. J.Chem. 1973. regimes. Eng., 51,31-38, 2.K.Kato, Y.Ozawa, H.Endo, M.Hiroyasu and T.Hanzawa, Particles andaxial hold-up pressure invertical reactor In:Fluidization Fundrop pneumatic transport (riser). V,Proc. Fifth Engineering dation K.Ostergaad andA.Sorensen New Conf. of Fluidization, Elsinore, Denmak, (eds). York: 265-272. Fundation, 1986, Engineering 3.H.Konno andS.Saito, J.Chem. Pneumatic ofsolid conveying through straight pipes. Eng. Jpn, 2(2), 211-217, 1969. 4.K.V.S.Reddy andD.C.T.Pei, insolid-gas Particle flow inavertical I&EC Fundynamics pipe. 1969. damentals, 8(3), 490-497, 5.S.L.Soo, G.J.Trezek, R.C.Dimick and G.F.Hohnstriiter, Concertration and mass flow distribuinagas-solid tions I&EC 1964. Fundermentals, suspension. 3(2), 98-106, 6.K.M.Luo, vertical PHD Illinois Institute ofTechnology, Experimental gas-solid Thesis, transport, 1987. Chicago, 7.G.B.Wallis, One-Dimensional Two-Phase Flow. New York: 1969. McGraw-Hill, 8.J.F.Richardson and W.N.Zaki, Sedimentation and Fluidization: I.Trans. Inst. Chem. part Eng., 1954. 32,35-53, 9.S.Ergun, Fluid flow Columns. Chem. 1952. through packed Eng. Prog., 48,89-94, 10.C.Y.Wen and Y.H.Yu, Mechanics offluidization. Chem. Eng. Prog. Series., Symp. 62, 100-111, 1966. 11.D.Gidaspow, Y.C.SeoandB.Ettehadieh, offluidization: and Hydrodynamic experimental theoretical bubble sizes inatwo-dimensional bed with ajet.Chem. Commun., Eng. 22, 253-272, 1983. 12.R.H.Wilhelm andM.Kwauk, Fluidization ofsolid Chem. 44(3), 201-217, particles. Eng. Prog. 1948.