Magnetically assisted gas-solid fluidization in a tapered vessel: Part Ⅱ Dimensionless bed expansion

Particuology 7 (2009) 183-192Contents lists available at ScienceDirectPARTICLJOLOGY |ParticuologyEL SEVIERjournal homepage: www. elsevier. com/locate/particMagnetically assisted gas-solid fluidization in a tapered vessel: Part IIDimensionless bed expansion scalingJordan HristovDepartment of Chemical Engineering. University of Chemnical Technology and Metallurgy, 1756 Sofa, 8 KL Ohridsky BIvd, BulgariaA RTICLE . INFOABSTRACTArticle history:The article presents an effort to create dimensionless scaling correlations of the overall bed porosity inReceived 7 October 2008the case of magnetically assisted fluidization in a tapered vessel with external transverse magnetic field.Accepted 6 January 2009This is a stand of portion of new branch in the magnetically assisted fluidization recently created con-cerning employment of tapered vessels. Dimensional analysis based on "pressure transform" of the initialset of variables and involving the magnetic granular Bond number has been applied to develop scaling. Funidizationrelationships of dimensionless groups representing ratios of pressures created by the fluid flow, gravityMagnetization FIRSTand the magnetic field over an elementary volume of the fluidized bed. Special attention has been paid onTapered bedthe existing data correlations developed for non-magnetic beds and the links to the new ones especiallyDimensional analysisPressure transform of variablesdeveloped for tapered magnetic counterparts. A special dimensionless variable Xp = (ArADbl)/3 V RgmQMagnetic granular Bond numbercombining Archimedes and Rosensweig numbers has been conceived for porosity correlation. Data cor-Rosensweig numberrelations have been performed by powe-law, exponential decay and asymptotic functions with analysisPorosity correlationsof their adequacies and accuracies of approximation.◎2009 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy ofSciences. Published by EIsevier B.V. All rights reserved.1. Introductionof Geldart's class B has been reviewed by Hristov (2002, 2003a,2003b, 2004, 2006a, 2007a). Recently, promising results concern-Tapered gas-solid fluidization (Mathur & Epstein, 1974) origi-ing magnetically assisted nano-fuidization have been reported bynally conceived for gas-fluidization of Geldart's D particles (Geldart,Hao, Zhu, Lei, and Li (2008a), Hao, Zhu, Jjiang, and Li (2008b) with1973) have encountered in many areas of chemical, mining anddirect mass transfer applications pertinent to CH4-CO2 reforming.food industries. Recently fluidization of cohesive particles (DeivaThe present article addresses scaling correlations of overallVenkatesh, Chaouki, & Klvana, 1996; Erbil, 1998) and drying of pas-porosity of magnetically assisted gas fluidized tapered beds withtas (Bacelos & Freire, 2008; Bacelos, Passos, & Freire, 2007) haveMagnetization FIRST mode. The first report in this new trendbeen developed in two- and three-dimensional tapered vessels. Inin magnetically assisted fluidization was presented by Hristovthis context, remotely controlled tapered fuidized gas- solids beds(2008). Dimensionless groups provided by the“pressure trans-have been conceived (Hristov, 2008, 2009) by means of magnetiz-form" approach (Hristov, 2006a, 2007b) are used to develop furtherable solids and external transverse magnetic field.results on porosity scaling relationship. A special standpoint isThe magnetically assisted fluidization deals with magneticdeveloped to link the existing empirical and semi-empirical corre-solids fluidized by liquid(Hristov, 2006a), gas (Hristov, 2002, 2003a,lations pertinent to non-magnetic tapered beds (DiFelice, Foscolo,2005, 2006b, 2007c) or gas-liquid flow (Hristov, 2007a) withGibilaro, Wallis, & Carta, 1991; Gui & Grace, 2008; Kmiec, 1975, 1977,two basic magnetization modes: Magnetization FIRST (preliminary1980) to the new situation arising in magnetically assisted taperedmagnetization of fixed bed and consequent fluidization) and Mag-beds (MATB) with controlled interparticle forces.netization LAST (magnetization of preliminarily fluidized beds).The hydrodynamic behaviour of such beds with coarse particles2. ExperimentalThe experimental set-up was described in details by Hristov(2008) so only some basic information pertinent to the results is tobe provided in this work. A conical vessel(15° opening angle, 30 mmfunction.ID-bottom diam中国煤化1eter) surrounded byE-mail address: jordan.hristov@mail.bg.saddle coils (Hnm ID and 400 mmURL: http://www.hristov.com/jordan.in height was uYHC N M H Gnted transversely to1674- 2001/$ - see front matter 0 2009 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by EIsevier B.V. All rights reserved.doi:10.1016/j.partic.2009.01.012184J Hristov / Particuology 7 (2009) 183- 192NomenclatureRep = pdbU/nj particle Reynolds numberAeddimensionless pre- factor in EDF expressed throughRg= (ρ2U2jμM2) Rosensweig number (basic definition),- -see Eq. (7)Rgm=(pjU21uoMsH) Rosensweig number expressed in termsdimensionless exponential pre-factor in EDIof Msexpressed through Xp- see Eq, (7)Rgme = ((16/r2D2)(pjQ2/uoMsH)) Rosensweig numberAaddimensionless asymiptote in the asymptotic func-expressed in terms of the volumetric flow rate Qtion expressed through Xp- see Eq. (8)Sb = πD7/4 cross-section area of the gas inlet orifice (m2 )ILfree term in the linear approximation y=al + brXpUsuperficial gas velocity (denoted also as Ug see theAr = (dPr(Ps - Pr)g)/nf Archimedes numbertext) (m/s)badimensionless pre-factor of the asymptotic functionJbsuperficial fluid velocity defined through the coneexpressed through- -see Eq. (8)Xpentance cross-see 1pre-factor in linear approximationy=an +bLXpMp/Ps volume occupied by the solids (m3)Bog-c= Pc/Pg = Pc/(Psgdp) Bond number of granular mate-(thbo/12)(D吃+ DrD, + Dg)(m3)rials with a natural cohesionindependent variable (see the text)Bog-m =(LoMH)/Psgdp magnetic Bond number for granularXp = (ArODb1)7/3√RgmQ general dimensionless indepen-materialsdent variableiarate of the Asymptotic Function, dimensionlessD，diameter of the flow entrance (m)Greek letterstop diameter of the bed (m)cone angle (°)△DbL =(DL- D,)/2hbo = tg(x/2) dimensionless ratio of bedporositygeometric characteristicsinitial bed porosityE0dpparticle diameter (m)Eaannular bed porosityFr = U2 /gdp particle diametfined Froude numberEmaxmaximum bed porosityGa = (dfgpp)/n子Galileo numberfree term in the exponential decay functionHgravity acceleration, 9.81 m/s2umagnetic permeability (Wb/Am) or (H/m)magnetic field intensity (A/m)μmagnetic permeability of vacuum (Wb/Am)hbnffluid dynamic viscosity (denoted also as η for sim-hboinitial bed height (m)plicity)(Pas)bed length at the wall (see Fig. 1) (m)fi.f2.f3.f4 general functionPffluid density (kg/m3)ρggas density (kg/m3)fzuJ2q general function0ssolid particle density (kg/m3 )fNIfN2 general function (Eq. (4))shear-stress in a liquid (Pa)fluid drag forces (Eq. (4)) (N)Fcgravity forces (Eq. (4)) (N)Subscripts°Mmagnetic forces (Eq. (B1))(N)KWg = (PjU})/(Psgdp) Kwauk numbergasMmagnetization (A/m)maxmaximumM、Msmagnetization at saturation (A/m)mfminimum fluidizationMpmass of particles charged into the vessel (kg)msminimum spoutingexponent in Eq. (3a,b)PparticlesolidsPg= Psgdp gravity pressure per unit surface of interparticlecontacts (see Appendix A) (Pa)Special symbolsPr=: n/(Uy/ldp) viscous pressure (shear stress) (see Appendix0(,)order of magnitudeA)(Pascale to, i.e. a~ b means“a scales to b"△Ppressure drop (Pa)equivalent(explanation: Rep = (PfdpUg/np)= Uf for .volumetric gas flow rate (m3/s)the reason that Pjdp/ηg= const.)volumetric gas flow rate at the onset of initial bedexpansion (onset of MSB)(m3 /:lmf-1volumetric gas flow rate at the fluidization onset inthe botom part of the bed (msudiatioothe cone axis of symmetry and the fluid flow direction (see Fig. 1). .Qmf0volumetric gas flow rate at theThe field was steady and the maximum field intensity attained inover the entire bed in absence of a field (m3 /s)Qmf-2 .volumetric gas flow rate at the fluidization onsetthese experiments was about 27 kA/m. Magnetite sand and ammo-nia catalyst“Haldorf Topsoe", KM-1 in narrow sieve fractions (seeover the entire bed (m3/s)Table 1) were used in the experiments. The bed porosity εa, knownQms-Uvolumetric gas flow rate at the minimum spoutingas annular porosity was calculated through (Hristov, 2008)(unstable) point (m3/s)Qms-s中国煤化工_. (M2)(stable) points (m3 /s)VbedRe = pfDbUf/nf Reynolds number defined through the coneMHCNMH GDp+DB) (Ps.(1)entrance diameterwhere DL = Db + 2hp tan(a/2).J. Hristov / Particuology 7 (2009) 183- 192185Fieldcauses "bed compression" by the magnetic pressure acting trans-Bedcross-sectionLinesversely to the field lines and oppositely to the fluid drags forces.Obviously, in the upper particle layers the field action predomi-nates and the expansion is lower than that observed near the gasentrance where the fluid drag forces predominates; with the samefield intensity over the entire bed height. More details and resultsare available elsewhere (Hristov, 2008).3.2. Dimensional correlations pertinent to bed expansionGeneral scaling relationships are developed through applica-tion of the "pressure transform”method (Hristov, 2006a, 2007b).Specific points of this dimensionless analysis are developed in theappendices for brevity and clarity of the main text.hp3.2.1. General dimensionless relationshipsIn accordance with the “pressure transform” developed inHristov (2008) for magnetically assisted tapered fluidization (seeh_Appendix A for details) the bed expansion can be expressed asεa =f(Ar, ODL>fzu(Rgm)(2a)orDbEa = f(Ar,△DbL )fzQ(RgmQ).(2b)Both RgM= (ρjU21μoMsH) (Appendices A and B) and RgmQ =C.(( 16/r2D)(0jQ2/μoMsH)) (Appendix C) are modified Rosensweignumbers. In general, the functions f2u should be a power-law(Barenblatt, 1996; Kline, 1965), i.e.Fig 1. Schematic experimental set-up.ε= (Rgm)" = U2m→ε= (Rgme)" = Q2m",(3a,b)because Rgm ~ U2. The function fi forms only the correlation pre-3. Resultsfactor since Ar and△DbL are constants for a given particle bed. .3.1. Porosity evolution curves3.2.2. Existing non-magnetic correlations and the link to theExperimental results concerning porosity evolution withmagnetic relationshipsincreasing the volumetric gas flow rate are shown in Fig. 2. In gen-Correlations pertinent to porosity of spouted beds are, in gen-eral, the porosity evolution demonstrates“saturation" at high gaseral, rare in the literature. The common form of these correlationsflow rate, thus reaching a plateau dependent on the field intensitycan be expressed as (KmieC, 1980)applied. This physical behaviour is strongly related to the vesselgeometry where the lift forces decreases as the gas superficialε= fv1 ( PD )fn2(initial bed geometry).(4)velocity decreases with increasing the local bed cross section. Thatis, the lower layers of the bed expand faster than the upper onesThe ratio of the fluid drag to gravity forces Fp/Fc introduced bywith the same gas flow rate. Furthermore, the field orientationKmiec (1975, 1977)is a function of Reynolds and Archimedes num-ber (Gui & Grace, 2008; Kmiec, 1980). i.e. Fp/Fc = 0.75Co(Re}/Ar).0.8(KM-1 (613- 800 )In the context of the pressure transform it can be simply expressedas Fo/Fc = (PjU2)/(psgdp), that is Fp/Fc = KWg. Furthermore, withG=2 kg .additional action of a magnetic field we may read (see Appendix B)hbo = 250 mmH(kA/m )? = RgmB0g-m，(5)0.702.38with the modified Rosensweig number Rgm =(pjU21μoMsH) (see0.654.77Appendix A) containing only one macroscopic variable, the veloc-9.55ity U. Then, the general form of the porosity correlation is出0.60ε=f3(Ar,Bog-m,ODbL). To circumvent the use of velocity U in RgMwe use a Rosensweig number RgmQ based on the volumetric flow+ 19.10.55十23.87Ppenalxits lower limit islence, the1.0 2.0 3.0 4.0 5.06.7.0 8.0general porosityYHc N M H G.Dol )<(RgMo) TheQ( m31s)x 10-3functions f2u(Rgequivalent. The mainproblem when ftting experimental data is to express the functionsFig. 2. Bed porosity curves of ammonia catalysts KM-1 (613- 800 μum),f1,f2.f3 and f4 adequately through dimensionless variables.186J. Hristov / Particuology 7 (2009) 183- 192Table 1Materials used in the experiments .MaterialFraction (um)Ps (kg/m3)Ms (kA/m)dp-Averageb (pum)Magnetite (FesO4) sand200 -31547736257.5315- -400357 .5Ammonia catalyst KM-1, H. Topsoe500-6135100236.34557.5613-800707.5a See Hristov (2002).b Average sieve diameter used in calculations.Table 2Dimensionless power-law correlations.CorrelationsRangesεa = ApxXp:Xp = (ArSDz)/3 V RgMQMagnetite (200-315)2kgP-10.5≤Xp s 1;3.7 x 10-5 s RgmQ≤1.5x 10-2: (Ar)V/3 - 102.26 (54 data points)SD=0.02041: P<0.0001SD=0.02041;P<0.0001Magnetite (315- 400)2kgP-21≤Xp≤10;2x 10 4≤RgMQ≤1.3x 10 2:(Ar)/3 = 102.26 (62 data point)Magnetite (315- 400) 3 kgP-3Ea= 0.517X;15:x2-0.00185;1≤Xp≤10;2.5x 10 4 s RgmQ s2.7x 10 2: (Ar)1/3 - 141.97 (53 data points)Magnetite (315- -400) all dataεq= 0.964X0.14; R2 =0.8594; P< 0.00011≤Xp≤ 10;4x 10-4< RgmQ≤3x 10-2; 100<(Ar)1/3 < 145 (119 data points)Magnetite (all fractions)No common correlation is available. See1≤Xp≤ 10:4x 10-5≤RgmQ≤3x 10 2: 100<(Ar)13 < 145 (173 data points)the equations above and Fig. 3b.KM-1 (500 613); 2kgP-5Ea = 01280:x20:0021≤Xp s 100; 4.2 x 10-3≤RgmQ≤0.4: (Ar)/3 =21.339 (48 data point)? -0.27446: P<0.0001KM-1 (613- 800)2kgP-6Ea = 0.505X9.0; x2 -0.00275;1≤Xp≤50;2.5x 10 3≤RgmQ s0.2; (Ar)]/3 =280.762 (83 data points)KM-1 (all fractions)P-7εa = 0.530X0.065; R2 =0.1755;1≤Xp≤100;2x 10-3 s RMQ≤0.5; 100<(Ar)13 <280 (131 data points)All materials used in this work1≤Xp≤100; 3x 10-5 s RgmQ s 0.5 102 <(Ar)]/3 < 280 (303 data points)Note: All the relationships use△Dbl =0.1316.Table3Dimensionless EDF correlations.Ea = 8max - Aed exp(- QeXp):Xp = (ArSDg)W/3V RgmEDF-1dEa =0.685 - 0.245 exp(-0.431Xp);0.5≤Xp≤1;3.7 x 10-5 s RgmQ s 1.5 x 10-2: (Ar)]/3 < 102.26 (53 data points)x2 = 0.00126; R2 = 0.9391; P<0.0001Magnetite (315- -400)2kgEDF-2dEa=0.741- 0, 467 exp(- 0.645X):1≤Xp≤10;2x 10-4≤RgmQ≤1.3 x 10 2: (Ar)13 < 102.26 (62 data points)y2 = 0.0017: R2 =0.69834: P< 0.0001E-0719-0329exp 0392X.:lagnetite (315 -400)3kgEDF-3d1≤Xp≤10;:4x 10-5≤RgMQ≤3x 10-2: 100<(Ar)V/3 < 145 (53 data points)Magnetite (315- 400) all dataEDF-4d1≤Xp≤10;4x 10-5≤Rg.MQ≤3x 10-2; 100 <(Ar)/3 < 145 (19 data points)Es=0.729- 0.407 exp(-0.536XpicoNo common correlation is available (see1≤Xp≤10;4x 10-5 s RgmQ≤3x 10 2: 100<(Ar)1/3 < 145 (172 data points)the separate crelations)t, see Figs.4aKM-1 (500- 613)2kgEDF-5dEa =0.731 - 0.584 exp( _0.052Xp);1≤Xp≤100;4.2x 10-3≤RgmQ≤0.4x 10-2: (Ar)1/3 - 221.339 (48 data points)x2 =0.00072; R2 = 0.76334; P<0.0001KM-1 (613-800)2 kgEDF-6dEa = 0.696 - 0.205 exp( -0.092Xp1≤Xp≤50; 1≤RgMQ≤50; (Ar)13 = 280.762 (83 data points)x2 = 0.00275; R2 - 0.2258; P< 0.0001EDF-7dεa= 0.675 - 0.211 exp(-0.128Xp);1≤Xp≤50;2.5x 10-3 < Rexn2: (Ar)1/3 = 280.762 (131 data points)x2 =0.00265; R2 =0.178; P< 0.0001中国煤化工AII materials used in this workNo common correlation is available (s1≤Xp≤100; 3>346 data points)the separate correlations). See Figs. 4a.MHCNMHGand 5a and bNote: All the relationships use ODou =0.1316.J. Hristov / Particuology 7 (2009) 183- 192187(A) 0.72卜KM-1(613-800);G=2kg3.2.3. Numerical results0.70The numerical results were obtained with three functional rela-H=2387.2A1mtionships: (i) power-law relationships; (i) exponential decay func-0.6tion; and (li) asymptotic function provided by standard libraries of0.66commercially available software such as Origin Version 6.5 0.643.2.3.1. Dimensionless power-law relationships. In accordance0.62with Eq. (2b) the practical relationship can be expressed as0.60εa =f(Ar,ODbl)f2q(RgMQ), by assuring equal orders of magnitudesy= 0.505Xpof both sides of the relationship are taken into account so that0.58ε ~ 0(1). Hence, the general relationship was expressed as，0.56εa =Apd[(ArQDbL)/3√Rgme]" = ApaX"P.(6)0.5In this_ way, a new complex dimensionless variable Xp =0.50(Ar△DbL)1/3√RgMQ was conceived.20 3040 5Table 2 summarises correlations for each particular materialstested and all data collected in this study (see Fig. 3a-c). With the .Xp(-)highest solids load in the experiments performed with 3 kg Mag-Eq.P-1Eq.P-2netite (315- 400) successful results are provided by a linear datacorrelation y=aL + bu Xp (see Fig. 3c) too. The logarithmic plots in(B) 1.0Fe3O4Eq. P-5Fig.3b reveal almost equal slopes of the lines approximating the0.9( 200-315 )(315.400 )expansion of the coarse particles (KM-1 and magnetite (315-400))KM-1( 500- 613)while the points of magnetite (200- 315) show a slightly different0.8Eq.P-6(A) 0.70 F KM-1(500-613); G=2kgKM-1(613 -800 )).6一0.66上. H= 4774.4 A/m0.64).50.4LL1wwwL1www.LLwwwwwL0.11000.560.54y= 0.673- 11.94 exp(-0.588 X)(C) 0.66C Fe3O4 (315-400); G= 3kg ..64H=4774.4A1m .0 2468101214161820二0.62y=0.517xg:15 .(B) 0.80Fe3O4 (315-400); G=2kgFH=19097A/m0.75必.，.y=0.432. x:0;163= 0.436 + 0.037Xp0.650.520.55y= 0.742 - 0.69 exp(- 0.96Xp)Fig. 3. Examples of dimensionless power law correlations εa = ApdXPp. (a) Exampleofsuccesstuldata htting inspecitic casesresembling thoseperformedwiththe directft by power law functionga=AgxX"p showing different trends depending on the .中国煤化工range ofX (see the text).(c) Power-law and linear function data correlations obtainedwith 3 kg Fe3O4(315- 400) of solids showing comparable accuracy.Fig. 4. Examples ofdYHC N M H Gtion (EDForrelations interms ofXp = (Ar△D/ KgMQ.[a) specinc correlations demonstrating satisfac-tory data fttings, comparable to those performed by the direct scaling (Fig. 3). (b)EDF shows better data fttings than those by PLF.188J. Hristov/ Particuology 7 (2009) 183- 192trend (see the equations in Table 2). Besides, these lines correspond(A) 0.75to different ranges ofXp (see the logarithmic abscissa ofFig 3b). TheFe3O4( 200 - 315 ); AII data pointsdimensionless variable Xp (Eq. (6)) and the relevant problems are0.70G=2kgexplained in the next section.G=3kg3.2.3.2. Complex dimensionless variable Xp. Among the constituentsofXp (see Tables 2 and 3), the only varying component is RgmQ, while0.65the Archimedes number Ar and△Dol = tg(ax/2) are non-varyingparameters for a given material. In turn, the exponent 1/3 might8.5%be replaced mechanistically, for instance, by 1/2 or 1/4 providing0.60the same order of magnitude of Xp. However, with Aril3~dp; itwis evident that 1/3 is more physically adequate exponent. Withsimultaneous variations of both the fluid flow and the field inten-0.55Eq. ( EDF-1d)sity Xp becomes strongly non-linear, since RgmQ = Q2 /MsH≈Q2/H2y= 0.685 - 0.245 exp(-0.431Xp)(remember Ms and H have equal dimension A/m!). Consecutively,since Ms= const. we have√RgMQ = Q/vH which is equivalent to0.500.60 0.65 0.70 0.75Xp= Q/vH. In turn, we have a complex dimensionless variableXp = dp(Q/vH). The principal requirement satisfied by Xp is itsEa (experimental )order of magnitude which varies fromXp ~ 0(1) toXp ~ 0( 100)(seealso Fig. 4a and b; Tables 2 and 3) with narrow sub-ranges (seeB) 0.75KM-1( 613 -800 )10.8%，1below) which enable correct scaling ofε~ 0(1).(A 0.75厂 KM-1(613- 800);G=2kg. H= 2387.2 A/m8.9 %1.2 %■y=0.255 X 0.26-17 %P 0.65, y= 0.640- 8512.57 C: XpC=0.5 .Eq. ( EDF- 6d)! y= 0.757 - 0.492 exp (- 0.046 Xp)y= 0.696 .0.205 exp(- 0.092 Xp )0.450.50 0.55Eεa (experimental )Fig. 6. Parity plots of EDF crrelations-examples.1060 70Xp(-)Because √RgmQ ~ Q/VH the lowest Xp values correspond to .(B) 0.80厂 Fe304(315.400); G=3kglow fluid flow rates and high field intensities, and then stronglymagnetically controlled bed regimes (MSB or smooth fluidizationH =23872A1m0.75with slits). With increasing the fluid flow the field H and the grav-ity effect (expressed by Ar) remain but do not vary, Xp increaseswith diminishing field effects and the upper values of the Xpcorrespond to fluidized regimes (bubbling or spouting). In this con-text, we especially refer to the range of 0.5