DETERMINATION OF MARGINALLY STABLE ZONE OF GAS-SOLID MAGNETICALLY FLUIDIZED BEDS

Journal of Hydrodynamics Ser. B A(2002 )7-14China Ocean Press, Beijing- Printed in ChinaDETERMINATION OF MARGINALLY STABLE ZONE OF GAS-SOLID MAGNETICALLY FLUIDIZED BEDSGui Ke-ting, Zhang Hui, Shi Ming-hengThe Key Laboratory of Clean Coal Power Generation and Combustion Technology of Ministry of Education SoutheastUniversity Nanjing 210018, ChinaReceived Oct. 11, 2000)ABSTRACT: The void fluctuation of magnetically fluidized beds 2. THE VOID FLUCTUATION IN MFBSas analyzed and their maginally stable zone was determined. The 2.1 Two-phase model of MFBsanalysis was based on the two-phase model of magnetically fluidizedIn order to study the void fluctuation on mfbs, webed and wave theory. The marginally stable zone determined by thispaper matches well withDetrimental resultsneed to model MFBs with a void fluctuation equationThe equation can be derived from the following two-phasKEY WORDS: fluidized bed magnetic field, marginally stalmodel of MFBs which was developed by Gui et al (4/actuation(1a)1. INTRODUCTIONThe Magnetically Fluidized Beds( MFBs )constitute xa new technology in the application of fluidization. As the+[(1-∈)]=0(1b)external magnetic fields suppress gas bubbles and improvethe contact between gas and solids in the beds MFBs willEfind many applications in industry, such as filtration, Pg at+uj十paration and synthesis reactions etc. Hence, it is impor-123)(1c)tant that mfbs can achieve stable fluidization with the aidof the magnetic fields. This type of MfBs has been calledMagnetically Stabilized fluidized Beds( MSBs ) by some p(1-eX31u ar=nffI-e )p:+nfneauthors, and numbers of research works on msbs havebeen reported. Among these works, the most distin-uished was that by rosensweig 2. His criterion of the(i=123)(1d)magnetically stable fluidization sets up a sharp theoreticalboundary between stable fluidization and unstable onewhere e is the values of voidage u the gas velocity,vHowever, because of the random property of gas-solid the particle velocity Ey the stress tensor of fluid, Pg , psflows in MFBS the experimental results reported by somethe density of gas and solid respectively nfi the dragauthors showed that it was not a sharp boundary but a force between gas and particles, nfmi the magnetic forcemarginally stable zone located between stable and unstable resulted from the external magnetic field and gi the accel-fluidization 33. In this paper, therefore we will analyze eration owing to gravity. The subscripts (j)=1 2 3void fluctuation of MFBs with the help of wave theory, denote three components in the direction X,Y,Z.Theand determine the zone of marginal stability based on the expressions of Ei, nfi and nfmi are also developed byFinally we will compare the theoretical zone of marginal Guf sassumption of stable propagation of void wave in the be中国煤化工CNMHGability with experPi oi(2)8nf:=(1-6)18usEu-vi )-(1-e oog: (3) tion takes place in gas-solid system each dependent vari-Considering the situation in which a small perturbaable in Eqs. (5), (6)and(9 )can be expressed as the2(1-c)M(4) sum of an average value and a small perturbed value,thatIn Eqs. (2)-(4), Pe is the gas pressure /u the vis- u;= U; +ui(1la)cosity do the particle diameter , di the Krone symbols e)the correcting coefficient of drag force due to voids, v:=v+(11b)uo the magnetic conductivity ,xs the magnetic susceptibility of solid and M, the solid magnetization. Substitutinge =Eo+e(1le)Eqs. (2)-(4)into Eq (1),we obtainf i+ Fi+ dfi(1ld)(5)where dfi is the effects of the small perturbation on gassolid system which can be expressed as(1-ε)(6)dfi,, dfif(12)uupgat +uj1Eq(1118and ignoring all the small quantities of order greater tha(7)one, we have18Pd6gE(14)3+(3-2e)yothe external forces exerted on gas and solids in MFBs. Pg at +dIn Eqs. (7)-(8), the terms on the right hand-ps at/(15Let fgi and fsi in Eqs. (7)-(8)represent the externalforces respectively and subtract(8)from(7 ), then weTaking the derivative of Eq 15 )with respect to rhavethe Eqs. 13 )and( 14 )with respect to t and r:, respectively and summing them together we have(B+,B)+23r(tat2fa-fs→x-f(9)aepUW)x0)(16)The external force fi is a function of u;, vi, e and中国煤化工that isCN MH Give of Eq. 12 )with re-spect to ri lead tof=从n,n,∈,)(10)x0)a(1019f).a2.2oid教equation in MFBs9afi Ui ai Vi dfithe void fluctuation equation of the MFBsPP),02∈cP2Vx ae /ar ar: d r(17)psi)+ ar drjAccording to the wave theory the force df; resultedfrom the small perturbation can be considered as the furtion of voidage e gas velocity u; and total flux j; of gasrade /arand solid phases 5, that isa2f i=f(ji e,ui)(23The relationship between total flux j; and voidageIn the above equation letcan be expressed asu i(1-ε)(19)A*=g+1-E0(24)Hence, it follows thatU1,P)(25)=(U;-V;)∈0xc/(26)afi dfi afB1(1,Of(27)(20)where Uoi defined by Eq. 25 )is in accordance withaverage velocity of kinetic wave5].Furthermore,defineSubstituting Eq. 20 ) into Eq (17)yieldthe relative velocity of kinetic wave asx0)=-102+2[U;-εoC oiUoj-Aiji()()+x00n)n1(21)(U1-V夏U1-V).0f(28)In Eq. 21), the coefficient of gradient ar. correWith the notations of Eqs、24)~(28),Fqs、23sponds well with the expression of the velocity of continu- can be expressed asous wave5], that isf中国煤化工CNMHGwhere Ube considered as the velocity of the continEq. 29) is the void fluctuant equation derived fromuous wave. Substituting Eq( 21)into Eq (16), leads to the two-phase model of MFBs. For this partial differentialequation it is not straightforward to obtain an analytic so- The square a2 of wave frequency in Eq. 35)shouldlution that describes the stability of MFBs. The next sec- be positive. The square w2 of relative velocity of voidtion will discuss the simplification of Eq. 29wave, therefore, must be located between C2 for thenetic wave and u for the continuous wave, If u>3. Marginally stable Zone of MFBsof stabilized fluidicthen u o > w>C. Hence, in Eq. 34 )is alwaysIn order to discuss the stability of MFBs with the greater than I and a is positive. This means that the pervoid fluctuant equation, we need to simplify Eq. 29 ). turbation of the void grows without any limitation and theBecause the direction of fluid flows in the vertically flu- state of fluidization is unstable. On the other hand, if u 2idized bed is vertical, we can assume r:= r w>u. Henceomit the suffix ij)in Eq. 29 ) With these simplifications, Eq (29)smaller thanI and a is negative This means that the disturbance of void will decay in the propagation and the flua2eidized beds will be maintained in stable stateThe relative magnitude of velocities of kinetic waveand continuous wave therefore can be considered as thecriterion of determining the stability of MFBs. If C>(30 )u2 MeBs, is in stable fluidization, otherwwise, It may bein unstable stateFor this simplified void fluctuant equation the gen-3.I Determination of marginally stable zoneeral solution isAccording to the analysis above, the condition ofstabilized fluidization can be expressed ase=npar+i【-正)(31)(36Eq (31)shows that e is a void wave resulted fromSubstituting Eq. 28 )and (3)into Eq. 36)yieldsvoid perturbation and propagating along the direction Zquency of the void wave the propagation velocity of the 2U..-A-U2>oIn Eq (31), the symbols w, W and a represent the fre(37)void wave and the amplifying factor of wave amplitudeIf we substitute the expressions of Uno U. and A intorespectively. The value a should be negative to guarantee Eq (37), we have an expression of the stablility condithat the wave amplitude is not infinite with evolutiontime. The relative velocity w of void wave and relative tionvelocity ue of continuous wave can be defined by subtractau egUing Uo from the void wave velocity W and the continuousU+Eowave velocity U respectively that is(32)(33)U(38)Substituting Eq (31 ) into Eq (30 )and separatingthe real part from the imaginary part we obtain the ex- In the fluidized beds, the following relationship exists be-pressions of a andtween中国煤化工(34)=CNMHGwhere u, is the terminal velocity of a particle. u, can becalculated by the Stokes drag formula because the size of4#方数据(35) particle in MFBs is small. n in eq, (39) is the richardson-Zaki constant With the value of about 2- 6]. Sub11stituting Eq 39 )into Eq 22), the velocity of continu- two-phase flows in MFBs, the particle velocity V is lessous wave in MFBs can be given asthan the half of gas velocity Ul 4 we supposeUn=U+ukan-∈(n+1)](40)(46)nother important parameter in Eq.( 38)To simplify the stablility criterion( 42), let us considx de/az). The force related to az in MFBS is magnetUtwo extreme conditions of v-0 and vforce, which is expressed by Eq (4 ). So the expressionIfv→0, the stablility criterion of I→0isXdE/aDD(x.)于tD(u)+1]f2161-0)Md dE/az) 3(41)Ifv→,, the stablility criterion of v→isSubstituting Eq (40 )and Eq (41 )into Eq. 39)and dividing U on the both sides of Eq (38 ) we have a Dm =[D(Xs ]I D(u,)]+ D(u,)+ystability criterion in the dimensionless form2-to/+uEln-E n+1)yeU(48)UpLet D be the abscissa and the ordinate, then theU2pol3(3-20)critical curve Co of stable fluidization of v-O is drawn inFig. I with Eq (47 ), and the critical stable curve C, of V{U+uEn-∈d(n+1)]}>0(42)is also drawn in Fig. I from Eq. 48). The rangeon the left hand side of curve Cp is the unstable fluidizedrange, that on the right hand side of Co is the stable oneand the range between two curves can beD(x)=3+(3-20)(43)marginally stable region of the MFBs. The zone ofmarginal stability reported from experiments by some authors is also shown in fig. 1 with the horizontal shadowD (ur)=veI n-edn+ D)(44)We see that the range between curve c and thecurve C, matches the marginally stable zone obtained fromexperiments 33Dm -o yau(45/ To simplify the above expression/letwhere D(xs )is a dimensionless group related to the suDeptibility xs,D ur )a dimensionless one related to the gm SD(x)]2terminal velocity u, and Dm also a dimensionless one butrepresenting the ratio of kinetic energy to magnetic ener- G中国煤化工(50)The symbol V in Eq. 42 )represents the average veCNMHGlocity of particles. Since the characteristics of particles or gncan pe considered as the dimensionlessgas in different fluidized beds are not the same, the aver- number related to the magnetic energy and voids, Gu, inage velocity V of particles may be anywhere between 0 Eq. 50 )is that related to the terminal velocity of partiand maximum particle velocity Vmax in different fluidized cles. With these dimensionless numbers, the criterion ofbeds. Accondn?6 the results of numerical simulation of marginally stable zone of MFBs can be expressedLet us compare the criterion Eq. 52 )with the crite-rion( 51)of the marginally stable zone derived in this pa-per. There are two differences between the two criteriaUnstable20 I fluidizationFirst, the criterion( 52 )is not a zone of marginal stabilStablefluidizationidization. The boundary is located in the zone of marginalstability shown in Fig. I with dotted line. Second, thereare two additional parameters of particle terminal velocityFixed bedu, and equivalent density p'in the criterion( 51According to the analysis of Geldart about fluidizetion in general, there are two important factors affecting80100the stability of fluidized beds. One is the difference ofdensities between the two phases and the other is the sizeof particles 7). Both effects of p, and Ps are included inFig.I The marginally stable zone of magnetically fluidized bed the equivalent density pI (51). The effects ofparticle size are reflected by the particle terminal velocityu, of criterion(51). The dimensionlestable(51a)in Rosensweig s criterion, however, only include thepartidensity pg and the effects of particle size are also nottaken into account in rosensweig s criterionmarginally stable(51b)4. EXPERIMENTS4. 1 Experimental faciliGm<(G2-g1Iy2UIn order to demonstrate the marginally stable zonederived from the wave theory we report an experimenton the stability of MFBs in this paper. A schematic diaunstable(51c)gram of the experimental facilities is shown in Fig. 2. Thebed column fabricated from plexiglass is 100mm in diamfrom rosensweig derived the stablility criterion of MFBS eter and the height of fixed bed is 115mm. The ferromag-m the basic equations of MFBS 21netic particles are made from cast iron and the averagesize of particles is 0. 56mm. To generate external magnetN(52a)ic field, a coil with 40Omm in length is used and the bedNN,= 1 marginally stable(52)is located in the middle range of the coil in which themagnetic flux is homogenous. The direction of the mag-netic field is along the axis of fluidized bed in which theNnN >1 unstable(52c) stable fluidization is easily obtained 2Based on the stochastic analysis to the void fluctuantsignal in MFBs, Gui et al. proposed the experimental cri7=(53)terion of stable fluidization in MFBs. This criterion in-uoMfrequency F, of auto-correlation function of void fluctuant4(3-2∈0中国煤化工 square-error o2ofXt1+(1-:0),](54)SignaC N M H Gagnetically fluidizedare measured with a mini-capacitance probe excluding theIn Eq (53), Nm is a dimensionless group represent- influence of magnetic field. The square-error o2and theng the ratio of kinetic energy and magnetic energy and N, auto-correlation function Rxx'( t )of void fluctuant signalin Eq. 53) is also a dimensionless group but related to X( t), are calculated by following equations and dominant frequency Fa are obtained from the reciprocal of de-instance, to H of Curve 5(88)as shown in Fig 4, Fa reaches its maximum valuebut o falls to a very small value. If we increase the magnetic intensity further, Fd decreases steeply with the o decreasing continuously. This indicates that the bubbles almost disappear and bed turns to particulate fluidization ataIIIIIIIIlthe moment of the void fluctuation becoming very smallin both amplitude and frequency. Consequently this extransition point from aggregate fluidization to particulatefluidization. This transition point can be used as the experimental criterion to distinguish the stable and unstablefluidizationtion system 2-Capacitance meter 3-Capacitance probe :4Fluidized beds :5-Ampere meter 6-Direct supply ; 7-Coil ;8-Valve :9-Ori-fice flowmeter 10-Bloweig 2 Schematic diagram of the experimental facilitiesay time in which the first peak of Rrr(t )beyond null XppearsN△X-fXt)(55)Rxx(t)=々!*XH/kAX■0 1(56)Fig 3 Relationship between a and HIn Eqs. (55 )and 56 ), the X: is the void fluctuantUsignal collected, EL X( t)] the average value of X,, NU1-1.4;2-1.53;3-1.654-1.78;51.8:62.0the amount of X; and the subscripts x j)the order of themeasurement valuesSix points shown in Fig. I correspond to six transi4.2 Experimental criterion of stable fluidization in MFBs tion points in Fig 4. They are located in the marginallyFig 3 and Fig. 4 illustrate the relationship betweenstable zone determined by wave theory. The theoreticalzone of marina stability of MFBs derived in this papermean square value a2, dominant frequency Fd and mag- therefore is demonstrated to coincide with the experimen-netic intensity H respectively. Six curves in the figurcorrespond to six different gas superficial velocities. Thevarying pattern of Fd with increasing magnetic intensity H5. CONCLUSIONis illustrated in Fig. 4: Fd rises slowly with increasing Hat first then it falls sharply after a certain point and tend中国煤化工heoy, we analyzed thevoidto zero however, fig 3 shows that g decreases continumaICNMHGcriterion of determiningously. This indicates that as H increases, the volume of the marginal stable zone includes both effects of ps and pgbubble turns to smaller and the quantity of bubblesand the effects of particle size. These two effects are im-creases,i.e., Fd increases and o decreases. The effectportant to the stability of gas-solid fluidization accordingof theetic field, therefore is demonstrated to cause to Geldert's analysis. The marginally stable zone deterbubbles splftthad las the magnetic intensity H increases to mined by the wave theory coincides well with the experi6. Foscolo P. U., Gibilaro L. G. 1984Predictive criteri.on for the Transition between Particulate and Aggregate fluidiza-2.5tion, Chem. Eng. Sci., 3( 2), 1667-16787. Geldart D., 1973: Types of Gas Fluidisation, Powder Tech-08. Gui Keting, Wang Rongnian, 1994: Determination of CriticalTransition Point by Void Fluctuant Signal in a Magnetically Fludized Bed Journal of Chemical Industry and Engineering(5),585~594.( in Chinese)9. Gui Keting, Zhang Xinyu Wang Rongnian, 1989: The Meaof the Stability of a Magnetized Gas-Solid Fluidized BedWith Capacitance Probes In: Proc. of the 2nd Int. Symp.orMultiphase Flow and Heat Transfer, Xi' an, China, HemispherePublishing Corp., 1271-12798H/kAXHmental results. The coicidence between theory and experiments proves that the wave theory is an important meansto analyze the gas-solid flows in MfBsREFERENCEStal and Practical Development of Magnetofluidized beds a Re-view, Powder Technol, 611),3-412. Rosensweig R. E, 1979: Magnetic Stabilization of the State ofUniform Fluidization, Ind. Eng. Chem. Fundam.,1& 3), 2603. Cohen A.H., Chi T. 1991: Aerosol Filtration in a Magnetical-ly Stabilized Fluidized Bed, Powder Technol., 6(1),1474. Gui Keting Chao Jianghui, Shi Mingheng et al., 1997: A Two-Phase model of Gas-Solid Magnetically Fluidized Beds, Joumalof Southeast University, 275), 36-45. in Chinese5. Wallis G, B. 1969: One-Dimensional Two-Phase Flow, McGraw-Hill, New York 184-356中国煤化工CNMHG