Modeling of segregation in magnetized fluidized bed with binary mixture of Geldart-B magnetizable and nonmagnetizable particles☆

Quanhong Zhu ,Hongzhong Li *,Qingshan Zhu 3,*,Qingshan Huang

1 Institute of Process Engineering,Chinese Academy of Sciences,Beijing 100190,China

2 Qingdao Institute of Bioenergy and Bioprocess Technology,Chinese Academy of Sciences,Qingdao 266101,China

3 University of Chinese Academy of Sciences,Beijing 100049,China

Keywords:Magnetized fluidized bed Binary mixture Segregation Model Agglomerate Force balance

A B S T R A C T For the magnetized fluidized bed(MFB)with the binary mixture of Geldart-B magnetizable and nonmagnetizable particles,the magnetically induced segregation between these two kinds of particles occurs at high magnetic field intensities(H),leading to the deterioration of the fluidization quality.The critical intensity(H ms)above which such segregation commences varies with the gas velocity(U g).This work focuses on establishing a segregation model to theoretically derive the H ms–U g relationship.In a magnetic field,the magnetizable particles form agglomerates.The magnetically induced segregation in essence refers to the size segregation of the binary mixture of agglomerates and nonmagnetizable particles.Consequently,the segregation model was established in two steps: first,the size of agglomerates(d A)was calculated by the force balance model;then,the H ms–U g relationship was obtained by substituting the expression of d A into the basic size segregation model for binary mixtures.As per the force balance model,the cohesive and collision forces were 1–2 orders of magnitude greater than the other forces exerted on the agglomerates.Therefore,the balance between these two for ceslargely determined d A.The calculated d A increased with increasing H and decreasing U g,agreeing qualitatively with the experimental observation.The calculated H ms–U g relationship agreed reasonably with the experimental data,indicating that the present segregation model could predict well the segregation behavior in the MFB with the binary mixture.

1.Introduction

The magnetic field has long been employed to improve thegas–solid fluidization quality and contact performance in the bubbling fluidized bed(BFB),thus creating the famous magnetized fluidized bed(MFB)[1].Both Powder Technology[2]and China Particuology[3]have published special issues on this subject.Previous research[4–12]indicates that apart from purely Geldart-Bmagnetizableparticles,the binary mixture of Geldart-B magnetizable and Geldart-B nonmagnetizable particles could also be used in the MFB,bringing this technique to a wider range of potential applications[4,6].

For the MFB with the binary mixture,the magnetic field does not necessarily improve the fluidization quality[12],because the magnetically induced segregation between the magnetizable and nonmagnetizable particles occurs under certain operating conditions.Under such circumstances,the fluidization quality as well as the gas–solid contact performance in the MFB deteriorates.Consequently,if the original intension of applying the magnetic field to the BFB with the binary mixture is to improve the fluidization quality,the magnetically induced segregation should be thoroughly explored and carefully avoided by controlling the operation parameters.

Under the action of the magnetic field,magnetizable particles exist in the form of agglomerates[12–14].Thus,the original binary mixture of magnetizable and nonmagnetizable particles in an MFB turns into one of agglomerates and nonmagnetizable particles.Moreover,the magnetically induced segregation mentioned above in fact denotes the segregation between the agglomerates and nonmagnetizable particles with the former playing the role of jetsam and the latter working as flotsam.

Furthermore,there mainly exist two modes to operate the MFB:Magnetization FIRST and Magnetization LAST[15–18].The Magnetization FIRST refers to applying the magnetic field to the fluidized bed first and then gradually increasing the gas flow to the desired velocity(Ug).The Magnetization LAST involves introducing the gas flow first and then gradually increasing the magnetic field intensity(H)to the desired value.

The magnetically induced segregation occurs under both operation modes in the MFB with the binary mixture.Under the Magnetization FIRST mode,the segregation usually commences after Ugexceeds the minimum bubbling velocity(Umb)[10,19,20].In this case,the segregation arises mainly from that the nonmagnetizable particles could no longer be trapped in the network formed by the agglomerates and are entrained to the upper part of the MFB by the high gas flow[21].Several experimental methods[9,22–26]were proposed to measure Umb,which increases as the applied H increases.In addition,two mathematical models were established to calculate Umb.The model proposed by Arnaldos et al.[4]is based on the Hagen–Poiseuille equation;it argues that Umbincreases exponentially with the increase of H.On the other hand,the model built by Ganzha and Saxena[5]rests on the Davies and Taylor[27]formulation of the bubbling velocity.The calculation implies that Umbis proportional to the 1.18th power of H.The calculated Umb–H relationships provide useful guidance on how to operate the MFB under the Magnetization FIRST mode.

Compared with the Magnetization FIRST,the Magnetization LAST mode was less adopted to operate the MFB with the binary mixture,let alone the segregation study under this operation mode[6].In our previous work[12],the magnetically induced segregation under this mode of operation was experimentally explored for the first time.The magnetizable and nonmagnetizable particles were iron and limestone particles,respectively.The reason why segregation occurred was found to differ from that under the Magnetization FIRST mode.In this case,the segregation results mainly from the size growth of the agglomerates with increasing H.When H is increased beyond a critical value(Hms),the agglomerates grow too big and heavy to mix well with the nonmagnetizable particles and begin to sink to the lower part of the MFB,eventually inducing the segregation.Values of Hmsat different gas velocities for the iron–limestone binary mixture(Admixture-I in[12])were experimentally measured,which indicates that Hmsincreases with the increase of Ug.At a higher Ug,the driving force for the mixing between the agglomerates and nonmagnetizable particles is greater.Therefore,the agglomerates need to grow bigger to sink to the lower part of the MFB and that requires a greater Hms.

The Hms–Ugrelationship is of great importance for avoiding segregation when the MFB with the binary mixture is operated under the Magnetization LAST mode.How ever,to the best of our know ledge,there is still no mathematical model to predict such a relationship,thus hindering the commercial application of the MFB to various chemical and biochemical processes.To resolve this problem,the present work aims to theoretically derive the Hms–Ugrelationship by establishing a magnetically induced segregation model.

2.Theoretical Analysis

2.1.Further explanation of segregation under the Magnetization LAST mode

As stated in Introduction,under the action of the magnetic field iron particles form agglomerates.Therefore,the magnetically induced segregation in essence refers to the segregation between the agglomerates and limestone particles.Furthermore,under the Magnetization LAST operation mode the segregation of the agglomerate–limestone binary mixture arises mainly from the size grow th of agglomerates with increasing H.Hence,such segregation in essence belongs to the type of size segregation from the segregation perspective.Based on this analysis,the magnetically induced segregation model could be established by quantitatively depicting the segregation with the basic size segregation model for binary mixtures.

2.2.Quantitative explanation of the Hms–Ug relationship

In order to further elaborate the modeling idea,the Hms–Ugrelationship should be quantitatively explained from the segregation perspective.Note that the experimental method of determining Hmsat a given Ugwas based on the pressure drop measurement,which is illustrated thoroughly in our previous work[12].As shown in Fig.1,Hmscorresponds to the point where ΔPup(pressure drop over the upper half bed)starts to decrease because the heavier iron particles in the form of agglomerates begin to sink to the lower half of the MFB.

The segregation degree of the agglomerate–limestone binary mixture could be quantitatively represented by the mixing index(M)whose definition[28]is given by

w here xJ,updenotes the mass fraction of jetsam in the upper half of the bed while xJ,averepresents the average mass fraction of jetsam in the whole bed.The value of M ranges from 0 to 1.0;the greater it is,the smaller the segregation degree.

Values of M under different combinations of(Ug,Hms)were measured and listed in Table 1[12].Obviously,the Hms–Ugrelationship is approximately the relationship between H and Ugat M=0.9.Note that due to the restriction of our experimental device,values of M in Table 1 deviate slightly(＜2.3%)from 0.9.

Table 1Values of M at different combinations of(U g,H ms)for the agglomerate–limestone binary mixture(Admixture-I in[12])

2.3.Basic size segregation model for binary mixtures

2.3.1.Segregation gas velocity

For a given binary mixture,the sizeratio(dJ/dF)and density ratio(ρJ/ρF)between the jetsam and flotsam are certain;therefore M depends solely on Ug.As shown in Fig.2[28],M increases from 0 to 1.0 with the increase of Ug,suggesting that segregation predominates at low gas velocities while mixing prevails at high gas velocities[28–31].In this paper,Ugcorresponding to M=0.9 is defined as the segregation gas velocity(Ugs).

Fig.2.Variation of M with U g for binary mixtures[28].

For a certain type of binary mixtures,ρJ/ρFis fixed;Ugsvaries as a function of dJ/dF.The basic size segregation model required to describe the segregation behavior of the agglomerate–limestone binary mixture in the MFB in fact is the Ugs–dJ/dFrelationship.

2.3.2.The Ugs–dJ/dFrelationship for binary mixtures

For different kinds of binary mixtures,the Ugs–dJ/dFrelationships are different given that the values of ρJ/ρFare different.How ever,the universal function between Ugsand dJ/dFwith ρJ/ρFas the controlling parameter is still unknown due to that the study on segregation is severely inadequate in the literature[28].As to the agglomerate–limestone binary mixture studied in this work,the Ugs–dJ/dFrelationship is no exception unavailable.

In order to solve this problem,it is assumed here that the agglomerate–limestone binary mixture has the same ρJ/ρFwith the iron–limestone binary mixture.Hence,the Ugs–dJ/dFrelationship(Eq.(2))derived from the latter binary mixture could be extended to depict the segregation behavior of the former.The experimental method and processof deriving Eq.(2)will be stated in detail in Appendix A.

2.4.Modeling idea and steps

For the agglomerate–limestone binary mixture in the MFB,when segregation occurs,the jetsam and flotsam are the agglomerates and limestone particles,respectively.When Eq.(2)is applied to describe this segregation behavior,dJand dFare practically the sizes of the agglomerates(dA)and limestone particles(dL),respectively.Hence,Eq.(2)can be rewritten into

For the agglomerate–limestone binary mixture,dLis fixed while dAvaries as a function of both H and Ug.If the specific function between dAand(H,Ug)could be theoretically derived,by substituting it into Eq.(3)we can obtain the relationship between Ugsand H.The obtained Ugs–H relationship is right the relationship between H and Ugat M=0.9 asper the definition of Ugs.Furthermore,according to the explanation in Section 2.2,the Ugs–H relationship is exactly the Hms–Ugrelationship,which is the ultimate goal of this work.Fig.3 summarizes the principal idea and steps of establishing the magnetically induced segregation model.

Based on the above analyses,the difficulty in obtaining the Hms–Ugrelationship arises mainly from determining the specific function between dAand(H,Ug),which will be derived in the following section by modifying the force balance model originally proposed by Zhou and Li[32,33].

3.Derivation of d A by Modifying the Force Balance Model

3.1.Assumptions

As a preliminary mathematical model,the following assumptions are made:

(1)The agglomerates of Geldart-B iron particles are spherical and of the same size.Consequently,their size can be simply represented by the diameter(dA).

(2)The agglomerates have the same density(ρA)with the iron particles(ρI).

(3)The agglomerates have the same magnetic permeability(μA)with the iron particles(μI).

3.2.The modified force balance model

The original force balance model was established to estimate the equilibrium size of the agglomerates formed by cohesive particles in a gas- fluidized bed[32,33].It basically assumes that the collisions between agglomerates determine their equilibrium size.Whether the size of the agglomerates increases or decreases is the consequence of such collisions.As shown in Fig.4,when two agglomerates come into collision,there will be three possible results:(a)the two agglomerates coalesce into a big one,leading to the increase of the agglomerate size;(b)the two agglomerates are separated with both remaining intact;the agglomerates keep their original size before the collision;(c)the two agglomerates are separated with one or both breaking up,resulting in the decrease of the agglomerate size.

Which of the three cases will happen depends upon the forces acted on the two colliding agglomerates.These forces include the gravitational force(Fg),the buoyancy(Fb)and drag force(Fd),the cohesive force(Fv)and collision force(Fc)between the two colliding agglomerates.Since Fgand Fbare constant for a given agglomerate,these two forces could be simply combined by their difference(Fbg=Fg?Fb).Among these forces,Fbgand Fvare the reasons for the coalescence of two colliding agglomerates while Fdand Fcare the causes of the breakup of agglomerates.If Fbg+Fv＞Fd+Fc,case(a)will happen.On the other hand,if Fbg+Fv＜Fd+Fc,case(c)will occur.Only when Fbg+Fv=Fd+Fc,will case(b)happen.

This force balance model will be modified in the following section to calculate the equilibrium size of the agglomerates formed by Geldart-B iron particles in the MFB.

3.2.1.The difference between gravitational force and buoyancy

As it is in the original force balance model[32,33],the difference(Fbg)between the gravitational force(Fg)and the buoyancy(Fb)can be calculated by

where UmfLdenotes the minimum fluidization velocity of the limestone particles.

Fig.3.Idea and steps of establishing the magnetically induced segregation model.

3.2.2.The drag force

According to Zou[34],dAcalculated with different drag models is basically the same.Hence,the drag model originally selected by Zhou and Li[32,33]is adopted here,which is given by

w here εbrepresents the bed voidage.For the MFB with Geldart-B iron particles,εbvaries hardly with H and depends mainly on Ug[14].At a given Ug,εbcould be calculated by

w here ΔPb(the bed pressure drop)could be measured by the pressure transducer and Δh(the height of the bed material)could be read directly through the rule attached to the transparent bed wall.According to the regression analysis of the experimental data[14],the relationship between εband Ugcould be correlated by

Fig.4.Three possible results of collision between two agglomerates.

By substituting Eq.(7)into Eq.(5),the explicit expression of Fdas a function of H,Ug,and dAcould be obtained as follow s:

3.2.3.The cohesive force

In the original force balance model[32,33],the cohesive force between the two colliding agglomerates is principally the van der Waals force.How ever,for the agglomerates of Geldart-B iron particles,the magnetic field induced attractive force will predominate over the van der Waals force and become the primary cohesive force.Therefore,the correlation for calculating Fvmust be modified accordingly.Eq.(9),initially proposed by Pinto-Espinoza[35],was most adopted in the literature[36–38]to calculate Fv.This correlation was established during dealing with the liquid–solid magnetized fluidized bed.How ever it is also applicable to the gas–solid magnetized fluidized bed because the physical model based on which Eq.(9)was derived does not restrict whether the space between two magnetizable particles/agglomerates is filled with gas or liquid.As long as the magnetic permeability of the gas and the distance between magnetizable particles/agglomerates could be obtained,Eq.(9)could be used to calculate Fv.

where χerepresents the effective magnetic susceptibility of the agglomerates and can be calculated according to

where χAdenotes the physical magnetic susceptibility of the agglomerates,whose definition is given by

where μAis the magnetic permeability of the agglomerates,which is supposed to equal that of the iron particles(μI).The values of μIat different magnetic field intensities were measured by the vibrating sample magnetometer(LDJ9600,LDJ Electronics,USA).The relationship between μIand H could be correlated by

By substituting Eqs.(11)and(12)into Eq.(10),we obtain the explicit expression of χeas a function of H and Ug:

Back to Eq.(9),VAtherein represents the volume of the agglomerate and can be calculated as per

B in Eq.(9)represents the magnetic flux density and can be calculated according to

whereμbrepresents the permeability of the bed material and can be calculated as per[5]

By substituting Eqs.(7),(12),and(16)into Eq.(15),we get the explicit expression of B in Eq.(9)as a function of H and Ug:

rAin Eq.(9)denotes the distance between the centers of two colliding agglomerates.The relationship between rAand εbcould be represented by

in which εb0represents the voidage of the settled bed.

By substituting Eq.(7)into Eq.(18),we obtain the explicit expression of rAas a function of H and Ug:

By substituting Eqs.(13),(14),(17),and(19)into Eq.(9),we acquire the explicit expression of Fv,which is given as Eq.(B1)in Appendix B.Since χe,VA,B,and rAare all functions of H,Ug,and dA,Fvis also a function of H,Ug,and dA.

3.2.4.The collision force

As it is in the original force balance model[32,33],the collision force could be calculated by

where Vrrepresents the relative velocity between the two colliding agglomerates.There still exists a great deal of controversy about the calculation of Vr.Four different methods were seen to calculate Vrin the literature.Eq.(21)was adopted by Zhou and Li[32]in the original force balance model.The calculated Vris greater than Ugin some cases,which is not consistent with the experimental data.To overcome this deficiency,Xu[39]proposed Eq.(22),the drawback of which lies in that it lacks a sound theoretical basis and there is no correlation to calculate ξ appearing therein.The other two methods simply assume Vrto be a constant:Morooka[40]reckons that Vrequals the minimum fluidization velocity(UmfA)of the agglomerates,while Lu[41]assigns Vra fixed value of 0.121 m·s?1.

w here UmfAcan be calculated as per[28]

In this work,Eq.(22)was finally selected to calculate Vrwithξ being equal to 0.1.By substituting Eqs.(22)and(23)into Eq.(20),we get the explicit expression of Fcas a function of H,Ug,and dAas follow s:

3.2.5.Parameters in the force balance model

Many parameters with a fixed value appear in the above equations,whose meanings and values are summarized in Table 2.

Table 2Parameters with a fixed value appearing in the force balance model

4.Results and Discussion

4.1.Effects of Ug,H,and d A on the forces exerted on the agglomerates

Figs.5–7 show the effects of Ug,H,and dAon the forces exerted on the agglomerates,respectively.Among the forces,Fvand Fcare always 1–2 orders of magnitude greater than Fdand Fbg.The balance between Fvand Fclargely determines the equilibrium size of the agglomerates.Therefore,the subsequent analysis focuses only on the impacts of Ug,H,and dAon Fcand Fv.As Ugincreases,Fcincreases while Fvdecreases(Fig.5),which simultaneously increases the breakup tendency of the two colliding agglomerates and eventually decreases their equilibrium size.On the other hand,Fvincreases while Fcremains constant with the increase of H(Fig.6),thus increasing the coalescence tendency of the two colliding agglomerates and finally increasing their equilibrium size.

Fig.5.Effect of U g on the forces exerted on the agglomerates.

Fig.6.Effect of H on the forces exerted on the agglomerates.

Fig.7.Effect of d A on the forces exerted on the agglomerates.

Moreover,when dAis small,Fcis smaller than Fv(Fig.7).Nevertheless,Fcincreases faster than Fvwith the increase of dAand eventually exceeds Fv.This phenomenon implies an important mechanism that the agglomerates could spontaneously balance the forces exerting on them by adjusting their size through coalescence or breakup.

4.2.Equilibrium size of the agglomerates

As stated in Section 3.2,the balance among the forces exerted on the agglomerates finally determines their equilibrium size.Hence,the explicit function between dAand(H,Ug)could be obtained by solving

How ever,when the specific expressions of Fd,Fc,Fv,and Fbgare substituted into Eq.(25),it turns into Eq.(B2)in Appendix B,which is strongly nonlinear with respect to dA.Hence,the explicit function between dAand(H,Ug)is difficult to obtain by analytically solving Eq.(B2).To solve this problem,values of dAat 100 combinations of(H,Ug)were first obtained by solving Eq.(B2).Then the explicit function between dAand(H,Ug)could be acquired via surface fitting of the above 100 data.

Fig.9.Values of d A calculated from the modified force balance model.

The value of dAat a given combination of(H,Ug)could be obtained by solving Eq.(B2)with the graphical method,an example of which is illustrated in Fig.8.Similarly,values of dAat the other 99 combinations of(H,Ug)could also be obtained,which are all shown in Fig.9.Appar-ently,the calculated dAincreases with the decrease of Ugand the increase of H,which is in qualitative agreement with the experimental observation.This phenomenon results from that the increase of Ugincreases the collision force while the increase of H enhances the cohesive force between the two colliding agglomerates(see Figs.5 and 6).

The relationship between dAand(H,Ug)was fitted by

Generally,the goodness of fitting could be quantitatively evaluated by the coefficient of determination(R2),whose value ranges from 0 to 1.0.The greater R2is,the better the fitting equation is.R2for the fitting Eq.(26)equals 0.998,indicating an excellent fitting performance.

4.3.The calculated Hms–Ug relationship

Recall the illustration in Section 2.4 that the function between Hmsand Ugcould be obtained by substituting the expression of dAinto the basic size segregation model for binary mixtures.Here by substituting Eq.(26)and dL=332μm into Eq.(3),we finally acquire the relationship between Ugsand H,which is given by Eq.(27).Based on the definitions of Hmsand Ugs,Eq.(27)can be rewritten into Eq.(28).

Ugis known under the Magnetization LAST operation mode;therefore Eq.(28)is a unitary quadratic equation with respect to Hms;the discriminant(Δ)is given by Eq.(29).In order to fluidizethebed,Ugshould be increased beyond Umf=0.078 m·s?1.Under such circumstances,Δ is always greater than 0.Therefore,Eq.(28)has two real roots(Hms1and Hms2),which are given by Eqs.(31)and(32),respectively.

Since Ugis always greater than Umf=0.078 m·s?1,Hms2is smaller than 0.Hence,it has no physical meaning and should be discarded.Hms1is the only reasonable solution of Eq.(28)and the subscript 1 could be removed.Finally,the relationship between Hmsand Ugcould be explicitly represented by Eq.(32).This equation represents exactly the magnetically induced segregation model for the MFB with the binary mixture of Geldart-B magnetizable and nonmagnetizable particles.

Fig.10.Comparison between the calculated and experimental values of H m s.

Fig.10 compares the calculation results of Eq.(32)with the experimental data from[12].The maximum of the relative error occurs at Ug=0.116 m·s?1with the calculated Hmsbeing 0.27 times greater than the experimental value.In general,there is a reasonable agreement,indicating that the segregation behavior of the MFB with the binary mixture could be well predicted by Eq.(32).

4.4.Discussion

As shown in Fig.3,the magnetically induced segregation model could be finally derived by substituting the expression of dAinto the basic size segregation model(Eq.(2))for binary mixtures.Such a size segregation model was generalized from our own experimental data based on the iron–limestone binary mixture(see Appendix A).The reason why the basic size segregation model has to be established by ourselves lies in the fact that there is a severe lack of investigation on particle segregation in the literature,much less the mathematical model of segregation[28].If the universal size segregation model for binary mixtures is to be established in the future,the magnetically induced segregation model could be simply acquired by replacing dJ/dFwith dA/dL.

Moreover,the reason why the iron–limestone binary mixture was selected to derive the basic size segregation model is that this binary mixture is assumed to have the same ρJ/ρFwith the agglomerate–limestone binary mixture.In fact,ρAis smaller than ρIsince there definitely exists some space between the adjacent particles within an agglomerate.ρAfor agglomerates formed by thousands of Geldart-C particles is relatively easy to measure,because the agglomerates could maintain its structure after being removed out of the fluidized bed.In this case,ρAis greater than the loosely packed density but smaller than the densely packed density.How ever,ρAfor agglomerates formed by Geldart-B particles is difficult to measure.For one thing,the agglomerates will decompose immediately after being removed out of the magnetized fluidized bed.For another,the in situ measurement is still impossible as far as we know.It is still a doubt whether the conclusion of ρAfrom Geldart-C particles could be extended to depict the Geldart-B case.Therefore,as a preliminary mathematical model,this work simply assumesρAto equalρI.This assumption has some rationality since the calculated dAis mostly not 3 times greater than the size of the original Geldart-B iron particles(see Fig.9).More efforts are still required to revise this mathematical model so that the assumptions will become closer to the actual situation.

As shown in Fig.10,the calculated Hmsis always slightly greater than the experimental value.This phenomenon arises partly from that during deriving dA,the force balance model considers only the magnetic attractive force between agglomerates while the magnetic repulsive force is ignored.Such an ignorance will increase the collision probability between agglomerates and further cause the calculated dAto be smaller than its actual value.Besides,the mathematical model here assumes that the agglomerates,which in fact are elliptical in shape,are spherical.This will cause the drag force to be greater than it actually is,increasing the breaking up tendency of the agglomerates.As a result,the calculated agglomerate size is smaller than it actually is.This suggests that only at a higher H could the agglomerates grow big enough to sink to the lower part of the bed.In other words,the critical magnetic field intensity(Hms)above which segregation occurs becomes higher.

On the other hand,note particularly that during the derivation of dA,the modified force balance model considers only the interaction between agglomerates while that between the agglomerates and nonmagnetizable particles is ignored.The ignorance of the collision of nonmagnetizable particles to the agglomerates will result in the calculated dAbeing greater and Hmsbeing smaller than their actual values.According to the modeling results,the ignorance of the magnetic repulsive force and the agglomerate shape prevails over the ignorance of the collision of nonmagnetizable particles to the agglomerates,eventually leading to the calculated dAbeing smaller and Hmsbeing greater than their actual values.

Segregation is found to occur under both the Magnetization FISRT and Magnetization LAST operation modes[15–18].Nevertheless,as stated in Introduction,the segregation mechanisms are different[12].Based on the modeling idea illustrated in Section 2,the magnetically induced segregation model established in this work is only suitable to predict the segregation behavior in the MFB operated under the Magnetization LAST operation mode.Aside from this,some other application restrictions of the magnetically induced segregation model should be pointed out.On the one hand,the basic size segregation model(Eq.(2))is generalized from the iron–limestone system.For other kinds of binary system,e.g.,iron–copper,nickel–sand,the density ratios between the two components are different.Therefore Eq.(2)could not be used and our segregation model is not applicable.On the other hand,even for the iron–limestone binary mixture,the magnetically induced segregation model has its own application range.As per Fig.9,within the scope of our experimental investigation the maximum dAof the agglomerates reaches about 700 μm while the minimum dAequals 172 μm,i.e.,the size of a single iron particle.On the other hand,the average size of the limestone particles is 332 μm.Under such circumstances,the size ratio between magnetic agglomerates and limestone particles ranges from 0.518 to 2.108.This represents an other application limitation of our segregation model.Beyond this range the segregation model should be used with care.

5.Conclusions

This paper presented a magnetically induced segregation model for the MFB with the binary mixture of Geldart-B magnetizable and nonmagnetizable particles.The segregation under the Magnetization LAST operation mode essentially belongs to size segregation of the binary mixture of the agglomerates and nonmagnetizable particles.The agglomerate size could be calculated by the force balance model.The cohesive force and the collision force are 10–100 times greater than the other forces acted on the agglomerates.To a large extent,the balance between these two forces determines the agglomerate size.The calculated dAincreases with the increase of H and the decrease of Ug.The calculated Hms–Ugrelationship is in reasonable agreement with the experimental results,implying that the magnetically induced segregation model established in this work could predict well the segregation behavior in the MFB.Therefore,it could be further used to guide the operation of the MFB,especially when the magnetic field is employed to enhance the fluidization quality of the binary mixture.

Nomenclature

B magnetic flux density,T

BFB bubbling fluidized bed

dAequilibrium size of the agglomerates formed by iron particles,m

dFaverage size of particles that work as flotsam,m

dIaverage size of the iron particles,m

dJaverage size of particles that work as jetsam,m

dLaverage size of the limestone particles,m

Fbbuoyancy exerted on the agglomerates by the fluidizing gas,N

Fbgdifference between Fgand Fb,N

Fccollision force between two colliding agglomerates,N

Fddrag force exerted on the agglomerates by the fluidizing gas,N

Fggravitational force exerted on the agglomerates by the earth,N

Fvcohesive force between two colliding agglomerates,N

g gravitational constant,m·s?2

H magnetic field intensity,k A·m?1

Hmscritical H above which segregation occurs,kA·m?1

Δh height of bed material above the distributor,m

k constant in Eq.(14),Pa?1

M mixing index

MFB magnetized fluidized bed

ΔPbbed pressure drop,kPa

ΔPuppressure drop over the upper half bed,kPa

rAdistance between the centers of two agglomerates,m

R2coefficient of determination

Ugsuperficial gas velocity,m·s?1

Ugssuperficial gas velocity at which M equals 0.9,m·s?1

Umbminimum bubbling velocity in the presence of the magnetic field,m·s?1

Umfminimum fluidization velocity of the binary mixture,m·s?1

UmfAminimum fluidization velocity of the agglomerates,m·s?1

UmfFminimum fluidization velocity of the flotsam,m·s?1

UmfJminimum fluidization velocity of the jetsam,m·s?1

UmfLminimum fluidization velocity of the limestone particles,m·s?1

VAvolume of the agglomerate,m3

Vrrelative velocity between the two colliding agglomerates,m·s?1

xJ,upthe mass fraction of jetsam in the upper half of the bed

xJ,avethe mass fraction of jetsam in the whole bed

εbbed voidage at a certain Ug

εb0voidage of the settled bed

μAmagnetic permeability of the agglomerates formed by iron particles,H·m?1

μbmagnetic permeability of the bed material asa whole,H·m?1

μgviscosity of air,Pa·s

μImagnetic permeability of the iron particles,H·m?1

μ0magnetic permeability of air,H·m?1

ξ parameters in Eq.(18)

ρAdensity of the agglomerates formed by iron particles,kg·m?3

ρFdensity of the particles that work as flotsam,kg·m?3

ρgdensity of air,kg·m?3

ρIdensity of iron particles,kg·m?3

ρJdensity of the particles that work as jetsam,kg·m?3

χAphysical magnetic susceptibility of the agglomerates

χeeffective magnetic susceptibility of the agglomerates

Acknowledgements

Our gratitude goes to the Supercomputing Center of USTC(University of Science and Technology of China)for the support.

Appendix A

This appendix aims to answer how the basic size segregation model for binary mixtures(Eq.(2))is experimentally derived on the basis of the iron–limestone binary mixture.For the iron–limestone binary mixture,ρJ/ρFis fixed and therefore Ugsdepends solely on dJ/dF.The objective of this appendix in essence is to correlate a relationship between Ugsand dJ/dFbased on experimental measurements.

A.1.Experimental

A.1.1.Materials

Six iron–limestone binary mixtures with different dJ/dFvalues were employed,the physical properties of which are listed in Table A1.In addition,air at ambient temperature and pressure was used as the fluidizing agent.

Table A1Physical properties of the six iron–limestone binary mixtures

A.1.2.Experimental setup

The experimental setup was simply a cylindrical fluidized bed with an inner diameter of 120 mm and a height of 600 mm.For easy visual observation,the fluidization column was constructed from transparent Plexiglass.The gas distributor was a sintered metal plate with a thickness of 4 mm.

A.1.3.Measurement of M

The iron–limestone binary mixture was first poured into the fluidized bed with a settled height of 200 mm.Then the fluidizing gas was introduced at adesired velocity.After fluidization for10min and the distribution of particles in the bed reached equilibrium,the fluidizing gas was suddenly shut down.Five minutes later,the particles in the upper half of the bed were sucked out by a vacuum cleaner.Then the iron and limestone particles were separated by using a magnet.Thereafter,the mass fraction(xJ,up)of jetsam(i.e.,iron particles)in the upper half of the bed was determined by weighing.Finally,the value of M could be calculated according to Eq.(1).

A.2.Basic size segregation model for binary mixtures

A.2.1.Determination of Ugs

Values of M for the iron–limestone binary mixture(case 5 in Table A1)at different gas velocities are shown in Fig.A1.M increases rapidly to 0.9 at low gas velocities and slowly to 1.0 at high gas velocities.Ugsequals 0.133 m·s?1for the binary mixture of case 5.Similarly,values of Ugsfor the other five binary mixtures could also be determined,which are all summarized in Table A1.

Fig.A1.Effect of U g on M for the iron–limestone binary mixture.

A.2.2.Fitting relationship between Ugsand dJ/dF

Based on the regression analysis,the relationship between Ugsand dJ/dFcould be fitted by Eq.(A1)with R2being equal to 0.974.As shown in Fig.A2,the Ugs–dJ/dFrelationship could be well represented by Eq.(A1).

Eq.(A1)is exactly the goal of this appendix—the basic size segregation model for binary mixtures.This mathematical model will be extended to depict the segregation behavior of the agglomerate–limestone binary mixture in the main body.

Fig.A2.Linear fitting relationship between(U gs–U mfF)/U m fF and d J/d F.

Appendix B

This appendix aims to explicitly show that Fvand Eq.(25)in the main body are simply functions of H,Ug,and dA.