Magnetically stabilized fluidization: modelling and application to mixtures

Powder

44 (1985)

Technology,

Magnetically J. ARNALDOS, Chemzcal

(Recewed

Stabilized J. CASAL,

Engrneering

July 15,1984;

Fluidization:

A. LUCAS

Department,

57

57 - 62

Modelling

to Mixtures

and L. PUIGJANER

Uniuersitat

Politicnicn

in revised form October

de Catalunya.

Barcelona

(Spam)

20,1984)

SUMMARY

A mathematical model of the magnetically stabilized fluidized bed, based on the arrangement of particles following field lines, is proposed. Magnetic stabilization has also been applied to fluidized beds wrth mixtures of magnetizable and non-magnetizable particles (sintered nrckel-silica, steeZ-copper, steelsilica)_ The behaviour of these systems is studied, as well as the influence of the magnetic material mass fraction on the delay of bubbling_ The results obtained show that the magnetic stabilization can be applied not only to the fluidization of magnetizable materials but also to non-magnetizable particles, if a certain fraction of magnetizable material is present_ Correlations are proposed to calculate the transition velocity as a function of gas-solid properties and operating conditions.

INTRODUCTION

When a system of magnetizable particles is fluidized in the presence of a magnetic field, particulate fluidization can be obtained at gas velocities higher than that corresponding to incipient fluidization. If gas velocity is gradually increased, a certain value will be reached at which, for a specific magnetic field intensity, bubbles will finally appear; this velocity is usually called the ‘transition velocity’, rib, and the system operated at u,f < u < ZQ, is termed a ‘magnetically stabilized fluldized bed’_ This kind of fluidization has been studied by several authors [l - 14]_ Nevertheless, there has been only a small amount of work on the subject of modelling [lo]. Most of 0032-5910/85/$3.30

and Application

these works are concerned wirh the fluidization of systems constituted only by magnetizable materials, and little attention has been paid to the fluidization of mixtures of magnetizable and non-magnetizable particles [8,12,13] _ These systems are, however, of the greatest interest from the point of view of the use of the magnetically stabilized fluidized bed in practice_ In this paper a mathematical model of this kind of bed, based on the arrangement of particles following field lines, IS proposed. The behaviour of fluidized mixtures of nickel-silica, steel-copper and steel-silica is also studied, and a correlation is proposed to calculate the transition velocity.

EXPERIMENTAL

ARRANGEMENT

The experimental set-up has been described in detail in previous work [9] _ The equipment consisted of a 51 mm id. column, surrounded by a coaxial COIL The fluidizing gas, air at essentially atmospheric pressure, was previously filtered and dried. The characteristics of the particles used are summarized in Table 1. Results were obtained at moderated values of N, in order to avoid the considerable channeling effect observed at higher values. Bed voidage was obtained from the measurement of pressure drop between two points of the bed, at decreasing velocity. A plot of E vs. u then gives the value of emf or E,,_ Mathematical modelling A theoretical expression for the minimum fluidization velocity may be derived by considering the gas flow through a bed of particles [ 123 _ The fluid velocity u, in the interstitial channels is given by (Fig. 1) @ Elsevler Sequoia/Printed

in The Netheriands

58

Fig. 1. FluId velocity mslde the bed. Fig. 2. Arrangement of particles in the bed. u u, = E

cos

(1)

$5

and the length I, of the interstitial channels is related to the interstitial velocity angle I,!J by 1

I, = -

(2)

4

cos

The hydraulic diameter of channels can be expressed as a function of voldage as follows:

2ed dh =

(3)

3(1 -E)

Using the Hagen-Poiseuille equation for the laminar flow through a tube, the fluid velocity is given by u,

=

c-d2

AE’

32~

1,

(4)

where d, and I, are the tube diameter and length respectnrely. From these relationships the following expression can be obtained for the gas velocity through the tube:

cos2$

lC= -

72

e3d2AP Zf.l(l -

(5)

E)2

Since the pressure drop for the fluidized is given by

AP = (1 - E)(P, --P)gZ

bed

(6)

eqn. (5) becomes

cos’+ u=

1 Ub

(p,-p)gd+

e3

-

72

constant as 5), 52.3” [12] and 45” for beds of spherical particles (Leva). As IS to be expected, for a bed in which particles are arranged in ‘chains’ following the vertical field lines [ 121 (Fig. 2), & has a zero value. Let us therefore assume the following [S] : a) the action of the magnetic field consists in a gradual particle arrangement (following field lines) as fluid velocity is increased; b) the maximum reordering which can be reached is a function of the intensity of the applied magnetic field; c) a ‘stabilized’ bed can be considered as a fixed bed, as shown by the fact that, if fluid velocity is decreased, pressure drop gradually decreases too; d) the apparition of bubbles (at transition velocity) corresponds in fact to the ‘real’ fluidization state of the system. Then the transition velocity has to be equal to the minimum fluidlzation velocity of a bed with the same porosity that exists in the stabrlized bed when it reaches its maximum reordering degree. This maximum is a function of the magnetic field applied. The dependence of the bed voidage on the magnetic field is essentially associated with the bed voidage function (e3/(1 - E)), thus affecting the ‘minimum fhndization velocity’ of the stabilized bed, called the ‘transition velocity’:

II

l--E

(7)

This espresslon IS only vahd at incipient fluidization. In these conditions, the value of u is that corresponding to minimum fluidization velocity Umf, bed voidage is emf, and $ = +O (randomly packed bed). Different values have been attributed to tie angle G for a bed of random particles: 53” (Bureau of Mines), 51” (taking the Kozeny

=

(p,

-

p)gd2

-

72

cc

cos2~eb3 1 -Eb

(8)

where eb is the bed voidage at transition conditions. Figure 3 shows the plot of ub experimental US. calculated values from eqn. (8), for spherical particles (9 = 45”), using eb experimental data. As can be observed, the correlation is very good, with a standard deviation of 6.4%. It is difficult to predict theoretically the variation of eb and $ (at transition condi-

59

tions) as a function of the magnetic field intensity. Nevertheless, it is obvious that it must be a function of particle magnetization, which will be responsible for the degree of arrangement following field lines- It will also depend on bed voidage in the absence of magnetic field. Therefore, the following equation can be written :

cos*I+b ~

s3

1 -Eb

=

f(%f,

M,

w

Bed magnetization will always be a function of magnetic field intensity and magnetic susceptibility of the material used:

co~*$e,,~/(l - E,,) versus magnetic field intensity can be obtained (Fig. 4). Correlation of these data by least squares leads to the following general expression:

(12) which shows a regression coefficient of 0.99. The accuracy obtained with this equation seems therefore appropriate to be introduced in eqn. (8), to obtain the following relationship :

cos2~/, (P, --Pkd2

3 cos”gJ

-5-

1 - Eb

=

f(Gnf*

H,

xl

(10)

where x, a property of the magnetic material used, is a linear function of N whenever saturatron is not reached. Therefore x=a+bH

(11)

a and b being specific constants for a given material, which express its capability of magnetization. The following expression is finally obtained:

cos*&!J-

G3

1 -Eb

=

f(cnf,

a, b, H)

The theoretical approach to the variation of eb as a function of H, for a stabilized bed, seems to be a cumbersome task. However, this relationship can be found experimentally. Using different magnetic materials, the plot of

ulJ=

72

X -wEta

cnf3 l--E,f

I-r

+ bH)p,HI

(13)

Now, when eqn. (7) is applied to the case of fluidization in the absence of magnetic field, eqn. (13) becomes u b=&-nf

=pEfa

+

bJ3f+Hl

(14)

This equation shows total agreement with the expsrimental correlations presented by Casal [ll] and Lucasetal. 181. Equation (14) as it has been obtained implies a certain error due to the variation of the angle I,!/ as a function of H. Nevertheless, at moderated values of H, when this kind of bed offers its most valuable applications and at which data presented in this article have been obtained, the error has been always found to be less than 4%. a value that is ouite acceptable in the calculation of z+,_

1

Fig.

3. Calculated

versus

experimental

values

of q,.

Fig.

4. Correlation

of eqn.

(12)

0

0

020-177 ,~rn 177-2io pm

I

200

Fxg. 5. Comparison of eqn. (14) data for sintered nickel particles

with

experimental

1000

2000

3oaJ H At/m

4000

Fig 6. Influence of mass percentage of magnetic terlal on the transition velocity for nickel-sihca mixtures.

ma-

l”b

This equation has been tested for the systems already quoted (nickel, steel), and plotted in Fig. 5 for the case of nickel. Agreement is very good (regression coefficient = 0.99, standard deviation = 7.6%). Applrcation to mixtures The bed stabilization is strongly influenced by the mass fraction of magnetizable material present. for the case of mixtures of magnetizable and non-magnetizable materials, as can be observed in Fig. 6; the transition velocity dependence on the magnetic field intensity is also shown in this figure for different mass percentages of magnetic material_ This behaviour is similar for the steel-copper and nickel-silica systems_ In all cases, particles were selected to have approximately the same mnnmum fluidization velocity; mixing was always good, and no segregation phenomena were observed. The general trend of curves IS similar to that observed for magnetizable mater&s [ 8,121, the effect of stabilization decreasing as the mass fraction of inert material IS increased. The dependence of transition velocity on the mass percentage of magnetrzable material is ah visualized in Fig. 7, in which several runs at different values of the magnetic field intensity H are presented_ Again a large influence of the magnetic field and the mass percentage of magnetizable material on ub is observed, the trend being the same for steelcopper and for nickel-silica systems. In al1 cases, no influence of H or x on the value of the minimum fluidization velocity was observed, this being in good agreement

mJi(Cm/sl

steel-copper

rmxtllres

3SO
3165 2535 1900 1270 635 H=OAVm 40’

0

20

mass

1

40 percentage

I

1

60 80 of magnetic

I

100 materet

Fig. 7. Influence of mass percentage of magnetic terial on the transition velocity for steel-copper mixtures_

ma-

with existing data from literature [ 1, 7, 8, 10,121. The simultaneous dependence of the transition velocity on magnetic field and on mass fraction of magnetizable material is enhanced by the representation of Fig. 8, in which a three-dimensional plot shows the value of z+, as a function of H and x for steel-copper mixtures. In order to obtain a quantitative approach to the phenomena described before, data were correlated to find the relationship between ub and the operating conditions. The model proposed in the first part of this work for ferromagnetic materials was used as a departure point. The same mechanism can be accepted for the fluidization - under the influence of a magnetic field - of an inert material, if a magnetizable material 1s being gradually added; magnetic particles will arrange themselves following field lines, thus creating a structural frame inside tine bed, which will produce stabilization.

Fig. 10. Comparison of the proposed correlation for the calculation of ub of nickel-copper and nickelsilica mixtures with experimental data.

Fig. 8. Transition velocity as a function of magnetic field intensity and mass fraction of magnetic material_

Fig. 11. Experrmental (a + bHlCt0. Fig. 9. Comparison of the proposed the calculation of u,, of steel-copper mixtures with experimental data.

correlation for and steel-silica

The preceding analysis leads us to consider that the transition velocity is a function of particle and gas properties, as well as of certain operating conditions. Thus, ub =

f(%s,

H, Ml

(15)

Also, the magnetization of the bed will be a function of the magnetic field intensity, magnetic susceptibility and mass fraction of the magnetic material. Then, ub = f(&nf, H, x, x)

(16) Following eqn. (ld), data were correlated according to an expression of the type ub = r&f exp[(u

+ b~&&k

This equation has been plotted 10 together with experimental

(17) in Figs. 9 and data for the

uersus calculated

values of

TABLE Particle properties IMaterial

Density

Shape

(I

Spherical Q = 0.87 @=0.96 Q = 0.70

40.00 46.24 -

(kg/m31 Steel Nickel tipper Silica

7500 5870 8890 2670

0.0377 0.0243 -

different systems studied_ In all cases, agreement is very good, regression coefficients ranging between 0.97 and 0.99. Furthermore, the values of (a + Mi)~e obtamed experimentally were compared with those calculated using data from literature [15] (see the Table). Agreement was good (standard deviation = 7.8%), as can be seen in Fig. 11.

62

pressure drop, N/m2 fluid velocity, m/s transition velocity, m/s minimum fluidization velocity, m/s interstitial fluid velocity, m/s mass fraction of magnetic material, -

It should be noted that eqn. (16) would apply for values of H less than those corresponding to saturation of the magnetizable material; on reaching saturation, eqn. (11) no longer holds and M reaches a constant value.

u, x

CONCLUSIONS

Greek E

Fluidized bed magnetic stabilization can be explained m terms of particle arrangement_ By considering this kind of bed as a fixed one, and from the Hagen-Poiseuille equation, a mathematical model has been developed, thus obtainmg eqn. (14), which allows the predictron of transition velocity ub_ This equation has been tested using experimental data, agreement being very good. Experimental data show that it is possible to achieve the magnetic stabilization of a fluidized bed consisting of non-magnetizable particles if a certain mass fraction of magnetizable materia1 is added. The application of this techmque is therefore no longer restricted to magnetizable materials, bringing this techmque to a wider range of potential applications m which good solid-gas contact is needed: catalytic reactions at moderated temperatures, aerosol collection, adsorption_ -4n equation has been obtained (eqn. (17)) which allows the calculation of the transition velocity for binary mixtures of magnetizable and non-magnetizable particles. Equation (17) is restricted to values of N below saturation, covering therefore the range at which the operation of this kind of fluidized bed offers its major interest, avoiding the channeling which would appear at higher values of H.

AP u ub

u mf

I& X ti #O

REFERENCES 1 M_ V_ Filipov,

a

b d d, H 1 L C M

constant in eqn. (II), constant in eqn. (II), (A- turn/m)-’ particle diameter, m tube diameter, m magnetic field intensity, A- turn/m bed height, m tube length, m length of interstitial channels, m magnetization, A- turn/m

Przk. Magnitogidrodin. Nank. Latv. SSR (USSR),

Fiz Akad. 215.

M_ V.

Filipov,

(USSR),

I2

Izv. Akad.

(1961)

Nank.

Tr. Inst. 12 (1960) Lntv.

SSR

47.

J. M. Kirko and M. V Filipov, Zir. Tekh Fia (USSR). 30 (1960) 1081. D. G. Ivanov and G. T. C. R. Grozev, Acad BuZg Scr ,23 (1970) 787. R. L. Sonohker, S. G. Ingle, J. R. Giradkar and P. S. Meue, Indran J. Technol., IO (1972) 377. P. J. Lucchesi, W. H. Hatch, F. X. Mayer and R. E. Rosensweig, Proc. 10th World Petroleum Congress,

Bucharest,

I4

(1979).

7 J_ Casal, J. Ripol and L. Puigjaner, Journees

Europeennes sur la Fluidisation. A&es. 1. 7.1. Toulouse (1981). 8 A. Lucas, J_ Casal and L. Puigjaner, in D. Kuneii and R. Toei (eds.), Fluidization, Engineering

Foundation, 9 J. Arnaldos, 10 11

LIST OF SYMBOLS

symbois bed voidage permeability of free space, 47r X lo-’ H/m (or T/(A-turn/m)) magnetic susceptibility, velocity angle at any value of H velocity angle in the random particles bed

12

13 14

15

New York, 1984, pp_ 129 - 136. J. Casal and L. Puigjaner, Powder Technol.. 36 (1983) 33. R E. Rosensweig, J. H. Slegall, W. K. Lee and T. Mikus, AIChE Symp. Ser., 77 (1981) 8. J. Casal, Tesr Doctoral, Universitat Politeenica de Barcelona (1982). J. Casal, Contribucio a L’estudi de la fluidztzacio homogenia, Arxius SecciB Ciencies, 77. Institut d’Estudis Catalans, Barcelona (1984). R. E. Rosensweig, United States Patent 4.115,927 (Sept. 26, 1978). W. Yang, E. Jaraiz, 2. Guo-tai, R. T. Chan and 0. Levenspiel, in D. Kuneii and R. Toei (eds.), Fluidization. Engineering Foundation_ New York. 1984, pp_ 241 - 248. E. W. Eashburn (ed_), International Critical Tables, Vol. VI, McGiaw-Hill, New York, 1st edn., 1929, pp_ 366 - 414.