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Magnetic stabilisation of a liquid fluidised bed
Powder Technology 124 (2002) 287 – 294 www.elsevier.com/locate/powtec
Magnetic stabilisation of a liquid f luidised bed Y.Y. Hou *, R.A. Williams Centre for Particle and Colloidal Engineering, School of Process, Environmental and Materials Engineering, University of Leeds, Leeds LS2 9JT, UK
Abstract The stability of a magnetically stabilised fluidised bed (MSFB) has been studied in an experimental investigation coupled with mathematical modelling. Experimental results indicate that the stability of the bed can be influenced by several factors, including liquid velocity, magnetic field strength, solid and liquid density, solid magnetic susceptibility and solid particle size distributions. A theoretical model based on Rosensweig’s mean field theory and the Richardson – Zaki equation has been established to predict the bed transition velocity when a fluidised bed starts to transform from a stable to an unstable state against an applied induction. The model predictions were found to have reasonable agreement with the experimental values obtained from bed pressure drop measurements. The results have applications to the design of magnetically stabilised fluidised bed bioreactors. D 2002 Elsevier Science B.V. All rights reserved. Keywords: Magnetically stabilised fluidised bed; Stability; Magnetic force; Modelling
1. Introduction Several types of reactors are used in biological and chemical engineering industries. Some of the most commonly employed ones include stirred tanks, packed bed, and conventional fluidised bed reactors. Unfortunately, all these reactors have some inherent disadvantages. A stirred tank reactor, for example, is limited by its ability to operate in a batch-mode, which can be problematic. Packed beds have a strong tendency to become plugged with various solid debris. Such malfunctioning is common in many bioengineering systems. It has been shown that fluidisation can provide an effective solution to many of the problems associated with plug flows and stirred tank reactors. A conventional fluidised bed, however, is often plagued by poor contact between the two phases and bypassing effects. In addition, the capacity of the bed is also heavily limited by substrate throughput. This paper explores an alternative means of reactor design using a magnetically stabilised fluidised bed reactor (MSFBR). The MSFBR is a fluidised bed reactor comprising of magnetic particles, or magnetic fluid, surrounded by an array of magnetic coils. Such a reactor has the potential capability of eliminating many of the difficulties mentioned above. If bed particles are made of magnetisable materials
*
Corresponding author. E-mail address: [email protected] (Y.Y. Hou).
[1], with introduction of an external magnetic field, the bed can then be stabilised to a quiescent and uniform condition. The hypothesis is that when in this state, the bed will possess properties that are almost an ideal combination of those exhibited by the packed bed and a conventional fluidised bed. This has been demonstrated for both gas [2,3] and liquid [4 – 6] medium systems. Some of the key features and benefits of a magnetic stabilised bed have been illustrated in detail in the literature [7– 9]. The stabilisation of a magnetic fluidised bed consisting of magnetic particles supported by a gas medium has been systemically studied by Rosensweig [10]. By neglecting the gas fluid density, the author developed a so-called ‘normalised magnetisation term’ to describe the stability of the gassolid fluidised bed. Unfortunately, this term is not applicable to liquid – solid fluidised beds, as the fluid density cannot be neglected in comparison with the particle density. Rosensweig and Ciprios [11] later modified the stability description model to accommodate its use in the liquid – solid fluidised bed applications, but they mainly concentrated on the case of fluidisation of nonmagnetic spheres with a magnetic ferrofluid, from which reasonable agreement between theory and experimental data was achieved. Fee [12], on the other hand, tried to give a suitable criterion for the stability of magnetically stabilised liquid-medium fluidised bed based on the particle magnetisation analysis approach. In his work, the liquid density was taken into account and the transition velocities that were predicted by Fee’s model were found to agree more closely with the
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experimental values than those from Rosensweig’s gas fluidised bed model. However, since Fee’s work ignored the effect of Reynolds number in the drag coefficient calculations, the theory failed to achieve quantitative predictions of the bed’s actual stability and behaviour. In the present paper, efforts have been made to deduce a more universal and applicable stability model for the liquidmedium magnetically stabilised fluidised bed based on the work given by Rosensweig [11]. A similar underlying assumption to that of Rosensweig’s will be adopted here, in which a normalised magnetic induction term is developed to yield a more reasonable criterion for evaluating the stability of a magnetically stabilised fluidised bed.
Rosensweig [13] has deduced the following dynamic equations to describe the evolution of a bed voidage perturbation impressed upon a magnetically fluidised bed, in which ef stands for bed voidage, and other terms are given in the nomenclature: @ 2 ef @ 2 ef @ef @ef @ 2 ef þ C þ D þ ðE þ FÞ þ B ¼ 0 ð1Þ @x@t @x @t @2t @2x ðe1 2ðe1 f 1Þqf f 1Þvqf ; B¼ ; qp qp ! g qf ef ð1 ef ÞbVðef Þ C¼ 1 1 2ef ef bðef Þ qp
A¼1þ
g D¼ ef v
! qf 1 ; qp
E¼
18v2p ð3 þ 3vp 2ef vp Þ
v 2 ðe1 f 1Þqf qp
3
s ¼ n ig
ð5Þ
Magnetic stabilisation is, therefore, achieved when n<0, let n=0 to satisfy the neutral stability condition. Defining another complex variable Z as: Bji þ D 2 Cji ðE þ FÞj2 Z ¼ X iY ¼ ; ð6Þ 2A A then rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 1 ðr þ X Þ i ðr X Þ Z¼ 2 2
The neutral stability condition means: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 D ðr þ X Þ ¼0 Res ¼ 2 2A
ð7Þ
ð8Þ
Solving X and Y from Eq. (6), and substituting them into Eq. (8) yields: AC 2 þ D2 ðE þ FÞ CBD ¼ 0:
ð1 e2f ÞB20 l0 qp
When vf ¼ 0 Obviously, Eq. (1) is a rather complex partial differential equation. One solution to the problem is to presume that the bed voidage perturbation can be represented in a Fourier components form, i.e.: ef ¼ const Reðest eijx Þ
ð3Þ
Our contribution has been focused on solving this equation for a range of practical conditions and further extending the theoretical work. Solving the above equation for s gives: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s Bji þ D 2 Cji ðE þ FÞj2 Bji þ D ð4Þ s¼F 2A 2A A
where r2 ¼ X 2 þ Y 2 :
Mp @Hp l ð1 ef Þ Mf @Hf þ F¼ 0 qp @ð1 ef Þ @ef ¼
As2 þ sðBji þ DÞ þ ½Cji ðE þ FÞj2 ¼ 0
Since s is known to be complex, we may arbitrarily define s to be the form of:
2. Theoretical modelling
A
negative, as a positive value would yield exponential growth, and, therefore, would have no physical meaning. Substituting ef of Eq. (2) into Eq. (1), and assuming ef > 0, we have a complex quadratic algebraic equation for s:
ð2Þ
Where Re denotes the real part, j is wave number, taken as real, and s is a complex number. The real part of s is usually called the growth factor. In order to achieve a stabilised fluidisation behaviour (i.e., the bed voidage perturbation approaches zero) it is clear that the growth factor has to be
ð9Þ
Expressions for A, B, D, E, F have already been given in Eq. (1). Parameter C, however, requires knowledge of the ratio bV(ef)/b(ef), which in turn is related to the bed expansion behaviour caused by changes in the superficial velocity of the fluidising liquid. By using the Richardson – Zaki equation, which states that efn=m/mt, where n depends on Reynolds number and now also the applied magnetic induction, and assuming the dynamic fluidised bed pressure drop DPf=b(ef)v fL is constant, the drag coefficient has the following relationship with the bed voidage: bðef Þ ¼ constð1 ef Þ=enf ; from which it can be seen: bVðef Þ n þ ef ð1 nÞ ¼ : bVðef Þ ef ð1 ef Þ
ð10Þ
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so-called normalised magnetisation term, as it can be considered as a ratio of magnetic field energy to fluid kinetic energy. Eq. (12) is virtually a mathematical description of the bed transition superficial velocity (u0) as a function of magnetic field induction (B0), particle physical properties (n, vp, qp) and substrate liquid properties (n, qf). From this equation, a criterion to separate the stable fluidising zone from the unstable one for any magnetically stabilised fluidised bed can be conveniently established for different magnetic and hydrodynamic conditions.
3. Experimental methods 3.1. Magnetic particle characteristics Fig. 1. A photomicrograph of magnetite embedded n-carrageenan beads.
Substituting the above expression into Eq. (1), C will yield: ! g qf 1 ð11Þ C¼ ½ð1 þ nÞð1 ef Þ ef qp Having acquired all the expressions for A, B, C, D, E, F and substituting them into Eq. (9), a stability criterion for a magnetically stabilised fluidised bed can be deduced: ( ð3 þ 3vp 2e0 vp Þ3 B20 ¼ ½n ð1 þ nÞe0 2 l0 qp u20 18v2p e20 ð1 e20 Þ ) ðe1 1Þq f 0 þ ½ð1 þ nÞð1 e0 Þ2 ð12Þ qp Where u0 is the bed transition superficial velocity, which equals emf 0. The left-hand side of the above equation is the
All the experiments were carried out using magnetiteembedded n-carrageenan beads as solid particles. The beads were prepared by first uniformly mixing 10 wt.% magnetite particles (<74Am) into 3% n-carrageenan gel at around 75 jC. The mixture was then poured into an oil phase with constant stirring and was allowed to cool down to below 10 jC. After removing all the oil phase, the magnetic ncarrageenan soft solid particles were hardened in 5% polyethylenimine solution for around 1 h, and then washed in room temperature, The products were then sieved to different sizes and stored in 0.1 M KCl solution at 4 jC, ready for experimental use. Microscopic inspection shows that almost all the beads are perfectly spherical. Specific gravity measurement indicates that the density of the beads is 1250– 1400 kg/m3. Magnetisation test on fully dried n-carrageenan and magnetite mixture has shown a susceptibility vp of 0.6 (SI). It is therefore estimated that the susceptibility vp of the fresh beads would be in the range of 0.04 – 0.05 (SI). Fig. 1
Table 1 Size distributions of beads in four sieved fractions used in the experiments Bead samples 125 – 180 Am
180 – 250 Am
250 – 355 Am
355 – 500 Am
Size (Am)
Volume (%)
Size (Am)
Volume (%)
Size (Am)
Volume (%)
Size (Am)
Volume (%)
56.4 63.3 71.0 79.6 89.3 100.2 112.5 126.2 141.6 158.9 178.3 200.0 224.4 251.8 282.5 317.0 355.7
0.02 0.06 1.83 3.79 6.34 9.09 11.56 13.13 13.47 12.48 10.39 7.74 5.08 2.83 1.26 0.35 0.02
100.2 112.5 126.2 141.6 158.9 178.3 200.0 224.4 251.8 282.5 317.0 355.7 399.1 447.7
0.09 0.74 2.66 6.29 11.16 15.87 18.15 17.56 13.07 8.39 4.32 1.35 0.32 0.02
126.2 141.6 158.9 178.3 200.0 224.4 251.8 282.5 317.0 355.7 399.1 447.7 502.4 563.7 632.5
0.05 0.32 1.63 4.31 8.51 13.23 16.81 17.69 15.41 11.19 6.57 3.04 1.04 0.18 0.02
178.3 200.0 224.4 251.8 282.5 317.0 355.7 399.1 447.7 502.4 563.7 632.5 709.6 796.2
0.05 0.41 1.87 4.82 9.25 14.15 17.45 17.85 15.06 10.37 5.7 2.35 0.61 0.05
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and Table 1 show a photomicrograph and the size distributions of the beads that are being used in the experiments. The size distributions were obtained by using laser diffraction (Malvern, 2000). 3.2. Fluidised bed reactor and pressure drop measurements The reactor consists of a plastic cylinder of 150 mm inner diameter and 300 mm in height. A grid is placed in the bottom of the cylinder to hold the particles and a baffle is added in the flow-input chamber to keep the flow input uniform and steady. The magnetic field was generated by three DC-powered copper wire coils surrounding the reactor, each of them having 500 turns, 200 mm ID, and spaced 70 mm apart, with the first coil being located 30 mm above the distributor. The schematic diagram in Fig. 2 illustrates the configuration of the rig. For each experiment, about 1 kg of beads was used. This corresponds to an initial bed height of approximately 40 mm. The fluid was introduced into the chamber from underneath the grid using a peristaltic pump, and circulating in the system via the storage tank. The volumetric flow rate of the recirculating fluid was measured by a flow meter incorporated in the flow circuit from which the superficial velocity was calculated. The entire reactor is fully filled with fluid at all times. The bed stability was monitored via pressure drop measurements throughout the experiment using an inclined
Fig. 2. Schematic diagram of experimental rig and pressure drop measurement.
manometer, with one measuring point located beneath the grid and the other at the top of the reactor. The arrangement was also illustrated in Fig. 2. Beginning with the system being stationary, when there is no fluid passing through, and denoting Lmf as the initial bed height, the initial pressure readings indicated by the levels at position A for points 1 ( P10) and 2 ( P20) should be equal. As the fluid starts to flow and the fluid velocity exceeds umf, the bed starts to fluidise, and the bed height changes from Lmf to L. Accordingly, an extra pressure drop DP can be observed from the manometer readings for pressure at points 1 ( P1) and 2 ( P2), respectively. This pressure drop has been used in the paper as a means of describing the bed stability. It has been noted that the pressure differences across the grid at all the flow conditions that were investigated are negligible (less than 5 Pa). Other process parameters which could have direct impact on the bed stability behaviour, such as magnetic field strength, particle size distributions, the density and viscosity of the substrate liquid, etc., [14] are also considered. Key results are summarised in this paper.
4. Experimental measurement of bed performance 4.1. Effect of magnetic field strength The investigations were carried out using 180– 250 and 355– 500-Am n-carrageenan-magnetite beads, respectively as solid particles and tap water as substrate fluid. Fig. 3 shows changes of (a) bed height and (b) bed pressure drop as a function of fluid velocity under different magnetic field strength for the 180 – 250-Am beads bed, the latter is the DP described in Fig. 2. It can be seen that the bed height starts to considerably increase as the fluid velocity reaches the minimum fluidisation velocity threshold. This has been confirmed by the pressure drop measurements which show that the threshold is also the point when the pressure drop starts to increase with the fluid velocity. It can also be seen from Figs. 3b and 4 that there exists another significant point at which a decline in the pressure drop was monitored. This is believed to be the transition point when the stabilised fluidised bed starts to collapse. At the transition point, it was observed that the upper bed surface was no longer smooth, the bed started to lose structure and changed from MSFB to partially stabilised fluidised bed (PSFB). Finally, beads would be flushed out from the reactor. Physically, the declination in pressure drop is caused by the drag force overcoming the magnetic binding force that exists between magnetic particles. Similar pressure decline phenomena have been recorded and explained as the results of bed channelling and rearrangement [15,16]. It is interesting to note that the bed transition velocity increases as the magnetic field gets stronger, indicating that use of magnetic field in a magnetic particle bed would help to expand the stability range of the fluidisation.
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Fig. 5. Effect of solid particle size on bed stability (5-mT field induction).
Fig. 3. Bed expansion (a) and pressure drop (b) (180 – 250-Am beads).
It should be noted that the pressure drop across both the conventional and magnetically fluidised bed, in our case, did not remain relatively constant after fluidisation. This is mainly because the pressure drop across the bed would only remain constant when the bed is fully fluidised, i.e., if all the particles are in a fully suspended state in the bed. Unfortunately, this has never been the case in our investigation in which even when the bed starts to collapse, there may still be a small portion of beads remaining on the grid. This is probably because the size distribution of the beads we used in the experiment was broad (see Table 1), and the density
Fig. 4. Effect of magnetic field strength on bed stability (355 – 500-Am beads).
difference between the beads and fluid was small. For the given size distribution that was used, the particle minimum fluidisation velocity that was calculated for the biggest particle is almost nine times larger than the smallest particle. Therefore, it is likely that at the transition liquid velocity level, smaller particles start to be gradually washed out from the reactor, but larger particles still remained on the grid. This is also the reason why our results show that even in a conventional fluidised bed without magnetic stabilisation, pressure drop across the bed does not remain relatively constant, but increases after fluidisation (see Figs. 3b and 4). The effect of external magnetic induction lies in the fact that it increases the liquid superficial velocity at which the particles of the bed start to escape from the reactor. This is not surprising, as the magnetic tension force becomes much stronger in this case. Similar phenomena were also observed and mentioned by Jaraiz et al. [17]. 4.2. Effect of particle size It is difficult to directly visualise the influence of particle size on the stability behaviour of the bed. However, it can be shown [18,19] that the magnetic force encountered by a magnetic particle in an external magnetic field is proportional to the second power of the diameter of that particle.
Fig. 6. Effect of particle magnetic susceptibility on bed stability for 15 mT of field induction (typically vhematite /vmagnetite=1/100).
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geenan hematite (Fe2O3) and n-carrageenan magnetite (Fe3O4) beads being loaded into the reactor, respectively. These beads were of the same size range (355 –500 Am) and density (1250 –1400 kg/m3) but possessed different magnetic susceptibility. Fig. 6 summarises the variation of the bed stability behaviour for the two types of beads. It can be noted that the magnetite beads exhibited a much higher bed transition velocity than the hematite bed, implying the significant advantage of a strongly magnetically stabilised fluidised bed. 4.4. Effect of fluid properties Fig. 7. Effect of substrate fluid property on bed stability (under 15 mT of field induction).
Moreover, it can be inferred that the larger the size of the particle, the higher the terminal velocity, and, consequently, the greater transition velocity of the fluidised bed. Fig. 5 plots the experimental results obtained from the investigation of the effect of particle size distribution on the bed stability. A total of four particle sizes were tested, with the field induction being fixed at 5 mT. It can be seen that, as expected, the bed transition velocity increases steadily with the mean particle size. The superficial velocity value at which the bed pressure drop starts to decrease sharply, for example, is more than doubled when the mean particle size increases from 146 to 404 Am.
The effects of fluid properties on bed stability were studied at room temperature by using three different concentrations of substrate solutions (A, B and C) with ncarrageenan-magnetite beads at 355– 500Am and at 15 mT field induction. At room temperature, the three substrates exhibit a density of 1000, 1049, 1116 kg/m3 and a viscosity of 1103, 8103, 12103 Pa s, respectively. The results (Fig. 7) indicate that the liquid viscosity and density changes can have a strong influence on the stability of the bed. This is not difficult to understand as the buoyancy force will increase steadily as the density of the substrate liquid increases, causing the whole bed to become unstable more easily under the same fluid hydrodynamic condition.
4.3. Effect of magnetic susceptibility
5. Comparison of experimental data with model predictions
Investigation of the effect of the bead magnetic susceptibility on the bed stability was performed with n-carra-
Based on Eq. (12), the bed transition velocities can be calculated for various solid particle and substrate liquid
Fig. 8. Stability behaviour predicted using Eqs. (12) and (13) for a magnetically stabilised fluidised bed consisting of mean sizes of 217 Am (the dotted line) and 404 Am (the solid line) magnetite embedded n-carrageenan beads, respectively, and the corresponding experimental results (o—404 Am and 4—217Am).
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properties, as well as fluid hydrodynamic conditions and magnetic field induction. Fig. 8 illustrates the calculated transition velocity variations as a function of magnetic field strength for the fluidised bed consisting of magnetite embedded n-carrageenan beads of 217 and 404-Am mean sizes, respectively. The results are presented in a normalised dimensionless term of u0/um for the bed transition velocity and B/(l0qp)1/2um for the magnetic field strength, respectively. The required velocity –voidage relation was acquired by using the Richardson – Zaki equation which can be written as:
e0 em
nþ1 ¼
u0 um
ð13Þ
The following numerical values in SI units have been adopted in the calculation: qp=1250 kg/m3, qf =1000 kg/m3, l0=4p107, vp=0.05, em=0.35[10], n = 4, for d p = 0.217 10 3 m; n=5, for d p = 0.404103 m. Fig. 8 shows that the calculated bed transition velocity increases with both magnetic field strength and particle sizes. This is consistent with the experimental observations. Shown in the figure are some of the data obtained in our experiments. It can be seen that they fit reasonably well with the calculated predictions. It would also be interesting to see how the bed transition velocity would change when the magnetic field induction is further increased, and to compare the experimental data with the theoretical predictions. In practice, this was limited by the capacity of the pump incorporated in our experimental apparatus.
6. Conclusions A new mathematical model to calculate the bed transition velocity for magnetically stabilised fluidised beds has been established based on the stability analysis by Rosensweig and use of the Richardson –Zaki equation. A wide range of experiments was conducted to investigate the effects of some key process parameters on the stability behaviour of the bed. It was found that the bed transition velocity is strongly influenced by all the parameters that were investigated, including magnetic field strength, particle size and liquid property. The model predicted transition velocities for the fluidised beds consisting of 180 –250 and 355 –500-Am n-carrageenan-magnetite beads respectively agreed well with the experimental data under various magnetic induction conditions. It is believed that the model can be used in conjunction with biocatalysis kinetics to assist in the
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assembly of an overall reactor model to predict scale-up operability. Nomenclature B0 Magnetic induction applied (T) dp Mean particle diameter (m) Mp Magnetisation of particle (A/m) Mf Magnetisation of fluid (A/m) u0 Superficial velocity (m/s) um Minimum fluidisation velocity (m/s) vt Particle terminal velocity (m/s) vf Local fluid velocity (m/s) X x axial (in Eq. (1))
Greek Letters ef bed fluid fraction, e.g., bed voidage e0 bed voidage at the point bed starts to transfer from stable to unstable fluidisation em bed voidage at the point bed starts to fluidise qf substrate fluid density (kg/m3) qp particle density (kg/m3) b(ef) drag coefficient bV(ef) @b/@e l0 permeability of free space (Henry/m) vp susceptibility of particle vf susceptibility of fluid
Acknowledgements The authors wish to thank Glaxo Wellcome PLC, Cerestar, EA Technology and DTI for the financial sponsorship of the research work, which is also partly funded by BBSRC Biochemical Engineering-Link grant.
References [1] R.A. Williams, Colloid and Surface Engineering: Applications in the Process Industries, Butterworth-Heinemann, Oxford, UK, 1992, pp. 314 – 319. [2] R.E. Rosensweig, Process for operating a magnetically stabilized fluidized bed, (1978) US Patent, 4, 115, 927. [3] S.C. Saxena, S. Shrivastava, Some hydrodynamic investigations of a magnetically stabilized air-fluidized bed of feeromagnetic particles, Powder Technology 64 (1991) 57 – 67. [4] J.H. Siegell, Liquid-fluidized magnetically stabilized beds, Powder Technology 52 (1987) 139 – 148. [5] G.M. Moffat, The development of a magnetically stabilized fluidized bed reactor for maltodextrin hydrolysis, PhD thesis, University of Manchester Institute of Science and Technology, 1995. [6] C. Webb, K.H. Kang, G. Moffat, R.A. Williams, A.M. Este´vez, J. Cue´llar, E. Jaraiz, M.A. Galan, The magnetically stabilized fluidized bed bioreactor: a tool for improved mass transfer in immobilized enzyme systems, Chemical Engineering Journal 61 (1996) 241 – 246. [7] M.A. Burns, D.J. Graves, Axial dispersion in a magnetically stabilized
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[11]
[12] [13]
Y.Y. Hou, R.A. Williams / Powder Technology 124 (2002) 287–294 fluidized bed liquid chromatography column, Powder Technology 64 (1991) 81 – 92. W.K. Lee, A review of the rheology of magnetically stabilized fluidized beds, Powder Technology 64 (1991) 69 – 80. J. Arnaldos, J. Casal, A. Lucas, I. Puigjaner, Magnetically stabilized fluidisation: modeling and application to mixtures, Powder Technology 44 (1985) 57 – 62. R.E. Rosensweig, Magnetic stabilization of the state of uniform fluidisation, Industrial and Engineering Chemistry Fundamentals 18 (3)(1979). R.E. Rosensweig, G. Ciprios, Ferroliquid stabilization of fluidisation in a bed of nonmagnetic spheres, Powder Technology 64 (1991) 115 – 123. C.J. Fee, Stability of the liquid-fluidized magnetically stabilized fluidized bed, AIChE Journal 42 (5)(1996). R.E. Rosensweig, G. Ciprios, Magnetohydrodynamics 24 (4) (1988) October – December.
[14] Y.Y. Hou, A systematic investigation of a liquid magnetically stabilised fluidised bed, PhD thesis, Chapter 6, University of Leeds, 2002. [15] Y.H. Cohen, C. Tien, Aerosol filtration in a magnetically stabilized fluidized bed. Powder Technology 64 (1991) 147 – 158. [16] R.K. Hamdy, Y.A. Liu, Studies in magnetochemical engineering: Part VI. An experimental study of screen-packed and conventional fluidized beds in axial and transverse magnetic fields, Powder Technology 64 (1991) 103 – 113. [17] E. Jaraiz, O. Levenspiel, T.J. Fitzgerald, Chemical Engineering Science 38 (1983) 107. [18] R.L. Sonolikar, Magneto-fluidized beds, in: L.K. Doraiswamy, A.S. Mujumdar (Eds.), Transport in Fluidized Particle Systems, Elsevier, Amsterdam, 1989, pp. 359 – 422. [19] F. Ouyang, Y.X. Wu, C. Guo, K. Mooson, Fluidisation under external forces: 1. Magnetized fluidisation, Journal of Chemical Industry and Engineering 5 (2)(1990).
Magnetic stabilisation of a liquid f luidised bed Y.Y. Hou *, R.A. Williams Centre for Particle and Colloidal Engineering, School of Process, Environmental and Materials Engineering, University of Leeds, Leeds LS2 9JT, UK
Abstract The stability of a magnetically stabilised fluidised bed (MSFB) has been studied in an experimental investigation coupled with mathematical modelling. Experimental results indicate that the stability of the bed can be influenced by several factors, including liquid velocity, magnetic field strength, solid and liquid density, solid magnetic susceptibility and solid particle size distributions. A theoretical model based on Rosensweig’s mean field theory and the Richardson – Zaki equation has been established to predict the bed transition velocity when a fluidised bed starts to transform from a stable to an unstable state against an applied induction. The model predictions were found to have reasonable agreement with the experimental values obtained from bed pressure drop measurements. The results have applications to the design of magnetically stabilised fluidised bed bioreactors. D 2002 Elsevier Science B.V. All rights reserved. Keywords: Magnetically stabilised fluidised bed; Stability; Magnetic force; Modelling
1. Introduction Several types of reactors are used in biological and chemical engineering industries. Some of the most commonly employed ones include stirred tanks, packed bed, and conventional fluidised bed reactors. Unfortunately, all these reactors have some inherent disadvantages. A stirred tank reactor, for example, is limited by its ability to operate in a batch-mode, which can be problematic. Packed beds have a strong tendency to become plugged with various solid debris. Such malfunctioning is common in many bioengineering systems. It has been shown that fluidisation can provide an effective solution to many of the problems associated with plug flows and stirred tank reactors. A conventional fluidised bed, however, is often plagued by poor contact between the two phases and bypassing effects. In addition, the capacity of the bed is also heavily limited by substrate throughput. This paper explores an alternative means of reactor design using a magnetically stabilised fluidised bed reactor (MSFBR). The MSFBR is a fluidised bed reactor comprising of magnetic particles, or magnetic fluid, surrounded by an array of magnetic coils. Such a reactor has the potential capability of eliminating many of the difficulties mentioned above. If bed particles are made of magnetisable materials
*
Corresponding author. E-mail address: [email protected] (Y.Y. Hou).
[1], with introduction of an external magnetic field, the bed can then be stabilised to a quiescent and uniform condition. The hypothesis is that when in this state, the bed will possess properties that are almost an ideal combination of those exhibited by the packed bed and a conventional fluidised bed. This has been demonstrated for both gas [2,3] and liquid [4 – 6] medium systems. Some of the key features and benefits of a magnetic stabilised bed have been illustrated in detail in the literature [7– 9]. The stabilisation of a magnetic fluidised bed consisting of magnetic particles supported by a gas medium has been systemically studied by Rosensweig [10]. By neglecting the gas fluid density, the author developed a so-called ‘normalised magnetisation term’ to describe the stability of the gassolid fluidised bed. Unfortunately, this term is not applicable to liquid – solid fluidised beds, as the fluid density cannot be neglected in comparison with the particle density. Rosensweig and Ciprios [11] later modified the stability description model to accommodate its use in the liquid – solid fluidised bed applications, but they mainly concentrated on the case of fluidisation of nonmagnetic spheres with a magnetic ferrofluid, from which reasonable agreement between theory and experimental data was achieved. Fee [12], on the other hand, tried to give a suitable criterion for the stability of magnetically stabilised liquid-medium fluidised bed based on the particle magnetisation analysis approach. In his work, the liquid density was taken into account and the transition velocities that were predicted by Fee’s model were found to agree more closely with the
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experimental values than those from Rosensweig’s gas fluidised bed model. However, since Fee’s work ignored the effect of Reynolds number in the drag coefficient calculations, the theory failed to achieve quantitative predictions of the bed’s actual stability and behaviour. In the present paper, efforts have been made to deduce a more universal and applicable stability model for the liquidmedium magnetically stabilised fluidised bed based on the work given by Rosensweig [11]. A similar underlying assumption to that of Rosensweig’s will be adopted here, in which a normalised magnetic induction term is developed to yield a more reasonable criterion for evaluating the stability of a magnetically stabilised fluidised bed.
Rosensweig [13] has deduced the following dynamic equations to describe the evolution of a bed voidage perturbation impressed upon a magnetically fluidised bed, in which ef stands for bed voidage, and other terms are given in the nomenclature: @ 2 ef @ 2 ef @ef @ef @ 2 ef þ C þ D þ ðE þ FÞ þ B ¼ 0 ð1Þ @x@t @x @t @2t @2x ðe1 2ðe1 f 1Þqf f 1Þvqf ; B¼ ; qp qp ! g qf ef ð1 ef ÞbVðef Þ C¼ 1 1 2ef ef bðef Þ qp
A¼1þ
g D¼ ef v
! qf 1 ; qp
E¼
18v2p ð3 þ 3vp 2ef vp Þ
v 2 ðe1 f 1Þqf qp
3
s ¼ n ig
ð5Þ
Magnetic stabilisation is, therefore, achieved when n<0, let n=0 to satisfy the neutral stability condition. Defining another complex variable Z as: Bji þ D 2 Cji ðE þ FÞj2 Z ¼ X iY ¼ ; ð6Þ 2A A then rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 1 ðr þ X Þ i ðr X Þ Z¼ 2 2
The neutral stability condition means: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 D ðr þ X Þ ¼0 Res ¼ 2 2A
ð7Þ
ð8Þ
Solving X and Y from Eq. (6), and substituting them into Eq. (8) yields: AC 2 þ D2 ðE þ FÞ CBD ¼ 0:
ð1 e2f ÞB20 l0 qp
When vf ¼ 0 Obviously, Eq. (1) is a rather complex partial differential equation. One solution to the problem is to presume that the bed voidage perturbation can be represented in a Fourier components form, i.e.: ef ¼ const Reðest eijx Þ
ð3Þ
Our contribution has been focused on solving this equation for a range of practical conditions and further extending the theoretical work. Solving the above equation for s gives: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s Bji þ D 2 Cji ðE þ FÞj2 Bji þ D ð4Þ s¼F 2A 2A A
where r2 ¼ X 2 þ Y 2 :
Mp @Hp l ð1 ef Þ Mf @Hf þ F¼ 0 qp @ð1 ef Þ @ef ¼
As2 þ sðBji þ DÞ þ ½Cji ðE þ FÞj2 ¼ 0
Since s is known to be complex, we may arbitrarily define s to be the form of:
2. Theoretical modelling
A
negative, as a positive value would yield exponential growth, and, therefore, would have no physical meaning. Substituting ef of Eq. (2) into Eq. (1), and assuming ef > 0, we have a complex quadratic algebraic equation for s:
ð2Þ
Where Re denotes the real part, j is wave number, taken as real, and s is a complex number. The real part of s is usually called the growth factor. In order to achieve a stabilised fluidisation behaviour (i.e., the bed voidage perturbation approaches zero) it is clear that the growth factor has to be
ð9Þ
Expressions for A, B, D, E, F have already been given in Eq. (1). Parameter C, however, requires knowledge of the ratio bV(ef)/b(ef), which in turn is related to the bed expansion behaviour caused by changes in the superficial velocity of the fluidising liquid. By using the Richardson – Zaki equation, which states that efn=m/mt, where n depends on Reynolds number and now also the applied magnetic induction, and assuming the dynamic fluidised bed pressure drop DPf=b(ef)v fL is constant, the drag coefficient has the following relationship with the bed voidage: bðef Þ ¼ constð1 ef Þ=enf ; from which it can be seen: bVðef Þ n þ ef ð1 nÞ ¼ : bVðef Þ ef ð1 ef Þ
ð10Þ
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so-called normalised magnetisation term, as it can be considered as a ratio of magnetic field energy to fluid kinetic energy. Eq. (12) is virtually a mathematical description of the bed transition superficial velocity (u0) as a function of magnetic field induction (B0), particle physical properties (n, vp, qp) and substrate liquid properties (n, qf). From this equation, a criterion to separate the stable fluidising zone from the unstable one for any magnetically stabilised fluidised bed can be conveniently established for different magnetic and hydrodynamic conditions.
3. Experimental methods 3.1. Magnetic particle characteristics Fig. 1. A photomicrograph of magnetite embedded n-carrageenan beads.
Substituting the above expression into Eq. (1), C will yield: ! g qf 1 ð11Þ C¼ ½ð1 þ nÞð1 ef Þ ef qp Having acquired all the expressions for A, B, C, D, E, F and substituting them into Eq. (9), a stability criterion for a magnetically stabilised fluidised bed can be deduced: ( ð3 þ 3vp 2e0 vp Þ3 B20 ¼ ½n ð1 þ nÞe0 2 l0 qp u20 18v2p e20 ð1 e20 Þ ) ðe1 1Þq f 0 þ ½ð1 þ nÞð1 e0 Þ2 ð12Þ qp Where u0 is the bed transition superficial velocity, which equals emf 0. The left-hand side of the above equation is the
All the experiments were carried out using magnetiteembedded n-carrageenan beads as solid particles. The beads were prepared by first uniformly mixing 10 wt.% magnetite particles (<74Am) into 3% n-carrageenan gel at around 75 jC. The mixture was then poured into an oil phase with constant stirring and was allowed to cool down to below 10 jC. After removing all the oil phase, the magnetic ncarrageenan soft solid particles were hardened in 5% polyethylenimine solution for around 1 h, and then washed in room temperature, The products were then sieved to different sizes and stored in 0.1 M KCl solution at 4 jC, ready for experimental use. Microscopic inspection shows that almost all the beads are perfectly spherical. Specific gravity measurement indicates that the density of the beads is 1250– 1400 kg/m3. Magnetisation test on fully dried n-carrageenan and magnetite mixture has shown a susceptibility vp of 0.6 (SI). It is therefore estimated that the susceptibility vp of the fresh beads would be in the range of 0.04 – 0.05 (SI). Fig. 1
Table 1 Size distributions of beads in four sieved fractions used in the experiments Bead samples 125 – 180 Am
180 – 250 Am
250 – 355 Am
355 – 500 Am
Size (Am)
Volume (%)
Size (Am)
Volume (%)
Size (Am)
Volume (%)
Size (Am)
Volume (%)
56.4 63.3 71.0 79.6 89.3 100.2 112.5 126.2 141.6 158.9 178.3 200.0 224.4 251.8 282.5 317.0 355.7
0.02 0.06 1.83 3.79 6.34 9.09 11.56 13.13 13.47 12.48 10.39 7.74 5.08 2.83 1.26 0.35 0.02
100.2 112.5 126.2 141.6 158.9 178.3 200.0 224.4 251.8 282.5 317.0 355.7 399.1 447.7
0.09 0.74 2.66 6.29 11.16 15.87 18.15 17.56 13.07 8.39 4.32 1.35 0.32 0.02
126.2 141.6 158.9 178.3 200.0 224.4 251.8 282.5 317.0 355.7 399.1 447.7 502.4 563.7 632.5
0.05 0.32 1.63 4.31 8.51 13.23 16.81 17.69 15.41 11.19 6.57 3.04 1.04 0.18 0.02
178.3 200.0 224.4 251.8 282.5 317.0 355.7 399.1 447.7 502.4 563.7 632.5 709.6 796.2
0.05 0.41 1.87 4.82 9.25 14.15 17.45 17.85 15.06 10.37 5.7 2.35 0.61 0.05
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and Table 1 show a photomicrograph and the size distributions of the beads that are being used in the experiments. The size distributions were obtained by using laser diffraction (Malvern, 2000). 3.2. Fluidised bed reactor and pressure drop measurements The reactor consists of a plastic cylinder of 150 mm inner diameter and 300 mm in height. A grid is placed in the bottom of the cylinder to hold the particles and a baffle is added in the flow-input chamber to keep the flow input uniform and steady. The magnetic field was generated by three DC-powered copper wire coils surrounding the reactor, each of them having 500 turns, 200 mm ID, and spaced 70 mm apart, with the first coil being located 30 mm above the distributor. The schematic diagram in Fig. 2 illustrates the configuration of the rig. For each experiment, about 1 kg of beads was used. This corresponds to an initial bed height of approximately 40 mm. The fluid was introduced into the chamber from underneath the grid using a peristaltic pump, and circulating in the system via the storage tank. The volumetric flow rate of the recirculating fluid was measured by a flow meter incorporated in the flow circuit from which the superficial velocity was calculated. The entire reactor is fully filled with fluid at all times. The bed stability was monitored via pressure drop measurements throughout the experiment using an inclined
Fig. 2. Schematic diagram of experimental rig and pressure drop measurement.
manometer, with one measuring point located beneath the grid and the other at the top of the reactor. The arrangement was also illustrated in Fig. 2. Beginning with the system being stationary, when there is no fluid passing through, and denoting Lmf as the initial bed height, the initial pressure readings indicated by the levels at position A for points 1 ( P10) and 2 ( P20) should be equal. As the fluid starts to flow and the fluid velocity exceeds umf, the bed starts to fluidise, and the bed height changes from Lmf to L. Accordingly, an extra pressure drop DP can be observed from the manometer readings for pressure at points 1 ( P1) and 2 ( P2), respectively. This pressure drop has been used in the paper as a means of describing the bed stability. It has been noted that the pressure differences across the grid at all the flow conditions that were investigated are negligible (less than 5 Pa). Other process parameters which could have direct impact on the bed stability behaviour, such as magnetic field strength, particle size distributions, the density and viscosity of the substrate liquid, etc., [14] are also considered. Key results are summarised in this paper.
4. Experimental measurement of bed performance 4.1. Effect of magnetic field strength The investigations were carried out using 180– 250 and 355– 500-Am n-carrageenan-magnetite beads, respectively as solid particles and tap water as substrate fluid. Fig. 3 shows changes of (a) bed height and (b) bed pressure drop as a function of fluid velocity under different magnetic field strength for the 180 – 250-Am beads bed, the latter is the DP described in Fig. 2. It can be seen that the bed height starts to considerably increase as the fluid velocity reaches the minimum fluidisation velocity threshold. This has been confirmed by the pressure drop measurements which show that the threshold is also the point when the pressure drop starts to increase with the fluid velocity. It can also be seen from Figs. 3b and 4 that there exists another significant point at which a decline in the pressure drop was monitored. This is believed to be the transition point when the stabilised fluidised bed starts to collapse. At the transition point, it was observed that the upper bed surface was no longer smooth, the bed started to lose structure and changed from MSFB to partially stabilised fluidised bed (PSFB). Finally, beads would be flushed out from the reactor. Physically, the declination in pressure drop is caused by the drag force overcoming the magnetic binding force that exists between magnetic particles. Similar pressure decline phenomena have been recorded and explained as the results of bed channelling and rearrangement [15,16]. It is interesting to note that the bed transition velocity increases as the magnetic field gets stronger, indicating that use of magnetic field in a magnetic particle bed would help to expand the stability range of the fluidisation.
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291
Fig. 5. Effect of solid particle size on bed stability (5-mT field induction).
Fig. 3. Bed expansion (a) and pressure drop (b) (180 – 250-Am beads).
It should be noted that the pressure drop across both the conventional and magnetically fluidised bed, in our case, did not remain relatively constant after fluidisation. This is mainly because the pressure drop across the bed would only remain constant when the bed is fully fluidised, i.e., if all the particles are in a fully suspended state in the bed. Unfortunately, this has never been the case in our investigation in which even when the bed starts to collapse, there may still be a small portion of beads remaining on the grid. This is probably because the size distribution of the beads we used in the experiment was broad (see Table 1), and the density
Fig. 4. Effect of magnetic field strength on bed stability (355 – 500-Am beads).
difference between the beads and fluid was small. For the given size distribution that was used, the particle minimum fluidisation velocity that was calculated for the biggest particle is almost nine times larger than the smallest particle. Therefore, it is likely that at the transition liquid velocity level, smaller particles start to be gradually washed out from the reactor, but larger particles still remained on the grid. This is also the reason why our results show that even in a conventional fluidised bed without magnetic stabilisation, pressure drop across the bed does not remain relatively constant, but increases after fluidisation (see Figs. 3b and 4). The effect of external magnetic induction lies in the fact that it increases the liquid superficial velocity at which the particles of the bed start to escape from the reactor. This is not surprising, as the magnetic tension force becomes much stronger in this case. Similar phenomena were also observed and mentioned by Jaraiz et al. [17]. 4.2. Effect of particle size It is difficult to directly visualise the influence of particle size on the stability behaviour of the bed. However, it can be shown [18,19] that the magnetic force encountered by a magnetic particle in an external magnetic field is proportional to the second power of the diameter of that particle.
Fig. 6. Effect of particle magnetic susceptibility on bed stability for 15 mT of field induction (typically vhematite /vmagnetite=1/100).
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geenan hematite (Fe2O3) and n-carrageenan magnetite (Fe3O4) beads being loaded into the reactor, respectively. These beads were of the same size range (355 –500 Am) and density (1250 –1400 kg/m3) but possessed different magnetic susceptibility. Fig. 6 summarises the variation of the bed stability behaviour for the two types of beads. It can be noted that the magnetite beads exhibited a much higher bed transition velocity than the hematite bed, implying the significant advantage of a strongly magnetically stabilised fluidised bed. 4.4. Effect of fluid properties Fig. 7. Effect of substrate fluid property on bed stability (under 15 mT of field induction).
Moreover, it can be inferred that the larger the size of the particle, the higher the terminal velocity, and, consequently, the greater transition velocity of the fluidised bed. Fig. 5 plots the experimental results obtained from the investigation of the effect of particle size distribution on the bed stability. A total of four particle sizes were tested, with the field induction being fixed at 5 mT. It can be seen that, as expected, the bed transition velocity increases steadily with the mean particle size. The superficial velocity value at which the bed pressure drop starts to decrease sharply, for example, is more than doubled when the mean particle size increases from 146 to 404 Am.
The effects of fluid properties on bed stability were studied at room temperature by using three different concentrations of substrate solutions (A, B and C) with ncarrageenan-magnetite beads at 355– 500Am and at 15 mT field induction. At room temperature, the three substrates exhibit a density of 1000, 1049, 1116 kg/m3 and a viscosity of 1103, 8103, 12103 Pa s, respectively. The results (Fig. 7) indicate that the liquid viscosity and density changes can have a strong influence on the stability of the bed. This is not difficult to understand as the buoyancy force will increase steadily as the density of the substrate liquid increases, causing the whole bed to become unstable more easily under the same fluid hydrodynamic condition.
4.3. Effect of magnetic susceptibility
5. Comparison of experimental data with model predictions
Investigation of the effect of the bead magnetic susceptibility on the bed stability was performed with n-carra-
Based on Eq. (12), the bed transition velocities can be calculated for various solid particle and substrate liquid
Fig. 8. Stability behaviour predicted using Eqs. (12) and (13) for a magnetically stabilised fluidised bed consisting of mean sizes of 217 Am (the dotted line) and 404 Am (the solid line) magnetite embedded n-carrageenan beads, respectively, and the corresponding experimental results (o—404 Am and 4—217Am).
Y.Y. Hou, R.A. Williams / Powder Technology 124 (2002) 287–294
properties, as well as fluid hydrodynamic conditions and magnetic field induction. Fig. 8 illustrates the calculated transition velocity variations as a function of magnetic field strength for the fluidised bed consisting of magnetite embedded n-carrageenan beads of 217 and 404-Am mean sizes, respectively. The results are presented in a normalised dimensionless term of u0/um for the bed transition velocity and B/(l0qp)1/2um for the magnetic field strength, respectively. The required velocity –voidage relation was acquired by using the Richardson – Zaki equation which can be written as:
e0 em
nþ1 ¼
u0 um
ð13Þ
The following numerical values in SI units have been adopted in the calculation: qp=1250 kg/m3, qf =1000 kg/m3, l0=4p107, vp=0.05, em=0.35[10], n = 4, for d p = 0.217 10 3 m; n=5, for d p = 0.404103 m. Fig. 8 shows that the calculated bed transition velocity increases with both magnetic field strength and particle sizes. This is consistent with the experimental observations. Shown in the figure are some of the data obtained in our experiments. It can be seen that they fit reasonably well with the calculated predictions. It would also be interesting to see how the bed transition velocity would change when the magnetic field induction is further increased, and to compare the experimental data with the theoretical predictions. In practice, this was limited by the capacity of the pump incorporated in our experimental apparatus.
6. Conclusions A new mathematical model to calculate the bed transition velocity for magnetically stabilised fluidised beds has been established based on the stability analysis by Rosensweig and use of the Richardson –Zaki equation. A wide range of experiments was conducted to investigate the effects of some key process parameters on the stability behaviour of the bed. It was found that the bed transition velocity is strongly influenced by all the parameters that were investigated, including magnetic field strength, particle size and liquid property. The model predicted transition velocities for the fluidised beds consisting of 180 –250 and 355 –500-Am n-carrageenan-magnetite beads respectively agreed well with the experimental data under various magnetic induction conditions. It is believed that the model can be used in conjunction with biocatalysis kinetics to assist in the
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assembly of an overall reactor model to predict scale-up operability. Nomenclature B0 Magnetic induction applied (T) dp Mean particle diameter (m) Mp Magnetisation of particle (A/m) Mf Magnetisation of fluid (A/m) u0 Superficial velocity (m/s) um Minimum fluidisation velocity (m/s) vt Particle terminal velocity (m/s) vf Local fluid velocity (m/s) X x axial (in Eq. (1))
Greek Letters ef bed fluid fraction, e.g., bed voidage e0 bed voidage at the point bed starts to transfer from stable to unstable fluidisation em bed voidage at the point bed starts to fluidise qf substrate fluid density (kg/m3) qp particle density (kg/m3) b(ef) drag coefficient bV(ef) @b/@e l0 permeability of free space (Henry/m) vp susceptibility of particle vf susceptibility of fluid
Acknowledgements The authors wish to thank Glaxo Wellcome PLC, Cerestar, EA Technology and DTI for the financial sponsorship of the research work, which is also partly funded by BBSRC Biochemical Engineering-Link grant.
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[14] Y.Y. Hou, A systematic investigation of a liquid magnetically stabilised fluidised bed, PhD thesis, Chapter 6, University of Leeds, 2002. [15] Y.H. Cohen, C. Tien, Aerosol filtration in a magnetically stabilized fluidized bed. Powder Technology 64 (1991) 147 – 158. [16] R.K. Hamdy, Y.A. Liu, Studies in magnetochemical engineering: Part VI. An experimental study of screen-packed and conventional fluidized beds in axial and transverse magnetic fields, Powder Technology 64 (1991) 103 – 113. [17] E. Jaraiz, O. Levenspiel, T.J. Fitzgerald, Chemical Engineering Science 38 (1983) 107. [18] R.L. Sonolikar, Magneto-fluidized beds, in: L.K. Doraiswamy, A.S. Mujumdar (Eds.), Transport in Fluidized Particle Systems, Elsevier, Amsterdam, 1989, pp. 359 – 422. [19] F. Ouyang, Y.X. Wu, C. Guo, K. Mooson, Fluidisation under external forces: 1. Magnetized fluidisation, Journal of Chemical Industry and Engineering 5 (2)(1990).