Ind. Eng. Chem. Res. 1989,28, 803-808 Broyden, C. G. A Class of Methods for Solving Nonlinear Simultaneous Equations. Math. Comput. 1965,19, 577-594. Crowe, C. M.; Nishio, M. Convergence Promotion in the Simulation of Chemical Process-The General Dominant Eigenvalue Method. AZChE J . 1975,21, 528-533. Dennis, J. E.; Schnabel, R. B. Least-Change Secant Updates for Quasi-Newton Methods. SIAM Rev. 1979,21, 443-459. Doherty, M. F.; Perkins, J. D. On the Dynamics of Distillation Processes. I V Uniqueness and Stability of the Steady State in Homogeneous Continuous Distillations. Chem. Eng. Sci. 1982,37, 381-392. Friday, J. R.; Smith, B. D. An Analysis of the Equilibrium Stage Separations Problem - Formulation and Convergence. AZChE J . 1964, IO, 698-706. King, C. J. Separation Processes; McGraw-Hill: New York, 1971. Lapidus, L.; Amundson, N. R. Stagewise Absorption and Extraction Equipment-Transient and Unsteady State Operation. Znd. Eng. Chem. 1950,42, 1071-1078. Lucia, A. Uniqueness of Solutions to Single-Stage Isobaric Flash Processes Involving Homogeneous Mixtures. AZChE J. 1986,32, 1761-1770. Michelsen, M. L. The Isothermal Flash Problem. Part 11. PhaseSplit Calculation. Fluid Phase Equilib. 1982, 9, 21-40. Orbach, 0.;Crowe, C. M. Convergence Promotion in the Simulation of Chemical Processes with Recycle-The Dominant Eigenvalue Method. Can. J . Chem. Eng. 1971,49, 509-513. Ortega, J. M.; Rheinboldt, W. C. Iterative Solution of Nonlinear
803
Equations in Seueral Variables; Academic Press: New York, 1970. Prausnitz, J. M.; Andersen, T. F.; Grens, E. A.; Eckert, C. A.; Hsieh, R.; O’Connell, J. P. Computer Calculations for Multicomponent Vapor-Liquid and Liquid-Liquid Equilibria; Prentice-Hall: Englewood Cliffs, NJ, 1980. Rosenbrock, H. H. A Theorem of Dynamic Conversion for Distillation. Trans. Znst. Chem. Eng. 1960, 38, 279-287. Rosenbrock, H. H. A Lyapunov Function with Applications to Some Nonlinear Physical Problems. Automatica 1962,1, 31-53. Sujata, A. D. Absorber-Stripper Calculations Made Easier. Hydrocarbon Process. Pet. Ref. 1961, 40, 137-140. Varga, R. S. Matrix Zteratiue Analysis; Prentice-Hall: Englewood Cliffs, NJ, 1962. Van Dongen, D. B.; Doherty, M. F.; Haight, J. R. Material Stability of Multicomponent Mixtures and the Multiplicity of Solutions to Phase Equilibrium Equation. 1. Nonreacting Mixtures. Znd. Eng. Chem. Fundam. 1983,22, 472-485. Wegstein, J. H. Accelerating Convergence of Iterative Process. ACM Comm. 1958, I , 9-13. Westerberg, A. W.; Hutchison, H. P.; Motard, R. L.; Winter, P. Process Flowsheeting; Cambridge University Press: Cambridge, England, 1979. Received for review July 29, 1988 Revised manuscript received February 11, 1989 Accepted February 17, 1989
Magnetic Filter for Solids: Theory and Experiment Shoichi Kimura and Octave Levenspiel* Chemical Engineering Department, Oregon State University, Corvallis, Oregon 97331
This paper introduces a new type of filter for removing ferromagnetic solids from a flow stream, one that freezes the particles to the tube wall by magnetic forces alone. Basic theory is developed t o tell how to find the parameters of any given system, and experiments are run to verify the theory. This is a report on a new type of filter that is designed to remove ferromagnetic solids from a slurry flowing in a tube or pipe. It acts by creating a strong magnetic field in the flow channel so as to freeze the solids to one side of the flow channel. Solids build up there, and eventually the flow channel becomes so restricted that no more solids can be captured. It is then time to turn off the magnetic field, which releases the captured solids to a collection chamber. The whole operation is then repeated. Asema (1988) presently offers such filters for removing suspended magnetic fines from industrial wastewaters. Another use for this device is for recovering ferromagnetic catalyst particles or catalyst particles having ferromagnetic cores from fluid streams leaving slurry reactors. At the beginning and close to the end of an operating cycle, a small fraction of the flowing solids may escape the magnetic filter. Hence, if all the solids are to be recovered, the magnetic filter will have to be followed with an ordinary filter. Nevertheless, with this magnetic filter, the duty of the mechanical filter will be greatly reduced because the magnetic filter can hold a large amount of solid with only a relatively small increase in pressure drop. In general, low pressure drop is a special advantage of the magnetic filter over its mechanic61 counterpart. This follows from the fact that it has no mechanical obstruction in its flow channel.
Theory Consider a pair of activated electromagnetic pole pieces snugly fitted to a paramagnetic tube through which is flowing a slurry containing ferromagnetic particles, as shown in Figure 1. With a high enough field intensity, 0888-5885/89/2628-0S03$01.50/0
particles can be arrested at the pole pieces, and since the magnetic flux density between the pole pieces is highest when the air gap is smallest, particles collect there first. Once a layer of particles is formed, the magnetic flux density increases because of the increased permeability of this solid layer. The solid layer grows progressively while reducing the cross-sectional area of the flow channel. This process continues until the magnetic forces just balance the net frictional and gravitational forces on the particles. This progression is shown in Figure 2. This section develops the basic theory which relates the capture current and the quantity of solids captured with the pole piece geometry, particle size, system geometry, and operating conditions. Balance of Forces. The pertinent forces influencing the action of a magnetic filter are the magnetic force, gravity, and hydrodynamic drag. Let us focus attention on a spherical particle that is touching a layer of already frozen particles. The net force tending to sweep the particle downstream is the result of gravity and drag, given by
where Cd is the drag coefficient and u is the mean velocity of the slurry. In eq 1,j = +1 for upflow of slurry, and j = -1 for downflow of slurry. Opposed to this is the vertical component of the magnetic force, F,, acting on the particle, or FZ = CfF, (2) 62 1989 American Chemical Society
804 Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989
field of intensity H, is given by (Ahanori, 1976; Eisenstein, 1977; Harpavat, 1974; Yoong et al., 1983)
F, = k p f l 2 d s n
where p, is the magnetic permeability of the particle material and the power n varies between 2 and 3. The magnetic field intensity during particle buildup between the pole pieces may be evaluated by applying Ampere's law. Thus, the reluctance of the magnetic circuit with its captured solids is given by (Davis and Levenspiel, 1985; Hayt, 1981)
Iron core of the
where Zi, Si, and pi are the length, cross-sectional area, and permeability of the materials along the path of the magnetic flux. Assuming that the permeability of the air gap and of the tube wall material are roughly equal to that of a vacuum, po, and that the cross-sectional areas of air gap and t u b wall are identical with that of the pole pieces Sp,eq 6 may be rewritten as
Figure 1. Sketch of the magnetic filter. pole
0
+ Zw)(l + +)
R = L(Za POSp
TOP View
-1
small current
(7)
where Z, and 1, are the thickness of the air gap and tube wall, respectively. The parameter is defined as
I:;
piece pole (.-,1
(5)
+
POSP m-2 zi + = 2(Za
,
+
large current
Upflow of Slurry
(8)
and it accounts for the reluctance of all parts of the magnetic circuit other than the tube wall and air gap. Usually 4 F2,the particle is swept downstream; if Fl < F2,the particle freezes in place. Particle buildup a t the filter will progressively restrict the flow channel, causing an increase in flow velocity and frictional resistance. Eventually, a steady state is reached where the filter cannot capture additional solids. This occurs when the net frictional force just balances the magnetic force on the last to be captured particle, or (3)
RS,B = NI (9) Noting that the flux density is related to flux intensity by B = pH,the flux intensity in the frozen particles, H,, is found by combining eq 7 and 9 or
Substituting eq 10 into eq 5 gives the magnetic force on a particle which just touches the clump of already frozen solids as
Continuity relates the velocity, u,to the cross-sectional area, A, of the channel still open to flow, or A0
= AUo
(4)
where A. and uo are the cross-sectional area and slurry velocity in the flow channel with no solid buildup. It should be noted that the balance of forces of eq 3 has a different meaning from that of conventional magnetic separations wherein hydrodynamic drag is given by Stokes' law and the flow velocity usually does not change (Oberteuffer, 1974; Takayasu et al., 1984; van Kleef et al., 1984). Magnetic Forces. The force acting on a spherical ferromagnetic particle of diameter d,, placed in a magnetic
where kuA2
Both the permeability of the particles and the factors that affect will influence the value of km. Quantity of Particles Collected by the Filter. The progress of particle buildup depends on the direction of slurry flow. As shown in Figure 2, particle buildup is gradual ahead of the pole pieces and breaks sharply a t some distance downstream due to the decreased magnetic
+
Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989 805 location of pole piece
--T
equivalent height of solids, h,
Table I. Experimental Conditions inner diameter of flow tube, mm length of pole pieces, mm length of air gap, mm thickness of tube wall, mm width of pole pieces, mm superficial velocity of slurry, m/s slurry concentration, kg/m3 av size of iron particles, mm density of particles, kg/m3 void fraction of freely settled bed of solid
Fm
I
\ Static boundary of solids buildup
t
Figure 3. Geometrical representation of the arrested particles and the forces on an outer-edge particle.
force and the strong eddy action in this region. On the basis of these observations, we approximate the frozen mass of solids by the volume enclosed by a cylinder and three planes, as shown in Figure 3. Referring to the dimensions on this figure, the mass of solids collected is given by
W = (A0 - A)h,p,(l - t)
(13)
where
+ hd + yh,
h, = h,
25.4 25.4-508.5 0.20-1.47 1.59 12.7 0.328-0.764 10-40 0.066,0.221 1930 0.281,0.244
50.8 50.8 0.40 3.18 25.4 0.191-0.432 10-40 0.221 1930 0.244
For any known system, k1 of eq 18 can be evaluated directly since all of its components are known or can be estimated. Finally, substituting eq 18 into eq 14 gives the total mass of particles trapped by the magnetic filter as
[ { (4 )'.
W = Woq 1- U O
122
N1 + 1,
j k 1 ) l 2 ] (21)
Equation 21 with NI t gives the most solid, Wmex= Woq,that can be collected. At the other extreme, putting W = 0 gives the minimum NI needed to just capture and hold the first particle of solid. Thus,
and where, by geometric considerations, the upstream buildup fraction is given by y = 0.4. Alternatively, we may write
w = woq(1-
2)
where the amount of solids that could freely and completely fill the flow tube to a height hm is
wo = A&mp,(l-
(15)
€0)
and where the actual amount collected compared to the free solids opposite the pole pieces is
In eq 16, the void fraction t may differ from to of a freely settled bed because particles collected in a magnetic field tend to arrange themselves in strings along the lines of magnetic flux. The fraction of the cross section at the pole pieces occupied by frozen solids, 1- A/Ao, is found by combining eq 3,4, and 11. Thus,
Solving for A/Ao yields
")'+
A = uo[k2( A0 4 + 1,
-112
ik,]
(18)
where
and
k2 =
8CflZ,d,fl-2 ?rcdpf
- magnetic force drag
(20)
Equation 21 suggests that at a given NI, less solids are captured with smaller particles, at higher flow velocity, and for downflow of slurry, while eq 22 suggests that the minimum power requirement increases with smaller d,, higher uo,and j = -1 or downflow. The effects of flow velocity and particle size on W are different from those in the high-gradient magnetic separator (Luborsky and Drummond, 1976; Liu and Oak, 1983). To use eq 21 and 22 for quantitative prediction and design requires determining experimentally the two parameters q and k2. This is the aim of the experimental program, described next.
Experimenta1 Sect ion The apparatus comprise of two parts, the magnetic filter and the circulating slurry system. Magnetic Filter System. This consisted of a horseshoe-type soft iron core with a 700-turn solenoid, a dc power supply, a digital ammeter, and a variable resistor. Two types of pole pieces were tested, as shown in Figure 5, the full circular shape and half-circular shape, both of which were fitted snugly to the wall of the slow tube. Table I gives the dimensions of the pole pieces. Slurry Circulation System and Operating Procedures. Figure 4 illustrates the slurry circulation system. At the start of an experiment, a known iron-water mixture was introduced into the slurry tank and kept suspended and uniform by rapid circulation around a loop which contains a flowmeter and by means of a water pump. Then, a known portion of the flow stream was shunted to the magnetic filter flow tube. After steady state was reached, a fixed current was applied to the magnetic circuit and solids began to be collected at the filter, as shown in Figure 2. When buildup of solids was complete, the feed stream was switched from slurry to plain water from the reservoir tank to wash away the excess slurry, keeping the flow velocity in the filter tube unchanged. The system was then shut down, the current turned off, and the solids drained, paper filtered, rinsed in alcohol,
806 Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989
20 gas cylinder
$
Water
*
IO
Figure 4. Schematic diagram of the slurry circulation system showing upflow of slurry.
Pole piece = No. 2 D, 25.4 mm
:.
I
*
I
I
I
I
2
0 20
from Eq. 21
-calculated ;:
'ddownflow I
4
(&)
t
I
I
6
x
I
8
I
IO
IO5 ,A-t/m
Figure 7. Plot of prediction that large particles are more easily captured than are small particles.
-
d,
pb b
221 pm
D, = 25.4 mm Downf low uo 0 . 3 2 8 m/s d, 66 pm
uo = 0 . 3 2 8 m/s I50
1000
500
0
1500
I
/
2000
NI, A-t Figure 5. Capture efficiency of two shapes of pole pieces. Halfcircular pole pieces are more efficient at freezing solids than the full circular design.
3
q
I
lI
50
20
g 3
0
10
4 (A)x pa+ QW
I UDflOW 0
2
- c a l c u l a t e d from Eq. 21
2
4
(&)xIO-~,
6
8
I IO
A-t/m
Figure 6. Effect of operating conditions on the amount of solids captured.
dried overnight, and weighed. The slurry concentration was monitored by draining small amounts of slurry from the circulation system during the experiment. Also, sodium sulfite was dissolved in the slurry water (-0.1 mol/L) to prevent oxidation of the iron particles. In addition, the air above the water in both tanks was purged with nitrogen to hinder oxidation of the iron. The conditions and particle sizes used in the experiments are summarized in Table I. Selection of Pole Pieces. Preliminary experiments showed that slurry concentrations in the range 10-40
6
8
10
A-t/m
Figure 8. Plot showing how longer pole pieces collect more particles. The bottom curve was calculated from eq 21. All the other curves were calculated from this bottom curve and were not drawn to fit the data.
kg/m3 did not affect the quantity of particles captured by the filter; hence, all the runs from then on used this concentration range. Next, Figure 5 compares the capture efficiency of two shapes of pole pieces. It shows that the half-circular shape collected more particles than the full circular shape for the entire range of currents tested. On the basis of these results, all further runs were made with the half-circular pole pieces. Effect of Operating Conditions. Equation 21 predicts that downflow with small particles and high velocity should capture less solid than upflow with large solids and low velocity. These predictions are verified in the experiments reported in Figures 6 and 7. Effect of Pole Piece Geometry. Figures 8 and 9 show how the length of pole pieces affects the quantity of solid captured. Thus, a t a given NI, longer pole pieces collect
Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989 807
c
6
D, = 25.4 mm
c
7 0 ,
/
/
-5
D, = 25.4"
d, = 221 pm -4
101.6
I .5
A A 700 o m 1400
-3 -2 - I
0
30
20
IO
hm,or length of pole pieces (inches)
Figure 9. Plot showing how longer pole pieces collect less solid per unit length but in total collect more solids.
0.5
D,: 508"
2"1D "Iter[
downf low uo = -0432 m/s upflow uo= 0 4 3 2 m/s
1
0
5
(-&-)'x
k
Figure 11, Graphical evaluation of
IO
I
IO6, A-t/m I)
and kl.
100
2
4
(&-)xIO-~,
6
A-t
8
IO
/m
are used, h,/h, is independent of power input and is close to 1 in both down- and upflow operations; consequently particle buildup beyond the pole pieces is negligible. On the contrary, when pole pieces are short, h, is roughly constant for NI/(la + 1,) > 2 X lo5 for upflow and is about 2.5 times the pole piece length. For downflow, h, increases gradually with power input up to 2.5hm. For large NI/(la +,),Z eq 21 may be written as
Figure 10. Quantity of particles collected by the 2-in.-i.d. filter compared to that collected by the 1-in.-i.d. filter.
more solids; however, the amount collected per unit length decreases. This pattern of behavior is roughly independent of the magnitude of current used. This result implies that the efficiency is determined by q alone and that kl and kz are independent of the length of pole pieces in the range of conditions investigated here. Figure 10 shows what happens when the whole unit is scaled up, tube diameter and pole pieces, but for the same particles. For a size ratio L z / L 1= 2, geometric similarity suggests that 8 times the amount of solid should be collected; however, Figure 10 shows that only 5-6 times as much solid is collected at identical NI/(1, + 1,). Thus, roughly, for a size ratio L2/L1,we find
Also, we find that much more solids can be trapped and held in upflow than in downflow, but this difference becomes smaller as the current goes up.
Analysis and Discussion Estimation of q and k 2 . In order to use eq 21 to estimate the amount of solids collected by a magnetic filter, we first need to evaluate q and k2. From Figure 3 and eq 16, we see that q concerns particle buildup at and beyond the pole pieces, h,/h,, due to the fringing of magnetic flux leaking from the pole pieces as well as the change in packing density (1- t ) / ( l - to) under the influence of the magnetic force. Observed data of hd and h, for 1-in. and 20-in. pole pieces show that, when long enough pole pieces
In addition, when q is independent of power input, a plot of W/Woversus (NI/(la l,))-I should give a straight line whose intercept and slope give q and kz. Figure 11illustrates such a plot for a fixed upflow velocity. A fairly good linear relationship holds a t large NI. The value of q is seen to vary with the length of the pole pieces, while k21/2/u0 is seen to be independent of pole piece length. Figure 9 also shows that q varies with the length of the pole pieces. We find that the value of k z is constant for all upflow conditions and for downflow operations with long pole pieces, with a value of k z = 1.5 X lo-" (m2/(s.A.turns))2. However, kz is not constant for downflow with short pole pieces. Since it is slurry upflow that is of practical interest, we did not further study the variation in k z in downflow operations. Comparing values of k z for different particle sizes and slurry upflow, we find from eq 20 that n = 2-2.25. Thus, the magnetic traction force may approximately be described by the second power of the particle size (Ahanori, 1976; Eisenstein, 1977; Jaraiz-M et al., 1984). The magnitude of q drops to as low as 0.46 for 20-in. pole pieces despite the fact that h,f h, remains close to unity. We may thus expect the void fraction, e, of the mass of captured solids to be much greater than that of a freely settled bed. Using values of 7 and h,, one may evaluate t by eq 16. We also find under the influence of magnetic force that the voidage is about double that of the freely settled bed and increases with the length of pole pieces.
+
808 Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989 15 I
1
*-
2
I
lo[
e \
N O
Figure 12. Plot showing how eq 23, from theory, predicts that the mass of solids collected at different slurry velocity, both upflow and downflow, in tubes of different diameter and using different power levels should fall on a single line.
Mass of the Particle Collected by the Magnetic Filter. Equation 21 may be rewritten as
system. The result is eq 21. Two parameters, 7 and k z , had to be evaluated in terms of the variables of the system, and we show how these quantities can be obtained from experiment. Extensive experiments were then run in a model system consisting of a slurry of porous iron particles in water in which particle size, slurry concentration, tube diameter, flow velocity and direction, filter length, and power input were all varied. These runs provided many tests of the theory. Good agreement was found in all cases, except in the prediction for very short pole pieces in downflow operations. Experiments showed that longer pole pieces collected more solids for a given power input so the optimal design is to use a pair of half-circle pole pieces which are as long as practical but which do not seriously saturate any part of the magnetic core of the circuit. As opposed to the mechanical filter, this filter does not remove all the solids in the flow stream. However, it can remove a much larger mass of solids with much smaller pressure drop and with no plugging or wear of the filter medium. There may well be situations where a combination of both types of filters can advantageously be used. Finally, it would be interesting to know under what conditions can powerful permanent magnets replace the electromagnets that are presently used in these devices.
Acknowledgment Thanks to Toshinori Ban of Fuji-Davidson Co. for helping with the experimental program. This study was done under NSF Grant CBT-8420034. or
Literature Cited y = kzx
Thus, a plot of y versus x should give a single straight line if the derived predictive equation is correct. This line should be able to represent all the data obtained a t different slurry flow rates, flow direction, particle size, size of pole pieces, and size of tube. This would be a robust test of the theory presented here. Figure 12 is such a plot for a particular particle size but a t different flow velocity and direction and for different pole piece length and tube size. All the data fall on a straight line, independent of all these factors. The slope of the straight line yields k , = 1.5 X lo-" m4/(s.A.turns)2, which is equal to the value obtained in Figure 11. Amount of Particles Collected by Different Lengths of Pole Pieces. Consider two lengths of pole pieces. A comparison of the amount of solids captured gives
Thus, knowing 7 for any one set of pole pieces, Figure 9 then allows us to evaluate 1) for any other pole piece length. By use of this procedure, the five upper curves of Figure 8 were calculated from the lowest curve on this figure. Agreement between prediction and experiment is good. Figure 9 shows that longer pole pieces have a lower collection efficiency but collect more solids than shorter pole pieces.
Final Comments Theory was developed to predict the amount of ferromagnetic solids that could be collected and removed from a flow stream as a function of the many properties of the
Ahanori, A. Traction Force on Paramagnetic Particles in Magnetic Separators. IEEE Trans. Magn. 1976, MAG-12, 234-235. Asema, S. A. Bulletin: Filmag 88/2, Salamanca, Spain, 1988. Davis, G . F.; Levenspiel, 0. Theory and Operational Characteristics of the Magnetic Valve for Solids (MVS). 111. External Coil Design Powder Technol. 1985,44, 19-26. Eisenstein, I. Magnetic Separations: Traction Force Between Ferromagnetic and Paramagnetic Spheres. IEEE Trans. Magn. 1977, MAG-13, 1646-1648. Harpavat, G. Magnetic Forces on a Chain of Spherical Beads in a Nonuniform Magnetic Field. IEEE Trans. Magn. 1974, MAG-IO, 919-922. Ha*, W. H., Jr. Engineering Electromagnetics;McGraw-Hill: New York, 1981; Chapter 9. Jaraiz-M, E.; Wang, Y.; Zhang, G. T.; Levenspiel, 0. Theory and Operational Characteristics of the Magnetic Valve for Solids. AiChE J. 1984, 30, 951-959. Liu, Y. A.; Oak, M. J. Studies in Magnetochemical Engineering Part 11: Theoretical Development of a Practical Model for High-Gradient Magnetic Separation. AIChE J. 1983,29, 771-779. Luborsky, F. E.; Drummond, B. J. Build-Up of Particles on Fibers in a High Field-High Gradient Separator. IEEE Trans. Magn. 1976, MAG-12,463-465. Oberteuffer, J. A. Magnetic Separation: A Review of Principles, Devices, and Applications. IEEE Trans. Magn. 1974, MAG-10, 223-238. Takayasu, M.; Hwang, J. Y.; Friedlaender, F. J.; Petrankis, L.; Gerber, R. Magnetic Separation Utilizing a Magnetic Susceptibility Gradient. IEEE Trans. Magn. 1984, MAG-20, 155-159. Van Kleef, R. P. A. R.; Myron, H. W.; Wyder, P.; Parker, M. R. Application of Magnetic Flocculation in a Continuous Flow Magnetic Separator. IEEE Trans. Magn. 1984, MAG-20, 1168-1170. Yoong, F. S.; Fletcher, D.; Parker, M. R. Magnetic Particle Capture and Build upon a Current-Carrying Wire. IEEE Trans. Magn. 1983, MAG-19, 2109-2111.
Received for review July 6 , 1988 Revised manuscript receiued March 13, 1989 Accepted March 26, 1989
