Performance of High Gradient Magnetic Filters with Granular Matrix

This article was downloaded by [Thammasat University Libraries]On 08 October 2014 At 1929Publisher Taylor amp FrancisInforma Ltd Registered in England and Wales Registered Number 1072954 Registered office MortimerHouse 37-41 Mortimer Street London W1T 3JH UK

Separation Science and TechnologyPublication details including instructions for authors and subscription informationhttpwwwtandfonlinecomloilsst20

Performance of High Gradient Magnetic Filters withGranular MatrixTEYMURAZ ABBASOV a SAADETDIN HERDEM a amp MUHAMMET KOumlKSAL ba ENGINEERING FACULTY INONU UNIVERSITY MALATYA 44069 TURKEYb ENGINEERING FACULTY INONU UNIVERSITY MALATYA 44069 TURKEYPublished online 19 Aug 2006

To cite this article TEYMURAZ ABBASOV SAADETDIN HERDEM amp MUHAMMET KOumlKSAL (1999) Performance of High GradientMagnetic Filters with Granular Matrix Separation Science and Technology 342 263-276 DOI 101081SS-100100649

To link to this article httpdxdoiorg101081SS-100100649

PLEASE SCROLL DOWN FOR ARTICLE

Taylor amp Francis makes every effort to ensure the accuracy of all the information (the ldquoContentrdquo) containedin the publications on our platform However Taylor amp Francis our agents and our licensors make norepresentations or warranties whatsoever as to the accuracy completeness or suitability for any purpose ofthe Content Any opinions and views expressed in this publication are the opinions and views of the authorsand are not the views of or endorsed by Taylor amp Francis The accuracy of the Content should not be reliedupon and should be independently verified with primary sources of information Taylor and Francis shallnot be liable for any losses actions claims proceedings demands costs expenses damages and otherliabilities whatsoever or howsoever caused arising directly or indirectly in connection with in relation to orarising out of the use of the Content

This article may be used for research teaching and private study purposes Any substantial or systematicreproduction redistribution reselling loan sub-licensing systematic supply or distribution in anyform to anyone is expressly forbidden Terms amp Conditions of access and use can be found at httpwwwtandfonlinecompageterms-and-conditions

SEPARATION SCIENCE AND TECHNOLOGY 34(2) pp 263ndash276 1999

Performance of High Gradient Magnetic Filterswith Granular Matrix

TEYMURAZ ABBASOV SAADETDIN HERDEM andMUHAMMET KO

umlKSAL

ENGINEERING FACULTYINONU UNIVERSITY44069 MALATYA TURKEY

ABSTRACT

The performance characteristics of high gradient magnetic filters composed ofspherical ferromagnetic granules are determined in terms of dimensionless parametersfor a wide range of system parameters The results obtained are used to overcomesome contradictions seen in the literature related to magnetic filters They are in agood agreement with the experimental data given in the literature for the filters usedin laboratories and industry

INTRODUCTION

Some liquids and gases used in certain technological processes must bepurified by eliminating dispersed fine particles to improve the quality ofproduction to the highest level Most technological liquids contain iron andits compounds because of corrosion The removal of these species is of greatimportance for high purification of the fluid Owing to their lower concentra-tions and fine dimensions conventional mechanical and chemical cleaningmethods are not sufficient to filter liquids containing these particles For thisreason in recent years high gradient magnetic filters (HGMF) have beenextensively used for purpose (1ndash3)

In magnetic filters the main part is a filter matrix which consists of a blockof ferromagnetic elements (spheres rods stainless steel wool etc) that can

To whom correspondence should be addressed

263

Copyright q 1999 by Marcel Dekker Inc wwwdekkercom

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS

264 ABBASOV HERDEM AND KOumlKSAL

be easily magnetized by an external uniform magnetic field A high gradientmagnetic field induced around the filter elements creates local capture zonesWhile the liquid or gas is flowing through the filter matrix the magneticparticles are captured and accumulated in these zones The strong mechanicaland thermal resistance of the filter matrix enables liquids and gases withdifferent chemical and physical properties to be purified with a high perfor-mance and filtration velocity (2ndash6) The convenience of using HGMF andits advantages have been proved by many experiments in the laboratory andsuccessful applications in industry (6ndash12) Although there are some system-atic approaches and results for the use and performance of HGMF (2 13) ageneral theory about magnetic filtration has not been developed Since thedependence of filter performance on the parameters of filtration systems indifferent applications is similar a general and unifying treatment of HGMFwhich will yield a formula for the performance characteristics is aimed at inthis paper The determination of the filter parameters which appear in thederived formula to optimize filter performance for special industrial applica-tions is also included

BASIC INTERACTION

Technological liquids used in different fields of industry especially inthermal and nuclear power stations generally contain micron-sized iron parti-cles (generally with magnetite properties and a diameter d of 1ndash10 mm) withvery low concentrations (7ndash14) For this reason the fraction of ferromagneticmaterials in the filter matrix must be relatively high (50ndash60) The capturezones are essentially created around the contact points of the filter elementsand the captured particles are accumulated around these points The materialsused as the filter matrix elements are usually ferromagnetic chips and spheresof 2ndash10 mm diameter (2 3 13)

In most industrial applications it is mainly magnetic filters that are usedfor magnetic separation processes in which the dimensions and concentrationsof magnetic particles are relatively higher than those in the condensates ofthermal and nuclear power stations These filters are generally made of finesteel wool with a packing fraction of 5 or less Filters with such low packingfractions are not effective in thermal and nuclear power stations since particleswith small dimensions and concentrations can not be captured efficiently Inorder to increase the efficiency either the packing fraction should be in-creased which necessitates oversqueezing the steel wool and causes a highpressure drop across the filter ends or a very high magnetic field intensity(up to 14 T) should be used (9) which may require the use of superconductors(superconducting HGMS filter) (15)

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS

265HIGH GRADIENT MAGNETIC FILTERS

As with steel wool matrix filters steel wire matrix filters also have lowpacking fractions and are widely used in magnetic separation processes tocapture particles with relatively high concentrations and sizes In the techno-logical liquids considered in this paper the concentrations and sizes of theparticles are very small and wire matrix filters (and steel wool filters) withlow packing fractions are not effective (8 9 15)

Experience on the tests and usage of HGMF reveals that filter performancedepends on the magnetic hydrodynamic and geometric parameters of thefilter system Hitherto the results obtained about their dependence have notbeen sufficient to explain completely the extent of the effect of each parameteron filter performance as noted below

The parameters affecting filter performance can be classified as active orpassive Such active parameters as external magnetic field intensity H0 filterlength L dimension of matrix elements d and filtration velocity (bulk velocityV) can be measured or determined very easily Passive parameters includethe magnetic properties of particles (susceptibility k magnetization Ms) theparticle size (diameter d) and the velocity of a liquid through pores (interstitialvelocity v) Due to the impossibility of accurately determining the passiveparameters some contradictory results have been reported According to Ref3 the performance of HGMF with spherical matrix elements is independentof the captured particle size and the optimum bulk velocity of the liquidbeing filtered is about 03 ms On the other hand results based on manyother experiments indicate that one of the main parameters affecting filterperformance is particle size and the optimum filtration velocity is less orequal to 01 ms (2) These results were obtained on the assumption of laminarflow through the pore in which case the Reynolds number calculated for thediameter of the spherical filter elements used was 700 or higher This isobviously impossible since the critical maximum Reynolds number for thelaminar flow of liquids in granular media is about 100ndash120

To overcome all of the above-mentioned conflicts instead of consideringeach system parameter separately a general expression of the performanceof HGMF with a ferromagnetic granular matrix is derived according to agroup of dimensionless parameters (VmV Red Ld) as in magnetic separationtheory (1) In fact these three dimensionless parameters involve all of theparameters of a filtration system The first dimensionless parameter Vm isthe magnetic velocity in magnetic separation theory and it expresses themagnetic geometric and rheological properties of the system and V is thefiltration velocity The second one Red is the Reynolds number expressedin terms of the diameter of the spheres Finally Ld 4 Ld is the dimensionlesslength of the filter matrix normalized with respect to the diameter d of thespheres This approach to the evaluation of filter performance makes it possi-

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS


ble to generalize and unify the theory of fine filtration and separation proc-esses of liquids and gases This is described in the following section

THEORY

Consider a HGMF with spherical ferromagnetic filter elements all withthe same magnetic property The force balance equation for particles carriedby liquids through pores can be written as

FW

i ` FW

m ` FW

D ` FW

g ` FW

A 4 0 (1)

In this equation the forces acting on the particle are inertial magnetic draggravitational and Archimedes forces respectively Due to the fine (small)dimensions of the particles only F

Wm and F

WD are dominant in the active zone

Considering only their radial components the necessary condition for thecapture of a particle is

Fm $ FD (2)

With an equality condition of these forces the capture radius of the activezone centered at the contact points of the spheres can be determined In orderto do this explicit expressions of Fm and FD are needed The magnetic forceacting on the fine particles in the pores between spherical magnetized granulesis expressed as (2)

Fm 4

wpkm0m2r (mr 1 1)H2

0 1ra2

a 31 ` 05(mr 1 1) 1ra2

2

43

asymp3wpkm0m138

r H20

d 1ra2

(3)

In this equation H0 (Am) and k are as defined before mr is the relativemagnetic permeability of the filter matrix elements m0 4 4p 2 1027 Hmis the magnetic permeability of free space d 4 2a is the diameter (m) of thespheres r is the radial distance (polar coordinate in m) from the contact pointsof the spheres and wp is the particle volume (pd36 in m3)

The general expression for the drag force FD is

FD 4Cdrpd2

8u2 (4)

which applies for all Reynolds numbers In this equation Cd is the drag coeffi-cient r is the liquid density (kgm3) and u 4 |vWp 1 vW| is the absolute relativevelocity of the particle with respect to liquid flow velocity (vW) in the poreWhen Fm 4 FD the particle is just suspended and the particle velocity vWp

4 0 hence u 4 |vW|

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS


The relation between the drag coefficient Cd and Reynolds number Red

can be expressed as (16)

Cd 4 CRe1ad 0 a 1 (5a)

Experience reveals that from the piecewise linearization of the logarithmicplot of Cd versus Red the coefficient C and the exponent a may be approxi-mated by

C 4 24 a 4 1 for Red 1

C 4 30 a 4 58 for 1 Red 103 (5b)

C 4 044 a 4 0 for 103 Red 2 2 105

for spherical shaped particles The Reynolds number is expressed by

Red 4 dun (5c)

where n is the kinematic viscosity of the liquidWith Eqs (3) and (4) and by using Eq (5) equilibrium condition Fm 4

FD can be expressed by

3m138r H2

0

dra4

34

C 1dun 2

1a

r1d

u2 (6)

This equality can be written in dimensionless form

Vm

u4

34

CraRe11ad 1dd2

a11

(7)

where

Vm 43km0m138

r H20d2

dh(7a)

ra 4 ra Red 4 dun h 4 nr (7bcd)

are the magnetic velocity dimensionless distance and Reynolds number withrespect to the diameter of the spheres and the dynamic viscosity respectively

In general the velocity u of the liquid carrying the particles through thepores of the filter matrix depends on the geometry of the pores It varies fromzero at the contacts of the spheres to a maximum value at the centers of thepores Since the dimensions of the pores are very small technically it is veryhard to determine the profile of this variance However as a result of manyexperiments performed by using laserndashDoppler technology this profile canapproximately be expressed as (2)

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS


u 4 kvr2aV (8)

where the filtration (bulk) velocity V of the liquid can be determined veryeasily by measurement Based on experiments the empirical value of kv variesbetween 10 and 20 Assuming kv 4 16 [ [10 20] and using Eqs (7) and(8) the expression for the limit value Ras of the capture radius ra is obtainedas

R3as 4

112C 1Vm

V 2 Rea11d 1dd2

a11

(9)

In principle this equation is similar to the relations computed in magneticseparation theory (1 4 8 9 12 14) It indicates that the radius of saturationof the capture zone depends on the flow regime of the fluid through the poresand the ratio of the particle size to that of the filter matrix elements This resultmakes it possible to explain some of the contradictory statements appearing inthe literature

As the final step for arriving at the desired filter performance equationfrom Eq (9) consider the formula

cl 4 1 1 e1aL (10)

a 434d

R3as (11)

which is derived by an investigation of the effective cross section and volumeof the capture zone (2) and where l is the ratio of the mass concentrationof ferromagnetic particles to that of all particles in the liquid and c is thefilter performance Inserting Eq (9) into Eq (11) and then Eq (11) into Eq(10)

cl

4 1 1 exp31 116C 1Vm

V 2 1Reddd2

a11

Ld4 (12)

is obtained for filter performance In the domain of laminar flow (C 4 24a 4 1) this performance can be simplified to

cl

4 1 1 exp3126 2 1023 1Vm

V 2 Ld4 (13)

which is similar to the general expression determining filter performance inthe processes of magnetic separation and filtration (8 9 12)

The formula in Eq (12) for filter performance is valid for a very widerange of Reynolds number By this formula the performance of a magneticfilter is expressed in terms of the dimensionless quantities VmV Red Ldand dd Although it is derived for spherical filter elements similar equations

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS


have been determined and experimentally verified for HGMF with periodi-cally ordered wire matrix elements (8 12) But one important property of theresult appearing in Ref 12 which is special to HGMF with spherical matrixelements is that the effects of the hydrodynamic property (Red) and the geo-metric property (dd) appear together in the product Red(dd) This obviouslymeans that the actual relative hydrodynamic property Red is scaled by thegeometric property (dd) so that Red(dd) is the effective Reynolds numberFor this reason (dd) behaves as a scaling factor of the Reynolds numberbut it is different from the known scaling reported in the literature (14)

RESULTS AND DISCUSSIONS

To justify the derived theoretical results about filter performance both theresults obtained experimentally on model filters tested in the laboratory (asphere size of 57 mm in diameter was used in the filter matrix) and theresults of tests on actual filters used in industry (with a sphere size of 5 mmin diameter) Eq (13) is plotted in Fig 1 which shows the variation of the

FIG 1 Variation of filter performance with VmV the continuous curve is computed fromthe theoretical result in Eq (13) The points indicate the experimental data with parameters H0

4 30 d 4 3ndash5 (V) d 4 7ndash9 (lowast) and d 4 10ndash15 (`) H0 4 52 d 4 10ndash15 (M) H0 475 d 4 10ndash15 (L) and finally H0 4 125 d 4 3ndash5 (2) d 4 10ndash15 (v) In all cases H0

in kAm and d in mm

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS


FIG 2 Variation of filter performance with VmV the continuous curve is computed fromthe theoretical result in Eq (13) The points were obtained by measurements made on actualfilters used in industry 2 and v are for liquid ammonia V is for hydroelectric turbine water

and L (M) is for the condensates used in thermal (nuclear) power plants

filter performance versus VmV On the same plot the experimental data forthe filtrations of artificial suspension containing magnetite (Fe3O4) particlesare also shown for different system parameters It is evident that in spite ofa wide range of variation of parameter values the experimental results arein a very good agreement with the theoretical curve

Similar agreement is observed when measurements on the actual filtersused in different industries are considered Figure 2 compares the theoreticalresults with the data of measurements made for the filter systems indicatedin the figure caption and where the parameters H0 Ld and V have the values80 kAm 1667 014 ms for 2 40 kAm 667 008 ms for v 90 kAm200 0056 ms for V 60 kAm 143 0056 ms for L and 75 kAm 14740056 ms for M respectively

Based on Eq (9) the variation of VmV versus the saturation radius Ras ofthe active zone around the contact points is plotted in Fig 3 for differentvalues of the Reynolds number An increase of the Reynolds number appar-ently causes a decrease of the saturation radius In laminar flow for exampled 4 1ndash2 mm it is necessary that (VmV) 10 to have particles captured in

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS


the pores up to the Ras 4 02ndash04 range which is a fact reported by manyauthors (2)

Dependence of filter performance on the dimensionless parameter dd isshown in Fig 4 For the region (dd) 1025 the effect of a change ofReynolds number on filter performance is relatively small ie the differencebetween the effects of turbulent and of laminar flows on filter performancebecomes smaller For the range 1025 (dd) 1022 there is quite a largedifference between filter performances corresponding to different values ofthe Reynolds number This range is the basic range for fine magnetic filtrationprocesses therefore consideration of the variation of the Reynolds numberin the evaluation of filter performance is inevitable Note that since filterperformance depends strictly on the product Red(dd) many possibilities existfor the values of Red and (dd) to have a fixed value of filter performanceas long as their product is the same Hence in many cases it is appropriateto investigate the variation of filter performance versus VmV by taking thisproduct as a parameter and this is shown in Fig 5

Since C 4 30 a 4 58 is chosen as in Fig 4 so the plots in Fig 5 arevalid for turbulent flow ie for the intermediate range of Red specified in

FIG 3 Variation of VmV with the saturation radius Ras for different values of the Reynoldsnumber

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS


FIG 4 Filter performance against (dd) for different Reynolds numbers (VmV) 4 16 Ld 4737

Eq (5b) Under these conditions the general formula in Eq (12) for filterperformance can be written as

cl

4 1 1 exp31bLd 1Vm

V 24 (14)

where

b 41

480 1Reddd2

138

(14a)

For each fixed b Eq (14) is exactly in the same form as Eq (13) thereforeit represents the filter characteristics for both laminar and turbulent flows

The practical importance of the expression in Eq (14) is not only its univer-sality for laminar and turbulent flows but also its shape-invariance when itis plotted against a logarithmic axis which is obviously seen in Fig 5 Forthis reason it is sufficient to give a single plot [for example for the value

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS


Red(dd) 4 1 the bold curve in the figure] and to shift this plot to the leftby the amount

138

log3Red 1dd24 (15)

to obtain the filter performance for any value of Red(dd) 1Although the plots shown in Figs 1 2 3 and 5 are given for the range

VmV 4 1021ndash103 investigation of the experimental results in different areasin industry reveals that depending on the area for high filter performance($80) VmV varies in specific ranges for thermal power stations this rangeis 2ndash3 for nuclear and hydraulic plants it is 15ndash16 and 17ndash18 respectivelyFigure 6 shows the ranges of values of particle concentrations (c) at the inputand output of HGMF and the number (N) of industrial experiments they havemeasured (2) for different technological processes in industry A detaileddiscussion of Fig 6 is beyond the aim of this paper however a literaturesurvey and research show that the indicated ranges of VmV as computed andarrived at are based on a sufficient number of experiments

FIG 5 Variation of filter performance versus VmV for different values of Red(dd) a filterlength of Ld 4 737 and turbulent flow are assumed in the plots

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS


FIG 6 Ranges of values of the inletndashoutlet particle concentrations and the number of industrialexperiments that they measured (a) thermal power station (b) hydroelectric power plant (c)nuclear plant (d) chemical industry Computed range of VmV (a) 25 (b) 17 (c) 15 (d) 3

CONCLUSIONS

A general formulation is presented for the performance of high gradientmagnetic filters which have a filter matrix composed of ferromagnetic spheri-cal granules The formula suggest that parameters affecting filter performance

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS


vary depending on the flow regime of the liquid through the pores In laminarflow the performance depends on the dimensionless quantities VmV and Ldwhile in turbulent flow it is also affected by the product Red(dd) The formu-lation presented leads to a single equation (Eq 14) for both laminar andturbulent flows Further it makes it possible to justify filter performance fordifferent values of this product (which includes many physical parameterssuch as interstitial velocity viscosity of liquid and diameters of the filterelements and particles) by simply shifting a template Hence depending onthe values of filter parameters and of VmV (for which a range of values isdetermined in this article on the basis of many industrial experimental resultsscattered in the literature) the unknown filter performance can be determinedor any filter can be designed to achieve the desired filter performance

NOMENCLATURE

a radius of matrix elements (spheres) (m)Cd drag coefficientC coefficient in Eq (5)d diameter of the spheres (m)FW

A Archimedes force (N)FW

D FD drag force (N)FW

g gravitational force (N)FW

i inertial force (N)FW

m Fm magnetic force (N)H0 magnetic field intensity (Am)k particle susceptibilitykv coefficient in Eq (8)L filter length (m)Ld dimensionless length of the filter matrix normalized with respect

to the diameter d of the spheresMs magnetization of the particle (Am)r radial distance (polar coordinate) from the contact points of the

spheres (m)ra dimensionless radial distanceRas limit (saturation) value of the capture radius ra

Red Reynolds number which is expressed in terms of the diameter ofthe particles

Red Reynolds number which is expressed in terms of the diameter ofthe spheres

u absolute relative velocity of the particle with respect to liquidflow velocity in the pore (ms)

V filtration velocity (bulk velocity) (ms)

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS


Vm magnetic velocity (ms)vp particle velocity (ms)wp particle volume (m3)

Greek Letters

a exponent in Eq (5)d particle size (diameter) (m)h dynamic viscosity (Nssdotm2)l ratio of the mass concentration of ferromagnetic particles to that

of all particles in the liquidm0 magnetic permeability of free space (Hm)mr relative magnetic permeability of the filter matrix elementsn kinematic viscosity of the liquid (m2s)r liquid density (kgm3)c filter performance

REFERENCES

1 J H P Watson J Appl Phys 44(9) 4209 (1973)2 A V Sandulyak Magnetic Filtration Liquids and Gases (in Russian) Ximya Moscow

19883 H G Heitmann Ind Water Eng (12) 31 (1969)4 W Leitermann F J Friedlaender R Gerber J Y Hwang and B B Emory IEEE Trans

Magn MAG-20(5) 1174 (September 1984)5 V Badescu and N Rezlescu Powder Technol 73 93 (1992)6 Li Zhengenan and J W P Watson IEEE Trans Magn 30(6) 4662 (November 1994)7 G Rupp Ibid MAG-20(5) 1192 (September 1984)8 H Greiner G Reger and H Hoffmann Ibid MAG-20(5) 1171 (September 1984)9 S J Liberman and D R Kelland Ibid MAG-20(5) 1195 (September 1984)

10 P Anand J E Etzel and F J Friedlaender Ibid MAG-21(5) 2062 (September 1985)11 M Tsuge J Yano E Shichi K Kawashima and S Matsumoto Ibid MAG-23(5) 2764

(September 1987)12 R Gerber and P Lawson Ibid 30(6) 4653 (November 1994)13 J Cuellar and A Alvaro Sep Sci Technol 30(1) 141 (1995)14 P C Wankat J Y Hwang D Beckemeyer and F J Friedlaender IEEE Trans Magn

MAG-20(5) 1177 (September 1984)15 G Gerard H Robert and L Claude Ibid MAG-20(5) 1210 (September 1984)16 R Clift J R Grace and M E Weber Bubbles Drops and Particles Academic Press

New York NY 1978

Received by editor November 25 1997Revision received March 1998

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

Order now

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081SS100100649

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request PermissionReprints Here link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

SEPARATION SCIENCE AND TECHNOLOGY 34(2) pp 263ndash276 1999

Performance of High Gradient Magnetic Filterswith Granular Matrix

TEYMURAZ ABBASOV SAADETDIN HERDEM andMUHAMMET KO

umlKSAL

ENGINEERING FACULTYINONU UNIVERSITY44069 MALATYA TURKEY

ABSTRACT

The performance characteristics of high gradient magnetic filters composed ofspherical ferromagnetic granules are determined in terms of dimensionless parametersfor a wide range of system parameters The results obtained are used to overcomesome contradictions seen in the literature related to magnetic filters They are in agood agreement with the experimental data given in the literature for the filters usedin laboratories and industry

INTRODUCTION

Some liquids and gases used in certain technological processes must bepurified by eliminating dispersed fine particles to improve the quality ofproduction to the highest level Most technological liquids contain iron andits compounds because of corrosion The removal of these species is of greatimportance for high purification of the fluid Owing to their lower concentra-tions and fine dimensions conventional mechanical and chemical cleaningmethods are not sufficient to filter liquids containing these particles For thisreason in recent years high gradient magnetic filters (HGMF) have beenextensively used for purpose (1ndash3)

In magnetic filters the main part is a filter matrix which consists of a blockof ferromagnetic elements (spheres rods stainless steel wool etc) that can

To whom correspondence should be addressed

263

Copyright q 1999 by Marcel Dekker Inc wwwdekkercom

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



BASIC INTERACTION



Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS






Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



THEORY


FW

i ` FW

m ` FW

D ` FW

g ` FW

A 4 0 (1)


Wm and F



Fm $ FD (2)


Fm 4

wpkm0m2r (mr 1 1)H2

0 1ra2

a 31 ` 05(mr 1 1) 1ra2

2

43

asymp3wpkm0m138

r H20

d 1ra2

(3)



FD 4Cdrpd2

8u2 (4)


4 0 hence u 4 |vW|

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS






C 4 24 a 4 1 for Red 1

C 4 30 a 4 58 for 1 Red 103 (5b)

C 4 044 a 4 0 for 103 Red 2 2 105


Red 4 dun (5c)



3m138r H2

0

dra4

34

C 1dun 2

1a

r1d

u2 (6)


Vm

u4

34

CraRe11ad 1dd2

a11

(7)

where

Vm 43km0m138

r H20d2

dh(7a)




Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS


u 4 kvr2aV (8)


R3as 4

112C 1Vm

V 2 Rea11d 1dd2

a11

(9)



cl 4 1 1 e1aL (10)

a 434d

R3as (11)


cl

4 1 1 exp31 116C 1Vm

V 2 1Reddd2

a11

Ld4 (12)


cl

4 1 1 exp3126 2 1023 1Vm

V 2 Ld4 (13)



Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS







in kAm and d in mm

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS







Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS






Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS




cl

4 1 1 exp31bLd 1Vm

V 24 (14)

where

b 41

480 1Reddd2

138

(14a)



Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



138

log3Red 1dd24 (15)




Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



CONCLUSIONS


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



NOMENCLATURE













Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



Greek Letters



REFERENCES







New York NY 1978


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

Order now








Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



BASIC INTERACTION



Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS






Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



THEORY


FW

i ` FW

m ` FW

D ` FW

g ` FW

A 4 0 (1)


Wm and F



Fm $ FD (2)


Fm 4

wpkm0m2r (mr 1 1)H2

0 1ra2

a 31 ` 05(mr 1 1) 1ra2

2

43

asymp3wpkm0m138

r H20

d 1ra2

(3)



FD 4Cdrpd2

8u2 (4)


4 0 hence u 4 |vW|

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS






C 4 24 a 4 1 for Red 1

C 4 30 a 4 58 for 1 Red 103 (5b)

C 4 044 a 4 0 for 103 Red 2 2 105


Red 4 dun (5c)



3m138r H2

0

dra4

34

C 1dun 2

1a

r1d

u2 (6)


Vm

u4

34

CraRe11ad 1dd2

a11

(7)

where

Vm 43km0m138

r H20d2

dh(7a)




Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS


u 4 kvr2aV (8)


R3as 4

112C 1Vm

V 2 Rea11d 1dd2

a11

(9)



cl 4 1 1 e1aL (10)

a 434d

R3as (11)


cl

4 1 1 exp31 116C 1Vm

V 2 1Reddd2

a11

Ld4 (12)


cl

4 1 1 exp3126 2 1023 1Vm

V 2 Ld4 (13)



Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS







in kAm and d in mm

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS







Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS






Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS




cl

4 1 1 exp31bLd 1Vm

V 24 (14)

where

b 41

480 1Reddd2

138

(14a)



Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



138

log3Red 1dd24 (15)




Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



CONCLUSIONS


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



NOMENCLATURE













Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



Greek Letters



REFERENCES







New York NY 1978


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

Order now








Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS






Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



THEORY


FW

i ` FW

m ` FW

D ` FW

g ` FW

A 4 0 (1)


Wm and F



Fm $ FD (2)


Fm 4

wpkm0m2r (mr 1 1)H2

0 1ra2

a 31 ` 05(mr 1 1) 1ra2

2

43

asymp3wpkm0m138

r H20

d 1ra2

(3)



FD 4Cdrpd2

8u2 (4)


4 0 hence u 4 |vW|

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS






C 4 24 a 4 1 for Red 1

C 4 30 a 4 58 for 1 Red 103 (5b)

C 4 044 a 4 0 for 103 Red 2 2 105


Red 4 dun (5c)



3m138r H2

0

dra4

34

C 1dun 2

1a

r1d

u2 (6)


Vm

u4

34

CraRe11ad 1dd2

a11

(7)

where

Vm 43km0m138

r H20d2

dh(7a)




Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS


u 4 kvr2aV (8)


R3as 4

112C 1Vm

V 2 Rea11d 1dd2

a11

(9)



cl 4 1 1 e1aL (10)

a 434d

R3as (11)


cl

4 1 1 exp31 116C 1Vm

V 2 1Reddd2

a11

Ld4 (12)


cl

4 1 1 exp3126 2 1023 1Vm

V 2 Ld4 (13)



Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS







in kAm and d in mm

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS







Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS






Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS




cl

4 1 1 exp31bLd 1Vm

V 24 (14)

where

b 41

480 1Reddd2

138

(14a)



Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



138

log3Red 1dd24 (15)




Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



CONCLUSIONS


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



NOMENCLATURE













Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



Greek Letters



REFERENCES







New York NY 1978


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

Order now








Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



THEORY


FW

i ` FW

m ` FW

D ` FW

g ` FW

A 4 0 (1)


Wm and F



Fm $ FD (2)


Fm 4

wpkm0m2r (mr 1 1)H2

0 1ra2

a 31 ` 05(mr 1 1) 1ra2

2

43

asymp3wpkm0m138

r H20

d 1ra2

(3)



FD 4Cdrpd2

8u2 (4)


4 0 hence u 4 |vW|

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS






C 4 24 a 4 1 for Red 1

C 4 30 a 4 58 for 1 Red 103 (5b)

C 4 044 a 4 0 for 103 Red 2 2 105


Red 4 dun (5c)



3m138r H2

0

dra4

34

C 1dun 2

1a

r1d

u2 (6)


Vm

u4

34

CraRe11ad 1dd2

a11

(7)

where

Vm 43km0m138

r H20d2

dh(7a)




Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS


u 4 kvr2aV (8)


R3as 4

112C 1Vm

V 2 Rea11d 1dd2

a11

(9)



cl 4 1 1 e1aL (10)

a 434d

R3as (11)


cl

4 1 1 exp31 116C 1Vm

V 2 1Reddd2

a11

Ld4 (12)


cl

4 1 1 exp3126 2 1023 1Vm

V 2 Ld4 (13)



Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS







in kAm and d in mm

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS







Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS






Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS




cl

4 1 1 exp31bLd 1Vm

V 24 (14)

where

b 41

480 1Reddd2

138

(14a)



Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



138

log3Red 1dd24 (15)




Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



CONCLUSIONS


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



NOMENCLATURE













Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



Greek Letters



REFERENCES







New York NY 1978


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

Order now








Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS






C 4 24 a 4 1 for Red 1

C 4 30 a 4 58 for 1 Red 103 (5b)

C 4 044 a 4 0 for 103 Red 2 2 105


Red 4 dun (5c)



3m138r H2

0

dra4

34

C 1dun 2

1a

r1d

u2 (6)


Vm

u4

34

CraRe11ad 1dd2

a11

(7)

where

Vm 43km0m138

r H20d2

dh(7a)




Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS


u 4 kvr2aV (8)


R3as 4

112C 1Vm

V 2 Rea11d 1dd2

a11

(9)



cl 4 1 1 e1aL (10)

a 434d

R3as (11)


cl

4 1 1 exp31 116C 1Vm

V 2 1Reddd2

a11

Ld4 (12)


cl

4 1 1 exp3126 2 1023 1Vm

V 2 Ld4 (13)



Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS







in kAm and d in mm

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS







Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS






Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS




cl

4 1 1 exp31bLd 1Vm

V 24 (14)

where

b 41

480 1Reddd2

138

(14a)



Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



138

log3Red 1dd24 (15)




Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



CONCLUSIONS


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



NOMENCLATURE













Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



Greek Letters



REFERENCES







New York NY 1978


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

Order now








Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS


u 4 kvr2aV (8)


R3as 4

112C 1Vm

V 2 Rea11d 1dd2

a11

(9)



cl 4 1 1 e1aL (10)

a 434d

R3as (11)


cl

4 1 1 exp31 116C 1Vm

V 2 1Reddd2

a11

Ld4 (12)


cl

4 1 1 exp3126 2 1023 1Vm

V 2 Ld4 (13)



Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS







in kAm and d in mm

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS







Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS






Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS




cl

4 1 1 exp31bLd 1Vm

V 24 (14)

where

b 41

480 1Reddd2

138

(14a)



Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



138

log3Red 1dd24 (15)




Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



CONCLUSIONS


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



NOMENCLATURE













Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



Greek Letters



REFERENCES







New York NY 1978


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

Order now








Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS







in kAm and d in mm

Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS







Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS






Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS




cl

4 1 1 exp31bLd 1Vm

V 24 (14)

where

b 41

480 1Reddd2

138

(14a)



Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



138

log3Red 1dd24 (15)




Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



CONCLUSIONS


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



NOMENCLATURE













Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



Greek Letters



REFERENCES







New York NY 1978


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

Order now








Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS







Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS






Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS




cl

4 1 1 exp31bLd 1Vm

V 24 (14)

where

b 41

480 1Reddd2

138

(14a)



Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



138

log3Red 1dd24 (15)




Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



CONCLUSIONS


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



NOMENCLATURE













Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



Greek Letters



REFERENCES







New York NY 1978


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

Order now








Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS






Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS




cl

4 1 1 exp31bLd 1Vm

V 24 (14)

where

b 41

480 1Reddd2

138

(14a)



Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



138

log3Red 1dd24 (15)




Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



CONCLUSIONS


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



NOMENCLATURE













Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



Greek Letters



REFERENCES







New York NY 1978


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

Order now








Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS




cl

4 1 1 exp31bLd 1Vm

V 24 (14)

where

b 41

480 1Reddd2

138

(14a)



Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



138

log3Red 1dd24 (15)




Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



CONCLUSIONS


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



NOMENCLATURE













Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



Greek Letters



REFERENCES







New York NY 1978


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

Order now








Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



138

log3Red 1dd24 (15)




Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



CONCLUSIONS


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



NOMENCLATURE













Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



Greek Letters



REFERENCES







New York NY 1978


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

Order now








Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



CONCLUSIONS


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



NOMENCLATURE













Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



Greek Letters



REFERENCES







New York NY 1978


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

Order now








Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



NOMENCLATURE













Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



Greek Letters



REFERENCES







New York NY 1978


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

Order now








Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

ORDER REPRINTS



Greek Letters



REFERENCES







New York NY 1978


Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

Order now








Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

Order now








Dow

nloa

ded

by [

Tha

mm

asat

Uni

vers

ity L

ibra

ries

] at

19

29 0

8 O

ctob

er 2

014

Documents

Performance of High Gradient Magnetic Filters with Granular Matrix