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The efficiency of particle capture by an infinite array of ferromagnetic wires at low reynolds numbers
Journal of Magnetism and Magnetic Materials 49 (1985) 291-300 North-Holland, Amsterdam
291
THE EFFICIENCY OF PARTICLE CAPTURE BY AN INFINITE ARRAY OF FERROMAGNETIC WIRES AT LOW REYNOLDS NUMBERS * Gerd REGER **, Richard GERBER ***, F.J. FRIEDLAENDER School of Electrical Engineering, Purdue University, West Lafayette, IN 47907, USA
and Horst HOFFMANN NWFII-Physik, Universit~t Regensburg, Universiti~tsstrasse31, 84 Regensburg, Fed. Rep. Germany Received 12 September 1984; in revised form 19 November 1984
A new theoretical model is presented to describe the capture and retention of weakly magnetic particles carried by a viscous fluid past an infinite array of ferromagnetic wires in the HGMS transverse configuration. While the forces involved in the capture process are expressed analytically, the equations of particle motion are solved numerically yielding the trajectories, singular points and capture efficiencies. The capture efficiencies, determined for a wide range of values of operational parameters, are calculated on the basis of a laminar flow model, which by contrast to the potential flow model used in previous studies, gives a very good description of fluid motion at low Reynolds numbers. The effect of particle buildup upon the capture efficiencies is also calculated and shown to decrease with increasing I Vm/Vo [.
1. Introduction
The simplest model in the hierarchy of models describing the behaviour of HGMS filters is the single wire model, which has been extensively studied by Zebel[1], Watson [2] and many others. The next more complicated model consists of an infinite array (row) of parallel equidistant wires. This model, analyzed by Uchiyama and Hayashi [3,4] and other [5,7], offers a better facility for theoretical investigations of HGMS but mathematically is more demand~g than the single wire model. The third type of model are wire matrices. They are two-dimensional wire arrays. Hence, as a model they further increase both the generality and mathematical complexity of the description of the HGMS process [8-14]. * Work supported by National Science Foundation and the North Atlantic Treaty Organization. ** Present address: NWFII-Physik, UniversitiR Regensburg, UniversitlRstr. 31, 84 Regensburg, Fed. Rep. Germany. *** On sabbatical leave from the Department of Pure and Applied Physics, Salford University, Salford M5 4WT, UK, supported in part by a NATO Senior Fellowship.
The modelling of HGMS at any of the three levels of complexity is far from being completed at present. The iuifinite array model, for instance, was investigated [3,4] only for clean (unloaded) wires surrounded by a potential flow of an ideal fluid. To make this model more realistic, it is desirable to modify it so that it considers a laminar flow of a viscous fluid and wires loaded with particles. Such a study, resulting in the determination of capture efficiencies of infinite wire arrays under these conditions, is the purpose of this paper.
2. Theory 2.1. The configuration Consider an infinite array of ferromagnetic wires, each of circular cross-section of radius a, which lie all parallel to the zo-axis in the (zo, Xo) plane of an orthogonal coordinate system {xo, yo, z a }, see fig. 1. The usual convention [15] of expressing spatial dimensions of physical quantities in terms of the wire radius a (e.g. x a ffi x/a,
0304-8853/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(7,. Reger et al. / Particle capture by ferromagnetic wires
292
expressed as the capture efficiency E¢ by E¢ ~
(
I
x:
f
',
ll&
-
i,
-
I
I
I1~
-I-
S,(x..,y_}
Itl&
- ii
100Rca//Ua
.
(1)
The calculation of this quantity for a range of values of IVm/V0 I and uo, where Vm is the magnetic velocity [2,15], is one of the main objectives of this paper.
2.2. The magneticforce
UA
tI
Fig. 1. The configuration of an infinite wire array. The particle trajectories are indicated only schematically.
y~ =y/a, etc.) is employed here. The wires are infinitely long and separated from each other by a distance 2uo = 2u/a. An external magnetic field H o, homogeneously magnetizing the wires to a magnetization M, is applied in the x,,-direction and an incompressible fluid of viscosity 71, density Of and susceptibility Xr flows with a background velocity Vo along the minus yo-direction towards the wire array. The fluid is a suspension of small spherical weakly magnetic particles of radius b (that is b ,~ a), volume Vp, susceptibility Xpand density pp. The particles are assumed to experience a magnetic force Fm, a viscous drag force Fv and a gravitational force Fs. These forces determine the particle trajectories which far away from the wire array are all paralld to fluid flow (i.e. to the yo-aXis) but closer to the wire array divide themselves into two categories, depending whether the particle is captured or not. The theoretical borderline case of a trajectory for which the particle is neither captured nor escapes, that is the trajectory for which the particle motion terminates in a singnlar point S '= {xo,, Yos}, is called the critical trajectory. The distance between the parallel part of the critical trajectory and the ya-aXis defines the critical entry coordinate (capture radius) Rea. When R ~ is found the capability of the wire array to extract particles from the fluid flow may be
Consider instead of the whole wire array just for a moment, only the single wire the axis of which passes through the origin of the orthogonal coordinate system {xo, Ya, za} shown in fig. 1. The magnetic scalar potential Oo outside the wire, which consists of the magnetic potentials due to the external magnetic field H 0 and due to the wire magnetization M, can be written [15] as 1
xo
Oo=a - H o x , + i M ~ ] .
1
(2)
Consider now the whole wire array. Generalizing expression (2) by including the magnetic potentials due to the magnetization M of all wires, the magnetic scalar potential • of the whole array (outside the wires) is obtained as
e~=a -Hox.+½M m - - Y'.- - o o (x.+2uom) 2 +y~2
"
(3) This formula can easily be rearranged to give
O=a[-Hox.
+AM
m-~-~o (x'+m)2+(y') 2
where x'= Xa/(2ua) and y' shown in the appendix that
=y,/(2u,). Now, it is
oo
E
x'+m
m--oo (x' + m)2 +(y') 2 _-
¢r sin(2~x') cosh(2~ry') - cos(2~rx')
(5)
G. Regeret al. / Particlecaptureby ferromagneticwires and hence
where
1- cosh(fiy.) fisin(fix.)_cos(fix.) ] (, 6 ) ¢ = a [ - H o x . + ~M
fmy =
where fi = "~/u.. The components of the magnetic field H in the space around the wire array are Hx~
1 8ff} a 0x.
- - - -
(7a)
and
Hy=
1 a0
(7b)
a 0yo'
(8)
rm = ½ 0XVpV'(n ),
where X = X p - X f is the difference between susceptibilities of the particles and the fluid. Therefore, since Vp = (4/3)~b 3, the components of the magnetic force are 2'nP,oXb3 0 3a OXa ( z ~ + H~)
(9a)
Fray 2'~#0Xb3 a (Hi2 + Hy2 ). 3a 0y.
(9b)
Combining eq. (6) with (7a, b) and (9a, b) yields, after some algebra, the final form for the components of the magnetic force
4~l~oX MHo b3 "~a fmx,
Fmx -~ --
(10a)
where
2
fi 2K ]
+sin (fix.) +--T--]
(llb)
and where K = M/(2Ho). Similar expressions may also be found in ref. [5].
Consider the configuration in fig. 1 but, for the time being, with the difference that the orthogonal system ( x, y, z ) has not yet been normalized with respect to the wire radius a. A steady laminar flow of incompressible fluid, with a background velocity V0 along the minus y-direction is assumed to pass through the infinite wire array. The distribution of the fluid velocity v(r) in this flow can be described by the Navier-Stokes equation [16]
fi 3 sin( fix, )
X [1 - cosh(fiy.) cos(fix.)
J-
J
(lla)
and
4~rPoXMHo b3 3a fmy,
(13)
where Re--2aptVo/71 is the Reynolds number. This number gives an estimate for the ratio of magnitudes of the two terms in (12) which contain the velocity o. It can be seen that not far away from the wire array [(v. grad) ol ~ VoX/a and [(~/pt)Av[ - ~/V0/(pta2). Hence the ratio
4(cosh( fiy. ) - cos( fix. ))3
2 fi 2K ] -sinh (fiy.) + - - g - - |
(12)
where p is the pressure in the fluid at position r. A particular flow pattern v(r) has to comply not only with (12) but also with the boundary conditions. Thus v(r) is determined unambiguously by V0, a, u and ,//pf. Consequently, a complete set of dimensionless parameters can be constructed as v/l/o, r/a, u/a, 2apfVo/*l, and the velocity distribution obtained by solving (12) can be expressed by a function of the form
O/Vo = / ( ra,u ,,Re),
I
FraY=
[ x [1 - c o s h ( f i y , ) cos(fix,)
(v.grad)v= -(1/pf)grad p +(n/pf)av,
and
fmx----
fi3 sinh(fiy. ) 4(cosh(fiyo) - cos(fix.))3
2. 3. The viscous drag force
The magnetic force can be written [15] as
fmx
293
(10b)
[(v" grad) v[/l ~/pfavl = ½ Re.
(14)
Eq. (13) indicates that in general the laminar flow of viscous fluid cannot be normalized with respect to the wire radius a since the similarity of
294
G. Reger et al. / Particle capture by ferromagnetic wires
the flow depends upon the Reynolds number as well. This complicates not only the calculation of the flow but also the analysis of capture in HGMS. Therefore the investigations in this paper will be restricted to the region of low Reynolds numbers. Then, because of (14), the 1.h.s. of (12) may be set approximately to zero and the Navier-Stokes equation is reduced to the Stokes equation
~lAv = grad p,
(15)
which together with the equation of continuity, div v = 0, describes the fluid motion. The application of cuff to (15) eliminates p and the equation of motion becomes Ao=0,
= 2iaw/a~.
(17)
Since to is .real, the general solution to A~0 = 0 is g i v e n as to = I 2 ( z ) 4 =
~(~),
(181
where D ( z ) is an analytic function of z and ~(~) is its conjugate function at 3. Since to must be periodic with respect to x, single-valued throughout the field of flow and vanish at infinity, the function D ( z ) and ~ ( ~ ) may be expected to be of the form d 2n
l'2(z) -*rV° ~ ( - 1 ) " a 2 n d - ~ t¢
--
--
w
_i------~o= 1 + a 0 [ l n ( - 2 i sin 41 + ln(2i sin ~ ) - ( ~ - 41 cot ~] [ d2n- 1
"~-pllX~(-1) ha2.L ~ a-d2ncot
(161
where ~ is the vorticity defined as ~ = cuff v. Since ~ does not appear explicitly in (16), the velocity distribution that satisfies (16) can be written as v / V o = f ( r , , Ua). Hence, for the case of low Reynolds numbers the flow may be normalized in terms of wire radius a only. The flow past an infinite wire array represents a solution to a two-dimensional problem. Hence a method of functions of a complex variable, described by Miyagi [17], may be used for solving (16). The position in the complex plane and its conjugate value are defined as z = x + iy and = x - iy, respectively. The complex velocity is defined as w = vx - ivy. Resulting from these definitions, the vorticity, ~ = 10,0,w I, is obtained as
~(~)
where ~ = ,rrz/(2u) and 4 = ~r~/(2u). Substituting (18) together with (19a, b) in (17) and integrating with respect to ~ leads to an expression which, after taking into account the boundary condition at infinity (that is w / ( - i V o ) ---, 1) and the periodicity of the flow field, yields the formula for the complex velocity
cot ~,
(19a)
cot 4,
(19b)
n-0
~
~rV° ~ ( - 1 ) ~ U n-O
d 2n
~
a 2 n d ~ 2n
n
+ ~ (-1)
cot 4
]
d 2n-1
b2nd-~S'~-lcot ~.
(20 /
n--1
The coefficients a2n and b2. are real and their values can be determined from the boundary condition at the surface of any of the wires. This condition is (w + iV0)/(-iV0) = - 1 at Izl = a or 4 = t2/~, where t = ~ra/(2u) = ~r/(2ua). The n u m e r i c a l v a l u e s o f a2n and b2n c a n be found as functions of t in Miyagi's paper [17]. Reader's attention is drawn to two miswritings in paper [17]. First, eq. (6) in that paper should read: w / U = 1 + (the expression on the r.h.s, of (6)). Second, boundary condition (7) should be: ( w U ) / U = - 1 at Izl = a or 4 = t e l l , t = ~ra/(2h). Having established the complex velocity w using (20), the components of the fluid velocity v can be obtained as real and imaginary parts of w, i.e. vx -- re(w) and ve = -ira(w). Two examples of the distribution of normalized fluid velocity components, vx/I Vo I and vy/I Vo l, are shown respectively for ua = 4.2 and uo = 7.4 in figs. 2a, b and 3a, b. The normalized velocity components, each curve for a particular value of y~, are plotted as functions of x a The symmetry and periodicity of the configuration of the infinite array in fig. 1 result in vx/I V01 being antisymmetric and Vy/] V01 being symmetric with respect to the vertical lines at x,, = ku,,, where k = 0, 1, 2, 3. . . . etc. This property is illustrated in figs. 2a, b where the velocity components are plotted over the whole interval of
G. Reger et al. / Particle capture by ferromagnetic wires x~ f r o m 0 to 2u~. T o a v o i d r e p e t i t i o n in figs. 3a, b, the curves are s h o w n there o n l y over the half interval ~0, uo). Since the particles are a s s u m e d to b e spherical, the viscous d r a g force is o b t a i n e d f r o m the Stokes e q u a t i o n F v - - - 6 " ~ * l b ( d r / d t - v), where ( d r / d t - v) is the p a r t i c l e velocity relative to the m o t i o n
I
,,,/Ivol
295
o f the fluid. I n t r o d u c i n g d i m e n s i o n l e s s t i m e = I Vo It~a, the Stokes e q u a t i o n b e c o m e s Fv = - 6 ~ b l Vo I(dra/d~" - v / I V0 I) a n d h e n c e the c o m p o n e n t s of the viscous d r a g force are
.rd~° IVol°X]
F,x=-6'~,#,IVo l-a-;
F"x=- 6'~'~blV°l[ ~'~
0,3
(21a)
,
vy ] IVol'
(21b)
where v/IVol and v/IVol are the normalized
0,2
fluid velocity c o m p o n e n t s at the p o s i t i o n ra.
0,1 0
~'~
,
567
,
I
i
?
i
Vx/IVo~
X&
i 2 s 4 ~
-0.1
0.3- 3
-0,2-
0.2. 0"10~ 012345678
-0,3. II&
a
4
Ill&
U&
Fig. 2.(a) The normalized Xo-component of fluid velocity,
v~/lVo[, plotted as a function of x a for various distances yo from the wire array the unit cell of which is u° ffi 4.2. No. Ya
1 0.0
2 1.0
3 2.0
4 4.0
5 6.0
6 8.0
, xs I
i
Fig. 3.(a) The normalized Xo-COmponent Qf fluid velocity, vx/IVol,plotted as a function of xo for various distances yo from the wire array the unit cell of which is u a = 7.4.
No.
1 0.0
2 1.0
3 2.0
4 4.0
5 6.0
6 8.0
7 10.0
v,/Iv~I b "21 .5.3,. i
-2-
oN
b
0 i23 s= =:_:
u.
I
8 7,.io u.
x.
:
Ca) The normalized yo-component of fluid velocity, v~,/lV0 I, plotted as a function of x a for various distances Ya from the wire array the unit cell of which is u a ffi 4.2.
(b) The normalized yo-component of fluid velocity, vylV0 [, plotted as a function of x a for various distances Yz from the wire array the unit cell of which is u a - 7.4.
No.
No. ~
1 0.0
2 1.0
3 2.0
4 4.0
5 6.0
6 8.0
1 0.0
2 1.0
3 2.0
4 4.0
5 6.0
6 8.0
7 10.0
296
(7. Reger et al. / Particlecapture by ferromagnetic wires
large ]Vm/Vo I) channels exist around these positions through which particles may escape.
2.4. The gravitational force The gravitational force is Fs = (0f - pf)Vpg and its components, since Vp = ( 4 / 3 ) ~ b , are Fv, = 0
(22a)
3.1. Trajectories
and
Fgy =
3. Numerical calculations
-
(4/3)~rb3(pp
-
(22b)
pf)g,
where g is the magnitude of gravitational acceleration and ( p p - Of) is the difference between the particle and fluid densities.
2.5. The equations of particle motion Since small particles dispersed in a liquid are considered here, the inertial force may be neglected [15] and the particle motion can be described by ym..l- Fv + Y8..~.O.
(23)
Substituting from (lOa, b), (21a, b) and (22a, b) in (23) gives the equations of particle motion in the x a- and ya-directions as follows =
d,
(24a)
Vm fmx
IVol
IVol
Particle trajectories are obtained as solutions of the equations of motion, (24a, b). A 6th-order Runge-Kutta method (provided by a NAG-Subroutine) was used to solve numerically these equations for various values of external parameters. An example of the results of such calculations is presented in fig. 4, where a family of particle trajectories is shown in the first unit cell of the wire array. Trajectories in other unit cells may be obtained by symmetry operations with respect to the vertical lines at x a = 0 , + % , +2u~, etc. The computed trajectories confirm that the influence of the wire array upon the particle motion falls off with the distance Ya and that the channels through which particles escape are located around the symmetry
14 -]Rca, '
t
and dyo
=
vy(ro)
Vm
Vs
(24b)
IVol IVol/m" IVoi' where Vm=2#oXMHob2/(9,a) and V g = 2 ( p p ot)gb2/(911), the magnetic velocity and the gravid,
tational velocity, fm~ and fray are given by eqs. (lla) and (b), respectively, and vx/IVol and vy/[ Vo [ are determined using (20). Eqs. (24a, b) are formally similar to those of the single wire configuration [15]. However, they take into account the structure of the wire array through the functions fmx, fray, Ox/IVol and oy/lVol. It will be seen later that for small values of IVm/Vo [ the capture radius R ~ increases with IVm/Vo I, as in the case of a single wire, but, by contrast, for larger values of I Vm//V0[ this increase peters out and R ~ is limited by the value of u~. Obviously the magnetic forces cancel out at mid-points between the wires of the array and finite (small for
10-
i
8-
°t
i
4-
2-
-4°
Fig. 4. Particle trajectories in the transverse configuration of an infinite wire array. The values of the input parameters are
Ivm/Vol=lO.6, Uo= 5.2, K= 0.302, [~/Vm[= 8.87x10-3.
G. Reger et al. / Particle capture by ferromagnetic wires
positions at the centre in between the wires. The critical trajectory, critical radius R~a and singular point S are also indicated in fig. 4.
3.2. Capture efficiencies To calculate the capture efficiency Ec of a wire array it is necessary, according to eq. (1), to determine the critical radius R~. For this purpose consider the critical trajectory. It starts at the entry point, the x a coordinate of which is equal to R~, and ends at the singular point S, where d x , J d ¢ - - d y a / d ¢ = O . Setting the 1.h.s. of the equations of motion, (24a, b), to zero a system of two algebraic transcendental equations is obtained which, when numericaUy solved, yields the values, xas and y~s, of the coordinates of singular point S. Integrating the equations of motion backwards (i.e. with negative time) starting from S, the whole critical trajectory is established and hence also Rca found. The substitution of R¢a in (1) gives the value of E c. The results of these calculations for paramagnetic particles and clean wire arrays are summarized in fig. 5. There E c is given as a function of [Vm/VoI for various values of u~, where u~ is the size of the unit cell. The short range constant
297
K-- 0.302 and the ratio of the gravitational velocity to the magnetic velocity [Vg/Vml ffi 8.87 × 10 -3 for all calculations carried out in this paper. When particles are being retained, the surface of the buildup, where the particle deposition takes place, moves gradually away from the wire surface into locations, where the magnetic force density is diminished. There the balance of forces is less favourable to particle capture, which manifests itself in the reduction of., Rca. The result is a decrease of the capture efficiency E c with an increasing amount of particles retained as buildup on individual wires of the array. To describe this effect quantitatively, it is assumed that, as far as the flow is concerned, the wire together with the particle buildup have a cross-section of elliptical shape, which grows along the horizontal direction twice as fast as along the vertical direction. That is a y a ffi ( a x a + 1)/2, where a,,, = ax/a and a y a ~ a y / a are t h e n o r l n a l i z e d buildup radii in the xo- and yo-directions, respectively. As the buildup develops, the fluid flow pattern changes accordingly. If an overall effect is to be described, the drag upon the particle may be taken into account by considering an effective radius acef of the elliptical cross-section which the
Ee 100908070605040 80 20 10 0 0
100
,01 8O 7O
lo
2b ab
4b
50
6o
7b
8o
9o 1oo lyre/V(~
Fig. 5. The particle capture efficiency E¢ of clean (unloaded) infinite wire arrays as a function of ] V m / Vo [ for various values of the unit cell size u a. No. ua
1 4.2
2 5.2
3 7.4
4 10.6
5 ~
o
lO
20
so
40
6o
60
7o
8o
90
lOO
IVm/V(~ Fig. 6. The particle capture efficiency E¢ of a loaded infinite wire arrays as a function of [ Vm/ Vo ] for various values of the buildup radii ax,, and aya. The unit cell size u,, - 4.2. No. axa ayo
1 1.0 1.0
2 1.8 1.4
3 2.6 1.8
4 3.0 2.0
5 3.4 ~2.2
6 3.6 2.3
7 3.8 2.4
298
G. R e g e r et al. / Particle capture by ferromagnetic wires
E c
100.
100
90. 80. 70. 60' 50"
90807060-
3020-
I00
0
10
20
30
40
50
60
70
80
90
100
50 40302010O, 0
~
10
2'0
30
40
gO
60
70
8'0
IVm/V~
90 160 ]Vm/V~
Fig. 7. The particle capture efficiency E c of a loaded infinite wire array as a function of I Vm/Iiol for various values of the buildup radii ax~ and ayo. The unit cell size u a = 5.2.
Fig. 8. The particle capture efficiency E c of a loaded infinite wire array as a function of I V m / V o l for various values of the buildup radii a,, a and aya. T h e unit cell size u a = 7.4.
No.
No. axa ay o
axa
ayo
1 1.0 1.0
2 1.8 1.4
3 2.6 1.8
4 3.0 2.0
5 3.4 2.2
6 3.8 2.4
7 4.2 2.6
8 4.6 2.8
particle at its current position r-- { x, y } "feels" through the fluid flow. The effective radius is given by the intersection of the elliptical surface with the position vector r, that is
[ cos20 a eff----L
sin20 ]-1/2 a--~"] '
a---~--+
1 1.0 1.0
2 1.8 1.4
3 2.6 1.8
4 3.4 2.2
5 4.2 2.6
6 5.0 3.0
7 5.4 3.2
8 5.5 3.4
where 0 = a r c t a n ( y / x ) . The effective radius is then associated with the parameter t by the formula t = ~raeff/(2u), which makes it possible, using the expressions in Miyagi's paper [17], to determine a2n and b2n for every stage of the buildup development. The values of a2, and b2, are then substituted in (20) and further computations proceed as in the case of a clean wire array. The results obtained, showing the effect of particle buildup upon the capture efficiency Ec, are given in figs. 6, 7, 8 and 9.
5O
Ect
3.3. Discussion
40
2O
10 C0
10
20
3'0
40
5'0
80
70
80
90 100 [Vm/V~
Fi 8. 9. The particle capture efficiency E¢ of a loaded infinite wire array as a function of [ V m / V o [ for various values of the buildup radii axa and ay a. The unit cell size u a ffi 10.6. No. a,, a aya
1 1.0 1.0
2 1.8 1.4
3 2.6 1.8
4 3.4 2.2
5 4.2 2.6
6 5.0 3.0
7 5.8 3.4
3 6.6 3.8
9 7.0 4.0
10 7.2 4.1
The results presented in this paper, which are based on a laminar flow model, may be compared with previous calculations [3,4], where the potential flow model was used to describe the motion of the fluid. Generally, the values of E¢ obtained from the laminar flow model are lower than the corresponding values of E¢ resulting from the potential flow calculations. The comparison between the two models can be exemplified as follows: It is known [3] that the single wire capture radius R s does not differ much from the capture radius Rca of an infinite wire array, providing that the flow is potential, I Vm/Vo [ > 2 and R~ea <~ u a.
G. Reger et al. / Particle capture by ferromagnetic wires
Table 1 Potential and laminar flow capture efficiencies, E~ and Ec of clean infinite wire arrays for various values of u~ at [Vm/Vo] =40 ua Ecs Ec
4.2 99.7 79.2
5.2 80.6 62.0
7.4 56.6 40.8
10.6 39.5 26.8
20.0 20.9 14.5
Consequently a single wire capture efficiency of a wire array EJ may be defined as E~ = lOOR~a//u ~,
for R ~ < u~
aod E~ = 100,
for R~a = u..
(25)
Thus E~ is the capture efficiency of such a wire array in which the magnetic and hydrodynamic interactions between the wires are "switched off" and individual wires behave as single wires inside their unit cells. Within the limits indicated above, E~ gives a good approximation of the capture efficiency of an infinite wire array for the case of potential flow. The capture radius R~a may be obtained from the formula [15] R:a = (3/4)V/3-
(I Vm/V01)'/311_ 2( Vm/V0)-2/3]. (26)
Substituting (26) in (25) gives E~s which may then be compared with the "laminar" E~ values shown in fig. 5. A comparison between these two quantities for ]Vm/V0 [ -- 40 can be found in table 1. As far as the theoretical results showing the reduction of E c due to the buildup of retained particles is concerned (figs. 6, 7, 8 and 9), it is interesting to note that this effect decreases with increasing [Vm/V0 [. Since there is a lack of relevant experimental data for comparison, further discussion of this matter will be left open at this stage. 4. Conclusions
An analytical theory was developed which describes the magnetic, viscous drag and gravita-
299
tional forces acting on particles in suspensions which pass, at low Reynolds numbers, through an infinite ferromagnetic wire array in the H G M S transverse configuration. Equations of particle motion were formulated and solved numerically yielding the particle trajectories, singular points and capture efficiencies. The simplifying assumptions made here, namely, a magnetically saturated collector array and low Reynolds numbers, do not pla~e a serious restriction on the usefulness of the solution. Most H G M S is carried out in the "saturated matrix" condition. The complexity of the problem is increased greatly when considering a matrix with magnetization at less than saturation, or when considering a flow regime when Reynolds numbers are not small - in fact, laminar flow conditions will not apply in that case. The laminar flow model used describes the fluid motion at low Reynolds numbers better than the potential flow model. It may, therefore, be expected that the efficiencies of capture established in this paper will predict the performance of an actual array filter more realistically, than the efficiencies found by previous calculations [3,4] which were based on the potential flow. A modification of the parametric description of the laminar flow made it possible to determine the capture efficiencies E c not only as functions of Ivm/Vol a n d u o, for clean wire arrays, but in addition to these variables also as a function of the extent of the buildup, for wire arrays with retained particles. The calculation of Ec for arrays loaded with particles is a major step forward, because hitherto only clean (unloaded) arrays or matrices have been considered in theoretical studies [3-11, 13, 14].
Acknowledgements
We are grateful to M. Takayasu, Francis Bitter National Magnet Laboratory, M.I.T., and W.J. Leitermann, NWFII-Physik, Universit~t Regensburg (currently at Purdue University) for many valuable discussions.
300
G. Reger et a L / Particle capture by ferromagnetic wires
Appendix
References
A convenient starting point [18] for proving eq. (5) is the formula
[1] G. Zebel, J. Colloid Sei. 20 (1965) 522. [2] J.H.P. Watson, J. Appl. Phys. 44 (1973) 4209. [3] S. Uchiyama and K. Hayashi, in: Industrial Applications of Magnetic Separation, IEEE Publ. No. 78CH14472MAG (1979) 169. [4] K. Hayashi and S. Uchiyama, IEEE Trans. Magn. MAG-16 (1980) 827. [5] M. Takayasu, F.J. Friedlaender, Y. Hiresaki and D.R. Kelland, IEEE Trans. Magn. MAG-17 (1981) 2810. [6] H.W. Richmond, Proc. London Math. Soe. 22 (1924) 389. [7] Y. Zimm¢ls, IEEE Trans. Magn. MAG-14 (1978) 21. [8] I. Eisenstein, IEEE Trans. Magn. MAG-14 (1978) 1148. [9] I. Eisenstein, IEEE Trans. Magn. MAG-14 (1978) 1155. [10] I. Eisenstein, J. Magn. Magn. Mat. 7 (1978) 293. [11] W.H. Simons and R.P. Treat, J. Appl. Phys. 51 (1980) 578. [12] W.F. Lawson and R.P. Treat, IEEE Trans. Magn. MAG-18 (1982) 1668. [13] H. Greiner and H. Hoffmann, J. Magn. Magn. Mat. 38 (1983) 187. [14] H. Greiner and H. Hoffmann, J. Magn. Magn. Mat. 38 (1983) 194. [15] R. Gerber and R.R. Birss, High Gradient Magnetic Separation (RSP-J. Wiley, Chichester, New York, 1983). [16] LD. Landau and E.M. Lifshitz, Fluid Mechanics (Pergamon Press, Oxford, London, New York, 1966). [17] T. Miyagi, J. Phys. Soc. Japan 13 (1958) 493. [18] M. Takayasu and R. Gerber are responsible for the formulation of the appendix. [19] T.J. I'A. Bromwich, An Introduction to the Theory of Infinite Series (MacMillan, London, New York, 1955).
1
m-- --oo Xr t_m
= .~ cot(,~x')
the proof of which may be found in Bromwich's book [19]. Then, since sin(2x r) =g i sinh(2y') c o t ( x ' + iy,) = cosh(2y') - cos(2x') ' it is possible to write oo
E
x'+m
m - - ~ (x' + m)2 +(y') 2 =½
--00
m + ( x ' - i y r) + m + ( x ' + i y ' )
= ½~r[cot ~r(x' - iy') + cot ~r(x' + iy')] ~r sin(2~rx') cosh(2"~y') - cos(2"~x') ' Q.E.D.
291
THE EFFICIENCY OF PARTICLE CAPTURE BY AN INFINITE ARRAY OF FERROMAGNETIC WIRES AT LOW REYNOLDS NUMBERS * Gerd REGER **, Richard GERBER ***, F.J. FRIEDLAENDER School of Electrical Engineering, Purdue University, West Lafayette, IN 47907, USA
and Horst HOFFMANN NWFII-Physik, Universit~t Regensburg, Universiti~tsstrasse31, 84 Regensburg, Fed. Rep. Germany Received 12 September 1984; in revised form 19 November 1984
A new theoretical model is presented to describe the capture and retention of weakly magnetic particles carried by a viscous fluid past an infinite array of ferromagnetic wires in the HGMS transverse configuration. While the forces involved in the capture process are expressed analytically, the equations of particle motion are solved numerically yielding the trajectories, singular points and capture efficiencies. The capture efficiencies, determined for a wide range of values of operational parameters, are calculated on the basis of a laminar flow model, which by contrast to the potential flow model used in previous studies, gives a very good description of fluid motion at low Reynolds numbers. The effect of particle buildup upon the capture efficiencies is also calculated and shown to decrease with increasing I Vm/Vo [.
1. Introduction
The simplest model in the hierarchy of models describing the behaviour of HGMS filters is the single wire model, which has been extensively studied by Zebel[1], Watson [2] and many others. The next more complicated model consists of an infinite array (row) of parallel equidistant wires. This model, analyzed by Uchiyama and Hayashi [3,4] and other [5,7], offers a better facility for theoretical investigations of HGMS but mathematically is more demand~g than the single wire model. The third type of model are wire matrices. They are two-dimensional wire arrays. Hence, as a model they further increase both the generality and mathematical complexity of the description of the HGMS process [8-14]. * Work supported by National Science Foundation and the North Atlantic Treaty Organization. ** Present address: NWFII-Physik, UniversitiR Regensburg, UniversitlRstr. 31, 84 Regensburg, Fed. Rep. Germany. *** On sabbatical leave from the Department of Pure and Applied Physics, Salford University, Salford M5 4WT, UK, supported in part by a NATO Senior Fellowship.
The modelling of HGMS at any of the three levels of complexity is far from being completed at present. The iuifinite array model, for instance, was investigated [3,4] only for clean (unloaded) wires surrounded by a potential flow of an ideal fluid. To make this model more realistic, it is desirable to modify it so that it considers a laminar flow of a viscous fluid and wires loaded with particles. Such a study, resulting in the determination of capture efficiencies of infinite wire arrays under these conditions, is the purpose of this paper.
2. Theory 2.1. The configuration Consider an infinite array of ferromagnetic wires, each of circular cross-section of radius a, which lie all parallel to the zo-axis in the (zo, Xo) plane of an orthogonal coordinate system {xo, yo, z a }, see fig. 1. The usual convention [15] of expressing spatial dimensions of physical quantities in terms of the wire radius a (e.g. x a ffi x/a,
0304-8853/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(7,. Reger et al. / Particle capture by ferromagnetic wires
292
expressed as the capture efficiency E¢ by E¢ ~
(
I
x:
f
',
ll&
-
i,
-
I
I
I1~
-I-
S,(x..,y_}
Itl&
- ii
100Rca//Ua
.
(1)
The calculation of this quantity for a range of values of IVm/V0 I and uo, where Vm is the magnetic velocity [2,15], is one of the main objectives of this paper.
2.2. The magneticforce
UA
tI
Fig. 1. The configuration of an infinite wire array. The particle trajectories are indicated only schematically.
y~ =y/a, etc.) is employed here. The wires are infinitely long and separated from each other by a distance 2uo = 2u/a. An external magnetic field H o, homogeneously magnetizing the wires to a magnetization M, is applied in the x,,-direction and an incompressible fluid of viscosity 71, density Of and susceptibility Xr flows with a background velocity Vo along the minus yo-direction towards the wire array. The fluid is a suspension of small spherical weakly magnetic particles of radius b (that is b ,~ a), volume Vp, susceptibility Xpand density pp. The particles are assumed to experience a magnetic force Fm, a viscous drag force Fv and a gravitational force Fs. These forces determine the particle trajectories which far away from the wire array are all paralld to fluid flow (i.e. to the yo-aXis) but closer to the wire array divide themselves into two categories, depending whether the particle is captured or not. The theoretical borderline case of a trajectory for which the particle is neither captured nor escapes, that is the trajectory for which the particle motion terminates in a singnlar point S '= {xo,, Yos}, is called the critical trajectory. The distance between the parallel part of the critical trajectory and the ya-aXis defines the critical entry coordinate (capture radius) Rea. When R ~ is found the capability of the wire array to extract particles from the fluid flow may be
Consider instead of the whole wire array just for a moment, only the single wire the axis of which passes through the origin of the orthogonal coordinate system {xo, Ya, za} shown in fig. 1. The magnetic scalar potential Oo outside the wire, which consists of the magnetic potentials due to the external magnetic field H 0 and due to the wire magnetization M, can be written [15] as 1
xo
Oo=a - H o x , + i M ~ ] .
1
(2)
Consider now the whole wire array. Generalizing expression (2) by including the magnetic potentials due to the magnetization M of all wires, the magnetic scalar potential • of the whole array (outside the wires) is obtained as
e~=a -Hox.+½M m - - Y'.- - o o (x.+2uom) 2 +y~2
"
(3) This formula can easily be rearranged to give
O=a[-Hox.
+AM
m-~-~o (x'+m)2+(y') 2
where x'= Xa/(2ua) and y' shown in the appendix that
=y,/(2u,). Now, it is
oo
E
x'+m
m--oo (x' + m)2 +(y') 2 _-
¢r sin(2~x') cosh(2~ry') - cos(2~rx')
(5)
G. Regeret al. / Particlecaptureby ferromagneticwires and hence
where
1- cosh(fiy.) fisin(fix.)_cos(fix.) ] (, 6 ) ¢ = a [ - H o x . + ~M
fmy =
where fi = "~/u.. The components of the magnetic field H in the space around the wire array are Hx~
1 8ff} a 0x.
- - - -
(7a)
and
Hy=
1 a0
(7b)
a 0yo'
(8)
rm = ½ 0XVpV'(n ),
where X = X p - X f is the difference between susceptibilities of the particles and the fluid. Therefore, since Vp = (4/3)~b 3, the components of the magnetic force are 2'nP,oXb3 0 3a OXa ( z ~ + H~)
(9a)
Fray 2'~#0Xb3 a (Hi2 + Hy2 ). 3a 0y.
(9b)
Combining eq. (6) with (7a, b) and (9a, b) yields, after some algebra, the final form for the components of the magnetic force
4~l~oX MHo b3 "~a fmx,
Fmx -~ --
(10a)
where
2
fi 2K ]
+sin (fix.) +--T--]
(llb)
and where K = M/(2Ho). Similar expressions may also be found in ref. [5].
Consider the configuration in fig. 1 but, for the time being, with the difference that the orthogonal system ( x, y, z ) has not yet been normalized with respect to the wire radius a. A steady laminar flow of incompressible fluid, with a background velocity V0 along the minus y-direction is assumed to pass through the infinite wire array. The distribution of the fluid velocity v(r) in this flow can be described by the Navier-Stokes equation [16]
fi 3 sin( fix, )
X [1 - cosh(fiy.) cos(fix.)
J-
J
(lla)
and
4~rPoXMHo b3 3a fmy,
(13)
where Re--2aptVo/71 is the Reynolds number. This number gives an estimate for the ratio of magnitudes of the two terms in (12) which contain the velocity o. It can be seen that not far away from the wire array [(v. grad) ol ~ VoX/a and [(~/pt)Av[ - ~/V0/(pta2). Hence the ratio
4(cosh( fiy. ) - cos( fix. ))3
2 fi 2K ] -sinh (fiy.) + - - g - - |
(12)
where p is the pressure in the fluid at position r. A particular flow pattern v(r) has to comply not only with (12) but also with the boundary conditions. Thus v(r) is determined unambiguously by V0, a, u and ,//pf. Consequently, a complete set of dimensionless parameters can be constructed as v/l/o, r/a, u/a, 2apfVo/*l, and the velocity distribution obtained by solving (12) can be expressed by a function of the form
O/Vo = / ( ra,u ,,Re),
I
FraY=
[ x [1 - c o s h ( f i y , ) cos(fix,)
(v.grad)v= -(1/pf)grad p +(n/pf)av,
and
fmx----
fi3 sinh(fiy. ) 4(cosh(fiyo) - cos(fix.))3
2. 3. The viscous drag force
The magnetic force can be written [15] as
fmx
293
(10b)
[(v" grad) v[/l ~/pfavl = ½ Re.
(14)
Eq. (13) indicates that in general the laminar flow of viscous fluid cannot be normalized with respect to the wire radius a since the similarity of
294
G. Reger et al. / Particle capture by ferromagnetic wires
the flow depends upon the Reynolds number as well. This complicates not only the calculation of the flow but also the analysis of capture in HGMS. Therefore the investigations in this paper will be restricted to the region of low Reynolds numbers. Then, because of (14), the 1.h.s. of (12) may be set approximately to zero and the Navier-Stokes equation is reduced to the Stokes equation
~lAv = grad p,
(15)
which together with the equation of continuity, div v = 0, describes the fluid motion. The application of cuff to (15) eliminates p and the equation of motion becomes Ao=0,
= 2iaw/a~.
(17)
Since to is .real, the general solution to A~0 = 0 is g i v e n as to = I 2 ( z ) 4 =
~(~),
(181
where D ( z ) is an analytic function of z and ~(~) is its conjugate function at 3. Since to must be periodic with respect to x, single-valued throughout the field of flow and vanish at infinity, the function D ( z ) and ~ ( ~ ) may be expected to be of the form d 2n
l'2(z) -*rV° ~ ( - 1 ) " a 2 n d - ~ t¢
--
--
w
_i------~o= 1 + a 0 [ l n ( - 2 i sin 41 + ln(2i sin ~ ) - ( ~ - 41 cot ~] [ d2n- 1
"~-pllX~(-1) ha2.L ~ a-d2ncot
(161
where ~ is the vorticity defined as ~ = cuff v. Since ~ does not appear explicitly in (16), the velocity distribution that satisfies (16) can be written as v / V o = f ( r , , Ua). Hence, for the case of low Reynolds numbers the flow may be normalized in terms of wire radius a only. The flow past an infinite wire array represents a solution to a two-dimensional problem. Hence a method of functions of a complex variable, described by Miyagi [17], may be used for solving (16). The position in the complex plane and its conjugate value are defined as z = x + iy and = x - iy, respectively. The complex velocity is defined as w = vx - ivy. Resulting from these definitions, the vorticity, ~ = 10,0,w I, is obtained as
~(~)
where ~ = ,rrz/(2u) and 4 = ~r~/(2u). Substituting (18) together with (19a, b) in (17) and integrating with respect to ~ leads to an expression which, after taking into account the boundary condition at infinity (that is w / ( - i V o ) ---, 1) and the periodicity of the flow field, yields the formula for the complex velocity
cot ~,
(19a)
cot 4,
(19b)
n-0
~
~rV° ~ ( - 1 ) ~ U n-O
d 2n
~
a 2 n d ~ 2n
n
+ ~ (-1)
cot 4
]
d 2n-1
b2nd-~S'~-lcot ~.
(20 /
n--1
The coefficients a2n and b2. are real and their values can be determined from the boundary condition at the surface of any of the wires. This condition is (w + iV0)/(-iV0) = - 1 at Izl = a or 4 = t2/~, where t = ~ra/(2u) = ~r/(2ua). The n u m e r i c a l v a l u e s o f a2n and b2n c a n be found as functions of t in Miyagi's paper [17]. Reader's attention is drawn to two miswritings in paper [17]. First, eq. (6) in that paper should read: w / U = 1 + (the expression on the r.h.s, of (6)). Second, boundary condition (7) should be: ( w U ) / U = - 1 at Izl = a or 4 = t e l l , t = ~ra/(2h). Having established the complex velocity w using (20), the components of the fluid velocity v can be obtained as real and imaginary parts of w, i.e. vx -- re(w) and ve = -ira(w). Two examples of the distribution of normalized fluid velocity components, vx/I Vo I and vy/I Vo l, are shown respectively for ua = 4.2 and uo = 7.4 in figs. 2a, b and 3a, b. The normalized velocity components, each curve for a particular value of y~, are plotted as functions of x a The symmetry and periodicity of the configuration of the infinite array in fig. 1 result in vx/I V01 being antisymmetric and Vy/] V01 being symmetric with respect to the vertical lines at x,, = ku,,, where k = 0, 1, 2, 3. . . . etc. This property is illustrated in figs. 2a, b where the velocity components are plotted over the whole interval of
G. Reger et al. / Particle capture by ferromagnetic wires x~ f r o m 0 to 2u~. T o a v o i d r e p e t i t i o n in figs. 3a, b, the curves are s h o w n there o n l y over the half interval ~0, uo). Since the particles are a s s u m e d to b e spherical, the viscous d r a g force is o b t a i n e d f r o m the Stokes e q u a t i o n F v - - - 6 " ~ * l b ( d r / d t - v), where ( d r / d t - v) is the p a r t i c l e velocity relative to the m o t i o n
I
,,,/Ivol
295
o f the fluid. I n t r o d u c i n g d i m e n s i o n l e s s t i m e = I Vo It~a, the Stokes e q u a t i o n b e c o m e s Fv = - 6 ~ b l Vo I(dra/d~" - v / I V0 I) a n d h e n c e the c o m p o n e n t s of the viscous d r a g force are
.rd~° IVol°X]
F,x=-6'~,#,IVo l-a-;
F"x=- 6'~'~blV°l[ ~'~
0,3
(21a)
,
vy ] IVol'
(21b)
where v/IVol and v/IVol are the normalized
0,2
fluid velocity c o m p o n e n t s at the p o s i t i o n ra.
0,1 0
~'~
,
567
,
I
i
?
i
Vx/IVo~
X&
i 2 s 4 ~
-0.1
0.3- 3
-0,2-
0.2. 0"10~ 012345678
-0,3. II&
a
4
Ill&
U&
Fig. 2.(a) The normalized Xo-component of fluid velocity,
v~/lVo[, plotted as a function of x a for various distances yo from the wire array the unit cell of which is u° ffi 4.2. No. Ya
1 0.0
2 1.0
3 2.0
4 4.0
5 6.0
6 8.0
, xs I
i
Fig. 3.(a) The normalized Xo-COmponent Qf fluid velocity, vx/IVol,plotted as a function of xo for various distances yo from the wire array the unit cell of which is u a = 7.4.
No.
1 0.0
2 1.0
3 2.0
4 4.0
5 6.0
6 8.0
7 10.0
v,/Iv~I b "21 .5.3,. i
-2-
oN
b
0 i23 s= =:_:
u.
I
8 7,.io u.
x.
:
Ca) The normalized yo-component of fluid velocity, v~,/lV0 I, plotted as a function of x a for various distances Ya from the wire array the unit cell of which is u a ffi 4.2.
(b) The normalized yo-component of fluid velocity, vylV0 [, plotted as a function of x a for various distances Yz from the wire array the unit cell of which is u a - 7.4.
No.
No. ~
1 0.0
2 1.0
3 2.0
4 4.0
5 6.0
6 8.0
1 0.0
2 1.0
3 2.0
4 4.0
5 6.0
6 8.0
7 10.0
296
(7. Reger et al. / Particlecapture by ferromagnetic wires
large ]Vm/Vo I) channels exist around these positions through which particles may escape.
2.4. The gravitational force The gravitational force is Fs = (0f - pf)Vpg and its components, since Vp = ( 4 / 3 ) ~ b , are Fv, = 0
(22a)
3.1. Trajectories
and
Fgy =
3. Numerical calculations
-
(4/3)~rb3(pp
-
(22b)
pf)g,
where g is the magnitude of gravitational acceleration and ( p p - Of) is the difference between the particle and fluid densities.
2.5. The equations of particle motion Since small particles dispersed in a liquid are considered here, the inertial force may be neglected [15] and the particle motion can be described by ym..l- Fv + Y8..~.O.
(23)
Substituting from (lOa, b), (21a, b) and (22a, b) in (23) gives the equations of particle motion in the x a- and ya-directions as follows =
d,
(24a)
Vm fmx
IVol
IVol
Particle trajectories are obtained as solutions of the equations of motion, (24a, b). A 6th-order Runge-Kutta method (provided by a NAG-Subroutine) was used to solve numerically these equations for various values of external parameters. An example of the results of such calculations is presented in fig. 4, where a family of particle trajectories is shown in the first unit cell of the wire array. Trajectories in other unit cells may be obtained by symmetry operations with respect to the vertical lines at x a = 0 , + % , +2u~, etc. The computed trajectories confirm that the influence of the wire array upon the particle motion falls off with the distance Ya and that the channels through which particles escape are located around the symmetry
14 -]Rca, '
t
and dyo
=
vy(ro)
Vm
Vs
(24b)
IVol IVol/m" IVoi' where Vm=2#oXMHob2/(9,a) and V g = 2 ( p p ot)gb2/(911), the magnetic velocity and the gravid,
tational velocity, fm~ and fray are given by eqs. (lla) and (b), respectively, and vx/IVol and vy/[ Vo [ are determined using (20). Eqs. (24a, b) are formally similar to those of the single wire configuration [15]. However, they take into account the structure of the wire array through the functions fmx, fray, Ox/IVol and oy/lVol. It will be seen later that for small values of IVm/Vo [ the capture radius R ~ increases with IVm/Vo I, as in the case of a single wire, but, by contrast, for larger values of I Vm//V0[ this increase peters out and R ~ is limited by the value of u~. Obviously the magnetic forces cancel out at mid-points between the wires of the array and finite (small for
10-
i
8-
°t
i
4-
2-
-4°
Fig. 4. Particle trajectories in the transverse configuration of an infinite wire array. The values of the input parameters are
Ivm/Vol=lO.6, Uo= 5.2, K= 0.302, [~/Vm[= 8.87x10-3.
G. Reger et al. / Particle capture by ferromagnetic wires
positions at the centre in between the wires. The critical trajectory, critical radius R~a and singular point S are also indicated in fig. 4.
3.2. Capture efficiencies To calculate the capture efficiency Ec of a wire array it is necessary, according to eq. (1), to determine the critical radius R~. For this purpose consider the critical trajectory. It starts at the entry point, the x a coordinate of which is equal to R~, and ends at the singular point S, where d x , J d ¢ - - d y a / d ¢ = O . Setting the 1.h.s. of the equations of motion, (24a, b), to zero a system of two algebraic transcendental equations is obtained which, when numericaUy solved, yields the values, xas and y~s, of the coordinates of singular point S. Integrating the equations of motion backwards (i.e. with negative time) starting from S, the whole critical trajectory is established and hence also Rca found. The substitution of R¢a in (1) gives the value of E c. The results of these calculations for paramagnetic particles and clean wire arrays are summarized in fig. 5. There E c is given as a function of [Vm/VoI for various values of u~, where u~ is the size of the unit cell. The short range constant
297
K-- 0.302 and the ratio of the gravitational velocity to the magnetic velocity [Vg/Vml ffi 8.87 × 10 -3 for all calculations carried out in this paper. When particles are being retained, the surface of the buildup, where the particle deposition takes place, moves gradually away from the wire surface into locations, where the magnetic force density is diminished. There the balance of forces is less favourable to particle capture, which manifests itself in the reduction of., Rca. The result is a decrease of the capture efficiency E c with an increasing amount of particles retained as buildup on individual wires of the array. To describe this effect quantitatively, it is assumed that, as far as the flow is concerned, the wire together with the particle buildup have a cross-section of elliptical shape, which grows along the horizontal direction twice as fast as along the vertical direction. That is a y a ffi ( a x a + 1)/2, where a,,, = ax/a and a y a ~ a y / a are t h e n o r l n a l i z e d buildup radii in the xo- and yo-directions, respectively. As the buildup develops, the fluid flow pattern changes accordingly. If an overall effect is to be described, the drag upon the particle may be taken into account by considering an effective radius acef of the elliptical cross-section which the
Ee 100908070605040 80 20 10 0 0
100
,01 8O 7O
lo
2b ab
4b
50
6o
7b
8o
9o 1oo lyre/V(~
Fig. 5. The particle capture efficiency E¢ of clean (unloaded) infinite wire arrays as a function of ] V m / Vo [ for various values of the unit cell size u a. No. ua
1 4.2
2 5.2
3 7.4
4 10.6
5 ~
o
lO
20
so
40
6o
60
7o
8o
90
lOO
IVm/V(~ Fig. 6. The particle capture efficiency E¢ of a loaded infinite wire arrays as a function of [ Vm/ Vo ] for various values of the buildup radii ax,, and aya. The unit cell size u,, - 4.2. No. axa ayo
1 1.0 1.0
2 1.8 1.4
3 2.6 1.8
4 3.0 2.0
5 3.4 ~2.2
6 3.6 2.3
7 3.8 2.4
298
G. R e g e r et al. / Particle capture by ferromagnetic wires
E c
100.
100
90. 80. 70. 60' 50"
90807060-
3020-
I00
0
10
20
30
40
50
60
70
80
90
100
50 40302010O, 0
~
10
2'0
30
40
gO
60
70
8'0
IVm/V~
90 160 ]Vm/V~
Fig. 7. The particle capture efficiency E c of a loaded infinite wire array as a function of I Vm/Iiol for various values of the buildup radii ax~ and ayo. The unit cell size u a = 5.2.
Fig. 8. The particle capture efficiency E c of a loaded infinite wire array as a function of I V m / V o l for various values of the buildup radii a,, a and aya. T h e unit cell size u a = 7.4.
No.
No. axa ay o
axa
ayo
1 1.0 1.0
2 1.8 1.4
3 2.6 1.8
4 3.0 2.0
5 3.4 2.2
6 3.8 2.4
7 4.2 2.6
8 4.6 2.8
particle at its current position r-- { x, y } "feels" through the fluid flow. The effective radius is given by the intersection of the elliptical surface with the position vector r, that is
[ cos20 a eff----L
sin20 ]-1/2 a--~"] '
a---~--+
1 1.0 1.0
2 1.8 1.4
3 2.6 1.8
4 3.4 2.2
5 4.2 2.6
6 5.0 3.0
7 5.4 3.2
8 5.5 3.4
where 0 = a r c t a n ( y / x ) . The effective radius is then associated with the parameter t by the formula t = ~raeff/(2u), which makes it possible, using the expressions in Miyagi's paper [17], to determine a2n and b2n for every stage of the buildup development. The values of a2, and b2, are then substituted in (20) and further computations proceed as in the case of a clean wire array. The results obtained, showing the effect of particle buildup upon the capture efficiency Ec, are given in figs. 6, 7, 8 and 9.
5O
Ect
3.3. Discussion
40
2O
10 C0
10
20
3'0
40
5'0
80
70
80
90 100 [Vm/V~
Fi 8. 9. The particle capture efficiency E¢ of a loaded infinite wire array as a function of [ V m / V o [ for various values of the buildup radii axa and ay a. The unit cell size u a ffi 10.6. No. a,, a aya
1 1.0 1.0
2 1.8 1.4
3 2.6 1.8
4 3.4 2.2
5 4.2 2.6
6 5.0 3.0
7 5.8 3.4
3 6.6 3.8
9 7.0 4.0
10 7.2 4.1
The results presented in this paper, which are based on a laminar flow model, may be compared with previous calculations [3,4], where the potential flow model was used to describe the motion of the fluid. Generally, the values of E¢ obtained from the laminar flow model are lower than the corresponding values of E¢ resulting from the potential flow calculations. The comparison between the two models can be exemplified as follows: It is known [3] that the single wire capture radius R s does not differ much from the capture radius Rca of an infinite wire array, providing that the flow is potential, I Vm/Vo [ > 2 and R~ea <~ u a.
G. Reger et al. / Particle capture by ferromagnetic wires
Table 1 Potential and laminar flow capture efficiencies, E~ and Ec of clean infinite wire arrays for various values of u~ at [Vm/Vo] =40 ua Ecs Ec
4.2 99.7 79.2
5.2 80.6 62.0
7.4 56.6 40.8
10.6 39.5 26.8
20.0 20.9 14.5
Consequently a single wire capture efficiency of a wire array EJ may be defined as E~ = lOOR~a//u ~,
for R ~ < u~
aod E~ = 100,
for R~a = u..
(25)
Thus E~ is the capture efficiency of such a wire array in which the magnetic and hydrodynamic interactions between the wires are "switched off" and individual wires behave as single wires inside their unit cells. Within the limits indicated above, E~ gives a good approximation of the capture efficiency of an infinite wire array for the case of potential flow. The capture radius R~a may be obtained from the formula [15] R:a = (3/4)V/3-
(I Vm/V01)'/311_ 2( Vm/V0)-2/3]. (26)
Substituting (26) in (25) gives E~s which may then be compared with the "laminar" E~ values shown in fig. 5. A comparison between these two quantities for ]Vm/V0 [ -- 40 can be found in table 1. As far as the theoretical results showing the reduction of E c due to the buildup of retained particles is concerned (figs. 6, 7, 8 and 9), it is interesting to note that this effect decreases with increasing [Vm/V0 [. Since there is a lack of relevant experimental data for comparison, further discussion of this matter will be left open at this stage. 4. Conclusions
An analytical theory was developed which describes the magnetic, viscous drag and gravita-
299
tional forces acting on particles in suspensions which pass, at low Reynolds numbers, through an infinite ferromagnetic wire array in the H G M S transverse configuration. Equations of particle motion were formulated and solved numerically yielding the particle trajectories, singular points and capture efficiencies. The simplifying assumptions made here, namely, a magnetically saturated collector array and low Reynolds numbers, do not pla~e a serious restriction on the usefulness of the solution. Most H G M S is carried out in the "saturated matrix" condition. The complexity of the problem is increased greatly when considering a matrix with magnetization at less than saturation, or when considering a flow regime when Reynolds numbers are not small - in fact, laminar flow conditions will not apply in that case. The laminar flow model used describes the fluid motion at low Reynolds numbers better than the potential flow model. It may, therefore, be expected that the efficiencies of capture established in this paper will predict the performance of an actual array filter more realistically, than the efficiencies found by previous calculations [3,4] which were based on the potential flow. A modification of the parametric description of the laminar flow made it possible to determine the capture efficiencies E c not only as functions of Ivm/Vol a n d u o, for clean wire arrays, but in addition to these variables also as a function of the extent of the buildup, for wire arrays with retained particles. The calculation of Ec for arrays loaded with particles is a major step forward, because hitherto only clean (unloaded) arrays or matrices have been considered in theoretical studies [3-11, 13, 14].
Acknowledgements
We are grateful to M. Takayasu, Francis Bitter National Magnet Laboratory, M.I.T., and W.J. Leitermann, NWFII-Physik, Universit~t Regensburg (currently at Purdue University) for many valuable discussions.
300
G. Reger et a L / Particle capture by ferromagnetic wires
Appendix
References
A convenient starting point [18] for proving eq. (5) is the formula
[1] G. Zebel, J. Colloid Sei. 20 (1965) 522. [2] J.H.P. Watson, J. Appl. Phys. 44 (1973) 4209. [3] S. Uchiyama and K. Hayashi, in: Industrial Applications of Magnetic Separation, IEEE Publ. No. 78CH14472MAG (1979) 169. [4] K. Hayashi and S. Uchiyama, IEEE Trans. Magn. MAG-16 (1980) 827. [5] M. Takayasu, F.J. Friedlaender, Y. Hiresaki and D.R. Kelland, IEEE Trans. Magn. MAG-17 (1981) 2810. [6] H.W. Richmond, Proc. London Math. Soe. 22 (1924) 389. [7] Y. Zimm¢ls, IEEE Trans. Magn. MAG-14 (1978) 21. [8] I. Eisenstein, IEEE Trans. Magn. MAG-14 (1978) 1148. [9] I. Eisenstein, IEEE Trans. Magn. MAG-14 (1978) 1155. [10] I. Eisenstein, J. Magn. Magn. Mat. 7 (1978) 293. [11] W.H. Simons and R.P. Treat, J. Appl. Phys. 51 (1980) 578. [12] W.F. Lawson and R.P. Treat, IEEE Trans. Magn. MAG-18 (1982) 1668. [13] H. Greiner and H. Hoffmann, J. Magn. Magn. Mat. 38 (1983) 187. [14] H. Greiner and H. Hoffmann, J. Magn. Magn. Mat. 38 (1983) 194. [15] R. Gerber and R.R. Birss, High Gradient Magnetic Separation (RSP-J. Wiley, Chichester, New York, 1983). [16] LD. Landau and E.M. Lifshitz, Fluid Mechanics (Pergamon Press, Oxford, London, New York, 1966). [17] T. Miyagi, J. Phys. Soc. Japan 13 (1958) 493. [18] M. Takayasu and R. Gerber are responsible for the formulation of the appendix. [19] T.J. I'A. Bromwich, An Introduction to the Theory of Infinite Series (MacMillan, London, New York, 1955).
1
m-- --oo Xr t_m
= .~ cot(,~x')
the proof of which may be found in Bromwich's book [19]. Then, since sin(2x r) =g i sinh(2y') c o t ( x ' + iy,) = cosh(2y') - cos(2x') ' it is possible to write oo
E
x'+m
m - - ~ (x' + m)2 +(y') 2 =½
--00
m + ( x ' - i y r) + m + ( x ' + i y ' )
= ½~r[cot ~r(x' - iy') + cot ~r(x' + iy')] ~r sin(2~rx') cosh(2"~y') - cos(2"~x') ' Q.E.D.