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Journal of Electrostatics, 5 (1978) 85--99 © Elsevier Scientific Publishing Company, A m s t e r d a m -- Printed in The Netherlands 85 Paper No. 2-A IMPACT CHARGING OF AN ISOLATED CYLINDER WITH SKEWED FIELD AND FLOW MARKUS ZAHN Department of Electrical Gainesville, Engineering, Florida University of Florida, 32611 ABSTRACT The charging of a cylinder due to impacting charged particulate when the imposed electric In a quasi-static field and inviscid limit and neglecting is analyzed fluid flow are in arbitrary directions. self-field effects the combined flow stream function is a constant along the particle trajectories. field and Separation streamlines which pass through points where the total force on the particle is zero divide those trajectories travel past. which deposit charge onto the cylinder from those which The cases of co-linear and crossed field and flow are particularly examined. I. INTRODUCTION Electrofluidized beds (1-3) represent a new class of precipitation where small charged particulates are collected on larger particles. collecting area of fluidized particles that of a plate collector distributed in a conventional results in a better collection efficiency, and a smaller size device. Furthermore, The effective over a volume is much larger than electrostatic precipitator. This the ability to collect submicron particles, the agglomerated can themselves be used to collect additional charged particulate particulate. Past analysis has been based on the Whipple and Chalmers model applied to the charging of raindrops, devices (4), originally where an applied electric field uniform at infinity is in the same or opposite direction to an imposed uniform flow field. Any charged particulate injected at infinity is propelled along the force lines. If a force line is incident upon an object, If the object is isolated, the particulate is deposited there. the charge build-up tends to diminish further charge collection due to Coulombic repulsion, until it reaches a saturation value of charge where all particle lines pass around the object so that the charge on the body remains constant thereafter. The Whipple and Chalmers model has been used with the limiting cases of viscous dominated (inertia-less) charging characteristics and inviscid (inertia dominated) flows around a sphere. for either flow are the same because for both cases the normal component of the fluid velocity on the sphere must be zero (3). The 86 Because the flow and electric depends on two spherical field are in the same direction the problem only coordinates allowing the definition of a stream-function. If the field and flow are not in the same direction a stream-function defined because the solutions now depend on all three coordinates. cannot be This problem is avoided in the case of impacting charge onto a cylinder with the electric field and flow at arbitrary angles as the solution remains two-dimensional dependence, on the axial coordinate. We choose to only consider flow to avoid the paradox concerning an inertia-less cylinder (5). with no inviscid fluid flow around an infinitely long However we suspect that the results are representative of any laminar flow around the cylinder as was found for spheres, because the boundary condition of zero normal fluid velocity at the cylinder wall is true for any fluid properties. II. GOVERNING EQUATIONS AND APPROXIMATIONS The infinitely long perfectly conducting cylinder of radius R in Fig. instantaneously carries a uniformly distributed unit length I. The imposed electric (I) surface charge with charge per field is at an angle % o to the imposed velocity field lim r ÷ ~ v = V i = Vo(ir cos 8-[ 8 sin 0) o y (I) = Eo(iy cos 8 o + ~ x sin Co) = Eo(ir = cos (e - eo)- ~0 sin (O - 8o)) Charged particulate injected at infinity with density Po and mobility b results in drift and convection currents with density = 0(bE + ~) (2) In the absence of further charge generation or recombination, charge conservation requires ~p V • 3+~-f = 0 (3) while the electric field E, charge density p, and permittivity e are related through Gauss's law v • ~ = £c (4) Fluid incompressibility also requires that V • v = 0 (5) Substituting eqn. ~P + (bE + ~ ) 8t (2) into eqn. 2 " Vp + bp__@_= 0 E (3) and using The chain rule of differentiation dp(x,y,z,t) eqn. (4) and eqn. (5) yields (6) operating on the charge density = ~ t dt + ~3-~xdx + ~-P-dy 3y + ~-~z dz = ~~P at + (d~ • V)p (7) g? shows that along those lines d~ = ~ + bE (8) dt eqn. (6) can be rewritten as an ordinary differential 2 d--e=!e+ dt equation (d~ . v)p = - h ° ~t dt (9) E with solution P p = d~ on ~ = bE o 1 + b p? + v (i0) to ) (t - C where Po is the charge density at some time to. Although eqn. the charge density decreases with time it is necessary find where the charge is. of convection concentric are necessary (6). solve Poisson's problem, or numerical techniques equation for the electric field is much larger than the self-fields generated by the charged particulate, we can neglect these self-fields and write Gauss's law as v • ~ ~ o (ll) In this limit eqn. (9) is also modified to d_p_p~ 0 ÷ p = Po dt (12) as the square of the charge density term on the right in eqn. self-field generated by the charge density. charge cloud keeps a constant with time because repel!). (8) to in the absence the charged particulate. if the imposed electric approximately eqn. geometries coaxial cylinders, For a two or three dimensional to self-consistently field which propels However, to integrate This is easy in one dimensional (~ = 0) such as between parallel planes, spheres (i0) tells us how (9) is due to the In the solution of eqn. (i0), a charge as it convects but its density decreases the self-field tends to make the cloud expand If we neglect self-field effects, this expansion (like charges is negligible for short times where bp°(t - to) << 1 (13) C At the cylinder's surface the normal component of fluid velocity while the perfectly conducting cylinder with uniformly distributed requires the electric which satisfies field to be normally incident. these boundary conditions ~ 1 v = V[(l ~ E = - ~l-z-) c o s e ir - (i + is zero surface charge The electric field and flow and approximations are - - ~7-) sin e ie] . (14) r 1 [(i + ~)r cos i__) (0-Co) + ~-- ] i r + (i - ~2 sin (8 - 80)i 0 (15) 88 w h e r e w e normalize all q u a n t i t i e s as - r ~ [, r V = v° = 1 ~ ~--' = o ~=__~ 2~E o R ' E bE v_ = v ' o t (16) o V-- o ' R Because the flow and field have zero d i v e r g e n c e it is convenient to define stream functions as v- = 1 ~_~ ~ ~8 r - ~2 ÷ ~ = - (r +i) sin 8 ir + ~ i [ e 8r i i (17) r + ~ i° f =- (~2+I) sin (e-e) - ~e (18) w h i c h a u t o m a t i c a l l y s a t i s f y the zero d i v e r g e n c e conditions of eqn. (5) and eqn. (Ii). U s i n g eqn. d~ + V~ -- m dt r r ~d0 = dt r ~e (17) and eqn. i -~ = -- -~ ~O +v@ = ~~ ~r (18) in eqn. (8) yields (~ + ~ ) (19) (X + ~) (20) Since X and ~ are functions of r, O, and t, we w r i t e their total d i f f e r e n t i a l as ~ ~ d(~+b =7~r ~ (~+~)d~ + ~~ In eqn. (~+f)dO + ~ (×+~)d~ (21) (17) X is only a function of r and 8 but not time. In eqn. (18) the im- posed field c o n t r i b u t i o n is also not a f u n c t i o n of time but I changes w i t h time as the charged p a r t i c u l a t e collects on the cylinder w i t h rate di --= dt I " ' I = I 2 p bE R oo (22) w h e r e I is the total current incident on the cylinder. If I changes slowly compared to the transit time of charged p a r t i c u l a t e injection point to the cylinder, time-independent electric field. This allows us to neglect the time d e p e n d e n c e of Z so that the last term on the right in eqn. (19) and eqn. (20) in eqn. from the each packet of injected charge sees an e s s e n t i a l l y (21) is zero. T h e n using eqn. (21) w e find that the sum of s t r e a m - f u n c t i o n s is constant a l o n g a flow trajectory dr d(X+E) = 0 + X + Z = constant o n - ~ = E + v dt (23) so that the governing e q u a t i o n of the charge trajectories is }2[V sin 8 + sin(8-8 o )] + r[%8-c] + [sin(8-8 ) - V sin 8] : 0 o (24) 89 where c is a constant of a given curve found by specifying a single point (r,8) w h i c h the curve passes through. Defining the coefficients = [V sin 8 + sln(8-8o) ] = V sin 8 - sin(8 - 8o) eqn. ~2 (24) is rewritten as + ~ _ ~ = 0 (26) w h i c h has solutions - --= r = . 2A - + 2A _ (27) A The necessary trajectory computations in plotting r vs. 8 from eqn. (27) for a given ~ constant c are easily performed by a pocket calculator. Two conditions are necessary for the trajectories of eqn. (27) to deposit particulate onto the cylinder: (i) The flux of ions must be directed cylinder, dr/dt I~=l < 0. zero because of the rigid wall boundary, that the radial component of electric If this condition (Er(r=l) radially inward at the surface of the Since the radial component of flow velocity this condition field be negative is equivalent to requiring ~ ~ ~ at r = i, Er(r=l) < 0. is not met the trajectory which emanates > 0) from the cylinder carries is already radially outwards no charge because no charge is injected at r = i. (2) Even if Er(r=l) < 0, the trajectory carries charge only if the streamline ~ starts ar r = = w h e r e charged particulate is injected. We will find cases where characteristic trajectories III. trajectories start and terminate on the cylinder, but these have zero charge because they do not begin in a charged region at COLIN-EAR FIELD AND FLOW Only that section of the cylinder w h i c h has a negative component can (but not necessarily 2 cos 8 + k < 0 ÷ 8 ~ 8 ~ 2~ - 8 S Thus, does) collect (8 S = cos -1 - ~ - 2 the entire cylinder can collect charge. be collected surface collects "window" becomes ) (28) on the cylinder while if Between these limits, charge. over the lower half of the cylinder charge this angular field S if k ~ 2 no further charge is collected a fraction of the cylinder's radial electric charge I~I ! 2, When ~ = 0, charge can (z/2 E 8 x 3/2 z). For positive smaller being zero when ~ ~ 2 while for 90 n e g a t i v e charge the w i n d o w increases b e i n g 2~ w h e n I ! -2. (I) V h 0 When the electric field and flow are in the same direction, s e p a r a t i o n streamlines are incident onto the cylinder at angles @ and s trajectories o u t s i d e this plume 2~ - @ as shown in Figure 2a. Characteristic s miss the cylinder carrying their charge out to infinity w h i l e those trajectories b e t w e e n the s e p a r a t i o n streamlines intersect the cylinder d e p o s i t i n g their charge. The streamlines emanating from the cylinder o u t s i d e the w i n d o w for charge collection 191 < @ do not carry any charge as they did not start in a region of charge. s The charging current is found by r e f l e c t i n g the s e p a r a t i o n streamline b a c k to y . . . . constant ~ c s = 2 sin 0 s From eqn. (27) w i t h O = 0 at r = 1 for o of the s e p a r a t i o n streamline is s IiI! 2, the trajectory + X0 (29) At @ = ~, r = ~ its d i s t a n c e x from the y axis is ~* x = r sin 9 lim + ~ (30) so that from eqn. ~* X CS - i~ (24) w i t h @ = 0 w e have o 2 sin @ S + l(Os-~) ~+l (31) ~+i The current per unit length at y = - ~ b e t w e e n the s e p a r a t i o n s t r e a m l i n e s is the current incident u p o n the cylinder * I = 2Po(Vo+bEo]X._ I + i = (V+l)x* = 2 sin 9 ; s I = 2PobEo R (32) + %(@ -~) s and is plotted in F i g u r e 3 versus I. This charging current is independent of the v e l o c i t y V for V ~ 0 b e c a u s e i n c r e a s i n g V increases the current d e n s i t y by the same factor as the area b e t w e e n separation streamlines decreases. > 2, { = O w h i l e (2) - 1 < V ~ 0 for ~ < - 2, ~ = - For ~. W h e n the field and flow are in o p p o s i t e d i r e c t i o n s w i t h the charges still entering the system at y = - ~ some streamlines b e g i n and end on the cylinder and thus carry no charge as in Figure 2b. Critical points of zero force (d#/dt=O) now exist outside the cylinder + E r = 0 ÷ V(r2-1) cos @ + (r2+l) cos @ + lr = 0 r (33) ~ v@ + E0 = 0 + V(r2+l) sin @ + (r2-1) sin @ = 0 91 Solutions to the pair of equations in eqn. (33) are sin 8 = 0 ÷ 0 = O, (34) 2 (V+I) 2 (V+I) [ "] - [ I+V~ ] and ~ , l - V • cos Oc = - ~ k (35) This second critical point at angle 8 c only has real values outside the cylinder when - 1 < V < 0 and is important which intersect for it separates the cylinder and those which don't. streamline with constant C w h i c h passes By reflecting the separation through the critical point back to c from the y axis is y = - o% its distance regions between those streamlines 1/2 ~* x Cc - ~ _ [4(I-V 2) - k2] ~+i so that the charging { = (V+I)x*= - k(~-Sc) (36) ~+i ~ current is [4(l-v2) - x231/2 - X(~-oc) ]~l ~ 2i f ~-~2 0 t _~ w h i c h is plotted (3) ~!-21-f[~- in Figure 3. V = - i If the imposed opposite (37) ~ m 21-~ ~V2 in direction fluid flow velocity to the electrical for charges at infinity. is equal in magnitude but drift velocity, All trajectories there is no net motion begin on the cylinder and thus carry zero charge. (4) V < - 1 direction If the flow velocity to the electric field, is larger in magnitude injected In this flow regime there are two critical (34) and shown in Figure 4. ~ Ccl = 0 Oc in points at 8 = 0 and O = ~ given by eqn. Each separation the critical points has a respective but opposite charges enter the System from y = + ~. streamline w h i c h passes trajectory through constant = 0 (38) Cc2 = ~ ec = ~ No streamlines intersect the cylinder for k ~ 0 so that the charging current 92 is zero. For I < 0, each streamline ~ -* Ccl xI = x~ = 0 ~+1 ; reflected Cc2 ~ O+l ~+l -* ~ *I) = - Cc2 ~ (x2-x independent : - IV. in Figure CROSSED 3 for various (40) the charged field particulate = r , 8 = 8 C + E are summarized values in Figure 5, and the charging of I and V. FIELD AND FLOW Again it is important r is of the flow velocity. When the electric v current ~ All the flow and field regimes current is at distance (39) from the y axis so that the charging =-(V+I) back to infinity ~ is perpendicular at infinity starts to find the critical from eqn. (14) and eqn. to the velocity flow field, out at angle O = ~ + y where points (0 =~/2), o tan y = I/V. of zero force at (15) C r = 0 + V(r2-1) cos 8 + (r2+l) v e + -E0 = 0 ÷ - -V(r2+l) - sin e + which yields tan O the equivalent sin @ + Ir = 0 -(r2-1) cos e -~ (41) (42) 0 set of equations (6-i) c (43) V(~+I) 84 _ ~63 _ 262 [i +--a(V2-1) ] -~6+i = 0 ~2+i where B = ~2 c ; ~ = (44) ~2+i The four roots of eqn. 61,2 = ~i { ~2 (43) are B I, 1/61 , 62 , I/B 2 where + Q +_ [(~ + Q)2 _ 411/2 ~ (45) with _)2 4~ 71/2 Q = [(~ + z - - ~ + i j As shown in Figure streamlines intersect constant passes C c angle 0cy I. (46) 6 for I = - i, all the charge contained the cylinder. through The first separation the critical Its perpendicular distance between streamline point and intersects two separation with trajectory the cylinder at X*c from the line at angle 8 = - (T-y) 98 at r = o~ is + Y,(~-y) C X~* = c lim(r sin(8+~-T)) = - C r + ~ O -~ 4 ~2+I The second separation radial electric 2 sin e s (47) (w - y) streamline intersects the cylinder at angle 8 s w h e r e the field is zero (48) + ~ = 0 ~ Its distance x from the line at angle 8 = w + y is S cs - ~(~+~) ~~ x = lira r sin(0-(~+y)) s = (49) ~ 4V2+I r*y 8+~+y Note that the line at 0 = w + y is the same line as O = - (w-y). principal values of the trigonometric functions in determining Using the C and C C us to keep 0 a continuous function along a given trajectory. requires S Thus along C we c in the third and fourth quadrants while along C ,8 s from ~. must consider 0 as negative is positive increasing Each of the trajectory c = constants (r - i__) ~ sin 8 - ~ + c r c (rc i___) r cos 0 c + X8 c C C + h0 S S The total charging and is plotted (50) C = - 2 cos 0 S is current in Figure is then 7. The separation There is a transition point when 8cy I = 8s as then Cs = ~c" streamline through the critical point is then just tangent entire surface area collects charge. of charge collection to the cylinder and the More negative values of ~ have the region bounded by both sides of this separation streamline = - ~w The results of eqns. (52) (47) - (52) are only valid the critical point appears 8 in the third quadrant is in the fourth quadrant S eqn. (48) become so that for negative ~. When ~ > 0 (w < 8 < 3/2 ~) while the angle (3/2 w < 8 < 2w) as in Figure 8 so that eqn. (47) and 94 ~, ~ X = + ~(~+~) C c V~2+I ., c (53) - ~ (~+~) S X = -- s / ~ 2+ 1 giving the charging current ~=~ + as = (54) As Soon as ~ reaches the value where C = C C (at some value less than 2), the S current is zero for larger values of %. A summary of the streamlines for various Z and V are shown in Figure 8. ACKNOWLEDGMENTS This work was performed while on sabbatical leave at the Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, Mass. 02139 with the support of a Faculty Development Grant from the University of Florida, a National Science Foundation Faculty Fellowship, and National Science Foundation Grant ENG 76-06605. Stimulating conversations with Professor J. R. Melcher are gratefully acknowledged. REFERENCES 1 K. Zahedi and J. R. Melcher, Electrofluidized Beds in the Filtration of a Submioron Aerosol, J. APCA, 26, No. 4, (1976) 345-352. 2 T. W. Johnson and J. R. Melcher, Electromechanics of Electrofluidized Beds, I&EC Fund., 14, No. 3, (1975) 146-153. 3 J. R. Melcher and K. S. Sachar, Final Report: Charged Droplet Scrubbing of Submicron Particulate, Contract No. 68-02-0250, EPA, Research Triangle Park, N.C., 1974. 4 F. J. W. Whipple and J. A. Chalmers, On Wilaon's Theory of the Collection of Charge by Falling Drops, Quart. J, of the Roy. Met. Soc. 70 (1944) 103-119. 5 H. Lamb, Hydrodynamics, 1932, Dover, N.Y., pp. 614-616. 6 M. Zahn, Transient Drift Dominated Conduction in Dielectrics, IEEE Trans. on Elec. Insul., EI-12, No. 2, (1977) 176-190. 95 + + ÷ + + + Coulombslmefer //////////////// ,njl~densi ctedoftyCh°rgepo ] v+bE ~; Vor,; Vo~r co, o -to ,,n O] lira E= Eo[TrcoS(~)-Oo)-Tesin{8-8o)] r-,~ Figure 1 An imposed electric field is at an angle 8o to an imposed velocity field incident upon an infinitely R instantaneously long perfectly conducting cylinder of radius carrying a charge per unit length ~. Y 8 y/ ~-Criticof Point V+bE =0 Y cc ;c =-.5, ~'= J,Oo = o FiBure 2 co-linear. = i,[=-4eo--O Separation streamlines when the electric and velocity fields are Field and flow in the (a) same or (b) opposite directions with charges entering the system at y = - ~. part of the cylinder's surface. Charge is deposited on the darkened 96 -2 Figure 3 co-linear 2 i" Charging current field and flow. ~ 0 or V ! - i. I versus cylinder charge per unit length ~ for The current does not depend on the velocity In between, if - i < V < 0, the current varies smoothly between these two limits. Ccl X2 CriticalPoint O, X = - ID, V = -1.5 Critical Figure 4 Separation oppositely directed with charge entering Point streamlines w h e n the electric and velocity the system at y = + ~. fields are Summary of the streamlines for co-linear field and flow as a function of charge ~ and flow velocity V. Fisure 5 --I 98 Critical Point (r c, e c) Cc XC L,s Figure 6 y Separation streamlines when the electric and velocity fields are perpendicular. ~ 6- \ I \ \ % \ _ -2 Figure 7 ~, V=OD 2 Charging current I versus ~ for crossed field and flow. current only slightly depends on the flow velocity. charge, the current varies linearly with ~. The For large enough negative 99 // V=O e8o=90 ° ~' = I X=2 ,,,,\ X = I k k -,,,, >,=-2 Figure 8 of I and V. Summary of the streamlines for crossed field and flow as a function