Ind. Eng. Chem. Fundam. 1981, 20, 109-110
boundary layer at the surface will exist. It is not clear in the paper by Lee et al. which value of E is being reported; in fact, it must be presumed that it is simply the potential difference per unit separation distance between the electrodes. If this is the case, then the field strength immediately adjacent to the fiitering surface preventing particles from entering the filter pores (using the filter dimensions given by Lee et al.) is (41.5 X 0.984E) V/m, Le., 0.41E V/cm, where E is the field strength reported by Lee et al. in V/cm. From eq 3 and 5 it is clear that as the radius of the axial cylindrical electrode is increased the electric field approaches a linear profile between the electrodes. This is shown in Figure 3 where the inner surface of the filtering electrodes is kept at a 1 cm radius while the diameter of the central electrode is varied. A method to reduce the voltage which must be applied to the filter is therefore to use larger diameter electrodes; this is particularly true in the case of the innermost electrode. Although the existence of a clear boundary layer at the filter surface is implicit in the qualitative descriptions of the phenomena given by previous workers, Lee et al. are the first to offer an approach to the analysis of the effect. It is an interesting analysis and worthy of further consideration. However, in the light of the foregoing discussion, the validity of eq 11, 12, 15, 18, and 19 for tubular crossflow electrofilters in their paper are limited to flow in the annular space between two large diameter cylinders. The governing equation for the solids concentration in the volume between the electrodes should be written more generally as
where the potential gradient is given by
E(r) =
*ama/r In (r,,/ri)
(7) from differentiation of eq 3, where ri is the outside radius of the inner electrode, r, in the inner radius of the outer (filtering) electrode, and amax is the potential applied between the electrodes. The sign in eq 7 is positive when the charge on the particle is positive and negative when the particle charge is negative, so that the product of the electrophoretic mobility and the potential gradient (EM).E(r) is always positive. The boundary condition stating the particles do not penetrate the filter pores is then written as
109
t---’l \
+
i
i
Figure 3. Effect of inner electrode radius distribution.
0196-43 1318 1/ 1020-0109$01.OO/O
on potential gradient
using the notation of Lee et al. Their eq 15, 18, and 19 can now be rewritten to take true account of the nonuniform electric field distributions in a tubular cross-flow electrofilter. It is now concluded that the filter geometry must be considered in some detail before potential gradients are quoted. The implicit suggestion here is that the potential gradient at the filter surface itself determines the critical voltage which must be applied for separations in a tubular geometry filter; this requires some clarification. The analysis of the clear boundary layer put forward by Lee et al. could be made more pertinent by inclusion of the electric field gradient profile.
Literature Cited Lee, C. H.; Gldaspow, D.; Wasan, D. T. Ind. Eng. Chem. Fundam. 1980, 19, 166. Moulik, S. P. Environ. Sci. Techno/. 1971, 5 , 771. Yukawa, H.; Kobayashi, K.; Tsukui, Y.; Yamano, S.; Iwata, M. J . Chem. Eng. Jpn. 1976, 9 , 396. Yukawa, H., Kobayashi, K., Yoshida, H., Iwata, M. In “Progress in Fikration and Separation”, Wakeman, R. J., Ed.; Elsevier: Amsterdam, 1979.
Department of Chemical Engineering University of Exeter Exeter, England EX4 4QF
Sir: We essentially agree with Wakeman’s (1980) comment that, in the absence of a significant electrically conductive cake formation on the platinum wire, it is better to report the electric field by means of the more exact logarithmic expression rather than by means of the potential difference divided by the electrode spacing, as was done in the paper by Lee et al. (1980). We also agree with Wakeman that the geometry of the location of the electrode is important. In fact, we have used the observation that the electric field is highest near the platinum wire to suggest to the Department of Energy (1979) that faster filtration may be obtained by withdrawing the clear fiitrate with porous electrodes placed into the center of the as-
i i
2 3 5 6 9 Radius from axis of tubular filter. mm
R. J. W a k e m a n
sembly. However, we do not agree with Wakeman that our analysis of the clear boundary layer is affected by the curvature. Since the clear liquid boundary layers near the outside electrode (filter) are thin, the curvature is unimportant, as is always assumed in the classical Leveque problems presented in standard texts on heat convection, e.g., Knudsen and Katz (1958). To obtain the outlet concentration of the concentrated slurry leaving the filter, it is not sufficient to consider the boundary layer only, as was done in the paper by Lee et al. (1980). To generalize Lee’s solution, we had (Liu, 1980) included the radial distribution of E in the concentration field eq 11 of Lee et al. (1980). We had also included the
0 1981 American Chemical Society
110
Ind. Eng. Chem. Fundam. 1981, 20, 110-111
effect of suction on the velocity profiles as given by Yuan and Finkelstein (1956). The velocity components in Lee's eq 11 are
u = uo
[ 1 - 7 - - ( -36 x
-4.(.,) A 83X2 Re R 1+-+18 5400 1
-&?- (166 - 7607 + 825$
-
+
u=uox2
10800
2x
[
7 - $2
-
6
1
3 0 0 ~ ~75q4 - 6q5)
10800 (- 417
(1)
+ 9q2 - 6q3 + q4) +
(1667 - 380 q2 + 2 7 5 -~ 75v4 ~ + 15v5-
(2) 00
where (3) In eq 3, v is the kinematic viscosity, uo is the maximum velocity at x = 0, and V , is the constant withdrawal velocity. The boundary conditions (B.C.) for the concentration field equation for a very thin central electrode and a leaky outer filter wall have been generalized to
B.C.1:
C(0,r) = C,
(4)
B.C.2:
C(x,O) = finite
(5)
B.C.3: D ( g ) r=R
04
06
08
10
Figure 1. Computer outlet slurry concentrations in a tubular cross-flow electrofilter.
where &J is the applied potential between the radii R and ro. An analytical solution had been obtained by Liu (1980) using the method of separation of variables. Figure 1 shows the calculated exit slurry concentrations as a function of radial position for various dimensionless filter lengths, x , for one set of dimensionless parameters, where (y
=
D EM*A4 V,R ' P = RV, In W / r J
-*
A more general solution for the annulus with deposition at the wire is in progress (Liu, 1980).
= [(l-mv, -
where K in eq 6 is a ratio of particle to fluid velocities leaving the porous filter. The filtration velocity, V,, in the absence of osmotic effects, can be related to the pressure drop across the thin porous tube (again neglecting curvature), AI', and to its thickness, 1, through Darcy's equation (7) where k is the permeability of the porous tubes. The electric field strength in Lee's eq 11 was evaluated using
E=
02
3
Literature Cited Gidaspow, D.; Wasan, D. T. "Separation of Particles from Coal Derived Liquids Based on Surface Charge Properties", Department of Energy Private Communication, Apr 4, 1979. Knudsen, J. G.; Katz, D. L. "Fluid Dynamics and Heat Transfer", McGraw-Hill: New York, 1958; pp 363-367. Lee, C.H.; Gidaspow, D.;Wasan, D. T. Ind. Eng. Chem. Fundam. 1980, 19, 166-175. Liu, Y. Ph.D. Thesis, Illinois Instiiute of Technology, 1980 (in progress). Wakeman, R. J . Ind. Eng. Chem. Fundam. 1980, preceding correspondence in this issue. Yuan, S. W.; Finkelstein, A. B. Trans. Am. SOC. Mech. Eng. 1956, 78, 719.
Department of Chemical Engineering Illinois Institute of Technology Chicago, Illinois 60616
Y. Liu D. Gidaspow* D. T.Wasan
Comments on "Cross-Flow Electrofilter for Nonaqueous Slurries"
Sir: Lee et al. (1980) are to be commended for their fresh approach to the problem of solid-liquid separation in the crude products of coal conversion processes. I should like to make two observations on this paper. Firstly, the phenomenon of dielectrophoresis is not mentioned, despite the fact that the electrofilter has the nonuniform (radial) electric field configuration commonly used in such studies. Dielectrophoresis and its application to solid-liquid separation in both aqueous and nonaqueous systems have been studied extensively over the past 30 years by Pohl (1978). The phenomenon is based on a 0196-4313/81/1020-0110$01.00/0
difference in polarizability between the particles and the fluid, and this difference is expressed theoretically in terms of the permittivities and conductivities of the particles and the fluid. Since permittivity and conductivity are bulk properties, dielectrophoresishas not generally been classed as an electrokinetic phenomenon, which involves surface processes. However, it was proposed by Lockhart (1980a), on the basis of dielectric studies of the surface and colloid characteristics of crude solvent-refined coal (without surfactant), that surface polarization of the particles would give rise to dielectrophoresis as well as electrophoresis. 0 1981 American Chemical Society
