Ind. Eng. Chem. Fundam. 1981, 20, 108-109
108
Miller’s claim that less than 6% of the kinetic energy contribution is effective is difficult to test. The low values of C1 have already made the impeller power input predominant. Miller’s justification for this relies on the work of Lehrer (1968) on bubble columns. Lehrer (1980) has informed us that he is not sure that his results could be transposed to stirred vessels without being independently justified. In our experiments the gas jet interacts with the stirrer within 7 nozzle diameters, after which it does not possess an independent identity. To investigate the effects of kinetic energy it is necessary to use pressurized vessels. Such data are time-consuming and expensive to obtain. By systematically varying pressure we have shown that the effect of kinetic energy is significant. Further details are available elsewhere (Sridhar and Potter, 1980b). We are not prepared to introduce surface aeration parameters because very little
information is available. Further work on larger pressure vessels is necessary before definitive scale-up criteria can be developed.
Literature Cited Hassan, I. T. M., Robinson, C. W., AIChE J., 20, 1233 (1974). Hassan, I. T. M., Robinson, C. W., AIChE J., 23,448 (1977). Lehrer, I. H., Ind. Eng. Chem. Process Des. Dev., 7, 226 (1968). Lehrer, I. H., personal communication, 1980. Miller, D. N., AIChE J., 20, 445 (1974). Miller, D. N., Ind. Eng. Chem. Fundam., preceding paper in this issue, 1980. Sridhar, T., Potter, 0. E., Ind. Eng. Chem. Fundam., 19, 21 (1980a). Sridhar, T., Potter, 0. E.,Chem. Eng. Sci., 35, 683 (1980b).
T.Sridhar* 0. E.Potter
Department of Chemical Engineering Monash University Clayton, Victoria 3168 Australia
Electrical Field Distributions and Clear Boundary Layers in Cross-Flow Electrofilters Sir: The use of electrical fields to improve the performance of filters for solid/liquid separations is attracting much attention, and in a recent article, Lee et al. (1980) describe a further possible application of the technique. The geometry of the filter tube was tubular, with the filter tube serving as one electrode and an axial wire as the other. When such a configuration is being used, much of the electrical field is dissipated in the region close to the central electrode, while the field strength near the filtering surface is considerably lower than would be expected if the total potential difference between the electrodes is simply divided by the electrode separation distance. That this is so can be verified by solving Laplace’s equation for the potential of a field between two conducting electrodes
V2@ = 0
(1)
when the volume between the electrodes is filled by a solution (neglecting the presence of particles in suspension, which is reasonable for dilute slurries such as those used by Lee et al. (1980)). Considering two parallel conductors extending to infinity which are kept a t constant potentials, the solution to eq 1 is @=ux+b (2) for two flat plates, and for two coaxial cylinders the solution is @ = a’ln r + b’ (3)
gstance frorp centre electrode
mm
Figure 1. D i s t r i b u t i o n of electrical potential.
35G1: 320,
‘
Although no infinitely extended conductors exist and there is fluid flow parallel to the electrodes, the field in the filter unit will be approximated by appropriate application of eq 2 or eq 3. The electric field strength between two plates is constant E=a (4) but between two cylinders it is a function the cylinder radii and the radial location between the electrodes E = a’/r (5)
Figure 2. D i s t r i b u t i o n of potential gradient.
CP and E are plotted in Figures 1 and 2 for the filter described by Lee et al. (curve A corresponds to the polarity chosen when the particle j- potential is measured to be positive, and curve C would be used for negatively charge particles). Also drawn are @ and E for two flat plates with the same separation distance as the coaxial cylinders.
It has been widely recognized (for example, Moulik (19711, Yukawa et al. (1976, 1979)) that the field strength parameter E is of major importance in this work, determining whether or not a particle will be enabled to reach the filtering surface and hence whether or not a clear
0196-4313/81/1020-0108$01.00/0
3 i!ance fro=, centre eiecxode, mm
1981 American Chemical Society
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 \
+ Figure 3. E f f e c t distribution.
i
i
of inner electrode radius on p o t e n t i a l 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. I n “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 as0196-43 1318 1/ 1020-0109$01 .OO/O
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
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