I n d . Eng. C h e m . R e s . 1988, 27, 170-179
170
Augmentation of Single-Stage Electrostatic Precipitation by Electrohydrodynamic Instability Richard S . Withers' and James R. Melcher* Laboratory for Electromagnetic and Electronic S y s t e m s , Massachusetts I n s t i t u t e of Technology, Cambridge, Massachusetts 02139
To demonstrate how electroconvection, driven by a n unstable electrical force distribution on charged aerosol particles, can augment high-density aerosol precipitation, deposited mass and current density are measured for an initially uncharged and quiescent aerosol simultaneously subjected t o uniform fields and ion fluxes. By use of 0.56q~m-radiusparticles with a number density of 0.6 X 10l2m-3, it is shown that these rates can be predicted numerically by laws that account for ion and aerosol migration, field and diffusional charging, and ion and aerosol space-charge field contributions. With the density increased t o 6 X 10l2m-3, these calculations predict precipitation times that are a factor of 4 or more longer than that measured. Although convection is implicated by visual observations, a self-consistent complete-mixing model cannot account for this augmentation, suggesting that field-induced flow structures rather than uniform mixing are responsible for circumventing the inhibiting effects of aerosol space charge on the precipitation rate. A widely used method of particle collection is singlestage electrostatic precipitation. While charged by exposure to ions, particles migrate toward the collecting surface in an applied electric field. It is well-known that the turbulent fluid motions present in high-Reynolds-number flow may participate in and even augment the aerosol transport (Robinson, 1968; Alexander et al., 1981; Ehrlich, 1979; Bart, 1986; Ehrlich and Melcher, 1987). Corona wind is also recognized as making contributions. Less wellknown and understood is the possibility of turbulent-like convection driven by the electrical forces on charged aerosol particles. The latter possibility has not been extensively investigated previously. Hoppel and Gathman (1970) investigated aerosol motions resulting from the bipolar charging due to ionizing radiation. They predicted that uniform ionization would produce a stable arrangement, while layered ionization would be unstable; some experimental confirmation was found. Presented here is a study of the increase in the rate of particle deposition caused by this electrically driven convection. I t is demonstrated conclusively that the convection having its origins in instability can increase the rate of deposition by factors of 4 or more. Work on electrocorivective contributions to transport of molecular ions through insulating liquids (Lacroix et al., 1975; Hopfinger and Gosse, 1971), also with its origins in electrohydrodynamic instability, gives considerable insight into the physical processes of interest in the present study. However, the gas ions used here are so mobile that they transfer a negligible momentum to the fluid, while the aerosol particles, which do transfer significant force to the fluid, are continuously charging in accordance with the local field and ion density. The experimental arrangement used for this study is depicted schematically in Figure 1. The lower region of the cell is almost filled with a monodisperse aerosol, bounded from below by a metal electrode and from above by a metallic mesh. Initially the mesh is held a t ground potential, while a set of points above it supplies a flux of negative atmospheric ions generated by corona discharge. When time t = 0, a dc potential V, is applied to the mesh, and ions from above are injected into the aerosol region. The initially quiescent aerosol cloud is charged and sub+ Present
address: Lincoln Laboratory, Massachusetts Institute
of Technology, Lexington, MA 02173. 0888-5885/88/2627-0170$01.50/0
sequently precipitated during the ensuing transient. The deposited aerosol mass and the current through the aerosol region are both recorded as functions of time. These experiments are compared to numerical models of the charging and precipitation. In two of these models, convection is neglected and the predicted behavior of the system is in the limit of pure migration. The two models are identical except for the way in which particles are pictured as charging in the regime which falls between the field charging and diffusion charging limits. The precipitation times predicted by the migration models are shown to be in good agreement with the experimental results for dilute aerosols, one predicting somewhat shorter times and other predicting somewhat longer times. At high aerosol densities, however, both pure migration models predict precipitation times which are in gross disagreement with the experimental values, as much as 5 times longer. The third numerical model brings in the electrically induced convective mixing. In this model, the aerosol is maintained in a state of complete mixing, as if by an externally driven flow field such as a high-Reynolds-number channel flow. The surprising result of these calculations is that this mixing has little effect upon the time required for precipitation; these calculations give times very close to those given by the migration theory. The complete mixing (with ad hoc approximations to be described) cannot account for the increased transport caused by aerosol-driven convection. These comparisons demonstrate that neither pure migration nor the seemingly opposite limit of complete mixing can account for the precipitation of dense aerosols. The flow field generated by the aerosol itself transports the charged aerosol particles toward the collecting electrode in a way that indiscriminate mixing cannot. On the basis of the numerical models for the interplay of aerosol distribution with charging processes, of limiting theories of electrohydrodynamic instability, and of high-speed motion pictures of the aerosol, mechanisms of this enhanced transport and the conditions for its existence are considered in the concluding sections.
Governing Equations and Numerical Predictions Migration Limit: Full Equations. In the migration limit (no fluid motion on scales larger than the particle radius), the system is one-dimensional and we look for solutions for the aerosol particle charge Q(z,t), aerosol Q 1988 American
Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988 171 would exist in the absence of one another: 5
MESH,
j t t i t t t .-
.-
- .- - .- ...-
EVAPORATION CONDENSATION OOP AEROSOL GENERATOR
AIR
CRYSTAL
where Q, = 127rtR2E. This rate tends to underestimate the actual rate. The second form of the charging rate is one which includes an enhancement of the diffusion charging rate by the applied field (Liu and Yeh, 1968): Figure 1. Schematic of the experimental apparatus, showing the precipitation chamber, aerosol generator and diagnostic equipment, power supplies, and current and deposited-mass measuring and recording devices. The X-Y plotter alternately plots the current or mass signal versus time. The fine mesh (10-pm openings) blocks the corona wind as well as regulates the ion current.
particle number density N(z,t),ion density n(z,t),and electric field E(z,t). The aerosol is assumed to be initially uncharged (Q = 0) and of uniform density No. The ion density is constrained to be a steady value no at the injecting boundary z = 0. (This latter condition is an attribute of the corona triode charging arrangement as demonstrated by Withers et al. (1978) and further elucidated by Misakian (1981).) Unipolar charging with positive ions is assumed, and without loss of generality, we take E > 0. The system is governed by four partial differential equations: (a) particle charging
where dQ/dt = f(Q,n,E)is the charging rate of a particle along its trajectory and is prescribed by the individual particle charging law assumed, and B = Q/6ml$ is the particle mobility; (b) conservation of aerosol particles
a (BEN)= 0 !at + az (c) conservation of ions
an a N dQ - + - (bEn) = -- (3) at az 4 dt The right-hand side of this equation represents the sink for ions provided by the aerosol particles. As b >> B, the ions quickly adjust to any change in aerosol density and charge; as we care only about dynamics on the aerosol time scale, a quasi-steady ion distribution may be assumed and the anlat term neglected: (d) Gauss' law (4) where e is the permittivity of the fluid medium. In the range of electric field intensities and particle sizes considered (3RVo/l(kT/q) l),both field and diffusion charging contribute significantly to the aerosol particle charging rate. This is unfortunate, as easily calculated yet highly accurate and theoretically consistent charging rates do not exist in this region. We have resorted to using two alternative formulations of the combined field and diffusion charging rate. The first (Doetsch et al., 1969) is the simple sum of the field and diffusion charging rates which
-
+
3nR2nqbE(1
-
n-)'
Q 1,
Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988 177 however, the ratio is larger than predicted by any model. Apparently the convection which occurs at the higher number densities breaks up the aerosol space charge accumulations which otherwise would thoroughly block the ion injection. Discussion of Mechanisms The experiments with low aerosol densities behave as predicted by the numerical migration models which contain no adjustable parameters. Based on this correlation of theory and experiment a t low density, it is concluded that electrohydrodynamic effects come into play at higher densities. Aerosol space charge alters the electric field distribution, quenching ion injection, unstable aerosol charge profiles form, and fluid flow begins. These aspects are now considered in that order, followed by insights on the resulting electrohydrodynamics. Space-Charge Effects. If the aerosol is dense enough to make (Q,,,Nl/t) > Vo/l where Q, is roughly the charge achieved in the imposed field, space-charge effects = Q, = are important. With field charging, Q, 1 2 ~ t $ ~ V ~and / l the criterion becomes N * ,E12.rrR21N> 1 (12) For diffusion charging (eq 5), the particle charge increases by roughly one “diffusion charge”, QD = 4r&kT/q, for every e folding of the product of the particle charging time and ion density. As the times of interest (aerosol precipitation times) are typically several orders of magnitude longer than the time required to reach one QD, it is reasonable to take Q, = 10QD. The criterion is then (10/P)N* > 1 (13) where 6 reflects the relative importance of field and diffusion charging. Thus, for IFD charging, eq 12 applies for lO/P < 1, and eq 13 for lo/@> 1. With FED charging, a reasonable fit to the data is Q, = 13QDp32over the range 1< < 10. Thus, space-charge effects can be seen at accordingly lower values of N than predicted by condition 13, Le., for 13 -N* >1 l l p l l o p0.68 Quenching of Ion Injection: Limitation of Particle Charge. The aerosol space charge is limited by quenching to a value only weakly dependent on the details of the particle charging. The quenched limit is approached as the space charge density becomes so great that the electric field a t the injector is reduced to zero. Creation of Unstable Charge Distribution. While the magnitude of the particle charge is only weakly dependent on the type of charging when the quenching condition is in effect (i.e., dense aerosol), the numerical results indicate that the spatial distribution of aerosol space charge is critically dependent on it. This space charge profile, in turn, is primarily responsible for the instability of the system. Profiles with
are potentially unstable (Turnbull and Melcher, 1969). The numerical results in the migration limit lead to the following conclusions. (1)When the aerosol density is low enough that its space charge is insignificant (i.e., the appropriate relation, eq 12 or eq 13 or 14, is not satisfied), the charge density profiles tend to be stable. (2) When diffusion charging dominates ( p < lo), potentially unstable profiles are produced when the space-
;‘.:
2 10
,
,I
,migration I
hypothetical, with convection Acornplate
I
mixing
~--.._____-_*.,‘C L t
Figure 12. Distributions of aerosol space charge for the pure migration limit, the complete mixing limit, and a convection model.
charge criterion, eq 13 or 14, is satisfied. The slope in the Q(z) curve is especially strong with FED charging, and the number density as well acquires steep gradients (both positive and negative), as is clear in Figure 2. (3) Significantly higher number densities than necessary to satisfy the inequality eq 12 are needed to produce unstable profiles with field charging; the monotonically increasing (in z ) field strength tends to impart a stable slope to QN. Clearly stable profiles are predicted even with No* = 2.5. Field charging can produce potentially unstable profiles at higher densities, No* > 5. Withers et al. (1978) exhibit the case No* = 20. (4) Shock waves of aerosol density are plausible, at least in the migration limit (Withers and Melcher, 1981). Electroconvection. The fluid velocity that results from this instability depends on the length scale (Melcher, 1981). On a scale of L that is large enough that inertial forces dominate those due to viscosity, the largest possible fluid velocity is obtained by equating the kinetic energy pmu2/2 with the electric pressure eE2/2. This results in a velocity that takes the form of an electrohydrodynamic mobility B- = (t/pm)’I2 multiplied by E. In air, BEm = 2.7 X 10+ m2/(V-s), while a typical submicrometer aerosol mobility is m2/(V.s). Thus, the “Mach-number-like”ratio MEHD B B / B = 27 is much greater than unity. This is indicative of the potential for increased mass transfer by aerosoldriven convection. As the length scale is shortened, viscous forces dominate over those due to inertia and the electrically driven motions tend to be characterized by a length-independent electroviscous time v/tE2 (Melcher, 1981). The transition between these regimes comes at a length, LIV, that is approximated by equating (perhaps one-fifth) the electroinertial time to the electroviscous time to obtain LIv = 5 7 / ( ~ p , ) ’ / ~ EFor . the experiments presented here, this length is on the order of 1 cm for the largest E. Complete Mixing a n d the Role of Inhomogeneities. Regardless of the charging, migration, and convection processes occurring in the bulk of the aerosol, the mass flux to the collector is the product BNE evaluated at the collector. Mixing will be effective in increasing the rate of transport if it increases the particle charge and hence mobility at the collector. Complete mixing can do this by bringing the more highly charged aerosol from smaller z , but the quenching condition imposes a severe limit on the value of QN since, according to the complete mixing model, this quantity is uniform throughout most of the volume. Three hypothetical distributions of the aerosol space charge are illustrated by Figure 12. The first is from the migration model, the second from the complete mixing. The last distribution is the average aerosol charge density N Q over a given z plane generated by a structured fluid convection which transports the highly charged aerosol directly from the region near the injector to the collector without greatly increasing the space charge density in the intervening region. In this way, the quenching condition allows higher particle charge levels than it does with complete mixing. The hypothetical distribution cannot exist without inhomogeneities in these planes, for in the absence of charging and gradients in mobility, N Q can be shown
178 Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988
Na
I
when t* = 0.01. The aerosol charge density which brings the injector field to zero is
t'
26
v,
(NQ)o= 6[2(1 - Z I O ) - 61
(16)
where 6 is the width of the charged layer. Now suppose that this charged layer is convected intact to the collector, resulting in the redistribution of Figure 13b. Then, in view of eq 16, the electric field a t the injector is
t
t'
d
0 0
e
z,
z
(b)
Figure 13. (a) Quenching charge distribution and electric field before onset of convection. (b) Charge distribution and electric field after convection.
to be a monotonically decreasing function along the particle trajectories (Withers, 1978, 1979). Even in the middle region where this average is low, there must be areas where the local value of QN is quite high; these are the areas where the aerosol is being convected toward the collector. This electroconvection is fundamentally different from a situation with gradient driven transport, such as thermal convection. In the latter case, thorough mixing creates a flat temperature profile throughout the volume, increasing the heat flux by confining the imposed temperature difference to thin boundary layers. With the constraints of the single-stage precipitation considered here, the effects of such mixing are limited and other distributions of the transported quantity are more effective. Relation to Experiments. Two features of the transients in Figure 9c give some insight into the transport mechanisms. These are the sudden delayed change in the mass flux and the small local maximum in the total current, both of which occur early in the transient. A plausible explanation for this delay and the hump in the current is that they are manifestations of the initial large-scale inertia-dominated motion. Their times of occurzence are indicative of the time required for aerosol which was initially near the injector to roll over to the collector. In the case of the mass curve, the mechanism is clear: Although there are aerosol particles present at the collector initially, they are uncharged and thus cannot be collected. Because the injected ions cannot reach these particles to charge them, collection occurs first when the highly charged aerosol near the injector is convected to the collector. This time corresponds to the breakpoint in the M ( t ) curves. For large-scale (inertia-dominated) convection, the delay time should be of the order of l/&(V/l). Not only is this time within a factor of 2-3 of the observed delay, but there is also the predicted inverse dependence of this delay on voltage as well. By more than an order of magnitude, the migration time l / B ( V / l ) is too long to account for the observed delay. In fact, one migration time is much longer than the time over which the entire transient occurs. To understand the hump in the I ( t ) curve, imagine the initial charge distribution to be idealized by Figure 13a. This layer of quenching charge near the injector is suggestive of the curves of example 1 (Figure 2b), for instance,
Thus, although the total amount of charge in the volume is unchanged, its motion toward the collector reduces its effect on the electric field, allowing the injector field to rise considerably, as illustrated in Figure 13.
Conclusions If the aerosol density is large (as defined by eq 12-14), it has been shown that space-charge effects cause the precipitation time predicted by the migration model to lengthen. The experiments show that electrohydrodynamic effects enter in such a way that, at least in the overall time dependence of the transient, they tend to compensate for the space-charge effects. Perhaps this is why these electrohydrodynamic effects in aerosols have remained neglected for so long. The crudest of all models, which pictures the charging as instantaneous, transport as pure migration, and the field unaffected by space charge, predicts the observed high-density deposition time more accurately than the elaborate numerical models which include space-charge effects but not convection. With the numerical predictions, the actual role of convection is now clear. Convection appears to bring the dense and mobile aerosol from the injector region to the collector. The complete mixing model has been used to argue that convection must occur without filling the intervening region with space charge, which would, through the quenching condition, reduce the particle charge. A kinematic model has been developed (Withers, 1978, 1979) which quantitatively explains the electrohydrodynamic contribution to the precipitation. This, together with visual evidence, serves as a gilide for future efforts to develop a self-consistent model for the three-dimensional processes that carry the aerosol from the injection to the collection regions. There are basically four issues which determine the importance of electrohydrodynamic transport in practical devices such as electrostatic precipitators and aerosol diagnostic devices: (1)the aerosol density, normalized as suggested by eq 12, 13, or 14 to indicate space charge effects; (2) the dominant charging mechanism (field, diffusion, or field-enhanced diffusion); (3) the parameter MB = ( E / ~ ~ ) ' ' which ~ / B ,sets an upper limit on the electrohydrodynamic effects; and (4) competing fluid motions such as channel turbulence or ion wind which are not included here. It is apparent from the literature (Withers, 1978, 1979) describing common industrial processes for which electrostatic precipitators are control candidates that the pollutants are sometimes sufficiently dense that spacecharge-generated electrohydrodynamic effects are implicated. It should be noted that the electrohydrodynamic transport, as hypothesized in the previous section, depends on having a convective pattern (averaged over a given z
Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988 179 plane) as illustrated in Figure 12. The highly charged aerosol moves from the region near the injector toward the collector in narrow, rapidly moving columns so that the intervening region has a low average space-charge density. One such column can be seen in Figure 8c. Pictures for single-stage instability have a similar appearance (Melcher, 1981, p 8.50; Withers, 1979, p 436). This pattern could be disturbed by random fluctuations (channel turbulence or ion wind) of somewhat lesser velocities than the convective velocity. Their effect would be to mix the charged aerosol into the middle region; the quenching condition then more severely limits the particle charge near the injector, and the enhancement of the transport is reduced. The numerical complete mixing model demonstrates the limit of this effect; the transport is little more than the pure migration limit. In conclusion, there is clearly the potential for electrohydrodynamic aerosol transport in many applications. High particle densities, lower fields, lower flow velocities, and, in particular, submicrometer particles are conducive to these effects. Because the transport effects tend to fortuitously cancel out the space-charge effects, both have been safely ignored in many instances.
N = aerosol particle number density N * = normalized N (Table 11) N o = initial N n = ion density no = n at injection boundary Q = aerosol particle charge Q* = normalized Q q = ion charge Q, = 127rcR'E R = aerosol radius rd = see eq 6 T = absolute temperature t = time t* = normalized time t (Table 11) u = fluid velocity Vo = voltage between mesh and collector z = distance z* = normalized z (Table 11)
Acknowledgment
Literature Cited
The authors wish to acknowledge the contributions of E. P. Warren and Profs. M. Zahn and W. V. R. Malkus. Joseph Bakewell of the Dynamics Research Corp. in Wilmington, MA, supplied the meshes used in the experimentation. The first author was supported by an NSF Graduate Fellowship, and the research was partially funded by NASA Contract NSG 1453.
Alexander, J. C.; Davey, K. R.; Melcher, J. R. Ind. Eng. Chem. Fundam. 1981,20, 207. Bart, F. B. Masters Thesis, Cambridge, MA, 1986. Doetsch, E.; Friedrichs, H. A.; Knacke, 0.;Krahe, J. Staub 29 1969, 7, 24. Ehrlich, R. M. Ph.D. Dissertation, MIT, Cambridge, MA, 1979. Ehrlich, R. M.; Melcher, J. R. IEEE Trans. Ind. Appl. 1987, IA-23, 103. Hayne, T. I E E E Trans. Ind. Appl. 1976, IA-12, 63. Hewitt, G. W. A I E E Trans. 1957, 76, 300. Hopfinger, E. J.; Gosse, J. P. Phys. Fluids 1971, 14, 1671. Hoppel, W. A.; Gathman, S. J . Appl. Phys. 1970, 41, 1971. Lacroix, J . C.; Atten, P.; Hopfinger, E. J. J . Fluid Mech. 1975, 69, 539. Liu, B. Y.; Yeh, H. C. J . Appl. Phys. 1968, 39, 1396. Misakian, M. J. Appl. Phys. 1981, 52, 3135. Melcher, J. R. Continum Electromechanics; MIT Press: Cambridge, MA, 1981; pp 8.20-8.24. Robinson, M. J. Air Pol. Con. Assoc. 1968, 18, 235. Turnbull, R. J.; Melcher, J. R. Phys. Fluids 1969, 12, 1160. Withers, R. S. Sc.D. Thesis, MIT, Cambridge, MA, 1978. Withers, R. S. Transport of Charged Aerosols; Garland: New York, 1979. Withers, R. S.; Melcher, J. R. J. Aerosol Sci. 1981, 12, 307. Withers, R. S.; Melcher, J. R.; Richmann, J. W. J . Electrostnt. 1978, 5 , 225.
Nomenclature B = aerosol particle mobility BEHD = EHD mobility, (c/p,)1/2 b = ion mobility ?i = mean thermal ion velocity CM = complete mixing E = z-directed electric field intensity FED = field-enhanced diffusion charging IFD = independent field and diffusion charging J* = normalized J (Table 11) JI,JA,JT = ion, aerosol, and total current density k = Boltzmann coefficient LIv = electroconvective length 1 = distance between mesh and collector M = deposited mass M * = normalized M (Table 11) MB Mb
= BEHD/B
= BEHD/b
Greek Symbols = see eq 6 = permittivity vc = fluid viscosity r = mass flux density r* = normalized r (Table 11) pm = fluid mass density
6
Received for reuiew July 7, 1986 Accepted August 31, 1987
