Afmospheric Environment, Pergamon Press 1967. Vol. 1, pp. 193-204. Printed in Great Britain.

A MODIFIED DEUTSCH EFFICIENCY EQUATION FOR

ELECTROSTATIC PRECIPITATION

MYRON ROBINSON

Research-Cottrell, Incorporated, Bound Brook, New Jersey, U.S.A.

(First received 13 January 1967 and in final form 23 February 1967)

Abstract-The Deutsch efficiency equation of electrostatic precipitation is modified to take into account the erosion of collected dust and the nonuniformity of dust concentration over the precipitator cross section. Recent attempts to obviate serious fallacies in the Deutsch equation have led to rather sophisticated and complex analyses. The present approach, however, represents an effort to retain the simplicity of the original exponential relation, yet at the same time to meet, in some measure, some of the more serious objections to the old theory.

Nomenchture:

zt see under “Variable Erosion” (dimensionless);

= precipitator collecting area (m’); A’ = cumulative collecting area from inlet up to given downstream distance L (ma); b see under ‘Wuiable Erosion” (dimensionless);

see under “Variable Erosion” (dimensionless); & = concentration of particles in gas near the collecting wall (kg/m3); CA = average concentration of particles at precipitator outlet @g/m’); C = concentration of particles averaged over precipitator cross section at L (kg/ms); CO = inlet particle concentration (kg/m3); E = erosion rate (kg/(m%ec)); E. = erosion rate at inlet (kg/(m%ec)); k = constant of proportionality (ms2); k’ = constant of proportionality (m-‘); K = Anderson precipitator constant, equation (1); L = distance downstream from precipitator inlet (m);

a see under “Variable Erosion” (dimensionless);

= gross precipitate flux (kg/(m2-set)); P” = net precipitate flux (kg/(m%ec)); P = net precipitate flux at A’ = 0 (kg/(m%ec)); P:’ = gross precipitate flux at A’ = 0 (keW-.=)h

= time rate of dust buildup (or erosion) on collecting plate at L due to effect I (kg/set); 2: = inlet dust feed rate (kg/@; & = time rate of gross precipitation (excluding erosion) (kg/&; Qz = time rate of a-erosion (kg/set); 0s = time rate of /&erosion (kg/set); r = pipe radius (m); t = gas treatment time (set); V = average gas velocity (m/set); vo = minimum gas velocity of series of tests (m/se@; Y = volume gas-Sow rate (m3/sec);

W = mass-transfer coetBcient (kg/(mz-se&/&g/m”) = m/set; WD = effective migration velocity of Deutsch relation (m/set); WI = effective mass-transfer coefhcient (m/see);

; = mass of dust eroded per unit mass precipitated (dimensionless); = mass of “problem” dust eroded per unit mass of dust precipitated at A’ = 0 (dimensionless);

?I = fractional etBciency (dimensionless); x = C,/Z, concentration uniformity ratio (dhnensionless).

193

194 MYRON ROBINSON

INTRODUCTION

THE WELL-KNOWN exponential efficiency equation of electrostatic precipitation was discovered by ANDERSON in 1919 and given in the empirical form :

?/=1-K’, (1)

where q (dimensionless) is the fractional efficiency, t (set) is the treatment time, and K is a “precipitator constant”. Three years afterward, DEUTSCH (1922) showed analytically that the precipitator constant for a wire-pipe system could be expressed in terms of physically more significant quantities:

K = e-2wDtr, (2)

where wh (m/set) is the “effective” particle migration velocity and I (m) is the radius of the precipitator pipe. Today, the efficiency equation is often written in the equivalent but more generally useful form, equally applicable to pipes and rectangular ducts:

q = 1 -e-A%/“, (3)

where A (m’) is the collecting surface area and Y (m3/sec) the volume of gas treated per unit time.

Equation (3) is widely employed in precipitator design, despite the experiments of numerous investigators indicating that precipitator dust loss 1 -q may only roughly be regarded as an exponential. Accumulating data from sources separated in both time and distance, and for diverse dusts and precipitator geometries, suggest that a better approximation to reality results in accepting not an exponential decay of the dust load of the gas with AwD/ V (which relation is required by equation (3)), but rather an exponential between the net flux of precipitate to the collecting walls, P,, (kg/(m2- set)), and distance downstream L (m) from the precipitator inlet. Letting A’ be the cumulative collecting surface traversed by the gas up to the position L, and k (mw2) a constant of proportionality, this relation becomes:

pnccequ. (4)

Experimental support for the foregoing expression is found in the data of KALASCH- NIKOW (1933), MIERDEL and SELIGER (1935), ROBINSON (1961), and others who have measured precipitate distribution on collecting walls. Most recently, studies in this laboratory of a fly-ash, duct-type precipitator of commercial dimensions (except for height) have provided still further confirmation. Despite the fact that this last-named work included both coarse and tie fly ashes, covered a gas velocity range of 1.5 to 4.5 m/set, involved rapping and rapping-free operation, and extended over inlet/outlet concentration ratios spanning more than three orders of magnitude, equation (4) was again substantiated, see FIG. 1.

We shall presently see that although equation (4) is a necessary consequence of equation (3) (assuming k = w/v>, the converse is not so. In other words, the validity of equation (4) does not require 1 -_tt to be equal to eSU’.

The weight of experience thus shows that, as a rule, a precipitation relation having the form of equation (4) holds tolerably well in practice both in laboratory and in- dustrial situations. Leaving the door open for exceptions in certain areas of applica- tion (e.g. cases in which special pains are taken to preserve laminar flow), we may

A Modified Deutsch Efficiency Equation for Electrostatic Precipitation 195

suppose that a precipitator efficiency equation based on physical arguments more rigorously meeting objections to Deutsch’s or similar derivations, will generally be compatible with equation (4) or a close approximation thereto. In connexion with a

Fro. 1. Dust collection as a function of downstream location (cumulative collecting area from inlet) in a parallel-plate precipitator of the following SpeciScations: Length, 4.0 m; plate-to- plate spacing, 25 cm; wire diameter, 0.28 cm; wire-to-wire spacing, 14 cm; voltage, 50 kV dc.; dust, fly ash 90 per cent < 30 cc. Coarser fly ash (90 per cent < 80,~) yielded equally well fitting exponential curves. The curves have been arbitrarily displaced along the ordinate for greater legibility. Each point represents dust collected in a hopper segment. Since bathed and bat&-free areas alternate on the plates, there is a tendency for points to occur in pairs. The average gas

velocity is Y.

recent attempt (WILLIAMS and JACKSON, 1962) to obviate one serious fallacy in the Deutsch equation which attempt, however, led to a rather sophisticated and complex analysis, C. J. Stairmand expressed the hope that an approach could be evolved “which would be as easy to understand as the Deutsch concept (including the use of an ‘effective drift velocity’ which, though fundamentally wrong, was nevertheless easy to appreciate and had served so long as a measure of precipitator performance).” This paper is offered as one step in that direction: an effort to retain the simplicity of the exponential relation, yet at the same time to meet, in some measure, some of the serious objections to the old theory. It remains to be seen, of course, whether or not this approach will ultimately lead to useful prediction and design equations, or whether the results of this work will merely confirm, as in the case of the Deutsch formula, the relationship between efficiency and certain relevant precipitation parameters.

196 MXRON ROBINSON

THE EFFICIENCY EQUATION

Consider the performance of an electrostatic precipitator over the collecting surface area dA’ (m”) extending between downstream distances L and L+dL from the pre- cipitator inlet. The net flux of precipitate to the collecting walls P, is equal to the algebraic sum of collections and losses at the walls from each of i effects:

where Qi (kg/se@ is the time rate of dust buildup on the collecting plate and C (kg/m3) is the concentration of dust in the gas averaged over the cross section. The volume flow rate of gas Y is assumed constant over the precipitator cross section.

Precipitation termj

Mass-transfer coeficient. We assume that the gross rate at which suspended dust precipitates onto collecting surface element dA’ is proportional to the dust concentra- tion in the gas immediately adjacent to the surface. Thus:

P = WC,,I (5)

where P (kg&m’-see)) is the gross precipitate flux at L, C, (kg/m3) is the corresponding dust concentration in the gas near the wall, and w (m/set) is a constant of propor- tionality. Although w has the dimensions of velocity we avoid identifying it with the particle migration velocity in the usual Deuts~b-Pauthenier sense, i.e. as the velocity with which a charged particle moves, under the action of a Coulomb force, to the collecting electrode (PAUTHENIER and MOREAU-HANOT, 1932). instead, we regard w phenomenologic~ly as a mass-transfer coefficient giving the gross precipitate flux per unit dust concentration at the wall. A semi-empirical approach for expressing R in terms of “fundamental” quantities, e.g. electric field, particle size, gas viscosity, by means of a dimensional analysis, is given elsewhere (ROBINSON, 1967u).

Cross-sectional concentration projcile. The familiar Deutsch efficiency equation :

q = 1 _.+-‘My (3)

(in which w is understood to be the particle migration velocity) is derived on the captions, among others, that the dust is uniformly di~~buted over the precipitator cross section and that there is no erosion from the walls of dust already precipitated. As early as 1925, Deutsch himself pointed out that a relatively dust-free space existed within a few diameters of the discharge electrode. He succeeded, in fact, in actually photo~aphing the trajectories of dust particles first approaching and then being repelled from the nei~~rhood of a corona wire. It was not till more recently, how- ever, that imperfect mixing of the dust was thought to give rise to significant departures from uniformity of the cross-sectional dust concentration. ROSE and WOOD (1956), COOPFBBSAN (1960), and W~L~MS and JACKSON (1962) cite reason for believing that turbulent eddies are often insufficient to achieve the degree of continuous remixing required for cross-sectional uniformity as dust is lost to the walls. Observations in ducts closely simulating actual precipitators clearly demonstrate narrow, relatively dust-free, wedge-shaped zones downstream of individual discharge wires (ROBPBON, 1%7b). ~antitative data on this phenomenon are, however, still lacking.

A Mod&d Deutsch E&iency Equation for Electrostatic Precipitation 197

To provide a measure of cross-sectional uniformity, it is useful to define the ratio:

Under the simplified assumptions of the Deutsch equation (3), x = 1. More realistic- ally, we should expect the sweeping out of particles by the electric field to tend to clear the region of the discharge electrodes first, a circumstance leading to the condi- tion x> 1 beyond the precipitator inlet. We shall assume, to a first approximation, that-although the dust (presumably) enters the precipitator evenly distributed over the inlet cross section-while the dust is still quite close to the entrance it distributes itself in such a way that even should the concentration profile change markedly with the downstream distance, the ratio x = C,,,/c remains essentially constant. Preliminary photographs of the cross-sectional dust distribution in a model duct-type precipitator do, indeed, suggest that, in practice, x is reasonably constant through the length of the precipitator and is probably not too far from unity (ROBINSON, 1967b). (We note, however, that in a precipitator in which the electric field is maintained more or less at its peak value over the length of the system, e.g. in a wire-pipe system (HIGNKIT, 1965) or in a duct-type system in which the wires are abnormally close together (ROSE and WOOD, 1956) the condition 1 % 1 does not appear to prevail.) The results of C~~PERMAN (1966), derived from a solution of the diffusion equation with an added term to account for the electrical drift of the particles, also lends credence to the supposition of moderately constant x as long as we are not too close to the mouth of the precipitator.

With the foregoing in mind we rewrite equation (6):

=wxc,

where 0, is the gross precipitation rate per unit time (kg/set).

(8)

Erosion term

Experimental wind tunnel. Quantitative data on erosion mechanisms have been obtained employing a wind tunnel 1.8 m long of 5cm square cross section. An even 1.3-cm thickness of a typical fly ash was spread manually on the floor of the tunnel before each test. A suction fan at the downstream end of the tunnel drew air through it at adjustable velocities. Tests were performed under room conditions. Entering air was either dust-free or contained a known concentration of fly ash of the same kind as deposited on the floor. The object of the experiment was to simulate the erosion and re-entrainment processes of an electrostatic precipitator by permitting airborne particles to impact against an already present surface deposit. In this experiment, however, the impacting particles fell to the bed under the influence of gravity as the system was not electrified.

Although we are here concerned with the similarities between gravity settling and electrostatic precipitation, we must not lose sight of basic differences in the two pro- cesses and the realization that the analogy not be pressed too far. We note, for example, that the gravity force is proportional to the particle’s volume, whereas the electro- static precipitating force is (usually) proportional to the particle’s surface area. Furthermore, while gravity functions wholly to drive particles to the collecting surface

198 MYRON ROBINSON

and retain them there, electrical effects serve both to attract to and repel from the surface not only particles that have already been pr~ipitated @&WE and LUCAS, 1953; SEMAN and PENNEY, 1965) but also particles that are still airborne (SEMAN and PENNEY, 1966). For these reasons and others, the wind-tu~el approach provides only a heuristic approximation to actual precipitator conditions.

The erosion rate E = d&dA’ (kg/m2-set)) is shown as a function of downstream distance and air velocity in FIG. 2. At low velocities or near the tunnel inlet, i.e. under conditions of relatively low airborne dust con~ntration~ E increases exponentially with downstream distance according to the equation:

E =: &ekX . (9)

Were E, is the erosion rate (kg/(m’-set)} at A’ = 0, and the coefficient k’ (me*) is independent of A’. The physical picture represented is one in which, on the average, each particle eroded at A’ causqs ek’ particles to be eroded one m2 further downstream, The link between erosion rate E and average dust concentration in the gas c thus appears to be in the bombardment or scraping of the surface deposit by impacting particles.

BAGNOLD (19541, in his extensive studies of the physics of blown sand and desert dunes, identses three interrelated but physically distinct processes of particle trans- port: (1) Surface creep, the movement of particles along the surface, the particles receiving their momentum by impact from saltation. (2) Saltation, the repeated impaction and rebounding of airborne particles in characteristic trajectories. Grains in saltation receive their moments directly from the pressure of the wind after they have risen into it. If sufficiency energetic, a saltating particle on impact will blast a tiny crater in the surface of the bed, splashing a number of particles upward into the gas stream. Particle transport may be very effectively multiplied by this process. (3) True suspension, the condition of particles so small that they are maintained aloft indefinitely by eddies. Surface creep and saltation (or closely corresponding pheno- mena) are commonly observed in precipitators. Suspension, as defined above, cannot account for the transport within the precipitator of a sizable fraction of precipitator dust, or else the close-to-one hundred per cent efficiency of the modern fly-ash pre- cipitator would not be attained. Suspension is, therefore, much less a matter of con- cern in precipitators than Bagnold found it to be with dust in the open air. The reason for this difference is, of course, the fact that the el~tric-field precipitation rate far exceeds the dust settling rate under gravity.

Variable em&n. We first direct our attention to the exponentially increasing seg- ments of the curves of FIG. 2. Equation (91, viewed in the light of the aforementioned saltation~reep processes, suggests an erosion mechanism of the following type. At a given distance down the wind tunnel the total flux of particles (airborne or creeping) is n per second. Over unit area AA’, bn of these particles “react” with the surface deposit and expel abn others, the net rate of removal from the surface dnfdA’ being bn(a - 1) particles~(m’-set). Consequently:

k’= b(a-1). 00)

In the wind tunnel, k’ is a positive number, i.e. erosion exceeds collection. In a pr~ipitator the reverse is true and k’ is negative. We shall assume the postulated erosion mechanism is su~ciently general to include both cases.

A Mod&d Deutsch EGciency Equation for ISSmstatic Precipitation 199

Without inquiring more deeply at this time into the physics of the processes in- volved, the hypothesis may plausibly be put forth that in the wind tunnel the erosion

FIG. 2. Erosion rate as a function of downstream location (cumulative floor area from inlet), gas velocity (given in term of the minimum test v&city Y& and inlet dust feed rate &.

rate E is proportional to the impaction rate (mass of dust due to all transport mechan- isms arriving in unit area per unit time) which is in turn proportional to the conam- tration of airborne dust. Assuming that an analogous erosion rn~an~rn exists in electrostatic precipitation we may write :

dQ2 _ap~ -= dA’ -@WG (11)

where the erosion coefXicient a (Dimensions) is the mass of dust eroded per unit mass precipitated (impacted).

Constant erosiin. At relatively high dust concentrations, RCL 2 shows that a new erosion rn~ha~srn becomes dominant in which the erosion rate (for a given inlet concentration) is independent of the cross-sectional average airborne dust concentra- tion. A kind of “emission-limited” erosion of the surface seems to be taking place. But whatever the exact mechanism, for present purposes we need only ask whether the steady-state process wnder consideration is suggestive of a possible precipitator erosion mechanism. Evidence for such a mechanism is provided by the observation that dusts sometimes possess au apparently unprecipitable fraction. For example, a very low resistivity dust component may be repeatedly precipitated and r~ntrain~ in its passage through a precipitator. A situation is thereby created in which the average concentration of the “problem” dust in the gas remains unchanged throughout the precipitator, regardless of its length. An erosion rate satisfying these requirements is given by the expression :

d& -= -j?P* = -pw&), dA’

&here PO ~g~(m2-~)) is the gross precipitate flux at A’ = 0, and 8, the dimensionIess erosion coefficient, is the mass of “problem” dust eroded per unit mass of total dust precipitated at A’ = 0.

200 MYRON ROEINSON

It is clear that since the total erosion cannot be greater than the precipitate flux, the erosion coefficients are subject to the condition:-

cc”t-fiQl* (131

~uations (81, (11)

(14)

~~~~t~on of the ~~~erentia~ e~~t~o~

We are now in a position to solve equation (5). Introducing and (12), there results:

P *=- vdC - = w$k%wxc+wxzfo. dA’

Integration yields :

where CA (kgfm3) is the dust ~~~~tration at A’ = A. The fractional efficiency q (dimensionless) of a precipitator of size A is defined:

,=I-~, whence:

q=l- l-$=-=- e ( -> -fl -a)pA/V B --t l-3

tW

(17)

We note from equation (17) that the presence of 8 erosion destroys the purely exponential decay of the t?vs.-A’ curve chara~te~stic of the Deutsch relation:

c 3 ~@“~~“~V. (18)

If #$ erosion is absent, however, the pure exponential is preserved, despite the presence of 01 and x terms. The oft-repeated statement that a uniform cross-sectional dust distribution is an essential condition for the derivation of any exponential efficiency equation is clearly not true.

The not wholly exponential equations (15) and (17) do, nevertheless, give rise to a net precipitate flux that is a pure exponential. Substituting equation (15) in equation (14) we find at any downstream distance L:

P, = (I- 01 -~)w~~~e-{~ -4)xwA’~y (191

This is reducible to Deutsch-Iike form by setting:

P 80 = (I-a-B)wxG, mv

and :

w, = (I- E)XW, (29

where P,,, is the net precipitate flux at A ’ = 0 and w,, is the net or effective mass- transfer coefficient. Thus :

A Mod&d Deutsch ESciency Equation for Electrostatic ~pi~tion 201

pa = p”oe-W~~vb (22)

in agreement with equation (4). Zero and total erosion. In a Deutschian precipitator x = I, tc = 0 and @ = 0.

Equation (17) then reduces to:

$7 = 1-e-WA/V, (3) as expected.

A precipitator fun~ioning as an agglomerator (e.g. a carbon-black pre~ipi~tor) is run under conditions yielding complete erosion. In such a case, equation (13) becomes :

wt-p = 1,

and equation (17) gives an efliciency :

B = 0, (23)

for all A. It is general practice among pr~ipitator designers to give equation (3) the broadest possible application, the effective migration velocity WD being adjusted to secure agreement with observation. In the case of the aforementioned agglomerator, WD is taken as zero, a view obliterating the fact the gross precipitate flux to the plates may be very high indeed.

Particle-transport mechanisms

~~n~ry-~~~er effectts. Particle collection in an electrostatic precipitator is essentially a process of mass transfer through a moving gas, in a direction, when observed relative to the moving gas, that is normal to the collecting surface. It is necessary to distinguish at least three forms of mass transfer from the main body of gas to the plates: (1) electrostatic drift under the action of a Coulomb force, (2) turbulent diffu- sion due to convective eddies, and (3) inertial drift. The electric-drift mechanism has been quantitatively investigated and its main features are quite clear. In contrast, the physical nature of mass transfer by turbulent convection is much more complicated and, in the present state of the problem as it relates to electrostatic precipitators, it has so far been impossible to set up general and rigorous quantitative definitions. (Noteworthy efforts in this direction have, however, been made by WILLIAMS and

JACKSON (1962) and COOPERMAN (1960, 1966)). Inertial drift, another complex phenomenon, is a consequence of a particle’s tendency, by virtue of its momentum, to continue moving in a straight line in the face of ohposing or deflecting forces. Thus, the more massive a particle, the less likely it is to closely follow the motion of an eddy in which it is entrained (SEMAN and PENN.W, 1965, 1966).

Pnrely electrostatic mass transfer to the walls occurs in stationary gases and under the influence of the laminar-flow regime. It is a characteristic feature of laminar flow that the direction of gas motion as a whole coincides with the direction in which any separate part of the gas moves. Therefore, there is no macroscopic motion of the gas in the transverse direction (i.e. normal to the collecting plate) under fully laminar conditions. In such circumstances, mass transfer to precipitator collecting plates can occur only by electrostatic drift.

The laminar-flow precipitator is, however, a laboratory novelty; all, or virtually all, commercial precipitators and all single-stage precipitators run at various levels of turbulence. Turbulent flow is most fully developed in the “core” of the duct

202 Mellon ROBINSON

(neglecting effects of plate baffles) and dies away as the surface of the collecting wall is approached. In the immediate vicinity of the wall, we assume that turbulence is damped out to such an extent that eddy diffusion no longer accounts for a sign&ant share of particle transport. (Experimental data on this point are, unfortunately, lacking. WHITE (1955) and WILLIAMS and JACKSON (1962), in their considerations of the problem invoke a laminar boundary layer of finite thickness; FRIJXDLANDEF~ (1959) and COOPERMAN (1960, 1966) put forth a contrary point of view.) Once a laminar boundary layer is assumed, it is clear that particles in the nearby turbulent region which acquire a sufficient component of momentum normal to the wall, are able, without benefit of electrostatic attraction, to penetrate the quiescent region and reach the collecting surface (FRIEDLANDER and JOHNSTONE, 1957). Still another contribution to nonelectrical particle transport may be provided by fluctuations in the laminar sublayer (LIN et al., 1956). And finally, effects due to asperities on the surface of the deposited dust or to collecting-plate bafIles may further modify the flow in the sup- posedly quiescent gas ti. We shall consider, however, that in the sublayer only electrostatic and inertial (“stopping-distance”) drifts contribute appreciably to particle collection.

In the core of the precipitator the two active transfer mechanisms, eddy diffusion and electrostatic drift, work in contrary directions. Electrostatic drift transfers particles to the walls, but turbulent diffusion, since it redistributes the dust more uniformly over the cross section of the duct, exerts an opposing effort. In other words, the higher the level of turbulence, the more effectively dust is kept from concentrating in the core near the walls, the closer x falls toward unity, and the lower the efficiency tends.

Effective mass-transfer coeficient. Since eddy diffusion decays gradually to zero as the core-boundary layer interface is approached from midstream, and then remains nil across the sublayer, the particle concentration assumes that value across the sub- layer that it possesses at the core-sublayer interface. The dust concentration at the wall, C,, that determines the precipitate flux is, therefore, the concentration in the laminar sublayer (equation (6)). The component of precipitate flux due to inertial drift through the sublayer depends on the concentration of particles of some minimum inertia close to the interface. Making the approximate assumption that the concentra- tion of massive particles is proportional to the total particulate concentration through- out the precipitator, we may rewrite equation (6):

P = WC, = P,+P, = (wL+w,)cw, (24)

where P, and Pi, and w, and wI, are the respective electrostatic and inertial precipitate fluxes and mass-transfer coefficients.

At first thought, inertial drift might be regarded a negligible quantity, or else the efficiency of an unenergized precipitator (mechanical efficiency) would not be so very low. But a better measure of the effects of inertial drift is given by the performance of an unenergized wet precipitator, one exhibiting low or zero erosion. In this case, the efficiency may be several tens of per cent, a magnitude to be reckoned with.

Recent results, reported by a number of investigators (KIRKWOOD, 1961; KOGLIN, 1962; WILLIAMS and JACKSON, 1962), and again confirmed in the present work (FIG. l), show a significant increase with gas velocity in the effective mass transfer coefficient w, :

A Modified Deutsch Efficiency Equation for Electrostatic Precipitation 203

W” = (l-cl)lW = (l-a&(w.+wJ, (25)

as long as the velocity is not so high that the increase in w, is masked by erosion. Since 1 - ct and x both decrease with gas velocity and w, is independent of velocity, the rise in wf must be pronounced.

In conclusion, it is worthwhile to consider the effect of the erosion parameters a and /I and the nonuniformity parameter x on various indicators of precipitator performance. Plausible values of these parameters in an assumed case lead to the results indicated in FIG. 3. Of particular interest is the ratio w,/w, where w, is the

0 2 4 6 6 IO 12 14 Aw/V

FIG. 3. Precipitator variables as a function of Aw/V. The introduction of a nonzero erosion parameter /I causes the Deutsch migration velocity wI) to decreax with collecting area A and

increase with gas flow rate V.

Deutsch migration velocity calculated from equation (3) using the efficiencies (based on equation (17)) given in the figure. The Deutsch velocity is seen to decrease with downstream distance, a phenomenon frequently observed in practice. This behavior has elsewhere been attributed to a decrease in mean particle size, or particle con- ductivity, or to a change in flow conditions. But it is clear from the foregoing that a decrease in w, at constant w may be merely a consequence of applying the Deutsch equation to physical circumstances which it does not properly fit.

REFERENCES

ANDERSON E. (1919) Engineering calculations for treater design with eficiency andpower consumption akta. Unpublished report, Western Precipitation Company, Los Angela.

BAGNOW R. A. (1954) The Physics of Blown Sand and Desert Dunes. Methuem, London. -PERMAN P. (1960) Turbulent gas pOw and electrical precipitation. Winter General Meeting,

American Institute of Electrical Engineers, New York. CQWBRMAN P. (1966) Bowu&ny hxyer effects in electrostatic precipitation. 59th Annual Meeting, Air

Pollution Control Association, Pittsburgh, Pennsylvania. DEUTXH W. (1922) Bewegung und Ladung der ElektrizitgtstrQer im Zylinderkondensator. An&

Phys. 68,335-344.

204 MYRON ROBINSON

DEUTXH W. (1925) Elektrische Gasreinigung. Z. tech. Z%ys. 6,423-447. F~_UEDLAEIDER S. K. and Jonnsrom H. F. (1957) Deposition of suspended particles from turbulent

gas streams. Znd. Engng Chem. it&. &f.) Edn 49,11X-1156. FRIEDLANDER S, K. (1959) Principles of gas-solid separation in dry systems. Ckem. Etzgng Prog.

Symp. Ser. 55,135-149. HIGNEIT E. T. (1965) Colloquium on electrostatic precipitators. Sraub 25, 197-198. KALAXHNMOW S. (1933) Der Ein5uss der Feldstlirke und der Verweildauer der Gase in Elektroliltern

auf der Reinigungsgrad. 2. tech. Pkys. 14,267-270. Kra~woo~ J. B. (1961) EIectrostutic precipitators for the collection of f5v ask from large pulverized

fuel-fired boilers. Electricity Commission of New South Wales, Australia. KOGL~M W. (1962) Die ~ekt~~lte~o~el fiir Tr~enelek~o~ter mit F~~~henelek~~en.

star& 22,189-190. LIN C. S., MOULTON R. W. and PUTNAM G. L. (1956) Mass transfer between solid wall and &rid

streams. Znd. Engng Ckem. ind. (hr.) Z%n 45,636-640. LOWE H. J. and Lucas D. H. (1953) The physics of electrostatic precipitation. Br. J. appf. Pkys.

Suppl. No. 2, S4O-S47. MIERDEL G. and Srnxxo~~ R. (1935) Untersuchungen iiber die physikalischen Vorgtinge bei der

Elektrotiltenmg. Arch. Elektroteck. 29, 149-172. PAUTH~NIER M. ‘M. and MOREAU-HANOT M. (1932) La charge des particules sphkriques dans un

champ ioni& f. Pkys. Radium, Paris 3,590-613. ROBINSON M. (1961) A miniature electrostatic precipitator for sampling aerosols. Theory and opera-

tion. Ana&. Ckem. 33,109-l 13. ROBINSON M. (1967a) The dimensionless parameters of electrostatic precipitation. Publication

pending. ROBINSON M. (196711) Tke role of turbulence in electrostatic precipitation. To be presented at 60th

Annual Meeting, Air Pollution Control Association, Pittsburgh, Pennsylvania. Rosa H. E. and WOOD A. 3. (1956) An Introduction to E[ectrostatfc Precipitation in Theory andPractice.

pp. 78-83. Constable, London. SUN G. W. and PI%NEY G. W. (1965) Photographic records of particie trajectories during electro-

static p~ipitation. Inst. Elec. E~ec#ron. erzgrs ht. Conv. Z&e. Part 7, pp. 69-72. E&MAN G. W. and PENNEY G. W. (1966) Photographic studies of particle behavior uuder varying

precipitating conditions. Winter Power Meeting, Institution Electrical and Electronic Engineers, New York.

WHITE H. J. (1955) Modern electrical precipitation. Znd. Engng Ckem. ind. (inr.) Edn 47, 932-939. WIUUMS J. C. and JACKSON R. (1962) The motion of solids in an electrostatic precipitator. Pru-

ceedings of the Symposium on Interaction Fluids Particles. Institute of Chemical Engineers, London, pp. 282-288, Discussion pp. 291-293,297-298.