The response of droplets and particles to electric fields, in the presence of ions

Tenth Symposium (International) on Combustion,

pp. 709-720,

The Combustion Institute, 1965

THE RESPONSE OF DROPLETS AND PARTICLES TO ELECTRIC FIELDS~ IN THE PRESENCE OF IONS K. GUGAN, J. LAWTON, AND F. J. WEINBERG

Department of Chemical Engineering and Chemical Technology, Imperial College of Science and Technology, London, England A theoretical study of the motion of charged molecular clusters, droplets, and particles is presented as an essential part of an investigation into electrical control of certain combustion processes. The information required is the charge and mobility of each such chargecarrier, in the presence of flame-ions, under the influence of large fields. The study is subdivided by particle size and by whether the particles are charged to a constant level, or are free to continue acquiring charge along their trajectories in a field. Subdivision by size follows the response to molecular collisions. For the smallest, the target area for impact is greater than geometrical, because of dipoles induced in neighboring molecules. In the next size range, such forces of attraction are negligible, while interaction with neutral gas can still be treated as individual collisions. Larger particles experience such collisions only as a viscosity--leading into the Stokes regime. Lastly, particles of sufficiently large Re are treated in terms of Newton's Law and it is shown that this regime includes the largest sizes likely to be encountered in practice. Particles carrying constant charge occur when, after a region of relatively plentiful space charge (corona discharge, flame) their trajectories lie where no smaller charge carriers exist, or where the theoretical equilibrium charge is no greater than initial. The mechanisms and rates of charging due to ion bombardment, diffusion, and thermionic emission are next examined and it is shown that in the most relevant case--the drifting of particles towards electrodes in the company of ions--the particles' charge, together with field intensity tends to increase continuously. The conclusions are analystical expressions for charge, mobility, and range of applicability for each combination of regimes. One noteworthy result is that the largest particles, when free to acquire charge, attain a constant mobility ~ 5 % of that of H30 +. Introduction

Theoretical and experimental investigations into the effects of applied fields on combustion processes demand knowledge of how the charge carriers involved respond to such fields. The information required is the charge and the mobility of each carrier. This information is available, to an adequate degree of accuracy, for ions (see, e.g., Ref. 1). I t is not, in general, available for large clusters of molecules, for small droplets and small particles, all of which may be charged. I t so happens that the presence of such charged particulate matter is particularly important in the context of electrical properties of flames, for several reasons. In the case of all luminous flames, behavior in applied fields seems to depend greatly on the charged carbon particles they contain. This attribute has been used ~ to influence the position, magnitude, and form of carbon deposition from such flames, and, more recently, 3 the process of carbon formation in the flame itself.

More generally, flames in practical use frequently burn dispersions of liquid or solid fuels and produce particles in the form of smoke or flyash. Attempts are being made 4 to manage such combustion processes by means of fields from the point where the fuel is electrostatically atomized and/or dispersed, via control of its movements throughout its burning history, to the point where any ash is precipitated electrostatically. The theory of maximum effects attainable b y applying fields to flames5 lends further significance to these cases. I t indicates that for certain effects, such as maximum rates of mass transport by "electrolysis" and drag by ions on gas, the process continues to improve as the size of the charge carrier increases. All this is in addition to the more spontaneous and universally applicable formation of charged particulate m a t t e r - - i o n clustering, attachment of charges to microscopic and submicroscopic particles present in unfiltered and undried air generally, and electron emission by particles of low work function at

709

710

ELECTRICAL PROPERTIES OF FLAMES

elevated temperatures. It is for these reasons that the study of the electrical aspects of flames containing particulate matter is desirable. This communication presents a purely theoretical investigation (which is being carried out in parallel with an experimental one) into the behavior of particles in the presence of flame ions, when subjected to large unidirectional fields. The study is subdivided by particle size and by whether the particles are charged to a constant value or are free to continue acquiring charge along their trajectory in a field. Charge and mobility are, of course, interrelated, since the latter depends on the force exercised by a field. Subdivision b y Particle Size and Reynolds Number Dependence on size is determined by the effect molecular collisions have on the particle. For the smallest, the target area for impact with molecules is greater than the geometrical target area because of dipole effects induced in neighboring molecules by the divergence of the field induced by the particle charge. This is followed by a range of sizes in which such forces of attraction are negligible, while the interaction with neutral gas can still be treated in terms of individual collisions. Larger particles experience such collisions only in the form of a gas viscosity--leading into the Stokes regime. Lastly, particles of sufficient size and/or velocity attain Reynolds Numbers large enough for the density of the gas to take the place of viscosity as the property determining drag. This allows the theory to be developed in terms of Newton's Law and it is shown that this regime extends to sizes large enough to include all cases likely to be encountered in practice. The range of interest overlaps--at its "small size, charge, and field end"--much previous work, which has been summarized in reviews (e.g., Ref. 1). Starting from the smallest radii, there is, first of all, a range of sizes of clusters so small that the theories of small ion mobility apply to them. Theory here gives ] k =

0.235[- ( M + Mi)/M~3~ [ - ( D - 1)0M-]~

(1)

For larger ions, the effects of divergence of the field, induced around them by their own charge, become negligible and the "classical" Langevin equation s k = O.815(qXi/Mc)[-(Mi + M)/M~] ~

applies.

(2)

When particles become large in comparison with the mean free path and no longer sense the gas as individual collisions, they acquire the velocities of bodies moving through fluids under the influence of an impressed force. Neglecting gravitational effects, this force (F) is, of course F = EeN,

(3)

where N is the number of unit charges (e) carried. The resistive force which opposes the above, and is equal to it under equilibrium conditions, is a function of velocity which depends on the Reynolds Number of the particle in the manner shown in Fig. 1. At low values of Re (up to ~--3, based on the diameter, the error in drag is less than 20%) Stokes' Law applies. Thus F = 61ryav, Re_< 3.

(4)

For 700 _< Re < 2 X 105, Newton's Law may be used: i.e., F = 0.227ra2pv2~

(5)

Beyond this, the dependence changes once more, but it will be shown that larger values of Re are irrelevant in the present context. Many of the sizes of interest here, unfortunately, fall in the transition region 3 < Re < 700. Although empirical laws will be applied to this regime, it will be more instructive to consider first the limiting values given by Stokes' and Newton's Laws. Figure 1 is a graph, on logarithmic scales, of drag coefficient for spherical particles against Reynolds Number, wherein the dashed lines are the asymptotes to the Stokes and Newton regimes, extended into the transition region. They intersect at about Re ---- 55. There the error due to using either law is a maximum. It falls off on either side provided Eq. (5) is used for Re > 55 and Eq. (4) for smaller Re. In calculating mobilities, two entirely different systems must be considered depending on whether the charge per particle is constant or whether it is free to increase due to further attachment along its trajectory. The former applies when the particles become charged in a region of relatively plentiful space charge (such as a region of corona discharge or a flame) and thereafter are considered in a space where no smaller charge carriers exist or, if they do, where the theoretical equilibrium charge that the particles would acquire is no greater than the initial value (this will be shown to apply if field intensity does not increase with distance). The second case, where the charge Ne on a particle increases with field strength is, in the present context, the usual one.

711

ELECTRICAL CONTROL OF DROPLETS AND PARTICLES I0 4 10 3 lO 2

% ioI LL

10o

\\\\\

I0-I 55 10-2 1

"3 Io•

ao l

II

I

xoo

lO 2

t

I

10 3

10 4

i

I

105 106

Re

FIG. 1. Drag coefficient vs Reynolds Number.

Mobilities of Particles Carrying a C o n s t a n t Charge Equating the electrical force [-Eq. (3)] to the drag, we obtain E e N = 67r~av

(6)

in the Stokes regime, givhlg a mobility k = v i e = eN/67r,a.

(6a)

a process of interpolation but, since X at STP is of the order 10-1 tt while collision radii of molecules are of the order 10-4 #, this regime will not be of great practical interest here. The upper limit of the Stokes regime is determined by the Re of the particle, a criterion which is most readily expressed in terms of the (constant) particle charge, since Re proves to be independent of diameter, in this regime.

E e N = 0.227ra2pv 2,

(7)

by Eq. (6). If we take 3 as the maximum then, approximately, for air at STP

so that the velocity is proportional to the square root of the field strength and the "mobility" depends on field strength k = v i e = a-l(eN/O.22~rpE) ~

(7~)

In both cases k is inversely proportional to the particle radius. Let us now consider the particle sizes for which each regime is relevant. The "Stokes mobility" may be used down to very small sizes and can be extrapolated to diameters of the order of a mean free path by the use of a semiempirical correction due to Cunningham 1 k -- N e ( 1 .-~ X / a ) 6vya

(8)

R e = v(2u)p/~l = E e N p / 3 w ~ ~,

In the Newton regime,

(6b)

where X is the mean free path. For smaller sizes there is a region of uncertainty until the regime of the "classical" Langevin equation [-Eq. (2)] is reached for particles too small to "see" an assembly of moving molecules as a viscous fluid, but large enough not to induce appreciable dipoles on neighboring molecules. Loeb suggests I

e N < (7 X l O - a ) / E .

(8a)

The maximum field, short of breakdown, is about 30 kV/cm = 100 esu/cm, so that the minimum value of charge at which Stokes' Law can become suspect is about 7 X 10-8 esu, which is, approximately, 15,000 electron charges. It must be borne in mind, however, that this value is inversely proportional to the field and rises, e.g., to 4.5 X l0 b electron charges at 10~ V/cm. The manner in which such charges are acquired will be discussed below. Suffice it to say that, under charging conditions permitting particles long times at field strengths close to breakdown conditions, particles of a few microns diameter can attain such charges. The Reynolds Number, computed in the Newton's Law regime, is again independent of radius for a given charge ,

=

'

(9)

712

ELECTRICAL

PROPERTIES

by Eq. (7). On taking the lowest acceptable Re as 700, the limiting charge becomes

eN > 88

2 0.22~rT(~/Ep) - 2/E;

(9a)

for air at STP at the maximum (breakdown) field, the minimum charge for Newton's Law to apply is therefore about 2 X 10-2 esu, which corresponds to approximately 5 X 107 electron charges. The smallest particle that can acquire such a charge (by charging to equilibrium in a breakdown field) is one of about 80 ~. By setting the limits of Re at 3 and 700, errors have been kept to within about 20%. From the academic point of view it is perhaps more satisfying to extend both limits to meet at Re 55 and leave the discussion there. This will always yield an order-of-magnitude estimate of mobility. When more accurate values are required for practical work, one of the empirical laws for the transition region 3 ~ Re ~ l0 g must be used. A relatively simple one, which is, in fact, the best straight line on the log-log graph of Fig. 1, is 2F/pv2(Tra2) = 18.5/Re ~ (10) This gives

k = v / / E - ~ O.121(Ne)O.715//a(Ep)~

(11)

w h e n a unidirectional field is applied to a flame, or other region in which particles and small ions coexist (and mobility is a significant quantity only in presence of a field), it will be shown that the field distribution set up under the effect of space charge favors continued charging along the trajectories of the particles. Since the charge so acquired is a function of particle diameter, a second function enters the dependence of mobility on particle size, so that the results are quite different from those in the case of a fixed particle charge. It therefore becomes necessary to investigate the form of this function. Charging of Particles a n d Droplets When particulate matter coexists with ions, it can acquire charge by attachment of ions or by simple charge transfer. In the absence of an applied field, the only forces acting are due to fields surrounding the charge carriers. Thus, a particle having collected charges of one kind will increasingly tend to repel similar charges. In the presence of both positive and negative ions, it will favor acceptance of charges of the opposite kind and achieve some neutralization in this manner. If the ion cloud is unipolar, however, the particle cannot normally lose charge until the field at its surface is large enough to cause breakdown. While it will repel slow ions with in-

OF FLAMES

creasing success, the Maxwellian velocity distribution in a gas ensures that there will always be some ions with sufficient energy to reach the surface. Their fraction will decrease with increasing particle charge so that the rate of charging decreases with time, even though it remains finite until the particle can discharge in some other manner. The variation with time t of the charge so attained is given 6 by

Ne = (akT/e) In [-1 ~- (~acne~t/kT)-],

(12)

where T is temperature, k the Boltzmann constant, and c the rms velocity. This "diffusion charging" can lead to the acquisition of a net negative charge by particles in flame regions where the negative charge carrier is a free electron and, particularly, where the electron temperature exceeds the equilibrium value. Since the velocity of the negative-charge carrier exceeds that of the positive one, under these circumstances, the rate of charging due to it exceeds that of positive-charge supply. Eventually, an equilibrium state must be attained and this happens when the field, due to the excess negative charge on the particle, so accelerates positive ions and slows down electrons that the two rates of charge supply become equal. This mechanism will not bc pursued further here because its theory is analogous to that of the Langmuir probe (see, e.g., Ref. 8). Work on diffusion charging in intense electric fields7 has shown that a higher rate of charge acquisition than is obtained under zero field conditions rEq. (12)-] is predictable theoretically; however, it will be shown that "diffusion" charging generally is unimportant by comparison with another type of charge acquisition, under these conditions. This mechanism in the presence of an applied field is quite different and more relevant here because the system with which we are concerned involves a flame between electrodes at a high potential difference so that unipolar ions exist in each space between flame and electrode and any particles drift through such regions. The concept of mobility, of course, becomes rather pointless unless a field is applied. In a uniform field, lines of force crowd into any conducting particle inducing a distribution of charge across its surface. The effect for a sphere can be shown 9,s to be equivalent to that which would be due to two "image charges" within the sphere, acting as a dipole of strength E0a3 (where E0 = field in absence of particle). Ions travelling along lines of force onto the surface will gradually tend to neutralize this effect, until an equilibrium charge is collected which allows lines of force to by-pass the particle. The equi-

ELECTRICAL

CONTROL OF DROPLETS

713

AND PARTICLES

IL Im b-

IP ~=

D'

ImL

B, PARTIALLY CHARGED

A, UNCHARGED

C, FULLY CHARGEO

FIG. 2. Field distribution around a conducting sphere. librium charge is thus a function of the unperturbed field intensity. Figure 2 shows the configuration of lines of force under three conditions: (a) uncharged, (b) partially charged, and (c) fully charged. At the surface of the particle E = [-3E0 cos 0 - - (Ne/a2)].

(13)

At 0 = 00, E = 0, i.e., cos 80 = Ne/3Eoa 2.

(14)

The total flux J (the product of field and area perpendicular to it) entering the particle between 0 = 0 and 0 = 80 is given by the surface integral J =

3Eo cos 0 - - (Ne/a~)]2vd sin 0 d~

(15)

i,e.~ J = 37ra2E0[1- (Ne/3Eoa2)'] ~.

(15a)

The flux originates from an area A in the undisturbed region in which there is a concentration of ions n~, of mobility kl. Ions which enter this area will terminate on the particle. This presupposes that the relevant dimensions are large in comparison with the mean free path so that, irrespective of individual ion inertia, ion trajectories follow lines of force. This assumption is consistent with the size of particles considered in this regime. Since ion mobility is always much greater than that of the particles, we can neglect the particle velocity by comparison. Thus, the current flowing to one particle at any time t is given by

kiAEon,~ = kJn~e = d(Ne)/dt

(16)

714

ELECTRICAL PROPERTIES OF FLAMES

i.e.,

d ( N e ) / d t = 3ra ~(lc,niEoe)[-1 -- (Ne/3Eoa2)]2 = 3~ra2ji[1- (Ne/3Eoa2)] 2,

(16a)

where jl is the current density due to the ions alone. In the equilibrium condition 00 = 0, i.e., no more current flows to the particle. Thus, setting cos 00 = 1 in Eq. (14} gives the equilibrium charge as (Ne) -~ (Ne)~ = 3Eoa ~.

(17)

Setting 3Eoa 2 = (Ne)l in Eq. (16a) d (We)/dt = 3~ra2j~{1 -- [Ne/(Ne)l-]}2

_- 3~a2j~(1 _ f)2,

(16b)

where f = ( N e ) / ( N e ) l , i.e., the fraction of the equilibrium charge attained at time t. For a nonconducting particle of dielectric constant K, the above results (both the equilibrium charge and the rate of charging) must be multiplied by K / ( K + 2). Most of the materials likely to be present as particulate matter either are conductors or may be considered conducting by comparison with the surrounding gas, at the ion densities and field strengths involved. Although the discussion will therefore continue to be based on K--+ ~ , the correction factor (if required) is a simple constant throughout. This mechanism of charging (sometimes termed "bombardment" charging) may now be compared with "diffusion" charging. Although the equilibrium charge due to the latter is potentially higher, as discussed, it takes so much longer to attain that the former is always the dominant mechanism at field strengths relevant here._ Typical "diffusion-acquired" charges for particles of 1 and 10 g radius have been given 6 as 110 and 1,150 electron charges, respectively, after 0.1 sec, 150 and 1,900 after 1 sec, and 190 and 2,300 after 10 sec. At field strengths approaching breakdown, equilibrium "bombardment" charges for the same two particle sizes are approximately 6 X 10 3 and 6 X I0 5 electron charges, while times of flight to electrodes several centimeters away are of the order of milliseconds or less. Hence, even at much smaller fields, "bombardment" charging alone need be considered, as will become apparent from the rates of approach to equilibrium, calculated below.

Particles of Variable Charge The equilibrium particle charge is thus proportional to the local field strength, for which we can now revert to symbol E, without the subscript. If the latter were to decrease along the

trajectory, the particle could not lose charge unless it found itself in the presence of charges of the opposite kind. In fact, however, Gauss' Law ensures that the field strength must continuously increase with distance in the presence of ions of the kind which are capable of increasing the particle charge further. In one dimension, x, ( d E ) / ( d x ) is 4~r times the charge density. Thus in the space between an ion source (such as a flame) and an electrode, a particle will tend to acquire more and more of the ion charges and its (Ne) will continuously increase with distance. In what follows, we shall consider first the case of particles which acquire their equilibrium charge at every point in the field. This represents the upper limit idealization--that of full charging-complementary to that of a fixed small charge. It is not necessarily a valid approximation in any real system because the time spent in each region may not allow the particle to pick up a sufficient number of charges there, or because a sufficient number of ions is not available in each particle's vicinity. However, it will be shown later that, although this approximation can never be exactly fulfilled, it is convenient and quite adequate for virtually all flames. In order to avoid much complicated mathematics, this will be demonstrated only after the laws relevant to Such an ideal case have been established. (Ne) now becomes equal to (3Ea 2) and the equations for particle velocity [Eqs. (6) and (7)] become (18)

v = E e N / 6 r ~ a = E2a/2v~7

in the Stokes' regime and = r EeN

'

]o.5__ [

LO.22~a~pJ

3 ]o.5 E _ 2.085E

Lo.~Y~J po.~

p0.5

(19) in that covered by Newton's Law. For particles freely acquiring charge as they progress, the mobility equations are thus quite different now. It is in the Stokes' regime that a "mobility dependent on field strength" appears; k = Ea/2~r~?. (18a) In the Newton regime, we not only have a true mobility but this is independent of radius and becomes a constant at any one gas density. For air at STP its value is 58 cm2 sec-1 esu volt-1 and this may be compared with values typical of small ions under similar conditions, which are about 103. It is remarkable that under these conditions large particles (the regime will be shown to extend to particles several millimeters in radius) show velocities as great as ~ of those of molecular ions, regardless of their size, provided they can always attain their equilibrium charge.

ELECTRICAL

CONTROL OF DROPLETS

Substituting from Eqs. (18) and (19), Reynolds Numbers in the two cases are Re = p (Ea)2/lr. 2

(20)

in the Stokes case, and Re = (4.16p~

(Ea)

(21)

for the Newton regime. The relationship is therefore most readily expressed in terms of (Ea). For air at STP, this product is approximately 1.55 X 10-~, 6.64 X 10-2, and 0.85 esu at Re 3, 55, and 700, respectively, using Eq. (20) for the first result, Eq. (21) for the last, and either for the middle one (the point of intersection when the two regimes are interpolated to meet). The upper limit of the Newton's Law regime occurs at about Re 2 X 105, where Ea becomes 242 esu. At the breakdown field of 100 esu/cm, this would represent a particle of a = 2.42 era. This is well beyond the range of conceivable possibilities. If trajectories (as distinct from mobilities) are calculated, gravitational forces must be taken into account long before this point. The calculation is only intended to show that the conclusions drawn for the Newton's Law regime will cover the largest particles encountered in practice. These conclusions are summarized, in terms of mobility, in Fig. 3. The generalization cannot be taken much beyond this point. At low field strengths, up to a few hundred volts/cm, Stokes' Law will cover

AND

715

PARTICLES

most of the particle and droplet sizes of interest. At high field strengths, up to breakdown, particles of more than a very few microns radius must be considered in terms of Newton's Law or some transition formula. Again, using Eq. (10), for practical purposes, the mobility in the intermediate regime can be shown to be k = v / E = 0.27(Ea)~176

~

again a function of (Ea). For air at STP, this becomes approximately k = 73 (Ea) ~

i01 ! F..J

100

~J I0-] t= lO-2

-.1

10-3.

I0-5

~o-~4 1o13 i0-3

lOi2' IloL1'oO 5.5 x I0

(22a)

The full line in Fig. 3 between Re 3 and 700 is a plot of this relationship. At Re 55 it is just under half of the result obtained by extrapolation of Stokes' and Newton's Laws. This completes the treatment of the complementary upper limit of a charge-acquiring particle; viz., when charging is to equilibrium everywhere. We must now consider how closely this condition is likely to apply under flame conditions. The continuous rise of field intensity with distance having already been mentioned, it remains to be considered to what extent the residence time of particles and the ion concentration in every zone is sufficient for charging'to approach equilibrium. To this end we shall use again the fraction (f) the actual particle charge is of the equilibrium value--i.e., particle charge = (3a~Ef)--and consider its variation in x, the distance coordinate between an ion source, such

10 2

r~ ! o u~

(22)

102

103(Ea) e.s.u, cm

2 X 105 (Re)

Fro.F3. Mobility vs field X radius, and vs Reynolds Number.

716

ELECTRICAL

PROPERTIES

as a plane flame, and either electrode. An example of a practical system which approximates to the theoretical mode] is the fiat counter-flow diffusion flame. 1~In this, the initial flow direction is perpendicular to the plane of the ion source, but, in general, the field-induced velocity of the charged particles, at the field strengths envisaged in the electrode spaces, makes the flow configuration and flow velocities unimportant for flames of the low combustion intensities considered here. The particle mobility kp, will be decreased in consequence to k~ = fkp./~l

and

kp -~ f~

(23)

in the Stokes and the Newton regimes, respectively. We shall consider the Newton regime first, since the greatest deviations of f from unity must occur there. This is because particles travel faster and therefore have less time to be charged to their equilibrium value as this increases by the passage of the particle through the ascending field. There are two parameters which do not vary with x. First, the flow rate of particles per unit area G must be conserved. Thus

so that the number of particles per unit volume, (24)

Second, the total current density jT made up of the current densities due to ion flow and particle flow (ji -]- j~) must be conserved. It follows that d j , / d x = -- djp/dx

(25)

;p = G.3a2Ef

(26a)

;i = nikiEe,

(26b)

and, since and that ji = jr

-

-

3Ga2Ef

(27)

and n~e = ( j r -

3Ga2Ef)/k,E.

(28)

All the relevant variables in x have now been expressed in terms of two: E and f. These are determined in terms of two differential equations. The first of these is Gauss' Law, which here becomes d E / d x = 4qr(n~e -]-- 3a2Efn~,).

takes on the form dE/dx = A/E-

B f - ] - Cf ~

(29a)

where A = 4qrj~./ki, B = 12vGa~/kl, and C = 12rra2G/kpl. The second equation required must express the variation of f with x in terms of the rate of charge attachment to particles, which has been considered above. In the section on particle charging, it was shown that each particle has a "catchment area" within which it collects all the ions that approach it and that this area decreases with the charge already accumulated. At fraction f of the equilibrium charge, the area is 31ra2(1 -- f)2 as can be seen from Eq. (16b), which gives the rate of charge acquisition by a single particle. It follows that, if we consider an element containing n~ particles per unit volume --djddx

= j,n~.3~ra2(1 -- f)2 = dj~,/dx.

(30)

Substituting from Eqs. (26), (24), and (27) this takes the form (d/dx) ( E l ) = r ( P / E f ~

-- Qf~ 5~(1 - f)~,

(30a) where P = ~jT/k~l

G - - const = npkp~f~

n~ = GkpiEf ~

O F F]_AMES

and

Q = 3~'Ga~/kpl.

Solving Eqs. (29a) and (30a) as simultaneous differential equations gives the distributions of E and f in x. Fortunately, it proves unnecessary to consider the perfectly general case. At first sight, there appear two quite separate reasons why f should fall short of unity. The first, which would apply even to a single particle, is that the particle moving too fast through a rapidly increasing E may never have time to charge up to the ever-increasing equilibrium value. The second presupposes a particle concentration so high that there are not enough ions to go around, so that jl falls to very low values near the electrode. It will be shown that in normal combustion systems only the former is limiting. As the particle flux G tends to zero, B and C in Eq. (29a) become negligible as compared with A--i.e., the space charge due to particles becomes small by comparison with that due to ions. Under these circumstances Eq. (29a) can be integrated directly to yield the field distribution as E 2 -- Eo 2 = 2 A x .

(31)

Here E0is the value of E at x = 0. In Eq. (30a), Q similarly tends to zero and that equation becomes

(29)

On substituting from Eqs. (28) and (24) this

multiplying by E and substituting from Eq.

ELECTRICAL CONTROL OF DROPLETS AND PARTICLES (31) gives (2Ax-]- Eo~)(df/dx) = ~-P(1 - - f ) 2 / f ~

Af. (32)

If f is close to unity, the first term on the righthand side tends to zero and f falls rapidly, until (df/dx) = O. If, on the other hand, f is less than its value at (df/dx) = O, ( d f / d x ) ) 0 and f increases. The implication, physically, is that if f for some reason becomes too small, the particles slow down so much that the charging rate gains on the rise in equilibrium charge, so that under all conditions f tends to a constant value given by (df/dx) = O. Introducing this condition into Eq. (32) gives the equation for the steady-state value of f as P(1 -- f)2 =

Aft,

(33)

or, putting P / A in terms of its constituent parameters (ki/4kpl) (1 -- f)2 = f]. Now k~/kpl -

(33a)

15 and the solution of

3.75=f~/(1--f)s

is f =

0.63.

Thus, so long as the particle concentration is sufficiently smaller than that of ions, f will tend to 0.63 regardless of its initial value and kp--) 0.63~

-- 0.8]~1

(34)

in the Newton regime. I t remains to be shown that the underlying assumption of a negligible contribution to space charge by the particles does not restrict the above result to special cases only. The condition that in Eq. (29) hie ) ~ 3aeEfnp reduces to j r ~ 38 Ga2E on expressing all variables in terms of E and f, and substituting f = 0.63 and (k~/kpl) = 15. Taking the worst case, this will be most difficult to satisfy at the highest field intensity. The greatest value is the breakdown field (3 X 104 V/cm) and the current under conditions which maximize practical effects 5 is approximately 2.5 X 10-4 A. This gives that (Ga s ) must be much smaller than 200. Ga s is

717

proportional to the surface area of particles which flow per unit time per unit area. If we draw the limit at Ga 2 = 20--i.e., the space charge due to particles not to exceed 10% of the total, the condition to be satisfied is that the flow of particle surface area sec-1 cm-2 must not exceed 250 cm 2. This appears to be a very large amount. To assess its magnitude let us consider the massflow rates of particles likely to be encountered in practice. Taking again a rather extreme case, let the particles consist of all the carbon in a stoichiometric (C + 02) mixture and assume a burning velocity as high as 500 em/sec. The mass flow rate in such a case is approximately 0.25 g sec-' cm-2. In order to exceed a surface area of 250 cm 2 sec-1 the particle radius should have to be less than 15 g. This would take it outside the Newton regime. In fact, however, an appreciable particle space charge affects the result only very gradually. The assumption that it is negligible was made here more to permit an analytical solution than for any other reason. A numerical solution was also carried out because it was thought that exceptions could arise in the following special cases. I n the combustion of finely-powdered metald dusts in oxygen, if all the particles were to be influenced by fields, the burning velocity might perhaps be so high t h a t the mass-flow rate of particles would be too great for the assumptions to apply rigorously. Again, if the electrode were a long way from the flame the same failure of the above theory could occur because the breakdown-limited ion current might then be too small? Last, in the case of pure hydrogen flames in which chemi-ionization is negligible, the current could be limited by ion formation rather than by space charge considerations. (This presupposes that it is desired to introduce some extraneous particles deliberately for charging in the system and t h a t their introduction does not provide sufficient impurities for ionization rates in the flame to increase appreciably.) To determine deviations, a detailed numerical analysis was carried out, in which the space charge due to particles was not assumed to be small. Table I gives results for particle space charges up to 3.2 times as great as those due to ions and shows how gradually these affect the result. The second assumption on which the above

TABLE I Results in dense particle space charges Particle space charge as percentage of that of ions Percentage decrease in mobility due to particle space charge Percentage decrease in f, due to particle space charge

4 1.3 3

19

2.6 6.4

130 12.5 22.3

320 21.3 36.5

718

ELECTRICAL PROPERTIES OF FLAMES

theory was based--the neglect of QJ89by comparison with PIEr89in Eq. (30a)--is less critical than that concerning space charge. On substituting the above numerical values, the term neglected is only about 1/600 of that retained, so that the calculated first criterion automatically includes the second. An analogous calculation could now be carried out for the Stokes regime. However, it has been shown that the particle velocity is always less in the Stokes than in the Newton regime for a given field and particle size. The charging time is accordingly greater and f is closer to unity. If the steady-state f in the Newton regime is deemed a sufficient approximation to full charging, we need therefore not carry out this calculation, because the approximation will be closer in the Stokes domain. Summarizing, the application of the theory of equilibrium charging to flames is never exact, because the (infinite) time required for full charging is never available to moving particles. A detailed analysis in the Newton regime reveals, however, that particles attain a charge everywhere proportional to the local hypothetical maximum. In that regime, the proportion is about 63% of the "infinite residence time charge" and the mobility is approximately 80% of the fully charged one. In the Stokes regime, the charge is even greater and always lies between 63% and 100% of the maximum. Particles of Low W o r k - F u n c t i o n The effect on the preceding theory of charge acquisition by electron emission, in the case of particles of low work-function subjected to flame temperatures, has been considered in detail. It constitutes a rather special case, because it is confined to only a few substances (metals, their oxides, and carbon) and because the effect is unsymmetrical in that only positive charge can be acquired in this manner. The main feature which sets it apart, however, is that, even for the lowest work-functions, the effect can be appreciable only in the zones of high temperature and the location of these is a property of the flame system, totally unrelated to the distribution of field strength. In the absence of any field, the emission current density j~ (e.g., from a filament forming part of a circuit) is given by Richardson's equation je = dT2exp

(--r

(35)

where A ---- a constant, r = the work function, and other symbols are as previously defined. Even in the absence of an externally applied field, however, this applies to a particle only initially, until the field due to the reaidual charge

on it becomes large enough to begin to retard emission. Under these conditions, the variation with time of the positive charge on the particle in vacuum is given by

d(Ne)Jdt = 4~ra~AT2exp (--dpe/kT)

exp

(--Ne2/akT), (36)

which gives

Nee = ak_Te In \(4~raer-tkT-bkT)

(36a)

~"= AT2exp (--che/kT),

(37)

where a function of temperature alone, for one material. The effect of equilibrium negative space charge, surrounding the particle, on the above theory has been discussed in Refs. 11-14. When an emitting particle traverses a region of an applied field under simultaneous ion bombardment, the theory must take into account both this bombardment and the variation of the field across the particle surface [-see Figs. 2(a)-(c)-], which varies with the charge acquired. This case will not he presented here, partly because it would make the paper unacceptably long, but mainly because it is difficult to make it correspond to a practical case within the frame of reference of this paper; under circumstances such that temperature does not fall to values at which emission can be neglected in the electrode spaces, the electrodes must themselves be hot and thermionically emitting as soon as they become coated with the particles, if not before. The process remains a significant one in the high-temperature zone of the flame. The field there is always less than elsewhere in the system but it (and hence bombardment charging) may be absolutely very small there, or it may not, depending on whether the applied potential is below or above the saturation value. 5 Thus, if the field is small, the effect of electron emission of predisposing particles towards acquiring positive charge may be dominant and the majority may then travel towards the cathode. Apart from thus being able to alter the proportion of particles the two electrodes would otherwise collect, electron emission has very little effect of mobility in the electrode spaces--which is the topic of the paper. Whether it competes against bombardment charging in the flame or whether it reinforces it (i.e., whether the particles emerge from the flame with a deficit or an excess over the local equilibrium charge) as soon as temperature falls, the rapidity of bombardment charging together with the rapid rise in the equilibrium charge ensures a speedy return to the theory derived.

ELECTRICAL CONTROL OF DROPLETS AND PARTICLES

Nomenclature a c

particle radius rms velocity of gas molecules d particle diameter e electron charge fraction of equilibrium particle charge f emission current density L j~ ion current density particle current density Jp total current density jr k Boltzmann constant, mobility k~ ion mobility k~ particle mobility ion concentration ni n~ number of particles per unit volume q ionic charge t time coordinate Y particle velocity X, distance coordinate A area in a region of undisturbed space charge, Richardson equation constant A, B, C derived parameters defined in text dielectric constant of the gas at 0~ Do and 1 atm E electric field strength electric field strength in absence of Eo particles F force G flux of particles (per unit area) total ion flux J particle dielectric constant K M molecular weight of gas Mi molecular weight of ion number of electron charges per particle N P,Q derived parameters Re Reynolds Number T absolute temperature gas viscosity semi angle subtended at the center of a 0 particle by a spherical cap angle defining the locus of zero surface 00 field at the particle

k kl p p0

719

gas molecule mean free path ion mean free path density of gas density of gas at 0~ and 1 arm derived temperature-dependent parameter thermionic work function

q-

Subscripts 1 e

full charge emission REFERENCES

1. LOEB, L. B.: Basic Processes of Gaseous Electronics, Chap. 1, University of California Press, 1961. 2. PAYNE, K. G. AND WEINBERO, F. J. : Proc. Roy. Soc. (London) A250, 1316 (1959). 3. PLACE, E. R. AND WEINBERG, F. J.: Aero Res.

Council, C. F., April, 1964. 4. GUGAN, K. AND WEINRERG, F. J.: In prep-

aration. 5. LAWTON, J. AND WEINBERG, F. J. : Proc. Roy.

Soc. (London) A277, 468 (1964). 6. WHITE, H. J.: Trans. AIEE 70, 1187 (1951). 7. MURPHr, A. T., ADLER, F. T., AND PENNY, G. W.: Trans. AIEE 78, 318 (1959). 8. LOEB, L. B.: Basic Processes of Gaseous Electronics, Chap. 4, University of California Press,

1961. 9. JEANS, J.: Mathematical Theory of Electricity

10.

11. 12. 13. 14.

and Magnetism, 5th ed., p. 192, Cambridge University Press, 1948. PANDYA, W. P. AND WEINBERG, F. J.: Proc. Roy. Soc. (London) A279, 544 (1964). ROSEN, G.: Phys. Fluids 5, 737 (1962). SODRA, M. S.: Brit. J. Appl. Phys. 1r 172 (1963). SODHA, M. S. ET AL.: Brit. J. Appl. Phys. 1r 916 (1963). Soo, L.: J. Appl. Phys. 3~4, 1689 (1963).

COMMENTS Prof. S. L. Soo (University of Illinois): To this comprehensive study we might add the effect of concentration of a particle cloud on the drag coefficient and on the dipole force for particles of given polarizability. The first effect limits the use of single-particle drag coefficient in the case of a dense cloud where the boundary-layer thicknesses merge (ScHLICHTING, H.: Boundary Layer Theory, p. 188, McGraw-Hill 1960), and their wakes disappear (Soo, S. L.: GasSolid Flow, Proc. of Symposium on Single and MultiComponent Flow Processes, Rutgers Engineering Centennial, Rutgers University, May 1, 1964).

The second effect gives the force acting on a particle in the absence of magnetic field as: qE + (0/0r)(p.]~), where r is the position vector, p the dipole moment. When applied to a cluster of charged particles with spherical symmetry, the forces due to self-field have the ratio Fa/E, = 41-(k -- 1)/(k -b 2)]pp/~p, where Fa and F, are forces due to dipole and electrostatic repulsion, respectively, k is the dielectric constant, p~ the density of cloud, and ~p is that of the particulate material; pp/~p is also [-1 -- (fraction void)] of the cloud [Soo, S. L.: Ind. Eng. Chem.,

720

ELECTRICAL PROPERTIES OF FLAMES

Fundamentals ~, 75 (1964)J. The dipole force might account for agglomeration of a cluster.

the ratio of Fa/F, (using Soo's nomenclature) is as small as 1/350.

Dr. F. J. Weinberg: We would like to express our thanks for the suggestion of extensions to our theory, which may be useful in special cases. I might emphasize that these would have to be rather unusual cases, because we axe not normally concerned with arbitrarily dense clouds (being tied by stoiehiometry) or with strongly divergent fields. To illustrate this, I have worked out the case of all the sulfur particles in stoichiometric, S + 0~ ~ SOs, hardly a very common system, but one in which the dielectric constant is as high as 4, while the density of the solid is much smaller than for a metal. I t transpires that, even there, the separation between particles is about 28 times their radius and

Dr. F. Briffa (Shell Research Ltd., Thornton): It seems to me that dust particles are not always spherical, so that it would be necessary to attribute to them an appropriate "shape factor" for estimating the Reynolds Number when calculating the drag coefficient. Further, in pratieal systems, turbulence (especially small-scale turbulence) may also be expected to influence the value of drag coefficients, so that the meaning of a Reynolds Number, based on the velocity of the particle relative to that of the ambient stream, is not clear. However, it is conceded that to allow for the above effects may be virtually impossible.