A model for the electric charging process in fuel filtration

A Model for the Electric Charging Process in Fuel Filtration PETER W. HUBER AND AIN A. SONIN Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received July 23, 1974; accepted April 22, 1975 The charge generation process in fuel filtration is analyzed in terms of an electrokinetic model and theoretical results are obtained for the streaming current and potential in terms of fluid properties, bulk filter characteristics, flow conditions, and a fixed charge density which is a physicochemical property of the filter/fluid interface. Experiments in which doped nheptane was passed through Millipore filters support the theoretical model. 1. I N T R O D U C T I O N

Electric charging is a problem of practical importance in the processing and handling of gasolines and other hydrocarbon liquids that have a very low electrical conductivity. Sufficient electric charges can be generated in flows through pipes and filters to produce spontaneous electrical discharges in the ambient atmosphere, resulting in ignitions and explosions (1-5). The problem is particularly severe in filtration, where very large solid/fuel interfaces are involved. An understanding of the charging characteristics of flows through filters is essential for evaluating hazards in fuel handling and processing operations. Previous work on this subject has been largely empirical (6-11). Apparently, the only scheme that has been put forward for predicting the charging characteristics in filtration is the one proposed by Gavis and Wagner (7). Using dimensional arguments and making several simplifying assumptions, these authors proposed a formula that appeared to correlate their own experimental data on the current generated when heptane doped with an ionizing additive was passed through various Millipore filters. They suggested that their correlating formula should apply to any hydrocarbon flow through any filter, provided

only that a single dimensionless correlating parameter falls in the same range as in their data. However, there is reason to question this claim since it is based, among other things, on the assumption that the charging current is independent of any physicochemical property of the filter or solid/fuel interface. This assumption cannot be generally valid if the charging process is electrokinetic, as has been commonly assumed. In the present paper we propose a relatively simple electrokinetic model for the charging mechanism in liquid hydrocarbon filtration. It resembles those used for flows of aqueous saline solutions through charged membranes (12-14), but differs in several key respects from the model suggested by Gavis and Wagner. Based on this model, a theoretical solution is derived for the charging process when certain simplifying assumptions apply, and the dependence of charging currents and potentials on fluid and filter properties and flow parameters is established. Our analysis takes into account some effects that are perhaps obvious but which appear to have received little attention in past works. First, a voltage may be generated across a filter in addition to a current being generated through it, and what must be established is a

415 Copyright (~ 1975 by Academic Press, Inc. All rights of reproduction in any form reserved.

Journal of Colloid and Interface Science, VoI. 52, No. 3, September 1975

416

HUBER AND SONIN

current-voltage characteristic rather than a simple charging current. Second, the placement of the measuring electrodes in an electric charging experiment must in general be carefully taken into account in interpreting the charging phenomena due to the filter. For example, if current flows between electrodes that are not contiguous with the filter surfaces, the constraint of zero voltage between the electrodes generally is not the same as zero voltage across the filter. The particular theOretical solution we present here is valid only under a limited, but well-defined, range of operating conditions. The range covers some of the situations that occur in practical filtration conditions. But, more important than that, our particular solution is in closed analytic form and can be compared conveniently with experiments to establish whether or not the basic model is sound. Once the basic model is confirmed one can, in principle, return to the governing equations and obtain mathematical solutions for whatever operating conditions are of interest. Experimental data are presented to support our model for the charging process. 2. MODEL AND ASSUMPTIONS The mechanism of electric charging in steady-state filtration is assumed to be the following. The filter is viewed as a porous structure carrying a fixed charge on its solid/ fluid interfaces. It is well established that most solid surfaces acquire a net charge when in contact with a solution containing ionic constituents (15). Whether the fixed charge actually resides on the solid/fluid interface or within the solid matrix of the filter does not affect the theory we present below. We assume further that the hydrocarbon fluid as a whole (including dissolved additives or impurities) behaves as a very weak electrolyte. Upstream of the filter, the fluid is electrically neutral and the positive and negative ions have equal concentrations. As the fluid enters the filter pores, the ion concentrations are thrown into imbalance because a net charge tends to be set

up in the fluid to neutralize the fixed charges associated with the solid surfaces. This charge in the fluid is convected downstream within the pores, giving rise to a streaming current. If a potential is allowed to develop across the filter as charge is convected into the downstream compartment, current will also be passed through the filter by conduction (i.e., migration due to an electric field). The conduction current can flow via the fluid within the pores and via the solid matrix of the filter if that matrix has a significant electronic conductivity and if the solid/fluid interfaces allow ionic discharge. Thus, the net effect is that the filter behaves as a flow-driven current generator in parallel with an internal resistance. In this model the property of the filter that gives rise to the charging phenomenon is the fixed charge on the solid/fluid interface. It is the fixed charge that causes a countercharge within the fluid in the pores and this, in turn, gives rise to a current when flow takes place. A filter without a fixed charge would not cause charging, and in general it is expected that the charging characteristics of any particular filter should depend on the fixed charge associated with its solid/fluid interface. Normally the filter characteristics are inferred from measurements made with respect to externally placed electrodes, as shown schematically in Fig. 1. Often the electrodes are remote from the filter faces. It is important to note that the net current generated by the filter will pass through the regions of fluid between the filter and the electrodes. This will give rise to a potential distribution in those regions and, as a result, the potential recorded between the measuring electrodes in general will not indicate the potential difference across the filter alone. Hence, if the current-voltage characteristic of a filter is to be determined as a function of the flow rate and the other operating variables, care must be taken either to account for the effect of the potential distribution outside the filter or to ensure that its effects are negligible. From a practical viewpoint it is of course important to establish the entire currentvoltage relation for the filter/electrode system

Journal of Colloid and Interface Science, Vol. 52, No. 3, September 1975

ELECTRIC CHARGING IN FILTRATION TYPICAL PORE, RADIUS a LENGTH ~

FILTER, /POROSITY E

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(2)

/

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FIG. 1. Model for filter system. rather than just the current at zero applied potential (the streaming current), which is the quantity usually measured (1-11). The complete current-voltage characteristic is required if the time history of the charging of a receiving vessel capable of storing charge is to be predicted. Initially, the potential difference between the feed line and the receiving vessel usually will be zero and the current will be equal to the streaming current. However, as a result of the charge transport a potential difference will build up across the filter and the net current will be reduced by back conduction through the filter. In the limiting case where no charge leakage can occur via an external circuit, the potential difference will continue to build up until the forward convection current becomes exactly balanced by the back conduction through the filter and the net current through the filter is reduced to zero. We shall call this limiting open-circuit potential the streaming potential. To put our analysis in quantitative terms, we model the filter as a solid material of thickness h, traversed by numerous pores of radius a and length l (Fig. 1). A particular filter is characterized by a pore tortuosity r~ = 1/h, regarded here as an empirically known quantity (16), and a porosity e. An

effective pore radius a can in principle be determined from the filter's porosity, tortuosity, and hydraulic permeability (14). However, in our present theory, the charging characteristics are independent of the pore size. The fixed charge is assumed to be uniformly distributed within the filter and its density p, defined for later convenience as the amount of fixed charge within the filter per unit volume of pore space, is regarded as an empirically determinable property of any particular filter/fluid combination. Based on this physical model, the equations governing the charging process can be set down in quite general terms. The solution we derive below is based on some further simplifying assumptions, mostly relating to operating conditions, which greatly simplify the mathematics but still leave us with a solution that applies in a practical range of filtration conditions. These assumptions are: (1) The effective pore radius a is smaller than the local Debye length XD in a pore,

a/x~ < 1.

E1-1

(2) The filter thickness is large compared with the Debye length, h/X~ >> 1.

E2]

Journal of Colloid and Interface Science, Vol. 52,[No. 3, September 1975

418

HUBER AND SONIN

(3) The Peclet number, based on the flow speed u in a pore, the Debye length, and the ion diffusion coefficient D, is small compared with unity,

UXD/D<< 1.

[-3]

(4) The hydrocarbon fluid behaves like a very weak electrolyte, where ion concentrations are everywhere in local thermodynamic equilibrium with the dissociating constituent, whether the latter is the hydrocarbon itself or a dissolved additive. Equations ]-1~-[-33 imply that except for short lengths of order XD near the entrance and exit of a pore, the ion concentrations and the electric potential within a pore are to a good approximation constant over a pore cross section (14, 17) and can be taken as functions -of s ohl)/ (s being the axial distance down a pore, as indicated in Fig. 1). This simplifies ~the analysis immensely. It also leads to the result that the filter's charging characteristics turn out to be independent of the pore radius a, or of the dimensionless group a/XD. Furthermore, as a result of Eq. [2-] there must be a balance between the fixed charge in the filter and the charge in the fluid in the pores except in the regions near the pore entrances and exits. Eq. [31 ensures that the region containing space charge does not extend into the fluid beyond the filter boundaries by more than a distance of order XD, which by Eq. [21 is small compared with the filter thickness. Thus, in this particular case, the current in the bulk of the fluid phase outside the filter is carried by ohmic conduction and not by convection. Note that the ratio of the charge relaxation time to the approximate convection time h/u through the filter is equal to (uXD/D)(XD/h), which by assumptions (2) and (3) is a very small quantity in the case we shall consider. Another consequence of these assumptions is that the potential and concentration jumps across the pore mouths (1-I and II-2 in Fig. 1) are related by the Donnan equilibrium condition (15). This allows for a very simple treatment of the pore mouth regions, which other-

wise could present a relatively complex theoretical problem. Assumption (4) closes the problem mathematically and allows for a simple solution. Equation [-13 is satisfied in most fuel filtration situations, except those where the fuel is very heavily doped with an ionizing additive to increase its conductivity. A typical undoped hydrocarbon liquid has a very low conductivity, of order 10-12-10-1I f2-I m -1. The corresponding Debye length is of order 100-30 ~m, to be compared with a typical filter pore radius of 1-10 #m (18, 19). Even when the conductivity is raised by another order of magnitude by ionizing additives, the Debye length is still of order 10 ~m, i.e., not smaller than the pore size. Eq. [-2~ is also satisfied in many practical situations, where filter thicknesses range from something of the order of 1 mm upward. The assumption that limits the practical utility of our solution most severely is Eq. [-33. This inequality may be satisfied, for example, in the clay percolation columns used in refineries to remove surfactants (19), but in high speed aircraft fueling operations, where the charging resulting from filtration presents the greatest problem, the parameter uXD/D generally has a value large compared with unity, as large as 105 or l0 s (8). However, as we have said, one main purpose of the present work is to try to establish a model for the charging mechanism by comparing experimental data with a theoretical solution that is both simple and fairly rigorous in the chosen range of operating conditions. Once the basic model is confirmed, mathematical solutions can in principle be obtained for whatever operating conditions are of interest. Finally, little can be said about the fourth assumption, since virtually no data are available on the dissociation or recombination rates of the ionizing constituents in hydrocarbon fluids. However, previous work has suggested that rate coefficients do not influence charging results, even under conditions where ion concentrations are lower and flow speeds are higher than in the experiments described in this paper (7). We use assumption (4) as a

Journal of Colloid and Interface Science, Vol. 52, No. 3, September 1975

419

ELECTRIC CHARGING IN FILTRATION

working hypothesis, giving it some justification a posteriori by the agreement of our theory with experiments.

charged species in the fluid, / dc~\ j p = F ( E Zici)u - F I E Z i D i - - ] i \ i ds/

3. S O L U T I O N

d¢ -- F(Yl. Zi~ciKi) - . i ds

(a) Current-Voltage Characteristic of the Filter

Within the fluid phase, the local flux of ions of species i gives rise to a current density ji = FZ~(c~v -- DiVc~ -- Zic~iV¢)

[-4]

where F is Faradays constant; Z~ is the charge number of the species; ci is the species concentration in moles per unit volume ; v is the bulk fluid flow velocity; $ is the electric potential; and Di and Ki are the diffusion coefficient and mobility, respectively, of the charged species in the fluid phase. The latter are taken as constants, since we are dealing with solutions that are very dilute in ions. The first term on the right represents the convection of species i, the second diffusion, and the third migration in the electric field. Note that although Eq. [-4-] allows for charge numbers with magnitudes greater than unity, it is highly unlikely that any molecular-sized ionic species will have multiple charge in a hydrocarbon liquid at room temperature; such liquids have a relative permitivity no larger than about 4. When Eq. [-1] applies, both ¢ and the ion concentrations ci are virtually uniform over any pore cross section in the filter interior, region I - I I in Fig. 1. In this region, the axial component of Eq. [-4] can be integrated over a pore cross section to yield, after division by ira~,

[-63

Assumptions (1)-(3), outlined in the previous section, imply that in the interior of the filter there must be a balance between the charge in the fluid in the pores and the fixed charge associated with the solid parts of the filter, leading to the quasineutrality condition

F ( 2 Zici) + Zso = O,

[-7]

i

where o is the magnitude of fixed charge density in the filter, expressed here as the amount of fixed charge per unit volume of pore space; and Zz is the sign of the fixed charge, defined as + 1 when the fixed charge is positive and - 1 when it is negative. The equations are further simplified by assumption (4). If assumption (4) holds, the ion species concentrations are completely determined by the condition of thermodynamic equilibrium and the quasineutrality condition, and since p is taken to be constant in region I-II, the ion concentrations are also constant in that region. Hence, the diffusion term in Eq. [-6-] is zero and the electrical conductivity in the last term is constant. Eq. [-6"] simplifies to Y~

=

z~ou - .,(d¢/ds)

-

[-S]

where %

=

F ( E Zi2ciKi)p

[-9]

i

dci dO) j i = F Z i ciu -- D , - - -- Z,ciKi-ds ds

[-5-]

in which ji is the average axial current density of species i in a pore; u is the average flow speed in a pore; and s is distance measured along the pore axis. The average total current density jp in the axial direction in a pore is obtained by summing Eq. [-5-] over all the

is the electrical conductivity of the fluid inside a pore. The latter is in general different from the conductivity eo of the fluid outside, because the ion concentrations in the pore are not the same as outside. To derive a working expression for the pore fluid conductivity, we consider the simplest example of a binary monovalent electrolyte where the positive and negative ions have equal mobilities ~. The fluid conductivity is then related to the positive and negative ion

Journat of Colloid and Interface Science, Vol. 52. No. 3, September 197S

420

HUBER AND SONIN

concentration by

obtain = F(c+ + c_)K.

[10]

Outside the filter, beyond points 1 and 2, the positive and negative ions have equal concentrations co, governed by the concentration of the dissociating constituent in the solution. The conductivity there is

(j,) = - - j , .

Similarly, the superficial flow speed through the filter is

Inside a pore, c+ and c_ are unequal because of the fixed charge in the filter,

Tp

Thus, Eq. 1,83 can be expressed as EO'p d(~

Zfp

( c + ) , - (c_), = - -

F

[-13]

Solving Eqs. [12] and 1,13] for c+ and c_ inside a pore, and substituting into Eq. [-103 to obtain ~,, we find GIo O"o

where p

P~

2Fco

~o

~= ---

(j,) = - &p(u)

E12J

However, if chemical equilibrium is also maintained everywhere in the fluid, then we can also write (20) : c+c_ = Co2.

1-173

(u) = - - u .

1-11"]

o'o = 2FCoK.

El6]

Tp

[15]

is the ratio of the fixed ion concentration in the filter to the free ion concentration in the fluid outside the filter. Equation [-8] is written in terms of the current density and flow speed in a pore. The more interesting quantities from the point of view of the overall process are the superficial current density (j), defined as the current per unit area of filter, and the superficial flow speed (u), defined as the volume flow rate per unit area of filter. If the filter is traversed by n pores per unit cross-sectional area, then the current flowing through unit area of filter via the pores is ( j p ) = j , . r r a 2 n . Now n can be related to the filter porosity e (ratio of pore volume to total filter volume), since in our model, e = n . 7ra~l/h. Finally, since 1= r , h , we

Tp 2 d x

E183

where we have written d s = r f l x , x being the distance measured across the filter. Now current can be carried through a filter via the solid matrix as well as via the fluid in the pores. We assume that in the solid parts the local current density obeys Ohm's law: [-193

j = --~,V¢

where ~, is the conductivity of the solid material. If we model the solid parts of the filter as a network of conducting fibers having a tortuosity r~, we can proceed as before and derive the result that the superficial current density associated with conduction through the solid is (1 -- e)z. d e

(j~) =

. rs 2

1-20-]

dx

The total superficial current density in the filter as a whole, (j) = ( j , ) + (j~)

[213

is now obtained as

de

(j) = -

Z:p(u) -- ~7-dx

[-22]

where ~p

~I = - Tp 2

(1-

~)~o

-t-

[-233 Ts 2

is the effective conductivity of the filter as a whole. In deriving Eqs. [-22] and 1,23] we have

Journal of Colloid and Interface Science, Vol. 52, No. 3, September 1975

ELECTRIC CHARGING IN FILTRATION

made the implicit assumption that at any given x, the potential in the solid parts of the filter is the same as that in the fluid, or at least that the difference in potential between the two phases, if it exists, remains constant with x. Equation [223 applies inside the filter, between points I and II. In the case of our plane filter, charge and mass conservation require that (j} and (u) be constant with x, and Eq. [22-] can be integrated across the filter to yield ,.I(¢I~

-

¢~)


[243 h

Assumptions (3) and (4) are needed to make the transitions across the pore mouths into the adjacent bounding fluid. If assumption (3) applies, it can be shown by an order of magnitude analysis of the terms in Eq. [4-] that within the transition regions 1-I and II-2, which are of order XD in length, the current and convection terms are negligible compared with the diffusion terms (the quantity uXD/D actually refers to the ratio of the convection term to the diffusion term, but in our present problem the current term never exceeds the convection term). Hence, the diffusion term is balanced by the migration term alone in the transition region. This leads to the Donnan equilibrium relation between the electrical potential and the ion concentrations (15), i.e., the potential jump across a pore month depends only on the change in concentrations. In our present case, where equilibrium applies and where the fixed charge is the same at points I and II, the concentration change between II and 2 is exactly equal and opposite to that between 1 and I. It follows that the potential jumps across the pore entrance and exit cancel each other and do not contribute to the overall potential rise across the filter, thus, ~ii

-

¢~

= 6~

-

[25]

¢~

and Eq. [-24] can be rewritten ~f(42

<.i> =

-

-

z~p(~)

h

4~)

•

[26]

421

Equation [241 expresses the current-voltage relation for the filter. Clearly, the filter acts as a current generator in parallel with an internal resistance. The strength of the generator depends on the superficial flow speed (u) and on the fixed charge density Zfp of the filter, while its internal resistance is controlled by the properties of the fluid (~o, K) as well as the thickness h and properties (Zip, E, rv, Ts) of the filter. The new quantities that are required in the application of our model are the fixed charge density Zfp and the tortuosities rp and r, of the pores and the solid matrix of the filter. The fixed charge density is a physicochemical property of a particular solid/liquid combination, and in principle, can be determined empirically by measuring the current generated in filtration when the filter is sandwiched between fine mesh electrodes that are shorted so that ¢~ - ¢1 = 0. The tortuosities refer to the interior architecture of the filter. In principle, these too can be determined empirically. Once the fixed charge density has been established, the tortuosity rv of the pores can be measured most conveniently by permeating the filter with a fluid which has so high a conductivity that % >> z~ and ~ << 1, in which case ~v ~ ~o (Eq. [-14]) and ~s ~'~ e~o/%2 (Eq. [23]). It follows from Eq. [-26] that the electrical resistance of the filter (without flow) will be a factor r2/e higher than that of a volume of the fluid with the same dimensions as the filter. Thus, rv can be readily determined. This technique of determining the tortuosity is basically the same as that advocated by Wyllie (21, 22), except that Wyllie obtained the ratio rp/e rather than rp2/e, using our notation (his definition of tortuosity was the square of ours). The difference apparently arises because he did not account for the tortuosity in the relation between the average current in a pore and the superficial current (see Eq. [-16"1). The tortuosity rs of the solid matrix can be determined by measuring the resistance of the filter in dry air, for example, or some other nonconducting

Journal of Colloid and Interface Science, Vol. 52, No. 3, September 1975

422

HUBER AND SONIN

fluid. The ratio of that resistance to the resistance of a solid chunk of filter material occupying the same space as the filter should be r~V(1 - e). Note that although Eq. [26~ was derived for a plane filter, in which the average current and flow (averaged, that is, over the tortuous path taken through the porous interior of the filter) were in one direction, a generalization can be made quite readily. The filter as a whole can be viewed as a homogeneous region in which the local current density is given by

the x-direction, then the same current density also passes through the regions of fluid outside the filter, but between the electrodes. In our particular case, the charge relaxation length is small compared with the filter thickness. This means that there is no net charge in the fluid outside the filter, and hence, outside the filter the current is carried entirely by ohmic migration in an electric field, not by convection of charge. Hence a current generated by the flow through the filter causes a linear Ohmic potential drop in the fluid outside the filter. The potential rise q~2 - ¢~ across the filter (j) = --Zsp(u) -- ~,-V4~ 11273 itself is thus related to the potential rise A4 and in which continuity of charge and mass between the electrodes by require that V.(i)

0

11281

V. ( u ) = 0.

[-291

=

A complete specification of the general problem also requires an equation of motion for the superficial velocity, such as Darcy's law, and a proper specification of the boundary conditions. Note also that although our derivation is based on a laminar flow equation, Eq. [-47, it is quite clear that our result can remain valid even when eddying or turbulent flow takes place inside the filter, since the diffusion term, which is the only component of the equation restricted by laminar flow, drops out.

(b) Current-Voltage Characteristic Measured at the Electrodes Finally, it is important in experiments to account for the fact that potentials usually are not imposed or measured directly across the filter, but between points remote from the filter surfaces. As an example, consider again the one-dimensional situation of Fig. 1, where a plane filter is placed in an insulating tube and one fine grid electrode is mounted a distance L~ upstream of the filter and another a distance L2 downstream. The downstream electrode has a potential &4~ with respect to the upstream one. If a current per unit crosssectional area (j} passes through the filter in

O"o

From Eq. ~-26~ one now obtains the currentvoltage relation for the system between the electrodes as

(j)

-zsp(u}

-

[-313

=

Finite electrode spacing thus tends to reduce the current. This point is best illustrated by considering a situation where the electrodes are shorted in an attempt to measure the streaming current generated by filtration. When any current passes through the regions of fluid between the filter and the electrodes, a potential drop invariably occurs, and to compensate for this drop a potential rise must be generated across the filter. This rise causes a back conduction of current through the filter, and the net current through the filter is thus reduced from the true streaming current --Zcp(u). Only when the resistance of the fluid between the electrodes and filter is small compared with that of the filter itself, that is, if o-s

(L1 + L~)

~o

h

<<

1,

Es2

is the effect of electrode spacing negligible. This point is discussed further in the next section.

Journal of Colloid and Interface Science, Vol. 52, No. 3, September 1975

ELECTRIC CHARGING IN FILTRATION Insulated wire wrapped on insulating electrode support O-ring seals

423

Moveable stainless steel electrode

neter

• flow out

cell wall

plates ~__ 5cm

FIG. 2. Scale diagram of experimental test cell. 4. EXPERIMENTS AND COMPARISON WITH THEORY Experiments were conducted in an apparatus (Fig. 2) which simulated the essentially Onedimensional filtration of Fig. 1, where a plane filter is mounted in an insulating constant-area tube. The liquid was n-heptane, its conductivity controlled by doping with Shell ASA-3 antistatic additive. Various cellulose ester fibers manufactured by the Millipore Company (23) were used (Table I), largely because they were available in controlled thicknesses, pore sizes, and chemical composition and are relatively inert to most organic solvents. Although the individual filters had virtually the same thickness, the total filter thickness was varied by using up to three filters sandwiched together in series. The test cell was made of Plexiglas, which is an insulator for our present purpose. Viton O-rings sealed the filter to the cell and all exterior tubing was made of cross-linked polyethylene. Current and potential were measured with a Keithley 610B electrometer between two movable stainless steel grid electrodes (30-mesh) on either side of the filter. The active filter diameter was 2.2 cm, and the electrodes could be moved up to 5 cm to either side of the filter. Fluid was passed to the cell from a stainless steel reservoir pressurized to about 1 atm with oil-free nitrogen, the flow rate being controlled by a valve down-

stream of the test cell, and collected in another stainless steel tank. The reservoir and the upstream electrode were grounded, while all parts downstream of the test cell, including the stainless steel grid which supported the filter on its downstream side, were well insulated from ground. The heptane used was quoted by the manufacturer as being 99 mole% pure. Without additives, it had a conductivity of about 10-I° ~-1 m-l, which is at least an order at magnitude higher than that of pure heptane (24). In all our experiments the conductivity was increased by a factor of four or more above the conductivity of the fluid as delivered by adding various amounts of Shell ASA-3 ionizing additive. The assumption was that all observed effects would then involve primarily the known controlled additive. Conductivities were measured in a 1000-cma cell consisting of two concentric stainless steel cylinders, similar to that used by Gavis and Wagner (6, 7). When a dc voltage was applied to the electrodes, the current decreased with TABLE

I

Filter type

Mean Dare diameter X10 7 (m)

Thickness h X l0 s (m)

Porosity

PH VC

3.0 ± 0.2 1.0 ± 0.08

1.5 -4- 0.1 1.3 ± 0.1

0.77 0.74

Source: Millipore Catalogue (23).

Journal of Colloid and Interface Science, Vol. 52, No. 3, September 1975

424

HUBER AND SONIN 5

I

v

% ×

3 --

bo o°

°

o

e 2 I

0

2 4 C o n c e n t r a t i o n of A S A - 5

6 (ppm)

8

FIG. 3. Conductivity of heptane as a function of the concentration of Shell ASA-3 antistatic additive, measured at 24°C. time from an initial value to a final s t e a d y value, the a m o u n t of decrease being small for voltages of the order of a few volts, b u t quite large for voltages of the order of several hundred volts. The decay was a p p a r e n t l y due to polarization or electrolysis (1). However, the conductivity based on the current recorded at zero time was independent of the applied voltage, and this value was accepted in all

cases. Most d a t a were obtained with an applied potential of 93 V. C o n d u c t i v i t y measurements were m a d e i m m e d i a t e l y after each experimental run, after first allowing a b o u t 1 hr for relaxation in the completely grounded conduct i v i t y cell. Figure 3 shows the r o o m - t e m p e r a t u r e cond u c t i v i t y of the heptane as a function of the concentration of Shell ASA-3 a n t i s t a t i c additive, in the range of concentrations used in our experiments (see Table I I ) . This additive, which is m a n u f a c t u r e d expressly for the purpose of increasing c o n d u c t i v i t y in fuels to reduce electrostatic hazards, is a mixture of equal p a r t s of chromium dialkylsalicylate, calcium didecyl sulfosuccinate, copolymer of lauryl m e t h a c r y l a t e and methylvinylpyridine, as a 50~v solution in a hydrocarbon solvent (25, 26). I t is interesting t h a t the c o n d u c t i v i t y of the doped heptane varies a p p r o x i m a t e l y linearly with concentration and not as the square root, as would be expected if ionization of the additive occurred b y b i n a r y dissociation of individual molecules. The linear dependence a p p a r e n t l y results from a bimolecnlar dissociation (20, 27) for which the binding energies of the particles are much weaker (e.g., coulombic) t h a n for a monomolecular dissociation. T h e results of Fig. 3 were quite reproducible.

TABLE II Run number

1 2 3 4 5 6 7 8 9 10

Temperature (°C)

26.8 23.6 25.5 28.2 25.5 23.0 23.0 25.0 23.6 24.4

Filter type

PH VC VC PH PH PH PH PH VC VC

Number of f i l t e r s

1 1 1 1 1 3 2 1 1 1

T o t a l filter thickness h X 104 (m)

F l u i d conductivity O-o X 10 x°

1.5 1.3 1.3 1.5 1.5 4.5 3.0 1.5 1.3 1.3

7.0 6.9 15 16 15 15 15 26 26 46

XD/a a

(XD/h) X l 0 s ~ M a x i m u m v a l u e b of

UXD/D

(~-1 112--1)

27 81 54 18 18 18 18 14 40 30

2.7 3.1 2.1 1.8 1.8 0.6 0.9 1.4 1.5 1.2

5.9 6.6 6.5 3.1 3.4 1.3 3.0 3.1 1.4 1.9

a Debye ratios are calculated from the bulk fluid conductivity and the nominal pore radius assuming D = 0.66 X 10-9 m2 sec-a and er = 1.97. b Peclet numbers are calculated from the Debye length based on bulk fluid conductivity with rp2/K = 1,0 X [Qa V secm -2. Journal of Colloid and Interface Science,

Vol. 52. No. 3, September 1975

ELECTRIC CHARGING iN PILTRATiON Maximum recorded variations in conductivity at any given concentration of ASA-3 were less than 200-/0. Before each run the filter a~ d test cell were soaked in the working fluid fo~ 24-48 hr with all metal parts grounded. Only after this procedure would the results be steady and reproducible. After any alteration in flow rate, electrode spacing, or external resistance between electrodes, a period of at least 5 min was required for a new steady state to be attained. A new filter was installed each time the fluid conductivity was changed. Transient phenomena were observed when a fresh dry filter was mounted in the cell and tested immediately, but these were not investigated methodically. For a constant flow rate through a fresh filter, the steaming current would usually decrease from zero to a negative value, stay there for a while, and then increase through zero and eventually reach a final steady positive value. The time scales involved were usually of the order of hours, but sometimes considerably longer. Special care was taken in all our tests to ensure that steady reproducible results were measured. The theoretical equation that is to be compared with the experiments is [-31-], with the effective filter conductivity ~s given in terms of filter and fluid properties by Eqs. [23-], [-14-], and 1-15-]. The application of these equations requires a knowledge of the fixed charge density p and the tortuosities r~ and r~, as well as the mobility K. Unfortunately, the fixed charge and tortuosities could not be determined in our present apparatus by the relatively direct methods suggested in the previous section. The reason was the extreme thinness of the filters, which made it difficult to place the electrodes close enough to the filter surfaces to be confident that the resistance of the fluid outside the filter was negligible and that Eq. [-32-] applied. Because of this difficulty, all tests were made in the opposite limit where the resistance of the fluid outside

425

the filter dominated, so that an (L1 + Z2) Cro

>>1.

[-33-]

h

Furthermore, by making direct measurements of the resistances of dry filters clamped between electrodes, it was determined that ~8 was of order 10-11 [1-1 m -1, o r about two orders of magnitude smaller than ~o in our experiment. In other words, the solid parts of the filter were in effect insulators and the effective filter conductivity ~f was governed by the fluid in the filter pores, EO'p

[34]

~z-------. Tp 2

Hence, the tortuosity and conductivity of the solid matrix do not enter the theory in this particular limit. Using Eqs. [-33-] and [-341 in Eqs. [-311, with av given by Eqs. V14-] and [-151, our model thus predicts the following current-voltage relation for our experiments: (j> = -

h

~o(u>

LI + L2

ne

Z#('r)--

Zx~ --

~ o - - - - .

[-351

L1 + L2 Here, -y = pK/zo is the dimensionless measure of fixed charge defined earlier, and

(1 + 72) ~ is a function shown in Fig. 4, its important characteristic being that f('r)~'~ "Y when -y < 0.3 and f(3,) ~'~ 1 when ~ > 3. The case 3' < 0.3 is the low fixed charge limit, where the fixed charge concentration is small compared with the ion concentration in the external fluid. When ~/> 3 the fixed charge is termed high. In practice, it is very likely that the situation will fall into one or the other of these limiting cases.

Journal of Colloid and Interface Science, Vol. 82, No. 3, September 1975

426

FIUBER AND SONIN l

--

I

Experiments were conducted with various filters at four nominal fluid conductivities, as shown in Table II. At each conductivity, the dependence of the current and voltage on +f ( 7 } : 7 flow rate, electrode spacing and external f+Y) / ~j/~f(7), Eq [36j resistance were recorded. The external resistance for the filtration system was provided by the internal resistance of the electrometer which was used to measure the current (see Fig. 1), and could be varied in steps from 10s~2 o I o 2 to 10uf~. In the potential measuring mode, the 7 meter resistance was 10149, which was effecFio. 4. T h e function f @ ) . tively infinite for our purposes. By measuring currents and potentials over this range of The high fixed charge limit is of particular meter resistance, it was possible to map out interest here, because we shall see that all complete current-voltage characteristics for our present experimental data can be explained given flow conditions and to establish zeroby assuming that our conditions fell into this potential streaming currents as well as openlimit. When 7 > 3, f ~ 1 and the current- circuit streaming voltages. Reference experivoltage relation becomes completely indepen- ments were made without filters and null dent of the fixed charge density or any other results were established to ensure that the observed effects were due entirely to the filters. physicochemical property of the filter: Figure 5 shows some typical results for the streaming potential, obtained with the elec(j) = z / - - trodes open-circuited. From Eq. [-37-] the L t + L2 ~e theoretical expression for this potential is, croAc~ (~ > 3). [-371 for 7 > 3, L1 + L2

o

i

Z/~']h(u)

This independence of the fixed charge is at first surprising. It arises because we are dealing here with a particular case where the fixed charge is high and the resistance of the fluid outside the filter (but between the electrodes) is very large compared with the internal resistance of the filter itself. As a result, the filter behaves as if the external resistance across it were infinite, that is, the emf which the filter develops across itself (between points 2 and 1 in Fig. 1) is essentially that which would arise with points 2 and 1 open-circuited. Eq. [-26-] shows that this potential is independent of p when the fixed charge is high, since ~s is then proportional to p. The current-voltage relation for the whole system between the electrodes simply results from this open-circuit emf of the filter in series with the resistance of the regions of fluid outside the filter.

=

E38-] KE

The results of Fig. 5 clearly demonstrate the linear dependence of A¢ on (u), as well as the linear dependence on the total filter thickness (the total filter thickness was varied by using up to three filters sandwiched together in series). The theoretical independence of pore size is also demonstrated. The results for both the VC and PH filter types, which have pore sizes differing by a factor of three, can be brought into excellent agreement with Eq. [-38-] by setting

Zz=-i and

[39-] rp2/K = 1.0 X 10s V sec m -2.

Taking r p ~ 1.6 (14), this would correspond to K---~2.5 >( 10-s m 2 V-1 sec-I, which is quite a

Journal of Colloid and Interface Science, Vol. 52, No. 3, September 1975

427

ELECTRIC CHARGING IN FILTRATION

,2sL

'/

7

' /h:.~.=x / /,o-'~,-,, /

I

o/,,:,.~x / <",, er/ 10"4 j • / P o i n t . . . . perimen, / :o =,.~ ,<,o-~,-,-, ~-, 2/ ~vc fi,er

l<~,,,le,~/ / {

IO.O~

J

/ ¢ 75/

]

J7 i ,~ 7

<3 ~ . 5.

/

/

l //

.,(E

t

/~:~.o~<,o -'<<,,, / {~""er~>

--

_

~}PH fillers

/

Lines: theory, ossurning = 1.0 X 108 V eec m-2

i

2.5~?/ o~

I I I 2 4 6 Reduced flow velocity

0

I I 8 I0 x 10 4 ( m sec-I)

FIO. 5. Streaming potential (potential measured with electrodes open-circuited) as a function of reduced flow rate for several filter thicknesses. Again, all the experimental data can be brought into excellent agreement with theory by setting Z i and rp2/K equal to the same values. Figure 7 shows the streaming current as a function of the electrode separation, everything else being kept invariant. The inverse dependence on L1 + L2 of Eq. [40] is satisfactorily confirmed. The linear dependence on the fluid conductivity ~o is verified by the results of Fig. 8. Figure 9 shows entire current-voltage characteristics (the solid lines) for several values of superficial flow speed. These characteristics

reasonable value for the mobility of a large ion (24). Assuming Einstein's relation holds, this mobility would correspond to a diffusion coefficient of 0.66 X 10.9 m 2 sec-1. Unfortunately, no data appear to be available in the open literature on the precise value of K or D for the ions of ASA-3 in heptane. Figure 6 shows similar data for the streaming current, measured with the electrodes shorted. For 7 > 3, our theory gives


h -

~2~o(U)

z j - -

[4o3

L1 + L2

Ke

,~.~1-

I /

Ih=4.5× / i#l [I;=I0 -4. m // _T bOi0.0

...../..

|

/

~ ~.~I-

I

|

~

/$ f?

/ 7/ I q~'

2.5

/

I

/

/ . -'~"

/~

/

/

J

/I /

h :3.0 x I0 m / I (2 f ers)

/

2"

I

-4

4~

/ /

~.oV ?1

o~

V /

o

xlO4 m

h=l.3 (I filler}

_

Poinls : experimenl

.o .,.=,,o-~-,=-,

ITcf'"=

~} PH filters

,,

. . . .

,,,o,, ......

r>s, zf=-I, ~ ~=,.o

,.

-

xlO 8 vsec m-2

7//

oF 0

I I I I I 2 4 6 8 ]0 Reduced flow velocity xlO 4 (rnsec -I) E

FIo. 6. Streaming current (current with electrodes shorted) as a function of flow rate for several filter thicknesses. Journal of Colloid and Interface Science,

VoL 52, No. 3, September 1975

428

HUBER

AND

SONIN

lIE m _ > 12.5

I

I

I

1

I

J

: theory, ossuming Y > 3 , Zf =-I,

'o ×

_

/

Line

/

b° IO0

^°] ^

Eo~

7.5

5.0

o/

_

E o

Points ~experiment

o~ / o°~ 2.5 - -

- -

.~=2.6 x fo-9~'% -I : .o21 ,-, sec-I

o~

Single VC filter

--

=o a:

0

//~f

0

f

I

S

r

2 4 6 8 ]0 Dimensionless reciprocol electrode spocing h / ( L I + L 2) x lO 3

Fro. 7. Reduced streaming current (current with electrodesshorted) as trode spacing. were generated by varying the external resistance between the electrodes. The broken lines link the points taken at the same external resistance. The lines shown are actually the theoretical expressions given by Eq. 1-37], with Zf and rp2/K as given in Eq. [39]. Again, the theory is substantiated. Finally, Figs. 10 and 11 show all our data on streaming potential and streaming current, respectively, as functions of flow rate. These

a

function of reciprocal elec-

data correspond to two filter types (the pore sizes of which differ by a factor of 3), several filter thicknesses, and a variety of electrode separations and fluid conductivities. All the data are in reasonable agreement with the theory, assuming again that 3' > 3, Z / = - 1 , rp2/K = 1.0 X 10 8 V sec m -2. A word is necessary about inequalities [1]-[-3], u p o n w h i c h our t h e o r e t i c a l s o l u t i o n is based. The D e b y e length can be expressed

IE

L _ J~

u

-o o1

FIG. 8.

I

I

I

Points: exper[menl ] 5.1 x 10-4msec-I< 5, Zf =-I,

I¸

J 7 "

/ ~

,.ox,o8v....- 2 / .25

_

o/~ 0

t/

/ F

I

1

I 2 5 Fluid c o n d u c l i v i t y o"o

I

[

4 5 x I 0 9 (Q-I m-I)

Reduced streaming current (current with electrodesshorted) as

Journal of Colloid and Interface Science, Vol. 52, No. 3, S e p t e m b e r 1975

a

function of fluid conductivity.

ELECTRIC CHARGING IN FILTRATION

'

~ .

/ "eat= u /-

'

~

I X ~

/

T:,[7 rliYn:s- f . . . . . .

,ant

l l i:t',o I× 110'O,Q,IC5 4.5x [0 -4 ¢

/ ~ , values ind cated o,o,e li . . . . , . . . . . . , o , Next, values indicated

I-I;o"alv-&:lo -~

o. . . .

,oo

>~,~,:-,,

z~KZ= 10 x /OBV s e ¢ r n 2

4~

~ ~ +

r ,

i

"s

~c:l

~.[ +~4~'7x10

_~.7.5 i \ g I i

L

-a ,

,oo

-

~

429

/\

~,z

~

~, [5]

",%,,..--~ _,o,Oa

(,5

,\ e "/

,%'

/

Lla-L2 = .015 m

\

~

%~

o

.°.,. i 2.5

0

Z5

5.0

I0.0 12.5 (L I + L 2 ) o-o



Reduced current

_

(V)

Fro. 9. Filter current-voltage characteristics at various flow rates.

as (1): xv

(..?)°

=

-

[41]

-

where er is the relative permittivity of the fluid; eo is the permittivity of free space; D is the diffusion coefficient of the ions involved; and ¢ is the fluid conductivity. In our calculations, we used er = 1.97 for heptane (24) and in the absence of more specific data took D = 0.66 X 10-9 m 2 sec -1, which is consistent with Eq. [39], as discussed earlier. Strictly speaking, the conductivity to be used for the

12.5

I

I

Debye length in Eq. [41] should be the conductivity of the fluid in the pores, which in our case is greater than conductivity Co Of the fluid outside. Unfortunately, our present experiments yielded no information on the fixed charge, and a rational estimate for ¢~ could not be made. However, for an approximate check on Eqs. [I]-[3], we can set ¢ equal to ¢o. Then, in our experiments,

l

I

".'S"


=~ ".°,V"

5,0

• ~.~

$

,~/°

2.5

0.01 < XD/h < 0.06

[43]

0 < UkD/D < 7.

[44-]

, /

[

• ,J"

>

g

[42]

/

IO.O

"~

I

Line: theory, ossuming Y > ~ , Zf =-l, Tp2 -~- = 1.0 x I0 8 V sec m-P-

14 < XD/a < 81

•

•

•

~* :

"

•

.

• " •

° Points: experiment 6 9 x io-IOD-I m-I _<¢o < 4. 6 x 10-gD-I m- I 1.3 x 10-4 m <- h < 4. 5 x 10-4 m VC ond PH Fillers

o 2 Reduced

I

I

I

I

l

4

6

8 h T

tO

12

flow velocity

x IO 8

14

(m2sec-l)

FIG. 10. Summary of streaming potential data compared with theory. Journal of Colloid and Interface Science, VoI. 52, N o . 3, S e p t e m b e r 1975

430

HUBER AND SONIN

~ b ° 12.5

_~ Line : Itheory,

I ma

assuming

I

I,

.

,

~

I_

7">3, Zf =-I, .r2 ]o.o - - @ =l.O xlO B Vsec

=

i "~

7.5

•-

5.0

~.•/~'/•• =" 6.9 x I0-10,0.-I m-I - % -< 4 . 6 x 10-99-1m-I

g

2.5 --

• " j.~/.

•

=,~ ,11

1.3 x

•

10 - 4 m -~ h -~ 4 . 5

x 10"4 m

.014m $ LI+L2~- .035 m VC and PH filters

- -

-o

0

/2.-

I

I

t

J

1

l

J

2

4

6

8

10

12

14

Reduced flow veloci'ly ,~u>h E

x l 0 8 (rnZsec-I)

FIG. 11. Summaryof streaming current data comparedwith theory. Thus, Eqs. ['1] and [-2] were satisfied reasonably well, as was Eq. [-3] for at least most of the data taken.

on pore size, and therefore, a dependence on pore size may be implicit. For example, if the fixed charge exists as a uniform distribution of surface charge on the interior surfaces of the filter, then for a given chemical composition 5. DISCUSSION of the solid matrix and a given fluid, p will be Our basic equation for the current generated proportional to a -1. Thus, there would be an by a filter between two electrodes is Eq. [-31]. apparent dependence on pore size, but that As discussed earlier, the filter itself behaves as dependence would actually come about only a current generator in parallel with an internal because of the dependence on the amount of resistance. Any fluid outside the filter, but fixed charge per unit pore volume. between the electrodes, adds a resistance in The restrictions on our analytical solution series with the filter. We see from Eq. ['31] that should be noted, as should the fact that the everything else being invariant, the current experiments reported in this paper were all generated is always proportional to (u), as is done with an electrode spacing much larger the potential developed when the electrodes than the filter thickness. The lack of an are open-circuited. In the special case where explicit dependence on pore size results from the electrode separation is sufficiently large our assumption that the Debye shielding and the fixed charge is sufficiently high, as in distance is larger than the pore radius. When our experiments, the current and voltage turn the conductivity is increased so much that XD out to be independent of the fixed charge decreases below the pore size, charge can no density, which is the only physicochemical longer be distributed uniformly over a pore property of the filter's solid/fluid interface cross section, but must exist only in more that enters into the equation. Eq. [-37] then confined regions near the pore walls. As a applies. In general, however, the fixed charge result, the streaming current will cease to be density will influence the charging characteristics. Note that the equation does not depend independent of a/XD and will begin to fall explicitly on the pore size of the filter if the below the value predicted under our current fixed charge density is regarded as given. assumption, other things being invariant. The However, the fixed charge density may depend observed linear dependence of the streaming .,rournal of Colloid and Interface Science. Vol. 52, No. 3. S e p t e m b e r 1975

ELECTRIC CHARGING IN FILTRATION current on conductivity follows partly from our assumption about the Debye length being large, and partly from the relatively large electrode spacing in our experiments, where err(L1 q- L2)/¢oh >> 1 (see Eq. [-31]). The implications of assumptions (2) and (3) are also very important. Our solution applies to cases where the free charge density in the fluid decays very rapidly after the fluid emerges from the downstream side of the filter, and quasineutrality is reestablished in the fluid in a distance that is very short compared with the filter thickness. The current generated by the filter is thus carried outside the filter entirely by ohmic conduction, the contribution due to convection of charge being negligible. Previous work (6-11) has been largely experimental and has dealt with conditions where one or more of the assumptions underlying our present solution do not apply, so that direct comparisons cannot be made. The work of Gavis and Wagner (6, 7) is of some interest, because they appear to be the only ones who have previously proposed a scheme for predicting charging currents in terms of filter and fluid properties. Gavis and Wagner dealt with the opposite limit where the charge relaxation distance is long compared with the filter thickness, not short as in our present theory and experiments. They did not derive a theory for the charging process. Rather, based on dimensional arguments and several assumptions about the charging mechanism, they proposed an empirical scaling law for correlating at least some of their data for streaming current. Although all their measurements were done with the same fluid (doped heptane) and filters of similar chemical composition (Millipore filters made of cellulose esters) and thickness, they suggested that their dimensionless scaling law should be generally applicable, provided only that their single scaling parameter he properly simulated. Since Gavis and Wagner's experiments were conducted under conditions very different from ours, a direct comparison of their conclusions

431

and ours cannot be made. However, we do have some reservations about the claim that their dimensionless correlation should be generally applicable. There are two main reasons for this. First, Gavis and Wagner's dimensional analysis of the charging problem is not general, but an oversimplified one that assumes a priori that the current is independent not only of the conductivity of the filter's solid matrix, but indeed also of any physicochemical property of the filter/fluid interface, such as our fixed charge density. That the current can under special circumstances be independent of something like the fixed charge density is conceivable, and indeed is illustrated by our own theory and experiments. However, that this should be the case in general, as Gavis and Wagner imply, is difficult to accept. Considering that all their measurements were made with the same fluid and filters of identical chemical composition, it is possible that Gavis and Wagner's data actually contain a dependence on the fixed charge density, but show it as an apparent dependence on pore size, as discussed above. The question of whether an observed dependence on pore size is actually a true geometrical effect or comes about because a bulk physicochemical property which influences the current depends on pore size is a very important one and it cannot be left unresolved. In one case, the current would depend on the interior geometry of the filter but not on its chemical composition. In the other, it would be influenced by the chemical composition of the material. As to the assumption that their filters behaved as perfect conductors compared with the fluid, we point out that the fluid conductivities in their experiments ranged from 2 X 10-13 to 4 X 10-9 Q-1 m-l, while the conductivity of cellulose esters, of which their Millipore filters were made, has been determined (28) to be of order 10-11 ~2-1 m -I, a value confirmed by direct measurement in our own tests. Thus, there is reason to believe that at least two additional parameters should appear in a more general correlation, one involving a

Journal of Colloid~and Interface Science, Vol. 52, No. 3, September 1975

432

HUBER AND SONIN

physicochemical surface p r o p e r t y like our fixed charge density and the other the ratio of the solid to fluid conductivities. The other difficulty with Gavis and Wagner's correlation lies in their assumption that the current was carried downstream of the filter entirely by convection of charge with negligible ohmic conduction. This assumption was necessary if the streaming current they measured was to be interpreted as a property of the filter, independent of the geometry of the filter holder and the placement of the measuring electrodes. Now, if the current is to be carried between two electrodes by convection alone, the charge relaxation length (U)•D2/D (see Section 2) must be large compared with the interelectrode spacing. I n Gavis and Wagner's experiments, (U)XD2/D ranged from about 10-2-500 cm, while their electrode separation, although n o t specifically given in the paper, appeared to be of the order of 10 cm. Thus, it is not clear that all their results are independent of the geometry of their specific system. 6. CONCLUDING REMARKS Our model for the charging process appears to be well supported by the experimental data. The specific analytical solution given in this paper is based on certain simplifying assumptions which limit its applicability (pore size smaller than the Debye length, charge relaxation length small compared with filter thickness). However, these assumptions were made largely for mathematical convenience, and it is in principle a straightforward matter to relax them and derive more general solutions based on the same model for the process. Work in this direction is necessary if the theory is to have broader engineering applicability. Also required will be information on the fixed charge densities for typical filter material and fluid combinations, and tortuosities of typical filters. Such information can be obtained empirically by measurements of bulk properties, as discussed earlier.

ACKNOWLEDGMENTS This research was sponsored by the National Science Foundation under Grant GK-35798X of the Fluid Mechanics Program. The first author was also supported in part by the Joseph Warren Barker Fellowship in Engineering while this work was in progress. REFERENCES 1. KLINKENBERG,A., AND VAN DER MINIXrE, J. L. (Eds.), "Electrostatics in the Petroleum Industry." Elsevier, Amsterdam, 1958. 2. KLII,rKENBERO, A., in "Advances in Petroleum Chemistry and Refining" (K. A. Kobe and J. J. McKetta, Eds.), Vol. VIII, Chapt. 2, p. 87, Wiley-Interscience, New York, 1964. 3. "Recommended Practice for Protection Against Ignitions Arising Out of Static, Lightning and Stray Currents," API RP2003 2nd ed. American Petroleum Institute, New York, 1967. 4. WINTER, E. F., J. Roy. Aeronaut. Soe. 66, 429 (1962). 5. BEUINZEEL C., J. Inst. Petrol. London 49, 125 (1963). 6. WAGNER,~. P., Ph.D. thesis, The Johns Hopkins University, 1966. 7. GAVIS,J., ANDWAGNER,J. P., Chem. Eng. Sci. 23, 381 (1968). 8. LEONARD,J. T., A~CDCARI:IART,H. W., J. Colloid Interface Sci. 32, 383 (1970). 9. LAUER,J. L., ANDANTAL,P. G., J. Colloid Interface Sci. 32, 407 (1970). 10. LAUER,J. L., AND A~TAL,P. G., Paper 52b, 64th Annual Meeting AIChE, San Francisco, 1971. 11. SI-IA~'ER,M. R., BAKER,D. W., ANDBENSON,K. R., J. Res. Nat. Bur. Stand. Sect. C 69C, 307 (1965). 12. SCI~LOGL,R., Z. Physlk, Chem. Neue Folge 3, 73 (1955). 13. OSTERLE,J. F., J. Appl. Mech. 31, 161 (1964); MORRISON, F. A. JR., AND OSTER]LE, ~. F., J. Chem. Phys. 43, 2111 (1965); GI~OSS,R. J., AND OSTERLE,J. F., Y. Chem. Phys. 49, 228 (1968). 14. JACAZlO,G., PROBSTEIN,R. F., SONIN,A. A., AND YI:NG, D., J. Phys. Chem. 76, 4015 (1972). 15. OVERBEEK,J. T. G., in "Colloid Science" (H. R. Kruyt, Ed.), Vol. I, Chapt. 4. Elsevier, Amsterdam, 1952. 16. SCHEIDEOOER,A. E., "The Physics of Flow Through Porous Media." University of Toronto Press, Toronto, 1963. 17. GAVlS,J., AI~rDKOSZMAN,I., J. Colloid Sel. 16, 375 (1961). 18. BENYON,L. R., Filtr. Separ., March/April 1966. 19. N~nSON,W. L., "Petroleum Refinery Engineering." McGraw-Hill, New York, 1958. 20. GERANT, A., "Ions in Hydrocarbons," WileyInterscience, New York, 1962.

Journal of Colloid and Interface Science, Vol. 52. No. 3. September 1975

ELECTRIC CHARGING IN FILTRATION 21. WYLLIE, M. R. J., AND SPANGLER, M. B., Am. Ass. Petrol. Geol. Bull. 36, 359 (1952). 22. WYLLIE, M. R. J., AND GARDNER,G. H. F., World Oil, March 1958. 23. "1974 Millipore Catalogue and Purchasing Guide," Catalogue No. LMC 0174/P, Millipore Co., Bedford, Mass., 1974. 24. WASI~BUI~N,E. W. (Ed.), "International Critical Tables of Numerical Data, Physics, Chemistry and Technology," Vol. VI, p. 144, Vol. V, p. 63, Vol. VI, p. 93. McGraw-Hill, New York, 1927.

433

25. "Shell Anti-Static Additive ASA-3," Shell Technical Bulletin ICSX : 69: 5, Revised, Charlmont Press, London, 1969. 26. Chem. Week, p. 45, December 26, 1964. 27. KLINKENBERG,A., in "Static Electrification, 1967," Proc. 2nd Conf. on Static Electrification, Conference Series No. 4, Inst. of Physics and Phys. Soc., London, 1967. 28. BII(ALES, N. M. (Ed.), "Encyclopedia of Polymer Science and Technology," Vol. 5, p. 524. WileyInterscience, New York, 1964.

3ournaI of Colloid and Interface Science, Vol. 52, No. 3, September 1975