Electric charging in flow of low-conductivity liquids through screens: A comparison of theory and experiments

Journal of Electrostatics, 25 (1990) 295-307

295

Elsevier

Electric charging in flow of low-conductivity liquids through screens: A comparison of theory and experiments I.N. Miaoulis, B. Abedian and M. Darnahal Mechanical Engineering Department, Tufts University, Medford MA 02155, U.S.A. (Received August 23, 1989; accepted in revised form August 25, 1990)

Summary This paper presents an experimental study of the charging of low-conductivityliquids by screens. Measurements were performed over a wide range of fluid conductivities and screen types. The relationship that was obtained between the current generated and the velocity of the fluid closely matched the theory. Potential measurements across the screen indicated that the charging process can be characterized by a charging electromotive force.

1. Introduction This paper describes a series of experiments that were conducted to test our theory for electric charging in flow of low-conductivity liquids through screens [ 1 ]. The experiments cover a wide range of fluid conductivities and velocities, and various screen materials. The relationship between the current and velocity for different conductivities as well as the relationship between the current and conductivity for different velocities is presented in graphical form for copper and stainless steel screens. The results of experiments where the potential across the screen was measured are described in Section 5. To the best of our knowledge, the only previous experimental study on the charging of low-conductivity fluids by screens is the one performed by Abedian and Miaoulis [2]. Yet that study was limited to one fluid conductivity, and only to current measurements. The authors are unaware of any experimental study of the potential across the screen. Experimental work is available on charging in artificial filters [3-5]. However, these studies are in terms of some filter properties which hardly can be defined for screens. Thus, any comparison between the present analysis and that for charging in filters appears difficult or impossible, despite the fact that there is a close resemblance between the two phenomena.

0304-3886/90/$03.50 © 1990--Elsevier Science Publishers B.V.

296

2. Experimental apparatus

The rate of charge generation from the screen can be measured either by measuring the current from the ground screen, or by measuring the rate at which charge leaves the screen in the flowing liquid using a charge collector. The first method was adopted for this investigation since it is more accurate and it is also independent of the charge relaxation time. A schematic of the apparatus is shown in Fig. 1. The system consists of a closed loop in which the working fluid is pumped through the screen which is located in the Teflon housing. The pump used is a 0.1 HP small electromagnetic pump which was resistive to corrosive liquids. The flow rate was measured with a Signet flow meter (model MK584). The pump speed was controlled by an autotransformer (Variac W20M). Two onefoot precollectors made from one-inch-diameter grounded copper pipe were placed before the screen so that they could relax the charge which the liquid attained flowing through the pump and flowmeter. Without the precollectors upstream of the screen, the charging prior to the screen can overwhelm the current generated by the screen especially at high flow rates [2 ]. A two-foot collector from the same material was placed downstream of the screen. Teflon bushings were used to electrically insulate the test section from the rest of the system. The inlet and drain pieces were made of PVC. Tygon tubes were used for the piping of the system. All wires used were Keithley 4803 low-noise cable. The wire connected to the screen was directly soldered. To minimize electrostatic noise from the surroundings, the entire test section was enclosed in a grounded Faraday cage made of aluminum screen. A Keithley 610C electrometer was used to measure the low current from the screen. The potential rise across the screen was measured using a Keithley 175 till inlet

flowmeter

::::::::::::::::::::::::::::::::::::: I I I I

I I I I ,/rPre

L......

collector

il

I

_ i

I

...............................

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Fig, 1. Schematic of the apparatus.

drgin

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--y----

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L Fig. 2. The screenmesh. Autoranging Multimeter, which measured the potential difference between the collector and precollector. The screens used were metal square-mesh screens made of solid cylindrical wires. The screen materials were stainless steel and copper. Figure 2 shows the geometry and dimensions of the screens. They both had 30 meshes per inch (30×30), H=0.593 mm and 2s=0.254 ram. The effective diameter of the screens was one inch. The electric conductivity of the liquid was measured before and after each experiment. The conductivity measurements were done in a "home made" cell similar to the one described by Koszman and Gavis [6 ]. 3. The hydocarbon liquid The low-conductivity liquid used was clear kerosene. At a pure state its conductivity was of the order of 10 -11 ~ - 1 m-1. Through the addition of small amounts of ethanol or propanol to the kerosene, it was possible to cover a range of conductivities up to 9.3 × 10- lo ~ - 1m - 1. The effect of the addition of ethanol and propanol on the the conductivity of the fluid is presented in Fig. 3. The ionic diffusivity of the fluids used is D = 1 0 -9 m 2 s -1 and the ionic mobility x _ - 5 × 1 0 - 8 m 2 V -1 s -1. 4. Electric current measurements and comparison with theory During the first set of experiments the convective current generated by the screen was measured. The electrodes (precollector-collector) and screen setup are shown in Fig. 4. The grounded precollector relaxes the liquid from the charge generated by the pump, flowmeter and tubing. Free of charge liquid passes through the screen and becomes charged. Charge which is equal in magnitude but opposite in sign, flows through the grounded electrometer where it is measured in the form of current. This charge is then relaxed in the grounded collector and the cycle is repeated.

298 12-0

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o []

2-0 o

i

0.0

0.0

I

tO

~

I

i

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VOL. CONC. %

Fig. 3. Volumetric concentration of propanol ([]) and ethanol (O) additive versus conductivity of kerosene.

~:Z::::

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I

J

I

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Fig. 4. Schematic of the set-up for current measurements.

It cannot be overemphasized that these measurements are extremely sensitive since the currents measured may be as low as 10- lo A. Although the Faraday cage placed around the test section prevented electrostatic noise from the surroundings from affecting our measurements, experiments were conducted on low-humidity days, and during the measurements no movement was allowed in the laboratory. Even a small movement of the experimenters could shift the current reading. Although one can see the difficulty in conducting these experiments, the measurements were very reproducible. The average of a series of measurements is plotted in the graphs that follow. The maximum fluctuations of the measurements never exceeded the 5% range. For the comparison of the experiments with the theory, the explicit analytical form for the electric current generated by the screen [1 ] was used:

I=4q,,, U e x p ( - H / A ) (I' +I")

(1)

where qw is the charge density at the wall of the screen, U the superficial ve-

299

locity, H the size of the mesh, 2 the Debye length of the liquid and I' and I" are defined in [ 1 ]. The charge density at the wall of the screen qw is described by the expression, qw = aFCo

(2)

where a is a constant, F is the Faraday constant, and Co is the concentration of positive and negative ionic species in an undisturbed fluid. For low-conductivity liquids, the charge density at the wall of the screen is also defined as, (3)

qw = ( a / D ) C

where a is the electric conductivity of the fluid, D the ionic diffusivity, and the zeta potential of the fluid-solid system. The conductivity a can be expressed by, a = 21cFCo

(4)

where K is the ionic mobility of the fluid. From eqns. (3)-(5) the constant c~ can be expressed by, (5)

o~= 21c~/D

Although the zeta potential, ~, was not measured, it was estimated to be of the order of 5 mV since this value is consistent with the zeta potentials measured 20.0

,

,

15.0 +4 3

10.0

5.0

2

0.0 0.0

0.5

1.0

v a l x ~ (m/s~

2.0

Fig. 5. Current-velocity relationship for various conductivities and comparison with theory. Copper screen; kerosene-ethanol mixture. Theory ( - - ) ; ( 1 ) (C) ) a = 0.044 X 10- lo ~ - 1 m - 1; (2) ( A ) a = 0 . 2 2 6 X 1 0 - ' ° i-1-1 m - l ; (3) ( + ) a=l.613><10-1°F2 -1 m - l ; (4) ( X ) a = 2 . 2 1 8 X 1 0 - 1 ° ~2-1 m -1.

300 20.0

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.

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.

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H

5.0

o

<>

o

x +

× +

+

0.0 0.0

0..5

1.0

1.5

2.0

(r~s) Fig. 6. Current-velocity relationship for various conductivities and comparison with theory. Copper screen; kerosene-propanol mixture. Theory (--); (1) (O) (7-~0.050X10-10 ~,~-1 m-l; (2) (A) a=0.087X10-1°~ - ' m-l; (3) ( + ) a--0.202×10-1°~ -1 m-l; (4) ( × ) a=0.282X10 -l° ~ - 1 m - 1 ; (5) ( ~ ) a=0.484X10 - l ° ~ - l m - 1 ; (6) (~7) a=0.673X10 - ~ ° ~ - l m - ~ ; (7) (F~) 6 = 1 . 6 5 3 × 1 0 - 1 o ~ - i m-i; (8) ( ~ ) 6--2.500X10-1°~ -1 m-i; (9) (*) a = 3 . 9 9 2 X 1 0 - i ° Q - 1 m-i; (10) ( ~ ) a=6.250X 10-10 ~-~-1m-1.

for other hydrocarbon liquids and metal systems. When these values are substituted in eqn (5), c~takes the value of 0.5. Kerosene with propanol as an additive was used for the first set of experiments presented in Fig. 5. Measurements were conducted for ten different conductivities that covered a range of two orders of magnitude (5×10 -126.25 X 10-lO ~-~-1 m - i ) . For each conductivity six measurements at different velocities were conducted and the current was measured for each velocity. A copper (30 × 30) screen was used. Using a finer screen would result in higher currents. Finer screens generated cavitation though, which made any reliable measurements impossible. The theoretical curves for each corresponding set of measurements are also plotted. One can observe an excellent agreement between the theoretical and the experimental values, considering that they are plotted in a normal scale. For the experiments described by Fig. 6, the additive used was ethanol. Four different conductivity sets of measurements were performed, yielding similar results to those using kerosene and propanol mix. No major difference was observed between using ethanol of propanol as an additive to increase the electric conductivity. This is in agreement with the results of Goodfellow and Graydon [ 7 ] of charging current of flow of different solutions through stainless steel tubes. In both experiments (Figs. 5 and 6) one can observe that the elec-

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20.0

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)<

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•

5

i

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5.0

X

X

X 3

A

A

2

|

0.0 0.0

0'~

~

12,

tO

2.0

(m/sl

Fig. 7. Current-velocity relationship for different conductivities and comparison with theory. Stainless steel screen; kerosene-propanol mixture. Theory ( - - ) ; ( 1 ) ( © ) a = 0.068 × 10- lo ~ - , m - l ; (2) ( A ) o-=0.085X10 -1° ~-~-1 m - l ; (3) ( + ) a=O.140×lO -1° ~ - 1 m-l; (4) ( × ) a = 0 . 2 7 4 × 1 0 - 1 ° ~ -1 m - l ; (5) (@) a = O . 4 4 4 X 1 0 - l ° ~ -~ m-~; (6) ( ~ ) a = 0 . 6 6 5 × 1 0 - 1 ° ~ -~ m - l ; (7) ([E) a=O.887XlO -1° ~ - 1 m-l; (8) (*) a=1.331X10 -~° ~ - 1 m - l ; (9) ( ~ ) a = 3 . 0 2 4 × 1 0 - 1 ° ~ -1 m - l ; (10) (@) o'=4.637X 10-I° ~ -1 m -1.

tric current increases with velocity. More charge is being convected at higher velocities so more charge is generated by the screen to redevelop the double layer. Increasing the conductivity, electric current increases as well. The increase in conductivity results in two antagonistic effects: (1) The higher the conductivity, the shorter the Debye length. This results in less current. (2) The higher the conductivity, the higher the number of ionic species in the liquid that interact with the solid surface. This results in higher current. Up to a certain conductivity, the latter effect is stronger, and the current increases with conductivity. As mentioned at a later point of the paper, after that conductivity level, the former effect results in a decrease in the charging current. In both experiments the assumption of ~ being 0.5 seems reasonable. In order to test the effect of different screen materials in the charging process, the copper screen was replaced by a stainless steel (30 ><30) screen. As in the previous experiments, kerosene and propanol were used. The results are presented in Fig. 7. In this case the theoretical values did not match as closely to the experimental findings. A probable cause is that during the theoretical curve calculation, the value of the constant ~ was kept at a value of 0.5 which was appropriate for a kerosene-copper system but not for a kerosene-stainless steel one. Stainless steel gives a larger current than copper. Koszman and Gavis [6] attributed the differences in charging currents that were obtained from

302 20.0

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v'l -1tO

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i

A

A

2

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w U

O1

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LOGa (1,/(fl.ml) Fig. 8. Current-conductivity relationship for different velocities and comparison with theory. Theory (--); (1) (©) V=0.319 m/s; (2) (ZX) V=0.639 m/s; (3) ( + ) V=0.958m/s; (4) (X) V=l.277m/s; (5) ( ~ ) V=l.597m/s; (6) (~7) V=l.756m/s.

their experiments with flow of n-heptane through tubes of different materials, to differences in surface roughness. They claim rougher surfaces induce higher charging currents. But since in our experiments the screens did not differ in surface roughness, this shows that the rate of charge generation depends on the chemical nature of the screen. Thus, each material has a different wall charge density qw, or a different constant c~. The relationship between electric current and conductivity for different velocities is presented in Fig. 8. Theoretical and experimental values for six velocities are plotted. The measurements were made from a copper (30X 30) screen and a kerosene-propanol mixture. The experimental values are close to the theoretical ones. One can observe that the current increases with increasing conductivity. After a maximum is reached for each velocity, the current starts decreasing with increasing conductivity. The physical reasoning is given earlier in this section. The phenomenon was predicted by the theory described in [1]. 5. E l e c t r i c p o t e n t i a l m e a s u r e m e n t s

The screen behaves as a flow-driven current generator with a parallel internal resistance. The apparatus was modified (Fig. 9) so that the electric potential generated across the screen could be measured. The precollector and collector were used as electrodes. A voltmeter recorded the potential differences

303

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Fig. 9. Schematic of the set-up for potential measurements. lo ~

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02,

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a

I t5

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(m/s)

Fig. 10. Potential-velocity relationship for different conductivities; copper screen. ( O ) a = 0.050 X 10- lO~ - 1 m - 1; (/x ) a = 0.087 × 10- lO~ - 1m - 1; ( + ) a = 0.202 X 10- lO~ - 1m - 1; ( × ) a = 0 . 2 8 2 X 1 0 - 1 ° ~ - 1 m - i ; ( ~ ) a = 0 . 4 8 4 X 1 0 - 1 ° ~ - 1 m - l ; ( V ) 6=0.673X 1 0 - 1 ° ~ - 1 m - l ; ([~) (T= 1.653 X 10-1° ~~-1 m - l ; ( , ) ~7--2.500X 10-1° ~-~-1 m - l ; ((~) a=3.992 × 10-1° ~~-1 m - l ; ((~) a = 6.250 X 10 -10 ~,~-1 m-1.

between the electrodes. Two sets of experiments were performed. For the first set a copper (30 × 30) screen was used, and for the second set a stainless steel (30 × 30 ) screen was used. Measurements were taken for ten different conductivities that covered a wide range. From Figs. 10 and 11, it is evident that the higher the conductivity, the higher the potential across the screen for a given velocity. Another interesting observation is that in the low-conductivity sets of measurements (5 × 10-12-2.82 × 10-11 ~ - 1 m - 1 for the copper screen and 6.8 X 10-12-2.74 X 10-11 ~ - 1 m - 1 for the stainless steel one ), the potential is independent of velocity, while in the higher conductivity range it is almost proportional. This is in agreement with the theoretical predictions [ 1 ]. After the fluid passes the screen, is stays charged downstream for a characteristic distance, the convective length. The convective length l is defined as, l=~U

(6)

304 .

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X

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o

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i

0

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0.0

03

o

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1.5

2.0

VELOC~ (m/s)

Fig. 11. Potential-velocity relationship for different conductivities; stainless steel screen. ~=0.068 X 10-'°~ - ' m-i; ( A ) 0=0.085× 10-'° ~ -1 m-i; ( + ) a=0.140X 10-i° ~ - ' m-l; a----0.274X 10-'° ~1-1 m-'; ( ~ ) a----0.444X 10-1°~ - 1m-'; (V) a= 0.665 X 10 -1° £2-' m-'; a----0.887X 10-'°~-1 m-'; ( , ) a= l.331><10- '° Fl- ' m-'; ( ~ ) a = 3 . 0 2 4 X l O - l ° F t - ' m - ' ; ~--4.637X 10-i° Q - ' m-i; ([3) a=9.274X 10-i°~ -1 m -1.

(O) (X) ([~) (•)

where T is t h e r e l a x a t i o n t i m e of t h e fluid a n d U the superficial velocity. T h e r e l a x a t i o n t i m e is large for low conductivities, t h u s t h e c o n v e c t i v e l e n g t h is large. It e x t e n d s far e n o u g h to r e a c h t h e collector, w h i c h in t u r n m a k e s t h e p o t e n t i a l rise i n d e p e n d e n t of t h e flow rate. As t h e c o n d u c t i v i t y increases, t h e p o t e n t i a l rise b e c o m e s m o r e d e p e n d e n t o n t h e flow rate. Using P o i s s o n ' s equation, V 2 ¢ = ----q

(7)

EEo

w h e r e q is a s s u m e d to be a spatially c o n s t a n t charge density, e t h e dielectric c o n s t a n t , a n d ¢o t h e p e r m i t t i v i t y , one c a n write t h e p o t e n t i a l difference across the c o n v e c t i v e l e n g t h l in o n e - d i m e n s i o n a l f o r m as, ql 2 A¢= --2eeo

(8)

T h e electric c u r r e n t w h i c h results f r o m t h e c o n v e c t i o n of t h e c h a r g e d liquid with the average d e n s i t y qa is equal to,

I=q~UA

(9)

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-4.0

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0

q

I, • + +

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,

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I

0.0

i

0.5

I

tO

,

I

t5

,

2.0

v~zoc~ (m/s) Fig. 12. Potential development for high and low conductivity ranges; copper screen. 6=0.050 × 1 0 - ' ° i'2-1 m - ' ; ( A ) a=0.087× 1 0 - ' ° ~ -1 m - i ; ( + ) a=0.202 × 1 0 - 1 ° L'~ - 1 m - i ; a=0.282X 1 0 - 1 ° ~ - 1 m - I ; (~,) a = 0 . 4 8 4 X 1 0 - ' ° i l - l m - 1 ; ( V ) a=O.673XlO-l°~-'m-i; a = l . 6 5 3 X 1 0 - 1 o ~ 1 m - i ; ( , ) o'=2.500X10-1° ~ -1 m - i ; (@) o-=3.992X10-1° ~ -1 m - l ; a=6.250X 10 -1° ~--~-1 m-1.

(O) (× )

([~) (q))

where A is the cross-sectional area of the screen. One can modify eqn. (9) to be I

q~-UA

(10)

From eqns. (8) and (10), //2

A¢=

2eeoUA

(11)

A dimensionless group defined as,

G= A¢eeo U/I

(12)

can be deduced from eqn. ( 11 ). Logarithmic plots of G versus velocity, for both the screen set-ups, are shown in Figs. (12) and (13). Again, one can see more clearly the behavior of the two conductivity ranges. Low-conductivity liquids tend to show no dependency between potential and flow rate after a certain velocity, while higher-conductivity liquids show dependency. We can also observe that there is a correlation between the convective current and the electric potential.

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0.5

+

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+

i 2.0

VELOCnY (m/s) Fig. 13. Potential development for high and low conductivity ranges; stainless steel screen. a = 0.068 × 10-1° f~ - l m - l ; ( £x ) a = 0.085 × 10 - l ° f~-l m - l ; ( + ) a=0.140 × 10 - l ° f~-i m - l ; a = 0 . 2 7 4 × 10-1° t2 -1 m - l ; ( ~ ) a = 0 . 4 4 4 × 10 -1° ~ - 1 m - l ; ( V ) a = 0 . 6 6 5 × 10-1° f~ -1 m - l ; a = 0 . 8 8 7 × 1 0 - 1 ° f ~ - I m - l ; ( , ) a = 1 . 3 3 1 × 1 0 - 1 ° ~ - 1 m - l ; ( ~ ) a = 3 . 0 2 4 × 1 0 - 1 ° f~-I m - l ; a = 4 . 6 3 7 × 10-~o ~ -1 m -1.

(O ) (× ) (~q) (~)

6. Concluding remarks

The theoretical expression for the charging current [ 1] was derived using the assumptions that the Debye length is much smaller than the screen mesh size and that the Peclet number of the flow, with respect to the Debye length is smaller than unity. The largest Debye length, corresponding to the lowest conductivity, was of the order of 0.1 mm, smaller than the screen mesh size. The Peclet number of the actual flow is larger than unity. Despite the fact that the theoretical expression was derived using the Peclet number assumption, there is a quite satisfactory agreement between the theory and our experimental measurements. This indicates either that the condition of a small Peclet number flow may not be critical for the application of the present theory, or the assumption that the charging takes place in the near vicinity of the walls of the screen where the velocity and Peclet number are very small [ 1 ] is correct. Charging current increases with flow velocity in a fashion predicted by the theory. Current also increases with conductivity up to a maximum and then drops. Experiments that involved the measurement of the potential difference across the screen indicated that modelling of the screen as a current generator is valid. Experiments showed that the prediction that the low-conductivity liquids would not show a potential-velocity relationship was a correct one.

307 T h e y also indicate a c o r r e l a t i o n b e t w e e n t h e c u r r e n t g e n e r a t e d b y t h e screen a n d t h e p o t e n t i a l across t h e screen. T h e fluid charge d e n s i t y at t h e solid-fluid i n t e r f a c e was c o n s i d e r e d to be 0.5FCo for b o t h c o p p e r a n d stainless steel. F o r t h e case of t h e copper, t h e ass u m p t i o n t h a t a is 0.5 a p p e a r s to be correct. In the stainless steel case, a is s h o w n to be higher, m e a n i n g t h a t t h e zeta p o t e n t i a l in the stainless s t e e l k e r o s e n e is h i g h e r t h a n in the c o p p e r - k e r o s e n e system. Since the surface r o u g h n e s s was n o t a f a c t o r in our e x p e r i m e n t s , this also shows t h a t d i f f e r e n t screen m a t e r i a l s affect t h e rate of charging.

Acknowledgements T h i s i n v e s t i g a t i o n was c o n d u c t e d u n d e r a g r a n t b y t h e N a t i o n a l Science Foundation.

References 1 I.N. Miaoulis, B. Abedian and M. Darnahal, Theory for electric charging in flow of low-conductivity liquids through screens, J. Electrostat., 25 (3) (1990) 287-294. 2 B. Abedian and I. Miaoulis, in: T.R. Hedrick and R.M. Reimer (Eds.), Mass Flow Measurements - - 1984 (ASME Winter Annual Meeting, New Orleans), ASME Spec. Publ. FED-Vol. 17, 1984, p. 149. 3 I. Gavis and J.J.P. Wagner, Chem. Eng. Sci., 23 (1968) 381. 4 P.W. Huber and A.A. Sonin, J. Colloid Interface Sci., 61 (1977) 126. 5 J.T. Leonard and H.F. Carhart, J. Colloid Interface Sci., 32 (1970) 383. 6 Koszman and I. Gavis, J. Colloid Interface Sci., 16 (1961) 375. 7 H.D. Goodfellow and W.F. Graydon, Chem. Eng. Sci., 23 (1968) 1267.