ENGINEERING, DESIGN, AND EQUIPMENT
e on the PHILIP E. BOCQUET’, CEDOMIR M. SLlEPCEVlCH2,
AND
DAVID F. BOHR
Univerrify o f Michigan, Ann Arbor, Mich.
A
BOUT 1860, Quincke (8) discovered that when he forced water through a clay diaphragm a potential developed across it. He realized that this effect was an inverse phenomenon to electroosmosis, nhich had been observed as early as 1807 by Reuss (IO). The next most significant contribution to a clarification of streaming potential-in fact, to the whole field of electrokinetics-was made by Helmholtz ( 4 ) in his classical theoretical analysis of the streaming potential concept. The basic premise of the Helmholtz development is that, vhen a liquid and solid come into contact, there is usually an accumulation of ions of the liquid near the surface of the solid t o produce a double la!-er. Helmholtz envisaged the double layer as t n o parallrl planes of oppositely charged ions. The fixed layer adheres to the surface, as schematically represented in Figure 1,A, and the movable layer exists nearby in the liquid. Subsequent development of the double layer theory by Gouy (3), Chapman (Z), Stern (12), and others has shown that the lalers of charge e u i d as dispersed ionic atmospheres because of the thermal agitation of the ions, as represented in Figure 1,B. Except for microcapillaries, however, these diffused la\rers are thin compared with the radius of the tube; and the Helmholtz development is still valid.
SOLID
SOLID
A
Figure 1.
B
Distribution of ions of double layer A.
B.
Helmholtz Govy, Chapman, Stern
Khen a liquid flows through a conduit, the ions of the movable la3 er are carried along with the liquid florv, If there is a sirplus of charge in the displaced lager (subsequently referred to a‘ displaceable charge), a streaming current is generated. This ionic convection current tends to causc a deficiency of charge a t the entrance of the conduit and a surplus of charge a t its exit. T o pre1-ent the forniation of the deficiency and surplus of charge a t the opposite end, a conduction current develops through any electrical path that mag be present. If no external path exists and the walls of the tube are a nonconductor, the only path for the conduction current is through the bulk of the solution. I n the development of the streaming potential equation, it is assumed that all conductances, other than that of the liquid, are negligible. The streaming potential i- the voltage generated when the ionic convection current is converted into the galvanic conduc1
2
Present address, Continental Oil Co., Ponca City, Okla Present address, University of Oklahoma, Norman, Okls.
February 1956
tion current which floas through the solution of the tube. The streaming current can be compared with the current which flo\$a in the interior of a battery between its terminals, while the conduction current would be the current which flows through the external circuit connected t o the same battery. In 1879, Helmholtz developed an equation from theoretical considerations shoving that the ratio of the streaming potential, E , to the pressure difference, AP, was a constant. The constant is directly proportional to the displaceable charge, 4 (which cannot be measured directly), and inversely proportimal t o the conductivity of the fluid, k , and viscority, i ~ . Thus _E = -
AP
+ 4 srpk
Later Pellat ( 7 ) , following a condenser analogy, modified this equation by substituting the product of the dielectric constant, E, and the zeta potential, c, for 4 , Thus
E - -Er _ 4P 4 rpk This equation is now generally referred to as the HelmholtzSmoluchowski equation, because Smoluchon-ski ( 1 1 ) generalized its validit,p for any shape conduit.. The zeta potential represents the difference in potential between the fixed and movable layers. Equation l a is used to evaluate the zeta potential from measurement of the streaming potential and pressure drop and known values of t’he dielectric constant, viscoity, and conductivity; however, some writers have questioned the use of liquid propert’ies obtained from measurements on the bulk liquid, as they believe t,hese properties are different within the double layer. The mathematical develGpments in Helmholtz’s original paper are difficrilt t o folloiv, pri.icipally because of undefined symbolization and confusing terminology. For example, in accord wit’h the accepted practice of his time, he considered electric potential arid charge as synonymous; however, his basic assumptions are adequately presented and are as follon-s: 1. The liquid motion is such that the act,ions of the rubbing liquid have time to be developed completely, so as to cause a steady state. (In terms of present-day fluid mechanics, the hoiindary layer fills t8herross section of the tube when thc flow becomes established.) 2. The displaced charged particles are moved with a constant, velocity parallel to the wall. Such a velocity can exist only if the entire liquid flow follows Poiseuille’s law. In 1928, Reichardt (9) showed in his doctoral research that it is not necessary to assume that the entire liquid flow be laminar in order that the dipplaced ions be moved with constant velocity parallel to the wall. The basis of his presentation was that a film exists a t any liquid-solid interface whose flow is always laminar, even if the flow in the interior of the liquid is turbulent; therefore, the necessary criterion is that the flow be “established” or be developed so as to cause a steady state (the velocity gradient near the wall is proportional to the pressure drop).
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ENGINEERING, DESIGN, AND EQUIPMENT
Figure 2.
Apparatus for measurement of streaming potential
Experimental equipment consists essentially of flow system and electrical measuring equipment
Measurements of the streaming potential by the different investigators have been characterized by a complete lack of agreement. The early investigations were handicapped by the polarization of electrodes and the lack of suitable instruments to measure the generated potential. The recent studies have been hampered by the difficulty of reproducing the surface conditions of the solids upon which the ions of a solution form a double layer. Most investigators have been primarily concerned with the use of the streaking potential as one of the means of evaluating the zeta potential, Emphasis in the current study is placed on t h e relation of fluid flow to the generation of the streaming potential. With the development of t h r electrometer vacuum tube as a measuring device and with the improved techniques for producing nonpolarizing or reversible electrodes, more precise measurements of the streaming potential are now possible; consequently the streaming potential provides a useful means for studying fluid flow. The experimental equipment used in this study is shown in the schematic diagram of Figure 2. The apparatuR consists essentially of the flow system and the electrical measuring equipment. Details of the equipment have been reported (I).
Table I. Tube No. 1 2 3a 3b 4
5 6 7
198
Composition and Dimensions of StreamingPotential Tubes Composition Quartz Borosilicate glass Borosilicate glass Borosilicate glass Borosilicate glass Teflon Teflon Butyl acetate Tap A Tap B Tap C Tap D Tap E Tap F Tap G Tap H Tap I
Diameter, Mm. 0.86 0.475
1.00 1.00 3.08 0.31 1.33 4.9
Length, Cm. 89.35 83 82 83.82 76.53 83.82 11.57 12.75 93.75 0
30.48 60.96 70.20 88.90 90.40 91.90 93 ;40 93.75
Flow System. The floiv system was composed of two borosilicate glass solution reservoirs (14-liter) located a t either end of a streaming potential tube. The tubes used are described in Table I. The solution reservoirs xvere connected individually t o a prcssurized nitrogen source, so that flow could be made to take place in either direction through the tubes. Teflon was used t o fabricate the two stopcocks and electrode chambers. KO sealing compounds were used in connecting the various components of the flow system, thereby eliminating a possible source of contamination of the solutions. The electrode chambers were large in diameter as compared with the diameter of the streaming potential tubes. The connections between the electrode chambers and streaming potential tube constitute a sharp-edged entrance. The silver-silver chloride electrodes were located in the electrode chambers rather than in the tube wall, so as not to disturb the flow in the tube itself. In the case of tube 7 , Table I, nine holes were drilled through the wall of the tube. The electrode chambers could be attached to any of these taps, so as to obtain the streaming potential along the tube length. The fluids used were dilute solutions of sodium or potassium chloride in distilled water. Electrical Equipment. A null technique was used to obtain precision measurements of the streaming potential by impressing a counterpotential originating in the variable (direct current) voltage supply. The null balance was detected through a specially designed electrometer detector and a high-gain amplifier (modifications of the Sanborn Twin-viso type). To observe consecutive variations in the streaming potential as the flow was changed from laminar to completely turbulent flow, the Sanborn amplifier and strip recorder were operated a t low gain. The entire flow system was shielded by '/4-inch mild steel plate, with the exception of one reservoir which was a t the grounded end of the electrical system. The apparatus permitted direct observations of the liquid level in the grounded reservoir. The difference in liquid levels in the reservoirs, added t o the nianometer reading of the nitrogen pressure difference between the reservoirs, constituted the pressure difference across the flow system. Equipment Innovations. While the equipment of this study conforms to the standards established as necessary by other investigators, it is different from their equipment in several
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
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ENGINEERING. DESIGN. AND EQUIPMENT
IO 3
102
Re
Figure 3.
Ratio of streaming potential to pressure difference for established flow vs. Reynolds number
+ 0
E B C / A P B ~ mv. , per
cm. Hg ( 7 ) Scale units, 10 per cm. Hg ( 9 )
aspects: The tubes can be interchanged with comparative ease; no sealing compounds are used in connecting the various components of the f l o system, ~ eliminating a possible source of contamination of the solutions; the electrometer detector and the amplifying system have a higher frequency response (up to 50 crcles per second) than those commerciallr available (up to 10 cycles per second); the amplifj ing sj-stem can be standardized to eliminate the effect of drift; and fixed resistors can be connected across the streaming potential tiihe by turning a switch, permitting the use of the streaming current as the source of energv to me:isure the electrical resistance of the flowing liquid in the tube. Experimental Procedure. Piior to each experimental run, the flow equipment was carefully cleaned by using sulfuric aciddichromate cleaning solution and detergent, followed by thorough rinping with double-distilled water for several hours. The rleaned tubes were then stored in distilled water until used. The stabilization of the electronic equipment usually required about 0 5 hour. Nitrogen mas then fed into one of the reqervoirs to obtain a steady-state flow through t h e tulles. The teniperature of the flon-ingfluid was measured by the therniocouple located in a glass tube which was 8 inserted int,o one of the electrode chamlwrs. The streaming potential for the steady flow rate was measured by estab, lishing a null point Jx-ith the varizible 7 voltage supply and reading the coiinterpotent,ial. Simultaneous presmre read0) ir?gs of t,he manometers attached to the I reservoirs were taken. In order to obtain the pressure drop between the res5 6 \ erooirs, the change in liquid level of \A \ 3 the grounded reservoir T T also ~ measE ured. As it was found tliat the pressurc drop in the delivery lines beta-een the p . 5 reservoirs and the ends of the strenming potent,ial tube was negligible, the pressure drop across the streaming potential tube n-as essentially equal to t,hat between the reservoirs. For tube 7 . it was not mact'ical because of electrical shielding and groiinding complications t o record siniultaneoiisly the streaming potential and pressure difference between the taps along the tube. These mmwrenients \Lele
February 1956
made separately as functions of the pressure drop between the reservoirs. From these values the relationship between pressurr drop and streaming potential between any two taps was oiitainctl Helmholtz-Smoluchowski equation i s valid for both laminar and turbulent flow
The results of the preliminary experiments on the streaming potential were not in agreement with current literature on electrokinetics. For example, contrary to accepted opinions, it was found that the streaming potential was proportional to theprr3sure drop for all flow rates, regardless of whether the bulk flow is laminar or turbulent. Helmholtz in his original paper indicated that his equation was valid only for fully established laminar floiv; however, in 1928, Reichardt experimentally shoxed that Eqiiation 1 was valid in both the laminar and turbulent, provided the f l o Tvas ~ fully established. Subsequent investigators ( 1 j not only have ignored
S Y M B O L - T U B E NO i C F - TABLEI) 0 I A 2 0 3
\ O\
o--o-o-o
f 0
-
A
-
a
-
4
5 6 7
< ul
W
4
Io2
lo3
lo4
Re
Figure 4. Ratio of streaming potential to pressure difference measured between points in front of tube entrance and beyond exit vs. Reynoldsnumber
INDUSTRIAL AND ENGINEERING CHEMISTRY
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ENGINEERING, DESIGN, AND EQUIPMENT Reichardt’s work but have overlooked an important emphasis in Helmholtz’s work by using laminar flow rather than fully established flow as a criterion for the validity of Equation 1. I n Figure 3 are shown both the data of the present investigation and those of Reichardt (9). I n this figure, the streaming potential and pressure drop were measured in current study across the middle third of the tube to ensure fully established flow, whereas Reichardt’s data were obtained on the exit half, Within the range of experimental conditions encompassed both by Reichardt and in the current study, the important factor is that the flow be fully established, not whether the flow is laminar or turbulent. I n Figure 4 the streaming potential and the pressure drop across the entire length of the tubes are correlated. Entrance and exit effects are consequently included in these data. It can be seen that, if the flow is not fully established because of entrance or exit effects, the ratio of the streaming potential to the pressure difference is not constant and Equation l a does not apply. This invalidity may also be true even in the laminar regions. The flat portions of the curves for tubes, 1, 2, 3, and 5 result from the fact that with low flow rates and small tube diameters the entrance and exit effects are substantially negligible, and the results are in agreement with Equation la. I n contrast, the entrance effects for tube 6 are appreciable even at a Reynolds number of 100. Reichardt had postulated that even in turbulent flow a viscous film existed adjacent to the wall; however, in 1949, Miller (6) questioned this belief on the basis that no satisfactory velocity measurements a t the wall have ever been made. It does not s e e m possible, however, that the data as shown in Figure 3 would have been obtained unless t h e
Conclusions. As a result of this study, the following conclusions are reached. The Helmholtz-Smoluchowski equation for the streaming potential is valid for both laminar and turbulent flow, provided the flow is fully established. The existence of a laminar film a t the wall in turbulent flow is further substantiated. Streaming potential can be utilized as a technique for studying fluid flow phenomena near the wall.
velocity near the wall was directly p r o p o r tional to the pressure
Acknowledgment
drop, a condition which can be fulfilled only if the eddy viscosity near the wall is small or approaching zero. This condition is in effect a sufficient c r i t e r i o n for laminar flow near the wall. As the flow changes
a
F
3
B
D
C TIME
Figure 6. Tape record of streaming potential as flow rate increases AB.
BC. DC.
Laminar Transition Turbulent
This investigation was supported in part by a research grant PHS H-670, from the National Heart Institute, National Institutes of Health, Public Health Service (1949-51). The study was supported during 1951-52 by a Michigan Heart Fellowship in the Department of Chemical and Metallurgical Engineering at the University of Michigan.
r? 0
5
4 E
literature cited
TIME
alternately from lamFigure 5. Tape record of streaminar t o t u r b u l e n t ing potential as flow changes flow within the transifrom transient to laminar flow tion region, the rapid decrease of the velocity in the center of the tube produces a surge in the velocity near the wall, This phenomenon was indirectly observed by sporadic pulsations in the streaming potential in the transition region, as shown in Figure 5, which is a photograph of the strip chart recording of the Sanborn amplifier and recorder. The fluctuations that occur near the wall during turbulent flow can also be indirectly observed by streaming potential measurements as shown in Figure 6. As the streaming potential is essentially a surface phenomenon, i t is apparent from the figure that the velocity fluctuations penetrate t o the wall. This turbulent effect in the laminar film has been observed by other methods
Bocquet, P. E., “Streaming Potential Concept,” Ph.D. dissertation, University of Michigan, 1952. Chapman, D. L., Phil.Mag. 25, (6) 475 (1913). Gouy, G., J. phys. 9, (4) 457 (1910). Helmholtz, H., Ann. Physik. 7, (3) 337 (1879); “Gesanimte Abhanelungen,” vol. I, p. 885, 1882; tr. by P. E. Bocquet, “Two Monographs on Electrokinetics,” Univ. Mich. Eng. Research Bull. 33 (1951). Lin, C. S., Moulton, R. W., and Putman, G. L., IND. ENQ. CHEM.45, 636 (1953). Miller, B., Trans. Am. SOC.Mech. Engrs. 71, 357 (1949). Perrin, J., J. chim. phys. 2, 601 (1904). Quincke, G., Ann. Physik 107, (2) 1 (1859). Reichardt, H., Z. physik. Chem. A174, 15 (1935). Reuss, F. F., MJm. sac. imperiale naturalistes Moscou 2, 327 (1809).
Smoluchowski, M., “Elektrische Endosmose und Stromungsstrijme,” “Handbuch der Elektrizitat und des Magnetismus,” vol. 11, p. 366, Graetz (ed.), Barth, 1921 ; tr. by P. E. Bocquet, “Two Monographs on Electrokinetics,” Univ. Nich. Eng. Research Bull. 33 (1951). Stern, O., Z. Elektrochem. 30, 508 (1924). RECEIVED for review June 17, 1955.
(6).
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ACCEPTEDIVovember 28, 1955.
OF ENGINEERING, DESIGN, AND EQUIPMENT SECTION
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