Streaming Potential Fluctuation around a Cylinder in Water Henry Liu University of Missouri, Columbia, Mo. 65201
Gilbert Binder Laboratoires de Me‘canique des Fluids, Universite’ de Grenoble, Grenoble, France
Jack
E.
Cermak
Colorado State University, Fort Collins, Colo. 80521
Streaming potential fluctuations around a circular cylinder held perpendicular to flows of distilled water were sensed by a pair of electrodes installed on the cylinder wall. From the intensities and spectra of the measured signals, information on the characteristics of turbulence in the diffuse layer around the cylinder was obtained. Free stream turbulence has a profound effect on turbulence in the wall region around a cylinder. The periodic shedding of vortices also affects the flow in this region to a large extent. A new method for measurement of vortex shedding frequencies has been found as a result of this study.
STREAMING
POTENTIAL is used here to denote the electrical potential difference between any two points in a liquid produced by the transport of charges in the diffuse layer of a solid-liquid interface by fluid motion. The flow referred to in the above definition need not be a pipe flow or flows through a membrane or porous material, although these are the streaming potential flows most commonly studied. When a flow becomes turbulent, fluctuations of charge transport due to turbulence cause the streaming potential to fluctuate. B y placing two electrodes on the wall in contact with fluid, potential fluctuation, may be measured between the electrodes. This potential fluctuation is hereafter termed the “streaming potential fluctuation’’ (SPF). I n the literature, the term “electrokinetic potential fluctuation” is used as a synonym. Since S P F is generated by fluctuations of diffuse-layer charges, measurements of it are expected to reveal information on turbulence characteristics near wall. This seems to be the primary purpose of most investigations in SPF. I n the literature, Bocquet (1952), Bocquet et al. (1956), Binder (1960), Binder and Cermak (1963), Chuang (1962), Chuang and Cermak (1967), Duckstein (1962), Duckstein and Cerinak (1964), Liu (1966, 1967), and Chartier et al. (1967) studied different aspects of the problem. Measurement of S P F in water is not easy, and even more difficult is often the proper interpretation of data. Although a differential equation governing S P F has been derived by Binder (1960) and modified by Liu (1966, 1967), the equation cannot be solved. This limits the usefulness of the equation to the prediction of variables upon which S P F depends. The form of the dependence must be determined experimentally; usually by varying one or two variables each time in a n otherwise fixed experimental system. I n the literature of SPF, much disagreement exists amoiig various investigators concerning observations and data interpretation. Contradictions of ten result from the difficulty of making good measurements and the lack of a theory that explains experimental data. Sometimes they
also arise from a failure to define the experimental system adequately. Despite the disagreement, certain common findings are reported and common opinions shared by all or most investigators. When measurements are proper and free of interference, the S P F between two closely spaced electrodes is believed to be caused by the disturbance of charges by turbulent velocity fluctuations in the diffuse layer of the electrode-water interface. The spectra of S P F resemble turbulence spectra. When the Reynolds number of a turbulent shear flow is increased, the frequency of the peak of S P F spectrum also increases. The intensity of S P F increases with turbulence intensity. S P F intensity decreases as the conductivity of fluid is increased, but the variation in conductivity has little effect on the shape of the S P F spectra. This paper reports data of S P F around a circular cylinder held perpendicular to flows of distilled water. The data are interpreted based on the first three of the four points mentioned above. Since no attempt was made in this study to change the conductivity of water, the fourth point is not discussed. I n this study, the S P F measured around a cylinder reflects (or transduces) turbulent velocity fluctuations in the diffuse layer around the cylinder. Since the thickness of the diffuse layer in distilled water is of a n order of only 100 -4.any information on the characteristics of turbulence in the diffuse layer around the cylinder is of interest because of its importance in understanding the variation of local heat transfer characteristics around a cylinder in turbulent flows, and the inaccessibility of ordinary turbulence transducers to a place so close to the cylinder wall. Experimental
Figure 1 is a schematic of the cylinder and the electrodes used. The cylinder was a 0.21-cm.-diameter stainless steel tubing approximately 10 em. long. Two 0.2-mm.-diaineter platinum electrodes, spaced 0.2 mm. apart and aligned along the circumference of the cylinder, were used to sense the Ind. Eng. Chem. Fundom., Vol. 9, NO. 2, 1970
21 1
ULATOR (Polyester Laminating Resin1
COPPER-WIRE CONNECTIONS
ELECTRODES
hf
SEC. A-A
0 21cm
Figure 1.
Test cylinder and electrodes
ELECTRODES ,ELECTRODE 2 PIPE WALL (Plastic)
ELECTRODE 1
?!-$+ FLOW
TEST CYLINDER (Stainless-steel tubing)
Results and Discussions
U
Figure 2.
Cylinder placed in pipe flow
Electrode I,Slngle Ended
.-C-.-€-.
..&..--e.. D i f f e r e n i i o R e =95,000
0
212
k.
1
'
potential fluctuation. The electrodes were held in the cylinder by cement, which also provided the necessary insulation between electrodes and the cylinder and between the electrodes themselves. At the location of the electrodes, the surface was carefully polished, to obtain a good cylindrical surface and to eliminate any irregularities. The cylinder was then placed in a horizontal pipe flow of distilled water, with its axis crossing the axis of the pipe a t right angles, and the electrodes located a t the center of the pipe (Figure 2 ) . The cylinder could be rotated around its own axis, making it possible for the electrodes to face in different directions. As shown in Figure 2, the angle e is 0 when the electrodes face upstream and 180" when they face downstream. The recirculating flow system consisted essentially of a Lucite pipe (id., 2.54 cm., length, 7 meters), a constant head tank, an orifice meter, a control valve, and a pump. The pipe is connected to the head tank through a well-rounded transition. The signal from the electrodes was fed to a lownoise preamplifier (Tektronix Type 122) which could be operated either single-ended (with one end grounded) or differentially (both ends floating). The preamplifier had a gain of 1000, a flat frequency response from 0.8 to 10,000 cycles per second, and an input impedance of 5 to 10 megohms for single-ended and differential operations, respectively. The amplified signal was fed t o a true root-mean-square voltmeter (Bruel and Kjaer, Type 2416), a wave analyzer (Bruel and Kjaer, Type 2109), aiid an oscilloscope. To eliminate extraneous signals, especially 60-cycle pickup, the flow system was housed in a metal shielding and several points of the apparatus, especially the test cylinder, the pump, and the shielding, had to be grounded. Xore detailed description of the apparatus and test procedure is given by Liu (1966).
1
30° 60° 90°
'
'
'
"
'
150°'800 210°
Ind. Eng. Chem. Fundam., Vol.
9, No. 2, 1970
'
'
270°
33Oo36O0
SPF Intensity around Cylinder. Figure 3 shows the SI'F intensities around the cylinder when it was placed at, x/D 180, a place where both the mean velocity profile and turbulence has become fully developed. The Reynolds number of the pipe flow, Re, was 95,000. The most salient feature of the single-ended signals is the presence of maxima a t e = 0" or 360°, 90" or 270°, and 180". Froin turbulence theory (Hinze, 1959)) it is known that a n increase in mean flow velocity generally proniot'es a decrease in turbulence energy, aiid vice versa. Since in the front part of t'he cylinder the mean velocity near the wall is a minimum at e = 0" (stagnation point), and increases as e increases, the intensity of turbulence near the wall is expected to be a maximum a t e = 0", and decreases as e increases. This is exactly what is revealed by the S P F data in Figure 3. At e = 90" and 270°, flow separation occurs and a sudden increase in flow disturbance is expected. This explains why the S P F intensity increases as 0 = 90" and 270" are approached. Finally, to understand why anot'her niaximuni occurred at e = B O " , one has only to realize that the turbulence int,ensity in the wake of the cylinder is a riiaxiniuni there. The differential signal between the two electrodes is also given in Figure 3. Siiice the differential signal is essentially the difference of the two single-ended signals, the fact that the former is considerably smaller than the latter suggests the existence of a high degree of correlation between fluctuations above the two electrodes. The fact that differential signals are not synimetric with respect to e = MOO, especially in t,he region behind separation points, indicat'es an unsteadiness in correlation between the fluctuations sensed by the two electrodes behind the cylinder. More data on SPF intensity around the cylinder a t x / D N 180 are given in Figure 4. Figure 5 gives the signal intensities around the cylinder
,
I50O1
,
,
,
8
,
,
,
,
I
I
I
(degree)
Figure 4. Variation of streaming potential fluctuation around a cylinder in turbulent flow x / O 'v 180, single-ended operation IO'
I o3
IO
FREQUENCY I
I u) 0,
(c ps
Figure 6. Streaming-potential-fluctuation a cylinder in flow with little turbulence
spectra around
Entrance of pipe, single-ended operation
.-E
m=
2 x 10-4mho /meter
. I -
O
noise ratio. Despite the irregularity, curves in Figure 5 seem to possess maxima a t 0 = go', 180', and 270". That the signal was a minimum a t 8 = 0' confirms the previous view that the signal in front of the cylinder was caused by free-stream turbulence. As a t x / D = 180, the maxima a t 90" and 270' are attributed to flow separations, and a t 180' to a turbulent wake.
0
> O
-I
a z*
:-
SPF
200
Spectra around Cylinder
t
0
> ..--
E
I00
0 Oo
60"
120°
180"
8
(degree)
240'
300'
360'
Spectra of single-ended signals were measured a t three positions around the cylinder: 8 = 0', 90', and 180'. Figures 6 and 7 give the spectra measured near the pipe entrance for t w o different values of V,. A hump is seen in every spectral distribution curve. The frequency corresponding to each hump was compared to the vortex shedding frequency of the cylinder calculated from the following equation-see, for instance, Figure 2.9 of Schlichting, 1960:
Figure 5. Variation of streaming potential fluctuation around a cylinder in flow with little turbulence
S (Strouhal number)
=
f6 d = 0.21
Entrance of pipe, single-ended operation
U
(1)
when when it was placed near the pipe entrance. The signal intensities in this case are much smaller than those measured a t x / D N 180. This reflects that little turbulence existed near the pipe entrance. The data in Figure 5 also show considerable irregularity, reasons for which are the occurence of intermittency near pipe entrance, and a poor signal-to-
re > 1,000
It can be seen from Figures 6 and 7 that the frequencies of these spectra peaks are almost identical to those calculated from Equation 1. Since the hump appeared in all the three measured positions, with the sharpest peak a t e = 90', it is Ind. Eng. Chem. Fundam., Vol. 9,
No. 2, 1970 213
O'O
0'
o-o--ao
U Q =262cm/sec
-A
Uc =3lOcm/sec
R, re fsc fsa
& O '
:82,000 =?,I70 =313 cps 5262 c p s
I
I I 102 ' 8 0
IO'
Id
lo3
FREQUENCY (c p s)
FREQUENCY
Figure 7. Streaming-potential-fluctuation spectra around a cylinder in flow with little turbulence Entrance of pipe, single-ended operation
1
200
( c p SI
Figure 9. Streaming-potential-fluctuation spectra around a cylinder in turbulent flow x/D
'v
180, single-ended operation
U Uc Re re
= 152 c m / s e c =182crn/sec
-52,000 -4,200
fsc= 184cps
f s a = 152 C P S
I
230
IO2
IO'
FREQUENCY
lo3
lo4
( c p s)
Figure 8. Streaming-potential-fluctuation spectra around a cylinder in turbulent flow x/D
214
'v
180, single-ended operation
Ind. Eng. Chem. Fundam., Vol.
9, No. 2, 1970
\I b
FREQUENCY
(cps)
Figure 10. Streaming-potential-fluctuation spectra around a cylinder in turbulent flow x/D
'v
180, single-ended operation
clear that vortex sheddings disturbed the flow in the wall region all around the cylinder, with maximum influence near the sides of the cylinder. This finding is consistent with the more recent findings by Son and Hanratty (1969). The spectra measured a t x / D 'V 180 (Figures 8 to 11) are different from those measured near the pipe entrance, in that the effect of frequency shedding can be seen only a t 8 = 90". This indicates that whereas the velocity field near a cylinder wall has a periodic component all around the cylinder when the outer flow is laminar, only the two sides of the cylinder are influenced by vortex shedding when the outer flow is turbulent. Another interesting feature about the spectral curves at x / D 'V 180 is the fact that each measured peak (when 8 = 90') is a t a frequency below the vortex shedding frequency calculated from Equation 1, using the centerline velocity, U,. For the two sets of curves a t lower Reynolds numbers-i.e., for R , = 31,000 and 53,000-the peaks are a t frequencies equal to the vortex shedding frequencies calculated from Equation 1 based on the mean velocity, U,. This suggests that despite the velocity variation along the radius of the pipe, Equation 1 is still valid if U , is substituted for U in the equation. However, this relationship does not hold a t higher Reynolds numbers. This can be seen from the two sets of curves at Reynolds numbers equal to 82,000 and 95,000, in which the peaks are a t frequencies even below the vortex shedding frequencies calculated from Equation 1 using U,. Thus, it seems that free stream turbulence decreases the Strouhal number to a value below 0.21. Conclusions a n d Discussion
From SPF sensed between a grounded metal cylinder and a small electrode on the cylinder wall, the following qualitative information on velocity fluctuations a t a distance within 1000 A. from the cylinder surface was found: \Then free stream turbulence is present, the turbulence in the wall region is a maximum a t the front stagnation point -Le., a t 8 = 0-and it decreases as e increases from 0" to about 60'. This is mainly a result of mean flow acceleration in that region. When 8 becomes greater than 60", disturbances caused by flow separation and shedding of vortices become important, causing the velocity fluctuation to reach a new peak a t 0 = 90". Behind the cylinder, turbulence intensity first decreases as one leaves the separation points but then increases as the influence of wake becomes increasingly important, reaching another peak a t the rear stagnation point. I n contrast to the four peaks of turbulence intensity around the cylinder as in the previous case, there are only three peaks when the free stream is laminar, two a t e = 90" and 2 i O o , caused by flow separations and vortex shedding, and one a t 8 = 180", caused by the turbulent wake. The magnitude of the turbulence intensity around the cylinder wall in this case is also much smaller than in the previous case with free stream turbulence. This difference in turbulence characteristics around the wall for the two cases may account for the difference in heat transfer coefficient around a cylinder held in flow with or without free stream turbulence. Without free stream turbulence, shedding of vortices causes a periodic velocity fluctuation not only on the sides of the cylinder but also around the entire circumference. On the other hand, with free stream turbulence flow periodicity can only be felt on the sides. When the mean velocity varies along the axis of a cylinder, the velocity in the Strouhal number used to calculate the
U,
=89cm/sec
Uc = i 0 7 c m / s e c
R e =31,000 r e = 2,475 fSC = l o s c p s fs,=89 c p s
O-O\
O\
I o2 FREQUENCY
IO'
Io3 (c ps)
4
IO
Figure 1 1. Streaming-potential-fluctuation spectra around a cylinder in turbulent flow x/O
'v
180, single-ended operation
vortex shedding frequency should not be the local mean velocity, but rather a n average over the length of the cylinder. Free stream turbulence has a definite effect on the frequency of vortex shedding from a cylinder. When the scale of the free stream turbulence is larger than the diameter of the cylinder, the effect tends to be a lowering of the Strouhal number. As a result of this study, a new method for the measurement of vortex-shedding frequency is found. This method is capable of measuring shedding frequency in flows with various degrees of free stream turbulence, if the electrode on the cylinder is situated near a separation point. Nomenclature
d
= diameter of the cylinder, cm.
D
=
f.
= vortex shedding frequency, cycles/sec. = vortex shedding frequency calculated from
pipe diameter, cm.
U,, cycles/ sec. fa0 = vortex shedding frequency calculated from U,, cycles/ sec. re = Reynolds number defined as U , d / v Re = Reynolds number defined as U , D / v S = Strouhal number, defined by Equation 1 U = mean velocity, cm./sec. U , = discharge velocity in pipe, cm./sec. U , = mean velocity a t center of pipe, cm./sec. z = distance from pipe entrance, meter fsa
GREEKLETTERS 8 = angle defined in Figure 2, degree Y = kinematic viscosity u = fluid conductivity, mhos/meter Ind. Eng. Chem. Fundam., Vol. 9, No. 2, 1970
21 5
literature Cited
Binder, G. J., Ph.D. dissertation, Colorado State University, Fort Collins, Colo., 1960. Binder, G. J., Cermak, J. E., Phys. Fluids 6 , 1192 (1963). Bocquet, P. E., Ph.D., dissertation, University of Michigan, Ann Arbor, 1952. Bocquet, P. E., Sliepcevich, C. >I., Bohr, D. F., Ind. Eng. Chem. 48, 197 (1956). Chartier, C., Dumargue, P., Rouchart, A., C. R. Acad. Sci. 264. Ser. A. 67. 142 11967). Chuang, H.,’ PhD. dissertat,ion, Colorado State University, Fort, Collins. 1962. Chuang, H., CGmak, J. E., A.I.Ch.E.J. 13, 266 (1967). Duckstein, L., Ph.D., dissertation, Colorado State University, Fort Collins, 1962.
Duckstein, L., Cermak, J. E., Znt. J . Heat Muss Transfer 7 , 159 (1964). Hinze, J. O., “Turbulence,” p. 66, McGraw-Hill, New York, 1959. Liu, H., A.I.Ch.E.J. 13, 644 (1967). Liu, H., Ph.D. dissertation, Colorado State University, Fort Collins, 1966. Schlichting, H., “Boundary Layer Theory,” McGraw-Hill, New York, 1960. Son, J. S., Hanratty, T. J., J . Fluid Mech. 35, 353 (1969). RECEIVED for review December 5 , 1968 RESUBMITTED December I ? , 1969 ACCEPTEDFebruary 12, 1970
Separation of Gas Mixtures by Pulsed Adsorption Theodore J. Jenczewski’ and Alan 1. Myers Department of Chemical Engineering, University of Pennsylvania, Philadelphia, P a .
A closed, thermal, pulsed adsorber was studied. Separations of gaseous mixtures were obtained experimentally for argon-propane and ethane-propane mixtures. No separation was observed for a propanepropylene mixture. An equilibrium model, the parameters of which are measured independently, provides good agreement with the experimental results.
SEPARATION
PROCESSES are of unique and fundamental importance to the chemical process industries. Consequently, researchers are continuously searching for new and improved separation techniques. Recently, there has been a renewal of interest in the application of coupled driving forces to effect separations. Of particular interest in this regard is the socalled parametric pumping process as proposed and developed by Wilhelm and his coworkers (Rolke and Wilhelm, 1969; Wilhelm, 1966; Wilhelm and Sweed, 1968; Wilhelm et al., 1966, 1968). The essence of the process concept is that an oscillating (in time and space) fluid phase mass velocity is coupled to a n oscillating fluid phase-adsorbate equilibrium relationship in such a manner as to produce a separation effect. Several means of accomplishing the coupling have been proposed and studied mathematically. The results have shown that significant separations can be obtained for the proper synchronization of the driving forces.
position I
T=
pos:tioi I 1
= Thigh
TIOW
1 7 - g phase ~
m-a d s o r b e d phase Figure 1. 2 16 Ind.
Eng.
Model of pulsed adsorber
Chem. Fundam., Vol. 9, No. 2, 1970
This report presents the results of an investigation into the application of pulsed adsorption to the separation of gaseous mixtures. The operating mode selected for study is a closed, thermal, pulsed adsorption system. The word “closed” denotes that the concentration gradient established in a n initially homogeneous system is studied and that separated components are not continuously withdrawn. The word “thermal” is used to indicate that it is the system temperature that is used to oscillate the fluid-adsorbate equilibrium relationship. Origin of Separation
A schematic diagram of a closed, thermal, pulsed adsorption system is shown in Figure 1 which illustrates the two halves of a n operating cycle. The fluid and adsorbed phases have been separated for clarity. The fluid phase is moved relative to the adsorbed phase, as indicated by positions I and 11. The system is in position I a t TI,,,,., pulsed to position 11, heated to Thigh, pulsed to position I, cooled to TI,,,, etc. The fluid-adsorbate equilibrium relationship is oscillated by oscillating the system temperature. The volume fraction of fluid displaced from the bed is arbitrarily chosen as one third. The temperature and fluid velocity may be considered to be driven by some external means. An operating cycle, assuming only one component of the binary mixture is adsorbed, is described in order to emphasize the characteristics of the cyclic process and the driving force for the separation effect. The system is initially in position 11, for which the temperature is uniform and equal to Thigh. The fluid and adsorbed 1 Present address, E. I. du Pont DeXemours & Co., Wilmington, Del. 19898
