verter and circuit time constants, a total transmission rate of 10 words/sec will be approached, limited of course by the 100-msec tirne constantof monostable 1. useof a fastercycling A D converter, and the appropriate time constant for monostable 1, will allow transmission rates of over 1000 charactedsec. No other changes in the external Circuitry need be made.
Literature Cited Hegedus, L. L., Petersen, E. E.. J. Chrometog. Sci., a, 551 (1971). “Serdex User’s Guide,” Analog Devlces, Route One, Industrial Park, Norwood. Mass. 02021.
Received for reuiew July 14,1975 Accepted November 3,1975 ‘The authors wish to acknowledge the support of the NSF for this work under grant GK-36495
COMMUNICATIONS
Flow of Dilute Polymer Solutions through a Packed Bed
The flow through a packed bed was evaluated experimentally as a simple qualitative method for the estimation
of the degree of elasticity in dilute polymer solutions. The pressure drops for the dilute polymer solutions (Separan and Polyox) were found to be about 2-9 times higher than those for Newtonian fluids with identical viscosities. Plots of relative pressure drop show distinct maxima for different polymer concentrations. Solutions of higher Separan concentrations, believed to exhibit more pronounced elastic effects, result in pressure peaks at lower flow velocities or Reynolds numbers. The effects of solution aging and polymer degradation were also studied.
Introduction The flow of dilute polymer solutions, in various geometries, is interesting from the point of view of both theoretical studies in rheology and practical applications. One of the latter is the drag reducing ability of small quantities of polymer additives on bodies moving inside liquids or fluids flowing inside tubes or other geometries. A recent review by Hoyt (1972) describing drag reducing studies lists 219 references. Since its appearance, this number has been a t least doubled, and a further future increase in scientific activity in this field is highly promising. Most of the papers dealing with experiments describing flows of dilute polymer solutions give very little data on the rheological properties of the fluids. This is not merely an oversight on the part of the investigators. The fact is that these fluids exhibit true Newtonian behavior under conditions of viscometric flows. Measurements of normal stresses, however, must be performed with sophisticated instruments, such as the Weissenberg Rheogoniometer. Such measurements either show nothing or their effect is masked by other effects such as surface tension or mechanical imperfections in the measuring instrument. Such results are due to the small absolute magnitude of the elastic effects themselves. The purpose of the present paper is to evaluate the flow through a packed bed as a simple method by means of which the investigator dealing with flows of dilute polymer solutions will be able to estimate the degree of elasticity in the fluid. It will be shown that the use of a packed bed for fluid characterization will allow classification of the fluids in order of increasing elastic effects. It may also indicate whether a newly prepared batch of fluid duplicates the viscoelastic properties of the previous batch. Effects such as 74
Ind. Eng. Chem., Fundam., Vol. 15, No. 1, 1976
the aging of the solution and degradation of the polymer can easily be followed by the use of the packed bed as a control instrument. Still, the proposed technique will not provide an absolute numerical value for the elastic effects, such as the time of relaxatiqn, normal stress difference, etc. Some recent investigations dealing with flow of polymer solutions through porous media have indicated that the same fluid solutions exhibiting drag reduction effects when flowing through a tube also exhibit at high concentrations of the polymers an increased pressure drop when flowing through a tube filled with a porous packing, as compared to a Newtonian fluid of the same viscosity (Kemblowski and Mertl, 1974b; Marshall and Metzner, 1967; Siskovic et al., 1971; Wissler, 1971). The difference between the flow through an empty tube and that through a tube filled with packing is in the fact that the first flow is viscometric while the latter is a stretching flow. Thus, by using strongly elastic fluids, flowing in laminar motion through a porous medium, Marshall and Metzner (1967) were able to show that increased elastic effects, as characterized by the relaxation time, result in an increase in pressure drop as compared to a nonelastic fluid (an aqueous Carbopol solution). Siskovic et al. (1971) obtained similar results with a molten polymer. We were interested in finding whether this effect of increased pressure drop also exists in the flow of dilute polymer solutions through a packed bed. In order to improve the odds in our favor, we have looked at fast flows as characterized by Re > 10 (transition from laminar to turbulent region), as opposed to the previous investigations which were performed for the laminar region. The effect of elasticity on flow may be described by the Deborah number, De = Ruo/d,. The higher De, the higher the effect. Thus, with dilute polymer solutions one has to
go to higher velocities before the appearance of elastic effects. This is exactly what has been done in the present investigation.
C
Literature Background The basic correlation describing the dependence between the pressure drop and fluid velocity for a Newtonian fluid is the Ergun equation (Ergun, 1952) which is herewith presented for a packed bed composed of spherical particles
This equation describes the motion of the fluid both in the laminar region and in the transition and turbulent regions. The first term on the right-hand side of eq 1 which applies to the laminar region only is commonly known as the Blake-Kozeny correlation (Kemblowski and Mertl, 1974a). Christopher and Middleman (1965) have modified this correlation for power law fluids. The basic flow model underlying the Blake-Kozeny equation assumes the flow taking place through a set of parallel channels of uniform cross sections. Thus in effect the flow through a BlakeKozeny type porous medium cannot exhibit elastic effects due to the absence of stretching flow. Even so, recent workers (Kemblowski and Mertl, 1974b; Marshall and Metzner, 1967; Sadowski and Bird, 1965) have found this correlation valuable in a sense, as it provided a reference, or base, from which departures due to elastic effects could be measured. Sadowski and Bird (1965) presented the Ellis number, El, as a criterion for departure from non elastic behavior. This number was defined as characteristic time for the fluid El=-- R (2) characteristic time for the flow system d,/UO They found that the departure occured for El = 0.10. The definition of El as given in eq 2 coincides with that of the Deborah number. Marshall and Metzner (1967) have assessed theoretically that viscoelastic fluids will deviate from the modified Blake-Kozeny correlation for De > 0.25. However, their experimental results showed that deviations occured already a t De = 0.05-0.06. Siskovic et al. (1971) found experimentally that the critical De was 0.19 or higher. This result is supported by the results of Kemblowski et al. (1974a), who have found this value to be 0.20 or higher. Slattery (1967) has considered the flow of Noll’s “simple” fluid through a porous medium. Using dimensional analysis he has found that the resistance of a packed bed to flow can be expressed by a Darcy type constant
k t = wo/dp2ko*
(3)
where ko* = h[u&/d,]. Here the bracketed expression is essentially the Deborah number. In summary, we note that there exists both theoretical and experimental evidence indicating that elastic effects, as expressed by De, affect the flow of a viscoelastic fluid through a porous bed. In the present paper we look a t this phenomenon from a different point of view: we use the deviation of a viscoelastic fluid from Newtonian behavior to point out the existence of elastic effects.
Experimental Section The experimental setup is schematically presented in Figure 1. The fluid to be characterized was introduced from the storage tank (C) into a 2-1. operating tank (A) located above the packed bed (B). The packed bed consisted of a 7.3 cm long Plexiglas tube of 1.6-cm internal diameter, packed with glass spheres of 0.1-cm diameter. The operat-
Figure 1. Experimental setup. I
I
I
I
I
’
1
J
710
Od-O‘
I
9/0
110
I30
i 150
Rep.?
Figure 2. Comparison between experimental results for a Newtonian fluid and Ergun’s correlation. ing pressure of the system was provided by means of a compressed air cylinder (D) equipped with a pressure regulating valve. The pressure in the tank (A) was measured by means of calibrated manometers. The packed bed was contained between two wire screens. The Plexiglas tube was terminated by means of a 2-in. ball valve (E), which when open provided a flow cross section larger than that of the empty tube, resulting in a zero resistance to flow. The experiment consisted in measuring the amount of fluid discharged at a given time and pressure. In order to avoid degradation of the polymer, the fluid was discarded after each experiment except in the case where the effect of degredation was actually measured. The packed bed was calibrated using water. The driving pressure for the calibration experiment was the static head of the water itself. The experiment was repeated for different heights. The results were used in Ergun’s correlation to calculate the porosity, t , of the bed, and a value of 0.447 was found. The experiment was repeated for several glycerine-water solutions. Figure 2 presents a typical glycerinewater experiment. The solid line was calculated from Ergun’s correlation using the porosity previously obtained from the low-pressure water calibration. As can be seen, the agreement between the theoretical line and the experimental points is fairly good. Experiments with Polymer Solutions. Dilute aqueous solutions of Separan AP-30 (Dow Chemical Co.) and Polyox WSR-301 (Union Carbide Corp.) were used in the experiments. For comparison purposes one experiment was performed using a 200 ppm Polyox-FRA solution (Union Carbide Corp.). The Separan solutions were prepared according to the manufacturer’s instructions, stabilized by means of a weak formaldehyde solution and aged for 2-3 Ind. Eng. Chem., Fundam., Vol. 15, No. 1, 1976
75
1
+
800 ppm SEPARAN p = 2 3 9 c . p 1200 PPn SEPARAN p ' 3 . 6 3 ~p. 2W ppm Wyoa FRA p=b06c.p I
I
50
I00
I
1
2 x
I50
250
Re P
Figure 3. Plot of relative pressure drop for dilute aqueous Separan AP-30 solution vs. Reynolds number (T,,, = 25OC).
IW
50
+ 50
ppm SEPARAN
-
p=123 c p
100ppm SEPARAN p.1 3 8 C P A 200 ppm SEPARAN p = I 5 9 E p
7
7-
-
4 0 0 ppm SEPARAN p -2 06 c p
* c 6-
* 14 'C
I
300
Rep
Figure 5. Plot of relative pressure drop for dilute aqueous Polyox WSR-301 solutions vs. Reynolds number.
pwAn~ 22=E lP
0'
%
250
8 O O m SEPARAN p - 2 73 C p 1 2 O O p p SEWRAN p ' 4 4 5 s p
. ,T
D
200
Ix)
-
5-
4r
"t-
1
so
I
1
100
Ix)
2w
. I
m
. I
300
P r a r u r c Drop = I 8 atm
Rep
Figure 4. Plot of relative pressure drop for dilute aqueous Separan AP-30 solutions vs. Reynolds number (T,,, = 14OC).
days. Fluid viscosities were measured using CannonFenske capillary viscometers. Figure 3 provides deviations in pressure drop, relative to those obtained from Ergun correlation for the same density and viscosity as the polymer solution as functions of Reynolds numbers for various fluids. The definition of Reynoldp number is the same as that used by Ergun, Re = (puod,)/h. As can be seen, the results are quite interesting. Not only are the pressure drops of the polymer solutions, especially for the more concentrated ones, about 4-5 times higher than those for Newtonian fluids with identical viscosities, but the plots exhibit distinct peaks of maximum deviation. The peaks move leftward on the Reynolds scale with increasing Separan concentration. Usually, the higher the concentration, the higher the peak. A notable exception is the peak of the 800 ppm Separan solution. By rechecking the data it was noted that this solution was aged for only 2 days as opposed to the 3 days aging for the other solutions (the Polyox solutions were prepared from a concentrated stock solution prepared several months before). Figure 4 presents results similar to those in Figure 3. In this case the experiments were conducted at a room temperature of 14OC (midwinter, unheated) while the previous ones reported in Figure 3 were performed at a room temperature of 25OC (midsummer). The viscosity of water for the two sets of experiments was 1.22 and 0.87 CP, respectively. This figure covers a broader range of Separan concentrations resulting in a broader range of pressure drop ratios, stretching from about 2 for 50 ppm Separan concentration to about 8 for 1200 ppm Separan concentration. One may also note that for equal concentrations the pressure drop ratios are higher in the winter than in the summer. Moreover, the peaks for the colder experiments occur a t lower 76
Ind. Eng. Chem., Fundam., Vol. 15, No. 1, 1976
AGING TIME l h r l
Figure 6. Aging experiment-plot function of solution age.
of relative pressure drop as a
flow rates or Reynolds numbers. Experiments with Polyox WSR 301 as depicted in Figure 5 show results similar to those of Separan in Figure 4. The effect of aging was examined more carefully with the results summarized in Figure 6. As can be seen, essentially no changes were observed in the flow properties of the solution after a period of aging of 5 days. Due to strong stretching occuring on passage through a packed bed, polymer degradation occurs. This phenomenon was investigated by repeatedly passing the same solution through the packed bed and measuring the pressure drop. The results are depicted for 2 concentrations in Figure 7. As is seen, the 800 ppm Separan AP-30 solution acts on its seventh pass identically with an undegraded 400 ppm solution.
Discussion Although much has been written on the characterization of viscoelastic fluids, the experimentalist interested in measuring or predicting flow phenomena is still very much a t a loss. For his purpose a simplistic description of a viscoelastic fluid that would allow him to characterize it by means of several material properties measured in simple
1
SEPARAN A P - 3 0
400ppm
0
SEPARAN A P - 3 0
BOOppm
trolling other experiments in viscoelastic fluids as it will show whether the elastic effects in a fluid have changed during the experiments or whether one fluid is more or less elastic than another. The results obtained in the present paper, in particular Figures 3,4, and 5, are extremely interesting. One is especially tempted to correlate the peaks of maximum relative pressure drop with the relaxation time or the Deborah number. As seen, the solutions of increasing Separan concentration, believed to have increased elastic effects, exhibit their pressure peaks at lower velocities (or Reynolds numbers). The Deborah number, De = Ruo/d,, contains the velocity and the reIaxation time as a product. Could it be that the pressure peaks in Figure 3 or 4 occur at constant Deborah numbers? Unfortunately, we were lacking the sophisticated instrumentation required for measuring relaxation times. Readers who possess such instrumentation are invited to perform the suitable experiments needed to calibrate the packed bed as an instrument for the quantitative measurements of elastic effects in dilute polymer solutions.
PRESSURE DROP. I 8 Olm
I1 1
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5
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10
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15
1
20
I
25
N U M B E R OF EXPERIMENTS l r m e botch1
Figure 7. Degradation experiment-relative pressure drop vs. number of experiments with the same batch of solution.
flow experiments and then to predict some of the phenomena associated with these fluids would be very satisfactory. The Maxwell type of the constitutive equation of 01droyd (1950) which was found by Marshall and Metzner (1967) to be representative for viscoelastic flows through a packed bed is given by the following expression
Nomenclature De = Deborah number, Ruo/d, d , = mean diameter of particles composing the packed bed El = Ellis number, defined by eq 2 1 = length of the packed bed R = relaxation time Re = Reynolds number, puOdp/p u o = superficial velocity Greek Symbols t = porosity of the packed bed t l J = rate of strain tensor, contravariant p = dynamic viscosity of the fluid p = specific density of the fluid ai; = stress tensor, contravariant ,
(4) here d,/dt is the convected differentiation given in the contravariant system with respect to time (Oldroyd, 1950). This model is simple enough, as it contains only two material properties, p , the viscosity and R , the relaxation time, to be measured. Still it explains such typical viscoelastic phenomena as the Weissenberg climbing effect. In dilute polymer solutions there is no problem to measure the constant p ; this is simply the dynamic viscosity which may be measured by any standard viscometer based on some simple viscometric flow. On the other hand, the measurement of relaxation times is extremely involved. Although in principle the Weissenberg Rheogoniometer suitably modified may be used for these measurements, in practice, due to the small value of the effect measured, instrument imperfections tend to mask or distort the measured results (Hartnett, 1974). As opposed to other flows, the effect of viscoelasticity on the flow of dilute polymer solutions through a porous medium is very prominent and, as demonstrated in the present paper, is easily measurable. Thus, one can use the packed bed as a measuring instrument for the detection of viscoelastic properties in fluids. The packed bed experiment is also very valuable in con-
Literature Cited Christopher, R. H.,Middleman, S.,I d . fng. Chem., Fundam., 4, 422 (1965). Ergun. S., Chem. fng. Prog., 48, 89 (1952). Hartnett, J. P., personal communication, University of Illinois at Chicago Circle, 1974. Hoyt, J. W., Trans. ASMf, J. Basic Eng., 94, D2, 258 (1972). Kemblowski, Z., Mertl. J.. Chem. Eng. Scb, 29, 213 (1974a). Kemblowski, Z., Mertl, J., Chem. Eng. Sci, 29, 1343 (1974b). Marshall, R. J., Metzner. A. B., I d . Eng. Chem., Fundam., 6,393 (1967). Oldroyd. J. G., Proc. Roy. SOC.London, Ser. A, 200, 523 (1950). Sadowski, T. J., Bird, R. B., Trans. SOC.Rheob. 9, (2), 243 (1965). Sadowski, T. J., Trans. SOC.Rheol., 9, (2), 251 (1965). Slattery, J. C., AlChf J., 13, (6). 1066 (1967). Siskovic, N.. Gregory, D. R.,Griskey, R. G.. AIChEJ., 17 (2). 281 (1971). Wissler, E. H.,Ind. Eng. Chem., fundam., 10, 411 (1971).
Faculty of Mechanical Engineering Technion-Israel Institute of Technology Haifa, Israel
Gabriel Laufer Chaim Gutfinger*' Nesim Abuaf
Received for review February 3,1975 Accepted October 22,1975 Address correspondence to this author at the Department of Mechanical Engineering, Carnegie-Mellon University, Pittsburgh, Pa. 15213.
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