Ind. Eng. Chem. Res. 1990,29, 2150-2151
2150
COMMUNICATIONS Flow of Polymer Solutions through Porous Media The flow of polymer solutions through porous media has been investigated a t different temperatures (278, 283, 288, 293,298, and 303 K) and flow rates (from 0.659 X lo* t o 22.29 X lo4 m3/s). The ranges of shear rates and pressure drops were 10.8-692.5 s-l and 0.414 X lo5 t o 2.72 X lo5 Pa, respectively. Three different kinds of polymers (polyacrylamide, 1000 ppm concentration) were used in a 100-mm column packed with 2-mm particle diameter glass beads. Two types of modifications for Darcy's law were tested and discussed in the form of modified friction factor-Reynolds number correlations. First, the modified Blake-Kozeny equation for the flow of power law fluids through porous media was tested experimentally; second, a viscosity-temperatureshear rate correlation was developed and applied in the BlakeKozeny equation for the flow of Newtonian fluids through porous media. Introduction Blake (1922) was first to introduce two dimensionless groups that describe the flow of Newtonian fluids through granular beds. These groups are given as follows: modified friction factor f*
AF'Dpc3
=
pVa2L(1- e ) modified Reynolds number
f*
150 =Rel*
(3)
where AF is the pressure drop due to friction, p the fluid density, V, the superficial velocity, D, the effective particle diameter, L the bed length, c the bed porosity, and I.L the viscosity of the fluid. Equations 1 and 2 are used in the majority of the papers published until now on Newtonian flow through granular beds. Previous studies of the flow of non-Newtonian fluids through porous media were investigated by Sadowski (1963), Sadowski and Bird (1969), and Christopher and Middleman (1965) using polymer solutions, while Gregory and Grisky (1967) used molten polymers. Recently, AlFariss and Pinder (1987) have presented a correlation describing the flow of non-Newtonian fluids with yield stress through a packed bed of sand. The present work provides a new method to describe the flow of polymer solutions (i.e., non-Newtonian fluids) through porous media using Blake's definition of Reynolds number (eq 2) while the viscosity term was substituted by a correlation developed by Al-Fariss (1988), which gives the viscosity of polymer solutions as a function of shear rate and temperature. Experimental results obtained for the flow of power law fluids through porous media were compared with the modified Blake-Kozeny equation developed by Christopher and Middleman (1965). Experimental Section Three types of non-Newtonian polyacrylamide type polymers (Pusher 500, Pusher 700, and Pusher 1000) were investigated in this study. The polymers tested are usually
used in enhanced oil recovery (EOR) processes. They were supplied by Dow Chemical RHEIN-WERK, GMBH, Germany. For each polymer, a 1000 ppm solution in distilled water was prepared. All samples tested in this study have shown shear-thinning behavior without yield stress, i.e., non-Newtonian behavior of the power law fit with an exponent of less than one. Rheological measurements were carried out by using a rotational viscometer (HAAKE Rotovisco Model RV-12, a rotating bob type co-axial cylinder viscometer). Figure 1 is a schematic diagram of the experimental setup used for the flow tests constructed in the KSU Chemical Engineering Department. Results and Discussion Modified Blake-Kozeny Equation. The derivation of the Blake-Kozeny equation through the capillary model takes as its starting point the Hagen-Poiseuille equation for the flow in a long straight capillary. A detailed derivation is given by Bird et al. (1960). A parallel treatment proposed by Christopher and Middleman (1965) for the power law fluid takes the analogue of the Hagen-Poiseuille equation as its starting point: D, Vo2-"p Re2* = (4) ped1 - e )
where beff =
(k/12)(9
+ 3/n)"(150Re)('-")~*
where K is the permeability of the medium, and it is usually given as K DP2e3/150(1- e)* (5) It is common to present the results for the flow in porous media in terms of a bed friction factor and a Reynolds number. The relationship between the above two terms for the flow of power law fluid through porous media was represented as f
*
= 150/Re2*
(6)
Proposed Method. The superposition method was used to develop a relationship for the viscosity of polymer so-
lution as a function of shear rate and temperature. The step-by-step approach to the correlation is given in detail by Al-Fariss (1988).
0888-5885/ 90 12629-2150$02.50/0 0 1990 American Chemical Society
Ind. Eng. Chem. Res., Vol. 29, No. 10, 1990 2151 o -PUSHER 500 ( 1000 PPM) -PUSHER 700 ( 1000 PPM) a -PUSHER loo0 (lo00 PPM)
7 -
LL
m 2 0
YlnOGCW CYLINDER
-4
-3
-2
-1
0
Log.
Figure 1. Schematic diagram of flow experiments.
1
o -PUSHER 500 (1000 PPM) -PUSHER 700 ( 1000 PPM)
1
a -PUSHER loo0 (1000 PPM)
6
*LL 9 5 A
4 . 3 . 2 1
-4
I
1
I
-3
-2
-1
I 0
Log. R,;
Figure 2. Modified friction factor-modified Reynolds number (eq 3) for the flow of polymer solutions through a packed bed of glass beads.
A modified form of the Eyring equation relating the viscosity (p)to the temperature (2') and shear rate (i.)with three constants ( A , B , and C) was given as (7) The rheological behavior of the polymer solutions tested was described well by the power law model: p = AeB/Tj.C
= k?" (8) where k and n are the rheological parameters of the power law model. To estimate the shear rate (i.) in the bed, one can use the definition suggested by Kemblowski and Michniewicz (1979), which was given as follows: 7
i., = 12(3n + 1)/4n(l - E / E ~ ) ( V ~ / D J (9) Figure 2 compares the modified friction factor (eq 3) with the friction factor equation that contains our definition for the viscosity (eq 7), while Figure 3 compares the modified Blake-Kozeny correlation (eq 6) with the experimental data. The ranges of absolute error calculated for eqs 6 and 3 were 3.4% and 6.1%, respectively; both equations fit the experimental data with a correlation coefficient of 98%, It is well-known that the power law gives a good approximation over a limited range of shear rates. Also it is known that polymer solutions approach Newtonian
Figure 3. Modified friction factor-modified Reynolds number (eq 6) for the flow of polymer solutions through a packed bed of glass beads.
behavior at very low shear rates. However, both methods presented here are based on the same rheological model, Le., power law model. The accuracy of the proposed method depends highly on the shear rate estimation in the packed bed. Equation 7 was obtained from a wide range of shear rates (10.8-692.5 s-9, while the shear rates estimated in the bed were in the lower range. One advantage of the method proposed here is that it contains the temperature effect on the rheological behavior of the polymer solution, while most studies done in this field do not include this parameter. Acknowledgment
The financial assistance of the Research Center, College of Engineering, King Saud University at Riyadh, Saudi Arabia, for this work is gratefully acknowledged. Registry No. Polyacrylamide, 9003-05-8.
Literature Cited Al-Fariss, T. F. Viscosity-Temperature-Shear Rate Correlation for Crude Oils and Polymers. J . Eng. Sci. 1988, 14 (2),231-244. Al-Fariss, T. F.; Pinder, K. L. Flow Through Porous Media of Shear-Thinning Liquid with Yield Stress. Can. J . Chem. Eng. 1987,65 (3), 391-406. Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; John Wiley and Sons Inc.: New York, 1960. Blake, F. C. The Resistance of Packing to Fluid Flow. Trans. AIChE 1922, 14,415-421. Christopher, R.; Middleman, S. Power-law Flow Through a Packed Tube. Ind. Eng. Chem. Fundam. 1965,4,422-426. Gregory, D.R.; Grisky, R. G. Flow of Molten Polymers Through Porous Media. AIChE J . 1967,13(l),122-125. Kemblowski, Z.; Michniewicz, T. A New Look at the Laminar Flow of Power Law Fluids Through Granular Beds. Rheol. Acta 1979, 18,730-739. Sadowski, T. J. Non-Newtonian Flow Through Porous Media. Ph.D. Thesis, Chemical Engineering Department, University of Wisconsin, Madison, 1963. Sadowski, T. J.; Bird, B. Non-Newtonian Flow Through Porous Media. Ind. Eng. Chem. 1969,61,118-147.
Tariq F. AI-Fariss Chemical Engineering Department College of Engineering King Saud University Riyadh 11421,Saudia Arabia Received for review November 6 , 1989 Revised manuscript received April 21, 1990 Accepted June 6,1990
