Ind. Eng. Chem. Fundam. 1983,22, 299-305
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Flow of Polymer Solutions in Porous Media: Inadequacy of the Capillary Model John L. Duda,' Seong-Ahn Hang,+ and Elmer E. Klaus Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802
Experimental measurements show that conventional capillary models are inadequate for the description of the flow of nonlinear purely viscous solutions in porous media. A theoretical analysis indicates that any model for the flow of purely viscous polymer solutions in porous medii must meet two criteria. First, the model must include expansion and contraction regions where excess pressure drops occur. Secondly, the rheological model for the fluid must include the Characteristic transition from Newtonian behavior at low shear rates to shear-thinning behavior at high shear rates.
Introduction The flow of polymer solutions in porous media is encountered in many diverse technologies such as enhanced oil recovery, filtration, and polymer processing. A primary interest in all these applications is how the pressure dropflow rate relationship for flow in the porous medium is related to the characteristics of the solution and of the porous medium. For example, the design of reservoir flooding with polymer solutions would be greatly facilitated if the pressure drop-flow rate relationship for polymer solutions in porous media could be predicted from the characteristics of the reservoir rock coupled with the characteristics of the bulk rheology of the polymer solution. An analogous situation exists when Newtonian solutions are flowing in porous media, and Darcy's law has proven to be an excellent model for this case v o = k- A- P P
L
Once the permeability, k, has been determined with one Newtonian solution, the pressure drop can be predicted as a function of the superficial velocity for any other Newtonian solution by utilizing the solution viscosity, p. Consequently, in this model the porous medium is characterized by the permeability and the solution by the viscosity. Unfortunately, the situation is not as simple for nonNewtonian fluids such as polymer solutions. In many cases, the size of the polymer molecules in solution is the same order of magnitude as the dimension of the pores. Consequently, complex phenomena related to polymer molecule-wall interaction such as adsorption, mechanical entrapment, and inaccessible pore volume preclude development of a theory to predict pressure drop-flow rate relationships based solely on the bulk properties of the solution. A rational approach for the development of a model for the flow of non-Newtonian solutions in porous media would be to first consider flow in a high-permeability porous medium where the molecule-pore wall interactions can be neglected. Then, this model could be extended to include the complex porous medium-solution interactions. Even if the limiting case for the flow of polymer solutions in high permeability porous media is considered, the situation is still fairly complex. I t has been established that some polymers such as polyacrylamide and poly-
'Gulf Research & Development Co., Pittsburgh, PA.
ethylene oxide form viscoelastic solutions, and excess pressure drops due to elastic phenomena can occur (Marshall and Metzner, 1967; Wang et al., 1979). Furthermore, at high flow rates the behavior of all solutions in porous media including Newtonian solutions is complicated by inertial effects. It is obvious that the starting point for the development of models for describing flow of polymer solutions in porous media would be the case in which elastic or relaxation effects as well as inertial and molecule-wall interactions can be neglected. This idealized situation will be approached when polymer solutions with short relaxation times are flowing at low velocities in high permeability porous media. Consequently, this study will be limited to the analysis of the flow of purely viscous non-Newtonian solutions in high permeability porous media at the creeping flow limit. It was natural for early investigators to attempt to modify Darcy's law to describe the flow of non-Newtonian solutions in porous media. If adsorption and other wall interactions are negligible, the permeability of a porous medium should be a fured parameter, and the only problem is to determine the effective viscosity in the porous medium flow field for a solution which has a shear rate dependent viscosity. A model of the porous medium is required in order to determine the appropriate solution viscosity, and the most successful approach has been to model the pore structure as a bundle of capillaries which exhibits the same resistance to flow as the porous medium. The flow is considered to be fully developed in the capillaries, and an empirical tortuosity factor is introduced. A clear description of this modeling technique is presented by Bird et al. (1960). The most widely accepted model leads to the well-known Blake-Kozeny-Carman equation when the solution is Newtonian. In this formulation, a tortuosity factor of 25/12 is used and the permeability is related to the porosity, E , and the characteristic particle diameter, D,, by
This approach can be modified to consider the flow of any purely viscous non-Newtonian solution for which the rheological constitutive equation has been defined. For example, Christopher and Middleman (1965) developed a model using the power-law relationship for the fluid. The results for the Newtonian case as well as those for a power-law fluid and an Ellis model fluid are presented in Table
0196-4313/83/1022-0299$01.50/00 1983 American Chemical Society
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Table I. Capillary Models for Flow of Purely Viscous Fluids in Porous Media
flu id
rheological model Newtonian
parameters
"
VO DpZe3AP
P
DpV0p
150fi(1- E ) ' L
150(1 - ~
) f i
power law = K j n-'
7)
Ellis model
I. In this formulation, the specific rheological constitutive equation will only alter the relationship between the pressure drop and the velocity in the capillaries. There have been numerous studies concerning the introduction of the tortuosity factor and the proper value of this empirical coefficient. Kemblowski and Michniewicz (1979) present a good review of this area. Details concerning the tortuosity are not germane to this study since the conclusions of this work are independent of this empirical factor. The conventional tortuosity factor of 25/12 was used in the models presented in Table I. Instead of considering the velocity vs. pressure drop, most investigators have examined the relationship between the dimensionless Reynolds number, N b , and the dimensionless friction factor (3) The common procedure has been to define the Reynolds number for a specific fluid model so that the product of this dimensionless group with the friction factor is equal to 1. This approach results in the NRedefinitions presented in Table I. Most studies of the flow of polymer solutions in porous media have used the power-law model presented in Table I. The common consensus has been that this model which is based on the coupling of the capillary model for the porous medium and the power-law model for the fluid rheology accurately describes the flow of purely viscous non-Newtonian polymer solutions in porous media. However, a few studies such as that by Sheffield and Metzner (1976) have indicated that this approach is inadequate. The common observation is that the experimental pressure drop is greater than that predicted, and some kind of real or apparent shear thickening behavior occurs (Dullien, 1979). In some cases, this behavior is clearly due to the viscoelastic properties of the polymer solutions. For example, Wang and co-workers (1979) showed that data for the flow of solutions of carboxymethyl cellulose, hydroxyethyl cellulose, xanthan gum, and cellulose sulfate ester in porous media are in reasonably good agreement with the capillary models, whereas solutions of polyacrylamide and polyethylene oxide showed pressure drops in large excess of those predicted by the model. All the available studies indicate that the polymer solutions which are essentially purely viscous in nature more closely follow the capillary models than solutions which exhibit strong elastic characteristics. However, it is not clear that these models are adequate for predictive and correlative purposes when purely viscous solutions are involved. Part of the problem in resolving this issue is related to the experimental problems associated with measuring the low
RESERVOIR
POROUS HOLDER
Figure 1. Schematic diagram of porous media viscometer. pressure drops that are characteristic of the creeping flow regime where the inertial effects are negligible. Also, the acceptance of resulta which are not in agreement with these models has been hampered by the lack of an explanation for disagreement when the solution is purely viscous and interactions with the pore walls are not important. The purpose of this study is to clearly establish the range of applicability of these capillary models for purely viscous solutions. This study includes a coupling of accurate experimental measurements over a wide range of conditions and a new theoretical development. Experimental Section The common procedure for studying flow in porous media is to measure the pressure drop across a section or bed of porous medium. Usually, a constant flow rate pump is utilized and manometers or pressure transducers are used to measure pressure differences. Consequently, the apparatus is rather complex, and relatively large samples of porous medium must be used to obtain accurate measurements a t low flow rates. In contrast, the instrument utilized in this study is based on the principles employed in simple capillary viscometers, and consequently, is referred to as a porous media viscometer. A schematic diagram of this device is shown in Figure 1. The cylindrical reservoir and efflux bulbs are constructed of glass, while the porous medium holder is machined from stainless steel. Information concerning the internal structure of the porous medium holder and details concerning construction, calibration, and operation of this instrument have been presented by Hong (1982) and Duda et al. (1981). During an experiment, the polymer solution flows from the reservoir through the porous medium and the connecting capillary to the efflux bulbs. The flow rate is determined from the time required for the fluid to fill an efflux bulb of known volume, and the pressure drop is determined
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Table 11. Characterization of the Polymer Solutions Used in This Study polymer symbol concn, 1) 0 , 71/21 in Figures wt % g/cm-s dyn/cm2 01 3-5 Carboxymethyl Cellulose Solutions
\
t
r l
O
1
L
L
U
L
L
_
10
_
L
~
,
d
100
1
.
_ i
IO00
SHEAR R A T E , s e c '
Figure 2. Power-law and Ellis model correlations for a solution of 0.40 w t % carboxymethyl cellulose in distilled water.
from measurements with a cathetometer of the liquid level in the reservoir and the levels at the entrance and the exit of the efflux bulb. By judicious choice of reservoir dimensions and efflux bulb dimensions, an essentially constant pressure drop can be maintained for each experiment. This device has been used by Duda and co-workers (1981) to study the influence of polymer molecule-wall interactions on the flow of polymer solutions in several types of porous media. This instrument is a very simple technique for obtaining accurate preasure drop vs. flow rate data. In all the experiments, the criteria of two consecutive runs which agree to within 1% of each other in flow rate were used to establish a steady-state condition. For these studies, sintered bronze disks obtained from the Pacific Sintered Metals Co. were used as the porous media. The diameter of these disks was 1 in. and in the thickness ranged from l / g to 1/4 in. Disks of different thickness were used to check for any end effects. The porosity of this porous medium is 0.26 and calibration with Newtonian solutions showed a permeability of 7.0 darcys. Carboxymethylcellulose (CMC; Hercules, cellulose gum) and xanthan gum (Kelzan MF from the Kelco Co.) were selected for this study because previous studies indicated that aqueous solutions of these polymers show purely viscous behavior during flow in porous media at low Reynolds numbers. The viscosity vs. shear rate behavior of the solutions used in this study was determined by utilizing a rotational viscometer (Mechanical Spectrometer by Rheometrics, Inc.) and several types of capillary viscometers. Details concerning the preparation and characterization of the polymer solutions used in this study have been presented by Unsal et al. (1978) and Hong (1982). All the characterization and flow experiments were conducted at 25 "C. Although most previous studies have utilized the power law to describe the rheology of the polymer solutions, the viscosity behavior of the CMC solutions (see Figure 2) indicates that many of the measurements of this study were obtained at shear rates outside the power-law region. As Figure 2 indicates, the CMC data approximately follow the power-law relationship for only one decade of shear rate. Consequently, the analyses presented in this study for both the xanthan gum and CMC solutions were based on the use of the Ellis model. It should be emphasized that there is nothing unique about the Ellis model and any rheological model which gives a good correlation of the data could be used. As will be shown, the critical characteristic of the rheological model is that it include the transition from Newtonian behavior a t low shear rates to shear thinning at high s h e a rates. Seven different solutions were used in this study and the characteristics of these golutions
0.10 0.40 0.50
0.440 2.47 4.11
10.4 38.7 58.3
1.81 1.99 2.15
= A 0
Xanthan Gum Solutionsa 0.10 0.20 0.30 0.40
0.596 1.97 32.4 150
2.42 2.37 3.16 9.72
2.41 2.77 3.51 5.63
0 0
A
o
a In the xanthan gum solutions, 200 ppm of NaCl was added to the distilled water.
REYNOLDS NUMBER, N,,
Figure 3. Capillary model correlation;key to symbols given in Table 11.
are presented in Table 11. This table also serves as a key for the symbols used in the following figures. The porous media viscometer was used to determine the pressure drop vs. flow rate relationship for the seven polymer solutions listed in Table I1 over a wide range of Reynolds numbers. Figure 3 shows a correlation of these results based on the Ellis model version of the capillary model as presented in Table I. Figure 3 is typical of many of the correlations which have appeared in the literature. A clustering of the data around the f.Nb = 1 line could be interpreted as indicating that the capillary model is a valid description of the flow of purely viscous non-Newtonian polymer solutions in porous media. The only obvious trend is that the friction factors for the CMC solutions are somewhat lower than the prediction, and conversely the measured pressure drops are a little high for the xanthan gum solutions. However, it could be argued that these discrepancies which appear small on this logarithmic scale are due to experimental errors. A similar conclusion would be reached if the power-law model were used instead of the Ellis model. Both of these rheological models appear to give an accurate description of the flow of the polymer solutions in porous media when the results are presented in the form used in Figure 3. Details of these comparisons are presented by Hong (1982). Sheffield and Metzner (1976) have demonstrated that the correlation of friction fador vs. Reynolds number such as that shown in Figure 3 is insensitive to deviations between the theory and experimental measurements, particularly for shear thinning fluids with a small power-law index, n. A more sensitive representation of the correlation is shown in Figure 4. As noted in this figure, the product f-NReis equal to the ratio of the experimental pressure
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251 ' I
I
I
0
I
I
30r I
- - - l -0 5,L 16' REYNOLDS NUMBER, NR,
Figure 4. Capillary model correlation; hp, = experimental pressure drop, AP, = pressure drop predicted by capillary model.
drop, AF',,, to the pressure drop predicted by the capillary model, AP,. This correlation clearly shows that the experimental results significantly deviate from the model predictions and the error in the predicted pressure drop is over 200% for the concentrated xanthan gum solutions at the higher Reynolds numbers. The distribution of the data points in Figure 4 indicates that the observed deviations are not due to random experimental errors. The qualitative trends of these data are in good agreement with the results of Sheffield and Metzner (1976). The magnitude of the predicted pressure drop, APc,would be altered by changing the tortuosity factor, but such modifications of the capillary model will not significantly improve the correlation. A significant result of this study is that the measured pressure drop is a stronger function of Reynolds number or flow rate than the model predicts. This is true if either the Ellis model or the power-law model is utilized (Hong, 1982). It also appears that higher pressure drops are realized with higher values of a or with solutions which show a higher degree of shear thinning behavior. A more revealing comparison of the theory and experimental measurements is obtained when the Ellis number is used as the correlating parameter as shown in Figure 5. The lines in this f i r e have no significance except to clarify the correlating characteristic of the parameter a. As this figure indicates, the solutions with the relatively low values of a (1.8-2.4) are clustered together. This correlation clearly shows that pressure drops in excess of those predicted by the capillary model occur at high Ellis numbers and high values of a. Hong (1982) has shown that these qualitative trends are independent of permeability and are common characteristics of different kinds of porous media such as Berea sandstone and filter media. This experimental study indicates that the capillary model is inadequate for the prediction of pressure drops during the flow of purely viscous polymer solutions in porous media. This inadequacy is particularly evident at high Ellis numbers and highly shear thinning fluids (large a). Furthermore, this inadequacy is not associated with a particular rheological model but is due to a fundamental deficiency of the capillary model formulation. None of the many modifcations of the capillary model which have been suggested will eliminate this basic problem. The low Reynolds numbers of these experiments and the high permeability of the porous media also eliminate inertial and wall interaction effects as probable causes of the observed deviation from theory. It could be argued that the inadequacy of the capillary model is due to elastic effects even though solutions of xanthan gum and CMC are commonly considered to be purely viscous in nature. However, this study will show that the observed deviations are consistent with the behavior of purely viscous fluids flowing in nonviscometric flow fields.
001
01
I
1
I
IO
10
100
ELLIS NUMBER (r(,V,/'T&)
Figure 5. Capillary model correlation with Ellis number and a: as correlating parameters.
Theoretical Analysis and Discussion The experimental results presented in the previous section clearly reveal the inadequacy of the capillary model. These results suggest that the problem may be associated with the limitations of the capillary model of the porous medium and not the model for the bulk rheology of the solutions. Most porous media do not consist of uniform channels but are composed of many abrupt constrictions. A more reasonable model for porous media may be cavities connected by pores. It is well-known that excess pressure drops associated with inertial or kinetic energy effects occur at these points of contraction and expansion when fluid velocities are high, These kinetic energy induced excess pressure drops are negligible at the low flow rates considered in this study. However, even under creeping flow conditions where the Reynolds number approaches zero, excess pressure drops associated with viscous forces can be significant at abrupt expansions and contractions. Several solutions are available in the literature for the complete equations of motion which describe the flow of Newtonian fluids in nonuniform conduits. Although significant pressure drops occur for Newtonian fluids in nonuniform flow channels, these pressure drops are linear functions of the fluid velocity in the creeping flow limit. Since the pressure drops associated with the fully developed laminar flow in uniform conduits are also linear functions of velacity, pressure drop vs. flow measurements cannot distinguish between flow in uniform conduits and flows through contractions and/or expansions when Newtonian fluids are involved at low Nw I t might be quite fortuitous that a capillary model can adequately describe the flow of Newtonian fluids in porous media. The socalled tortuosity correction term may be correcting for the excess pressure drop associated with abrupt changes in the flow field as well as the tortuosity of the flow path. Duda and Vrentas (1972) pointed out that any model of bulk rheology which does not possess a characteristic time constant such as a Newtonian model or the power-law model will predict that the excess pressure drop at a sudden contraction or expansion is proportional to the fully developed pressure drop at low Reynolds numbers. This point has recently been demonstrated by Boger et al. (1978),who solved the equations of motion for a power-law fluid flowing in a sudden tubular contraction in the creeping flow limit. Consequently, the excess pressure drops associated with the flow of the power-law fluid in a nonuniform conduit cannot explain the deviation from theory, as shown in Figure 4 and 5. However, all polymer solutions show a transition from Newtonian flow at low
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I
n
n
n
Figure 6. Schematic diagram of pore model.
shear rates to a shear-thinnii behavior at high shear rates.
This transition region is traversed in all flows in conduits since a zero shear rate exists a t the center line. Any rheological model which correlates this characteristic transition will have a characteristic time parameter. Such rheological models will predict that the relationship between excess pressure drop and the flow rate is different than the relationship between the fully developed pressure drop and the flow rate in a uniform conduit. For example, Duda and Vrentas (1973) used a finitedifference technique to solve the complete equations of motion for the creeping flow of a Powell-Erying fluid in a sudden tubular contraction. The Powell-Erying model has a characteristic time constant and shows a transition from Newtonian behavior at low shear rates to a shear thinning behavior at high shear rates. These calculations showed that the excess pressure drop increases more rapidly with flow rate or shear rate than a pressure drop associated with fully developed flow in a tube. This trends agrees with the experimental measurements of polymer solution flow in porous media. If the failure of the capillary models is related to the excess pressure drop near expansion and contraction regions, then the experimental results should be consistent with a model of the porous media which incorporates this expansion-contraction nature of porous media. As a first approximation, the porous media will be modeled as a bundle of nonuniform conduits with the periodic geometry as depicted in Figure 6. This particular configuration is not unique, and a sinusoidal type geometry could also be considered. However, the excess pressure drops of interest should be more dramatic in the abrupt converging and diverging geometry of this model. Previous investigators have considered nonuniform conduits as models of porous media. Most investigators such as Dullien (1979) and Payatakes et al. (1973) have restricted their analysis to Newtonian fluids. Others such as Sheffield and Metzner (1976) have considered nonNewtonian fluids, but they have assumed that the relationship between the flow rate and pressure drop in a converging or diverging pore is the same as that which occurs during fully developed flow in a cylindrical pore. Recently, Deiber and Schowalter (1981) determined both theoretically and experimentally that viscoelastic fluids flowing in a tube with sinusoidal axial variations in diameter exhibit excess pressure drops. However, previous studies have not presented a model for flow in porous media which incorporates excess pressure drops in converging and diverging regions associated with a purely viscous fluid which shows a transition from Newtonian behavior at low shear rates to a shear-thinning behavior at high shear rates. To evaluate this hypothesis, the complete equations of motion which describe the steady flow of an incompressible Ellis model fluid in the periodic flow field shown in Figure 6 were solved for the limiting case of zero Reynolds number. It was further assumed that the flow is axisymmetric
25
1
20 NEWTONIAN LIMIT
15001
01
10
100
m
ELLIS NUMBER ( #&!Ul/TaR,)
Figure 7. Prediction of excess pressure drop occurring during flow of an Ellis model fluid in the pore model shown in Figure 6; AP,= total pressure drop; U r d = pressure drop associated with fully developed flow in the cylindrical segments of the pore model; j3 = 4, y = 2, 6 = 3.
and that there is no circumferential velocity. The finitedifference technique used to solve these equations follows directly from the technique developed by Duda and Vrentas (1973), and the details of this solution are presented by Hong (1982). In the creeping flow limit, the dimensionless pressure drop is a function of the Ellis number, NEl,the shear thinning parameter, a,and the three geometric parameters, 8, y, 6, defied in Figure 6. The Ellis number dictates the rate at which the viscosity initially decreases with increasing deformation rate. Newtonian behavior is approached as NEI 0. For sufficiently large rates of deformation, the Ellis model has a power-law region with a power-law index, n = l/a. The method of solution employed gives direct information concerning the distribution of vorticity and the stream function in this periodic flow field. This information can be used to determine the distribution of the two velocity components and the pressure in this nonuniform pore. However, the total pressure in the axial direction is required for comparison with experimental data. The method developed by Vrentas and Duda (1973) which involves the integration of the vorticity field was used to determine the total pressure drop, AP,,in this nonuniform model pore. For the evaluation of this porous media model, this total pressure drop should be compared to the pressure drop which would be realized if a fully developed velocity field existed in every cylindrial segment of this nonuniform pore. Such a comparison is shown in Figure 7, where hpfdis the hypothetical pressure drop associated with fully developed flow in the cylindrial segments of this nonuniform conduit, and AP, is determined from the finite-difference solution. The difference between these two pressure drops is the excess pressure drop associated with the converging and diverging regions of this pore model. Figure 7 shows that the excess pressure drop associated with the converging and diverging regions is significant. At the Newtonian limit, N m 0, the excess pressure drop is a constant factor. However, as Nu increases, this excess pressure drop is a stronger function of the characteristic velocity, VI, than the pressure drop associated with fully developed flow in a uniform conduit. This analysis also shows that for most values of the Ellis number, this excess pressure drop is more prevalent for fluids which show a strong shear-thinning behavior. Evidence is presented by Hong (1982) that the crossover in Figure 7 between the
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a = 2 line and the a = 4 line is not an artifact of the numerical analysis but is a result of the characteristics of the Ellis model. The magnitude of the pressure drop in this pore model is also a function of 0,y,and 6, but these qualitative trends are independent of specific changes in pore geometry, and results for other cases are presented by Hong (1982). The similarities between the experimental and theoretical results are quite obvious when Figures 7 and 5 are compared. Both figures show how the pressure drop which is in excess of that associated with fully developed flow in a cylindrical conduit is related to the Ellis number and a. A direct comparison of absolute magnitudes is not meaningful, but both the experimental results and the theoretical analysis indicate that the capillary models which are based on fully developed flow in uniform conduits are more prominently inadequate at high values of Ellis number and a. The analysis presented in this study is the only one available which gives results which are consistent with the experimental observations. As the recent paper of Deiber and Schowalter (1981) indicates, there is no question that the anomalously high pressure drops exhibited during flow through porous media of solutions of some polymers such as polyacrylamide are due to elastic effects. However, the vast majority of experimental studies which utilize polymer solutions which are commonly considered purely viscous in nature such as solutions of xanthan gum and CMC appear to confirm the applicability of capillary models. For example, the recent paper of Kumar and Upadhyay (1981) states that: “It has been shown that with appropriate choice of viscosity, the results for both Newtonian and non-Newtonian fluids can be expressed by correlations based on the capillary tube bundle model.” The results presented here may help clarify why deviations between experimental measurements and the capillary model predictions have not been commonly observed. The answer to this question has many facets. First of all, the appropriate experiments are difficult and it is easy to have deviations between experimental measurements and theoretical predictions masked by experimental errors. A related factor is that, as Sheffield and Metzner (1976) demonstrated, the conventional, f vs. N,, method of correlating data conceals the deviations of interest. It is particularly interesting that this analysis shows that the largest deviations occur as the fluid becomes progressively more shear thinning, and conversely the conventional correlating procedure has diminishing sensitivity as this limit of fluid behavior is approached. A second consideration is that great care must be taken to design experiments to eliminate other phenomena which are not incorporated in the conventional model theories, such as inertial effects and polymer molecule-wall interactions. When the experiments have not been carefully designed to be unambiguous, it is natural to consider these phenomena as the source of disagreement with theory. Finally, it is evident that the rheological characteristics of the polymer solutions utilized in most studies have biased the results. It is difficult to conduct measurements at high Ellis numbers while remaining in the flow region where inertial effects are negligible when solutions of polymers of conventional molecular weights are used. For example, CMC can be considered typical of most conventional polymer solutions and has been the most widely used polymer for such studies. As Figure 5 shows, the deviations observed with the CMC solutions are rather small, and it would be difficult to build a case condemning the capillary model using only the CMC data.
This is consistent with the study of Kumar and Upadhyay (1981) in which CMC solutions were used exclusively. By coincidence, the high molecular weight polymers which are being produced specifically for oil recovery processes such as the xanthan gum used in this study are ideal for demonstrating the inadequacies of the capillary models. The economics of oil recovery processes dictate the use of high molecular weight polymers whose chains are greatly extended in solution so that the amount of mobility control obtained with a quantity of polymer is maximized. Such polymers produce solutions which deviate from Newtonian behavior at very low shear rates and show enhanced shear-thinning behavior. This analysis indicates that large deviations from capillary model predictions will occur for these solutions which are characterized by large values of qo/T1/2 and large values of a. Furthermore, since xanthan gum solutions do not exhibit any significant elastic behavior under the flow conditions of interest, they are ideal solutions for the evaluation of theories describing the flow of non-Newtonian purely viscous fluids in porous media. It is interesting to note that solutions of polyacrylamide, which is the other major polymer for oil recovery applications, show very pronounced elastic behavior.
Conclusions This study shows that capillary models of porous media which utilized relationships between pressure drop and flow rate which are derived for fully developed flow in uniform conduits are inadequate for the description of the flow of purely viscous polymer solutions in porous media. It is concluded that a model to describe the flow of purely viscous polymer solutions in porous media must meet two criteria: (1)the model for the porous medium must include expansion and contraction regions where excess pressure drops are developed; (2) the rheological model for the fluid must include the characteristic transition from Newtonian behavior at low shear rates to shear-thinning behavior at high shear rates. This study further indicates that the proposed model of flow in porous media which incorporates the Ellis rheological model and a nonuniform pore model has the potential of being useful for characterizing flow in porous media. Not only could this model be used to predict the pressure drop associated with the flow of purely viscous solutions in porous media, but such a model should also be useful for the analysis of the flow of viscoelastic fluids where excess pressure drops in the converging and diverging regions are even more dominant. The recent work of Deiber and Schowalter (1981) supports this conclusion. Another potential application of the model developed in this study is the characterization of porous media. By adjusting pore geometry characteristics in the theoretical analysis so that predictions agree with experimental measurements, the converging-diverging nature of a porous media could be determined. Such a porous media characteristic may be important in correlating other phenomena in porous media such as dispersion, the trapping of fluids, and multiphase flow. For example, it seems reasonable that the residual oil saturation in reservoir rock after water flooding could be related to the convergingdiverging characteristic of the rock. At the present time, the flow of a Newtonian fluid is used to determine the characteristic permeability of a porous medium. This work indicates that a method of characterizing the convergingdiverging nature of the porous media could be based on an analysis of the flow of a purely viscous non-Newtonian polymer solution through the porous medium. This study gives a new perspective to our understanding
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of the flow of non-Newtonian fluids in porous media. It is obvious that more studies are needed to further evaluate this point of view and to more completely exploit the basic concepts presented here. Although capillary models may be quite adequate for many cases, this work suggests that large deviations from these conventionally accepted theories will occur in the analysis of the flow of polymer solutions in enhanced oil recovery processes. The importance of this energy related application suggests that further studies are highly desirable. Acknowledgment This research was conducted as a part of the Penn State Cooperative Research Project in Enhanced Oil Recovery which is financially supported by the following petroleum and chemical companies: Amoco, Arco, DOW,Exxon, Marathon, Mobil, Sohio, Texaco, and Witco. Also, the authors are indebted to Professor J. S. Vrentas for his valuable suggestions. Nomenclature hp = pressure drop D, = characteristic particle diameter of the porous media as defined by eq 2 f = friction factor defined by eq 3 k = permeability K = parameter in power law model L = length of porous media n = parameter in power-law model NE,= Ellis number NRe= Reynolds number defined in Table I R1 = radius in the pore model shown in Figure 6 VI = average velocity in small radius section of the pore model shown in Figure 6 Vo = superficial velocity Greek Symbols
p =
viscosity of Newtonian fluid
= porosity p = density t
a = parameter in Ellis model
P = geometric parameter defined in Figure 6 y = geometric parameter defined in Figure 6 6 = geometric parameter defined in Figure 6 T
~ =/parameter ~
+ = shear rate
in Ellis model
viscosity of a nowNewtonian fluid vo = parameter in Ellis model r = shear stress Registry No. Sodium carboxymethyl cellulose, 9004-32-4; xanthan gum, 11138-66-2. Literature Cited 7 =
Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. "Transport Phenomena"; Wiley: New York, 1960. Booer, D. V.; Gupta, R.; Tanner, R. I. J. Non-Newtonian F/uMMech. 1978. 4, 239. Christopher, R. H.; Middleman, S. Ind. Eng. Chem. Fondem. 1965, 4 , 422. Deiber, J. A.; Schowatter. W. E. AIChE J. 1981, 27, 912. Duda, J. L.; Vrentas, J. S. Can. J . Chem. Eng. 1872, 50, 671. Duda. J. L.; Vrentas, J. S. Trans. SOC.Rheol. 1873, 17, 89. Duda. J. L.; Klaus, E. E.; Fan, S. K. SOC.Pet. Eng. J. 1981, 27, 613. Dullien, F. A. "Porous Media-Fluid Transport and Pore structure"; Academic Press: New York, 1979. Hong, S. A. Ph.D. Thesis, The Pennsylvania State University, University Park, PA, 1982. Kemblowski, Z.; Michniewicz, M. Rhml. Acta 1979, 78, 730. Kumar, S.; Upadhyay, S. N. Ind. Eng. Chem. Fundam. 1981, 2 0 , 186. Marshall, R. J.; Metzner, A. B. Ind. Eng. Chem. Fundam. 1967, 6 , 393. Payatakes, A. C.; Tien, C.; Turian, R. M. AIChE J . 1873. 79, 58. SheffieM, R. E.; Metzner, A. B. A I C E J. 1878, 22, 736. Unsai, E.; Duda, J. L.; Klaus, E. E. "Chemistry of Oil Recovery"; ACS Sympw sium Series 91, 1978; Chapter 8. Vrentas, J. S.; Duda, J. L. Appl. Sc/. Res. 1973, 28, 241. Wang, F. H. L.; Duda, J. L.; Klaus, E. E. Society of Petroleum Engineering Paper 8418, 1979.
Received for review February 28, 1982 Revised manuscript received January 27, 1983 Accepted February 28, 1983
Simplified Solution Technique for Moving Boundary Problems in Gas-Solid Noncatalytic Reactionst P. A. Ramachandran' Chemical Engineering Division, National Chemical Laboratory, Poona 4 1 7 008 Indla
I n many gas-solid noncatalytic reactlons the solid reactant is depleted near the surface. The prediction of the conversion-time behavior of these systems then involves the solution of a moving boundary problem, for which a simplified procedure has been proposed in this paper.
Introduction Gas-solid noncatalytic reactions are of importance in many metallurgical and chemical processes. The model equations describing these systems generally require a numerical solution due to the interaction of diffusion with reaction and also due to the inherent transient nature of these systems* The local rate Of at any given psition in the pellet is Often represented by the kinetic model NCL Communication Number: 3011.
* Currently on leave at Washington University, St. Louis, MO 63130. 0198-4313/83/1022-0305$01.50/0
-db/de = ~ f ( b ) (1) Equation 1 is based on the assumption of a first-order reaction with respect to the gas which is observed in many practical systems.The termf ( b) representsthe dependency of the rate on the solid concentration (b). This may be an empirical power law of the type: f ( b ) = b" (volume reaction models) or the form could be complex derived from suitable structural models such as the grain model (Calvelo and Smith, 1970; Szekely and Evans, 1970) or the random pore model (Bhatia and Perlmutter, 1980). For many of the commonly encountered rate forms the solid reactant is completely depleted at some positions in the pellet after a finite time (for example for n < 1in case 0 1983 American Chemical Society
