Bogue’s theory and Zapas’ form of the BKZ theory correlate data u p to moderate shear rates. One adjustable parameter is involved i n both theories; however, it is possible that these parameters will prove to be constant for classes of materials. T h e latter theories predict nonlinear stress behavior from linear data (the relaxation spectrum) assuming prior knowledge of a material constant. Possibly another constant is required to predict the second normal stress difference. Acknowledgment
T h e authors express appreciation to the National Science Foundation for its support under NSF GP2005. T h e terminal work of one of the authors (JOD) was supported by the National Aeronautics and Space Administration under NASA NsG 671. T h e support of this agency is also gratefully acknowledged. Nomenclature = dimensionless material constant = dimensionless material constant = a decay function of time = shear loss modulus, F/L2 = continuous relaxation spectrum, F/L2 = first invariant o f the Finger strain tensor = second invariant of the Finger strain tensor = shear rate, T-‘ = decay function = function defined by Equation 7 = function defined by Equation 8 = a backward-running time index, T = a n arbitrary scalar in the BKZ theory, F / L 2 = decay function
of time
q
= material function (the viscosity), F T / L 2
q,,
= material constant in Oldroyd’s theory,
XI, X u
FT/L2
= dynamic viscosity, F T / L 2
q’
~
Bernstein, B., Kearsley, E. A., Zapas, L. J., J. Res. Natl. Bur. Stds. B68B, 103 (1964). Bernstein, B., Kearlsey, E. A., Zapas, L. J., Trans. SOG.Rheol. 5 , 391 (1963). Bird, R. B., Turian, R. M., Chem. Eng. Sci. 17, 331 (1963). Bogue, D. c.,IND. ENG.CHEM. FUNDAMENTALS 5,253 (1966). Bogue, D. C., Doughty, J. O., IND.ENG. CHEM.FUNDAMENTALS 5,243 (1966). De Vries, A. J., “Proceedings of 4th International Congress on Rheology,” Part 3, pp. 321-44, Interscience, New York, 1963. Doughty, J. O., Ph.D. dissertation, University of Tennessee, Knoxville, Tenn., 1966. Ferry, J. D., “Viscoelastic Properties of Polymers,” Wiley, New York, 1961. Ginn, R. F., Metzner, A. B., “Proceedings of 5th International Congress on Rheology,” Part 2, pp. 583-601, Interscience, New York, 1963. Huseby, T. \V., Blyler, L. L., Jr., “Steady Flow and Dynamic Oscillatory Experiments,” Society of Rheology, Atlantic City, N.J., October 1966; Trans.Sot. Rheol., in press. Markovitz, H., Trans.SOG.Rheol. 1, 37 (1957). Oldroyd, J. G., Proc. Roy. SOC.A200, 523 (1950). Oldrovd. J. G.. Proc. Rov. SOC. A245. 278 (1958). Pao, Y . ,J. A/$. Phys. 28, 591 (1959). Pao, Y . , J . Polymer Sci. 61, 413 (1962). Spriggs, T. W., Chem. Eng. Sci.20, 931 (1965). Spriggs, T. \V., Bird, R. B., IND.ENG. CHEM.FUNDAMENTALS 4. 182 11965). Spriggs, T. \V.; Huppler, J. D., Bird, R. B., Trans. SOG.Rheol. 10 ( l ) , 191 (1966). Staverman, A. J., Schwarzl, F., in Stuart, H. A,, “Die Physik der Hochpolymeren,” Vol. IV,Springer-Verlag, Berlin, 1956. Van Wazer, J. R., Lyons, J. I V . , Kim, K. Y., Colwell, R. E., “Viscosity and Flow Measurement,” p. 113, Wiley, New York, 1963. Williams, M. C., Bird, R. B., IND.ENG.CHEM.FUNDAMENTALS 3, 42 (1964). Williams, M. C., Bird, R. B., Phys. Fluids 5 , 1126 (1962). Zapas, L. J., “Correlations of an Elastic Fluid with Experiments,” Society of Rheology, Pittsburgh, Pa., 1964, and personal communication. Zapas, L. J., J . Res. Nat. Bur. Stds. 70A, 525 (1966). .
GREEK LETTERS a(s)
Literature Cited
X2
= material constants i n Oldroyd’s theory,
RECEIVED for review September 22, 1966 ACCEPTED March 29, 1967
T
= relaxation t h e , T i
j
= components of the deviatoric stress tensor,
I
F/L2 in
Cartesian coordinates = angular frequency, T-l
Symposium on Mechanics on Rheologically Complex Fluids, Society of Petroleum Engineers, AIME, Houston, Tex., December 1966.
FLOW OF VISCOELASTIC FLUIDS THROUGiH POROUS MEDIA R. J.
M A R S H A L L A N D A. 6 . M E T Z N E R
University of Delaware, iVewark, Del. 19711 HILE the pragmatic: significance of studies of flows through porous media requires no discussion, there is a very strong motivation for such studies from a strictly theoretical point of view: Flows in this geometry provide a n excellent opportunity for studying the behavior of viscoelastic fluids a t high Deborah number levels (Metzner et al., 1966b). This dimensionless group, representing a ratio of time scales of the material and the flow process, may be defined as (Astarita, 1967; Metzner et al., 1966a, b) ;
in which denotes the relaxation time of the fluid under the conditions of interest in the problem under consideration
and IId represents the second invariant of the deformation rate tensor (Bird et al., 1960; Fredrickson, 1964). This latter term depicts the intensity or magnitude of the deformation rate process, and the dimensionless group defined by Equation 1 may be considered to represent the ratio of the time interval required for the fluid to respond to a change in imposed conditions of deformation rate to the time interval between such changes. I t is thus an index of the extent to which the velocity field is unsteady from the viewpoint of an observer moving with a given fluid element as it proceeds along its course or trajectory in a process, using the relaxation time of the fluid as a unit of time. For perfectly steady flows-e.g., under laminar flow conditions in a very long tube-the Deborah number is identically zero; for highly unsteady processes it may be large. VOL. 6
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This work was undertaken to define the conditions under which the Deborah number may become large enough to cause significant deviations from the usual drag coefficient-Reynolds number relationships for purely viscous fluids flowing through porous media under noninertial conditions. The analysis suggests that major effects may b e expected a t Deborah numbers in the range 0.1 to 1.0. Experimental studies using a porous medium support this analysis and yield a critical value of the Deborah number a t which viscoelastic effects were first found to b e measurable. The influences of a high Deborah number level upon the uniformity of the flow and upon the distribution of the fluid residence times in porous media are considered briefly.
I t has been shown that quantitative mathematical descriptions of the properties of viscoelastic fluids rather generally predict a fluid-like response whenever the Deborah number is sufficiently low, and that these materials may exhibit a n essentially solid-like response whenever the Deborah number becomes large (Astarita, 1965; Metzner and White, 1965; Metzner et al., 1966a; Pipkin, 1966; Reiner, 1964; Tokita and White, 1966). I n the case of dilute polymeric solutions in steady laminar shearing flows (ND,,, = 0) the fluid-like response is well known. T h a t the same materials may behave as elastic solids when deformed sufficiently suddenly (Nneb large) may be demonstrated dramatically by impacting a blunt object suddenly upon a pool of such a “fluid”: In this case the material may deform very appreciably (sheets 6 to 20 inches in diameter are readily formed) but it retracts elastically to its initial configuration, rather than flowing or splashing as a Newtonian fluid does. High speed motion pictures of this phenomenon are available (Metzner and Johnson, 1966). This characteristic of viscoelastic materials, which behave as low viscosity fluids under steady flow conditions to become somewhat solid-like in their behavior whenever the Deborah number describing the system becomes sufficiently large, has important implications in boundary layer flows (Metzner and Astarita, 1967; Metzner and White, 1965), in accelerating (converging) velocity fields (Metzner, 1967), in turbulent flows (Astarita, 1965; Savins, 1964; Seyer and Metzner, 1967), and, a t least in the case of more viscous systems, in the stability of the velocity field imposed on the material (Tokita and White, 1966). T h e phenomenon of drag reduction under turbulent conditions and the iack of any dependence of the heat transfer rates in laminar boundary layers upon the velocity of the fluid, a t least under certain conditions, appear to be dramatic effects predictable from a consideration of the influences of the Deborah number of the system upon the kinds of responses offered by the viscoelastic material. I t has been suggested (Christopher and Middleman, 1965; Gaitonde, 1966; Sadowski, and Bird, 1965) that in flow of viscoelastic fluids through porous media elastic effects should arise, though their precise nature is not considered. More recently Metzner et al. (1966b) have shown that for dilute polymeric solutions this geometry is one of only a few in which the velocity field is sufficiently unsteady from a Lagrangian viewpoint to enable one to expect major departures from a purely viscous response, and that one aspect of this departure would be a great increase in the expected pressure drop through the porous medium. The purpose of the present investigation was to consider this problem in greater detail. Background and Analysis
The most definitive study of flow of Newtonian fluids through porous media appears to be that of Ergun (1952), which provides a basis for consideration of non-Newtonian effects. Good discussions of this analysis and its extension to purely 394
I&EC FUNDAMENTALS
viscous non-Newtonian fluids are provided by Bird, Stewart, and Lightfoot (1960), Christopher and Middleman (1965), and Gaitonde and Middleman (1967). As discussed by Christopher and Middleman, a number of equations are derivable for purely viscous non-Newtonian fluids, all of which reduce correctly to the Blake-Kozeny or Ergun form in the case of Newtonian fluids, but differ in the non-Newtonian case. As the choice made by Christopher and Middleman is both reasonable conceptually and supported by moderately extensive experimental measurements, it is chosen as the basis for further consideration of viscoelastic influences. Their equations may be summarized as:
APD,ea p = LGZ(l - e) and
N R= ~
D,G2 -nP“ 150 H ( l
-I
-
e)
(3)
H denotes a viscosity level parameter defined by (1 -n)/2
(4) in which K and n are the usual power-law rheological parameters. The particle diameter, D,, and the fractional void volume, e, are related to the permeability, k , of the bed by the equation :
k =
DP2e3 150 (1
- €)Z
(5)
I t has been assumed that the usual power law formulation may be employed for depicting the (non-Newtonian) properties of the fluids used. While this is likely to be adequate for purely viscous fluids, care must be exercised to ensure that the deformation rates a t which the power law parameters, K and n, are evaluated correspond to those actually encountered in the bed. For this purpose the average shear rate in the bed is estimated as:
rav =
3n+I
____
4n
12G
pd150e
for conditions of laminar (noninertial) flow-Le., olds numbers of 0.05.
(6 1 below Reyn-
The mass velocity, G, is based upon the superficial velocity in an empty bed:
G = pV,
(7)
which in turn is related to the pressure drop using a “capillary model” for the porous bed as :
for laminar flow conditions. Through combination of Equations 2 and 3, employing the definitions given in Equations 4, 5, 7, and 8. it may be shown that: f=-
1
Equations 14 give, for the second invariant defined by Equation 12 :
IId = or
fNRe =
l.OO(NRe 6 0.05)
1
2
trd2
(9)
NRe
Equation 9 is important in that it defines a "base" from which deviations due to viscoelasticity are measured. As major viscoelastic effects are encountered primarily under Lagrangian unateady flow conditions, the capillary model of the porous medium used to obtain Equations 6 and 8 may no longer suffice. I n its place the porous medium will be modeled by consideration of flows in the frusta of right cylindrical cones, converging and diverging sections being used to represent flows into and out of constrictions in the porous medium, respectively. T h e principal mode'; of deformation to which a material element is subjected as the flow converges into a constriction in the porous medium involve both a shearing of the material element and a stretching or elongation in the direction of flow. To compute the deformation rate components of interest a spherical coordinate system with its origin a t the apex of the conical section is employed, with 4 and 0 denoting angular positions across and around the cone, respectively. T h e flow of Newtonian fluids in such conical sections is discussed by Ackerberg (1962) and Langlois (1964). Rosenhead (1963) discusses the analogous Hamel flow problem (flow between converging or diverging flat plates). As a first approximation the secondary flows which occur in the conical geometry even with Newtonian fluids are neglected-i.e., it is assumed that = UT
UT
+,
=
( X I , X+)
ve = 0
(loa)
and the material derivative reduces to:
DIDt =
b br
V I -
To take the material derivative of the square root of the second invariant of the deformation rate tensor, required for evaluation of the Deborah number (Equation l ) , one may now introduce the usual order-of-magnitude arguments of boundary layer theory (Schlichting, 1960) to obtain a n explicit though approximate result through evaluation of the derivatives in Equations 15 and 16 by means of finite differences. Distances along the direction of the primary flow, xr, may be associated with a length scale, L1, of the particles which make up the porous medium, and distances across the conical channel with a secoiid length scale, LZ. These length scales will in general differ, being equal only in the special case of isotropic porous media. Introducing these considerations, vT
r
4
z E
m
Av'
Ar
A$
V
(17 4
LI
(1 7b)
E
z E
Lz/L1
(17c)
Substitution of Equations 17 into Equations 15 and 16 gives for the material deiivative of Equation 1 :
(1Ob)
Using Equations 10 the continuity equation reduces to:
a
--
i)r
(rzzf) = 0
The second invariant of the deformation rate tensor, d , is given by the relation:
IId =
-l/2
trd2
(12)
and the contravariant components of d may be evaluated employing the methods outlined by Hawkins (1963) or McConnell (1957), to be a.s follows for the velocity field described by Equations 10:
cP1 = dv'/br d12 = avr/2 r2b+ d22 =
vT/r3
d33 = u3/r3 sin2 4
T h e first term under the square root in Equation 18 arises from fluid stretching deformations, the second from shearing. T h e comparative magnitudes of these terms will depend upon the ratio LZ/L1, but fluid stretching is more important unless L2 is substantially smaller than L I , which would be true only if the porous medium were to be made up of elongated particles aligned in the direction of the velocity field. Introduction of Equation 18 into Equation 1 gives the Deborah number as:
(1 3 4 (13b) (1 3c) (1 3 4
with all other terms identically zero. T h e physical components corresponding to Equations 13 are:
Equation 19 again emphasizes the importance of the length scale, L I , in the direction of the flow, as distinguished from the distance, L z , across the pore. I n the special case of a porous medium composed of spherical particles, as used in the present experimental study, one may take, as a n approximation, L1 = Lz = D p / 2 . I n this case, and only in this case, Equation 19 reduces to:
This Deborah number appears to have the form of a fluid relaxation time multiplied by a shear rate, but reference to Equation 19 shows this similarity to be illusory, as the principal deformation rate term is that due to fluid stretching rather than to shearing. I t is also clear that the frequently VOL. 6
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AUGUST 1 9 6 7
395
used capillary tube models of porous media, which d o not have a meaningful length scale, L1, cannot be used to define a Deborah number or predict effects of fluid elasticity. I n view of the fact that the elongation or stretching of fluid elements is of interest, it seems pertinent to inquire into the form of the stress-strain rate relationships for such deformations. I t is readily shown that viscoelastic materials may not be subjected to infinite stretch rates since the stresses involved become infinite a t a finite and rather low stretch rate, which is dependent on the characteristic time of the fluid (Astarita, 1967; Lodge, 1964; Metzner, 1967). If one uses the contravariant convected “Maxwell” model of the fluid as a realistic, simple approximation to the behavior of real materials in both steady and unsteady velocity fields (Etter and Schowalter, 1965; Ginn, 1963; Ginn and Metzner, 1965; Kapoor et al., 1965; Vela et al., 1965; White and Metzner, 1963), as defined by the equations:
frusta were precise. Thus both the magnitude of the excess pressure losses and the critical value of the Deborah number a t which such excess pressure losses are first observed may be expected to be functions of the uniformity of the porous medium. 5. As the pressure drop increases to levels appreciably in excess of that defined by the base line, f ~ = i1.00,~ because ~ portions of the medium become partially restricted, a sharp change in the uniformity of flow through the medium, measurable by means of suitable residence time distribution studies, is to be expected. 6. In the case of anisotropic media, with L1 either much smaller or much larger than Lz, limiting forms of Equations 19 may be of interest. As LL/L2 + 0 (highly anisotropic media with flow across the short axes of the particles), the limiting Deborah number is essentially indistinguishable from that defined by Equation 19a or 19b as the LdL2 term of Equation 19 contributes little even in the case of isotropic media. On the other hand, as L1/L2 -L m (flow in the direction of the major axis of highly anisotropic particles), the Deborah number approaches the limiting form: V
1
and the axial length scale still is important. and
Experimental
the stresses in a simple steady stretching experiment are preapproaches dicted to rise to infinity as the group 2 O~~bv,/bx unity. (This general conclusion does not appear to depend upon the constitutive equation chosen, though the numerical constant does. I n particular, the usual Rivlin-Ericksen approximations to the behavior of simple fluids yield a similar result except when incorrectly truncated after only a small number of terms.) If the stretch rate, bv,/dx, in the flow process through a n isotropic porous medium is approximated by 2 V/D,, this suggests that when a Deborah number simply defined as
approaches a value of about l / 4 , the pressure drop required to pump the viscoelastic fluid through-the porous medium should also rise to infinity. T h e model used is too crude to enable such a prediction to be precise, but the following key features must be recognized in the case of flow of viscoelastic fluids through porous media:
1. Very low values of the Deborah number imply a velocity field in which viscoelastic effects are negligible: The fluid is able to respond to its local state of deformation essentially instantly-i.e., it does not “remember” its earlier configurations or deformation rates. In this case the ChristopherMiddleman result (Equation 9) should apply equally to viscoelastic and to purely viscous fluids. 2. As the flow rate or the Deborah number is increased from very low values, a point should be reached a t which the solid-like characteristics of viscoelastic materials begin to make themselves felt. In view of the above comments, these should take the form of a pressure drop which is larger than that predicted by Equation 9. The precise value of the Deborah number, B,lV/D,, a t which measurable effects are first obtainable is not known, but it Ivould presumably occur a t values appreciably lower, perhaps a n order of magnitude lower, than the value of 1/4 a t which major effects are to be expected. 3. As the Deborah number is further increased into the range 0.1 to 1.0, pressure drop requirements far in excess of those predicted by Equation 9 are likely to be encountered. 4. Since most porous media contain a distribution of pore radii, rather than a single one, a sharp cutoff of flow, or a pressure drop approaching infinity, would not be expected a t O,LV/D, equal to l / 4 , even if the analysis of flow in conical 396
I&EC FUNDAMENTALS
The porous medium used consisted of a sintered bronze porous disk 0.50 inch in diameter and 0.25 inch in length. T h e spherical particles comprising the porous medium had a number-average diameter of 0.01266 cm., as determined by microscopic measurement; the distribution of diameters about this mean value was narrow, the mean squared deviation being 14%. The bed was attached to the bottom of a brass reservoir, mounted vertically, and connected to a dry nitrogen constant pressure system. With this arrangement pressure drop-flow rate measurements were obtained in the standard manner. The bed porosity was calculated from a knowledge of the slopes of the pressure drop-flow rate curves of the Newtonian fluids used, employing Equations 7 and 8, knowing the particle diameter, D,, and employing only data for which AVll, < 0.05. Having values for the porosity and the particle diameter, the bed permeability was obtained through use of Equation 5 . The values of these three geometric parameters may be summarized as follows for the porous medium used in this work :
D , = 0.01266 cm. E
= 0.486
k
= 46.4
x
10-8 sq. cm.
Experimental data were obtained using two Newtonian and three non-Newtonian fluids, the former for calibrating the bed and determining the adequacy and accuracy of the experimental techniques. The Newtonian fluids employed were 50 and 75% (by weight) aqueous solutions of glycerol. The polymeric non-Newtonian solutions were chosen to obtain a wide variation in the degree of viscoelasticity as measured by the magnitudes of the relaxation times of these materials; the specific systems chosen were as follows: Polymeric Material
Carbopol934 Polyisobutylene LlOO ET-597
Concn. in Soln. 0 . 2 7 , in water
5 . OY0 in Decalin 0 . 2 5 7 , in water
Comparative Reiaxation Times
Very low Intermediate High
Measurements were also made with ET-597 solutions over a concentration range from 0.1 to 0.45y0. Available measurements of the relaxation times of these solutions as a function of
concentration indicate them to be only weakly dependent upon concentration within the range of interest (Seyer and hfetzner, 1967; Shertzer, 1965; Uebler, 1966). As a result, the general behavior of these systems did not change greatly with concentration level. The results for the 0.25% solution, for which the most detailed normal stress (relaxation time) measurements were available, are the only ones reported in detail for this polymeric system. Rheological measurements for the characterization of the fluid properties were made as follows: Shear stress-shear rate rurges were obtained using either a capillary tube or a coneand-plate viscometer, using standard techniques to reduce the data. Normal stress measurements obtained using both a rheogoniometric device and a capillary thrust apparatus are available from Shertzer (1965), Uebler (1966), and Seyer (1367) for ET-597 solutions, and Ginn (1963, 1965, 1967) and Uebler (1964) f t x polyisobutylene solutions. Savins (1 966) provided normal stress measurements for Carbopol. These data are frequently sensitive to the method of fluid preparation. As a result, only those batches of fluid for which the shear stress-shear rate curves were similar to those obtained on samples used in the normal stress measurements were employed in this study. .4 number of recent papers, in which inability to correlate pressure drop-flow rate measurements in porous media was noted, have invoked conjectures of either positive or negative slip a t the solid surfaces to rationalize the results. Several of these report complicatisms which appear to be real, although others did not use proper equations to represent non-Newtonian behavior in porous media and are evidently invalid. Since capillary tube measurements using tubes having diameters comparable to those of the particles used in the present work reveal no effects with the fluids employed, such hypotheses are clearly inapplicable to the results a t hand and are not considered herein. O n the other hand, if porous media are employed in which the voids have dimensions of the same order of magnitude as those of the macromolecules used, molecular adsorption onto the surfaces of the porous medium may affect its permeability appreciably and must be taken into consideration. Thus additional complications, beyond those considered in the present analysis. may be of importance in studies involving media of significantly lo\ver permeability. T h e present study was designed to provide an unambiguous analysis of the importance of viscoelastic effects; it should be free from aberrations due to adsorption. I n porous media having low permeabilities the latter effects may be of great significance, hoivever: and studies separately directed to their understanding are a further important requirement. Results Figure 1 shows the friction factor-Reynolds number results for the Newtonian fluids and the Carbopol solution. The Sebvtonian data serve to calibrate the bed and technique; thus their general agreement with the Blake-Kozeny equation (-VRe < 0.05) is not surprising and shows only the accuracy and consistency of the tivo sets of data. ,4dditionally, however, they also conform to the Ergun equation at Reynolds numbers above 0.05, at. which inertial effects are beginning to be felt, thus providing an additional check on the correctness of the results. This was felt to be especially important in view of the unusually high porosity of the medium used in this kvork. The Carbopol solution is characterized by very small Deborah numbers and would not be expected to show appreci-
N "E 10-5
IO-'
IO-'
10-2
10-1
I IO'
A
IO'
75% G
IO'
f IO'
IO2
IO'
10
f
IO'
I
lo-'
10-
10-7
10-5
NR, Figure 1. Drag coefficient-Reynolds number relationships for fluids showing only viscous and inertial responses
able viscoelastic effects; the data fall on the laminar line as expected for purely viscous fluids. This provides a n additional check on the experimental techniques employed and extends somewhat the experimental verification of Middleman and his co\vorkers (1965, 1967) of the correctness of the non-Newtonian terms in Equations 3 and 6. In previous studies the values of the flow behavior index ranged from 0.46 to 1.00: while the Carbopol data of Figure 1 are for a flow behavior index of 0.40. The range of Reynolds numbers included in Figure 1 is greater than that used by prior workers by a factor of 100. The check obtained bet\veen the Carbopol and the Sewtonian data also supports the earlier comments concerning the expected absence of polymer adsorption effects in a porous medium having the permeability level used in the present study. The data for the more highly viscoelastic fluids, when plotted on the f-.VIte coordinates of Figure 1, generally fall well above the laminar curve, and the friction factor is not so greatly dependent on the Reynolds number. In essence a series of branching curves, of l o ~ vslope and departing from the curves of Figure 1 a t a series of well defined but lo\v Reynolds numbers, was obtained. In the case of several of the more dilute ET-597 solutions, Lvith which very high flow rates could be attained, the ratio of the experimental value of the friction factor to that given by the Blake-Kozeny-Ergun equations was as great as 80 and ratios above 10 were readily and frequently obtained. The results obtained a t .\-Re < 0.05 and for which detailed relaxation time measurements are available are shown in Figure 2 using the f-Vlle us. 6',,V/D, coordinates suggested by the theoretical analysis. The ordinate, representing the ratio of the actual pressure drop to that predicted for purely viscous fluids, though not so large or dramatic as the figure of 80 quoted earlier, nevertheless rises to nearly 20 for these fluids. The product f-Y1le depends only on the measured pressure drop and flow rate in the packed bed, the bed characteristics, and the viscous properties of the fluids used. Thus, for a given fluid the ordinate of Figure 2 shows directly how rapidly the pressure drop increases as BflV/D, increases. The relaxation VOL. 6
NO. 3
AUGUST
1967
397
100
IO'
0 CARBOPOL A POLY I S 0 B U T Y L E NE 0 ET-597
10
IO* 0 0.25% ET-597 2 0 % CAREOPOL
AP 10
100
f NRE
I
AP I
10
0.I ' O I
I i i i i i i i i i i i i i i i i iiil IO-e
I
10-1
IO
I 10-3
10-2
lo-'
I
10
100
0
(0'1 V ) / Dp Figure 2. Dependence of viscoelastic effects 1-00)on the Deborah number of the flow process
(fN,,
>
All data in region N R < ~ 0.05
AP, p.5.i.; Q,
time, is obtained from measurements of the normal and shearing stresses in simple laminar shearing flow experiments using the relation:
which is derivable directly from Equations 20 (Seyer and Metzner, 1967; White and Metzner, 1963). As in the case of the viscosity function, p , B f l is not a constant except over limited ranges of conditions, but is dependent upon the fluid deformation rate. i n the present work it was largely immaterial whether this term was evaluated a t deformation rates defined by Equation 15 or 6 ; the simpler Equation 6 was chosen. Since O f l decreases rapidly as V / D , increases, the abscissa of Figure 2 has the effect of contracting experimental pressure drop-flow rate results. The ranges of the variables included in Figure 2 are summarized in Table I. Though the relaxation times listed in Table I are comparable for all three fluids, the flow rates used (Reynolds numbers) were different; a t equal deformation rates the relaxation times are approximately equal for the PIB and the ET-597 solutions and one order of magnitude larger than those of the Carbopol. The principal virtue of the ET-597 is thus not so much that it is more highly elastic than the other fluids but that this elasticity is developed a t such low polymer concentrations as to enable very high flow rates in view of the comparatively low viscosity levels of such dilute solutions. The critical value of the Deborah number a t which appreciable influences of the fluid elasticity are first felt is approximately 0.05 to 0.06> in agreement with the values expected from the analysis of this problem. As the Deborah number
Table 1.
Ranges of Flow Conditions and Fluid Properties Included in Figure 2 Rejnolds 'Vumbers, Flu id ell, Millisec. Dimensionless 2.16 X 10-8 Carbopol 0,005-27 7
Polyisobutylene
0,96-34
ET-597
0.95-6 , 8
ox
I&EC F U N D A M E N T A L S
10-3
2.25 X 10-8 2.87 x 10-4 6.05 x 10-4 0.035
398
Figure 3. Experimental pressure drop-flow surements in porous medium
rate mea-
cc./sec.
rises to above unity, the experimental pressure drops increase asymptotically toward infinity; this increase would be expected to be somewhat less rapid in beds having a wide range of pore diameters. However, it may also be even greater for more highly elastic solutions and is only partly dependent on the contraction afforded by the particular abscissa used in Figure 2. These facts are emphasized in Figure 3, in which the experimental pressure drop-flow rate measurements for the Carbopol and ET-597 solutions depicted in Figure 2 are shown directly and further compared with the results obtained using a more concentrated ET-597 solution. The slope of the AP - Q curve for Carbopol compares well with the shear stress-shear rate curve for the same material. By contrast, the flow behavior index of the ET-597 solutions is in the range 0.6 to 0.7, while the AP - Q curves for the packed bed never exhibit a slope below 0.94. The startling increase in slope of the A P - Q curve for the more concentrated ET-597 solution at high flow rates indicates, tentatively. a complete transition from viscous flow throughout the bed to flow conditions determined by fluid stretch rate limitations. I t is fruitful to consider two further questions: Are the effects noted in Figures 2 and 3 real or due to some spurious effect; if real, why have specific viscoelastic effects not been reported by prior investigators? O n the first it is important that only one bed length was used and if a large end effect were present it would be included in the calculated results. Gaitonde (1966) tabulated data taken with several bed lengths (approximately 60 and 120 particle diameters) and found only small end effects under his conditions. The present bed, of length 50 diameters, does not suffer obviously from this difficulty. I n a similar vein end corrections for flow into capillary tubes, though very large for viscoelastic materials (Astarita and Metzner, 1966; Feig, 1966; La Nieve, 1966; Metzner and White, 1965; Pruitt and Craivford, 1965), indicate maximal values which, though not negligible, are but a fraction of the effects noted in Figure 2, and it is not clear that any end effect in the packed bed should be larger than that a t the abrupt entry to a capillary. Additionally, the end effect \vould need to be 2 to 3 orders of magnitude greater than the pressure drop past a single particle in the bed to account for the present results, while effects of comparable magnitude \\auld be expected if the bed itself were not constricted. Finally, several runs were made with polyisobutylene solutions a t high flow
rates using a bed twice as deep as that used in the quantitative studies. Though these deep bed tests were not so detailed nor so accurate as the data reported, they would have revealed any large effects, yet none were found. Thus the absence of end effects, for all fluids used, has not been demonstrated conclusively but there is good evidence that this is not a special problem. Spurious effects due to a progressive plugging of the bed or to adsorption of m,acromolecular aggregates must also be considered. Adsorption may be ruled out by the fact that flows through capillary tubes of comparable diameters revealed no problems (Cebler, 1964). Secondly, the solution concentrations employed were chosen partly on considerations of optical clarity-i.e., the “fish eyes” so commonly observed in aqueous polymeric solutions appeared to be absent. Finally, the sequence of the experiments was such that some of the data a t the highest flow rates were always taken early in a series of runs, and some of the lowest flow rates were always last, so that any progressive plugging of the bed would have been revealed directly ‘either as scatter on a plot such as Figure 2 or as a trend with time in the original data. T h e absence of both aberrations appears to rule out such effects. Other investigation3 in which polymeric solutions were employed are those of Christopher, Gaitonde, and Middleman (1965, 1966, 1967), Dauben (1966), McKinley and coworkers (1966), Pye (1964), Sadowski (1965), and Sadowski and Bird (1965). The latter investigators believed they had found significant viscoelastic effects, but Christopher, Gaitonde, and Middleman showed convincing evidence but this was not the case, as their analysis for purely viscous fluids correlated not only their own data but also those of Sadowski and Bird. I n view of the large particle diameters (hence, low shear rates) and the comparatively low elasticity levels of the cellulose derivatives used by Sadowski-see, for example Metzner et al. (1964) and Oliver (1966) for data on similar though not identical fluids-this appears to be reasonable : Deborah numbers as defined by Equations 19b and 21 appear to be small for the fluids used by Sadowski, as Equation 21 would yield much smaller values of the relaxation time than the alternative definition used by Bird and Sadowski. Similarly, if one approximates th.e relaxation times of the materials used by Gaitonde and Middleman through use of the normal stress measurements reported by Ginn (1963), one obtains values for Oii which are lower than those given by the semitheoretical approximation employed by Gaitonde and Middleman for the same purpose. In fact, Ginn’s relaxation times give maximal Deborah numbers in the Gaitonde-Middleman study which are approximately equal to those a t which measurable viscoelastic effects were first noted in Figure 2. Thus Gaitonde and Middleman may have reached but not entered the region in which appreciable viscoelastic effects are observable; no evident inconsistency exists between their work and the present study. I n a similar manner McKinley and coworkers (1966) were able to interpret their measurements using relationships for purely viscous fluids. O n the other hand both Pye (1964) and Dauben (1966) observed major departures from the behavior of purely viscous fluids similar to those reported in Figure 2. No viscoelastic measurements are available for the fluids used by Pye, but the data of Dauben would appear capable of a n interpretation similar to that embodied in Figure 2. Neither of these studies was known to the present investigators when this paper was first prepared; similarly there has been no opportunity to study the even more contemporary work of Hermes (1966), Slattery (1966), and Jones and Maddock (1966) in the frame-
work of the present analysis. However, limitations of the viscometric measurements of the latter authors will make their results difficult to incorporate into any quantitative analysis. Applications
T h e threshold values of the Deborah number a t which viscoelastic effects first appear, 0.05 to 0.06, correspond to linear velocities of 60 feet per day in a porous medium having a permeability of 5 X 10-8 sq. cm. and about 20 feet per day a t a permeability level of 0.5 x 10-8 sq. cm., using fluids having a relaxation time of 0.01 second a t the deformation rate levels employed (50 to 100 sec.-l). Relaxation times of the order of 0.01 to 0.1 second a t low shear rates are not difficult to obtain (Uebler, 1966). I t thus seems possible that measurable effects could be obtained in petroleum production and in waste disposal operations with fluids characterized by very large relaxation times. Conclusions
An analysis of flow in converging channels suggests that the pressure drop should increase to values well above those expected for purely viscous fluids a t Deborah number levels of the order of 0.1 to 1.O. An experimental investigation reveals measurable increases beyond Deborah numbers of about 0.05 and a n order-of-magnitude effect a t N D =~ 1.0. ~ These results appear to be consistent with those of earlier investigators, who found no effects but who worked a t significantly lower levels of the Deborah number; several other recent studies show effects of comparable magnitude. The dependence of the observed behavior upon the homogeneity and the isotropy of the porous medium, and its effect on flow uniformity, would appear to be important and to require study. I n particular, the large void volume of the porous medium used in the present study suggests that the medium may not have been perfectly isotropic as assumed and considerable work may be necessary to relate the effects noted to pore geometry in a quantitative manner. Acknowledgment
The assistance of J. V. Bishop and the Dow Chemical Co. in arranging for a donation of the experimental polymer used (ET-597) is appreciated. Helpful discussions with R. F. Burdyn, R . A. Hermes, S. Middleman, and J. G. Savins are acknowledged with thanks. The assistance of J. G. Savins and S. Middleman in making unpublished results available is especially appreciated. Nomenclature
d D, D/Dt
f 8
= deformation
rate tensor, having contravariant components d t j and physical components d ( * j );
= material derivative = drag coefficient or friction factor
= metric tensor, having contravariant components
k
g%j(Hawkins, 1963; McConnell, 1957) = mass velocity = viscosity level parameter (Equation 4) = permeability of porous medium (Equation
K
=
L
=
L1, Lz
=
n
= = =
G
H
AP
Q
5)
fluid consistency variable in power law formulation length or depth of porous medium, a macroscale of the bed microscales of porous medium, related to pore structure flow behavior index (power law exponent) pressure drop flow rate VOL. 6
NO. 3
AUGUST 1967
399
r
= = = =
R t V”,m 07,
ve, vd =
V
=
V,
=
xr, xo, xd = hiDeb = NR~ = cy =
r
=
I.1
= = = = =
T, T’
=
nd
=
E
P
*rz 0, I
radial position coordinate radius of time or conduit time covariant derivative of vi with respect to x m coordinate velocity components (contravariant) in directions of r , 0, and 6 coordinates mean velocity of flow through porous medium or conduits superficial mean velocity based on total crosssectional area coordinate labels, spherical coordinate system Deborah number, Equations 1 and 19-19c Reynolds number, Equation 3 isotropic pressure (arbitrary for incompressible fluids) deformation rate or shear rate fractional void volume density relaxation time of fluid characteristic process time viscosity of fluid stress tensors; T denotes total stresses and Q’ stresses relative to isotropic pressure, cy second invariant of deformation rate tensor, Equation 12
literature Cited Ackerberg, R. C., Ph.D. thesis, Johns Hopkins University, Baltimore, Md., 1962. Astarita, G., IND.ENG.CHEY.FUNDAMENTALS 4, 354 (1965). Astarita. G.. IND.ENG.CHEM.FUNDAMENTALS 6. 257 (1967). Astarita; G.; Metzner, A. B., Atti. Accad. Lincei 40, 606 (1966). Bird, R. B., Stewart, \Y. E., Lightfoot, E. N., “Transport Phenomena,’’ pp. 196-207, Wiley, New York, 1960. Christopher, R. H., Middleman, Stanley, IND. ENG. CHEM. FUNDAMENTALS 4, 422 (1965). Dauben, D. L., Ph.D. thesis, University of Oklahoma, Norman, Okla.. 1966. Ergun, Sabri, Chem. Eng. Progr. 48, 89 (1952). Etter, Irwin, Schowalter, W. R., Trans. SOC.Rheol. 9 (2), 351 (1965). Feig, J. L., M.Ch.E. thesis, University of Delaware, Newark, Del., 1966. Fredrickson, .4.G., “Principles and Applications of Rheology,” Prentice-Hall, Englewood Cliffs, N. J., 1964. Gaitonde, N. Y., M.S. thesis, University of Rochester, Rochester, N. Y., 1966. Gaitonde, N. Y., Middleman, Stanley, IND.ENG.CHEM.FUNDAMENTALS 6, 147 (1967). Ginn, R. F., M.Ch.E. thesis, University of Delaware, Newark, Del., 1963; Ph.D. thesis in preparation, 1967. Ginn, R. F., Metzner, A . B., Proceedings of 4th International Congress on Rheology, p. 583, Interscience, New York, 1965. Hawkins, G. A , , “Multilinear Analysis for Students in Engineering and Science,” Lt’iley, New York, 1963.
Hermes, R. A., Meeting, Society of Petroleum Engineers, December 1966. Jones, W.M., Maddock, J. L., Paper 1686, Society of Petroleum Engineers, December 1966. Kapoor, N. N., Kalb, J. W., Brumm, E. A., Fredrickson, A. G., IND.ENG.CHEM.FUNDAMENTALS 4,186 (1965). Langlois, \Y. E., “Slow Viscous Flow,” Macmillan, New York, 1964. LaNieve, H. L., Ph.D. thesis, University of Tennessee, Knoxville, Tenn., 1966. Lodge, A. S., “Elastic Liquids,” Academic Press, New York, 1-O,h4, .
McConnell, A. J., “Applications of Tensor Analysis,’’ Dover, New York, 1957. McKinlev. R. M.. Jahns. H. 0.. Harris. W. W.. Greenkorn. R. A,. A.I.Ch.E. J . 12,’17 (1966). ’ Metzner, A. B., A.I.Ch.E. J . 13, 316 (1967). Metzner, A . B., Astarita, G., A.I.Ch.E. J., in press. Metzner, A. B., Houghton, W. T., Hurd, R. E., Wolfe, C. C., Proceedings of International Symposium on Second-Order Effects in Elasticity, Plasticity, and Fluid Dynamics, p. 650, Pergamon Press, Oxford, 1964. Metzner, A. B., Johnson, M., annual meeting, Society of Rheology, 1966. Metzner, A. B., White, J. L., A.I.Ch.E. J. 11, 989 (1965). Metzner, A. B., \Yhite, J. L., Denn, M. M., A.I.Ch.E. J . 12, 863 Il966ai. Metzner,’A. B., White, J. L., Denn, M. M., Chem. Eng. Progr. 62 (12), 81 (1966b). Oliver, D. R., Can. J . Chem. Eng. 44, 100 (1966). Pipkin, A. C., Quart. J . Appl. Math. 23, 297 (1966). Pruitt. G. T.. Crawford. H. R.. Report to David Taylor Model B a s k Cloniract Nonr-4306(00) Cl$65). Pyi, D. J.: J . Petrol. Technol. 1 6 , 911 (1964). Reiner, M., Physics Today 17, 62 (January 1964). Rosenhead, L., “Laminar Boundary Layers,” Oxford University Press, Oxford, 1963. Sadowski, T. J., Trans. SOC. Rheol. 9 (2), 251 (1965). Sadowski, T . J., Bird, R. B., Trans. Soc. Rheol. 9 (2), 243 (1965). Savins, J. G., private communication, 1966. Savins, J. G., SOC.Petrol. Eng. J . 4, 203 (1964). Schlichting, Hermann, “Boundary Layer Theory,” 4th ed., McGraw-Hill, New York, 1960. Seyer, F. A,, Metzner, A. B., Can. J . ChPm. Eng., in press. Shertzer, C. R., Ph.D. thesis, University of Delaware, Newark, Del., 1965. Slatterv, J. C., Paper 1684, Society of Petroleum Engineers, December 1966. Tokita, N., White, J. L., J . Appl. Polymer Sci.10, 1011 (1966). Uebler. E. A,. M.Ch.E. thesis. University of Delaware, 1964. Uebler; E. A,; Ph.D. thesis, University of Delaware, 1966. Vela, Saul, Kalb, 3. LY., Fredrickson, A. G., A.I.Ch.E. J . 11, 288 (1965). White, J. L., Metzner, A. B., J . Appl. Polymer Sci. 7 , 1867 (1963). RECEIVED for review November 1, 1966 ACCEPTED April 25, 1967 Work supported by the Water Resources Center, University of Delaware.
HOLDUP IN IRRIGATED RINGIPACKED
TOWERS BELOW T H E LOADING POINT J.
E. B U C H A N A N
University of hrew South Wales, Kensington, N.S. W., Australia
holdup may well be considered as the basic liquidside dependent variable in packed tower operation. Holdup has been shown to have a direct influence on liquidphase mass transfer (2),on loading behavior (72), on gasphase pressure gradient (72), and on mass transfer (9). Ir. itself it is important only in the consideration of unsteadystate behavior of a tower-e.g., in batch distillation ( 8 ) . IQIJID
400
l&EC FUNDAMENTALS
Many workers (3, 4, 7, 70-72) have measured holdup, with or without gas flow, and have produced empirical descriptions of their results. Only the correlation of Otake and Okada (7) is in dimensionless form and can claim any generality. This correlation fits the available experimental data very well but it is derived by essentially empirical methods. I t is desirable therefore to justify this form of relation theo-
