FLOW OF VISCOELASTIC FLUIDS THROUGH POROUS MEDIA The laminar flow of toluene solutions of polyisobutylene i s studied in beds packed with uniform glass spheres, sand, and binary mixtures of glass spheres. The data may be correlated by a power-law modification o f the Blake-Kozeny equation. No evidence of viscoelastic effects i s observed.
rates achieved in the porous medium. Table I shows the solution properties, including the zero-shear viscosities of the solutions and the viscosity-average molecular weight of PIB
EARLIER study by Christopher and Middleman (5) of the applicability of a non-Newtonian generalization of the Blake-Kozeny equation is extended here to the laminar flow of non-Newtonian polymer solutions through porous media. T h e earlier work, done with dilute aqueous solutions of carboxymethylcellulose, indicated that the model put forth correlated data with an average error of 18Yc and a standard deviation of the friction factor of 0.21 over a range of three orders of magnitude in a modified Reynolds number. A survey of the literature available a t that time (8) indicated some evidence of viscoelastic effects, in contrast with the conclusions drawn by Christopher and Middleman. Thus it was decided to extend the earlier study by examining a more elastic fluid, polyisobutylene dissolved in toluene. I n addition, data were obtained for tubes packed with sand, and for tubes packed with binary mixtures of glass spheres. We still fail to see viscoelastic effects. The basic experimental techniques, and the methods of reducing and correlating the data, are as described by Christopher and Middleman ( 5 ) and in more detail by Christopher (4) and by Gaitonde (7). Polyisobutylene (PIB L-100, Enjay Co.) solutions in toluene were run a t 25.0' C. Powerlaw parameters were obtained from capillary viscometry performed over a range of shear rates corresponding to the shear
AN
10'
I
i
I
L-100. Correlation of Data
Figure 1 shows the friction factor-Reynolds number correlation of the data obtained using tubes packed with glass spheres of a narrow size range. The data are correlated with an average error of 9.8YCand a standard deviation of 0.14 by
f = 1/Re
(1 )
where
AP D, ea p = LGZ (1
-
e)
PIB 1-100 in Toluene at 25.0' C., Mq K , Dynes-Sei.n V?, c, n Poises Wt. yo Sq. cm.
Table I.
8.7
112 0
8.25 9.2 9.0
61 .O 286.0 95.2
~
1 i l l 1
I
I
I
I""I
12.2 23 8
0.55 0.51 0.564 0.46 0.52
75.0
0 . 0_
I
I
I I
io:$ 15.0 13.7
= 7.3 X
lo5
71
See. -l 0,0052 0.0086 0.0047 0.0061 0.0056
Ill4
A
8 . 2 5 % PI8 I 8.7
A 9.0 e
9.2
.10-6
10-6
IO-'
10-3
lo-e
Re Figure 1. solutions
Test of modified Blake-Kozeny equation for polyisobutylene
VOL. 6
NO. 1
FEBRUARY 1967
145
and R e = DPG2-n P n - l 150 H (1 - e)
(3)
T h e factor H i s given as
H
K (9 12
= -
u
t
l5
K
+ 3,1’n)~(150 k ~ ) ( ’ - ~ ) / *
. IO
(4)
1
These results are consistent with those reported by Christopher
(5). As a test for viscoelastic effects, these data were replotted in Figure 2 a s f R e us. VOr1/Dp. This latter group plays the role of a viscoelastic parameter. 71 is a molecular relaxation time, given by Bueche (3) as
,$== 0
L
0
0
0
0
0
0
=
0
0.5
I
I
I
02
04
06
I IO
I
OB rl V o I Dp
I 12
14
Figure 2. Data of Figure 1 replotted as a test for viscoelastic effects
T h e viscosity average molecular weight, M,,was used in place of M in calculating 7 1 (6). Values of T I are included in Table
I. D p / V ois a characteristic time for the acceleration induced in the fluid because of its passage around and between spheres. When the relaxation time for the fluid is long in comparison with the time scale of flow disturbances, one might expect to see viscoelastic phenomena. In this case the viscoelastic phenomena would be observed as a failure of a purely viscous flow model to correlate the data successfully. Figure 2 reveals no such failure for values of Vor1/Dpup to 1.2. Sadowski (8, 9 ) introduced a similar parameter into his study of this problem. H e used the Ellis number:
where r1,2 is an Ellis model parameter which represents the shear stress at which the apparent viscosity falls to one half of its zero-shear value, ‘lo, Dunleavy (6) has shown, in a study of the shear behavior of solutions of polyisobutylene, that
10-
10-
IO-)
Re
Figure 3. Test of modified Blake-Kozeny equation for polyisobutylene solutions and nonuniform beds
(7) Hence our group V o ~ 1 / D is p very nearly equal to the Ellis number. Yet Sadowski observes, for El > 0.1, significant deviations between his theory and experimental results. Sadowski’s theory is based upon the Ellis model rather than the power law model, but it is nevertheless a purely viscous theory. Sadowski’s results may indeed be due to viscoelasticity, perhaps arising from a peculiarity of the fluid for which this effect was observed, Natrosol 250-H. If this is the case, then our failure to see similar results a t comparable, and even higher, Ellis numbers indicates that the Ellis number is not really the appropriate viscoelastic parameter with which anomalous behavior can be correlated. Astarita (7) comments that q o / ~ l , sis not necessarily a viscoelastic parameter, in contrast to the arguments of Bird ( 2 ) .
lo-’
the proposed model to flow of non-Newtonian fluids through beds of sand. As a final test it was decided to prepare a bed composed of a binary mixture of spheres. I n addition to providing a nonuniform particle size distribution, binary sphere mixtures allow one to extend the data to porosity values of e as low as 0.27. Two beds were prepared, one with a sphere diameter ratio of 3 to 1, and the other with a ratio of 6 to 1. I n both beds tests with Newtonian fluids indicated that the Blake-Kozeny equation held if D , was replaced by the volume-average particle size, (D,),. Figure 3 shows results for non-Newtonian fluids in binary sphere mixtures. When k is measured with a Newtonian fluid, and when ( D p ) z Iis used for the particle diameter, the modified Blake-Kozeny equation is adequate to describe the flow of non-Newtonian fluids through binary sphere mixtures.
Flow through Nonuniform Packing
Having thus established the applicability of the power-law model to flow through a packing composed of uniform spheres, it was decided to investigate somewhat more complex media. Figure 3 shows results for flow through beds packed with sand of fairly uniform grain size. As in the other cases, the permeability of the sand was first determined by using a Newtonian fluid. The experimental results indicate the applicability of 146
I&EC FUNDAMENTALS
Nomenclature G
D, El
f
G H
concentration of polymer solution, g./100 cc. particle diameter, cm. Ellis number (Equation 6) friction factor (Equation 2) = mass velocity, gram-cm.-*-sec.-l = non-Newtonian bed factor (Equation 4) dynes-sec.ncm .--l--n
= = = =
k
= bed permeability, sq. cm.
K M M,
= power law parameter: T = K(+)n, dynes-sec.n-cm.-2 s molecular weight of polymer = viscosity average molecular weight
- power law parameter A P / L = pressure drop per unit length, dynes/cc. gas constant = 8.31 X lo7 g.-sq. cm. sec.-2 g.-mole-’ O K.-l bed Reynold:?number (Equation 3) absolute temperature, K. superficial velocity in bed, cm./sec. shear rate, sec.-l void fraction zero-shear vigcosity, poises fluid density, gram,’cc. shear stress, dynes/sq. cm. molecular relaxation time (Equation 5), sec. shear stress a.t which viscosity falls to ‘/z )lo, dynes,’ sq. cm.
n
literature Cited (1) A;tarita, G., Can. J . Chem. Eng. 44, 59 (1966). ( 2 ) Bird, R. B., Zbid.,43, 161 (1965). (3) Bueche, F., J . Chem. Phys. 22, 1570 (1954).
14) Christopher, R. H., M.S. thesis, University of Rochester, Rochester. N. Y . . 1965. ( 5 ) Christopher, R: H., Middleman, S., IND.ENG. CHEM.FUNDAM E N T A L S 4, 422 (1965). ( 6 ) Dunleavy, J. E., M.S. thesis, University of Rochester, Rochester, N. y.,1965. (7) Gaitonde, N. Y . , M.S. thesis, University of Rochester, Rochester, N. Y . , 1966. 181 ~, Sadowski. T. J.. Ph.D. thesis, Universitv of Wisconsin, Madison, Wis., i963. ’ ( 9 ) Sadowski, T. J., Trans. SOC. Rheol. 9, 251 (1965). N. Y. GAITONDE STANLEY MIDDLEMAN University of Rochester Rochester, h? Y. RECEIVED for review May 9, 1966 ACCEPTED September 14, 1966
DETERMI NATION OF COMPRESSIBILITY FACTORS USING SONIC VELOCITY MEASUREMENTS Equations of state for gases would b e more reliable representations of experimental data if measurements of specific volumes could be avoided. This may b e accomplished by introduction of the sonic velocity, which is relatively easy to measure experimentally, into the thermodynamic network. The specific volume may be replaced by an integral, evaluated along an isotherm, involving the sonic velocity, the specific heat ratio, and the pressure. Compressibility factors evaluated with this modified equation of state agree almost exactly with those obtained from the standard form.
of z through direct application of the equation pv = z R T is complicated by difficulties in obtaining accurate experimental values for u. This may be circumvented by employing sonic velocity measurements a t constant temperature and a series of pressures. A closed resonance tube with fixed ends, described elsewhere (2) for a different experimental purpose, is a suitable device. A lowamplitude, variable-frequency signal, emitted a t one end of the tube and detected a t the other, defines a sequence of standing waves which permits calculation of the sonic velocity in the contained gas a t a given temperature and pressure. T h e pressure is altered by addition or deletion of gas, maintaining constant temperature T1, a n d the socic measurement is repeated. Hence a set of sonic velocities a = a(p,T1) is rapidly obtained. T h e sonic velocity is related to the state variables by ETERMINATION
a2 =
@p/dp)s = r(ap/aP)r
(11
where s is the entropy and y the specific heat ratio, C,/C,. Differentiation of the equation of state, p = pzRT, gives
and combining Equations 1 and 2 yields
(az/qJ)*
- z/p
=
- (yRT/a2) (2”p)
(3)
This expression may be linearized by the change of variable u = 2-l and the resulting standard, first-order differential equation integrated to give
(4)
This solution satisfies Equation 3 and also the required condition that lim z = 1 [making use of the property P’0 lim (yRT/a2) = 1 1. P+O
l 1
y / a ? dP has the units of density p.
In
fact, Equation 4 may be regarded as a modified equation of state wherein p(p,T1) =
y / a 2 dP, provided that the inte-
gration is performed along the isotherm TI (as indicated). T h e integral may be evaluated graphically for any p, using the experimentally determined set a ( p , TI) plus appropriate values for 7, thus determining the set z(p,Tl). Another run of sonic measurements a t a new temperature, TI, permits calculation of another set z(p, Tp), etc. I t would be desirable and pertinent a t this point to determine the applicability of Equation 4 using existing sonic measurements on a variety of gases. Unfortunately, suitable data are severely limited. T h e literature records many investigations of the sonic properties of gases, but only a few data sets show a pressure variation sufficient for computing the integral in Equation 4. Typically (7) previous workers have been interested in fine molecular structure, a type of measurement favored by conditions which minimize intermolecular forces. Usually operation through a narrow pressure range (ambient VOL. 6
NO. 1
FEBRUARY 1967
147
