Conclusions
GREEKLETTERS
Numerical solutions developed for laminar flow of variable viscosity Newtonian fluids in the entrance region are the first with variable viscosity to take inertial and radial convection flow development phenomena into account. In heating of viscous materials and in constant viscosity flow, the flow development effect is important only in a region very near the entrance. However, in cooling of viscous materials the flow development effect is important over a wide range of conditions. The solutions agree with experimental measurements and correlations of experimental data in the ranges where measurements are available, and are more reliable than empirical correlations of experimental data even in ranges of the parameters covered by the experiments. T h e numerical method used is direct and requires relatively little computation bv modern standards.
cy
Nomenclature a
D
= =
Gz h
=
h,
=
k
=
=
Nu = Nu, =
P p Pr
R r 5’ T U u
= = =
= = = =
= =
=
V u
w X x
= = =
= =
tube radius tube diameter Graetz number, ( w c ) / ( k x ) heat transfer coefficient based o n mixed mean temperature heat transfer coefficient based on average of inlet and outlet mixed mean temperatures thermal conductivity local Nusselt number, hD/k average Nusselt number, h,D/k dimensionless pressure, (pa)/(.ii2pvl) pressure Prandtl number dimensionless radial distance, r / a radial coordinate dimensionless viscosity ratio, p/pi temperature dimensionless axial velocity, u/a axial velocity mean axial velocity dimensionless radial velocity, (va)/cy radial velocity mass flow rate dimensionless axial distance, ( x a ) / ( v i a 2 ) axial coordinate
thermal diffusivity temperature difference parameter, (Ti T,J/Tu = dimensionless temperature ( T - T w ) / T ( , - Tu) = viscosity = kinematic viscosity = =
e
6 p Y
-
SUBSCRIPTS b = bulk i = inlet k = radial position w = wall SUPERSCRIPTS = axial position
n
literature Cited
(1) Bodoia, J. R., Osterle, J. F., Appl. Sei. Res. A10,255 (1961). (2) Christiansen, E. B., Craig, S. E., A.Z.CI1.E. J . 8, 154 (1962). (3) Collins, Morton, Schowalter, W. R., Zbid., 9, 804 (1963). (4) Graetz, L., Ann. Phys. Chem. 2 5 , 337 (1885). (5) Kays, W. M., Trans. Am. SOC.Mech. Engrs. 77,1265 (1955). (6) Koppel, L. B., Smith, J. M., J . Heat Transfer 84, 157 (1962). (7) Langharr, H. L., J . Appl. Mech. 9 , 55 (1942). (8) Leveque, J., Ann. Mines 12, 201, 301, 381 (1928). (9) Pfenninger, W., reported in (76). (10) Piqford, R. L., Chem. Eng. Progr. Symp. Ser. No. 17, 51, 79 (1955). (111 Prandtl, L.. Tietiens. 0. G.. “ADulied Hvdro and Aero’ mechanics,’” p.’25, DLver, New York, ij34. (12) Reshotko, E., reported in (76). (13) Rosenberg, D. E., M.S. thesis, Rice University, Houston, Tex., 1963. (14’1 Sieder. E. N.. Tate. G. E.. Znd. Eng. Chern. 44. 1683 (1936). ’ (15) Siegel,’R., Sparrow, E. M., A.Z.Ch.E. J . 5 , 73 (1959). (16) Sparrow, E. M., Lin, S. H., Lundgren, T. S., Phys. Fluids 7, 338 (1964). (17) Wang, Y . L., Longwell, P. A., A.Z.Ch.E. J. 10, 323 (1964). (18) Wendel, M. M., Whitaker, S., Appl. Sei. Res. A l l , 313 (1962). (19) Yarnagata, K., Mem. Fac. Eng. Kyushu Imp. Univ. 8, 365 (1940). \
,
RECEIVED for review October 27, 1964 ACCEPTEDApril 27, 1965 Work partially supported by the National Science Foundation under Grant GP 661.
POWER-LAW FLOW T H R O U G H A PACKED T U B E ROBERT H . C!HRISTOPHER1 AND STANLEY M I D D L E M A N
Department of Chemical Engineering, University of Rochester, Rochester, N . Y . HE flow of non-Newtonian liquids through packed and T p o r o u s media is a phenomenon of increasing familiarity as the technological importance of high polymeric materials grows. Examples of such flows include the filtration of polymer solutions and slurries, and the movement of aqueous polymer solutions through sand in secondary oil recovery operations. Just as in the case of the corresponding Newtonian flows, the complex geometry of porous media precludes any detailed solution of the equations of motion. Instead one seeks simple correlative techniques for relating the imposed pressure drop across a system to the flow rate through the system. 1 Present address, Experimental Station, E. I. du Pont de Nemours Br Co., Inc., rvilmington, Del.
422
I&EC FUNDAMENTALS
T h e most successful semi-empirical expression describing the laminar flow of Newtonian fluids through packed beds is the Blake-Kozeny equation (7) :
v, =
AP Dp2 t3 150 L p(1 - e)’
Here V, is the superficial velocity, defined in terms of the mass velocity (mass flow rate per unit area of empty bed) :
pVo = G
(2)
The mean particle diameter is defined in terms of the “specific surface” of the packing, a,:
Dp
= G/a,
(31
The “capillary model” i s used to develop a modified Blake-Kbzeny equation application to the laminar flow of non-Newtonian fluids through packed and porous media. It i s assumed that the power law can be used to represent the rheological behavior of the fluid. The theory is tested for the flow of dilute pdlynier solutions through a pipe packed with small glass spheres. The data can be correlated with an average error of 18%, over a range of three orders of magnitude in a modified Reynolds number. Data obtained in an independent study by Sadowski can be correlated with an average error of 1270 over another range of three orders of magnitude in Reynolds number.
where a, is the ratio of the total particle surface area to the total volume of particles in the bed. I n the case of a bed packed with uniform spheres, D , is just the diameter of the packing. T h e “permeability,” k, of the medium is often introduced as a characteristic property of the medium. I t is usually given as
k =
Dp2 e3 150(1
- e)’
(4)
T h e Blake-Kozeny equation may be derived through the so-called “capillary tube,” or “hydraulic radius,” model (7). T h e factor of 150 appearing in the denominator has been established experimentally. T h e effect of the confining walls of the bed, and any entrance and exit effects, are not accounted for in the development and so the Blake-Kozeny equation is valid only in beds which are long and wide compared to a particle dimension. In principle one can modify the “capillary tube” approach to account for departure from Newtonian flow. One must, of course, select some model for non-Newtonian flow which is appropriate to the conditions of interest. I n this study we have examined the applicability of the simple power-law model to flow through a tube packed with spherical particles. Modified Blake-Kozeny Equation
T h e derivation of thiz Blake-Kozeny equation through the capillary model takes as its starting point the Hagen-Poiseuille equation for flow in a long straight capillary. A detailed derivation is given by Bird, Stewart, and Lightfoot (7). Agreement with experimeni is found if the “tortuosity” of the capillaries is accounted for by increasing L by a factor of 25/12. A parallel treatment for the power-law fluid takes as its starting point the analog of the Hagen-Poiseuille equation. Thus one begins with (5) where K and n are defined by the relationship between shear stress, T , and shear rate, y, for a power law fluid. For flow in a capillary, this can be written as
I n the limit of Newtonian behavior (n = 1, K = p ) , Equation 7 reduces to Equation 1, the Blake-Kozeny equation. If Equation 4 for the permeability is compared with the Blake-Kozeny equation: it is seen that
If it is assumed that the permeability of the porous medium is the same for all identical bed configurations, independent of flow conditions in the bed, then Equation 8 may be used to establish a value for k by performing a n experiment with a Newtonian fluid. If the permeability is now introduced into the modified Blake-Kozeny equation, the result is
k AP
where H is a factor which accounts for the additional dependence of l/, on k and e due to non-Newtonian behavior. By inspection, H i s given as
n H = - (9 f 3 / n ) n (150 kc)(1-n)12 12
This result is identical to that presented by Bird, Stewart, and Lightfoot (7, p. 207) except for a factor of (25/12)n-1. T h e difference arises from correcting ( V ) of Equation 5 by 25/12, rather than L. We believe L should be corrected. I n the Newtonian case, of course, it makes no difference. By introducing the permeability into the modified BlakeKozeny equation we eliminate the need to determine Dp. For nonregular packing, this would have required a measurement of the specific surface, a,, and this is a difficult measurement to make. If the mass velocity is now introduced, we find
k AP I n order to use Equation 11, one must measure k with a Newtonian fluid, measure the porosity? E, and determine the rheological parameters, K and n. I t is common to present results for flow in porous media in terms of a bed friction factor and a Reynolds number. W e use Ergun’s (4) definition of the friction factor,
A P D p e3p = LG2 (1 - e)
r = K-jn
T h e power-law model is found to describe the flow behavior of many non-Newtonian fluids. Objections to the use of this model will be discussed later. If the hydraulic radius and superficial velocity are introduced as in the Newtonian case, and if L is again replaced by 25L/12, the final result is
(9)
Vo =
T h e Reynolds number for this system is arbitrarily defined so that
f = 1/Re After some algebraic manipulation it follows that Re =
DpG2-n
pn-l
150H(l
- e)
(1 3)
When n = 1, this reduces to the usual bed Reynolds number
(7) VOL. 4
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Table 1.
1
Power-law Parameters
1
CMC in water, 25.0" C.
;
IO3
c
P r,
PIB in toluene, 25 . O o C.
IOL
+
io3
(set'')
Figure 1. Typical flow curve obtained with capillary viscometer
Re =
DPG 150p(l - E)
The modified Blake-Kozeny equation is tested by presenting experimental results plotted according to Equation 13, with f and R e defined by Equations 12 and 14. Since the power law is known to be a good approximation only over a limited range of shear rates, it is necessary to estimate the shear rates in the bed, and then fit a power law to rheological data obtained over comparable shear rates. For simple pipe flow of a power-law fluid, the wall shear rate is given by (7)
+
3n 18(V) Y w = ~4n D
32.0 41.2 21.3 32.4 2.72
0.53 0.53 0.50 0.53 0.54 0.56 0.96
size 20 and collected on a sieve of size 25. Hence, the diameters were in the range 710 to 840 microns. Each tube was filled in a standard manner ( 3 ) . A 70-mesh screen supported the bed at both ends of the tube. T h e tubes were attached vertically to the bottom of the reservoir of the capillary viscometer, and pressure-flow rate data were obtained in the same manner as in the viscometry experiments. For a given flow rate, the measured pressure drop included not only the drop across the packed section, but also the drop across the fittings connecting the bed to the reservoir, and any entrance and exit effects at the ends of the bed. Hence; the data were treated according to the method of Fredrickson ( 5 ) . At the same flow rate, the difference in required pressure for two beds of different lengths is just the pressure drop across a hypothetical packed bed whose length is the difference between the lengths of the two beds used. This pressure drop is then free of all effects except the friction due to fully developed laminar flow, and, of course, this is the only pressure drop accounted for by the modified Blake-Kozeny equation. T h e permeability of the bed was measured with the glycerolwater solution, using Equation 8. The value of k was calculated to be 4.45 X 10-6 sq. cm. Subsequent determinations of k gave the same result. T h e bed porosity was determined by water displacement, and independently by weighing the beads filling a given tube. Both methods gave a value of E of 0.370. With these values of k and E , an average D, was calculated from Equation 4, and found to be 0.0723 cm.
Figure 2 shows f us. Re, according to our modification of the Blake-Kozeny correlation. T h e data scatter about the
IO6
1
+1 Yw = 4n
17.8
Discussion of Results
(16)
For a n estimate of the shear rate in the bed, ( V ) can be replaced by G/pc, and D by four times the hydraulic radius, as in the capillary model, with the result 3n
16.0
1 .OO 1 .oo 1.25 1.50 1.75 2.00 6.00
E
I111111
I
I I 1 /Ill
1
I I Illll
I
12 G p(150 ke)'i2 1.25%CMC I. 50% CMC 1.75% CMC A 2.00% CMC A 6.00% PIE
where, again, we have introduced the bed permeability, k.
0
'
Experimental Materials and Techniques
A detailed description of the experimental features of this study appears elsewhere ( 3 ) . We summarize the most important points here. T h e Newtonian fluid used in the determination of the bed permeability was an aqueous solution of glycerol, with a viscosity of 0.54 poise at 25.0' C. T h e viscosity was determined by an Ostwald viscometer. The non-Newtonian fluids investigated were aqueous solutions of carboxymethylcellulose (CMC Type 70 high molecular weight, Hercules Powder Co.), and a toluene solution of polyisobutylene (PIB L 100, Enjay Co.). Shear stress-shear rate diagrams were obtained from pressure-flow rate data taken with a capillary viscometer designed by Grant (6). Data were taken in the shear rate range of 100 to 6000 set.-'. A typical curve is shown in Figure 1. Parameters for the fluids investigated are given in Table I. T h e packed beds were I-inch-i.d. brass pipes with lengths of 3, Y/*, and 10 inches. T h e particles used were Superbrite glass beads (3M Co.) sized through a U. S. Standard sieve of 424
I&EC FUNDAMENTALS
I-
10-6
10'~
10-5
IO-^
RI
Figure 2. Test of modified Blake-Kozeny equation for power law fluids
CAREOWAX ELVANOL 0 NATROSOL o
io3 f
IOe
IO' 10-4
IO-^
IO-e
10-1
Re Figure 3. Test of modified Blake-Kozeny equation with data of Sadowski
expected line with a n average error of 18%. No trends are observed with respect to Reynolds number or concentration. These results appear to confirm the utility of the correlation developed here. Some objections to the use of the power law should be pointed out. Most real polymer solutions approach Newtonian behavior a t very low shear rates. T h e power law does not predict this behavior. Indeed, the power law predicts an infinite viscosity in the limit of vanishing shear rate. This is a fundamental failure, and provides for a valid objection to the general applicability of the power-law model. From a practical point of view, however, one must simply be careful not to extrapolate power-law parameters beyond the range of shear rates over which they were determined. I n the present case, this places a n obvious premium on the ability to estimate the range of shear rates to be encountered in the porous medium. Apparently Equation 17 provides reasonable estimates. A more serious objection lies in the fact that the power law is an inelastic model, whereas C M C and PIB both exhibit viscoelastic effects (8),which manifest themselves particularly in flows involving local xceleration. The capillary model for flow in a packed bed is deceptive, because such a flow actually involves continual accelrration and deceleration as fluid moves through the irregular interstices between particles. Hence, for flow of non-Newtonians in porous media, we might expect to observe viscoelastic effects which do not show u p in the steady-state spatially homogeneous floivs usually used to establish the rheological parameters. Sadowski (9) has pointed this out, and claims to have observed elastic effects in a study similar to ours. Although many details of Sadowski's work differ from the details of this study, we can examine his data as a further test of our theoretical development. Firsi, however, in order to put Sadowski's work in perspective, we point out the major differences between his work and ours. Sadowski, objecting to the power law a t low shear rates, used the Ellis model y = Mr
+ Nrm
(18)
where M , K, and rn are rheological parameters. H e also replaced the factor of 150 in the Blake-Kozeny equation with
the value of 180 given by Carman (2). Sadowski used solutions of Carbowax, Elvanol, and Natrosol for his non-Newtonian studies. His experimental techniques were probably more precise than ours, judging from the lower scatter observed in his data. Sadowski developed a modified friction factor-Reynolds number correlation by using the capillary model for the Ellis fluid. In testing his correlation he found very good agreement, except for the high molecular weight Natrosol data a t high Reynolds numbers. Sadowski argued that this was a viscoelastic effect, and introduced a modified correlation which used a viscoelastic parameter-i.e., a parameter with the dimension of time. H e found improved agreement between the modified correlation and his data. We wish to argue now that Sadowski's conclusion could be a n artifact of his modification of the Blake-Kozeny equation for the Ellis model. To do this we have taken the original data in Sadowski's thesis, and tested our theory with his data. This required fitting his rheological data with a power law. We have used his values for bed properties. Figure 3 shows the result. First of all, the range of variables is continuous with, and partially overlaps, the range obtained with our own data. Thus, xve have tested the power-law modification of the BlakeKozeny equation for Reynolds numbers in the five-decade range of 10-6 to 10-I. We correlate Sadowski's data with a n average error of 127c, as compared with the 18% error in our own data. For both Sadowski's data and our data, 210 data points in all, the average error is 15%. However, the Natrosol data deviate systematically from the theory, but now the deviation is a t low Reynolds numbers, whereas in Sadowski's correlation the deviation was a t high Reynolds numbers. This raises the possibility that other effects give rise to the observed deviation, but we are unable to explain them. I t seems possible that neither study has provided a critical test of the importance of viscoelastic effects in the flow of non-Newtonian fluids through porous media. We intend to pursue this question by examining the behavior of more highly viscoelastic polymer solutions than have been previously studied. I n the meantime a simple modification of the Blake-Kozeny correlation, requiring the measurement of power-law parameters, bed permeability, and void fraction, correlates data for dilute polymer solutions over five orders of magnitude in a modified Reynolds number, with an accuracy which is probably acceptable for most engineering design purposes.
Nomenclature a,,
=
sDecific surface of a Darticle. cm. -l
I j = capillary diameter, cm. D, = particle diameter, cm.
f
= friction factor (Equation 12) = mass velocity, gram-cm. -2-sec.-1 H = non-Xewtonian bed factor (Equation lo), dynessec.n-cm.-l-n k = bed permeability, sq. cm. K = power-law parameter (Equation 6), dynes-sec."-cm. -* n = power-law parameter (Equation 6) A P / L = pressure drop per unit length, dynes-cm.-3 Re = bed Reynolds number (Equation 14) = superficial velocity in bed, cm./sec. = average velocity in capillary model, cm./sec. - void fraction E I * = Newtonian viscosity, poise P = fluid density, gram/cc. r -- shear stress + = shear rate G
v,
VOL. 4
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425
Acknowledgment
The financial assistance of the Union Carbide Corp. and the Shell Oil Co. is gratefully acknowledged. literature Cited
(1) Bird, R. B., Stewart, W. E., Lightfoot, E. N., “Transport Phenomena,” pp. 196-8, Wiley, New York, 1960. 2) Carman, P. C., Trans. I m t . Chem. Engrs., London 15,150 (1937). j3) Christopher, R. H., M.S. thesis, University of Rochester, Rochester, N. Y., 1965.
(4) Ergun, S., Chem. Eng. Progr. 48,89 (1952). (5) Fredrickson, A. G., “Principles and Applications of Rheology,” p. 199, Prentice-Hall, Englewood Cliffs, N. J., 1964. (6) Grant, R. P., Ph.D. thesis, University of Rochester, Rochester, N. Y., 1965. ( 7 ) McKelvey, J. M., “Polymer Processing,” p. 69, Wiley, New York, 1962. (8) Middleman, S., Gavis, J., Phys. Fluzds 4, 963 (1961). (9) Sadowski, T. J., Ph.D. thesis, University of Wisconsin, Madison, Wis., 1963. RECEIVED for review January 13, 1965 ACCEPTEDJune 21, 1965
DYNAMICS OF MULTIPHASE FLOW SYSTEMS S
. L. S00,
University of Illinois, Urbana, Ill.
General motion of nonreactive gas-solid systems in potential fields with distribution in size of solid particles and including interactions among particles was formulated through the introduction of “multiphase” generalization, applicable to mean interparticle spacing greater than two diameters. Cases of adiabatic potential flow, laminar boundary layer motion, and electrohydrodynamic slug flow are illustrated and physical significance of results i s discussed. Results demonstratethe feasibility of treating such a flow system rigorously,
THE significance
of multiphase (gas-solid, gas-liquid, or other combinations of a particulate phase and a fluid phase) flow (including fluidization) with regard to chemical and nuclear processes, rocketry, and air pollution control is recognized. A substantial number of theoretical and experimental studies have been reported. However, one basic issue has been avoided: the distribution in the size of particles in the theoretical formulation. Except in very isolated cases of large particles (millimeter size), distribution in the size of particles is unavoidable (25) in most physical systems, including a suspension. Rigorous formulation of gas-solid flow with a distribution in the size of particles is considered in this presentation. The basic theoretical approach includes extending the earlier formulation of the flow system as a continuum (78) with recognized qualifications (79). The rationale is that continuum mechanics of a single-phase fluid amounts to a successful simplification of the classical kinetic theory of a flow system by replacing the coordinates of the phase space with configuration space and transport properties. Any generalized formulation of a gas-solid nonreactive system has as its basis the recognized physical concepts concerning a single particle and a cloud of particles as outlined in Table I. Apart from the obvious definition of a multiphase system (phases of solid, liquid, and gas), it is further recognized here that, from the point of view of “continuum” mechanics of a cloud of particles, particles of different sizes constitute different “phases” ; the nonreactive suspension may consist of one gas and one type of solid material. General Formulation
As the solid particles of different size range constitute different phases from the point of view of continuum mechanics, the previous basic formulations (78) may be generalized. 426
l&EC FUNDAMENTALS
Here we illustrate with the case of nonreactive suspension in which the solid particles are sufficiently numerous; and we define the density of a cloud of particles of size range s as p P ( ’ ) [the density of the solid material is denoted as P ~ ( ~ ) ] ; (1)
p p ( S ) = n(s)m,
where n(’) is the number of solid particles per unit volume, m, is the mass of each particle, and the over-all density of the cloud of solid particles is given by
Table 1.
A.
Basic Concepts Relating to Multiphase Flow Concept for Author Date Ref. No.
Single particle 1. Drag of sphere Apparent mass in oscillatory motion 3. Relative acceleration in turbulent fluid
2.
B.
Newton Stokes Oseen
1686 1881 1911
(8) (8) (8)
Stokes
1898
(8)
Lin Tchen Gilbert
1943 1947 1600 1963
(70) (28) (30) (22, 23)
4. Charge collection 5. Thermal electrification so0 Cloud of particles 1. Brownian motion Brown 1882 (6) 2. Dispersion and attenuation of Sewell 1910 (74) sound wave 3. Apparent thermodynamic properties Tangren 1949 so0 1961 1962 So0 4. Multiplicity of streamlines Multifurcation of states of tur65.. so0 1962 bu 1ence 1962 Slip velocity at boundary So0 1962 7. Integrals of phase interaction So0 1964 so0
