GENERALIZED PLANE COUETTE FLOW OF A NON=NEWTONIAN FLUID RAYMOND W . FLUMERFELT,l MARK W . P I E R I C K , * STUART L. COOPER, AND R. BYRON B I R D Department of Chemical Engineering, University of Wisconsin, Madison,Wis. 63706
A non-Newtonian fluid is in steady-state, laminar flow between two infinite parallel flat surfaces, one of which is in motion; in addition there is a pressure gradient acting in the same (or opposite) direction to the motion of the bounding surface. For this problem the velocity profiles, volume rate of flow, and force on the surface are found. A power-law function is used to describe the nowNewtonian viscosity. A numerical table facilitates numerical calculations.
THE solutions
of simple flow problems for elementary rheological models often find use in the approximate description of portions of industrial equipment. Since the flow called “generalized Couette flow” seems to arise approximately in a number of situations, including screw-extruder operation, it is worthwhile to summarize the analysis of this system for a power-law non-Xewtonian viscosity. We imagine a fluid contained between two parallel surfaces of infinite extent located at x = + B / 2 and x = - B / 2 . The plate a t x = - B / 2 is fixed, whereas that at x = + B / 2 moves in its own plane in the positive z-direction with a uniform velocity V . A pressure gradient (6,- @ L ) / Lis also present over a distance L in the z-direction; here 6 is defined by 6 = p - pgh, where h is the distance above any chosen equipotential gravity reference plane. The local velocity in the z-direction is called v, and depends on x alone. This steady-state flow is often referred to as “generalized Couette flow” (Figure 1). The analytical solution for this problem for Kewtonian fluids is well known (Schlichting, 1955), and is simply the superposition of the solution of two problems: flow between two parallel walls, one of which is moving, with no pressure gradient, and flow between two fixed parallel walls because of a pressure gradient. For non-Xewtonian fluids such a simple superposition is not possible rigorously, and is even a poor approximation over a wide range of variables. There are several published non-Newtonian solutions. Skelland (1967) has presented a solution for the power law which is incomplete in that it is applicable only when there is no maximum or minimum in the velocity profile. Wadhwa (1966) has given a complete solution for the Ellis fluid. I n neither case are the necessary auxiliary numerical tables provided. Here we present the complete solution for a power-law function describing the non-Sewtonian viscosity, along with the numerical table which is needed to make the results useful.
This may be integrated to give (with
=
z/B)
where X is a dimensionless constant of integration. The force of the fluid on the boundary surface a t x = + B / 2 is then
F,
=
72,
lcp.1,2
LW
=
- (So- SL)BW(X - +)
(3)
Furthermore the volume rate of flow for a slit of width W (in the y-direction) is: W
Q=
l,,, +BIZ
v,(z)dxdy
=
WBVQ
(4 1
in which +1/2
=
11,* 4df
is the dimensionless volume flow rate; here 4 = v,/V. These equations are valid for all fluids I n order to obtain a velocity distribution, we introduce a rheological equation of state. For steady shearing flows (sometimes called “viscometric flows”) the general elasticoviscous fluid of Oldroyd (1950, 1965) gives results indistinguishable from those obtained from the equation (Bird, 1969; Bird and Carreau, 1968) : ’5
-7y
- -;( 0
+ 26)(y*yl+ $0 (WWY
(5 1
where 7 is the non-Kewtonian viscosity, and 0 and 0 are the normal-stress functions; D/Dt is the Jaurnann derivative. For engineering purposes we often take: ?I =
m.j”-l
,
0=
ml+n’-l
,
P=O
(6 1
where rri, n,m’, and n’are constants and? = d$ (y :y) is the magnitude of the rate of deformation tensor. For the problem at hand, Equations 5 and 6 lead to:
Notation and Problem Statement
For the generalized Couette flow of a n incompressible fluid the equation of motion can be simplified to:
dTrz - - --6 o - @L dx L
(1 )
Present address, Department of Chemical Engineering, University of Houston, Houston, Tex. *Present address, Cities Service Oil Co., East Chicago, Ind. 354
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FUNDAMENTALS
dx Sometimes we use s = l / n in subsequent equations; also the abbreviation A = [ ( S o- 6 3 ) B / m L ](B/V)”is used. A may be positive or negative. Equations 2 and 7 may be combined to give:
Figure 1.
1 1
-------IAI 0. o o c
Case
0.02' 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.4 3.8 4.2 4.6
(S
5.0
6.0
8.0 10.O d 20.0d m d
Generalized Couette flow
I
Table I. Values of X in Terms of A and n n _ _ _ _ _ _ _ _ _ _ _ I
0.1
0.2
0.3
0.4
0.5O
0.6
0.7
0.8
0.9
1.0b
m
m
m
m
m
m
m
m
m
m
49.999 9.9938 4.9875 3.3146 2.4750 1.9687 1.6290 1.3846 1.1996 1.0543 0.9368 0.8394 0.7571 0.6865 0.6249 0.5225 0.4400 0.3721 0.3157 0.2687 0.2294 0.1966 0.1693 0.1275 0.0981 0.0771 0.0617 0.0503
49.999 9.9958 4.9917 3.3208 2.4833 1.9791 1.6415 1.3991 1.2162 1.0730 0.9574 0.8620 0.7817 0.7130 0.6534 0.5543 __-0.4747 0.4089 0.3540 0.3082 0.2697 0.2373 0.2099 0.1668 0.1352 0.1115 0.0934 0.0793 0.0553 0.0312 0.0200 0.0050
49.999 9.9973 4.9945 3.3250 2.4889 1.9860 1.6499 1.4089 1.2274 1.0856 0.9715 0.8776 0.7987 0.7315 0.6733 0.5772 0,5005 0.4370 0.3&10 0.3396 0.3023 0.2706 0.2436 0.2005 0.1680 0 1430
49.999 9.9983 4.9965 3.3280 2.4928 1.9910 1.6559 1.4159 1.2355 1.0947 0.9817 0.8888 0.8110 0.7449 0.6878 0.5941 0.5198 0.4587 0.4077 0.3650 0.3289 0.2982 0.2718 0.2293 0.1967 0.1712 .. _ _ -~ 0.1507 0.1341 0,1036 0.0689 0.0501 0.0186
49.993 49.997 49.999 9.9628 9.9834 9.9903 4.9268 4.9668 4.9806 3.2266 3.2838 3.3042 2.3628 2.4343 2.4612 1.8357 1.9186 1.9515 1.4787 1.5700 1.6086 1.2199 1.3171 1.3609 1.0233 1.1244 1.1729 0.8686 0.9720 1.0246 0.7436 0.8481 0.9042 0.6404 0.7451 0.8041 0.5536 . ~ . . . 0.6580 0.7193 0.6464 0.5828 0.4159 0.3112 0.4112 0.4771 0.2288 0.3260 0.3925 0.1623 0.2567 0.3232 0.1077 0,1995 0.2659 0.0642 0.1524 0,2182 0.0339 0.1144 0.1788 0.0170 0.0848 0.1465 0.0086 0.0626 0.1204 0.0025 0.0347 0.0824 O.ooO8 0.0201 0.0579 0.0003 0.0122 0.0418 O.OOO1 0.0078 0.0310 O.ooO1 0.0051 0.0235 0.0 0.0021 0.0128 0.0 0.0005 0.0049 0.0 0.0002 0.0023 0.0 0.0 0.0002 0.0 0.0 0.0
0.0320
0.0156 0.0089 0.0016 0.0
0.0
0.1234
0.1077 0.0797 0.0495 0.0342 0.0108 0.0
49.999 50.000 50.oM) 9.9990 9.9996 10.000 4.9979 4.9991 5.oooO 3.3302 3.3320 3.3333 2.4958 2.4982 2.5000 1.9948 1.9977 2.oooO 1.6604 1.6639 1.6667 1.4212 1.4253 1.4286 1.2415 1.2462 1.2500 1.1015 1,1069 1.1111 0.9893 0.9952 1.oooO 0.8972 0.9038 0,9091 0.8203 0.8276 0.8333 0.7550 0.7629 0.7692 0.6988 0.7074 0.7143 0.6070 0.6170 0.6250 0.5347 0.6463 0.5556 0.4757 0.4892 0.5000 0.4545 0.4267 0.44% 0.3856 0.4026 0.4167 0.3509 0.3692 0.3846 0.3212 0.3406 0.3571 0.2957 0.3160 0.3333 0.2541 0.2755 0.2941 0.2218 0.2438 0.2632 0.2381 0 1962 0. 2184 .. _ ___ . 0.1754 0.1975 0.2174 0.1582 0.1802 0.2000 0.1262 0.1473 0.1667 0.0882 0.1071 0.1250 0.0668 0.0836 0.1000 0.0281 0.0387 0.0500 0.0
0.0
0.0
0.0
-~ a
For n = 1/2 and Case I, I A I =
b F o r n = 1, I A 1 =
1 -
Since the right side is always positive, we conclude that X 2 4 if A 2 0 and X 5 -4 if A 5 0. When the nth root of both sides is taken, the differential equation can be integrated from -1/2 to 4 to give:
IAI
For small 1 A I, there i s an asymptotic expression I A I
J
1
* For large 1 A I, there is an asymptotic expression I X I
'p-1 --
~ i :
IAlid'
The constant X (introduced in Equation 2 ) is obtained from the condition that 4 = 1 a t 6 = 1/2: which is to be solved for t'he boundary conditions:
I 15
Solution for Case I [ A
(S
+ l)l's]
Equations 11 and 12 give the dimensionless velocity distribution. The dimensionless flow rate is then obtained from Equation
4:
When there is no maximum or minimum in the velocity profile in the range -B/2 5 B / 2 , d+/d[ is everywhere positive. Then Equation 8 becomes:
[I+
VOL.
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where X = X (A, s) is given by Equation 12. These equations are all valid for both positive and negative A. Solution for Case I1
[I A
12
(S
+ 1) '1
where X = h(A, s ) is given by Equation 16. This result is valid for both positive and negative values of A. Numerical Calculations
When the velocity profile has a maximum or minimum in the range - B / 2 5 .$I + B / 2 , Equation 8 has to be written separately for .$I X and for .$ 2 A, and we designate the corresponding dimensionless velocity profiles by +< and +>, respectively.
If the results (velocity distribution, volume flow rate, force on fluid boundary) are to be useful, we need a table for X = X(A, s ) . We note from both Equations 12 and 16 t h a t X (-A, s) = --X (A, s ) ; hence 1 X I can be given numerically as a function of 1 A 1 and n, with the understanding that X is then found from X = I X 1 sgn A. Such a t,abulation is given in Table I. Cases I and I1 are separated by a heavy line running across the table. Because h is an odd function of A, the results in Equations 13 and 17 can be writ,ten as 0 = ( 1 / 2 ) a' (sgn A ) ; 0' is then plotted in Figure 2. This figure, along with Table I, enables one to compute the volume rate of flow for any value of the pressure drop, whether the pressure drop be working with or against the motion of the moving boundary. The curves exhibit linear asymptotes on the log-log plots, and the analytical expressions for these asymptotes are given on the graph. These asymptotes cannot be superposed to obtain a n approximate representation of the results. There are two important assumptions in the numerical results: I t is assumed that there are no appreciable viscous heating effects xhich would cause the rheological properties to be position-depeadeiit. I t is assumed that the power law is appropriate; in Case 11, if an appreciable portion of the slit cross section is in the neighborhood of zero velocit,y gradient, the inadequacies of the power law niay become apparent [such inadequacies have been discu Bird, and Lescarboura (1965) in coiinect annular flow].
+
These equations are easily integrated to give :
The constant of integration, A, in this case has the significance that .$ = X is the location of the maximum or minimum in the velocity profile; hence X is determined by the requirement that +< = +> at = A :
Finally the dimensionless volume flow rate is:
Acknowledgment
The authors acknon-ledge the financial assistance provided
10
1.0
Figure 2.
Dimensionless flow rate
a'
O/WSV, as a function of A and n (or l/s)
D
=
0. I
0.01 0. I
356
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FUNDAMENTALS
1.0
IO
IO0
by NSF grant GK-1275 and a n unrestricted research grant from the Petroleum Research Fund of the American Chemical Society. They thank A. H. P. Skelland for reviewing the manuscript and offering critical comments.
X
Nomenclature
+= dimensionless velocity profiles for 5 _< X and f Case I1 Q = dimensionless volume rate of flow = dimensionless quantity given in Figure 2 Q’ S/Dt = Jaumann derivative
B F, Q
= gap between parallel surfaces = force of fluid on moving surface = gravitational acceleration
h L
= distance above gravity reference plane = length m,n = rheological constants associated with q m’,n’ = rheological constants associated with 0 6 = defined by p pgh = pressure = volume rate of flow 8 = defined by l / n Ti = uniform velocity of moving surface
Ashare, E., Bird, R. B., Lescarboura, J. A., A.I.Ch.E. J . 11, 910-16 (19$Ei).
Bird, R. B., Lectures in Transport Phenomena,” Chap. 1 of “Today Series,” R. B. Bird, W. E. Stewart, E. N. Lightfoot, and T. W. Chapman, A.1.Ch.E. Meeting, March 1969, New Orleans, La. Bird, R. B., Carreau, P. J., Chem. Eng. Sci. 23, 427-34 (1968) (Equation 22). Oldrovd. J. G.. Proc. Rou. Soe. London A200.523-41 (1950):A283.
= local velocity in z-direction
W*
= width s, y, z = rectangular coordinates
aoundary Layer Theory,” pp. 60-2, McGraw.ark, 1955. I. P., “Non-Newtonian Flow and Heat Transfer,” k, 1967. 12, 890-3 (1966).
GREEKLETTERS
4
= normal stress function associated with = shear rate
7w
- ryu
7
= shear viscosity
6
= normal stress function associated with rZr rz2 = dimensionless group defined in Equation 7
A
2 X,
literature Cited
+
uz
= dimensionless integration constant = dimensionless position in s-direction P = density 1 = stress tensor rZE = component of stress tensor = dimensionless velocity in z-direction @ (
-
RECEIVED for review January 21, 1969 February 17, 1969 ACCEPTED
COMMUNICATIONS ZONE REFINING CONSIDERED AS A MULTISTAGE SEPARATION METHOD Some aims are suggested for a science of separations which seeks to unify the understanding of diverse multistage separation methods. Zone refining, examined in this context, is found to be an unusual type of equilibrium separation method in which multistage separation is achieved without conventional countercurrent flow and reflux.
W E examine a typical multistage separation method, zone refining, and consider how a concentration difference is produced and how it is multiplied. Examining such matters is one task of a science of separations, the purpose of which is to establish a common basis for understanding the many and diverse methods of purification and separation in use today. A few steps by way of classification have been taken toward such a unifying science. For example, Benedict and Pigford (1957) distinguished between separation methods based on the irreversible flow of heat or matter, such as diffusion methods, and those using potentially reversible processes, such as distillation and liquid-liquid extraction. Pfann (1966a) in a cursory survey suggested kinetic for the former group and change-of-phase (which is not broad enough) for the latter. Pratt (1967) suggested nonequilibrium for the former, and equilibrium for the latter. At this point we suggest the terms “kinetic” and “equilibrium” for these two classes of separation methods. Examples of methods of the two classes are shown in Table I (after Pratt). [We question Pratt’s classifying the gas centrifuge as an equilibrium process. Surely it is akin to electrodiffusion (electrostatic potential gradient), thermal diffusion (temperature gradient), and mass diffusion (concentration gradient).]
Suggested goals of a science of separation include: to learn how to perform single-stage separations more effectively, to seek new methods of producing a concentration difference, to seek new methods of utilizing countercurrent flow and reflux, to seek and define the underlying unity of the various disciplines involved, and to express this underlying unity in basic theoretical form (Pfann 1966a). I n a typical equilibrium process involving two phases, such as multistage distillation, separation is achieved by producing the two phases, bringing them into contact so that equilibration can occur, and moving the phases in countercurrent fashion. Equilibrium is approached by means of diffusional processes in both phases, with transfer of components across the interface. Separation methods based on crystallization Table 1.
Classification of Multistage Separation Processes
Equilibrium Distillation Absorption Liquid-liquid extraction Adsorption Chemical exchange
Kinetic Mass diffusion Sweep diffusion Thermal diffusion Electrodiff usion Molecular distillation Irreversible electrolysis
Ion exchange
Countercurrent gas centrifuge VOL.
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