offers a conceptual basis for the synthesis of control systems based on the modal character of the process. It represents a potentially useful approach to a fundamental problem of long standing : the rational selection of small numbers of measured and manipulated variables for the control of large interacting processes in the presence of restrictions on manipulation. Acknowledgment
Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. Nomenclature
B B,
objective function for manipulation = process input matrix, n x m = process input matrix consisting of
p
columns of B ,
B, E n = n-dimensional Euclidian space F = feedback matrix, B,TKS’, 1 x 1 k , = controller gain for mode i K = diagonal gain matrix, diag[kl, k ~. ,. .,knl 1 = number of controlled modes m = number of process inputs M y = the set of Y selected measured variables n = number of state variables st = measurement vector for mode i, dimension Y
s
.
u
= [Sl,S?, * .,,si1 = manipulation vector for mode i, dimension p = [tl, t z , til = process input vector, dimension m
up
= control vector consisting of p elements of u
V,
= left eigenvectors of = [VI, V Z , v,], n X
T V
. . .,
W
state matrix A , i = 1, . . ., n n right eigenvectors of state matrix A , i = 1, . . ., n [ w ~W,Z , . . ., w n l , n X n
. . .,
wL =
state vector, dimension n
= state vector expressed in terms of right eigenvectors
as bases, dimension n
GREEKLETTERS = measurement criterion (value of measurement jective function at optimum) p = manipulation criterion Yr = l(sr,w,>/(sl,w~)l h( = eigenvalues of state matrix A , i = 1, . . ., n A = d i a g h , XZ, . . , X,l Y = number of measured variables p = number of manipulated variables CY
ob-
SUBSCRIPTS i, j = running indices for modes
=
n X P = set of p selected manipulated inputs
t,
=
SUPERSCRIPTS T = transpose * = projection
a(s) = objective function for measurement A = state matrix, n x n
b(t)
x z
literature Cited
Davison, E. J., Trans. Inst. C h a . Eng. 45(6), T299 (1967). Davison, E. J., GEldberg, R. W., Autmatica 5,335 (1969). Franklin, J. N., Matrix Theory,” p 188, Prentice-Hall, Englewood Cliffs, N. J., 1968. Howarth, B. R., Ph,D. Dissertation, University of California, Berkeley, 1970. Howarth, B. R., Grens, E. A., FOSS,A. S., IND.ENG. CHEM., FUNDAM. 11,403 (1972). Loscutoff, W. V., Ph.D. Dissertation, University of California, Berkeley, 1968.’ Levy, R. E., Ph.D. Dissertation, University of California, BerkA l C2V 7.
1967.
Levy, R. E., FOSS, A. S., Grens, E. A,, IND ENG.CHEM.,FUNDAM. 8,765 (1969). Rosenbrock, H. H., C h a . Eng. Progr. 58(9), 43 (1962). Simon, J. D., Mitter, S. K., X7rlfoTrn. Control. 13,316 (1968). Zadeh, L. A., Desoer, C. A., Linear Systems Theory: the StateSpace Approach,” McGraw-Hill, New York, N. Y., 1963. RECEIVED for review August 6, 1971 ACCEPTED May 11, 1972
Predicting Non-Newtonian Flow Behavior in Ducts of Unusual Cross Section Chester Miller Engineering Service Division, E . I . du Pont de Nemours & Co., Inc., Wilmington, Del. 19898
A simple technique has been devised for estimating the flow behavior of nowNewtonian fluids in ducts of unusual cross section, Application of this technique requires knowledge of a geometric shape factor for the channel and the shear stress-shear rate relationship in some simple geometry such as a circular tube. Compared with methods previously reported in the literature, the present approach is somewhat simpler to employ.
T h e pressure dropflow rate relationship for laminar flow of non-Newtonian fluids in ducts of unusual cross section is of interest for a variety of industrial applications. Flow channels of unusual shape are frequently found in heatexchange equipment, spinnerets, and the interior of film casting and coating dies, to cite a few examples. Several methods have appeared in the literature for predicting non-Sewtonian flow behavior in ducts of arbitrary 524 Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 4, 1972
cross section. Wheeler and Wisder (1965) solved the complicated nonlinear partial differential equations describing flow of a power-law fluid in a rectangular channel by using a numerical technique. Schechter (1961) adopted a variational approach for this same problem. Illisushima, et al. (1965), and Mitsuishi, et al. (1968), used the variational procedure for the case of flow in elliptical and triangular ducts. Kozicki, et al. (1966), and Kozicki and Tiu (1967) have
recently suggested a less complicated method for predicting the pressure dropflow rate relationship. The technique is based on knowledge of only two geometric parameters characteristic of the flow geometry and a function of shear stress characterizing the fluid model. The geometric parameters are evaluated from analytical solutions and experimental measurements conducted on Newtonian fluids. Although this method is less difficult to apply than solving the detailed differential equations for non-Kewtonian flow, i t nonetheless involves a complicated integration as well as determination of two geometric constants. I n the present article, an extremely simple technique is outlined for estimating the flow behavior of non-Xewtonian fluids in ducts of arbitrary cross section. The technique is based on knowledge of only a single geometric shape factor and the shear stress-shear rate relation for fluid flow in some simple geometry such as a circular tube. The validity of the proposed technique is partially verified by comparing its predictions with results reported in the literature.
Table 1. Values of X for Various Geometries - P
Geometry
16.0
Circular Cylinder (Knudsen and K e n , 1958)
24.0
Parallel Plates (Knudsen and Ksrz, 1958) 20 = 10“ =2 P
Isosceles Triangles (Sparrow, 19621
---
i
i
bE3
Annular Ducts
13.1
40° 50° 60° 700 800
13.15 13.25 13.33 13.25 13.2 13.15
90°
24.0‘ [ ( l - O 3 5 l b / s ) ( l c b/a)l’ 24 0 15.7
.
b/s = 0.0 0.5 = 1.0
14.3 16 (1
-
K=R$/R,
K
,,v)
Star-Shaped Conduits (Shih, 1967)
,--,3-polnred sm,
\
‘-,.-
16.0
0.0 0.1
22.4 23.4 24.0 24.0 24.0
0.3
= 0.5 = 0.7 1.0
No. of P0l”W
6.50 6.61 6 63 6 64
3 4 5 6 8
6 63
No. of
Sides 3 4
13.33 14.23 14.74 15.05 15.41
5 6 6 Blh
Rhombic Conduits (Shih, 1968)
14.22 14.00 13.62 13.1 12.75 12.15
1 = 2/3 i
.
4A
--
112
= 1/3
= -
P
/
with A being the channel cross-sectional area and P the wetted perimeter. For a circular duct, Dh is equal to the actual diameter. To define the pressure dropflow rate relationship completely for a particular fluid flowing in a duct of specified geometry, it is necessary to provide, in addition to eq 1, an expression describing the dependence of the average wall shear stress on material, geometrical, and flow parameters. For a Newtonian fluid, me can readily show t h a t such a n expression may be written in the form
(3) where Q is the volumetric rate of flow, 7 is the viscosity (a material constant), and is a shape factor t h a t depends only on duct cross-sectional geometry (but is constant for geometrically similar channels). The shape factor is defined here in a manner such t h a t its numerical value is equal to the product of Reynolds number (based on average velocity and hydraulic diameter) and friction factor for a given channel. Note that in most literature articles which consider analytically the pressure drop for Newtonian fluids in ducts of various cross sections, this friction factor-Reynolds number product is commonly reported as part of the theoretical results. Thus, we are provided with a convenient means for obtaining the shape factor directly from the literature. Values for the parameter X compiled from a number of literature references are shown in Table I. Now, in the flow of Sewtonian fluids, it is well known t h a t shear stress is proportional to shear rate, with the constant
- K)’
(Knudscn and Katr, 1958)
Regular P d y g m a l Conduits [Shih, 1967)
Dh
30°
(Bouchcr and Alveo, 19631
Flow Analysis
where p is the pressure, L is the axial length of the duct, 7 is the average wall shear stress, and Dh is the ‘‘hydraulic diameter” defined by
12.5 12.8
b/e 5 1.0
Recringulir Ducts
i
Consider the isothermal laminar flow of an incompressible fluid in a duct of arbitrary cross section. If we carry out an overall force balance to determine the pressure drop in the duct (escluding entrance effects), we find that
Cross-Sectional
J
1/4 1/10
b/a 5 1.0
Rounded Rectangular Ducts (Lahi!, 19631
b/a
1
L
24. U.lb/s; 0.4iblaJ’ 24 0 17.8 16.0
0.0 0.5 1.0
=
i
Haif-Rounded Recrangular Ducrs (Lahri, 1963)
b/a 5 1 . 0 b/e
24’ 1 + 1.24b/e - 0 63fb
0.0 = 0.5 1.0 2.0 i
T-
Elliptical Ducts
i
b/e 5 1.0
81’
24 0 16 4 14 9 14.8
=
19 7 x 1
- 0.310b,’s + O.l2Ofb/si’l’
b / s = 0.0 c 0.25 = 0.667 = 1.0
19 7 5 18.25 16 4 16.0
‘ Approximare
of proportionality being the viscosity 7. If we refer then to eq 3, we note that the term in brackets can be identified as the average shear rate a t the wall of the flow channel. T h a t is, we can write i =
77
(4)
where the average wall shear rate 7 is given by
7=-
2ADh
(5)
Equations 4 and 5 are valid solely for the flow of Newtonian fluids. We next seek to modify these expressions to yield corresponding equations applicable to non-n’ewtonian materials (fluids with viscosities t h a t depend on shear rate). Ind. Eng. Chem. Fundom., Vol. 1 1 , No. 4, 1972
525
and “apparent wall shear rate” 1 might, to a good approximation, be regarded as entirely independent of duct geometry for a wide variety of fluids and ducts. Under such circumstances, we could then write i=
0.4
0.8 Power Law Bponent n
0.6
1.0
Figure 1 . i s ( $ / i c ( y ) as a function of the power-law exponent n for a power-law fluid
t(1)
(9)
where the function t(1)is assumed to depend only on the properties of the fluid and not on the channel configuration. This expression implies t h a t the curve of t us. 1 would be the same for all duct geometries. Equation 9 and its implications are the central result of this article. Let us next evaluate the validity of eq 9 by comparing its predictions with analytical and experimental results reported in t h e literature. Verification of Results
Wheeler and Wissler (1965) considered both theoretically and experimentally the pressure dropflow rate relationship for non-Sewtonian fluid flow in a duct of rectangular cross section. They assumed t h a t the shear behavior of the aqueous polymer solutions studied could be described adequately by the power-law viscosity model ( T = m y “ ) . In terms of the present notation, the results of their theoretical analysis for the case of a square duct (A = 14.3) can be expressed in the form
‘ 4 = 0,936 rby
’
7, (1) TU2
Figure 2. Dimensionless shear stress vs. dimensionless shear rate for axial annular flow of an Ellis fluid (a = 2)
For this purpose, let us begin by considering two special cases: the flow of a non-Newtonian fluid in a circular tube and the flow between two infinite parallel plates. I n the former instance, one can show (Metzner, 1959) t h a t the shear stress a t the duct wall i can be represented as a unique function of the “apparent wall shear rate” 7 , where t and 7 are defined as in eq 1 and 5, respectively, with X = 16 for circular tube flow. T h a t is i =
ic(T)
in which the function i c ( T ) can either be measured directly or calculated analytically from shear viscosity data. Similarly, for non-Xewtonian flow between parallel plates, we again find that i is uniquely related to 1,in this case with h = 24 for the parallel-plate geometry
t = t,(T)
(7)
where t,(T) is the shear-stress function characteristic of flow between infinite parallel plates. These findings for circular cylinders and parallel plates suggest strongly t h a t the flow characteristics of non-Xewtonian fluids in channels of arbitrary cross section might in general be described by a relation of the form
t = i,(Y)
(8)
where i, refers to the average wall shear stress function in each particular geometry. Now, i t can readily be shown analytically that, for various non-Xewtonian fluid models, the functions i c ( q ) and t,(1) are approximately equal to one another over a wide range of material parameters. This then leads us to postulate further t h a t the relationship between average wall shear stress 7 526 Ind.
Eng. Chern. Fundam., Vol. 1 1 , No. 4, 1972
+
0.560(1.7330 5.8606n)I” 3n+1
[
(lo)
where is(y) is the average wall shear stress function for flow in a square channel, and n is the exponent in the powerlaw model. Figure 1 shows a graphical representation of eq 10, with iS(T),/fc(T) plotted as a function of the “flonbehavior index” n. We note from this figure t h a t is(?) and io(?) differ by less than 5y0 over the range of n from n = 0.4 to n = 1.0, in approximate agreement with the predictions of eq 9. This range of values for n includes a wide variety of actual non-Kewtonian fluids. The experimental results of Wheeler and Wissler for flow of non-Xewtonian polymer solutions in square ducts xere found to confirm their theoretical predictions. McEachern (1966) analyzed the axial annular flow of a fluid whose non-Newtonian viscosity behavior is characterized by the “Ellis model.’’ For this model, in simple shear flows, local shear stress is related to local shear rate by the equation
where q0 is the viscosity at zero shear rate, is a characteristic stress, and CY is a constant. If we consider the flow of a n Ellis fluid in a channel of circular cross section, we can show t h a t the wall shear stress function i c ( ?is) given in dimensionless form by the implicit equation (12)
S o w if the method proposed in this article for predicting average wall shear stress in ducts of arbitrary cross section is valid, then eq 12 should also describe, to a good approximation, the flow behavior of an Ellis fluid in axial annular flow. Figure 2 shows McEachern’s annular flow results / ~a&/ ~ 1 (with ) ~ X for the case of a = 2 replotted as i(1)l ~ l us. taken from Table I for annular channels). The circled points were estimated from one of NcEachern’s figures, reproduced
-
0 Circular tube 0 E 1.0
A E . U3 OE.U5 N
i
a - 2
0.01
0. 1
1
10
'
100
.
-5 $102
Ir'
.
1wo
or6 &,2
Figure 3. Axial annular flow of an Ellis fluid (McEachern, 1966). [From the American lnstitute of Chemical Engineers Journal, with permission]
IO1 101
lo3
lo2
7, sec-1 :1
Figure 5. Rectangular-duct data of Mitsuishi, et a/. ( 1 968)) for a 1.1 1 % solution of CMC (carboxymethylcellulose) replotted as i vs. 7
t
pi;!
4
0 Circular tube
A Equilateral triangle 0 Right isosceles triangle
10'
10'
.
Cl/meCl
7. sec-l
Figure 4. Figure from Mitsuishi, et a/. ( 1 968), for non-Newtonian flow in rectangular ducts and circular cylinders. [From lnterna/ional Chemical Engineering, with permission]
Figure 6. Triangular-duct data of Mitsuishi, et a/. ( 1 968), for a 3.79% solution of CMC replotted as i vs. 7
here as Figure 3. The results are for values of K , the ratio of the radius of the inner core to the radius of the outer cylinder, ranging from K = 0.1 to K = 0.9. A11 the findings appear to fall on a single curve. Such was not the case in XcEachern's original figure (Figure 3 ) , where a series of curves was obtained with K as a parameter. I n terms of the present' notation, the ordinate in Figure 3 can be identified as TI/?(^ - K ) ] and the abscissa as ( q O ? / n / J (4(1 KZ)(1 - K ) , ' A ) . Comparison of Figures 2 and 3 emphasizes the ability of the present theory to predict the most suitable form for the parameters. The solid curve in Figure 2 is a graphical representation of eq 12 for a = 2 . The close agreement between eq 12 and NcEachern's results indicates the validity of eq 9 for axial annular flow of a n Ellis fluid. llitsuishi, et al. (1968), have considered analytically and esperimentally the pressure drop-flow rate relation for non-Xewtonian fluids in circular tubes, rectangular ducts, and isosceles triangular channels. Typical results from their experiments on flow of a specific polymer solution in circular tubes and rectangular channels of various aspect ratios are shown in Figure 4. This figure is t,aken directly from the original reference. The data farthest to the right in the figure represent a plot of io(?) us. 1 for ducts of circular cross section. The other sets of data correspond to the rectangular flow channels. I n terms of the present notation,
X t s u i s h i , et al. (1968), plotted the data for rectangular conduits in Figure 4 as i us. 167;(1 + l / E ) 'A, where E is the aspect ratio (defined in Table I as b / a ) . The solid lines are the theoretical predictions of these investigators. Figure 5 shows some of the points in Figure 4 replotted as i us. 7 (a few points were estimated a t each value of E ) . A11 the data in Figure 5 appear to superimpose on a single curve, in agreement with the present theoretical predictions. Similar results are obtained for the data on flow in isosceles triangular channels (Figure 6). The accuracy of the present method for predicting and correlating non-Sewtonian pressure drop data in ducts of unusual cross section depends on both the specific fluid being considered and the particular flow geometry. K'ow, for most non-Newtonian fluids one finds that the shear stressshear rate behavior can be approximated very closely over limited ranges of shear rate (nearly a decade in most cases) by the power law equation. Therefore, we can obtain a good estimate of the accuracy of the technique by considering its predictions for power lam fluids in various types of ducts. As we have already seen, for the case of square ducts, the method is accurate, in terms of pressure drop predictions, to within 5% over the range of f l o ~behavior indes n from 0.4 to 1.0. Similarly, for t h e case of flow between infinite parallel plates, one finds a n accuracy of better than 3.570 over the same range of n. This kind of accuracy is usually Ind. Eng. Chern. Fundarn., Voi. 1 1 , No. 4, 1972
527
14.6 as the average between h = 16.0 for a circular cylinder and h = 13.2 for a 45’ isosceles triangular channel. I n view of the narrow diversity of values used in taking the average, this estimate should be quite accurate. Nomenclature
A
=
E
= aspect ratio for rectangular channel, dimensionless
m n P
= parameter in power-law model, dyn/cmz set' = exponent in power-law model, dimensionless = wetted perimeter of conduit, cm
$
= volumetric flow rate, cm3/sec
L Figure 7. Conduit comprising circular cylinder and isosceles triangle
45”
better than the degree of scatter in the data for most pressure drop-flow rate measurements. Discussion
As we have seen above, literature results on flow of nonNewtonian fluids in channels of arbitrary cross section generally confirm the predictions of the present theoretical development. Thus i t is reasonable to assume that, for flow of a non-Newtonian fluid in a n arbitrary duct, the average wall shear stress 5 is, to a good approximation, a unique function of the “apparent wall shear rate” f , The functional relationship is characteristic of each specific fluid but independent of duct cross-sectional geometry. Moreover, we can determine the relationship between i and ?, using any convenient geometry (such as flow in a circular cylinder), and obtain results applicable to all other duct cross sections. To use the procedure outlined in this article for predicting the pressure drop-flow rate equation for non-Newtonian flow in a chrtnnel of unusual cross section, it is essential to have a n accurate estimate of the shape factor A. Generally, if X is not available from Table I, i t can be determined either by solving the equations of Newtonian flow in the channel geometry of interest or by making experimental flow measurements with a Newtonian fluid. Thus applicability of the present procedure depends on knowledge of the pressure drop-flow rate behavior of a Newtonian fluid in the specific channel. I n practice, it is not convenient to tabulate values of the shape factor h for the infinite iiumber of duct cross sections possible; i t is often equally inconvenient to solve t h e Kewtonian flow equations for each particular flow channel. Thus one would desire a simple procedure for estimating the shape factor h , based on shape factors already tabulated. I n the case of a duct comprising a combination of previously tabulated cross sections, we recommend t h a t an average value of h be used as a rough approximation. Considering, for example, the duct shown in Figure 7 , one would use A =
528 Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 4, 1972
= cross-sectional area of conduit, cm2
Dh
hydraulic diameter of conduit, cm
= length of conduit, cm
- pressure, dyn/cm2
GREEKLETTERS parameter in Ellis fluid model, dimensionless local shear rate, sec-I 1 apparent average wall shear rate, sec-’ 7 Newtonian viscosity, P viscosity at zero shear rate, P ;I” = geometric shape factor, dimensionless r = local shear stress, dyn/cm2 7 = average wall shear stress, dyn/cm2 !(?) = average wall shear-stress function, dyn/cmz rc(?) = wall shear-stress function for circular tube flow, dyn/cmz i,(?)= wall shear-stress function for parallel plate flow, dyn/cniz ?a(?) = average wall shear-stress function for an arbitrary channel, dyn/cmz i s ( ?= ) average wall shear-stress function for a square duct, dyn/cmz = characteristic stress in Ellis fluid model, dyn/cm2 TI/^ a y
= = = = =
Literature Cited
Boucher, D. F., Alves, G. E., “Chemical Enginers’ Handbook,” 4th ed, Section 5, J. H. Perry, Ed., RIcGraw-Hill, New York, N. Y., 1963. Knudsen, J. G., Katz, D. L., “Fluid Dynamics and Heat Transfer,” p 97, McGraw-Hill, New York, N. Y., 1988. Kozicki, W., Chou, C. H., Tiu, C., Chem. Eng. Sci. 21, 665 (1966). Kozicki, W., Tiu, C., Can. J . Chem. Eng. 45,127 (1967). Lahti, P., J . SOC.Plastics Engrs. 19, 619 (1963). McEachern, D. W., A.1.Ch.E. J . 12(2),328 (1966). Metzner, A. B., Processing of Thermoplastic Alaterials,” p 41, E. C. Bernhardt, Ed., Reinhold, New York, N. Y., 1959. Mitsuishi, N., Kitayama, Y., Aoyagi, Y . , Int. Chem. Eng. 8(1), 168 (1968). Mizushima, T., Mitsuishi, N., Nakamura, R., Kagaku Kogaku 28,648 (1965). Schechter, R. S.,A.I.Ch.E.J. 7(1),445 (1961). Shih, F. S., CCZLJ . Chem. Eng. 45,285 (1967). An Analysis of Laminar Flow in Rhombohedral Shih, F. S Conduits’,’” presented at 61st Annual Xeeting of A.I.Ch.E., Los Angeles, Calif., Dec 1968. Sparrow, E. M., A.I.Ch.E. J. 8 ( 5 ) , 509 (1962). Wheeler, J. A , , Wissler, E. H., A.1.Ch.E. J . 11(2)207 (1965). RECEIVED for review August 23, 1971 ACCEPTEDMay 15, 1972
