Shannon, P. T., Tory, 13. M., IND.ENG.CHEM.FUXDAMENTALSTory, E. M., privatecommunication, 1968. 4, 367 (1965b). Yoshioka, M., Hotta, Y . , Tanaka, S., Naito, S., Tougami, S., Shannon, P. T., Tory, E,. M., Trans. SOG.Mining Engrs. 235, 375 Kagaku Kogaku 21, 66 (1957). iDec. 1966). Talniage, I V . P., Fitch, E. B., Znd. Eng. Chem. 47, 38 (1955). Tory, E. hl., Ph.D. thesis, Purdue UniLersity, Lafayette, Ind., RECEIVED for review August 21, 1967 1961. ACCEPTED June 14, 1968
EXCESS PRESSURE DROP IN LAMINAR FLOW THROUGH SUDDEN CONTRACT ION Non-New ton ian Liquids G l A N N l A S T A R I T A , G U l D O GRECO, J R . , A N D L U l G l PELUSO Zstitiito dz Elettrochimica, Unitersitci di .Vapolz, ,Vaples, Italy Excess pressure drops for laminar flow of non-Newtonian liquids, both weakly and highly elastic, have been measured. When plotted in the appropriate dimensionless form, the data show that there are no conspicuous non-Newtonian or elastic effects. The dimensionless pressure drop is inversely proportional to the Reynolds number, the constant of proportionality being a function of the flow index. Values both larger and smaller than the corresponding Newtonian values have been observed.
and Greco (1968) analyzed the phenomenon of laminar flo\v of Ne..vtonian liquids through a sudden contraction connecting tivo circular tubes of different diameter. This ivork presents the extension of the analysis to non-Newtonian liquids, both purely viscous and viscoelastic. Entrance effects for laminar flow of non-Seivtonian liquids have been discussed (BN+pe, 1959; Collins and Schowalter, 1963). These analyses have been based on the same assumptions as corresponding analyses for Neivtonian liquids, and thus suffer the shortcomings discussed by Astarita and Greco (1968). Furthermore, these analyses have not taken into consideration elastic effects, \\-hich m,ay be important for the floiv pattern considered (Astarita, 1966). Experimental data for purely viscous liquids have been reported by Jastrzebski (1967). More recently, bouncI,u-y layer analyses of rather qualitative character have been publislied concerning elastic liquids (Hermes and Fredrickson, 1967 ; Marrucci and Astarita, 1966 ; Metzner and Astarita, 1967). The tube entrance effect has been explicitly discussed by A4staritaand Metzner (1966) and by Uebler (1967); d a t a on excess pressure drop for such liquids are reported by Feig (1966) and Pruitt and Craivford (1965). STARITA
A
In Equation 2, the Reynolds number needs to be properly defined; it is presumed that an appropriate definition would permit Equation 2 to be put in the form of Equation 1, ivith the additional complication that K and R' are expected to depend on both the geometry of the system and the rheological properties of the liquid considered. The rheological properties need to be expressed through appropriate dimensionless parameters, relative to both the purely viscous behavior of the fluid (say to the nonlinearity of the shear stress-shear rate curve) and its elastic properties. Appropriate parameters could be the poiver-law index and the \\.eissenberg number. Kow restrict attention to purely viscous non-Se\vtonian liquids. A generalized Reynolds number can be defined folloiving Metzner and Reed (1955) as
(3) In Equation 3, n and 17 are obtained from capillary viscometer data as
Theory
(4)
The dimensionless excess pressure drop, 2 A p / p C 2 , for laminar flow of Newtonian liquids through sudden contractions is a unique function of the Reynolds number, which can be cast in the form (Astarita and Greco, 1968) :
(5)
24 5 --
PU2
-K+-
K' Re
where K and R' are, as a first approximation, constants for a given geometrical configuration. In the case of non-Xe\vtonian liquids, which cannot be characterized by only one rheological parameter such as viscosity, dimensional considerations show that = f (Re, rheological properties)
P
u2
Both n and 7 are local values, and may depend on the value of the shear stress, T ~ O ;thus Equations 3,4, and 5 do not necessarily imply the validity of the power law. The definition in Equation 3 is such that, for well developed laminar flow through a circular tube, the friction factor is related to the Reynolds number by the Newtonian equation
I n applying this definition of R e to the problem considered here, the question arises as to the proper value of the shear VOL. 7
NO. 4
NOVEMBER 1 9 6 8
595
stress at which 7 and n need to be evaluated. I t seems a natural though not necessarily a useful choice to evaluate the Reynolds number at the fully developed downstream tube-wall shear stress; this is consistent with the usual convention for Newtonian liquids, where Reynolds number is defined for the downstream tube conditions. If the value of T,O for well developed flow in the downstream tube is measured, the Reynolds number can thus be calculated directly as Re =
8pU2
(11) T h e values of u a and uB can be calculated if the normal stress distribution in fully developed laminar flow is known. T h e term u A - uB//3 in Equation 10 is presumably positive; in fact, at least as a first approximation, normal stresses are proportional to the square of the velocity gradient, and hence
(7)
~
Two
At sufficiently low flow rates creeping flow is expected to prevail and the excess pressure drop should depend on velocity in the same way as the shear stress depends on the shear rate, say d log A p / d log U = n. This, in turn, implies that 2Ap/pUz should be inversely proportional to the Reynolds number as given by Equation 3, and that the Couette correction should have the form R’/Re predicted by Equation 1. Now turning attention to elastic liquids, the definition given above for the Reynolds number is presumably still satisfactory, though K and E(’ are expected, at least in principle, to depend also on the elastic properties of the liquid. In other \cords. two liquids having the same shear stress-shear rate behavior in viscometric flow, but characterized by different elastic properties, may present different values of Ap. The implication of Bird (1965) that no such t\ro liquids may exist has been discussed (Astarita, 1966; Astarita and Metzner, 1966); in fact. liquids exist which exhibit a Neivtonian shear stress-shear rate behavior, yet are elastic in character (Hershey and Zakin, 1967). The direction and the magnitude of the difference \vhich may be observed among elastic and nonelastic liquids, as far as the value of Ap is concerned, are open to some question. TVhile Astarita and Metzner (1966) imply that Ap is expected to be larger in elastic liquids, their analysis is perhaps oversimplified on this point. The macroscopic momentum and energy balances for purely viscous non-Newtonian liquids are the same as for Newtonian liquids :
MOMENTUM. (Ap),Ir
= pU2(aB -
b . 4 )
+
sp
7,dA
-
SAB
TWodA ( 8 )
Factors cy and a‘ are not 4/3 and 2, respectively; their value depends on the radial velocity distribution in well developed laminar flow. Equation 8 is based on the assumption that p , = pc, discussed earlier (Astarita and Greco, 1968), and found to be true also for non-NeLvtonian liquids. When elastic effects are considered, additional terms arise in both the momentum and energy balances. For incompressible elastic liquids the pressure at any given cross section cannot be defined; from a pragmatic viewpoint, however, we interpret pressure as the reading of a pressure tap mounted flush to the wall; in the case of a circular tube, the pressure is the negative of the normal stress in the radial direction at the tube wall. IVith this definition, the momentum balance for a viscoelastic liquid yields
where (Ap),, is the excess pressure drop, (Ap),, is the value of Ap calculated from Equation 8, and 596
I&EC FUNDAMENTALS
uB is positive if the normal stresses are a pull in the flow direction, as in this case. This does not, however, imply that (LIP)“, < (AP)~~,,because the integral of the \Val1 shear stress may be very different in the case of viscoelastic liquids. In fact, the
value of the term
SAB
7,dA \vhich appears in the expression of
(l,~ is influenced ) , ~ mainly by the rheological behavior of the fluid in the region of the developing velocity profile, \\,here the elastic character of the liquid is particularly important. In the energy balance, two additional terms arise
( A P ) V E=
(AP)pv
4- AEe - AE’
(13)
xvhere E , is the stored elastic energy, irhile lE’ needs to be discussed further. For viscoelastic liquids, the work done on the fluid to enter the system is not equal to the product of the pressure as pragmatically tlefined above times the volumetric flow rate. Let LC’ be the work required to push a unit volume of liquid into the system; then
E’ = p - w =
(-TrT)?=R
+2
Astarita and Metzner (1966) hint that AE, - AE’ should be positive, and hence conclude that ( A P ) ~ . > ~ (AP),~. This conclusion does not seem entirely justified; while for purely viscous liquids the difference in mechanical energy equals the dissipated energy, for elastic liquids the accumulated elastic energy also needs to be taken into account (AEe # 0). But, in the case of elastic liquids, w # p-Le., the usual definition of en1 halpy fails, because the pressure times volume term is replaced by a term which accounts for the local normal stress pattern-and the term AE’ arising as a consequence of this is positive and cannot be directly compared with AE,. Knowledge of rheology does not allow any firm conclusion to be drawn concerning the relationship among normal stresses in developed laminar flow and stored elastic energy; even if a specific constitutive equation is assumed, the former can be calculated but the latter cannot. Thus if there is any appreciable change in Ap due to elasticity of the liquid, it needs to be found experimentally, and if observed, its value could not be used to infer elastic properties of the liquid quantitatively, unless our theoretical understanding of thermodynamics of elastic liquids proceeds well beyond the present state. Experimental
The experimental arrangement used was described by Astarita and Greco (1968). Eight liquids were tested: five aqueous solutions of ET-597, a drag-reducing additive manufactured by the Dow Chemical Co., and three aqueous solutions of Carbopol (carboxypolymethylene). The former are
A
3 00
0.5%
0 1.0% 2.0%
V
2.5%
0 3.0%
100
30 N
E v
u
N
8
13
5 30
$ 10
2
0
%
k 10 1
5
31
Figure 1.
1’0
3’0 l i e
\n
100
3b0
Presssure drop in ET-597 solutions
believed to be elastic; the latter are only weakly so (Oliver, 1966). Two major difficulties have been encountered. First, polymer solutions are always rather viscous, and experiments could be carried out at large Reynolds numbers only for extremely dilute solutions, which are almost Newtonian in character. Thus, no data could be obtained concerning the Hagenbach correction for highly non-Newtonian liquids ; the few data obtained for very dilute solutions suggest a value of K slightly smaller than for Kewtonian liquids. The second difficulty is that only a limited range of values of the Reynolds number can be investigated for any single liquid. I n fact, while an upper limit is set by the maximum total pressure drop allowed through the test section, a lower limit is set by the necessity of obtaining a significant value for the excess pressure drop, Ap. I n practice, no values of Ap lower than about 0.6 cin. of the liquid tested could be measured with suflicient confideace. I n the case of Kewtonian liquids, this difficulty is overcome by using liquids with different viscosities to cover a wider range of Reynolds numbers; in the case of non-Newtonian. liquids, a change of viscosity level also implies a change of other rheological properties, and thus the data for t\vo different liquids cannot be used together. I n vieiv of these difficulties, only a value for the Couette constant, K’, could bse obtained for each liquid. The data relative to the five so1ui:ions of ET-597 are reported in Figure 1 ; the expected inverse dependency of 2 A p / p U 2 on the Reynolds number is confirmed, substantiating the definition given by Equation 3. Values of K’ obtained from the data in Figure 1 are reported in Table I. While a consistent trend is observed-Le., K’ decreases with increasing polymer concentration-the values observed are both larger and smaller than the Newtonian value, K’ = 795 (Astarita and Greco, 1968). This implies that, at extremely low concentrations, the trend is reversed. Unfortunately, at concentrations lower than 0.570, the vis-
Table 1. Values of K’and n for Highly Elastic liquids (ET 597) Concentration, P.P.M. K’ n
1120 920 720 500 3000
3
400
0.875 0.74s ..
0.590 0.540 0.530
cosity level is so low that data can be obtained only at large Reynolds number, and no value of K’ can be obtained. T h e pressure drop in fully developed flow is obtained for all experimental conditions, and thus a side result of this work is the shear stress-shear rate curves plotted in Figure 2 . Over the range of flow rates investigated, these curves are straight on a log-log plot; thus the value of n is constant (Table I). Similar data were obtained with the Carbopol solutions used; again the dimensionless pressure drop, 2 A p / p U 2 , was found to be inversely proportional to the Reynolds number defined by Equation 7. Detailed data are available (Peluso, 1967) ; values of K’ and n are reported in Table 11. In the case of Carbopol solutions there is no definite trend of K’ with concentration; the highest value of K’ is observed at the intermediate concentration level. Values of n are larger than in the case of the ET-597 solution, though always smaller than unity. The results suggest the possibility that the value of K‘ may be a unique function of the flow behavior index, n, as dimensional considerations indicate is true in the absence of large elastic effects. Figure 3 is a plot of K’ us. n; a single curve seems to correlate all the data within experimental accuracy. T h e results are not conclusive, because no data have been obtained for the ET-597 solutions at n values exceeding 0.85-Le., on the right of the maximum point-yet it may be concluded that the present data contradict the hypothesis that Ap is larger in the case of elastic liquids than for purely viscous fluids. In fact, if there is any significant effect of elasticity, it appears to be in the opposite direction. I t is unfortunate that the data for highly elastic liquids (ET-597) and weakly elastic ones (Carbopol) overlap over only
Table II. Values of K’ and n for Weakly Elastic Liquids (Carbopol) Concentration, P.P.M. K’ n
1060
1130 1230 1170
1360
1580
VOL. 7
NO. 4
0.89 0.86 0.79
NOVEMBER 1 9 6 8
597
diction with the present results is left unexplained, and may be due to the much larger values of the shear rate in Feig’s and in Pruitt and Crawford’s experiments.
1500.
r
Nomenclature
Newtonians
1‘0
0
Carbopol solutions
0
ET.597
’I
0 1
I
.6
Figure 3.
n
.8
1
Couette constants K’ vs. flow index n
A B
= = = = = =
a
= 2
a’
=
0
= doivnstream-to-upstream diameter ratio = consistency, dynes sec.n/sq. cm. = dimensionless radial position
section A, far upstream section B, far downstream D downstream tube diameter, cm. E‘ p - w , dynes/sq. cm. stored elastic energy, dynes/sq. cm. Ec f/2 friction factor K , K‘ = constants n = flow behavior index = pressure, as measured at tap mounted flush to the wall, p dynes/sq. cm. Ap = excess pressure drop, dynes/sq. cm. pV = purely viscous behavior assumed VE = viscoelastic behavior R e = Reynolds number U = velocity w = Ivork per unit volume, dynes/sq. cm.
s,
1
a limited range of n values in the K’n plot of Figure 3. This leaves the question open as to whether there are any important effects of elasticity; the purely viscous K’(n) curve could be very different from the one drawn in Figure 3 at low values of n. In fact, there are t\vo additional indications in favor of the hypothesis that there are no important elastic effects. For all the liquids tested the pressure on the annular region normal to the flow direction, p c , was undistinguishable from the extrapolated radial pressure at the same section, p A . Incidentally, this has been used in writing Equation 8, which would otherwise contain an additional term proportional to P A - p,. Important elastic effects would be expected to result in anomalies in the value ofp,. No significant or consistent differences have been observed, for all the liquids tested, as far as the entry length is concerned. Quantitative data on entry lengths are always subject to some question, due to the inevitable amount of subjectivity in the procedure of curve fitting required; but any major difference in behavior between t\vo types of fluids should be easily observed, and none has been detected. Complete pressure profiles for all the runs performed are available (Peluso, 1967). Finally a comparison with previous published results is in order. The data of Jastrzebski (1967) on purely viscous nonNeivtonian suspensions are in qualitative agreement with the O present ones (Ap depends on U in the same way as T ~ depends on shear rate); quantitative agreement is dificult to assess. Uebler’s (1967) observation of major anomalies in the case of elastic liquids concerns the kinematics of motion upstream of the contraction, and therefore does not contrast with the present results concerning Ap. Older data of non-Newtonian liquids are well summarized by Skelland (1967), and are again in qualitative agreement with the present ones. Results apparently in contrast are those of Feig (1966) and Pruitt and Crawford (1965). Both claim that viscoelastic liquids exhibit much larger values of Ap at the same value of ~ ~ compared 0 , with nonelastic liquids. I n view of Equation 7 and the form of the Couette correction, this is the same as comparing 2 A p / p U z at equal Reynolds number. The contra-
598
l&EC FUNDAMENTALS
7
t
p u T,O
2
P2t4
p t d t
= density, g./cc.
= see Equation 11, dynes/sq. cm. = shear stress at \Val1 in well developed flow (downstream
tube), dynes/sq. cm. = physical components of stress tensor along axial, radial, and tangential directions p(t) = dimensionless velocity distribution r,,,
T,,, 7 0 0
literature Cited
Astarita, G., Can. J . Chem. Eng. 44, 59 (1966). Astarita, G., Greco, G., IND.ENG.CHEM.FUNDAMENTALS 7, 27 (1968). Astarita, G., Metzner, A . N., Rend. Acc. .Vaar. Lincei (Rorne),VIII-46, 74 (19661. --, Bird, R . B., Can. J . Chem. Eng. 43, 161 (1965). Bogue, D. C., Ind. Eng. Chem. 51, 874 (1959). Collins. M.. Schowalter, \V. R., A.I.Ch.E. J . 9, 98, 804 (1963). Feig, J: L.,’M.Ch.E. thesis, University of Delaware, 1966. Hermes, R. E., Fredrickson, A . G., A . I . C h . E . J . 13, 253 (1967). Hershey, H. C., Zakin, J. L., IND.ENG.CHEM.FUNDAMENTALS 6, 381 (1967). Jastrzebski, Z. D., IND.ENG.CHEM.FUKDAMENTALS 6, 445 (1967). Marrucci, G., Astarita, G., Rend. Acc. N a r . Lincei (Rome) VIII-41, 355 (1966). Metzner, A.B., .ktarita, G., A.I.Ch.E. J . 13, 350 (1967). Metzner, A . B., Reed, J. C., A . I . C h . E . J . 1, 434 (1955). Oliver, R. D., Can. J . Chem. Eng. 44, 100 (1966). Peluso, L., Chem. Eng. thesis, University of Naples, 1967. Pruitt, G. T., Crawford, H. R., Report to David Taylor Model Basin, Contract nonr-4306(00) (1965). Skelland, .A. H. P., “Non-Newtonian Flow and Heat Transfer,” pp. 121-5, Wiley, New York, 1967. Uebler, E. A, Ph.D. thesis, University of Delaware, 1967. \ -
RECEIVED for review November 13, 1967 ACCEPTED July 15, 1968 Work supported by the Consiglio Nazionale delle Ricerche, Grant 115.1729.0.0434.
