literature Cited
(1) Baddour, R. F., Yoon, C. Y., Chem. Eng. Progr. Symp. Ser., No. 32, 57, 35 (1962). (2) Deissler, R. G., Eian, C.S., Natl. Aeron. Space Admin. NACA RM E52C05 (1952). (3) Eckert, E. R. G., Dra.ke, R. M., Jr., “Heat and Mass Transfer,” McGraw-Hill, New York, 1959. (4) Fischer, R. A,: J . A~gr.Sci. 16, 492 (1926). (5) Grootenhuis, P.: Mackworth, R. C. A., Saunders, 0. A,, “Proceedings of Genera.1 Discussion on Heat Transfer,” p. 363, Inst. Mech. Engrs., London, 1951. (6) Kannuluick, LV. G.: Martin, L. H., Proc. Roy. SOL. (London) A141. 144 (1933). (7) Kennard. E. 8.. “Kinetic Theory of Gases,” McGraw-Hill, New York, 1938. (8) Kling, G.. Forsch. Gebiete Zngenieurw. 9, 28 (1938). (9) Kunii, Daizo, Smith, J. M., A.Z.Ch.E. J . 6, 71 (1960). (10) zbzd., 7 , 29 (1961).
(11) Masamune, Shinobu, Smith, J. M., J . Chem. Eng. Datu 8, 54 (1963). (12) Rose, Walter, J . Appl. Phys. 29, 687 (1958). (13) Schotte, W., A.Z.Ch.E. J . 6, 63 (1960). (14) Schumann, T. E., Voss, V., Fuel 13, 249 (1934). (15) W,addems, A. L., J . Soc. Chem. Znd. 63,337 (1944). (16) heidmann, M. L., Trumpler, P. R., Trans A S M E 68, 57 (1946). (17) Wilhelm, R. H., Wynkoop, W. C., Collier, D. W., Chem. Eng. Progr. 44, 105 (1948). (18) Willhite, G. P., Kunii, Daizo, Smith, J. M., A.Z.Ch.E.J. 8, 340 (1962). (19) Yagi, Sakae, Kunii, Daizo, Zbid., 3, 373 (1957). RECEIVED for review August 13, 1962 ACCEPTEDDecember 26, 1962 Financial assistance provided by National Science Foundation Grant G-17765.
VELOCITY PROFILES IN TURBULENT PIPE FLOW Newtonian and Non-NewtonianFluids D . C. BOGUEI AND A. B. M E T Z N E R
Universit.v o j Delaware, Newark, Del. Velocity profiles were measured in the turbulent core region of viscous Newtonian and non-Newtonian fluids flowing through smooth round tubes. Prior art data for Newtonian velocity profiles were comprehensively reviewed, and an empirical correction to the traditional correlation was devised. The nonNewtonian data were obtained using fluids free of significant viscoelastic effects. Reynolds numbers ranged from transitional values to beyond 100,000, and flow behavior indices were varied between 0.45 and 0.90. Turbulent core profiles for the non-Newtonian fluids were essentially the same as those for Newtonian flulids when normalized with respect to the mean velocity or, equivalently, when compared on the basis of the velocity defect parameter. Recasting the correlation in terms of generalized u + - y + parameters leads to a difference between the Newtonian and nowNewtonian cases because of differences between frictiion factors.
the classical work of Nikuradse (20). velocity profiles for the turbulent flow of fluids through smooth round tubes have been correlated by the equation (26) : ATING mob1
21+
5.75 logy+
=
+ 5.5
(1)
T h e form of this correlation is wrong in one respect: it fails to predict a zero velocity gradient a t the centerline. I n addition, there are small, but consistent, deviations of the data from the correlation; in the case of Sikuradse’s data it is found that in each run the points are higher than the curve near the centerline. These deviations are accounted for in treatments by Millikan (79): Reichardt (24),and Hinze ( 9 ) . I n connection with the present work, .a comprehensive review of the data was undertaken, and a n empirical correction function, similar to that used by these investigators but including a small effect of Reynolds number (or friction factor), was introduced ( 7 ) . A summary of the various correlations follows : Millikan: Reichardt:
u+ U+
.-
C~W(.$) =
-
C R ( ~ )=
5.75 l o g y + 5.75 logy+
+ 5.0 + 5.5
(2; (3)
Present address, Department of Chemical and Metallurgical Engineering, University of Tennessee, Knoxville, Tenn.
+
Hinze: u + - GH(E) = 5.61 logy” constant 5.57 Present work: u + - c ( E , f ) = 5.57 logy+
+
(4)
(5)
The empirical correction functions are shown for the four cases in Figure 1. T h e curves for G(, f) can be represented by the equation:
&f)
=
0.05 4 - e ~ ~
- ( E - 0.8)’ o.15
(6)
The constants in Equation 5 were so selected that this equation would, upon integration, yield the usual friction factor correlation for smooth pipes (12, 20) :
dT
= 4.0 log [NR$’~]
- 0.4
(7)
The prior a r t data of Nikuradse (20) and Laufer (74) are displayed in Figure 2 in support of Equation 5. Nikuradse’s data a t low values of y+, which are in dispute (78), are not shown. Reference to any of the several common texts which display the original data of Laufer and Nikuradse reveals the significant improvement in the correlation obtained by introduction of the correction function. Subsequent discussion makes use of Equation 5 as the best available Newtonian VOL. 2
NO. 2 M A Y 1 9 6 3
143
32
I .2
--
28
c
i
0.8
24
3
20
0.4
16
‘2
I00
0,000
1,000
100,003
Y+
E
Figure 2.
Figure 1. Empirical correction (Equations 2 through 5)
functions
=
dfp[5.57 log- $ + c ( t , , f )
+ 3.631 + 0.984
(8)
and
,4 different approach to the correlation of velocity profiles has been taken by Pai ( Z ) ,who assumes a polynomial form for the mean velocity profile. Pai‘s correlation allows one to inspect the Reynolds stress terms conveniently, but for graphical display of data, a modification of the classical u - - y + correlation appears to be more convenient. Non-Newtonian Background
The present work is concerned with those non-Seivtonian fluids which are purely viscous-that is, fluids whose rheological behavior is free of “memory” or viscoelastic effects 14,76). A frequently-used one dimensional rheological equation for these fluids is the “power lalv” : T
=
K
($)”
A Reynolds number ivhich arises naturally from the power law equation is (29) :
Another related Reynolds number is that based directly and rigorously on capillary tube viscometry (77) :
If n and K are constants. the two Reynolds numbers (Equations 11 and 12) are identical. In the present discussion, no distinction is made betLveen them, although in fact Equation 12 was used to calculate the Reynolds numbers presented. Any differences are of negligible consequence under turbulent flow conditions. A comprehensive study of turbulent non-Newtonian fluids, involving friction factor measurements on three types of material [Carbopol solutions, clay suspensions, and carboxymethyl cellulose (CMC) solutions] is available (5, 6 ) . The first two behaved similarly, whereas C M C gave much lower 144
I&EC
FUNDAMENTALS
-
y + correlation
Because of crowding, only about one third of the data points ore shown
correlation inasmuch as it is based on an extensive survey of available data. Alternate forms of Equation 5 are: u/V
Newtonian u f
friction factors. These lower friction factors have since been shown to be common to viscoelastic systems ( E ) ,and the decreased pressure drop possible with these fluids has been extensively exploited in petroleum drilling (75, 27, 23). The use of viscoelastic fluids as drag-reducing agents in naval architecture has also been discussed (7. 8). The viscoelastic properties of CMC solutions have been quantitatively measured (3, 10, 73). The Carbopol solutions and clay suspensions, on the other hand, show little if any viscoelastic properties in loiv concentrations ( 7 7 , 25). From the combined experience of these investigators, it was concluded that Carbopol solutions (in the low concentrations used) and clay suspensions are, for practical purposes. purely viscous fluids. Dodge and Metzner’s friction factor correlation for purely viscous fluids is (5: 6) :
Another turbulent study is that of Shaver and Merrill ( 2 7 ) , Lvho measured friction factors for four non-Newtonian fluids (CMC, alginate and Carbopol aqueous solutions, and polyisobutylene in cyclohexane) ; and velocity profiles for two ( C U C and alginate solutions). Their friction factor correlation predicts lower friction factors than does Equation 13. Polyisobutylene solutions are the classical example of strongly viscoelastic materials (2) and. as discussed previously, the C M C solutions are also elastic. Thus, rhese t\vo materials may not be included in any friction factor correlation \\-ithout the inclusion of additional parameters which can account quantitatively for the effects of the viscoelastic properties of these fluids. The elastic behavior of alginate has apparently not been studied, but its friction factors suggest that it should be grouped with CMC. While the velocity profiles of Shaver and Merrill are discussed qualitatively in a later section, it is obvious that no quantitative application of their results is possible. Inspection and discussion of a recent report to the contrary (28) reveal that the rheological data employed by Thomas to arrive a t his conclusions are subject to more than a single interpretation. Accordingly, the conclusions drawn by Thomas are controversial. Experimental
Velocity profiles were obtained for clay suspensions (.4ttagel) and Carbopol solutions of various concentrations. Additionally, two Newtonian fluids (water and aqueous sugar
solutions of various concentrations) were studied for check purposes. The viscosities of the Newtonian fluids varied between 0.8 and 80 c p . ; the apparent viscosities of the nonSewtonian fluids at the wall shear stresses employed ranged from 2.0 to 200 cp. A summary of the runs made and the variables covered is presented in Table I. 4 run consisted of establishing a flo\c rate? as indicated on a Foxboro magnetic meter: and of measuring the profile and the friction factor a t that condition. Some readings within a run were checked for reproducibility. Rheological measurements \cere made with a capillary tube viscometer, with standard corrections (76) being applied to the measured pressure differences ( 7 ) . The measurements Lvere made in the same range of shear stresses as were used in the large scale equipment. although extrapolations ivere necessary in the case of some of the high shear stress runs. The only run in which the extrapolation is questionable is one involving the clay suspensions at very high shear stresses (Run 37). for \chich case the bounds of any reasonable extrapolation were estimated and a range of Reynolds numbers calculated (Table I). Some degradation of the Carbopol occurred over a period of time, and runs during which there was a significant change of properties \cere not used. The clay suspensions clearly had time-dependent properties until thoroughly mixed and sheared, but finally remained constant. Good rheological measurements (i.e,! in agreement \vith the laminar pipe flow friction factors) were obtained by whipping the sample vigorously by hand and testing it in the viscometer immediately. The test section consisted of either a 1-inch or 2-inch commercially smooth brass pipe. T h e Pitot tube traverse was made 150 pipe diameters downstream from the inlet in the
Table 1.
Run *\To.
1 6 7 5 4 2 8 25 24 32 31
12 26 28 14 13 22 21 34 35 11 9
33 37 36 18 20 17 19 15
Summary of Velocity Profile Runs
.4pprox.
Material Sugar solution Sugar solution Sugar solution Sugar solution Sugar solution Sugar solution Sugar solution IVater Water Water Water
n’
1 .0 1 .0 1 .0 1 0 1. 0 1 .0 1 .0 1 .0 1. o 1. o 1. o
Carbopol (0.3%) Carbopol (0.1 yo) Carbopol (0.1yo) Carbopol (0.2yo) Carbopol 10.257,) Carbopol (0.2%) Carbopol ( 0 . 2 57,) Carbopol ( 0 3%) Carbopol ( 0 . 3yo) Carbopol ( 0 4%) Carbopol (0.4y0) Carbopol ( 0 , 4y0) Clay susp. (12%)
0.745 0.895 0.895 0.825 0.825 0.80 0.80 0.70 0.70 0.65 0.615 0.59 0.6-0.8
Clay susp. [lZyo) Clay susp. :14y0) Clay susp. 8:14%) Clay susp. (14’%) Clay susp. 8:1457,) Carbopol ( 0 .6yc)
0.53 0.465 0,465 0,465 0.47 0.445
former case and 90.5 pipe diameters downstream in the latter. Preceding the pipe entrance was a n enlargement containing a coarse screen to assure that disturbances were present to trigger the onset of turbulence. Velocity profiles were obtained with a traversing Pitot tube connected to a micrometer mechanism for determining its position. T h e positioning of the probe was checked visually from the end of the pipe, and a n electrical contacting device was used to sense the exact reading a t which the Pitot tube touched the wall. Two Pitot tubes were used : one terminating in a 0.048-inch outer diameter tip, the other in a 0.022-inch outer diameter tip. The probes were purged \vith Lvater to minimize plugging. The final technique consisted of using the purge to establish the neighborhood of the reading and then allowing the manometer to equilibrate without any purge. The final manometer reading was approached from both a high and loiv displacement of the fluid. Plugging ivas a recurring problem, but with several repetitions reproducible results were obtained. Various U-tube manometers and a Prandtl type micromanometer were used to measure the pressure differences. Possible Pitot tube corrections were considered in mme detail ( 7 ) . Such corrections can be grouped as follows: contributions from the turbulent, fluctuating velocity components; a correction for the radial pressure gradient for the case in which the static tap is at the wall and only the impact tube traverses; and, finally, corrections for the disturbed flow pattern caused by inserting a solid object in a flowing stream. In the last case, the available Newtonian theories and data for predicting viscous corrections in a free stream \cere reviewed and extended in an approximate manner to non-Newtonian fluids. I t was concluded that all of the possible corrections
.VIR“
800
2.000 2.600 3.300 6.800 13,390 26.000 91 .700 197.000 118.000 119.000 3.660 24.960 95.000 28.500 107,500 29.450 94,000 11.700
12.100 17.500 23.240 6.100 46.50085,000 17.400 7,880 12.900 17.150 19.500 8.300
Pipe Diam., In. 1 1 1 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 2 2 1 1 2
Pitot Tube Diam., In.
0.048 0 048 0.048 0 048 0.048 0.048 0.048 0.048 0.048 0.022 0.048 0.048 0.048 0.048 0,048 0,048 0.048 0.048 0.022 0.048 0 048 0.048 0,048 0.048 0.048 0,048 0.048 0.048 0.048 0.048
.Mean Velocity
-
Flozv
Intqrated, f t . /sec. 3 . 6 (est.) 8 . 6 (est.) 12.7 14.15 26.5 29.8 37.7 9.05 19.3 7.51 7.34 5.35 10.4 33.9 15.3 46.0 16.7 43.2 12.9 12.9 35.1 44.3 15.6 Profile incomplete 30.0 19.3 25.9 30.2 33.9 47.2
VOL. 2
metpi‘;
f t . !sec.
3.54 8.53 12.5 13.8 27,l 3c.7 38.0 9.15 19.3 7.26 7.26 5.26 10.5 34.3 14.8 45.8 16.5 44.0 12.7 12.7 35.3 44.9 15.3 51.5 30.0 18.9 26.1 31.4 34,I 47.5
Error,
%
+1.6 f2.5 -2.2 -2.9 -0.8 -1.1 0 +3.4 +1.1
$1.7 -1 . 0 -1.2 $3.4 C0.4 +I . 2 -1.8 +1.5 +1.5 -0.6
-1.3 +2.0
... 0
$2.1 -0.8 -3.5 -0.6 -0.6
NO. 2 M A Y 1 9 6 3
145
1.6 Turbulent
0.4
"
2.0
I
(Newtonian)I
1
sugar sol'n. 26
0
0.2
0
,
x sugar sol'n. 6.8-13.3
0.4
0.6
Larnhar ( n ='0.825)
1
15 25- 43
0.8
1.0
0.4 k / O
Carbopol
0.80 - 0.895 24,9604
t Figure 3.
Newtonian velocity profiles
E Figure 4. Non-Newtonian velocity profiles: region of high flow behavior index (mildly nowNewtonian fluids)
were negligibly small in the turbulent runs, except the correction for the disturbed flow when the Pitot tube is near the wall. T h e data obtained very near the wall showed no consistent pattern with Pitot tube size, and it was decided to retain only those points which, judging from the Newtonian check runs, were far enough from the wall to be accurate. In the 1-inch pipe with the 0.048-inch Pitot tube, the closest retained point was a t y / R = 0.15; in the 2-inch pipe with the same Pitot tube, y / R = 0.075; and in the 2-inch pipe with the 0.022-inch tube, y / R = 0.05. Small viscous corrections were indicated in the case of the laminar Newtonian check runs and were made in accordance with prior art data based on a Pitot tube Reynolds number (7). Excepting these laminar runs, then, the point mean velocities were calculated from the equation :
The velocity profiles were numerically integrated across the pipe, using a semilogarithmic interpolation formula between measured points. T h e integrated mean velocities were compared with those measured using the magnetic flowmeter. Four runs in which the integration error was in excess of 4.570 were discarded. T h e errors in the retained runs were small (Table I). Complete details of equipment and procedure are available ( 7). Results
Pressure Drop Measurements. Experimental values of the friction factors are not available for some of the earlier runs a t high Reynolds numbers. For other runs, such data were obtained and were in good agreement with the prior art correlations (5, 6 ) . Excepting one clay suspension point a t a low Reynolds number, the maximum deviation was 5.870, and the average absolute error of the 22 turbulent points was 2.770. I n calculating parameters involving the wall shear stress, the experimental friction factor was used when available; otherwise the friction factor was obtained from Equation 13 (Runs 8; 9, 11, 13, 14, 17, 19, 21, 22). Velocity Profiles. The Newtonian velocity profiles of the present study are shown in Figure 3. T h e curves shown in these figures are plots of Equation 8,with the two lines representing the bounds determined by the experimental friction factors. 146
l&EC FUNDAMENTALS
All of the non-Newtonian data of the present work are shown in Figures 4-6. The curves shown are again plots of Equation 8, the turbulent h'ewtonian correlation, with the two lines bounding the experimental friction factors. For comparative purposes the laminar velocity profiles for the various values of n are shown. These curves were calculated from simple laminar theory (29): 3n
+1
[ I - (r/R)(n+l)'n]
Considering all of the data together, neither a Reynolds number nor a non-h'ewtonian effect is discernible. It was concluded that the velocity profiles are not substantially different from the Newtonian ones when compared on the basis of u / V or, equivalently, on the basis of ( U O- u ) / u * . More precisely, if all of the non-Newtonian data are pooled a t some value of y / R (0.2) and compared with Sewtonian data a t the same position, the following quantitative statement is possible: There is no significant difference between the Newtonian and non-Newtonian data until dropping to the 45% confidence level; the difference here, if real, is a small one, 0.21 units of (Uo - u ) / u * or 4%. This conclusion that the turbulent velocity profile is insensitive to the non-Newtonian character of the fluid (i.e., to the flow behavior index) must, of course, be restricted to the range of flow behavior indexes studied: 1.0 (Newtonian) to 0.45. At very low flow indexes (below about 0.10), a corresponding conclusion would require that the profile become steeper on becoming turbulent. As this is perhaps unlikely, the extrapolation of the present conclusions to such highly non-Newtonian systems (if they were actually encountered) cannot be recommended. I t is of interest to compare the present velocity profiles with those of Shaver and Merrill (27). Their C M C velocity profiles a t low flow behavior indexes are substantially steeper than those of the present study. As might be expected, the viscoelastic behavior of CMC suppresses the turbulent momentum transport and causes the profiles to be more nearly laminar (steeper). This mechanism would also be expected to produce more nearly laminar (lower) friction factors, as observed by both Shaver and Merrill (27) and Dodge and Metzner (5, 6). At higher values of the flow behavior index, there is no discernible difference between the turbulent
2 .o
I
I
I
I
~
Laminar (n.0.46)
I
-t\
1.6
I .6
1.2
1.2
>
>
\ 3
\
3
lent (Newtonian)
0.8
0.8 0 Carbopol 0.59 - 0.70
6,10023,24 0 Carbopol e 0.6- 0.8 =46,50085.0 O Y J 1 , 0 0.2 0.4 0.6 0.8 1.0
I/
1
r r b o p o l , 0.445 8,330 0 Clay SUSP. 0.46 - 0.53 7,880 -
19,500
0
0
0.2
0.6
0.4
&
0.8
1.0
E Figure 6. Non-Newtonian velocity profiles: low values of flow behavior index (highly nowNewtonian fluids)
Figure 5. Non-Newtonian velocity profiles: intermediate values of flow behavior index
Sewtonian curves and the C M C data. I t appears that a t these lower concentrations the elastic cffects are negligible. The same is apparently true of the alginate solutions, which were studied only in the range of high flow behavior index. The form of the velocity profile correlation proposed by Dodge and Metzner (!;\ 6) could in principle be used to interpret the results. Fundamentally, however, the use of friction factors is a n insensitive \vay to deduce information about the velocity profiles, inasinuch as one sees only an integrated effect in the case of the friction factors. I n the case of the Dodge and Metzner analysis, and the Millikan analysis for Kexvtonian fluids from which it was generalized (7Q), it is necessary to neglect arbitrarily one of the terms (the g-function) to carry the analysis through to completion. Inclusion of this very general function complicates the method considerably, and in the non-Newtonian case masks the physical fact that the turbulent core is the same as that for Newtonian fluids. For that reason, a direct analysis was used in the present work in lieu of the previous model.
LNNewtonian
0
IO
100 yu*p/p,,
1.0
I 10,000
1,000
= (y+)”n
Figure 7. Correlation of previous figures recast into u+ - y + coordinates
Correlation
Consequences of the Correlation
I t has been concluded that for purely viscous non-Xeivtonian fluids one may use the Nelvtonian correlation, Equation 8 (or, in an alternate form, Equation 9). Additionally, Equation 8 may be cast in terms (of the more familiar u+ - y + parameters by algebraic manipulation. Rewriting Equation 8 as follows:
The generalized correlation, Equation 16, approaches the Newtonian case at large Reynolds number (of the order of 100,000). T o understand the physical significance of this behavior, it is useful to analyze the parameters more closely. Introducing the apparent viscosity a t the wall :
U f
=
5.57 log
( V + y
5.57 log
+G(E,f)
~RnU*2-np)1’n
gcK
-
+ 0.984 mj + 3.63
(15)
and employing the friction factor-Reynolds number correlation, Equation 13, one obtains: u+
-
c($,
+
f) = 5.57 log ( ~ + ) l ’ ~I(n, N ’ R ~ )
(16)
where the function Z(n,,,+’’Re) is given in the following tabulation : N k e
n
5000
10,000
50,000
1. o
5.57 6.01 6.78 8.39
5.57
5.57 5.69 5.89 6.27
0.8 0.6 0 4
5.92
6.51 7.70
it can be shown that
700,000
5.57 5.58
5.60 5 60
Equation 16 is the most general statement of the velocity profile correlation. I t is plotted in Figure 7 .
Thus, the parameter (y-)”” may be thought of as the conventional j + into which has been introduced an apparent viscosity evaluated a t the wall shear stress. T h e above correlation, extrapolated to Reynolds numbers above 105. predicts that the flow situation is entirely characterized by a single apparent viscosity term, pa=. Physically, this might appear to be quite reasonable as the layer near the wall, in which viscosity plays a part, may be so thin under these conditions that the variation of the shearing stress, hence VOL. 2
NO. 2
MAY
1963
147
viscosity, within it is negligible. At lower Reynolds numbers, the radial variation of viscosity must be considered. As one moves away from the wall> the fluid becomes more viscous (if the flo\v behavior index is less than unity), the turbulent motion is Therefore partially damped out, and the friction factors are lobver than in the case of Nelvtonian fluids, as is found to be the case (see below). Recasting the previous friction factor-Reynolds number correlation (5. 6) into one tvhich employs the Reynolds number Dtipt’i.raZc. instead of that defined by Equation 12, is instructive. Using the apparent viscosity at the wall, Equation 17, the t\vo Reynolds number: can be related by the equation:
where f is given implicitly by Equation 13. T h e resulting correlation is shoivn in Figure 8. I t brings the friction curves for various values of n closer together and suggests that they will gradually merge a t high Reynolds numbers. Such a recorrelation is perhaps of minor interest from a design point of view, as a trial-and-error procedure is involved in obtaining paE. However. the suggestion is of interest from the viewpoint of understanding turbulent transport processes in these systems and suggests an interesting area for further investigation. For the present, the behavior a t high Reynolds number must be accepted Lvith reservation as this would involve extrapolation of the friction factor correlation (Equation 13. which was used in the development of Figure 8) to Reynolds numbers well above those studied to date.
From the present work i t was concluded that turbulent velocity profiles for purely viscous non-Navtonian fluids are essentially the same as those for Sewtonian fluids when normalized to the mcan velocity or Ivhen expressed as the velocity defect, (L‘o - u ) , u*. T h e range of flow behavior indexes included in the present \vork, over which this conclusion may be expected to be valid. extends from unity to 0.45. On the other hand, recasting the correlation in terms of the parameters u+ and y + leads to a difference between the Newtonian and non-Sejvtonian cases; these parameters implicitly make use of the friction factor-Reynolds number relationship in which the flow behavior near the wall (and thus the rheological character of the fluid) is an important factor. T h e apparent viscosity evaluated at the wall shear stress is a useful parameter from a mechanistic point of view. At lo\v Reynolds numbers its use reduces thP dependence of the friction factor correlation on model-based parameters (i.e., on the flo\v behavior index n). Extrapolation suggests the interesting possibility that the apparent viscosity Lvould alone provide sufficient characterization at high Reynolds numbers. T h e parameter is of less interest from a design point of \Tie\.\;, since a trial-and-error procedure is involved in its calculation.
Nomenclature
empirical correction function defined by Equation 5 and 6, dimensionless pipe diameter, ft. Fanning friction factor, ~ ~ / ( p I i * / 2 9 , )dimension, less lb. mass ft. conversion constant, Ib. force s e c 2 rheological parameter, defined by Equation 10: lb. force seen sq. ft. rheological parameter [see (76) or ( 1 7 ) ] , lb. force set."' sq. ft. rheological parameter, defined by Equation 10, dimensionless d log T , dimensionrheological parameter = d log (8 V / D ) ’ less ~
Conclusions
T h e traditional semilogarithmic correlation of Newtonian velocity profiles (Equation 1) is not accurate in detail, the data shoiving small consistent deviations. T h e deviations are a function primarily of radial position and: to a small extent, of friction factor. An empirical correction term was devised w.hich has resulted in a n improvement of the correlation. T h e correction is similar to those suggested by Millikan (19); Reichardt (24) and Hinze (Q)! but in the present work a more comprehensive revie\$- of the available data \vas made in evaluating it.
_ .
Newtonian Reynolds number,
dimensionless
Dn ’ V 2 - n l ( N ’ R e ) , l ’= non-Newtonian Reynolds number,
P
g,K18nl-l,
dimensionless (1)”R~)n
or N ’ R ~= non-Newtonian Reynolds number (see discussion 0.03
following Equation 12), mensionless P o
0.01
= stagnation pressure at center of Pitot tube, Ib. force/
sq. ft.
f
Pw
R
= static pressure at \\.all of pipe, lb. force/sq. ft. = pipe radius, ft.
U
=
U* U+
= friction velocity, v’r.g
