A Correlation for Velocity and Eddy Diffusivity for the Flow of Power-Law Fluids Close to a Pipe Wall W. B. Krantz Department of Chemical Engineering, University of Colorado, Boulder, Colo. 80501
D. T. Waean Department of Chemical Engineering, Illinois Institute of Technology, Chicago, Ill. 60616
Mean velocity and eddy diffusivity distributions are derived for the fully developed turbulent pipe flow of power-law fluids in the wall region. These distributions satisfy the equations of motion and boundary conditions in the wall region and provide a smooth and continuous transition to the universal velocity distribution of Bogue and Metzner valid in the turbulent core.
I n order to predict rates of heat and mass transfer in turbulent pipe flows, it is necessary to know the mean velocity and eddy diffusivity distributions in the vicinity of the pipe wall. These distributions must be inferred theoretically since experimental difficulties associated with hot wire and hot film anemometry a t present preclude the possibility of measuring these quantities in the wall region. Kestjn and Richardson (1963) and Zandi and Rust (1965) give excellent reviews of the available correlations for the mean velocity and eddy diffusivity for the turbulent pipe flow of Newtonian fluids. The conventional approach to obtaining the mean velocity and eddy diflusivity distributions in the wall region for Newtonian fluids has been to assume an empirical form for either the mean velocity or eddy diffusivity. Once a form of the distribution for either of these quantities has been assumed, the other can be obtained by a straightforward calculation since they are not independent in the constant stress region near the wall. This approach has a disadvantage in that the “constants” in the assumed form of the distribution must be determined experimentally. The latter has been the source of the controversy regarding the power dependence in y a t which the Reynolds stress UU vanishes near the wall. That is, several of the assumed or derived forms of the eddy diffusivity, or equivalently, the Reynolds stress distribution, include empirical constants which would have to vanish in order to satisfy the equations of motion and boundary conditions in the wall region. This disadvantage is compounded in the case of non-Newtonian fluids in that the empirical “constants” become unknown functions of the Reynolds number and rheological parameters such as the flow behavior index, which a t present precludes the possibility of determining them experimentally. I n this note the authors develop mean velocity and eddy diffusivity distributions near the pipe wall for the turbulent flow of power-law fluids which satisfy the equations of mean motion and the boundary conditions in the wall region. The resulting distributions contain empirical “constants.” However, their functional dependence on the generalized Reynolds number and flow behavior index are obtained by demanding that the mean velocity distribution in the wall region give a smooth and continuous transition to the universal law of the wall developed for power-law fluids by Bogue and Metzner (1963). The resulting mean velocity and
eddy diflusivity distributions for power-law fluids include Newtonian fluids as a special case. Theoretical Analysis
The continuity equation for the fluctuating velocity components and the two relevant components of the equations of mean motion are given as du -
d~
+ -1r- (dTbor)
+ -r1 dw -= 0 ae
(1)
The terms T ~ rrr, ~ and , TOOare the time-averaged components of the viscous stress tensor, which for the case of powerlaw fluids is given by (4)
+
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where A is the symmetrical deformation tensor, K is the consistency index, and n is the flow behavior index. A solution to the system of eq 1-4 is intractable unless some simplification in the stress tensor is afforded. By virtue of the fact that the region of interest is very close to the wall the terms T ~ T ~~ and , ~ ,788 can be simplified greatly. The method of simplifying these terms will be shown in detail for the component T , which ~ is given in its exact form as
where the bar indicates a time-averaged quantity. Since the region of interest is very close to the pipe wall, it is convenient to define a new variable y = a - r, where a is the pipe radius, and to introduce the dimensionless variables defined by Dodge and Metzner (1959)
where U*
= (7ro/p)1/2 is
the friction velocity. Hence after
considerable simplification eq 5 becomes eq 7 where Q
1 E -
2
(A:A)
sublayer t o yield a relationship between the Reynolds stress UV and the mean velocity
-
with the rate of deformation tensor, A, written in terms of the dimensionless variables, and
Following the idea of Elrod (1960), the velocities in the wall region can be expanded in Taylor series form
s=
(9) The absolute value sign does not appear in eq 7 since it was assumed t h a t Q 1) rheological behavior as 1%-ellas Newvtoiiian fluids (n = 1). ;Zs the flow behavior index decreases yc+ increases. For pseudoplastic and Xeewtonian fluids ycc decreases as t'he generalized Reynolds number increases, whereas for dilatant' fluids it increases as the generalized Reynolds number increases. At high Reynolds nunibers (SR, > lo5), yc+ approaches a constant value of 19.6 independent of Reynolds number and flow beharior index. The viscous sublayer thickness for Newtoniaii fluids is a n-eak function of the Reynolds number and varies between 19.5 and 20.6. This range of values corresponds closely n-ith the value of 26 commonly assumed for 426
Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
It is shown in Figure 3 for characteristic values of the flow behavior index a t Reynolds numbers of 5000 and 50,000, corresponding to two definitions of the power-law kinematic viscosity. The solid curves correspond to the kinematic viscosity defined by hIetzner and Reed (1955)
The above definition of the kinematic viscosity is consistent with the kinematic viscosity in the generalized Reynolds number defined by Lfetzner and Reed (1955). This Reynolds number offers the advantage of satisfying the familiar laminar ~ , is flow relationship for the friction factor, j = 1 6 / i V ~ and used widely in the literature. The dimensionless variables given by eq 6 when substituted into the equations of mean
good, and has been presented elsewhere in the literature by Krantz and Wasan (1970). Acknowledgment
William B. Krantz is grateful to the National Science Foundation, which provided partial support of this work through a n Engineering Research Initiation grant, NSF GK5149. Nomenclature a
= = I(n,N R e ) = K = n = N ~e = P = r = U = = L7a = L'* = 21 = W = X = Y = Ye =
f 0
2
4
6
8
IO
12
14
16
18
20
22
24
26
28
Y'
Figure 3. Eddy diffusivity distribution as a function of flow behavior index and Reynolds number; solid lines refer to eq 28; dashed lines refer to eq 29
motion suggest the following definition for the kinematic viscosity y*
E
(K/p)'/"(a*)2("-1)/"
If this definition is used in eq 26 the factor
(29)
becomes unity. The corresponding eddy diff usivity is indicated by the dashed curves in Figure 3. The eddy diffusivity defined in accord with eq 28 increases with increasing Reynolds number for dilatant and Newtonian fluids, whereas it decreases for pseudoplastic fluids. However, the eddy diffusivity corresponding t o the kinematic viscosity defined by eq 29 appears to be nearly independent of Reynolds number for dilatant fluids and increases slightly with Reynolds number for pseudoplastic and Xewtonian fluids. One would have to conclude t h a t the kinematic viscosity as defined by eq 29 is to be preferred in developing these universal eddy diffusivity distributions. It is interesting t o note t h a t whereas neglecting the terms in 4 for Newtonian fluids as suggested by Hinze (1959) has a negligible effect on the mean velocity profile, it can result in errors in excess o€ 20% in determining the local Reynolds stress or eddy diffusivity. -1s was mentioned in the introduction, the reason for developing the mean velocity and eddy diffusivity distributions in the wall region is the fact that experimental difficulties a t present preclude the possibility of measuring these quantities in the wall region. Hence there are no data available to check the predictions of eq 18 and 19 in detail. I n addition, no other mean velocity or eddy diffusivity distributions in the wall region for power-law fluids have been proposed in the literature, which precludes the possibility of a check with esiating theory. However, a n indirect check on the validity of eq 18 and 19 can be made using the abundant heat transfer data available in the literature. To this end the equations developed here for the continuous variation of mean velocity and eddy diffusivitp have been used to develop the heat and mass transfer analogies for power-law fluids. The comparison of these rerultr with the available heat transfer data is quite $I
c
radius of tube Fanning friction factor semiempirical function defined by eq 21 consistency index defined by eq 4 flow behavior index defined by eq 4 generalized Reynolds number (2a) C,/Y time-averaged local pressure radial coordinate fluctuating component of axial velocity time-averaged local axial velocity bulk average velocity friction velocity ( r z 0 / p ) ' / 2 fluctuating component of radial velocity fluctuating component of azimuthal velocity axial coordinate radial coordinate measured from wall viscous sublayer thickness
GREEKLETTERS rate of strain tensor eddy diffusivity for momentum azimuthal coordinate kinematic viscosity defined by eq 28 kinematic viscosity defined by eq 29 dimensionless radial coordinate ( y / a ) density component of viscous stress tensor representing transfer of j momentum in i direction viscous stress tensor shear stress at the wall function defined by eq 26 function defined by eq 20 function defined by eq 7
SUPERSCRIPT^ = denotes time-averaged quantity = denotes dimensionless variable
+
literature Cited
Bogue, D. C., Metzner, A. B., IND.ENG.CHEM.,FUNDAM. 2 , 143 (1963). Dodge, D . W., Metzner, A. B., A.1.Ch.E. J . 5 , 189 (1959). Elrod, H. G., J . Aerospace Sci. 24, 468 (1957); erratum, 27, 145 (1960). Hinze, J. O., "Turbulence," p 516, McGraw-Hill, Kew York, N.Y., 1959. Kestin, J., Richardson, P. D., Int. J . Heat Mass Transfer 6 , 147 (1963). Krantz, W. B., Wasan, D. T., Proceedings of the Fluid Dynamics Symposium, Hamilton, Ontario, Canada, Aug 25-27, 1970. Laufer, J., "The Structure of Turbulence in F d l y Developed Pipe Flow," NACA TR-1174 (1954). LIetzner, A. B., Reed, J. C., A.I.Ch.E. J . 1, 434 (1955). Sirkar, K. K., Hanratty, T. J., IND.ENG.CHEM.,FUNDAM. 8, 189 (1 969). Wasan, D. T., Tien, C. L., Wilke, C. R., A.1.Ch.E. J . 9, 567 (1963). Zandi, I., Rust, R. H., Proc. Amer. Soc. Civzl Eng. HY6, 37 ( 1965). RECEIVED for review November 4, 1970 ACCEPTED April 14, 1971
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