Dispersion in the Laminar Flow of Power-Law Fluids through Straight Tubes George S. Booras and William B. Krantz* Department of Chemical Engineering, University of Colorado, Boulder, Colorado 80309
The generalized dispersion model of Gill and Sankarasubramanian is applied to the laminar tube flow of powerlaw fluids. The time-independent dispersion coefficient, when appropriately nondimensionalized, is within f2% of the Newtonian dispersion coefficient for the range of flow behavior indices 0.5 In 5 1.2. In all cases the time-dependent dispersion coefficient is within 95 % of its time-independent value at a dimensionless time of 7 = 0.30. The Taylor-Aris dispersion theory results appear to apply with negligible error when T > 0.60. This analysis suggests that the widely used generalized dispersion model may not be applicable at very small 7 , at least for high Peclet number flows.
Dispersion theory is concerned with the dispersal of a solute in a flowing fluid due to the combined action of a nonuniform velocity profile, molecular diffusion, and eddy diffusion in turbulent flows. Numerous papers have discussed dispersion in a variety of laminar and turbulent flows since Sir Geoffrey Taylor (1953) and Aris (1956) published the first papers on the subject. However, most papers have been confined to dispersion in Newtonian fluids. Developments in the areas of polymer processing, biomedical engineering, and biochemical processing have contributed to the ever-increasing interest in the flow and properties of non-Newtonian fluids. Typical occurrences of this dispersion phenomenon in applications involving non-Newtonian fluids include the behavior of dyes in injection molding processes, the determination of the residence time of tracer solutes injected into the bloodstream, the transport of slurries and polymer solutions, and the design of flow reactors for biological systems. Relatively few papers have considered dispersion in nonNewtonian fluids. Taylor-Aris dispersion theory has been extended to the laminar tube flow of power-law, Bingham plastic, and Ellis fluids by Fan and co-workers (1965, 1966). Erdogan (1967) applied Taylor-Aris dispersion theory t o non-Newtonian fluids which obey the Casson constitutive equation and compared his results with those of Fan and coworkers (1965, 1966). Harlacher and Engel (1970) predicted the steady-state concentration distribution resulting from a step change a t some point along the axis for laminar flow tube flow of power-law fluids. However, Gill and Sankarasubramanian (1972) have shown that Harlacher and Engel’s results are useful only for relatively large distances from the continuous source. Taylor-Aris dispersion theory has been extended to the turbulent tube flow of power-law fluids by Wasan and Dayan (1970) and Krantz and Wasan (1974). This brief review indicates that all analyses of dispersion in non-Newtonian fluids have followed Taylor-Aris dispersion theory. Ananthakrishnan e t al. (1965) have shown that Taylor-Aris dispersion theory for Newtonian fluids applies only for sufficiently large values of the dimensionless time T (= Dtla’) ranging from 0.80 a t Pe = 500 to 20 a t Pe = 1. This limitation of Taylor-Aris dispersion theory led Gill (1967) and Gill and Sankarasubramanian (1970) to develop a method of analysis for unsteady convective diffusion in Newtonian fluids which applies to small values of T , and which reduces to the Taylor-Aris results a t larger values of T. Gill and co-workers [Gill and Sankarasubramanian (1971,1972);Sankarasubramanian and Gill (1972,1973)] have extended this analysis to both non-uniformly distributed and time-variable sources as well as to time-variable laminar flows including mass transfer
a t the tube wall. Indeed, the generalized dispersion theory model of Gill and co-workers has permitted us to consider dispersion phenomena in a wide variety of flows which hitherto were far too complex to solve analytically. In order to appreciate more fully the rather limited domain of TaylorAris dispersion theory let us estimate the transit time T or equivalently the tube length L necessary for Taylor-Aris theory t o apply; we will assume a typical liquid phase diffusivity of cm2/s and P e = 500. The criterion of Ananthakrishnan et al. then indicates that for tube radius a = 0.01 cm, T > 8 s a n d L > 4 cm; for a = 0.1 cm, T > 800 s a n d L > 40 cm; and for a = 1 cm, T > 80 000 s and L > 400 cm. This clearly indicates that many dispersion phenomena of practical interest occur within the time period r < 0.8 and thus are properly described by the generalized dispersion analysis of Gill and co-workers, not by Taylor-Aris theory. Clearly it is desirable to apply the generalized dispersion theory approach of Gill and co-workers to the dispersion process in power-law fluids as well. This problem is of interest from both a practical and a fundamental point of view. In the latter case, considering the power-law constitutive equation allows one to investigate the effect of variation in the velocity profile on the dispersion process since the velocity profiles become progressively more blunt as the degree of pseudoplasticity increases. This property of the power-law constitutive equation has very interesting implications in that the analysis presented here suggests limitations in the generalized dispersion theory approach which were not revealed in prior analyses which were confined to Newtonian fluids.
Purely Convective Dispersion Let us first consider the purely convective dispersion of a power-law fluid for the case of a plug of solute having concentration Co initially confined to a length x, described by
Define a dimensionless concentration, axial coordinate, radial coordinate, and time, respectively, as 0 = C/C,; X = Dx/a2Uo;y = rla;
T
= Dt/a2
(2)
where a is the tube radius, D is the molecular diffusivity, and C‘o is the centerline or maximum fluid velocity. These dimensionless variables arise naturally from the full convective diffusion equation; they are introduced here to facilitate comparing the results of the purely convective dispersion solution with those of the full convective diffusion equation. Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
249
The velocity profile for the fully developed laminar tube flow of power-law fluids is given by
u = UO[l- y ( n + l ) / n ]
(3)
Note here that the flow behavior index n is defined as in the form of the power-law constitutive equation given by Bird et al. (1960). It then follows that for T < X , the mean concentration 8, defined by
(X
8, = 1 - [l - ( X
< -l/2Xs)
(10)
(54
Upon substituting eq 9 and 10 into eq 7 and equating the coefficients of 6k8,/6XIh we obtain an infinite coupled set of linear, partial, nonhomogeneous differential equations to be solved for the f h ’ s givm by
+ 1/2Xs)/7]2n/(n+l) < X < T - ‘/2Xs) (T - ‘/2X, < X < ‘/2X,) (-Y2Xs
8, = 1
8, = [1-
where fO(T,y) = 1,and assume that the process of distributing 8, is diffusive in nature a t all times. Hence
(4)
is given for purely convective dispersion by 8, = 0
Following the method of Gill and Sankarasubramanian (1970) we formulate the solution as a series expansion such that
(5b) (5c)
(x- 1/2Xs)/T]2n’(n+1) < %X,+ T ) (l/2XS+ T < X < a) ( % X S< X
8, = 0
(5d)
+ [Pe+
(5e)
- K2(T)] (12)
For T > X , the axial distribution of the mean cbncentration in purely convective dispersion is given by
(X< -1/.X,)
8, = 0 8, = (1- [l - ( X
(64
+ 1/*Xs)/T]2n/(n+l)J (-l/2Xs
W
S
)
=0
By multiplying eq 11 by y and integrating from y = 0 to y = 1 we obtain K 1 = 0. A similar treatment of eq 12 and 13 yields
K > ( T =) Pec2 K K + ~ ( T= )2
J1
+2
s,’
y‘2n+1)’nfh+l
~ ( 2 n + l ) / ~ dy fl
dy
Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
(k = 1, 2 , . . .) (16)
In order to obtain the unsteady solution for K z ( r ) it is necessary to obtain the complete solution of eq 11 for f l . This solution can be expressed in the form f l = F , ( y ) F,(T,Y) where F,(y) is the steady-state solution of eq 11 which independently satisfies the conditions given by eq 14b and 14c. The resulting solution for F , ( T J ) , which must vanish as 7 a,involves a homogeneous differential equation with homogeneous boundary conditions and hence constitutes an eigenvalue problem which can readily be solved by the method of separation of variables. The complete solution for f l is then given by
+
-
(84 (8b) (8c)
3.”) n - - -y(:in+l)/n 2 3n+1
+ 8n2 - (3n + 1)(5n +.l) 4(3n 1)(5n 1)
+
+ m=2 1 A,Jo(X,y) 250
(15)
+
I
exp(-Xm2T) (17)
where the Am's are given by
and the eigenvalues Am must satisfy
T h e definite integral in eq 18 is given by
K
Figure 1. Dispersion coefficient as a function of dimensionless time for power-law fluids.
Note that when (1 - p ) / 2 is zero or a negative integer the r function becomes infinite. This occurs for values of the flow behavior index n given by 1, ll?,. . . 1/(2r - 3) for r 2 2 and a positive integer. In these cases the summation in eq 20 has nonzero contributions only when [(l- p)/2 i] is also zero or a negative integer. Hence only terms up t o i = ( p - 1)/2 contribute and eq 20 simplifies t o
+
Equation 19 also has been used in arriving at the above simplified form. Equation 17 can be substituted into eq 15 t o yield the general solution for K ~ ( T )
K ~ ( T=)Pe-2
It will be seen in the next section that for all values of n , I is significantly smaller than KZ(m).Thus it appears ) all values of T; hence reasonable to assume IKx(i)1
