Literature Cited Box, G. E. P., Hill, W. J., Technometrics 9,57 (1967). Froment, F. G., Mezaki, R., Chem. Eng. Sci. 25, 293 (1970). Gerheim, C. C., Ph.D. Thesis, Purdue University, 1966. Hill, W. J., Hunter, W. G . , Wichern, D. W., Technometrics I O , 145 (1968). Hougen,'O. A.. Watson, K. M., "Chemical Process Principles Part 1 1 1 , " p 906. Wiley, New York, N. Y.. 1943. Kahney, R. H., Ph.D. Thesis, Purdue University, 1969.
Kittrell, J. R.. Advan. Chem. Eng. 8, 97 (1970). Mezaki, R., Happel, J., Catal. Rev. 3, 241 (1969). Sane, P. P., Ph.D. Thesis, Purdue University, 1972. Sane, P. P., Eckert, R. E., Woods, J. M., Chem. Eng. Sci. 27, 1611 (1972).
Received for reczew February 5 , 1973 Accepted October 18, 1973
Axial Dispersion in the Turbulent Flow of Power-Law Fluids in Straight Tubes William B. Krantz" Department of Chemical Engineering, University of Colorado. Boulder, Cola 80302
Darsh T. Wasan Department of Chemical Engineering, Illinois Institute of Technology, Chicago, Ill. 60616
The theory of Taylor for axial dispersion in turbulent flow in straight tubes is extended to power-law fluids. The effects of molecular and turbulent Schmidt number and soiute holdup in the wall region are incorporated in the analysis. The results agree well with axial dispersion data for the special case of Newtonian liquids. The predictions indicate that for Reynolds numbers larger than approximately 6000, pseudoplastic fluids will exhibit reduced axial dispersion, whereas dilatant fluids will exhibit enhanced axial dispersion relative to Newtonian fluids at the same flow conditions; for Reynolds numbers extending from the critical Reynolds number to 6000 the reverse trend is predicted. Presumably at larger Reynolds numbers the effect of the more blunt velocity profiles associated with decreasing flow behavior index predominates that of the thicker viscous sublayer, whereas at smaller turbulent Reynolds numbers the converse is true.
If a pulse of solute is injected into a fluid flowing in a straight tube it does not remain concentrated as a pulse but is axially dispersed relative to a reference frame moving at the average velocity. This dispersion is due to both axial diffusion and the velocity of the fluid particles relative to the average velocity. The former contribution is small compared to the latter for most fluids under a wide range of flow conditions. As the velocity profile approaches that corresponding to plug flow, the dispersion is reduced due to the decreased relative motion of the fluid particles. Radial diffusion also diminishes the axial dispersion since it reduces the radial concentration gradient. Interest in axial dispersion was generated by the need to measure the flow rates in long pipelines and to ascertain the degree of interdispersion when two liquids are transported successively in a pipeline. Flow rates in large water mains or in petroleum products pipelines can be determined by measuring the transit time of a tracer injected into the flow. Axial dispersion of the tracer yields an output concentration distributed in time; thus it is necessary to identify that point on the concentration-time curve which corresponds to the average velocity. Different liquids, for example regular and premium gasoline, are oftentimes transported in long pipelines successively without any separation. Intermixing due to axial dispersion produces a slug of liquid which must be purified or discarded in order to maintain product quality; it is of interest to predict the extent of the contaminated slug of material. Axial dispersion is also of importance in the design of tubular reactors since it decreases the driving force. The 56
Ind. Eng. Chem., Fundam., Vol. 13, No. 1, 1974
recent work of Gill and Sankarasubramanian (1971, 1972) on the dispersion of nonuniformly distributed time-variable sources in time-dependent laminar flow in straight tubes should permit many new applications of axial dispersion theory, a few of which are indicated by these authors. Axial dispersion in both laminar and turbulent flow of Newtonian fluids has been extensively studied. Sir Geoffrey Taylor (1953, 1954a) and later Aris (1956) developed the theory for axial dispersion in Newtonian fluids for laminar flow in straight tubes. Axial dispersion in turbulent flow was first analyzed by Taylor (1954b); however, his results are restricted to Reynolds numbers greater than 20,000 and to fluids having a Schmidt number of unity. Improved forms of the mean velocity profiles for both the turbulent core and the wall region which have been developed in recent years have allowed Taylor's analysis to be extended to the full range of turbulent Reynolds numbers. Tichacek, et al. (1957), extended Taylor's results to include large Schmidt numbers and used an improved form of the mean velocity profile based on averaging the velocity profiles of many investigators. Nonetheless their results may be inaccurate a t smaller turbulent Reynolds numbers since experimental velocity profiles are not available for the viscous sublayer in the region very close to the wall. Although this region is small in extent, it contributes greatly to the axial dispersion; thick viscous sublayers can hold up a considerable amount of solute, thus greatly increasing the axial dispersion. Flint and Eisenklam (1969)
used a modified form of the universal mean velocity profiles of Reichardt (1951) and obtained results in substantial agreement with Tichacek, et al., except for Reynolds numbers less than 8000. This is to be expected since the value obtained for the axial dispersion coefficient is particularly sensitive to the form of the mean velocity profile in the wall region at these smaller turbulent Reynolds numbers. The transport of slurries and polymer solutions as well as the design of flow reactors for biological systems have prompted an interest in axial dispersion in non-Newtonian fluids. Relatively little work has been done until recently on axial dispersion in non-Newtonian fluids for either laminar or turbulent flow. Axial dispersion of a pulse of solute injected in steady laminar flow in straight tubes has been analyzed by Fan, et al. (1965, 1966), for power-law, Ellis, and Bingham plastic fluids. Their results indicate that the axial dispersion coefficient decreases as the non-Newtonian nature of the fluid causes the velocity profile to become progressively more blunt. Harlacher and Engel (1970) considered the steady-state concentration distribution resulting from a step change in solute concentration at some point along the axis in a straight tube for the case of steady laminar flow of powerlaw fluids. An error in their results was corrected by Gill and Sankarasubramanian (1972). Recently, Wasan and Dayan (1970) presented an analysis for axial dispersion in the turbulent flow of power-law fluids; however, their analysis yielded contradictory results. They predict that the axial dispersion coefficient D* increases with increasing Reynolds number for all values of the flow behavior index R. Since Newtonian fluids (R = 1) are a special case of power-law fluids, their results contradict the theories of Taylor (1954b), Tichacek, et al. (1957), and Flint and Eisenklam (1969). Wasan and Dayan also predict that D* increases as the flow behavior index n decreases for all values of the Reynolds number. This is a surprising result since the mean velocity profile approaches that of plug flow as n decreases; these more blunt velocity profiles should be associated with reduced axial dispersion. It is possible that the thicker viscous sublayers associated with smaller values of the flow behavior index could give rise to enhanced axial dispersion; however, this effect should be insignificant at large Reynolds numbers. These anomalous results may be due to applying the universal velocity profile proposed by Bogue and Metzner (1963) in the wall region where it is not valid. An interesting application of turbulent axial dispersion studies in non-Newtonian fluids was proposed by Kenny and Thwaites (1971), who suggested that such studies could be used to provide additional information on the mechanism by which certain polymer solutions reduce drag in turbulent flow. Since the axial dispersion coefficient is heavily weighted by the mean velocity distribution in the viscous sublayer, such measurements could possibly be used to gain information on the wall region without disturbing the flow. Kenny and Thwaites used a semiempirical modification of the approach of Tichacek, et al.. and obtained reasonable agreement with measured axial dispersion coefficients for drag-reducing polymer solutions. They found that the viscous sublayer was considerably thicker than for Newtonian fluids, resulting in slightly larger values for the axial dispersion coefficient. Their approach suffers from the fact that the rheological parameters characterizing the non-Newtonian fluid do not enter into the determination of the axial dispersion coefficient, which precludes extension of their predictions to fluids for which no data are available.
A new expression for the turbulent axial dispersion coefficient for power-law fluids is developed in this paper. The universal mean velocity profile developed by Bogue and Metzner is used in the turbulent core and the mean velocity and eddy diffusivity profiles developed by Krantz and Wasan (1971) are used in the wall region. The latter profiles are consistent with the equations of motion in the wall region and provide a smooth and continuous transition to the velocity profile of Bogue and Metzner in the turbulent core. The resulting expression for the turbulent axial dispersion coefficient includes Newtonian fluids as a special case.
Theoretical Development The. analysis presented here follows that of Taylor (1954b). His development assumes that the radial concentration gradient is small, that the local axial concentration gradient in the convected coordinate system is constant, and that axial diffusion can be neglected relative to axial convection. An .alternate solution to the axial dispersion problem has been developed by Gill (1967) and Gill and Sankarasubramanian (1970). Their method of solution obviates the assumptions of Taylor’s analysis and allows for time-dependent dispersion coefficients as well. Taylor’s analysis yields the same results as does the more general solution of Gill and Sankarasubramanian for moderately large Peclet numbers and for all but the smallest values of time. Under the assumptions of Taylor’s analysis, axial dispersion is due to convection of solute relative to the bulkaverage velocity V. Radial diffusion will affect the axial dispersion insofar as it determines the radial concentration gradient; the latter can be determined from the timeaverage convective diffusion equation F i t t e n in a coordinate system moving a t the bulk average velocity
where c is the concentration in arbitrary units, r is the radial coordinate, x is the axial coordinate in a coordinate system moving at the bulk-average velocity, t is the time coordinate, U is the mean or time-average velocity, and cy = D td is the total diffusivity in the radial direction where D is the molecular diffusion coefficient and t d is the radial eddy diffusivity for mass transfer. If the characteristic radial diffusion time is much smaller than the characteristic longitudinal convection time, the radial. concentration variation will be small and the axial concentration gradient in the convected cmrdinate system will be nearly constant; it then follows that & / a t = 0. We will seek a solution to eq 1 subject to these conditions and will assume that the resulting relationship between the mass transfer rate relative to the bulk-average velocity and the axial concentration gradient will also apply to the case where & / a x is not precisely constant. These approximations have been justified in detail for axial dispersion in laminar flow by Taylor (1954a) and have been justified by Tichacek, et al. (1957), for the turbulent flow of Newtonian fluids. If &/ax is constant, the solution to eq 1 must be of the form c = cx C r (2) where c x is a linear function of x, and cr is a function of r. The complete solution is then given by
+
+
where co is the concentration at the axis of the tube, and Ind. Eng. Chem.,
Fundam., Vol. 13,No.
1, 1974
57
the respective superscript primes denote dummy integration variables. We seek the net transfer of solute across a plane moving a t the bulk-average velocity, defined by Q and given by
Q2n
elR [l' r( U - V )
{ irt r"(U - V)dr" )dr']
&i
Application to Power-Law Fluids The constitutive equation for power-law fluids is given by 7
dr
where R is the tube radius. Since we are assuming that the radial concentration variations are small, it is permissible to replace c by cm, the bulk-average concentration along a plane moving a t the bulk-average velocity. If we consider a material balance in this convected coordinate system for the case where &/ax is not a constant, we obtain
=
- {mlJS)[-'}
A
(12)
where 7 is the symmetrical shear stress tensor, A is the symmetrical deformation tensor, m is the consistency index, and n is the flow behavior index. The universal mean velocity profile for power-law fluids, valid in the turbulent core, was developed by Bogue and Metzner (1963)
where (14) Substituting eq 4 into eq 5 we obtain the familiar form of Fick's second law of diffusion if we identify the apparent axial dispersion coefficient D, as
D,
=
f is the Fanning friction factor and Re is the generalized non-Newtonian Reynolds number defined by Dodge and Metzner (1959) as Re E 2RV/v where the power-law kinematic viscosity is given by v = [(3n 1)/4nIn(4V/ R)n-lm/p. The functions C(z,f) and Z(n,Re) are defined by
+
z = r / R ;U '
=
U / U * ; a += a / R U *
(7)
where U, = (7w/p)1/2 is the friction velocity, 7\*. is the wall shear stress, and p is the density; the dimensionless axial dispersion coefficient defined by Dc+ = D c / R U , is then given by
{ i2' z"(U'
dz'] dz
(8)
The axial dispersion coefficient D,+ includes only the convective contribution; axial eddy and molecular diffusion also contribute to the axial dispersion. An estimate of this contribution to the axial dispersion coefficient is given by
The total axial dispersion coefficient is then given by Dt' = D,' -I- Dd' (10) An alternate definition of the axial dispersion coefficient frequently appearing in the literature is given by
K
= D,+/2V+
(11)
where V+is the dimensionless bulk-average velocity. Equations 8-11 are generally applicable to both Newtonian and non-Newtonian fluids. The rheological behavior of the fluid is introduced through the equations chosen for the mean velocity ( U + )and diffusivity (a+)distributions. The resultant value for the axial dispersion coefficient is particularly sensitive to the form of these distributions in the wall region. This follows from the fact that the viscous sublayer in the wall region can hold up a considerable amount of solute. Prior investigators have used mean velocity and eddy diffusivity distributions which do not satisfy the equations of motion in the wall region. 58
Z(n,Re)
-2.42 l,n { ( F ) x [Re(l/2)1"n'/'R"';':")
Ind. Eng. Chem., Fundam.,Vol. 13, No. 1, 1974
+ 3.63 i- 0.984
(16)
The parameter zc defines a dimensionless measure of the turbulent core thickness and is given by the equation developed by Krantz and Wasan (1971) fc
- V'idz''}
e x d 4 0 . 2 - ~ ) ~ / 0 . 1 5 ] (15)
c(z,f ) E 0.05
It is convenient to nondimensionalize the above using the following dimensionless variables
= l-n*y,
+
(17)
where yc+ is determined from the equation
1.09 = 2.42 In yc+
+ I(n,Re)
(18)
and is the value of the independent variable y + = lyU, (2-n)'n]/(rn/p)1lntwhere y is the dimensional distance from the wall, which defines the extent of the wall region wherein molecular transport of vorticity or mass is significant. The universal mean velocity profile given by eq 13 is valid for all Reynolds numbers larger than the critical Reynolds number for transition to turbulence. It has been verified for flow behavior indices in the range 0.4-1.6, although in theory it should be valid for all physically realizable flow behavior indices. The total mass diffusivity in the turbulent core is given by
where Sct is the turbulent Schmidt number. This equation is derived from the equation for the eddy diffusivity for vorticity transfer. The latter neglects the molecular diffusion of vorticity in the turbulent core. This assumption is imperative since as Krantz and Wasan (1971) have pointed out, there is no theoretically justifiable form for the molecular diffusion of vorticity in the turbulent core
available for power-law fluids. Since we have ignored the molecular transport of vorticity in the turbulent core we will also ignore the molecular transport of mass in this region. This restricts the results to reasonably large Peclet numbers. The corresponding mean velocity and diffusivity distributions for power-fluids in the wall region have been developed by Krantz and Wasan (1971)
I
A D
-
1 1 1 1 1 1
A L L E N 8 TAYLOR FOWLER 8 BROWN
SMITH 8 S C H U L Z E TAYLOR KRANTZ 8 WASAN
104
Re
Figure 1. Axial dispersion coefficient as a function of Reynolds number and Schmidt number for Newtonian fluids
where Uq+ and I-Jb+ are universal functions defined by
and
I
0
lo3
dY+
I
A
a, the Reynolds stress, is given by +
(2n3- 9n2 10n 3, 24
[5Uj+- ( 4 n - l ) U , + --
$41
n(y+)4
(24) Equations 20-24 are the only equations proposed for power-law fluids in the wall region. They include Newtonian fluids, for which n = 1, as a special case. They are in improvement over other equations proposed for Newtonian fluids in the wall region in that they satisfy the mean equations of motion and boundary conditions in the wall region and provide a smooth and continuous transition to the universal mean velocity profile valid in the turbulent core. Equations 8 and 9 were integrated numerically using Simpson’s method, employing eq 13 and 19 in the turbulent core and eq 20 and 21 in the wall region. This incorporated an adjustable step size in an iterative technique which converged on the correct values for D,+ and Dd+ such that subsequent iteration resulted in less than 1% change in the calculated values.
Axial Disperison in Newtonian Liquids The predictions of the theory developed here for the dispersion coefficient K defined by eq 11 are compared with axial dispersion for Newtonian liquids in Figure 1. These include both laboratory data and field data on long pipelines. The laboratory data include those of Allen and Taylor (1923), Fowler and Brown (1943), Kenny, and Thwaites (1971), Kohl and Newacheck (1953), and Taylor (1954b). The field data include those of Hull and Kent (1952) and Smith and Schulze (1948). The data of Fowler and Brown and Smith and Schulze only include those data considered by Tichacek, et al. (1957), in their analysis. The latter investigators discarded a considerable number of data which involved fluids of substantially different viscosity, nonfully developed turbulent flow, and coils or bends in the pipe. Since the Schmidt numbers were not reported for these data the theory developed here is presented in Figure 1 for Schmidt numbers of 10, 100, and lo00 corresponding to a turbulent Schmidt number of unity. The theory of Taylor valid for Sc = 1 is also shown in Figure 1 in order to illustrate the pronounced effect of in-
creasing Schmidt number, which is seen to correspond to a marked increase in the dispersion coefficient for Reynolds numbers less than 100,OOO. This is reasonable since larger Schmidt numbers correspond to a decrease in the molecular transfer of mass relative to the transfer of vorticity; reduced molecular diffusion sustains sharper radial concentration gradients and hence enhances axial dispersion. At Reynolds numbers greater than 100,000 the magnitude of the Schmidt number has little effect on the axial dispersion coefficient. This is consistent with the fact that the mean velocity profile approaches that of plug flow at these larger Reynolds numbers, thus giving rise to relatively little axial dispersion. The theory developed here and that of Taylor appear to approach different asymptotic limits as the Reynolds number becomes very large; this behavior is due to the different forms assumed for the mean velocity and diffusivity distributions in the wall region. This illustrates very clearly the sensitivity of these calculations to the forms assumed for these distributions. The theory developed by Tichacek, et al. (1957), incorporating smoothed experimental velocity profiles is also shown in Figure 1. Although Tichacek, et al., included the Schmidt number in their analysis, they found little effect of increasing Schmidt number for Sc > 100; thus they only report their calculations for Sc = 1 and Sc = 100, the latter of which is shown in Figure 1. The Schmidt number behavior reported by these investigators is in sharp contrast to that displayed by the theory developed here which shows a pronounced increase in the axial dispersion coefficient when the Schmidt number is increased from 100 to 1OOO. It is of interest to note that the theory of Tichacek and coworkers appears to predict nearly the same values for the axial dispersion coefficient a t large Reynolds numbers as does the theory developed here. This is confirmation of the fact that the mean velocity distribution given by eq 13 accurately conforms to the data for the mean velocity profile employed by Tichacek and coworkers in their development. The theory developed here appears to predict the axial dispersion coefficient as well as the theory of Tichacek, et al. A detailed comparison is not possible owing to the scatter in the data; this is in part due to not knowing the Schmidt numbers for the data, although they certainly fall in the range 10 < Sc < 1OOO. A more probable source of scatter, particularly a t the higher Reynolds numbers, is the unknown roughness factors for the tubes used in the various studies. Roughness increases the radial eddy diffusion of vorticity making the mean velocity profile less blunt; however, it also increases the radial eddy diffusion Ind. Eng. Chem., Fundam., Vol. 13, No. 1, 1974 59
of mass. The former effect enhances axial dispersion whereas the latter reduces it. The higher Reynolds number data in Figure 1 suggest that the enhancement mechanism predominates. Flint and Eisenklam (1969) employed a modified form of the universal mean velocity profiles of Reichardt (1951) and also accounted for the Schmidt number dependence. Their results are in substantial agreement with those of Tichacek, et al., and those developed here except for Re < 8000 at which point their theory predicts considerably smaller values for the axial dispersion coefficient although their predictions still fall within the scatter of the data. This may be due to the inapplicability of the mean velocity profiles of Reichardt at these small turbulent Reynolds numbers. At large Reynolds numbers (Re = lo6) they report a value of D+ = 7.3 for Sc = 1 which is close to the value 7.8 predicted by the theory developed here. The effect of turbulent Schmidt number has not been considered in prior analyses. Several investigations indicate that the local turbulent Schmidt number for flow in conduits varies from 0.6 to 0.95; hence the theory developed here is shown in Figure 1 corresponding to Sct = 0.7 and Sc = 100. This reduced value of Sct correspqnds to a reduction in the axial dispersion coefficient relative to that observed for Sct = 1 and Sc = 100. This is reasonable since the decrease in eddy transport of vorticity relative to that of mass reduces the radial concentration profile as does a decrease in molecular Schmidt number. As opposed to a decrease in Sc, a decrease in Sct has a significant effect even a t large Reynolds numbers. This follows from the fact that eddy transport is significant throughout the flow for all Reynolds numbers, whereas molecular transport is only significant (under the assumptions of this analysis) in the wall region; the latter becomes negligibly thin a t large Reynolds numbers. The line corresponding to Sct = 0.7 does not agree as well with the data a t large Reynolds numbers as does the similar curve for Sct = 1. One might infer from this that the best value for Sct is unity since values larger than this are not physically significant. However, such an assumption is unwarranted and may be considerably in error a t smaller Reynolds numbers, The local turbulent Schmidt number assumes a value near unity in the turbulent core and only becomes small in the region very near the wall. Thus a value of unity works relatively well a t large Reynolds numbers for which the viscous sublayer is very thin. Lower values of Sct may be more appropriate at smaller Reynolds numbers for which the viscous sublayer is thicker; in particular, values of Sct less than unity may be applicable to pseudoplastic, power-law fluids for which the viscous sublayer is considerably thicker at smaller turbulent Reynolds numbers.
Axial Dispersion in Power-Law Fluids The preceding discussion indicates that for Newtonian fluids, which are a special case of the theory developed here, the predictions conform to the available data for axial dispersion as well as do those of the two other theories developed for fluids having large Schmidt numbers. Hopefully, the theory will predict axial dispersion in non-Newtonian, power-law fluids. The predictions of the theory developed here for the axial dispersion coefficient K are shown in Figure 2 for Schmidt numbers of 10, 100, and 1000, respectively, for flow behavior indices of 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, and Reynolds numbers extending from the critical Reynolds number to l,OOO,OOO. The critical Reynolds number Rec, corresponding to that at which transition flow begins, was determined using the equation of Hanks and Christiansen 60
Ind. Eng. Chem., Fundam., Vol. 13, No. 1 , 1974
I
I
\
I
I 11111
I
I
I
\
I I Ill1
,n 1.6 /, n = 1.4
I
1
I
I I l l
4I
sc = 10
i
I
//,"= 1.2
\
\
\\\
.\
sc
A\\ \
100
sc = 1000
1
'//,n = 0.8
t
lo-ll
103
I
I
I
1 I
IIII
I
I 1 4 1 1 1 1
105
104
I
I
I
I
I I I I I
IO6
Ro
Figure 2. Axial dispersion coefficient as a function of generalized Reynolds number and flow behavior index for power-law fluids (1961)
Since there are no measurements of the turbulent Schmidt number in power-law fluids, Sct was assumed to be unity for all the curves in Figure 2. For fixed Schmidt number and Reynolds numbers greater than approximately 6oO0, the axial dispersion coefficient is seen to decrease markedly with decreasing flow behavior index. Although it is not shown in these figures, the axial dispersion coefficient Dt+ also decreases with decreasing flow behavior index. This is to be contrasted with the results of Wasan and Dayan (1970) which indicate that K decreases slightly as n decreases and that Dt+ increases as n decreases. The behavior predicted by the theory developed here is reasonable in view of the fact that the mean velocity profiles become progressively more blunt as the flow behavior index decreases. This, however, is not the only effect associated with decreasing the flow behavior index. Krantz and Wasan (1971) have shown that the viscous sublayer thickness increases as the flow behavior index decreases; this would correspond to increased axial dispersion owing to the increased holdup of
material in the wall region, thus counteracting the decreased axial dispersion associated with the more blunt mean velocity profiles. Figure 1 suggests that the effect of the mean velocity profile predominates that of the thicker viscous sublayer for Reynolds numbers larger than 6OOO. Further evidence in support of this interpretation can be obtained by considering a measure of the viscous sublayer thickness y c + . At Re = 2500, yc+ = 37.4 for n = 0.4, and yc+ = 20.9 for n = 1; however a t Re = 100,OOO, yc+ = 19.5 for all values of the flow behavior index considered here; hence the viscous sublayer thickness cannot account for the differences in the axial dispersion coefficient observed for larger Reynolds numbers. For this reason the trend in the Dt+-n relationship observed by Wasan and Dayan cannot be explained by a viscous sublayer effect; this effect was not incorporated in their analysis since they applied the universal velocity profile of Bogue and Metzner across the entire flow including the wall region. An increase in molecular Schmidt number corresponds to a n increase in K for all values of n. The effect of Schmidt number becomes less pronounced as the flow behavior index decreases. This behavior is reasonable since the decrease in molecular diffusion of solute relative to vorticity associated with larger Schmidt numbers tends to sustain sharper radial concentration gradients which enhance axial dispersion. This effect becomes less significant as the flow behavior index is decreased since the velocity profiles become progressively more blunt. A decrease in the turbulent Schmidt number holding all other parameters constant would cause a decrease in K , although this effect is not shown in Figure 2. The predictions of the theory developed here must be interpreted with caution in the case of pseudoplastic fluids (n < 1).The increased holdup in the thicker viscous sublayer region, which was observed to be significant a t lower turbulent Reynolds numbers, may invalidate one of the assumptions in the theory. The analysis of Taylor followed here implies that the average solute concentration will be distributed normally in time a t a fixed observation point provided the criterion of Levenspiel and Smith (1957) is satisfied, that is L/R > 20K where L is the distance between the injection and observation points. Excessive holdup of solute in the thicker viscous sublayers associated with pseudoplastic fluids may give rise to an asymmetrical axial concentration distribution having a long “tail.” Such cases may be handled using time-dependent dispersion coefficients following the method of solution developed by Gill and Sankarasubramanian (1970). Kenny and Thwaites were the only investigators to report axial dispersion data for non-Newtonian fluids; however, these investigators did not study power-law fluids. Hence a detailed check of the predictions of the theory developed here is not possible, although a partial confirmation of the theory is provided in that it agrees well with the data for the special case of Newtonian fluids. The prediction that pseudoplastic fluids will exhibit reduced axial dispersion relative to Newtonian fluids at the same flow conditions for Re > 6000 may suggest some interesting applications. Consider, for example, inserting a plug of pseudoplastic fluid between two fluids transported successively in a long pipeline in order to reduce interdispersion of the two fluids. However, before the theoretical predictions can be applied with confidence it will be necessary to check them in detail with axial dispersion data for power-law fluids. Acknowledgment The authors gratefully acknowledge the helpful comments of Professor William N. Gill (State University of New York a t Buffalo).
Nomenclature
c = local solute concentration cm = mean solute concentration co = solute concentration at r = 0 cr = functionofr c x = linear function of x C(z,f) = function defined by eq 15 D = molecular diffusion coefficient D, = convective contribution to axial dispersion coefficient Dd = diffusive contribution to axial dispersion coefficient Dt = axial dispersion coefficient f = Fanning friction factor I(n,Re) = function defined by eq 16’ K = dimensionless axial dispersion coefficient defined by eq 11 L = length between injection and measurement points m = consistency index n = flow behavior index Q = net transport of solute past reference plane with velocity V r = radial coordinate R = radius of tube Re = generalized Reynolds number (2R)V/Y Re, = critical Reynolds number defined by eq 25 Sc = Schmidt number u / D Sct = turbulent Schmidt number € / t d t = time coordinate u = fluctuating component of axial velocity U , = friction velocity ( ~ ~ / p ) ’ / ~ U = time-averaged local axial velocity U, = function defined by eq 22 Us = function defined by eq 23 u = fluctuating component of radial velocity V = bulk average velocity x = axial coordinate in coordinate system convected a t velocity V y = radial coordinate measured from wall y c = viscous sublayer thickness z = dimensionless radial coordinate zc = dimensionless turbulent core thickness Greek Letters (Y = total diffusivity (D + t d ) A = rate of strain tensor e = eddy diffusivity for vorticity Ed = eddy diffusivity for mass u = kinematic viscosity [(3n 1)/4n]G(8V/2R)”-1rn/p p = density T = viscous stress tensor T~ = shear stress at the wall IC/ = functiov defined by eq 14 Superscripts + = denotes dimensionless variable
+
Literature Cited Allen, C. M., Taylor, E. A.. Amer. SOC.Mech. Eng. 45, 285 (1923). Aris, R., Roc. Roy. SOC.,Ser. A 235,67 (1956). Bogue, D. C., Metzner, A . B., lnd. Eng. Chern., Fundarn. 2,143 (1963). Dodge, D . W., Metzner, A . B., A.I.Ch.E. J. 5 , 189 (1959) Fan, L. T. Hwang, W. S., Roc. Roy. SOC., Ser. A 283,576 (1965). Fan, L. T., Wang, C. E., Proc. Roy. SOC.,Ser. A 292, 203 (1966). Flint, L. F., Eisenklam, P., Can. J. Chem. Eng. 47, 101 (1969). Fowler, F. C., Brown, G. G., Trans. Amer. Inst. Chem. Eng. 39, 491 (1943). Gill, W . N., Proc. Roy. Soc., Ser. A 298,335 (1967). Gill, W. N., Sankarasubramanian, R., Proc. Roy. SOC.,Ser. A 316, 341 (1 970) Gill, W. N., Sankarasubramanian, R., Proc. Roy. SOC.,Ser. A 3 2 2 , 101 (1971). Gill, W. N., Sankarasubrarnanian, R., Proc. Roy. SOC.,Ser. A 327, 191 (1972). Hanks, R. W.,Christiansen, E. B . , A . l . C h . E .J. 7,519 (1961). Harlacher, E. A., Engel., A. J., Chem. Eng. Sci. 2 5 , 717 (1970). Hull, E. E., Kent, J. W., Ind. Eng. Chern. 44, 2745 (1952). Kenny, C. N., Thwaites, G. R., Chem. Eng. Sci.26,503 (1971) Kohl, J., Newacheck, R. L., paper presented a t the fall meeting of the American Society of Mechanical Engineers, Petroleum Division, 1953. Krantz, W. 6.; Wasan, D . T., lnd. Eng. Chem., fundam. 10,424 (1971). Smith, W. K., Chem. Eng. Sei. 6 ,227 (1957). Levenspiel, O.,
Ind. Eng. Chem., Fundam., Vol. 13, No. 1, 1 9 7 4
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Reichardt, H., Angew. Math. 31, 208 (1951). Smith, S.S., Schulze, R. K.,Petrol. Eng. 330 (1948). Ser. A 219,186 (1953). Taylor, G. I., Proc. Roy. SOC., Ser. A 223,446 (1954a). Taylor, G. I., Proc. Roy. SOC., Taylor, G. I., Proc. Roy. SOC., Ser. A 225,473 (1954b).
Tichacek, L. J., Barkelew, C . H., BaronT., A . / . C h . f . J. 3,439 (1957). Wasan, D. T., Dayan, J., Can. J. Chem. Eng. 48,129 (1970).
Received for review February 23, 1973 Accepted October 1,1973
On the Prediction of Non-Newtonian Flow Behavior in Ducts of Noncircular Cross Section Richard W. Hanks Department of Chemical Engineering, grigham Young University, Provo, Utah 84602
The analytical basis of Miller’s (1972) method for estimating pressure loss-flow rate values for noncircular ducts from pipeline data is examined for the power-law and Bingham plastic models. Several restrictions to the original method are observed and its limits of applicability are defined. In particular it is shown that one must know the laminar-nonlaminar transition conditions for Newtonian flow in the noncircular duct in order to use the method safely to estimate non-Newtonian flow behavior. Fluids having, yield stresses create special problems which render use of the method somewhat uncertain. Similar problems may arise for highly viscoelastic fluids which exhibit anomalous transition behavior in pipes. Additional methods are proposed to supplement Miller’s basic technique and render it more useful and reliable for engineering design purposes.
In a recent paper published in this journal, Miller (1972) proposed an interesting technique for estimating the pressure drop-flow rate behavior of non-Newtonian fluids in ducts of unusual cross section. The importance of such a technique to the design engineer who must deal with such fluids in the design of a multitude of types of industrial process equipment is obvious. Because of the simplicity of Miller’s method, the engineer is spared the often difficult task (Hanks, 1963a,b; Kozicki and Tiu, 1967; Kozicki, et al., 1966; Mitsuishi, et al., 1968; Mizsushima, et al., 1965; Schechter, 1961; Wheeler and Wissler, 1965) of solving the complex equations describing the motion. This is a highly desirable simplification. However, there is inherent in Miller’s simplification an equally important limitation which is not obvious from Miller’s paper. The purpose of the present paper is to point out this limitation and to suggest a possible solution to the problem raised thereby. Miller (1972) suggests that one should choose as his coordinates for representation of laminar flow pressure loss-flow rate data (1) the average wall shear stress, i defined as where - A p / L is the mean axial pressure gradient, Dh = 4A/P is the hydraulic diameter of the duct having cross sectional area A and wetted perimeter P, and (2) the average apparent wall shear rate, 7, defined as where (v) = Q / A , Q is the volumetric flow rate, and h is a geometric shape factor. If one uses such coordinates, Miller (1972) shows that data obtained in pipes, parallel plate ducts, triangular ducts, rectangular ducts, and concentric annuli can apparently be superimposed successfully upon a single curve. Thus, if one can measure the t vs. 9 curve for a pipe (this may easily be done using a simple pipeline 62
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Eng. Chem., Fundam., Vol. 13, No. 1, 1974
capillary viscometer), one may readily predict the similar quantities for apparently arbitrary geometry ducts. (He actually only proved this for circles, triangles, rectangles, and concentric annuli for certain types of fluids.) The key to Miller’s method is the geometry factor X used in defining T. He defined it in such a manner that X = fRe, where f = 27/p ( u ) is ~ the friction factor and Re = Dh ( u ) p/g is the Reynolds number for flow of a Newtonian fluid of density p and viscosity p through the duct in question. Miller (1972) presented a fairly comprehensive list of h factors gathered from a number of literature sources. Additional ones may be found in the paper by Sparrow and Haji-Shiekh (1966) or computed by the method given in their paper. An important but subtle limitation inherent in Miller’s method is found when one asks the question: over what range of 4 may this method be safely applied? We shall show below that this is not an easily answered question. In particular, we shall show that whereas the t-7 curves for various geometry ducts are relatively insensitive to the influence of certain types of geometry-rheology variations, the upper limits ( T C , Tc) of these curves are extremely sensitive to the interaction of geometric and rheological factors. We shall propose a solution to the problem.
Analytical Basis of Miller’s Method In his paper Miller argued by analogy that since nonNewtonian flow in a pipe could be represented as a unique function t p ( T ) , and similarly, non-Newtonian flow between parallel plates could be represented by a different function t p p ( ~one ) , might‘ expect in general that a unique relation t ( T ) might exist between t and T for ducts having arbitrary geometry. He observed that one may show ana= l for various non-Newtonian lytically that ?p(T)/?pp(T) models, although he did not present any analytical results to substantiate his observation. On the basis of this observation he then hypothesized that tp(T)/t(T) 1 for arbi-
