Ind. Eng. Chem. Res. 1999, 38, 851-854
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Dispersion in Flows with Streamwise Variations of Mean Velocity: Radial Flow† H. A. Stone* Division of Engineering & Applied Sciences, Harvard University, Cambridge, Massachusetts 02138
H. Brenner Department of Chemical Engineering, MIT, Cambridge, Massachusetts 02139
The familiar example of Taylor dispersion in pipe flow is extended to the case of laminar flows with velocity variations in the streamwise direction. The effective dispersivity is shown to depend on the streamwise coordinate. The generic scheme is illustrated by the simple example of radial source (sink) flows in two and three dimensions, and the distinctive results for the two cases are contrasted. 1. Introduction Taylor-Aris dispersion describes the cross-sectionally averaged axial redistribution of a dissolved solute occurring in laminar pipe-flow configurations for axial transport times that are long compared with crossstream diffusion times.1,2 Most analyses treat only those flows without velocity variations in the streamwise direction, i.e., unidirectional flows. In such cases, as Taylor first showed and Aris formally demonstrated, all initial pulse-like concentration distributions tend to a Gaussian profile whose downstream spread, above that due to purely molecular diffusion alone, increases proportionally to t1/2. Relative to the mean solvent flow the dispersive process is quantified by a positionindependent dispersion coefficient that is proportional to 〈u〉2a2/D, where 〈u〉 is the cross-stream average velocity, a the pipe radius, and D the coefficient of molecular diffusion. As such, there exists the feature that for flows with velocity gradients, such as the parabolic velocity variations characteristic of pipe and channel flows, those solutes possessing the smallest diffusivities manifest more highly dispersed concentration profiles than do those with larger molecular diffusivities. Numerous variants of this classical problem have been studied, many of which are summarized by Brenner and Edwards.3 There appear, however, to be few analyses of dispersion in flows for which the velocity varies in the streamwise direction, despite the common occurrence of such flow configurations. These include, for example, flows with diverging or converging streamlines. A discussion of some general features of this class of problems has been given recently by Bryden and Brenner.4 Here, we provide an example of Taylor-Aris dispersion arising during radial outflow (inflow) for cylindrically symmetric, pressure-driven flow between two parallel plates. The average concentration profile in such circumstances is characterized by a dispersion coefficient that varies radially. This has the consequence that the ultimate solute concentration distribution is both non-Gaussian and subdiffusive, in the sense that * To whom correspondence should be addressed. E-mail:
[email protected]. Fax: 617 495-9837. † Submitted in honor of Professor Roy Jackson.
the solute spreads relative to the local mean pulse position at a rate proportional to t1/4. A practical example of a radial outflow configuration occurs in microfabricated devices proposed for use as bioartificial livers.5 The remainder of this paper effects the dispersion calculation via two independent approachessone informal and the other formal. In section 2, the ad hoc approach of area averaging in conjunction with orderof-magnitude estimates in the spirit of Taylor’s original paper are used to establish the form of the dispersivity. The same final result is then formally derived using a paradigm based upon the methods of “generalized Taylor dispersion theory” (macrotransport processes). In the concluding remarks, dispersive phenomena occurring in two- and three-dimensional flows characterized by streamwise velocity variations are briefly contrasted. The specific radial flow exercise presented herein is chosen primarily to illustrate in as simple a context as possible how easily classical Taylor dispersion arguments, involving unidirectional flows (the mean velocities of which are independent of axial position), can be extended to encompass flow configurations characterized by axial variations in the mean velocity. In contrast to the more formal approach of Bryden and Brenner4 to this latter class of problems, we believe that the intuitive nature of the present scheme together with its obvious historical connection to Taylor’s original uniform velocity case will make the broader streamwise velocity variation ideas and conclusions (e.g., nonGaussian solute distributions) more accessible to a nonspecialist audience. 2. Dispersion Calculation via Area Averaging Consider steady, pressure-driven radial flow between two parallel plates spaced a distance 2h apart (see Figure 1). An axisymmetric cylindrical (r,z) coordinate system is utilized, with radial outflow occurring at a volumetric flow rate Q* (inflow is given by changing the sign of Q*). In the calculation reported in this section we use an “averaging” analysis, wherein it is convenient to work with the cross-sectional average 〈‚‚‚〉 ) (2h)-1 h ∫-h ‚‚‚ dz ≡ h-1 ∫h0 ‚‚‚ dz.
10.1021/ie980355f CCC: $18.00 © 1999 American Chemical Society Published on Web 02/13/1999
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As we are only interested in the long times characteristic of cross-stream diffusion being rapid relative to streamwise transport, i.e., ∆r . 〈u〉h2/D, and because u′ ) O(〈u〉) while c′ , 〈c〉, then eq 6 simplifies to
∂〈c〉 ∂2c′ ≈D 2 ∂r ∂z
u′
(7)
whose form was first argued by Taylor1 (his eq 19). Since u′ is known, the solution of this equation satisfying ∂c′/ ∂z ) 0 at z ) 0 is straightforward: Figure 1. Schematic of radial flow between two planar surfaces.
The low-Reynolds-number velocity profile for flow in this geometry is purely radial:
3〈u〉 z2 u(r,z) ) 12 h
[ ( )]
〈 〉
∂〈c〉 ∂〈c〉 ∂〈c〉 ∂c′ ∂c′ ∂c′ + + 〈u〉 + u′ + 〈u〉 + u′ ) ∂t ∂t ∂r ∂r ∂r ∂r D ∂ ∂〈c〉 r + D∇2c′ (3) r ∂r ∂r
( )
Cross-sectional averaging of the latter equation leads to the mean convective-diffusion equation:
( ) 〈 〉
∂〈c〉 ∂〈c〉 D ∂ ∂〈c〉 ∂c′ + 〈u〉 ) r - u′ ∂t ∂r r ∂r ∂r ∂r
(4)
The mechanical contribution to the effective diffusion of the solute arises from the “fluctuation”-generated flux 〈u′∂c′/∂r〉. These steps eqs 2-4 emphasize that the analysis is equivalent to that originally outlined by Taylor,1 with the small variant that the mechanical contribution to the flux now includes a variation in the streamwise direction, embodied in the detailed form of u′(r, z). In present circumstances
z2 Q* 1-3 8πhr h
( )]
(8)
(
) ( )
(9)
where contributions from c′(r, 0, t) do not contribute since 〈u′〉 ) 0. It therefore follows from eq 4 that the average concentration is governed by the one-dimensional convective-diffusion equation:
( )
∂〈c〉 Q* ∂〈c〉 D ∂ ∂〈c〉 r + + ) ∂t 4πhr ∂r r ∂r ∂r
(2)
In introductory discussions of Taylor dispersion it is conceptually useful to work in terms of averages and deviations therefrom, and so write u(r,z) ) 〈u〉(r) + u′(r,z) and c(r,z,t) ) 〈c〉(r,t) + c′(r,z,t). Substituting into eq 2 gives
]
Q*2 1 ∂ 1 ∂〈c〉 ∂c′ 2 )∂r 105 16π2D r ∂r r ∂r
u′
∂c ∂c + u(r,z) ) D∇2c ∂t ∂r
[
[
Q* z4 1 ∂〈c〉 z2 - 2 + c′(r,0,t) 16πhD 2h r ∂r
Thus, the mechanical contribution to the flux is
(1)
where 〈u〉(r) ) Q*/4πhr is the cross-sectionally averaged velocity. The significant difference here from usual investigations of Taylor-Aris dispersion lies in the fact that this flow possesses an average velocity that is a function of the streamwise (r) coordinate. The axisymmetric concentration field c(r,z,t) ≡ (2π)-1 π ∫-π c(r,φ,z,t) dφ of a dissolved solute evolves according to the convective-diffusion equation:
u′(r,z) )
c′(r,z,t) )
(
) ( )
2 Q*2 1 ∂ 1 ∂〈c〉 (10) 105 16π2D r ∂r r ∂r
The phenomenological coefficients appearing above (e.g., 2/105) are those that one might “guess” on the basis of knowledge of Taylor-Aris dispersion in unidirectional parabolic channel flow between parallel plates, albeit modified by the caveat that the mean velocity varies with r-1. We note that the second term on the right-hand side may be written
(
)
∂〈c〉 1 ∂ Drr r ∂r ∂r
Dr )
where
( )
2 Q* 2h2 105 4πhr D
(11)
which corresponds precisely to the familiar two-dimensional channel flow dispersion coefficient written in terms of the average velocity. In a more general context, Bryden and Brenner4 discuss classes of flow configurations characterized by streamwise mean velocity variations that also display comparable streamwise dependence of the dispersion coefficient (see sections 3 and 4 of the present paper). Provided r is not too large, so that the mean velocity is not too small, i.e., r , O(Q*/10 D), the spread of the concentration distribution is dominated by the second term on the right-hand side of eq 10. In such circumstances the average concentration profile may be obtained from the fact that
(5)
It remains to derive the equation governing c′. To do so, subtract eq 4 from eq 3 to obtain
〈 〉
∂〈c〉 ∂c′ ∂c′ ∂c′ ∂c′ + u′ + 〈u〉 + u′ ) D∇2c′ + u′ ∂t ∂r ∂r ∂r ∂r
(6)
(
〈c〉(r,t) ≡ C r2 -
Q*t , t 2πh
)
(12)
where C(s,t) satisfies an ordinary one-dimensional dif˜ has units fusion equation (s has units length2 and D length4/time):
Ind. Eng. Chem. Res., Vol. 38, No. 3, 1999 853
∂C ∂2 C )D ˜ 2 ∂t ∂s
with
( )
2 Q*2 105 4π2D
D ˜ )
∂2g z2 1 ) 1-3 2 2r h ∂z
[
and
s ) r2 -
Q*t (13) 2πh
For the outflow case it is clear from the above that for a pulse-like initial condition not only does the mean pulse position increase proportionally to t1/2 owing to the radial dependence of the average velocity, but there also occurs a spreading about the mean position that increases proportionally to t1/4. Similar conclusions can be deduced for inflow because the dispersion depends on Q*2, which is invariant to a change in the algebraic sign of Q*. In particular, for the case in which a pulse is introduced into this radial channel flow, with the total h ∫∞0 c(r,z,t)rdrdz, the mass injected given by A ) 2π∫-h long-time distribution is
〈c〉(r,t) )
{
)}
1 2 Q*t 2 A exp r 1/2 4D ˜t 2πh 2hπ (D ˜ t) 3/2
(
In this section we demonstrate that the ad hoc analysis of section 2, which combines a method of averaging with order-of-magnitude estimates, is in perfect agreement with more formal, general macrotransport multiple timescale analyses of the dispersion process (e.g., see refs 3 and 6). The detailed calculations given below follow the generic notation used by Bryden and Brenner4 (section 8.3) (hereafter denoted BB),who describe the microscale convective-diffusive transport process using general orthogonal curvilinear coordinates (q1, q2, q3). In particular, we choose circular cylindrical coordinates with q1 ) r, q2 ) φ, and q3 ) z, and recall that the three metrical coefficients characterizing this coordinate system are, respectively, h1 ) 1, h2 ) r-1, and h3 ) 1 (see ref 7, eq A-9.5). With regard to generalized Taylor dispersion terminology,3 we note that the “local” coordinates φ and z span the respective finite ranges 0 e φ < 2π and -h < z < h, whereas the “global” coordinate r spans the semi-infinite region 0 < r < ∞. Moreover, again in their generic notation, the curvilinear duct wall surface corresponds to the pair of planar surfaces F(z) ) const ) (h. Comparison with BB’s eq 8.5 yields (in their notation) a function proportional to velocity:
q(z) )
z2 3Q* 18πh h
[ ( )]
(15)
We also list here a few equivalences explicitly giving BB’s variables (Q, Q h, A h , χ, v) for the specific radial flow parallel-plate geometry under discussion:
Q ) Q*
Q h )
Q* 4πh
χ(r) ) 4πhr
A h (r) ) 4πhr v(z) )
z2 3 1(16) 2 h
[ ( )]
The convective contribution D h c to the dispersion process involves a function g(r, z) that satisfies the following equation (BB, eq 8.15):
(17)
(compare with eq 5) and the boundary conditions
∂g )0 ∂z
on
z ) (h
and
∫-hhg dz ) 0
(18)
The solution is readily found to be
g(r,z) )
h2 z 2 1 z 4 7 4r h 2h 30
[( )
()
]
(19)
(compare with eq 8). It is now straightforward to calculate the Taylor dispersivity D h c(r) from either BB eqs 8.13 or 8.18. (Note that there is a misprint in BB’s eqs 8.18, as the coefficient Q h 2/D is missing in front of the integral sign.) As a result of the requisite integration one obtains
(14)
3. Calculation via Generalized Taylor Dispersion Theory
( )]
D h c(r) )
hQ*2 210πDr
(20)
After substituting into BB’s eq 8.10 (see also their eq 8.19), the equation for the area-averaged “concentration”, 〈P〉(r,t|r′), or macroscale conditional probability density, is found to be
( ) ) ( )
∂〈P〉 Q* ∂〈P〉 D ∂ ∂〈P〉 + ) r + ∂t 4πhr ∂r r ∂r ∂r δ(r - r′)δ(t) Q*2 1 ∂ 1 ∂〈P〉 2 (21) + 2 105 16π D r ∂r r ∂r 4πhr
(
which agrees exactly with the form of eq 10. The last term on the right-hand side of eq 21 represents an impulsive unit source input introduced at time t ) 0 on the circular surface r′, so thatsas usual in generalized Taylor dispersion analysess〈P〉(r,t|r′) represents the area-averaged Green’s function or equivalently the areaaverage conditional probability density.8 As such, it provides the evolutionary connection between the current mean solute concentration 〈c〉(r,t) of section 2 and the initial mean solute concentration 〈c〉(r,0) at t ) 0:
〈c〉(r,t) ) 4πh
∞ 〈P〉(r,t|r′)〈c〉(r′,0)r′ dr′ ∫r′)0
(22)
It is also evident from the basic similarity of the intermediate results given in sections 2 and 3 that the area-averaging approach is equivalent to the leadingorder formal multiple timescale calculation based upon use of the probability distribution of a Brownian tracer to characterize the dispersive process. 4. Concluding Remarks The results reported here in conjunction with those of Bryden and Brenner4 demonstrate that the effective dispersion term appearing in the averaged convectivediffusion equation governing the transport of a dissolved solute (e.g., eq 10) depends on the explicit features characterizing the streamwise variations of velocity. The structure of the different results can be deduced from an examination of the dispersive, or “fluctuation-generated” flux 〈u′∂c′|∂x〉, where x denotes the appropriate streamwise variable. Hence, we can contrast: (i) axial flow with no streamwise variations of velocity; (ii) two-
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Table 1. Summary of the Order-of-Magnitude Estimates Used for Characterizing the Respective Dispersive Contributions in a Macrotransport Analysisa
u′ ordinary pipe flow
〈u′∂c′∂x 〉
c′
(
)
(
)
u′a2 ∂〈c〉 D ∂z
O(〈u〉)
O
(Q* rh )
O
( )
O
a ) pipe radius two-dimensional radial source flow
O
2h ) plate separation three-dimensional conical source flow
O
Q* r2
u′h2 ∂〈c〉 D ∂r
(
)
u′r2 ∂〈c〉 D ∂r
(
)
(
( ))
(
)
O
O
O
〈u〉2a2 ∂2〈c〉 D ∂z2
Q*2 1 ∂ 1 ∂〈c〉 D r ∂r r ∂r 2 Q*2 ∂ 〈c〉 Dr2 ∂r2
a It is convenient to arrive at these results by evaluating the “fluctuation-generated” flux 〈u′∂c′/∂x〉, where x denotes the streamwise direction. This flux enters the corresponding form of eq 4, which describes the redistribution of the average solute concentration. For pipe flow, x ) z denotes the axial coordinate, and 〈u〉 the average fluid velocity; for radial source flow, x ) r, where r is the radial variable in cylindrical coordinates; and for conical source flow, x ) r, where r is the radial variable in spherical coordinates; for the latter two cases Q* is the volume flux of fluid. The order-of-magnitude of c′ follows from the form of eq 7.
dimensional configurations with radially directed streamlines, for which the velocity varies as r-1; and (iii) threedimensional,conicalflowswithradiallydirectedstreamlines, for which the velocity varies as r-2. The results for the form of the dispersive contribution to the macrotransport process are given in Table 1. Acknowledgment We are pleased to have this opportunity to contribute to this special issue honoring Roy Jackson upon the occasion of his retirement from academic life. Each of us has benefited directly from Roy’s research contributions, as well as from his thoughtful, soft-spoken, and insightful comments during conferences, seminars, and private conversations. Roy is a classical engineering scientist in the best traditions of the genre, and his formal departure as an active participant in its ongoing development coincides closely with the end of an era in the field. We wish him bon voyage. H.A.S. acknowledges support from the Army Research Office (DAAG55-971-0114). H.B. was supported by grants from the Office of Basic Energy Sciences as well as the Mathematical, Information and Computational Sciences Division of the Department of Energy. Michael Brenner is thanked for helpful comments on an early draft of the paper. Literature Cited (1) Taylor, G. I. Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. A 1953, 219, 186-203.
(2) Aris, R. On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc. A 1956, 235, 67-77. (3) Brenner, H.; Edwards, D. A. Macrotransport Processes; Butterworth-Heinemann: Boston, 1993. (4) Bryden, M. D.; Brenner, H. Multiple-time scale analysis of Taylor dispersion in converging and diverging flows. J. Fluid Mech. 1996, 311, 343-359. (5) Ledezma, G. A.; Folch, A.; Bhatia, S. N.; Balis, U. J.; Yarmush, M. L.; Toner, M. Numerical model of fluid flow and oxygen transport in a microfabricated bioartificial liver device. Biomech. Eng. 1999, (Feb), in press. (6) Pagitsas, M.; Nadim, A.; Brenner, H. Multiple time scale analysis of macrotransport processes. Physica 1986, 135A, 533550. (7) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; Martinus Nijhoff: Hingham, MA, 1983. dz, (8) Explicitly, 〈P〉(r,t|r′) ) (2h)-1∫h-hP(r,z,t|r′,z′) where the integrand denotes the microscale Green’s function for the axisymmetric convective-diffusion eq 2. That is, P represents the solution c of eq 2, satisfying the prescribed no-flux boundary conditions on the surfaces z ) (h, vanishing at large radial distances |r - r′| f ∞, and representing an initial impulsive, instantaneous unit source situated at the point (r′, z′) at time t ) 0, namely P(r,z,0|r′,z′) ) (2hr)-1 δ(r - r′)δ(z - z′)δ(t). This microscale Green’s function constitutes the kernel in the relation h ∞ c(r,z,t) ) ∫z′)-h ∫r′)0 P(r,z,t|r′,z′)c(r′,z′,0)r′ dr′ dz′ describing the evolution of the solute concentration field from its prescribed initial state c(r, z, 0) to its current state c(r, z, t).
Received for review June 2, 1998 Revised manuscript received August 12, 1998 Accepted August 19, 1998 IE980355F
