Local and global dispersion effects in Couette-Taylor flow—II. Quantitative measurements and discussion of the reactor performance

Pergamon

Chemical Engineering Science, Vol. 51, No. 8, pp. 1299-1309, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0009 2509/96 $15.00 + 0.00

0009-2509(95)00382-7

LOCAL AND GLOBAL DISPERSION EFFECTS IN COUETTE-TAYLOR FLOW--II. QUANTITATIVE MEASUREMENTS AND DISCUSSION OF THE REACTOR PERFORMANCE G. DESMET, H. VERELST and G. V. BARON* Department of Chemical Engineering, Vrije Universiteit Brussel, Pleinlaan, 2, B-1050 Brussels, Belgium (First received 28 November 1994; revised manuscript received 20 October 1995; accepted 25 October 1995)

Abstract--For the first time, the extent to which laminar and turbulent Couette Taylor flow (CTF) combine a limited axial dispersion with a high local mixing intensity has been quantified by simultaneously measuring the inter-vortex flux and the tangential dispersion. The correlations for the tangential dispersion indicate that the local mixing intensity of CTF is of the same order of magnitude as in coiled pipe flow. The axial dispersion was measured in such a way that the dispersion enhancing or reducing effects which originate from the slow intra-vortex transport were eliminated. This allowed the comparison of the measurements in this study with recent theoretical calculations of the inter-vortex flux in 2D-periodicflows. 1/2~shows that in the laminar regime the inter-vortex flux can be The variation of the inter-vortex flux with Dmo described by the penetration model. A mechanism for the axial dispersion in the turbulent regime is also proposed. By comparing the ratios of the local and global dispersion coefficients,it is quantitatively shown that for CTF the local mixing intensity can be increased with a relativelysmaller increase in axial dispersion than for conventional flow types.

l . INTRODUCTION

A large number of chemical and mass transfer processes benefit from a minimal axial dispersion combined with a maximal local mixing intensity. Due to its cellular vortex structure, the Couette-Taylor flow (CTF) is claimed (Kataoka et al., 1975) to be one of the rare flow types combining intense local mixing with a limited axial dispersion. In what follows, the extent to which both mixing characteristics are combined will be referred to as the reactor performance. Other advantages of CTF are found in the enhanced transfer of heat and mass to the cylinder walls (Kataoka et al., 1977) and/or its gentle way of mixing. A description of the flow properties and the dispersion mechanism of CTF is given in Part I. The ratio of the governing inertia (centrifugal) and viscosity forces is usually given by the Taylor number (Ta): T a = ~o. d 3/2 . Rli/Z/v.

(1)

In 1951, Croockewit et al. showed that for turbulent CTF the axial dispersion coefficient does not vary with the net axial flow rate when the imposed axial velocities are low compared to the circumferential rotation speed of the inner cylinder. For laminar CTF, similar conclusions were drawn from tracer experiments, by Kataoka et al. (1975), and from visualization experiments by Gu and Fahidy (1985b). This implies that it is possible, under the condition of small

*Corresponding 32.2.629.32.48.

author. Tel.: 32.2.629.32.46.

Fax:

net axial flow rates, to exploit the typical mixing characteristics of the CTF during a continuous operation. Since the publication of these results, a vast number of possible applications of the unique reactor performance of CTF have been proposed, covering the field of catalytic (Cohen and Moalem Maron, 1983, 1991), biocatalytic (Baron and Van Capellen, 1990; Desmet et al., 1991), electrochemical [e.g. Mizushina (1971), Gabe (1974), Gu and Fahidy (1985a) and Coeuret and Legrand (1981)], photochemical (Haim and Pismen, 1994) and polymerization (Sinevic et al., 1986; Kataoka and Okubu, 1995) reactions, as well as the field of mass transfer operations, such as countercurrent extraction (Leonard et al., 1981) and tangential filtration (Holechovsky and Cooney, 1991). For laminar CTF, the performance calculations of these applications were all based upon simplifying assumptions, such as the absence of vortex-intermixing or the representation of the intra-vortex mixing behaviour with an ideal stirred tank, which have been shown to be incorrect in the laminar regime (see Part I) and which lead to an overestimation of the theoretical reactor productivities. For turbulent CTF, there exists a reliable set of measurements of the axial dispersion (Tam and Swinney, 1987; Enokida et al., 1989; Moore and Cooney, 1995), but measurements of the intra-vortex mixing have not been reported yet. This lack of accurate quantitative information on the mixing mechanisms has been the incentive for the present study. Additional motivations are found in Part I, where a mixing behaviour with strong intervortex flux and poor intra-vortex mixing has been

1299 CE$ 51:8-I

1300

G. DESMETet al.

found, which is the opposite of the ideal behaviour (i.e. combining perfect plug flow with perfect local mixing) claimed in literature. Furthermore, single-parameter models have been shown to be invalid to represent the axial dispersion in laminar CTF. In the present study, a measurement method based upon a two-parameter model is used. To estimate the local degree of micromixing, simultaneous measurements of the tangential dispersion will be used. Combining these measurements with the inter-vortex flux data will then finally allow to quantify the extent to which CTF really combines perfect plug flow with perfect local mixing. 2. EXPERIMENTAL

The experimental conditions and set-up are described in Part I. It was observed that, over the entire range of investigated T a numbers, the number of vortices (N) occupying the cylinder gap was either 30 or 32. This implies that the axial width (A) of the vortices was always approximately equal to the annulus width (d). This finding is in agreement with all previous studies of the vortex width in both the laminar [e.g. Gu and Fahidy (1985a) and Legrand and Coeuret (1987)] and the turbulent regime (Koschmieder, 1979). The measurement conditions have been selected in such a way that for laminar CTF the inter-vortex flux (represented by the inter-vortex exchange coefficient Ki) is determined independently from the dispersion enhancing or reducing effects which originate from the slow intra-vortex transport. It has been shown (Part I) that these effects make the comparison with axial dispersion correlations obtained in other CTFreactors impossible. However, it was found that when the intra-vortex mixing time (Tv) is smaller than the characteristic time the tracer requires to reach the reactor end-walls (l'ax), the inter-vortex flux can be separately determined from the rate of concentration decrease in the tail of the response curves. From a simulation study, it was found that the range of operating conditions in which this dispersion regime can be established is largest when the cross-section of the vortices is injected homogeneously and when the detection probe is positioned near a vortex interface (see Part I, Fig. 2). For this case and for the same number of vortices as in the present set-up (i.e. N = 32), the simulation yielded: zv <~ Zax'c*'Ki.A <~ 1000.Dintra (N =32).

Ki. A.

1.o exp. curve model response,eq. (7)

C(-)

0

400

800 1200 time (s)

1600

1.0 exp. curve (7) modetresponse, eq.

C(-)

0.5

(2)

When eq. (2) is satisfied, the values of Dax which are derived from the concentration decrease in the tail of the response exclusively represent the inter-vortex flux. They are related to K~ and the vortex width(A) by (see Part I): Dax =

The tracer was injected with a micropipette (with a volume -~ 100 #1). For the most viscous mixtures, a homogeneous tracer distribution in the (r, x)-plane of the injected vortex was achieved by moving the injection device radially over the entire annulus width. To obtain simultaneous measurements of the tangential dispersion, the duration of the injections was kept as short as possible, such that the initial tracer block only occupied a part of the reactor circumference. A typical example of the response curves which are obtained with the proposed type of experiment is shown in Fig. 1. Due to the tangential recirculation, the initial part of the response curves is marked by a pattern of decaying peaks (Fig. 1), whose rate of decay will be used to measure the tangential dispersion. To minimize peak interference (and thus loss of accuracy for the determination of the tangential dispersion coefficient), all injected pulses had tangential dimensions which were smaller than 10% of the circumference. Also for reasons of accuracy, the diameter of the inner cylinder has been chosen sufficiently large (the larger the available space over which the tracer may disperse in the tangential direction, the more pronounced will be the effect upon the recorded signal). When the tangential

(3)

For turbulent CTF, the global axial dispersion is fully determined by the inter-vortex flux, from the first moments of the experiment on. The intra-vortex transport occurs so fast that it does not influence the rate of dispersion.

0.0 kJ 0

I

25

I

5°time (s) 75

100

Fig. 1. Top: example of experimental response curve for laminar CTF (co = 16.3 rad/s, v = 1.5 x 10- 5 mA/s, Ta = 840, Sc = 9.7 x 104).The mathematical model is given in Section 3. The parameters of the fitted curve are: Vtan=0.25 m/s, Da~ =4.2 x 10-6 m2/s, Dtan = 1.8 × 10-4 m2/s, h =0.015 m, 01 = ~/20. Bottom: expanded view of the initial part of the response curve.

Local and global dispersion effects--II mixing is completed, the tail of the response is fully determined by the axial dispersion. 3. MATHEMATICAL DISPERSION MODEL

As mentioned in Section 2, the inter-vortex flux can, even for laminar CTF, be quantified with a single dispersion coefficient (Dax or Ki) derived from the tail of the obtained responses when eq. (2) is satisfied. In the tail, the two-zone model then reduces to the conventional continuous axial dispersion model. The latter model is preferred since it allows to establish an analytical solution. Since the degree of tangential dispersion does not vary with the tangential coordinate, the tangential dispersion can also be represented by a single dispersion coefficient (Dtan). Since the crosssection in the (r, x)-plane of the injected vortex is marked homogeneously (Fig. 2), the radial dispersion effect is eliminated. The mass fluxes and the shape of the initial tracer block which determine the tracer responses are represented in Fig. 2. The global mass balance is given by 0C

/)tan 0C

&

Rm" 00 +

Dta n 02C R2 " 002

~2C ~-D.~. 0x 2 .

(4)

The first term of the right-hand side of eq. (4) represents the tangential recirculation of the tracer curve (convection). The second and the third term represent, respectively, the dispersion in the tangential and the axial direction. Since no information on the radial variation of the tangential velocity is available, the tangential convection is represented by its mean value (/)t,n) and by its mean tangential path length (with mean radius Rm). This simplification was found to be sufficiently accurate over the entire range of investigated experimental conditions.

1301

Since the cross-section of the vortices is injected homogeneously, the axial width of the initial tracer block was always equal to the width of a vortex. In the tangential direction, at least 5% of the circumference is injected with tracer (see Fig. 2), due to the fast tangential convection and the finite duration of the injections. The dimensions of the initial tracer distribution can hence not be neglected compared to the tangential and axial reactor dimensions. Therefore, the conventional Dirac-impulse approximation, which has a zero spatial width and predicts an infinite concentration at the injection point, will necessarily introduce interpretation errors, especially considering that in the present experiments the tracer concentration is recorded at the injection point. Instead, the difference of two Heaviside functions, representing a rectangular block with a non-zero axial and tangential width and a finite concentration (Fig. 2), will be used as the initial condition: t=O:

C=Co.[n(h+x)-n(h-x)].

[H(O, + O) - H(O, - 0)].

The axial width of the initial tracer block (2h) was always taken equal to the vortex width(2). The tangential dimension (01) of the initial tracer block had to be estimated visually. Although this seems rather inaccurate, a simulation study pointed out that as long as 01 <0.2n, an error upon the value of 0~ did not influence the estimated value of Dta n. As represented in Fig. 2, the initial tracer block is situated exactly in the middle of the reactor. In the absence of a net axial flow, the boundary conditions in the axial direction are hence given by L x=_+~-:

0C ~=0.

0 =--7~

i

t

0 //L=

/

001y /

0 +TZ

r=R° l r I=" ~R~ i

~

~ [ , " '~ x=-h

/

/

,/ / / / /

x=0

(5)

/

/

il

'' ' ~

Initial tracer block~

x=h

Fig. 2. Schematic 3D-representation of the tracer experiment (injected pulses are represented by the shaded block). The injected pulses only occupy a fraction of the tangential dimension. The tracer is initially distributed homogeneously over the entire injected tracer block.

(6a)

G. DESMETet al.

1302

In the absence of a net tangential flow, the symmetry of the tangential dispersion problem should be expressed by 0C/00 = 0 at 0 = _+n. Since in our experiments a tangential recirculation is present, one has to express that this point is recirculated along the tangential coordinate with an angular velocity m. The boundary conditions in the tangential direction [eq. (6b)] are hence given by 0= +n+w.t:

OC

ff~-=0.

(6b)

Due to the linearity of eqs (4)-(6), the original problem can be decoupled in its tangential and axial components by applying a iemma discussed by Carslaw and Jaeger (1959, p. 33). The solution to eqs (4)-(6) can be written as the product of the solution of two one-dimensional problems:

C(x, o, t) = c~.(x, t). c,..(o, t).

(7)

In eq. (7), C,, is the solution of the following axial dispersion problem: 0Cax

+ erf(h + x - n . L ] ]

(16)

\ 27-~-]j

The tangential dispersion equation [eq. (11)] has first been solved with the boundary conditions for an infinite medium without recirculation [eq. (14a)]. One obtained Co [erf(.0Z + Oz w.t ~

Cm.(O,t)= 2"L

\ 2 . ~

/I (17)

(8) This solution is then summed over an infinite number

with

C,,=Co.[H(x+h)-H(x-h)]

t=0:

x = +

L

- 2

:

OCa, =0 0x

(9) (10)

and Cta, is the solution of the following tangential dispersion problem:

0C,..(0, t) = Dr... OCt..(O, t____)+ m.OCm.(O, t) Ot 002 O0

(11)

with

Ri + Ro

=--'-~--'

/~tan Dt.n

=-~'~ffz and

#u,.

W=R--~

(12)

and t=0:

Co +v~ ~' f ,/'h-- x + n.L'~ Cax(X,t)=--, 2., / e r i / . . . . / 2 .=_+L \ : . ~ /

+ erf(OZ-O +w't)~ \ 2 . ~ /J"

0 2 fax

e--t- = Oa.. Ox-----/-

Rm

using the method of images, discussed by Carslaw and Jaeger (1959). They showed that in the absence of a net axial flow rate, the solution in a closed reactor can be obtained by reflecting the solution of the infinite problem at the reactor boundaries. The solution of the original axial dispersion problem [eqs (8)-(10)1 is hence given by

Cmo=Co.[n(o+ox)-n(o-o,)]

0 = + oo : Ct,. =0

(13)

(infinite medium condition) (14a)

or

0C

O= + n + ~J.t: ~--~=0 (condition valid in present case).

(14b)

The axial dispersion equation [eq. (8)1 has first been solved for the given initial conditions [eq. (9)1, but in an infinite medium (C,. = 0 at x = + oo). Using the Laplace transform technique, this yields: Co 1- re/" h + x

c.,(x,,) = T [ e_

h-x_']]. LTo )"~ + ere(\2.x/D,,.t] J (15)

From this solution, the solution satisfying the boundary condition presented in eq. (10) can be constructed

of periods (with Tp,r = 2. n. Rm/vt,.). The summing procedure reflects the physical reality in which, due to the tangential recirculation, a fixed observer measures at a given time not only the signal at 0 = 0 , but also at 0 = 2n, 4n, 6n, ... r +~F /0 +O+2nn--w.t'~ Ct,,(O, t) = t~o. ~, / e f t ( 2.=oL \ 2__~ /!

+ erf(_0,-O-2nn + m.t)].

(18)

This solution satisfies the original boundary condition [eq. (14b)]. This can be noted from the fact that for large times, when the tangential mixing is complete, the final concentration predicted by eq. (18) equals Co.0ffn, showing that the condition of mass conservation in the tangential direction (which is also expressed by 8C/80 = 0 at 0 = + n + 'm. t) is satisfied. According to eq. (7), the product ofeqs (16) and (18) yields the solution to the dispersion problem [eqs (4)-(6)1. Figure 3 illustrates how this problem can be decomposed in its tangential and axial effect, allowing the simultaneous, independent determination of D,, and Dtan. The good agreement between the solution of the model and the experimental curve presented in Figs l(a) and (b) illustrates the validity of the proposed dispersion model. For the determination of Dtan, the smallest relative weight was attributed to the peaks at the beginning of the curve, since it was our experimental experience that their height was distorted by deviations from the perfect-block initial condition. The mean tangential fluid velocity (vta.) can easily be determined from the period of oscillations (see Fig. 3). When the tangential mixing is completed, D+~is determined by comparing the rate of concentration decrease

Local and global dispersion effects--II

1303

4. I N T E R - V O R T E X

1.0 C(-) 0.8

Vta n = 0 . 1 2 m / s l~x=2.10

•

-4 (m2/s)

Dtan=1.10 "2(m2/s)

0.6 0.4

~),

0.2 0.0 0

10

,

20

30 40 time (s)

50

0.3 C(-) 0.2

~

0.1

t

0.0 0

Vtan.d

_,

20

~

i

30 40 time (s)

10-1 oo

o

50

eQ. ( z / I [ S c = l . O 10"1

•

10-2

10!

i • al r % ,% ~T u r b u l e n t *r e g i o n Laminar region, I ~, ~ ~ ~a~% ~ o • AAA ~ • • el o OOoo~ t ,~ , 102

Reran=

~t. Z a

v

with a =0.48 (+0.02).

Dax/(~andl

10-5

or

v

1

10-4

~.d3/2.R1/2 ~.

Fig. 3. Decomposition of a tracer response in its axial and tangential dispersion effect.

10-3

There exists at present no experimental evidence for the fact that the combination of the geometrical parameters and the angular velocity of the inner cylinder used in the definition of Ta I-eq. (1)] is suited to transpose the mixing parameters from one experimental set-up to another. To correlate the axial dispersion coefficients, we preferred a tangential Reynolds number (Retan), based upon the experimentally obtained values of Via,, rather than the commonly used Ta number. We believe that since Retan reflects the actually measured flow conditions, it is more suited to predict the variation of the dispersion behaviour with the geometry and to compare with correlations for pipe flow. The established correlations can however easily be converted in terms of Ta. For the present set-up, the following experimental linear relation between Vtan and co was found to be valid in the entire range of 1×10 -6 < v < 4 × 1 0 - 4 m2/s:

. 7)

q

10

FLUX

103

ld

Sc=1.4 SC=2.2 Sc=3.9 × Sc=6.6

10' I 10' I 10' IV I

•

Sc=1.4 lff I SC=3,5 10" I

t, • o •

S¢=9.7 10" I SC=5.1 lff I Sc=1.210' I Sc---4.01ffl ,

Reran

1 05

Fig. 4. Axial dispersion vs Reran for different water/glycerol mixtures.

in the tail of the response with the rate predicted by eq. (15). The validity of eq. (2) was tested by analysing the tail (C < 1.1 Ca) of the responses with both the axial dispersion model and the two zone model. For most of the experimental conditions, both models yielded approximately the same value for Ks. For the mixtures with Sc = 1.4 x 04 and Sc = 3.5 × 104 (see Fig. 4), eq. (2) was however violated. For these conditions, the inter-vortex fluxes were relatively large, whereas the mixing in the core of the vortices still mainly occurred by the slow molecular diffusion.

(19)

As in the remainder of the text, the figure between brackets represents the standard deviation on the regressed parameters. A similar linear relation has been obtained by Legrand and Coeuret (1986). It should be noted that the value of a will be different for each reactor geometry. This is due to the fact that there exists no unique analytical expression for the influence of the geometrical parameters (R~ and Ro) upon the relation between the rotation speed of the inner cylinder (co) and Vtan (Kreith, 1968). Since this relation cannot be derived analytically from the Navier-Stokes equations, the proportionality constant will have to be determined experimentally for each combination of the geometrical parameters. The experimental values for Da~ ranged from 5 x 10 - s to 8 x 10 -4 m2/s (Fig. 4). As mentioned in Section 3, eq. (2) was violated for the experiments with S c = l . 4 x l 0 4 and S c = 3 . 5 x 1 0 4 . These data have therefore not been considered in the parameter fitting leading to eqs (20) and (27). Just as for the tangential dispersion (see Fig. 6), a transition between laminar and turbulent behaviour occurs around Reran --=103. Accounting for the Schmidt-number (Sc), the data in the laminar regime (Retan < 103) correlate as (Fig. 5): vt~,.d = Peax =2.5 Relt~/2. Sc 1/2

(20)

Dax Rearranging eq. (20) gives D~x =0.4

"1/2 r~l/2 ./1/2 Utan • l J m o l • t~

(21)

which can be rearranged with eq. (3), and with 2 ~ d, to yield an expression for the inter-vortex exchange coefficient: Ki

=

Da,Jd~ nl/2 ~Vmo I . t., ~ V t a n /,t~1/2 /~ ! •

(22)

G. DESMET et al.

1304

vection is much smaller than for diffusion, the intervortex exchange coefficient is given by

105

Pe ax

Ki = 1.04 V r , a tool

° ~

.

(26)

° o°

(20) I ~ Sc=9.7 104 ]A Sc--5.1 105 I Sc--12 t0 [" Se--~_.0106 •

~'@~ /~ ~///~ ~ 104 103

/

.... 104

(RetanSC)1/2 105

Fig. 5. Axial Pe number versus (Retan.Sc) 1/2 in the laminar regime. A regression analysis yielded a slope of 1.02 (+0.05).

Equation (22) suggests that the inter-vortex flux can be represented by the penetration model, which predicts that (Sherwood et al., 1975): K i ~ (Dmol/tc) 1/2.

(23)

The validity of this model can be established from the flow pattern presented in Part I. As the fluid particles are convected along their helicoidal streamlines, mass is exchanged between fluid particles of adjacent vortices when they make contact during their radial passage along the vortex boundaries. The secondary convection ensures the continuous renewal of the fluid particles which make contact along the interfaces. Due to the high velocities of the secondary rotation, tracer particles only penetrate the adjacent vortex over a thin layer during their passage along the interfaces. In this layer, just as along the central axis of a conventional jet, the velocity gradients are negligible (Marcus, 1984). All the conditions for the applicability of the penetration model are hence satisfied. The concept of the penetration layer provides a physical meaning for the width 6 of the outer zone in the two-zone model (Part I). The time (4) during which fluid particles of adjacent vortices make contact, is determined by the radial velocity at the interfaces: tc

= d/1)rad.

(24)

From a dimensional analysis, one might assume that Vr~d is directly proportional to o9 (Batchelor, 1961), and hence, considering eq. (19), also directly proportional to v,~,. Hence: tc = d/Vrad ~ d / v

....

(25)

Combining eqs (23) and (25) shows that the experimental correlation [eq. (22)] can indeed be explained by the penetration model. Another theoretical validation for the experimentally obtained correlation is provided by the theoretical calculations (McCarty and Horsthemke, 1988) of the inter-vortex flux in 2D-periodic laminar flows, such as, e.g., Rayleigh-B6nard flow. They found that when the characteristic time for intra-vortex con-

In eq. (26), vr,a represents the mean velocity of the secondary rotation along the interface. Considering that Vt,n is directly proportional to o9 [eq. (19)], and since a dimensional analysis shows that the velocity of the secondary rotation is also directly proportional to co (Batchelor, 1960), one might state that vr,a varies linearly with yr,,. Hence, by rewriting eq. (26) in terms of v,a., an expression is obtained which is, apart from the proportionality constant, identical to the experimentally obtained correlation [eq. (22)]. Our D,x-values are at least an order of magnitude smaller than the values obtained by Kataoka and Takigawa (1981), Pudjiono et al. (1992, 1993) and Moore and Cooney (1995) in the same range of T a numbers ( T a < 1000). This is due to the fact that in their experiments the tracer was introduced along the outer vortex layers, which drastically enhances (see Part I) the observed axial dispersion as long as the vortices have not become perfectly mixed. The relatively large axial dispersion coefficients obtained by the above-mentioned authors should therefore nor be explained by the existence of a convective interchange across the vortex boundaries, as was suggested by Moore and Cooney (1995). For R e t , , > 10 a, the axial dispersion occurs by a turbulent mechanism. Our data (Fig. 4) agree with the measurements of Croockewit et al. (1951), who obtained values for Dax ranging from 3 x 1 0 -5 to 7 x 10 -4 m2/s in the range of 1300 < T a <4.3 x 104. They also agree with the measurements of Tam and Swinney (1987), who obtained values for Dax ranging from 1 x 10 -4 to 8 x 10-*m2/s in the range of 2500< T a < 5 x 104. The values obtained by Moore and Cooney for R e ~ > 2 x 103 are also of the same order of magnitude. Correlating our data with Reta, gives (Fig. 4): Peax = 2 . R e ~ ,

with m =0.28 (_+0.02)

(27)

or D,x

0.7 l~ta n.

(28)

Equation (28) is in agreement with the correlations established by Tam and Swinney (1987), who obtained powers in the range of 0.69 < m < 0.86 for the relation D~ ~ v,'~, and with the correlations obtained by Enokida et al. (1989), who obtained a power of m =0.8. With the insights in the dispersion mechanisms obtained in Part I of this study, we can now resolve the lack of agreement which exists in literature about the axial dispersion mechanism in turbulent CTF, and more precisely concerning the role of the vortex structure in the axial transport (Tam and Swinney, 1987), and concerning the validity of some of the correlations presented in literature (Moore and Cooney,

Local and global dispersion effects--II 1995). Video recordings clearly showed that for turbulent CTF the characteristic times for the intra-vortex dispersion and for the secondary convection are of the same order of magnitude. Rosenbluth et al. (1987) have shown theoretically that in this case the intervortex exchange coefficient K~ in 2D-periodic flows is given by Ki = Dl..... /d.

Dloe, t....

~(P

....

)m,

with 0.6 < m <0.9.

(30)

Translating this result to CTF, one might infer that the relation between Dlo¢,a~ and the velocity of the secondary rotation (Vrad) along the interfaces will be similar to eq. (30): D~ . . . . . ~ (Urad) m.

(31)

Considering furthermore the linear relation between V,adand Vta. [eq. (25)], the combination of eqs (29) and (31) yields Ki ~ (Utah)m.

(32)

Our visual observations showed furthermore that turbulent CTF might be considered as a series connection of perfectly stirred vortex units, for which the transport across the vortex interfaces is the rate-limiting step for the axial dispersion. For this case of fast intra-vortex mixing, the global axial dispersion is according to the two-zone model given by (see Part I) D~=K~.2

(2~d).

(33)

With K~ given by eq. (32), eq. (33) yields a semitheoretical validation for the experimentally obtained relation between D~, and vt~, [eq. (28)]. Equation (33) suggests that for a given Reran, which fixes the K r value, the axial dispersion is directly proportional to the vortex width. This is in agreement with the experimental findings of Tam and Swinney (1987). 5. T A N G E N T I A L

100 l/Pe~n

DISPERSION

All experimentally obtained tangential mixing times, determined by the moment at which the peak amplitude is reduced to 5% of its initial value, ranged from 5 to 900 s. Comparing these values with the mean residence time of a vortex in a continuous reactor, which is typically of the order of hours when a non-disturbing axial flow is imposed, one might conclude that the tangential macro-mixing inside a vortex is perfect. This fact, together with the exist-

tJ S c = I . 0 103 D

x+

Sc=l.4 Sc=2.2 + Sc=3.9 x Sc=6.6

Am

10 -1

~.'2 o

* oA •

° Turbulent region

A

e ° • o •

(29)

In eq. (29), D~.... ~ represents the local degree of axial dispersion along the vortex interfaces. To estimate this degree of dispersion, one will use the equivalence between the axial dispersion in the radially oriented in- and outflowing turbulent jets which form the vortex interfaces in the (r, x)-plane (see Fig. 4, Part I) and the dispersion transversal to the direction of mean flow in the central portion of turbulent pipe flow. According to Sherwood et al. (1975, p. 125), the transversal dispersion in the central portion of a turbulent pipe flow is given by

1305

e°o 10-2

10-3 10 ]

A

o Laminar

........

, 102

.......

........ l0 3

L

,

1 04

•

103 103 103 103

o •

Sc=1.4 I ~ Sc=3.5 104

• o •

Sc=9.7 Sc=5.| Sc=1.2 Sc=4.0

......... Ret~

Fig. 6. Tangential dispersion vs Reta, water/glycerol mixtures.

10~ 105 1~ 1~

,

1 05

for different

ence of a strong secondary circulation, has lead some authors [e.g. Kataoka et al. (1975)] to the suggestion that the intra-vortex flow shows an ideal stirred tank contacting (i.e. micromixing) behaviour. In order to test the validity of this concept, the micromixing ability of CTF was estimated by comparing the correlations for the tangential dispersion coefficient (Dtan) with correlations for pipe flow. The experimental values for Dtan are plotted in Fig. 6, as a tangential Peclet number (Peta,) vs Reran. For Retan > 103, Dtan is directly proportional to Vtan and is independent of Dmo~. This is the region of turbulent mixing. Peta n remains constant in this region: 1/Peta, ~- 0.35

(+0.015).

(34)

Equation (34) is identical to the expression for the axial dispersion in turbulent pipe flow given by Wen and Fan (1975). One might hence assume that the degree of micromixing in the vortex bulk is of the same order of magnitude as in turbulent pipe flow. For Reran < 103, Dtan depends o n DmoI. This is the laminar region. The transition point (Reta. = 103) is equal to the transition point obtained for Dax (Fig. 4). The fact that it is of the same order of magnitude as the transition point in pipe flow, emphasizes the analogy with conventional pipe flow. For Reta, < 103 (laminar region), two regions, with an opposite influence of Reran, can be discerned (Fig. 7): l/Pet~, =1.7 x 106.(Retan.Scl/3) - 2 " 2

Retan. Sc 1/3 < 104

(35a) 1/Peta, =4 x 10- s. (Reta,. Sc 1/3)1.5 Retan. Sc 1/3 > 104.

(35b) The values of Dtan are much smaller than the axial dispersion values given by Wen and Fan (1975) for the Taylor diffusion regime in straight pipes. This deviation from pipe flow can be explained from the action of the secondary rotation. Studying the axial dispersion in coiled pipe flow, Ruthven (1971) showed that the secondary rotation in the pair of axially oriented

1306

G. DESMET et al.

100

..... eq. (35a) - - eq. (35b). Sc--9.7104 : Sc=5.1105 Sc=1.2 106 • Sc--M.0 106

1/Pe t ~

°° A A

10-1 •

o

•

o 2.

O'A .

.

.

.

.

.

.

I 0"2103

.

I 4

'

10

,

,

,

,

Re.Sc It3

,

,,

105

Fig. 7. Tangential dispersion vs Re,.,.Sc 1/3 (laminar region).

_

.

.. . . .

(

V a x = O

value is reached (Shetty and Vasuveda, 1977). This deviation originates from the difference in boundary conditions for the tangential velocity (Fig. 8). In coiled pipe flow, the difference in velocity between fluid particles located near the lower or near the upper boundary is relatively small, whereas in CTF the tangential velocity varies from vt,, = co. R~ near the inner to Vta, = 0 near the outer cylinder (Fig. 8). In both laminar flow types, the secondary rotation strongly reduces the velocity gradient in a broad region around the tube centre and creates a high-frequency exchange of fluid particles located near the upper and lower boundary. In coiled pipe flow, the latter effect does not significantly contribute to the dispersion, whereas in CTF, the fluid particles which are situated near the upper and lower boundary are continuously exchanged between a position with a large and with a small tangential velocity. In the second regime of Fig. 7 [represented by eq. (35b)], this effect seems to outweigh the dispersion reducing effect obtained in the vortex centre.

Vax/Vax,mean 6. REACTOR PERFORMANCE

..... Vax 0 pipe flow: small difference in velocity between fluid particles situated near lower and upper boundary ( ~ coiled pipe flow,- ...... straight pipe flow). ~

,~,~:~.~,~, , .... ~

~

Vtan = 0

Vtan/Vtan,mean ~ ~.~.......V t a n = 0 5 . 1 ~ .

tangential annular flow with moving lower boundary: large difference in velocity between fluid particles situated near lower and upper boundary ( - - CTF,. ...... pure Couette-flow).

Fig. 8. Fully developed laminar velocity profiles in the presence of a secondary rotation. The velocity profiles for CTF are based upon the experimental study of Simmers and Coney (1979). vortices reduces the radial variation of the axial velocity, which in turn reduces the axial dispersion. Considering that CTF is marked by a series of tangentially oriented, counterrotating vortex pairs (see Part I, Fig. 1), CTF can be regarded as a juxtaposition of a series of tangentially oriented, recirculating coiled pipe flows, for which the action of the secondary rotation reduces the tangential dispersion. Although this analogy with coiled pipe flow seems straightforward, it has apparently never been proposed before in literature. There exists however a remarkable deviation from this analogy. For laminar CTF, there exists a range in which Petan increases with increasing Reta, [eq. (35b)], whereas for laminar coiled pipe flow, Peax decreases monotonically with increasing Reax until a constant

It is now obvious that CTF does not combine perfect plug flow with perfect intra-vortex mixing, as was originally claimed by Kataoka et al. (1975). In order to quantify the extent to which CTF deviates from this ideal behaviour, the ratio of axial to tangential, or more generally, the ratio of axial to transversal dispersion coefficients will be used [eq. (36)-]. This ratio quantifies the ability of a flow system to minimize the turbulent effect, if present, in the axial direction and to maximize it in the other (transversal) directions: @ = Da,,/Dt .... •

(36)

A minimal value of @ represents the necessary condition for a flow system to combine a minimal spread in residence time with a maximal degree of local mixing. For CTF, we only dispose of data for the tangential dispersion to quantify the transversal (or local) dispersion. Since for turbulent flows, the dispersion in the streamwise (i.e. tangential for CTF) direction is directly related to the degree of turbulence (Sherwood et al., 1975), Dta. can indeed be used as an estimate for the degree of intra-vortex micromixing in turbulent CTF. For laminar CTF, Dr,, essentially reflects the macromixing-effect which is due to the radial variation of the tangential velocity (Fig. 8), and is hence not a proper measure for the micromixing ability. For laminar CTF, one has therefore used Dt,a,s = Dmo~in eq. (36), expressing that the micromixing in the vortex bulk occurs only by molecular diffusion. For turbulent CTF, • decreases slightly, but continually with increasing Ret,n (Fig. 9), even after the disappearance of the cellular vortex structure (Reta, > 5 x 103). For laminar CTF, the ¢~-values are orders of magnitude larger than for turbulent CTF. This shows that, opposite to what has been proposed in literature [e.g. Cohen and Moalem Maron (1983)],

Local and global dispersion effects--II 105

10 3

i

102

Lan~nsx i

region 10 ! 100

:z Sc=l.0 103 • S c = 1 . 4 103 Sc=2.2 103 ÷ Sc=3.9103 x Sc=6.61(~ Sc=9.7 10~

t • S c = 5 . 1 1 0~ t~Turbulent o Sc=1.2 1~ • S c = 4 . 0 11~ ~region P

10-I 10-2 10

2 10

3 10

4 5 I0 Retan l 0

Fig. 9. • vs Reran for laminar and turbulent CTF. The Ovalues are calculated from the experimental correlations for Dax and Dta n obtained in the present study. Table 1. Comparison of reactor performance with conventional flow types Reactor type Laminar pipe flow Laminar packed bed flow Turbulent pipe flow Turbulent packed bed flow Stirred tank flow Turbulent Couette-Taylor flow Turbulent rotating disk flow

•

(u. d/Dmo02/192 150 100-150 l0 1 ~<0.1 0.01

the turbulent regime is much more suited then the laminar regime to combine a small axial dispersion with an intense local mixing. By using dispersion coefficients rather than the more obvious mixing times, (I) only depends upon the governing flow conditions and is independent from reactor geometry and tracer injection conditions. Equation (36) is hence very suited to compare CTF with other flow types. In Table 1, we have summarized the typical values of (I) for some other flow types. For stirred tank flow, assuming isotropic turbulence, (I) is near unity. For pipe flow and packed bed flow, the transversal direction which has to be considered is of course the radial direction [Dt . . . . = Drao in eq. (36)]. Typical values for Draa and Da~ were estimated from the graphs presented by Sherwood et al. (1975) for pipe flow and by Wen and Fan (1975) for packed bed flow. Table 1 shows that turbulent CTF excels the conventionally used flow types by at least one order of magnitude. This is due to the fact that for CTF the turbulence is created by a transversally oriented flow, which is within certain limits independent of the net axial flow rate, and due to the fact that the axial effect of the generated turbulence remains more or less restricted to a single vortex tube. The rotating disk flow (RDF) is another flow type in which the turbulence is also created by a transversal rotation. The value of (I)xDF,calculated from correlations published by Schweitzer (1979), is even an order of magnitude

1307

smaller than ~CXF. This gain in reactor performance results of course from the presence of impermeable physical boundaries which strongly limit the mass transfer between adjacent mixing units. One might therefore think of artificial compartimentation as a possible means to increase the reactor performance of CTF.

7. CONCLUSIONS Correlations for the tangential dispersion coefficient and the inter-vortex flux, which serve, respectively, as a measure for the degree of mixing in the vortex bulk and for the global axial dispersion, have been established for both laminar and turbulent CTF. Unlike in other axial dispersion studies, the experiments were conducted in such a way that exclusively the inter-vortex flux was measured, independently from the dispersion enhancing or reducing effects which originate from the slow intra-vortex transport. This allowed to compare our measurements with recent theoretical calculations of the inter-vortex flux in 2D-periodic flows. In the laminar regime, the variation of the inter-vortex flux with D~/~ shows that the inter-vortex mass transfer can be described by the penetration model. A mechanism for the inter-vortex flux is also proposed for turbulent CTF. We believe that the proposed concept, based upon the analogy between the inter-vortex transport and the radial dispersion along the central axis of a turbulent jet, allows to resolve the present disagreement in literature. Simultaneously, the tangential dispersion has been quantified. For laminar CTF, all tangential dispersion phenomena can be explained by assuming a streamline flow regime. This suggests that the mixing in the core of the vortices occurs by pure molecular diffusion, which can hence not be represented by an ideal stirred tank as has been proposed in literature [e.g. Kataoka and Takigawa (1981), Pudjiono and Tavare (1993)]. The relatively small tangential mixing times which are obtained for CTF are due to the Taylor dispersion which originates from the parabolic tangential velocity profile. It is this (macroscopic) residence time spreading effect which has mistakenly lead to the assumption of highly effective micromixing. For turbulent CTF, measurements of the tangential dispersion have not been performed before. The obtained correlation suggests that the degree of micromixing in the vortex bulk is of the same order of magnitude as in turbulent pipe flow. Although it is now obvious that CTF does not combine perfect plug flow with perfect intra-vortex mixing, as originally claimed by Kataoka et al. (1975), CTF still has its advantages over conventionally used flow types. Laminar CTF provides a simple and effective means to eliminate the parabolic velocity profile, which otherwise leads to the unwanted axial Taylor dispersion. The reactor performance measure defined in eq. (36) quantifies the ability of CTF to yield a given degree of micromixing with a relatively smaller axial dispersion than conventional flow types. The gain in

1308

G. DESMETet al.

reactor performance results from the fact that the local mixing can, within certain limits, be increased independently from the applied net axial flow rate. And since the velocity has no net axial component along the vortex interfaces, the effect of the created turbulence remains more or less restricted to a single vortex tube.

C d

Dax Dintra Dloc Drool

Dtan F, F'

H(x)

NOTATION tracer concentration (M/L 3, arbitrary units in Figs 1 and 3) annulus width L axial dispersion coefficient L Z / T intra-vortex dispersion coeff., L~/T local dispersion coeff., transversal to mean flow, L2/T molecular diffusion coeff., L2/T reduced tangential dispersion: (= Dta,/R2m), l/T, see eq. (12) mass fluxes determining the tracer responses, M/T, see Fig. 2 axial dimension of tracer pulse, L, see Fig. 2 and eq. (5) Heaviside function: [ = 0 ( x <0), and = 1

(x > 0)] Ki

L N Pea~

inter-vortex mass transfer coeff., L / T (see Part I) reactor length, L n u m b e r of vortices axial Peclet n u m b e r based on via,,

(= Petan

v,...

dlO.,)

tangential Peclet n u m b e r based on /)tan,

(= v,... dlO,..) r Ri, Ro

R= Rea, Reran Sc t tc Ta Tac Tper /)ax /)rad /)tan

radial coordinate, L, see Fig. 2 radius inner and outer cylinder, L mean radius: [ = (Ri + Ro)/2], L axial Reynolds number, (= vax. d/v) tangential Reynolds number, ( = v,~,. d/v) Schmidt number, ( = v/Dmol) time coordinate, T contact time, T, see eq. (24) Taylor number, see eq. (1) critical Taylor number, ( ~ 50) period of tangential fluid rotation, l/T, see Figs 1 and 3 axial velocity related to net axial flow, L / T mean velocity of secondary rotation along vortex interfaces, L/T, eqs (26) and (31) tangential velocity, L/T, [ = n. (Ri + Ro)/Tp.,] axial coordinate, L, see Fig. 2

Greek letters ct proportionality constant in eq. (19) 0 tangential coordinate, see Fig. 2 01 tangential dimension of tracer pulse, see Fig. 2 and eq. (5) 2 axial vortex width, L v kinematic viscosity, L2/T z,x global axial mixing time, T, see Part I

% •

09

intra-vortex mixing time, T, see Part I reactor performance, see eq. (36) reduced tangential fluid velocity: (= Vtan/Rm), l/T, see eq. (12) angular velocity of the inner cylinder, 1/T

Subscripts 0 initial value oo final value ax axial rad radial tan tangential trans transversal

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