Transient and stationary axial dispersion in vortex array flows—I. Axial scan measurements and modeling of transient dispersion effects

Pergamon

Chemical Enyineerinq Science, Vol. 52, No. 14, pp. 2383 2401, 1997

PII: S0009-2509(97)00048-1

1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain o009 ~2509/97 $17.00 4 0.00

Transient and stationary axial dispersion in vortex array flows--l. Axial scan measurements and modeling of transient dispersion effects G. Desmet, H. Verelst and G. V. Baron* Vrije Universiteit Brussel, Department of Chemical Engineering, Pleinlaan, 2, B-1050 Brussels, Belgium (Received 24 July 1996; accepted in revised form 23 January 1997) Abstract--Whereas conventional RTD experiments yield incomplete and potentially misleading information, the axial scan method is shown to be a powerful technique to analyze the transient dispersion effects in vortex array flows (VAFs). Applying different initial tracer distributions to a given vortex, and making axial scans of the spreading tracer distribution, allowed, for the first time, to quantify and classify the complete set of strongly different transient dispersion modes. As a model system, the laminar Couette-Taylor flow has been selected. By working under high viscosity conditions, the time scale of the different acting phenomena has been enlarged to such an extent that even the fastest dispersion events could be extensively studied and quantified. It is shown that in laminar VAFs effective axial dispersion coefficients can be obtained which vary over orders of magnitude, just by applying different initial tracer distributions to a given vortex. A first principles two-dimensional model (valid when the mixing along the streamlines occurs fast) with which all observed transient dispersion effects can be accurately represented is proposed. The insights obtained in the present study are especially useful for the development of VAF reactors for the treatment of strongly viscous fluids. ~'7, 1997 Elsevier Science Ltd. All rights reserved Keywords:

Dispersion; vortex; flow; tracer; axial scan; mass transfer.

l. INTRODUCTION

The present study deals with the axial dispersion in vortex array flows (VAFs), defined as flows consisting of a spatial repetition of a large number of identical vortex units with closed internal streamlines. The main focus is on the transient dispersion effects (i.e. dispersion modes for which the apparent axial dispersion coefficient is time dependent) which are observed on time scales larger than the characteristic time for mixing along the streamlines. The complex nature of the transient dispersion regime became apparent in a previous study (Desmet et al., 1996a, b), in which the axial dispersion in laminar Couette-Taylor flow (CTF) was investigated. CTF is a three-dimensional, axisymmetric flow consisting of an even number of toroidal, square vortices [Fig. l(a)], generated by the rotation of the inner of two concentric cylinders. In the (r,x)-plane, an array of square convection cells with closed internal streamlines is obtained [Fig.

* Corresponding author.

l(b)]. In Desmet et al. (1996a, b), the axial dispersion was investigated in the absence of a net axial flow rate by following the rate of tracer dispersion at the injection point. Although this approach allowed to determine the inter-vortex flux from long-time limit experiments, it was not suited to quantify the transient dispersion effects. The transient axial dispersion in VAFs has thus far only been the subject of a heuristic theoretical analysis (Pomeau et al., 1988; Young, 1988; Young et al., 1988; Dykhne et al., 1994), and to our knowledge, no relevant experimental data exist. This can be to the use of the conventional RTD methods (Kataoka and Takigawa, 1981; Pudjiono and Tavare, 1993), with which inconsistent measurements cannot be avoided: since the transient axial dispersion depends extremely strongly upon the initial tracer distribution (Desmet et al., 1996a), a correct interpretation of the data is only possible when the initial condition is accurately monitored. Furthermore, considering the transient nature of the process, evaluation of the tracer distribution at different instants is required. Both requirements are difficult to meet with the conventional RTD

2383

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G. Desmet et al.

(0

Ca) Fig. l(a). Toroidal vortex structure of Couette-Taylor flow.

k~--d O (b) Fig. l(b). Corresponding array of identical vortex units in the (r, x)-plane.

methods, but, as will be shown, can easily be satisfied with the axial scan technique (see Section 2). Another problem which has inhibited the progress in this research field, is the requirement of a model which allows to calculate the complex relation between the initial condition and the resulting tracer distribution. The numerical solution of the continuous convection-diffusion equation over a large number of vortex units requires large amounts of computation time. A simple, but sufficiently accurate two-dimensional model is therefore proposed (see Section 4). It should be noted that, in order to be able to accurately monitor even the fastest dispersion phenomena, our experiments have been performed with strongly viscous fluids. In the more frequently used aqueous fluids, the transient dispersion effects are less pronounced and are more rapidly eliminated, but remain important on the scale of most practical reactor residence times (see Section 7). Although our study has exclusively been performed in laminar CTF, it should be noted that other VAF-types exist. When a net axial flow is present, two major types (respectively moving and stationary VAFs) can be distinguished. With moving VAFs [e.g. CTF or Rayleigh-B6nard flow combined with a moderate axial flow (Gu and Fahidy (1985)], we denote those VAFs which are generated independently from the applied net axial flow rate and in which the vortex units move in a single file along the reactor axis under the action of this net axial flow. With stationary VAFs (multiple impeller flow, flow through pipes with periodic array of cavities, etc.), we denote those VAFs which remain in place under the action of the net axial flow rate, and in which the fluid particles are convected through the

successive vortex units. Strictly speaking, the results of the present study can only be used to predict the RTD of moving VAFs. Such VAFs are encountered in chemical engineering [e.g. cultivation of animal cells (Janes et al., 1987), combined reaction-separation bioreactors (Belfort, 1989), tangential filtration systems (Holechovsky and Cooney, 1991), continuous crystallization (Pudjiono et al., 1992), emulsion polymerization (Kataoka et al., 1995), photochemical conversions (Sczechowski et al., 1995) etc.], and also in the field of magneto-hydrodynamics and plasma physics (Rosenbluth et al., 1987), and, on a very large scale, in astro- and geophysics (Sagues and Horsthemke, 1986). However, for stationary VAFs which satisfy the description of the presently investigated flows (Section 4) and for which the dispersion parameters are independent of the net axial flow rate, the proposed methods and insights are also useful: they can be used to separately determine the dispersion parameters in the absence of a net axial flow. Such measurements have a much larger accuracy and yield a much more detailed description of the dispersion phenomena than is possible with the conventional RTD methods which are based upon the presence of a net axial flow. In the remainder of the text, the mechanisms causing the mass transfer across the vortex interfaces will be referred to as the inter-vortex transport and are represented by an inter-vortex exchange coefficient. As will be shown, the inter-vortex exchange coefficient is determined by the transport properties (velocity, diffusivity) in the penetration layers near the interfaces. All transport phenomena which occur outside these penetration layers will be referred to as the intra-vortex transport.

2. EXPERIMENTALSETUP Whereas the conventional single- and two-point RTD methods only yield information around a given residence time, the axial scan technique allows to perform successive scans of the entire tracer distribution at any desired moment, and is hence much more suited to study transient dispersion effects. Since CTF is of the Moving VAF-type (Section 1), this VAF is ideally suited to be studied at a zero net axial flow rate. The advantages of this measurement condition are already partially given elsewhere (Desmet et al., 1996a). For the present study, it is important to note that the presence of a net axial flow rate limits the possible number of successive axial scans, by the simple fact that the tracer progressively leaves the flow system. This is especially problematic when the infinite reactor condition has to be satisfied, i.e. no tracer may reach the end walls (which is one of the necessary conditions of the dispersion theorem presented in Part II). Compared to conventional RTD methods, where the axial velocity is fixed by the applied axial flow, the scan velocity (u) in axial scan measurements can be selected such that measurements with a large effective Bodenstein number (Bo)

Dispersion in vortex array flows I

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homt

step m¢

Fig. 2. Schematic representation of the experimental setup. The shaded areas represent the tracer distribution some time after a homogeneous injection in a given vortex.

Table I. Experimental conditions Ri = 0.044 m, d = 0.0225 m, L = 0.70 m; 5 x 10-~ m/s < u < 5 x 10-3 m/s 4.12x10 ~mZ/s
can be performed. This avoids curve tailing and the subsequent measurement errors. A schematic view of the experimental setup is given in Fig. 2. The experimental conditions are summarized in Table 1. To dispose of a maximum of free space in both directions, the tracer (Rhodamin B) was always injected in one of the two vortices situated in the middle of the reactor. For a more complete description of the injection procedure, the reader is referred to Section 4, Part If. The tangential dispersion effect has been eliminated by using injection times which are equal to two or three times the tangential fluid recirculation time. The tracer concentration was recorded with a fiber-optic spectrophotomettic probe (beam diameter = 2 mm), connected to a photodiode array (MCPD 1000, Otsuka Electronics). Using a transparent outer cylinder (plexiglass) and a highly polished inner cylinder to reflect the light beam, the spectrophotometric detection method allows nonintrusive measurements. The detection probe was translated along the reactor axis by means of a commercial axial positioning system. The scan velocity (u) could be varied between 4x10

5 ~ < u ~ < 2 x 1 0 -2m/s.

(I)

The fastest dispersion phenomenon (i.e. the dispersive bypass, see Fig. 10) occurred with a rate of the order of a few cm/s. This was just within the limits of our setup. The much faster intra-vortex convection (order 10 1 m/s and more) could not be monitored with our setup and has therefore been investigated with video recordings. The selection of the optimal scan velocity

depends upon a number of factors. The minimum scan velocity is of course determined by the rate of the phenomenon to be studied. Using large scan velocities leads to large Bo numbers, with the resulting minimum tailing, but the required sampling frequency is then so large that only small sampling times (during which the measurement signal is integrated) can be used. This leads to very noisy signals. Slower scan velocities allow larger sampling times, leading to large S/N ratios, but the smaller Bo numbers can lead to extensive tailing of the peaks. A suitable axial scan velocity has been selected for every experimental condition. In order to accurately determine the initial condition, a first axial scan is always performed immediately after the injection. By operating the detection unit near its maximum sensitivity, the base-line of the measurement signals was found to be extremely sensitive to variations of the optical path length. In order to reduce these variations, the probe was slightly pressed against the outer cylinder. The remaining variations in optical path length, due to the tolerances on the mechanical construction, were eliminated by subtracting a signal obtained by axially scanning the reactor without injecting tracer. This also allowed to eliminate other deterministic disturbances of the measurement signal, such as those caused by the presence of imperfections (scratches) on the inner and outer cylinder. 3. OBSERVED DISPERSION EFFECTS

As is shown schematically in Fig. 3, three different injection types (types A-C) are conceivable when considering block-shaped initial tracer distributions (i.e. initial concentration is either Co or 0). They can be characterized by the dimensionless parameters a a n d b ( 0 ~ < a ~ < l : 0 ~ h ~ < 1).

(2)

The case in which the entire vortex cross-section is marked homogeneously (type D) arises as the limiting

2386

G. Desmet et al.

I j, type (A)

type (B)

type (C)

type (D) 'homogeneous' injection'

Fig. 3. Survey of the different conceivable block-shaped initial tracer distributions.

case (a = 0, b = 0) of the three other types. Experimentally, the different block-shaped initial conditions can be easily imposed, just by radially moving the injection needle tip between position a and b during the injection process. The two-dimensional shape (in the (r,x) plane) of the initial tracer distribution is then created by the fast intra-vortex convection. 3.1. Axial scans of global axial dispersion in the case of slow intra-vortex dispersion Figs 4-8 show examples of successive axial scans which are typically obtained for each of the injection modes presented in Fig. 3. It should be noted that all responses have been obtained for the same hydrodynamical condition (~o = 21.21 rad/s, v = 1.09 x 10 -4 m2/s, Sc = 2.85 x 106). Compared to scans performed at later moments, some first scans show a relatively large scatter in the injected vortex. This is due to the occasional presence of long persisting strings of unmixed tracer in the poorly mixed vortex bulk. The asymmetrical curves are due to the fact that the scan velocity was not infinitively fast as compared to the dispersion phenomena. The succession of peaks and valleys obtained for type (A) injections (Fig. 4) is due to the fact that the detection probe measures the radial average of a tracer distribution such as depicted by the shaded areas in Fig. 2. The open-cellular structure of this tracer distribution (see also Fig. 10) is caused by the so-called dispersive bypass process (see Section 3.2), which is a typical dispersion phenomenon observed in laminar VAFs when tracer is injected near the vortex interfaces. Its effect upon the global axial dispersion has already been discussed in a previous study (Desmet et al., 1996a) and is more extensively treated in Figs 9 and 10. The steep intra-vortex concentration gradients reflect the slow nature of the intra-vortex dispersion. This clearly refutes the concept of a perfectly mixed vortex bulk, an assumption which is very popular in studies of laminar CTF (see e.g., Kataoka and Takigawa, 1981; Pudjiono and Tavare, 1993). For type (B)-injections, the spreading of the tracer distribution is initially limited to the vortex core. The responses are then marked by the gaussian profiles

which are typical for pure diffusion or dispersion phenomena (Fig. 5). It is only when the tracer fluid elements reach the vortex interface that they are subjected to the dispersive bypass mechanism (see Section 3.2) and are transported along the system of outer fluid layers (see discussion Fig. 8). Injections of type (C) initially yield similar spreading gaussian profiles, but, as the recorded concentrations are radially averaged, the concentration profiles now show a local minimum around the vortex centre (Fig. 6). The local minimum then corresponds to the presence of an unmarked region near the vortex centre. The slow spreading (see time scale) of the tracer distributions (Figs 5 and 6) and the slow 'filling' of the central gap (Fig. 6) reconfirms the weak intra-vortex dispersion. In Fig. 7, the response to a so-called homogeneous injection (type D) is given. Just as for type (A) injections, the presence of sharp peaks, caused by the dispersive bypass along the outer layers of the vortices, is the most apparent effect. For injections of type (B) and (C), in contrast to injections of type (A) and (D), where high concentrations in the outer vortex layer are imposed by the injection mode, the tracer concentration in the outer layers is at a very low level when the dispersive bypass begins to spread the tracer over the outer layers of the nearby vortices. At the interfaces of the injected vortex, the supply of tracer (coming from the vortex core) occurs more slowly than the withdrawal. As long as the system behaves as axially infinite, the tracer concentration in the outer vortex layers then never exceeds this very low (and most probably undetectable) concentration level. This complicates the recording of RTD curves. In Fig. 8, this effect is illustrated by considering a limiting case (b --, 0) of type (B) injections. Since in this case the initial tracer block occupies nearly the entire vortex width, the initial gaussian-like spreading (see, e.g. Fig. 5) can only proceed over a very short distance before the outer vortex layers are reached. There, the tracer is immediately removed by the dispersive bypass process and its concentration thus remains at a very low level. As long as no tracer has reached the reactor end walls (i.e. as long as the system behaves infinite), the concentra-

Dispersion in vortex array flows--I

2387

0.12 Cobs(')

0.10

vortex interfaces

o.o8l scan (a)

0.06t 0,04 0.02 t ] i

0,00 L __..^~ ^~

,,

0

- -

10

20

~_~.~

.~._:.. r~

time (s)

30

40

0.05 .......................................................... Cobs(') 0.04

0.03

scan (b)

0.02

0.01

0.00

~'~.-"-'~--'.

170 0.035I

.

~

'

190

210

.

..

,

. . . . . . . . .

~_r_

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230

..................................................................................................................................................................

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s c a n

°°'tooo " 0.010l[

~

0.000 ,_~__~.a~._:._..... 460

I~¢ ~I/

500

II 540

(c)

& ~.I._A~.~,.._~ ~,_~ 580 time (s)

Fig. 4. Axial scans(u = 3.0 x I0 3m/s) of a responseto an injection of type (A), with a ~ 0.95. tion never exceeds this level, and one records an apparently nonspreading peak [see Fig. 8(a)]. Careless interpretation of RTD curves would then lead to the erroneous conclusion of having measured a perfect plug flow. In the presently used experimental

apparatus, the first amounts of tracer reach the (closed) reactor end walls after they have travelled a distance of some 14 to 16 vortices in each direction. The process of unhindered dispersion over the system of outer fluid layers (dispersive bypass) is then stopped

2388

G. Desmet et al. 0.08

Cobs(-)

- - t--275 s 4 - t=5545 s --t=12476 S ---t=22B74 S .-- t--28470 S

/~ c'~/ /~'~

0.06

/ \

I

0.04

'~ /

~ex i ' t ~ "

'

~ i n t e ~ .

,..

0.02

0.00 -0.01

0.00

0.01

x (m)

Fig. 5. Axial scans (u = 5.0 x 10-4 m/s) of a response to an injection of type (B), with b ~ 0.77.

0.16 - -4- t=67.5 s -.~t=7304 s --t=18212 s ,~ t=29192 s -=-t=86611 s

Cobs(-) 0.12

0.08

0.04

vortex interface

vortex interface

I 0.00

¥ ~ -

-0.01

0.00

-r

0.01

x (m)

Fig. 6. Axial scans (u = 5.0 x 10-4 m/s) of a response to an injection of type (C), with a = 0.51 and b -~ 0.09.

and the tracer concentration begins to build up in the outer layers [Fig. 8(b)]. After a given time, the concentration profiles in all the noninjected vortices of the array become nearly identical and the tracer fronts only proceed very slowly towards the vortex centers. Again, this clearly shows that the dispersive bypass occurs much more rapidly than the intra-vortex dispersion.

3.2. Dispersive bypass effect Figure 9 shows a series of photographs of the different transport phenomena involved in the dispersive bypass effect observed in Figs 4 and 7. During the initial stages, this effect completely determines the global tracer dispersion for type (A) and type (D)

injections when the dispersion in the direction perpendicular to the streamlines is weak. In order to obtain a maximally detailed view, a reactor with a larger annulus width (d = 0.052 m) was used and the photographs were taken in the very first seconds after an injection in the outer vortex layers. The different phenomena can be identified from the schematic twodimensional description given in Fig. 10. In Fig. 9, the positions of the vortex interfaces are marked by the white lines. The photographs clearly show how tracer, which is injected at position A (see Fig. 10), is convected along the inner cylinder [Fig. 9(a)] and flows radially [Fig. 9(b)] along the vortex interface towards the outer cylinder (B). During this radial passage, tracer is exchanged with fluid elements of the vortex at the left

Dispersion in vortex array flows

I

2389

0.05 ......................l ..............h-omogeneousiY..........................................................

~

Cobs(') 0.04

//

injected vortex

0.03

scan (a)

/

0.02

~J

0.01 +

I I

o.ool ....... o

F

100

50

150

-

2ootime

(s)

-

25o

0.08

Cobs(') 0.06

scan (b)

0.04

0.02

0.00 800

900

1000

1100

time (s)

-T

0.08

Cobs(') 0.06

o.j

scan (c)

J

l

0.02 t

J

0.00 2050

,

I

2150

2250

2350

245°time

(s)

2550

Fig. 7. Axial scans (u = 9.0 × 10 -4 m/s) of a response to an injection of type (D).

of the initially injected vortex. The tracer which remains in the injected vortex is then further convected [Fig. 9(c)] along the streamlines of this vortex and moves along the outer cylinder (C~) towards the interface with the vortex at the right. There, while moving

radially (C2) towards transferred crossed the streamlines interface at

the inner cylinder, some tracer is to this vortex. The tracer which has interface is then convected along the of its 'new' vortex (C3) and reaches the position C4, where the tracer is able to

G. Desmet et al.

2390 0.600

t=275 s t=11399 s ..... t=29219 s

Cobs(-)

- -

0.500 f

0.400

0.300

0.200

o.loIo 0.000

(a)

.---

/ ~-'~-

I

-0.05

"

'1

-

-0.025

I

I

I -

0.0

0.025

0.05

x (m)

Fig. 8(a). Axial scans (u = 5.0 x 10- 3 m/s) of a response to an injection of type (B) with b ~ 0 (b ~ 0.04). The peak width appears to remain constant.

0.020 0.018 ",Jl~obs/"~ ~

t=11399 s t=29219 S --t=87060

0.016

S

0,014 0.012 0.010 0.008 0.006

/ill i:!

0.004

~!i i~ 0.002 0.000 (b)

-0.10

-0.05

0.0

0.05

0.10

x (m)

Fig. 8(b). Detailed view of global tracer distribution at long times in a closed reactor system.

invade the following vortex. The transferred tracer is then further convected along the outer cylinder [-Fig. 9(d), position D in Fig. 10] and flows on to the following interface. Figure 9 thus clearly shows how tracer is transported over a distance equal to a multiple of vortex widths in a very short time. This dispersion process, in which the outer vortex layers behave as serially connected mixing units, will further be referred to as dispersive bypass. The fastest tracer particles travel a distance of n times the vortex width in a time equal to n/2 times the circulation time in the outer vortex layer. Considering the very large velocities in the outer vortex layers, which were observed from our video recordings to lead to circula-

tion times of the order of 0.1 to 0.5 s, it is obvious that the initial tracer dispersion caused by the dispersive bypass occurs very rapidly. Due to the small intravortex dispersion rates, nearly all tracer remains initially in the outer vortex layers. The global dispersion is then entirely determined by the dispersive bypass effect. 3.3. Determination of local mixing enhancing effects The axial scan technique is also very well suited to investigate some more detailed aspects of the intravortex transport. F o r example, during most of our laminar C T F experiments, it was observed that, as soon as a given vortex is reached, the tracer fluid

Dispersion in vortex array flows

(d) Fig. 9. Dispersive bypass mechanism (o) = 19.23 rad/s, v = 1.09 x 10 4 m2/s, d = 0.052m) in three-dimensional CTF [IC = inner cylinder: OC - outer cylinder). The photographs are taken at t = 0.1 s (a), t = 0.3 s (b), t = 2.1 s/c) and t = 2.4 s (d). The white curved lines represent the vortex interfaces. The vertical streaks in photographs (c) and (d) are caused by the tracer which is present near the outer cylinder, and which, due to the three-dimensional nature of the flow, apparently covers the entire vortex.

immediately marks a layer which is much broader (1 to 2 mm) than what can be predicted from the penetration model (see Part II) when assuming pure molecular diffusion. It is as if the intra-vortex dispersion occurs in two distinct phases: the tracer invades nearly immediately the outer few millimeter of the vortices, whereas the subsequent dispersion over the rest of the vortex occurs much slower. Since the inflowing and outflowing jets are strictly laminar, as evidenced in Part II, the initial strong penetration can only be explained by the presence of some local mixing enhancing effects near the inner and/or outer cylinder. This hypothesis can be supported by existing hydrodynamical studies. Barcilon et al. (1979) showed

I

2391

that for a flow past a concave wall (which is the case for the outer cylinder), centrifugal instabilities give rise to the formation of Goertler vortices as soon as T a > 300 (i.e. for the present reactor geometry). The authors claim that it is precisely the presence of these Goertler vortices which causes the typical herringbone flow pattern observed in so many visualization studies of C T F [e.g. Legrand and Coeuret (1987)]. On a convex wall, Goertler vortices can only be generated when this wall is moving IWei et al., 1992). Since this is the case in CTF, the presence of Goertler vortices or other instabilities near the inner cylinder can be expected. For the presently used reactor geometry, the relation proposed by Wei et al. (1992) shows that the boundary layer flow near the inner cylinder becomes unstable from Ta = 200 on, and predicts values of the order of 1 to 2 mm for the Goertler vortex thickness at the inner cylinder, corresponding closely to our observations. To investigate these mixing enhancing effects, the following experiment was undertaken. An experimental condition (i.e. large co) was selected for which the outer layer turbulence could be clearly detected: due to the loss of axial resolution caused by the relatively large width {2 mm) of the detection beam, a relatively large value of (,) is required in order to have a sufficiently broad turbulent zone. First the vortex bulk was marked with tracer. The resulting peak served as a measure for the slow dispersion in the bulk. After one hour, during which the peak remained nearly unchanged, tracer was injected in the outer vortex layers, leading to the two peaks situated at the vortex interfaces [Fig. 1 l(a)]. Unlike the central peak, the height of the outer two peaks decreased very rapidly. This is of course mostly due to the dispersive bypass mechanism, but the fact that the peaks are broadening much more rapidly than the central peak shows that the dispersion in the outer layers is indeed much stronger than in the vortex bulk. In Fig. I ltbt this is supported by the fact that the fitted curve (a), which has been obtained by assuming pure molecular diffusion in the outer-vortex layers, does not fit the peak broadening at the interfaces, whereas curve (bl, which has been obtained by gradually varying the intra-vortex dispersion coefficient tsee Section 4) over two orders of magnitude in the outer few mm of the vortex, fits perfectl). The fact that in the outer layers a varying intra-vortex dispersion coefficient has to be used increases the titting problem. The value for the inter-vortex exchange coefficient used in the simulation leading to both curves (a) and (b) has been taken from the correlation for strictly laminar inter-vortex transport established in Part I1. Another illustration of the potential of the axial scan technique is given in Fig. 12, where the dispersion effects arising from the injection of a few air bubbles in the vortex center are investigated. The injected air bubbles had a diameter of about 1 ram, being about 20 times smaller than the vortex diameter. It was found that injected air bubbles preferentially remain in the vortex core and do not migrate to

2392

G. Desmet et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

-

D

......

.... t

................

scanning detection probe

Fig. 10. Two-dimensional schematic representation of the dispersive bypass mechanism. The letters refer to the description given in the text. The rectangle corresponds to the dimensions of the photographs. The radial averaging of the regions which are marked by the tracer (shaded areas) yields the peaked concentration profiles such as in Figs 4 and 7.

0.20 --t=

Cobs(')

14.5 s

t=3435 t=3627 - - t=3668 - - t=3868 .... t=4268

0.16

s s s s s

0.12 initial pe~k

0.08

0.04 .'

',

,

I

(a)

-0.01

0.00

0.01

x (m)

Fig. ll(a). Evidence of the enlarged rate of tracer dispersion in the outer vortex layers due to turbulent phenomena near inner and outer cylinder (co = 51 rad/s, v = 1.09 × 10 -4 m2/s). The central peak allows the comparison with the dispersion rates in the vortex center.

£?,:1

exp. curve at t= 3868

0.20

0.15

curve~ 0.I0

il

c u ~ e (b) I

~

i!

0.05

0.00

(b)

-0.01

f

I

0.00

0.01

x (m)

Fig. 1 l(b). Fitting of the concentration profile obtained at t = 3860 s (bold line). The existence of turbulent effects near cylinders can be derived by comparing the best fitted curve obtained by assuming pure molecular diffusion in the outer zones (curve a) with the best fitted curve obtained by increasing the dispersion rate gradually when approaching the vortex boundary (curve b).

2393

Dispersion in vortex array flows--I 0.12 --t=69 s

Cobs(-)

.... t= 1025 s --I=4132 s

0.09 tracer injection in outer vortex layers 0.06

O O3 0.00

j

:":

injected air bubbles in vortex core

"l

-0.10

-0.05

0,0

0.05

x (m)

0.10

Fig. 12. Influence of the presence of injected air bubbles in the center of a given vortex upon the rate of intra-vortex dispersion (~o = 21.21 rad/s v = 1.09 x 10 -4 m2..'s, u = 5.1 x 10 -4 m/s).

another vortex over a period which is orders of magnitude longer than the period considered in Fig. 12. A similar effect has recently been reported by Lismonde et al. (1996). Bubble coalescence was also found to be negligible over the same period. As can be denoted, the intra-vortex dispersion in the injected vortex is strongly enhanced. Fitting the M Z model (see Section 4) to the experimental concentration profiles yielded a value of Dintr a ~-- 9 x 10 -8 mZ/s (for definition of Dintra, see Section 5), being about 2000 times larger than the value in the noninjected vortices. Other similar experiments all yielded Dintr a values of the same order of magnitude, but a relatively large scatter on the Dintr a values existed. We believe this is due to the variation in bubble diameters and due to the occasional presence of smaller 'satellite' bubbles whose formation could not be avoided during the bubble injection.

(molecular diffusion in vortex core, outer layer turbulence, etc.) is represented by the coefficients K i (i = 2 . . . . . N.). By varying the K~ values according to a certain pattern with the index, the model also allows to simulate local turbulent dispersion effects such as those shown in Figs 11 and 12. Writing the mass balance for each zone yields a system of N~ O D E s for each of the N vortices: Outer zone (i = 1): dC1 j = K I S I ( ~ j + K 2 S 2 ( C 2 , j - CI,j),

j = 1,N (3)

(~j = CI, 2 -- C I . 1 ,

j = 1

!4a)

~j=CI,j+I-2CI,jq-CI,j_I,

j=2, N-I

(4b)

¢bj = Cl.~v=-I - Cl.u=,

j = N

(4c)

Vip'S'-

with

Intermediate zones (i = 2, N= - 1): 4. M U L T I - Z O N E

MODEL

In the present study, we have generalized the twozone model used in Desmet et al. (1996a) by increasing the number of internal, perfectly mixed zones from 2 to N~, in which N= may vary from 2 to infinity. The increased refinement of this multi-zone model (MZ model) allows a much more accurate representation of the initial tracer distribution. In the M Z model, the mass transfer mechanisms occurring at the vortex interfaces are represented by a localized mass transfer coefficient K v The dependency of Ka upon the flow conditions is investigated in Part II. The convection pattern is represented by the shape of the internal zones (see below). The perfectly mixed nature of the nested zones corresponds to the fact that we only look at phenomena occurring on time scales larger than the characteristic time for mixing along the streamlines. The intra-vortex dispersion in the direction perpendicular to the streamlines

dCi,j Vi ~ = K i S i ( C i - 1 , i - Ci,j) 1,N

(5)

1,a - CN=j), j = 1,N

(6)

~- K i + l S i + l ( C i + l , j - C i , j )

,

j=

Central zone (i = N=}: dCu=j dr - K%Sv:(C.~,=

VN=

Equations (4a) and (4c) represent the condition of impermeability at the reactor end walls. In the present study, all fittings have been performed with an M Z model (model A, Fig. 13) which respects the shape of the streamlines derived from the most simple stream function with which the boundary conditions of the flow field (i.e. velocity is zero at r = 0 and r = d, and velocity is maximal x = 0 and x = d) can be satisfied: sin (rcx/d)r/d(1 - r/d) = t<.

(7)

G. Desmet et al.

2394

model A

--V 1 K ,K2 model B

model C

~. (L=d for square vortices)

Fig. 13. Lay-out of the multi-zone model and different investigated zone shapes: stream function model (model A), square-zone model (model B) and circular-zone model (model C).

To investigate the sensitivity of the model predictions to the shape of the internal zones, model A has been compared (see Sections 5 and 6) with two MZ models with an extreme zone shape: the square-zone model (model B, Fig. 13), which can be assumed to yield a good approximation of the intra-vortex transport in the outer vortex layers, and the circular-zone model (model C, Fig. 13), which can be assumed to yield a good representation of the dispersion in the vortex center. The width of the zones has been characterized by the axial width (6) obtained at half the vortex height (see Fig. 13). Throughout the present study, all calculations have, for reasons of simplicity, been performed with equally spaced zones (i.e. with a constant f-value). As will be shown in Part II, this condition is however not restrictive. In the remainder of the text, the intra-vortex dispersion in the direction perpendicular to the streamlines will be referred to as Din,a, given by D~ .... =

K~6 (i>~ 2).

(8)

It should be noted that Di,tra can be varied with the distance from the vortex core to represent local mixing enhancing effects. A constant value of Dintra is obtained by considering constant Ki (i >1 2)- and 3-values for each zone. It was found that all curves presented in Figs 4-8 could be accurately fitted with the MZ model (with a discretization according to the stream function model), using a unique set of Ki values (i = 1,Nz). In all the simulations presented in this study, a number of Nz = 100 internal zones has been used. A best fit is obtained when the K~ values which represent the intra-vortex dispersion (i.e., for i>~ 2) are increased gradually (i.e. for decreasing /-values) over two to three orders of magnitude in the outer few mm of the vortices. An accurate description of the dispersion in

the outer-vortex layers was found to be most stringent for injections of type (A) and (D). The necessity to use Ki-values (i >~ 2) which vary with the index seriously increases the fitting problem: it was our experience that, for a given degree of discretization, the experimental curves could be satisfactorily fitted with more than one set of Ki values. We believe that the impossibility to discriminate between the large number of possible fitting parameter sets is due to the relatively large beam width of the presently used detection probe. For the same reason, the curves presented in Figs 4, 7 and 8 (essentially determined by the dispersion effects in the outer vortex zones) did not allow to discriminate between the square zone model and the stream function model and the curves presented in Figs 5 and 6 (essentially determined by the dispersion in the vortex core) did not allow to discriminate between the circular zone model and the stream function model. 5. INTRA-VORTEX TRANSPORT

In order to quantify the dispersion rates in the vortex center, responses to type (B) injections have been used. Three different water/glycerol mixtures have been investigated (see Table 2). For each mixture, three different values of the angular rotation speed of the inner cylinder (co) were investigated. All

Table 2. Intra-vortex dispersion experiments (Di.... = D~o~)

Series 1 Series2 Series 3

v (m2/s)

Di.... (m2/s)

4.12×10 -5 6.80x10-5 1.09× 1 0 - 4

7.49 x 10-11 5.31x10-11 3.83 x l0 ix

Dispersion in vortex array flows- 1

2395

25° I

/ ~

A~t2 (s2)

Series 1,0):7.41; 19.21; 30.12 rad/s

200

Series 2 ,~0)=6.02; 19.86; 43.44 rad/s

Seliesy

o co=0 radts

Series 1 /

: m=O rad/s

~

~

"

~

Series 3

= [u=u IdtUb

I

0 ~

0.00

1.1012

2.1012

3.1012

4.1012

5.10 ~2

2.At/u 2 (s3/m 2) Fig. 14. Ao,z vs 2At/u 2 for each of the three investigated water/glycerol mixtures. The straight line relationships are independent of the angular velocity of the inner cylinder.

4.0 10-11

i

0.5 A(~tZ.uZ/At (mZ/s) 4.0 10-11

i ,exp. values(m>0) exp. values(m=O)

./

3.9 10-11

model C, Fig. 13

n

t'

3.8 10-11

u

D

3.7 10-1!

model A, Fig. 13

.....................................

4m. . . . . . . . . . .

i ..........

........

3.6 10-11 3.5 1011

100

I000

10000

time (s)

I00000

¸ Fig. 15. 0.5 Aa;'u 2 .'At vs At for the experimental data of series 3. The circular-zone model (model C. Fig. 13) accurately fits the data of the co = 0-cases. The stream-function model (model A, Fig. 13) is able to represent the influence of the intra-vortex convection upon the effective axial dispersion in the c~ > 0-cases.

values of co were chosen such that the l a m i n a r C T F regime was established. The dispersion rates were recorded by m e a n s of a series of successive axial scans (cf. Fig. 5). The experiments were stopped as soon as the tracer c o n c e n t r a t i o n s near the vortex interface became significant, since then the inter-vortex transport begins to influence the observed dispersion rates. To o b t a i n a maximal time range in which the responses are i n d e p e n d e n t from the inter-vortex transport, the injected tracer blocks all h a d a dimension h > 0.85. F o r each of the three investigated viscosities, the experimental results were represented as the difference in second-order time m o m e n t between two successive axial scans (Aa/~) vs the c o r r e s p o n d i n g difference in first-order m o m e n t (At) (Fig. 14). Since the o b t a i n e d relations between k o 2 and At were found to be i n d e p e n d e n t of (o, the dispersion rates were also m e a s u r e d when the fluid was completely at rest ({o = 0). F o r this purpose, a vortex core

was h o m o g e n e o u s l y filled with tracer while the C T F regime was established. After the initial scatter h a d decayed, a sharply defined, h o m o g e n e o u s l y m a r k e d initial tracer block was obtained. The r o t a t i o n of the inner cylinder was then stopped a n d the dispersion rate was recorded as in the (o > 0-cases. Figure 14 shows that, for a given viscosity, the dispersion data o b t a i n e d in the (o = 0-case can be represented by the same linear relationship which already represented the dispersion for all the (.,) > 0-cases. F o r the circular zone model (used to represent the convection-free diffusion p r o b l e m o b t a i n e d in the (rJ = 0-caset, it can easily be s h o w n t h a t the proportionality c o n s t a n t between Aa { and At/'lt 2 exactly equals 2K/6 when a c o n s t a n t Ki-value {i >~ 2) is used a n d when the zones are equally spaced (i.e. c o n s t a n t ,5): A a ff = 2 D i . , r ~ A t / u 2

with Di,,m, = K i d (i >~ 2)

(9)

G. Desmet et al.

2396

Equation (9) has been used to derive the Di,tr~-values from the co = 0-data presented in Fig. 14. The obtained Dintr a values are presented in Table 2. In Fig. 15, the values of 0.5 A a Z u 2 / A t are plotted vs At for each point of the experiments of series 3 (see Table 2). Whereas Fig. 14 does not allow to distinguish between the co = 0-case and the co > 0-case, the manner of representation in Fig. 15 clearly shows how the presence of a given flow pattern alters the observed axial dispersion in the vortex core. In Fig. 15, the experimental data are also compared with simulations performed with the MZ model. The plain line is obtained with the circular-zone model (model C, Fig. 13) with a constant K~ value (i >/2, with Ki = Dintra/(~), the dashed line is obtained with the stream function model (model A, Fig. 13) and using the same K~- and f-value as in the calculation of the plain line (i.e. considering the same degree of diffusion in the direction perpendicular to the streamlines as the degree of diffusion in the co = 0-case). The good agreement between experiment and model shows that the zone shape derived from the stream function given in eq. (7) is a sufficient approximation to the actual flow pattern. For the two other investigated viscosities, graphs similar to Fig. 15 are obtained. From the fact that the dispersion data obtained for the different og-values (including the 09 = 0-case) can all be represented with a single, constant K~-value (i ~> 2), it can be conjectured that the transport in the vortex core in laminar CTF occurs by pure molecular diffusion [thus refuting the popular assumption of a perfectly mixed vortex bulk, see, e.g. Moore and Cooney (1995)]. This is justified by the fact that the obtained Di,,ra-values agree with the molecular diffusion coefficients (Dmol) given in literature for similar water/glycerol mixtures (Kataoka and Takigawa, 1981). In the remainder of the study, the Dintra-values given in Table 2 have therefore been used as the Dmovvalues. 6. GLOBAL AXIAL DISPERSION

In order to compare the different observed global axial dispersion modes on a quantitative basis, one used the second-order time moments (a~) of the successive axial scans made during experiments as those shown in Figs 4-8. In Fig. 16(a), one has represented the difference in second-order moment between a given axial scan and the first scan ( A a ~ i = a~i - a ~ l with i = 2, total number of scans) vs the corresponding difference in first-order moment ( A t i = t i - tl) with

always yield a linear relationship in the case of type (D)-injections since the influence of the intra-vortex transport can only be eliminated by subtracting two different a~-values (see mathematical proof in Part II), and, on the other hand, the representation in terms of the difference between two successive scans (A~L: ~ , - ,~t,i-1 VS Atm,i = tm,i -- tin.i-1) which always yields a linear relation for type (D)-injections, but which does not yield a proper representation of the variation of the dispersion rate over the entire time scale (e.g. when the axial scans are performed with fixed time intervals, all experimental points have the same abscissa value). Considering that all Aa~-values represented in Fig. 16(a) have been obtained for the same set of hydrodynamical parameters (see Section 3), both the large variety of observed dispersion modes as the time scale over which they extend is striking. In Fig. 16(a), one has also compared the experimental Aa{-values with those predicted by the stream function MZ model, using a unique set of parameter values obtained from other, independent experiments: the different curves are obtained by imposing different initial conditions to the model. The shape of the initial condition was always estimated from the first axial scan of each series. The Kl-value was obtained from the experiments presented in Part II, the Krvalues in the vortex bulk were obtained from the experiments shown in Section 5, and the Ki-values in the outer fluid layers were varied in a similar way as has been proposed in Fig. 1 l(b). The thus obtained agreement between the experimental points and the theoretical predictions demonstrates the validity of the MZ model. The model can hence be used with confidence to extrapolate the recorded dispersion behavior over the entire transient range. This was the only way in which the end phase of the transient regime could be studied, since, as can be seen from the time scale of Fig. 16(a), an experimental study of the entire time range requires extremely long experimentation times. As a consequence, also unpractically long reactor lengths are required, since the condition of unhindered axial dispersion has to be satisfied in order to reach the final stationary dispersion regime (see Part II). The use of less viscous fluids, with which the stationary regime is reached more rapidly, would not have allowed to study the dispersive bypass mechanism in as much detail. Figure 16(b) shows the data o f F @ 16(a) in terms of apparent axial dispersion coefficients (Dax,app), as they would be obtained when performing conventional two-point RTD experiments:

t = M,,1/Me,o

and

Oax'aPP:2u2 At" aZ,

=

(M,,2/M,,o)

- [Mt,1/M,,o]

2.

(10)

This representation compromises between, on one hand, the representation of a~i vs ti, which allows to represent the variation of the dispersion rate over the entire experimental time scale, but which does not

(11)

In the remainder of the text, the Dax,app representation is preferred, since it yields the graphs with the largest resolution. The large diversity of possible D.x,.ppvalues which can be obtained for one given hydrodynamical condition [Fig. 16(b)] clearly demon-

Dispersion in vortex array flows

I

2397

0.5 A~3tZ.uz (m 2) 1.10-2

(a) ,

1.10 .4

A

1.10-6

1.10-8

1.10-10 10

1.102

1.103

1.104

1.105

1.106

At (s)

(a)

Fig. 16(a). Influence of initial tracer distribution upon transient dispersion behavior. The different injection types are defined in Fig. 3. Two type (A)-injections (curve a: a = 0.97, curve a': a = 0.75), three type (B)-injections (curve b: b = 0.91, curve b': b = 0.25, curve b" and curve b'": b < 0.05), one type (C)-injection (curve c: a = 0.29, b = 0.61) and one type (D)-injection (curve d) have been considered. The symbols represent the experimental data and the plain lines are predicted by the MZ model. The difference between curve b" and curve b'" is only due to the use of different K~-values in the outer zones of the MZ model (see text). The plain lines have been obtained with the stream function MZ model, using a unique set of dispersion parameters (K1 = 1.21 x 10 5 m/s. Ki(i >/ 5) = 1.30 x 10 ~ m's, & = 1.125 x 10 -4 m).

1.10 .5 ,

(a)

Dax,app(m2/s) 1"10"5 f [

,

• "4

1.10 -11 r ' 1.

10

1.102

1.103

(b)

1.104

1.105

1.106

1.107

.10 8

At (s)

Fig. 16(b). Corresponding apparent dispersion coefficients. The meaning of Fm,x is given in Section 8.

strates that the use of conventional two-point R T D methods in the transient dispersion regime is completely out of order. In order to investigate the influence of the shape of the internal zones of the M Z model, the theoretical dispersion curves presented in Fig. 16(b) (plain lines, obtained with the stream function model defined in Fig. 13) were compared with those obtained with the square-zone model (model B, Fig. 13). The influence was found to be rather small, such that it be concluded that for the presently investi-

gated conditions, the exact knowledge of the intravortex flow pattern is only of secondary importance. Fig. 16(b) clearly shows that three major types of transient behavior can be distinguished, depending upon whether Dax,app decreases with the time (Section 6.1), increases with the time (Section 6.2), or is independent from the time (Section 6.3). The different dispersion modes can be classified according to the different types of initial tracer distributions which are distinguished in Fig. 3. It should be noted that for

2398

G. Desmet et al.

initial conditions which deviate from the blockshaped initial conditions shown in Fig. 3 (for example, applying a linear gradient between C = Co in the vortex center and C = 0 near the interfaces), a variety of other dispersion modes will be observed. However, it will always be possible to explain the observed effects from the responses to the block-shaped initial conditions, since these can be considered as the extreme cases of possible initial tracer distributions. In the long-time limit all curves converge to the same stationary dispersion regime, in which all vortex units behave as perfectly mixed. The system then behaves as a series connection of ideal stirred tanks with symmetrical backmixing, for which it can be shown [see, e.g. Desmet et al., (1996)] that Acr~ varies according to (12a)

Atr 2 = 2 K l ~ At/u 2

~Dax,app = K i d

(2 = d for square vortices).

(12b)

6.1. Transient dispersion modes f o r which D .... pp decreases with the time

injected tracer is readily involved in the dispersive bypass process. The remaining, relatively large fraction of tracer (i.e. the tracer situated in the most inner layers of the injected zone) is initially only subjected to the slow molecular diffusion (Section 6.2). The contribution of this fraction to Aa~ is hence negligible, leading to lower average dispersion rates. 6.2. Transient dispersion modes f o r which Dax,,pp increases with the time For type (B) and (C)-injections, all obtained dispersion curves are situated below line (d). F o r these injection types, the dispersion process is initially limited to the core of the injected vortex. The initially (small) constant Dax,app-values reflect the fact that in the vortex core the transport is a pure (slow) dispersion process. The initial dispersion process is hence described by the linear relation between Aa 2 and At given by eq. (9). As soon as a significant fraction of tracer has reached the vortex interface, Dax,appbegins to increase by the increasing amount of tracer which is involved in the process of invading new vortices along the apparently serially connected system of outer vortex layers. Comparing curve (b) with curve (b'), curve (c) and curve (b"), it can be denoted that the deviation from the first constant regime is most sharply when b~l. The large influence of the outer-layer turbulence (cf. Section 3.3) upon the axial dispersion can be derived by the large difference between curves (b") and curves (b") in Fig. 16(a). In the fitting leading to curve (b"), the turbulent dispersion phenomena in the outer layer are represented by increasing the Ki-values over two orders of magnitude in the very outer zones of the MZ model. Curve (b'") is predicted by the MZ model for a similar injection condition as for curve (b"), but by assuming pure molecular diffusion in the outer layers (Ki is constant for i ~> 2). For b ~> 0.2, the influence of the outer-layer turbulence upon the dispersion curve was found to be negligible (curves b, b' and c).

For type (A)-injections, all dispersion curves are situated above line (d), which represents the Dax,appvalue obtained in the long-time limit regime. This can be explained by the fact that for type (A)-injections, tracer is present near the vortex interfaces from t = 0 on. The tracer can hence immediately be transferred to the adjacent vortices according to the dispersive bypass mechanism described in Figs 9 and 10. For the presently considered case of weak intra-vortex dispersion, the tracer initially only penetrates the very outer layers of the freshly 'invaded' vortices. Due to the rapid circulation around the vortex streamlines, the outer vortex layer can be represented by a perfectly mixed zone, corresponding to the outer zone (index i = 1) in the MZ model. The axial dispersion is hence initially determined by the system of outer vortex layers which behaves as a series of ideal stirred tank reactors with symmetrical backmixing. The axial dispersion can thus be represented with a constant Dax,app-value. Compared to the case in which the 6.3. Dispersion modes .for which Dax,app is constant entire vortices are perfectly mixed (such as in the long f r o m t = 0 on time limit), the D~,,app-values are initially enhanced by As can be denoted from Fig. 16(b), the time scale on a factor which is equal to the ratio of the total vortex which the transient dispersion regimes exist strongly volume to the volume of the outer vortex layers over depends upon the initial conditions, such that for which the tracer is able to penetrate during the initial a ~ 0 and/or for b ~ 0, these transient regimes might stages of the dispersion process. This explains the occur undetectably rapid. In the limiting case of a = 0 upward shift, compared to line (d), of the initial part of and b = 0, i.e. for type (D)-injections, the transient all type (A)-dispersion curves. As the time increases, dispersion regimes no longer exist and a strictly conan increasing fraction of tracer is involved in the stant Dax,app-value is obtained [line d, Fig. 16(b)], process of filling up the cores of the vortices whose which equals the the long-time limit value which is outer layers have yet been invaded. Since this tracer exclusively determined by K1 [eqs (12a) and (12b)]. fraction no longer increases the AaE-value, the disper- For homogenous injections, this independence from sion rate decreases. This continues until the long-time the intra-vortex transport phenomena is however limit regime is obtained in which all intra-vortex valid from t = 0 on. The observation of this dispersion gradients have vanished from the system. behavior in the presence of strong intra-vortex conComparing curve (a) with curve (a') shows that the centration gradients (see Fig. 7) is unexpected, since initial dispersion enhancement is most apparent when a constant Dax.app-value is a typical characteristic of a ~ 0. This is because in case (a') only a fraction (i.e. a series connection of perfectly mixed tanks. A maththe fraction situated in the most outer layer) of the ematical proof for this unexpected behavior is given in

Dispersion in vortex array flows---I Part II, where the phenomenon is also further experimentally validated. 7. ' A M P L I T U D E '

AND DURATION OF THE TRANSIENT DISPERSION REGIME

A simulation study with the M Z model is used to describe the variation of the main characteristics (duration, maximal difference in apparent axial dispersion coefficient) of the transient dispersion regime with the dispersion parameters K1, 2 and Di.tra. The 'amplitude' of the transient dispersion regime is characterized by the ratio of the maximal and minimal Dax,app-values which can be obtained for a given experimentation time and under identical hydrodynamical conditions, but by applying the two most extreme initial conditions to a given vortex: 1-- =

(13)

D ax,app/ A "Dax,appB

In eq. (13), the superscript A corresponds to a type (A)-injection with a--, 0, and the subscript B corresponds to a type (B)-injection with b ~ 0. It is obvious that F depends upon the instant (t~v, see Fig. 17) at which Dax.app is evaluated. Now, it can easily be derived from Fig. 16(b) that F is maximal (F = Fmax) when tov ~ 0 , and gradually decreases until in the long-time limit F = 1 (i.e. for tev --* oo ). A simulation study pointed out that Fmax is directly proportional to the ratio of the transport on the supra-vortex scale to the intra-vortex transport given by the Sh-number: Fmax ~ Sh(=Kad/Dintra)

(Sh > 5).

(14)

F r o m Fig. 16(b), it can be observed that for the strongly viscous fluids (Dmoj = 3.8 x 10 t l m2/s, yielding Sh = 9 x 104) for which the presented experimental data have been obtained, Fmax is of the order of 1 x 105. Equation (14) can now be used to estimate F . . . . for VAFs of aqueous fluids (Dmo I =

2399

l × 10 -9 mZ/s), for which the present experimental setup did not allow to establish similar laminar conditions. Considering identical Re-numbers for the intra-vortex flow, the experimental correlation presented in Part II shows that for the aqueous fluid case (v = 1 x 10 -6 mZ/s) a value of K l = 5.9 x 10 5 m/s has to be considered instead of the value of K1 = 1.2 x 10 - 6 m / s in the viscous fluid case (v = 1.1 x 10-4ma/sl. This yields a Sh-number of Sh = 130. Using eq. (14), it can now be conjectured that, in a laminar VAF of an aqueous fluid, V.... is of the order of 103 (i.e. for the system with d 2.25 x 10 2 m), which is still large (see also Fig. 171. Another important characteristic of the transient regime is its duration. Now, as can be physically expected, a simulation study, pointed out that the duration is fully determined by the characteristic time for intra-vortex dispersion and is hence inversely proportional to Dintra (=Drool in laminar flows). For the viscous fluid case, the transient effects become unimportant after approximatively 400h [see Fig. 16(b@ It can therefore be conjectured that for aqueous fluids (Dmol = l x l 0 - g m 2 / s ) , the transient effects only vanish after a period of the order of hours (i.e. for the system with d = 2.25 x 10 -2 m), which is still very long (see also Fig. 17). To illustrate the entire variation of F with the experimentation time in the case of a pure laminar VAF of an aqueous fluid, a simulation study with the M Z model has been performed for K 1 - 5.9x 1 0 - 5 m / s , Dintra = 10-gmZ/s and d = 2.25x 10 Zm (Fig. 17). It can clearly be denoted that the regime in which F = Fm,x = 2 x 103 persists over a period of the order of 102 s. After a time of 3 x 103 s, F still approximately equals 103 . Considering that typical residence times in chemical engineering processes involving aqueous fluids are usually less than 1 h, Fig. 17 clearly demonstrates that, even for aqueous fluids, the axial

1.10-41 Dax,app (mZ/s) 1.10-5

type (A)-injeetion (a=0.97)

1.10-6 1.10.7

type (D)-injection (a=0, b=0)

Fma x

1.10.8 1.10-9

type (B)-injection (b=0.91)

1.10-10 10

1.102

1.103

1.104

1.105

1.106

time (s)

Fig. 17. 'Duration' and 'amplitude' of the transient dispersion regime in a laminar VAF of an aqueous fuid from a simulation with the MZ model (Di,tr, = Dmo~= 10 -gmz/s. K1 = 5.9× 10 5m/s and d = 2.25 x l0 -z m).

2400

G. Desmet et al.

dispersion in VAF-reactors is fully determined by the transient effects, emphasizing the importance of the study of the transient dispersion regime.

8. CONCLUSIONS An axial scan technique has been developed which allowed to describe quantitatively and classify all possible dispersion modes which may exist in VAFs. The dispersion rates which are observed when the intravortex mixing time is larger than the residence or experimentation time vary in a strongly nonlinear way with the time and are very sensitive to the exact form of the initial tracer distribution. It is shown that in laminar VAFs, just by changing the initial tracer distribution in a given vortex, differences in apparent dispersion coefficients of up to five orders of magnitude can be obtained for the same hydrodynamical condition and experimentation time. This finding should be an incentive for a more accurate control and description of the injection conditions when RTD studies of VAFs are performed, and to pay special attention to the design of the entrance section when VAF-type reactors are developed. Unlike the conventionally used single-parameter dispersion models, and despite of its relative simplicity, the MZ model is able to represent the extremely sensitive relation between initial tracer distribution and the resulting residence time distribution for all two-dimensional VAFs in which the mixing along the vortex streamlines occurs rapidly, and provided the shape of the internal zones sufficiently approximates the actual flow pattern. As an ultimate validation test, it has been shown that the MZ model is able to accurately predict the variation of the apparent axial dispersion coefficient with the experimentation time, departing from an accurate description of the initial condition and using a set of model parameters obtained from independent experiments. Since the MZ model allows to distinguish between the different inter- and intra-vortex phenomena, it is now possible to determine all the dispersion phenomena (inter-vorrex flux, dispersion in vortex core, turbulent dispersion in outer vortex layers, etc.) separately, allowing to relate them in a unique way to the hydrodynamical conditions. As an unexpected dispersion effect, it was found that when the cross-section of a given vortex is injected homogeneously with tracer, the second-order moment of the tracer distribution varies with the residence time as if all the vortices are perfectly mixed, regardless of the presence of strong intra-vortex concentration gradients. In Part II, this feature will be exploited to make independent measurements of the inter-vortex exchange coefficient. The 'importance' of the transient dispersion regime has been quantified as a function of the different system parameters. Although it has been shown that, even in aqueous fluids, the transient effects persist over a long time scale and have a large amplitude, the effects are most pronounced when viscous fluids are

considered. Now, these are precisely the fluids for which the application of VAFs in continuous chemical engineering processes is most beneficial. When such fluids are processed in continuous tubular reactors, the flow conditions are often such that a laminar flow is established: the creation of a secondary VAF then serves as an excellent means to suppress the strong axial dispersion resulting from the radial velocity profile (Desmet et al., 1996a) and/or as a means to increase the heat transfer and mass transfer. The results of the present study are hence directly applicable to the development of novel VAF-type reactors for the treatment of viscous liquids. NOTATION geometrical parameters describing the a,b shape of the initial tracer concentration, see Fig. 3 Bodenstein number ( = u 2 At/Dax.,pp) Bo normalized concentration in zone~.j of MZ Ci.j model, see Fig. 13 observed, radially averaged concentration, Cobs normalized to Co concentration in initial tracer block, M/L 3 Co d annular gap width in CTF, L apparent axial dispersion coefficient, L2/T, Oax'app eq. (11) intra-vortex dispersion transversal to Olntra mean streamlines, L2/T, eq. (8) molecular diffusion coefficient, L2/T Drool inter-vortex exchange coefficient, L/T K1 K~ (i >~ 2) intra-vortex exchange coefficients, L/T, see Fig. 13 reactor length, L L ruth-order moment in time domain, T m+ 1 M,,m number of vortices N number of internal zones in MZ model, see Nz Fig. 13 radial coordinate, L r radius inner cylinder, L R~ Re-number based upon the exp. deterRetan mined mean tangential velocity in CTF outer surface of zone i in MZ model, see Si Fig. 13 Schmidt number ( = v/Drool) Sc Sherwood number, eq. (14) Sh time, or normalized first order moment in t time domain, eq. (10) time at which F is evaluated, T tev axial scan velocity, L/T u volume of zone i in MZ model, see Fig. 13 Vi axial coordinate, L x Greek letters F ratio of maximal to minimal Dax,app, eq. (13) 6 axial width inner zones MZ model, L, see Fig. 13 A difference stream function constant, eq. (7)

Dispersion in vortex array flows--I 2 V

a/-

O)

axial vortex width, L kinematic viscosity, Le/T second-order moment around first-order moment in time domain, T 2, eq. (10) driving force for inter-vortex transport, eqs (4a)-(4c) angular rotation speed inner cylinder, 1/T

Subscripts zone number in MZ model, or reference i number of axial scan in Section 6 inter inter-vortex intra intra-vortex vortex number in MZ model .J order of moment m

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