Visual 3-D Modelling of Stirred Vessel Mixing for an Inclined-Blade Impeller

0263±8762/00/$10.00+0.00 q Institution of Chemical Engineers Trans IChemE, Vol 78, Part A, April 2000

VISUAL 3-D MODELLING OF STIRRED VESSEL MIXING FOR AN INCLINED-BLADE IMPELLER M. RAHIMI, P. R. SENIOR and R. MANN (MEMBER) Department of Chemical Engineering, UMIST, Manchester, UK

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he 3-D visualization of passive tracer mixing inside a stirred vessel offers a powerful means of validating computational models. Taken together, the forward elevation and underneath plan views (acquired by a 458 mirror) of the mixing of a visible tracer in space and time, captured on video, present approximately 0.1 Mb of data per video frame. This density of potentially quantitative concentration ®eld information is very suitable for validation of computational ¯uid mixing (CFM) predictions for models comprised of the order of 105 voxels. New results are presented for an axial pumping 458 inclined-blade turbine in a semi-tech 30 dm3 vessel equipped for 3-D visualization. Passive scalar mixing tests have been evaluated using an improved networks-of-zones model with 32,000 zones. Conformal stretching of the radial ¯ow con®guration of existing software provides a simple way of accommodating the predominantly axial ¯ow pattern. Good agreement between theory and experiment has been demonstrated using AVS graphics, giving see-through close to photo-realism reconciliation of mixing images for two different injection positions. Keywords: computational ¯uid mixing; macro-mixing; networks-of-zones

INTRODUCTION

scale (so-called macro-mixing) in typical semi-batch stirred reactor operation, especially for long addition times. Several advanced experimental techniques are being concurrently developed (and driven by continuously improving CFD capabilities) for validation of these CFD computational predictions. These include laser-doppler anemometry LDA (Mao et al., 19974), laser-doppler velocimetry LDV (Tiljander et al., 19975), laser induced ¯uorescence LIF (Distelhoff et al., 19976), and digital particle imaging velocimetry DPIV (Myers et al., 19977, Sheng et al., 19988). However, all these techniques are experimentally complex and require relatively expensive equipment. A potentially simpler approach is to use visualized 3-D imaging of the mixing of a visible tracer pulse, which requires only an everyday video camera (Holden and Mann, 19969). Such simple visualization results can be readily used to validate a simpler description of the ¯uid mechanics represented by networks-of-backmixed-zones (Mann et al., 199410) in conjunction with 3-D visualization graphics. This theoretical approach can also be more easily exploited to perform simulations of semi-batch reaction simulations without the need for very large computing resources (Mann et al., 199711), especially for cases with multiple reactions. The quanti®cation of macro-mixing using passive scalar pulse injection was previously demonstrated for a Rushton Turbine using 3-D visualization9. In this work it will be shown how a simple 3-D networks-of-zones model10 can be adapted to describe the axial ¯ow pattern of an inclinedblade impeller. The model parameters for this `impellertype’ adaptation can be estimated from 3-D video images of a visible pulse of tracer. The network model can then be straightforwardly applied to predict the macro-mixing behaviour from another quite different (widely separated)

Macro-mixing refers to concentration gradients in a ¯uid which are of a scale comparable to the size of the vessel within which mixing is taking place. Macro-mixing effects can be inferred by making visual observations in 3-D of the unsteady timewise evolution of a pulse of a passive tracer. In semi-batch operation, typical of chemical manufacturing carried out in a stirred vessel, when the mixing is accompanied by chemical reaction, the macro-mixing effects are manifested as a partial or macro-segregation of the reagents. The degree of segregation may often be crucial in determining chemical selectivity for multiple reactions. Thus it is important to be able to understand and control macro-mixing if by-product reactions are to be suppressed and the yield of desired components maximized. Computational ¯uid dynamics (CFD) relies upon theoretical solutions of the ®nite volume or ®nite difference algebraic forms of the Navier-Stokes equations. These can be derived and solved using turbulence theories of varying degrees of complexity, some of which, like the k « theory, ê are largely empirical and contain `constants’ whose values are uncertainly related to typical stirred vessel ¯uid mechanics. In principle, CFD provides information on the fundamentals of turbulent ¯uid ¯ow patterns, power consumption and the spatial distribution of turbulent energy dissipation. Recent examples are available from Bakker et al., 19971 and Wechsler et al., 19992. Increasingly, the details of the ¯uid mechanics inside the impeller envelope can be measured, for example by LDA (Schafer et al., 19983). However, the computational intensity implicit in these CFD approaches makes it dif®cult to proceed to simultaneously compute concentration ®elds at the vessel 348

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Figure 2. Inclined-blade turbine.

Figure 1. 3-D visualization of mixing.

injection point. This distinguishing and quanti®cation of the macro-mixing characteristics for different injection points is one of the key factors in subsequently determining the impact of injection point on semi-batch chemical yield performance in typical manufacturing. That macro- mixing can affect selectivity for a triplet reaction has already been established (Mann and El Hamouz, 199512). PASSIVE TRACER VISUALIZATION EXPERIMENTS The experimental arrangement is shown schematically in Figure 1(a). This provides for 3-D visualization by viewing simultaneously a forward (elevation) view and an underneath (plan) view using a 458 inclined plane mirror positioned below a cylindrical stirred vessel of standard geometry. An actual view is then shown alongside in Figure 1(b) (These typical images depict a Rushton and novel geometry impeller respectively, rather than an inclined blade impeller). The way in which these visualization experiments for pulse injection can be image reconstructed by AVS graphics is shown in Figure 1(c). Results have been obtained using the six-inclined-blade impeller shown in Figure 2 (D = H/3). A pair of sample experimental images are then presented in Figure 3, showing how a pulse of blue nigrosine tracer (injected at t = 0) has been visualized and captured on video after 2.3 s. The stirrer speed is 100 rpm with the impeller positioned at the mid-point with c = H/2. The left view shows the tracer Trans IChemE, Vol 78, Part A, April 2000

cloud for a top injection close to the liquid surface (at H = T) and the right view an injection below the impeller. The macro-mixing pattern is quite different for these two cases (although it will be shown later that the volume occupied by tracer at this time is identical in each case). This paper will examine the question of whether a simple formulation of a network-of-zones model can reproduce this effect of variation in injection position, simultaneously providing a prediction of the concentration ®eld in 3-D, thereby identifying the differences in macro-mixing behaviour. If the two different patterns of macro-mixing could be reconciled by the model, this would provide some con®dence in prediction of the effect of addition point for semi-batch operation with reaction(s), when one reagent is added continuously to another contained in the vessel. In particular it would then be possible to quantify the reactioninduced macro-segregation effects manifested during the addition period. NETWORK-OF-ZONES MODEL FOR AN INCLINED-BLADE IMPELLER The basic ideas underlying the assembly of networks-of zones to simulate 3-D mixing have already been presented elsewhere9,10. These previous formulations were for a Rushton turbine impeller. In principle, this approach can be adapted for any ¯ow pro®le by a conformal-mapping of the zone volumes into a suitable adjusted shape and ¯ows. The basic topology of an `i ´ j ´ k’ network-of-zones thus remains unchanged, but the zone volumes are individually stretched or compressed to provide an adapted convection pro®le appropriate to the particular impeller type. Figure 4(a) shows diagrammatically how the ¯ows are con®gured for the general (i, j, k) zone. The axial position z has its origin at the base of the vessel. Figure 4(b) shows the detail of the adjusted geometry of the set of zones for a 458 inclined-blade impeller. These adjustments create a dominant top-to-bottom ¯ow loop, with a smaller recirculating loop below the impeller. A 458 angled plane separates the two ¯ow loops. Each individual zone can be imaged by the AVS graphics (AVS,199313), thus providing a 3-D image-reconstructed view of the instantaneous concentration ®eld at any moment as mixing proceeds (as indicated in Figure 1(c)). The visualization is achieved using a modi®ed Scatt-bub version of the standard Scatterdots module. This represents each zone by a bubble which can be coloured to any intensity and opacity. Figure 4(c)

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Figure 3. Equal volume occupied by tracer from two different positions.

shows the colour intensity and associated opacity values used in the image reconstruction of the 3-D tracer clouds generated during mixing.

The basic parameters are then the network size and the magnitude of the overall axial-radial convective ¯ow. This is stipulated from Q = KND3 with this ¯ow allocated uniformly amongst the individual ¯ow loops as before9. The remaining parameters are then the turbulent exchange coef®cient b and the swirl ¯ow factors b L and b R . In this adaptation b R is set equal to b giving local isotropic turbulence. The net swirl is then b L b in the direction ê of rotation (the subscript L refers to leftwards or clockwise looking down from above and the front). The basic computer code is easily adapted to allow b L (r,z) to vary with both radial and axial position. With both the impeller constant K and the tangential swirl ¯ow speci®ed, the only remaining parameter is b which is judged from a matching of experimental and theoretical AVS visualizations of the pulse tests. As in previous work, b is magni®ed by a factor 10 in the impeller envelope to reproduce the locally intensive highly uniform redistribution that invariably takes place as tracer mixes through the impeller region. MODEL VALIDATION FROM IMAGES OF TOP INJECTION

Figure 4. Adapting the networks-of-zones model for inclined-blade impeller.

Simulation results have been obtained with i = 16, j = 32 and k = 64. This is termed a 2 ´ (16 ´ 16) ´ 64 network. It comprises some 32,000 zones. A set of experimental results from initial injection to 16 seconds are shown in Figure 5. The top injection point corresponds to i = 3, j = 31 with k = 1. A 1.2 ml of blue dye is injected into water in a fraction of a second at t = 0. This volume is assumed not to signi®cantly disturb the ¯ow pattern adjacent to the injection point. The image reconstructed AVS simulations are designed to appear in exactly the same format. The tangential swirl ¯ow parameters used are depicted in Figure 6. This swirling ¯ow is most intense in the impeller plane (z/H = 0.5) and between i = 2 and 4 corresponding to the blade tips. This ¯ow pattern is consistent with experimental velocity pro®les (Zhou and Kresta, 199614) and provides for a swirling ¯ow pattern that very closely Trans IChemE, Vol 78, Part A, April 2000

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Figure 5. Model visualization by AVS imaging of tracer mixingÐtop injection

Figure 6. Tangential b

L

values and zone positions.

matches the experimental tangential spreading of the blue tracer pulse. In Figure 5, the isotropic turbulent exchange factor b = 0.12 is ratiod to the local loop ¯ow q and K = 2.9. It is clear from Figure 5 that the network-of-zones model provides for a close match in 3-D with the experimental visualizations. The ®nal experimental images at 16 s appear to show incomplete mixing at the vessel walls. However, this is not the case. This effect is produced by highlights on the external vessel walls, which are equally present in the whole set of images showing the frontal view from 0 to 16s. Each theoretical image in Figure 5 contains of the order of 105 data values for C (i,j,k) which are rendered into close to photo-realism images by the AVS graphics by speci®cation of colour and optical opacity (see Figure 4(c)). Lighting was by 5 ¯oodlights positioned around the vessel giving as uniform illumination as

Figure 7. Model visualization by AVS imaging of tracer mixingÐbase injection.

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possible. Images were captured using a standard VHS video camera. Both the experimental and theoretical images are each approximately 100 kB in size. If the model as speci®ed is adequate, it should be capable of straightforwardly predicting the effect of changing the position of the injection point without any further adjustments whatsoever. IMAGE RECONSTRUCTIONS WITH BASE INJECTION It is straightforward to change the point of dye injection in the software and then image reconstruct the predicted behaviour. This is shown in the upper part of Figure 7 where the 3-D views are for base injection at i = 6, j = 15 and k = 1. This position is just below the impeller tip and three-quarters of the impeller clearance above the vessel base. This theoretical mixing sequence can be compared directly with the experimental visualizations shown in the lower part of Figure 7. It is evident that the agreement is very good, con®rming that the networks-of-zones should be able to predict mixing behaviour from any injection point within the ¯uid. Since the ¯ow ®eld is unchanging, this is achieved by simply repeating the numerical integration with new i , j , k values. Figure 8(a) then shows the results as a set of timewise evolving perspective isometric views from the front. The pulse of tracer is convected and mixed in the downward ¯owing stream leaving the impeller. After 1 s, the tracer is predicted to have reached the base, beginning to spread inwards along the ¯at base

and upwards along the wall. After 2 s, the dye has just reached the upper surface and is looping upwards just below the impeller. At 4 s, the swirl has begun to obscure the view of the impeller blades on the right-hand side (from the forward view). After 6s, the swirling tangential movement and mixing has now fully obscured the view of the impeller. Figure 8(b) then shows another set of isometric views after 1s, indicating how the dispersing cloud of tracer appears from different viewing positions around the top of the vessel at this instant of time. COMPARISON OF DIFFERENT INJECTION POINTS That the model satisfactorily predicts the effect of changing injection point can also be judged by comparing the 3-D AVS predictions/simulations in Figure 9 with the corresponding experimental results already presented in Figure 3. In each ®gure, the left and right hand results are for top and base injection respectively after an elapsed time of 2.3 s. The 3-D pattern of concentration distribution is quite different in each case. For top injection, the blue tracer has convected downwards and passed through the impeller without reaching the wall. On passing through the impeller, signi®cant tangential swirl distribution takes place. In contrast, for injection below the impeller, after the same time, the tracer is still being returned towards the impeller but shows high concentrations along the base and up the walls. In this case, also, there is a lesser tangential swirl mixing. In relation to understanding the role of injection point on the concentrations ®elds for complex reactions, Figure 9 also shows the volume fraction that has C(i,j,k) greater than 0.1 of the ®nal fully mixed concentration C¥ , here shown as a dimensionless concentration of unity. This measure of the integral extent of mixing that has been achieved at each instant of time shows that the top injection achieves a faster overall level of mixing because the tracer reaches the impeller sooner than for base injection. However, the ®gure shows that initially top injection produces marginally lower mixing and the integral degree of mixing is actually then identical after 2.3 s. These observations are relevant to the mixing achieved in semi-batch operation. Thus Figure 9 and Figure 3 show that although the S C(x,y,z) is similar for both injection positions, the detailed patterns of C(x,y,z) are wholly different and moreover represent different `histories’ of mixing. CONCLUSIONS

Figure 8. Image reconstruction for base tracer injection.

· A prior version of the networks-of-zones model for a radial ¯ow impeller has been adapted by a simple conformal mapping of zone volumes so as to simulate the ¯ow pattern of an inclined-blade impeller. · In this version of the networks of zones, the two major parameters continue to be the overall axial ¯ow constant K given by Q = KND3 and an isotropic local turbulence parameter b . The tangential swirl ¯ows can then be represented by b L (r,z), so that the ¯ow tangentially around the vessel is most intense around the impeller and diminishes towards the impeller shaft, the walls, the base and the upper ¯uid surface. · These parameter values have been identi®ed from Trans IChemE, Vol 78, Part A, April 2000

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Figure 9. Tracer disposition from two different injection positions (simulation corresponds to Figure 3).

experimental video 3-D images of pulse tracer mixing, corroborated by AVS see-through close to photo-realism image reconstructions of the unsteady mixing. · This modelling approach independantly predicted the mixing achieved when the point of injection was moved from the upper surface to below the downward pumping impeller. · These capabilities of the model are important for answering questions of choice of injection point for semibatch stirred reactors. NOMENCLATURE C C¥ c D H i, j, k K N Q q r T t x, y, z

tracer concentration ®nal mixed concentration impeller clearance impeller diameter liquid depth zone positions overall ¯ow constant impeller speed overall ¯ow rate loop ¯ow (in network) radial coordinate vessel diameter time Cartesian co-ordinates

Greek symbols b turbulent exchange factor b L,R swirl ¯ow factors Superscript * refers to injection zone

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ADDRESS Correspondence concerning this paper should be addressed to Professor R. Mann, Department of Chemical Engineering, UMIST, Manchester M60 1QD, UK. (E-mail: [email protected]). The manuscript was received 5 October 1999 and accepted for publication after revision 22 February 2000. The work was originally presented at the Fluid Mixing 6 Symposium, held 7±8 July 1999 at the University of Bradford, UK.