An optimised split-and-recombine micro-mixer with uniform ‘chaotic’ mixing

Abstract

A second generation micro-mixer, being a further optimised version of a first prototype, relying on the consequent utilisation of the split-and-recombine principle is presented. We show that the mixing can be characterized by a positive finite-time Lyapunov exponent although being highly regular and uniform. Using computational fluid dynamics (CFD) we investigate the mixing performance for Reynolds numbers in the range of about 1 to about 100. In particular for low Reynolds numbers (Re < 15) the CFD results predict an almost ideal multi-lamination. Thus, the developed mixer is especially suited for efficient mixing of highly viscous fluids. Furthermore, the numerical results are experimentally validated by investigations of mixing of water–glycerol solutions. The experimental results are found to be in excellent agreement with the numerical data and prove the high mixing efficiency.

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Introduction

Since mixing is the most commonly practised unit operation in today's chemical industrial processes and is also important for analytical tasks, the realisation of optimised micro-mixers is one of the paramount tasks of microchemical process engineering (see e.g.refs 1–3) as well as in the development of μ-TAS and Lab-on-a-chip systems (see e.g.refs 4–7). To name a few examples of use, micro-mixers are constituent parts of set-ups for micro-encapsulation for drug delivery,⁸ for extraction in the framework of mini-plant technique,⁹ for organic synthesis with unstable intermediates,¹⁰ for high-throughput kinetic screening of chiral homogeneous catalysts in multi-phases,¹¹ for micro-plants with recycle loops,¹² for powder production,¹³ and for analytical techniques such as time-resolved NMR¹⁴ and time-resolved FTIR.¹⁵

Despite this wide use of micro-mixers, now including even commercial off-the-shelf catalogue sales and industrial lab- and production-scale applications, the basic features of micro-mixers and their flow still are not well understood and thus there is a demand for further in-depth investigations. This is not only the case for complex micro-channel-flow configurations employed for multi-phase processing in micro-mixers, but also concerns standard blending and mixing of miscible liquids. Among the class of micro-mixers for the latter task, all passive micro-devices basically can be classified according to their functional principle which relies either on multi-lamination by interdigital lamella arrangement (see e.g.refs 1,16,17), on chaotic mixing^4–6 or on a successive split-and-recombine (SAR) approach (see e.g.ref. 18), ideally yielding multi-lamination patterns as well. The latter is especially appealing since it allows the achievement of uniform mixing properties due to a fine homogeneous multi-lamination under moderate pressure drops without the need for high fabrication requirements.

In the last years, different SAR mixers have been fabricated (see e.g.refs 18–20). However, all these mixers were mainly experimentally validated with regard to global process results such as yield, effective reaction times, or properties of produced particles. However, there is, to the best of our knowledge, neither an in-depth simulation on the course and efficiency nor an experimental validation of the mixing process itself, e.g. no SAR flow patterns were visualised so far. While a global experimental yield evaluation is the most straightforward way in order to analyse the mixing performance with respect to concrete applications it does not allow for a detailed inspection and hence optimisation of the SAR process itself.

For this reason, both a theoretical analysis on SAR mixing and a respective flow-pattern visualisation are given here for a special IMM micro-mixer. In particular, a novel, favourable mixer design is identified by means of CFD simulations. Subsequently, these numerical results are supplemented by experimental validation.

Results and discussion

Split-and-recombine principle

In contrast to the commonly applied interdigital multi-lamination approach used in micro-reaction technology, SAR mixing relies on a multi-step procedure. According to its name the basic operations are: splitting of a bi- or multi-layered stream perpendicular to the lamella orientation into sub-streams and recombination of these. Usually, these basic steps are accomplished by one or more re-shaping steps. The mixing mechanism is schematically displayed in Fig. 1a. Fig. 1b shows an idealised multi-lamination pattern of a SAR mixer developed at IMM which is the first-generation prototype of the optimised mixer presented here.²¹

Fig. 1 (a) Schematic of a SAR approach: initial configuration, splitting, recombination, reshaping (from left to right). (b) Idealized realization of the SAR approach by means of structured channel walls.²¹

In the framework of chaotic advection, the mixing performance is commonly characterized by interfacial stretching:

where L₀ and L(t) denote the characteristic dimensions of the interfacial area at t = 0 and at finite time t, respectively (see e.g.refs 22, 23 and 25 and references therein). In case of a 2D, incompressible flow the diffusive mass transport, which determines the mixing performance, depends quadratically on the interfacial stretching λ.^22,24 Chaotic flows ensure particularly efficient mixing since they imply an exponential increase of stretching over time. Accordingly, the finite-time Lyapunov exponent σ may be defined viaeqn. (2):

Although the repeated application of the SAR principle leads to a highly regular lamella pattern, as sketched in Fig. 1, it exhibits an exponential increase of interfacial area as well. The corresponding stretching factor is given by eqn. (3):

where n, u, and l denote the number of SAR steps, the mean velocity and the length of one SAR unit (cf.Fig. 1b), respectively. According to eqn. (3) the SAR mixer has a finite-time Lyapunov exponent of σ = ln2(u/l). Thus, even in case of a highly regular flow pattern a positive finite time Lyapunov exponent can be achieved – a characteristic feature of chaotic advection.^22,26 Whereas chaotic advection generally induces regions of regular flows and poor mixing besides regions of high chaos where mixing predominantly occurs,²² here, in case of an ideal SAR multi-lamination, a spatial homogeneous mixing is obtained

Since the final lamella dimension does not solely depend on the channel width, but also on the number of SAR steps, thorough mixing can be achieved under moderate pressure drops. Specifically, a linear total pressure drop Δp = n·p₀, where p₀ denotes the pressure drop of one SAR step, leads to an exponential decrease of the lamella dimension L_n = L₀/2ⁿ.

Layout/simulation of SAR flows

The starting point for the development of an optimised SAR mixer was the numerical investigation of flow patterns of the so-called caterpillar mixer realized in 1999²¹ which is now a commercial off-the-shelf product of IMM (brand name: caterpillar micro-mixer; CPMM-R1200/8). The mixer has been successfully applied in various applications (e.g. see ref. 27). While showing a very good mixing performance in high Reynolds number applications, mixing is of minor quality in case of liquid flows below 2 l h⁻¹, i.e. for Re below about 500.²⁸

A part of the CFD model used in the simulations, the first SAR step, is shown in Fig. 2. For theses initial simulations a structured grid comprising 220 000 hexahedral cells was used. Using the commercial solver CFX4 (ANSYS/CFX, Canada) the Navier–Stokes equation was solved and simultaneously a convection equation for a user scalar describing a non-diffusive tracer. Generally, mixing quantification of liquid/liquid systems requires elaborate means to minimize the discretization error.²⁹ Here, the tracer was solely used for lamella visualization and hence the diffusivity was set to zero. A higher order differencing scheme (QUICK³⁰) was used for discretization and the SIMPLEC algorithm was applied for pressure-velocity coupling.³¹ Liquid properties of water were chosen.

Fig. 2 CFD grid of one SAR segment according to the geometry shown in Fig. 1b. The inlet dimensions are 1.2 mm × 1.2 mm. The length is 2.4 mm.

Simulation results showing cross sectional views in the flow direction are given in Fig. 3. As noted above, the caterpillar mixer was successfully applied in various high-Reynolds-number applications. However, as can be seen in the figure for Re of about 1 the SAR principle is imperfectly realized. Due to internal friction the horizontal splitting taking place in the first half of the geometry does not preserve the initial lamination. This leads to an S-shaped lamella configuration at the centre of the geometry, i.e. at x = 1.2 mm. A similar problem occurs in the recombination step (lower row in Fig. 3). Again, due to the internal friction the lamella configuration is changed, ending up with a more pronounced S-shaped structure rather than a multi-lamella configuration.

Fig. 3 Cross sections taken from a CFD simulation for Re = 1, displaying the lamella reshaping in the SAR mixer shown in Figs. 1b and 2. The initial configuration is shown in the small cross section in the upper left. The other cross sections in the upper row show the splitting at x = 0.6 mm and x = 1.2 mm. The cross sections in the lower row show recombination at x = 1.8 mm and x = 2.4 mm at the end of the first cell.

Fig. 4 Left: geometry model of an optimised SAR mixer – a) slanted view with the splitting layer shown in black; b) side and c) top views. Right: scanning-electron micrograph of the lower channel part. The image shows one step of a SAR mixer made of stainless steel with a minimum channel cross section of 1 mm and a length of 6 mm.

In case of high Re the situation is different. Inertial forces induce a secondary flow which causes strong tilting and entanglement of the lamella, and hence, a considerable increase of interfacial area is achieved.

In the present framework, however, we aim at an ideal implementation of the SAR principle in order to realize an optimised multi-step mixer. According to the results shown in Fig. 3, there are basically three measures which have to be taken to improve the SAR functionality.

First, before upper and lower channel parts are horizontally separated the flow has to be split by means of an intermediate layer. In that way the S-shaped lamella configuration is avoided. Second, after flow splitting both sub-streams have to be independently guided in separate channels. Third, favourably, recombination is realized without any viscous drag between the merging sub-streams. These design rules ensure a minimised lamella reshaping due to internal friction.

The CFD model of a corresponding geometry is shown in Fig. 4a–c. The CFD simulations were performed on an unstructured grid with about 1.4 × 10⁶ tetrahedral cells. The flow solver CFX 5.6 (ANSYS/CFX, Canada) was used. Cross sectional views depicting the lamella configurations for Re = 3.45 are given in Fig. 5. According to the CFD simulation results the new design leads to an almost perfect lamella configuration after passing one SAR step. The same holds true if a series of three SAR steps is simulated. Thus, by subsequent implementation of the above design rules a vertically aligned multi-lamination pattern is achieved.

Fig. 5 Cross sectional views of lamella configurations within a SAR step shown in Fig. 4 for Re = 3.45. a) splitting; b) rearrangement of sub streams and recombination; c) reshaping.

For increased flow rates, however, the CFD simulations show more and more deviations from an ideal SAR multi-lamination pattern. Since inertial forces come into play a secondary flow pattern is superimposed on the SAR velocity profile of the creeping flow regime. Fig. 6 shows a top view of streamlines seeded at the initial lamella interface for various Reynolds numbers. Fig. 7 shows the lamella pattern right at the outlet for the same set of Reynolds numbers (the cross section for Re = 3.45 was shown already in Fig. 5). Further simulations showed that for Re above about 15 the centre lamellae are thinned out until they detach from the top and bottom walls for Re about 30 (cf. centre image of Fig. 7).

Fig. 6 Top view of streamlines, seeded at the initial lamella interface for Re = 3.45 (top left), 8.63 (bottom left), 34.5 (top right), and 104 (bottom right).

Fig. 7 Cross sectional views of multi-lamination patterns corresponding to the streamline patterns shown in Fig. 6 for Re = 8.63, Re = 34.5 and Re = 104 (from left to right). The former is almost identical with the multi-lamination pattern for Re = 3.45 shown in Fig. 5c.

Concerning an appropriate dimensioning of the SAR mixer geometry (Fig. 4) we consider the Dean number K commonly used to characterize secondary flows induced by inertial forces

where R and d denote the radius of curvature and the hydrodynamic diameter, respectively. The latter is defined as 4 times the cross sectional area over its circumference. In order to generate homogeneous multi-lamination by SAR, the secondary flow and hence K should be minimized. On the other hand, with respect to industrial applications, a maximization of throughput is desirable.

For simplicity we restrict the consideration to mixer geometries where the lateral off-sets of both, the upper and the lower s-shaped channels (cf. left part of Fig. 4a and 4c), are each build up from two circular channel segments. Using basic geometry the radius of curvature of each segment, R, can be expressed by the magnitude of the lateral off-set which is about b/2 for each channel, where b denotes the channel width. The Dean number can be written as

Here, Φ denotes the volume flow rate, ν τhe kinematic viscosity, g a constant geometric factor and a channel height of xb was assumed. So far, only the first part of the geometry, i.e. the splitting part, has been regarded. However, similar arguments apply also for recombination and reshaping. According to eqn. (5) large channel widths and heights imply low Dean numbers. From this argument the SAR mixer should be realized on a macroscopic scale. On the other hand, in order to achieve a fast mixing by diffusion the dimension of final lamellae should be of the order of microns and a compact mixer format implies a moderate number of SAR steps, of the order of 10. Thus, an inlet width of a few millimetres is appropriate; the precise value, however, depends on the requirements of the concrete process of the mixing application. Concerning the channel height, eqn. (5) suggests large dimensions as well in order to achieve high volume flows under low Dean numbers. However, depending on the fluid properties of the lamellae, instabilities may develop under certain conditions. The susceptibility of the lamellae to instabilities decreases with smaller channel dimensions. Again, a channel height of a few millimetres is a reasonable compromise.

Fabrication of SAR prototypes

Based on the above considerations an 8-step SAR mixer with a channel width of 2 mm, a height of 4 mm and a total length of 960 mm was fabricated out of PMMA to allow for an optical inspection of the mixing process. Furthermore, the same mixing geometry scaled by a factor of 0.5 was realised in stainless steel. As shown in Fig. 8 the main channel is formed by assembly of two identically plates (1a,b), structured by milling using a Picomax 60 CNC (Fehlmann, Switzerland) milling machine with a cutting bit diameter: 1 mm (0.4 mm), revolution speed: 18000 rpm (38000 rpm) and lateral velocities: 500–1000 mm min⁻¹ (110–220 mm min⁻¹) for PMMA (stainless steel). One SAR step of a structured stainless steel plate is shown in the electron micrograph given in Fig. 4d. The continuous change of brightness of the channel bottom reflects the unruffled surface and the smooth change of inclination angles. The obviously curved channel edges caused by the spherical shape of the cutting bit are expected to have no negative impact on the mixing performance compared to ideally rectangular channels. Anyway, in case of the PMMA mixer which was experimentally evaluated (see next section) the question of curved channel edges is less important due to its larger dimension.

Fig. 8 Left: design drawing of a 8-step mixer according to the SAR geometry shown in Fig. 4. The SAR channel is build up from two identically structured plates (1a,b). (2a,b) and (3) denote the graphite gaskets and the splitting layer, respectively. Right: prototype of the 8-step SAR mixer made of PMMA.

The horizontal splitting layer (3 in Fig. 8) was realized by a 0.1 mm stainless steel sheet inserted between the PMMA (stainless steel) plates. The layer was structured by means of pulsed sublimation cutting (pulse energy: 10.5 mJ; pulse frequency: 1 kHz; pulse length: 10 ns) using a Nd : YAG laser from Haas (Trumpf), Germany with a processing velocity of 1 mm s⁻¹. To structure the graphite gaskets which seal the sandwich system (2a,b in Fig. 8) pulsed laser cutting with modified parameters (pulse energy: 2.625 mJ; pulse frequency: 4 kHz; same pulse length; processing velocity: 4 mm s⁻¹) was used. A steel housing serves for compression and provides the fluidic connections.

Fig. 9 Optical inspection of multi-lamination in the SAR mixer. The dyed (water-blue) and transparent lamellae of an 85% glycerol–water solution are shown in dark and light grey, respectively. The applied total volume flow rate of 0.2 l h⁻¹ corresponds to Re = 0.22.

The development of the finally fabricated layout did not involve a strict optimisation in the mathematical sense, moreover the CFD-based design process included already fabricational aspects. Especially the width of the fin-structure separating both channels (cf.Fig. 4c) was chosen to be 0.5 mm in order to facilitate fabrication. From the theoretical point of view the width would be chosen infinitesimally small.

Experimental validation

With respect to eqn. (4) most mixing experiments were performed at relatively low Re. To prevent any negative impact of pump pulsations on the flow patterns an 85% glycerol–water solution was used, with dynamic viscosity and density of about 100 mPa s and 1.2 kg l⁻¹, respectively,³² which thereby allows applications of higher flow rates compared to the usage of aqueous solutions. Generally, flow patterns, as given here, can also be realised using aqueous solutions, however, at the expense of considerably increased experimentation expenditure. Total flow rates in the range of 0.2 l h⁻¹ (Re = 0.22) and 2 l h⁻¹ (Re = 2.2) were applied. For the given dimensions CFD simulation show that for Re above about 15, corresponding to a total flow rate of 13.5 l h⁻¹, secondary flow induced by inertial forces has a notable effect. Thus the applied flow rates were well below the critical one and indeed, pronounced secondary flow effects were not observed.

As introduced in ref. 1, the combination of two complementary flow-pattern imaging approaches allows for a reliable experimental quantisation of the mixing quality. Firstly, mixing of a transparent and a dyed sub-stream is optically inspected by means of transmitted light microscopy. Secondly, the usage of two transparent sub-streams of yellowish iron ion (Fe³⁺) and transparent rhodanide (SCN⁻) solutions which form a deep coloured complex after mixing allows to identify those regions where the first method mistakenly suggests good mixing due to layered lamella configurations.

Fig. 9 shows a series of micrographs obtained by the first method, usage of a transparent and blue dyed glycerol–water solutions, with a total flow rate of 0.2 l h⁻¹. Highly regular lamella patterns are observed up to the 8th SAR step. To our best knowledge, this is the first experimental proof on the existence of a SAR-flow pattern by giving the corresponding multi-lamination patterns. According to the low Reynolds number of 0.22 secondary flow effects are suppressed and homogeneous mixing is to be expected. As outlined above, mixing itself can be quantified using the iron–rhodanide reaction. Corresponding experimental results, again for a total flow rate of 0.2 l h⁻¹, are given in Fig. 10. Starting from two almost transparent solutions in the first SAR step a uniform distribution of the brown iron–rhodanide complex is derived in the last mixing step.

Fig. 10 Optical inspection of mixing in the SAR mixer. The applied total volume flow rate of 0.2 l h⁻¹ corresponds to the experiment shown in Fig. 9. Starting from a bi-lamination of yellowish iron ion (Fe³⁺) and transparent rhodanide (SCN⁻) solutions a homogeneous mixing is achieved in the 8th mixing step indicated by the deep brown colour of the formed iron–rhodanide complex.

All in all, the layout concept and CFD simulations are excellently confirmed by the above experiments. The good agreement is also highlighted in Fig. 11, where the simulated streamline pattern of for Re = 0.22 is superimposed on the micrograph of the first SAR step as shown in Fig. 10. Besides the global SAR performance the CFD simulations indicate a certain curvature of the lamella interface as shown in Figs. 5–7. By the integral imaging used in the experiments the curvature translates into grey shaded regions between the lamellae in case of dyed solutions (cf.Fig. 9) and into brown coloured regions in case of iron–rhodanide solutions (cf.Fig. 10). A close inspection of the latter allows a direct quantification of mixing in each step.³³

Fig. 11 Comparison between the CFD result (black lines) and the experimental image showing the multi-lamination for 85% glycerol–water solutions at a flow rate of 0.2 l h⁻¹. The black lines denote the numerically computed streamlines seeded at the lamella interface at the inlet (cf. first cross section of Fig. 5).

A close inspection of the lamella configurations shown in Fig. 9 and 10 reveals also discrepancies from an optimum configuration. Lamellae of slightly increased dimensions are found adjacent to the channel walls. This has to be attributed to the smaller flow velocity near the wall. A detailed analysis of flow patterns and the impact on the mixing is left for a further study being in preparation.³³

Summary and conclusion

Based on a detailed inspection of multi-lamination by means of CFD an improved mixer layout was derived which assures an almost ideal realization of SAR mixing. To our best knowledge, this is the first time that a thorough validation of the SAR mixing concept is given based on both, numerical and experimental data. Only the combination of both approaches allows for a reliable and efficient system development. Moreover, both approaches are found to be in good agreement

Generally, a positive finite-time Lyapunov exponent which is a generic feature of multi-step SAR mixers, is associated with chaotic advection. The corresponding exponential stretching of the interfacial area together with the uniform lamella distribution opens up a route to efficient homogeneous mixing especially at low Reynolds numbers. This is achieved without the need of microstructures inside the mixing channel and hence conventional milling equipment can be used for fabrication. Thus, the developed mixer is particularly suited for the mixing of highly viscous fluids, as e.g. polymer melts.

For higher Reynolds numbers, above about 15 for the chosen dimensions, secondary flow effects develop disturbing the development of a uniform multi-lamination. However, moderate scaling up shifts the critical volume flow to larger values. With respect to micro-technological applications, secondary flows driven by inertial effects can be minimized by changing the flow profile towards a more flat, plug-flow-like profile. To this end electrokinetic actuation or mixing channels filled with a porous medium might enable even more ideal lamella patterns possibly for a wide range of throughputs. Furthermore, the particular guidance of fluid lamellae causes a wall/fluid contact between the fin structure separating the channels and new generated fluid lamellae. Accordingly, similar structures might be utilised for improved heat transfer in mini heat exchangers.

References

1. V. Hessel, S. Hardt, H. Löwe and F. Schönfeld, AIChE J., 2003, 49, 566.
2. T. Bayer, K. Himmler and V. Hessel, Chem. Eng., 2003, 5, 2.
3. S. Sugiura, M. Nakajima and M. Seki, Proceedings of the 5th International Conference on Microreaction Technology, Microreaction Technology - IMRET 5, 2001, Berlin, Springer-Verlag, 252.
4. A. D. Stroock, S. K. W. Dertinger, A. Ajdari, I. Mezic, H. A. Stone and G. M. Whitesides, Science, 2002, 295, 647.
5. X. Niu and Y.-K. Lee, J. Micromech. Microeng., 2003, 13, 454.
6. C.-P. Jen, C.-Y. Wu, Y.-C. Lin and C. Y. Wu, Lab Chip, 2003, 3, 73.
7. J. Mengeaud, J. Josserand and H. H. Girault, Anal. Chem., 2002, 74, 4279.
8. S. Freitas, A. Walz, H. P. Merkle and B. Gander, J. Microencapsulation, 2003, 20(1), 67.
9. K. Benz, K.-J. Regenauer, K.-P. Jäckel, J. Schiewe, W. Ehrfeld, H. Löwe and V. Hessel, Chem. Eng. Technol., 2001, 24(1), 11.
10. S. Suga, M. Okajima, K. Fujiwara and J.-I. Yoshida, J. Am. Chem. Soc., 2001, 123(32), 7941.
11. C. de Bellefon, N. Pestre, T. Lamouille and P. Grenouillet, Adv. Synth. Catal., 2003, 345, 19.
12. T. Fukuyama, M. Shinmen, S. Nishitani, M. Sato and I. Ryu, Org. Lett., 2002, 4(10), 1691.
13. R. Schenk, V. Hessel, B. Werner, A. Ziogas, C. Hofmann, M. Donnet and N. Jongen, Chem. Eng. Trans., 2002, 1, 909.
14. M. Kakuta, D. A. Jayawickrama, A. M. Wolters, A. Manz and J. V. Sweedler, Anal. Chem., 2003, 75(4), 956.
15. M. Kakuta, P. Hinsman, A. Manz and B. Lendl, Lab Chip, 2003, 3, 8.
16. P. Hinsman, J. Frank, P. Svasek, M. Harasek and B. Lendl, Lab Chip, 2001, 1, 16.
17. A. M. Guzman and C. H. Amon, J. Fluid Mech., 1996, 321, 25.
18. N. Schwesinger, T. Frank and H. Wurmus, J. Micromech. Microeng., 1996, 6, 99.
19. J. P. Branebjerg, J. Gravesen, J. P. Krog and C. R. Nielsen IEEE-MEMS '96, San Diego, USA, 1996, p. 441.
20. H. Mensinger, T. Richter, V. Hessel, J. Döpper and W. Ehrfeld, in Micro Total Analysis System, eds. A. van den Berg and P. Bergfeld, Kluwer Academic Publishers, Dordrecht 1995, 237.
21. H. Löwe, W. Ehrfeld, V. Hessel, T. Richter and J. Schiewe, Proceedings of 4th International Conference on Microreaction Technology, IMRET 4, Atlanta, USA, 2000, p. 31.
22. J. M. Ottino, F. J. Muzzio, M. Tjahjadi, J. G. Franjione, S. C. Jana and H. A. Kusch, Science, 1992, 257, 754.
23. D. M. Hobbs, P. D. Swanson and F. J. Muzzio, Chem. Eng. Sci., 1998, 53, 1565.
24. J. M. Ottino, The Kinematics of Mixing: Stretching, Chaos, and Transport, Cambridge University Press, Cambridge, 1990.
25. D. R. Sawyers, M. Sen and H.-C. Chang, Chem. Eng. J., 1996, 64, 129.
26. D. M. Hobbs and F. J. Muzzio, Chem. Eng. Sci., 1998, 53, 3199.
27. V. Hessel, H. Löwe, C. Hofmann, F. Schönfeld, D. Wehle and B. Werner, Proceedings of the 6th International Conference on Microreaction Technology, IMRET 6, AIChE Pub. No. 164, New Orleans, USA, 2002, 39.
28. W. Ehrfeld, V. Hessel and H. Löwe, Microreactors, WILEY-VCH, Weinheim, 2000, p. 63.
29. S. Hardt and F. Schönfeld, AIChE J., 2003, 49, 578.
30. B. P. Leonard, Locally modified QUICK scheme for highly convective 2D and 3D flows, in Numerical Methods in Laminar and Turbulent Flow, eds. C. Taylor, W. G. Habashi and M. M. Hafez, Vol. 5, Part 1, p. 35.
31. B. Noll, Numerische Strömungsmechanik, Springer, Berlin 1993.
32. D. R. Lide, Handbook of Chemistry and Physics, 79th edn., CRC Press, New York, 1998.
33. V. Hessel, S. Hardt, C. Hofmann and F. Schönfeld, in preparation for AIChE J..

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