INFLUENCE OF THE INLET CONDITIONS ON THE DEGREE OF MIXING OF A T-SHAPED MICRO-MIXER

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1 HEFAT01 9 th International Conference on Heat Transfer, Fluid Mechanics and Therodynaics July 01 Malta INFLUENCE OF THE INLET CONDITIONS ON THE DEGREE OF MIXING OF A T-SHAPED MICRO-MIXER Galletti C.* Brunazzi E. and Mauri R. *Author for correspondence Departent of Cheical Engineering, Industrial Cheistry and Materials Science University of Pisa, Pisa, I-5616, Italy E-ail: chiara.galletti@diccis.unipi.it ABSTRACT The degree of ixing of a T-shaped icro-ixer depends strongly on the inlet flow conditions. Specifically, through a series of nuerical siulations, we copared the case where the flow at the icro-ixer confluence is fully developed with that when it is not, and found that in the forer case engulfent occurs at saller Reynolds nuber, with a different flow pattern and with uch larger ixing efficiencies than in the latter case. In particular for fully developed flow conditions the engulfent shows S-shaped orphology, whereas for nonfully developed inlets a syetry breaking occurs, leading to a C-shaped engulfent pattern. This is characterized by uch lower ixing efficiencies. INTRODUCTION Recently, great attention has been devoted to understanding the governing echaniss and optiising the perforance of icro-ixers. Aong the, passive icroixers are especially interesting as they proote ixing without the help of any external power, through stretching and recobining of the flow fields [1]. T or Y shapes, in which the inlets join the ain channel with T- or Y-shaped branches, are the siplest icro-ixer geoetries. These are also suitable to shed light into fundaental aspects; oreover the T-shaped icro-ixer is a junction eleent often involved in ore coplex icrosystes []. The efficiency of T-shaped icro-ixers for liquid ixing has been proved by [3] who used such devices to quench down a cheical reaction in a very short tie. Lately, [4] [5] and [6] showed that three different flow regies occur in the icroixer, depending on the Reynolds nuber, Re, naely: the lainar, vortex and engulfent flow regies. NOMENCLATURE A [ ] area c [-] noralised concentration c [-] ean concentration D h_in [] hydraulic diaeter of the inlet channels G [Pa/] pressure gradient H [] depth of all the channels k [-] index L [] ixing channel length L [] inlet channel length Le [] length to reach fully developed conditions Pe [-] Peclet nuber Re [-] Reynolds nuber in the ixing channel Re in [-] Reynolds nuber in the inlet channel Sc [-] Schidt nuber v [/s] velocity v [/s] ean velocity W [] width of the ixing channel W [] width of the inlet channels x, y, z [] spatial coordinates Y [] conduit width Z [] conduit height Greek sybols δ [-] degree of ixing ( c) [-] ean square concentration difference, η [-] conduit aspect ratio µ [kg/( s)] dynaic viscosity ρ [kg/ 3 ] density σ [-] cup ixing flow variance c Subscripts i indices x,y,z directions ax axiu Abbreviations CFD Coputational Fluid Dynaics FD Fully Developed O order of 6

2 In the lainar flow regie the inlet streas coe into contact with one another in the ixing zone and then flow side by side through the ixing channel, resulting in a copletely segregated flow (stratified flow). In the vortex regie, above a critical Re, a secondary flow in the for of a double vortex pair occurs, due to the instabilities induced by the centrifugal forces at the confluence. In the engulfent regie occurring at higher Re, fluid eleents reach the opposite side of the ixing channel, thus largely increasing the degree of ixing. For a icro-ixer having a ixing channel with W = W and W = H = 100 µ, the engulfent occurred at Re = 146 [4] [6]. Subsequently, in order to ensure fast and efficient liquid ixing it is clearly fundaental to work in the engulfent flow regie and thus at relatively high Re. This is why several investigations were aied at assessing the effect of different geoetric and operating paraeters on the engulfent process [7] [8]. In recent research contributions, Coputational Fluid Dynaics (CFD) was extensively used to evaluate ixing perforance or propose design upgrading [6] [9] [10] [11]; in fact, CFD calculations have been found to give encouraging results when copared to experiental data, ainly because in lainar conditions and with sall size devices no additional odels have to be used and basically the CFD solver is eployed as a Navier-Stokes equation integrator. However, a few researchers highlighted the iportance of a careful choice of the discretization process to obtain reliable results and indicated the need of very fine (and thus CPU tie costing) coputational grids, even though lots of investigations were perfored with coarser grids [1], thus yielding unrealistic predictions. The present paper ais at analyzing the effect of the inlet flow conditions on the engulfent, as this has not been investigated in literature, yet. Our otivation stes fro the fact that in the engulfent regie the inlet channels could not be sufficiently long to ensure that at the junction of the T-ixer the inlet flow is fully developed, as it is generally assued to be. For exaple, in the case of icro-channel networks, of which T-shaped icro-ixers are the basic eleents, there can be liitations on the inlet channel a length, preventing the inlet flows fro being fully developed. TEST CASES The T-shaped icro-ixer and the reference coordinate syste are depicted in Figure 1. The icro-ixer has a ixing channel length L = 3000 µ, width W = 300 µ and depth H = 00 µ, while the inlet channels have a width W = W/ = 150 µ and a length L = 1500 µ. The liquid is injected at the two inlets with the sae flow rates. The inlet flow ean velocity was varied between 0.4 and. /s, corresponding to ixing channel Reynolds nuber between 96 to 58. Two hypotheses are ade: - the flow enters the inlets after a conduit that is long enough to reach fully developed (FD) flow conditions (FD case); - the flow is injected at the inlets with an unifor velocity profile (non-fd case). Figure 1 - Sketch of the T-shaped icro-ixer NUMERICAL METHOD Siulations were perfored with the coercial Ansys 13 code, using the FLUENT fluid dynaics package. Coputational doain and grid The grid was created with the software ICEM. A structured grid with regular cubical eleents of 4 µ edge (4. illion eleents) was chosen based on a grid independency study on the velocity field. Such a grid, although rather coputationally expensive, was found to be necessary to capture the flow features. In a cross section of the ixing channel there are 76x50 eleents, thus in fully agreeent with recoendations by [1]. Physical odel The liquid is water at abient conditions (density ρ = 998 kg/ 3, viscosity µ = kg/ s). A tracer was injected fro one inlet with a olecular diffusivity of /s, corresponding to the value of the crystal violet dye. Since the Schidt nuber Sc = O(10 4 ), the Peclet nuber, Pe = Sc Re is of O(10 6 ), indicating that ass transport is doinated by convection, so olecular diffusion can be neglected altogether and the solute can be considered as a passive tracer. For all practical purposes, the proble is equivalent to studying the ixing process of one water strea into with the other [13]. Therefore, in the following text c will denote either the noralized concentration (i.e. ranging fro 0 to 1) of the solute or that of one of the two water streas. At the outlet of the ixing channel, the pressure was set equal to the abient pressure, while no-slip boundary conditions were applied to the fixed walls. In addition, as entioned above, velocity boundary conditions at both inlet sections. were: an unifor velocity profile (non-fd inlet case); a fully developed velocity profile in a rectangular conduit (FD inlet case). The latter is found by solving the Navier-Stokes equations with no-slip boundary conditions at the walls in a rectangular conduit [14-15], obtaining: 7

3 G µ (1) 4GY 1 y z kπ z sin kπ Cosh kπ Tanh Sinh kπ 3 3 µπ k odd k Y Y η Y v( y, z) = y( Y y) where Y and Z are the sizes of the conduit (corresponding to W and H in our case), while η is the conduit aspect ratio. The pressure gradient G can be derived as a function of the ean velocity v, finding:. G 1µ v 19 1 kπ 1 Tanh = η 5 5 Y π k odd k η The velocity profile of Eq. (1) was iposed at the inlet boundaries by defining an ad hoc C++ subroutine. Figure illustrates the ipact of the two types of boundary conditions on the velocity profile at the entrance of the ixing zone (x = -400 µ) for Re = 480. Such a region was evaluated fro the analysis of the FD case as the region at which strealines start to bend and deviate fro the Poiseuille flow. It can be noticed that the non-fd case exhibits a flat region in the centre of the conduit, with a peak velocity which is about 15% lower than that for the FD case. Such a difference is due to the fact that a 1500 µ inlet channel is not long enough to ensure a fully developed flow for Re = 480. Indeed, the conduit length to hydraulic diaeter ratio that is needed to reach fully developed conditions, for rectangular conduits can be evaluated as [16]: Le D h _ in 1 ( 0.148Re ) Re = 0.379exp in in Figure - Longitudinal velocity (v x ) profiles along the vertical centerline of the inlet channels at x = -400 µ. Re = 480. For the present case L e /D h_in = 0 and thus L e = 3800 µ, which is uch larger than the actual length of the inlet channel L = 1500 µ, corresponding to a L /D h_in = 8.8 (actually the reduced length of the inlet channels, which represents the () (3) distance between the inlet and the edge of the ixing region, should be used instead thus leading to L / D h_in = 7.3). Thus the non-fd case presents conditions far fro a fully developed flow and, consequently, the resulting velocity profile is ore blunt that the Poiseuille parabolic curve. Solver control Most of the siulations were run with a steady solver. A second order discretisation schee was used for all equations, as higher order discretization schees were found to provide no appreciable differences. A sufficient nuber of iterations was perfored to obtain residuals for all equations of less than and a stationary solution with iterations. Most of the ties, equations residuals of 10-5 or 10-6 are considered sufficiently sall to reach steady solutions in CFD probles. However, the onitor of soe physical variables indicated that for the present case residuals of even 10-7 were not sufficient to obtain stable solutions (leading to copletely different ixing patterns than the converged ones), so that residuals should be driven to values less than Practically any ties, residuals were found to firstly decrease down to about (finding an unstable solution of the flow field), then to increase again (changing the flow pattern) and only after that to decrease down to (corresponding to a stable solution) [17]. The siulation for the FD case at Re = 480 was also perfored with a transient solver, but this indicated a steady flow. RESULTS Figure 3 depicts velocity vectors (colored by the longitudinal velocity coponent, i.e. v y ) at different crosssections of the ixing channel for the non-fd case. It can be observed (Figure 3a) the foration of pair of vortices (on the right- and left- hand sides) which initially are very siilar. Moreover the flow field is syetric with respect to the xy id-plane. While proceeding along the ixing channel, the engulfent occurs as one strea tends to enter through the central part of the cross-section. This leads to an engulfent flow with a C-shaped pattern, i.e. displaying a) a irror syetry with respect to the xy id-plane; b) no interchangeability between the two inlet fluids (see Figure 3d). This is also evident fro the distribution of tracer concentration on the sae planes reported in Figure 4. At the beginning of the ixing channel, the two fluids are well segregated (see Fig. 4a for y = -500 µ;); then the tracer, coing fro the righthand side, crosses the yz id-plane near the center, whereas the strea coing fro the left-hand side crosses the plane near the walls, thus originating a pair of vortices, as it can be evinced fro Fig. 4d, with y = -500 µ. The authors knowledge this kind of engulfent (with C-shape pattern or kidney-vortex structure) has not been observed previously in icro-ixer, even though it is known for other configurations, such as crossflow jets [18]. Additional siulations have shown that the fluid that tends to concentrate at the center of the ixing channel could be the red fluid, coing fro the lefthand inlet channel, as reported here or, with equal probability, the blue fluid, coing fro the right-hand side. For the non- FD case the Re = 480 of the present results correspond to the onset of the engulfent as shown in our previous work [17]. 8

4 a) (b) (a) (b) (c) (d) (c) (d) Figure 3 - Velocity vectors on xz planes (coloured by v y, /s) for non-fd case at: (a) y = -500 µ; (b) y = µ; (c) y = µ; (d) y = -500 µ. Re = 480. Figure 5 - Velocity vectors on xz planes (coloured by v y, /s) for FD case at: (a) y = -500 µ; (b) y = µ; (c) y = µ; (d) y = -500 µ. Re = 480. Figure 4 Tracer concentration distribution along the ixing channel for the non-fd case at: (a) y = -500 µ; (b) y = µ; (c) y = µ; (d) y = -500 µ. Re = 480. Figure 6 Tracer concentration distribution along the ixing channel for the FD case at: (a) y = -500 µ; (b) y = µ; (c) y = µ; (d) y = -500 µ. Re =

5 For lower Re a lainar flow regie was observed resulting in a segregated flow. Siilarly Figure 5 reports velocity vectors for the FD case. It can be observed that in this case engulfent occurs as one strea enters preferentially below the xy ixer id-plane, whereas the other strea enters above it, leading to a big pair of vortices and other inor vortical structures. Thus the flow is in the fully engulfent regie. The engulfent for the FD case shows a S-shaped pattern, i.e. displaying a) a point syetry with respect to the center of the cross section; b) interchangeability between the two inlet fluids. It was found that for the present icro-ixer, the engulfent occurred at uch lower Reynolds nuber, i.e. around Re = 19 in case of FD case [17]. Moreover Re = 480 is also above the range of Reynolds nubers for which periodic instabilities were observed [9, 17]. For the present icro-ixer geoetry such instabilities were found to occur between Re = 0 and Re = 400. However for higher Re, stable steady state solutions were found. Figure 6 illustrates the distribution of the tracer at different cross sections along the ixing channel for the FD case, indicating that the engulfent patterns are siilar to those observed by other researchers. For instance, the tracer distribution at y = -500 µ (Fig. 6a) is alike the S-shaped engulfent pattern observed experientally by [4] and nuerically by [6] for a different T-icro-ixer. Figure 7 copares the tracer concentration distribution in the xy (horizontal) id-plane for the non-fd and FD cases. It can be observed how the presence of a flat region in the inlet velocity profile (non-fully developed conditions) daps the ixing process. Moreover the FD case shows a syetric pattern with respect to the yz id-plane, whereas for the non- FD case the two streas are not interchangeable. To copare the ixing efficiency in the two cases, a bulk concentration (also referred to as cup ixing concentration or flow average concentration ), that is the flow-weighted ean concentration, is defined as: of the cross section and c=1 in the other half) we have: ( c) = 1 ax 4. For sake of convenience, it will be used the following definition of degree of ixing, δ = 1 σ (6) c which varies fro 0, when the two streas are perfectly unixed, to 1, when they are copletely ixed. Figure 8 copares the degree of ixing, along the ixing channel in the FD case with that in the non-fd showing that it is uch larger in the forer. It can also be observed that in both cases as expected, the degree of ixing augents along the ixing channel. In the outlet section the degree of ixing for the FD case is δ = 0. 5, whereas it is just δ = for the non-fd case, thus confiring the ost effective ixing obtained with a parabolic inlet flow. cv 1 1 c = ; cv = cv da; v v da, v A = A A A (4) where the overbar indicates the cup ixing average, while the brackets denote the volue (i.e. un-weighted) average on a cross section of area A. The cup ixing is preferred over the ore usual volue, average concentration, because it, describes the concentration that one would easure if the tube were chopped off and if the fluid issuing forth were collected in a container and thoroughly ixed [19] In the sae way, a bulk, or cup ixing, ean square concentration difference can be defined as: ( ) ( ) c c v c = ; σ c = ( c) ( c), ax v where its axiu value is used as a noralizing factor. Such axiu ean square concentration difference occurs when the two streas are copletely unixed (i.e., c=0 in half (5) (a) (b) Figure 7 - Tracer concentration distribution in the xy id-plane for (a) non-fd case and (b) FD case (branches have been cut in the iage). Re = 480. CONCLUSION The effect of inlet flow conditions on the ixing process of a T-shaped icro-ixer has been investigated. In particular it was found that when fully developed conditions are ensured at the inlets, the engulfent occurs at relatively low Reynolds nubers and with typical S-shaped flow patterns. On the other hand, when the fluid flow at the confluence is non-fully 30

6 developed and presents even a sall plug flow region at the center, engulfent occurs at uch larger Reynolds nubers and with a C-shaped engulfent pattern (with the tracer crossing the ixing channel id-plane either near the center or at the walls). This is also characterised by a uch lower ixing efficiency. Therefore the issue of fully (or not) developed flow at inlets ay affect strongly the ixing process, thus it should be taken into account for the design of ixers, especially when there can be liitations on the length of the inlet channels, as for instance in case of icro-channel networks. Figure 8 - Degree of ixing along the ixing channel for the non-fd and the FD cases. Re = 480. REFERENCES [1] Kuar V., Paraschivoiu M., and Niga K.D.P., Single-phase fluid flow and ixing in icrochannels, Cheical Engineering Science, Vol. 66, 011, pp [] Bothe D., Lojewski A., and Warnecke H.-J., Coputational analysis of an instantaneous cheical reaction in a T-icroreactor, Aerican Institution of Cheical Engineers Journal, Vol. 56, 010, pp [3] Bökenkap D., Desai A., Yang X., Tai, Y. Marzluff E., and Mayo S., Microfabricated silicon ixers for subillisecond quenchflow analysis, Analytical Cheistry, Vol. 70, 1998, pp [4] Hoffann M., Schlüter M., and Räbiger N., Experiental investigation of liquid liquid ixing in T-shaped icro-ixers using µ-lif and µ-piv, Cheical Engineering Science, Vol. 61, 006, pp [5] Engler M., Kockann N., Kiefer T., and Woias P., Nuerical and experiental investigations on liquid ixing in static icroixers, Cheical Engineering Journal, Vol.101, 004, pp [6] Bothe D., Steich C., and Warnecke H.-J., Fluid ixing in a T-shaped icro-ixer, Cheical Engineering Science, Vol. 61, 006, pp [7] Soleyani A., Yousefi H., Turunen I., Diensionless nuber for identification of flow patterns inside a T-icroixer, Cheical Engineering Science, Vol. 63, 008, pp [8] Reddy Cherlo S.K., and Pushpavana S., Effect of depth on onset of engulfent in rectangular icro-channels, Cheical Engineering Science, Vol. 65, 010, pp [9] Dreher S., Kockann N., and Woias P., Characterization of lainar transient flow regies and ixing in T-shaped icroixers, Heat Transfer Eng. 30, 009, pp [10] Soleyani A., Kolehainen E., and Turunen I., Nuerical and experiental investigations of liquid ixing in T-type icroixers, Cheical Engineering Journal, Vol. 135, 008, pp. S19-S8. [11] Aoki N., Uei R., Yoshida A., and Mae K., Design ethod for icroixers considering influence of channel confluence and bend on diffusion length, Cheical Engineering Journal, Vol.167, 011, pp [1] Hussong J., Lindken R., Pourquie M., and Westerweel J., Nuerical study on the flow physics of a T-shaped icro ixer, in: B.V. M. Ellero et al. (eds.), IUTAM Syposiu on Advances in Micro- and Nanofluidics, IUTAM Bookseries 15, Springer Science + Business Media, 009. [13] Ottino J.M., and Wiggins S., Introduction: ixing in icrofluidics, Philos. Trans. R. Soc. London, Ser. A 36,004, pp [14] Chatwin P.C., and Sullivan P.J., The effect of aspect ratio on the longitudinal diffusivity in rectangular channels, Joural of Fluid Mechanics, Vol. 10, 198, pp [15] Happel J., and Brenner H., Low Reynolds Nuber Hydrodynaics, Prentice Hall, 1965; Eq. (-5.4). [16] Dobrowski N., Foueny E.A., S. Ookawara, and A. Riza, The influence of Reynolds nuber on the entry length and pressure drop for lainar pipe flow, Canadian Journal of Cheical Engineering, Vol. 71, 1993, pp [17] Galletti C., Roudgar M., Brunazzi E., and Mauri R., Effect of inlet conditions on the engulfent pattern in a T-shaped icro-ixer Cheical Engineering Journal, Vol , 01, pp [18] Haven B.A., and Kurosaka M., Kidney and anti-kidney vortices in crossflow jets. Journal of Fluid Mechanics, Vol. 35, 1997, pp 7-64 [19] Bird R.B., Stewart W.E., and Lightfoot E.N., Transport Phenoena, Wiley, 1960, pp. 97 and