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Ind. Eng. Chem. Res. 2009, 48, 8678–8684
Screening, Selecting, and Designing Microreactors Siva Kumar Reddy Cherlo, Sreenath K., and Pushpavanam S.* Chemical Engineering Department, I.I.T. Madras, Chennai-600036, India
Microreactors can be used to carry out hazardous organic reactions like nitration of benzene and toluene safely. A theoretical analysis of the performance of these systems is usually based on using a computational fluid dynamics approach incorporating the effect of reactions. These simulations help us identify different flow regimes of the system as stratified flow, vortex flow, and engulfment flow. Though it provides realistic estimates of the performance this is a computationally intensive approach. In this work we discuss two strategies for modeling, which exploits the inherent features of single phase flows in microchannels. The first is based on a one-dimensional model where lateral mixing is incorporated using a “pseudo” mass transfer coefficient. In the second we use an effective dispersion coefficient to represent lateral mixing in microchannels. In both approaches we assume the fluid flows as a plug, that is, with a uniform velocity across the cross-section. We consider two different sets of reactions, (i) a set of parallel reactions and (ii) a set of series parallel reactions, and determine how the flow regime in the channel can be exploited to carry out reactions and simultaneously separate the products. The 1D results are compared with those of the 2D model where the transport in the lateral direction is modeled using a dispersion coefficient. The results of the two simplified models are compared with rigorous two-dimensional simulations using Fluent. The approach proposed here can be used to screen and analyze the performance of different reactions in microchannels and examine if the flow features can be used to carry out reactions and separate the products simultaneously. The promising candidates can then be analyzed rigorously for a realistic evaluation of the performance incorporating the prevailing flow structures using computational fluid dynamics. 1. Introduction Miniaturized components play a vital role in biological and chemical synthesis. Several components like micromixers, microreactors have been analyzed with the view of developing devices for synthesis of chemicals,1 environmental monitoring,2 pharmaceuticals, and polymerization reaction.3 The performance of mixers has attracted interest of many investigators as it is relevant for many applications. Microreaction engineering involves carrying out many industrially important chemical reactions in microchannels. The progress of these reactions in microchannels is determined by the interaction of hydrodynamics and reaction kinetics. Here the flow is characterized by extremely low Reynolds numbers (Re), that is, the flow is essentially laminar. There is no lateral convective flow across the cross section except in the developing region. Any mixing in the lateral direction in the fully developed region occurs primarily due to molecular diffusion and is very slow. The dependence of mixing in the lateral direction on Re has been analyzed experimentally and theoretically in various geometries. T-shaped microchannels were investigated and three different regimes in the developing region of laminar flow, that is, stratified flow, vortex flow, and the engulfment regimes were determined. These flow regimes at different Reynolds numbers have been investigated recently by many researchers4,5 Kockmann et.al6 has discussed how these regimes can be exploited recently for the high through-put production of chemicals in microreactors. Mixing in microchannels was found to be enhanced by secondary vortices which were induced by bends.7The performance of several micromixers has been reviewed recently by Nguyen and Wu.8 Two phase flow systems similarly exhibit different flow regimes as slug flow, stratified flow, etc.9 * To whom correspondence should be addressed. E-mail: spush@ iitm.ac.in. Fax: +91-44-22570509. Phone: +91-44-22574161.
Investigations on mixing at lower Reynolds numbers have been performed.10,11 These studies have focused on flow behavior alone without giving any consideration to reactions and separations. Aoki et. al12 considered multiple chemical reactions under parallel flows and determined the influence of stream width on yield and selectivity. They discuss how separation and reaction can occur simultaneously in a microchannel. Polona and Igor13 have considered extraction in a microchannel under stratified flow conditions. A simple bimolecular reaction was analyzed experimentally and theoretically under stratified flow conditions recently by Charles et.al.14 A good agreement between the experimental values and the theoretical predictions was found. The separation of components exiting the reaction is an important challenge in microreactors.15 Computational fluid dynamics (CFD) has been used extensively to investigate and understand flows in these systems.16,17 However these are computationally intensive and are prone to numerical diffusion. Accurate simulations require a large number of grids to compensate these artifacts and the primary issue in using CFD as a design tool is the extremely large amount of time required for the simulations. This computational time depends on the accuracy of the solution we seek. Separation of products after the partial conversion of reactants is an important step when product purity is a critical parameter. The hydrodynamics prevailing in a microchannel render it feasible to carry out reaction and separation simultaneously. Numerical simulations based on CFD can be used to simulate the performance of the system to investigate the optimum conditions in a microreactor. However, this is computationally intensive. We need efficient tools to screen conditions which give optimum performance of a microsystem before analyzing these using rigorous CFD models. Hence it is desirable to have a simplified approach to quickly screen different designs and operating conditions which is based on a simplified model incorporating the important physics of the system. These would
10.1021/ie900306j CCC: $40.75 2009 American Chemical Society Published on Web 08/04/2009
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product and U is the undesired product. The reactions are assumed to be nonelementary with the reaction rates given as rD ) kDCACB2 and rU ) kUCA2CB where ri is the rate of generation of species i; (ii) the elementary series-parallel reactions A + B f R; R + B f S. 2. One Dimensional Approach Based on Pseudo-mass-transfer
Figure 1. Schematic representation of the T-shaped microchannel being analyzed (flow top view, the two fluids flow side by side).
consist of simple models incorporating the essential physics and capable of capturing the behavior of the real system. In the present work, we discuss a methodology which provides an efficient tool for designing microreactors using mathematical models which have been simplified on the basis of the physics prevailing in the system. We compare the performance of these simple models with rigorous CFD simulations. The proposed methodology can be used to screen different designs and to choose the optimum operating conditions. The selected design can then be analyzed rigorously via CFD to determine the performance of a given design. Our focus is on single phase flows in microchannels. The flow behavior is well understood here and we investigate the stratified flow regime. We discuss two modeling approaches which incorporate the essential physics of the system and are not based on rigorous CFD simulations. In the first approach, mixing in the stratified flow in the microchannel is captured using a “pseudo” mass transfer coefficient. In the second approach, we study the effect of mass transfer as arising due to an effective dispersion coefficient. Both approaches assume that the fluid behaves as a plug flow with uniform velocity across the channel. In these simplified approaches, entrance effects are neglected. We compare these results with the rigorous CFD simulations using Fluent. Here the developing flow field is solved along with the species balance equations. The simulation predictions based on the simplified models as well as Fluent are found to be comparable. We suggest the use of the simplified models to screen designs and evaluate the performance of various reactions in different microsystems to elegantly examine their effectiveness. The designs which yield promising results based on the simplified approach are determined so that they can be subject to a more detailed analysis using CFD. To increase the selectivity and separation of the products simultaneously we analyze how the stratified flow regime prevailing at low Re in a microchannel can be exploited to carry out parallel reactions. The microchannel geometry that we consider is T-shaped (see Figure 1). The two reactants enter through the two symmetric limbs. We focus only on the mixing channel where the reaction occurs. For low flow rates the two fluid streams flow in a stratified manner in the mixing channel and there is no effective mixing. The primary focus of this work is to discuss modeling approaches to determine conditions under which the poor mixing can be exploited to achieve reaction and separation. We consider two sets of reactions: (i) parallel reactions, A + B f D; A + B f U, where D is the desired
As a first step we use a one-dimensional model where all the concentrations are assumed to vary only with “x” the axial direction, and the concentration in the other directions are assumed to be uniform. We consider convective transport due to flow in each half at an average velocity. This velocity is assumed to be the same in both halves. The governing species balance equations are written for each half and the diffusive convective flux of a species in the transverse “y” direction from one-half to the other is approximated by a “pseudo-mass-transfer coefficient”. The steady-state concentration profile of the various species in each half is governed by ordinary differential equations which describes their variation along the axial direction. The species concentrations in the first (0 < y* < 170 µm) and second half (170 < y* < 340 µm) are denoted by subscripts 1 and 2, respectively. Here the superscript * is used to denote the coordinate with dimensions. The behavior of the parallel reaction system is governed by the following eight ordinary differential equations which describe the variation of the dimensionless concentration of various species along the length: dCA1 dx
) -Da1CA1CB12 - Da2CA12CB1 - P[CA1 - CA2] (1a)
dCB1 dx
) -Da1CA1CB12 - Da2CA12CB1 - P[CB1 - CB2] (1b)
dCA2 dx
) -Da1CA2CB22 - Da2CA22CB2 - P[CA2 - CA1] (1c)
dCB2 dx
) -Da1CA2CB22 - Da2CA22CB2 - P[CB2 - CB1] (1d) dCD1 dx dCU1 dx dCD2 dx dCU2 dx
) Da1CA1CB12 - P[CD1 - CD2]
(1e)
) Da2CA12CB1 - P[CU1 - CU2]
(1f)
) Da1CA2CB22 - P[CD2 - CD1]
(1g)
) Da2CA22CB2 - P[CU2 - CU1]
(1h)
In all the equations the term on the left represents the convective flux. The nonlinear terms on the right represent the reaction kinetics while the linear term accounts for the lateral mass transfer between the two halves CA1 (CA2) represent concentration of “A” in the first (second) lateral half of the channel, respectively. The dimensionless parameters occurring in the above equations are
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Da1 )
kDCA02L uj
,
Da2 )
kUCA02L uj
,
P)
km uj
Here kd (ku) represent the rate constant of the reaction giving the desired (undesired) product. km is the pseudo-mass-transfer coefficient which characterizes mixing in the lateral direction between the two halves, CA0 is the initial concentration of A (i.e., feed concentration), ujj is the average velocity, L is the length of the microchannel. The parameter P can be viewed as a dispersion number or the reciprocal of a Pe number. This becomes clear when km is interpreted as the diffusion coefficient divided by a length scale in the lateral direction (hydraulic diameter in this case). These equations are subject to the initial condition CA1 ) CB2 ) 1 at x ) 0, and the concentration of all other species is zero at the inlet x ) 0. These conditions are valid only when the feed is stochiometric. Axial dispersion effects can be neglected and the plug flow velocity field can be used when “D/uL” < 0.001. In our simulations we have ensured that these conditions are satisfied. Figure 2 shows the variation of the dimensionless exit concentrations of desired product (D) and undesired product (U) in each half of the microchannel as we vary the pseudo-masstransfer coefficient, that is, parameter P, which is proportional to the lateral mixing. The exit concentrations (at x ) 1) are obtained by numerically integrating eq 1a. We see that at P ) 7, that is, an intermediate value, the difference in the concentration values between the products D and U goes through a maximum. In the first half we have more of the undesired product (Cu1 ) 0.19 and CD1 ) 0.09) while in the second half the reverse situation prevails (CU2 ) 0.09 and CD2 ) 0.19). We see that as P increases there is good mixing between the two halves so that the concentration of the two products D and U are equal in each half and are equal to each other. The parameter P in this approach is a measure of the transverse mixing. These results can be understood by noting that the local selectivity is given by rD kDCB ) rU kUCA
(2)
Consequently we have higher selectivity toward the desired product when the concentration of B is high and that of A is low. The low Re flow in a microchannel creates an intrinsic stratification in concentrations of A, B. Thus when we have low Re, the first half of the channel has high concentration of A (0 < y < 1/2) and the second half has a high concentration of B (1/2 < y < 1). In the first half which is rich in A, B enters by diffusion or convective flow in the transverse direction from the other half. Here the formation of U is preferentially favored as we have a high concentration of A and a low concentration of B. Similarly, we can argue that in the lower half, the formation of D is preferred. Thus a natural separation of the two products can be achieved in the two halves of the channel, and this is possible only since the flow in the microchannel is stratified. This explains the results in Figure 2 where CD2 > CD1 and CU2 < CU1 for intermediate P. In Figure 3 we show how the dimensionless exit concentration of D and U vary as we change Da1 for P ) 35 and Da2 ) 100. For Da1 < 100 the reaction giving the desired product is slower and so we have high values of CU in both halves while for Da1 > 100 the value of CD is higher in both the halves. The values of CD, CU in each half is equal as the value of the effective mass transfer parameter P chosen is high representing good mixing between the two halves. In Figure 4 we show how the dimensionless exit concentrations of D, U varies with Da2 in each half of the channel, for P
Figure 2. Variation of dimensionless exit concentration of D and U in each half of the microchannel as a function of the dimensionless mass transfer coefficient P for Da1 ) Da2 ) 100.
Figure 3. Variation of dimensionless exit concentration of D and U in each half of the microchannel as a function of Da1, for Da2 ) 100, P ) 35.
Figure 4. Variation of dimensionless exit concentration of D and U in each half of the microchannel as a function Da2 at P ) 5 and Da1 ) 100.
) 5 at Da1 ) 100. For low Da2 the concentration of U is low in each half as the reaction rate favoring its production is low. As we increase Da2 the reaction rate increases and more U is formed. This results in an increase in the concentration of U in both halves. The concentration of D decreases as the parameter
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Da2 increases since now more reactants are consumed to produce U and so less is available to obtain D. The concentration of D in the first half CD1 is lower than that in the second half. This follows as CB is lower in the first half since now the pseudomass-transfer coefficient is low. Hence here we have a preferential formation of U and this results in CD1 being much lower than CD2. The essential difference between Figure 3 and Figure 4 arises from the values of the pseudo-mass-transfer coefficient. In Figure 4 this value is low and hence the concentration of various species in each half is different, while in Figure 3 the value is high and the concentration of species in each half is the same due to good lateral mixing. 3. Mass Transfer Using a Transverse Dispersion Coefficient: A 2D Model The flow in a microchannel is laminar in the fully developed region as the prevailing Reynolds number is low. In the developing region vortices can form in each half and can aid in the transport of species. This transport of species in the transverse direction in this developing region is now represented using an effective diffusion coefficient in the lateral direction. The transport in the axial direction is represented using an average convective velocity. This is different from the situation which arises in the modeling of packed bed reactors where a nonuniform flow is represented as a plug flow with an axial dispersion coefficient (Fogler15). The system behavior is now governed by species balance equations which are second-order partial differential equations. These equations are solved for the dependence of the steady-state concentration profile along the dimensionless x and y directions. Convection is considered in the axial direction (x) and diffusional transport is considered along the direction normal to the flow (i.e., y). To reduce the complexity of the problem, we assume the velocity to be uniform in the y direction, that is, plug flow. The species balance equation for A throughout the channel is Vj
∂2CA* ∂CA* ) DAM - (kDCA*CB*2 + kUCA*2CB*) ∂x* ∂y*2
(3a)
Here the variations in the third direction z are neglected rendering the model essentially two-dimensional. In dimensionless form the above equation which determines the steady-state profile can be written as 2 ∂CA 1 ∂ CA ) - (Da1CACB2 + Da2CA2CB) ∂x PeA ∂y2
(3b)
Similar equations are written for the other species B, D, and U. The steady-state behavior is governed by three dimensionless parameters Da1, Da2, and Pe. These dimensionless numbers are defined as kDCAo2dh Da1 ) , Vj
kUCAo2dh Da2 ) , Vj
PeA )
Vjdh DAM
CA ) CA* /CA0 CB ) CB* /CA0 Here dh is the hydraulic diameter of the channel and CA0 is the feed stream concentration. The initial conditions chosen for the simulations is at x ) 0
CA ) 1 for 0 < y < (1/2) CB ) 1 for (1/2) < y < 1 CD ) CU ) 0 for all y
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(4a)
The corresponding boundary conditions in the y direction are the no-flux boundary conditions ∂CA ) 0 at y ) 0 and y ) 1 ∂y
(4b)
The T-shaped microchannel of width 340 µm and of length 5000 µm is considered for simulations of the simplified 2D model. We simulate the system with an explicit finite difference scheme in MATLAB 7 to analyze the behavior of system numerically. The parameters chosen for the various simulations are a dispersion coefficient of 10-9 m2/sec, and rate constants k1 ) k2 ) 6.5 × 10-4 (m3/gmol)2 /sec. The velocity (V) of the fluid through the channel is taken as 0.2 mm/s. With the above choice of parameters as base values Peclet numbers (Pe) are in the range of 50-200 and Damkholar numbers (Da) are in the range of 0.85-85. We first discuss results where we have considered both the Da numbers to be equal to 0.85 and the Pe numbers for all species to be 68. Figure 5 shows the concentration of D and U at x ) 4.5 mm when the diffusivities of all species are constant. We see that the concentration of D in the first half (0 < y< 1/2) is lower than that of U. This arises because of the form of the rate expression chosen for the reactions. In this half the concentration of A is high and the concentration of B is low as it arrives here by diffusional transport, and this results in preferential production of U resulting in CU > CD. The reverse situation prevails in the second half (1/2 < y < 1). We see that the variation of the dimensionless concentrations of species D and U is from 0.12 to 0.19 in the transverse direction. We now analyze the case when the diffusivities of the various species are different. For this we choose PeA ) 68, PeB ) 450, PeD ) PeU ) 500; Da1 ) Da2 ) 0.85. The variation of the concentrations of D and U for this set of parameters is shown in Figure 6. We see that in the first half (0 < y < 1/2) the concentrations of both products are low and equal, while in the second half (1/2 < y < 1) there is a significant difference in the concentrations of D, U. Here for the choice of the diffusional constants species B cannot diffuse from the lower half to the other half as a result of which no significant reaction occurs in the upper half. However species A can diffuse to the lower half so that the reaction occurs here, and we see a substantial difference in the concentration of D and U. Since B is in excess in this half in view of the form of the rate expressions chosen as discussed earlier we have CD > CU. 4. CFD Simulations Based on Fluent (2D case) So far we have assumed that the fluid flow behaves as a plug flow with an average uniform velocity. To take into account the actual situation including entrance effects arising in the channels, simulations are performed using the commercial CFD software FLUENT 6.2.16. This rigorous approach helps us incorporate the developing flow field and the associated flow structures. The lengths of T-arms are taken as 3 mm each and length of mixing channel was 5 mm. From one inlet we introduce a mixture of A, C with a mass fraction of 0.5 each and from the other side we send a mixture of B, C with equal mass fractions of 0.5 each. Here C is an inert species. All input parameters as rate constants, diffusivities were maintained at the same values as in the 2D-simplified model. We depict the
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Figure 5. Variation of the dimensionless concentrations of D and U along the transverse direction for x ) 4.5 mm when the diffusivities are equal using a simplified model. PeA ) 68 ) PeB ) PeD ) PeU, for constant diffusivity.
Figure 6. Variation of the dimensionless concentration of D and U along the transverse direction for x ) 4.5 mm when the diffusivities are not equal using a simplified model. PeA ) 68, PeB ) 450, PeD ) PeU ) 500.
variation of the dimensionless concentrations of the desired and undesired products along the y direction at x ) 4.5 mm. The number of cells considered for the 2d simulations is 43075 for the T-shaped geometry. The boundary conditions used for the simulations are the no flux condition at the walls and the outflow condition at the outlet. For specifying mass diffusivity, the dilute approximation is considered and different diffusivities are specified for different species. The velocity chosen for simulations is the one corresponding to the simplified model, that is, 0.2 mm/sec at the inlet boundary. We have carried out simulations using a constant diffusivity of 10-9 m2/sec for each of the species and analyzed the system behavior using FLUENT. This approach takes into account entrance effects prevailing in the system and does not assume the velocity profile to be laminar throughout the channel. The concentration profiles predicted at x ) 4.5 mm is shown in Figure 7. We observe that there is a good qualitative as well as quantitative match between the predictions using the detailed model and the one using our idealized model (Figure 5). As a further test and validation of the results using our simplified model we have also considered the case where the dispersion coefficients of the different species are different. For solving the mass balance equations for species A, B, D, and U the specified coefficients are 10-9 m2/sec, 0.15 × 10-9 m2/sec, 0.136 × 10-9 m2/sec and 0.136 × 10-9 m2/sec (corresponding to the
Figure 7. Variation of the dimensionless concentration of D and U for the same conditions as in Figure 5 using the more rigorous simulations based on Fluent.
Figure 8. Variation of the dimensionless concentration of D and U for the same conditions as in Figure 6 using the more rigorous simulations based on Fluent.
values used in the 2D model). The concentration profiles obtained for this case are shown in Figure 8. We again see that there is a good qualitative as well as quantitative agreement between the results of the simplified (Figure 6) and the detailed models (Figure 8). We now discuss a comparison of the predictions using the three approaches: the 1D model, 2D model, and the Fluent simulations. For the case of the parallel reaction these predictions are compared in Figure 9. The transversely averaged concentrations from the 2D model and Fluent simulations are compared with the 1D model and we see a good agreement between the predictions. For this comparison study of the two models the total length of the channel was chosen as 5 mm. The mass transfer coefficient used in the 1D model is selected so that it matches with the Pe of the 2D model using film theory (km ) Dab/dh). The diffusivity and the hydraulic diameter used are the same in both models. The nonzero value at the inlet of the Fluent simulations arises because some reaction occurs before the two fluid streams enter the straight length of the channel. 5. The Series Parallel Reactions We now apply the above approach and illustrate it on a seriesparallel reaction system A + B f R; R + B f S. For the case when all the diffusivity of all the species is assumed to be equal, the profiles of the products R, S obtained from the simplified
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Figure 11. Variation of the concentration of R and S along the transverse direction for x ) 4.5 mm when the diffusivities are not equal using a simplified model. PeA ) 450, PeB ) 68, PeR ) PeS ) 500. Figure 9. Comparison of axial profiles of 1D and transversely averaged profiles of 2D model and Fluent simulations for parallel reactions for PeA ) PeB ) PeD ) PeU ) 68 and Da1 ) 0.85 ) Da2.
Figure 12. Comparison of 1D and transversely averaged profiles of 2D model and Fluent simulations for series-parallel reactions for PeA ) PeB ) PeD ) PeU ) 68 and Da1 ) 0.03 ) Da2. Figure 10. Variation of the concentration of R and S along the transverse direction for x ) 4.5 mm when the diffusivities are not equal using a simplified model. PeA ) 68, PeB ) 68, PeR ) PeS ) 68.
2D model and the rigorous Fluent simulations are shown in Figure 10. We see that the two predictions are in reasonable agreement, and the concentrations are uniform along the cross section indicating that molecular diffusion is sufficiently strong giving a uniform lateral concentration profile. When the diffusivities are assumed to be different for various species the profiles obtained for the concentrations of R and S along the cross section are shown in Figure 11. We see that the predictions are in good agreement for this case as well. We see that when the diffusivities are high, that is, when the Pe is low, the separation of the products across the channel is poor, that is, the concentrations of the species across the channel are uniform (Figure 10). When the diffusivities are low or Pe is high, we have a good separation of the two products across the channel (Figure 11). Here the concentration of R in the first half of the channel is high as here the amount of A present is high. The B coming in by virtue of a higher diffusion coefficient is consumed to give rise to R but there is not sufficient amount of B to give rise to S. In the second half there is a high concentration of B and less concentration of A so we have a
low concentration of R. Since there is less amount of R present the amount of S in this half is low. The deviation observed between the Fluent and the simplified model predictions is because in Fluent we consider the flow field to be developing from the entrance to the exit and this effect is neglected in the simplified models. Figure 12 shows a comparison of the predictions of the idealized 1D and 2D (averaged concentrations) models with Fluent predictions of species R along the axial length for the series-parallel system. We see that the two models give similar predictions. 6. Summary and Conclusions We conclude that the behavior of the microsystems can be captured elegantly by using simplified models which incorporate the essential physics. For stratified flows we can use a pseudomass-transfer coefficient to determine the lateral mixing. In this approach we lose the information on the concentration variation in the lateral direction. Alternatively we can use a diffusion coefficient to measure lateral mixing and capture the effect of representing a developing flow profile with a plug flow profile. Here we obtain the variation of the concentration profile in the transverse direction which can be compared with the predictions
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of a more detailed model based on CFD simulations. The approach here is different from the axial dispersion model since here we are incorporating the dispersion in the direction transverse to the flow. Our results show that the simplified models can be used to get a good estimate of the performance of real reactors. They can be used to screen different designs effectively, and the selected designs can then be analyzed more rigorously using CFD tools. We observed that the more rigorous analysis does not show any significant difference in the results obtained from the simplified idealized models. The simplified models were applied for parallel reactions, series parallel reactions. These models give us a reliable quantitative estimate of the system performance. The results of the work here show a promising methodology to save computational time and evaluate reactor performance. The work here can also be extended to the vortex flow regime in microchannels but now the diffusion coefficient and mass transfer coefficient would be enhanced. We can exploit the stratified flow regimes to devise microchannel systems wherein reaction and separation can be carried out in a single system. Literature Cited (1) Watts, P.; Haswell, S. J. The application of microreactors for organic synthesis. Chem. Soc. ReV. 2005, 34, 235. (2) Leanne, M.; Gillian, M. G. Microfluidic devices for environmental monitoring, TrAC. 2005, 24, 795. (3) Takeshi, I.; Jun-ichi, Y. Free radical polymerization in microreactors. Significant improvement in molecular weight distribution control. Macromolecules 2005, 38, 1159. (4) Engler, M.; Kockmann, N.; Kiefer, T.; Woias, P. Numerical and experimental investigations on liquid-liquid mixing in static micromixer. Chem. Eng. J. 2004, 101, 315.
(5) Wong, S. H.; Michael, C. L. W.; Christopher, W. W. Micro T-mixer as a rapid mixing micromixer. Sens. Actuators, B 2004, 100, 359. (6) Kockmann, N.; Kiefer, T.; Engler, M.; Woias, P. Silicon microstructures for high throughput mixing devices. Microfluid. Nanofluid. 2006, 2, 327. (7) Schonfeld, F.; Hardt, S. Simulation of helical flow in microchannels. AIChE J. 2004, 50, 771. (8) Nguyen, N. T.; Wu, Z. MicromixerssA review. J. Micromech. Microeng. 2005, 15, R1–R16. (9) Zhao, Y.; Chen, G.; Yuan, Q. Liquid-liquid two-phase flow patterns in a rectangular microchannel. AIChE. 2006, 52, 4052. (10) Glasgov, I.; Aubry, N. Enhancement of microfluidic mixing using time pulsing. Lab Chip 2003, 3, 114. (11) Wang, H.; Iovenitti, P.; Harvey, E.; Masood, S. Optimizing layout of obstacles for enhanced mixing in microchannels. Smart Mater. Struct. 2002, 11, 662. (12) Aoki, N.; Hasebe, S.; Mae, K. Mixing in microreactors: Effectiveness of lamination segments as a form of feed on product distribution for multiple reactions. Chem. Eng. J. 2004, 101, 323. (13) Polona, Z. P.; Igor, P. Steroid extraction in a micro channel systemsmathematical modelling and experiments. Lab Chip 2007, 7, 883. (14) Baroud, C. N.; Okkels, F.; Menetrier, L.; Tabeling, P. Reactiondiffusion dynamics: confrontation between theory and experiment in a microfluidic reactor. Phys. ReV. E, 2003, 67, 060104-1-060104-4. (15) Fogler, H. S. Elements of Chemical Reaction Engineering, 4th ed.; Eastern Economy ed.; Prentice-Hall of India: New Delhi, India, 2008. (16) Bothe, D.; Stemich, C.; Warnecke, H. J. Fluid mixing in a T-shaped micromixer. Chem. Eng. Sci. 2006, 61, 2950. (17) Hardt, S.; Schonfeld, F. Laminar mixing in different interdigital micromixers. II: Numerical simulations. AIChE J. 2003, 49, 578.
ReceiVed for reView February 23, 2009 ReVised manuscript receiVed June 19, 2009 Accepted July 7, 2009 IE900306J
