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Experimental Investigation of Flow Regimes in an SMX Sulzer Static Mixer K. Hirech, A. Arhaliass, and J. Legrand* GEPEA-UMR CNRS 6144, University of Nantes, CRTT-IUT Saint Nazaire, BP 406, 44602 Saint Nazaire Cedex, France
An experimental study of the flow pattern in the wall region of an SMX static mixer was performed by using electrochemical shear rate sensors. Electrochemical signals were related to the local shear stress and the fluctuating rate of the velocity gradient. Signal fluctuation analysis allowed one to define the flow regimes inside the static mixers. The laminar flow was characterized by a time-constant evolution of the velocity gradient, while the turbulent flow corresponded to the stabilization of the fluctuating rate of the velocity gradient for high values of pore Reynolds number, defined by considering the static mixer as a porous medium and by using a capillary model. The transition from a laminar flow to an intermediate flow occurred at a pore Reynolds number of about 200. The turbulent flow was observed at a pore Reynolds number of between 1500 and 3000. Introduction Mixing or dispersing operations are necessary in many processes and at different scales, from laboratory to industrial scale. However, the choice of agitated systems to realize these operations depends essentially on their capacity and efficiency to ensure mixing or dispersing action with respect to their energy consumption. In this context, the static mixer technology is increasingly used in most areas of chemical engineering such as homogenization, liquid-liquid dispersion, heat and mass transfer, and chemical reactions. These mixers present at the same time an alternative and a complementarity to the more traditional stirred vessel.1 The SMX mixer consists of a series of inclined baffles. Elements of the mixer are rotated by 90° and arranged successively in a tube. The more important properties of static mixers over conventional dynamic mixers are no moving parts, simple design, lower capital and operating costs, lower energy requirements, etc. However, despite their performances and growing uses in different operations, the working mode of these mixers as well as the physical phenomena governing the mechanisms of these operations remain to be elucidated. Many works have been devoted to the determination of flow regimes in porous media. Different criteria have been used to characterize flow regimes. Jolls and Hanratty,2 Wegner et al.,3 and Dybbs and Edwards4 followed the streamlined evolution from a visualization technique with a dye tracer. Jolls and Hanratty,2 Latifi et al.,5 Rode et al.,6 and Seguin et al.7,8 have analyzed the time evolution of the local mass transfer measured by an electrochemical method. Hall and Hyatt9 have determined the turbulence intensity measured by laser anemometry, while Mickly et al.10 and Van der Merve and Gauvin11 have used the film anemometer technique. On the contrary, there is practically no experimental work concerning the local flow investigation in an SMX static mixer. However, numerical simulations of the flow * To whom correspondence should be addressed. Tel: 33 240 17 26 33. Fax: 33 240 17 26 18. E-mail: jack.legrand@gepea. univ-nantes.fr.
through SMX static mixers are available in the literature. Fradette et al.12 have calculated the energy dissipation and the elongation inside the SMX mixer, and Rauline et al.13 have determined the axial shear rate evolution. This paper deals with an experimental study of the flow pattern in the wall region of an SMX Sulzer static mixer by an electrochemical technique. The goal is to better understand the hydrodynamic phenomena governing the flow in static mixers through flow-regime characterization, spectral analysis, and mean velocity gradient evolution. The limits of laminar and turbulent regimes determined from both local (electrochemical technique) and global (pressure drop) measurements will be compared. The results were analyzed by considering the static mixer as a porous medium.14 Experimental Section Experimental Setup. A schematic view of the experimental setup used in the present work is given in Figure 1. The working fluid stored in a tank (1) was pumped by a centrifugal pump (2). At the exit of the flowmeter (3), the fluid entered the test section, which consisted of a honeycomb (4), playing the role of a calming section, and a test cell containing the static mixer element (5). At the exit of the test cell (5), the fluid was returned to the storage tank (1). The working fluid temperature was controlled thanks to heating (8) and cooling (9) systems. The test cell (5) contained a four Sulzer SMX static mixer elements of 51.6 mm nominal diameter. The geometrical characteristics of the SMX static mixer are14 ) 0.90, dp ) 15.15 mm, and τ ) 1.46. Nineteen electrochemical sensors were mounted flush to the surface of the static mixer housing tube (Figure 1b). Their axial locations with respect to the static mixer inlet are given in Table 1. Experimental Technique. The electrochemical technique allows the determination of the local flow, i.e., the local wall shear rate and the turbulence characteristics. Its principle is based on the determination of mass transfer by diffusion-controlled conditions of an active ion to a nonintrusive electrochemical probe. It consists of measuring the mass flux at the electrode where a
10.1021/ie0206195 CCC: $25.00 © 2003 American Chemical Society Published on Web 03/05/2003
Ind. Eng. Chem. Res., Vol. 42, No. 7, 2003 1479 Table 1. Axial Position of the Electrochemical Probes with Respect to the Static Mixer Inlet probe 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 location (mm) 17 25 33 41 50 60 65 77 90 102 110 118 126 134 142 151 161 166 178 reduced distance (L/D) 0.34 0.5 0.66 0.82 1 1.2 1.3 1.54 1.8 2.04 2.2 2.36 2.52 2.68 2.84 3.02 3.22 3.32 3.56
transfer function. This experimental technique was described in detail by Reiss and Hanratty,15 Deslouis et al.,16 Rode et al.,6 and Seguin et al.7,8 To cover a wide range of Reynolds numbers, we have used two test fluids of different viscosities. The first fluid was a solution of potassium ferricyanide (5 × 10-3 M) and ferrocyanide (5 × 10-3 M) with sodium nitrate (1 M) as the supporting electrolyte. The physical properties at 30 °C were F ) 1050 kg/m3, µ ) 8.6 × 10-4 Pa‚s, and Dc ) 7.45 × 10-10 m2/s. The second fluid was a solution of potassium ferricyanide (5 × 10-2 M) and ferrocyanide (5 × 10-2 M) and sodium nitrate (1 M), prepared with 62% (w/w) glycerol to increase the viscosity. The same amount of potassium ferri- and ferrocyanide and sodium nitrate as that for the first solution was added. The physical properties of the second solution at 30 °C were F ) 1210 kg/m3, µ ) 8.3 × 10-3 Pa‚s, and Dc ) 5.6 × 10-11 m2/s. The electrical current was recorded by a digital audio tape (Tekelec RD-145 T) with a sampling frequency of 6000 Hz. The signal processing was performed with Labview for the spectral analysis of the fluctuating signals. The spectral density of the diffusional current is related to the one of the wall shear stress by means of a transfer function established by Deslouis et al.16 The power spectral density, Wi, of the current allows one to calculate the turbulence intensity, FRc:
x2∫ W (f) df FR ) ∞
i
0
c
(1)
hI
Results and Discussion Determination of the Wall Velocity Gradient in the Static Mixer. The mean wall velocity gradient S h is related to the measured limiting current hI by the following equation:15
S h)
Figure 1. Experimental setup: (a) hydraulic circuit; (b) test section.
rapid redox reaction occurs. The electrolyte was a solution of potassium ferri- and ferrocyanide in a supporting solution of sodium nitrate in dionized water. A small probe embedded in a nonconducting wall was taken as the working electrode in an electrochemical cell, polarized at a potential corresponding to the diffusional plateau. The probe acted as a perfect mass sink; the concentration of the reacting species on the probe surface was equal to zero. The mean value of the diffusional current, hI, delivered by the microelectrodes leads to the mean wall velocity gradient, and the power spectral density of the current fluctuations is linked to that of the velocity gradient fluctuations through a
(
hI 0.677nFC0Dc2/3d5/3
)
3
(2)
where n is the number of electrons evolved in the electrochemical reaction, F Faraday’s constant, C0 the bulk concentration of the ferricyanide ions, Dc the diffusion coefficient of reacting species, and d the microelectrode diameter. Figure 2 shows the axial evolution of the mean velocity gradient along the mixer. It can be noted that curves exhibit maximal and minimal values. This specific evolution of the velocity gradient is observed whatever the flow regimes and is more pronounced for high Reynolds number values. These extrema correspond to the large velocity variations due to the geometry of the mixer device as well as to the flow reorientation. Indeed, static mixer elements consist of wide crossing lamella opposing the flow, and each element is rotated by 90° to the other one. This disposition induces flow distortion and leads to flow contraction followed by flow expansion. Then, a series of convergent (divergent) flow types are obtained, inducing an increase (a decrease) of the velocity. The ratio between maximal and minimal values of the velocity gradient gives an
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Figure 2. Mean velocity gradient evolution along the static mixer.
overview of the acceleration undergone by the fluid. This ratio varies between 2 and 12 according to the Reynolds number values. In the literature, only a few works are devoted to the evolution of the velocity in the SMX static mixer. One of the works is the study of Fradette et al.,12 who performed numerical simulation flow through an SMX static mixer by using POLY3D software. Their numerical simulation is restricted to the viscous flow regime, in which the pressure drops are proportional to the flow velocity. Fradette et al.12 reported important acceleration effects along the mixer, which they attributed to the complicated geometry of the SMX mixer. The ratio between the maximum velocity and the mean flow velocity is on the order of 4. Furthermore, Mokrani17 studied another kind of static mixer, high-efficiency vortex (HEV) static mixer, and he also observed the influence of the geometry and the disposition of the tab on the velocity field evolution. The HEV mixer consists of four tabs located around the circumference of the pipe. Each tab, angled with its tip leaning downstream, creates a complex vortex field that promotes rapid mixing. From the evolution of the axial mean velocity profiles along the mixer, obtained by laser Doppler anemometry (LDA) measurements, Mokrani17 observed that the velocity acceleration is linked to a reduction of the apparent flow cross section, caused by the presence of the tabs. Characterization of the Flow Regime. The efficiency of the mixing, dispersing, or reaction process is dependent on the nature of the flow regime. As an exemple, drop breakage in the laminar flow is the result of viscous shearing action, while in the turbulent flow the dynamic pressure forces of the turbulent motions are responsible for the breakup.18 Then, it is important to characterize the nature of the flow regimes in the static mixer and to delimit them in order to further model the formation of the liquid-liquid dispersion. The flow-regime determination is obtained from the analysis of the turbulent intensity, calculated from the time evolution of the diffusional current or the velocity gradient fluctuation:
FRc
xi2 × 100 ) hI
FRvelocity gradient
xs2 × 100 ) S h
(3)
(4)
Figure 3. Examples of time variation of recorded limiting currents.
This experimental approach is similar to that used by Jolls et al.,2 Latifi et al.,5 Rode et al.,6 and Seguin et al.7,8 to identify and delimit flow regimes in various porous media. Its principle is based on the analysis of the time evolution of the current and velocity gradients for a wide range of Reynolds numbers. The attenuation of the current fluctuations in response to the velocity perturbations is due to the presence of a diffusional layer under the hydrodynamic layer, acting as a highpass filter. A correction of the recorded signal in the Fourier domain is made by using a transfer function to properly characterize fluctuating flows.16 The dimensionless criterion used in this study to establish the flowregime transition is the pore Reynolds number, Rep, defined by considering the static mixer as a porous medium and by using a capillary model.14
Updp Rep ) F µ
(5)
where Up ) Uτ/ is the pore velocity and τ is the tortuosity factor. (i) Laminar Flow Regime. The laminar flow is characterized by the absence of significant fluctuations.
Ind. Eng. Chem. Res., Vol. 42, No. 7, 2003 1481 Table 2. Limits of Flow-Regime Transitions along the Static Mixer
static mixer element no. 1
2
3
4
Figure 4. Examples of current fluctuating rate evolution as a function of the pore Reynolds number.
Examples of time evolution of the diffusional current are given in Figure 3. For small values of pore Reynolds numbers, the recorded signal is characterized by a constant evolution, without fluctuations. For more important pore Reynolds numbers, as indicated in Figure 3, we can notice the appearance of increasingly important fluctuations, corresponding to the end of the laminar flow and the beginning of the intermediate flow. Furthermore, to analyze the flow regime and the transition from laminar to intermediate flow, we report in Figure 4 the evolution of the current fluctuating rate as a function of the pore Reynolds number. Despite the discontinuity between each mixer element, no specific evolution of the current fluctuations is observed. Three different zones can be distinguished (Figure 4): (1) the first zone, for low values of the pore Reynolds number, is characterized by the absence of significant fluctuations; (2) the second zone corresponds to an asymptotic increase of fluctuations as a function of the pore Reynolds number; (3) the last zone corresponds to the stabilization of the current fluctuations for high values of the pore Reynolds number.
reduced position (L/D)
laminar flow Rep
turbulent flow limit with standard deviation (%)
0.34 0.50 0.66 0.82 1.00 1.20 1.30 1.54 1.80 2.04 2.20 2.52 2.68 2.84 3.02 3.22 3.32 3.56
400 300 300 230 230 260 260 200 200 200 230 200 200 200 200 200 200 200
16 ( 0.25 34 ( 1.10 30 ( 0.44 39 ( 0.89 20 ( 0.46 49 ( 0.50 56 ( 1.30 45 ( 0.68 34 ( 0.52 20 ( 4.07 70 ( 7.37 44 ( 0.28 59 ( 0.76 15 ( 0.17 20 ( 0.16 50 ( 0.79 70 ( 1.00 31 ( 0.41
turbulent flow Rep 3100 3100 3800 4150 2100 2450 3100 4150 1500 3100 2100 2450 2450 1500 3100
The limit of laminar flow is determined from the intersection between the initial plateau and the increasing part of the curve, corresponding to the onset of fluctuations. The results obtained, according to the probe positions, are given in Table 2. We can notice that the critical pore Reynolds number decreases approximately from 400 at the inlet of the first static mixer element to 260 at a reduced position of 1.3. From a reduced position of 1.54, we observe a stabilization of the critical pore Reynolds number around a homogeneous value of 200. This result shows that the flow regime is not established at the inlet. The establishment length is equal to about 1.5 mixing element. Nevertheless, the critical pore Reynolds number characterizing the upper limit of the laminar flow is defined by the critical value of Rep for the established flow regime in the static mixer, i.e., Rep ) 200. (ii) Turbulent Flow Regime. Figure 5 shows the evolution of the velocity gradient fluctuating rate as a function of the pore Reynolds number, for two different probe positions. The turbulent flow corresponds to the stabilization of the fluctuating rate of the velocity gradient for high values of the pore Reynolds number. However, the determination of the beginning of the turbulent flow regime is not accurately defined. Then, the procedure to delimit the onset of the turbulent flow is the following: in a first time, the limiting fluctuation rate, FRlimit, is calculated for a pore Reynolds number greater than or equal to 4160. A pore Reynolds number of 4160 is chosen because of the low standard deviation between the fluctuating rate calculated for Rep ) 4160 and that calculated for higher Reynolds number values; in a second time, the deviation (eq 6) between FRlimit and the fluctuating rate for a pore Reynolds number inferior to 4160 is calculated:
deviation )
|FRlimit - FRvelocity gradient| × 100 (6) FRvelocity gradient
Finally, the critical pore Reynolds number corresponding to the beginning of the turbulent flow is calculated by assuming a deviation, with respect to FRlimit less than or equal to 5%, i.e., deviation e 5%. The results obtained are summarized in Table 2. The pore Reynolds number corresponding to the onset of the turbulent flow is not constant along the static
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Figure 6. Examples of the power spectral density of velocity gradient fluctuations for Rep ) 2460, 3120, 4520, and 6020.
Figure 5. Examples of velocity gradient fluctuating rate evolution as a function of the pore Reynolds number.
mixer; it is between 1500 and 3000. This spatial dispersion certainly does express the complexity of the flow pattern in the static mixer, which is also confirmed by the spectral analysis of the fluctuating value of the velocity gradient (Figure 6). The more significant characteristics of the power spectral density are as follows(Figure 6): (1) absence of a characteristic peak or dominant frequency; (2) existence of an inertial subrange of less than a decade; (3) existence of a region between large structure plateaus and a viscous dissipation subrange that seems to be substituted into the wellknown -5/3 power law in isotropic turbulence corresponding to an inertial subrange. This region is representative of more than a decade by a power law between -8/3 and -12/3. This region is certainly related to the particular eddy production in the case of the static mixer. The type of transition to turbulence in static mixers is different than the one in channel with eddy promoters. In the latter case, Kapat et al.19 have observed a peak in the spectrum at the primary vortex shedding
Figure 7. Map of flow regimes in SMX static mixers.
frequency, and the spectrum has a broad-band appearance, which indicates the existence of an energycascading characteristic of turbulent flow. The laminarturbulent transition was obtained19 for Reynolds number values between 400 and 900 according to the eddy promoter configuration. The spectral analysis of velocity gradient fluctuations in the static mixer reveals that the energy-transfer
Ind. Eng. Chem. Res., Vol. 42, No. 7, 2003 1483 Table 3. Flow-Regime Transitions Obtained from Both Local and Global Measurements authors Li et al.21 Streiff et al.22 Moranc¸ ais et al.14 present work: local measurements
Reynolds number
viscous flow
ReD ) FUD/µ Reh ) ReDdh/µD ReD Repore Rep ) ReDτdpore/D ReD Rep ) ReDτdpore/D ReD
15 20 61 29 4.3 9
mechanisms are different from those corresponding to isotropic turbulence. In the case of isotropic turbulence, the energy cascade was described by Kolmogorov.20 Between production scales, corresponding to the contribution of the energy input to the system, and the viscous dissipation subrange, where the viscous dissipation occurs, there is a region corresponding to an energy transfer between large and small eddies. So, in the static mixer case, the concept of inertial subrange or spectral equilibrium is not respected. The spectral deviation, in the inertial subrange, can be due to the fact that the production mechanism of eddies is more important than the dissipation mechanism. Eddies are convected and broken, and the energy transfer is not strictly made from large to small eddies because of the interaction with the static mixer networks. (iii) Comparison of Local and Global Measurements. To our knowledge, only a few works have been devoted to the determination of the transition between the different flow regimes from pressure drop measurements. Li et al.21 realized their experimental study in a SMX static mixer of 16 mm nominal diameter, under variable operating conditions, for Newtonian and nonNewtonian fluids. Streiff et al.22 assumed that, in static mixers, the flow transition occurs at the same Reynolds number as the one corresponding to an empty pipe flow pattern. In our previous work,14 we delimited viscous and inertial flow regimes from a pressure drop model. The flow-regime transitions proposed by different authors are summarized in Table 3. However, because the definition of the dimensionless criterion (Reynolds number) differs from one author to another, we have expressed the different results as a function of the Reynolds number relative to an empty pipe, i.e.
ReD )
FUD µ
(7)
In the so-called “Darcian” flow regime, the viscous contribution is very important with respect to the inertial one. In this case, pressure drops vary linearly with the mean flow velocity (∆P/L ∝ U). For inertial flow, the viscous contribution is very low with respect to the inertial contribution. When the Reynolds number increases, a nonlinear term appears in the expression of the pressure drop, due to the inertial eddies. This term becomes gradually predominant for high values of the Reynolds number. Pressure drops vary with the square velocity (∆P/L ∝ U2). From Table 3, we notice a good agreement between the results, relative to the viscous flow regime, obtained by Li et al.21 and Moranc¸ ais et al.,14 while the limit proposed by Streiff et al.22 is slightly superior. We can also notice from Table 3 that the Reynolds number corresponding to the beginning of the inertial flow obtained from global measurements differs according to the authors. The limit defined by Li et al.21 is largely
laminar flow
turbulent flow
200 420
1000 2300 7050 3360 1570 3300 1500-3100 3150-6500
inferior to that given by Moranc¸ ais et al.14 and Streiff et al.22 The onset of turbulent flow defined by these authors,14,21 from pressure drops, is approximately close to the one obtained from local measurements. The pressure drop results allow one to estimate both viscous and inertial contributions, through the pressure drop model14 expressed in terms of the friction factor (eq 8):
f)
16 + 0.194 Rep
(8)
We have defined in a previous work,14 on the one hand, q1, representing the proportion of the viscous contribution and, on the other hand, q2, the proportion of the inertial contribution, as follows:
q1 )
16 16 + 0.194Rep
(9)
q2 )
0.194Rep 16 + 0.194Rep
(10)
Figure 7 shows the flow-regime map in the static mixer determined from both local and global measurements as well as their corresponding viscous and inertial contributions. This type of representation is similar to that proposed by Comiti et al.23 in the case of flow in porous media. Conclusion The objective of this paper was to study the flow pattern in an SMX Sulzer static mixer from flow-regime characterization, power spectral density analysis, and mean velocity gradient evolution. For that purpose, an electrochemical technique was performed. Flow-regime transitions were determined from the analysis of the time evolution of the current and velocity gradients for a wide range of pore Reynolds numbers. The critical pore Reynolds number characterizing the upper limit of the laminar flow was of about 200. The onset of the turbulent flow was not constant along the static mixer; it was between 1500 and 3000. This spatial dispersion certainly expresses the complexity of the flow pattern in the static mixer and turbulent intermittency along the mixer. The complexity of the flow pattern in the static mixer was also confirmed from the spectral analysis of the fluctuating value of the velocity gradient. It was shown that the energy-transfer mechanisms were different from those corresponding to isotropic turbulence. The axial evolution of the mean wall velocity gradient along the mixer axis has shown large velocity variations linked to the apparent flow cross-section variation, caused by the presence of the SMX mixing elements. Finally, the association of the local and global
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results was allowed to establish flow-regime maps in an SMX Sulzer static mixer. Nomenclature C0 ) concentration D ) pipe diameter d ) microelectrode diameter Dc ) diffusion coefficient of ferricyanide ion dh ) hydraulic diameter dp ) pore diameter f ) friction factor FRc ) current fluctuations rate FRlimit ) limiting fluctuation rate defined by eq 3 FRvelocity gradient ) velocity gradient fluctuation rate hI ) average limiting diffusional current i ) fluctuating current L ) static mixer length q1 ) proportion of the viscous contribution q2 ) proportion of the inertial contribution ReD ) Reynolds number relative to an empty pipe Reg ) generalized Reynolds number Rep ) pore Reynolds number S h ) mean velocity gradient s ) fluctuating velocity gradient U ) mean flow velocity UP ) pore velocity Wi ) power spectral density of the diffusional current ∆P ) pressure drop ) porosity µ ) dynamic viscosity F ) fluid density τ ) tortuosity
Literature Cited (1) Godfrey, J. C. Static mixer. In Mixing in the process industries; Harnby, N., Edwards, M. F., Nienow, A. W., Eds.; Butterworth & Heinmann: Oxford, U.K., 1992. (2) Jolls, K. R.; Hanratty, T. J. Transition to turbulent for flow through a dumped bed of spheres. Chem. Eng. Sci. 1966, 44, 2501. (3) Wegner, T. H.; Karabelas, A. J.; Hanratty, T. J. Visual studies of flow in a regular array of spheres. Chem. Eng. Sci. 1971, 8, 59. (4) Dybbs, A.; Edwards, R. V. A new look at porous media fluid mechanics. Darcy to turbulent. In Fundamentals of Transport Phenomena in Porous Media; Bear, J., Corapciaglu, Y., Eds.; Martinus Nijhoff Publishers: New York, 1984. (5) Latifi, M. A.; Midoux, N.; Storck, A. The use of microelectrodes in the study of flow regimes in packed bed reactor with single phase liquid flow. Chem. Eng. Sci. 1989, 21, 1185. (6) Rode, S.; Midoux, N.; Latifi, M. A.; Storck, A.; Saatdjian, E. Hydrodynamic of liquid flows in packed beds: An experimental study using electrochemical shear rate sensors. Chem. Eng. Sci. 1994, 49, 889.
(7) Seguin, D.; Montillet, A.; Comiti, J. Experimental characterization of flow regimes in various porous mediasI: Limit of laminar flow regime. Chem. Eng. Sci. 1998, 53, 3751. (8) Seguin, D.; Montillet, A.; Comiti, J.; Huet, F. Experimental characterization of flow regimes in various porous mediasII: Transition to turbulent regime. Chem. Eng. Sci. 1998, 53, 3897. (9) Hall, M. J.; Hyatt, J. P. Measurement of pore scale flows within and exiting ceramic foams. Exp. Fluids 1996, 20, 433. (10) Mickly, H. S.; Smith, K. A.; Korchak, E. I. Fluid flow in packed bed. Chem. Eng. Sci. 1965, 20, 237. (11) Van der Merve, D. F.; Gauvin, W. H. Velocity and turbulence measurements of air flow through packed beds. AIChE J. 1971, 17, 519. (12) Fradette, L.; Li, H. Z.; Choplin, L.; Tanguy, P. 3D finite element simulation of fluid flow through a SMX static mixer. Comput. Chem. Eng. 1998, 22, S759. (13) Rauline, D.; Tanguy, P. A.; Le Ble´vec, J. M.; Bousquet, J. Numerical investigation of the performance of several static mixers. Can. J. Chem. Eng. 1998, 76, 527. (14) Moranc¸ ais, P.; Hirech, K.; Carnelle, G.; Legrand, J. Friction factor in static mixer and determination of geometric parameters of SMX Sulzer mixers. Chem. Eng. Commun. 1999, 171, 77. (15) Reiss, L. P.; Hanratty, T. J. An experimental study of the unsteady nature of the viscous sublayer. AIChE J. 1963, 8, 154. (16) Deslouis, C.; Gil, O.; Tribollet, B. Frequency response of electrochemical sensors to hydrodynamic fluctuation. J. Fluid Mech. 1990, 215, 85. (17) Mokrani, A. Analyse expe´rimentale et nume´rique de deux proce´de´s comple´mentaires de me´lange et de transfert thermique en e´coulement tridimensionnel ouvert: Advection chaotique laminaire et e´coulement turbulent Eule´rien. Ph.D. Thesis, University of Nantes, Saint Nazaire, France, 1997. (18) Hinze, J. O. Fundamentals of the hydrodynamic mechanism of splitting in dispersion processes. AIChE J. 1963, 1, 289. (19) Kapat, J. S.; Ratnathicam, J.; Mikic, B. B. Experimental determination of transition to turbulence in a rectangular channel with eddy promoters. J. Fluids Eng. 1994, 116, 484. (20) Tennekes, H.; Lumley, J. L. A first course of turbulence; MIT Press: Cambridge, MA, 1972. (21) Li, H. Z.; Fasol, C.; Choplin, L. Pressure drop of Newtonian and non-Newtonian fluids across a Sulzer SMX static mixer. Inst. Chem. Eng. 1997, 75, 792. (22) Streiff, F. A.; Jaffer, S.; Schneider, G. The design and application of static mixer technology. 3rd International Symposium on Mixing in Industrial Processes (ISMIP 3), Osaka, Japan, Sep 1999; pp 107-114. (23) Comiti, J.; Sabiri, N. E.; Montillet, A. Experimental characterization of flow regimes in various porous mediasIII: Limit of Darcy’s or creeping flow regime for Newtonian and purely viscous non-Newtonian fluids. Chem. Eng. Sci. 2000, 55, 3057.
Received for review August 9, 2002 Revised manuscript received January 23, 2003 Accepted January 29, 2003 IE0206195
