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Liquid–liquid mixing using micro-fluidised beds
chemical engineering research and design 9 1 ( 2 0 1 3 ) 2235–2242
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Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd
Liquid–liquid mixing using micro-fluidised beds Elham Doroodchi ∗ , Mayur Sathe, Geoffrey Evans, Behdad Moghtaderi University of Newcastle, Chemical Engineering, Callaghan, NSW 2308, Australia
a b s t r a c t This study experimentally investigates the application of a solid–liquid micro-fluidised bed as a micro-mixing device. The experiments were performed in a borosilicate capillary tube with an internal diameter of 1.2 mm (i.e. near the upper-limit dimension of a micro-fluidic system) using borosilicate particles with a mean diameter of 98 m. Refractive index matching technique using sodium iodide solution was employed to achieve a transparent fluidised bed. Mixing performance of the micro-fluidised bed in terms of mixing time was investigated using a dye dilution technique. Experiments were carried out in the creeping flow regime at Reynolds numbers ranging between 0.27 and 0.72. It was demonstrated that the micro-fluidised bed mixing time sharply decreases as the Reynolds number increases. That is because at relatively high Reynolds numbers, the particle oscillation is stronger creating larger disturbances in the flow. The energy dissipation rate in micro fluidised bed was estimated to be four orders of magnitude less than other passive micro mixers which operate in the turbulent regime. It was also demonstrated that the ratio of mixing time and the energy dissipation rate for fluidised bed micro-mixer was comparable to K-M, Tangential IMTEK, and interdigital micro-mixers. However, the fluidised bed micro-mixer was found to operate at much lower Reynolds numbers compared to other passive mixers, with a mixing time of the order of few seconds. © 2013 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Micro-fluidised bed; Micro-mixer; Liquid–liquid mixing; Mixing time; Refractive index matching technique; Dye dilution technique, Energy dissipation
1.
Introduction
Application of micro-fluidic devices for processing of multiphase flows in areas such as medical diagnostics, chemical analysis, power generation and fuel processing, invariably relies upon the physical or chemical interaction between at least two fluid phases. Such interaction is achieved through effective mixing. However, at length-scales associated with micro-fluidic devices where systems operate in a laminar regime with Reynolds numbers typically less than 1, the mixing process is rather poor as it is principally governed by molecular diffusion. Generally, active or passive techniques are employed to achieve effective mixing at micro-scales (Hessel et al., 2005; Nguyen, 2012). In active mixing external forces including ultrasound, acoustic, electrokinetic, and magneto-hydrodynamic forces are used to induce mixing. In these mixers the transversal disturbances generated by external fields lead to instabilities at the interface between the two mixed phases.
∗
Passive mixing on the other hand is accomplished by maximising the mixing contact area and/or reducing the mixing path through “Streaming Techniques” where the flow path is restructured using geometrically-based methods. There are two main mechanisms namely the diffusive transport and chaotic advection which drive mixing in passive micro-mixers. Diffusive mixing is often improved by decreasing the striation thickness through optimisation of the geometrical designs to create parallel or sequential laminations, sequential segmentation and focusing of fluid flow. Similarly, chaotic advection is generated by modifying the channel shape for stretching, folding and breaking of the laminar flow. The aim is to promote the streamlines to cross each other periodically (i.e. a spatially periodic flow). For example, the use of obstacles in passive mixers is demonstrated to be effective in generating such advections (Nguyen, 2012; Wang et al., 2002). At Reynolds numbers greater than 100, vortices are generated as the liquid is passing by an obstacle. These vortices disrupt the laminar flow pattern and induce transversal
Corresponding author. Tel.: +61 2 4033 9066; fax: +61 2 4033 9095. E-mail address: [email protected] (E. Doroodchi). Received 5 December 2012; Received in revised form 17 June 2013; Accepted 20 June 2013 0263-8762/$ – see front matter © 2013 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cherd.2013.06.024
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Nomenclature C1 Cy d D I Ix,∞ I* L Lx M Pe Re Qd QT Qc tm tdiff x y
proportionality constant in Eq. (3) (m−1 ) fitting parameter in Eq. (6) capillary internal diameter (m) molecular diffusivity (m/s2 ) grey scale intensity of individual pixel in shadow image intensity for perfect mixing normalised intensity of individual pixel in shadow image length of capillary (m) chord length (m) percentage mixing index Peclet number Reynolds number dye stream flow rate (L/min) total flow rate (L/min) clear stream flow rate (L/min) mixing time (s) diffusion time, d2 /D (s) distance from the centre of capillary in horizontal direction (m) distance from the inlet of capillary in vertical direction (m)
Greek symbols effective effective shear rate (1/s) total shear rate (1/s) total ı0 striation thickness (m) pressure drop (kPa) P I,0 deviation from concentration profile for perfect mixing at inlet (y = 0) deviation from concentration profile for perfect I,y mixing at y deviation from concentration profile for perfect I,∞ mixing at infinity ε energy dissipation rate (w/kg) mixing efficiency kinematic viscosity (m/s2 )
transport, in turn, improving the mixing. At Reynolds numbers less than 100, the lateral mass transport induced by the obstacles leads to mixing. Besides the fabrication complexity, the main drawback associated with the presence of obstacles, however is the rise in pressure drop across the bed. Clearly, the existing passive and active techniques suffer from fabrication complexity which limits their mass productions, high fabrication and maintenance costs, and high energy dissipation rates (Hessel et al., 2005; Nguyen, 2012). To overcome the shortcomings of the above techniques, a fluidising-based micro-mixer system is considered here. It is well-established that fluidisation provides efficient mixing and intensification of mass and heat transfer. In recent years, a number of studies focusing on miniaturised fluidised beds have emerged (Doroodchi et al., 2012; Potic et al., 2005). The major focus of these studies was to establish the fluidisation hydrodynamic characteristics of the fluidised beds both experimentally and theoretically. The focus of this work however is to experimentally investigate the mixing performance of two miscible fluids in a miniaturised solid–liquid fluidised bed in terms of mixing time and mixing efficiency. The relative
performance of the fluidising-based micro-mixer compared with typical existing passive micro-mixers is also established. It should be highlighted that apart from non-reacting flows the abovementioned miniaturised fluidised-beds can be also used as effective platforms in reacting flow systems, for example micro-reactors, where the mixing agents undergo a series of chemical reactions.
2.
Experimental
A schematic diagram of the experimental setup is presented in Fig. 1. The setup consisted of (i) a 30 cm glass capillary tube with an internal diameter of 1.2 mm (i.e. a dimension near the upper-limit dimension of 1 mm for micro-fluidic systems), (ii) a fluid reservoir, (iii) two syringe pumps, (iv) a LED light source and focusing lens, (v) a CCD camera, and (vi) a data acquisition system. The bed material was clear borosilicate glass sphere from Cospheric LLC with a size range of 90–106 m and mean diameter of 98 m, particle density of 2230 kg/m3 , and refractive index of 1.47–1.48. A 52 m wire mesh was used as a distributor. For optical diagnostics, the refractive index of the fluidising medium (i.e. sodium iodide solution) and bed material were matched (forming a transparent fluidised bed) minimising the noise generated by the light reflections from the surface of particles. Dye dilution technique (Nguyen, 2012) was used to determine the mixing performance at various operating conditions given in Table 1. In this work, the tracer dye stream (i.e. dye + sodium iodide solution) was fed in the centre from the bottom of the capillary tube through a 30G½ needle whilst the sodium iodide solution (56 wt%) was fed annularly passing through the fluid reservoir. The two streams have similar density (1800 kg/m3 ) and viscosity (0.0018 Pa.s). A LED lamp combined with a biconvex focusing lens was used to illuminate the region of interest using a back lighting technique. A cuvette filled with the sodium iodide solution was placed around the capillary tube to reduce the light refraction at the curved surface of the tube. An IDT XS3 high speed camera was then used to capture the dye dilution profile across the channel cross section at a rate of 50 Hz. 2000–3000 images were captured using IDT Motion Studio software whilst MATLAB software was employed for image analysis. Background images were also obtained at dye flow rates of zero at the start of each experiment. For benchmark purposes, the same experiments were carried out in the empty capillary tube (i.e. fluidised bed with no particles). The variation of the dye concentration profile along the length of capillary tube has been calculated by processing the images recorded using the procedure described above. The criterion used to evaluate the performance of the liquid–liquid mixing within micro-mixers was the uniformity of the dye intensity. The concentration of the dye was estimated from the shadow image generated by the diffusing jet of opaque dye stream using image processing in MATLAB software. The background images obtained at the start of each experiment was averaged and subtracted from individual mixing images. In a shadow image, the pixel intensity is inversely proportional to the concentration of dye at that point. Therefore a negative of each image was taken after the background subtraction. The background intensity was non-uniform. This non-uniformity implies that direct calculation of local concentration using extinction of the backlight is erroneous. To compensate for the gradient in backlight, normalisation of image intensity based
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Fig. 1 – Schematic representation of the experimental setup. on the material balance of tracer was adopted. The greyscale intensity of each pixel was divided by the average grey scale intensity along each horizontal line. This normalisation was based on the consideration that the liquid flows only in the vertical direction, and the integration of radial concentration profile of the dye with respect to radial coordinate should be constant along the height. The normalised intensity was then calculated by,
I(x, y)
I∗ (x, y) = 1/d
d/2
−d/2
(1)
I(y)dx
where I(x,y) is the intensity of each pixel. The term in the denominator is the average intensity of each horizontal line. The normalisation ensured that the area under the concentration profile remains constant along y direction. It should be noted that by this scaling only the magnitude of concentration profile was corrected to fit the material balance. The shape of the concentration profile however was unchanged. The corrected profiles were then used to calculate the deviation from concentration profile corresponding to perfect mixing. The concentration profile for perfect mixing was derived as follows. The intensity recorded by the CCD camera is proportional to the integral optical density along the line of sight. The absorption of light is directly proportional to the concentration and the optical path length according to the Lambert–Beer law. For the case of perfect mixing the concentration profile is uniform over the entire cross section. Hence, the profile of the absorption path length defines the intensity profile for perfect mixing. Since the capillary cross
section is circular the absorption path length at any particular radial location is equal to the chord length of the circle. The chord length at any radial distance x from the centre of capillary is given by:
Lx =
d 2 2
− x2
(2)
Since the dye concentration used in the current work is not very high then Lambert–Beer’s law is valid and the absorption of light is linear with respect to the absorption path length. Hence, the radial intensity profile in the image recorded by the camera can be expressed as:
Lx,∞ = C1
d 2 2
− x2
(3)
Eq. (3) indicates a semicircular profile for the radial intensity profile. The extent of mixing was calculated using the deviation of radial intensity profile from this semicircular profile, which is characteristic of a capillary cross-section. For the case of micro-channels with rectangular cross section, the profile should become rectangular. In the final step of the analysis the deviation of concentration profile at a particular height from the profile corresponding to perfect mixing was calculated as:
d/2
I,y =
|I∗ − Ix,∞ |dx
(4)
−d/2
As the relationship between intensity and concentration is linear then I,y also quantifies the deviation of the
Table 1 – Operating conditions. Total flow rate (QT ) (L/min) Reynolds number (Re) Dye stream flow rate (Qd ) (L/min) Clear stream flow rate (Qc ) (L/min) Flow ratio, Qd /QT a
10 0.18 0a 10 0a
Calibration case for background subtraction.
20 0.35 2 8 0.2
0a 20 0a
8 12 0.4
0a 35 0a
35 0.62 7 28 0.2
17.5 17.5 0.5
15.27 0.27 4.58 10.69 0.3
23.75 0.42 7.13 16.63 0.3
32.23 0.57 9.67 22.56 0.3
40.72 0.72 12.21 28.50 0.3
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Fig. 2 – Time-averaged concentration profiles for (a) capillary tube, (b) micro-fluidised bed [QT = 20 L/min, Qd/ QT = 0.4]. concentration profile at any height y from a concentration profile corresponding to perfect mixing. The extent of mixing was then calculated as:
M=
1−
I,y − I,∞ I,o − I,∞
× 100
(5)
The value of I,∞ is 0 since the two liquid streams are completely mixed at y = ∞. M is the mixing index, I,o is the variance between the concentration profile at the inlet (just above the central nozzle) and the concentration profile for complete mixing.
3.
Results and discussion
Typical time-averaged normalised concentration profiles of the dye in the capillary tube and the micro-fluidised bed are presented in Fig. 2a and b, respectively. As Fig. 2a shows, in the capillary tube the interface between the two streams remains undisturbed with dye and clear streams following their distinct core and annular flow paths, respectively. Conversely, under the same operating conditions enhanced mixing was observed when the two fluid streams were introduced into the micro-fluidised bed (Fig. 2b).
Fig. 3a and b illustrates the normalised concentration distribution profile at different elevations along the height of the capillary tube and the micro-fluidised bed, respectively. The bottom of the capillary is considered as zero elevation. The flatness of the profile corresponds to the degree of mixing (i.e. a flat profile represents complete mixing). In the capillary tube the concentration profile remained almost unchanged over the height of the tube (Fig. 3a) with the shape of the profile representing non-mixed regions along the examined elevations of 0.82, 1.3, 2.3, 3.3 and 4.3 mm. In the micro-fluidised bed however the concentration profile became flatter as the height increased approaching the average concentration profile (i.e. the dashed line in Fig. 3b). The flattening of the concentration profile indicates an increase in the homogeneity of the mixture and hence significant improvement in mixing. Fig. 4 shows M versus residence time for both the capillary tube and the micro-fluidised bed. Generally M increases as the time increases indicating an increase in the mixing efficiency. Clearly the rate of mixing achieved in the micro-fluidised bed is much greater than that of the capillary tube. The length of capillary tube required to achieve the same M as the fluidised bed is more than the corresponding length of the fluidised bed. The field of view in the current experiments however was limited and did not permit recording of the length of capillary tube required to achieve the same extent of mixing as those of
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1.3 mm
3.3 mm
4.3 mm
2.3 mm
(b) 1.2
3
Normalized Concentraon (-)
Normalized Concentraon (-)
(a)
0.82 mm
2.5 2 1.5 1 0.5 0
500
0
0.82 mm
1.3 mm
3.3 mm
4.3 mm
2.3 mm
1 0.8 0.6
0.4 0.2 0
1000
0
500
x (μm)
1000
x (μm)
Fig. 3 – Concentration profiles at different elevations for (a) capillary tube, (b) micro-fluidised bed [QT = 35 L/min, : Normalised concentration for complete mixing. Qd/ QT = 0.2]. 120
the fluidised bed. In order to facilitate quantitative comparison of the mixing performance for these two geometries, a first order model was fitted to the M data for the micro-fluidised bed and the capillary tube. The model equation for M as a function of axial coordinate is: Cy
}
80
(6)
where Cy is the curve fitting parameter. The fitted curves are shown as solid lines in Fig. 4. The length of capillary tube required to achieve the mixing index of 90% was calculated by extrapolating the data based on the fitted model. At the total flow rate of 35 L/min and flow ratio of 0.2, a mixing index of 90% can be achieved in the micro-fluidised bed in 2.3 s. In contrast, 82 s are required for the capillary tube to achieve the same mixing index. Further improvement was achieved as the dye flow rate to total flow rate ratio increased from 0.2 to 0.5. For the flow rate ratio of 0.5 the mixing index of 97% was obtained at less than 2.5 s.
M
−y
M(y) = 100 × {1 − e
100
60
40
20
0
0
5
20
25
30
35
t, s
: QT = 35 L/min; 100
80
M
15
Fig. 5 – Mixing index versus residence time [Qd/ QT = 0.2].
120
60
40
20
0
10
0
2
4
6
8
10
t,s
Fig. 4 – Mixing index versus residence time [QT = 35 L/min].
: Empty tube, Qd/ QT = 0.2;
bed, Qd/ QT = 0.2;
: fluidised bed, Qd/ QT = 0.5.
: fluidised
: 10 L/min.
Fig. 5 shows the effect of an increase in total flow rate on mixing performance of the micro-fluidised bed. As the flow rate increases from 10 L/min to 35 L/min, the rate of mixing increases reducing the mixing time for achieving mixing efficiency of 92% by more than half. The above results show the advantage of using a micro-fluidised bed as a micro-mixer over a core–annular micro-mixer. In a core–annular micro-mixer (i.e. the capillary tube), mixing is mainly driven by molecular diffusion which causes transversal mass transport. As a result the mixing is extremely slow and inefficient requiring long residence time and hence great mixing length. In the micro-fluidised bed however, the mixing is considered to be driven by passive chaotic advection with a three dimensional orbit which in turn leads to secondary transversal mass transport. Also, the bed materials (i.e. particles) act as obstacles and hence lead to periodic changes in the flow directions causing splitting and recombining effects with streamlines crossing each other. Such unique characteristics can improve mixing significantly at low Reynolds numbers.
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1.E+00
1.E+01
6 5 4
1.E-01
3 2
1.E+00
1.E-04
5.5E-04
5.0E-04
1.E-01
1.E-02
1.E-05
1.E-06 0.1
1
10
100
1000
10000
100000
1.E-03 1.E-04
Reynolds Number, Re (-) Fig. 6 – tm /tdiff as a function of Reynolds number. Fluidised bed;
0 4.5E-04
1.E-03
1 4.0E-04
Mixing me (s)
tm/tdiff
1.E-02
,
,
,
: Starlam mixer;
: IMM slit inter digital mixer; Tangential IMTEK;
: K-M;
: : T-mixer;
: T-mixer;
1.E-02
1.E+00
1.E+02
Energy Dissipaon, ε(w/kg)
:
: caterpillar, - - tm /tdiff ∝ (ln(Pe))/Pe.
The above results provide an insight into the driving mechanism for mixing enhancement in the micro-fluidised bed. It is interesting to compare the mixing performance of the fluidised bed relative to typical micro-mixers. Comparative studies of micro-mixers are typically carried out in the context of mixing time. Falk and Commenge (2010) have recently performed a detailed comparison of mixing time for different mixers as a function of Reynolds number and energy dissipation. The data presented in their work was used here to assess the performance of the fluidising-based micro-mixer relative to typical mixers. Fig. 6 shows the ratio of mixing time to the diffusion time versus Reynolds number for different passive mixers (solid symbols) and the examined miniaturised fluidised bed (open symbols). It can be seen that the majority of the collected experimental data follow the theoretical linear relationship between tm /tdiff and the Reynolds number (dashed line) especially at lower values of Reynolds numbers. This relationship is based on the equation proposed by Baldyga and Bourne (1984) with some simplifications introduced by Falk and Commenge (2010) by using the expression for pressure drop in Poiseuille flow. For the micro-fluidised bed examined in this study, the Reynolds number varies between 0.1 and 1 and the corresponding Péclet number is approximately 300–3000. For this range of Péclet number convection dominates molecular diffusion. From Fig. 6 it can be observed that the mixing time for the micro-fluidised bed was two orders of magnitudes less than the theoretical prediction. This low mixing time further confirms that molecular diffusion is not the only mechanism governing the mixing in micro-fluidised beds. It is believed that the oscillating motion of particles promote mixing by generating periodic spatial flow advections and flow instabilities. From visual observation the particles were found to move around their mean position by the distance equivalent to their diameter. However, the oscillation was slow with the frequency ranging between 1 and 10 Hz. Also, as Fig. 6 shows, the micro-fluidised bed mixing time sharply decreases as the Reynolds number increases. This is because at relatively high Reynolds numbers the particles vibrate more rigorously and
Fig. 7 – Mixing time as a function of energy dissipation rate.
: Fluidised bed;
IMTEK;
: T-mixer;
: Triangular interdigital,
: Tangential : IMM caterpillar;
,
, : Starlam; ––: tm = 0.15ε−0.45 ; • • • •: = 2%; - -: = 3%;- - -: = 10%.
create larger disturbances in the flow. However, it should be noted that the operation of such devices are limited by the fluidising characteristics of particles specifically their expansion behaviour. Moreover, Fig. 6 demonstrates that comparable mixing performances to those of K-M, Tangential IMTEK, and interdigital micro-mixers can be obtained in a fluidising-based micro-mixer at much lower Reynolds numbers. This indicates that for a given fluid throughput, more efficient mixing can be potentially achieved in a micro-fluidised bed with the exception of a caterpillar mixer which was found to outperform the examined fluidising-based mixer. Baldyga and Bourne (1984, 1986) have provided a relationship between the mixing time and the effective shear rate. The equation is as follows:
tm =
1 2effective
arcsinh
0.76effective ı2o D
(7)
where effective is the effective shear rate, ı0 is the striation thickness, and D is the molecular diffusivity. The effective shear rate is defined as effective = total , where is the mixing efficiency and total = ε/2v. ε is the energy dissipation rate and is the kinematic viscosity. The mixing efficiency is determined by fitting the experimental data of mixing time, tm , as a function of energy dissipation rate, ε. Fig. 7 shows the mixing time as a function of energy dissipation rate for different micro-mixer geometries. In Fig. 7, the dashed lines represent the mixing time calculated using Eq. (7) for different energy efficiency, values with 3% efficiency giving the best fit for all mixer geometries. An alternate expression for the mixing time as a function of energy dissipation rate is the empirical power law given by Falk and Commenge (2010): tm = 0.15ε−0.45
(8)
chemical engineering research and design 9 1 ( 2 0 1 3 ) 2235–2242
70
To demonstrate explicitly how the pressure drop affects the performance of the fluidising-based micro-mixer, the findings of this study were compared with those of a micro-channel equipped with fixed obstacles of 60 m arranged in different configurations (Wang et al., 2002). Fig. 8 shows the mixing efficiency in terms of mixing index versus pressure drop per unit length. The diamond symbols represent the literature data reported by Wang et al. (2002) and the square symbols represent our experimental data obtained at L = 1.2 mm. Clearly, the miniaturised fluidised bed outperforms the micro-channel mixer with fixed obstacles by obtaining greater mixing efficiencies at significantly lower pressure drops. This finding in turn implies that the energy dissipation rate for the fluidisingbased micro-mixer is much lower than the micro-channel mixer with fixed obstacles.
60 50
M
40 30 20 10 0 10
100
1000
ΔP/L (kPa/m) Fig. 8 – Mixing index as function of pressure drop per unit length.
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: Fluidised bed;
: Patterned microchannel.
Eq. (8) has been plotted as a solid line in Fig. 7, where it can be seen that the energy dissipations associated with the micro-fluidised bed mixer are significantly lower in comparison with the other laminar flow-based mixers (Falk and Commenge, 2010). The lower energy dissipation however comes at a cost of an increased mixing time. These characteristics make such devices especially suitable for processing of relatively small volumes of fluids with reaction times in the order of seconds. It should be noted that the mixing time in Fig. 7 has been plotted as a function of overall energy dissipation rate. In this work we have not attempted detailed modelling of impact of ε on mixing time. In a fluidised bed shear rate in the vicinity of particles is higher than the bulk liquid, which implies non-uniform ε. Such non-uniform energy dissipation has significant impact on the micro-mixing process, as reported by Baldyga and Bourne (1988) who found that the product distribution for rapid azo-coupling reactions in three sizes of mechanically agitated vessels could be successfully modelled when spatial distribution of ε was properly taken into account. In the present work, we are currently not able to attempt such a modelling approach due to lack of reliable velocity and shear rate data in the micro-fluidised bed. Eq. (7) provided the best fit for mixing time data of the fluidised bed at 2% mixing efficiency, which is shown in the inset of Fig. 7 as the dotted line. This efficiency is comparable to the majority of other passive mixers. The low efficiency exhibited by the micro-fluidised bed is expected since the majority of the energy is used to keep the particles in suspension. In general, the increase in turbulence does not always lead to enhancement in mixing. That is mainly because not all of the turbulent flow structures enhance mixing. Only the structures at the interface of two fluids which result into splitting/stretching of fluid lamellae improve the mixing while the flow structures within either fluid have no effect on the mixing process. Overall, the increase of turbulence intensity by increasing the pressure drop, either by changing the geometry of mixer or increasing the velocity of fluid, increases both mixing and non-mixing flow structures. Consequently, the mixing time is reduced with an increase in pressure drop whilst the mixing efficiency remains unaffected.
4.
Conclusions
The novel idea of using a fluidised bed with micro-scale dimensions as a micro-mixer for mixing of two miscible liquids was examined experimentally. The results of the study were bench marked against mixing behaviour in a capillary tube and typical passive mixers. Generally, the fluidisation process was found to achieve high mixing efficiencies reduced mixing time well below those for a capillary tube. An increase in the flow rate was found to increase the mixing rate within the micro-fluidised bed resulting in a reduced mixing time. The enhancement in mixing was considered to be due to destabilisation of the diffusion layer between the two fluids by fluidised particles and hence periodic changes in the flow directions causing splitting and recombining effect with streamlines crossing each other. The micro-fluidised bed was found to have comparable mixing performances to those of KM, Tangential IMTEK, and interdigital micro-mixers at much lower Reynolds numbers. The implication here is that for a given fluid throughput a more efficient mixing can be potentially achieved in a miniaturised fluidised bed. The analysis of energy dissipation rate and mixing time suggested that the fluidising-based micro-mixer is suitable for processing of significantly small volumes of fluids with reaction times of a few seconds. Finally, the miniaturised fluidised bed was also found to outperform the micro-channel mixer with fixed obstacles by obtaining greater mixing efficiencies at significantly lower pressure drops.
Acknowledgements Thanks to Miss Andrea Boyes for assisting with the collection of experimental data. This work was supported by the Australian Research Council and the University of Newcastle Priority Centre for Advanced Particle Processing and Transport.
References Baldyga, J., Bourne, J.R., 1984. A fluid mechanical approach to turbulent mixing and chemical reaction. Part III: computational and experimental results for the new micromixing model. Chem. Eng. Commun. 28, 259–281. Baldyga, J., Bourne, J.R., 1986. Principles of micromixing. In: Cheremisinoff, N.P. (Ed.), Encyclopedia of Fluid Mechanics, vol. 1. Gulf Publishing Company, Houston, TX, pp. 145–201. Baldyga, J., Bourne, J.R., 1988. Calculation of micromixing in inhomogenous stirred tank reactors. Chem. Eng. Res. Des. 66, 33.
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Doroodchi, E., Peng, Z., Sathe, M., Abbasi-Shavazi, E., Evans, G., 2012. Fluidisation and packed bed behaviour in capillary tubes. Powder Technol. 223, 131–136. Falk, L., Commenge, J.M., 2010. Performance comparison of micromixers. Chem. Eng. Sci. 65, 405–411. Hessel, V., Löwe, H., Muller, A., Kolb, G., 2005. Chemical Micro Process Engineering: Processing and Plants. Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany.
Nguyen, N.T., 2012. Micromixers: Fundamentals, Design and Fabrication. Elsevier, Waltham, USA. Potic, B., Kersten, S.R.A., Ye, M., van der Hoef, M.A., Kuipers, J.A.M., van Swaaij, W.P.M., 2005. Fluidization with hot compressed air in micro-reactors. Chem. Eng. Sci. 60, 5982. Wang, H., Iovenitti, P., Harvey, E., Masood, S., 2002. Optimization layout of obstacles for enhanced mixing in microchannels. Smart Mater. Struct. 11, 662–667.
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Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd
Liquid–liquid mixing using micro-fluidised beds Elham Doroodchi ∗ , Mayur Sathe, Geoffrey Evans, Behdad Moghtaderi University of Newcastle, Chemical Engineering, Callaghan, NSW 2308, Australia
a b s t r a c t This study experimentally investigates the application of a solid–liquid micro-fluidised bed as a micro-mixing device. The experiments were performed in a borosilicate capillary tube with an internal diameter of 1.2 mm (i.e. near the upper-limit dimension of a micro-fluidic system) using borosilicate particles with a mean diameter of 98 m. Refractive index matching technique using sodium iodide solution was employed to achieve a transparent fluidised bed. Mixing performance of the micro-fluidised bed in terms of mixing time was investigated using a dye dilution technique. Experiments were carried out in the creeping flow regime at Reynolds numbers ranging between 0.27 and 0.72. It was demonstrated that the micro-fluidised bed mixing time sharply decreases as the Reynolds number increases. That is because at relatively high Reynolds numbers, the particle oscillation is stronger creating larger disturbances in the flow. The energy dissipation rate in micro fluidised bed was estimated to be four orders of magnitude less than other passive micro mixers which operate in the turbulent regime. It was also demonstrated that the ratio of mixing time and the energy dissipation rate for fluidised bed micro-mixer was comparable to K-M, Tangential IMTEK, and interdigital micro-mixers. However, the fluidised bed micro-mixer was found to operate at much lower Reynolds numbers compared to other passive mixers, with a mixing time of the order of few seconds. © 2013 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Micro-fluidised bed; Micro-mixer; Liquid–liquid mixing; Mixing time; Refractive index matching technique; Dye dilution technique, Energy dissipation
1.
Introduction
Application of micro-fluidic devices for processing of multiphase flows in areas such as medical diagnostics, chemical analysis, power generation and fuel processing, invariably relies upon the physical or chemical interaction between at least two fluid phases. Such interaction is achieved through effective mixing. However, at length-scales associated with micro-fluidic devices where systems operate in a laminar regime with Reynolds numbers typically less than 1, the mixing process is rather poor as it is principally governed by molecular diffusion. Generally, active or passive techniques are employed to achieve effective mixing at micro-scales (Hessel et al., 2005; Nguyen, 2012). In active mixing external forces including ultrasound, acoustic, electrokinetic, and magneto-hydrodynamic forces are used to induce mixing. In these mixers the transversal disturbances generated by external fields lead to instabilities at the interface between the two mixed phases.
∗
Passive mixing on the other hand is accomplished by maximising the mixing contact area and/or reducing the mixing path through “Streaming Techniques” where the flow path is restructured using geometrically-based methods. There are two main mechanisms namely the diffusive transport and chaotic advection which drive mixing in passive micro-mixers. Diffusive mixing is often improved by decreasing the striation thickness through optimisation of the geometrical designs to create parallel or sequential laminations, sequential segmentation and focusing of fluid flow. Similarly, chaotic advection is generated by modifying the channel shape for stretching, folding and breaking of the laminar flow. The aim is to promote the streamlines to cross each other periodically (i.e. a spatially periodic flow). For example, the use of obstacles in passive mixers is demonstrated to be effective in generating such advections (Nguyen, 2012; Wang et al., 2002). At Reynolds numbers greater than 100, vortices are generated as the liquid is passing by an obstacle. These vortices disrupt the laminar flow pattern and induce transversal
Corresponding author. Tel.: +61 2 4033 9066; fax: +61 2 4033 9095. E-mail address: [email protected] (E. Doroodchi). Received 5 December 2012; Received in revised form 17 June 2013; Accepted 20 June 2013 0263-8762/$ – see front matter © 2013 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cherd.2013.06.024
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Nomenclature C1 Cy d D I Ix,∞ I* L Lx M Pe Re Qd QT Qc tm tdiff x y
proportionality constant in Eq. (3) (m−1 ) fitting parameter in Eq. (6) capillary internal diameter (m) molecular diffusivity (m/s2 ) grey scale intensity of individual pixel in shadow image intensity for perfect mixing normalised intensity of individual pixel in shadow image length of capillary (m) chord length (m) percentage mixing index Peclet number Reynolds number dye stream flow rate (L/min) total flow rate (L/min) clear stream flow rate (L/min) mixing time (s) diffusion time, d2 /D (s) distance from the centre of capillary in horizontal direction (m) distance from the inlet of capillary in vertical direction (m)
Greek symbols effective effective shear rate (1/s) total shear rate (1/s) total ı0 striation thickness (m) pressure drop (kPa) P I,0 deviation from concentration profile for perfect mixing at inlet (y = 0) deviation from concentration profile for perfect I,y mixing at y deviation from concentration profile for perfect I,∞ mixing at infinity ε energy dissipation rate (w/kg) mixing efficiency kinematic viscosity (m/s2 )
transport, in turn, improving the mixing. At Reynolds numbers less than 100, the lateral mass transport induced by the obstacles leads to mixing. Besides the fabrication complexity, the main drawback associated with the presence of obstacles, however is the rise in pressure drop across the bed. Clearly, the existing passive and active techniques suffer from fabrication complexity which limits their mass productions, high fabrication and maintenance costs, and high energy dissipation rates (Hessel et al., 2005; Nguyen, 2012). To overcome the shortcomings of the above techniques, a fluidising-based micro-mixer system is considered here. It is well-established that fluidisation provides efficient mixing and intensification of mass and heat transfer. In recent years, a number of studies focusing on miniaturised fluidised beds have emerged (Doroodchi et al., 2012; Potic et al., 2005). The major focus of these studies was to establish the fluidisation hydrodynamic characteristics of the fluidised beds both experimentally and theoretically. The focus of this work however is to experimentally investigate the mixing performance of two miscible fluids in a miniaturised solid–liquid fluidised bed in terms of mixing time and mixing efficiency. The relative
performance of the fluidising-based micro-mixer compared with typical existing passive micro-mixers is also established. It should be highlighted that apart from non-reacting flows the abovementioned miniaturised fluidised-beds can be also used as effective platforms in reacting flow systems, for example micro-reactors, where the mixing agents undergo a series of chemical reactions.
2.
Experimental
A schematic diagram of the experimental setup is presented in Fig. 1. The setup consisted of (i) a 30 cm glass capillary tube with an internal diameter of 1.2 mm (i.e. a dimension near the upper-limit dimension of 1 mm for micro-fluidic systems), (ii) a fluid reservoir, (iii) two syringe pumps, (iv) a LED light source and focusing lens, (v) a CCD camera, and (vi) a data acquisition system. The bed material was clear borosilicate glass sphere from Cospheric LLC with a size range of 90–106 m and mean diameter of 98 m, particle density of 2230 kg/m3 , and refractive index of 1.47–1.48. A 52 m wire mesh was used as a distributor. For optical diagnostics, the refractive index of the fluidising medium (i.e. sodium iodide solution) and bed material were matched (forming a transparent fluidised bed) minimising the noise generated by the light reflections from the surface of particles. Dye dilution technique (Nguyen, 2012) was used to determine the mixing performance at various operating conditions given in Table 1. In this work, the tracer dye stream (i.e. dye + sodium iodide solution) was fed in the centre from the bottom of the capillary tube through a 30G½ needle whilst the sodium iodide solution (56 wt%) was fed annularly passing through the fluid reservoir. The two streams have similar density (1800 kg/m3 ) and viscosity (0.0018 Pa.s). A LED lamp combined with a biconvex focusing lens was used to illuminate the region of interest using a back lighting technique. A cuvette filled with the sodium iodide solution was placed around the capillary tube to reduce the light refraction at the curved surface of the tube. An IDT XS3 high speed camera was then used to capture the dye dilution profile across the channel cross section at a rate of 50 Hz. 2000–3000 images were captured using IDT Motion Studio software whilst MATLAB software was employed for image analysis. Background images were also obtained at dye flow rates of zero at the start of each experiment. For benchmark purposes, the same experiments were carried out in the empty capillary tube (i.e. fluidised bed with no particles). The variation of the dye concentration profile along the length of capillary tube has been calculated by processing the images recorded using the procedure described above. The criterion used to evaluate the performance of the liquid–liquid mixing within micro-mixers was the uniformity of the dye intensity. The concentration of the dye was estimated from the shadow image generated by the diffusing jet of opaque dye stream using image processing in MATLAB software. The background images obtained at the start of each experiment was averaged and subtracted from individual mixing images. In a shadow image, the pixel intensity is inversely proportional to the concentration of dye at that point. Therefore a negative of each image was taken after the background subtraction. The background intensity was non-uniform. This non-uniformity implies that direct calculation of local concentration using extinction of the backlight is erroneous. To compensate for the gradient in backlight, normalisation of image intensity based
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Fig. 1 – Schematic representation of the experimental setup. on the material balance of tracer was adopted. The greyscale intensity of each pixel was divided by the average grey scale intensity along each horizontal line. This normalisation was based on the consideration that the liquid flows only in the vertical direction, and the integration of radial concentration profile of the dye with respect to radial coordinate should be constant along the height. The normalised intensity was then calculated by,
I(x, y)
I∗ (x, y) = 1/d
d/2
−d/2
(1)
I(y)dx
where I(x,y) is the intensity of each pixel. The term in the denominator is the average intensity of each horizontal line. The normalisation ensured that the area under the concentration profile remains constant along y direction. It should be noted that by this scaling only the magnitude of concentration profile was corrected to fit the material balance. The shape of the concentration profile however was unchanged. The corrected profiles were then used to calculate the deviation from concentration profile corresponding to perfect mixing. The concentration profile for perfect mixing was derived as follows. The intensity recorded by the CCD camera is proportional to the integral optical density along the line of sight. The absorption of light is directly proportional to the concentration and the optical path length according to the Lambert–Beer law. For the case of perfect mixing the concentration profile is uniform over the entire cross section. Hence, the profile of the absorption path length defines the intensity profile for perfect mixing. Since the capillary cross
section is circular the absorption path length at any particular radial location is equal to the chord length of the circle. The chord length at any radial distance x from the centre of capillary is given by:
Lx =
d 2 2
− x2
(2)
Since the dye concentration used in the current work is not very high then Lambert–Beer’s law is valid and the absorption of light is linear with respect to the absorption path length. Hence, the radial intensity profile in the image recorded by the camera can be expressed as:
Lx,∞ = C1
d 2 2
− x2
(3)
Eq. (3) indicates a semicircular profile for the radial intensity profile. The extent of mixing was calculated using the deviation of radial intensity profile from this semicircular profile, which is characteristic of a capillary cross-section. For the case of micro-channels with rectangular cross section, the profile should become rectangular. In the final step of the analysis the deviation of concentration profile at a particular height from the profile corresponding to perfect mixing was calculated as:
d/2
I,y =
|I∗ − Ix,∞ |dx
(4)
−d/2
As the relationship between intensity and concentration is linear then I,y also quantifies the deviation of the
Table 1 – Operating conditions. Total flow rate (QT ) (L/min) Reynolds number (Re) Dye stream flow rate (Qd ) (L/min) Clear stream flow rate (Qc ) (L/min) Flow ratio, Qd /QT a
10 0.18 0a 10 0a
Calibration case for background subtraction.
20 0.35 2 8 0.2
0a 20 0a
8 12 0.4
0a 35 0a
35 0.62 7 28 0.2
17.5 17.5 0.5
15.27 0.27 4.58 10.69 0.3
23.75 0.42 7.13 16.63 0.3
32.23 0.57 9.67 22.56 0.3
40.72 0.72 12.21 28.50 0.3
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Fig. 2 – Time-averaged concentration profiles for (a) capillary tube, (b) micro-fluidised bed [QT = 20 L/min, Qd/ QT = 0.4]. concentration profile at any height y from a concentration profile corresponding to perfect mixing. The extent of mixing was then calculated as:
M=
1−
I,y − I,∞ I,o − I,∞
× 100
(5)
The value of I,∞ is 0 since the two liquid streams are completely mixed at y = ∞. M is the mixing index, I,o is the variance between the concentration profile at the inlet (just above the central nozzle) and the concentration profile for complete mixing.
3.
Results and discussion
Typical time-averaged normalised concentration profiles of the dye in the capillary tube and the micro-fluidised bed are presented in Fig. 2a and b, respectively. As Fig. 2a shows, in the capillary tube the interface between the two streams remains undisturbed with dye and clear streams following their distinct core and annular flow paths, respectively. Conversely, under the same operating conditions enhanced mixing was observed when the two fluid streams were introduced into the micro-fluidised bed (Fig. 2b).
Fig. 3a and b illustrates the normalised concentration distribution profile at different elevations along the height of the capillary tube and the micro-fluidised bed, respectively. The bottom of the capillary is considered as zero elevation. The flatness of the profile corresponds to the degree of mixing (i.e. a flat profile represents complete mixing). In the capillary tube the concentration profile remained almost unchanged over the height of the tube (Fig. 3a) with the shape of the profile representing non-mixed regions along the examined elevations of 0.82, 1.3, 2.3, 3.3 and 4.3 mm. In the micro-fluidised bed however the concentration profile became flatter as the height increased approaching the average concentration profile (i.e. the dashed line in Fig. 3b). The flattening of the concentration profile indicates an increase in the homogeneity of the mixture and hence significant improvement in mixing. Fig. 4 shows M versus residence time for both the capillary tube and the micro-fluidised bed. Generally M increases as the time increases indicating an increase in the mixing efficiency. Clearly the rate of mixing achieved in the micro-fluidised bed is much greater than that of the capillary tube. The length of capillary tube required to achieve the same M as the fluidised bed is more than the corresponding length of the fluidised bed. The field of view in the current experiments however was limited and did not permit recording of the length of capillary tube required to achieve the same extent of mixing as those of
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1.3 mm
3.3 mm
4.3 mm
2.3 mm
(b) 1.2
3
Normalized Concentraon (-)
Normalized Concentraon (-)
(a)
0.82 mm
2.5 2 1.5 1 0.5 0
500
0
0.82 mm
1.3 mm
3.3 mm
4.3 mm
2.3 mm
1 0.8 0.6
0.4 0.2 0
1000
0
500
x (μm)
1000
x (μm)
Fig. 3 – Concentration profiles at different elevations for (a) capillary tube, (b) micro-fluidised bed [QT = 35 L/min, : Normalised concentration for complete mixing. Qd/ QT = 0.2]. 120
the fluidised bed. In order to facilitate quantitative comparison of the mixing performance for these two geometries, a first order model was fitted to the M data for the micro-fluidised bed and the capillary tube. The model equation for M as a function of axial coordinate is: Cy
}
80
(6)
where Cy is the curve fitting parameter. The fitted curves are shown as solid lines in Fig. 4. The length of capillary tube required to achieve the mixing index of 90% was calculated by extrapolating the data based on the fitted model. At the total flow rate of 35 L/min and flow ratio of 0.2, a mixing index of 90% can be achieved in the micro-fluidised bed in 2.3 s. In contrast, 82 s are required for the capillary tube to achieve the same mixing index. Further improvement was achieved as the dye flow rate to total flow rate ratio increased from 0.2 to 0.5. For the flow rate ratio of 0.5 the mixing index of 97% was obtained at less than 2.5 s.
M
−y
M(y) = 100 × {1 − e
100
60
40
20
0
0
5
20
25
30
35
t, s
: QT = 35 L/min; 100
80
M
15
Fig. 5 – Mixing index versus residence time [Qd/ QT = 0.2].
120
60
40
20
0
10
0
2
4
6
8
10
t,s
Fig. 4 – Mixing index versus residence time [QT = 35 L/min].
: Empty tube, Qd/ QT = 0.2;
bed, Qd/ QT = 0.2;
: fluidised bed, Qd/ QT = 0.5.
: fluidised
: 10 L/min.
Fig. 5 shows the effect of an increase in total flow rate on mixing performance of the micro-fluidised bed. As the flow rate increases from 10 L/min to 35 L/min, the rate of mixing increases reducing the mixing time for achieving mixing efficiency of 92% by more than half. The above results show the advantage of using a micro-fluidised bed as a micro-mixer over a core–annular micro-mixer. In a core–annular micro-mixer (i.e. the capillary tube), mixing is mainly driven by molecular diffusion which causes transversal mass transport. As a result the mixing is extremely slow and inefficient requiring long residence time and hence great mixing length. In the micro-fluidised bed however, the mixing is considered to be driven by passive chaotic advection with a three dimensional orbit which in turn leads to secondary transversal mass transport. Also, the bed materials (i.e. particles) act as obstacles and hence lead to periodic changes in the flow directions causing splitting and recombining effects with streamlines crossing each other. Such unique characteristics can improve mixing significantly at low Reynolds numbers.
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1.E+00
1.E+01
6 5 4
1.E-01
3 2
1.E+00
1.E-04
5.5E-04
5.0E-04
1.E-01
1.E-02
1.E-05
1.E-06 0.1
1
10
100
1000
10000
100000
1.E-03 1.E-04
Reynolds Number, Re (-) Fig. 6 – tm /tdiff as a function of Reynolds number. Fluidised bed;
0 4.5E-04
1.E-03
1 4.0E-04
Mixing me (s)
tm/tdiff
1.E-02
,
,
,
: Starlam mixer;
: IMM slit inter digital mixer; Tangential IMTEK;
: K-M;
: : T-mixer;
: T-mixer;
1.E-02
1.E+00
1.E+02
Energy Dissipaon, ε(w/kg)
:
: caterpillar, - - tm /tdiff ∝ (ln(Pe))/Pe.
The above results provide an insight into the driving mechanism for mixing enhancement in the micro-fluidised bed. It is interesting to compare the mixing performance of the fluidised bed relative to typical micro-mixers. Comparative studies of micro-mixers are typically carried out in the context of mixing time. Falk and Commenge (2010) have recently performed a detailed comparison of mixing time for different mixers as a function of Reynolds number and energy dissipation. The data presented in their work was used here to assess the performance of the fluidising-based micro-mixer relative to typical mixers. Fig. 6 shows the ratio of mixing time to the diffusion time versus Reynolds number for different passive mixers (solid symbols) and the examined miniaturised fluidised bed (open symbols). It can be seen that the majority of the collected experimental data follow the theoretical linear relationship between tm /tdiff and the Reynolds number (dashed line) especially at lower values of Reynolds numbers. This relationship is based on the equation proposed by Baldyga and Bourne (1984) with some simplifications introduced by Falk and Commenge (2010) by using the expression for pressure drop in Poiseuille flow. For the micro-fluidised bed examined in this study, the Reynolds number varies between 0.1 and 1 and the corresponding Péclet number is approximately 300–3000. For this range of Péclet number convection dominates molecular diffusion. From Fig. 6 it can be observed that the mixing time for the micro-fluidised bed was two orders of magnitudes less than the theoretical prediction. This low mixing time further confirms that molecular diffusion is not the only mechanism governing the mixing in micro-fluidised beds. It is believed that the oscillating motion of particles promote mixing by generating periodic spatial flow advections and flow instabilities. From visual observation the particles were found to move around their mean position by the distance equivalent to their diameter. However, the oscillation was slow with the frequency ranging between 1 and 10 Hz. Also, as Fig. 6 shows, the micro-fluidised bed mixing time sharply decreases as the Reynolds number increases. This is because at relatively high Reynolds numbers the particles vibrate more rigorously and
Fig. 7 – Mixing time as a function of energy dissipation rate.
: Fluidised bed;
IMTEK;
: T-mixer;
: Triangular interdigital,
: Tangential : IMM caterpillar;
,
, : Starlam; ––: tm = 0.15ε−0.45 ; • • • •: = 2%; - -: = 3%;- - -: = 10%.
create larger disturbances in the flow. However, it should be noted that the operation of such devices are limited by the fluidising characteristics of particles specifically their expansion behaviour. Moreover, Fig. 6 demonstrates that comparable mixing performances to those of K-M, Tangential IMTEK, and interdigital micro-mixers can be obtained in a fluidising-based micro-mixer at much lower Reynolds numbers. This indicates that for a given fluid throughput, more efficient mixing can be potentially achieved in a micro-fluidised bed with the exception of a caterpillar mixer which was found to outperform the examined fluidising-based mixer. Baldyga and Bourne (1984, 1986) have provided a relationship between the mixing time and the effective shear rate. The equation is as follows:
tm =
1 2effective
arcsinh
0.76effective ı2o D
(7)
where effective is the effective shear rate, ı0 is the striation thickness, and D is the molecular diffusivity. The effective shear rate is defined as effective = total , where is the mixing efficiency and total = ε/2v. ε is the energy dissipation rate and is the kinematic viscosity. The mixing efficiency is determined by fitting the experimental data of mixing time, tm , as a function of energy dissipation rate, ε. Fig. 7 shows the mixing time as a function of energy dissipation rate for different micro-mixer geometries. In Fig. 7, the dashed lines represent the mixing time calculated using Eq. (7) for different energy efficiency, values with 3% efficiency giving the best fit for all mixer geometries. An alternate expression for the mixing time as a function of energy dissipation rate is the empirical power law given by Falk and Commenge (2010): tm = 0.15ε−0.45
(8)
chemical engineering research and design 9 1 ( 2 0 1 3 ) 2235–2242
70
To demonstrate explicitly how the pressure drop affects the performance of the fluidising-based micro-mixer, the findings of this study were compared with those of a micro-channel equipped with fixed obstacles of 60 m arranged in different configurations (Wang et al., 2002). Fig. 8 shows the mixing efficiency in terms of mixing index versus pressure drop per unit length. The diamond symbols represent the literature data reported by Wang et al. (2002) and the square symbols represent our experimental data obtained at L = 1.2 mm. Clearly, the miniaturised fluidised bed outperforms the micro-channel mixer with fixed obstacles by obtaining greater mixing efficiencies at significantly lower pressure drops. This finding in turn implies that the energy dissipation rate for the fluidisingbased micro-mixer is much lower than the micro-channel mixer with fixed obstacles.
60 50
M
40 30 20 10 0 10
100
1000
ΔP/L (kPa/m) Fig. 8 – Mixing index as function of pressure drop per unit length.
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: Fluidised bed;
: Patterned microchannel.
Eq. (8) has been plotted as a solid line in Fig. 7, where it can be seen that the energy dissipations associated with the micro-fluidised bed mixer are significantly lower in comparison with the other laminar flow-based mixers (Falk and Commenge, 2010). The lower energy dissipation however comes at a cost of an increased mixing time. These characteristics make such devices especially suitable for processing of relatively small volumes of fluids with reaction times in the order of seconds. It should be noted that the mixing time in Fig. 7 has been plotted as a function of overall energy dissipation rate. In this work we have not attempted detailed modelling of impact of ε on mixing time. In a fluidised bed shear rate in the vicinity of particles is higher than the bulk liquid, which implies non-uniform ε. Such non-uniform energy dissipation has significant impact on the micro-mixing process, as reported by Baldyga and Bourne (1988) who found that the product distribution for rapid azo-coupling reactions in three sizes of mechanically agitated vessels could be successfully modelled when spatial distribution of ε was properly taken into account. In the present work, we are currently not able to attempt such a modelling approach due to lack of reliable velocity and shear rate data in the micro-fluidised bed. Eq. (7) provided the best fit for mixing time data of the fluidised bed at 2% mixing efficiency, which is shown in the inset of Fig. 7 as the dotted line. This efficiency is comparable to the majority of other passive mixers. The low efficiency exhibited by the micro-fluidised bed is expected since the majority of the energy is used to keep the particles in suspension. In general, the increase in turbulence does not always lead to enhancement in mixing. That is mainly because not all of the turbulent flow structures enhance mixing. Only the structures at the interface of two fluids which result into splitting/stretching of fluid lamellae improve the mixing while the flow structures within either fluid have no effect on the mixing process. Overall, the increase of turbulence intensity by increasing the pressure drop, either by changing the geometry of mixer or increasing the velocity of fluid, increases both mixing and non-mixing flow structures. Consequently, the mixing time is reduced with an increase in pressure drop whilst the mixing efficiency remains unaffected.
4.
Conclusions
The novel idea of using a fluidised bed with micro-scale dimensions as a micro-mixer for mixing of two miscible liquids was examined experimentally. The results of the study were bench marked against mixing behaviour in a capillary tube and typical passive mixers. Generally, the fluidisation process was found to achieve high mixing efficiencies reduced mixing time well below those for a capillary tube. An increase in the flow rate was found to increase the mixing rate within the micro-fluidised bed resulting in a reduced mixing time. The enhancement in mixing was considered to be due to destabilisation of the diffusion layer between the two fluids by fluidised particles and hence periodic changes in the flow directions causing splitting and recombining effect with streamlines crossing each other. The micro-fluidised bed was found to have comparable mixing performances to those of KM, Tangential IMTEK, and interdigital micro-mixers at much lower Reynolds numbers. The implication here is that for a given fluid throughput a more efficient mixing can be potentially achieved in a miniaturised fluidised bed. The analysis of energy dissipation rate and mixing time suggested that the fluidising-based micro-mixer is suitable for processing of significantly small volumes of fluids with reaction times of a few seconds. Finally, the miniaturised fluidised bed was also found to outperform the micro-channel mixer with fixed obstacles by obtaining greater mixing efficiencies at significantly lower pressure drops.
Acknowledgements Thanks to Miss Andrea Boyes for assisting with the collection of experimental data. This work was supported by the Australian Research Council and the University of Newcastle Priority Centre for Advanced Particle Processing and Transport.
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