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Numerical analysis of an MHD micro-device with simultaneous mixing and pumping capability
Journal of Industrial and Engineering Chemistry 38 (2016) 23–36
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Numerical analysis of an MHD micro-device with simultaneous mixing and pumping capability Xuejiao Xiao a, Chang Nyung Kim b,* a b
Department of Mechanical Engineering, Graduate School, Kyung Hee University, Yong-in, Kyunggi-do 446-701, Republic of Korea Department of Mechanical Engineering, College of Engineering, Kyung Hee University, Yong-in, Kyunggi-do 446-701, Republic of Korea
A R T I C L E I N F O
Article history: Received 21 December 2015 Received in revised form 1 April 2016 Accepted 3 April 2016 Available online 21 April 2016 Keywords: Magnetohydrodynamic (MHD) Micro-mixer Micro-pump Numerical simulation
A B S T R A C T
In this study, a new magneto-hydrodynamic (MHD) micro-device with simultaneous mixing and pumping capability is proposed, and a numerical investigation for the performance of the device is performed. The present MHD micro-device utilizes the arrays of electrodes deposited on duct walls. By judiciously applying different potentials to different electrode pairs, the Lorentz-force can be created, directing the liquid to move in a desired path for pumping without a need of separate pumps, and simultaneously inducing complex flows for mixing. The results show that the micro-device proposed in this study can achieve highly-efficient, simultaneous mixing and pumping for electrolytes. ß 2016 The Korean Society of Industrial and Engineering Chemistry. Published by Elsevier B.V. All rights reserved.
Introduction In recent years, microfluidic systems and devices have emerged as a necessary tool for Laboratory-on-a-Chip (LOC) in different fields such as bio-detection, biotechnology, chemical reactors and environmental monitors [1]. Usually, it is necessary to pump the fluid in one region of the device to another region in many different applications. Meanwhile, in order to promote chemical and biological reactions, full mixing of various reagents is needed. However, the characteristic length associated with micro-device is very small, which means that the diffusion effect alone cannot provide a sufficiently fast way for mixing. Furthermore, turbulence that may enhance mixing is not available because of flows with low Reynolds number (Re 1) in micro-devices [2]. The importance of the convective stirring mechanism relative to the molecular diffusion in a given mixing problem is measured by a nondimensional number called Peclet number. In microfluidic devices the Peclet number is very large, whose value is typical in the range of 103–105, so that concentration change is achieved mainly by the convection term, rather than by the diffusion term. Therefore, the stirring process must be utilized with artificial vortices created by appropriate mechanism in the stirring channel [3]. Magneto-hydrodynamics (MHD) can provide us with a relatively convenient way for pumping and mixing. Firstly, MHD
* Corresponding author. Tel.: +82 312012578. E-mail address: [email protected] (C.N. Kim).
pumps can produce a big force that is suitable for high volumetric pumping [4]. Recently, a number of researchers constructed MHD micro-pumps with silicon [5,6] and ceramic substrates [7], and demonstrated that these pumps are able to move liquids in microchannels. Secondly, a great deal of attention has been paid to the design of more efficient mixers in microfluidic systems [8–11], which triggers the formation of cross-sectional vortexes and velocity recirculation by exploiting the coupling of momentum transport with MHD effects. Gleeson et al. [10] examined the MHD mixing of two fluids in an annular micro-channel using asymptotic analysis and numerical simulation. Cerbelli et al. [11] developed a quantitative analysis of regimes in an annular MHD-driven micromixer proposed by Gleeson et al. [10]. Kabbani et al. [12] introduced a new approximate solution for MHD flows in micro-channels, which can be used for MHD micropumps and MHD microfluidic networks under DC or AC fields. Ho [13] provided an analytic model based on the steady-state, incompressible and fully developed laminar flow theory, and analyzed the MHD flow characteristics. His analytic solution was compared with experimental results with an excellent agreement. Meanwhile, the numerical solution method based on computational fluid dynamics has become an important and effective way to analyze MHD flows. Lim and Choi [14] numerically investigated the continuous MHD micro-pump with side-walled electrodes by using commercial CFD code CFD-ACE, and predicted the velocity profiles of the working fluid in the micro-channel under various operation currents and magnetic flux intensities. Lee et al. [15] proposed a chaotic mixing mechanism suitable for the micro-mixer of LOC, and
http://dx.doi.org/10.1016/j.jiec.2016.04.001 1226-086X/ß 2016 The Korean Society of Industrial and Engineering Chemistry. Published by Elsevier B.V. All rights reserved.
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Fig. 1. Duct geometry for one set of electrodes and the coordinate system.
Fig. 2. Schematic depiction of an MHD mixer and pump in the present study.
X. Xiao, C.N. Kim / Journal of Industrial and Engineering Chemistry 38 (2016) 23–36
investigated the fluid flow by numerical simulation and experiment. It was confirmed that the proposed micromixer was able to achieve high mixing efficiency. Also, Akbar et al. [16,17] showed that the magnetic field can control the flow of fluids with finite electric
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conductivity under a magnetic field, for example, in the flow of Eyring–Powell fluid and CNT suspended nanofluid, respectively. In the present study, we propose a new micro-device under a uniform applied magnetic field, where judicious application of
Fig. 3. Validation of the present numerical model.
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different electric potentials to different electrodes can create a Lorentz-force that can direct the liquid to move along a desired path for pumping without a need of separate pumps, and induce complex flows for mixing. The formation of the cross-sectional double velocity recirculations obtained in this study leads to the enhancement of mixing performance. Three cases with different combinations of input voltage of electrodes, together with the consideration of the subcases with one set, two-sets of electrodes in each case, are numerically examined for the comparison of the mixing capability and pumping performance. The obtained results including the information of velocity, current density, pressure and mass fraction of one fluid are visualized, with the values of mixing index, in detail. It can be noted that the present numerical model is validated against an existing numerical work with a good agreement. Problem formulation PBS (phosphate buffered saline) solution, which is an electrolyte, is generally used in LOC as a working fluid in real experiments for the analysis of proteins, and it is also used as medium for the fixation of protein. The properties of PBS solution are outlined in Table 1. A micro-device with rectangular cross-section proposed in this study is shown in Fig. 1, where the uniform magnetic field is applied in the y-direction with the intensity of 0.2 T. Eight electrodes, which are denoted by L1, L2, L3, L4, R1, R2, R3, and R4, respectively, are positioned on the left and right vertical walls of the conduit as one set of electrodes. The length of each electrode is 2 mm and the width is 1 mm. In the present study, firstly, one set of electrodes is installed for simultaneous MHD pumping and mixing. Fig. 2(a) shows the input voltages and the schematic depiction of the electrodes on the left vertical wall. The electric current (denoted by J M ! ) flows from L1 to L2 (in the x-direction) and from L4 to L3 (in the negative xdirection), which induces the z-directional Lorentz force (denoted by F M ! ) in the region between the electrodes L1 and L2 (the first pair on the left wall) and the negative z-directional Lorentz force in the region between the electrodes L3 and L4 (second pair of on the left wall). Fig. 2(b) shows the schematic depiction of the electrodes on the right vertical wall, where the situation similar to that in the left vertical wall is depicted. The positive and negative z-direction Lorentz force in the gap between the two electrodes of each pair may enhance the mixing performance. Meanwhile, as the average electric potential of the electrodes R1 and R2 is higher than that of the electrodes L1 and L2, the electric current (denoted by J P ! ) also can move in the negative zdirection, which yields the x-directional Lorentz force (denoted by F P ! ) responsible for the pumping of the fluids in the x-direction as shown in Fig. 2(c). Also, the average electric potential of the electrodes R3 and R4 is higher than that of the electrodes L3 and L4, which also works for pumping. Fig. 2(d) shows the locations of electrodes and different cross-sections. In the present study, three cases (Case 1, Case 2 and Case 3) with different combinations of the input voltage of electrodes are considered, as shown in Table 2. In each case, two subcases (Case 11 and Case 1-2 for Case 1, Case 2-1 and Case 2-2 for Case 2, Case 3-1
Table 2 Input voltage of electrodes (unit V).
Case 1 Case 2 Case 3
L1, R4
L2, R3
L3
L4
R1
R2
2.2 1.1 0.5
2.2 1.1 0.5
2.3 1.2 0.6
2.1 1.0 0.4
2.3 1.2 0.6
2.1 1.0 0.4
and Case 3-2 for Case 3) with one-set, two-sets of electrodes are considered, with an aim to investigate the difference in mixing performance. The ratio of the inertial effect to viscous effect in the flow field can be described by the Reynolds number, Re = rUL/m, where r is density of the fluid, U is the average flow velocity in the outlet, L is the characteristic length, and mf is the dynamic viscosity of the fluid. In the present study the Reynolds numbers in different cases are different from each other because of different outlet velocities (which will be shown later in Table 4). The ratio of the electromagnetic force to ffi viscous force is measured by Hartmann qffiffiffiffiffiffiffiffiffiffiffiffiffi number, M ¼ B0 L s f =mf , where B0 is the external magnetic field strength, sf is the electrical conductivity of the fluid, the Hartmann number in the present study is 0.267. The ratio of the characteristic time of diffusion effect to the characteristic time of convection effect is denoted by the Peclet number, whose values in different cases are also different from each other (which will also be shown in Table 4) because of different outlet velocities in the present study. Here, Pe = UL/D, where D is the diffusion coefficient. Governing equations The system of governing equations for the flow of an electricallyconducting fluid in a magnetic field includes the Navier–Stokes equation with Lorentz force, conservation of mass, conservation of charge and Ohm’s law, which can be referred to Ref. [5]:
ruru ¼ rp þ mf r2 u þ JB
(1)
ru ¼ 0
(2)
Table 1 Material property of PBS solution. PBS solution Density (kg/m3) Conductivity (S/m) Dynamic viscosity (kg/s m) Relative permeability Relative permittivity pH Diffusion coefficient (m2/s)
1000 1.5 6 104 1 72 7.4 1 1010
Fig. 4. Three-dimensional streamlines of current moving from L1 to L2, L4 to L3, R1 to R2 and R4 to R3 in Case 1-1.
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rJ ¼ 0
(3)
J ¼ s f ðrf þ uBÞ
(4)
where u, r, p, mf, sf, J, and f are the velocity vector, fluid density,pressure, dynamic viscosity of the fluid, electrical conductivity of the fluid, current density vector, and electric potential, respectively.
Fig. 5. Flow features in y–z plane at x = 0.00325 m in Case 1-1.
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In a homogenous fluid behaving under a diluted concentration and with an assumption of electrical neutrality for the electrolyte, the concentration field of species is governed by the following transport equation, which is given in Ref. [18]: urC ¼ Dr2 C
(5)
where C is the mass concentration and D is the diffusion coefficient.
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Fig. 5. (Continued ).
In order to figure out the mixing performance in different cases, the mixing index shown in the below (see Ref. [18]) can be considered. Mixing index ¼ 1
s s max
(6)
where s is the standard deviation of the concentration of the Fluid A in a monitoring plane. smax is the maximum standard deviation of the concentration of the Fluid A that can be found before the two fluids enter the conduit, whose value is 0.5. If the two fluids are ideally well mixed, then s 0, so that the mixing index would be 1.0. Boundary conditions At the inlet and the outlet, the pressure is considered to be zero. At the fluid–solid interface the no-slip and no-penetration conditions are applied. In the left portion of the inlet, Fluid A is supposed to come in, while in the right portion of the inlet Fluid B is ready to get in, as depicted in Fig. 1. The concentration of species A, CA, is imposed to be 1 at the inlet for Fluid A and to be 0 at the inlet for Fluid B. And, the boundary condition for the concentrations at the walls is @C/@n = 0, where n is the unit vector outer normal to the side wall. It is considered that there is no normal current at the inlet, outlet, and fluid–solid interface (except the electrodes), which yields the insulating boundary condition for the electric potential @f/@n = 0. Numerical method The present study uses a structured grid system chosen after a series of grid independence tests. In the current simulation, a grid system with 30,800 grids is used for one set of electrodes. Underrelaxation is employed in the iteration procedure for the coupled
governing equations. The second-order upwind scheme is used to discretize the convective terms and the central difference scheme for diffusion terms. A multi-grid accelerated Incomplete Lower Upper factorization technique [19,20] is employed for the solution method for discretized equations. The Rhie–Chow interpolation method [21] is used for the pressure–velocity coupling in CFX code. Validation Numerical simulations of three types of MHD flows generated in a micro-channel for pumping and/or mixing are performed to validate the accuracy and reliability of the present numerical model against the numerical results of La [22]. The duct geometry is shown in Fig. 3(a), and the applied magnetic field used in the simulations is applied in the z-direction with the intensity of 18.7 mT. Also, PBS solution is chosen as the working fluid. The applied voltage conditions for the three types of the MHD flow considered in La’s simulations [22] are as follows: Firstly, for the axial flow (for pumping only) the voltages applied to the electrodes are V w1 ¼ 1:8 V, V w2 ¼ 0 V, and there is no current between Vb1 and Vb2, that is, Jb1 = Jb2 = 0 A m2. Secondly, for the sinusoidal flow (for pumping and mixing) the voltages applied to the electrodes are V w1 ¼ V b1 ¼ 1:8 V and V w2 ¼ V b2 ¼ 0 V. Thirdly, for the multivortical flow (for pumping and mixing), the voltages applied to the electrodes are V w1 ¼ 0:9 V, Vb1 = 1.8 V and V w2 ¼ V b2 ¼ 0 V. The concentration distributions at the exit of the microchannel of La’s numerical results [22] and the present simulation results for (i) axial flow, (ii) sinusoidal flow, and (iii) multi-vortical flow are compared in Fig. 3(b) and (c), respectively. And, they demonstrate a good agreement. The standard deviations of the cross-sectional concentration values for the cases of the axial, sinusoidal, and multi-vortical flows in La’s numerical results are reported to be 0.343, 0.293, and 0.096, respectively. Meanwhile, in the present simulation results, these values are 0.367, 0.312, and 0.098, respectively. The differences between La’s numerical results and
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the present simulation results are 6.9%, 6.5% and 2.1%, respectively. With smaller differences shown above, the present numerical model is validated. Results and discussion Case 1 One set of electrodes (Case 1-1) Fig. 4 shows the three-dimensional streamlines of current moving from electrode L1 to L2, from L4 to L3, from R1 to R2 and from
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R4 to R3 in Case 1-1, where the curved paths of the current flows are shown, as expected (see Fig. 2(b)). Fig. 5(a) shows the Lorentz vectors in the y–z plane at x = 0.00325 m (inside the gap between L1 and L2), where the Lorentz force is induced in the positive zdirection, associated with the direction of applied magnetic field and the current therein ðJM ! ~ BÞ, as mentioned before. The Lorentz force near the left and right vertical walls in the middle fluid regions is higher than that in the other fluid region, due to the higher current density therein. The positive z-directional Lorentz force induces the highest pressure in the fluid region near the right vertical wall in the middle fluid region, while the lowest pressurein
Fig. 6. Flow features in y–z plane at x = 0.00875 m in Case 1-1.
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Fig. 7. Flow features in x–y plane at z = 0.0005 m in Case 1-1.
Fig. 8. Flow features in x–y plane at z = 0.0015 m in Case 1-1.
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the fluid region near the left vertical wall in the middle fluid region (see Fig. 5(b)). The distribution of the plane velocity in the same plane is shown in Fig. 5(c). Because of the Lorentz force, in the middle part of the fluid region, the fluid generally moves from the left to the right, and reaches the fluid–right vertical wall interface, then bifurcates to the right-top part and right-bottom part of the fluid region, which is based on the direction of the gradient of the pressure. Therefore, two large-scaled velocity circulations are generated in this plane. Meanwhile, two smaller vortexes are also induced near the top and bottom walls. This kind of complex flow can enhance the mixing performance. The potential difference between the group of electrodes L1, L2 and the group of electrodes R1, R2 induces the electric current to move from the right vertical wall to the left vertical wall (J P ! ), as shown in Fig. 5(d), which induces the x-direction Lorentz force ðJ P ! BÞ ! used for pumping. Fig. 6 shows the flow features in the y–z plane at x = 0.00875 m (inside the gap between L3 and L4). Based on the direction of the current around L3, L4, R3 and R4 (see in Fig. 4), the Lorentz force heads leftward in this plane (Fig. 6(a)), whose direction is opposite to that in the y–z plane at x = 0.00325 m (Fig. 5(a)). The highest pressure is induced in the fluid region near the left vertical wall in the middle fluid region, while the lowest pressure is induced in the fluid region near the right vertical wall in the middle fluid region (Fig. 6(b)). With the influence of the combination of Lorentz force and pressure gradient, two large-scaled velocity circulations are also induced in this plane, with the directions of the velocity being in the way opposite to those in Fig. 5(c). Moreover, the plane current ðJP ! Þ also moves to the left, which works for pumping. Fig. 7 shows the flow features in the x–y plane at z = 0.0005 m. Here, as shown in Fig. 7(a), the lowest pressure is observed in the fluid region around L1 and L2, while the highest pressure is seen in the fluid region around L3 and L4 (see Figs. 5(a) and 6(b) also). Fig. 7(b) shows the distribution of the plane velocity in this plane. With the pumping effect, the fluid is sucked at the inlet and the flow generally moves from the inlet to the outlet. Meanwhile, because of the large pressure gradient in the fluid regions around the electrodes L1 and L2 and around the electrodes L3 and L4, the flow pattern is complicated with eight velocity circulations. The direction of four velocity circulations around electrodes L1 and L2 are opposite to the directions of those around L3 and L4. These kinds of complex flow also can be beneficial to mixing. Fig. 8 shows the flow features in the x–y plane at z = 0.0015 m. Here, the highest pressure is observed in the fluid region around R1 and R2, while the lowest pressure is seen in the fluid region around R3 and R4 (Fig. 8(a)). Fig. 8(b) shows the velocity distribution in this plane, with four velocity circulations around R1 and R2 and four velocity circulations around R3 and R4, with the swirling directions opposite to those in Fig. 7(b). Fig. 9(a) showsthe mass fraction of Fluid A near the inlet (in the y–z plane at x = 0.0001 m), where the two fluids are not mixed yet, practically. So, the mass fraction of Fluid A in the left fluid region is near 1, and that in the right fluid region is near zero. Fig. 9(b) shows the mass fraction of Fluid A in the monitoring plane located at x = 0.014 m. The cross-section for the mass-fraction display (monitoring plane) is chosen to be located at 4 mm downstream of the trailing edge of the latest electrode in the mixer. The hydraulic diameter of the conduit is Dh = 4A/P = 8/3 mm = 2.667 mm. Therefore, the axial distance (4 mm) between the trailing edge of the latest electrode and the cross-section for the mass-fraction display corresponds to 1.5Dh. If the axial distance for the mass-fraction display is too short, then in the cross-section for the mass-fraction display the axial velocity is not well distributed with some notable magnitude of the velocity components perpendicular to the main flow. Therefore, in the above situation it is highly possible for the value of the mass-fraction of a fluid in the cross-section for the massfraction display not to be well distributed. Also, if the axial distance
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Fig. 9. Mass fractions of Fluid A in the different y–z plane in Case 1-1.
for the mass-fraction display is too long, then the diffusion effect (caused by molecular diffusion) in the region ranging from the trailing edge of the latest electrode to the cross-section for the massfraction display can be meaningful. In Fig. 9(b) the two fluids are mixed in the inner part of fluid region. However, in the fluid region near the corners, the two fluids are not mixed so well because of the relatively low velocity therein and no-slip condition. Fig. 10 shows the axial velocity distribution along the central line of the duct. The axial velocity starts to increase right after the entrance, and after some velocity fluctuations, the axial velocity reaches a value of around 0.0032 m/s in the fluid region near the exit. Meanwhile, the Reynolds number is 14.24 based on the average velocity in the monitoring plane and the hydraulic diameter of the duct.
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Fig. 10. Axial velocity distribution along the central line of the duct in Case 1-1.
Also, a case with two sets of electrodes is also considered in the present study (donated as Case 1-2). Fig. 11 shows the arrangement of the electrodes in Case 1-2. The distance between adjacent electrode sets is 3 mm in multiple-set subcases, which is obtained by considering the feasibility of the electrical influence of one set on another set. The cross-sectional position for the mass-fraction display in Case 1-2 can also be seen in Fig. 11, which is also located 4 mm downstream from the trailing edge of the latest electrode. Fig. 12 shows the duct geometry and the three-dimensional streamlines of the current moving from L1 to L2, from L4 to L3, from R1 to R2 and from R4 to R3, which shows the curved paths of the current flows similar to those in Case 1-1. Fig. 13 shows the mass fraction of Fluid A in the y–z plane at x = 0.025 m in Case 1-2. Witness that in the inner fluid region of the conduit the mass fraction in Case 1-2 appears almost uniform, indicating that the two fluids are reasonably well-mixed. In the current section, the effect of the electrode voltage on the mixing performance is considered. In Case 2 and Case 3, as shown in Table 2, a relatively lower potential difference is considered,
compared to that in Case 1. The potential difference between the electrodes with highest and lowest potential for each set in Case 1 is 4.4 V, while the potential difference in Case 2 is 2.2 V, and the potential difference in Case 3 is 1.0 V. As observed before, the potential differences between L1 and L2 and between R1 and R2, the potential difference between L3 and L4 and between R3 and R4 work mostly for mixing. This means that Case 2 and Case 3 have a mixing capability quite smaller than that in Case 1. However, the potential difference for pumping in each case is the same. More precisely, the potential differences between electrodes L1 and R1, L2 and R2, L3 and R3, L4 and R4 are always 0.1 V in each case. This means that Case 2 and Case 3 may have a pumping capability similar to that in Case 1 as mentioned before. Two subcases (Case 2-1, Case 2-2) in Case 2 and two subcases (Case 3-1, Case 3-2) in Case 3 are also designed to investigate the effect of the number of electrode set on the mixing/pumping performance. Fig. 14 showsthe axial velocity distributions along the central lines for the two subcases in Case 1. In different subcases for each case, the axial velocity distribution associated with the configuration
Fig. 11. Locations of each electrode and the position (in blue) for displaying the mixing performance in Case 1-2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
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Fig. 13. Mass fraction of Fluid A in the y–z plane at x = 0.025 m in Case 1-2. Fig. 12. Three-dimensional current streamlines from L1 to L2, L4 to L3, R1 to R2 and R4 to R3 in Case 1-2.
of the electrodes shows the same tendency. More precisely, in Case 12 the velocity profile around the second set of electrodes is almost the same as that around the first set of electrodes. Since each set of electrodes has pumping capability, the exit velocity in Case 1-2, with
increased flow resistance associated with increased pumping capability, is around 1.2 times the velocities in Case 1-1. Fig. 15 shows the contours of the mass fraction of Fluid A in the two subcases in Case 2. It is obvious that the mixing performance increases with an increase in the number of electrode set.
Fig. 14. Axial velocity distributions along the central lines for two subcases in Case 1.
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Fig. 16. Mass fractions of Fluid A in the different y–z plane near the exit of Case 3.
Fig. 15. Mass fractions of Fluid A in the different y–z plane near the exit of Case 2.
However, the mixing performance in Case 2 is lower than that in Case 1, which is because of relatively weak cross-sectional velocity circulation in association with relatively weak Lorentz force for mixing. Fig. 16 shows the contours of the mass fraction of Fluid A in the two subcases in Case 3, presenting poor mixing. Also, the mixing performance increases with an increase in the number of electrode set. Fig. 17 shows the axial velocity distributions along the central lines in Case 1-1, Case 2-1 and Case 3-1. The inlet and outlet velocities in each case are very similar, since the pumping capabilities of Case 1, Case 2 and Case 3 are almost the same. The velocity fluctuation in Case 1-1 is the strongest in line with the fact that the absolute values of the electric potential of the electrodes are the highest in Case 1-1.
Table 3 shows the mixing indexes in the subcases. As mentioned before, the mixing index increases with an increase in the number of the electrode set. It can also be seen that the mixing index in Case 1-2 is the highest, whose value is 0.9393, showing that the proposed device is able to achieve high mixing efficiency. Also, the mixing indexes in Case 1 are generally higher than those of Case 2 and Case 3, as explained before.
Table 3 Mixing index for Case 1, Case 2 and Case 3. Case 1
s Mixing index
Case 2
Case 3
Case 1-1
Case 1-2
Case 2-1
Case 2-2
Case 3-1
Case 3-2
0.2462 0.7533
0.0607 0.9393
0.316 0.684
0.228 0.772
0.538 0.462
0.449 0.551
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Fig. 17. Axial velocity distributions along the central lines in Case 1-1, Case 2-1 and Case 3-1.
Table 4 Mass flow rate, Reynolds number and Peclet number for Case 1, Case 2 and Case 3. Case 1
Qm (kg/s) Re Pe
Case 2
Case 3
Case 1-1
Case 1-2
Case 2-1
Case 2-2
Case 3-1
Case 3-2
1.291 105 14.24 1.068 104
1.534 105 16.91 1.268 104
1.339 105 14.69 1.101 104
1.583 105 17.44 1.3083 104
1.364 105 14.99 1.125 104
1.605 105 17.62 1.322 104
Table 4 shows the mass flows in subcases of Case 1, Case 2 and Case 3. Cases with two sets of electrodes induce a mass flow rate higher than those with one set of electrode. More precisely, the mass flow rate of Case 1-2 is 1.188 times of that of Case 1-1. The mass flow rate of Case 2-2 is 1.182 times of that of Case 2-1. And the mass flow rate of Case 3-2 is 1.177 times of that of Case 3-1. The Reynolds number and Peclet number in each case can also be seen in Table 4. Since the Reynolds number and Peclet number are in association with the average velocity, the Reynolds number and Peclet number in each case show the same tendency as the mass flow rate.
Conclusion This study numerically investigates the pumping and mixing performance in a newly designed magneto-hydrodynamic mircomixer with pumping capability. Commercial software CFX, based on a finite-volume method, is adopted to solve the governing equations. By judiciously applying different electric potentials to different electrode pairs, the Lorentz-force is obtained that can direct the liquid to be pumped and mixed simultaneously.
In this study, the mixing index is used in order to estimate the mixing performance in different cases. The effect of the magnitude of the voltages imposed on the electrodes on the mixing performance is investigated. For the case with relatively higher potential difference, the mixing index is relative higher, which means a higher mixing performance. Meanwhile, the subcases with one- and two-sets of electrodes are considered to investigate the difference in the mixing performance. The mixing index increases with an increase in the number of the electrode set used in each case, which means that the use of multiple sets of electrodes can enhance the mixing performance. In summary, the present study proposes a new magnetohydrodynamic mixer with pumping capability, and the present numerical analysis shows that the proposed device can have high mixing performance. Acknowledgements This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 20110022679).
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J.P. Gleeson, O.M. Roche, et al. SIAM J. Appl. Math. 64 (4) (2004) 747. S. Cerbelli, A. Adrover, et al. Microfluid. Nanofluid. 6 (2009) 747. H. Kabbani, M.J. Mack, et al. J. Fluids Eng. 130 (2008) 091204. J.E. Ho, J. Mech. Sci. Technol. 15 (4) (2007) 315. S.S. Lim, B.Y. Choi, J. Mech. Sci. Technol. 23 (2009) 739. S.H. Lee, H.J. Kang, et al. J. Microsyst. Technol. 15 (2) (2009) 269. N.S. Akbar, A. Ebaid, Z.H. Khan, J. Magn. Magn. Mater. 382 (2015) 355. N.S. Akbar, Z.H. Khan, S. Nadeem, J. Mol. Liquids 196 (2014) 21. M. Yi, H.H. Bau, Int. J. Heat Fluid Flow 24 (5) (2003) 645. B.R. Hutchinson, G.D. Raithby, Numer. Heat Transf. 9 (1986) 511. W. Raw, Robustness of coupled algebraic multigrid for the Navier–Stokes equations, A9618260, AIAA Meeting Paper 96-0297, 1996. [21] W.Z. Shen, J.A. Michelsen, J.N. Sorensen, AIAA J. 39 (12) (2001) 2406. [22] M. La, W. Kim, et al. Int. J. Precis. Eng. Manuf. 15 (3) (2014) 463.
Contents lists available at ScienceDirect
Journal of Industrial and Engineering Chemistry journal homepage: www.elsevier.com/locate/jiec
Numerical analysis of an MHD micro-device with simultaneous mixing and pumping capability Xuejiao Xiao a, Chang Nyung Kim b,* a b
Department of Mechanical Engineering, Graduate School, Kyung Hee University, Yong-in, Kyunggi-do 446-701, Republic of Korea Department of Mechanical Engineering, College of Engineering, Kyung Hee University, Yong-in, Kyunggi-do 446-701, Republic of Korea
A R T I C L E I N F O
Article history: Received 21 December 2015 Received in revised form 1 April 2016 Accepted 3 April 2016 Available online 21 April 2016 Keywords: Magnetohydrodynamic (MHD) Micro-mixer Micro-pump Numerical simulation
A B S T R A C T
In this study, a new magneto-hydrodynamic (MHD) micro-device with simultaneous mixing and pumping capability is proposed, and a numerical investigation for the performance of the device is performed. The present MHD micro-device utilizes the arrays of electrodes deposited on duct walls. By judiciously applying different potentials to different electrode pairs, the Lorentz-force can be created, directing the liquid to move in a desired path for pumping without a need of separate pumps, and simultaneously inducing complex flows for mixing. The results show that the micro-device proposed in this study can achieve highly-efficient, simultaneous mixing and pumping for electrolytes. ß 2016 The Korean Society of Industrial and Engineering Chemistry. Published by Elsevier B.V. All rights reserved.
Introduction In recent years, microfluidic systems and devices have emerged as a necessary tool for Laboratory-on-a-Chip (LOC) in different fields such as bio-detection, biotechnology, chemical reactors and environmental monitors [1]. Usually, it is necessary to pump the fluid in one region of the device to another region in many different applications. Meanwhile, in order to promote chemical and biological reactions, full mixing of various reagents is needed. However, the characteristic length associated with micro-device is very small, which means that the diffusion effect alone cannot provide a sufficiently fast way for mixing. Furthermore, turbulence that may enhance mixing is not available because of flows with low Reynolds number (Re 1) in micro-devices [2]. The importance of the convective stirring mechanism relative to the molecular diffusion in a given mixing problem is measured by a nondimensional number called Peclet number. In microfluidic devices the Peclet number is very large, whose value is typical in the range of 103–105, so that concentration change is achieved mainly by the convection term, rather than by the diffusion term. Therefore, the stirring process must be utilized with artificial vortices created by appropriate mechanism in the stirring channel [3]. Magneto-hydrodynamics (MHD) can provide us with a relatively convenient way for pumping and mixing. Firstly, MHD
* Corresponding author. Tel.: +82 312012578. E-mail address: [email protected] (C.N. Kim).
pumps can produce a big force that is suitable for high volumetric pumping [4]. Recently, a number of researchers constructed MHD micro-pumps with silicon [5,6] and ceramic substrates [7], and demonstrated that these pumps are able to move liquids in microchannels. Secondly, a great deal of attention has been paid to the design of more efficient mixers in microfluidic systems [8–11], which triggers the formation of cross-sectional vortexes and velocity recirculation by exploiting the coupling of momentum transport with MHD effects. Gleeson et al. [10] examined the MHD mixing of two fluids in an annular micro-channel using asymptotic analysis and numerical simulation. Cerbelli et al. [11] developed a quantitative analysis of regimes in an annular MHD-driven micromixer proposed by Gleeson et al. [10]. Kabbani et al. [12] introduced a new approximate solution for MHD flows in micro-channels, which can be used for MHD micropumps and MHD microfluidic networks under DC or AC fields. Ho [13] provided an analytic model based on the steady-state, incompressible and fully developed laminar flow theory, and analyzed the MHD flow characteristics. His analytic solution was compared with experimental results with an excellent agreement. Meanwhile, the numerical solution method based on computational fluid dynamics has become an important and effective way to analyze MHD flows. Lim and Choi [14] numerically investigated the continuous MHD micro-pump with side-walled electrodes by using commercial CFD code CFD-ACE, and predicted the velocity profiles of the working fluid in the micro-channel under various operation currents and magnetic flux intensities. Lee et al. [15] proposed a chaotic mixing mechanism suitable for the micro-mixer of LOC, and
http://dx.doi.org/10.1016/j.jiec.2016.04.001 1226-086X/ß 2016 The Korean Society of Industrial and Engineering Chemistry. Published by Elsevier B.V. All rights reserved.
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Fig. 1. Duct geometry for one set of electrodes and the coordinate system.
Fig. 2. Schematic depiction of an MHD mixer and pump in the present study.
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investigated the fluid flow by numerical simulation and experiment. It was confirmed that the proposed micromixer was able to achieve high mixing efficiency. Also, Akbar et al. [16,17] showed that the magnetic field can control the flow of fluids with finite electric
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conductivity under a magnetic field, for example, in the flow of Eyring–Powell fluid and CNT suspended nanofluid, respectively. In the present study, we propose a new micro-device under a uniform applied magnetic field, where judicious application of
Fig. 3. Validation of the present numerical model.
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different electric potentials to different electrodes can create a Lorentz-force that can direct the liquid to move along a desired path for pumping without a need of separate pumps, and induce complex flows for mixing. The formation of the cross-sectional double velocity recirculations obtained in this study leads to the enhancement of mixing performance. Three cases with different combinations of input voltage of electrodes, together with the consideration of the subcases with one set, two-sets of electrodes in each case, are numerically examined for the comparison of the mixing capability and pumping performance. The obtained results including the information of velocity, current density, pressure and mass fraction of one fluid are visualized, with the values of mixing index, in detail. It can be noted that the present numerical model is validated against an existing numerical work with a good agreement. Problem formulation PBS (phosphate buffered saline) solution, which is an electrolyte, is generally used in LOC as a working fluid in real experiments for the analysis of proteins, and it is also used as medium for the fixation of protein. The properties of PBS solution are outlined in Table 1. A micro-device with rectangular cross-section proposed in this study is shown in Fig. 1, where the uniform magnetic field is applied in the y-direction with the intensity of 0.2 T. Eight electrodes, which are denoted by L1, L2, L3, L4, R1, R2, R3, and R4, respectively, are positioned on the left and right vertical walls of the conduit as one set of electrodes. The length of each electrode is 2 mm and the width is 1 mm. In the present study, firstly, one set of electrodes is installed for simultaneous MHD pumping and mixing. Fig. 2(a) shows the input voltages and the schematic depiction of the electrodes on the left vertical wall. The electric current (denoted by J M ! ) flows from L1 to L2 (in the x-direction) and from L4 to L3 (in the negative xdirection), which induces the z-directional Lorentz force (denoted by F M ! ) in the region between the electrodes L1 and L2 (the first pair on the left wall) and the negative z-directional Lorentz force in the region between the electrodes L3 and L4 (second pair of on the left wall). Fig. 2(b) shows the schematic depiction of the electrodes on the right vertical wall, where the situation similar to that in the left vertical wall is depicted. The positive and negative z-direction Lorentz force in the gap between the two electrodes of each pair may enhance the mixing performance. Meanwhile, as the average electric potential of the electrodes R1 and R2 is higher than that of the electrodes L1 and L2, the electric current (denoted by J P ! ) also can move in the negative zdirection, which yields the x-directional Lorentz force (denoted by F P ! ) responsible for the pumping of the fluids in the x-direction as shown in Fig. 2(c). Also, the average electric potential of the electrodes R3 and R4 is higher than that of the electrodes L3 and L4, which also works for pumping. Fig. 2(d) shows the locations of electrodes and different cross-sections. In the present study, three cases (Case 1, Case 2 and Case 3) with different combinations of the input voltage of electrodes are considered, as shown in Table 2. In each case, two subcases (Case 11 and Case 1-2 for Case 1, Case 2-1 and Case 2-2 for Case 2, Case 3-1
Table 2 Input voltage of electrodes (unit V).
Case 1 Case 2 Case 3
L1, R4
L2, R3
L3
L4
R1
R2
2.2 1.1 0.5
2.2 1.1 0.5
2.3 1.2 0.6
2.1 1.0 0.4
2.3 1.2 0.6
2.1 1.0 0.4
and Case 3-2 for Case 3) with one-set, two-sets of electrodes are considered, with an aim to investigate the difference in mixing performance. The ratio of the inertial effect to viscous effect in the flow field can be described by the Reynolds number, Re = rUL/m, where r is density of the fluid, U is the average flow velocity in the outlet, L is the characteristic length, and mf is the dynamic viscosity of the fluid. In the present study the Reynolds numbers in different cases are different from each other because of different outlet velocities (which will be shown later in Table 4). The ratio of the electromagnetic force to ffi viscous force is measured by Hartmann qffiffiffiffiffiffiffiffiffiffiffiffiffi number, M ¼ B0 L s f =mf , where B0 is the external magnetic field strength, sf is the electrical conductivity of the fluid, the Hartmann number in the present study is 0.267. The ratio of the characteristic time of diffusion effect to the characteristic time of convection effect is denoted by the Peclet number, whose values in different cases are also different from each other (which will also be shown in Table 4) because of different outlet velocities in the present study. Here, Pe = UL/D, where D is the diffusion coefficient. Governing equations The system of governing equations for the flow of an electricallyconducting fluid in a magnetic field includes the Navier–Stokes equation with Lorentz force, conservation of mass, conservation of charge and Ohm’s law, which can be referred to Ref. [5]:
ruru ¼ rp þ mf r2 u þ JB
(1)
ru ¼ 0
(2)
Table 1 Material property of PBS solution. PBS solution Density (kg/m3) Conductivity (S/m) Dynamic viscosity (kg/s m) Relative permeability Relative permittivity pH Diffusion coefficient (m2/s)
1000 1.5 6 104 1 72 7.4 1 1010
Fig. 4. Three-dimensional streamlines of current moving from L1 to L2, L4 to L3, R1 to R2 and R4 to R3 in Case 1-1.
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rJ ¼ 0
(3)
J ¼ s f ðrf þ uBÞ
(4)
where u, r, p, mf, sf, J, and f are the velocity vector, fluid density,pressure, dynamic viscosity of the fluid, electrical conductivity of the fluid, current density vector, and electric potential, respectively.
Fig. 5. Flow features in y–z plane at x = 0.00325 m in Case 1-1.
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In a homogenous fluid behaving under a diluted concentration and with an assumption of electrical neutrality for the electrolyte, the concentration field of species is governed by the following transport equation, which is given in Ref. [18]: urC ¼ Dr2 C
(5)
where C is the mass concentration and D is the diffusion coefficient.
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Fig. 5. (Continued ).
In order to figure out the mixing performance in different cases, the mixing index shown in the below (see Ref. [18]) can be considered. Mixing index ¼ 1
s s max
(6)
where s is the standard deviation of the concentration of the Fluid A in a monitoring plane. smax is the maximum standard deviation of the concentration of the Fluid A that can be found before the two fluids enter the conduit, whose value is 0.5. If the two fluids are ideally well mixed, then s 0, so that the mixing index would be 1.0. Boundary conditions At the inlet and the outlet, the pressure is considered to be zero. At the fluid–solid interface the no-slip and no-penetration conditions are applied. In the left portion of the inlet, Fluid A is supposed to come in, while in the right portion of the inlet Fluid B is ready to get in, as depicted in Fig. 1. The concentration of species A, CA, is imposed to be 1 at the inlet for Fluid A and to be 0 at the inlet for Fluid B. And, the boundary condition for the concentrations at the walls is @C/@n = 0, where n is the unit vector outer normal to the side wall. It is considered that there is no normal current at the inlet, outlet, and fluid–solid interface (except the electrodes), which yields the insulating boundary condition for the electric potential @f/@n = 0. Numerical method The present study uses a structured grid system chosen after a series of grid independence tests. In the current simulation, a grid system with 30,800 grids is used for one set of electrodes. Underrelaxation is employed in the iteration procedure for the coupled
governing equations. The second-order upwind scheme is used to discretize the convective terms and the central difference scheme for diffusion terms. A multi-grid accelerated Incomplete Lower Upper factorization technique [19,20] is employed for the solution method for discretized equations. The Rhie–Chow interpolation method [21] is used for the pressure–velocity coupling in CFX code. Validation Numerical simulations of three types of MHD flows generated in a micro-channel for pumping and/or mixing are performed to validate the accuracy and reliability of the present numerical model against the numerical results of La [22]. The duct geometry is shown in Fig. 3(a), and the applied magnetic field used in the simulations is applied in the z-direction with the intensity of 18.7 mT. Also, PBS solution is chosen as the working fluid. The applied voltage conditions for the three types of the MHD flow considered in La’s simulations [22] are as follows: Firstly, for the axial flow (for pumping only) the voltages applied to the electrodes are V w1 ¼ 1:8 V, V w2 ¼ 0 V, and there is no current between Vb1 and Vb2, that is, Jb1 = Jb2 = 0 A m2. Secondly, for the sinusoidal flow (for pumping and mixing) the voltages applied to the electrodes are V w1 ¼ V b1 ¼ 1:8 V and V w2 ¼ V b2 ¼ 0 V. Thirdly, for the multivortical flow (for pumping and mixing), the voltages applied to the electrodes are V w1 ¼ 0:9 V, Vb1 = 1.8 V and V w2 ¼ V b2 ¼ 0 V. The concentration distributions at the exit of the microchannel of La’s numerical results [22] and the present simulation results for (i) axial flow, (ii) sinusoidal flow, and (iii) multi-vortical flow are compared in Fig. 3(b) and (c), respectively. And, they demonstrate a good agreement. The standard deviations of the cross-sectional concentration values for the cases of the axial, sinusoidal, and multi-vortical flows in La’s numerical results are reported to be 0.343, 0.293, and 0.096, respectively. Meanwhile, in the present simulation results, these values are 0.367, 0.312, and 0.098, respectively. The differences between La’s numerical results and
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the present simulation results are 6.9%, 6.5% and 2.1%, respectively. With smaller differences shown above, the present numerical model is validated. Results and discussion Case 1 One set of electrodes (Case 1-1) Fig. 4 shows the three-dimensional streamlines of current moving from electrode L1 to L2, from L4 to L3, from R1 to R2 and from
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R4 to R3 in Case 1-1, where the curved paths of the current flows are shown, as expected (see Fig. 2(b)). Fig. 5(a) shows the Lorentz vectors in the y–z plane at x = 0.00325 m (inside the gap between L1 and L2), where the Lorentz force is induced in the positive zdirection, associated with the direction of applied magnetic field and the current therein ðJM ! ~ BÞ, as mentioned before. The Lorentz force near the left and right vertical walls in the middle fluid regions is higher than that in the other fluid region, due to the higher current density therein. The positive z-directional Lorentz force induces the highest pressure in the fluid region near the right vertical wall in the middle fluid region, while the lowest pressurein
Fig. 6. Flow features in y–z plane at x = 0.00875 m in Case 1-1.
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Fig. 7. Flow features in x–y plane at z = 0.0005 m in Case 1-1.
Fig. 8. Flow features in x–y plane at z = 0.0015 m in Case 1-1.
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the fluid region near the left vertical wall in the middle fluid region (see Fig. 5(b)). The distribution of the plane velocity in the same plane is shown in Fig. 5(c). Because of the Lorentz force, in the middle part of the fluid region, the fluid generally moves from the left to the right, and reaches the fluid–right vertical wall interface, then bifurcates to the right-top part and right-bottom part of the fluid region, which is based on the direction of the gradient of the pressure. Therefore, two large-scaled velocity circulations are generated in this plane. Meanwhile, two smaller vortexes are also induced near the top and bottom walls. This kind of complex flow can enhance the mixing performance. The potential difference between the group of electrodes L1, L2 and the group of electrodes R1, R2 induces the electric current to move from the right vertical wall to the left vertical wall (J P ! ), as shown in Fig. 5(d), which induces the x-direction Lorentz force ðJ P ! BÞ ! used for pumping. Fig. 6 shows the flow features in the y–z plane at x = 0.00875 m (inside the gap between L3 and L4). Based on the direction of the current around L3, L4, R3 and R4 (see in Fig. 4), the Lorentz force heads leftward in this plane (Fig. 6(a)), whose direction is opposite to that in the y–z plane at x = 0.00325 m (Fig. 5(a)). The highest pressure is induced in the fluid region near the left vertical wall in the middle fluid region, while the lowest pressure is induced in the fluid region near the right vertical wall in the middle fluid region (Fig. 6(b)). With the influence of the combination of Lorentz force and pressure gradient, two large-scaled velocity circulations are also induced in this plane, with the directions of the velocity being in the way opposite to those in Fig. 5(c). Moreover, the plane current ðJP ! Þ also moves to the left, which works for pumping. Fig. 7 shows the flow features in the x–y plane at z = 0.0005 m. Here, as shown in Fig. 7(a), the lowest pressure is observed in the fluid region around L1 and L2, while the highest pressure is seen in the fluid region around L3 and L4 (see Figs. 5(a) and 6(b) also). Fig. 7(b) shows the distribution of the plane velocity in this plane. With the pumping effect, the fluid is sucked at the inlet and the flow generally moves from the inlet to the outlet. Meanwhile, because of the large pressure gradient in the fluid regions around the electrodes L1 and L2 and around the electrodes L3 and L4, the flow pattern is complicated with eight velocity circulations. The direction of four velocity circulations around electrodes L1 and L2 are opposite to the directions of those around L3 and L4. These kinds of complex flow also can be beneficial to mixing. Fig. 8 shows the flow features in the x–y plane at z = 0.0015 m. Here, the highest pressure is observed in the fluid region around R1 and R2, while the lowest pressure is seen in the fluid region around R3 and R4 (Fig. 8(a)). Fig. 8(b) shows the velocity distribution in this plane, with four velocity circulations around R1 and R2 and four velocity circulations around R3 and R4, with the swirling directions opposite to those in Fig. 7(b). Fig. 9(a) showsthe mass fraction of Fluid A near the inlet (in the y–z plane at x = 0.0001 m), where the two fluids are not mixed yet, practically. So, the mass fraction of Fluid A in the left fluid region is near 1, and that in the right fluid region is near zero. Fig. 9(b) shows the mass fraction of Fluid A in the monitoring plane located at x = 0.014 m. The cross-section for the mass-fraction display (monitoring plane) is chosen to be located at 4 mm downstream of the trailing edge of the latest electrode in the mixer. The hydraulic diameter of the conduit is Dh = 4A/P = 8/3 mm = 2.667 mm. Therefore, the axial distance (4 mm) between the trailing edge of the latest electrode and the cross-section for the mass-fraction display corresponds to 1.5Dh. If the axial distance for the mass-fraction display is too short, then in the cross-section for the mass-fraction display the axial velocity is not well distributed with some notable magnitude of the velocity components perpendicular to the main flow. Therefore, in the above situation it is highly possible for the value of the mass-fraction of a fluid in the cross-section for the massfraction display not to be well distributed. Also, if the axial distance
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Fig. 9. Mass fractions of Fluid A in the different y–z plane in Case 1-1.
for the mass-fraction display is too long, then the diffusion effect (caused by molecular diffusion) in the region ranging from the trailing edge of the latest electrode to the cross-section for the massfraction display can be meaningful. In Fig. 9(b) the two fluids are mixed in the inner part of fluid region. However, in the fluid region near the corners, the two fluids are not mixed so well because of the relatively low velocity therein and no-slip condition. Fig. 10 shows the axial velocity distribution along the central line of the duct. The axial velocity starts to increase right after the entrance, and after some velocity fluctuations, the axial velocity reaches a value of around 0.0032 m/s in the fluid region near the exit. Meanwhile, the Reynolds number is 14.24 based on the average velocity in the monitoring plane and the hydraulic diameter of the duct.
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Fig. 10. Axial velocity distribution along the central line of the duct in Case 1-1.
Also, a case with two sets of electrodes is also considered in the present study (donated as Case 1-2). Fig. 11 shows the arrangement of the electrodes in Case 1-2. The distance between adjacent electrode sets is 3 mm in multiple-set subcases, which is obtained by considering the feasibility of the electrical influence of one set on another set. The cross-sectional position for the mass-fraction display in Case 1-2 can also be seen in Fig. 11, which is also located 4 mm downstream from the trailing edge of the latest electrode. Fig. 12 shows the duct geometry and the three-dimensional streamlines of the current moving from L1 to L2, from L4 to L3, from R1 to R2 and from R4 to R3, which shows the curved paths of the current flows similar to those in Case 1-1. Fig. 13 shows the mass fraction of Fluid A in the y–z plane at x = 0.025 m in Case 1-2. Witness that in the inner fluid region of the conduit the mass fraction in Case 1-2 appears almost uniform, indicating that the two fluids are reasonably well-mixed. In the current section, the effect of the electrode voltage on the mixing performance is considered. In Case 2 and Case 3, as shown in Table 2, a relatively lower potential difference is considered,
compared to that in Case 1. The potential difference between the electrodes with highest and lowest potential for each set in Case 1 is 4.4 V, while the potential difference in Case 2 is 2.2 V, and the potential difference in Case 3 is 1.0 V. As observed before, the potential differences between L1 and L2 and between R1 and R2, the potential difference between L3 and L4 and between R3 and R4 work mostly for mixing. This means that Case 2 and Case 3 have a mixing capability quite smaller than that in Case 1. However, the potential difference for pumping in each case is the same. More precisely, the potential differences between electrodes L1 and R1, L2 and R2, L3 and R3, L4 and R4 are always 0.1 V in each case. This means that Case 2 and Case 3 may have a pumping capability similar to that in Case 1 as mentioned before. Two subcases (Case 2-1, Case 2-2) in Case 2 and two subcases (Case 3-1, Case 3-2) in Case 3 are also designed to investigate the effect of the number of electrode set on the mixing/pumping performance. Fig. 14 showsthe axial velocity distributions along the central lines for the two subcases in Case 1. In different subcases for each case, the axial velocity distribution associated with the configuration
Fig. 11. Locations of each electrode and the position (in blue) for displaying the mixing performance in Case 1-2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
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Fig. 13. Mass fraction of Fluid A in the y–z plane at x = 0.025 m in Case 1-2. Fig. 12. Three-dimensional current streamlines from L1 to L2, L4 to L3, R1 to R2 and R4 to R3 in Case 1-2.
of the electrodes shows the same tendency. More precisely, in Case 12 the velocity profile around the second set of electrodes is almost the same as that around the first set of electrodes. Since each set of electrodes has pumping capability, the exit velocity in Case 1-2, with
increased flow resistance associated with increased pumping capability, is around 1.2 times the velocities in Case 1-1. Fig. 15 shows the contours of the mass fraction of Fluid A in the two subcases in Case 2. It is obvious that the mixing performance increases with an increase in the number of electrode set.
Fig. 14. Axial velocity distributions along the central lines for two subcases in Case 1.
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Fig. 16. Mass fractions of Fluid A in the different y–z plane near the exit of Case 3.
Fig. 15. Mass fractions of Fluid A in the different y–z plane near the exit of Case 2.
However, the mixing performance in Case 2 is lower than that in Case 1, which is because of relatively weak cross-sectional velocity circulation in association with relatively weak Lorentz force for mixing. Fig. 16 shows the contours of the mass fraction of Fluid A in the two subcases in Case 3, presenting poor mixing. Also, the mixing performance increases with an increase in the number of electrode set. Fig. 17 shows the axial velocity distributions along the central lines in Case 1-1, Case 2-1 and Case 3-1. The inlet and outlet velocities in each case are very similar, since the pumping capabilities of Case 1, Case 2 and Case 3 are almost the same. The velocity fluctuation in Case 1-1 is the strongest in line with the fact that the absolute values of the electric potential of the electrodes are the highest in Case 1-1.
Table 3 shows the mixing indexes in the subcases. As mentioned before, the mixing index increases with an increase in the number of the electrode set. It can also be seen that the mixing index in Case 1-2 is the highest, whose value is 0.9393, showing that the proposed device is able to achieve high mixing efficiency. Also, the mixing indexes in Case 1 are generally higher than those of Case 2 and Case 3, as explained before.
Table 3 Mixing index for Case 1, Case 2 and Case 3. Case 1
s Mixing index
Case 2
Case 3
Case 1-1
Case 1-2
Case 2-1
Case 2-2
Case 3-1
Case 3-2
0.2462 0.7533
0.0607 0.9393
0.316 0.684
0.228 0.772
0.538 0.462
0.449 0.551
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Fig. 17. Axial velocity distributions along the central lines in Case 1-1, Case 2-1 and Case 3-1.
Table 4 Mass flow rate, Reynolds number and Peclet number for Case 1, Case 2 and Case 3. Case 1
Qm (kg/s) Re Pe
Case 2
Case 3
Case 1-1
Case 1-2
Case 2-1
Case 2-2
Case 3-1
Case 3-2
1.291 105 14.24 1.068 104
1.534 105 16.91 1.268 104
1.339 105 14.69 1.101 104
1.583 105 17.44 1.3083 104
1.364 105 14.99 1.125 104
1.605 105 17.62 1.322 104
Table 4 shows the mass flows in subcases of Case 1, Case 2 and Case 3. Cases with two sets of electrodes induce a mass flow rate higher than those with one set of electrode. More precisely, the mass flow rate of Case 1-2 is 1.188 times of that of Case 1-1. The mass flow rate of Case 2-2 is 1.182 times of that of Case 2-1. And the mass flow rate of Case 3-2 is 1.177 times of that of Case 3-1. The Reynolds number and Peclet number in each case can also be seen in Table 4. Since the Reynolds number and Peclet number are in association with the average velocity, the Reynolds number and Peclet number in each case show the same tendency as the mass flow rate.
Conclusion This study numerically investigates the pumping and mixing performance in a newly designed magneto-hydrodynamic mircomixer with pumping capability. Commercial software CFX, based on a finite-volume method, is adopted to solve the governing equations. By judiciously applying different electric potentials to different electrode pairs, the Lorentz-force is obtained that can direct the liquid to be pumped and mixed simultaneously.
In this study, the mixing index is used in order to estimate the mixing performance in different cases. The effect of the magnitude of the voltages imposed on the electrodes on the mixing performance is investigated. For the case with relatively higher potential difference, the mixing index is relative higher, which means a higher mixing performance. Meanwhile, the subcases with one- and two-sets of electrodes are considered to investigate the difference in the mixing performance. The mixing index increases with an increase in the number of the electrode set used in each case, which means that the use of multiple sets of electrodes can enhance the mixing performance. In summary, the present study proposes a new magnetohydrodynamic mixer with pumping capability, and the present numerical analysis shows that the proposed device can have high mixing performance. Acknowledgements This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 20110022679).
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