Journal of Mechanics http://journals.cambridge.org/JOM Additional services for Journal of Mechanics: Email alerts: Click here Subscriptions: Click here Commercial reprints: Click here Terms of use : Click here
Numerical Investigation on the Effect of the Size And Number of Stages on the Tesla Microvalve Efficiency K. Mohammadzadeh, E. M. Kolahdouz, E. Shirani and M. B. Shafii Journal of Mechanics / FirstView Article / May 2013, pp 1 8 DOI: 10.1017/jmech.2013.29, Published online: 24 May 2013
Link to this article: http://journals.cambridge.org/abstract_S1727719113000294 How to cite this article: K. Mohammadzadeh, E. M. Kolahdouz, E. Shirani and M. B. Shafii Numerical Investigation on the Effect of the Size And Number of Stages on the Tesla Microvalve Efficiency. Journal of Mechanics, Available on CJO 2013 doi:10.1017/ jmech.2013.29 Request Permissions : Click here
Downloaded from http://journals.cambridge.org/JOM, IP address: 2.180.38.144 on 29 May 2013
NUMERICAL INVESTIGATION ON THE EFFECT OFTHE SIZE ANDNUMBER OF STAGES ON THE TESLA MICROVALVE EFFICIENCY K. Mohammadzadeh
E. M. Kolahdouz
Foolad Institute of Technology Foolad Shahr Esfahan, Iran
University at Buffalo Department of Mechanical Engineering Buffalo, U.S.A
E. Shirani ∗
M. B. Shafii
Foolad Institute of Technology Foolad Shahr Esfahan, Iran
Department of Mechanical Engineering Sharif University of Technology Tehran, Iran ABSTRACT
In the present study, the effect of the number of stages of Tesla Micro-Valve (TMV), as well as the dependency of Reynolds number, Re, on the valve performance has been analyzed. For this purpose, different layouts include one to four-stage with different sizes are investigated numerically. The main criterion for evaluation of valves performance is diodicity, Di. Unsteady and steady flow in valve have been simulated and compared. It is shown that although there are some difference but the trend is similar for both responses. Finally, 2-D and steady state computations of the fluid flow have been utilized that reveal a strong dependence of Di on Re and pressure drop, ΔP. The results showed that the maximum Di of the two-stage microvalve is approximately 1.45 times of that of one-stage. Additional stages increase the complexity, and they do not change Di appreciably. It is concluded that two-stage layout of Tesla type valve is the best option. Also, the two-stage valve performance for three different sizes is compared with Nozzle-Diffuser type Micro-Valve (NDMV). Comparisons, which are performed based on calculation Di in applicable range of Re, showed that Di as a function of Re is independent of the valve size. Also, the superiority of the Tesla type valve at higher Re and its weakness at lower Re is observed. Keywords: Micropump, Tesla microvalve, Number of stages, Diodicity. 1.
INTRODUCTION
No-Moving-Part (NMP) valves are used in valve-less reciprocating micropumps to convert the nondirectional flow to directional flow. In NMP valves, instead of external mechanism, fluid flow behavior is responsible for the variations of pressure drop, ΔP, between forward and reverse flows. Two main types of these fixed microvalves are microdiffusers and Tesla Micro-Valves, TMVs. Wear and fatigue in these valves are eliminated since they have no moving parts and the risk of clogging is reduced, too. Complex geometry of TMV can make it possible to show different resistance in forward and backward directions like Nozzle-Diffuser Micro-Valves, NDMVs. In TMV, the main diodicity, Di, is due to the difference between conjunction angles in two parts of the valve that causes larger resistance in backward direction compared to that of forward direction (Fig. 1). Y-junction and T-junction locations are specified in Fig. 1. *
(a) Backward direction
Fig. 1
(b) Forward direction
TMV and flow configurations
Numerous experimental and numerical studies have been conducted on valveless micropumps [1-6]. The idea of using NMP microvalves was pioneered by Stemme and Stemme [7]. They incorporated a nozzlediffuser piezoelectric micropump. Foster et al. [8], introduced TMVs, which has higher efficiency in comparison to NDMV, in construction of valveless micro-
Corresponding author (
[email protected])
Journal of Mechanics DOI : 10.1017/jmech.2013.29 Copyright © 2013 The Society of Theoretical and Applied Mechanics, R.O.C.
1
pumps. Diodicity capability and simplicity in fabrication are two effective parameters in valve selection. Many studies are performed regarding optimization of TMV geometry. Tesla [9] designed a set of these valvular conduits in sequence and multistage and used them in a channel for pumping gases. However, he did not optimize the number of stages and he claimed that adding stages increase Di, but according to the complex relation between valve efficiency and number of stages, the design should be conducted based on considering type, space limitation and system cost. Reed and Fla [10] were able to eliminate some disadvantages of Tesla type valve, including the veer angle of the main channel, to achieve a more optimal geometry. They experimentally proved that for a conventional Tesla valve, increasing the number of valves from 1 to 2, Di becomes double, and additional stages increases the complexity and ΔP, but they do not change Di enormously. Eventually, Forster et al. [11], for the first time, proposed the use of Tesla valve in microscale. In his patent, he used these valves in piezoelectric micropumps, and compared its efficiency with NDMVs designed by Stemme and Stemme [12]. Also, Forster and Williams [13] proposed a Tesser valve which is a combination of NDMVs and TMVs. By the means of parametric design, the performance of Tesser valve was compared with NDMVs, and TMVs that proposed until that time for 2-D steady state flow using FLUENT software; however, they did not prove superiority of new valves in a spectrum of flow regime to previous valves. Their study showed that the Tesser valve was superior at low Reynolds number (Re), 10 < Re < 100, and the Tesla-type valve superior at high Re, 100 < Re < 2000. Liao [14] proposed microvalve with 135° (instead of 45°) veer of the main channel, and called it T135. He considered this as other possible option of TMV. He proved that T135 has higher Di in comparison to other TMVs. Gamboa et al. [15] used numerical method and dynamic models for analysis of TMV and new values were determined for geometrical parameters. They experimentally proved that the optimized model has higher Di, especially at higher Re. Their experimental data demonstrated the ability of 2-D modeling to improve valve performance. Kolahdouz [16] recently proposed a TMV that called it MT135 with a better efficiency for a Di in comparison to previous models by numerical analysis of flow in valve as shown in Fig. 1. Also, he used the two-stage layout as an ideal case in structure of piezoelectric micropump. The simulation results showed that the Tesla micropump is more efficient than conventional nozzle-diffuser micropump in terms of both maximum pressure head and flow rate. The number of stages is one of the geometrical parameters in Tesla valve that effects on Di and consequently micropump performance, as mentioned in the previous experimental works [9,10]. The empirical works cannot obtain the relation between valve efficiency and number of stages by experiments accurately and thus only some global results such as maximum 2
flow rate or ΔP with some degree of accuracy are presented. In the present study, numerical simulation of fluid flow in TMV has been considered and the effect of the number of stages on valve performance has been studied. According to the knowledge of authors, this case has not been investigated so far. In this paper the performance of TMV in different sizes are compared with a NDMV. In this regard, Di versus nominal ΔP and Re are extracted in the operational range for current micropump designs. In this paper, the performance principle of valveless micropump is introduced, then the governing equations and the problem formulation and numerical solution of the fluid flow in mirovalves are given. Finally, the description and detailed discussion of the numerical results and conclusions are presented. 2.
METHODS OF SOLUTION
2.1 Performance Principle of Valveless Micropump In this section, applying NDMV in structure of a valveless diaphragm micropump is described. Obviously the performance of the Tesla valve is similar to the current case. The micropumps performance can be divided to two half-cycles of supply and discharge. Output velocity of the nozzle is extremely high and is usually assumed as a free jet. So, larger ΔP is created compared to the diffuser one. Therefore, a diffuser/nozzle element has a lower flow rate in the nozzle direction than in the diffuser direction for the same ΔP. During the supply mode, as the pump diaphragm deflects upward, chamber volume increased and two identical ΔP are provided on both sides of valves and causes the fluid motion through valves into the chamber. In this mode, a larger volume flow rate is transported through the inlet into the chamber than through the output. In discharge mode, as the pump diaphragm deflects downward, chamber volume decreased and causes the fluid motion through valves out of the chamber. Here, larger volume flow rate is transported through the outlet of the chamber than through the input. Therefore, during a complete pump cycle a net volume is pumped from the inlet side to the outlet side. 2.2
Formulation and Governing Equations
An estimate of the efficiency of the NMP valves which is related to the net flow rate, can be obtained by using pressure loss data of fluid flow inside valves. Obviously, a microvalve has the highest efficiency when Di is maximum. Di in the form of nondimensional pressure is expected to be a function of Re. In this study, the variation of Di with Re and ΔP is investigated. The figure of merit that characterizes the valve ability to pass flow in the forward direction while inhibiting the flow in the reverse direction is Di [8].
Di = (ΔPBackward / ΔPForward )Q = (ΔPNozzle / ΔPDiffuser )Q
(1)
in which the ratio of ΔP is for identical volume flow rates in both directions. Journal of Mechanics
ΔP that creates Di is due to inertial and viscous forces. The inertial loss is proportional to the square of the velocity while the viscous forces are proportional to the velocity, and they become significant in laminar flow. According to dimensions of studied microvalves, it can be concluded that the characteristic lengths are much larger than mean free path of water which confirms the continuum regime in microchannels. Therefore, Navier-Stokes equations are used for modeling and no-slip boundary conditions imposed on the walls. The range of Re is 10 < Re < 300 and the working fluid is assumed to be water. The similar trend of results for steady and unsteady state solutions allow us to use relatively inexpensive steady-state simulations instead of computationallyintensive transient simulations [17]. To show this matter and also to simulate the flow more accurately, two cases for the flow in valve are considered here. In Case 1, the flow is assumed steady state and in the second case, transient flow with sinusoidal inlet velocity is simulated. The flow is assumed to be laminar and incompressible and the fluid is Newtonian. Based on these assumptions and neglecting the body forces, conservation equations for mass and momentum are:
∂U j / ∂x j = 0
(2)
ρ f (∂U i / ∂t ) + U j ∂U i / ∂x j = −∂P / ∂xi + ∂ ( μ f (∂U i / ∂x j + ∂U j / ∂xi ) ) / ∂x j
(3)
where Ui denotes velocity in ith-direction, P is pressure. x demands for direction. μf and ρf are viscosity and density of the fluid, respectively. Since 2-D simulations have been proved to be capable of predicting the flow characteristic in the previous works [8,13 to 16], 2-D simulations is used in the present study to evaluate the valve performance 2.3
Numerical Solution of Flow in Microvalves
2.3.1 Microvalve Geometries Figure 2 shows the different layouts of TMV with the base geometry of MT135. For all layouts, the channel width is 120μm. For comparing the performance of MT135 with the conventional NDMV, the NDMV fabricated by Yamahata [18], is considered (Fig. 3). He tested the performance of a single diffuser in a separate device and compared the experimental data with the numerical 2-D simulation. The dimensions are L = 2.3 mm, W1 = 100μm, W2 = 500μm and Ө = 9.5°. The microvalve depth is 0.5mm. 2.3.2 Numerical Method The structured and unstructured grids are used to make the models. 2-D elements that are used are quadrilateral and trilateral. The grid density is refined near walls and at channel connections, where large velocity gradients are expected. Moreover, adaptation of mesh by solver in areas with high velocity and pressure gradient are used. Discretization is based on the finite volume technique and the SIMPLE and PISO schemes are used for Journal of Mechanics
Fig. 2 Different layout used for MT135 microvalve: a. One-stage, b. Two-stage, c. Three-stage, d. Four-stage
Fig. 3
Geometries of NDMV fabricated by [18]
velocity-pressure coupling algorithm. The discretization method used for momentum equations, is first order upwind scheme. Also, a central difference scheme is used to discretize the diffusion terms. Second order implicit scheme for time derivative is used for transient case. Grid independent solution as well as the proper time step size has been obtained. For different cases, different inlet boundary conditions have been used. In Case 1, Pressure boundaries are used at both inlet and outlet of the valve. First, backward flow is evaluated for different pressure differences. After solving flow field, flow rate of backward flow is considered as initial condition for forward flow and the outlet pressure is set to zero. For the Case 2, the oscillating velocity is used as the inlet boundary condition and atmospheric pressure is imposed at the outlet. The velocity amplitude is Vmax = (π/2) × V and time varying inlet sinusoidal velocity is based on the diaphragm displacement and is V(t) = Vmax Sin(2πf (τ/4−t)) where f is frequency in terms of Hz, V is average velocity, is and τ is the time period for full oscillating of the diaphragm (τ = 1/f). In this Case, four different inlet velocities for V (0.085, 0.165, 0.25 and 0.35m/s) that correspond to different Re (Re = 52, 100, 152 and 213) are used. The time average of Di in microvalve is used as a criterion to determine the optimum step size so that any corresponding physical changes in the flow field can be mentioned. For evaluting the average Di ( Di ), the 3
flow in microvalve must be converged to periodic manner (quasi-steady condition), i.e. the integral of pressure curve versus time does not change at inlet and outlet of the valve. Then Di can be evaluated from the last cycle by dividing the average ΔP in reversed flow (when V(t) < 0) to the average ΔP in forward flow (when V(t) > 0).
3.
RESULTS AND DISCUSSION
3.1 Comparing the Steady and Unsteady Simulations The flow in microvalve becomes periodic after about 6 cycles and Di is then evaluated. Figure 4 shows the effect of time step on the calculation of Di in onestage microvalve at Re = 150 and f = 100Hz. This figure shows the same Di is obtained for both time steps of 0.01τ and 0.02τ. Therefore time step 0.02τ is used. Figure 5 shows the results for Cases 1 and 2. The maximum difference of the results is about nine percent and so transient effects are not negligible. But, for the Re higher than 100, the trend of the results for steady state and transient flow are similar. Also, for better understanding the transient effects on valve performance, the results presented in Fig. 5 are reported in Table 1. As can be seen, in the transient flow, the ΔP for both forward and backward direction is higher compared to that of steady state. Moreover, ΔP is higher in backward flow compared to that of forward flow. 3.2
Reynolds number
Steady state response ΔPBackward
ΔPForward
Unsteady state response Di
ΔPBackward ΔPForward
Di
Re = 52
684.07
664.55
1.03
783.29
741.90
1.06
Re = 100
1875.06
1717.45
1.09
2275.42
1898.30
1.20
Re = 152
3841.95
3166.54
1.21
4824.36
3565.27
1.35
Re = 213
6500.20
4786.71
1.36
7917.93
5346.65
1.48
Fig. 4
Effect of time step size on the calculation of Di in One-stage microvalve at Re = 150 and f = 100Hz
Velocity and Pressure Field
The velocity field for different layouts of MT135 is shown in Figs. 6 and 7 for forward and reverse flows with the nominal pressure drop of 5000 Pa (Npd = 5000 Pa), respectively. For both forward and backward directions, flow separation and laminar jets occur in three locations: Y-junction, T-junction and at the valve exit. These figures reveal that by increasing the number of stages, the pressure loss increases in backward flow and it leads to higher Di. This suggests that the losses generated by adding more stages may be important for maximizing Di. In Figs. 8 and 9, the pressure contours are plotted for different layout of MT135for forward and backward directions, respectively. The pressure field showed that a large ΔP occurs exactly after Y-junction in reverse-flow, but not in forward flow. 3.3 The Effect of the Number of Stages on the Performance of Tesla Microvalve To validate the present model, the numerical results for MT135 microvalve are compared with the numerical results of Kolahdouz [16]. As can be seen in Fig. 10, there was a good agreement between them. Different arrangements include one, two, three and fourstage are investigated numerically. Numerical implementation is the same as the one in section 2.3.2. Di 4
Table 1 Pressure loss in forward and backward flow for steady and unsteady simulation for different Reynolds number
Fig. 5
Di of one-stage valve versus Re for Case 1 and 2
versus Re for different arrangements has been shown in Fig. 11. The results are also reported in Table 2. Re is based on mean velocity in microchannel and the width of the microchannel. Figure 11 shows that Di for all arrangements is increased by Re and it becomes prominent at higher Re due to the augment the inertial losses. The main pressure loss in TMV is due to inertial losses. It is observed that Di is enhanced by increasing the number of stages for a given Re. But it is not the case for Re < Table 2 reveals that maximum Di enhancement 50. Journal of Mechanics
Fig. 6
Velocity vectors (m/s) for MT135 in forward direction in nominal ΔP 5000Pa for different layouts: upper: vector plot, lower: velocity contour
Fig. 7
Velocity vectors (m/s) for MT135 in backward direction in nominal ΔP 5000Pa for different layouts: upper: vector plot, lower: velocity contour
Fig. 8
Fig. 9 Table 2
Pressure contours (Pa) for MT135 in forward direction in nominal ΔP 5000Pa
Pressure contours (Pa) for MT135 in backward direction in nominal ΔP 5000Pa Diodicity parameter for various Reynolds number and different number of stages
Layout
Re = 10
Re = 20
Re = 50
Re = 70
Re = 100
Re = 200
One-stage
1.001
1.005
1.024
1.050
1.104
1.312
Two-stage
1.002
1.006
1.032
1.080
1.205
1.826
Three-stage
1.002
1.007
1.041
1.107
1.302
2.275
Four-stage
1.002
1.007
1.048
1.125
1.364
2.643
Journal of Mechanics
5
Fig. 10
Di as a function of ΔP
Fig. 12 Di of different arrangements of MT135 microvalve versus nominal ΔP Table 3 Diodicity parameter for various nominal pressure drop (Npd) and different number of stages Layout
Npd = 0Pa
Npd = 5000Pa
Npd = 10000Pa
Npd = 50000Pa
Npd = 70000Pa
One-stage
1
1.242
1.376
1.746
1.898
Two-stage
1
1.316
1.659
2.547
2.759
Three-stage
1
1.317
1.660
2.868
3.148
Four-stage
1
1.290
1.660
3.047
3.393
head of the pump in one period. Therefore, it can be concluded that the two-stage valve is a better option as demonstrated by experiments [4]. 3.4 Comparing the Performance of NDMVs and TMVS Fig. 11 Di of different arrangements of MT135 microvalve versus Re decreases by increasing the number of stages. To determine the appropriate number of stages, the Di is obtained for different nominal ΔP and is shown in Fig. 12. Table 3 also provides the results. Figure 12 shows that Di is increased with nominal ΔP for all arrangements and these changes are slower for higher nominal ΔP. Di is enhanced by increasing the number of stages in a specified nominal ΔP. Table 3 shows that for the same flow condition, the maximum Di of the two-stage microvalve is approximately 1.45 times of that of one-stage. However, additional stages (more than two-stages) increase the complexity and they do not change Di notably. Two-stage arrangement of TMV has the following advantages: Flow direction becomes parallel in outlet and inlet of valve, easing fabrication process and its maintenance and application in different working conditions. Increasing the number of stages results in difficulty of fabrications and decreases the absolute Journal of Mechanics
To compare and choose the most efficient microvalve for different flow conditions, Di is considered and compared. Since, the use of NDMVs has been more industrialized, their comparison with the Tesla valves can be revealed the capability of Tesla valves. In this section, the validity of the numerical results that previously presented at Fig. 10 is examined by comparing them with other available numerical and experimental results. A specified pressure at the inlet and outlet are imposed for the simulations in the diffuser direction and the conditions for the simulation in the nozzle direction are inverted. A no-slip condition for the fluid in contact with the walls is imposed. Figure 13 shows the flow rate in both the diffuser and nozzle direction, compared with available numerical simulation and experimental data. The general trend in variation of flow rate versus pressure is similar for all cases. To compare the two NMP valves, first the TMV width is set to be equal to the nozzle throat, 100μm, then it is equal to the averaged diameter, 300μm, and finally, 500μm. Figures 14 and 15 illustrate the pressure-flow rate curve for NMP valves for forward and 6
Fig. 13 Flow rate in nozzle and diffuser as a function of ΔP
Fig. 14 Flow rate as a function of ΔP for NMP valves in forward direction
Fig. 15 Flow rate as a function of ΔP for NMP valves in backward direction Journal of Mechanics
backward direction, respectively. According to these figures the following conclusions can be drawn: 1. ΔP is increased with decreasing channel width and for a given pressure drop, the flow rate increases with channel width. 2. The overall flow rate for TMVs with widths of 300μm and 500μm, unlike for 100μm, incur the lower ΔP than NDMV and thus passes more flow rate for a given ΔP. 3. By increasing channel width, difference between flow rates becomes prominent. The similarity of the characteristic curves, pressure-flow rate diagram, suggests that the TMV with a width of 300μm and NDMV are equally good. Finally, after calculating and plotting the pressureflow rate curve for microvalves, Di can be easily extracted for each of the four microvalves and analyzed to compare the performance of these microvalves. Figures 16 and 17 show Di versus nominal ΔP and Re, respectively for three TMV in terms of width channel in comparison with the NDMV. Re is calculated based on W1 for NDMV. Figure 16 shows Di of TMV is increased with channel width for given nominal ΔP and these changes are slower at higher nominal ΔP. It can be concluded from Fig. 17 that Di in the form of non-dimensional ΔP as a function of Re is independent of valve size. The NDMV curve crosses the TMV curve approximately in Re = 200 and efficiency of four valves is similar and Di at this point is almost equal to 1.55. For Re < 200, NDMV is more efficient than TMV and for Re > 200, TMV is superior. In both Figs. 16 and 17, it is observed that TMV has no extermum point and has upward trend. But the results for NDMV has one maximum around Re = 130 and at this point Di is approximately 1.65 and efficiency of valve is maximum. For Re < 130, Di enhances with Re, otherwise it decreases. The decrease in Di of NDMV with increasing Re is due to flow separation in the diffuser direction.
Fig. 16 Di as a function of nominal ΔP for TMVs with widths of 100, 300 and 500μm and NDMV 7
4.
5.
6.
7. 8. Fig. 17 Di as a function of Re for TMVs with widths of 100, 300 and 500μm and NDMV 4.
CONCLUSIONS
Numerical simulation of fluid flow inside a TMV (MT135), that have different geometry in comparison to conventional NDMV, is performed for different arrangements of these valves in terms of number of stages. First, unsteady response of the valve studied and concluded that the trend is similar to the steady case. The good agreement between results of numerical simulation of some cases with available numerical results confirmed the validity of used numerical method. So, the method used in this study can be a valuable method for proper design of NMP microvalves. Performance curves, Di versus Re and nominal ΔP, for the NDMV and different layouts of MT135 valve in the applicable range for current micropump designs. Numerical results showed that a two-stage layout of theMT135valve is an acceptable case in terms of increase in Di. This arrangement besides being compact, has the adaptability of the various functions. Then, the performance of proposed valve for three different size is compared with a NDMV. The comparisons indicate the superiority of Tesla valve for higher Re and its weakness at lower Re than NDMV. The efficiency of valves is similar at Re = 200. There is a maximum Di in the applied range of Re for NDMV that occurred at Re = 130, but for TMV no maximum point is found. ΔP and Di of TMV increase with increasing the channel width. It is found that Di is independent of the valve size.
9. 10. 11.
12. 13.
14.
15.
16.
17.
REFERENCES 1. Laser, D. J. and Santiago, J. G., “A Review of Micropumps,” Journal of Micromechanics and Microengineering, 14, pp. 35−64 (2004). 2. Woias, P., “Micropumps—Past, Progress and Future Prospects,” Sensors and Actuators B: Chemical, 105, pp. 28−38 (2005). 3. Nisar, A., Afzulpurkar, N., Mahaisavariya, B. and Tuantranont, A., “MEMS-Based Micropumps in Drug Delivery and Biomedical Applications,” 8
18.
Sensors and Actuators B: Chemical, 130, pp. 917−942 (2008). Iverson, B. D. and Garimella, S. V., “Recent Advances in Microscale Pumping Technologies: A Review and Evaluation,” Microfluidics and Nanofluidics, 5, pp. 145−174 (2008). Nabavi, M., “Steady and Unsteady Flow Analysis in Microdiffusers and Micropumps: A Critical Review,” Microfluidics and Nanofluidics, 7, pp. 599−619 (2009). Amirouche, F., Zhou, Y. and Johnson, T., “Current Micropump Technologies and Their Biomedical Applications,” Microsystem Technologies, 15, pp. 647−666 (2009). Stemme, E. and Stemme, G., “A Valveless Diffuser Nozzle-Based Fluid Pumps,” Sensors and Actuators A: Physical, 39, pp. 159−167 (1993). Forster, F., Bardell, R., Afromowitz, M., Sharma, N. and Blanchard, A., “Design, Fabrication and Testing of fixed-Valve Micropumps,” In Proceedings of the ASME Fluids Engineering Division, San Francisco (1995). Tesla, N., “Valvular Conduit,” U. S. Patent, 329, p. 559 (1920). Reed, J. L. and Fla, O., “Fluidic Rectifier,” U. S. Patent, 265, p. 636 (1993). Forster, F. K., Bardell, R. L., Blanchard, A. P., Afromowitz, M. A. and Sharma, N. R., “Micropumps with Fixed Valves,” U. S. Patent, 876, p. 187 (1999). Stemme, E. and Stemme, G., “Valve-Less Fluid Pump,” Swedish Patent Application, 300, pp. 604−607 (1993). Forster, F. and Williams, B., “Parametric Design of Fixed-Geometry Microvalves-The Tesser Valve,” Proceedings of IMECE2002, ASME International Mechanical Engineering Congress and Exposition, New Orleans, Louisiana, U.S.A (2002). Liao, P. F., “A Study on No-Moving-Part Valves for Flows in Microchannels,” M. S. Thesis, Department of Aeronautics and Astronautics, National Cheng Kung University, China (2004). Gamboa, A. R., Morris, C. J. and Forster, F. K., “Improvement in Fixed-Valve Micropump Performance Through Shape Optimization of Valves,” Journal of Fluids Engineering, 127, pp. 339−346 (2005). Kolahdouz, E. M., “Design and Fabrication of a Piezoelectrically Actuated Micropump Using Modified Tesla Type Micovalves,” M. S. Thesis, Isfahan University of Technology, Isfahan, Iran (2010). Bardell, R., “The Diodicity Mechanism of TeslaType-No-Moving-Part Valves,” Ph. D. Thesis, Mechanical Engineering Department, University of Washington, U.S.A (2000). Yamahata, C., “Magnetically Actuated Micropumps,” Ph.D. Thesis, Laboratory for Microsystems, Institute of Microelectronics and Microsystems, Swiss Federal Institute of Technology, Lausanne, EPFL (2005). (Manuscript received August 25, 2012, accepted for publication January 14, 2013.)
Journal of Mechanics
