sensors Article
Equivalent Circuit Modeling for a Valveless Piezoelectric Pump Jianhui Zhang 1, *, Yuan Wang 2 and Jun Huang 3, * 1 2 3
*
College of Mechanical and Electrical Engineering, Guangzhou University, Guangzhou 510006, China College of Communication Engineering, Army Engineering University of PLA, Nanjing 210007, China;
[email protected] National Research Center of Pumps, Jiangsu University, Zhenjiang 212013, China Correspondence:
[email protected] (J.Z.);
[email protected] (J.H.); Tel.: +86-020-3936-6923 (J.Z.)
Received: 30 June 2018; Accepted: 16 August 2018; Published: 31 August 2018
Abstract: Various kinds of the models had been proposed to explain the relationship between the performance and the structural parameters of valveless piezoelectric pumps, so as to evaluate the functional performance such devices. Among the models, the equivalent circuit model, which converts the multi-field problem of a valveless piezoelectric pump system into a simple circuit problem, is the most simple and clear one. Therefore, the proposed structure and working principle of the valveless piezoelectric pump with multistage Y-shape treelike bifurcate tubes are analyzed; then, the equivalent circuit model of the valveless piezoelectric pump is established based on the working principles of this pump and liquid-electric analogy theory. Finally, an experimental study of the pump is carried out, with a comparative analysis of the experimental results and the simulation results of the generated equivalent circuit. The experimental results show that with a driving voltage of 100 V and frequency of 6 Hz, the maximum flow rate of the valveless piezoelectric pump is 1.16 mL/min. Meanwhile, the output current of equivalent circuit also reaches its peak at the frequency of 6 Hz, therefore, indicating a good predictive ability of this model in calculating the maximum output flow rate and best working frequency of valveless piezoelectric pumps. Keywords: piezoelectric; pump; valveless; treelike bifurcate; equivalent circuit
1. Introduction With the technical development of MEMS, various sensors and actuators, such as piezoelectric micropumps, ultrasonic motors and piezoelectric energy harvesters, have been invented and successfully applied in various kinds of micro-systems [1–13]. Valveless piezoelectric pumps, as a kind of micropump without moving parts inside, have drawn widespread attention due to their simple structure and absence of electromagnetic interference. Moreover, their functional integration has been continuously developed, especially in the research and application of piezoelectric pump water-cooling systems [14–19]. Among the various types of the valveless piezoelectric pumps, the valveless piezoelectric pumps with Y-shaped tubes, which have a simple and unique bifurcation structure, have been widely studied [20–22]. The natural treelike bifurcation system, due to its small drag and ideal dimensions, which can be used for minimizing the energy loss, has been designed into microstructures for energy transmission systems and thermal conversion systems. Therefore, based on Y-shape tubes, the tubes with treelike bifurcation structure were constructed for application in the microfluidic transmission and heat conduction fields. With the in-depth ongoing research of valveless piezoelectric pumps domestically and abroad, various kinds of the models had been proposed to explain the relationship between the performance Sensors 2018, 18, 2881; doi:10.3390/s18092881
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and the structural parameters of valveless piezoelectric pumps, so as to evaluate the functional performance of the devices. Ullmann and Stemme et al. [23,24] proposed a dynamic model for predicting the maximum flow rate of a micro-pump based on the known volume changes caused by the movement of a piezoelectric vibrator. Then Olsson [25] and Ullmann [26] modified the dynamic model, taking into consideration the inertia effect of the acceleration speed of the fluid in the tubes. However, this model was incapable of describing the dynamic changes inside the piezoelectric pump. Therefore, Ullmann et al. [27] proposed an improved dynamic model to determine the dynamic behavior of the piezoelectric pump. Since the changes of fluid flow and pressure in the chamber caused by the vibration of piezoelectric vibrator were not correctly considered in these studies, Azarbadegan [28] developed a fluid-solid coupling model to evaluate the response characteristics of the valveless piezoelectric pump under resonance. Singh et al. [29] presented analytical modeling of a planar valveless micropump to predict the natural frequency and flow rate performance of the proposed micropump. However, the abovementioned models are solid in theory but complicated in calculation; moreover, as a result of multi-physics field coupling problems such as mechanical-electric coupling and fluid-solid coupling, these models must be simplified in adopting mathematical models to analyze practical problems, resulting in errors. To decrease the complexity of the analytical model and reduce the calculation time, based on liquid-electric analogy theory, Morganti [30], Bourouina [31] and Hsu et al. [32,33] developed the equivalent circuit model of valveless piezoelectric pump with tapered tubes, and translated the multi-field problem of valveless piezoelectric pump systems into a circuit problem for solution. Therefore, in this study, the equivalent circuit model of the valveless piezoelectric pump with multistage Y-shape treelike bifurcate tubes is also proposed based on liquid-electric analogy theory to demonstrate its effectiveness, so as to lay a foundation for the optimization and improvement of the valveless piezoelectric pump. 2. Construction of the Device Based on the fractal model of biological vascular network [34], our research group has designed the multistage Y-shape tube in 2013, and successfully developed a valveless piezoelectric pump with multistage Y-shape treelike bifurcate tubes [35,36]. A sketch of the multistage Y-shape treelike bifurcate tube and a photograph of the piezoelectric pump are shown in Figure 1. The width of mother tube of the whole tube is defined as a, and the outlet widths of daughter tubes of each level is half of that of the upper level. The length of mother tube is defined as l, and the lengths of daughter tubes of each level is half of that of upper level (except for the level I sub-tubes and the end level sub-tubes); the bifurcation angle of the tube is defined as 2α, and the depth of the whole tube is defined as h. The geometric parameters and material properties of tubes and piezoelectric vibrator are shown in Table 1. Taking the tube as the valve body of valveless piezoelectric pump that is without moving parts, the valveless piezoelectric pump with multistage Y-shape treelike bifurcate tubes consists of a piezoelectric vibrator, pump chamber and a pair of multistage Y-shape treelike bifurcate tubes. When the fluid flows in the tube, the energy consumption is different if different outlets are taken, namely taking the 2i (i = 1, 2, 3 . . . ) daughter tubes as outlet (defined as dividing flow) and the mother tube as outlet (defined as merging flow), it means there are flow resistance differences between dividing flow and merging flow. The piezoelectric vibrator will perform a vertical reciprocating motion when loaded with an alternating voltage, resulting in periodic volume changes of the pump chamber and hence the movement of fluid in the chamber. When the vibrator uplifts outward, the volume of the pump chamber increases, and the fluid flows into the chamber through the Y-shape treelike bifurcate tubes; when the vibrator depresses inward, the volume of chamber decreases, and the fluid flows out of the chamber through the tubes. Due to the different flow resistance of the merging flow and dividing flow of the fluid in the Y-shaped treelike bifurcate tube, the tube now works as a valve, resulting in different flow rates between chamber inflow and outflow from both the Y-shape treelike bifurcate
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tubes during a vibration period. With the continual motion of the piezoelectric vibrator, the fluid in the pump chamber macroscopically shows that it flows from inlet to outlet, resulting in a flow in a single direction.
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Figure treelike bifurcate bifurcate tubes. tubes. Figure 1. 1. Valveless Valveless piezoelectric piezoelectric pump pump with with multistage multistage Y-shape Y-shape treelike Table 1. Parameters of the valveless piezoelectric pump with multistage Y-shape treelike bifurcate tubes. Table 1. Parameters of the valveless piezoelectric pump with multistage Y-shape treelike bifurcate tubes. Parameter Parameter Diameter of pump chamber (mm) Depth ofofpump Diameter pumpchamber chamber(mm) (mm) Width mother tube (mm) Depth ofof pump chamber (mm) Width ofofmother Length mothertube tube(mm) (mm) Length of mother (mm) Bifurcation angle tube of tubes (°) Bifurcation angle tubes3)(◦ ) Density of PZTof(kg/m 3 Density RadiusofofPZT PZT(kg/m (mm) ) Radius of PZT (mm) Thickness of PZT (mm) Thickness of PZT (mm) Radius of metal substrate (mm) Radius of metal substrate (mm) Thickness of metal metal substrate substrate (mm) (mm) Thickness of Elastic modulus of metal substrate (GPa) Elastic modulus of metal substrate (GPa) 3 Density of metal substrate (kg/m Density of metal substrate (kg/m3 )) Poisson ratio ratio of of metal metal substrate substrate Poisson
Symbol Symbol d d h h a a l l 2α 2αρPZT ρPZT rPZT rPZT hPZT hPZT rmembrane rmembrane h membrane hmembrane E membrane Emembrane ρ membrane ρmembrane νmembrane νmembrane
Value Value 40 40 2 2 4 4 10 1060 60 7800 7800 15 15 0.2 0.2 50 50 0.2 0.2 120 120 7850 7850 0.33 0.33
The piezoelectric vibrator will perform a vertical reciprocating motion when loaded with an 3. Experimental Setup alternating voltage, resulting in periodic volume changes of the pump chamber and hence the movement in the of chamber. the vibrator uplifts outward, of of the100 pump Figure of 2 isfluid a diagram the flowWhen experiments. For the pumping flowthe test,volume a voltage V is chamber increases, and the fluid flows into the chamber through the Y-shape treelike bifurcate tubes; used to drive the piezoelectric vibrator, and deionized water is utilized as working fluid. By changing when the vibrator depresses inward, the volume of chamber decreases, fluid flows out of the the driving frequency, the output mass flow of the piezoelectric pumpand in the a unit time is measured, chamber through the tubes. Due to the different flow resistance of the merging flow and dividing thereby the test values of flow that changes with frequency under voltage of 100 V are obtained for the flow of the fluid in the Y-shaped treelike bifurcate tube, the tube now works as a valve, resulting in piezoelectric pump. different flow rates between chamber inflow and outflow from both the Y-shape treelike bifurcate tubes during a vibration period. With the continual motion of the piezoelectric vibrator, the fluid in the pump chamber macroscopically shows that it flows from inlet to outlet, resulting in a flow in a single direction. 3. Experimental Setup Figure 2 is a diagram of the flow experiments. For the pumping flow test, a voltage of 100 V is used to drive the piezoelectric vibrator, and deionized water is utilized as working fluid. By changing
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the piezoelectric pump.
Figure 2. Schematic diagram of the flow experiments. Figure 2. Schematic diagram of the flow experiments.
4. Modeling
4. Modeling 4.1. Definition of Basic Parameters
4.1. Definition of Basic Parameters
According to liquid-electric analogy theory, when a fluid flows through a fluidic element, the According analogy theory, when a fluid flows a fluidic element, pressure drop to ∆p liquid-electric across the element is defined as the voltage U, and thethrough flow volume QV passing thethrough pressure ∆p can across the element is defined as the U, and the resistance, flow volume thedrop element be treated as current I. Therefore, thevoltage conception of fluid fluidQV passing through element can be treated asas current Therefore, the conception of fluid resistance, inductance andthe fluid capacitance are defined followsI. [37].
fluid inductance and fluid capacitance are defined as follows [37]. 4.1.1. Fluid Resistance
4.1.1. Fluid Resistance
When fluid is flowing in the tubes, it shows a resistance due to the viscosity force, resulting in flowing tubes,drop. it shows a resistance dueexpression to the viscosity force, resultingthe in an an When energyfluid loss,isshown as in a the pressure Therefore, like the of electric resistance, energy loss, shown as a pressure drop. Therefore, like the expression of electric resistance, the fluid fluid resistance of a certain fluid part in steady flow is defined as the ratio between the pressure drop at both ends the flow through it, whichasis:the ratio between the pressure drop at resistance of a and certain fluidvolume part in passing steady flow is defined
both ends and the flow volume passing through it, which is: R p / QV
(1)
R =vibrator ∆p/QVin a valveless piezoelectric pump is small in(1) Because the vibration of the piezoelectric scale, it can be assumed that the flow in the pump is a laminar flow. According to Poiseuille’s law, the vibration of the piezoelectric vibrator in follows: a valveless piezoelectric pump is small in theBecause relationship between pressure drop and flow rate is as scale, it can be assumed that the flow in the pump is a laminar flow. According to Poiseuille’s law, 32 lv p rate 2is as follows: the relationship between pressure drop and flow (2) D 32µlv where v is the flow rate of the fluid, and the ∆p flow =volume is:
D2 πD 2 v QV where v is the flow rate of the fluid, and the flow volume is: 4
(2) (3)
2 v is: Therefore, the fluid resistance of tube in laminar πDflow
QV =
1284 l R 4 πDflow Therefore, the fluid resistance of tube in laminar is: where μ is the viscosity coefficient, l is the length of the tube and D is diameter of the tube. 128µl
R=
4.1.2. Fluid Capacitance
πD4
(3) (4)
(4)
where µ is the viscosity coefficient, l is the length of the tube and D is diameter of the tube.
Fluid is compressible, which can be shown under large pressure changes for liquids. In terms of the Fluid container, the internal fluid body would have the capacitive damping against the compressive 4.1.2. Capacitance
Fluid is compressible, which can be shown under large pressure changes for liquids. In terms of the container, the internal fluid body would have the capacitive damping against the compressive
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deformation. According to the definition of electric capacitance, the fluid capacitance can be defined as the ratio of changes of flow volume and changes of pressure that results in its deformation, namely:
R
QV dt ∆p
C=
(5)
The compressibility of fluid can be described by the bulk modulus, namely: κ=−
d(∆p) dV/V
(6)
where V is the volume of the fluid. Therefore, the fluid capacitance of the tube can be represented as: C=
V κ
(7)
4.1.3. Fluid Inductance In the fluid transmission unit, high speed transient flow occurs for the flow field inside. The fluid matter is accelerated or decelerated due to fluid inertia, resulting in a pressure variation. Therefore, the expression of fluid inductance is represented as the ratio of pressure changes aroused by both ends of fluid part and the change rate of flow volume, namely: L=
∆p dQV /dt
(8)
According to [37], this expression can be transformed into: L=
ρl A
(9)
4.2. Equivalent Circuit of Piezoelectric Vibrator As a structural unit, the piezoelectric vibrator will have elastic deformation when an alternating current is applied, resulting in an elastic effect and an inertial effect. Therefore, electrical inductance and capacitance effects are introduced. Meanwhile, the fluid motion in pump chamber caused by the vibration of the vibrator can be treated as a source of voltage. According to [30], the expression of electric capacitance of piezoelectric vibrator can be described as follows: Cv = k=
(γA)2 k
64 γπEh3 2 12(1 − υ ) ( D/2)2
(10) (11)
where k is the equivalent spring constant; E is elastic modulus of the vibrator; ν is the Poisson ratio of vibrator; h is the thickness of the vibrator; γ is the equivalent coefficient, which is related to the loading method and boundary conditions of the vibrator. Hence, the expression electrical inductance of piezoelectric vibrator can be described as follows: Lv =
meff
(γA)2
meff = γ(mmembrane + mactuator )
(12) (13)
where meff is the equivalent mass of the vibrator; mmembrane is the mass of the base; and mactuator is the mass of the piezoelectric ceramics.
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From the analysis above, the equivalent circuit of a piezoelectric vibrator can be obtained, as shown in Figure 3. Sensors 2018, 18, x FOR PEER REVIEW 6 of 13 Sensors 2018, 18, x FOR PEER REVIEW
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Figure 3. Equivalent circuit of a piezoelectric vibrator. Figure 3. Equivalent circuit of a piezoelectric vibrator. Figure 3. Equivalent circuit of a piezoelectric vibrator.
4.3. Equivalent Circuit of the Fluid Domain 4.3. Equivalent Circuit of the Fluid Domain 4.3. Equivalent Circuit of the Fluid Domain The components of the valveless piezoelectric pump with multistage Y-shape treelike bifurcate The components of the valveless piezoelectric pump with multistage Y-shape treelike bifurcate of the pump with multistage Y-shape treelike bifurcate tubesThe arecomponents analyzed based onvalveless the basic piezoelectric parameters defined by liquid-electrical analogy method, so as tubes are analyzed based on the basic parameters defined by liquid-electrical analogy method, so as to tubes are analyzed based on the basic parameters defined by liquid-electrical analogy method, so as to obtain the equivalent circuit of the valveless piezoelectric pump. obtain the equivalent circuit of the valveless piezoelectric pump. to obtain circuitarea of the valveless pump. Sincethe theequivalent cross-sectional of the pumppiezoelectric chamber is far larger than that of the tubes, only the Since the cross-sectional area of the pump chamber is far larger than that of the tubes, only the the cross-sectional area of theare pump chamber larger than thatequivalent of the tubes, onlypart, the fluidSince capacitance and flow tube effects considered in is thefarliquid-electrical of that fluid capacitance and flow tube effects are considered in the liquid-electrical equivalent of that part, fluid and flow resistance tube effects considered in the liquid-electrical equivalent that part, whilecapacitance the influence of fluid is are ignored. Therefore, the fluid capacitance and fluidofinductance while the influence of fluid resistance is ignored. Therefore, the fluid capacitance and fluid inductance while influence of fluid resistance is ignored. Therefore, the fluid capacitance and fluid inductance of the the pump chamber are defined as follows: of the pump chamber are defined as follows: of the pump chamber are defined as follows: Ah Cc A c hc (14) c c A K CC (14) c c= c c (14) K K where Ac and hc are the cross-sectional area and depth of the pump chamber: where the cross-sectional area and depth of the pump chamber: where AAcc and andhc hare c are the cross-sectional area and depth of the pump chamber: h Lc ρhcc (15) Ahcc (15) LLc = (15) c A Acc Hence, the equivalent circuit of that part is shown as in Figure 4. Hence, the equivalent circuit of that part is shown as in Figure 4. Hence, the equivalent circuit of that part is shown as in Figure 4.
Figure 4. Equivalent circuit of the pump chamber. Figure 4. Equivalent circuit of the pump chamber. Figure 4. Equivalent circuit of the pump chamber.
Different from moving part valves, tapered flow tubes, Tesla tubes and Y-shaped tubes and Different from moving part type valves, tapered tubes,actions Tesla tubes Y-shaped pumps tubes and other flow resistance differential tubes have flow no switch whenand piezoelectric are Different from moving part valves, tapered flow tubes, Tesla tubes and Y-shaped tubes and other other flowtherefore, resistancethe differential type tubes no switch actions piezoelectric are working, fluid capacitive effecthave is ignored and only thewhen influences of fluid pumps inductance flow resistance differential type tubes have no is switch actions when piezoelectric pumps working, working, capacitive ignored the influences of fluidare inductance and fluidtherefore, resistancethe arefluid considered. Ateffect the same time, and due only to different loss coefficients along the therefore, the fluid capacitive effect is ignored and only the influences of fluid inductance and and fluidand resistance considered. At fluid the same time,types due of to tubes, different loss coefficients alongfluid the positive negativeare directions of the in these different fluid resistances are resistance arenegative considered. At the same duethese to different loss coefficients along positive and positive and directions of thetime, fluid types of tubes, different fluidthe resistances are shown. Therefore, the equivalent circuit of thisinpart is as shown in Figure 5. negative directions the of the fluid in these types of tubes, resistances are shown. Therefore, shown. Therefore, equivalent circuit of this part isdifferent as shownfluid in Figure 5. the equivalent circuit of this part is as shown in Figure 5. Based on the analogy results of fluid and electrical models of parts of the valveless piezoelectric pump mentioned above, as well as the mathematical model of the valveless piezoelectric pump, the equivalent circuit diagram of the valveless piezoelectric pump is obtained, as shown in Figure 6. The subscripts ‘in’ and ‘out’ represent the inlet pipe and outlet pipe of the valveless piezoelectric pump, respectively. As can be seen from Figure 1, the piezoelectric pump consists of a piezoelectric Figure 5. Equivalent circuit of tubes with fluid resistance differences. Figure 5. Equivalent circuit of tubes with fluid resistance differences.
Based on the analogy results of fluid and electrical models of parts of the valveless piezoelectric Based on the analogy ofthe fluid and electrical models of parts of thepiezoelectric valveless piezoelectric pump mentioned above, asresults well as mathematical model of the valveless pump, the pump mentioned asof well the mathematical model of is the valveless pump, the equivalent circuit above, diagram theas valveless piezoelectric pump obtained, aspiezoelectric shown in Figure 6. The
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Figure 4. Equivalent circuittreelike of the pump chamber. vibrator, circular pump chamber, multistage Y-shape bifurcate tubes, square collection chamber and inlet/outlet tubes. According to the working principle of the piezoelectric pump, when the Different moving tapered flow tubes, Teslathe tubes and Y-shaped and vibrator upliftsfrom outward, the part fluidvalves, flows into the chamber through inlet/outlet tubes, tubes collection other flow resistance differential type tubes have no switch actions when piezoelectric pumps chambers and Y-shape treelike bifurcate tubes. This process can be equivalent to the input voltageare of Sensors 2018,therefore, 18, x FOR REVIEW 7 of 13 working, thepositive fluid capacitive is ignored and only the influences of fluid inductance the circuit model inPEER the half cycle,effect and the current flows through the above equivalent elements and fluid resistance considered. At the samethe time, to out different loss coefficients the respectively. When theare vibrator depresses inward, fluiddue flows of the chamber throughalong the tubes, subscripts ‘in’ and ‘out’ represent the inlet pipe and outlet pipe of the valveless piezoelectric pump, positive negative directions of the fluid these typescan of tubes, differenttofluid resistances collectionand chambers and the inlet/outlet tubes.inThis process be equivalent the input voltageare of respectively. As can be seen from Figure 1, the piezoelectric pump consists of a piezoelectric vibrator, shown. Therefore, the equivalent circuit of this part is as shown in Figure 5. the circuit model in the negative half cycle, and the current flows through each equivalent component. circular pump chamber, multistage Y-shape treelike bifurcate tubes, square collection chamber and inlet/outlet tubes. According to the working principle of the piezoelectric pump, when the vibrator uplifts outward, the fluid flows into the chamber through the inlet/outlet tubes, collection chambers and Y-shape treelike bifurcate tubes. This process can be equivalent to the input voltage of the circuit model in the positive half cycle, and the current flows through the above equivalent elements respectively. When the vibrator depresses inward, the fluid flows out of the chamber through the tubes, collection chambers and the inlet/outlet tubes. This process can be equivalent to the input Figure 5. Equivalent circuit of tubes with fluid resistance differences. Figure 5. in Equivalent circuithalf of tubes with differences. voltage of the circuit model the negative cycle, andfluid the resistance current flows through each equivalent component. Based on the analogy results of fluid and electrical models of parts of the valveless piezoelectric It can circuit diagram that when the the input voltage is in is theinpositive halfcan be beconcluded concludedfrom fromthe the circuit diagram that when input voltage the positive pump mentioned above, as well as the mathematical model of the valveless piezoelectric pump, the period, the existence of conduction and diode blocking results in a higher electric resistance for half-period, the existence of conduction and diode blocking results in a higher electric resistance the for equivalent circuit diagram of the valveless piezoelectric pump is obtained, as shown in Figure 6. The branch circuits at the outlet section compared with the the branch circuits at the outlet section compared with theinlet inletsection, section,which whichcreates createsaa larger larger branch current at the inlet section compared to the outlet section. Therefore, Therefore, this half period can be treated as the suction period of the valveless piezoelectric pump. Similarly, Similarly, when the input voltage is in the negative negative half-period, half-period, the the existence existence of of conduction conduction and and blocking blocking of of diode diode results results in in a higher electric resistance for the branch branch circuits circuits at at the theinlet inletsection sectioncompared comparedwith withthe theboutlet boutletsection, section,which whichmakes makesa alarger largerbranch branchcurrent currentatatthe theoutlet outletsection sectioncompared comparedto tothe theinlet inletsection. section. Therefore, Therefore, this half period can be treated as the discharge period of the valveless piezoelectric pump. At this time, the current flowing through Rout out, , L Lout andCCoutoutcan canbebeequivalent equivalenttotothe theoutput outputflow flowof of the the piezoelectric piezoelectric pump. pump. outand According to the unidirectional conduction characteristics of the diode, when the input voltage voltage is in the negative half-period, and the corresponding equivalent circuit circuit model model is is shown shown in in Figure Figure 7. 7.
Figure 6. Equivalent circuit diagram of valveless piezoelectric pump with multistage Y-shape treelike Figure 6. Equivalent circuit diagram of valveless piezoelectric pump with multistage Y-shape treelike bifurcate tubes: (a) Piezoelectric vibrator; (b) Pump chamber; (c) Multistage Y-shape treelike bifurcate bifurcate tubes: (a) Piezoelectric vibrator; (b) Pump chamber; (c) Multistage Y-shape treelike bifurcate tube; (d) Collection chamber; (e) Inlet/Outlet tube. tube; (d) Collection chamber; (e) Inlet/Outlet tube.
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Figure 7. 7. Equivalent Equivalent circuit circuit diagram diagram when when the the input input voltage voltage is is in in the the negative negative half-period. half-period. Figure Figure 7. Equivalent circuit diagram when the input voltage is in the negative half-period.
A commercial commercial EDA EDA software software package package (Multisim (Multisim 12.0, 12.0, National National Instruments, Instruments, Austin, Austin, TX, TX, USA) USA) is is A A commercial EDA software package (Multisim 12.0, National Instruments, Austin, TX, USA) is adopted for for the the simulation simulation of of the the equivalent equivalent circuit circuit of of the the valveless valveless piezoelectric piezoelectric pump. pump. The The results results adopted adopted for the simulation of the equivalent circuit of the valveless piezoelectric pump. The results are compared with experimental pumping flow results to demonstrate the feasibility and validity of are compared with experimental pumping flow results to demonstrate the feasibility and validityare of compared with experimental pumping flow results to demonstrate the feasibility and validity of the the proposed proposed equivalent equivalent circuit. circuit. the proposed equivalent circuit. 5. Results Results and and Discussion Discussion 5. 5. Results and Discussion By substituting substituting the the related related parameter parameter values values in in Table Table 11 into into the the equations, equations, the the values values of of the the By Bycomponents substituting the related parameter values inpump Table 1with into multistage the equations, the values of the circuit circuit of the valveless piezoelectric Y-shape treelike bifurcate circuit components of the valveless piezoelectric pump with multistage Y-shape treelike bifurcate components of the valveless pump with multistage Y-shape treelike bifurcate tubes are tubes are are obtained. obtained. However,piezoelectric the loss loss coefficient coefficient for dividing dividing flow and merging merging flow of of multistage multistage tubes However, the for flow and flow obtained. However, the loss coefficient for dividing flowsimulation and merging flow of multistage Y-shape treelike treelike bifurcate tubes are different, different, therefore, for multistage multistage Y-shapeY-shape treelike Y-shape bifurcate tubes are therefore, simulation for Y-shape treelike treelike bifurcate tubes are different, therefore, simulation for multistage Y-shape treelike bifurcate bifurcate tubes with ANSYS CFX is carried out in this research. bifurcate tubes with ANSYS CFX is carried out in this research. tubesThe with ANSYS CFX model is carried out in thisofresearch. finite-element of flow flow field multistage Y-shape Y-shape treelike treelike bifurcate bifurcate tubes tubes is is shown shown in in The finite-element model of field of multistage The finite-element model of flow field of multistage Y-shape treelike bifurcate tubes is shown in Figure 8. By importing this model into ANSYS CFX for flow field simulation, the relation curve of Figure 8. By importing this model into ANSYS CFX for flow field simulation, the relation curve of Figure 8. difference By importing this model into ANSYS CFX foroutlet flow field simulation, the relation curve of pressure between the two two ends of tubes tubes and flow in in positive positive flowing flowing and negative negative pressure difference between the ends of and outlet flow and pressure ends flowing is isdifference obtained,between as shown shownthe in two Figure 9. of tubes and outlet flow in positive flowing and negative flowing obtained, as in Figure 9. flowing is obtained, as shown in Figure 9.
Figure 8. 8. Finite Finite element element model model of of the the tube. tube. Figure
It can be seen from Figure 9 that the relation curve of mass flow and pressure drops is similar to a straight line. Therefore, according to Equation (1), the flow resistance of the multistage Y-shape treelike bifurcate tube is the cotangent function of angle of the “straight line” in Figure 9. The equivalent circuit parameters have been listed in Table 2.
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Figure 9. Curves of mass flow versus pressure drop.
It can be seen from Figure 9 that the relation curve of mass flow and pressure drops is similar to a straight line. Therefore, according to Equation (1), the flow resistance of the multistage Y-shape treelike bifurcate tube is the cotangent function of angle of the "straight line" in Figure 9. The Figure 9. Curves of mass flow versus pressure drop. Figure 9. Curves of mass flow versus pressure drop. equivalent circuit parameters have been listed in Table 2. TableFigure 2. Calculation results for equivalent It can be seen from 9 that the relation curve ofcircuit mass components. flow and pressure drops is similar to Table 2. Calculation results for equivalent circuit components. a straight line. Therefore, according to Equation (1), the flow resistance of the multistage Y-shape Name Symbol Value Name Symbol Valuein Figure 9. The treelike bifurcateComponent tube Component is the cotangent function of angle of the "straight line" Vibrator capacitance C 37.87 pF Vibrator capacitance 37.87 pF equivalent circuit parameters have been listed in Table 2. Cvv
Vibrator inductance 4371.135 H H Vibrator inductance LLvv 4371.135 Chamber inductance L 1592.4 H c Chamber inductance 1592.4 H Table 2. Calculation results for equivalentLccircuit components. Chamber capacitance 1.1418 fF fF Chamber capacitance CCc c 1.1418 Y-shape tube resistance Name (high resistance state) RSymbol 242 MΩ Value h Component Y-shape tube resistance (high resistance state) Rh 242 MΩ Y-shape tube resistance (Low resistance state) R 240 MΩ 37.87 pF Vibrator capacitance Y-shape tube resistance (Low resistance state) Rl l Cv 240 MΩ Y-shape tube inductance Lt 4.6875 MH Vibrator inductance Lv 4371.135 H Y-shape tube inductance L t 4.6875 Collection chamber inductance Lcc 0.2 MH MH Chamber inductance L c 1592.4 H Collection chamber inductance LCcccc Collection chamber capacitance 0.15 0.2 fF MH Chamber capacitance C c 1.1418 fF Collection chamberchamber capacitance CRcccc 0.15 fF Collection resistance 1.8577 MΩ Y-shape tube resistance (high resistance state) R h 242 MΩ Collection chamber resistance R/L cc out 1.8577 Inlet/Outlet tube inductance Lin 37.689 MH MΩ Y-shape tube resistance (Lowcapacitance resistance state) Rl MΩ Inlet/Outlet tube 0.4824 fF 240 Inlet/Outlet tube inductance LCinin/L/Coutout 37.689 MH Inlet/Outlet tube resistance R /R 178.4 MΩ Y-shape tube inductance L t 4.6875 out Inlet/Outlet tube capacitance Cinin/Cout 0.4824 fF MH Collectiontube chamber inductance Inlet/Outlet resistance Rin/RoutLcc 178.4 0.2 MΩMH Collection chamber capacitance Ccc 0.15 fF Thetransient transientflow flowrate ratecurves curves ofthe theoutlet outlettube tubebranches branchesin inthe the simulation ofthe the equivalent The Collection chamberofresistance Rcc simulation of 1.8577 equivalent MΩ circuitof ofthe thevalveless valveless piezoelectric pumpare areshown shownin inFigure Figure10. 10. circuit piezoelectric pump Inlet/Outlet tube inductance Lin/Lout 37.689 MH Inlet/Outlet tube capacitance Cin/Cout 0.4824 fF Inlet/Outlet tube resistance Rin/Rout 178.4 MΩ
The transient flow rate curves of the outlet tube branches in the simulation of the equivalent circuit of the valveless piezoelectric pump are shown in Figure 10.
Figure 10. Transient flow rate curves of branches of outlet tube.
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Figure10. 10.Transient Transientflow flowrate ratecurves curvesof ofbranches branchesof ofoutlet outlettube. tube. Figure
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FromFigure Figure10 10ititcan canbe beseen seenthat thatthe theoutput outputflow flowrate ratein innegative negativehalf-period half-periodisislarger largerthan thanthat that From From Figure 10 it can be seen that the output flow rate in negative half-period is larger than that inthe thepositive positivehalf-period, half-period,therefore, therefore,the thenet netflow flowof ofthe thevalveless valvelesspiezoelectric piezoelectricpump pumpwith withmultistage multistage in in the positive half-period, therefore, the net flow of the valveless piezoelectric pump with multistage Y-shape treelike treelike bifurcate bifurcate tubes tubes can can be be obtained obtained by by subtracting subtracting the the integration integration values values of of the the two two Y-shape Y-shape treelike bifurcate tubes can be obtained by subtracting the integration values of the two currentfunctions. functions. current current functions. Figure 11 11 shows shows the the variation variation of of the the flow flow rate rate with with the the change change in in time time at at the the outlet outlet and and the the Figure Figure 11 shows the variation of the flow rate with the change in time at the outlet and the velocity velocityof ofthe thepiezoelectric piezoelectricvibrator vibratorfor foraadriving drivingfrequency frequencyof of66Hz. Hz.The Theflow flowrate ratecurve curveof ofthe thepump pump velocity of the piezoelectric vibrator for a driving frequency of 6 Hz. The flow rate curve of the pump and the andthe thevelocity velocitycurve curveof ofthe thevibrator vibrator are arein inthe thesame samephase. phase. and velocity curve of the vibrator are in the same phase. By changing changing the the drive drive frequency frequency of of equivalent equivalent circuit circuit of of the the valveless valveless piezoelectric piezoelectric pump pump with with By By changing the drive frequency of equivalent circuit of the valveless piezoelectric pump with multistage Y-shape Y-shape treelike treelike bifurcate bifurcate tubes, tubes, the the output output current current of of outlet outlet tube tube isis obtained. obtained. Then Then its its multistage multistage Y-shape treelike bifurcate tubes, the output current of outlet tube is obtained. Then its integratedvalues valuesare arethe theoutflow outflowof ofthe thepiezoelectric piezoelectricpump. pump.The Theexperimental experimentaland andtheoretical theoreticalcurves curves integrated integrated values are the outflow of the piezoelectric pump. The experimental and theoretical curves ofoutflow outflowof ofthe thepump pumpthat thatchanges changes with withfrequency frequencyare areshown shownin inFigure Figure12. 12. of of outflow of the pump that changes with frequency are shown in Figure 12.
Figure11. 11.Curves Curvesof offlow flowrate rateand andvibrator vibratorvelocity. velocity. Figure 11. Curves of flow rate and vibrator velocity. Figure
Figure12. 12.Comparison Comparisonof ofexperimental experimentalflow flowrate ratewith withsimulation simulationflow flowrate. rate. Figure 12. Comparison of experimental flow rate with simulation flow rate. Figure
The experimental maximum flow rate of the valveless piezoelectric pump 1.16 mL/min under The Theexperimental experimentalmaximum maximumflow flowrate rateof ofthe thevalveless valvelesspiezoelectric piezoelectricpump pumpisis is1.16 1.16mL/min mL/minunder under 100 V (6 Hz) power supply. The theoretical maximum flow rate of the pump is 0.88 mL/min, while 100 V (6 Hz) power supply. The theoretical maximum flow rate of the pump is 0.88 mL/min, while 100 V (6 Hz) power supply. The theoretical maximum flow rate of the pump is 0.88 mL/min, while the the largest relative error (85.71%) between the theoretical calculation results and experimental results the largest relative error (85.71%) between theoretical calculation results and experimental results largest relative error (85.71%) between thethe theoretical calculation results and experimental results of of the pump appear at the frequency of 2 Hz. This deviation is due to the longer multistage Y-shape of the pump appear at the frequency of 2 Hz. This deviation is due to the longer multistage Y-shape the pump appear at the frequency of 2 Hz. This deviation is due to the longer multistage Y-shape treelike bifurcate tubes and larger volume; while the fluid capacitance of the tube neglected in the treelike treelikebifurcate bifurcatetubes tubesand andlarger largervolume; volume;while whilethe thefluid fluidcapacitance capacitanceof ofthe thetube tubeisis isneglected neglectedin inthe the
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equivalent circuit circuitmodel, model, resulting in a relatively large deviation between and theoretical and equivalent resulting in a relatively large deviation between theoretical experimental experimental results at low frequency. outflow resultsoutflow at low frequency. Figure 13 shows the relation curve curve between between the the simulation simulation flow flow rate rate and and the the input input voltage voltage when when Figure 13 shows the relation the frequency is 6 Hz. It is seen that the flow rate is positively related to the driving voltage. the frequency is 6 Hz. It is seen that the flow rate is positively related to the driving voltage.
Figure 13. Simulation flow rate with various driving voltage inputs at 66 Hz. Hz.
Using the the Laplace Laplace transform transform analysis analysis method method to to analyze analyze the the circuit circuit shown shown in in Figure Figure 7, 7, the the system Using system transfer function of the output of the equivalent model can be obtained： transfer function of the output of the equivalent model can be obtained:
2 Rd 2R 2 R+e 2R R+h R 2+sL t Rl e d h 2sLt + Rl ) [([(RR sL sCout 1]{[+ 1]{[ H (sH ) ( s= +out sL)out )sCout outout R + R + R + sL e d l Rd Re Rl sLt t (16) + ( Rd + Re + Rh + sLt )sCc ][ sC1 v + s( Lv + Lc )] + 1} (16) 1 ( Rd Re Rh sLt ) sCc ][ s( Lv Lc )] 1} sCv 1 Rd = ( Rcc + sLcc ) k (17) sCcc 1 Rd ( R1cc sLcc ) (17) 1 cc sL out ) k = ( RoutsC+ (18) Re = ( Rin + sLin ) k sCin sCout 1 1 Because of the higher order circuit system, calculation is too complicated. Re of ( Rinthe sL ( Routthe sLmanual (18) in ) out ) sCout of the equation, the poles of The simulation software Multisim 12.0 is used sC to incalculate the pole-zero the system areof located in theorder left half plane of thesystem, complex thatcalculation is, the system hascomplicated. good robustness. Because the higher of the circuit theplane, manual is too The simulation software Multisim 12.0 is used to calculate the pole-zero of the equation, the poles of the 6. Conclusions system are located in the left half plane of the complex plane, that is, the system has good robustness. In this paper, based on the liquid-electric analogy theory, the equivalent circuit model of a 6. Conclusions valveless piezoelectric pump with multistage Y-shape treelike bifurcate tubes is developed. Based on the structural dimensions and material parameters of the piezoelectric pump, the parameters for In this paper, based on the liquid-electric analogy theory, the equivalent circuit model of a circuit components of equivalent circuit of the valveless piezoelectric pump are determined, and the valveless piezoelectric pump with multistage Y-shape treelike bifurcate tubes is developed. Based on flow resistance of the tube is obtained by finite element software calculations; the simulation results the structural dimensions and material parameters of the piezoelectric pump, the parameters for show that at the driving frequency of 6 Hz, the theoretical maximum flow rate is 0.88 mL/min. circuit components of equivalent circuit of the valveless piezoelectric pump are determined, and the The electrical model is verified by experimental results. The experimental maximum flow rate of the flow resistance of the tube is obtained by finite element software calculations; the simulation results valveless pump is 1.16 mL/min under 100 V (6 Hz) power supply. The largest relative error between show that at the driving frequency of 6 Hz, the theoretical maximum flow rate is 0.88 mL/min. The theoretical calculation results and experimental results of the pump is observed at the frequency of electrical model is verified by experimental results. The experimental maximum flow rate of the 2 Hz, and calculated to be 85.71%. The above results show that the circuit model can predict the output valveless pump is 1.16 mL/min under 100 V (6 Hz) power supply. The largest relative error between performance of valveless piezoelectric pumps quickly and effectively. theoretical calculation results and experimental results of the pump is observed at the frequency of 2 Hz, andContributions: calculated to beData 85.71%. The above results show that analysis, the circuit model can predict the output Author curation, Y.W. and J.H.; Formal Y.W.; Funding acquisition, J.H.; Investigation, Y.W. andpiezoelectric J.H.; Methodology, Y.W. and J.H.; administration, J.Z.; Supervision, J.Z.; performance J.Z., of valveless pumps quickly and Project effectively. Writing—original draft, J.H.; Writing—review & editing, J.Z.
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Funding: This research was funded by [National Natural Science Foundation of China] grant number [51605200], [Jiangsu Provincial Natural Science Foundation of China] grant number [BK20150518], [Guangzhou Municipal University research projects] grant number [1201610315], and [Industry-Academia-Research project of Guangzhou University] grant number [2018A001]. Conflicts of Interest: The authors declare no conflict of interest.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
Khosla, A.; Hesketh, P.J. Microfluidics, MEMS/NEMS, sensors and devices. J. Electrochem. Soc. 2014, 161, Y1. [CrossRef] Guedes, A.; Macedo, R.; Jaramillo, G.; Cardoso, S.; Freitas, P.P.; Horsley, A.D. Hybrid GMR sensor detecting 950 pT/sqrt(Hz) at 1 Hz and room temperature. Sensors 2018, 18, 790. [CrossRef] [PubMed] Munas, F.R.; Melroy, G.; Abeynayake, C.B.; Chathuranga, H.L.; Amarasinghe, R.; Kumarage, P.; Dau, V.T.; Dao, D.V. Development of PZT actuated valveless micropump. Sensors 2018, 18, 1302. [CrossRef] [PubMed] Sviliˇci´c, B.; Wood, G.S.; Mastropaolo, E.; Cheung, R. Thermal- and piezo-tunable flexural-mode resonator with piezoelectric actuation and sensing. J. Microelectrom. Syst. 2017, 26, 609–615. [CrossRef] Gupta, V.; Sharma, M.; Thakur, N. Optimization criteria for optimal placement of piezoelectric sensors and actuators on a smart structure: A technical review. J. Intell. Mater. Syst. Struct. 2010, 21, 1227–1246. [CrossRef] Hwang, W.S.; Park, H.C. Finite element modeling of piezoelectric sensors and actuators. AIAA J. 2015, 31, 930–937. [CrossRef] Gajasinghe, R.W.R.L.; Toprak, A.; Senveli, S.U.; Yildirim, Y.A.; Jones, M.; Ince, T.; Tigli, O. Rare cell immobilization on MEMS scale sensors using micro-electromagnets. IEEE Sens J. 2016, 16, 7572–7580. [CrossRef] Wang, D.; Liu, N. A shear mode piezoelectric energy harvester based on a pressurized water flow. Sens. Actuators A Phys. 2011, 167, 449–458. [CrossRef] Yu, H.; Zhou, J.; Deng, L.; Wen, Z. A vibration-based mems piezoelectric energy harvester and power conditioning circuit. Sensors 2014, 14, 3323. [CrossRef] [PubMed] Wang, Y.; Huang, W. A piezoelectric motor with two projections using two orthogonal flexural vibration modes. Sens. Actuators A Phys. 2016, 250, 170–176. [CrossRef] Zhang, J.; Wang, Y.; Fu, J.; Yan, K.; Li, Z.; Zhao, C. Advances in technologies of piezoelectric pumping with valves. Trans. Nanjing Univ. Aero. Astronaut. 2016, 33, 260–273. Liu, C.; Li, J.; Che, L.; Chen, S.; Wang, Z.; Zhou, X. Toward large-scale fabrication of triboelectric nanogenerator (TENG) with silk-fibroin patches film via spray-coating process. Nano Energy 2017, 41, 359–366. [CrossRef] Zhou, X.; Li, J.; Sun, H.; Hu, Y.; Che, L.; Wang, Z. Controlled cell patterning on bioactive surfaces with special wettability. J. Bionic Eng. 2017, 14, 440–447. [CrossRef] Singhal, V.; Garimella, S.V.; Raman, A. Microscale pumping technologies for microchannel cooling systems. Appl. Mech. Rev. 2004, 57, 191–221. [CrossRef] Ma, H.; Chen, B.; Gao, J.; Lin, C. Development of an OAPCP-micropump liquid cooling system in a laptop. Int. Commun. Heat Mass Transf. 2009, 36, 225–232. [CrossRef] Pires, R.F.; Vatanabe, S.L.; de Oliveira, A.R.; Silva, E.C.N. Water cooling system using a piezoelectrically actuated flow pump for a medical headlight system. Ind. Commer. Appl. Smart Struct. Technol. 2007, 6527, 65270P. Uchino, K. Piezoelectric actuators 2006-Expansion from IT/robotics to ecological/energy applications. J. Electroceram. 2008, 20, 301–311. [CrossRef] Chen, C.; Wang, H. Droplet generation and evaporative cooling using micro piezoelectric actuators with ring-surrounded circular nozzles. Microsyst. Technol. 2015, 21, 2067–2075. [CrossRef] Chen, H.; Cheng, W.; Peng, Y.; Zhang, W.; Jiang, L. Experimental study on optimal spray parameters of piezoelectric atomizer based spray cooling. Int. J. Heat Mass Transf. 2016, 103, 57–65. [CrossRef] Zhang, J.; Lu, J.; Xia, Q.; Kou, J.; Ren, G. Application of valve-less piezoelectric pump with Y-shape tubes for transporting cells and macromolecule. Chin. J. Mech. Eng. 2008, 44, 92–99. [CrossRef] Zhang, J.; Li, Y.; Xia, Q. Analysis of the pump volume flow rate and tube property of the piezoelectric valveless pump with Y-shape tubes. Chin. J. Mech. Eng. 2007, 43, 136–141. [CrossRef] Fadl, A.; Demming, S.; Zhang, Z.; Büttgenbach, S.; Krafczyk, M.; Meyer, D.M.L. A multifunction and bidirectional valve-less rectification micropump based on bifurcation geometry. Microfluid. Nanofluid. 2010, 9, 267–280. [CrossRef]
Sensors 2018, 18, 2881
23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.
13 of 13
Stemme, E.; Stemme, G. A valveless diffuser/nozzle-based fluid pump. Sens. Actuators A Phys. 1993, 39, 159–167. [CrossRef] Ullmann, A. The piezoelectric valve-less pump–performance enhancement analysis. Sens. Actuators A Phys. 1998, 69, 97–105. [CrossRef] Olsson, A.; Stemme, G.; Stemme, E. A numerical design of the valveless diffuser pump using a lumped mass model. J. Micromech. Microeng. 1999, 9, 34–44. [CrossRef] Ullmann, A.; Fono, I.; Taitel, Y. A piezoelectric valveless pump-dynamic model. J. Fluid. Eng. 2001, 123, 92–98. [CrossRef] Ullmann, A.; Fono, I. The Piezoelectric valve-less pump-improved dynamic model. J. Microelectromech. Syst. 2002, 11, 655–664. [CrossRef] Azarbadegan, A.; Eames, I.; Wojcik, A. Fluid-structure coupling in valveless micropumps. J. Micromech. Microeng. 2011, 21, 085033–085043. [CrossRef] Singh, S.; Kumar, N.; George, D.; Sen, A.K. Analytical modeling, simulations and experimental studies of a PZT actuated planar valveless PDMS micropump. Sens. Actuators A Phys. 2015, 225, 81–94. [CrossRef] Morganti, E.; Fuduli, I.; Montefusco, A.; Petasecca, M.; Pignatel, G.U. SPICE modelling and design optimization of micropumps. Int. J. Environ. Anal. Chem. 2005, 85, 687–698. [CrossRef] Bourouina, T.; Grandchamp, J.P. Modeling micropumps with electrical equivalent networks. J. Micromech. Microeng 1996, 6, 398. [CrossRef] Hsu, Y.C.; Le, N.B. Inertial effects on flow rate spectrum of diffuser micropumps. Biomed. Microdevices 2008, 10, 681–692. [CrossRef] [PubMed] Hsu, Y.C.; Le, N.B. Equivalent electrical network for performance characterization of piezoelectric peristaltic micropump. Microfluid. Nanofluid. 2009, 7, 237–248. [CrossRef] West, G.B.; Brown, J.H.; Enquist, B.J. A general model for the origin of allometric scaling laws in biology. Science 1997, 276, 122–126. [CrossRef] [PubMed] Huang, J.; Zhang, J.; Xun, X.; Wang, S. Theory and experimental verification on valveless piezoelectric pump with multistage Y-shape treelike bifurcate tubes. Chin. J. Mech. Eng. 2013, 26, 462–468. [CrossRef] Huang, J.; Zhang, J. A valveless piezoelectric pump with multistage Y-shape treelike bifurcate tubes. Robot. Biomim. (ROBIO) 2012, 212–216. Ogata, K. System Dynamics; Machinery Industry Press: Beijing, China, 2005. © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
