Abstract
Background
Impedance pump is a simple valveless pumping mechanism, which transports fluid through the mismatch of impedance in the system. Mismatch in impedance occurs when an asymmetrical periodic excitation is exerted on the elastic tube. Periodic asymmetrical excitation will then produce a unidirectional flow. Considering a multistage system with a single and constant power source, it is well expected that there would be pumping limitation to which the enhancement can reach.
Methods
A multistage openloop impedance pump is developed in the current work. The current work experimentally analyzes and evaluates the pumping performance of a multistage openloop impedance pump, with emphasis on the flow rates induced and pumping limitation. Analyses of flow rates, pressure head and excitation frequencies are performed in its nondimensional form.
Results
Taking a singlestage system as benchmark, enhancement of 35 and 33.3% is shown in the flow rate and pressure head respectively for a twostage system. Enhancement of 110 and 60% is shown in flow rate and pressure head of a threestage system in comparison to the singlestage system. For fourstage system, however, only 27 and 46.7% increments are demonstrated in the flow rate and pressure head, respectively.
Conclusions
The implementation of multistage system with single constant power input is demonstrated to be limited only to the threestage system where the declined pumping performance is exhibited in a fourstage system.
Background
Impedance pump is a simple pumping mechanism, which transports liquid through the mismatch of impedance in the system. It is very simple in design and offers a promising new technique for producing and enhancing flow rates for both macro and microscale devices (Avrahami and Gharib 2008; Rinderknecht et al. 2005).
Impedance pump is a type of valveless pump which does not require vanes or blades to operate. In addition, it offers a lownoise and lowenergy alternatives to current pumping system (Lee et al. 2011). A typical openloop impedance pump consists of an elastic tube, connected to rigid tubing, where the rigid section is connected to a reservoir. Excitation at a single location (not in the middle) of the fluidfilled elastic tube will result in unidirectional flow. This is due to the mismatch in impedance in the elastic tube. Such pumping mechanism has shown to be highly sensitive towards the impedance in the tube, the location, and excitation frequency (Avrahami and Gharib 2008; Lee et al. 2015; Manopoulus et al. 2006; Hickerson 2005; Timmermann and Ottesen 2009).
The first demonstration of valveless impedance pumping was performed by Gerhart Liebau in 1954, using an elastic tube connected to reservoirs at different heights (Liebau 1954, 1955). Following Liebau’s work in 1954, there have been several works studying the underlying physics of the system, be it numerical or experimental. The major contributions include Hickerson in 2005, Loumes in 2007, Rosenfeld in 2010, Meier in 2011, and Lee in 2014. All these works show significant milestones in the advancement of impedance pump, emphasizing on the exploration and application of impedance pump in different domains. To date, three designs of flow rate enhancement impedance pumping system were reported.
Loumes introduced a concept on multilayer bioinspired impedance pump. Flow output and inner wall motion are found to be highest at the resonant frequency. Excitation force needed to produce a significant flow was demonstrated to be relatively smaller than a singlelayer pump. Having said, with the similar excitation force, a larger flow rate can be induced. This design is operating at a singlestage system configuration. In 2010, Rosenfeld’s studied on the effects of sequential excitations on a single elastic tube and showed promising results where increase in flow rates was observed (Rosenfeld and Avrahami 2010). Similarly, Lee worked on a twostage system, integrated from two singlestage systems (Lee et al. 2013, 2014), and also showed an enhancement in the flow rates. However, Rosenfield’s design showed higher complexities, where synchronizations of locations and excitation frequencies are prerequisites for an efficient pumping. Lee’s design, on the other hand, is much simpler as the operating conditions are similar to a singlestage system. Locations and excitation frequencies for pumping are shown to be unaffected by the multistage configuration. The interim reservoir in between stages has also showed to be a valuable part of the system, serving as the driving mechanism in flow rate enhancement (Lee and Chong 2016).
While enhancement in flow rates is shown possible, there lies a question to which the enhancement can sustain under single and constant power source. The current work analyzes and evaluates the pumping performance of such a multistage openloop impedance pump, with emphasis on the flow rates induced and pumping limitation.
Methods
Experiments were conducted for the analysis and evaluation of the pumping performance for the multistage system. Four systems will be tested, that is

(i)
A conventional singlestage system

(ii)
A twostage system

(iii)
A threestage system, and

(iv)
A fourstage system
The schematic diagram of a full fourstage system is shown in Fig. 1. Location of compression is characterized in its nondimensional form for the ease of comparison. x _{ i } will be used to denote the instantaneous location of the mechanism while L _{ i } represents the total length of the elastic tube. Subscript, i, of 1–4 will be used to represent four different tubes. All four excitation mechanism is fixed at tube location \( \raisebox{1ex}{${x}_i$}\!\left/ \!\raisebox{1ex}{${L}_i$}\right. \) = 0.1. Detailed experimental methods and materials are available in Ref (Lee et al. 2014). Water is used as the working fluid in all experiments.
Analyses of the flow dynamics are in its nondimensional form. Excitation frequencies are presented as the Womersley number (α), as expressed in Eq. (1)
Flow rates (Q) are normalized against the highest induced flow rate in a singlestage system for better illustration of the enhancement (Eq. 2). Pressure head (H) is normalized with respect to the initial pressure head at height of 250 mm (Eq. 3).
With reference to Ref (Timmermann and Ottesen 2009), the theoretical resonant frequency is expressed as a function of the wave propagation, c, as shown in Eq. 4,
Substituting the wave propagation, c, as a function of the tube and fluid properties, the resonant frequency hence becomes
where
r is the radius of the tube
ρ is the water density
ω is the excitation frequency
μ is the coefficient of liquid viscosity
E is Young’s modulus of the elastic tube, and
h is the thickness of the tube
Resonant frequency is theoretically calculated to be 4.8 Hz using Eq. (5). Hence, a range of frequencies around the resonant frequency will be experimented to obtain the excitation frequency nearest to the resonance.
Results and discussion
Pumping performance of a multistage openloop impedance pump was analyzed and evaluated. This section consists of five subsections. First four subsections discuss on the pumping performance of each stages. The fifth subsection discusses the performance analyses and evaluation of a multistage system.
A singlestage system
A singlestage system experiments were conducted with selective range of the Womersley number from 75.15 to 99.41, which corresponds to frequencies of 4 to 7 Hz, which was deduced to be the resonance range for efficient pumping as calculated theoretically using Eq. (5). Experimental results are shown in Figs. 2 and 3. The trend shows a nonlinear dependency of pump’s responses towards the Womersley number. It is observed that the highest achievable height difference is 167 mm with deliverable pumping rate of 7.64 L min^{−1} at the Womersley number of 79.7 which corresponds to frequency of 4.5 Hz. This point of interest will serve as the benchmark pumping rate for comparison with the multistage systems of twostage, threestage, and fourstage systems. As shown in Fig. 2, the highest pumping rate occurs at the Womersley number of 79.9 (4.5 Hz), marking this number to be the resonance condition which is predicted theoretically to be 4.8 Hz with a difference of 6.3% due to the physical limitation of the actuators. Due to the instrumental limitation of hardware, the actuator was unable to work with frequency of 4.8 Hz. Hence, frequency interval of 0.5 is used instead. Frequency for efficient resonant pumping was located to be at 4.5 Hz. A plot of flow rate against pressure head (Q − H diagram) is shown in Fig. 4, indicating the flow development in a singlestage system.
The relation between Q and H is portrayed as a sinuslike curve. Q is normalized such that the maximum achievable flow rate is denoted as 1 while H is normalized with respect to the initial pressure head at water level of 250 mm in the reservoir. Q _{1} is the flow rate observed in the left reservoir, shown to be the source reservoir while Q _{2} is the flow rate observed in the right reservoir, the deliverable reservoir. Q _{1} shows a negative flow rate indicating a decrement in the pressure head while Q _{2} shows a positive flow rate indicating an increment in the pressure head. This shows that the transportation of water occurred in a unidirectional from left to right. The highest flow rate in the decreasing reservoir, Q _{1}, is observed to be at the pressure head of 0.25 while for the increasing reservoir, Q _{2}, is observed to be at 0.5; indicating the pumping work in the decreasing reservoir is higher than the increasing reservoir. This phenomenon is due to the impedance difference along the tube. At the compression location, \( \raisebox{1ex}{${x}_i$}\!\left/ \!\raisebox{1ex}{${L}_i$}\right. \) of 0.1, the impedance at the right end tube is smaller as compared to the left end tube. Therefore, the water would tend to move towards the region with lesser impedance. As the water flow through the tube, pressure gradient between both reservoirs changes as well, creating a higher pressure in the right reservoir. With higher pressure in the right reservoir, the forces required to overcome the fluid inertia in the reservoir hence increases, thus required longer time to reach a higher pressure head.
A twostage system
In accordance to Ref (Lee et al. 2014), it was shown that effective excitation frequency of the twostage system at resonance, similarly, is at 4.5 Hz. This shows that the resonant frequency for a multistage system is independent of the number of stages. Pumping is most efficient at the compression location synchronization of \( \raisebox{1ex}{${x}_i$}\!\left/ \!\raisebox{1ex}{${L}_i$}\right. \) = 0.1. With this, Q − H diagram of a twostage system is plotted and shown in Fig. 5. Q _{1} represents the flow rate measured in the left reservoir, Q _{2} is the flow rate in the middle reservoir, and Q _{3} is the flow rate measured in the right reservoir. The flow rates are normalized based on the highest flow rate found in a singlestage system that serves as benchmark of 1.0. In Fig. 5, the main transportation work is observed to be from the left reservoir to the right reservoir. It is shown such that Q _{1} is in the negative direction showing water flowing out of the reservoir while Q _{3} is in the positive direction indicating water flowing in. Q _{2} on the other hand is shown to be oscillating at the initial level with pressure head of ±0.1 and flow rate of ±0.8. This shows that with a twostage system, the middle reservoir serves as a transportation medium between the two end reservoirs. Similar phenomenon is also shown numerically in Ref (Lee and Chong 2016). The highest flow rate for Q _{1} and Q _{3} is observed to be at pressure head of 0.3 with flow rates of −1.5 and 1.35, respectively. The maximum pressure head of ±0.8 is shown where the flow rate reaches zero reading, giving a total pressure head of 1.6.
A threestage system
Flow rates of a threestage system were measured and presented in Fig. 6. Flow rate in the first reservoir, Q _{1}, is observed to increase drastically to the maximum flow rate at pressure head of −0.16, where gradient is shown to be extremely steep. The flow rate is then gradually decreases to −0.64 with very small gradient and stops at the maximum pressure head of −0.8. Similarly, for flow rate in the second reservoir, Q _{2} stops at pressure head of −0.8, the formation to the maximum flow rate is however different such that the flow rate is observed to gradually increase to maximum at a pressure head of −0.64. Drastic drop is then observed from −0.64 to −0.8. Based on the figure, both Q _{1} and Q _{2} are shown to have negative flow rates indicating them as the source reservoirs for the system.
Flow rate in the third reservoir, Q _{3}, is observed to have positive flow indicating water flowing into the reservoir. The maximum flow rate is observed at pressure head of 0.16; the flow rate is then shown to decrease and stop at pressure head of 0.48. Based on the curve, flow rate in the third reservoir is observed to be minimal. Looking into Q _{4}, high positive flow rate is observed to reach a maximum of 2.1 at pressure head of 0.16. Flow rate is then observed to decrease gradually until pressure head of 1.12. Based on the curves, it is observed that the main deliverable pumping work is from Q _{1} and Q _{2} to the last two reservoirs where flow rates are shown to be very high. Deliverable rates are observed to have double increment of 1.1 than the singlestage system. Normalized increment of 0.7 is observed between the two and threestage systems.pt?>
A fourstage system
Figure 7 illustrates the Q − H diagram for a fourstage system. Based on the figure, it is shown that the source of water transportation is from Q _{1} and Q _{2}. The main source is shown to be Q _{1}, while Q _{2} is shown to be the secondary source similar to the threestage system. The curve trend is however different such that Q _{1} reduces more drastically as compared to the threestage system; maximum flow rate is observed to be at the similar pressure head of −0.16. Q _{2} is observed to have similar trend for both three and fourstage system where the maximum flow rate is observed to be at −0.48. Flow rate at the intermediate reservoir, Q _{3}, is observed to oscillate around the initial pressure head, similar to the curve shown for twostage system as shown in Fig. 5.
Q _{4} and Q _{5} are observed to be the deliverable reservoir where positive flow rates are shown in Fig. 7. Q _{4} shows minimal flow rate and stops at pressure head of 0.48, similar to the curve shown in the threestage system. Q _{5} on the other hand shows a higher flow rate as compared to Q _{4}. Q _{5} represents the final deliverable flow rate for a fourstage system and based on the flow rate shown; Q _{5} is obviously lower than the threestage system, indicating that for a multistage openloop impedance pump, the maximum number of stages that can be implemented with notable and significant enhancement is only limited to threestages. It should be noted that this limitation is subjected only to a single and constant power source.
Performance analysis and evaluation
Performance of different stages in multistage openloop impedance pump as function for pressure heads and flow rates was compared and analyzed. The performance was compared nondimensionally where the pressure head is with respect to the initial pressure head at water level of 250 mm and the flow rate is with respect to the maximum flow rate induced (7.64 L min^{−1}) in a conventional singlestage system. Performance of a singlestage system was made the benchmark of comparison where the highest flow rate achieved is normalized to the number of 1. The pressure head on the other hand was observed to reach the highest increment of normalized value 0.6 giving a total pressure head difference of 1.2. The Q − H diagram of a singlestage system charting the performance of singlestage system is shown in Fig. 4.
Extensive experiments were performed for the twostage impedance pump on the complex parameters governing its performance. A maximum pressure head of 0.8 was obtained giving a total pressure head difference of 1.6; this indicates an improvement of 33.3% on the achievable pressure head difference. The highest flow rate of 1.35 was induced in the twostage system, showing an improvement of over 35% on the instantaneous flow rate in the system. Water was however observed to be oscillating at its initial water level in the interim level for a twostage system. The interim reservoir also shown to be the transportation and driving mechanism for flow rate enhancement (Lee and Chong 2016).
The experiments on multistage were furthered with the investigation on a threestage system. As shown in Fig. 6, maximum incremental pressure head was observed to be 1.12, giving a total pressure head difference between the first and last reservoir of 1.92. A percentage increment of 60% was observed in the pressure head with respect to a singlestage system. The maximum flow rate of 2.1 was observed for the threestage system, showing an increment of 110% as compared to the singlestage system. Comparison with the twostage system shows an improvement of 20 and 55.6% in the pressure head and flow rate, respectively (Table 1).
Figure 7 shows the Q − H diagram of a fourstage system. A maximum pressure head of 0.96 with a total pressure head difference of 1.76 and maximum flow rate of 1.27. Comparing with the previous trend of performance of multistage system, the fourstage system had obviously decreases in its performance. A performance increment of 46.7 and 27% was observed in the pressure head and flow rate respectively as compared to a singlestage system. Improvement of 10% in pressure head was shown comparing with twostage system; it was however shown a decrease of 5.9% comparing flow rate in four and twostage system. A complete decrease in performance was observed when comparing with the threestage system where a percentage decrement of 8.3 and 39.5% was observed for the pressure head and flow rate, respectively.
Based on the experimental studies conducted, for an even number of multistage system, it is shown that the interim reservoir serves as a medium of fluid transportation where water is observed to oscillate around the initial water level. A separate study (Lee and Chong 2016) showed that the oscillation is due to the flow circulation within the reservoir, which works as driving mechanism in increasing the flow rates. For odd number of multistage system, transportation of fluid is observed to be the reflection of each other with respect to the symmetric of the system. Main work of fluid transportation is observed to be from the first reservoir from the left end of the system for all cases. As the studies on multistage system work with one single and constant power source, the performance of different stages can be explicitly measured and compared. Actuators are drawing from only one power source through a parallel connection; therefore, less power will be drawn to the actuators as the number of stages increased. This will create less compression forces on the elastic tubes. With less compression induced from the mechanism due to lower power input, lower flow rate is generated. As the compression force and excitation frequencies are closely related to the speed of wave, it is therefore essential that the compression force is maintained such that the speed of wave is maintained at optimum for the highest efficiency for the system. Speed of wave of approximately 30 cm/s is observed in the tube, with the power input of 720 W and excitation frequency of 4.5 Hz. Performance of the pump is observed to increase as the number of stages increases. This can be observed in both the pressure heads and flow rates induced in this multistage system. Based on the studies performed, it is however shown that the maximum number of stages that the pump can go with power source of 720 W is threestage as decline in performance is observed in the fourstage system; indicating the source power input is incapable to produce the sufficient wave speed for the system to work at its finest. Q − H performance with respect to number of stages is presented in Fig. 8.
Conclusions
For a multistage system drawing power from a single constant power source, it is well expected that there would be a limitation to which the enhancement can reach. Analysis and evaluation on the pumping performance of such a multistage system are conducted in the current work to study the highest possible enhancement with the increase of stages, as well as the system’s limitation. All studies are conducted using only a single power source of 720 W. A singlestage system was made the benchmark of multistage system here, where a normalized flow rate of 1 and normalized pressure head of 0.6 are established. With the increase in the number of stages, an improvement in the performance is observed. Enhancement of 35 and 33.3% is shown in the flow rate and pressure head respectively for a twostage system. Enhancement of 110 and 60% is shown in flow rate and pressure head of a threestage system in comparison to the singlestage system. For fourstage system, however, only 27 and 46.7% increments are demonstrated in the flow rate and pressure head, respectively. The implementation of multistage system with single constant power input is demonstrated to be limited only to the threestage system where the declined pumping performance is exhibited in a fourstage system.
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Acknowledgements
The authors would like to thank the Ministry of Science, Technology and Innovation of Malaysia for providing the financial support in this project; Faculty of Engineering and Science (Curtin University Sarawak) and Faculty of Engineering (University of Nottingham Malaysia Campus) for providing the facilities and equipment for conducting this research.
Authors’ contributions
VCC conducted the experimental work, and contributed the major part of the paper. YA and KC participated in the analysis and evaluation of experimental data. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Lee, V.C.C., Abakr, Y.A. & Woo, K.C. Performance evaluation of multistage openloop impedance pump. Int J Mech Mater Eng 12, 11 (2017). https://doi.org/10.1186/s4071201700791
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DOI: https://doi.org/10.1186/s4071201700791
Keywords
 Valveless
 Multistage
 Impedance pump
 Performance evaluation
 Limitation