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Sensors and Actuators A: Physical
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / s n aEvaluation of the performance of a valveless micropump by CFD
and lumped-system analyses
Yeng-Yung Tsui
∗, Shiue-Lin Lu
Department of Mechanical Engineering, National Chiao Tung University, Hsinchu 300, Taiwan, ROC
a r t i c l e i n f o
Article history: Received 18 March 2008
Received in revised form 29 June 2008 Accepted 30 June 2008
Available online 15 July 2008 Keywords: Valveless micropumps Nozzle/diffuser Lumped-system analysis CFD analysis
a b s t r a c t
The flow in a valveless micropump is analyzed using both the CFD method and the lumped-system method. In the multidimensional simulation of the CFD model, the Navier–Stokes equations are solved using a finite volume method suitable for the use of unstructured grids. The moving membrane is modeled by imposing a reciprocating velocity boundary condition. It is seen that a good agreement with measurements can be obtained for various back pressures by adopting an appropriate membrane shape blending the parabolic and the trapezoidal profiles. The multidimensional predictions serve as benchmark solutions to the lumped-system analysis. In the latter analysis two correlations for the loss coefficients of the nozzle and the diffuser are employed. The results show that with a more accurate one of the two correlations, a better agreement with the multidimensional calculations is yielded. The performance of the pump can be evaluated by considering the pumping efficiency. The pumping efficiency can be approximately formulated in two different ways, depending on the average ratios of the outlet flow rate to the inlet flow rate in the pumping and supply stages. In the averaging process to determine the mean ratios, the transient region between the pumping stage and the supply stage is excluded. This leads to even closer agreement to the multidimensional calculations in the pumping efficiency.
© 2008 Elsevier B.V. All rights reserved.
1. Introduction
Microfluidic systems are of interest in medical testing, drug delivery, chemical analysis, chip cooling and many others. Microsystems have the advantages of small volume, cheap cost, high precision and fast reaction time. Micropumps are essential devices in the microfluidic systems, which provide momentum to cause fluid flow. The development of micropumps has been based on various principles[1–3]. They can be divided into two categories: mechanical and non-mechanical[1]. Most mechanical pumps use oscillatory movement of mechanical parts to drive the flow. Some examples include check valve micropumps, peristaltic micropumps and rotary micropumps. In the non-mechanical category, the cause of fluid flow is mainly via electric–fluid interactions, such us elec-trokinetic micropumps, electrohydrodynamic micropumps, etc.
In general, mechanical micropumps have higher flow rates than the non-mechanical micropumps. Among the various mechani-cal micropumps the valveless membrane micropumps have drawn much attention in recent years. The actuation of the reciprocating membrane can be achieved by different ways, such as piezoelectric discs, pneumatic actuators, thermomechanic actuators, to name
∗ Corresponding author. Tel.: +886 3 5131556; fax: +886 3 5720634. E-mail address:[email protected](Y.-Y. Tsui).
a few. Most of the membrane micropumps are constructed with intake and exhaust valves to direct the fluid flow. During minia-turization of these pumps, the presence of the moving valves leads to manufacturing difficulties and suffers the risk of wear and fatigue. Furthermore, if particles are present in the pumps, there is a concern about valve clogging. To avoid these drawbacks, a novel idea is to replace the moving valves with fluidic diodes, such as nozzle/diffuser elements or the valvular conduits. Fluidic diodes have different flow resistances when the direction of the flow through the devices is reversed. In the positive direction through a diffuser the velocity is reduced and static pressure is gradually recovered. As the fluid flows in the reversed direction, the device functions as a nozzle and larger loss is caused. Thus, for a recip-rocating flow through such a device, it performs the function of directing a net flow in the positive direction. Incorporation of the nozzle/diffuser elements into micropumps was realized first by Stemme and Stemme[4]and then by Gerlach and Wurmus[5]. Ols-son et al.[6]described a flat, planar design with the nozzle/diffuser arranged in the same plane as the pump chamber. This kind of pla-nar micropumps can be fabricated using a commercially available injection-molding machine for compact disc manufacture[7]. This means that it can be produced in high volumes at low costs.
A simple way to analyze the flow in valveless micropumps is the use of the lumped-system method. This approach ignores spa-tial variation and focuses mainly on the time variation. Ullmann 0924-4247/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
Nomenclature
AL, At areas of the two ends of the nozzle/diffuser
f oscillating frequency
Fc, Fd convective and diffusive fluxes
h deflection of the membrane
K loss coefficient
Kd, Kn loss coefficients of the diffuser and the nozzle ˙
m mass flux
P pressure
Pc pressure in the chamber
Pin, Pout pressures at the inlet and the outlet
Q volumetric flow rate
Q1, Q2 volumetric flow rates at the inlet and the outlet
r radial distance
ro radius of the pump chamber
rl radius of the piezo disc
Sf surface vector of considered face
t time
T one oscillating period
Vm half the maximum volume swept by the membrane
w weighting factor
W oscillating velocity of the membrane
Z time-dependent variation of the deflection of the
membrane
Greek symbols
ˇ ratio of Q2to Q1
ıpc distance vector connecting the principal node p and the neighboring node c
pumping efficiency R real efficiency 1,2,3 approximate efficiencies fluid viscosity fluid density Subscripts c chamber d diffuser en entrance ex exit in inlet
max maximum value
n nozzle
out outlet
p pumping stage or parabolic profile
s supply stage
t trapezoidal profile
Superscripts
t total value
[8]proposed a simple model based on the principle of mass con-servation over the pump chamber. With this method a variety of single-chamber and double-chamber micropumps were analyzed to find the flow output rate. In the lump model of Olsson et al.[9], the conservation of both the mass and energy was considered. A sub-model, based on momentum balance over the membrane, was included to account for the interaction between the membrane and the fluid flow. In the study of Pan et al.[10], the movement of the membrane was modeled by a partial differential equation taken from the clamped-thin-plate theory. The equations were approx-imately solved by the small parameter perturbation method and
the Galerkin method. However, the flow unsteadiness within the micropump was not taken into account. Pan et al.[11]showed that this inertial force is not negligible, compared with the viscous loss, in the dynamic coupling analysis.
The recent development of Navier–Stokes equations solvers has made three-dimensional calculations possible in complicated geometry. This multidimensional analysis approach can provide accurate solutions as well as detailed flow field. Nguyen and Huang [12]reported a numerical simulation of pulse-width-modulated micropumps with nozzle/diffuser elements. The vibration of the membrane was modeled by either a moving wall or moving veloc-ities as the boundary condition. In both models, the interaction between the structure analysis and flow analysis is neglected. Yang et al.[13]evaluated the performance of micropumps with two chambers arranged in parallel or series combination. In the series arrangement with 90◦ phase angle difference between the two membranes, an eightfold increase of flow rate was yielded, relative to the single-chamber pump. A study of complete coupling between the electrical, mechanical and fluid systems in a piezoelectric-actuated micropumps was conducted by Fan et al.[14]using finite element method and computational fluid dynamics. The behavior of the membrane at different frequencies, which may affect the pumping rate, was investigated. It was found that the pumping rate as well as the deflection amplitude of the membrane increases with the increasing frequency of the vibrating membrane for frequencies less than 7.5 kHz. Further increase of frequency leads to degradation of pumping rate due to the undesirable deflection way of the mem-brane. The fluid–structure interaction and the electro-mechanical coupling were also employed in the studies by Yao et al.[15]and Jeong and Kim[16].
It could be found from the above numerical studies that most CFD simulations were performed using commercial codes, espe-cially the CFD-ACE. Recently, a computational procedure based on the use of the fully conservative finite volume method and the unstructured mesh was developed for both compressible and incompressible flows by the group of the present authors[17,18]. In order to analyze the periodic flow encountered in micropumps, this method is modified to cope with the unsteady state. In addition, a way to implement the pressure boundary condition, easily applica-ble to unstructured grids, is addressed. A lumped-system analysis is also conducted. The lump model is assessed by comparing with the results obtained from multidimensional calculations.
2. Multidimensional method
A drawing of the micropump under consideration is shown in Fig. 1. Only half of the pump is considered in calculations due to its symmetric geometry. The membrane is placed on the top of the main chamber with a diameter of 6 mm. A piezo disc of diameter 4 mm is attached to cause vibration of the membrane. The open-ing on the left chamber is the inlet and that on the right chamber the outlet. Both openings have a diameter 1 mm. Nozzle/diffuser elements are used to connect the inlet and the outlet chambers on both sides of the main chamber. The length of this element is 1 mm with a diffusion angle of 7◦and the width at the throat is 0.1 mm. The height of the micropump is 0.2 mm.
The dynamics of the flow in the micropump is modeled by the incompressible Navier–Stokes equations which can be cast into the following form:
∂
∂t +
∇
· ( V) =∇
2 + S (1)
where represents Cartesian velocity components and the source
S includes the pressure gradients. Integrating the equation over a
Fig. 1. Configuration of the valveless micropump.
divergence theorem lead to: V t ( − o)+
s V · d−→S = s∇
· dS + S V (2)where V is the volume of the considered cell, o denotes the velocity component at old time step. Here the fully implicit scheme is used for the time discretization. The convection and diffusion terms have been transformed into a surface integral form by the divergence theorem. The convective flux through the surface of the control volume can be approximated by
Fc=
( V · S)ff (3)
where the subscripts f stand for face values, Sfis the surface vec-tor, and the summation is taken over all the faces of the control volume surface. The face valuefis approximated using a scheme blending the upwind and the central differences with a weighting factor of 0.9 biased toward the central difference. The diffusive flux is approximated by Fd=
S 2 f ıpc· Sf (c− p)+
∇
f Sf− S2 f ıpc· Sf ıpc (4) The subscripts p and c denote the centroids of the principal cell and the neighboring cell on the two sides adjacent to the face f, ıpcis the distance vector directed from node p to node c (seeFig. 2)∇
f represents the gradient at the face obtained by linear interpolation from the two nodes p and c.Fig. 2. Illustration of a typical control volume with neighboring cells.
The coupling between the momentum and the continuity equa-tions is treated in a manner similar to the SIMPLE algorithm. The velocities and pressure are collocated on the centroid of each control volume. To avoid checkerboard oscillations result-ing from the decouplresult-ing between the velocity and the pressure, the momentum interpolation method is adopted. Details about the discretization and the method can be found in the studies [17,18]. However, for unsteady flow the use of SIMPLE algorithm requires iteration in each time step, which is time consuming. Therefore, the non-iterative, predictor–corrector procedure of PISO algorithm [19]is employed. In the predictor step, the momen-tum equation is solved using the prevailing pressure field. It is followed by a corrector step in which the velocity and pressure are adjusted such that the mass is conserved. Although the new velocity field after the first corrector step satisfies the continu-ity constraint, the momentum equation is not adequately solved. The PISO algorithm relies on a second corrector to make the pres-sure field get rid of the mass imbalance left by the predictor step and give letter approximation to the momentum conservation. It is this step to make the PISO different from the SIMPLE and result in higher computational efficiency even in steady flow calcula-tions.
For the micropump under investigation, appropriate boundary conditions must be imposed on both the membrane and the inlet and outlet. It is usual to specify a pressure difference across the inlet and the outlet and the flow rate through these boundaries are sought. To determine the mass flux, or the velocity, at an open boundary with a specified pressure, one approach is to make an approximation to the momentum equation in a manner similar to that used at an internal face. Although this method has been successfully implemented in steady flow calculations [20,21], it may not be appropriate in the non-iterative procedure of the PISO algorithm because the mass is not conserved. In the following, a method, ensuring conservation of mass, is described.Fig. 3 illus-trates a control volume P next to an open boundary. The boundary pressure Pb is prescribed at the centroid of this cell. As usually done for non-staggered grid calculations, an extrapolation prac-tice is undertaken to find the pressure on the boundary node B. With this boundary face pressure, the velocity at the node P can be solved for in the momentum predictor step in the same way as the other internal nodes. After the mass fluxes through all the internal faces are calculated using the momentum interpolation method mentioned above, the mass flux through the open bound-ary ˙mbis then obtained via conservation of mass in this boundary
Fig. 3. Illustration of implementation of the pressure condition at a boundary cell. cell. ˙ mb+
˙ mi= 0 (5)where ˙midenotes the mass flux through an internal face and the summation is taken over all the internal faces.
Since the main concern of this study is the fluid flow in the pump, the interaction between the fluid and the structure is neglected. The deflection of the membrane can be modeled by prescribed shapes. The plate-and-shell theory of Timoshenko and Woinowsky-Krieger[22]leads to a fourth-order polynomial for the deflection of a clamped circular plate with uniformly distributed load. The fluid–structure interaction calculations by Jeong and Kim [16]showed that the displacement of the piezoelectric membrane varies in a trapezoidal-like profile. In the present study, the deflec-tion of the membrane is modeled by a combinadeflec-tion of two profiles. One is of parabolic profile:
hp(r) = hmax
1−r2 r2 o (6) The other is of trapezoidal profile:ht(r) = hmax· Max
1, ro− r ro− r1(7) Here hmaxis the maximum deflection of the membrane at the cen-ter, rothe radius of the pump chamber, r1the radius of the piezo disc, and r is the coordinate in the radial direction. These two profiles are combined in a linear manner:
h(r) = (1 − w)hp(r) + wht(r) (8)
wherew is a weighting factor. The determination of the weighting factor depends on the pressure difference between the inlet and the outlet, which will be given later in Section5. Because of the small membrane deflection (hmax= 1m) compared with the chamber height (0.2 mm), the influence of the variation of the pump chamber can be ignored. Instead of the moving wall boundary condition, a velocity in harmonic motion is imposed on the membrane which is assumed to be fixed at its neutral position. The time-dependent variation of the deflection is given by
Z(r, t) = −h(r) cos(2f · t) (9)
The oscillating velocity is then obtained as
W(r, t) = 2f · h(r) · sin(2f · t) (10)
Here f is the frequency of the harmonic motion.
The flow in the pump is assumed to be stagnant initially. Cal-culations proceed until the variation of the flow becomes periodic.
Fig. 4. Illustration of the three modes for the valveless pump. (A) Pump mode:
Pin< Pout< Pc; (B) supply mode: Pc< Pin< Pout; (C) transition: Pin< Pc< Pout.
It will be seen that at least three periods are required to reach the fully periodic state.
3. Lumped-system method
As shown inFig. 4, the pumping process is divided into three modes: pump mode (Pin< Pout< Pc), supply mode (Pc< Pin< Pout), and transition mode (Pin< Pc< Pout). Here Pcis the chamber pres-sure, Pinthe inlet pressure, and Poutis the outlet pressure. The inlet pressure Pin is always smaller than the outlet pressure Pout. The volumetric flux through a nozzle/diffuser element is related to the pressure difference across this element by
P =
2At2
KQ2 (11)
where K is the loss coefficient and At is the cross-sectional area at the throat. With the mass conservation over the chamber, the equations governing the three modes can be given in the following: (1) Pump mode (Pin< Pout< Pc): The fluid emerges from the cham-ber due to the high chamcham-ber pressure Pc The intake element functions as a nozzle and the outflow element behaves like a diffuser. −Q1+ Q2= 2f · Vmsin(2f · t) (12a) Pc= Pout+ 2At2 Kt dQ 2 2 (12b) Pc= Pin+ 2At2 Kt nQ12 (12c)
(2) Supply mode (Pc< Pin< Pout): The fluid is forced to flow into the chamber. Comparing with the pump mode, the functions of the intake and outflow elements are reversed.
Q1− Q2= 2f · Vmsin(2f · t) (13a)
Pc= Pout− 2At2
Kt
Pc= Pin− 2At2 Kt dQ 2 1 (13c)
(3) Transition mode (Pin< Pc< Pout): In this case, the fluid outside the outlet flows into the chamber and the outflow element works as a nozzle. In the mean time, the fluid inside the cham-ber flows out though the intake element, which also works as a nozzle. −Q1− Q2= 2f · Vmsin(2f · t) (14a) Pc= Pout− 2At2 Kt nQ22 (14b) Pc= Pin+ 2At2 Kt nQ12 (14c) In the above,Kt
nandKdtdenote the total loss coefficients of the noz-zle and the diffuser, respectively, and Vmis one half of the maximum volume swept by the membrane. It is noted that the flow rates Q1 and Q2are regarded as positive when they are in the direction of
x-axis, i.e., in the direction from the left (inlet) to the right (outlet).
The above equations can be solved to find Q1, Q2and Pcif the total loss coefficientsKt
dandKnt are known.
The pressure loss across a nozzle or a diffuser consists of three parts. The major part is that due to the flow in the noz-zle/diffuser itself. Other losses result from the sudden contraction at the entrance as well as the sudden expansion at the exit. The total loss coefficients can be expressed as[23]
Kt d= Kd,en+ Kd+AAt LKd,ex (15a) Kt n=
A t AL2 Kn,en+ Kn+ Kn,ex (15b)
where At/ALis the area ratio of the two ends of the nozzle/diffuser with At being the smaller one and ALthe larger one. For sharp-edge entrances, as in the present study, Kd,en= Kn,en= 0.4 and for exits, Kd,ex= Kn,ex= 1 regardless of the shape of the exit region[24]. The major loss coefficients, in general, depend upon the Reynolds number, or the flow rate. To simplify the problem, both Kdand Kn are assumed to be a function of one half of the instantaneous flow rate displaced by the membrane:
Kd(t) = Kd(Q(t) = fVm|sin(2ft)|) (16a)
Kn(t) = Kn(Q(t) = fVm|sin(2ft)|) (16b)
Three-dimensional simulations have been conducted to find the correlation between the loss coefficients and the flow rate. The geometry of the nozzle/diffuser element is the same as that used in the present micropump. The results (model A) are shown inFig. 5. It is not surprised that the total loss coefficient for the nozzleKt n is greater than the total loss coefficient for the diffuser Kt
d. The relationships can be expressed as
Kt
d= 1.315 × 10−7Q−0.921+ 0.5981 (17a)
Kt
n= 1.173 × 10−6Q−0.8112+ 1.204 (17b)
In determination of the above correlations, three levels of grid with 15,000, 33,600, and 53,760 cells were tested. It was revealed that only minor differences were detected among the results for different grids.
A model was proposed by Yang et al. [25] to correlate the pressure loss and the Reynolds number for the flow through a micronozzle/diffuser. Employing this model leads to the following correlations:
Kt
d= 1.5838 × 10−8Q−0.9995+ 0.602 (18a)
Fig. 5. Variation of the loss coefficients against the flow rate for the nozzle and
diffuser for the two models. Kt
n= 1.5838 × 10−8Q−0.998+ 1.081 (18b)
They are also shown inFig. 5as model B. The correlations for the two models are valid for Reynolds numbers less than 400. The flows in the present micropump fall in this range.
4. Pumping efficiency
The real efficiency of the membrane pump is defined by R= Q
2Vm
(19) HereQ is the net volume of the flow through the pump in one period T and 2Vmis the volume swept by the membrane from the top dead center (TDC) to the bottom dead center (BDC). The change rate of volume of this harmonic motion can be written as
dV
dt = 2fVmsin(2ft) (20)
The net flow volume through the pump in one period T is obtained from
Q =12
T 0(Q2(t) − Q1(t)) dt (21)
where Q1is the flow rate into the chamber at the inlet and Q2that out of the chamber at the outlet. We assume that Q1 varies in a harmonic manner similar to the membrane:
Q1(t) = Qmsin(2ft) (22) Then Q =Qm 2
T 0 (ˇ(t) − 1) sin(2ft) dt (23)whereˇ is the ratio of the outlet flux to the inlet flux. ˇ(t) =Q2(t)
Q1(t)
(24) The change rate of the chamber volume dV/dt is equal to 2Q1 approximately. Thus,
Vm= Qm
Substituting Eqs.(23) and (25)into Eq.(19)yields an approximate pumping efficiency denoted by1:
1=f 4
T 0(ˇ(t) − 1) sin(2ft) dt (26)
The formulation can be further simplified if the flow rate can be represented by some constants. It was seen that the variation of the ratio is considerably flat in either the pumping stage or the supply stage except near the transition region. Two average flow ratios corresponding to the pumping stage and the supply stage are defined as ˇp= 5 2T
9T/20 T/20 ˇ(t) dt (27a) ˇs= 5 2T 19T/20 11T/20 ˇ(t) dt (27b)The reason for the integration over 2T/5 only is to avoid the tran-sition region, which will become clearer in the case tests shown later. Eq.(26)can then be integrated to yield:
2= f 4
T/2 0 (ˇp− 1) sin(2ft) dt + T T/2 (ˇs− 1) sin(2ft) dt= 14(ˇp− ˇs) (28)
This pumping efficiency only depends on the constant ratiosˇp andˇs. It is noted that in general,ˇp> 1 andˇs< 1.
Another simplified form of pumping efficiency can be derived from the following assumptions for Q1 and Q2 in the first half period: Q1(t) = Qm1sin(2ft) (29a) Q2(t) = Qm2 sin(2ft) (29b) Thus, Q =
T/2 0 (Q2(t) − Q1(t)) dt = 1 f(Qm2− Qm1) (30) Since Vm= 1 2f(Qm1+ Qm2), (31)The pumping efficiency can be given by 3=Qm2− Qm1 Qm2+ Qm1 (32) Let ˇp=Qm2 Qm1 (33) It is obtained 3=ˇp− 1 ˇp+ 1 (34) In the multidimensional and lumped-system analyses,ˇpis esti-mated using Eq.(27a). It is interesting to notice that the same expression for the pumping efficiency3can be found in the study of Olsson et al.[4]by replacingˇpby the square root of efficiency of the nozzle/diffuser1/2nd where
nd=Kn Kd
(35)
Fig. 6. The net flow rates in each period for different levels of grid for the case Pb= 0.
Knand Kdare loss coefficients of the nozzle and the diffuser, respec-tively. It can be noticed from Eq.(11)that for a constant pressure loss across a nozzle/diffuser, K1/2∼ 1/Q.
5. Results
The settings of the geometry of the micropump are given in Fig. 1. It is assumed that the membrane reciprocates in a harmonic motion with a frequency of 2200 Hz. The maximum amplitude of the vibration is 1.0m. The large ratio of the chamber height to the vibration amplitude justifies the use of moving velocity instead of moving surface as the boundary condition. Zero pressure is spec-ified at the inlet and various back pressures Pb, ranging from 0 to 5900 Pa, are set at the outlet. Calculations were undertaken for a
Fig. 7. Variation of the flow rates at the inlet (Q1) and outlet (Q2) for (a) Pb= 0 and
Fig. 8. The plots of streamlines and pressure contours in the two nozzle/diffuser regions at t = T/4.
number of periods. The net flow rates averaged over each period for different levels of grid are given inFig. 6for the case with Pb= 0. It can be observed that it takes at least three periods for the flow to become fully periodic.
The pumping effect caused by the valveless micropump becomes clear by viewing the variation of the flow rates−Q1 (neg-ative value of Q1) and Q2, as shown inFig. 7. For the case Pb= 0 the flow rate at the outlet Q2is higher than the flow rate at the inlet−Q1 in the first half of a period (the pumping stage) due to the diffuser function of the element connecting the outlet chamber and the noz-zle function of the element connecting the inlet chamber. In the second half of the period, the absolute value of Q2becomes lower than that of Q1because the flow direction is reversed (the supply stage). The flow rate Q2can be approximately expressed by a sinu-soidal function a + b sinωt and the flow rate −Q1by−a + b sin ωt. The constant a represents the net flow rate. For the case Pb= 5310 Pa the variations of the flow rates −Q1 and Q2 are very close to each other and the net flow rate, or the constant a, approaches zero.
To illustrate the flow field, the plots of streamlines and pressure contours in the two nozzle/diffuser regions at t = T/4 and 3T/4 are shown inFigs. 8 and 9, respectively. At t = T/4 in the pumping stage, the flow is directed from the main chamber toward the inlet and the outlet chambers. The discharge element on the left functions as a nozzle and that on the right as a diffuser. The pressure decreases
gradually is the nozzle whereas the pressure drops sharply at the entrance of the diffuser and then recovers gradually. It can be seen that there exists a recirculation zone near the entrance corner in the nozzle and another recirculation zone before the fluid enters the nozzle. The former is caused by the sharp entrance while the latter is due to the fact that as seen inFig. 9for t = 3T/4, the recircula-tion formed by the flow emerging from the nozzle/diffuse element in the supply stage persists and moves down toward the central region during the pump stage. The recirculating flow leads to a low pressure region at the entrance of the nozzle and, thus, causes addi-tional losses. The flow type is reversed in the supply stage at t = 3T/4. It can be detected that both the velocity and pressure fields inFig. 9 are very similar to those inFig. 8except that the roles of the two nozzle/diffuser elements are interchanged.
The multidimensional solution procedure is validated by com-paring the predicted net flow rates with the measurements of Olsson et al.[7], as shown inFig. 10. The calculations were per-formed using the parabolic, trapezoidal, and blending profiles, as given by Eq.(7). As expected, the net flow rates decrease when the back pressure increases. In general, the flow rates are overpredicted by using the trapezoidal profile, especially for sufficiently large Pb, and underpredicted when the parabolic profile is employed. The higher flow rate obtained by using the trapezoidal profile is due to its larger volume displacement. The strategy in the blending pro-file is to use the trapezoidal propro-file at Pb= 0, the parabolic profile
Fig. 10. Comparison of predicted net flow rates using different membrane profiles
with measurements at various back pressures.
at Pb= 5900 Pa, and a linear combination of the two in between. As seen from the figure, the resulting flow rates become much closer to the experiment data.
Calculations were conducted using grids with 65,288, 117,232 and 210,258 cells. The computational time step is T/400 for the two coarse meshes and is reduced to T/800 for the finest mesh to suppress instabilities which may lead to divergence of the solu-tion procedure. As shown inFig. 6, moderate differences in the results for different meshes could be detected. No further efforts were taken to find fully grid-independent solutions due to too much computer resources required. However, it can be concluded from the above tests that with a suitable membrane profile accu-rate results can be obtained. The multidimensional results serve as benchmark solutions for the lumped-system analysis to com-pare. Two different expressions have been described to correlate the loss coefficients and the flow rates for the nozzle/diffuser ele-ment. Eqs.(17a) and (17b), termed model A, were obtained from the multidimensional simulation for the nozzle/diffuser configura-tion considered in the present study. Eqs.(18a) and (18b), termed as model B, were adopted from the study of Yang et al.[25]. The resulting flow rates at the outlet (Q2) in one period are presented inFig. 11for back pressures Pb= 0 and 5310 Pa. The first half of the period corresponds to the pumping stage and the second half the supply stage. The positive Q2indicates the flow out of the outlet. It is seen that the curves by the models follow the multidimen-sional results closely for the case with Pb= 0. It can be detected that the values of Q2 at t/T = 0 and 1 are not zero in the multi-dimensional calculations due to the inertial effect of the flow at the end of the supply mode. This effect is not accounted for in the lump models. The differences between the multidimensional simu-lation and the model results increase with increasing back pressure. There appears non-smoothness in the variation of Q2obtained by the lumped-system analyses in the transition region between the pumping stage and the supply stage. This is also ascribable to the inertial effect not taken into account. It is noted that the driving momentum of the membrane is low in the transition stage and, thus, the inertial effect becomes more prominent. When a pressure difference exists between the inlet and the outlet, the variation of the pressure in the chamber becomes not so regular, resulting in the kinks seen inFig. 11b. Comparing with the multidimensional
Fig. 11. Variation of the flow rates at the outlet in one period predicted by the 3D
simulation and the two lump models for (a) Pb= 0 and (b) Pb= 5310 Pa.
simulation, closer agreement is obtained by the model A. This is evident inFig. 12by examining the real efficiencyR.
The variation of the flow rate ratioˇ(=Q2/Q1) is shown inFig. 13. Generally, the value ofˇ is greater than 1 in the first half of the period (the pumping stage) and less than 1 in the second half (the supply stage) in the case Pb= 0, implying a net flow from the inlet to
Fig. 12. Comparison of the real efficiencies predicted by the 3D simulation and the
Fig. 13. Comparison of the flow rate ratiosˇ in one period obtained by the 3D simulation and the two lump models for (a) Pb= 0 and (b) Pb= 5310 Pa.
the outlet. In the pumping stage, the curve obtained by the multidi-mensional simulation is of concave type and those by the two lump models are of convex type. The curve types for the multidimen-sional simulation and the models are interchanged in the supply stage. The different types of curves are believed to be caused by the inertial effect being absent in the lump models. With Pb= 5310 Pa, there is almost no net flow. This is reflected in the fact that the val-ues ofˇ become close to one, as seen inFig. 13b. In contrast to the case Pb= 0, the curves for the multidimensional simulation and the two models are of the same type.
It can be seen from the above figure that the value ofˇ may approach positive or negative infinity and completely different characteristics inˇ appear in the multidimensional calculations and the model predictions in the transition region. This is the rea-son why the two mean flow ratios ˇpand ˇs, appearing in the simplified formulations for pumping efficiency (2and3), are esti-mated by averaging over 4T/5 only in the corresponding first and second halves of a period to avoid the transition region as done in Eqs.(27a)and(27b). The variations ofˇpandˇsfor different back pressures are displayed inFig. 14. Obviously, by using the model A the agreement with the multidimensional simulation is closer. As shown inFig. 15, the approximate efficiencies2and3obtained
Fig. 14. Comparison of the flow rate ratiosˇpandˇsobtained by the 3D simulation
Fig. 15. Comparison of the predicted efficiencies2and3by the 3D simulation
and the two lump models.
from the multidimensional simulation are only marginally lower than the real efficienciesR. The under-estimate of the approxi-mate efficiencies by the lumped-system analysis is considerable, with the model A being the better. The comparison between the two approximate efficiencies indicates that3is slightly better at high back pressures. It is of interest to notice that comparing with the real efficienciesRobtained from the lumped models shown in Fig. 12, the approximate efficiencies2and3of the lump models are even closer to theR obtained by the multidimensional cal-culations. This is ascribed to the smart ignorance of the transition regions in calculatingˇpandˇs.
6. Conclusions
A CFD solution method has been developed to examine the unsteady flow field prevailing in the valveless micropump. The con-servative equations are solved using the finite volume approach within the frame of unstructured grid. The pressure
bound-ary conditions specified at the inlet and outlet openings are tackled using a mass conservation treatment, which is more appropriate for unsteady flow calculations. The vibration of the membrane is modeled by a reciprocating velocity boundary con-dition derived from a harmonic motion. The deflection of the membrane during vibration is assumed to be of a shape blend-ing the parabolic and trapezoidal curve profiles. The different characteristics of the nozzle and the diffuser results in a net pumping flow from the inlet to the outlet. The agreement in the net flow rate between predictions and measurements is satisfac-tory.
The performance of the micropump has also been analyzed using the lumped-system method. In the lump model the relations between the loss coefficients and the flow rates for the nozzle and the diffuser are expressed in two descriptions. One of the two cor-relations is obtained by the multidimensional simulation for the considered nozzle/diffuser element. With this specially tailored correlation, a better agreement with the CFD calculations has been achieved.
The pumping performance depends upon the ratio of the outlet flow rate to the inlet flow rate. Comparing with the multidimen-sional calculations, the variation of the flow rate ratio is quite different in the lumped-system analysis, especially in the transi-tion region between the pumping stage and the supply stage, due to the inertial effect not taken into account in this simple method. The pumping efficiency of the pump has been formulated in two simplified expressions which are functions of the mean ratios of the two flow rates. In the averaging process to determine the mean ratios in the pumping and supply stages, the transition regions are ignored to reduce the influence of the inertial effect not accounted for in the lumped-system analysis. This simply results in better agreement with the CFD calculations in terms of the pumping efficiency.
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Biographies
Yeng-Yung Tsui received MSc from University of Tennessee Space Institute, U.S.A
in 1981 and PhD from Imperial College of Science and Technology, U.K., in 1987. He joined the Department of Mechanical Engineering, National Chiao Tung Univer-sity, Taiwan, ROC, since then and is currently a professor of the department. His major research interests lie in the computational fluid dynamics. Recently, he was involved in the studies of molecular pumps, mixers with agitators, two-fluid flows, and microfluid flows.
Shiue-Lin Lu received his BSc from Tamkang University in 2005 and MSc from
National Chiao Tung University, Taiwan, ROC, in 2007. He is currently in the military service. His research during the graduate study was concerned with the analysis of the flow in micropumps.
