## Reliable Modeling Approach for MPs with Fixed Valves

A modeling approach is considered reliable, which divides the overall design methodology into two parts. The first part must consist of a low-order, linear model to maximize the resonant behavior through optimization of all the design parameters. However, such model cannot predict net pressure and flow characteristics, other than the relationship between the geometrical parameters and material properties of the pump discrete components needed to meet the resonant frequency and oscillation requirements. Discrete elements for a fixed-valve MP or a straight-channel system are shown in Figure 7.1. The second part of the above model must determine the optimum shape and size to optimize the nonlinear pump performance in terms of net pressure and flow rate. Linear modeling can be used to optimize pump membrane thickness for a given valve size and to improve the resonant frequency and, therefore, the pump performance in terms of net pressure and flow rate. Regardless of the models used for MP design, experimental data involving critical performance parameters must be obtained to verify the accuracy and reliability of the model obtained through the computer simulations.

It is important to mention that the finite analysis offers exact solution for an oscillatory flow in straight rectangular channels to obtain appropriate values for the fluidic elements. However, a straight-channel fluid flow behavior is considered an approximate solution of the complex flow in actual values, because the differential behavior of fixed-valve MP is small compared to that of mechanical valve MP.

(b) actuator

Figure 7.1 Fixed-valve MP. (a) Schematic diagram showing the critical elements, (b) linear graph representing the device elements, and (c) diagram showing a typical Tesla-type valve and two definitions of valve length. For L ' Leq segments were combined as parallel and series impedances, i.e., Leq = Lt 1 (1/L2 11/L3)"1 1 •••. For L ' Lave the shortest and longest path through the valve were averaged, i.e., Lave = (Lt 1 L3 1 ••• 1 L1 1 L2 1 •• )/2.

Figure 7.1 Fixed-valve MP. (a) Schematic diagram showing the critical elements, (b) linear graph representing the device elements, and (c) diagram showing a typical Tesla-type valve and two definitions of valve length. For L ' Leq segments were combined as parallel and series impedances, i.e., Leq = Lt 1 (1/L2 11/L3)"1 1 •••. For L ' Lave the shortest and longest path through the valve were averaged, i.e., Lave = (Lt 1 L3 1 ••• 1 L1 1 L2 1 •• )/2.

Both the viscous and inertial effects must be taken into account to obtain exact solution of the Navier-Stokes equation. These effects are critically important, when the pump operating frequency (fo) is in the vicinity of the valve cutoff frequency (fcutofj).

7.3.2.1 Electrical and Mechanical Parameters for Low-Order Model

The low-order model is best suited to describe the dynamic behavior of the fixed-valve MP or straight-channel device as illustrated in Figure 7.1 [1]. The inlet and

 Properties Pyrex Silicon Epoxy PZT-5A Young's modulus (GPa) 61 130 5.2 70 Poisson's ratio 0.20 0.22 0.30 0.28

outlet ports of a fixed-valve MP are connected to open fluid reservoirs through the tubes that are significantly larger than the a—b and c-d values shown in Figure 7.1b. The linear graph shown in the figure is dependent on 15 discrete elements between the electrical (piezoelectric actuation), mechanical (membrane), and fluid domains. Parametric values of pump electronics can be derived from at least 36 geometrical dimensions and physical properties [1]. Matlab and MathCad software programs can be used to solve the output variables in the frequency domain. The centerline velocity-amplitude of the membrane must be the primary output variables due to the dynamic response of the membrane rather than the flow rate of pumped fluid. Important information on the mechanical parameters ofthe MP can be obtained from the free-air resonance frequency (/n) of the membrane or the pump containing no fluid. The electrical- and mechanical-lumped parameters for a single-large piezoelectric bimorph can be determined from finite element analysis (FEA) using ANSYS 5.5 software program, which uses the equations best suited for an analytical solution.

The bimorph model uses lead-zirconate-titanate (PZT)-5A piezoelectric material bonded to a Pyrex backing plate with conductive epoxy and the membrane support structure includes the silicon pump housing. Material properties for various materials for a single PZT-actuator layer are summarized in Table 7.1.

7.3.2.2 Mathematical Expression for Critical Pump Parameters

The spring constant (k) can be calculated using chamber pressure (Pc) acting on the internal surface of the membrane of area Ac. The expression for the parameter k can be written as k =

(FsMcPc

where Fs is the shape factor and Smem is the centerline displacement of the membrane.

AVm,

where A Vmem indicates the volume displacement by the membrane, which is equal to the volume displaced by a piston having the same centerline displacement. Now the volume swept by the elastic membrane surface as a result of voltage applied across the piezoelectric actuator or the displacement per volt can be written as

where V is the actuation voltage applied across the piezoelectric actuator.

The product of k8V is called the gyrator coupling, which couples the electrical to mechanical energy. But the product of FsAc is known as the gyrator coupling coefficient, which couples the mechanical component (or membrane) to the fluidic component.

The fundamental free-air resonant frequency or the natural frequency (fn) of the membrane system can be written as fn =

where m is the mass of the membrane system and k is the spring constant. Multiplying the numerator and denominator in Equation 7.1 by the chamber area Ac, one gets k =

FsAc Pc

^mem

FA2 Pc

D Vm where A Vmem is the product of membrane displacement and chamber area.

7.3.2.3 Chamber Parameters

The chamber capacity is equal to the sum of the chamber capacity (Cc) with fluid and chamber housing capacity and can be written as

The housing capacity can be calculated using the FEA model and can be expressed as

TAVhl

where A Vh indicates the change in housing volume due to chamber pressure Pc.

From Equation 7.5, one can write as

"AVnem"

A2]

L Pc J

The total change in the chamber volume is equal to the sum of change in membrane volume and change in housing volume and thus, it can be written as

where the subscript c stands for chamber and h stands for housing. It is important to point out that the capacity ofthe pump fluid must include the compressibility effect of the liquid and gas, if any. Thus, the chamber capacity with fluid can be written as

where V is the chamber volume and Gbm is the bulk modulus of the fluid. Similarly, the contribution to the chamber capacity due to adiabatic compression of the gas can be written as

KVeff gPc

where Veff represents the effective volume of gas per unit volume of liquid or fluid and g represents the ratio of specific heat for constant pressure to constant volume. The total chamber capacity is equal to the sum of chamber capacity due to fluid and chamber capacity due to gas, and can be written as

Gbm gPc

The first term in the equation will be zero, ifthe pump fluid is filled with gas and the value of parameter Vff is unity. The bulk modulus of water can be assumed as 2.19 GPa and the specific heat ratio can be assumed as 1.4 for air. If the fluid is other than water and the gas is other than air, then appropriate values for the parameters must be used to get the exact value for the total chamber capacity.

7.3.2.4 Fluidic Valve Parameters and Their Typical Values

The impedance of the fluid channel with rectangular configuration on each side of a circular chamber will define all the parameters of the fluid. The impedance of the valve consists of two components, namely, the resistance and the inductance of

 mm3 (Cc) mm5/N 1 0.009 2 0.019 3 0.027 4 0.036

Source: From Morris, C.J. and Forster, F.K., J. MEMS, 12, 325, 2003. With permission.

the inlet and outlet valves. The valve resistance for a steady, viscous flow is a function of absolute viscosity, channel length, channel depth, channel output ratio (width/ depth), and a constantpn, which is equal to 4.712,7.854, 10.996, and 14.137 when n is equal to 1, 2, 3, and 4, respectively. The valve inductance is proportional to mass density of the fluid and channel length, but inversely proportional to channel width and channel output ratio. Computed values of chamber capacity with fluid obtained using Equation 7.10 are summarized in Table 7.2.

The lumped parameters for the valves are strictly determined by the length and the transverse dimensions of a straight rectangular channel also called duct, fluid density (2.205 lb/mm3, if water is used as fluid), absolute viscosity of the fluid, and frequency of oscillation. Typical geometrical parameters for various straight-channel MPs are summarized in Table 7.3.

The pressure drop across each branch is dependent on the shortest and longest path through the valve as illustrated in Figure 7.1c. The fluidic load parameters such as the inlet and outlet capacities, namely, C; and Co, and the open reservoir inlet and outlet capacities, namely, Cri and Cro are attached to the MP. The reservoir capacity with water can be computed using the following expression involving various reservoir parameters:

 Pump No. Chamber Diameter (mm) (mm3) Membrane Thickness (mm) PTZ-Wafer Diameter (mm) 1 3 0.110 0.777 0.250 2.5 2 3 0.152 1.074 0.250 2.5 3 6 0.148 4.185 0.500 5.0 4 6 0.115 3.252 0.500 5.0

Reservoir capacity(Cr)

pfl2'

4Pg where

D is the reservoir diameter p is the density g is the acceleration due to gravity (10.04 m/s2)

Assuming a diameter of 1 mm, water density of 62.43 lb/ft3 and standard value of acceleration, one gets

7.3.2.5 Description of Micropumps with Straight-Channel Configurations

MP devices with straight-channels or valves are widely used for various scientific, clinical research, and medical applications. The valves as well as the reservoirs can be etched together to reduce the fabrication costs. Four distinct MPs with Tesla-type valves or straight-channel widths are available. These MPs can be used to predict the resonance behavior under various flow conditions and to investigate the effects of fluid composition. Typically, the MP's straight channels are 120 mm wide and 1.4 mm long for a 3 mm-diameter chamber and roughly 2.25 mm long for a 6 mm-diameter chamber. The membrane or the Pyrex cover is bonded to the pump structure, while a piezoelectric actuator is attached onto the Pyrex with a conductive epoxy. After attaching the piezoelectric actuator, the thickness of the epoxy layer is reduced to 20 mm.

Various category and geometrical parameters of Tesla-type valves are shown in Figure 7.2. Critical parameters of some selected Tesla-type valves are summarized in Table 7.4 with emphasis on cell category.

Member thickness plays a key role in determining the mechanical parameters, namely, the spring constant (k) and the velocity per volt (V^olt). The chamber diameter affects the chamber capacity and the mechanical parameters. Resonant frequency tests can be performed using air-amplitude with inlet and outlet reservoirs, which are typically 15 cm long and 1 mm in diameter. The actuator voltages must be high enough to achieve a good velocity signal and low enough to minimize entrance effects in the straight channel. The actuation voltage can vary from 5 to 25 V depending on the spring constant and other valve parameters.

Chamber Etch Membrane PZT

Pump Valve diameter depth thickness diameter

T45/2

3 mm pump

3 mm pump

T45/2

T45/2

T45/2

T45/2

4 87 I 6 141 500 5.0

Note: Type 1 means single-cell, Tesla-type valve pump Type 2 means double-cell, Tesla-type valve pump

Figure 7.2 Geometric parameters for various Tesla-type valves. (From Morris, C.J. and Forster, F.K., J. MEMS, 12, 325, 2003. With permission.)

 Tesla-Type Valve Equivalent Path Length (mm) Average Path Length (mm) Cell Category Type I 1.43 1.90 Single-cell Type II (a) 1.91 2.64 Dual-cell Type II (b) 2.13 3.03 Dual-cell

Source: From Morris, C.J. and Forster, F.K., J. MEMS, 12, 325, 2003.

Source: From Morris, C.J. and Forster, F.K., J. MEMS, 12, 325, 2003.

Free-air resonance frequency also known as natural resonance frequency is the most important parameter to evaluate the dynamic performance of an MP, which can be computed using Equation 7.4. Computed values of resonant frequency as a function of spring constant, mass, and dimensional parameters of device and chamber are summarized in Table 7.5.

7.3.2.6 Impact of Viscosity and Membrane Parameters on Valve Performance

Membrane velocity as a function offrequency appears linear in liquid-free MPs. The membrane thickness and valve width can be optimized for maximum deflection per unit actuation voltage. The valve length to valve width ratio of 1.5 yields optimum pump performance in terms of peak velocity flow rate and peak Reynolds number. Both the peak centerline velocity and peak Reynolds number are dependent on membrane thickness and valve width. It is important to point out that linear modeling is most efficient and practical to optimize the multitude of pump design parameters. Experimental verification is needed to verify both the size

 Pump No. Chamber Diameter/Membrane Thickness (mm/mm) Spring Constant k (N/m) Mass m (N) Natural Frequency (kHz) 1 3/0.165 4.62 x 106 0.462 x 10~s 159.2 2 3/0.250 7.89 x 106 0.509 x 10~s 198.2 3 6/0.165 1.37 x 106 1.53 x 10~s 47.6 4 6/0.500 8.56 x 106 2.18 x 10~s 99.6

and shape parameters of the valve obtained through simulations. This model provides excellent results for fixed-valve MPs, when the valves are replaced by straight channels with appropriate dimensional parameters. Computation errors are possible in straight-channel approximation due to valve fluidic characteristics. An MP can be designed using a linear model for optimum resonant behavior through adjustments of geometrical parameters and material properties. However, determination ofoptimum valve size and prediction ofoptimum pump performance in terms of net pressure and flow rates would require fluid dynamics analysis. In summary, fluid dynamics analysis is essential, if optimum performance and accurate prediction of valve shape and size are the principal requirements.