Efficiency Analysis

The performance of our design is based on the following factors: the membrane length L, the membrane width W, the pump-cavity height H, the amplitude of vibration of the membrane a, the frequency of vibrations $\omega$, the minimum valve clearance (the gap between the closed-valve and the top wall) g, the time-lapse in between the opening and the closing of the valves (see equation (5.15), and figure 5.16, bottom) $\delta^{-1}$, the dynamic viscosity of the fluid $\mu$, and the fluid density $\rho$. There are nine variables associated with the performance of the micro-pump, with-dimensions of length, time and mass. This corresponds to six non-dimensional variables: $\frac{a}{L}$, $\frac{W}{L}$, $\frac{H}{L}$, $\frac{g}{L}$,$\frac{\delta}{\omega}$ and $\frac{\rho a^2 \omega}{\mu}$. In this study, we have fixed $\delta=0.15 \omega$. The geometric length-scales are set as H=0.4L and g=0.025L. The parameters $\frac{W}{L}$ and $\frac{a}{L}$, and $\frac{\rho a^2 \omega}{\mu}$ are varied.

The magnitude of the membrane velocity $u \simeq \omega a$. Therefore the parameter $\frac{\rho a^2 \omega}{\mu} = \frac{\rho u a}{\mu}$is the Reynolds number. Since, the ratio of the dynamic viscosity to the fluid density is the kinematic viscosity $\nu=\mu/\rho$, the Reynolds number can be simplified as $Re = \frac{ a^2 \omega}{\nu}$.

The volumetric flow rate per channel width (W) can be calculated by using equation (5.13)

The suction stage of the micro-pump happens while $-\frac{L}{2c} \le t \le \frac{L}{2c}$. Therefore, average volumetric flow for a given period $T = \omega^{-1}$ is

The average flow rate is

This simple analysis indicates that the volumetric flow rate is proportional to the Reynolds number, the width of the micro-pump membrane W, and the $\frac{L}{a}$ ratio. Our analysis has assumed no leaks from both the inlet valve during ejection stage and from the exit valve during the suction stage. Therefore, equation (5.17) gives the maximum theoretical volumetric flow rate of the micro-pump system. This value will be used in the next section in determining the efficiency of the micro-pump, while leakage effects due to the imperfect motion of inlet and exit valves will be considered.