Abstract
System analysis is essential for accurate design of complex microfluidic systems. For instance, lumped-parameter system models of micropumps with no-moving-parts (NMP) valves need fluid elements that represent the separated, oscillating, laminar flow that occurs in these valves. No experimental or analytical method is currently available to determine the impedance of this type of flow. A numerical method for deriving NMP valve impedance has been developed, based on an extension of analytical methods for channels of non-varying cross-section. The resistance and inertance calculated are 20% lower than for a straight rectangular channel of equivalent length.
Keywords: microfluidic, micropump, microvalve, impedance, inertance
1. Introduction
As applications for microfluidic systems become increasingly complex, design techniques based on system analysis are essential to ensure pressure and flow requirements of system components are met. Analytical models for lumped-parameter fluid elements for simple, straight channels are available for steady laminar flow [1]. But these models are inappropriate for separated flow, in which recirculation regions appear downstream of a change in the cross-sectional flow area, (eg. cavities, projections, T and Y-junctions), as has been demonstrated experimentally in flows with Reynolds numbers as low as Re = 0.41 [2].
In addition to separated flow, microfluidic components may also experience unsteady flow from sources such as pumps and opening or closing valves. An example is the oscillatory flow experienced in no-moving-parts (NMP) valves used in micropumps [3]. In this case, if the oscillations are sinusoidal and the valve diodicity is small, the forced response of a pump with NMP valves can be approximated once the fluid impedance of the valves is known.
Fortunately, at the micro-scale, analytical methods can be replaced by numerical simulation with computational fluid dynamics (CFD), because realistic flows are laminar, and the exact governing equations can be solved.
2. Methods
A three step process was followed to numerically calculate the impedance of an NMP valve with oscillating, separated laminar flow. First, a transient numerical solution method was developed to match an existing analytical solution for a straight channel with oscillating pressure boundaries. Second, a CFD valve model was developed, and its computational grid refined based on steady-flow experimental data. Third, the transient numerical method of step 1 was combined with the computational grid of step 2 to predict fluid impedance in an NMP
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In the first step, an existing analytical solution for oscillating flow in a 2-D channel [4] was used to demonstrate fluid impedance could be calculated from a transient CFD simulation. As the radian frequency of the oscillations, W, increases, the viscous diffusion length becomes small compared to the height of the channel, 2h , and the flow in the center of the channel is no longer in phase with the flow near the wall. This is the case if the dimensionless parameter l = h/Ö{n/W} > 2 . The dimensionless solution for the velocity profiles is
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A finite-volume method code, (CFX-F3D 4.1c, AEA Technology, www.aeat.com/cfx), was used for the CFD simulations. The discretization algorithms used for the velocities were central differencing where the mesh Peclet number, (the ratio of local convection to viscous diffusion coefficients), Pe = u Dx/n < 2 , and upwind differencing where Pe > 2 . Central differencing was used for the pressure [5]. The time-stepping scheme was time-centered Crank-Nicolson with time steps equal to 30° phase angles. Decreasing the time step size by a factor of 3 did not affect the solution.
Since both the pressure gradient and the flow rate were sinusoidal, the forced response of the oscillating flow was modeled, using the electrical/hydraulic system analogy, as a resistor and inertance in series. The impedance Z , resistance R , and inertance I were determined in the complex plane from the amplitudes of the driving pressure gradient DP , the resulting volume flow rate Q , and the constant phase angle q of the pressure gradient relative to the flow rate.
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(2) |
The velocity was integrated over a cross-section to determine Q .
In the second step, CFD simulations of NMP valves were developed to model steady, separated, laminar flow. Since the most important characteristic of valves is their diodicity, the accuracy of the computational grid, discretization, and boundary conditions was determined by comparing valve diodicities computed numerically with values obtained by driving steady flow through the valve with an infuser pump and measuring pressure drop across the valve. In steady-flow, the ratio of pressure to flow rate is the fluid resistance, R = DP/Q . The valve diodicity is the ratio of the resistances, Di = Rreverse flow/Rforward flow .
In the third step, the CFD valve model from step 1was modified to use the time-stepping techniques from step 2. The simulation was run through 6 cycles, which ensured the velocity solution was periodic in time within 1%. Valve impedance was calculated with Eq. 2.
3. Results and Discussion
Figure 1(a) shows the velocity profiles from the numerical and analytical solutions at 60° phase angle increments for a 1 mm long, 90 mm deep 2-D channel with a pressure gradient amplitude of 1 atm oscillating at 10 kHz . For this case, l = 11.28 , which is high enough to
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cause a phase shift between the wall flow and center flow. The resistance and inertance per unit depth calculated from the numerical solution were R = 6.83×107 Pa s/m3 and I = 1.194×104 Pa s2/m3 , which are within 0.5% of the analytical solution.
A 3-D steady-flow simulation of a Tesla-type NMP valve that has both a T-junction and a Y-junction is shown in Figs. 2 and 3. The regions with dormant or recirculating flow change dramatically depending on flow direction. The CFD prediction of Di = 1.69 is within 18% of the experimental value at 2000 ml/min, a typical RMS flow rate during pump operation. Grid independence was demonstrated by a less than 1% change in the calculated mass flow rate when grid density was varied by 20%.
Direct measurement of the chamber pressure in a pump with NMP valves has shown that sinusoidal excitation of the pump results in a sinusoidal pressure gradient in the valves [3].
Figure 2: Laminar forward flow with separated regions in a Tesla-type NMP valve, left to right, at Re=494 based on hydraulic diameter.
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Figure 3: Laminar reverse flow with separated regions in a Tesla-type NMP valve, left to right, at Re=343 based on hydraulic diameter.
Thus, the transient NMP valve simulation used oscillating pressure boundaries at the typical system resonance ( 3085 Hz) of a 10 mm diameter pump with this Tesla-type valve ( 120 mm etch depth), resulting in l = 8.14 based on the hydraulic radius, and a phase shift between wall and center flows as can be seen in Fig. 1(b). The oscillating-flow impedance for a pressure amplitude of 0.5 atm was calculated by Eq. 2 as R = 2.70×1011 Pa s/m3 and I = 6.65×107 Pa s2/m3 . These values are within 20% of standard lumped-parameter approximations of resistance based on Pousielle flow [1] and inertance based on r L/A , where A is the cross-sectional area of a straight rectangular channel of equivalent length, width, and aspect ratio.
We have shown that transient computational techniques can be used to determine fluid impedance in a channel of arbitrary geometry exhibiting flow separation and oscillation. In the case of NMP valves, impedance values enable selection of pump components and valve geometries to optimize the magnitude and quality of the system resonance, resulting in a higher-performing micropumps [6], and facilitating their incorporation into general fluidic systems.
Acknowledgement: This work was supported by DARPA Microflumes Contract # N660001-97-C-8632
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