Universal Curve for Mixing with Laminar Flow in Microfluidic Devices

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1 Universal Curve for Mixing with Laminar Flow in Microfluidic Devices Bruce A. Finlayson Professor Emeritus of Chemical Engineering University of Washington CPAC Meeting, July 17, 2008

2 Characterize Mixers Flow is laminar and slow - inertial effects are not important (Reynolds number < 1-10) Mixers are passive - no mechanical stirrers Perform the same characetization on all mixers

3 Same curve holds in 2D and 3D Daniel Kress, Sp, 2007

4 Serpentine Mixer Lab on a Chip (2004)

5 Spring Chem. Engr. 499 Undergraduate Research

6 Mixers to Characterize

7 Variance Variance/ Diff. 1.3e-9/ 1e-8 m^2/s 1.7e-4/ 1e-9 1.3e-3/ 7e e-3/ 5e e-2/ 2e e-2/ 1e-10

8 Questions to ask A. Do the variances collapse onto one curve if properly presented? B. Do your results follow the same curve of variance vs. as for a T-sensor? C. How different are the mixing cup and optical variances? Is this difference important? D. How do 2D and 3D results compare? E. What would you need to do in your device to reach a variance of 0.01? 0.001? F. What is the effect of Reynolds number? (This is pertinent only to a few of the geometries.)

9 Equations Re = ρu sx s η u u = p'+ 1 Re 2 u Pe = u sx s D u c = 1 Pe 2 u Navier-Stokes Equation Convective Diffusion Equation c mixing cup avg = A A c u da u da 2 σ mixingcup = A [c c mixingcup ] 2 u da A u da Mixing cup average concentration; variance from average Optical average and optical variance are the same formulae without the velocity - pertinent to measurement via fluorescence

10 Why should the curves superimpose? This is expected because the flow is basically straight down the device, except for the short entrance region, with diffusion sideways, and there is no convection sideways. Thus, diffusion controls the mixing, and the time in the device determines how far the material can diffuse. The parameter z' Pe = z x s D = z / u s u s x s x 2 s / D = t flow t diffusion is a ratio of the characteristic time for flow in the axial direction to the time for diffusion in the transverse direction.

11 Alternatively, one can examine the convective diffusion equation when there is no transverse velocity and deduce that axial diffusion term can be neglected compared with the axial convection term since their ratio is proportional to 1/Pe. w(x, y) c z = D 2 c x c y c z 2

12 "Characterizing Mixing in a Lithographed Flow Device" by Vann Brasher Hinsmann, Lab Chip, 1 16 (2001)

13 "Mixing in Flow Devices: Spiral Channels" by Ha Dinh Sudarson, Lab Chip, 6 74 (2006)

14 Micro-mixing by Rectangular Expansion Channel by Ho Hack Song Sudarson, Lab Chip, 6 74 (2006)

15 "Mixing Efficiency in Rough Channels" by Francis Ninh Kiplik, Phys. Fluids A, (1993) Variance Across 3D Channel at Varying Peclet Numbers Comparision of Mixing Efficiency of Rough Channel to T-Sensor and Flat Plates 1.E E+00 Pe E-01 Pe 200 Pe E-01 Variance 1.E-02 Pe 400 Pe 500 Pe 600 Variance 1.0E-02 Pe E-03 1.E-03 Pe 800 Pe 900 Pe E-04 1.E-03 1.E-02 1.E-01 1.E+00 Z/Pe 1.0E Z/Pe T-Sensor Rough Channel Flat Plates

16 "Micropillars Mixing in Microfluidic Devices" by Andy Aditya

17 "Flow in a Cross" by Adam Field 3-D Mixing in a Cross 1 Pe = 100 Pe = 200 Pe = 300 Pe = 500 Pe = 700 Pe = 1000 Variance Z/Pe

18 Evaluation of Concentration Variance as a Function of z'/pe by Jordan Flynn Holden, Sensors Actuators B, (2003) Pe from 10 to 1,000

19 Self Circulating Mixer Chamber by Cindy Yuen Chung, Lab Chip, 4, 70 (2004).

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21 "Mixing Properties of an Optimized SAR Mixer" by Lisa Dahl Schonfeld, Lab Chip, 4 65 (2004) Comparison Between the Concentration Variances in One Step 0 Mixing Cup Variance Optical Variance Peclet Number Pe

22 "Microfluidic Research: Mixing Effectiveness of Modified Tesla Structures" by Curtis Jenssen Hong, Lab Chip, (2004) Convergence of Error Pe=100 Pe=200 Pe=300 Pe=400 Pe=500 Pe=700 Pe=1000 3D Pe=100 3D Pe=500 3D Pe=1000 Pathlength/Pe(non-dimensional)

23 "Folding Flow Mixers" by Andrew Nordmeier micronit.com 2d case, multiple mixers variance Pe = 100 Pe=200 Pe=300 Pe=500 Pe=700 Pe= z/pe

24 Conclusions The variance for each geometry, for Re = 1, fell on one curve as a function of. The curve was similar in all cases, but shifted a bit for each device. The optical variances differed from the mixing cup variance somewhat, but not significantly on a logarithmic scale. Oftentimes the 2D simulations give a good representation of the 3D simulations; the cases when this doesn t hold is when the flow is particularly 3D in nature to induce mixing. If the device is similar to a T-sensor, increasing the Reynolds number makes little difference. The mixing is improved with increasing Reynolds number for geometries that induce laminar vortices based on inertial effects.

25 Thanks to: Dreyfus Foundation for Senior Mentor Grant - paid part of tuition of students And to: Vann Brasher Ha Dinh Ho Hack Song Francis Ninh Andy Aditya Adam Field Jordan Flynn Cindy Yuen Lisa Dahl Curtis Jenssen Andrew Nordmeier

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