Universal Curve for Mixing with Laminar Flow in

Universal Curve for Mixing with Laminar Flow in Microfluidic Devices

Bruce A. FinlaysonProfessor Emeritus of Chemical Engineering

University of Washington

CPAC Meeting, July 17, 2008

Characterize Mixers Flow is laminar and slow - inertial effects are not

important (Reynolds number < 1-10)Mixers are passive - no mechanical stirrersPerform the same characetization on all mixers

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

Serpentine MixerLab on a Chip 4 342-350 (2004)

Spring Chem. Engr. 499 Undergraduate Research

Mixers to Characterize

Variance

Variance/ Diff.

1.3e-9/ 1e-8 m^2/s

1.7e-4/ 1e-9

1.3e-3/ 7e-10

5.1e-3/ 5e-10

4.5e-2/ 2e-10

9.4e-2/ 1e-10

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.)

u •∇u = −∇p '+1Re

∇2u u •∇c =1Pe

∇2u

Navier-Stokes Equation Convective Diffusion Equation

Mixing cup average concentration; variance from average

Optical average and optical variance are the same formulae without the velocity - pertinent to measurement via fluorescence

cmixing cup avg =c u • dA

A∫

u • dAA∫

EquationsRe =

ρusxs

ηPe =

usxs

D

σmixingcup2 =

[c− cmixingcup]2 u • dA

A∫u • dA

A∫

Why should the curves superimpose?

z 'Pe

=zxs

Dusxs

=z / us

xs2 / D

=t flow

tdiffusion

This is expected because the flow is basically straight down thedevice, except for the short entrance region, with diffusionsideways, and there is no convection sideways. Thus, diffusioncontrols the mixing, and the time in the device determines how farthe material can diffuse. The parameter

is a ratio of the characteristic time for flow in the axial direction to the time for diffusion in the transverse direction.

Alternatively, one can examine the convective diffusionequation when there is no transverse velocity and deducethat axial diffusion term can be neglected compared withthe axial convection term since their ratio is proportionalto 1/Pe.

w(x, y)∂c∂z

= D∂2c∂x2 +

∂2c∂y2 +

∂2c∂z2

⎡

⎣⎢

⎤

⎦⎥

"Characterizing Mixing in a Lithographed Flow Device" by Vann Brasher

Hinsmann, Lab Chip, 1 16 (2001)

"Mixing in Flow Devices: Spiral Channels" by Ha Dinh

Sudarson, Lab Chip, 6 74 (2006)

“Micro-mixing by Rectangular Expansion Channel”by Ho Hack Song

Sudarson, Lab Chip, 6 74 (2006)

"Mixing Efficiency in Rough Channels" by Francis Ninh

Kiplik, Phys. Fluids A, 6 1333 (1993)

Variance Across 3D Channel at Varying Peclet Numbers

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E-03 1.E-02 1.E-01 1.E+00

Z/Pe

Varia

nce

Pe 100

Pe 200

Pe 300

Pe 400

Pe 500

Pe 600

Pe 700

Pe 800

Pe 900

Pe 1000

Comparision of Mixing Efficiency of Rough Channel toT-Sensor and Flat Plates

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0.001 0.01 0.1 1

Z/Pe

Varia

nce

T-Sensor Rough Channel Flat Plates

"Micropillars Mixing in Microfluidic Devices" by Andy Aditya

"Flow in a Cross" by Adam Field

3-D Mixing in a Cross

0.01

0.1

1

0.0001 0.001 0.01 0.1 1

Z/Pe

Varia

nce

Pe = 100Pe = 200Pe = 300Pe = 500Pe = 700Pe = 1000

“Evaluation of Concentration Variance as a Function of z'/Pe”by Jordan Flynn

Holden, Sensors Actuators B, 92 199 (2003)

Pe from 10 to 1,000

“Self Circulating Mixer Chamber” by Cindy Yuen

Chung, Lab Chip, 4, 70 (2004).

"Mixing Properties of an Optimized SAR Mixer" by Lisa Dahl

Schonfeld, Lab Chip, 4 65 (2004)

Comparison Between the Concentration Variances in One Step

-6

-5

-4

-3

-2

-1

0

0 100 200 300 400 500 600 700 800 900 1000

Peclet Number Pe

Mixing Cup Variance

Optical Variance

"Microfluidic Research: Mixing Effectiveness of Modified Tesla Structures"

by Curtis Jenssen

Hong, Lab Chip, 4 109 (2004)

Convergence of Error

0.000001

0.00001

0.0001

0.001

0.01

0.1

10.01 0.1 1

Pathlength/Pe(non-dimensional)

Pe=100Pe=200Pe=300Pe=400Pe=500Pe=700Pe=10003D Pe=1003D Pe=5003D Pe=1000

"Folding Flow Mixers" by Andrew Nordmeier

micronit.com

2d case, multiple mixers

0.0001

0.0010

0.0100

0.1000

1.00000.001 0.01 0.1

z/Pe

varia

nce

Pe = 100Pe=200Pe=300Pe=500Pe=700Pe=1000

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.

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

http://courses.washington.edu/microflo/