Universal Curve for Mixing with Laminar Flow in ...
Transcript of 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/