Characterization techniques - Micromixers Fundamentals Design and Fabrication - page 313

The dimensionless variance correlates with time as an exponential function and can therefore be taken
as the mixing index M :

2 2

N

¼ s

s

M (8.34)

2

s

0 s

2
N

h

Since mixing index varies from 1 for unmixed state to 0 for fully mixed state, a mixing efficiency can

be introduced as

2 2

¼ s

0 s

h ¼ M

1 (8.35)

0 s 2

N

s

If the standard deviation is taken for the analysis, the corresponding mixing index and mixing
efficiency are

¼ s s N
s 0 s N

M (8.36)

s 0 s
s 0 s N

h ¼ 1 M ¼ (8.37)

Because good mixing is understood as the homogeneity of the mixed results, the distribution of the
intensity values or the histogram H ( I ) of an image can be used for evaluating the degree of mixing. As
mentioned in the previous section, intensity values are proportional to the concentration of the fluo-
rescent dye molecules. The gray-scale histogram actually represents the probability distribution
function (PDF) of the concentration.
The probability values can be obtained by normalizing the pixel number of each intensity value by
the total number of the pixels in the evaluated region:

NðIÞ
P

c Þ¼ I Þ¼

P ð P ð P ð I Þ¼ : (8.38)

I max

N ð I Þ

I min

The following example shows the implementation of PDF in MATLAB for evaluating the concen-
tration field:
Example 8.2 ( Measurement of Probability Distribution Function of the Concentration Field ).
Using the intensity array I as shown in Example 8.1, the PDF of a region of interest is obtained by the
following MATLAB code:
%Defining the region of interest
rect

getrect(image_handle);
%Reducing the image to the region of interest
ROI

¼

imcrop(I, rect);
%Open a new window
%Display the region of interest
imshow(ROI, [cmin cmax]);
%Obtaining the normalized concentration field
ROInorm ¼ (double(ROI)-double(cmin))/(double(cmax)-double(cmin));
%Determine the number of discrete values
N ¼ double(cmax)-double(cmin);
for i ¼ 1:N þ 1
edges (i) ¼ (i-1) /N;

¼