Mixing is a unit operation in which a relatively uniform mixture is obtained
from two or more components.
Mixing has no preservative effect (on food) but is meant only as a processing aid,
though in some cases mixing is required to promote some other objective
such as mass transfer or chemical reaction.
The degree of uniformity achievable varies widely. It is easy to achieve
virtually complete homogeneity when mixing miscible liquids or mixing
a soluble solids into a liquid, but it can be difficult to achieve
a homogeneous result when mixing two solids, mixing two highly viscous liquids,
or mixing items with widely varying densities, especially if the amount
of one component is very small compared to the amounts of the others.
The efficiency of mixing depends on the efficient use of energy
to generate flow of the components. Important aspects in the design
of a mixer include:
-- provision of adequate input energy (for an appropriate time)
-- design of the mechanism for introducing the energy
-- properties of the components
How can the degree of mixing be quantified?
You would expect small samples taken from various locations in the container
at the start of mixing to contain close to 100% of either one or another
of the components, so a plot of composition as a function of sample location
would be very uneven. As mixing proceeds, successive plots of the composition
of samples taken from these same locations should become more uniform.
Ideally, when mixing is complete all samples should contain the same percent
of the each ingredient as the percent of that ingredient added to the vessel at the start.
What size sample should be taken?
It should be large enough so the required amount of the smallest ingredient
is easily measured, yet small enough so that you are confident that the quantities
normally used will contain the ingredients in the required concentration
(for example, pack size or less). [See Reilly et al, 1994, Sections 1.3 and 3]
The "standard deviation", 
, provides a satisfactory way
of quantifying the extent to which the fractional concentration of a component
scatters about its mean value in the various samples.
The subscript m has been added as a reminder that these measures change
with the mixing time.
where
xm (with an over-bar) is the mean fractional concentration
xi = the fractional concentration of the component in the i-th sample
n = the number of samples taken
For a perfect mix, (or 2)
would equal zero. Note: 2 is called
the variance of the mixture.
Many different mixing indices may be used to monitor
the extent of mixing and to compare alternative types of equipment.
Each one is based on measurements of standard deviations from the mean.
The subscript m has been added to the indices in the equations below
as a reminder that the indices change with the mixing time.
where
0 = standard deviation at the start (time = 0)
m = standard deviation at time = tm
= standard deviation
of a "completely random" sample
( can be taken as zero for mixtures of liquids
and for solids where the particle size is very small
relative to the sample size)
= mean concentration of the samples
np = the number of particles in the sample
Vc = volume fraction or mass fraction of a component
in the mixture
The first three mixing indices are described in Fellows, 2000, Section 5.1.1.
M1 and M3 may be more suitable when approximately
equal masses are mixed, and/or at comparatively low mixing rates.
M2 is more likely to be suitable when a small amount of one component
is mixed with a large amount of another, or at high mixing rates.
Fellows suggests calculating all three and then using the most appropriate one
(see Sample Problem 5.1 in Fellows).
M4 is a simplified index. The following table shows what values
of this index are found for bad through excellent homogeneity:
Quality of Mixture M4
Bad 0.70
Unsatisfactory 0.70 to 0.80
Fairly good 0.80 to 0.90
Good 0.90 to 0.94
Very good 0.94 to 0.96
Excellent:
For granular materials >0.96
For fluids >0.98
M5 is an index recommended by Reilly et al (1994) after
reviewing nine different mixing indices
Whichever formulation of M you elect to use, you can compute
the goal mixing index (Mgoal) from the standard deviation
that would be acceptable (to your customer, for example)
for samples taken from the final mixture.
How long should ingredients be mixed?
For mixing indices 1, 2, 3, and 5, the mixing index will decrease
with mixing time according to the relationship
ln M = - k tm
where k is the mixing rate constant [1/s] and tm is mixing time [s].
The rate constant k can be obtained by taking and analysing samples
after various mixing times, and plotting ln M vs mixing time. This should be
a straight line, and its slope is -k. Once you have obtained k you can to calculate
the required mixing time to get to the desired (goal) mixing index from
tgoal = - (1 / k) ln Mgoal
Earle (1983) also discusses use of a "mixing index" to examine the thoroughness
of mixing and rates of mixing.