The true composition of a mixture µ is often not known but an estimate y may be found by sampling.
If we have N samples of composition y1 to yy in one component the estimate of the mixture composition is given.
Standard deviation and variance.
The true standard deviation σ, and the true variance σ2, of the composition of the mixture are quantitative measures of the quality of the mixture.
The true variance is usually not known but an estimate S2 can be obtained as:
Theoretical limits of variance.
For a two-component system the theoretical upper and lower limits of mixture variance are:
Where p and (1-p) are the proportions of the two components determined from samples and n in the number of particles in each samples.
Actual values of mixture variance lie between these two extreme values
A measure of the degree of mixing is the Lacey mixing index (Lacey, 1954).
In practical terms the Lacey mixing index is the ratio of “mixing achieved” to “mixing possible”. A Lacey mixing index of zero would represent complete segregation and a value of unity would represent a completely random mixture. Practical value of this mixing index, however, are found to lie in the range 0.75 to 1.0 and so the Lacey mixing index does not provide sufficient discrimination between mixtures.
A further mixing index suggested by Poole et al. (1964) is defined as: (fig.)
This index gives better discrimination for practical mixtures and approaches unity for completely random mixtures.
When the sample compositions have a normal distribution the sampled variance values will also have a normal distribution. The standard deviation of the variance of the sample compositions is known as the “standard error” of the variance E(S2).
Test for precision of mixture composition and variance.
The mean mixture composition and variance which we measure from sampling are only samples from the normal distribution of mixture compositions and variance values for the mixture.
We need to be able to assign a certain confidence to this estimate and to determine its precision.
Based on N samples of mixture composition with mean and estimated standard deviation S, the true mixture composition µ may be stated with precision: (fig.)
where t is from Student’s t-test for statistical significance. The value of t depends on the confidence level required.
For example , at 95% confidence level, t=2.0 for N=60, and so there is a 95% probability that the true mean mixture composition lies in the range y±0.258S.
In other words, 1 in 20 estimates of mixture variance would lies outside this range.
a) When more than 50 samples are taken (i.e. n>50), the distribution of variance values can also be assumed to be normal and the Student’s t-test may be used. The best estimate of the true variance σ2 is then given by: (fig. 1).
The standard error of the mixture variance required in this test is usually not known but is estimated from: (fig. 2).
The standard error decreases as 1/√N and so the precision increases as √N.
b) When less than 50 samples are taken (i.e. n>50), the variance distribution curve may not be normal and is likely to be a Χ2 (chi squared) distribution. In this case, the limit of precision are not symmetrical. The range of values of mixture variance is defined by lower and upper limits:
Where α is the significance level [for a 90% confidence range, α=0.5(1-90/100)=0.05; for a 95% confidence range α=0.5(1-95/100)=0.025]. The lower and upper Χ2 values, Χ2α and Χ21-α for a given confidence level are found in Χ2 distribution tables.
21. Liquid Mixing