Crop Process Engineering

Lesson 19. Mixture Quality: Mixing Index and Rate

The degree of uniformity of a mixed product may be measured by analysis of a number of spot samples. Food powder mixers act on two or more separate materials to intermingle them. Once a material is randomly distributed through another, mixing may be considered to be complete. Based on that, the well-known statistical parameters of mean and standard deviation of component concentration can be used to characterize the state of a mixture. If spot samples are taken at random from a mixture and analyzed, the standard deviation of the analyses s about the average value of the fraction of a specific powder   is estimated by the following relation:

\[s =\sqrt{{{\sum\limits_{i=1}^N{\left({x_i -\bar x}\right)^2}}\over{N-1}}}\]..................................................................................................................(18.1)

where x_i is every measured value of fraction of one powder and N is the number of samples.

The standard deviation value on its own may be meaningless, unless it can be checked against limiting values of either complete segregation s₀ or complete randomization s_r. The minimum standard deviation attainable with any mixture is s_r, which represents the best possible mixture. Furthermore, if a mixture is stochastically ordered, s_r would equal zero. Based on these limiting values of standard deviations, Lacey (1954) defined a mixing index M₁ as follows:

\[M_1={{s_0^2-s^2}\over{s_0^2-s_r^2}}\]..........................................................................................................................................................................(18.2)

The numerator in Eq. (18.2) would be an indicator of how much mixing has occurred, while the denominator would show how much mixing can occur. A Lancey mixing index M₁ of zero would represent complete segregation, and a value of unity would represent a completely random mixture.Practical values of this mixing index, however, are found to lie in the range 0.75–1. Thus, the Lancey mixing index does not provide sufficient discrimination between mixtures. In practice, however, the values of s, even for a very poor mixture, lie much closer to s_r than      to s₀.  Poole et al. (1964) suggested an alternative mixing index:

\[M_2={s\over{s_r}}\].....................................................................................................................................................................................................(18.3)
Equation (18.3) clearly indicates that for efficient mixing or increasing randomization M₂ would approach unity. The values of s₀ and s can be determined theoretically. These values would be dependent on the number of components and their size distributions. This index gives better discrimination for practical mixtures and approaches unity for completely random mixtures. Equations (18.2) and (18.3) can be used to calculate mixing indices defined by Eq. (18.1).

 References:

1. Uhl, V.W. and Gray, J.B. (1986). Mixing, Vols. 1–3. Academic Press, Orlando, FL.

2. Gustavo V. Barbosa-C´anovas et.al. (2005) FOOD POWDERS Physical Properties, Processing, and Functionality, Kluwer Academic/Plenum Publishers, New York.