## Lesson 19. Mixture Quality: Mixing Index and Rate

Introduction

Assessing the extent of mixing is of great interest for both equipment manufacturers and food powder processors. Mixing indices have been proposed to assess the extent of mixing. Mixing indices intend to provide a measure of the performance of a piece of equipment (a blender) or a process as related to ideal desired conditions. Considering some previously discussed aspects of the mixing process, it can be gathered that food mixing is a complicated task not easily described by mathematical modelling. Mixture quality results from several complex mechanisms operating in parallel, which are hard to follow or to fit to a particular model. The scale and intensity of segregation is defined as the quantities necessary to characterize a mixture. The scale of segregation is a description of unmixed components, while the intensity of segregation is a measure of the standard deviation of composition from the mean, taken over all points in the mixture. In practice, it is difficult to determine these parameters, since they require concentration data from a large number of points within the system. However, they provide a sound theoretical basis for assessing mixture quality. Taking into account the complexity of components and interactions in food solids mixing, it would be difficult to define a unique criterion to assess mixture quality. In fact, there over 30 criteria have been developed to express the degree of mixedness. A mixing endpoint or optimum mixing time can also be considered as a very relative definition due to the segregating tendency of food powder mixing.

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 xi 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 sr. The minimum standard deviation attainable with any mixture is sr, which represents the best possible mixture. Furthermore, if a mixture is stochastically ordered, sr 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 sr 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.