Striation Thickness and Laminar Mixing

The striation thickness concept was first introduced by Mohr et al. (1957) and analyzed extensively by Ranz (1979) and Ottino et al. (1979, 1981). The striation thickness, S, is defined as one-half (Ottino et al., 1979) of the thickness of the repeating unit (i.e., one-half of the sum of the thickness of two adjacent layers of components A and B). Note that in some references the striation thickness is defined as the sum 

FIGURE 6.8 Interfacial element Ao in a Cartesian coordinate system at time t = 0. 

As mixing progresses, the interfacial area per unit volume increases and the striation thickness decreases. If there is a distribution of striation thickness then not only the mean but also the variance should be taken into consideration for the quality of mixing calculations. In Section 6.3.1 the interfacial area growth, or equivalently the striation thickness reduction, is calculated from geometrical arguments. 

---

STRIATION THICKNESS AND LAMINAR MIXING 165 Equations 6.40 and 6.41 can be combined to give... 

Most of this handbook treats spatial mixing. Suppose that a sample of fluid is collected and analyzed. One may ask Is it homogeneous Standard measures of homogeneity such as the striation thickness in laminar flow or the coefficient of variation in turbulent flow can be used to answer this question quantitatively. In this chapter we look at a different question that is important for continuous flow systems When did the particles, typically molecules but sometimes larger particles, enter the system, and how long did they stay This question... 

First and foremost, the laminar mixing flow created in the reactive processing equipment, must reduce the striation thickness to a level where the diffusion characteristic time, tD = r2/ )AB, is small compared to the reaction characteristic time. Since the molecular diffusivities of low molecular weight components in polymeric melts (see Section 8.3) are very small and of the order of 10 6 cm2/s, the striation thickness must be reduced to the micron level in order to get a characteristic time t of the order of 1 s. Shear flow can accomplish this in reasonable mixing times because the striation thickness is inversely proportional to the total shear (see Section 7.3)... 

This study (3) was done to produce an ABS type resin by dry and melt blending SAN and a nitrile rubber in a motionless mixer. In polyblends of two semicompatible polymers, the particle size of the dispersed phase is an important factor concerning final properties, particularly if a rubber is dispersed to improve impact strength. Motionless mixers should give precise control over the final particle size since for laminar flow the number of fluid layers and the striation thickness can be predicted mathematically. The hypothesis that the impact strength should peak out at a precise number of mixing elements was thus investigated. 

According to the expression for the shear in a tube with laminar flow and assuming that the initial striation thickness is equal to half of the channel diameter d, the mixing time is given by the flowing expression as a function of the Peclet number Pe ... 

Polymer mixtures containing deformable components tend to contain streaky or layered structures resulting from laminar shear flow. In such cases, the striation thickness (the distance between the streaks) can be measured to provide an indication of degree of mixing. As mixing progresses, a decrease in striation thickness is accompanied by an increase in interfacial area between the components. Reductions in striation thickness will depend on the level of imposed shear strain and the orientation of the components relative to the direction of shear [38]. 

The mixing process in a single-screw extruder is laminar. The materials being mixed exist as discrete layers, striations. The process of mixing involves a reduction in the striation thickness, this being related to an increase in interfacial area between the major and minor components of the mixture. 

It can be shown that the change in striation thickness on mixing, and hence in the degree of laminar mixing, is a simple function of the total shear strain imposed on the system. However, at the end of the process the components of the mixture still exist as discrete components. The total shear strain exerted on the melt is a function of the residence time of the melt in the process. As a result of the complex velocity profile of the melt in the screw channel, the residence time of the melt in the channel varies as a function of the position of the melt in the screw channel as well as the down-channel velocity of the melt. 

The real power of using stretching computations to characterize chaotic flows lies in the fact that stretching is the link between the macro- and micromixing intensities in laminar mixing flows. In this section we describe the method for computing striation thickness distribution in our 3D example, the Kenics mixer. 

---