Residence Time and Strain Distributions

Obviously, the residence time and its distribution only partially determine the chance of degradation in an extruder. The other factors that play an important role are the actual stock temperatures and the strain rates to which the polymer is exposed. The actual stock temperatures and strain rates are closely related. In the extruder, there are two major areas of concern the screw channel and the flight clearance. Janssen, Noomen, and Smith [65] studied temperature distribution of the polymer melt in the screw channel. Temperature distribution of the polymer right after the end of the screw was measured, for instance, by Anders, Brunner, and Pan-haus [66]. The temperature variations in the screw channel at the end of the screw were reported to be less than 5 to 10°C and relatively close to the barrel temperature. More recently, Noriega et al. [145] measured melt temperature distribution with a thermocomb and found temperature variations as high as 20 to 30°C. 

Fig. 9.14 Values of F(y) for a 6-in-diameter, 20 1 IVD extruder at constant flow rate (500 lb/h) with screw speed as a parameter. Simulation was made for a square pitched screw with a constant channel depth of 0.6 in. [Reprinted by permission from G. Lidor and Z. Tadmor, Theoretical Analysis of Residence Time Distribution Functions and Strain Distribution Functions in Plasticating Extruders, Polym. Eng. Sci., 16, 450-462 (1976).]...

If we accept the premise that the total strain is a key variable in the quality of laminar mixing, we are immediately faced with the problem that in most industrial mixers, and in processing equipment in general, different fluid particles experience different strains. This is true for both batch and continuous mixers. In the former, the different strain histories are due to the different paths the fluid particles follow in the mixer, whereas in a continuous mixer, superimposed on the different paths there is also a different residence time for every fluid particle in the mixer. To quantitatively describe the various strain histories, strain distribution functions (SDF) were defined (56), which are similar in concept to the residence time distribution functions discussed earlier. 

Continuous Mixers In continuous mixers, exiting fluid particles experience both different shear rate histories and residence times therefore they have acquired different strains. Following the considerations outlined previously and parallel to the definition of residence-time distribution function, the SDF for a continuous mixer/(y) dy is defined as the fraction of exiting flow rate that experienced a strain between y and y I dy, or it is the probability of an entering fluid particle to acquire strain y. The cumulative SDF, F(y), defined by... 

D. Bigg and S. Middleman, Mixing in Screw Extruders A Model for Residence Time Distribution and Strain, Ind. Eng. Chem. Fundam., 13, 66 (1974). 

The situation changes significantly in the case of strains like C. necator, where autocatalytic growth of biomass is followed by a phase of linear PHA production. In this case, biomass production should occur in the first step in a CSTR which is coupled to a subsequent plug flow reactor (PFR). The combination CSTR-PFR not only ensures higher productivity, but also minimizes the loss of substrates and co-substrates. Furthermore, product quality can be enhanced by the fact that the PFR features a narrow residence time distribution, leading to higher uniformity of cell populations. This should also have positive impacts on the distribution of the PHA molecular masses and the composition of polyesters [128]. 

As discussed in Chapter 6, the quality of mixing is related to the increase in interfacial area, which is proportional to strain. The calculation of the average strain, y, requires the residence time distribution function,/(f). We present the analysis for isothermal Newtonian flow in a single-screw extruder only, which is due to Pinto and Tadmor (1970). We give only a descriptive analysis for twin-screw extruders. 

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