Distributive Mixing in Screw Extruders

The shear rate in the polymer melt is found by taking the first derivative of the velocity. The velocity profiles for Newtonian fluids were derived in Section 7.4.1. From Eq. 7.197 we can determine the down-channel shear rate  

When the pressure gradient is positive the shear rates increase toward the barrel surface, when the pressure gradient is zero the shear rate is constant, and when the pressure gradient is negative the shear rates reduce toward the barrel surface. 

The cross-channel shear rate is determined the same way  

In the bottom of the channel the fluid is exposed to negative shear rates, and in the top of the channel the fluid is exposed to positive shear rates. This has important consequences for the mixing that occurs in screw extruders. At the top of the channel the fluid elements travel in the direction of the barrel, while at the bottom of the channel the fluid elements travel across the channel. The position of a fluid element in the upper portion of the channel (y) corresponds to a complementary position in the lower portion of the channel (yc) as shown in Fig. 7.124. 

The time that a fluid element spends in the upper portion of the channel (t ) is determined by the width of the channel and the cross-channel velocity  

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