Temperature and Pressure Effects in Flow

The analysis of extrusion in the preceding chapter was based on the assumption that the temperature and pressure dependence of physical properties, especially the viscosity, could be neglected. This assumption simplifies the analysis, especially in the case of temperature dependence, because it introduces an uncoupling between the fluid mechanics and the heat transfer. The assumption is dangerous if not used with care, however, as we shall demonstrate in this chapter. 

The viscosity of organic liquids depends on both temperature and pressure molecular motion becomes more difficult as free volume is reduced, and the viscosity increases. To a first approximation, the viscosity of polymer melts can be written  

T]° is the viscosity at atmospheric pressure (p = 0) and the reference temperature To T]° may depend on the shear rate, fi is typically 1-5 x 10 Pa , while a is typically 1-8 X 10 K. Thus, temperature differences of 10 degrees can have a significant effect on the viscosity, and we expect the pressure dependence to become important at a pressure of about 5 x 10 Pa (50 atm), which can be reached in extrusion and is routinely seen in injection molding. The density change at these elevated pressures is small, so compressibility is rarely important, and it usually suffices to retain the incompressible form of the continuity equation, even when accounting for the pressure dependence of the viscosity. For illustrative purposes, it suffices to consider only the case of die flow, where the wall velocity F = 0 and the exit 

We start by considering an isothermal flow, where T = To everywhere. As before, we assume that the flow is rectilinear, so the only nonzero velocity component is v, and is a function only of y. The x component of the momentum equation then becomes 

To proceed further, we need to assume that p = p(x) that is, the pressure is independent of the transverse coordinate. This is at best an approximation because it readily follows from the y component of the momentum equation that p cannot be a function of x alone. The transverse dependence of p can be shown to be very small, however, and in fact an exact solution without the approximation p = p(x) is available for flow in a cyhndrical cross section. We will therefore proceed by taking p to be independent of y. Equation 4.2 can then be written 

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