Newtonian Fluids with Negligible Viscous Dissipation

1 Newtonian Fluids with Negligible Viscous Dissipation 

For a Newtonian fluid (n = 1) the momentum balance equation can be written as  

The moving surface, the barrel, is at temperature T and the stationary surface, the screw, is at temperature T. The momentum balance equation can be Integrated to yield the velocity gradient  

The integration constant corresponds to the normal coordinate where the shear stress is zero and the velocity profile has an extremum. If it is assumed that the viscosity is determined by a linear temperature profile the velocity gradient can be written as  

The velocity profile can be obtained by integrating the velocity gradient this results in the following expression  

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Figures 4-15-4-20 show some of the results obtained in the cases of Table 4-2. Figure 4-15 shows the effect of thermal expansion [17] for a non-Newtonian fluid (/I = 0.25) with negligible viscous dissipation and constant fluid properties. The various e values represent the average value of Te across the tube. As can be seen, the effect of thermal expansion or compressibility cooling is to depress the point temperatures in the center of the tube.

The viscous dissipation term p4> is generally negligible except where extremely large velocity gradients exist. It will be omitted in the discussions to follow. Equation (5.6-9) is the equation of energy change for a Newtonian fluid with constant k in terms of the fluid temperature T. 

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