Figure 8. Exit pressure values for corn grits extruded under different conditions. Source: Ref. 76.

(105). This method has been used successfully to measure the normal stress difference for polymer solutions and melts (90,96,106-111). Typically, a slit die with holes or slots of different geometries is used to estimate the normal stress differences. The common hole geometry is a rectangular slot with its length transverse to the direction of the flow. This geometry is known as the transverse slot.

The original equations relating hole pressure and the normal stress difference (112) were later modified by Baird (109) to give the HPB equations. The HPB equations were reinterpreted to give the HPBL equations (90). Depending on the geometry of the hole, the HPBL equations are given by:

(Parallel slot) (7b)

(Circular hole) (7c)

where PHE signifies the elastic contribution to the hole pressure. Despite violations of some assumptions, reasonably good comparisons of Nx obtained from standard geometries and slit die were obtained (90,96,111). However, recent numerical studies have shown the HPBL equations to be valid (113,114).

This method is presently under evaluation in the authors' laboratory and has yielded promising results. A po tential problem with this method is that for a fluid with yield stress, an error in pressure transmission could occur. For a fluid that has yield stress, higher hole pressure values will be obtained (95). Thus the effect of yield stress on hole-pressure values will have to be accounted for.

Extensional Viscosity

When a fluid flows from a larger to a smaller diameter tube (eg, from the barrel of the extruder to the die), a pressure drop is encountered. For a polymer melt or solution the magnitude of this pressure drop is significantly higher than that obtained for a Newtonian fluid of the same viscosity (87,115). Similar results were reported for corn grits during extrusion cooking (75). The excess pressure drop at the entrance was originally believed to be due to the elastic properties of the fluid. Recent studies for plastic polymers indicate that the flow in the die entry region cannot be explained by melt elasticity alone (116) and that the extensional viscosity is an important parameter that must be considered.

Cogswell (117) was the first to propose a method for obtaining extensional viscosity from the entrance pressure drop method. The entrance pressure drop is calculated by linear extrapolation of the pressure readings along the walls from the fully developed flow region of the capillary or slit to the die entrance. This reading is subtracted from that of the transducer located at the barrel exit or a reservoir. The basis for Cogswell's analysis is that the large entrance pressure drop in the converging flow region is due to the extensional nature of the flow. In addition, the presence of the wall introduces a shearing component. Thus, the entrance pressure drop is assumed to consist of two components: due to shear flow and due to extensional flow. The expression for the shear component of the pressure drop due is given by:

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