## Rheology

Although, rheology is defined as the science of deformation and flow of matter, it typically is used to describe the flow of non-Newtonian fluids (65). The flow properties of the fluid can affect important extrusion parameters such as velocity and pressure profiles in the extruder, heat transfer between the walls and the fluid, pressure drop in the die, and energy requirements. These parameters determine the product quality, extruder and die design, and production rate.

During the extrusion process the flow patterns are complex (Fig. 5) and involve six major flow regions (66). These regions are:

1. Metering section, where the material is fully plasti-cized.

2. The entrance region to the die, where a converging flow field is developed generating large stresses.

3. The region within the die, where the disturbance caused by the entry flow gradually disappears.

4. The steady flow region in the die, where a fully developed flow exists.

5. The exit region, where the vapor bubble nucleation and growth and velocity profile rearrangement occur within the die followed by the expansion of the extrudate outside the die.

\ Metering section hAP*

^Exitlffgion -Po

Free stream region

Figure 5. Flow pattern in an extruder barrel and die system. Source: Adapted with permission from Ref. 66.

\ Metering section hAP*

Entrance region

Entrance effect _ _Rela2LaIjon region APd'ie'

Viscometric flow region

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### Free stream region

The three types of extensional flows are (1) uniaxial, (2) biaxial, and (3) planar. A cylindrical sample when stretched along its axis results in uniaxial extension. Compression of a cylindrical sample along its axis results in biaxial extension. Stretching a rectangular sheet of a sample along one direction, while keeping one of the remaining dimensions constant, results in planar extension (there are two viscosities in planar extension; see Ref. 67 for details). The ratio of the elongational viscosity (rje) to the shear viscosity (t]s) is defined as the Trouton ratio. For Newtonian fluid, this ratio is a constant and equal to 3 for uniaxial extension, 4 for planar extension, and 6 for biaxial extension. For non-Newtonian fluids, this ratio is generally much higher and may vary with extensional rate.

The measurement of these rheological parameters during extrusion processing poses special problems. The rotational instruments used in the measurements of rheological parameters of liquids foods cannot be used because the shear rates obtained in these rheometers are several orders of magnitude below those encountered during extrusion and it is difficult to reproduce extrusion-like conditions in the rheometer. Thus on-line measurements of rheological parameters are necessary to avoid these problems.

Figure 5. Flow pattern in an extruder barrel and die system. Source: Adapted with permission from Ref. 66.

6. The free stream region, where the extrudate expansion reaches an equilibrium value.

These flow situations of viscoelastic fluids necessitates the measurement of rheological parameters such as steady shear viscosity, primary and secondary normal stress differences, and elongational viscosity.

Two types of flows occur during extrusion cooking. The first is the shear flow that occurs owing to the presence of the walls, as in the screw channel and the die. The second is the extensional flow that is present farther away from the wall where the stream lines converge or diverge, such as at the entrance and exit to the die. The stress (<r) components in shear flow, are given by:

where 1 is the direction in which the fluid flows, 2 is the direction of the velocity gradient, and 3 is the remaining neutral direction. The shear viscosity (tjB) and the normal stress differences (N1( N2) are functions of shear rate. For Newtonian fluids the shear viscosity is a constant and under simple shear flow Nx and N2 = 0.

For extensional flows, extensional viscosity is defined by:

Extensional viscosity is constant for a newtonian fluid, but it may be a function of extensional rate for non-Newtonian fluids.

### Shear Viscosity

The rheological property that has received the most attention by scientists is shear viscosity. Some of the published viscosity studies are summarized in Table 3. A die or a viscometer (capillary or slit cross section with multiple pressure transducers located along the wall) attached to the extruder is often used to measure the viscosity. Alternatively, a single pressure transducer mounted near the exit of the barrel, along with capillary dies of different lengths but having the same radius and entry geometry, can be used to obtain the shear viscosity (77).

The first investigators to evaluate the viscosity of doughs during extrusion found that doughs exhibited shear thinning behavior (68). The viscosity decreased exponentially with temperature and was found to follow Ar-rhenius kinetics. Moisture was found to act as a plasticizer. Increased moisture content (M) decreased the dough viscosity An empirical model of the form below was proposed:

This model is one of the simplest and the most popular. The following model was proposed for constant moisture dough (69):

In equation 4 it is assumed that the temperature is a function of time is known. Reactions such as gelatinization and denaturation can lead to network formation and affect viscosity (parameters k and AEk control the reactions). A log polynomial model was used to express viscosity as a function of moisture content, shear rate, and temperature (71).

Product |
Method |
Moisture (% w.b.) |
Temperature (°C) |
Reference |

Cooked cereal dough |

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