/ Recovery

R heodestruction

No recovery

Constant shear rate

Figure 3. Idealized rheograms for thixotropic and completely rheodestructive dispersions.

Figure 3. Idealized rheograms for thixotropic and completely rheodestructive dispersions.

Shear rate

Figure 4. Idealized rheograms of time-dependent flow in continuous up curve and down curve experiments.

Figure 4. Idealized rheograms of time-dependent flow in continuous up curve and down curve experiments.

Figure 5. Velocity profile for an ideal viscous fluid in steady laminar shearing flow between two parallel plates.

to the (movable) upper plate, it will move with a velocity v relative to the (stationary) lower plate. The shear strain (y) is a measure of the relative distortion of the sample (x/y, or in differential form, dx/dy); the shear rate, or more precisely the shear strain rate, is the change in strain as a function of time (dy/dt or dv/dy); and the shear stress is the force per unit area (F/A = a); the apparent viscosity (defined earlier) is the ratio of shear stress to shear rate. The fluid next to the upper plate moves with it at a velocity v and the fluid next to the lower plate is stationary. Be tween the plates there exists a velocity gradient; for a Newtonian material the gradient is linear as shown in Figure 5 but the gradient will have a more complex shape for other types of flow behavior.

While parallel plate viscometers are available commercially, in practice this is sometimes not a convenient form of the instrument. Coaxial cylinder viscometers are more commonly used than are parallel plate viscometers. It is possible to imagine constructing a coaxial cylinder viscometer from a parallel plate viscometer by forming one of the plates into a cylinder and then shaping the second plate into a cylinder around the first one. Then one cylinder is rotated while the other is stationary and the fluid fills the gap between the cylinders. Parameters are obtained in basically the same way as discussed in the preceding paragraph for the parallel plate viscometer, except the factors that account for geometric effects are somewhat more complicated. A rotating bob immersion viscometer could be regarded as a simplified version of the coaxial cylinder viscometer in which the inner cylinder rotates and the outer cylinder has been replaced with the wall of the container. If the container is large enough, it can be assumed that the gap between the rotating bob and the container wall is infinitely large. In fact, with a 2-cm diameter bob, this assumption is fair even if the container is a medium size (say a 250-mL) beaker. If the assumption can be made, the data analysis is somewhat simplified. Such immersion viscometers are simple to use and very popular in industry. While it is more difficult to obtain rheologically rigorous parameters from immersion viscometers, their readings can be very useful in quality assurance and process control.

Temperature effects can be critical in rheological measurements. Some viscometers have a built-in means for controlling the sample temperature, such as a thermostatic jacket. Sometimes the measurements can be made in a room where the temperature is well controlled. At any rate, temperature control is necessary; this is illustrated in Figure 6, which shows the effects of temperature on the consistency coefficient of plain and salted liquid egg yolk. Note that the ordinate scale in Figure 6 is logarithmic, showing that the consistency coefficients for these samples decrease exponentially with increasing temperature, so a relatively small error in temperature control could lead to a much larger (proportionally) error in the rheological parameters. In general, the viscosities of Newtonian liquids decrease logarithmically as the temperature is raised, and this dependence can be expressed as an Arrhenius-type relationship:

where E is the activation energy of flow (.E is an energy barrier that must be overcome before the elementary flow processes can occur and is related to the coherence of the molecules in the liquid), A is a constant, R is the gas constant (8.314 J/K mole or 1.987 cal/K mole), and T is the absolute temperature (K) (6,15).

An Arrhenius-type plot is shown in Figure 7 for a 40% sucrose solution over a temperature range of 0-80°C. In this example, the activation energy of flow (E) is 23.8 kJ/ mole. If data can be fitted in this manner, it is possible to use equation 3 to predict the viscosity of a Newtonian fluid at any temperature in the range. Moreover, because E represents molecular associations within a fluid, this may be a useful tool in the study of interactions in fluid systems, and their changes with the application of heat. (It should be remembered that equation 3 is an empirical equation and the activation energy is not derived from fundamental molecular theory, even though it is thought to represent molecular-level phenomena.)

For non-Newtonian fluids, temperature effects on flow behavior may be more complex. An approach to analyzing such effects was indicated by the egg yolk data (Fig. 6) where the temperature dependence of the consistency coefficient (m) was shown. Similarly, the temperature dependence of the flow index parameter (n) could be determined and, from equation 1, the apparent viscosity could be calculated at any temperature-shear rate combination; if the yield value were nonzero, it would also need to be determined as a function of temperature.

SOLID FOODS Viscoelasticity

The rheology of solid foods is usually considered in the area of food texture (2,7,8,16,17). Most solid foods are viscoelas-

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