Mixers for Determining Flow Properties

Quantitative shear stress–shear rate data can be obtained with agitators having complex geometries if one assumes that the shear rate is directly proportional to the rotational speed of the agitator and if the flow behavior of the fluid can be described by the power law model. The procedure is based on the studies of Metzner and Otto (1957), Bongenaar et al. (1973), and Rieger and Novak (1973). It has been described in detail by Rao (1975) and is considered only in brief here.

It is assumed that the impeller (agitator) exerts an average shear rate that is directly proportional to the rotational speed:

γ = cn (5.65)

where c is the proportionally constant between the shear rate and the rotational speed of the impeller. For any impeller, the constant c can be determined from a plot of 1–n versus log [p/(knⁿ⁺¹d³)]; the slope of the line is equal to −log c (Rieger and Novak, 1973). It is clear that the tests for a given impeller must be conducted such that the following data are obtained; p, the power (Nm/s); n, the rotational speed (s⁻¹); d, the diameter of the impeller (m); and the power law parameters of several test fluids.

The method proposed by Rieger and Novak (1973) to determine the magnitude of the proportionally constant c is relatively simple. However, because the magnitude of the constant is equal to the antilogarithm of the slope of the plot indicated earlier, a small error in determining the magnitude of the plot indicated earlier, a small error in determin-ing the magnitude of the slope will result in a large error in the magnitude of c (Rao and Cooley, 1984). Compared to the method of Rieger and Novak (1973), the method of Metzner and Otto (1957) involves more work. It is contingent on finding a Newtonian and a non-Newtonian fluid such that the power consumed by an impeller at a given speed is identical in the two fluids. Also, the experimental conditions, such as the dimensions of the mix-ing vessel, must be the same in the experiments with the Newtonian and non-Newtonian fluids.

5.4.3.1.1 Power Law Parameters Using a Mixer

Once the proportionality constant in Equation 5.64 is known for a specific impeller, one can determine the power law parameters of a test suspension (x). Because the shear stress and the shear rate are directly proportional to the torque (t) and the rotational speed (n), respectively, a plot of log (t) versus log (n) would yield the magnitude of the flow behav-ior index nx. The consistency index of a test fluid (Kx) can be calculated using the torque values at known rotational speeds of the test fluid and another fluid (y) whose power law parameters (Ky , ny) are known:

where the subscripts x and y refer to the test and standard fluids, respectively. Because of the assumption that the impeller exerts an average shear rate, the method is approximate.

Nevertheless, it is the only one available for the quantitative study of food suspensions.

Because viscoelastic fluids climb up rotating shafts (the Weissenberg effect), mixer viscom-eters may not be suitable for studying these fluids.

5.4.3.1.2 Yield Stress with a Mixer

As stated earlier, yield stress is also an important property of many foods whose determi-nation requires considerable care. In the relaxation method using concentric cylinder or cone-and-plate geometries, it can be determined (Van Wazer et al., 1963) by recording the shear stress level at a low rpm at which no stress relaxation occurs on reducing the rpm to zero (Vitali and Rao, 1984b). This method, however, is time-consuming and requires great care to obtain reliable results. For the most part, magnitudes of yield stress were deter-mined by extrapolation of shear rate–shear stress data according to several flow methods such as those of Casson (Equation 5.9), Herschel and Bulkley (Equation 5.8), and Mizrahi and Berk (Equation 5.10) (Rao and Cooley, 1983; Rao et al., 1981a).

Dzuy and Boger (1983) employed a mixer viscometer for the measurement of yield stress of a concentrated non-food suspension and called it the “vane method.” This method is relatively simple because the yield stress can be calculated from the maximum value of torque recorded at low rotational speeds with a controlled shear-rate viscometer. The maxi-mum torque (t_m) value recorded with a controlled shear-rate viscometer and the diameter (d_V) and height (H) of the vane were used to calculate the yield stress (σV) using the equation

t d H

Equation 5.67 was derived by conducting a torque balance on the surface of the impel-ler (Dzuy and Boger, 1983).

Yield stress of applesauce samples was determined according to the vane method by Qiu and Rao (1988) using two vanes. With both impellers, the maximum torque value increased slightly with rotational speed over the range 0.1–2.0 rpm. In the derivation of Equation 5.66, the shear stress was assumed to be uniformly distributed everywhere on the cylinder and the test material yields at the impeller surface. In a comprehensive review of the vane method for yield stress, Dzuy and Boger (1985) concluded that these assump-tions are satisfactory. In contrast, there is limited experimental evidence (Keentok, 1982) to suggest that some materials may yield along a diameter (ds) that is larger than the actual diameter of the vane; the ratio ds/dV can be as large as 1.05 for some greases and is appar-ently dependent on the plastic, thixotropic, and elastic properties of the material. However, for inelastic and plastic substances, the data published by these workers suggest that ds/ dV is very close to 1.0.

5.4.3.1.3 Role of Structure on Yield Stress

A sample with undisrupted structure has a higher value of yield stress, called static yield stress (σ0s), whereas that whose structure has been disrupted by shear has a lower magni-tude, called dynamic yield stress (σ0d) (Rao, 1999; Yoo et al., 1995). In two different foods with equal magnitudes of yield stress (σ0d), it is very likely that they are due to contribu-tions of different magnitudes by differing forces. In turn, these forces are affected by the composition of the two foods and their manufacturing methods.

RHeologiCAl PRoPeRties oF Fluid Foods

From an energy balance, the shear stress necessary to produce deformation at a con-stant shear rate of a dispersion of particle clusters or flocs is (Metz et al., 1979; Michaels and Bolger, 1962):

est = esb+esy +esn (5.68)

where e_st is the total energy dissipation rate to produce deformation, e_sb is the energy dis-sipation rate required to break bonds, e_sv is the energy dissipation rate due to purely vis-cous drag, and e_sn is the energy dissipation rate required to break the aggregate network.

Because e = σ^γ yield stress (σ0) is calculated from vane mixer data it can be written as

σ0 = σb +σv+σn (5.69)

where σb, σ_v , and σn are components of yield analogous to the energy components.

The failure stress of an undisturbed food dispersion is:

σ

σ =0 0s (5.70)

The total energy required for sample deformation at yield can be calculated:

est = σ γ0s (5.71)

The subscript s of the energy terms is used to denote shear-based quantity. The continu-ous phase of a typical food dispersion is an aquecontinu-ous solution of solutes, such as sugars, and polymers, such as amylose or pectins. In the special case of a starch dispersion, it arises from association of amylose and a few amylopectin molecules into double-helical junctions, with further association of helices into aggregated assemblies (Genovese and Rao, 2003b). Bonding, sometimes called adhesivity, in a food dispersion can be associated with bridging between particles (e.g., irregularly shaped insoluble-in-water particles) and their interactions with the continuous phase. The stress required to break the bonds, σb, can be calculated as follows:

σ_b =σ_0s −σ_0d (5.72)

Values of e_b can be calculated as

eb = σ γb (5.73)

The viscous stress component σv is given by (Metz et al., 1979)

σv = η γ∞ (5.74)

where ή∞ is the viscosity of the dispersion at infinite shear rate. Therefore, e_v, the energy dissipation rate due to purely viscous drag, is calculated as

ev = σ γv = η γ∞² (5.75)

It is known that at zero shear rate, the network yield stress σn equals σ0; however, no relationship to estimate it at low finite values of shear rate exists (Metz et al., 1979), and it can only be estimated by difference

σn = σ0s−(σb +σv) (5.76)

Based on a study on three starch dispersions (Genovese and Rao, 2003b) and com-mercial samples of mayonnaise and tomato ketchup and concentrates (Genovese and Rao, 2004), it can be said that, in general, compared to the contributions of bonding (σb) and network (σn), the contribution of the viscous component (σv) would be small.

5.4.3.1.4 Effective Shear Rate in a Brabender Viscograph

Wood and Goff (1973) determined the effective shear rate to be 40 s⁻¹ in a Brabender visco-graph having the following characteristics: model, VSK 4; bowl speed, 75 rpm; bowl inter-nal diameter, 8.8 cm; length of bowl pin, 7.0 cm; length of sensing pin, 9.75 cm; and depth of liquid with 450 g water, 7.5 cm. It is necessary to note these characteristics because the viscograph is not an absolutely standard instrument.

5.4.3.1.5 Effective Shear Rate in a Rapid Visco Analyzer

Using the principle of mixer viscometry, Lai et al. (2000) determined the average shear rate in the mixing system (impeller-cup combination) of the RVA. A relationship between the impeller Reynolds number and the power number was established with Newtonian standards. Using the matching viscosity technique and non-Newtonian fluids consisting of various aqueous solutions of guar gum and methylcellulose, the average value of the mixer viscometer constant (c) was 20.1 per revolution over speeds of 1.0 to 3.5 r/s (60 to 210 r/min). Hence, the average shear rate in the RVA can be estimated as 20.1 multiplied by the angular velocity given in r/s.