Flow models Herschel-Bulkley

This analysis can readily be extended to the laminar flow of Herschel-Bulkley model fluids (equation 1.17), and the resulting final expressions for... 

Two protocols are presented for non-Newtonian fluids. Basic Protocol 1 is for time-independent non-Newtonian fluids and is a ramped type of test that is suitable for time-independent materials. The test is a nonequilibrium linear procedure, referred to as a ramped or stepped flow test. A nonquantitative value for apparent yield stress is generated with this type of protocol, and any model fitting should be done with linear models (e.g., Newtonian, Herschel-Bulkley unithit). 

Table 8-2 contains expressions for the velocity profiles and the volumetric flow rates of the three rheological models power law, Herschel-Bulkley, and the Bingham plastic models. 

In the MEB equation, kinetic energy losses can be calculated easily provided that the kinetic energy correction factor a can be determined. In turbulent flow, often, the value of a = 2 is used in the MEB equation. When the flow is laminar and the fluid is Newtonian, the value of a = 1 is used. Osorio and Steffe (1984) showed that for fluids that follow the Herschel-Bulkley model, the value of a in laminar flow depends on both the flow behavior index ( ) and the dimensionless yield stress ( o) defined above. They developed an analytical expression and also presented their results in graphical form for a as a function of the flow behavior index ( ) and the dimensionless yield stress ( o)- When possible, the values presented by Osorio and Steffe (1984) should be used. For FCOJ samples that do not exhibit yield stress and are mildly shear-thinning, it seems reasonable to use a value of a = 1. 

A comprehensive example for sizing a pump and piping for a non-Newtonian fluid whose rheological behavior can be described by the Herschel-Bulkley model (Equation 2.5) was developed by Steffe and Morgan (1986) for the system shown in Figure 8-2 and it is summarized in the following. The Herschel-Bulkley parameters were yield stress = 157 Pa, flow behavior index = 0.45, consistency coefficient = 5.20 Pas". 

In Equation (2), n is the flow behavior index (-),K is the consistency index (Pa secn), and the other terms have been defined before. For shear-thinning fluids, the magnitude of nshear-thickening fluids n>l, and for Newtonian fluids n=l. For PFDs that exhibit yield stresses, models that contain either (Jo or a term related to it have been defined. These models include, the Bingham Plastic model (Equation 3), the Herschel-Bulkley model (Equation 4), the Casson model (Equation 5), and the Mizrahi-Berk model (Equation 6). 

Yield stresses can also be obtained by extrapolation of shear rate-shear stress data to zero shear rate according to one of several flow models. The application of several models was studied by Rao et al. (AS.) and Rao and Cooley (Al) The logarithm of the yield stresses predicted by each model and the total solids (TS) of the concentrates were related by quadratic equations. The equations for the yield stresses predicted by the Herschel-Bulkley model (Equation 4) which described very well the flow data of Nova and New Yorker tomato cultivars were ... 

COJ of 65 °Brix is a mildly shear-thinning fluid 160) with magnitudes of flow behavior index of the power law model (n) (Equation 2) of about 0.75 that is mildly temperature dependent. In contrast, the consistency index (K) is very sensitive to temperature for example, Vital and Rao (hi) found for a COJ sample magnitudes of 1.51 Pa sec11 at 20 °C and 27.63 Pa secn at -19 °C. Mizrahi and Firstenberg (hi) found that the modified Casson model (Equation 5) described the shear rate-shear stress data better than the Herschel-Bulkley model (Equation 4). 

Casson models were used to compare their yield stress results to those calculated with the direct methods, the stress growth and impeller methods. Table 2 shows the parameters obtained when the experimental shear stress-shear rate data for the fermentation suspensions were fitted with all models at initial process. The correlation coefficients (/P) between the shear rate and shear stress are from 0.994 to 0.995 for the Herschel-Bulkley model, 0.988 to 0.994 for the Bingham, 0.982 to 0.990 for the Casson model, and 0.948 to 0.972 for the power law model for enzymatic hydrolysis at 10% solids concentration (Table 1). The rheological parameters for Solka Floe suspensions were employed to determine if there was any relationship between the shear rate constant, k, and the power law index flow, n. The relationship between the shear rate constant and the index flow for fermentation broth at concentrations ranging from 10 to 20% is shown on Table 2. The yield stress obtained by the FL 100/6W impeller technique decreased significantly as the fimetion of time and concentration during enzyme reaction and fermentation. 

Based on the magnitude of n and to, the non-Newtonian behavior can be classified as shear thinning, shear thickening, Bingham plastic, pseudoplastic with yield stress, or dilatant with yield stress (see Fig. 2 and Table I). The Herschel-Bulkley model is able to describe the general flow properties of fluid foods within a certain shear range. The discussion on this classiflcation and examples of food materials has been reviewed by Sherman (1970), DeMan (1976), Barbosa-Canovas and Peleg (1983), and Barbosa-Canovas et al. (1993). 

The power law, Herschel-Bulkley equation, and Casson model are simple and easy to use. However, these equations only work for modeling steady shear flows rather than transient or elongational flows. Thus, many other models have been proposed to fit experimental data more closely for food materials. Among these, it is worth mentioning the Ree-Eyring equation which has three constants... 

A simple generalisation of the Bingham plastic model to embrace the nonlinear flow curve (for tyx > Tq ) is the three constant Herschel-Bulkley fluid... 

Laminar flow conditions cease to exist at Rcmod = 2100. The calculation of the critical velocity corresponding to Rcmod = 2100 requires an iterative procedure. For known rheology (p, m, n, Xq) and pipe diameter (D), a value of the wall shear stress is assumed which, in turn, allows the calculation of Rp, from equation (3.9), and Q and Qp from equations (3.14b) and (3.14a) respectively. Thus, all quanties are then known and the value of Rcmod can be calculated. The procedure is terminated when the value of x has been found which makes RCjnod = 2100, as illustrated in example 3.4 for the special case of n = 1, i.e., for the Bingham plastic model, and in example 3.5 for a Herschel-Bulkley fluid. Detailed comparisons between the predictions of equation (3.34) and experimental data reveal an improvement in the predictions, though the values of the critical velocity obtained using the criterion Rqmr = 2100 are only 20-25% lower than those predicted by equation (3.34). Furthermore, the two... 

Khalkhal and Carreau (2011) examined the linear viscoelastic properties as well as the evolution of the stmcture in multiwall carbon nanotube-epoxy suspensions at different concentration under the influence of flow history and temperature. Initially, based on the frequency sweep measurements, the critical concentration in which the storage and loss moduli shows a transition from liquid-like to solid-like behavior at low angular frequencies was found to be about 2 wt%. This transition indicates the formation of a percolated carbon nanotube network. Consequently, 2 wt% was considered as the rheological percolation threshold. The appearance of an apparent yield stress, at about 2 wt% and higher concentration in the steady shear measurements performed from the low shear of 0.01 s to high shear of 100 s confirmed the formation of a percolated network (Fig. 7.9). The authors used the Herschel-Bulkley model to estimate the apparent yield stress. As a result they showed that the apparent yield stress scales with concentration as Xy (Khalkhal and Carreau 2011). 

At high particle concentrations, slurries are often non-Newtonian. For non-Newtonian fluids, the relationship between the shear stress and shear rate, which describes the rheology of the slurry, is not linear and/or a certain minimum stress is required before flow begins. The power-law, Bingham plastic and Herschel-Bulkley models are various models used to describe the flow behaviour of slurries in which these other types of relationships between the shear stress and shear rate exist. Although less common, some slurries also display time-dependent flow behaviour. In these cases, the shear stress can decrease with time when the shear rate is maintained constant (thixotropic fluid) or can increase with time when the shear rate is maintained constant (rheopectic fluid). Milk is an example of a non-settling slurry which behaves as a thixotropic liquid. 

In the Herschel-Bulkley model, the yield stress, consistency index k, and the flow behaviour index n characterize the slurry. 

Flow of Bingham and Herschel-Bulkley fluids through a contraction, using the biviscosity model, is treated in... 

Herschel-Bulkley general model Many systems show a dynamic yield value followed by a shear thinning behavior. The flow curve can be analyzed using the Herschel-Bulkley equation [45] ... 

When Op = 0, equation (4.27) reduces to the power fluid model. The Herschel-Bulkley equation Ats most flow curves with a good correlation coefficient and hence it is the most widely used model. 

Estimates need to be made for the parameters defined in the flow models. As the Herschel-Bulkley model can be reduced to the Newtonian, power law and Bingham plastic models, a least squares regression analysis can first be performed on the (x, y ) data to obtain XyHB, K and n. It may then be possible to simplify the model by setting the XyHB to zero if the estimate is close to zero and/or setting n to 1 if the estimate is close to unity. 

Two principal models are used to describe much of the observed behavior seen in PE and other complex fluids the Herschel-Bulkley and Maxwell models. The first is an empirical model that describes the flow of a yield stress fluid (tq) in response to varying shear rates, according to following equation [30] ... 

This model accounts for a yield stress combined with power law behavior in stress as a function of shear rate. Besides, this model predicts a viscosity that diverges continuously at low shear rates and is infinite below the yield stress. When n = 1, the Herschel-Bulkley model reduces to the Bingham fluid model where the flow above the yield stress would be purely Newtonian and the constant k would represent the viscosity [28]. 

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