Herschel-Bulkley general model

Many systems show a dynamic yield value followed by a shear thinning behaviour [9]. The flow curve can be analysed using the Herschel-Bulkley equation  

When (Tp = 0, Equation (20.14) reduces to the Power Fluid Model, but when n = 1, Equation (20.14) reduces to the Bingham model. When = 0 and n= 1, Equation 

Several other models have been suggested, of which the following is worthy of mentioning. 

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FIGURE 11 Generalized correlation of drag coefficient for Herschel-Bulkley model fluids Qff is defined by Eq. (165) and reduces to appropriate parameters for Bingham plastic, power law, and Newtonian fluid limits. 

A general model for shear rate-shear stress data that under specific assumptions reduces to the Herschel-Bulkley, the Casson, and other models was presented by Ofoli et al. (1987) ... 

A more general mathematical description of shear thinning materials comprising an apparent yield stress is given by the Herschel-Bulkley model ... 

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). 

Eq. (3). In addition, Fp is the particle (granule) volume, mp = FpPp is the granule mass, and r(y°) is some characteristic strength of the granule. In the most general case, this stress can be taken according to the Herschel-Bulkley model... 

It can be observed that equation (5 a) described shear stress in terms of strain whereas equation (Sb) in terms of shear strain rate. It can also be observed that equations (S) are but generalization of the Herschel-Bulkley model (3). There are more sophisticated constitutive equations and models developed but it will not be possible to cover them in this chapter. A detailed description could be found in a review by White [45]. 

In other terms, above a critical shear stress, it flows as a Newtonian fluid of (constant) viscosity t). It follows that a fluid obeying the Herschel-Bulkley model is sometimes called a generalized Bingham fluid, since with n=1 and K=r in Equation 5.3, one obviously obtains Equation 5.4. The three fit parameters of the Herschel-Bulkley equation can be reduced to two, when considering that n=0.5. This was in fact the approach used by Casson in proposing the following model ... 

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