Bingham plastic rheology model

Since the power-law and the Bingham plastic fluid models are usually adequate for modelling the shear dependence of viscosity in most engineering design calculations, the following discussion will therefore be restricted to cover just these two models where appropriate, reference, however, will also be made to the applications of other rheological models. Theoretical and experimental results will be presented separately. For more detailed accounts of work on heat transfer in non-Newtonian fluids in both circular and non-circular ducts, reference should be made to one of the detailed surveys [Cho and Hartnett, 1982 Irvine, Jr. and Kami, 1987 Shah and Joshi, 1987 Hartnett and Kostic, 1989 Hartnett and Cho, 1998]. 

One simple rheological model that is often used to describe the behavior of foams is that of a Bingham plastic. This appHes for flows over length scales sufficiently large that the foam can be reasonably considered as a continuous medium. The Bingham plastic model combines the properties of a yield stress like that of a soHd with the viscous flow of a Hquid. In simple Newtonian fluids, the shear stress T is proportional to the strain rate y, with the constant of proportionaHty being the fluid viscosity. In Bingham plastics, by contrast, the relation between stress and strain rate is r = where is... 

The rheological properties of a particular suspension may be approximated reasonably well by either a power-law or a Bingham-plastic model over the shear rate range of 10 to 50 s. If the consistency coefficient k is 10 N s, /m-2 and the flow behaviour index n is 0.2 in the power law model, what will be the approximate values of the yield stress and of the plastic viscosity in the Bingham-plastic model ... 

While the Bingham plastic model is an adequate approximate description of foam rheology, it is by no means exact, especially at low strain rates. More detailed models attempl to relate the rheological properties of foams to the structure and behavior of the bubbles. 

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. 

Rheology is the study of the deformation and flow of fluids. Four different models are used to characterize the flow of fluids Newtonian, Bingham plastic, power law, and viscoelastic In the characterization, models have been developed to relate the observed effects that shear rate has on foam. Several scientists who have studied foamed fluid rheology categorize foam into various models. 

Figure 10. Pressure dependence of parameters from various models of the rheology of invert emulsion oil-based drilling fluids at various temperatures. Casson high shear viscosity Bingham plastic viscosity consistency, power law exponent, and yield stress from Herschel-Bulkley model. (Reproduced with permission from reference 69. Copyright 1986 Society of Petroleum Engineers.)...

Figure 2 7. Comparison of fit of power law, Bingham plastic and Robertson-Stiff rheological models to experimental data from bentonite drilling fluid. (Data from reference 106.)...

Approach used in section 3.2 for power-law and Bingham plastic model fluids can be extended to other fluid models. Even if the relationship between shear stress and shear rate is not known exactly, it is possible to use the following approach to the problem. It depends upon the fact that the shear stress distribution over the pipe cross-section is not a function of the fluid rheology and is given simply by equation (3.2), which can be re-written in terms of the wall shear stress, i.e. 

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... 

The rheological behaviour of a coal slurry (1160kg/m ) can be approximated by the Bingham plastic model with Tq = 0.5 Pa and /ng = 14mPa-s. It is to be pumped through a 400 mm diameter pipe at the rate of 188kg/s. Ascertain the nature of the flow by calculating the maximum permissible velocity for laminar flow conditions. 

A similar procedure can, in principle, be used for other rheological models by inserting an appropriate expression for shear stress in equation (3.62). The analogous result for the laminar flow of Bingham plastic fluids in this geometry is given here ... 

A polymer solution (of density 1000 kg/m ) is flowing parallel to a plate (300 mm x 300 mm) the free stream velocity is 2 m/s. In the narrow shear rate range, the rheology of the polymer solution can be adequately approximated by both the power-law (m = 0.3 Pa-s" and n =0.5) and the Bingham plastic model (tq = 2.28 Pa s and /xg = 7.22 mPa-s). Using each of these models, estimate and compare the values of the shear stress and the boundary layer thickness 150 mm away from the leading edge, and the total frictional force on each side of the plate. 

The following rheological data for milk chocolate at 313 K are available. Determine the Bingham plastic (equation 1.16) and Casson model (equation 1.18) parameters for this material. What are the mean and maximum deviations for both these models ... 

---