Bingham viscosity model

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... 

Of the models Hsted in Table 1, the Newtonian is the simplest. It fits water, solvents, and many polymer solutions over a wide strain rate range. The plastic or Bingham body model predicts constant plastic viscosity above a yield stress. This model works for a number of dispersions, including some pigment pastes. Yield stress, Tq, and plastic (Bingham) viscosity, = (t — Tq )/7, may be determined from the intercept and the slope beyond the intercept, respectively, of a shear stress vs shear rate plot. 

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 ... 

The Bingham plastic model can describe acrylic latex paint, with a yield stress of 112 dyn/cm2, a limiting viscosity of 80 cP, and a density of 0.95 g/cm3. What is the maximum thickness of this paint that can be applied to a vertical wall without running ... 

You must determine the horsepower required to pump a coal slurry through an 18 in. diameter pipeline, 300 mi long, at a rate of 5 million tons/yr. The slurry can be described by the Bingham plastic model, with a yield stress of 75 dyn/cm2, a limiting viscosity of 40 cP, and a density of 1.4 g/cm3. For non-Newtonian fluids, the flow is not sensitive to the wall roughness. 

You want to predict how fast a glacier that is 200 ft thick will flow down a slope inclined 25° to the horizontal. Assume that the glacier ice can be described by the Bingham plastic model with a yield stress of 50 psi, a limiting viscosity of 840 poise, and an SG of 0.98. The following materials are available to you in the lab, which also may be described by the Bingham plastic model ... 

The Bingham plastic model usually provides a good representation for the viscosity of concentrated slurries, suspensions, emulsions, foams, etc. Such materials often exhibit a yield stress that must be exceeded before the material will flow at a significant rate. Other examples include paint, shaving cream, and mayonnaise. There are also many fluids, such as blood, that may have a yield stress that is not as pronounced. 

A pipeline is installed to transport a red mud slurry from an open tank in an alumina plant to a disposal pond. The line is 5 in. sch 80 commercial steel, 12,000 ft long, and is designed to transport the slurry at a rate of 300 gpm. The slurry properties can be described by the Bingham plastic model, with a yield stress of 15 dyn/cm2, a limiting viscosity of 20 cP, and an SG of 1.3. You may neglect any fittings in this pipeline. 

As discussed in Chapter 3, the Carreau viscosity model is one of the most general and useful and reduces to many of the common two-parameter models (power law, Ellis, Sisko, Bingham, etc.) as special cases. This model can be written as... 

Bingham fluids that are either shear-thinning or shear-thickening above their yield stresses have corresponding power-law expressions incorporated into their viscosity models. 

A more convenient extrapolation technique is to approximate the experimental data with a viscosity model. The Power Law, shown in Eq. 6, is the most commonly used two-parameter model. The Bingham model, shown in Eq. 7, postulates a linear relationship between x and y but can lead to overprediction of the yield stress. Extrapolation of the nonlinear Casson model (1954), shown in Eq. 8, is straightforward from a linear plot of x°5 vs y05. Application of the Herschel-Bulkley model (1926), shown in Eq. 9, is more tedious and less certain although systematic procedures for determining the yield value and the other model parameters are available (11) ... 

Rheocalic V2.4. The Bingham mathematic model was used to determine viscosity. The Bingham equation is t = Tq + ly. Where t is the shear stress applied to the material, y is the shear strain rate (also called the strain gradient). To is the yield stress and p is the plastic viscosity. 

The Bingham body model describes materials with an apparent yield strength above which Newtonian flow is observed. This is illustrated in Figs. 4 and 5, which show a typical flow curve and viscosity as a function of shear strain rate, respectively. 

The Bingham plastic model usually provides a good representation for the viscosity of concentrated slurries, suspensions, sediments, emulsions, foams, etc. Such materials often exhibit a yield stress,... 

When x < x, deformation does not occur (Fig. IX-14). Since the parameter of Bingham s model, r B, defines the derivative, dx/dy=r B, this constant value is referred to as the differential viscosity, in contrast to a variable effective viscosity, x/y = r ef (y). 

Three different additives (Additive 1, 2 and 3) were used in order to see their effects on Crude oil-5. The viscosity of the crude oil is decreased considerably by treatment with the additives. Measurements of the shear stress-shear rate relationships were performed at constant temperature and experimental data were fitted to the Bingham plastic model using a linear regression program. When the amount of additive was increased from 500 ppm to 1000 ppm, the pour point of the crude oil was decreased significantly (Table 3). Denis and Durand [25] determined the low temperature properties of petroleum products that had been improved either by refinery processing or by adding wax... 

To represent the above-named Bingham viscosity in the simulation model the possibility to characterize Non-Newtonian fluids in the CFD-code is necessary. Most commercial CFD-Software allocate a so-called Power-Law-Model whereby (26) may be possibly specified. Otherwise the Bingham model has to be implemented in the actual internal expression language, e.g. an executive Fortran-Routine in ANSYS/FLOTRAN or a definition in the CFX-Expression Language (CEL). In either case it is inevitable for the CFD-code to have dynamic access to the (system-) variable for the shear strain rate y. 

Another model in which the viscosity is described as a function of shear stress is the Bingham model [20]. This model is used for fluids with a yield stress Xq. Below this yield stress, the viscosity is infinite (no motion) above the yield stress, the viscosity is finite (motion occurs). The Bingham Fluid model is written as ... 

In the Bingham plastic model, the yield stress Ty and the plastic viscosity Hp (the slope of the line on the shear stress-shear rate plot in Figure 4.4) characterize the slurry. 

Rheological properties of foams (elasticity, plasticity, and viscosity) play an important role in foam production, transportation, and applications. In the absence of external stress, the bubbles in foams are symmetrical and the tensions of the formed foam films are balanced inside the foam and close to the walls of the vessel [929], At low external shear stresses, the bubbles deform and the deformations of the thin liquid films between them create elastic shear stresses. At a sufficiently large applied shear stress, the foam begins to flow. This stress is called the yield stress, Tq- Then, Equation 4.326 has to be replaced with the Bingham plastic model [930] ... 

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