Flow models Bingham

Numerous examples of polymer flow models based on generalized Newtonian behaviour are found in non-Newtonian fluid mechanics literature. Using experimental evidence the time-independent generalized Newtonian fluids are divided into three groups. These are Bingham plastics, pseudoplastic fluids and dilatant fluids. 

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 ... 

What will be the pressure drop, when the suspension is flowing under laminar conditions in a pipe 200 m long and 40 mm diameter, when the centre line velocity is 1 m/s, according to the power-law model Calculate the centre-line velocity for this pressure drop for the Bingham-plastic model. 

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. 

For greater concentrations of fine particles the suspension is more likely to be non-Newtonian, in which case the viscous properties can probably be adequately described by the power law or Bingham plastic models. The pressure drop-flow relationship for pipe flow under these conditions can be determined by the methods presented in Chapters 6 and 7. 

The methods presented in Sections 3.1 to 3.6 are general and do not require the assumption of any particular flow model. While the flow of power law fluids and Bingham plastics can be treated by those methods, some results specific to these materials will be considered in this and the next sections. 

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. 

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. 

Parameters of the Bingham Model from Measurements of Pressure Drops in a Line 107 Pressure Drop in Power-Law and Bingham Flow 110 Adiabatic and Isothermal Flow of a Gas in a Pipeline Isothermal Flow of a Nonideal Gas 113 Pressure Drop and Void Fraction in Liquid-Gas Flow Pressure Drop in Flow of Nitrogen and Powdered Coal 120... 

In time-independent liquid food products, the flow curve is linear but intersects the shear stress axis at a positive value of shear stress. This value is known as a yield stress. The significance of the yield stress is that it is the stress that must be exceeded before the material will flow. This type of flow can be characterized by the following rheological equation (for the Bingham-Schwedoff model) ... 

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). 

Figure 28. Comparison of calculated frictional pressures for flow of a drilling fluid in a pipe using (a) power law and (b) Bingham plastic models. (Data from reference 106.)...

Figure 29. Calculation of frictional pressures in turbulent flow using the power law and Bingham plastic models.

The use of the modified rheoviseometer, which enables handling of unstable mineral suspensions, has recently revealed that the Casson Equation fits the flow curve for the magnetite suspension better than the typically used Bingham plastic model [see the special issue of Coal Preparation entirely devoted to magnetite dense media [Coal Preparation ( 99Qi). 8(3 ).]... 

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

For the region Rp < r < R, the value of shear stress will be greater flian the yield stress of the fluid, and the Bingham fluid model for pipe flow is given by (equation (1.16) in Chapter 1) ... 

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. 

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