Bingham equation fluids

For Newtonian fluids the dynamic viscosity is constant (Equation 2-57), for power-law fluids the dynamic viscosity varies with shear rate (Equation 2-58), and for Bingham plastic fluids flow occurs only after some minimum shear stress, called the yield stress, is imposed (Equation 2-59). 

Thus, equation 3.127, which includes three parameters, is effectively a combination of equations 3.121 and 3.125. It is sometimes called the generalised Bingham equation or Herschel -Bulkley equation, and the fluids are sometimes referred to as having/n/re body. Figures 3.30 and 3.31 show shear stress and apparent viscosity, respectively, for Bingham plastic and false body fluids, using linear coordinates. 

Starting with the equations for r = fn(j>) that define the power law and Bingham plastic fluids, derive the equations for the viscosity functions for these models as a function of shear stress, i.e., rj = fn(r). 

Equation (11) states that the conventional Fanning friction factor, which may be used through Eq. (10) to calculate pipe-line pressure drops, is a unique function of two dimensionless groups for Bingham-plastic fluids. Newtonian fluids represent that special case for which r , and hence the second dimensionless group, is equal to zero. 

Fluids that show viscosity variations with shear rates are called non-Newtonian fluids. Depending on how the shear stress varies with the shear rate, they are categorized into pseudoplastic, dilatant, and Bingham plastic fluids (Figure 2.2). The viscosity of pseudoplastic fluids decreases with increasing shear rate, whereas dilatant fluids show an increase in viscosity with shear rate. Bingham plastic fluids do not flow until a threshold stress called the yield stress is applied, after which the shear stress increases linearly with the shear rate. In general, the shear stress r can be represented by Equation 2.6 ... 

Non-Newtonian Flow For isothermal laminar flow of time-independent non-Newtonian liquids, integration of the Cauchy momentum equations yields the fully developed velocity profile and flow rate-pressure drop relations. For the Bingham plastic fluid described by Eq. (6-3), in a pipe of diameter D and a pressure drop per unit length AP/L, the flow rate is given by... 

A Bingham-plastic fluid (yield stress 14.35 N/m2 and plastic viscosity 0.150 Ns/m2) is flowing through a pipe of diameter 40 mm and length 200 m. Starting with the rheological equation, show that the relation between pressure gradient —AP/l and volumetric flowrate Q is ... 

Other expressions for concentric cylinder geometry include that for Bingham plastic fluids where the yield stress must be taken into account which leads to the Reiner-Riwlin equation ... 

Table 8-2 Velocity Profile and volumetric flow rate equations for power law, Herschel-Bulkley, and Bingham plastic fluids...

Hanks and Christiansen (97) have shown that equations 47 and 48 are not applicable to Bingham plastic fluids or those that exhibit a finite yield stress. Hanks (98) has developed the concept of the critical Reynolds number for Bingham plastics. From equation 33, the Bingham fluid Reynolds number can be defined by... 

For smooth pipes, the relationship between friction factor and Reb is close to that found for Newtonian fluids (99). From equation 33, the friction factor for a Bingham plastic fluid is given by... 

In the same way as there are many equations for predicting friction factor for turbulent Newtonian flow, there are munerous equations for time-independent non-Newtonian fluids most of these are based on dimensional considerations combined with experimental observations [Govier and Aziz, 1982 Heywood and Cheng, 1984]. There is a preponderance of correlations based on the power-law fluid behaviour and additionally some expressions are available for Bingham plastic fluids [Tomita, 1959 Wilson and Thomas, 1985], Here only a selection of widely used and proven methods is presented. 

Despite the fact that equation (3.37) is applicable to all kinds of time-independent fluids, numerous workers have presented expressions for turbulent flow friction factors for specific fluid models. For instance, Tomita [1959] applied the concept of the Prandtl mixing length and put forward modified definitions of the friction factor and Reynolds number for the turbulent flow of Bingham Plastic fluids in smooth pipes so that the Nikuradse equation, i.e. equation (3.37) with n = 1, could be used. Though he tested the applicability of his method using his own data in the range 2000 < Reg(l — 4>f 3 — )< 10, the validity of this approach has not been established using independent experimental data. 

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 ... 

The flow of viscoplastic fluids through beds of particles has not been studied as extensively as that of power-law fluids. However, since the expressions for the average shear stress and the nominal shear rate at the wall, equations (5.41) and (5.42), are independent of fluid model, they may be used in conjimction with any time-independent behaviour fluid model, as illuslrated here for the streamline flow of Bingham plastic fluids. The mean velocity for a Bingham plastic fluid in a circular tube is given by equation (3.13) ... 

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